NAMIC Wiki - User contributions [en]

File:Kernel scatter viz2.png

2015-01-09T16:13:17Z

Anuja:

Distribution modeling

2015-01-09T16:13:02Z

Anuja:

__NOTOC__
<gallery>
Image:PW-2015SLC.png|[[2015_Winter_Project_Week#Projects|Projects List]]
Image:Genu_fa_3d_conte_vs_krabbe.png
Image:Avg_vs_avgcurve.png
Image:Neo_Avg_leave1out_CDF.png
Image:Neo_AvgCurves_leave1out_CDF.png
Image:kernel_scatter_viz2.png
Image:Neo_DIST_leave1out_CDF.png
</gallery>

==Key Investigators==
Anuja Sharma, Guido Gerig

==Project Description==
<div style="margin: 20px;">
<div style="width: 27%; float: left; padding-right: 3%;">
<h3>Objective</h3>
* Evaluate the ability of regression methods utilizing distribution-valued measurements to differentiate between healthy controls and patients with pathology.
</div>
<div style="width: 27%; float: left; padding-right: 3%;">
<h3>Approach, Plan</h3>
* Model probability distributions along DTI fiber tracts instead of scalar summary measurements to create a tract profile (e.g. cross-sectional averages).
* Our spatiotemporal modeling method utilizes these distribution-valued data to create growth trajectories such that the complete probability distribution is estimated continuously in space and time.
* Using these methods, we can estimate a spatiotemporal growth trajectory for the healthy infant population. We then compare our method's ability to detect clinical differences in infants with Krabbe's disease in reference to a control population.
</div>
<div style="width: 27%; float: left; padding-right: 3%;">
<h3>Progress</h3>
* We found that our method had better sensitivity towards finding differences in growth trajectories.
* We could find statistically significant differences in scenarios where the differences were small enough to be missed by conventional methods.
</div>
</div>

File:Avg vs avgcurve.png

2015-01-09T15:38:17Z

Anuja:

Distribution modeling

2015-01-09T15:38:00Z

Anuja:

__NOTOC__
<gallery>
Image:PW-2015SLC.png|[[2015_Winter_Project_Week#Projects|Projects List]]
Image:Genu_fa_3d_conte_vs_krabbe.png
Image:Avg_vs_avgcurve.png
Image:Neo_Avg_leave1out_CDF.png
Image:Neo_AvgCurves_leave1out_CDF.png
Image:Neo_DIST_leave1out_CDF.png
</gallery>

==Key Investigators==
Anuja Sharma, Guido Gerig

==Project Description==
<div style="margin: 20px;">
<div style="width: 27%; float: left; padding-right: 3%;">
<h3>Objective</h3>
* Evaluate the ability of regression methods utilizing distribution-valued measurements to differentiate between healthy controls and patients with pathology.
</div>
<div style="width: 27%; float: left; padding-right: 3%;">
<h3>Approach, Plan</h3>
* Model probability distributions along DTI fiber tracts instead of scalar summary measurements to create a tract profile (e.g. cross-sectional averages).
* Our spatiotemporal modeling method utilizes these distribution-valued data to create growth trajectories such that the complete probability distribution is estimated continuously in space and time.
* Using these methods, we can estimate a spatiotemporal growth trajectory for the healthy infant population. We then compare our method's ability to detect clinical differences in infants with Krabbe's disease in reference to a control population.
</div>
<div style="width: 27%; float: left; padding-right: 3%;">
<h3>Progress</h3>
* We found that our method had better sensitivity towards finding differences in growth trajectories.
* We could find statistically significant differences in scenarios where the differences were small enough to be missed by conventional methods.
</div>
</div>

File:Neo DIST leave1out CDF.png

2015-01-09T15:36:38Z

Anuja:

File:Neo AvgCurves leave1out CDF.png

2015-01-09T15:36:10Z

Anuja:

File:Neo Avg leave1out CDF.png

2015-01-09T15:35:51Z

Anuja:

Distribution modeling

2015-01-09T15:35:19Z

Anuja:

__NOTOC__
<gallery>
Image:PW-2015SLC.png|[[2015_Winter_Project_Week#Projects|Projects List]]
Image:Genu_fa_3d_conte_vs_krabbe.png
Image:Neo_Avg_leave1out_CDF.png
Image:Neo_AvgCurves_leave1out_CDF.png
Image:Neo_DIST_leave1out_CDF.png
</gallery>

==Key Investigators==
Anuja Sharma, Guido Gerig

==Project Description==
<div style="margin: 20px;">
<div style="width: 27%; float: left; padding-right: 3%;">
<h3>Objective</h3>
* Evaluate the ability of regression methods utilizing distribution-valued measurements to differentiate between healthy controls and patients with pathology.
</div>
<div style="width: 27%; float: left; padding-right: 3%;">
<h3>Approach, Plan</h3>
* Model probability distributions along DTI fiber tracts instead of scalar summary measurements to create a tract profile (e.g. cross-sectional averages).
* Our spatiotemporal modeling method utilizes these distribution-valued data to create growth trajectories such that the complete probability distribution is estimated continuously in space and time.
* Using these methods, we can estimate a spatiotemporal growth trajectory for the healthy infant population. We then compare our method's ability to detect clinical differences in infants with Krabbe's disease in reference to a control population.
</div>
<div style="width: 27%; float: left; padding-right: 3%;">
<h3>Progress</h3>
* We found that our method had better sensitivity towards finding differences in growth trajectories.
* We could find statistically significant differences in scenarios where the differences were small enough to be missed by conventional methods.
</div>
</div>

Distribution modeling

2015-01-09T14:43:33Z

Anuja: /* Project Description */

__NOTOC__
<gallery>
Image:PW-2015SLC.png|[[2015_Winter_Project_Week#Projects|Projects List]]
</gallery>

==Key Investigators==
Anuja Sharma, Guido Gerig

==Project Description==
<div style="margin: 20px;">
<div style="width: 27%; float: left; padding-right: 3%;">
<h3>Objective</h3>
* Evaluate the ability of regression methods utilizing distribution-valued measurements to differentiate between healthy controls and patients with pathology.
</div>
<div style="width: 27%; float: left; padding-right: 3%;">
<h3>Approach, Plan</h3>
* Model probability distributions along DTI fiber tracts instead of scalar summary measurements to create a tract profile (e.g. cross-sectional averages).
* Our spatiotemporal modeling method utilizes these distribution-valued data to create growth trajectories such that the complete probability distribution is estimated continuously in space and time.
* Using these methods, we can estimate a spatiotemporal growth trajectory for the healthy infant population. We then compare our method's ability to detect clinical differences in infants with Krabbe's disease in reference to a control population.
</div>
<div style="width: 27%; float: left; padding-right: 3%;">
<h3>Progress</h3>
* We found that our method had better sensitivity towards finding differences in growth trajectories.
* We could find statistically significant differences in scenarios where the differences were small enough to be missed by conventional methods.
</div>
</div>

Distribution modeling

2015-01-05T11:28:22Z

Anuja: Created page with '__NOTOC__ <gallery> Image:PW-2015SLC.png|Projects List </gallery> ==Key Investigators== Anuja Sharma, Guido Gerig ==Project Description== …'

__NOTOC__
<gallery>
Image:PW-2015SLC.png|[[2015_Winter_Project_Week#Projects|Projects List]]
</gallery>

==Key Investigators==
Anuja Sharma, Guido Gerig

==Project Description==
<div style="margin: 20px;">
<div style="width: 27%; float: left; padding-right: 3%;">
<h3>Objective</h3>
* Model probability distributions along DTI fiber tracts instead of scalar summary measurements to create a tract profile (e.g. cross-sectional averages).
* Our spatiotemporal modeling method utilizes these distribution-valued data to create growth trajectories such that the complete probability distribution is estimated continuously in space and time.
* Create a spatiotemporal growth trajectory for the healthy infant population. Evaluate our method's ability to detect clinical differences in infants with Krabbe's disease when compared with the control population.
</div>
<div style="width: 27%; float: left; padding-right: 3%;">
<h3>Approach, Plan</h3>
*
</div>
<div style="width: 27%; float: left; padding-right: 3%;">
<h3>Progress</h3>
*
</div>
</div>

2015 Winter Project Week

2015-01-05T11:27:48Z

Anuja: /* Image-Guided Therapy */

__NOTOC__

[[image:PW-2015SLC.png|300px]]

Welcome to the 20th Project week page!

== Introduction ==
Founded in 2005, the National Alliance for Medical Image Computing (NAMIC), was chartered with building a computational infrastructure to support biomedical research as part of the NIH funded [http://www.ncbcs.org/ NCBC] program. The work of this alliance has resulted in important progress in algorithmic research, an open source medical image computing platform [http://www.slicer.org 3D Slicer], built using [http://www.vtk.org VTK], [http://www.itk.org ITK], [http://www.cmake.org CMake], and [http://www.cdash.org CDash], and the creation of a community of algorithm researchers, biomedical scientists and software engineers who are committed to open science. This community meets twice a year in an event called Project Week.

[[Engineering:Programming_Events|Project Week]] is a semi-annual event which draws 80-120 researchers. As of August 2014, it is a MICCAI endorsed event. The participants work collaboratively on open-science solutions for problems that lie on the interfaces of the fields of computer science, mechanical engineering, biomedical engineering, and medicine. In contrast to conventional conferences and workshops the primary focus of the Project Weeks is to make progress in projects (as opposed to reporting about progress). The objective of the Project Weeks is to provide a venue for this community of medical open source software creators. Project Weeks are open to all, are publicly advertised, and are funded through fees paid by the attendees. Participants are encouraged to stay for the entire event.

Project Week activities: Everyone shows up with a project. Some people are working on the platform. Some people are developing algorithms. Some people are applying the tools to their research problems. We begin the week by introducing projects and connecting teams. We end the week by reporting progress. In addition to the ongoing working sessions, breakout sessions are organized ad-hoc on a variety of special topics. These topics include: discussions of software architecture, presentations of new features and approaches and topics such as Image-Guided Therapy.

Several funded projects use the Project Week as a place to convene and collaborate. These include [http://nac.spl.harvard.edu/ NAC], [http://www.ncigt.org/ NCIGT], [http://qiicr.org/ QIICR], and [http://ocairo.technainstitute.com/open-source-software-platforms-and-databases-for-the-adaptive-process/ OCAIRO]. The next events in this ongoing series will occur in [http://wiki.na-mic.org/Wiki/index.php/2015_Winter_Project_Week Salt Lake City, Utah in January of 2015], followed by one in Boston, MA in June of 2015.

A summary of all previous Project Events is available [[Project_Events#Past|here]].

This project week is an event [[Post-NCBC-2014|endorsed]] by the MICCAI society.

Please make sure that you are on the [http://public.kitware.com/mailman/listinfo/na-mic-project-week na-mic-project-week mailing list]

== '''Logistics''' ==
*'''Dates:''' January 5-9, 2015.
*'''Location:''' Salt Lake City, Utah
*'''REGISTRATION:''' Please click [http://umarket.utah.edu/ecom/checkout.tpl?App_Type=0046&Item_Num=WPW2015 '''here'''] to register online before December 24, 2014. All participants must pay a registration fee 550 USD, which covers our catering and facilities costs.
*'''Venue:''' The venue for the meeting is the Marriott City Center, Salt Lake City, Utah. You can book online, by clicking [http://www.marriott.com/meeting-event-hotels/group-corporate-travel/groupCorp.mi?resLinkData=2015%20Winter%20Project%20Week^slccc%60namnama%60149%60USD%60false%601/4/15%601/10/15%6012/05/14&app=resvlink&stop_mobi=yes here.] The room rate for the meeting is 149 USD per night.

== Agenda==
{| border="1" cellpadding="5"
|- style="background:#eeeeee; font-size:125%; color:#0063B6" align="center"
| style="width:5%"| '''Time'''
| style="width:19%" | '''Monday, January 5'''
| style="width:19%" | '''Tuesday, January 6'''
| style="width:19%" | '''Wednesday, January 7'''
| style="width:19%" | '''Thursday, January 8'''
| style="width:19%" | '''Friday, January 9'''
|-
| style="background:#eeeeee; color:black"|
| style="background:#dbf3ff; color:#522200"| '''[[2015_Winter_Project_Week|Project Activities]] ''' <br><font color="#44AA88">''(Olympus B)''</font>
| style="background:#dbf3ff; color:#522200"| '''[[2015_Winter_Project_Week|Project Activities]] '''<br><font color="#44AA88">''(Olympus B)''</font>
| style="background:#dbf3ff; color:#522200"| '''[[2015_Winter_Project_Week|Project Activities]] ''' <br><font color="#44AA88">''(Olympus B)''</font>
| style="background:#dbf3ff; color:#522200"| '''[[2015_Winter_Project_Week|Project Activities]] ''' <br><font color="#44AA88">''(Olympus B)''</font>
| style="background:#dbf3ff; color:#522200"| '''[[2015_Winter_Project_Week|Project Activities]] ''' <br><font color="#44AA88">''(Olympus B)''</font>
|-
| style="background:#eeeeee; color:black"|'''7:30-8:00'''
| style="background:#ffffff; color:black"|
| style="background:#dbf3ff; color:black"| Breakfast <br><font color="#BB9933">''(Olympus A)''</font>
| style="background:#dbf3ff; color:black"| Breakfast <br><font color="#BB9933">''(Olympus A)''</font>
| style="background:#dbf3ff; color:black"| Breakfast <br><font color="#BB9933">''(Olympus A)''</font>
| style="background:#dbf3ff; color:black"| Breakfast <br><font color="#BB9933">''(Olympus A)''</font>
|-
| style="background:#eeeeee; color:black"|'''8:00-10:00'''
| <br><font color="#9966cc"></font>
|9-10am: [[2015 Winter Project Week COPD Breakout|Breakout Session:COPD]] (Raul San Jose) <br><font color="#44AA88">''(Amethyst 1)''</font> <br>
|9-10am: [[2015 Winter Project Week QIICR and DICOM Breakout|Breakout Session:QIICR and DICOM]] (Andrey Fedorov, Steve Pieper) <br><font color="#44AA88">''(Amethyst 1)''</font> <br>
| '''[[2015_Winter_Project_Week|Project Activities]] ''' <br><font color="#44AA88">''(Olympus B)''</font>
| style="background:#dbf3ff; color:#522200"|[[2015_Winter_Project_Week|9am: Project Presentations]]<br><font color="#44AA88">''(Olympus B)''</font>
|-
| style="background:#eeeeee; color:black"|'''10:00-10:30'''
| style="background:#ffffff; color:black"|
| style="background:#dbf3ff; color:black"| Coffee <br>''(General area)''
| style="background:#dbf3ff; color:black"| Coffee <br>''(General area)''
| style="background:#dbf3ff; color:black"| Coffee <br>''(General area)''
| style="background:#dbf3ff; color:black"| Coffee <br>''(General area)''
|-
| style="background:#eeeeee; color:black"|'''10:30-12:00'''
|
|
|
| [[2015_Winter_Project_Week|Project Activities]] ''' <br><font color="#44AA88">''(Olympus B)''</font>
|
|-
| style="background:#eeeeee; color:black"|'''12:00-1:00'''
| style="background:#dbf3ff; color:black"| Lunch <br><font color="#BB9933">''(Olympus A)''</font>
| style="background:#dbf3ff; color:black"| Lunch <br><font color="#BB9933">''(Olympus A)''</font> <br>
| style="background:#dbf3ff; color:black"| Lunch <br><font color="#BB9933">''(Olympus A)''</font><br>
| style="background:#dbf3ff; color:black"| Lunch <br><font color="#BB9933">''(Olympus A)''</font>
| style="background:#dbf3ff; color:black"| Boxed Lunch and Adjourn <br><font color="#BB9933">''(Olympus A)''</font>
|-
| style="background:#eeeeee; color:black"|'''1:00-3:00'''
| style="background:#dbf3ff; color:#522200"| [http://ferenc.jolesz.muchloved.com/frame.aspx? A tribute to Ferenc Jolesz] <br> [[2015_Winter_Project_Week|Project Presentations]]<br><font color="#44AA88">''(Olympus B)''</font>
| style="background:#dbf3ff; color:#522200"|
| <font color="#44AA88">''(Olympus B)''</font>
|
|
|-
| style="background:#eeeeee; color:black"|'''3:00-3:30'''
| style="background:#dbf3ff; color:black"| Coffee <br>''(General area)''
| style="background:#dbf3ff; color:black"| Coffee <br>''(General area)''
| style="background:#dbf3ff; color:black"| Coffee <br>''(General area)''
| style="background:#dbf3ff; color:black"| Coffee <br>''(General area)''
| style="background:#ffffff; color:black"|
|-
| style="background:#eeeeee; color:black"|'''3:00-5:00'''
|
|3:15-4pm: [[2015_Winter_Project_Week_Segmentations_Breakout|Breakout Session: Segmentations]] <br><font color="#44AA88">''(Amethyst 1)'' <br> 
| style="background:#ffffff; color:black"|
|[[2015_Winter_Project_Week|Project Activities]] ''' <br><font color="#44AA88">''(Olympus B)''</font>
| style="background:#ffffff; color:black"|
|-
| style="background:#eeeeee; color:black"|'''05:00-07:00'''
| style="background:#dbf3ff; color:black"|
| style="background:#dbf3ff; color:black"|
| style="background:#dbf3ff; color:black"|
| style="background:#dbf3ff; color:black"|'''6:00''' Optional: [http://www.murphysbarandgrillut.com/ Beer at Murphy's] (like last year)
| style="background:#ffffff; color:black"|
|}

=Projects=
* [[2015_Project_Week:_Template | Template for project pages]]

==Image-Guided Therapy==
* [[2015_Winter_Project_Week:_Neurosurgery_Case_Review| Review of data from recent AMIGO US/MR neurosurgery]] (Steve Pieper, Jim Miller, Alireza Mehrtash, Sandy Wells, Tina Kapur, Ron Kikinis)
* [[2015_Winter_Project_Week:_Benchtop_Nuerosurgery|Tracked ultrasound benchtop experimental system for neurosurgery]] (Steve Pieper, Jim Miller)
* [[User:Yli|Registration of pre-op and intra-op DTI to correlate parameters with post-op prognosis]] (Li Ye, Steve Pieper, Alireza Mehrtash, Lauren O'Donnell)
* [[New Distance|A new modified Frechet distance for measuring the similarity between fiber tracts]] (Ruizhi Liao, Lauren O'Donnell)
* [[UKF Edema|Performance of UKF tractography in edema]] (Ruizhi Liao, Lauren O'Donnell)
* [[ProstateSegmentation | Prostate segmentation and biopsy]] (Peter Behringer, Andriy Fedorov)
* [[NeedleFinder | Image-based Needle Detection from MRI]] (Andre Mastmeyer, Guillaume Pernelle, Tina Kapur, Steve Pieper, Ron Kikinis)
* [[Needle Finder Tutorial | Needle Finder Tutorial]] (Gao Yang, Andre Mastmeyer, Guillaume Pernelle, Tina Kapur)
*[[NeedlePlace | Workflow for percutaneous needle place]] (Bamshad Azizi, Li Ming, Li Ye, Kevin Cleary)
*[[VR Radiology | Application of consumer virtual reality devices in creating environments for diagnostic radiology]] (Franklin King, Steve Pieper)
*[[Distribution modeling | Summary statistics versus distributions: evaluating improved ability to detect clinical differences using DTI]] (Anuja Sharma, Guido Gerig)

==Huntington's Disease==
* [[PREDICT-HD Longitudinal Shape Analysis | PREDICT-HD Longitudinal Shape Analysis]] (Regina EY Kim, James Fishbaugh, Hans Johnson)

== COPD ==
* [[Slicer CIP | SlicerCIP Extension]] (Alex Yarmakovich, Jorge Onieva, Raul San Jose)
* [[Nodule Sizing | Lung nodule sizing tool]] (Raul San Jose)
* [[Density Inspector | Density Inspector]] (Alex Yarmakovich, Raul San Jose, Jorge Onieva)
* [[PA/A Tool | Pulmonary Artery/Aorta measuring tool]] (Jorge Onieva, Rola Harmouche, German Gonzalez)
* [[Picassa Snap | Picasa Snap: tagging your favorite slicer snapshots]] (Jorge Onieva)
* [[Feature Extraction with Particles | Feature Extraction on ROIs based on Particles]] (Raul San Jose, James Ross)
* [[ CIP and Nipype | CIP analysis pipelines in Nipype]] (Rola Harmouche, James Ross, Alex Yarmakovich)
* [[Supervised Fissure Enhancement | Supervised fissure enhancement]] (James Ross, German Gonzalez, Rola Harmouche)
* [[Organ Detection | Organ detection with OpenCV]] (German Gonzalez, James Ross, Raul San Jose)
== Lung ==

*Small lung nodule differential diagnosis using 3D Slicer (Li Ming, Jay Jagadeesan)

==[http://qiicr.org QIICR]==
* [[2015_Winter_Project_Week:_Multiframe_DICOM | Segmentation object and enhanced multiframe object IO in DCMTK and Slicer]] (Steve, Andrey, Michael)
* [[T1 mapping for variable flip angle | T1 mapping for variable flip angle]] (Artem, Xiao, Andrey)
* [[2015_Winter_Project_Week:_DICOM_representation_of_QIICR_Iowa_DBP_data|Representation of Iowa QIN data using DICOM]] (Andrey, Christian Bauer, Steve Pieper)
* [[Whole Body PET/CT Reference Region segmentation]] (Christian Bauer)
* [[2015_Winter_Project_Week:Bolus_Arrival_Time_Estimation_in_PK_Modelling|Bolus Arrival Time (BAT) Estimation in PK Modelling]] (Alireza Mehrtash, Andriy Fedorov, Jim Miller)
* [[2015_Winter_Project_Week:DICOM_Module_Improvements|DICOM Module Improvements]] (Alireza Mehrtash, Andriy Fedorov, Ron Kikinis, Steve Pieper)
* [[2015_Winter_Project_Week:Slice_View_Annotations|Slice View Annotations]] (Alireza Mehrtash, Andriy Fedorov, Steve Pieper)

==Feature Extraction==
* [[3D_SIFT_VIEW| 3D SIFT Feature Visualization in Slicer]] (Matthew Toews, Steve Pieper, Nicole Aucoin, Andriy Fedorov, Raul San Jose, William Wells)

==Slicer4 Extensions==
* [[PET Tumor Segmentation]] (Christian Bauer)

==TMJOA RO1 - Collaboration with NAMIC==
* Improved model display (Francois Boudin, Steve Pieper)

==Infrastructure==

* [[2015_Winter_Project_Week:SlicerMicroMacroScale | Micro and Macro Scale in Slicer]] (Nicole Aucoin, Bradley Lowenkamp)
* [[2015_Winter_Project_Week:Markups | Markups]] (Nicole Aucoin)
* [[2015_Winter_Project_Week:Segmentations|Segmentations]] (Csaba Pinter, ?)
* [[2015_Winter_Project_Week:DICOM_References|DICOM references]] (Andrey Fedorov, Csaba Pinter, Steve Pieper)
* [[2015_Winter_Project_Week:OpenAtlas|Open Atlas]] (Bill Lorensen)
* automatic indexing of CLI modules in Slicer's extension store for nice dashboards based on the [https://github.com/commontk/ctk-cli-indexer ctk-cli-indexer] (JC, maybe Hans Meine remotely)

== '''Registrants''' ==

Do not add your name to this list - it is maintained by the organizers based on your paid registration.

# Nicole Aucoin ; Brigham and Women's Hospital
# Bamshad Azizi Koutenaei ; Children's National Health System
# Christian Bauer ; University of Iowa
# Peter Behringer ; Brigham and Women's Hospital
# Francois Budin ; UNC
# Byunghyun Cho; Koh Young Technology Inc.
# Xiao Da ; Athinoula A. Martinos Center for Biomedical Imaging
# Andriy Fedorov ; Brigham and Women's Hospital
# JC Fillion-Robin; Kitware, Inc.
# James Fishbaugh; University of Utah
# Yang Gao ; Brigham and Women's Hospital
# German Gonzalez ; Brigham and Women's Hospital
# Rola Harmouche ; Brigham and Women's Hospital
# Hans Johnson ; University of Iowa
# Tina Kapur ; Brigham and Women's Hospital
# Ron Kikinis ; Brigham and Women's Hospital
# Eun Young Regina Kim ; University of Iowa
# Sangyong Kim, Kohyoung Techonology
# Franklin King, Queen's University and Brigham and Women's Hospital
# Ming Li ; Brigham and Women's Hospital
# Ye Li ; Brigham and Women's Hospital
# Ruizhi Liao; Brigham and Women's Hospital
# Julia Lopinto ; The University of Michigan
# Bill Lorensen; Noware
# Bradley Lowenkamp; Medical Science Computing
# Lucie Macron ; The University of Michigan
# Artem Mamonov ; MGH Martinos
# Andre Mastmeyer ; University of Keil and Brigham and Women's Hospital
# Alireza Mehrtash ; Brigham and Women's Hospital
# James Miller ; GE Global Research
# Lauren O'Donnell; Brigham and Women's Hospital and Harvard Medical School
# Jorge Onieva ; Brigham and Women's Hospital
# Guillaume Pernelle; Imperial College London
# Steve Pieper ; Isomics Inc.
# Csaba Pinter ; Queens University
# Adam Rankin ; Robarts Research Institute
# James Ross ; Brigham and Women's Hospital
# Raul San Jose ; Brigham and Women's Hospital
# Anuja Sharma; University of Utah
# Matthew Toews; Brigham and Women's Hospital
# Junichi Tokuda ; Brigham and Women's Hospital
# Bo Wang ; SCI Institute University of Utah
# William Wells; Brigham and Women's Hospital
# Alexander Yarmarkovich; Brigham and Women's Hospital

Algorithm:Utah2

2014-10-15T23:19:47Z

Anuja: /* Tract-based longitudinal modeling of DTI data */

Back to [[Algorithm:Main|NA-MIC Algorithms]]
__NOTOC__
= Overview of Utah 2 Algorithms (PI: Guido Gerig) =

The Utah II group, guided by Guido Gerig and Marcel Prastawa, is interested in providing new methodology for analysis of DTI (including group analysis, fiber tract parametrization and quantification, longitudinal analysis), of image data including pathology (lesions (MS, lupus), tumor, trauma etc.), and methodology for registration and segmentation of serial/longitudinal image data. Methodology development will focus on subject-specific analysis in personalized medicine applications, and on providing normative data in the form of spatial altlases and descriptive information on geometric and appearance change in the presence of pathology.

= Utah 2 Projects =

{| cellpadding="10" style="text-align:left;"

|-
| style="width:15%" | [[Image:Caudate_snapshot.png|200px]]
| style="width:85%" |

== [[Projects:DiffeoMixedEffects|Diffeomorphic Shape Trajectories for Improved Longitudinal Segmentation and Statistics]] ==

<font color="red">'''Ongoing: '''</font> The goal of longitudinal studies is to quantify biological shape variability within and across individuals, and also to distinguish between normal and disease populations. However, data variability is influenced by outside sources such as image acquisition, image calibration, human expert judgment, and limited robustness of segmentation and registration algorithms. In this project, we propose a new method for the statistical analysis of longitudinal shape.
[[Projects:DiffeoMixedEffects|More...]]

''Muralidharan, P., Fishbaugh, J., Johnson, H., Durrleman, S., Paulsen, J., Gerig, G., Fletcher, P.T. Diffeomorphic shape trajectories for improved longitudinal segmentation and statistics. Proc. of Medical Image Computing and Computer Assisted Intervention (MICCAI '14). (2014).''

|-
| style="width:15%" | [[Image:Geodesic_subcort_panel.png|200px]]
| style="width:85%" |

== [[Projects:GeodesicShapeRegression|Geodesic Regression for Anatomical Shape Complexes]] ==

<font color="red">'''Ongoing: '''</font> Shape regression is of crucial importance for statistical shape analysis. It is
useful to find correlations between shape configuration and a continuous scalar parameter such as age, disease progression, drug delivery, or cognitive scores. In this project, we develop a parametric growth model for anatomical shape complexes which is the extension of scalar linear regression.
[[Projects:GeodesicShapeRegression|More...]]

''Fishbaugh, J., Prastawa, M., Gerig, G., Durrleman, S. Geodesic regression of image and shape data for improved modeling of 4D trajectories. IEEE International Symposium on Biomedical Imaging (ISBI '14). (2014).''

''Fishbaugh, J., Prastawa, M., Gerig, G., Durrleman, S. Geodesic image regression with a sparse parameterization of diffeomorphisms. Geometric Science of Information (GSI '13). LNCS vol 8085, pp. 95-102. (2013)''

''Fishbaugh, J., Prastawa, M., Gerig, G., Durrleman, S. Geodesic shape regression in the framework of currents. Proc. of Information Processing in Medical Imaging (IPMI '13). Vol 23, pp. 718-729. (2013)''

|-
| style="width:15%" | [[Image:Ace_modes_final.png|200px]]
| style="width:85%" |

== [[Projects:LongitudinalShapeAnalysis|Analysis of Longitudinal Shape Variability]] ==

<font color="red">'''Ongoing: '''</font> Statistical analysis of longitudinal imaging data is crucial for understanding normal anatomical development as well as disease progression. This fundamental task is challenging due to the difficulty in modeling longitudinal changes, such as growth, and comparing changes across different populations. In this project, we develop a new methods for the statistical analysis of longitudinal shape variability.
[[Projects:LongitudinalShapeAnalysis|More...]]

''Fishbaugh, J., Prastawa, M., Durrleman, S., Piven, J., Gerig, G. Analysis of Longitudinal Shape Variability via Subject Specific Growth Modeling. In N. Ayache et al. (Eds.): MICCAI 2012, Part I, LNCS 7510, pp. 730--737. (2012)''

|-

| style="width:15%" | [[Image:Segmentation_TBI_u.png|200px]]
| style="width:85%" |

== [[Projects:PathologyAnalysis|Analysis of Brain Images with Pathological Changes]] ==

<font color="red">'''Ongoing: '''</font> Quantification, analysis and display of brain pathology as observed in MRI is important for diagnosis, monitoring of disease progression, improved understanding of pathological processes and for studying treatment strategies.
We conduct research on developing novel methodologies that covers registration, segmentation, visualization of pathological structures
with the aim of quantifying changes due to lesions, bleeding, and deformations over time.
[[Projects:PathologyAnalysis|More...]]

''Bo Wang, Wei Liu, Marcel Prastawa, Andrei Irimia, Paul M. Vespa, John D. Van Horn, P. Thomas Fletcher, and Guido Gerig . 4D Active Cut: An Interactive Tool for Pathological Anatomy Modeling, In Biomedical Imaging (ISBI), 2014 IEEE 11th International Symposium on, pp. 529-532. IEEE, 2014.''

|-

| style="width:15%" | [[Image:namic_tract_longitudinal_main.png|200px]]
| style="width:85%" |

== [[Projects:TractLongitudinalDTI|Tract-based longitudinal modeling of DTI data]] ==

This project develops a methodology to explore subject-specific, DTI data obtained from brain's white matter tracts, available at multiple but often sparsely present timepoints. The aim is to have a continuous representation of the longitudinal diffusion changes along tract definitions.
[[Projects:TractLongitudinalDTI|More...]]

''A. Sharma, P.T. Fletcher, J.H. Gilmore, M.L. Escolar, A. Gupta, M. Styner, G. Gerig. “ Parametric Regression Scheme For Distributions: Analysis of DTI Fiber Tract Diffusion Changes In Early Brain Development,” In Proceedings of IEEE ISBI 2014, pp. 559-562. 2014.''

''A. Sharma, P.T. Fletcher, J.H. Gilmore, M.L. Escolar, A. Gupta, M. Styner, G. Gerig. “Spatiotemporal Modeling of Discrete-Time Distribution-Valued Data Applied to DTI Tract Evolution in Infant Neurodevelopment,” In Proceedings of IEEE ISBI 2013, pp. 684--687. 2013.''

''A. Sharma, S. Durrleman, J.H. Gilmore, G. Gerig. “Longitudinal Growth Modeling of Discrete-Time Functions with Application to DTI Tract Evolution in Early Neurodevelopment,” In Proceedings of IEEE ISBI 2012, pp. 1397--1400. 2012.''
|-

| style="width:15%" | [[Image:ColorFA-4D.png|200px]]
| style="width:85%" |

== [[Projects:longitudinaldwi|Longitudinal analysis of DWI]] ==

This project develops methodology to analyze serial/longitudinal DWI data, e.g. as baseline and follow-up in trauma, serial follow-up scans as acquired in the Huntington PREDICT study, or subject-specific white matter maturation in early brain development (see picture). [[Projects:longitudinaldwi|More...]]

''Sharma, A., Durrleman, S. , Gilmore, J.H. , Gerig, G. Longitudinal growth modeling of discrete-time functions with application to DTI tract evolution in early neurodevelopment. Proc. of 9th IEEE International Symposium on Biomedical Imaging (ISBI May'12), p.1397-1400.''
''N. Sadeghi, M. Prastawa, J. H. Gilmore, W. Lin, and G. Gerig, "Spatio-Temporal Analysis of Early Brain Development," in Proceedings IEEE Asilomar Conference on Signals, Systems & Computers, Nov. 2010, in print''

|-

| style="width:15%" | [[Image:DTIFiberTractStatistics.png|200px]]
| style="width:85%" |

== [[Projects:dtistatisticsfibers|Quantitative Description of White Matter Fiber Tracts]] ==

As part of this project, we are processing DTI data and computing statistics along white matter fiber tracts. For this, we are building a command line tool which allows the user to study the behavior of water diffusion along the length of the tracts.
[[Projects:dtistatisticsfibers|More...]]

|-

| | [[Image:Intracranial_evo.png|200px]]
| |

== [[Projects:Utah2ShapeRegression|Smooth Growth Trajectories from Time Series Shape Data]] ==

<font color="red">'''Ongoing:'''</font> Clinical research is interested in the spatiotemporal analysis of changes of anatomical shapes and structures, which potentially leads to improved understanding of the rate of change, locality and growth trajectory of structures of interest. This project develops a new methodology for the generation of a continuous growth model generated from a sparse set of shapes.
[[Projects:Utah2ShapeRegression|More...]]

''Fishbaugh, J., Durrleman, S., Gerig, G. Estimation of Smooth Growth Trajectories with Controlled Acceleration from Time Series Shape Data. Proc. of Medical Image Computing and Computer Assisted Intervention (MICCAI '11). September 2011.''

|-

| | [[Image:UtahAtlasSegmentation.png|200px]]
| |

== [[Projects:UtahAtlasSegmentation|Atlas Based Brain Segmentation]] ==

<font color="red">'''Ongoing: '''</font> Automatic segmentation can be performed reliably using priors from brain atlases and an image generative model. We have developed a tool that provides an automatic segmentation pipeline in a modular framework. Input are arbitrary number of image channels and a normative statistical brain atlas representing the population.
[[Projects:UtahAtlasSegmentation|More...]]

|-

| style="width:15%" | [[Image:Cbg-dtiatlas-tracts.png|200px]]
| style="width:85%" |

== [[Projects:DTIPopulationAnalysis|Group Analysis of DTI Fiber Tracts]] ==

Analysis of populations of diffusion images typically requires time-consuming manual segmentation of structures of interest to obtain correspondance for statistics. This project uses non-rigid registration of DTI images to produce a common coordinate system for hypothesis testing of diffusion properties. [[Projects:DTIPopulationAnalysis|More...]]

''Casey B. Goodlett, P. Thomas Fletcher, John H. Gilmore, Guido Gerig. Group Analysis of DTI Fiber Tract Statistics with Application to Neurodevelopment. NeuroImage 45 (1) Supp. 1, 2009. p. S133-S142.''

|-

| | [[Image:reg_main.png|200px]]
| |

== [[Projects:RegistrationEvaluation|Evaluation of Registration Algorithms]] ==

We are interested in comparing existing registration packages to determine how registration in Slicer3 can be improved. This work focuses on examining various packages researchers are currently using for registration and comparing results on a set of examples representative of common registration tasks.
[[Projects:RegistrationEvaluation|More...]]

|-

| | [[Image:regtestbedmain.png|200px]]
| |

== [[Projects:RegistrationTestbed|A Framework for Registration Evaluation and Exploration]] ==
In response to the difficulties of image registration, we propose an environment where registration applications can be explored, tested, and compared.
[[Projects:RegistrationTestbed|More...]]

|-

| | [[Image:LesionSegmentation.png|200px]]
| |

== [[Projects:LesionSegmentation|Lesion Segmentation]] ==

Quantification, analysis and display of brain pathology such as white matter lesions as observed in MRI is important for diagnosis, monitoring of disease progression, improved understanding of pathological processes and for developing new therapies.
[[Projects:LesionSegmentation|More...]]

Marcel Prastawa and Guido Gerig. Automatic MS Lesion Segmentation by Outlier Detection and Information Theoretic Region Partitioning. 3D Segmentation in the Clinic: A Grand Challenge II Workshop at Medical Image Computing and Computer Assisted Intervention (MICCAI) 2008.

|-

| | [[Image:DTINoiseStatistics.png|200px]]
| |

== [[Projects:DTINoiseStatistics|Influence of Imaging Noise on DTI Statistics]] ==

Clinical acquisition of diffusion weighted images with high signal to noise ratio remains a challenge. The goal of this project is to understand the impact of MR noise on quantiative statistics of diffusion properties such as anisotropy measures, trace, etc. [[Projects:DTINoiseStatistics|More...]]

|-

|-

| | [[Image:Meningiomasim_iter1.jpg|200px]]
| |

== [[Projects:UtahTumorSimulation|Tumor Simulation for Validating Change Tracking Applications]] ==

Determining extent of pathology as it changes over time is an important clinical task.
However, there is a lack of a reliable, objective ground truth for evaluating automatic tracking methods. We have developed a simulation tool that can generate MR images with known
tumor and edema.
[[Projects:UtahTumorSimulation|More...]]

|-

|}

Algorithm:Utah2

2014-10-15T23:18:52Z

Anuja: /* Tract-based longitudinal modeling of DTI data */

Back to [[Algorithm:Main|NA-MIC Algorithms]]
__NOTOC__
= Overview of Utah 2 Algorithms (PI: Guido Gerig) =

The Utah II group, guided by Guido Gerig and Marcel Prastawa, is interested in providing new methodology for analysis of DTI (including group analysis, fiber tract parametrization and quantification, longitudinal analysis), of image data including pathology (lesions (MS, lupus), tumor, trauma etc.), and methodology for registration and segmentation of serial/longitudinal image data. Methodology development will focus on subject-specific analysis in personalized medicine applications, and on providing normative data in the form of spatial altlases and descriptive information on geometric and appearance change in the presence of pathology.

= Utah 2 Projects =

{| cellpadding="10" style="text-align:left;"

|-
| style="width:15%" | [[Image:Caudate_snapshot.png|200px]]
| style="width:85%" |

== [[Projects:DiffeoMixedEffects|Diffeomorphic Shape Trajectories for Improved Longitudinal Segmentation and Statistics]] ==

<font color="red">'''Ongoing: '''</font> The goal of longitudinal studies is to quantify biological shape variability within and across individuals, and also to distinguish between normal and disease populations. However, data variability is influenced by outside sources such as image acquisition, image calibration, human expert judgment, and limited robustness of segmentation and registration algorithms. In this project, we propose a new method for the statistical analysis of longitudinal shape.
[[Projects:DiffeoMixedEffects|More...]]

''Muralidharan, P., Fishbaugh, J., Johnson, H., Durrleman, S., Paulsen, J., Gerig, G., Fletcher, P.T. Diffeomorphic shape trajectories for improved longitudinal segmentation and statistics. Proc. of Medical Image Computing and Computer Assisted Intervention (MICCAI '14). (2014).''

|-
| style="width:15%" | [[Image:Geodesic_subcort_panel.png|200px]]
| style="width:85%" |

== [[Projects:GeodesicShapeRegression|Geodesic Regression for Anatomical Shape Complexes]] ==

<font color="red">'''Ongoing: '''</font> Shape regression is of crucial importance for statistical shape analysis. It is
useful to find correlations between shape configuration and a continuous scalar parameter such as age, disease progression, drug delivery, or cognitive scores. In this project, we develop a parametric growth model for anatomical shape complexes which is the extension of scalar linear regression.
[[Projects:GeodesicShapeRegression|More...]]

''Fishbaugh, J., Prastawa, M., Gerig, G., Durrleman, S. Geodesic regression of image and shape data for improved modeling of 4D trajectories. IEEE International Symposium on Biomedical Imaging (ISBI '14). (2014).''

''Fishbaugh, J., Prastawa, M., Gerig, G., Durrleman, S. Geodesic image regression with a sparse parameterization of diffeomorphisms. Geometric Science of Information (GSI '13). LNCS vol 8085, pp. 95-102. (2013)''

''Fishbaugh, J., Prastawa, M., Gerig, G., Durrleman, S. Geodesic shape regression in the framework of currents. Proc. of Information Processing in Medical Imaging (IPMI '13). Vol 23, pp. 718-729. (2013)''

|-
| style="width:15%" | [[Image:Ace_modes_final.png|200px]]
| style="width:85%" |

== [[Projects:LongitudinalShapeAnalysis|Analysis of Longitudinal Shape Variability]] ==

<font color="red">'''Ongoing: '''</font> Statistical analysis of longitudinal imaging data is crucial for understanding normal anatomical development as well as disease progression. This fundamental task is challenging due to the difficulty in modeling longitudinal changes, such as growth, and comparing changes across different populations. In this project, we develop a new methods for the statistical analysis of longitudinal shape variability.
[[Projects:LongitudinalShapeAnalysis|More...]]

''Fishbaugh, J., Prastawa, M., Durrleman, S., Piven, J., Gerig, G. Analysis of Longitudinal Shape Variability via Subject Specific Growth Modeling. In N. Ayache et al. (Eds.): MICCAI 2012, Part I, LNCS 7510, pp. 730--737. (2012)''

|-

| style="width:15%" | [[Image:Segmentation_TBI_u.png|200px]]
| style="width:85%" |

== [[Projects:PathologyAnalysis|Analysis of Brain Images with Pathological Changes]] ==

<font color="red">'''Ongoing: '''</font> Quantification, analysis and display of brain pathology as observed in MRI is important for diagnosis, monitoring of disease progression, improved understanding of pathological processes and for studying treatment strategies.
We conduct research on developing novel methodologies that covers registration, segmentation, visualization of pathological structures
with the aim of quantifying changes due to lesions, bleeding, and deformations over time.
[[Projects:PathologyAnalysis|More...]]

''Bo Wang, Wei Liu, Marcel Prastawa, Andrei Irimia, Paul M. Vespa, John D. Van Horn, P. Thomas Fletcher, and Guido Gerig . 4D Active Cut: An Interactive Tool for Pathological Anatomy Modeling, In Biomedical Imaging (ISBI), 2014 IEEE 11th International Symposium on, pp. 529-532. IEEE, 2014.''

|-

| style="width:15%" | [[Image:namic_tract_longitudinal_main.png|200px]]
| style="width:85%" |

== [[Projects:TractLongitudinalDTI|Tract-based longitudinal modeling of DTI data]] ==

This project develops a methodology to explore subject-specific, DTI data obtained from brain's white matter tracts, available at multiple but often sparsely present timepoints. The aim is to have a continuous representation of the longitudinal diffusion changes along tract definitions.
[[Projects:TractLongitudinalDTI|More...]]

''A. Sharma, P.T. Fletcher, J.H. Gilmore, M.L. Escolar, A. Gupta, M. Styner, G. Gerig. “ Parametric Regression Scheme For Distributions: Analysis of DTI Fiber Tract Diffusion Changes In Early Brain Development,” In Proceedings of the 2014 IEEE 11th International Symposium on Biomedical Imaging (ISBI), pp. 559-562. 2014.''

''A. Sharma, P.T. Fletcher, J.H. Gilmore, M.L. Escolar, A. Gupta, M. Styner, G. Gerig. “Spatiotemporal Modeling of Discrete-Time Distribution-Valued Data Applied to DTI Tract Evolution in Infant Neurodevelopment,” In Proceedings of the 2013 IEEE 10th International Symposium on Biomedical Imaging (ISBI), pp. 684--687. 2013.''

''A. Sharma, S. Durrleman, J.H. Gilmore, G. Gerig. “Longitudinal Growth Modeling of Discrete-Time Functions with Application to DTI Tract Evolution in Early Neurodevelopment,” In Proceedings of IEEE ISBI 2012, pp. 1397--1400. 2012.''
|-

| style="width:15%" | [[Image:ColorFA-4D.png|200px]]
| style="width:85%" |

== [[Projects:longitudinaldwi|Longitudinal analysis of DWI]] ==

This project develops methodology to analyze serial/longitudinal DWI data, e.g. as baseline and follow-up in trauma, serial follow-up scans as acquired in the Huntington PREDICT study, or subject-specific white matter maturation in early brain development (see picture). [[Projects:longitudinaldwi|More...]]

''Sharma, A., Durrleman, S. , Gilmore, J.H. , Gerig, G. Longitudinal growth modeling of discrete-time functions with application to DTI tract evolution in early neurodevelopment. Proc. of 9th IEEE International Symposium on Biomedical Imaging (ISBI May'12), p.1397-1400.''
''N. Sadeghi, M. Prastawa, J. H. Gilmore, W. Lin, and G. Gerig, "Spatio-Temporal Analysis of Early Brain Development," in Proceedings IEEE Asilomar Conference on Signals, Systems & Computers, Nov. 2010, in print''

|-

| style="width:15%" | [[Image:DTIFiberTractStatistics.png|200px]]
| style="width:85%" |

== [[Projects:dtistatisticsfibers|Quantitative Description of White Matter Fiber Tracts]] ==

As part of this project, we are processing DTI data and computing statistics along white matter fiber tracts. For this, we are building a command line tool which allows the user to study the behavior of water diffusion along the length of the tracts.
[[Projects:dtistatisticsfibers|More...]]

|-

| | [[Image:Intracranial_evo.png|200px]]
| |

== [[Projects:Utah2ShapeRegression|Smooth Growth Trajectories from Time Series Shape Data]] ==

<font color="red">'''Ongoing:'''</font> Clinical research is interested in the spatiotemporal analysis of changes of anatomical shapes and structures, which potentially leads to improved understanding of the rate of change, locality and growth trajectory of structures of interest. This project develops a new methodology for the generation of a continuous growth model generated from a sparse set of shapes.
[[Projects:Utah2ShapeRegression|More...]]

''Fishbaugh, J., Durrleman, S., Gerig, G. Estimation of Smooth Growth Trajectories with Controlled Acceleration from Time Series Shape Data. Proc. of Medical Image Computing and Computer Assisted Intervention (MICCAI '11). September 2011.''

|-

| | [[Image:UtahAtlasSegmentation.png|200px]]
| |

== [[Projects:UtahAtlasSegmentation|Atlas Based Brain Segmentation]] ==

<font color="red">'''Ongoing: '''</font> Automatic segmentation can be performed reliably using priors from brain atlases and an image generative model. We have developed a tool that provides an automatic segmentation pipeline in a modular framework. Input are arbitrary number of image channels and a normative statistical brain atlas representing the population.
[[Projects:UtahAtlasSegmentation|More...]]

|-

| style="width:15%" | [[Image:Cbg-dtiatlas-tracts.png|200px]]
| style="width:85%" |

== [[Projects:DTIPopulationAnalysis|Group Analysis of DTI Fiber Tracts]] ==

Analysis of populations of diffusion images typically requires time-consuming manual segmentation of structures of interest to obtain correspondance for statistics. This project uses non-rigid registration of DTI images to produce a common coordinate system for hypothesis testing of diffusion properties. [[Projects:DTIPopulationAnalysis|More...]]

''Casey B. Goodlett, P. Thomas Fletcher, John H. Gilmore, Guido Gerig. Group Analysis of DTI Fiber Tract Statistics with Application to Neurodevelopment. NeuroImage 45 (1) Supp. 1, 2009. p. S133-S142.''

|-

| | [[Image:reg_main.png|200px]]
| |

== [[Projects:RegistrationEvaluation|Evaluation of Registration Algorithms]] ==

We are interested in comparing existing registration packages to determine how registration in Slicer3 can be improved. This work focuses on examining various packages researchers are currently using for registration and comparing results on a set of examples representative of common registration tasks.
[[Projects:RegistrationEvaluation|More...]]

|-

| | [[Image:regtestbedmain.png|200px]]
| |

== [[Projects:RegistrationTestbed|A Framework for Registration Evaluation and Exploration]] ==
In response to the difficulties of image registration, we propose an environment where registration applications can be explored, tested, and compared.
[[Projects:RegistrationTestbed|More...]]

|-

| | [[Image:LesionSegmentation.png|200px]]
| |

== [[Projects:LesionSegmentation|Lesion Segmentation]] ==

Quantification, analysis and display of brain pathology such as white matter lesions as observed in MRI is important for diagnosis, monitoring of disease progression, improved understanding of pathological processes and for developing new therapies.
[[Projects:LesionSegmentation|More...]]

Marcel Prastawa and Guido Gerig. Automatic MS Lesion Segmentation by Outlier Detection and Information Theoretic Region Partitioning. 3D Segmentation in the Clinic: A Grand Challenge II Workshop at Medical Image Computing and Computer Assisted Intervention (MICCAI) 2008.

|-

| | [[Image:DTINoiseStatistics.png|200px]]
| |

== [[Projects:DTINoiseStatistics|Influence of Imaging Noise on DTI Statistics]] ==

Clinical acquisition of diffusion weighted images with high signal to noise ratio remains a challenge. The goal of this project is to understand the impact of MR noise on quantiative statistics of diffusion properties such as anisotropy measures, trace, etc. [[Projects:DTINoiseStatistics|More...]]

|-

|-

| | [[Image:Meningiomasim_iter1.jpg|200px]]
| |

== [[Projects:UtahTumorSimulation|Tumor Simulation for Validating Change Tracking Applications]] ==

Determining extent of pathology as it changes over time is an important clinical task.
However, there is a lack of a reliable, objective ground truth for evaluating automatic tracking methods. We have developed a simulation tool that can generate MR images with known
tumor and edema.
[[Projects:UtahTumorSimulation|More...]]

|-

|}

Projects:TractLongitudinalDTI

2014-10-15T23:11:21Z

Anuja: /* Parametric linear regression for distribution-valued data */

Back to [[Algorithm:Utah2|Utah 2 Algorithms]]
__NOTOC__

== Tract-based longitudinal modeling of DTI data ==

This project develops a methodology to explore subject-specific, DTI data obtained from brain's white matter tracts, available at multiple but often sparsely present timepoints. The challenge is to develop a continuous spatio-temporal growth model, given discrete 4D DWI images. This would enable comparison of growth trajectories across subjects and along tracts which are biologically of interest in developmental and pathological changes.

== Background ==

We use the arc length parametrization scheme initially proposed by Corouge et al. It represents white matter fiber tracts obtained via streamline tractography in the brain's atlas tensor image as a function of arc length. (Atlas construction uses unbiased atlas building schemes followed by back transformations to subjects' DTI images to obtain identical fiber tract geometry across subjects, populated with subject specific diffusion data). We use the mean diffusion scalar invariants derived from these individual fiber bundle cross sections, as our input longitudinal diffusion profiles.

{| border="0" style="background:transparent;"
|[[Image:Tract-stats.png|thumb|700px|White matter diffusion properties along fiber tract: Left: A fiber tract with an origin plane defined for arc length parametrization, Right: FA mean and standard deviation as function of arc-length. Dots mark location of coordinate origin.]]
|-
|}

== Subject-specific spatiotemporal continuous growth model ==

We propose the use of Verhulst-Pearl logistic equation to capture temporal changes, while using a non-parametric kernel along arc length to account for biologically motivated functional along-tract relationship in the diffusion data.

{| border="0" style="background:transparent;"
|[[Image:logistic_eq.png|thumb|600px|Logistic equation with P as the population value at a given time, r being the growth rate of the population and K being the carrying capacity (also the limiting asymptote value).]]
|[[Image:logistic_simulation_growthrate_r.png|thumb|400px|Effect of growth rate r on the logistic curve.]]
|[[Image:logistic_simulation_asymptote_K.png|thumb|400px|Effect of carrying capacity K on the logistic curve.]]
|-
|}

From the definition of the logistic function, the parameter r represents the growth rate of the diffusion invariant and the parameter K represents the asymptote value. The overall function shape intuitively follows the growth pattern we expect to see during brain maturation. The diffusion invariants start with an initial diffusion profile along tract, and have a non linear temporal growth trajectory showing maximum changes in early childhood and then slowing down or almost saturating at a certain age. (For instance, observed diffusion changes are much more in neonates than in an adult brain). The temporal trajectories may also differ along the tract's length giving localized changes. Since our method gives us continuous along-tract, growth trajectories all along time, we can compare subjects with respect to differences in diffusion profiles at birth, growth rates at any given age as well as the asymptote saturation values. This gives us important information to understand delayed or abnormal brain maturation by comparing a normative growth surface with an individual's or by comparing the model's parameters across subjects.

== Results ==

Below are some results using synthetic data. For more validation results and extension of the framework to jointly estimate individual subject trajectories together with a normative trajectory, refer to [http://ieeexplore.ieee.org/xpl/login.jsp?tp=&arnumber=6235829&url=http%3A%2F%2Fieeexplore.ieee.org%2Fxpls%2Fabs_all.jsp%3Farnumber%3D6235829 Sharma et al. (ISBI '12)].

{| border="0" style="background:transparent;"
|[[Image:namic_tract_longitudinal_main.png|thumb|400px|Longitudinal modeling of synthetic data. The data is generated from a template neonate FA mean curve and uses known logistic parameters to generate the three timepoint curves (red). Our method faithfully recovers the original curves (blue) while simultaneously creating a continuous longitudinal growth profile (green) without having any prior knowledge of the logistic parameters used for data generation. It also estimates an unbiased time=0 template curve (black) along with parameters r and K to avoid biasing the estimation with any assumed initialization at time=0.]]
|[[Image:namic_tract_longitudinal_3d.png|thumb|400px|Results on real data (mean FA curves for a subject at three timepoints:neonate, 1 year and 2 year).]]
|-
|}
{| border="0" style="background:transparent;"
|[[Image:namic_tract_longitudinal_p1p2only.png|thumb|400px|For the above real data (right), these are the final along-tract initial values and the final estimates for parameters-growth rate r(mentioned as p1 here) and asymptote K(mentioned as p2 here).]]
|[[Image:namic_tract_longitudinal_alphaonly.png|thumb|400px|The complete continuous growth grid as estimated for the real data above.]]

|-
|}

The below images show jointly estimated personalized trajectories for 15 control subjects along with the average growth trajectory. The estimated model parameters for 15 subjects as well as the average trajectory are also shown. The normative trajectory is colored by the local standard deviation. It points to the fact that despite individual variability in the maturation process, there is a strong agreement in the asymptote FA values seen around a gestational age of 2.5 years indicating a relative stabilization of the white matter changes across subjects. The framework thus quantifies patient-specific changes in serial diffusion data given discrete-time diffusion curves.

{| border="0" style="background:transparent;"
|[[Image:CONTE_Final_3d.png|thumb|400px|The jointly estimated individual growth trajectories for 4 of the 15 control subjects.]]
|[[Image:CONTE_Norm_Params.png|thumb|400px|Estimated model parameters for 15 subjects together with the normative growth trajectory colored by the local variability and the estimated normative model parameters.]]

|-
|}

== Experiments with Huntington's data ==

During the NAMIC Winter project week, we worked on registering subjects with Huntington's disease in the same coordinate space as control subjects. We then applied the above framework [http://ieeexplore.ieee.org/xpl/login.jsp?tp=&arnumber=6235829&url=http%3A%2F%2Fieeexplore.ieee.org%2Fxpls%2Fabs_all.jsp%3Farnumber%3D6235829 Sharma et al. ISBI'12] to quantify the differences in normal expected aging versus the accelerated white matter changes expected in HD. Details of the data pre-processing steps and image registration are available on the [http://www.na-mic.org/Wiki/index.php/2012_Winter_Project_Week:_DTI_Change_Modeling Winter AHM page]. Some results are summarized here for FA value along the genu tract.

Two subjects- one being a HD patient (10027) with a high burden factor (higher factor value (of factor 12) implying that the subjects is closer to the onset) and the other being a control subject (10004) are chosen for concept demonstration. Both have a similar baseline scan age (42 and 43 years respectively) as well as a similar time separation between follow up scans. This allows an intuitive comparison of the estimated growth trajectories.

{| border="0" style="background:transparent;"
|[[Image:ORIG_hd_cn.png|thumb|300px|center|FA curves corresponding to the genu tract (middle clipped portions) for one HD case (10027) and one control case (10004). Visual inspection clearly shows the huge decrease relative to the Control in the HD subject. On the other hand, the control shows normal expected decrease due to aging. Red-timepoint 1, Green-timepoint 2, Blue-timepoint 3. The left is HD case and the right plot is Control.]]
|[[Image:HD_vs_CN_3d.png|thumb|300px|center|The red trajectory corresponds the the HD subjects while the blue is a control subject with healthy aging. HD patient clearly shows a much sharper FA decrease along time than the Control.]]
|[[Image:HD_vs_CN_params.png|thumb|300px|center|The estimated model parameters (Red-HD case, Blue-Control case) for the two subjects. The top plot is the rate of change per unit time, per unit FA value. The middle plot is the function asymptote and the last plot is the common baseline curve estimation for providing a common reference frame for model estimation and comparison. The asymptote clearly shows a much lower FA range for HD when compared to the Control which already shows promising early prediction capabilities for HD cases where white matter decay is much faster than healthy patients. Also, a higher per unit rate of change quantifies the sharper FA decrease compared to control.]]
|-
|}

== Experiments with Krabbe's data ==

We have also applied the framework to study white matter changes in infants with Krabbe's disease. Here we show some preliminary results for the FA tract profile along the genu tract for a single subject with Krabbe's disease. In theory, FA non-linearly increases with time along genu owing to a healthy brain maturation process. However, in Krabbe's case, the FA does not follow a monotonic change pattern.

{| border="0" style="background:transparent;"
|[[Image:Genu_fa_3d_conte_vs_krabbe.png|thumb|400px|center|FA curves corresponding to the genu tract for the Krabbe's patient are shown (Neonate:black, 6months:cyan, 1year:magenta). The other FA profiles are for the 15 healthy control subjects as discussed previously (neonate:red, 1year:green, 2year:blue). This also highlights the fact that the framework does not assume uniformity or correspondence in the scan timings across subjects.]]
|[[Image:Genu_fa_3dnorm_conte_vs_krabbe.png|thumb|400px|center|The black trajectory is the normative FA growth estimated using 15 controls as discussed previously. The three FA curves for the Krabbe case are also plotted and clearly show how the patient is lagging behind on expected brain maturation.]]
|-
|}

== Future work ==

As we see in the Krabbe experiment, the model assumes a monotonic change along time. Neighboring locations along the tract can have different, yet correlated time changes. For eg. we can see localized increases or decreases along the tract length with respect to the time evolution. However, owing to the definition of the logistic function, any non-monotonic behavior along time at any given tract location is not modeled accurately. Therefore, in cases like the Krabbe's subject, we need a more relaxed and flexible semi-parametric or non-parametric model to remove the constraints coming in from the parametric nature of the current model. This is currently work under progress.
To utilize the quantification of growth trajectories enabled by this method, we also require a systematic hypothesis testing scheme to draw conclusive results of individuals differing from normative patterns or populations differing in their evolution behaviors. This too is currently being worked upon.

= References =

* Sharma, A., Durrleman, S. , Gilmore, J.H. , Gerig, G. Longitudinal growth modeling of discrete-time functions with application to DTI tract evolution in early neurodevelopment. Proc. of 9th IEEE International Symposium on Biomedical Imaging (ISBI May'12), p.1397-1400.
* Corouge, I., Fletcher, P.T., Joshi, S., Gouttard, S., Gerig, G., 2006. Fiber tract-oriented statistics for quantitative diffusion tensor MRI analysis. Med Image Anal, pp. 786-798.
* Goodlett, C.B., Fletcher, P.T., Gilmore, J.H., Gerig, G., 2009. Group analysis of DTI fiber tract statistics with application to neurodevelopment. Neuroimage, pp. S133-142.

==Key Investigators==
* Utah: Anuja Sharma, Stanley Durrleman, Guido Gerig
* INRIA/ICM, Pitie Salpetriere Hospital: Stanley Durrleman
[http://www.na-mic.org/Wiki/index.php/Algorithm:Utah2 Back to Utah 2 Algorithm Core]

== Regression of probability distributions obtained along white matter tracts to create spatio-temporal growth models ==

Temporal modeling frameworks often operate on scalar variables by
summarizing data at initial stages as statistical summaries of the underlying
distributions. For instance, DTI analysis often employs
summary statistics, like mean, for regions of interest and properties
along fiber tracts for population studies and hypothesis testing.
This reduction via discarding of variability information may introduce
significant errors which propagate through the procedures. Moreover, the data variability information is lost at
all the intermediate interpolated times.

{| border="0" style="background:transparent;"
|[[Image:DistributionRegression_Synthetic.png|thumb|600px|center|Left:Two synthetically generated distributions along time
(blue:t0, orange:t1) with translated locations and opposite skews.
Right: Simple linear interpolation using classic single-valued summary
statistics (means: red, medians: blue) and ignoring data variability
(shown as error bars) at t0 and t1. The decision to choose mean versus median can heavily influence the final regression estimates. Additionally, once the inherent variability is discarded for an early decision regarding a summary statistics, it cannot be recovered for the interpolated time points.]]
|-
|}

We propose a novel framework which uses distribution-valued variables
to retain and utilize the local variability information. We propose two different regression approaches to achieve our goals. The first is a non-parametric moving average method where minimal assumptions are involved in terms of the growth trajectory evolution (Sharma ISBI 2013). The second is a parametric regression scheme where the expected temporal trend is linear (Sharma ISBI 2014). It provides a compact representation in terms of model parameters for further statistical inference and has a closed form solution.

Our driving application is the modeling of age-related changes along DTI
white matter tracts. Results are shown for the spatiotemporal population
trajectory of genu tract estimated from a group of healthy infants and compared with an infant with Krabbe's disease.

{| border="0" style="background:transparent;"
|[[Image:DWIalongTime.png|thumb|400px|center|Model age related DTI changes in early neurodevelopment.]]
|[[Image:genu_distributions_along_time.png|thumb|600px|center|Distributions of DTI-derived scalar diffusion parameters like FA are obtained along the length of the genu tract. The distributions correspond to the cross-sections of the 3D tract geometry and are a function of the arc-length along the tract's total length as we move from one end of the tract to the other. This provides a functional curve which is parametrized by a feature of the local tract geometry (arc-length along the tract in this case) and is attributed by probability distributions (instead of scalar values like the 'mean' FA obtained from these local distributions).]]
|-
|}

=== Non-parametric regression for distribution-valued data ===

The proposed method
presents a completely nonparametric regression framework for spatial-temporal evolution of
statistical distributions. This avoids any parametric assumptions regarding the nature of the
underlying noise model in the DTI data and retains and utilizes the variability information to
estimate the regression trend. The estimated trajectory is completely characterized by statistical
distributions in space and time enabling a very powerful statistical inference framework. Both
population and individual trajectories can be estimated. The concept of ’distance’ between distributions and an ’average’ of distributions is employed. The dissimilarity metric employed is the Mallow's distance between distributions. It is the L2 norm of the generic Wasserstein metric and captures translation, changing width and shape of the participating distributions.

{| border="0" style="background:transparent;"
|[[Image:dist2_3d_bary_300_revrev.png|thumb|400px|center|Temporal evolution of the synthetic distributions between two timepoints as estimated by the proposed method.]]
|[[Image:bary_stat_300.png|thumb|400px|center|Since the regression estimates the complete probability distribution continuously along time, we now have access to the complete variability information at the interpolated time points. This can enable statistical inference using statistical summary measures as well as distribution differences at time points where the original image data is not available.]]
|-
|}

{| border="0" style="background:transparent;"
|[[Image:conte_krabbe_median_3d_300_2.png|thumb|400px|center|The medians of the estimated probability distributions are shown here for the healthy population (in red/pink) versus a single infant with Krabbe's disease (blue/black). Note that the method can applied to estimate normative population trajectories as well as subject-specific evolutions. Each point on these statistical manifolds are characterized by the complete distribution information (instead of just a scalar value).]]
|[[Image:conte_vs_krabbe_quartiles.png|thumb|600px|center|Highlighting the quantiles obtained at one of the arc length locations in the left plot. The Krabbe's subject seems to lag behind the normative population and the difference can be quantified in terms of distribution-based inferences.]]
|-
|}

=== Parametric linear regression for distribution-valued data ===

We build on the motivation as highlighted in the previous distribution-based regression. However,
the previous method is completely nonparametric leading to an intensive and complex statistical
inference framework based on comparison of statistical manifolds. Therefore, we simplify the
analysis for situations where a linear time-course trajectory can be assumed. We apply the classic
linear regression technique, extended to work with distribution-valued data. The advantage lies in
the compact description of the spatial-temporal trajectory in terms of the estimated regression
coefficients which can now be conveniently used for statistical inference. A synthetic validation
experiment is conducted to show the improved reliability and robustness of this method when
compared with methods working with statistical summaries of the underlying data.

{| border="0" style="background:transparent;"
|[[Image:Baryhist3d1.png|thumb|400px|center|The dependent variable is the barycentric FA distribution estimated at each arc length location along the tract, using the original FA distributions available from all subjects within the corresponding age group. The regression estimates the continuous spatiotemporal trends in terms of the model parameters using the respective pairs of age and FA distributions as the 'observed' data.]]
|[[Image:Agedist.png|thumb|400px|center|The independent variable is the age distribution of healthy infants within each age group (age groups are clustered around 2 months, 1 year and 2 year in the given population).]]
|-
|}

{| border="0" style="background:transparent;"
|[[Image:PopuCN_dist_3d1.png|thumb|300px|center|Continuous spatiotemporal normative growth trajectory: Estimated mean FA (gray), Age range where control
data is originally available (green, orange, purple).]]
|[[Image:PopuvsKrabbe_dist_3d_2_edit1.png|thumb|300px|center|Krabbe's subject's scans (solid lines): (14 days, 6 mo., 1 yr.) with respect to the estimated healthy trajectory.]]
|[[Image:PopuvsKrabbe_dist_3d31.png|thumb|300px|center|Krabbe's subject's scans:mean FA (solid lines).Control population:estimated mean FA (dashed lines) +/- 3*std.dev. bounds (calculated for Gaussian age distributions centered at matched timepoints with std.dev.=5 days).]]
|-
|}

= References =

* A. Sharma, P.T. Fletcher, J.H. Gilmore, M.L. Escolar, A. Gupta, M. Styner, G. Gerig. “ Parametric Regression Scheme For Distributions: Analysis of DTI Fiber Tract Diffusion Changes In Early Brain Development,” In Proceedings of the 2014 IEEE 11th International Symposium on Biomedical Imaging (ISBI), pp. 559-562. 2014.
* A. Sharma, P.T. Fletcher, J.H. Gilmore, M.L. Escolar, A. Gupta, M. Styner, G. Gerig. “Spatiotemporal Modeling of Discrete-Time Distribution-Valued Data Applied to DTI Tract Evolution in Infant Neurodevelopment,” In Proceedings of the 2013 IEEE 10th International Symposium on Biomedical Imaging (ISBI), pp. 684--687. 2013.
* A. Sharma, S. Durrleman, J.H. Gilmore, G. Gerig. “Longitudinal Growth Modeling of Discrete-Time Functions with Application to DTI Tract Evolution in Early Neurodevelopment,” In Proceedings of IEEE ISBI 2012, pp. 1397--1400. 2012.
* Corouge, I., Fletcher, P.T., Joshi, S., Gouttard, S., Gerig, G., 2006. Fiber tract-oriented statistics for quantitative diffusion tensor MRI analysis. Med Image Anal, pp. 786-798.
* Goodlett, C.B., Fletcher, P.T., Gilmore, J.H., Gerig, G., 2009. Group analysis of DTI fiber tract statistics with application to neurodevelopment. Neuroimage, pp. S133-142.

Projects:TractLongitudinalDTI

2014-10-15T23:09:18Z

Anuja: /* Parametric linear regression for distribution-valued data */

Back to [[Algorithm:Utah2|Utah 2 Algorithms]]
__NOTOC__

== Tract-based longitudinal modeling of DTI data ==

This project develops a methodology to explore subject-specific, DTI data obtained from brain's white matter tracts, available at multiple but often sparsely present timepoints. The challenge is to develop a continuous spatio-temporal growth model, given discrete 4D DWI images. This would enable comparison of growth trajectories across subjects and along tracts which are biologically of interest in developmental and pathological changes.

== Background ==

We use the arc length parametrization scheme initially proposed by Corouge et al. It represents white matter fiber tracts obtained via streamline tractography in the brain's atlas tensor image as a function of arc length. (Atlas construction uses unbiased atlas building schemes followed by back transformations to subjects' DTI images to obtain identical fiber tract geometry across subjects, populated with subject specific diffusion data). We use the mean diffusion scalar invariants derived from these individual fiber bundle cross sections, as our input longitudinal diffusion profiles.

{| border="0" style="background:transparent;"
|[[Image:Tract-stats.png|thumb|700px|White matter diffusion properties along fiber tract: Left: A fiber tract with an origin plane defined for arc length parametrization, Right: FA mean and standard deviation as function of arc-length. Dots mark location of coordinate origin.]]
|-
|}

== Subject-specific spatiotemporal continuous growth model ==

We propose the use of Verhulst-Pearl logistic equation to capture temporal changes, while using a non-parametric kernel along arc length to account for biologically motivated functional along-tract relationship in the diffusion data.

{| border="0" style="background:transparent;"
|[[Image:logistic_eq.png|thumb|600px|Logistic equation with P as the population value at a given time, r being the growth rate of the population and K being the carrying capacity (also the limiting asymptote value).]]
|[[Image:logistic_simulation_growthrate_r.png|thumb|400px|Effect of growth rate r on the logistic curve.]]
|[[Image:logistic_simulation_asymptote_K.png|thumb|400px|Effect of carrying capacity K on the logistic curve.]]
|-
|}

From the definition of the logistic function, the parameter r represents the growth rate of the diffusion invariant and the parameter K represents the asymptote value. The overall function shape intuitively follows the growth pattern we expect to see during brain maturation. The diffusion invariants start with an initial diffusion profile along tract, and have a non linear temporal growth trajectory showing maximum changes in early childhood and then slowing down or almost saturating at a certain age. (For instance, observed diffusion changes are much more in neonates than in an adult brain). The temporal trajectories may also differ along the tract's length giving localized changes. Since our method gives us continuous along-tract, growth trajectories all along time, we can compare subjects with respect to differences in diffusion profiles at birth, growth rates at any given age as well as the asymptote saturation values. This gives us important information to understand delayed or abnormal brain maturation by comparing a normative growth surface with an individual's or by comparing the model's parameters across subjects.

== Results ==

Below are some results using synthetic data. For more validation results and extension of the framework to jointly estimate individual subject trajectories together with a normative trajectory, refer to [http://ieeexplore.ieee.org/xpl/login.jsp?tp=&arnumber=6235829&url=http%3A%2F%2Fieeexplore.ieee.org%2Fxpls%2Fabs_all.jsp%3Farnumber%3D6235829 Sharma et al. (ISBI '12)].

{| border="0" style="background:transparent;"
|[[Image:namic_tract_longitudinal_main.png|thumb|400px|Longitudinal modeling of synthetic data. The data is generated from a template neonate FA mean curve and uses known logistic parameters to generate the three timepoint curves (red). Our method faithfully recovers the original curves (blue) while simultaneously creating a continuous longitudinal growth profile (green) without having any prior knowledge of the logistic parameters used for data generation. It also estimates an unbiased time=0 template curve (black) along with parameters r and K to avoid biasing the estimation with any assumed initialization at time=0.]]
|[[Image:namic_tract_longitudinal_3d.png|thumb|400px|Results on real data (mean FA curves for a subject at three timepoints:neonate, 1 year and 2 year).]]
|-
|}
{| border="0" style="background:transparent;"
|[[Image:namic_tract_longitudinal_p1p2only.png|thumb|400px|For the above real data (right), these are the final along-tract initial values and the final estimates for parameters-growth rate r(mentioned as p1 here) and asymptote K(mentioned as p2 here).]]
|[[Image:namic_tract_longitudinal_alphaonly.png|thumb|400px|The complete continuous growth grid as estimated for the real data above.]]

|-
|}

The below images show jointly estimated personalized trajectories for 15 control subjects along with the average growth trajectory. The estimated model parameters for 15 subjects as well as the average trajectory are also shown. The normative trajectory is colored by the local standard deviation. It points to the fact that despite individual variability in the maturation process, there is a strong agreement in the asymptote FA values seen around a gestational age of 2.5 years indicating a relative stabilization of the white matter changes across subjects. The framework thus quantifies patient-specific changes in serial diffusion data given discrete-time diffusion curves.

{| border="0" style="background:transparent;"
|[[Image:CONTE_Final_3d.png|thumb|400px|The jointly estimated individual growth trajectories for 4 of the 15 control subjects.]]
|[[Image:CONTE_Norm_Params.png|thumb|400px|Estimated model parameters for 15 subjects together with the normative growth trajectory colored by the local variability and the estimated normative model parameters.]]

|-
|}

== Experiments with Huntington's data ==

During the NAMIC Winter project week, we worked on registering subjects with Huntington's disease in the same coordinate space as control subjects. We then applied the above framework [http://ieeexplore.ieee.org/xpl/login.jsp?tp=&arnumber=6235829&url=http%3A%2F%2Fieeexplore.ieee.org%2Fxpls%2Fabs_all.jsp%3Farnumber%3D6235829 Sharma et al. ISBI'12] to quantify the differences in normal expected aging versus the accelerated white matter changes expected in HD. Details of the data pre-processing steps and image registration are available on the [http://www.na-mic.org/Wiki/index.php/2012_Winter_Project_Week:_DTI_Change_Modeling Winter AHM page]. Some results are summarized here for FA value along the genu tract.

Two subjects- one being a HD patient (10027) with a high burden factor (higher factor value (of factor 12) implying that the subjects is closer to the onset) and the other being a control subject (10004) are chosen for concept demonstration. Both have a similar baseline scan age (42 and 43 years respectively) as well as a similar time separation between follow up scans. This allows an intuitive comparison of the estimated growth trajectories.

{| border="0" style="background:transparent;"
|[[Image:ORIG_hd_cn.png|thumb|300px|center|FA curves corresponding to the genu tract (middle clipped portions) for one HD case (10027) and one control case (10004). Visual inspection clearly shows the huge decrease relative to the Control in the HD subject. On the other hand, the control shows normal expected decrease due to aging. Red-timepoint 1, Green-timepoint 2, Blue-timepoint 3. The left is HD case and the right plot is Control.]]
|[[Image:HD_vs_CN_3d.png|thumb|300px|center|The red trajectory corresponds the the HD subjects while the blue is a control subject with healthy aging. HD patient clearly shows a much sharper FA decrease along time than the Control.]]
|[[Image:HD_vs_CN_params.png|thumb|300px|center|The estimated model parameters (Red-HD case, Blue-Control case) for the two subjects. The top plot is the rate of change per unit time, per unit FA value. The middle plot is the function asymptote and the last plot is the common baseline curve estimation for providing a common reference frame for model estimation and comparison. The asymptote clearly shows a much lower FA range for HD when compared to the Control which already shows promising early prediction capabilities for HD cases where white matter decay is much faster than healthy patients. Also, a higher per unit rate of change quantifies the sharper FA decrease compared to control.]]
|-
|}

== Experiments with Krabbe's data ==

We have also applied the framework to study white matter changes in infants with Krabbe's disease. Here we show some preliminary results for the FA tract profile along the genu tract for a single subject with Krabbe's disease. In theory, FA non-linearly increases with time along genu owing to a healthy brain maturation process. However, in Krabbe's case, the FA does not follow a monotonic change pattern.

{| border="0" style="background:transparent;"
|[[Image:Genu_fa_3d_conte_vs_krabbe.png|thumb|400px|center|FA curves corresponding to the genu tract for the Krabbe's patient are shown (Neonate:black, 6months:cyan, 1year:magenta). The other FA profiles are for the 15 healthy control subjects as discussed previously (neonate:red, 1year:green, 2year:blue). This also highlights the fact that the framework does not assume uniformity or correspondence in the scan timings across subjects.]]
|[[Image:Genu_fa_3dnorm_conte_vs_krabbe.png|thumb|400px|center|The black trajectory is the normative FA growth estimated using 15 controls as discussed previously. The three FA curves for the Krabbe case are also plotted and clearly show how the patient is lagging behind on expected brain maturation.]]
|-
|}

== Future work ==

As we see in the Krabbe experiment, the model assumes a monotonic change along time. Neighboring locations along the tract can have different, yet correlated time changes. For eg. we can see localized increases or decreases along the tract length with respect to the time evolution. However, owing to the definition of the logistic function, any non-monotonic behavior along time at any given tract location is not modeled accurately. Therefore, in cases like the Krabbe's subject, we need a more relaxed and flexible semi-parametric or non-parametric model to remove the constraints coming in from the parametric nature of the current model. This is currently work under progress.
To utilize the quantification of growth trajectories enabled by this method, we also require a systematic hypothesis testing scheme to draw conclusive results of individuals differing from normative patterns or populations differing in their evolution behaviors. This too is currently being worked upon.

= References =

* Sharma, A., Durrleman, S. , Gilmore, J.H. , Gerig, G. Longitudinal growth modeling of discrete-time functions with application to DTI tract evolution in early neurodevelopment. Proc. of 9th IEEE International Symposium on Biomedical Imaging (ISBI May'12), p.1397-1400.
* Corouge, I., Fletcher, P.T., Joshi, S., Gouttard, S., Gerig, G., 2006. Fiber tract-oriented statistics for quantitative diffusion tensor MRI analysis. Med Image Anal, pp. 786-798.
* Goodlett, C.B., Fletcher, P.T., Gilmore, J.H., Gerig, G., 2009. Group analysis of DTI fiber tract statistics with application to neurodevelopment. Neuroimage, pp. S133-142.

==Key Investigators==
* Utah: Anuja Sharma, Stanley Durrleman, Guido Gerig
* INRIA/ICM, Pitie Salpetriere Hospital: Stanley Durrleman
[http://www.na-mic.org/Wiki/index.php/Algorithm:Utah2 Back to Utah 2 Algorithm Core]

== Regression of probability distributions obtained along white matter tracts to create spatio-temporal growth models ==

Temporal modeling frameworks often operate on scalar variables by
summarizing data at initial stages as statistical summaries of the underlying
distributions. For instance, DTI analysis often employs
summary statistics, like mean, for regions of interest and properties
along fiber tracts for population studies and hypothesis testing.
This reduction via discarding of variability information may introduce
significant errors which propagate through the procedures. Moreover, the data variability information is lost at
all the intermediate interpolated times.

{| border="0" style="background:transparent;"
|[[Image:DistributionRegression_Synthetic.png|thumb|600px|center|Left:Two synthetically generated distributions along time
(blue:t0, orange:t1) with translated locations and opposite skews.
Right: Simple linear interpolation using classic single-valued summary
statistics (means: red, medians: blue) and ignoring data variability
(shown as error bars) at t0 and t1. The decision to choose mean versus median can heavily influence the final regression estimates. Additionally, once the inherent variability is discarded for an early decision regarding a summary statistics, it cannot be recovered for the interpolated time points.]]
|-
|}

We propose a novel framework which uses distribution-valued variables
to retain and utilize the local variability information. We propose two different regression approaches to achieve our goals. The first is a non-parametric moving average method where minimal assumptions are involved in terms of the growth trajectory evolution (Sharma ISBI 2013). The second is a parametric regression scheme where the expected temporal trend is linear (Sharma ISBI 2014). It provides a compact representation in terms of model parameters for further statistical inference and has a closed form solution.

Our driving application is the modeling of age-related changes along DTI
white matter tracts. Results are shown for the spatiotemporal population
trajectory of genu tract estimated from a group of healthy infants and compared with an infant with Krabbe's disease.

{| border="0" style="background:transparent;"
|[[Image:DWIalongTime.png|thumb|400px|center|Model age related DTI changes in early neurodevelopment.]]
|[[Image:genu_distributions_along_time.png|thumb|600px|center|Distributions of DTI-derived scalar diffusion parameters like FA are obtained along the length of the genu tract. The distributions correspond to the cross-sections of the 3D tract geometry and are a function of the arc-length along the tract's total length as we move from one end of the tract to the other. This provides a functional curve which is parametrized by a feature of the local tract geometry (arc-length along the tract in this case) and is attributed by probability distributions (instead of scalar values like the 'mean' FA obtained from these local distributions).]]
|-
|}

=== Non-parametric regression for distribution-valued data ===

The proposed method
presents a completely nonparametric regression framework for spatial-temporal evolution of
statistical distributions. This avoids any parametric assumptions regarding the nature of the
underlying noise model in the DTI data and retains and utilizes the variability information to
estimate the regression trend. The estimated trajectory is completely characterized by statistical
distributions in space and time enabling a very powerful statistical inference framework. Both
population and individual trajectories can be estimated. The concept of ’distance’ between distributions and an ’average’ of distributions is employed. The dissimilarity metric employed is the Mallow's distance between distributions. It is the L2 norm of the generic Wasserstein metric and captures translation, changing width and shape of the participating distributions.

{| border="0" style="background:transparent;"
|[[Image:dist2_3d_bary_300_revrev.png|thumb|400px|center|Temporal evolution of the synthetic distributions between two timepoints as estimated by the proposed method.]]
|[[Image:bary_stat_300.png|thumb|400px|center|Since the regression estimates the complete probability distribution continuously along time, we now have access to the complete variability information at the interpolated time points. This can enable statistical inference using statistical summary measures as well as distribution differences at time points where the original image data is not available.]]
|-
|}

{| border="0" style="background:transparent;"
|[[Image:conte_krabbe_median_3d_300_2.png|thumb|400px|center|The medians of the estimated probability distributions are shown here for the healthy population (in red/pink) versus a single infant with Krabbe's disease (blue/black). Note that the method can applied to estimate normative population trajectories as well as subject-specific evolutions. Each point on these statistical manifolds are characterized by the complete distribution information (instead of just a scalar value).]]
|[[Image:conte_vs_krabbe_quartiles.png|thumb|600px|center|Highlighting the quantiles obtained at one of the arc length locations in the left plot. The Krabbe's subject seems to lag behind the normative population and the difference can be quantified in terms of distribution-based inferences.]]
|-
|}

=== Parametric linear regression for distribution-valued data ===

We build on the motivation as highlighted in the previous distribution-based regression. However,
the previous method is completely nonparametric leading to an intensive and complex statistical
inference framework based on comparison of statistical manifolds. Therefore, we simplify the
analysis for situations where a linear time-course trajectory can be assumed. We apply the classic
linear regression technique, extended to work with distribution-valued data. The advantage lies in
the compact description of the spatial-temporal trajectory in terms of the estimated regression
coefficients which can now be conveniently used for statistical inference. A synthetic validation
experiment is conducted to show the improved reliability and robustness of this method when
compared with methods working with statistical summaries of the underlying data.

{| border="0" style="background:transparent;"
|[[Image:Baryhist3d1.png|thumb|400px|center|The independent variable is the barycentric FA distribution estimated at each arc length location along the tract, using the original FA distributions available from all subjects within the corresponding age group. The regression estimates the continuous spatiotemporal trends in terms of the model parameters using the respective pairs of age and FA distributions as the 'observed' data.]]
|-
|}

{| border="0" style="background:transparent;"
|[[Image:PopuCN_dist_3d1.png|thumb|300px|center|Continuous spatiotemporal normative growth trajectory: Estimated mean FA (gray), Age range where control
data is originally available (green, orange, purple).]]
|[[Image:PopuvsKrabbe_dist_3d_2_edit1.png|thumb|300px|center|Krabbe's subject's scans (solid lines): (14 days, 6 mo., 1 yr.) with respect to the estimated healthy trajectory.]]
|[[Image:PopuvsKrabbe_dist_3d31.png|thumb|300px|center|Krabbe's subject's scans:mean FA (solid lines).Control population:estimated mean FA (dashed lines) +/- 3*std.dev. bounds (calculated for Gaussian age distributions centered at matched timepoints with std.dev.=5 days).]]
|-
|}

= References =

* A. Sharma, P.T. Fletcher, J.H. Gilmore, M.L. Escolar, A. Gupta, M. Styner, G. Gerig. “ Parametric Regression Scheme For Distributions: Analysis of DTI Fiber Tract Diffusion Changes In Early Brain Development,” In Proceedings of the 2014 IEEE 11th International Symposium on Biomedical Imaging (ISBI), pp. 559-562. 2014.
* A. Sharma, P.T. Fletcher, J.H. Gilmore, M.L. Escolar, A. Gupta, M. Styner, G. Gerig. “Spatiotemporal Modeling of Discrete-Time Distribution-Valued Data Applied to DTI Tract Evolution in Infant Neurodevelopment,” In Proceedings of the 2013 IEEE 10th International Symposium on Biomedical Imaging (ISBI), pp. 684--687. 2013.
* A. Sharma, S. Durrleman, J.H. Gilmore, G. Gerig. “Longitudinal Growth Modeling of Discrete-Time Functions with Application to DTI Tract Evolution in Early Neurodevelopment,” In Proceedings of IEEE ISBI 2012, pp. 1397--1400. 2012.
* Corouge, I., Fletcher, P.T., Joshi, S., Gouttard, S., Gerig, G., 2006. Fiber tract-oriented statistics for quantitative diffusion tensor MRI analysis. Med Image Anal, pp. 786-798.
* Goodlett, C.B., Fletcher, P.T., Gilmore, J.H., Gerig, G., 2009. Group analysis of DTI fiber tract statistics with application to neurodevelopment. Neuroimage, pp. S133-142.

File:PopuvsKrabbe dist 3d31.png

2014-10-15T23:08:38Z

Anuja:

File:Agedist.png

2014-10-15T23:07:51Z

Anuja:

Projects:TractLongitudinalDTI

2014-10-15T23:07:39Z

Anuja: /* Parametric linear regression for distribution-valued data */

Back to [[Algorithm:Utah2|Utah 2 Algorithms]]
__NOTOC__

== Tract-based longitudinal modeling of DTI data ==

This project develops a methodology to explore subject-specific, DTI data obtained from brain's white matter tracts, available at multiple but often sparsely present timepoints. The challenge is to develop a continuous spatio-temporal growth model, given discrete 4D DWI images. This would enable comparison of growth trajectories across subjects and along tracts which are biologically of interest in developmental and pathological changes.

== Background ==

We use the arc length parametrization scheme initially proposed by Corouge et al. It represents white matter fiber tracts obtained via streamline tractography in the brain's atlas tensor image as a function of arc length. (Atlas construction uses unbiased atlas building schemes followed by back transformations to subjects' DTI images to obtain identical fiber tract geometry across subjects, populated with subject specific diffusion data). We use the mean diffusion scalar invariants derived from these individual fiber bundle cross sections, as our input longitudinal diffusion profiles.

{| border="0" style="background:transparent;"
|[[Image:Tract-stats.png|thumb|700px|White matter diffusion properties along fiber tract: Left: A fiber tract with an origin plane defined for arc length parametrization, Right: FA mean and standard deviation as function of arc-length. Dots mark location of coordinate origin.]]
|-
|}

== Subject-specific spatiotemporal continuous growth model ==

We propose the use of Verhulst-Pearl logistic equation to capture temporal changes, while using a non-parametric kernel along arc length to account for biologically motivated functional along-tract relationship in the diffusion data.

{| border="0" style="background:transparent;"
|[[Image:logistic_eq.png|thumb|600px|Logistic equation with P as the population value at a given time, r being the growth rate of the population and K being the carrying capacity (also the limiting asymptote value).]]
|[[Image:logistic_simulation_growthrate_r.png|thumb|400px|Effect of growth rate r on the logistic curve.]]
|[[Image:logistic_simulation_asymptote_K.png|thumb|400px|Effect of carrying capacity K on the logistic curve.]]
|-
|}

From the definition of the logistic function, the parameter r represents the growth rate of the diffusion invariant and the parameter K represents the asymptote value. The overall function shape intuitively follows the growth pattern we expect to see during brain maturation. The diffusion invariants start with an initial diffusion profile along tract, and have a non linear temporal growth trajectory showing maximum changes in early childhood and then slowing down or almost saturating at a certain age. (For instance, observed diffusion changes are much more in neonates than in an adult brain). The temporal trajectories may also differ along the tract's length giving localized changes. Since our method gives us continuous along-tract, growth trajectories all along time, we can compare subjects with respect to differences in diffusion profiles at birth, growth rates at any given age as well as the asymptote saturation values. This gives us important information to understand delayed or abnormal brain maturation by comparing a normative growth surface with an individual's or by comparing the model's parameters across subjects.

== Results ==

Below are some results using synthetic data. For more validation results and extension of the framework to jointly estimate individual subject trajectories together with a normative trajectory, refer to [http://ieeexplore.ieee.org/xpl/login.jsp?tp=&arnumber=6235829&url=http%3A%2F%2Fieeexplore.ieee.org%2Fxpls%2Fabs_all.jsp%3Farnumber%3D6235829 Sharma et al. (ISBI '12)].

{| border="0" style="background:transparent;"
|[[Image:namic_tract_longitudinal_main.png|thumb|400px|Longitudinal modeling of synthetic data. The data is generated from a template neonate FA mean curve and uses known logistic parameters to generate the three timepoint curves (red). Our method faithfully recovers the original curves (blue) while simultaneously creating a continuous longitudinal growth profile (green) without having any prior knowledge of the logistic parameters used for data generation. It also estimates an unbiased time=0 template curve (black) along with parameters r and K to avoid biasing the estimation with any assumed initialization at time=0.]]
|[[Image:namic_tract_longitudinal_3d.png|thumb|400px|Results on real data (mean FA curves for a subject at three timepoints:neonate, 1 year and 2 year).]]
|-
|}
{| border="0" style="background:transparent;"
|[[Image:namic_tract_longitudinal_p1p2only.png|thumb|400px|For the above real data (right), these are the final along-tract initial values and the final estimates for parameters-growth rate r(mentioned as p1 here) and asymptote K(mentioned as p2 here).]]
|[[Image:namic_tract_longitudinal_alphaonly.png|thumb|400px|The complete continuous growth grid as estimated for the real data above.]]

|-
|}

The below images show jointly estimated personalized trajectories for 15 control subjects along with the average growth trajectory. The estimated model parameters for 15 subjects as well as the average trajectory are also shown. The normative trajectory is colored by the local standard deviation. It points to the fact that despite individual variability in the maturation process, there is a strong agreement in the asymptote FA values seen around a gestational age of 2.5 years indicating a relative stabilization of the white matter changes across subjects. The framework thus quantifies patient-specific changes in serial diffusion data given discrete-time diffusion curves.

{| border="0" style="background:transparent;"
|[[Image:CONTE_Final_3d.png|thumb|400px|The jointly estimated individual growth trajectories for 4 of the 15 control subjects.]]
|[[Image:CONTE_Norm_Params.png|thumb|400px|Estimated model parameters for 15 subjects together with the normative growth trajectory colored by the local variability and the estimated normative model parameters.]]

|-
|}

== Experiments with Huntington's data ==

During the NAMIC Winter project week, we worked on registering subjects with Huntington's disease in the same coordinate space as control subjects. We then applied the above framework [http://ieeexplore.ieee.org/xpl/login.jsp?tp=&arnumber=6235829&url=http%3A%2F%2Fieeexplore.ieee.org%2Fxpls%2Fabs_all.jsp%3Farnumber%3D6235829 Sharma et al. ISBI'12] to quantify the differences in normal expected aging versus the accelerated white matter changes expected in HD. Details of the data pre-processing steps and image registration are available on the [http://www.na-mic.org/Wiki/index.php/2012_Winter_Project_Week:_DTI_Change_Modeling Winter AHM page]. Some results are summarized here for FA value along the genu tract.

Two subjects- one being a HD patient (10027) with a high burden factor (higher factor value (of factor 12) implying that the subjects is closer to the onset) and the other being a control subject (10004) are chosen for concept demonstration. Both have a similar baseline scan age (42 and 43 years respectively) as well as a similar time separation between follow up scans. This allows an intuitive comparison of the estimated growth trajectories.

{| border="0" style="background:transparent;"
|[[Image:ORIG_hd_cn.png|thumb|300px|center|FA curves corresponding to the genu tract (middle clipped portions) for one HD case (10027) and one control case (10004). Visual inspection clearly shows the huge decrease relative to the Control in the HD subject. On the other hand, the control shows normal expected decrease due to aging. Red-timepoint 1, Green-timepoint 2, Blue-timepoint 3. The left is HD case and the right plot is Control.]]
|[[Image:HD_vs_CN_3d.png|thumb|300px|center|The red trajectory corresponds the the HD subjects while the blue is a control subject with healthy aging. HD patient clearly shows a much sharper FA decrease along time than the Control.]]
|[[Image:HD_vs_CN_params.png|thumb|300px|center|The estimated model parameters (Red-HD case, Blue-Control case) for the two subjects. The top plot is the rate of change per unit time, per unit FA value. The middle plot is the function asymptote and the last plot is the common baseline curve estimation for providing a common reference frame for model estimation and comparison. The asymptote clearly shows a much lower FA range for HD when compared to the Control which already shows promising early prediction capabilities for HD cases where white matter decay is much faster than healthy patients. Also, a higher per unit rate of change quantifies the sharper FA decrease compared to control.]]
|-
|}

== Experiments with Krabbe's data ==

We have also applied the framework to study white matter changes in infants with Krabbe's disease. Here we show some preliminary results for the FA tract profile along the genu tract for a single subject with Krabbe's disease. In theory, FA non-linearly increases with time along genu owing to a healthy brain maturation process. However, in Krabbe's case, the FA does not follow a monotonic change pattern.

{| border="0" style="background:transparent;"
|[[Image:Genu_fa_3d_conte_vs_krabbe.png|thumb|400px|center|FA curves corresponding to the genu tract for the Krabbe's patient are shown (Neonate:black, 6months:cyan, 1year:magenta). The other FA profiles are for the 15 healthy control subjects as discussed previously (neonate:red, 1year:green, 2year:blue). This also highlights the fact that the framework does not assume uniformity or correspondence in the scan timings across subjects.]]
|[[Image:Genu_fa_3dnorm_conte_vs_krabbe.png|thumb|400px|center|The black trajectory is the normative FA growth estimated using 15 controls as discussed previously. The three FA curves for the Krabbe case are also plotted and clearly show how the patient is lagging behind on expected brain maturation.]]
|-
|}

== Future work ==

As we see in the Krabbe experiment, the model assumes a monotonic change along time. Neighboring locations along the tract can have different, yet correlated time changes. For eg. we can see localized increases or decreases along the tract length with respect to the time evolution. However, owing to the definition of the logistic function, any non-monotonic behavior along time at any given tract location is not modeled accurately. Therefore, in cases like the Krabbe's subject, we need a more relaxed and flexible semi-parametric or non-parametric model to remove the constraints coming in from the parametric nature of the current model. This is currently work under progress.
To utilize the quantification of growth trajectories enabled by this method, we also require a systematic hypothesis testing scheme to draw conclusive results of individuals differing from normative patterns or populations differing in their evolution behaviors. This too is currently being worked upon.

= References =

* Sharma, A., Durrleman, S. , Gilmore, J.H. , Gerig, G. Longitudinal growth modeling of discrete-time functions with application to DTI tract evolution in early neurodevelopment. Proc. of 9th IEEE International Symposium on Biomedical Imaging (ISBI May'12), p.1397-1400.
* Corouge, I., Fletcher, P.T., Joshi, S., Gouttard, S., Gerig, G., 2006. Fiber tract-oriented statistics for quantitative diffusion tensor MRI analysis. Med Image Anal, pp. 786-798.
* Goodlett, C.B., Fletcher, P.T., Gilmore, J.H., Gerig, G., 2009. Group analysis of DTI fiber tract statistics with application to neurodevelopment. Neuroimage, pp. S133-142.

==Key Investigators==
* Utah: Anuja Sharma, Stanley Durrleman, Guido Gerig
* INRIA/ICM, Pitie Salpetriere Hospital: Stanley Durrleman
[http://www.na-mic.org/Wiki/index.php/Algorithm:Utah2 Back to Utah 2 Algorithm Core]

== Regression of probability distributions obtained along white matter tracts to create spatio-temporal growth models ==

Temporal modeling frameworks often operate on scalar variables by
summarizing data at initial stages as statistical summaries of the underlying
distributions. For instance, DTI analysis often employs
summary statistics, like mean, for regions of interest and properties
along fiber tracts for population studies and hypothesis testing.
This reduction via discarding of variability information may introduce
significant errors which propagate through the procedures. Moreover, the data variability information is lost at
all the intermediate interpolated times.

{| border="0" style="background:transparent;"
|[[Image:DistributionRegression_Synthetic.png|thumb|600px|center|Left:Two synthetically generated distributions along time
(blue:t0, orange:t1) with translated locations and opposite skews.
Right: Simple linear interpolation using classic single-valued summary
statistics (means: red, medians: blue) and ignoring data variability
(shown as error bars) at t0 and t1. The decision to choose mean versus median can heavily influence the final regression estimates. Additionally, once the inherent variability is discarded for an early decision regarding a summary statistics, it cannot be recovered for the interpolated time points.]]
|-
|}

We propose a novel framework which uses distribution-valued variables
to retain and utilize the local variability information. We propose two different regression approaches to achieve our goals. The first is a non-parametric moving average method where minimal assumptions are involved in terms of the growth trajectory evolution (Sharma ISBI 2013). The second is a parametric regression scheme where the expected temporal trend is linear (Sharma ISBI 2014). It provides a compact representation in terms of model parameters for further statistical inference and has a closed form solution.

Our driving application is the modeling of age-related changes along DTI
white matter tracts. Results are shown for the spatiotemporal population
trajectory of genu tract estimated from a group of healthy infants and compared with an infant with Krabbe's disease.

{| border="0" style="background:transparent;"
|[[Image:DWIalongTime.png|thumb|400px|center|Model age related DTI changes in early neurodevelopment.]]
|[[Image:genu_distributions_along_time.png|thumb|600px|center|Distributions of DTI-derived scalar diffusion parameters like FA are obtained along the length of the genu tract. The distributions correspond to the cross-sections of the 3D tract geometry and are a function of the arc-length along the tract's total length as we move from one end of the tract to the other. This provides a functional curve which is parametrized by a feature of the local tract geometry (arc-length along the tract in this case) and is attributed by probability distributions (instead of scalar values like the 'mean' FA obtained from these local distributions).]]
|-
|}

=== Non-parametric regression for distribution-valued data ===

The proposed method
presents a completely nonparametric regression framework for spatial-temporal evolution of
statistical distributions. This avoids any parametric assumptions regarding the nature of the
underlying noise model in the DTI data and retains and utilizes the variability information to
estimate the regression trend. The estimated trajectory is completely characterized by statistical
distributions in space and time enabling a very powerful statistical inference framework. Both
population and individual trajectories can be estimated. The concept of ’distance’ between distributions and an ’average’ of distributions is employed. The dissimilarity metric employed is the Mallow's distance between distributions. It is the L2 norm of the generic Wasserstein metric and captures translation, changing width and shape of the participating distributions.

{| border="0" style="background:transparent;"
|[[Image:dist2_3d_bary_300_revrev.png|thumb|400px|center|Temporal evolution of the synthetic distributions between two timepoints as estimated by the proposed method.]]
|[[Image:bary_stat_300.png|thumb|400px|center|Since the regression estimates the complete probability distribution continuously along time, we now have access to the complete variability information at the interpolated time points. This can enable statistical inference using statistical summary measures as well as distribution differences at time points where the original image data is not available.]]
|-
|}

{| border="0" style="background:transparent;"
|[[Image:conte_krabbe_median_3d_300_2.png|thumb|400px|center|The medians of the estimated probability distributions are shown here for the healthy population (in red/pink) versus a single infant with Krabbe's disease (blue/black). Note that the method can applied to estimate normative population trajectories as well as subject-specific evolutions. Each point on these statistical manifolds are characterized by the complete distribution information (instead of just a scalar value).]]
|[[Image:conte_vs_krabbe_quartiles.png|thumb|600px|center|Highlighting the quantiles obtained at one of the arc length locations in the left plot. The Krabbe's subject seems to lag behind the normative population and the difference can be quantified in terms of distribution-based inferences.]]
|-
|}

=== Parametric linear regression for distribution-valued data ===

We build on the motivation as highlighted in the previous distribution-based regression. However,
the previous method is completely nonparametric leading to an intensive and complex statistical
inference framework based on comparison of statistical manifolds. Therefore, we simplify the
analysis for situations where a linear time-course trajectory can be assumed. We apply the classic
linear regression technique, extended to work with distribution-valued data. The advantage lies in
the compact description of the spatial-temporal trajectory in terms of the estimated regression
coefficients which can now be conveniently used for statistical inference. A synthetic validation
experiment is conducted to show the improved reliability and robustness of this method when
compared with methods working with statistical summaries of the underlying data.

{| border="0" style="background:transparent;"
|[[Image:Agedist.png|thumb|400px|center|The independent variable is the age distribution of the healthy infants clustered around 2 months, 1 year and 2 year.]]
|[[Image:Baryhist3d1.png|thumb|400px|center|The independent variable is the barycentric FA distribution estimated at each arc length location along the tract, using the original FA distributions available from all subjects within the corresponding age group. The regression estimates the continuous spatiotemporal trends in terms of the model parameters using the respective pairs of age and FA distributions as the 'observed' data.]]
|-
|}

{| border="0" style="background:transparent;"
|[[Image:PopuCN_dist_3d1.png|thumb|400px|center|Continuous spatiotemporal normative growth trajectory: Estimated mean FA (gray), Age range where control
data is originally available (green, orange, purple).]]
|[[Image:PopuvsKrabbe_dist_3d_2_edit1.png|thumb|400px|center|Krabbe's subject's scans (solid lines): (14 days, 6 mo., 1 yr.) with respect to the estimated healthy trajectory.]]
|[[Image:PopuvsKrabbe_dist_3d31.png|thumb|400px|center|Krabbe's subject's scans:mean FA (solid lines).Control population:estimated mean FA (dashed lines) +/- 3*std.dev. bounds (calculated for Gaussian age distributions centered at matched timepoints with std.dev.=5 days).]]
|-
|}

= References =

* A. Sharma, P.T. Fletcher, J.H. Gilmore, M.L. Escolar, A. Gupta, M. Styner, G. Gerig. “ Parametric Regression Scheme For Distributions: Analysis of DTI Fiber Tract Diffusion Changes In Early Brain Development,” In Proceedings of the 2014 IEEE 11th International Symposium on Biomedical Imaging (ISBI), pp. 559-562. 2014.
* A. Sharma, P.T. Fletcher, J.H. Gilmore, M.L. Escolar, A. Gupta, M. Styner, G. Gerig. “Spatiotemporal Modeling of Discrete-Time Distribution-Valued Data Applied to DTI Tract Evolution in Infant Neurodevelopment,” In Proceedings of the 2013 IEEE 10th International Symposium on Biomedical Imaging (ISBI), pp. 684--687. 2013.
* A. Sharma, S. Durrleman, J.H. Gilmore, G. Gerig. “Longitudinal Growth Modeling of Discrete-Time Functions with Application to DTI Tract Evolution in Early Neurodevelopment,” In Proceedings of IEEE ISBI 2012, pp. 1397--1400. 2012.
* Corouge, I., Fletcher, P.T., Joshi, S., Gouttard, S., Gerig, G., 2006. Fiber tract-oriented statistics for quantitative diffusion tensor MRI analysis. Med Image Anal, pp. 786-798.
* Goodlett, C.B., Fletcher, P.T., Gilmore, J.H., Gerig, G., 2009. Group analysis of DTI fiber tract statistics with application to neurodevelopment. Neuroimage, pp. S133-142.

Projects:TractLongitudinalDTI

2014-10-15T23:07:23Z

Anuja: /* Parametric linear regression for distribution-valued data */

Back to [[Algorithm:Utah2|Utah 2 Algorithms]]
__NOTOC__

== Tract-based longitudinal modeling of DTI data ==

This project develops a methodology to explore subject-specific, DTI data obtained from brain's white matter tracts, available at multiple but often sparsely present timepoints. The challenge is to develop a continuous spatio-temporal growth model, given discrete 4D DWI images. This would enable comparison of growth trajectories across subjects and along tracts which are biologically of interest in developmental and pathological changes.

== Background ==

We use the arc length parametrization scheme initially proposed by Corouge et al. It represents white matter fiber tracts obtained via streamline tractography in the brain's atlas tensor image as a function of arc length. (Atlas construction uses unbiased atlas building schemes followed by back transformations to subjects' DTI images to obtain identical fiber tract geometry across subjects, populated with subject specific diffusion data). We use the mean diffusion scalar invariants derived from these individual fiber bundle cross sections, as our input longitudinal diffusion profiles.

{| border="0" style="background:transparent;"
|[[Image:Tract-stats.png|thumb|700px|White matter diffusion properties along fiber tract: Left: A fiber tract with an origin plane defined for arc length parametrization, Right: FA mean and standard deviation as function of arc-length. Dots mark location of coordinate origin.]]
|-
|}

== Subject-specific spatiotemporal continuous growth model ==

We propose the use of Verhulst-Pearl logistic equation to capture temporal changes, while using a non-parametric kernel along arc length to account for biologically motivated functional along-tract relationship in the diffusion data.

{| border="0" style="background:transparent;"
|[[Image:logistic_eq.png|thumb|600px|Logistic equation with P as the population value at a given time, r being the growth rate of the population and K being the carrying capacity (also the limiting asymptote value).]]
|[[Image:logistic_simulation_growthrate_r.png|thumb|400px|Effect of growth rate r on the logistic curve.]]
|[[Image:logistic_simulation_asymptote_K.png|thumb|400px|Effect of carrying capacity K on the logistic curve.]]
|-
|}

From the definition of the logistic function, the parameter r represents the growth rate of the diffusion invariant and the parameter K represents the asymptote value. The overall function shape intuitively follows the growth pattern we expect to see during brain maturation. The diffusion invariants start with an initial diffusion profile along tract, and have a non linear temporal growth trajectory showing maximum changes in early childhood and then slowing down or almost saturating at a certain age. (For instance, observed diffusion changes are much more in neonates than in an adult brain). The temporal trajectories may also differ along the tract's length giving localized changes. Since our method gives us continuous along-tract, growth trajectories all along time, we can compare subjects with respect to differences in diffusion profiles at birth, growth rates at any given age as well as the asymptote saturation values. This gives us important information to understand delayed or abnormal brain maturation by comparing a normative growth surface with an individual's or by comparing the model's parameters across subjects.

== Results ==

Below are some results using synthetic data. For more validation results and extension of the framework to jointly estimate individual subject trajectories together with a normative trajectory, refer to [http://ieeexplore.ieee.org/xpl/login.jsp?tp=&arnumber=6235829&url=http%3A%2F%2Fieeexplore.ieee.org%2Fxpls%2Fabs_all.jsp%3Farnumber%3D6235829 Sharma et al. (ISBI '12)].

{| border="0" style="background:transparent;"
|[[Image:namic_tract_longitudinal_main.png|thumb|400px|Longitudinal modeling of synthetic data. The data is generated from a template neonate FA mean curve and uses known logistic parameters to generate the three timepoint curves (red). Our method faithfully recovers the original curves (blue) while simultaneously creating a continuous longitudinal growth profile (green) without having any prior knowledge of the logistic parameters used for data generation. It also estimates an unbiased time=0 template curve (black) along with parameters r and K to avoid biasing the estimation with any assumed initialization at time=0.]]
|[[Image:namic_tract_longitudinal_3d.png|thumb|400px|Results on real data (mean FA curves for a subject at three timepoints:neonate, 1 year and 2 year).]]
|-
|}
{| border="0" style="background:transparent;"
|[[Image:namic_tract_longitudinal_p1p2only.png|thumb|400px|For the above real data (right), these are the final along-tract initial values and the final estimates for parameters-growth rate r(mentioned as p1 here) and asymptote K(mentioned as p2 here).]]
|[[Image:namic_tract_longitudinal_alphaonly.png|thumb|400px|The complete continuous growth grid as estimated for the real data above.]]

|-
|}

The below images show jointly estimated personalized trajectories for 15 control subjects along with the average growth trajectory. The estimated model parameters for 15 subjects as well as the average trajectory are also shown. The normative trajectory is colored by the local standard deviation. It points to the fact that despite individual variability in the maturation process, there is a strong agreement in the asymptote FA values seen around a gestational age of 2.5 years indicating a relative stabilization of the white matter changes across subjects. The framework thus quantifies patient-specific changes in serial diffusion data given discrete-time diffusion curves.

{| border="0" style="background:transparent;"
|[[Image:CONTE_Final_3d.png|thumb|400px|The jointly estimated individual growth trajectories for 4 of the 15 control subjects.]]
|[[Image:CONTE_Norm_Params.png|thumb|400px|Estimated model parameters for 15 subjects together with the normative growth trajectory colored by the local variability and the estimated normative model parameters.]]

|-
|}

== Experiments with Huntington's data ==

During the NAMIC Winter project week, we worked on registering subjects with Huntington's disease in the same coordinate space as control subjects. We then applied the above framework [http://ieeexplore.ieee.org/xpl/login.jsp?tp=&arnumber=6235829&url=http%3A%2F%2Fieeexplore.ieee.org%2Fxpls%2Fabs_all.jsp%3Farnumber%3D6235829 Sharma et al. ISBI'12] to quantify the differences in normal expected aging versus the accelerated white matter changes expected in HD. Details of the data pre-processing steps and image registration are available on the [http://www.na-mic.org/Wiki/index.php/2012_Winter_Project_Week:_DTI_Change_Modeling Winter AHM page]. Some results are summarized here for FA value along the genu tract.

Two subjects- one being a HD patient (10027) with a high burden factor (higher factor value (of factor 12) implying that the subjects is closer to the onset) and the other being a control subject (10004) are chosen for concept demonstration. Both have a similar baseline scan age (42 and 43 years respectively) as well as a similar time separation between follow up scans. This allows an intuitive comparison of the estimated growth trajectories.

{| border="0" style="background:transparent;"
|[[Image:ORIG_hd_cn.png|thumb|300px|center|FA curves corresponding to the genu tract (middle clipped portions) for one HD case (10027) and one control case (10004). Visual inspection clearly shows the huge decrease relative to the Control in the HD subject. On the other hand, the control shows normal expected decrease due to aging. Red-timepoint 1, Green-timepoint 2, Blue-timepoint 3. The left is HD case and the right plot is Control.]]
|[[Image:HD_vs_CN_3d.png|thumb|300px|center|The red trajectory corresponds the the HD subjects while the blue is a control subject with healthy aging. HD patient clearly shows a much sharper FA decrease along time than the Control.]]
|[[Image:HD_vs_CN_params.png|thumb|300px|center|The estimated model parameters (Red-HD case, Blue-Control case) for the two subjects. The top plot is the rate of change per unit time, per unit FA value. The middle plot is the function asymptote and the last plot is the common baseline curve estimation for providing a common reference frame for model estimation and comparison. The asymptote clearly shows a much lower FA range for HD when compared to the Control which already shows promising early prediction capabilities for HD cases where white matter decay is much faster than healthy patients. Also, a higher per unit rate of change quantifies the sharper FA decrease compared to control.]]
|-
|}

== Experiments with Krabbe's data ==

We have also applied the framework to study white matter changes in infants with Krabbe's disease. Here we show some preliminary results for the FA tract profile along the genu tract for a single subject with Krabbe's disease. In theory, FA non-linearly increases with time along genu owing to a healthy brain maturation process. However, in Krabbe's case, the FA does not follow a monotonic change pattern.

{| border="0" style="background:transparent;"
|[[Image:Genu_fa_3d_conte_vs_krabbe.png|thumb|400px|center|FA curves corresponding to the genu tract for the Krabbe's patient are shown (Neonate:black, 6months:cyan, 1year:magenta). The other FA profiles are for the 15 healthy control subjects as discussed previously (neonate:red, 1year:green, 2year:blue). This also highlights the fact that the framework does not assume uniformity or correspondence in the scan timings across subjects.]]
|[[Image:Genu_fa_3dnorm_conte_vs_krabbe.png|thumb|400px|center|The black trajectory is the normative FA growth estimated using 15 controls as discussed previously. The three FA curves for the Krabbe case are also plotted and clearly show how the patient is lagging behind on expected brain maturation.]]
|-
|}

== Future work ==

As we see in the Krabbe experiment, the model assumes a monotonic change along time. Neighboring locations along the tract can have different, yet correlated time changes. For eg. we can see localized increases or decreases along the tract length with respect to the time evolution. However, owing to the definition of the logistic function, any non-monotonic behavior along time at any given tract location is not modeled accurately. Therefore, in cases like the Krabbe's subject, we need a more relaxed and flexible semi-parametric or non-parametric model to remove the constraints coming in from the parametric nature of the current model. This is currently work under progress.
To utilize the quantification of growth trajectories enabled by this method, we also require a systematic hypothesis testing scheme to draw conclusive results of individuals differing from normative patterns or populations differing in their evolution behaviors. This too is currently being worked upon.

= References =

* Sharma, A., Durrleman, S. , Gilmore, J.H. , Gerig, G. Longitudinal growth modeling of discrete-time functions with application to DTI tract evolution in early neurodevelopment. Proc. of 9th IEEE International Symposium on Biomedical Imaging (ISBI May'12), p.1397-1400.
* Corouge, I., Fletcher, P.T., Joshi, S., Gouttard, S., Gerig, G., 2006. Fiber tract-oriented statistics for quantitative diffusion tensor MRI analysis. Med Image Anal, pp. 786-798.
* Goodlett, C.B., Fletcher, P.T., Gilmore, J.H., Gerig, G., 2009. Group analysis of DTI fiber tract statistics with application to neurodevelopment. Neuroimage, pp. S133-142.

==Key Investigators==
* Utah: Anuja Sharma, Stanley Durrleman, Guido Gerig
* INRIA/ICM, Pitie Salpetriere Hospital: Stanley Durrleman
[http://www.na-mic.org/Wiki/index.php/Algorithm:Utah2 Back to Utah 2 Algorithm Core]

== Regression of probability distributions obtained along white matter tracts to create spatio-temporal growth models ==

Temporal modeling frameworks often operate on scalar variables by
summarizing data at initial stages as statistical summaries of the underlying
distributions. For instance, DTI analysis often employs
summary statistics, like mean, for regions of interest and properties
along fiber tracts for population studies and hypothesis testing.
This reduction via discarding of variability information may introduce
significant errors which propagate through the procedures. Moreover, the data variability information is lost at
all the intermediate interpolated times.

{| border="0" style="background:transparent;"
|[[Image:DistributionRegression_Synthetic.png|thumb|600px|center|Left:Two synthetically generated distributions along time
(blue:t0, orange:t1) with translated locations and opposite skews.
Right: Simple linear interpolation using classic single-valued summary
statistics (means: red, medians: blue) and ignoring data variability
(shown as error bars) at t0 and t1. The decision to choose mean versus median can heavily influence the final regression estimates. Additionally, once the inherent variability is discarded for an early decision regarding a summary statistics, it cannot be recovered for the interpolated time points.]]
|-
|}

We propose a novel framework which uses distribution-valued variables
to retain and utilize the local variability information. We propose two different regression approaches to achieve our goals. The first is a non-parametric moving average method where minimal assumptions are involved in terms of the growth trajectory evolution (Sharma ISBI 2013). The second is a parametric regression scheme where the expected temporal trend is linear (Sharma ISBI 2014). It provides a compact representation in terms of model parameters for further statistical inference and has a closed form solution.

Our driving application is the modeling of age-related changes along DTI
white matter tracts. Results are shown for the spatiotemporal population
trajectory of genu tract estimated from a group of healthy infants and compared with an infant with Krabbe's disease.

{| border="0" style="background:transparent;"
|[[Image:DWIalongTime.png|thumb|400px|center|Model age related DTI changes in early neurodevelopment.]]
|[[Image:genu_distributions_along_time.png|thumb|600px|center|Distributions of DTI-derived scalar diffusion parameters like FA are obtained along the length of the genu tract. The distributions correspond to the cross-sections of the 3D tract geometry and are a function of the arc-length along the tract's total length as we move from one end of the tract to the other. This provides a functional curve which is parametrized by a feature of the local tract geometry (arc-length along the tract in this case) and is attributed by probability distributions (instead of scalar values like the 'mean' FA obtained from these local distributions).]]
|-
|}

=== Non-parametric regression for distribution-valued data ===

The proposed method
presents a completely nonparametric regression framework for spatial-temporal evolution of
statistical distributions. This avoids any parametric assumptions regarding the nature of the
underlying noise model in the DTI data and retains and utilizes the variability information to
estimate the regression trend. The estimated trajectory is completely characterized by statistical
distributions in space and time enabling a very powerful statistical inference framework. Both
population and individual trajectories can be estimated. The concept of ’distance’ between distributions and an ’average’ of distributions is employed. The dissimilarity metric employed is the Mallow's distance between distributions. It is the L2 norm of the generic Wasserstein metric and captures translation, changing width and shape of the participating distributions.

{| border="0" style="background:transparent;"
|[[Image:dist2_3d_bary_300_revrev.png|thumb|400px|center|Temporal evolution of the synthetic distributions between two timepoints as estimated by the proposed method.]]
|[[Image:bary_stat_300.png|thumb|400px|center|Since the regression estimates the complete probability distribution continuously along time, we now have access to the complete variability information at the interpolated time points. This can enable statistical inference using statistical summary measures as well as distribution differences at time points where the original image data is not available.]]
|-
|}

{| border="0" style="background:transparent;"
|[[Image:conte_krabbe_median_3d_300_2.png|thumb|400px|center|The medians of the estimated probability distributions are shown here for the healthy population (in red/pink) versus a single infant with Krabbe's disease (blue/black). Note that the method can applied to estimate normative population trajectories as well as subject-specific evolutions. Each point on these statistical manifolds are characterized by the complete distribution information (instead of just a scalar value).]]
|[[Image:conte_vs_krabbe_quartiles.png|thumb|600px|center|Highlighting the quantiles obtained at one of the arc length locations in the left plot. The Krabbe's subject seems to lag behind the normative population and the difference can be quantified in terms of distribution-based inferences.]]
|-
|}

=== Parametric linear regression for distribution-valued data ===

We build on the motivation as highlighted in the previous distribution-based regression. However,
the previous method is completely nonparametric leading to an intensive and complex statistical
inference framework based on comparison of statistical manifolds. Therefore, we simplify the
analysis for situations where a linear time-course trajectory can be assumed. We apply the classic
linear regression technique, extended to work with distribution-valued data. The advantage lies in
the compact description of the spatial-temporal trajectory in terms of the estimated regression
coefficients which can now be conveniently used for statistical inference. A synthetic validation
experiment is conducted to show the improved reliability and robustness of this method when
compared with methods working with statistical summaries of the underlying data.

{| border="0" style="background:transparent;"
|[[Image:Baryhist3d1.png|thumb|400px|center|The independent variable is the age distribution of the healthy infants clustered around 2 months, 1 year and 2 year.]]
|[[Image:Baryhist3d1.png|thumb|400px|center|The independent variable is the barycentric FA distribution estimated at each arc length location along the tract, using the original FA distributions available from all subjects within the corresponding age group. The regression estimates the continuous spatiotemporal trends in terms of the model parameters using the respective pairs of age and FA distributions as the 'observed' data.]]
|-
|}

{| border="0" style="background:transparent;"
|[[Image:PopuCN_dist_3d1.png|thumb|400px|center|Continuous spatiotemporal normative growth trajectory: Estimated mean FA (gray), Age range where control
data is originally available (green, orange, purple).]]
|[[Image:PopuvsKrabbe_dist_3d_2_edit1.png|thumb|400px|center|Krabbe's subject's scans (solid lines): (14 days, 6 mo., 1 yr.) with respect to the estimated healthy trajectory.]]
|[[Image:PopuvsKrabbe_dist_3d31.png|thumb|400px|center|Krabbe's subject's scans:mean FA (solid lines).Control population:estimated mean FA (dashed lines) +/- 3*std.dev. bounds (calculated for Gaussian age distributions centered at matched timepoints with std.dev.=5 days).]]
|-
|}

= References =

* A. Sharma, P.T. Fletcher, J.H. Gilmore, M.L. Escolar, A. Gupta, M. Styner, G. Gerig. “ Parametric Regression Scheme For Distributions: Analysis of DTI Fiber Tract Diffusion Changes In Early Brain Development,” In Proceedings of the 2014 IEEE 11th International Symposium on Biomedical Imaging (ISBI), pp. 559-562. 2014.
* A. Sharma, P.T. Fletcher, J.H. Gilmore, M.L. Escolar, A. Gupta, M. Styner, G. Gerig. “Spatiotemporal Modeling of Discrete-Time Distribution-Valued Data Applied to DTI Tract Evolution in Infant Neurodevelopment,” In Proceedings of the 2013 IEEE 10th International Symposium on Biomedical Imaging (ISBI), pp. 684--687. 2013.
* A. Sharma, S. Durrleman, J.H. Gilmore, G. Gerig. “Longitudinal Growth Modeling of Discrete-Time Functions with Application to DTI Tract Evolution in Early Neurodevelopment,” In Proceedings of IEEE ISBI 2012, pp. 1397--1400. 2012.
* Corouge, I., Fletcher, P.T., Joshi, S., Gouttard, S., Gerig, G., 2006. Fiber tract-oriented statistics for quantitative diffusion tensor MRI analysis. Med Image Anal, pp. 786-798.
* Goodlett, C.B., Fletcher, P.T., Gilmore, J.H., Gerig, G., 2009. Group analysis of DTI fiber tract statistics with application to neurodevelopment. Neuroimage, pp. S133-142.

Projects:TractLongitudinalDTI

2014-10-15T23:06:05Z

Anuja: /* Parametric linear regression for distribution-valued data */

Back to [[Algorithm:Utah2|Utah 2 Algorithms]]
__NOTOC__

== Tract-based longitudinal modeling of DTI data ==

This project develops a methodology to explore subject-specific, DTI data obtained from brain's white matter tracts, available at multiple but often sparsely present timepoints. The challenge is to develop a continuous spatio-temporal growth model, given discrete 4D DWI images. This would enable comparison of growth trajectories across subjects and along tracts which are biologically of interest in developmental and pathological changes.

== Background ==

We use the arc length parametrization scheme initially proposed by Corouge et al. It represents white matter fiber tracts obtained via streamline tractography in the brain's atlas tensor image as a function of arc length. (Atlas construction uses unbiased atlas building schemes followed by back transformations to subjects' DTI images to obtain identical fiber tract geometry across subjects, populated with subject specific diffusion data). We use the mean diffusion scalar invariants derived from these individual fiber bundle cross sections, as our input longitudinal diffusion profiles.

{| border="0" style="background:transparent;"
|[[Image:Tract-stats.png|thumb|700px|White matter diffusion properties along fiber tract: Left: A fiber tract with an origin plane defined for arc length parametrization, Right: FA mean and standard deviation as function of arc-length. Dots mark location of coordinate origin.]]
|-
|}

== Subject-specific spatiotemporal continuous growth model ==

We propose the use of Verhulst-Pearl logistic equation to capture temporal changes, while using a non-parametric kernel along arc length to account for biologically motivated functional along-tract relationship in the diffusion data.

{| border="0" style="background:transparent;"
|[[Image:logistic_eq.png|thumb|600px|Logistic equation with P as the population value at a given time, r being the growth rate of the population and K being the carrying capacity (also the limiting asymptote value).]]
|[[Image:logistic_simulation_growthrate_r.png|thumb|400px|Effect of growth rate r on the logistic curve.]]
|[[Image:logistic_simulation_asymptote_K.png|thumb|400px|Effect of carrying capacity K on the logistic curve.]]
|-
|}

From the definition of the logistic function, the parameter r represents the growth rate of the diffusion invariant and the parameter K represents the asymptote value. The overall function shape intuitively follows the growth pattern we expect to see during brain maturation. The diffusion invariants start with an initial diffusion profile along tract, and have a non linear temporal growth trajectory showing maximum changes in early childhood and then slowing down or almost saturating at a certain age. (For instance, observed diffusion changes are much more in neonates than in an adult brain). The temporal trajectories may also differ along the tract's length giving localized changes. Since our method gives us continuous along-tract, growth trajectories all along time, we can compare subjects with respect to differences in diffusion profiles at birth, growth rates at any given age as well as the asymptote saturation values. This gives us important information to understand delayed or abnormal brain maturation by comparing a normative growth surface with an individual's or by comparing the model's parameters across subjects.

== Results ==

Below are some results using synthetic data. For more validation results and extension of the framework to jointly estimate individual subject trajectories together with a normative trajectory, refer to [http://ieeexplore.ieee.org/xpl/login.jsp?tp=&arnumber=6235829&url=http%3A%2F%2Fieeexplore.ieee.org%2Fxpls%2Fabs_all.jsp%3Farnumber%3D6235829 Sharma et al. (ISBI '12)].

{| border="0" style="background:transparent;"
|[[Image:namic_tract_longitudinal_main.png|thumb|400px|Longitudinal modeling of synthetic data. The data is generated from a template neonate FA mean curve and uses known logistic parameters to generate the three timepoint curves (red). Our method faithfully recovers the original curves (blue) while simultaneously creating a continuous longitudinal growth profile (green) without having any prior knowledge of the logistic parameters used for data generation. It also estimates an unbiased time=0 template curve (black) along with parameters r and K to avoid biasing the estimation with any assumed initialization at time=0.]]
|[[Image:namic_tract_longitudinal_3d.png|thumb|400px|Results on real data (mean FA curves for a subject at three timepoints:neonate, 1 year and 2 year).]]
|-
|}
{| border="0" style="background:transparent;"
|[[Image:namic_tract_longitudinal_p1p2only.png|thumb|400px|For the above real data (right), these are the final along-tract initial values and the final estimates for parameters-growth rate r(mentioned as p1 here) and asymptote K(mentioned as p2 here).]]
|[[Image:namic_tract_longitudinal_alphaonly.png|thumb|400px|The complete continuous growth grid as estimated for the real data above.]]

|-
|}

The below images show jointly estimated personalized trajectories for 15 control subjects along with the average growth trajectory. The estimated model parameters for 15 subjects as well as the average trajectory are also shown. The normative trajectory is colored by the local standard deviation. It points to the fact that despite individual variability in the maturation process, there is a strong agreement in the asymptote FA values seen around a gestational age of 2.5 years indicating a relative stabilization of the white matter changes across subjects. The framework thus quantifies patient-specific changes in serial diffusion data given discrete-time diffusion curves.

{| border="0" style="background:transparent;"
|[[Image:CONTE_Final_3d.png|thumb|400px|The jointly estimated individual growth trajectories for 4 of the 15 control subjects.]]
|[[Image:CONTE_Norm_Params.png|thumb|400px|Estimated model parameters for 15 subjects together with the normative growth trajectory colored by the local variability and the estimated normative model parameters.]]

|-
|}

== Experiments with Huntington's data ==

During the NAMIC Winter project week, we worked on registering subjects with Huntington's disease in the same coordinate space as control subjects. We then applied the above framework [http://ieeexplore.ieee.org/xpl/login.jsp?tp=&arnumber=6235829&url=http%3A%2F%2Fieeexplore.ieee.org%2Fxpls%2Fabs_all.jsp%3Farnumber%3D6235829 Sharma et al. ISBI'12] to quantify the differences in normal expected aging versus the accelerated white matter changes expected in HD. Details of the data pre-processing steps and image registration are available on the [http://www.na-mic.org/Wiki/index.php/2012_Winter_Project_Week:_DTI_Change_Modeling Winter AHM page]. Some results are summarized here for FA value along the genu tract.

Two subjects- one being a HD patient (10027) with a high burden factor (higher factor value (of factor 12) implying that the subjects is closer to the onset) and the other being a control subject (10004) are chosen for concept demonstration. Both have a similar baseline scan age (42 and 43 years respectively) as well as a similar time separation between follow up scans. This allows an intuitive comparison of the estimated growth trajectories.

{| border="0" style="background:transparent;"
|[[Image:ORIG_hd_cn.png|thumb|300px|center|FA curves corresponding to the genu tract (middle clipped portions) for one HD case (10027) and one control case (10004). Visual inspection clearly shows the huge decrease relative to the Control in the HD subject. On the other hand, the control shows normal expected decrease due to aging. Red-timepoint 1, Green-timepoint 2, Blue-timepoint 3. The left is HD case and the right plot is Control.]]
|[[Image:HD_vs_CN_3d.png|thumb|300px|center|The red trajectory corresponds the the HD subjects while the blue is a control subject with healthy aging. HD patient clearly shows a much sharper FA decrease along time than the Control.]]
|[[Image:HD_vs_CN_params.png|thumb|300px|center|The estimated model parameters (Red-HD case, Blue-Control case) for the two subjects. The top plot is the rate of change per unit time, per unit FA value. The middle plot is the function asymptote and the last plot is the common baseline curve estimation for providing a common reference frame for model estimation and comparison. The asymptote clearly shows a much lower FA range for HD when compared to the Control which already shows promising early prediction capabilities for HD cases where white matter decay is much faster than healthy patients. Also, a higher per unit rate of change quantifies the sharper FA decrease compared to control.]]
|-
|}

== Experiments with Krabbe's data ==

We have also applied the framework to study white matter changes in infants with Krabbe's disease. Here we show some preliminary results for the FA tract profile along the genu tract for a single subject with Krabbe's disease. In theory, FA non-linearly increases with time along genu owing to a healthy brain maturation process. However, in Krabbe's case, the FA does not follow a monotonic change pattern.

{| border="0" style="background:transparent;"
|[[Image:Genu_fa_3d_conte_vs_krabbe.png|thumb|400px|center|FA curves corresponding to the genu tract for the Krabbe's patient are shown (Neonate:black, 6months:cyan, 1year:magenta). The other FA profiles are for the 15 healthy control subjects as discussed previously (neonate:red, 1year:green, 2year:blue). This also highlights the fact that the framework does not assume uniformity or correspondence in the scan timings across subjects.]]
|[[Image:Genu_fa_3dnorm_conte_vs_krabbe.png|thumb|400px|center|The black trajectory is the normative FA growth estimated using 15 controls as discussed previously. The three FA curves for the Krabbe case are also plotted and clearly show how the patient is lagging behind on expected brain maturation.]]
|-
|}

== Future work ==

As we see in the Krabbe experiment, the model assumes a monotonic change along time. Neighboring locations along the tract can have different, yet correlated time changes. For eg. we can see localized increases or decreases along the tract length with respect to the time evolution. However, owing to the definition of the logistic function, any non-monotonic behavior along time at any given tract location is not modeled accurately. Therefore, in cases like the Krabbe's subject, we need a more relaxed and flexible semi-parametric or non-parametric model to remove the constraints coming in from the parametric nature of the current model. This is currently work under progress.
To utilize the quantification of growth trajectories enabled by this method, we also require a systematic hypothesis testing scheme to draw conclusive results of individuals differing from normative patterns or populations differing in their evolution behaviors. This too is currently being worked upon.

= References =

* Sharma, A., Durrleman, S. , Gilmore, J.H. , Gerig, G. Longitudinal growth modeling of discrete-time functions with application to DTI tract evolution in early neurodevelopment. Proc. of 9th IEEE International Symposium on Biomedical Imaging (ISBI May'12), p.1397-1400.
* Corouge, I., Fletcher, P.T., Joshi, S., Gouttard, S., Gerig, G., 2006. Fiber tract-oriented statistics for quantitative diffusion tensor MRI analysis. Med Image Anal, pp. 786-798.
* Goodlett, C.B., Fletcher, P.T., Gilmore, J.H., Gerig, G., 2009. Group analysis of DTI fiber tract statistics with application to neurodevelopment. Neuroimage, pp. S133-142.

==Key Investigators==
* Utah: Anuja Sharma, Stanley Durrleman, Guido Gerig
* INRIA/ICM, Pitie Salpetriere Hospital: Stanley Durrleman
[http://www.na-mic.org/Wiki/index.php/Algorithm:Utah2 Back to Utah 2 Algorithm Core]

== Regression of probability distributions obtained along white matter tracts to create spatio-temporal growth models ==

Temporal modeling frameworks often operate on scalar variables by
summarizing data at initial stages as statistical summaries of the underlying
distributions. For instance, DTI analysis often employs
summary statistics, like mean, for regions of interest and properties
along fiber tracts for population studies and hypothesis testing.
This reduction via discarding of variability information may introduce
significant errors which propagate through the procedures. Moreover, the data variability information is lost at
all the intermediate interpolated times.

{| border="0" style="background:transparent;"
|[[Image:DistributionRegression_Synthetic.png|thumb|600px|center|Left:Two synthetically generated distributions along time
(blue:t0, orange:t1) with translated locations and opposite skews.
Right: Simple linear interpolation using classic single-valued summary
statistics (means: red, medians: blue) and ignoring data variability
(shown as error bars) at t0 and t1. The decision to choose mean versus median can heavily influence the final regression estimates. Additionally, once the inherent variability is discarded for an early decision regarding a summary statistics, it cannot be recovered for the interpolated time points.]]
|-
|}

We propose a novel framework which uses distribution-valued variables
to retain and utilize the local variability information. We propose two different regression approaches to achieve our goals. The first is a non-parametric moving average method where minimal assumptions are involved in terms of the growth trajectory evolution (Sharma ISBI 2013). The second is a parametric regression scheme where the expected temporal trend is linear (Sharma ISBI 2014). It provides a compact representation in terms of model parameters for further statistical inference and has a closed form solution.

Our driving application is the modeling of age-related changes along DTI
white matter tracts. Results are shown for the spatiotemporal population
trajectory of genu tract estimated from a group of healthy infants and compared with an infant with Krabbe's disease.

{| border="0" style="background:transparent;"
|[[Image:DWIalongTime.png|thumb|400px|center|Model age related DTI changes in early neurodevelopment.]]
|[[Image:genu_distributions_along_time.png|thumb|600px|center|Distributions of DTI-derived scalar diffusion parameters like FA are obtained along the length of the genu tract. The distributions correspond to the cross-sections of the 3D tract geometry and are a function of the arc-length along the tract's total length as we move from one end of the tract to the other. This provides a functional curve which is parametrized by a feature of the local tract geometry (arc-length along the tract in this case) and is attributed by probability distributions (instead of scalar values like the 'mean' FA obtained from these local distributions).]]
|-
|}

=== Non-parametric regression for distribution-valued data ===

The proposed method
presents a completely nonparametric regression framework for spatial-temporal evolution of
statistical distributions. This avoids any parametric assumptions regarding the nature of the
underlying noise model in the DTI data and retains and utilizes the variability information to
estimate the regression trend. The estimated trajectory is completely characterized by statistical
distributions in space and time enabling a very powerful statistical inference framework. Both
population and individual trajectories can be estimated. The concept of ’distance’ between distributions and an ’average’ of distributions is employed. The dissimilarity metric employed is the Mallow's distance between distributions. It is the L2 norm of the generic Wasserstein metric and captures translation, changing width and shape of the participating distributions.

{| border="0" style="background:transparent;"
|[[Image:dist2_3d_bary_300_revrev.png|thumb|400px|center|Temporal evolution of the synthetic distributions between two timepoints as estimated by the proposed method.]]
|[[Image:bary_stat_300.png|thumb|400px|center|Since the regression estimates the complete probability distribution continuously along time, we now have access to the complete variability information at the interpolated time points. This can enable statistical inference using statistical summary measures as well as distribution differences at time points where the original image data is not available.]]
|-
|}

{| border="0" style="background:transparent;"
|[[Image:conte_krabbe_median_3d_300_2.png|thumb|400px|center|The medians of the estimated probability distributions are shown here for the healthy population (in red/pink) versus a single infant with Krabbe's disease (blue/black). Note that the method can applied to estimate normative population trajectories as well as subject-specific evolutions. Each point on these statistical manifolds are characterized by the complete distribution information (instead of just a scalar value).]]
|[[Image:conte_vs_krabbe_quartiles.png|thumb|600px|center|Highlighting the quantiles obtained at one of the arc length locations in the left plot. The Krabbe's subject seems to lag behind the normative population and the difference can be quantified in terms of distribution-based inferences.]]
|-
|}

=== Parametric linear regression for distribution-valued data ===

We build on the motivation as highlighted in the previous distribution-based regression. However,
the previous method is completely nonparametric leading to an intensive and complex statistical
inference framework based on comparison of statistical manifolds. Therefore, we simplify the
analysis for situations where a linear time-course trajectory can be assumed. We apply the classic
linear regression technique, extended to work with distribution-valued data. The advantage lies in
the compact description of the spatial-temporal trajectory in terms of the estimated regression
coefficients which can now be conveniently used for statistical inference. A synthetic validation
experiment is conducted to show the improved reliability and robustness of this method when
compared with methods working with statistical summaries of the underlying data.

{| border="0" style="background:transparent;"
|[[Image:Timedist_rev1.png|thumb|400px|center|The independent variable is the age distribution of the healthy infants clustered around 2 months, 1 year and 2 year.]]
|[[Image:Baryhist3d1.png|thumb|400px|center|The independent variable is the barycentric FA distribution estimated at each arc length location along the tract, using the original FA distributions available from all subjects within the corresponding age group. The regression estimates the continuous spatiotemporal trends in terms of the model parameters using the respective pairs of age and FA distributions as the 'observed' data.]]
|-
|}

{| border="0" style="background:transparent;"
|[[Image:PopuCN_dist_3d1.png|thumb|400px|center|Continuous spatiotemporal normative growth trajectory: Estimated mean FA (gray), Age range where control
data is originally available (green, orange, purple).]]
|[[Image:PopuvsKrabbe_dist_3d_2_edit1.png|thumb|400px|center|Krabbe's subject's scans (solid lines): (14 days, 6 mo., 1 yr.) with respect to the estimated healthy trajectory.]]
|[[Image:PopuvsKrabbe_dist_3d31.png|thumb|400px|center|Krabbe's subject's scans:mean FA (solid lines).Control population:estimated mean FA (dashed lines) +/- 3*std.dev. bounds (calculated for Gaussian age distributions centered at matched timepoints with std.dev.=5 days).]]
|-
|}

= References =

* A. Sharma, P.T. Fletcher, J.H. Gilmore, M.L. Escolar, A. Gupta, M. Styner, G. Gerig. “ Parametric Regression Scheme For Distributions: Analysis of DTI Fiber Tract Diffusion Changes In Early Brain Development,” In Proceedings of the 2014 IEEE 11th International Symposium on Biomedical Imaging (ISBI), pp. 559-562. 2014.
* A. Sharma, P.T. Fletcher, J.H. Gilmore, M.L. Escolar, A. Gupta, M. Styner, G. Gerig. “Spatiotemporal Modeling of Discrete-Time Distribution-Valued Data Applied to DTI Tract Evolution in Infant Neurodevelopment,” In Proceedings of the 2013 IEEE 10th International Symposium on Biomedical Imaging (ISBI), pp. 684--687. 2013.
* A. Sharma, S. Durrleman, J.H. Gilmore, G. Gerig. “Longitudinal Growth Modeling of Discrete-Time Functions with Application to DTI Tract Evolution in Early Neurodevelopment,” In Proceedings of IEEE ISBI 2012, pp. 1397--1400. 2012.
* Corouge, I., Fletcher, P.T., Joshi, S., Gouttard, S., Gerig, G., 2006. Fiber tract-oriented statistics for quantitative diffusion tensor MRI analysis. Med Image Anal, pp. 786-798.
* Goodlett, C.B., Fletcher, P.T., Gilmore, J.H., Gerig, G., 2009. Group analysis of DTI fiber tract statistics with application to neurodevelopment. Neuroimage, pp. S133-142.

File:Timehistrev11.png

2014-10-15T23:04:25Z

Anuja:

Projects:TractLongitudinalDTI

2014-10-15T23:03:25Z

Anuja: /* Parametric linear regression for distribution-valued data */

Back to [[Algorithm:Utah2|Utah 2 Algorithms]]
__NOTOC__

== Tract-based longitudinal modeling of DTI data ==

This project develops a methodology to explore subject-specific, DTI data obtained from brain's white matter tracts, available at multiple but often sparsely present timepoints. The challenge is to develop a continuous spatio-temporal growth model, given discrete 4D DWI images. This would enable comparison of growth trajectories across subjects and along tracts which are biologically of interest in developmental and pathological changes.

== Background ==

We use the arc length parametrization scheme initially proposed by Corouge et al. It represents white matter fiber tracts obtained via streamline tractography in the brain's atlas tensor image as a function of arc length. (Atlas construction uses unbiased atlas building schemes followed by back transformations to subjects' DTI images to obtain identical fiber tract geometry across subjects, populated with subject specific diffusion data). We use the mean diffusion scalar invariants derived from these individual fiber bundle cross sections, as our input longitudinal diffusion profiles.

{| border="0" style="background:transparent;"
|[[Image:Tract-stats.png|thumb|700px|White matter diffusion properties along fiber tract: Left: A fiber tract with an origin plane defined for arc length parametrization, Right: FA mean and standard deviation as function of arc-length. Dots mark location of coordinate origin.]]
|-
|}

== Subject-specific spatiotemporal continuous growth model ==

We propose the use of Verhulst-Pearl logistic equation to capture temporal changes, while using a non-parametric kernel along arc length to account for biologically motivated functional along-tract relationship in the diffusion data.

{| border="0" style="background:transparent;"
|[[Image:logistic_eq.png|thumb|600px|Logistic equation with P as the population value at a given time, r being the growth rate of the population and K being the carrying capacity (also the limiting asymptote value).]]
|[[Image:logistic_simulation_growthrate_r.png|thumb|400px|Effect of growth rate r on the logistic curve.]]
|[[Image:logistic_simulation_asymptote_K.png|thumb|400px|Effect of carrying capacity K on the logistic curve.]]
|-
|}

From the definition of the logistic function, the parameter r represents the growth rate of the diffusion invariant and the parameter K represents the asymptote value. The overall function shape intuitively follows the growth pattern we expect to see during brain maturation. The diffusion invariants start with an initial diffusion profile along tract, and have a non linear temporal growth trajectory showing maximum changes in early childhood and then slowing down or almost saturating at a certain age. (For instance, observed diffusion changes are much more in neonates than in an adult brain). The temporal trajectories may also differ along the tract's length giving localized changes. Since our method gives us continuous along-tract, growth trajectories all along time, we can compare subjects with respect to differences in diffusion profiles at birth, growth rates at any given age as well as the asymptote saturation values. This gives us important information to understand delayed or abnormal brain maturation by comparing a normative growth surface with an individual's or by comparing the model's parameters across subjects.

== Results ==

Below are some results using synthetic data. For more validation results and extension of the framework to jointly estimate individual subject trajectories together with a normative trajectory, refer to [http://ieeexplore.ieee.org/xpl/login.jsp?tp=&arnumber=6235829&url=http%3A%2F%2Fieeexplore.ieee.org%2Fxpls%2Fabs_all.jsp%3Farnumber%3D6235829 Sharma et al. (ISBI '12)].

{| border="0" style="background:transparent;"
|[[Image:namic_tract_longitudinal_main.png|thumb|400px|Longitudinal modeling of synthetic data. The data is generated from a template neonate FA mean curve and uses known logistic parameters to generate the three timepoint curves (red). Our method faithfully recovers the original curves (blue) while simultaneously creating a continuous longitudinal growth profile (green) without having any prior knowledge of the logistic parameters used for data generation. It also estimates an unbiased time=0 template curve (black) along with parameters r and K to avoid biasing the estimation with any assumed initialization at time=0.]]
|[[Image:namic_tract_longitudinal_3d.png|thumb|400px|Results on real data (mean FA curves for a subject at three timepoints:neonate, 1 year and 2 year).]]
|-
|}
{| border="0" style="background:transparent;"
|[[Image:namic_tract_longitudinal_p1p2only.png|thumb|400px|For the above real data (right), these are the final along-tract initial values and the final estimates for parameters-growth rate r(mentioned as p1 here) and asymptote K(mentioned as p2 here).]]
|[[Image:namic_tract_longitudinal_alphaonly.png|thumb|400px|The complete continuous growth grid as estimated for the real data above.]]

|-
|}

The below images show jointly estimated personalized trajectories for 15 control subjects along with the average growth trajectory. The estimated model parameters for 15 subjects as well as the average trajectory are also shown. The normative trajectory is colored by the local standard deviation. It points to the fact that despite individual variability in the maturation process, there is a strong agreement in the asymptote FA values seen around a gestational age of 2.5 years indicating a relative stabilization of the white matter changes across subjects. The framework thus quantifies patient-specific changes in serial diffusion data given discrete-time diffusion curves.

{| border="0" style="background:transparent;"
|[[Image:CONTE_Final_3d.png|thumb|400px|The jointly estimated individual growth trajectories for 4 of the 15 control subjects.]]
|[[Image:CONTE_Norm_Params.png|thumb|400px|Estimated model parameters for 15 subjects together with the normative growth trajectory colored by the local variability and the estimated normative model parameters.]]

|-
|}

== Experiments with Huntington's data ==

During the NAMIC Winter project week, we worked on registering subjects with Huntington's disease in the same coordinate space as control subjects. We then applied the above framework [http://ieeexplore.ieee.org/xpl/login.jsp?tp=&arnumber=6235829&url=http%3A%2F%2Fieeexplore.ieee.org%2Fxpls%2Fabs_all.jsp%3Farnumber%3D6235829 Sharma et al. ISBI'12] to quantify the differences in normal expected aging versus the accelerated white matter changes expected in HD. Details of the data pre-processing steps and image registration are available on the [http://www.na-mic.org/Wiki/index.php/2012_Winter_Project_Week:_DTI_Change_Modeling Winter AHM page]. Some results are summarized here for FA value along the genu tract.

Two subjects- one being a HD patient (10027) with a high burden factor (higher factor value (of factor 12) implying that the subjects is closer to the onset) and the other being a control subject (10004) are chosen for concept demonstration. Both have a similar baseline scan age (42 and 43 years respectively) as well as a similar time separation between follow up scans. This allows an intuitive comparison of the estimated growth trajectories.

{| border="0" style="background:transparent;"
|[[Image:ORIG_hd_cn.png|thumb|300px|center|FA curves corresponding to the genu tract (middle clipped portions) for one HD case (10027) and one control case (10004). Visual inspection clearly shows the huge decrease relative to the Control in the HD subject. On the other hand, the control shows normal expected decrease due to aging. Red-timepoint 1, Green-timepoint 2, Blue-timepoint 3. The left is HD case and the right plot is Control.]]
|[[Image:HD_vs_CN_3d.png|thumb|300px|center|The red trajectory corresponds the the HD subjects while the blue is a control subject with healthy aging. HD patient clearly shows a much sharper FA decrease along time than the Control.]]
|[[Image:HD_vs_CN_params.png|thumb|300px|center|The estimated model parameters (Red-HD case, Blue-Control case) for the two subjects. The top plot is the rate of change per unit time, per unit FA value. The middle plot is the function asymptote and the last plot is the common baseline curve estimation for providing a common reference frame for model estimation and comparison. The asymptote clearly shows a much lower FA range for HD when compared to the Control which already shows promising early prediction capabilities for HD cases where white matter decay is much faster than healthy patients. Also, a higher per unit rate of change quantifies the sharper FA decrease compared to control.]]
|-
|}

== Experiments with Krabbe's data ==

We have also applied the framework to study white matter changes in infants with Krabbe's disease. Here we show some preliminary results for the FA tract profile along the genu tract for a single subject with Krabbe's disease. In theory, FA non-linearly increases with time along genu owing to a healthy brain maturation process. However, in Krabbe's case, the FA does not follow a monotonic change pattern.

{| border="0" style="background:transparent;"
|[[Image:Genu_fa_3d_conte_vs_krabbe.png|thumb|400px|center|FA curves corresponding to the genu tract for the Krabbe's patient are shown (Neonate:black, 6months:cyan, 1year:magenta). The other FA profiles are for the 15 healthy control subjects as discussed previously (neonate:red, 1year:green, 2year:blue). This also highlights the fact that the framework does not assume uniformity or correspondence in the scan timings across subjects.]]
|[[Image:Genu_fa_3dnorm_conte_vs_krabbe.png|thumb|400px|center|The black trajectory is the normative FA growth estimated using 15 controls as discussed previously. The three FA curves for the Krabbe case are also plotted and clearly show how the patient is lagging behind on expected brain maturation.]]
|-
|}

== Future work ==

As we see in the Krabbe experiment, the model assumes a monotonic change along time. Neighboring locations along the tract can have different, yet correlated time changes. For eg. we can see localized increases or decreases along the tract length with respect to the time evolution. However, owing to the definition of the logistic function, any non-monotonic behavior along time at any given tract location is not modeled accurately. Therefore, in cases like the Krabbe's subject, we need a more relaxed and flexible semi-parametric or non-parametric model to remove the constraints coming in from the parametric nature of the current model. This is currently work under progress.
To utilize the quantification of growth trajectories enabled by this method, we also require a systematic hypothesis testing scheme to draw conclusive results of individuals differing from normative patterns or populations differing in their evolution behaviors. This too is currently being worked upon.

= References =

* Sharma, A., Durrleman, S. , Gilmore, J.H. , Gerig, G. Longitudinal growth modeling of discrete-time functions with application to DTI tract evolution in early neurodevelopment. Proc. of 9th IEEE International Symposium on Biomedical Imaging (ISBI May'12), p.1397-1400.
* Corouge, I., Fletcher, P.T., Joshi, S., Gouttard, S., Gerig, G., 2006. Fiber tract-oriented statistics for quantitative diffusion tensor MRI analysis. Med Image Anal, pp. 786-798.
* Goodlett, C.B., Fletcher, P.T., Gilmore, J.H., Gerig, G., 2009. Group analysis of DTI fiber tract statistics with application to neurodevelopment. Neuroimage, pp. S133-142.

==Key Investigators==
* Utah: Anuja Sharma, Stanley Durrleman, Guido Gerig
* INRIA/ICM, Pitie Salpetriere Hospital: Stanley Durrleman
[http://www.na-mic.org/Wiki/index.php/Algorithm:Utah2 Back to Utah 2 Algorithm Core]

== Regression of probability distributions obtained along white matter tracts to create spatio-temporal growth models ==

Temporal modeling frameworks often operate on scalar variables by
summarizing data at initial stages as statistical summaries of the underlying
distributions. For instance, DTI analysis often employs
summary statistics, like mean, for regions of interest and properties
along fiber tracts for population studies and hypothesis testing.
This reduction via discarding of variability information may introduce
significant errors which propagate through the procedures. Moreover, the data variability information is lost at
all the intermediate interpolated times.

{| border="0" style="background:transparent;"
|[[Image:DistributionRegression_Synthetic.png|thumb|600px|center|Left:Two synthetically generated distributions along time
(blue:t0, orange:t1) with translated locations and opposite skews.
Right: Simple linear interpolation using classic single-valued summary
statistics (means: red, medians: blue) and ignoring data variability
(shown as error bars) at t0 and t1. The decision to choose mean versus median can heavily influence the final regression estimates. Additionally, once the inherent variability is discarded for an early decision regarding a summary statistics, it cannot be recovered for the interpolated time points.]]
|-
|}

We propose a novel framework which uses distribution-valued variables
to retain and utilize the local variability information. We propose two different regression approaches to achieve our goals. The first is a non-parametric moving average method where minimal assumptions are involved in terms of the growth trajectory evolution (Sharma ISBI 2013). The second is a parametric regression scheme where the expected temporal trend is linear (Sharma ISBI 2014). It provides a compact representation in terms of model parameters for further statistical inference and has a closed form solution.

Our driving application is the modeling of age-related changes along DTI
white matter tracts. Results are shown for the spatiotemporal population
trajectory of genu tract estimated from a group of healthy infants and compared with an infant with Krabbe's disease.

{| border="0" style="background:transparent;"
|[[Image:DWIalongTime.png|thumb|400px|center|Model age related DTI changes in early neurodevelopment.]]
|[[Image:genu_distributions_along_time.png|thumb|600px|center|Distributions of DTI-derived scalar diffusion parameters like FA are obtained along the length of the genu tract. The distributions correspond to the cross-sections of the 3D tract geometry and are a function of the arc-length along the tract's total length as we move from one end of the tract to the other. This provides a functional curve which is parametrized by a feature of the local tract geometry (arc-length along the tract in this case) and is attributed by probability distributions (instead of scalar values like the 'mean' FA obtained from these local distributions).]]
|-
|}

=== Non-parametric regression for distribution-valued data ===

The proposed method
presents a completely nonparametric regression framework for spatial-temporal evolution of
statistical distributions. This avoids any parametric assumptions regarding the nature of the
underlying noise model in the DTI data and retains and utilizes the variability information to
estimate the regression trend. The estimated trajectory is completely characterized by statistical
distributions in space and time enabling a very powerful statistical inference framework. Both
population and individual trajectories can be estimated. The concept of ’distance’ between distributions and an ’average’ of distributions is employed. The dissimilarity metric employed is the Mallow's distance between distributions. It is the L2 norm of the generic Wasserstein metric and captures translation, changing width and shape of the participating distributions.

{| border="0" style="background:transparent;"
|[[Image:dist2_3d_bary_300_revrev.png|thumb|400px|center|Temporal evolution of the synthetic distributions between two timepoints as estimated by the proposed method.]]
|[[Image:bary_stat_300.png|thumb|400px|center|Since the regression estimates the complete probability distribution continuously along time, we now have access to the complete variability information at the interpolated time points. This can enable statistical inference using statistical summary measures as well as distribution differences at time points where the original image data is not available.]]
|-
|}

{| border="0" style="background:transparent;"
|[[Image:conte_krabbe_median_3d_300_2.png|thumb|400px|center|The medians of the estimated probability distributions are shown here for the healthy population (in red/pink) versus a single infant with Krabbe's disease (blue/black). Note that the method can applied to estimate normative population trajectories as well as subject-specific evolutions. Each point on these statistical manifolds are characterized by the complete distribution information (instead of just a scalar value).]]
|[[Image:conte_vs_krabbe_quartiles.png|thumb|600px|center|Highlighting the quantiles obtained at one of the arc length locations in the left plot. The Krabbe's subject seems to lag behind the normative population and the difference can be quantified in terms of distribution-based inferences.]]
|-
|}

=== Parametric linear regression for distribution-valued data ===

We build on the motivation as highlighted in the previous distribution-based regression. However,
the previous method is completely nonparametric leading to an intensive and complex statistical
inference framework based on comparison of statistical manifolds. Therefore, we simplify the
analysis for situations where a linear time-course trajectory can be assumed. We apply the classic
linear regression technique, extended to work with distribution-valued data. The advantage lies in
the compact description of the spatial-temporal trajectory in terms of the estimated regression
coefficients which can now be conveniently used for statistical inference. A synthetic validation
experiment is conducted to show the improved reliability and robustness of this method when
compared with methods working with statistical summaries of the underlying data.

{| border="0" style="background:transparent;"
|[[Image:Timehistrev1.png|thumb|400px|center|The independent variable is the age distribution of the healthy infants clustered around 2 months, 1 year and 2 year.]]
|[[Image:Baryhist3d1.png|thumb|400px|center|The independent variable is the barycentric FA distribution estimated at each arc length location along the tract, using the original FA distributions available from all subjects within the corresponding age group. The regression estimates the continuous spatiotemporal trends in terms of the model parameters using the respective pairs of age and FA distributions as the 'observed' data.]]
|-
|}

{| border="0" style="background:transparent;"
|[[Image:PopuCN_dist_3d1.png|thumb|400px|center|Continuous spatiotemporal normative growth trajectory: Estimated mean FA (gray), Age range where control
data is originally available (green, orange, purple).]]
|[[Image:PopuvsKrabbe_dist_3d_2_edit1.png|thumb|400px|center|Krabbe's subject's scans (solid lines): (14 days, 6 mo., 1 yr.) with respect to the estimated healthy trajectory.]]
|[[Image:PopuvsKrabbe_dist_3d31.png|thumb|400px|center|Krabbe's subject's scans:mean FA (solid lines).Control population:estimated mean FA (dashed lines) +/- 3*std.dev. bounds (calculated for Gaussian age distributions centered at matched timepoints with std.dev.=5 days).]]
|-
|}

= References =

* A. Sharma, P.T. Fletcher, J.H. Gilmore, M.L. Escolar, A. Gupta, M. Styner, G. Gerig. “ Parametric Regression Scheme For Distributions: Analysis of DTI Fiber Tract Diffusion Changes In Early Brain Development,” In Proceedings of the 2014 IEEE 11th International Symposium on Biomedical Imaging (ISBI), pp. 559-562. 2014.
* A. Sharma, P.T. Fletcher, J.H. Gilmore, M.L. Escolar, A. Gupta, M. Styner, G. Gerig. “Spatiotemporal Modeling of Discrete-Time Distribution-Valued Data Applied to DTI Tract Evolution in Infant Neurodevelopment,” In Proceedings of the 2013 IEEE 10th International Symposium on Biomedical Imaging (ISBI), pp. 684--687. 2013.
* A. Sharma, S. Durrleman, J.H. Gilmore, G. Gerig. “Longitudinal Growth Modeling of Discrete-Time Functions with Application to DTI Tract Evolution in Early Neurodevelopment,” In Proceedings of IEEE ISBI 2012, pp. 1397--1400. 2012.
* Corouge, I., Fletcher, P.T., Joshi, S., Gouttard, S., Gerig, G., 2006. Fiber tract-oriented statistics for quantitative diffusion tensor MRI analysis. Med Image Anal, pp. 786-798.
* Goodlett, C.B., Fletcher, P.T., Gilmore, J.H., Gerig, G., 2009. Group analysis of DTI fiber tract statistics with application to neurodevelopment. Neuroimage, pp. S133-142.

File:PopuvsKrabbe dist 3d 2 edit1.png

2014-10-15T23:01:58Z

Anuja:

File:PopuCN dist 3d1.png

2014-10-15T23:01:46Z

Anuja:

File:Baryhist3d1.png

2014-10-15T23:01:30Z

Anuja:

File:Timehist rev1.png

2014-10-15T23:01:03Z

Anuja:

Projects:TractLongitudinalDTI

2014-10-15T23:00:41Z

Anuja: /* Parametric linear regression for distribution-valued data */

Back to [[Algorithm:Utah2|Utah 2 Algorithms]]
__NOTOC__

== Tract-based longitudinal modeling of DTI data ==

This project develops a methodology to explore subject-specific, DTI data obtained from brain's white matter tracts, available at multiple but often sparsely present timepoints. The challenge is to develop a continuous spatio-temporal growth model, given discrete 4D DWI images. This would enable comparison of growth trajectories across subjects and along tracts which are biologically of interest in developmental and pathological changes.

== Background ==

We use the arc length parametrization scheme initially proposed by Corouge et al. It represents white matter fiber tracts obtained via streamline tractography in the brain's atlas tensor image as a function of arc length. (Atlas construction uses unbiased atlas building schemes followed by back transformations to subjects' DTI images to obtain identical fiber tract geometry across subjects, populated with subject specific diffusion data). We use the mean diffusion scalar invariants derived from these individual fiber bundle cross sections, as our input longitudinal diffusion profiles.

{| border="0" style="background:transparent;"
|[[Image:Tract-stats.png|thumb|700px|White matter diffusion properties along fiber tract: Left: A fiber tract with an origin plane defined for arc length parametrization, Right: FA mean and standard deviation as function of arc-length. Dots mark location of coordinate origin.]]
|-
|}

== Subject-specific spatiotemporal continuous growth model ==

We propose the use of Verhulst-Pearl logistic equation to capture temporal changes, while using a non-parametric kernel along arc length to account for biologically motivated functional along-tract relationship in the diffusion data.

{| border="0" style="background:transparent;"
|[[Image:logistic_eq.png|thumb|600px|Logistic equation with P as the population value at a given time, r being the growth rate of the population and K being the carrying capacity (also the limiting asymptote value).]]
|[[Image:logistic_simulation_growthrate_r.png|thumb|400px|Effect of growth rate r on the logistic curve.]]
|[[Image:logistic_simulation_asymptote_K.png|thumb|400px|Effect of carrying capacity K on the logistic curve.]]
|-
|}

From the definition of the logistic function, the parameter r represents the growth rate of the diffusion invariant and the parameter K represents the asymptote value. The overall function shape intuitively follows the growth pattern we expect to see during brain maturation. The diffusion invariants start with an initial diffusion profile along tract, and have a non linear temporal growth trajectory showing maximum changes in early childhood and then slowing down or almost saturating at a certain age. (For instance, observed diffusion changes are much more in neonates than in an adult brain). The temporal trajectories may also differ along the tract's length giving localized changes. Since our method gives us continuous along-tract, growth trajectories all along time, we can compare subjects with respect to differences in diffusion profiles at birth, growth rates at any given age as well as the asymptote saturation values. This gives us important information to understand delayed or abnormal brain maturation by comparing a normative growth surface with an individual's or by comparing the model's parameters across subjects.

== Results ==

Below are some results using synthetic data. For more validation results and extension of the framework to jointly estimate individual subject trajectories together with a normative trajectory, refer to [http://ieeexplore.ieee.org/xpl/login.jsp?tp=&arnumber=6235829&url=http%3A%2F%2Fieeexplore.ieee.org%2Fxpls%2Fabs_all.jsp%3Farnumber%3D6235829 Sharma et al. (ISBI '12)].

{| border="0" style="background:transparent;"
|[[Image:namic_tract_longitudinal_main.png|thumb|400px|Longitudinal modeling of synthetic data. The data is generated from a template neonate FA mean curve and uses known logistic parameters to generate the three timepoint curves (red). Our method faithfully recovers the original curves (blue) while simultaneously creating a continuous longitudinal growth profile (green) without having any prior knowledge of the logistic parameters used for data generation. It also estimates an unbiased time=0 template curve (black) along with parameters r and K to avoid biasing the estimation with any assumed initialization at time=0.]]
|[[Image:namic_tract_longitudinal_3d.png|thumb|400px|Results on real data (mean FA curves for a subject at three timepoints:neonate, 1 year and 2 year).]]
|-
|}
{| border="0" style="background:transparent;"
|[[Image:namic_tract_longitudinal_p1p2only.png|thumb|400px|For the above real data (right), these are the final along-tract initial values and the final estimates for parameters-growth rate r(mentioned as p1 here) and asymptote K(mentioned as p2 here).]]
|[[Image:namic_tract_longitudinal_alphaonly.png|thumb|400px|The complete continuous growth grid as estimated for the real data above.]]

|-
|}

The below images show jointly estimated personalized trajectories for 15 control subjects along with the average growth trajectory. The estimated model parameters for 15 subjects as well as the average trajectory are also shown. The normative trajectory is colored by the local standard deviation. It points to the fact that despite individual variability in the maturation process, there is a strong agreement in the asymptote FA values seen around a gestational age of 2.5 years indicating a relative stabilization of the white matter changes across subjects. The framework thus quantifies patient-specific changes in serial diffusion data given discrete-time diffusion curves.

{| border="0" style="background:transparent;"
|[[Image:CONTE_Final_3d.png|thumb|400px|The jointly estimated individual growth trajectories for 4 of the 15 control subjects.]]
|[[Image:CONTE_Norm_Params.png|thumb|400px|Estimated model parameters for 15 subjects together with the normative growth trajectory colored by the local variability and the estimated normative model parameters.]]

|-
|}

== Experiments with Huntington's data ==

During the NAMIC Winter project week, we worked on registering subjects with Huntington's disease in the same coordinate space as control subjects. We then applied the above framework [http://ieeexplore.ieee.org/xpl/login.jsp?tp=&arnumber=6235829&url=http%3A%2F%2Fieeexplore.ieee.org%2Fxpls%2Fabs_all.jsp%3Farnumber%3D6235829 Sharma et al. ISBI'12] to quantify the differences in normal expected aging versus the accelerated white matter changes expected in HD. Details of the data pre-processing steps and image registration are available on the [http://www.na-mic.org/Wiki/index.php/2012_Winter_Project_Week:_DTI_Change_Modeling Winter AHM page]. Some results are summarized here for FA value along the genu tract.

Two subjects- one being a HD patient (10027) with a high burden factor (higher factor value (of factor 12) implying that the subjects is closer to the onset) and the other being a control subject (10004) are chosen for concept demonstration. Both have a similar baseline scan age (42 and 43 years respectively) as well as a similar time separation between follow up scans. This allows an intuitive comparison of the estimated growth trajectories.

{| border="0" style="background:transparent;"
|[[Image:ORIG_hd_cn.png|thumb|300px|center|FA curves corresponding to the genu tract (middle clipped portions) for one HD case (10027) and one control case (10004). Visual inspection clearly shows the huge decrease relative to the Control in the HD subject. On the other hand, the control shows normal expected decrease due to aging. Red-timepoint 1, Green-timepoint 2, Blue-timepoint 3. The left is HD case and the right plot is Control.]]
|[[Image:HD_vs_CN_3d.png|thumb|300px|center|The red trajectory corresponds the the HD subjects while the blue is a control subject with healthy aging. HD patient clearly shows a much sharper FA decrease along time than the Control.]]
|[[Image:HD_vs_CN_params.png|thumb|300px|center|The estimated model parameters (Red-HD case, Blue-Control case) for the two subjects. The top plot is the rate of change per unit time, per unit FA value. The middle plot is the function asymptote and the last plot is the common baseline curve estimation for providing a common reference frame for model estimation and comparison. The asymptote clearly shows a much lower FA range for HD when compared to the Control which already shows promising early prediction capabilities for HD cases where white matter decay is much faster than healthy patients. Also, a higher per unit rate of change quantifies the sharper FA decrease compared to control.]]
|-
|}

== Experiments with Krabbe's data ==

We have also applied the framework to study white matter changes in infants with Krabbe's disease. Here we show some preliminary results for the FA tract profile along the genu tract for a single subject with Krabbe's disease. In theory, FA non-linearly increases with time along genu owing to a healthy brain maturation process. However, in Krabbe's case, the FA does not follow a monotonic change pattern.

{| border="0" style="background:transparent;"
|[[Image:Genu_fa_3d_conte_vs_krabbe.png|thumb|400px|center|FA curves corresponding to the genu tract for the Krabbe's patient are shown (Neonate:black, 6months:cyan, 1year:magenta). The other FA profiles are for the 15 healthy control subjects as discussed previously (neonate:red, 1year:green, 2year:blue). This also highlights the fact that the framework does not assume uniformity or correspondence in the scan timings across subjects.]]
|[[Image:Genu_fa_3dnorm_conte_vs_krabbe.png|thumb|400px|center|The black trajectory is the normative FA growth estimated using 15 controls as discussed previously. The three FA curves for the Krabbe case are also plotted and clearly show how the patient is lagging behind on expected brain maturation.]]
|-
|}

== Future work ==

As we see in the Krabbe experiment, the model assumes a monotonic change along time. Neighboring locations along the tract can have different, yet correlated time changes. For eg. we can see localized increases or decreases along the tract length with respect to the time evolution. However, owing to the definition of the logistic function, any non-monotonic behavior along time at any given tract location is not modeled accurately. Therefore, in cases like the Krabbe's subject, we need a more relaxed and flexible semi-parametric or non-parametric model to remove the constraints coming in from the parametric nature of the current model. This is currently work under progress.
To utilize the quantification of growth trajectories enabled by this method, we also require a systematic hypothesis testing scheme to draw conclusive results of individuals differing from normative patterns or populations differing in their evolution behaviors. This too is currently being worked upon.

= References =

* Sharma, A., Durrleman, S. , Gilmore, J.H. , Gerig, G. Longitudinal growth modeling of discrete-time functions with application to DTI tract evolution in early neurodevelopment. Proc. of 9th IEEE International Symposium on Biomedical Imaging (ISBI May'12), p.1397-1400.
* Corouge, I., Fletcher, P.T., Joshi, S., Gouttard, S., Gerig, G., 2006. Fiber tract-oriented statistics for quantitative diffusion tensor MRI analysis. Med Image Anal, pp. 786-798.
* Goodlett, C.B., Fletcher, P.T., Gilmore, J.H., Gerig, G., 2009. Group analysis of DTI fiber tract statistics with application to neurodevelopment. Neuroimage, pp. S133-142.

==Key Investigators==
* Utah: Anuja Sharma, Stanley Durrleman, Guido Gerig
* INRIA/ICM, Pitie Salpetriere Hospital: Stanley Durrleman
[http://www.na-mic.org/Wiki/index.php/Algorithm:Utah2 Back to Utah 2 Algorithm Core]

== Regression of probability distributions obtained along white matter tracts to create spatio-temporal growth models ==

Temporal modeling frameworks often operate on scalar variables by
summarizing data at initial stages as statistical summaries of the underlying
distributions. For instance, DTI analysis often employs
summary statistics, like mean, for regions of interest and properties
along fiber tracts for population studies and hypothesis testing.
This reduction via discarding of variability information may introduce
significant errors which propagate through the procedures. Moreover, the data variability information is lost at
all the intermediate interpolated times.

{| border="0" style="background:transparent;"
|[[Image:DistributionRegression_Synthetic.png|thumb|600px|center|Left:Two synthetically generated distributions along time
(blue:t0, orange:t1) with translated locations and opposite skews.
Right: Simple linear interpolation using classic single-valued summary
statistics (means: red, medians: blue) and ignoring data variability
(shown as error bars) at t0 and t1. The decision to choose mean versus median can heavily influence the final regression estimates. Additionally, once the inherent variability is discarded for an early decision regarding a summary statistics, it cannot be recovered for the interpolated time points.]]
|-
|}

We propose a novel framework which uses distribution-valued variables
to retain and utilize the local variability information. We propose two different regression approaches to achieve our goals. The first is a non-parametric moving average method where minimal assumptions are involved in terms of the growth trajectory evolution (Sharma ISBI 2013). The second is a parametric regression scheme where the expected temporal trend is linear (Sharma ISBI 2014). It provides a compact representation in terms of model parameters for further statistical inference and has a closed form solution.

Our driving application is the modeling of age-related changes along DTI
white matter tracts. Results are shown for the spatiotemporal population
trajectory of genu tract estimated from a group of healthy infants and compared with an infant with Krabbe's disease.

{| border="0" style="background:transparent;"
|[[Image:DWIalongTime.png|thumb|400px|center|Model age related DTI changes in early neurodevelopment.]]
|[[Image:genu_distributions_along_time.png|thumb|600px|center|Distributions of DTI-derived scalar diffusion parameters like FA are obtained along the length of the genu tract. The distributions correspond to the cross-sections of the 3D tract geometry and are a function of the arc-length along the tract's total length as we move from one end of the tract to the other. This provides a functional curve which is parametrized by a feature of the local tract geometry (arc-length along the tract in this case) and is attributed by probability distributions (instead of scalar values like the 'mean' FA obtained from these local distributions).]]
|-
|}

=== Non-parametric regression for distribution-valued data ===

The proposed method
presents a completely nonparametric regression framework for spatial-temporal evolution of
statistical distributions. This avoids any parametric assumptions regarding the nature of the
underlying noise model in the DTI data and retains and utilizes the variability information to
estimate the regression trend. The estimated trajectory is completely characterized by statistical
distributions in space and time enabling a very powerful statistical inference framework. Both
population and individual trajectories can be estimated. The concept of ’distance’ between distributions and an ’average’ of distributions is employed. The dissimilarity metric employed is the Mallow's distance between distributions. It is the L2 norm of the generic Wasserstein metric and captures translation, changing width and shape of the participating distributions.

{| border="0" style="background:transparent;"
|[[Image:dist2_3d_bary_300_revrev.png|thumb|400px|center|Temporal evolution of the synthetic distributions between two timepoints as estimated by the proposed method.]]
|[[Image:bary_stat_300.png|thumb|400px|center|Since the regression estimates the complete probability distribution continuously along time, we now have access to the complete variability information at the interpolated time points. This can enable statistical inference using statistical summary measures as well as distribution differences at time points where the original image data is not available.]]
|-
|}

{| border="0" style="background:transparent;"
|[[Image:conte_krabbe_median_3d_300_2.png|thumb|400px|center|The medians of the estimated probability distributions are shown here for the healthy population (in red/pink) versus a single infant with Krabbe's disease (blue/black). Note that the method can applied to estimate normative population trajectories as well as subject-specific evolutions. Each point on these statistical manifolds are characterized by the complete distribution information (instead of just a scalar value).]]
|[[Image:conte_vs_krabbe_quartiles.png|thumb|600px|center|Highlighting the quantiles obtained at one of the arc length locations in the left plot. The Krabbe's subject seems to lag behind the normative population and the difference can be quantified in terms of distribution-based inferences.]]
|-
|}

=== Parametric linear regression for distribution-valued data ===

We build on the motivation as highlighted in the previous distribution-based regression. However,
the previous method is completely nonparametric leading to an intensive and complex statistical
inference framework based on comparison of statistical manifolds. Therefore, we simplify the
analysis for situations where a linear time-course trajectory can be assumed. We apply the classic
linear regression technique, extended to work with distribution-valued data. The advantage lies in
the compact description of the spatial-temporal trajectory in terms of the estimated regression
coefficients which can now be conveniently used for statistical inference. A synthetic validation
experiment is conducted to show the improved reliability and robustness of this method when
compared with methods working with statistical summaries of the underlying data.

{| border="0" style="background:transparent;"
|[[Image:Timehist_rev1.png|thumb|400px|center|The independent variable is the age distribution of the healthy infants clustered around 2 months, 1 year and 2 year.]]
|[[Image:Baryhist3d1.png|thumb|400px|center|The independent variable is the barycentric FA distribution estimated at each arc length location along the tract, using the original FA distributions available from all subjects within the corresponding age group. The regression estimates the continuous spatiotemporal trends in terms of the model parameters using the respective pairs of age and FA distributions as the 'observed' data.]]
|-
|}

{| border="0" style="background:transparent;"
|[[Image:PopuCN_dist_3d1.png|thumb|400px|center|Continuous spatiotemporal normative growth trajectory: Estimated mean FA (gray), Age range where control
data is originally available (green, orange, purple).]]
|[[Image:PopuvsKrabbe_dist_3d_2_edit1.png|thumb|400px|center|Krabbe's subject's scans (solid lines): (14 days, 6 mo., 1 yr.) with respect to the estimated healthy trajectory.]]
|[[Image:PopuvsKrabbe_dist_3d31.png|thumb|400px|center|Krabbe's subject's scans:mean FA (solid lines).Control population:estimated mean FA (dashed lines) +/- 3*std.dev. bounds (calculated for Gaussian age distributions centered at matched timepoints with std.dev.=5 days).]]
|-
|}

= References =

* A. Sharma, P.T. Fletcher, J.H. Gilmore, M.L. Escolar, A. Gupta, M. Styner, G. Gerig. “ Parametric Regression Scheme For Distributions: Analysis of DTI Fiber Tract Diffusion Changes In Early Brain Development,” In Proceedings of the 2014 IEEE 11th International Symposium on Biomedical Imaging (ISBI), pp. 559-562. 2014.
* A. Sharma, P.T. Fletcher, J.H. Gilmore, M.L. Escolar, A. Gupta, M. Styner, G. Gerig. “Spatiotemporal Modeling of Discrete-Time Distribution-Valued Data Applied to DTI Tract Evolution in Infant Neurodevelopment,” In Proceedings of the 2013 IEEE 10th International Symposium on Biomedical Imaging (ISBI), pp. 684--687. 2013.
* A. Sharma, S. Durrleman, J.H. Gilmore, G. Gerig. “Longitudinal Growth Modeling of Discrete-Time Functions with Application to DTI Tract Evolution in Early Neurodevelopment,” In Proceedings of IEEE ISBI 2012, pp. 1397--1400. 2012.
* Corouge, I., Fletcher, P.T., Joshi, S., Gouttard, S., Gerig, G., 2006. Fiber tract-oriented statistics for quantitative diffusion tensor MRI analysis. Med Image Anal, pp. 786-798.
* Goodlett, C.B., Fletcher, P.T., Gilmore, J.H., Gerig, G., 2009. Group analysis of DTI fiber tract statistics with application to neurodevelopment. Neuroimage, pp. S133-142.

File:Timehist rev.png

2014-10-15T22:58:42Z

Anuja:

Projects:TractLongitudinalDTI

2014-10-15T22:58:23Z

Anuja: /* Parametric linear regression for distribution-valued data */

Back to [[Algorithm:Utah2|Utah 2 Algorithms]]
__NOTOC__

== Tract-based longitudinal modeling of DTI data ==

This project develops a methodology to explore subject-specific, DTI data obtained from brain's white matter tracts, available at multiple but often sparsely present timepoints. The challenge is to develop a continuous spatio-temporal growth model, given discrete 4D DWI images. This would enable comparison of growth trajectories across subjects and along tracts which are biologically of interest in developmental and pathological changes.

== Background ==

We use the arc length parametrization scheme initially proposed by Corouge et al. It represents white matter fiber tracts obtained via streamline tractography in the brain's atlas tensor image as a function of arc length. (Atlas construction uses unbiased atlas building schemes followed by back transformations to subjects' DTI images to obtain identical fiber tract geometry across subjects, populated with subject specific diffusion data). We use the mean diffusion scalar invariants derived from these individual fiber bundle cross sections, as our input longitudinal diffusion profiles.

{| border="0" style="background:transparent;"
|[[Image:Tract-stats.png|thumb|700px|White matter diffusion properties along fiber tract: Left: A fiber tract with an origin plane defined for arc length parametrization, Right: FA mean and standard deviation as function of arc-length. Dots mark location of coordinate origin.]]
|-
|}

== Subject-specific spatiotemporal continuous growth model ==

We propose the use of Verhulst-Pearl logistic equation to capture temporal changes, while using a non-parametric kernel along arc length to account for biologically motivated functional along-tract relationship in the diffusion data.

{| border="0" style="background:transparent;"
|[[Image:logistic_eq.png|thumb|600px|Logistic equation with P as the population value at a given time, r being the growth rate of the population and K being the carrying capacity (also the limiting asymptote value).]]
|[[Image:logistic_simulation_growthrate_r.png|thumb|400px|Effect of growth rate r on the logistic curve.]]
|[[Image:logistic_simulation_asymptote_K.png|thumb|400px|Effect of carrying capacity K on the logistic curve.]]
|-
|}

From the definition of the logistic function, the parameter r represents the growth rate of the diffusion invariant and the parameter K represents the asymptote value. The overall function shape intuitively follows the growth pattern we expect to see during brain maturation. The diffusion invariants start with an initial diffusion profile along tract, and have a non linear temporal growth trajectory showing maximum changes in early childhood and then slowing down or almost saturating at a certain age. (For instance, observed diffusion changes are much more in neonates than in an adult brain). The temporal trajectories may also differ along the tract's length giving localized changes. Since our method gives us continuous along-tract, growth trajectories all along time, we can compare subjects with respect to differences in diffusion profiles at birth, growth rates at any given age as well as the asymptote saturation values. This gives us important information to understand delayed or abnormal brain maturation by comparing a normative growth surface with an individual's or by comparing the model's parameters across subjects.

== Results ==

Below are some results using synthetic data. For more validation results and extension of the framework to jointly estimate individual subject trajectories together with a normative trajectory, refer to [http://ieeexplore.ieee.org/xpl/login.jsp?tp=&arnumber=6235829&url=http%3A%2F%2Fieeexplore.ieee.org%2Fxpls%2Fabs_all.jsp%3Farnumber%3D6235829 Sharma et al. (ISBI '12)].

{| border="0" style="background:transparent;"
|[[Image:namic_tract_longitudinal_main.png|thumb|400px|Longitudinal modeling of synthetic data. The data is generated from a template neonate FA mean curve and uses known logistic parameters to generate the three timepoint curves (red). Our method faithfully recovers the original curves (blue) while simultaneously creating a continuous longitudinal growth profile (green) without having any prior knowledge of the logistic parameters used for data generation. It also estimates an unbiased time=0 template curve (black) along with parameters r and K to avoid biasing the estimation with any assumed initialization at time=0.]]
|[[Image:namic_tract_longitudinal_3d.png|thumb|400px|Results on real data (mean FA curves for a subject at three timepoints:neonate, 1 year and 2 year).]]
|-
|}
{| border="0" style="background:transparent;"
|[[Image:namic_tract_longitudinal_p1p2only.png|thumb|400px|For the above real data (right), these are the final along-tract initial values and the final estimates for parameters-growth rate r(mentioned as p1 here) and asymptote K(mentioned as p2 here).]]
|[[Image:namic_tract_longitudinal_alphaonly.png|thumb|400px|The complete continuous growth grid as estimated for the real data above.]]

|-
|}

The below images show jointly estimated personalized trajectories for 15 control subjects along with the average growth trajectory. The estimated model parameters for 15 subjects as well as the average trajectory are also shown. The normative trajectory is colored by the local standard deviation. It points to the fact that despite individual variability in the maturation process, there is a strong agreement in the asymptote FA values seen around a gestational age of 2.5 years indicating a relative stabilization of the white matter changes across subjects. The framework thus quantifies patient-specific changes in serial diffusion data given discrete-time diffusion curves.

{| border="0" style="background:transparent;"
|[[Image:CONTE_Final_3d.png|thumb|400px|The jointly estimated individual growth trajectories for 4 of the 15 control subjects.]]
|[[Image:CONTE_Norm_Params.png|thumb|400px|Estimated model parameters for 15 subjects together with the normative growth trajectory colored by the local variability and the estimated normative model parameters.]]

|-
|}

== Experiments with Huntington's data ==

During the NAMIC Winter project week, we worked on registering subjects with Huntington's disease in the same coordinate space as control subjects. We then applied the above framework [http://ieeexplore.ieee.org/xpl/login.jsp?tp=&arnumber=6235829&url=http%3A%2F%2Fieeexplore.ieee.org%2Fxpls%2Fabs_all.jsp%3Farnumber%3D6235829 Sharma et al. ISBI'12] to quantify the differences in normal expected aging versus the accelerated white matter changes expected in HD. Details of the data pre-processing steps and image registration are available on the [http://www.na-mic.org/Wiki/index.php/2012_Winter_Project_Week:_DTI_Change_Modeling Winter AHM page]. Some results are summarized here for FA value along the genu tract.

Two subjects- one being a HD patient (10027) with a high burden factor (higher factor value (of factor 12) implying that the subjects is closer to the onset) and the other being a control subject (10004) are chosen for concept demonstration. Both have a similar baseline scan age (42 and 43 years respectively) as well as a similar time separation between follow up scans. This allows an intuitive comparison of the estimated growth trajectories.

{| border="0" style="background:transparent;"
|[[Image:ORIG_hd_cn.png|thumb|300px|center|FA curves corresponding to the genu tract (middle clipped portions) for one HD case (10027) and one control case (10004). Visual inspection clearly shows the huge decrease relative to the Control in the HD subject. On the other hand, the control shows normal expected decrease due to aging. Red-timepoint 1, Green-timepoint 2, Blue-timepoint 3. The left is HD case and the right plot is Control.]]
|[[Image:HD_vs_CN_3d.png|thumb|300px|center|The red trajectory corresponds the the HD subjects while the blue is a control subject with healthy aging. HD patient clearly shows a much sharper FA decrease along time than the Control.]]
|[[Image:HD_vs_CN_params.png|thumb|300px|center|The estimated model parameters (Red-HD case, Blue-Control case) for the two subjects. The top plot is the rate of change per unit time, per unit FA value. The middle plot is the function asymptote and the last plot is the common baseline curve estimation for providing a common reference frame for model estimation and comparison. The asymptote clearly shows a much lower FA range for HD when compared to the Control which already shows promising early prediction capabilities for HD cases where white matter decay is much faster than healthy patients. Also, a higher per unit rate of change quantifies the sharper FA decrease compared to control.]]
|-
|}

== Experiments with Krabbe's data ==

We have also applied the framework to study white matter changes in infants with Krabbe's disease. Here we show some preliminary results for the FA tract profile along the genu tract for a single subject with Krabbe's disease. In theory, FA non-linearly increases with time along genu owing to a healthy brain maturation process. However, in Krabbe's case, the FA does not follow a monotonic change pattern.

{| border="0" style="background:transparent;"
|[[Image:Genu_fa_3d_conte_vs_krabbe.png|thumb|400px|center|FA curves corresponding to the genu tract for the Krabbe's patient are shown (Neonate:black, 6months:cyan, 1year:magenta). The other FA profiles are for the 15 healthy control subjects as discussed previously (neonate:red, 1year:green, 2year:blue). This also highlights the fact that the framework does not assume uniformity or correspondence in the scan timings across subjects.]]
|[[Image:Genu_fa_3dnorm_conte_vs_krabbe.png|thumb|400px|center|The black trajectory is the normative FA growth estimated using 15 controls as discussed previously. The three FA curves for the Krabbe case are also plotted and clearly show how the patient is lagging behind on expected brain maturation.]]
|-
|}

== Future work ==

As we see in the Krabbe experiment, the model assumes a monotonic change along time. Neighboring locations along the tract can have different, yet correlated time changes. For eg. we can see localized increases or decreases along the tract length with respect to the time evolution. However, owing to the definition of the logistic function, any non-monotonic behavior along time at any given tract location is not modeled accurately. Therefore, in cases like the Krabbe's subject, we need a more relaxed and flexible semi-parametric or non-parametric model to remove the constraints coming in from the parametric nature of the current model. This is currently work under progress.
To utilize the quantification of growth trajectories enabled by this method, we also require a systematic hypothesis testing scheme to draw conclusive results of individuals differing from normative patterns or populations differing in their evolution behaviors. This too is currently being worked upon.

= References =

* Sharma, A., Durrleman, S. , Gilmore, J.H. , Gerig, G. Longitudinal growth modeling of discrete-time functions with application to DTI tract evolution in early neurodevelopment. Proc. of 9th IEEE International Symposium on Biomedical Imaging (ISBI May'12), p.1397-1400.
* Corouge, I., Fletcher, P.T., Joshi, S., Gouttard, S., Gerig, G., 2006. Fiber tract-oriented statistics for quantitative diffusion tensor MRI analysis. Med Image Anal, pp. 786-798.
* Goodlett, C.B., Fletcher, P.T., Gilmore, J.H., Gerig, G., 2009. Group analysis of DTI fiber tract statistics with application to neurodevelopment. Neuroimage, pp. S133-142.

==Key Investigators==
* Utah: Anuja Sharma, Stanley Durrleman, Guido Gerig
* INRIA/ICM, Pitie Salpetriere Hospital: Stanley Durrleman
[http://www.na-mic.org/Wiki/index.php/Algorithm:Utah2 Back to Utah 2 Algorithm Core]

== Regression of probability distributions obtained along white matter tracts to create spatio-temporal growth models ==

Temporal modeling frameworks often operate on scalar variables by
summarizing data at initial stages as statistical summaries of the underlying
distributions. For instance, DTI analysis often employs
summary statistics, like mean, for regions of interest and properties
along fiber tracts for population studies and hypothesis testing.
This reduction via discarding of variability information may introduce
significant errors which propagate through the procedures. Moreover, the data variability information is lost at
all the intermediate interpolated times.

{| border="0" style="background:transparent;"
|[[Image:DistributionRegression_Synthetic.png|thumb|600px|center|Left:Two synthetically generated distributions along time
(blue:t0, orange:t1) with translated locations and opposite skews.
Right: Simple linear interpolation using classic single-valued summary
statistics (means: red, medians: blue) and ignoring data variability
(shown as error bars) at t0 and t1. The decision to choose mean versus median can heavily influence the final regression estimates. Additionally, once the inherent variability is discarded for an early decision regarding a summary statistics, it cannot be recovered for the interpolated time points.]]
|-
|}

We propose a novel framework which uses distribution-valued variables
to retain and utilize the local variability information. We propose two different regression approaches to achieve our goals. The first is a non-parametric moving average method where minimal assumptions are involved in terms of the growth trajectory evolution (Sharma ISBI 2013). The second is a parametric regression scheme where the expected temporal trend is linear (Sharma ISBI 2014). It provides a compact representation in terms of model parameters for further statistical inference and has a closed form solution.

Our driving application is the modeling of age-related changes along DTI
white matter tracts. Results are shown for the spatiotemporal population
trajectory of genu tract estimated from a group of healthy infants and compared with an infant with Krabbe's disease.

{| border="0" style="background:transparent;"
|[[Image:DWIalongTime.png|thumb|400px|center|Model age related DTI changes in early neurodevelopment.]]
|[[Image:genu_distributions_along_time.png|thumb|600px|center|Distributions of DTI-derived scalar diffusion parameters like FA are obtained along the length of the genu tract. The distributions correspond to the cross-sections of the 3D tract geometry and are a function of the arc-length along the tract's total length as we move from one end of the tract to the other. This provides a functional curve which is parametrized by a feature of the local tract geometry (arc-length along the tract in this case) and is attributed by probability distributions (instead of scalar values like the 'mean' FA obtained from these local distributions).]]
|-
|}

=== Non-parametric regression for distribution-valued data ===

The proposed method
presents a completely nonparametric regression framework for spatial-temporal evolution of
statistical distributions. This avoids any parametric assumptions regarding the nature of the
underlying noise model in the DTI data and retains and utilizes the variability information to
estimate the regression trend. The estimated trajectory is completely characterized by statistical
distributions in space and time enabling a very powerful statistical inference framework. Both
population and individual trajectories can be estimated. The concept of ’distance’ between distributions and an ’average’ of distributions is employed. The dissimilarity metric employed is the Mallow's distance between distributions. It is the L2 norm of the generic Wasserstein metric and captures translation, changing width and shape of the participating distributions.

{| border="0" style="background:transparent;"
|[[Image:dist2_3d_bary_300_revrev.png|thumb|400px|center|Temporal evolution of the synthetic distributions between two timepoints as estimated by the proposed method.]]
|[[Image:bary_stat_300.png|thumb|400px|center|Since the regression estimates the complete probability distribution continuously along time, we now have access to the complete variability information at the interpolated time points. This can enable statistical inference using statistical summary measures as well as distribution differences at time points where the original image data is not available.]]
|-
|}

{| border="0" style="background:transparent;"
|[[Image:conte_krabbe_median_3d_300_2.png|thumb|400px|center|The medians of the estimated probability distributions are shown here for the healthy population (in red/pink) versus a single infant with Krabbe's disease (blue/black). Note that the method can applied to estimate normative population trajectories as well as subject-specific evolutions. Each point on these statistical manifolds are characterized by the complete distribution information (instead of just a scalar value).]]
|[[Image:conte_vs_krabbe_quartiles.png|thumb|600px|center|Highlighting the quantiles obtained at one of the arc length locations in the left plot. The Krabbe's subject seems to lag behind the normative population and the difference can be quantified in terms of distribution-based inferences.]]
|-
|}

=== Parametric linear regression for distribution-valued data ===

We build on the motivation as highlighted in the previous distribution-based regression. However,
the previous method is completely nonparametric leading to an intensive and complex statistical
inference framework based on comparison of statistical manifolds. Therefore, we simplify the
analysis for situations where a linear time-course trajectory can be assumed. We apply the classic
linear regression technique, extended to work with distribution-valued data. The advantage lies in
the compact description of the spatial-temporal trajectory in terms of the estimated regression
coefficients which can now be conveniently used for statistical inference. A synthetic validation
experiment is conducted to show the improved reliability and robustness of this method when
compared with methods working with statistical summaries of the underlying data.

{| border="0" style="background:transparent;"
|[[Image:timehist_rev.png|thumb|400px|center|The independent variable is the age distribution of the healthy infants clustered around 2 months, 1 year and 2 year.]]
|[[Image:baryhist3d.png|thumb|400px|center|The independent variable is the barycentric FA distribution estimated at each arc length location along the tract, using the original FA distributions available from all subjects within the corresponding age group. The regression estimates the continuous spatiotemporal trends in terms of the model parameters using the respective pairs of age and FA distributions as the 'observed' data.]]
|-
|}

{| border="0" style="background:transparent;"
|[[Image:PopuCN_dist_3d.png|thumb|400px|center|Continuous spatiotemporal normative growth trajectory: Estimated mean FA (gray), Age range where control
data is originally available (green, orange, purple).]]
|[[Image:PopuvsKrabbe_dist_3d_2_edit.png|thumb|400px|center|Krabbe's subject's scans (solid lines): (14 days, 6 mo., 1 yr.) with respect to the estimated healthy trajectory.]]
|[[Image:PopuvsKrabbe_dist_3d3.png|thumb|400px|center|Krabbe's subject's scans:mean FA (solid lines).Control population:estimated mean FA (dashed lines) � 3� bounds (calculated for Gaussian age distributions centered at matched timepoints with std.dev.�=5 days).]]
|-
|}

= References =

* A. Sharma, P.T. Fletcher, J.H. Gilmore, M.L. Escolar, A. Gupta, M. Styner, G. Gerig. “ Parametric Regression Scheme For Distributions: Analysis of DTI Fiber Tract Diffusion Changes In Early Brain Development,” In Proceedings of the 2014 IEEE 11th International Symposium on Biomedical Imaging (ISBI), pp. 559-562. 2014.
* A. Sharma, P.T. Fletcher, J.H. Gilmore, M.L. Escolar, A. Gupta, M. Styner, G. Gerig. “Spatiotemporal Modeling of Discrete-Time Distribution-Valued Data Applied to DTI Tract Evolution in Infant Neurodevelopment,” In Proceedings of the 2013 IEEE 10th International Symposium on Biomedical Imaging (ISBI), pp. 684--687. 2013.
* A. Sharma, S. Durrleman, J.H. Gilmore, G. Gerig. “Longitudinal Growth Modeling of Discrete-Time Functions with Application to DTI Tract Evolution in Early Neurodevelopment,” In Proceedings of IEEE ISBI 2012, pp. 1397--1400. 2012.
* Corouge, I., Fletcher, P.T., Joshi, S., Gouttard, S., Gerig, G., 2006. Fiber tract-oriented statistics for quantitative diffusion tensor MRI analysis. Med Image Anal, pp. 786-798.
* Goodlett, C.B., Fletcher, P.T., Gilmore, J.H., Gerig, G., 2009. Group analysis of DTI fiber tract statistics with application to neurodevelopment. Neuroimage, pp. S133-142.

Projects:TractLongitudinalDTI

2014-10-15T22:46:34Z

Anuja: /* Parametric linear regression for distribution-valued data */

Back to [[Algorithm:Utah2|Utah 2 Algorithms]]
__NOTOC__

== Tract-based longitudinal modeling of DTI data ==

This project develops a methodology to explore subject-specific, DTI data obtained from brain's white matter tracts, available at multiple but often sparsely present timepoints. The challenge is to develop a continuous spatio-temporal growth model, given discrete 4D DWI images. This would enable comparison of growth trajectories across subjects and along tracts which are biologically of interest in developmental and pathological changes.

== Background ==

We use the arc length parametrization scheme initially proposed by Corouge et al. It represents white matter fiber tracts obtained via streamline tractography in the brain's atlas tensor image as a function of arc length. (Atlas construction uses unbiased atlas building schemes followed by back transformations to subjects' DTI images to obtain identical fiber tract geometry across subjects, populated with subject specific diffusion data). We use the mean diffusion scalar invariants derived from these individual fiber bundle cross sections, as our input longitudinal diffusion profiles.

{| border="0" style="background:transparent;"
|[[Image:Tract-stats.png|thumb|700px|White matter diffusion properties along fiber tract: Left: A fiber tract with an origin plane defined for arc length parametrization, Right: FA mean and standard deviation as function of arc-length. Dots mark location of coordinate origin.]]
|-
|}

== Subject-specific spatiotemporal continuous growth model ==

We propose the use of Verhulst-Pearl logistic equation to capture temporal changes, while using a non-parametric kernel along arc length to account for biologically motivated functional along-tract relationship in the diffusion data.

{| border="0" style="background:transparent;"
|[[Image:logistic_eq.png|thumb|600px|Logistic equation with P as the population value at a given time, r being the growth rate of the population and K being the carrying capacity (also the limiting asymptote value).]]
|[[Image:logistic_simulation_growthrate_r.png|thumb|400px|Effect of growth rate r on the logistic curve.]]
|[[Image:logistic_simulation_asymptote_K.png|thumb|400px|Effect of carrying capacity K on the logistic curve.]]
|-
|}

From the definition of the logistic function, the parameter r represents the growth rate of the diffusion invariant and the parameter K represents the asymptote value. The overall function shape intuitively follows the growth pattern we expect to see during brain maturation. The diffusion invariants start with an initial diffusion profile along tract, and have a non linear temporal growth trajectory showing maximum changes in early childhood and then slowing down or almost saturating at a certain age. (For instance, observed diffusion changes are much more in neonates than in an adult brain). The temporal trajectories may also differ along the tract's length giving localized changes. Since our method gives us continuous along-tract, growth trajectories all along time, we can compare subjects with respect to differences in diffusion profiles at birth, growth rates at any given age as well as the asymptote saturation values. This gives us important information to understand delayed or abnormal brain maturation by comparing a normative growth surface with an individual's or by comparing the model's parameters across subjects.

== Results ==

Below are some results using synthetic data. For more validation results and extension of the framework to jointly estimate individual subject trajectories together with a normative trajectory, refer to [http://ieeexplore.ieee.org/xpl/login.jsp?tp=&arnumber=6235829&url=http%3A%2F%2Fieeexplore.ieee.org%2Fxpls%2Fabs_all.jsp%3Farnumber%3D6235829 Sharma et al. (ISBI '12)].

{| border="0" style="background:transparent;"
|[[Image:namic_tract_longitudinal_main.png|thumb|400px|Longitudinal modeling of synthetic data. The data is generated from a template neonate FA mean curve and uses known logistic parameters to generate the three timepoint curves (red). Our method faithfully recovers the original curves (blue) while simultaneously creating a continuous longitudinal growth profile (green) without having any prior knowledge of the logistic parameters used for data generation. It also estimates an unbiased time=0 template curve (black) along with parameters r and K to avoid biasing the estimation with any assumed initialization at time=0.]]
|[[Image:namic_tract_longitudinal_3d.png|thumb|400px|Results on real data (mean FA curves for a subject at three timepoints:neonate, 1 year and 2 year).]]
|-
|}
{| border="0" style="background:transparent;"
|[[Image:namic_tract_longitudinal_p1p2only.png|thumb|400px|For the above real data (right), these are the final along-tract initial values and the final estimates for parameters-growth rate r(mentioned as p1 here) and asymptote K(mentioned as p2 here).]]
|[[Image:namic_tract_longitudinal_alphaonly.png|thumb|400px|The complete continuous growth grid as estimated for the real data above.]]

|-
|}

The below images show jointly estimated personalized trajectories for 15 control subjects along with the average growth trajectory. The estimated model parameters for 15 subjects as well as the average trajectory are also shown. The normative trajectory is colored by the local standard deviation. It points to the fact that despite individual variability in the maturation process, there is a strong agreement in the asymptote FA values seen around a gestational age of 2.5 years indicating a relative stabilization of the white matter changes across subjects. The framework thus quantifies patient-specific changes in serial diffusion data given discrete-time diffusion curves.

{| border="0" style="background:transparent;"
|[[Image:CONTE_Final_3d.png|thumb|400px|The jointly estimated individual growth trajectories for 4 of the 15 control subjects.]]
|[[Image:CONTE_Norm_Params.png|thumb|400px|Estimated model parameters for 15 subjects together with the normative growth trajectory colored by the local variability and the estimated normative model parameters.]]

|-
|}

== Experiments with Huntington's data ==

During the NAMIC Winter project week, we worked on registering subjects with Huntington's disease in the same coordinate space as control subjects. We then applied the above framework [http://ieeexplore.ieee.org/xpl/login.jsp?tp=&arnumber=6235829&url=http%3A%2F%2Fieeexplore.ieee.org%2Fxpls%2Fabs_all.jsp%3Farnumber%3D6235829 Sharma et al. ISBI'12] to quantify the differences in normal expected aging versus the accelerated white matter changes expected in HD. Details of the data pre-processing steps and image registration are available on the [http://www.na-mic.org/Wiki/index.php/2012_Winter_Project_Week:_DTI_Change_Modeling Winter AHM page]. Some results are summarized here for FA value along the genu tract.

Two subjects- one being a HD patient (10027) with a high burden factor (higher factor value (of factor 12) implying that the subjects is closer to the onset) and the other being a control subject (10004) are chosen for concept demonstration. Both have a similar baseline scan age (42 and 43 years respectively) as well as a similar time separation between follow up scans. This allows an intuitive comparison of the estimated growth trajectories.

{| border="0" style="background:transparent;"
|[[Image:ORIG_hd_cn.png|thumb|300px|center|FA curves corresponding to the genu tract (middle clipped portions) for one HD case (10027) and one control case (10004). Visual inspection clearly shows the huge decrease relative to the Control in the HD subject. On the other hand, the control shows normal expected decrease due to aging. Red-timepoint 1, Green-timepoint 2, Blue-timepoint 3. The left is HD case and the right plot is Control.]]
|[[Image:HD_vs_CN_3d.png|thumb|300px|center|The red trajectory corresponds the the HD subjects while the blue is a control subject with healthy aging. HD patient clearly shows a much sharper FA decrease along time than the Control.]]
|[[Image:HD_vs_CN_params.png|thumb|300px|center|The estimated model parameters (Red-HD case, Blue-Control case) for the two subjects. The top plot is the rate of change per unit time, per unit FA value. The middle plot is the function asymptote and the last plot is the common baseline curve estimation for providing a common reference frame for model estimation and comparison. The asymptote clearly shows a much lower FA range for HD when compared to the Control which already shows promising early prediction capabilities for HD cases where white matter decay is much faster than healthy patients. Also, a higher per unit rate of change quantifies the sharper FA decrease compared to control.]]
|-
|}

== Experiments with Krabbe's data ==

We have also applied the framework to study white matter changes in infants with Krabbe's disease. Here we show some preliminary results for the FA tract profile along the genu tract for a single subject with Krabbe's disease. In theory, FA non-linearly increases with time along genu owing to a healthy brain maturation process. However, in Krabbe's case, the FA does not follow a monotonic change pattern.

{| border="0" style="background:transparent;"
|[[Image:Genu_fa_3d_conte_vs_krabbe.png|thumb|400px|center|FA curves corresponding to the genu tract for the Krabbe's patient are shown (Neonate:black, 6months:cyan, 1year:magenta). The other FA profiles are for the 15 healthy control subjects as discussed previously (neonate:red, 1year:green, 2year:blue). This also highlights the fact that the framework does not assume uniformity or correspondence in the scan timings across subjects.]]
|[[Image:Genu_fa_3dnorm_conte_vs_krabbe.png|thumb|400px|center|The black trajectory is the normative FA growth estimated using 15 controls as discussed previously. The three FA curves for the Krabbe case are also plotted and clearly show how the patient is lagging behind on expected brain maturation.]]
|-
|}

== Future work ==

As we see in the Krabbe experiment, the model assumes a monotonic change along time. Neighboring locations along the tract can have different, yet correlated time changes. For eg. we can see localized increases or decreases along the tract length with respect to the time evolution. However, owing to the definition of the logistic function, any non-monotonic behavior along time at any given tract location is not modeled accurately. Therefore, in cases like the Krabbe's subject, we need a more relaxed and flexible semi-parametric or non-parametric model to remove the constraints coming in from the parametric nature of the current model. This is currently work under progress.
To utilize the quantification of growth trajectories enabled by this method, we also require a systematic hypothesis testing scheme to draw conclusive results of individuals differing from normative patterns or populations differing in their evolution behaviors. This too is currently being worked upon.

= References =

* Sharma, A., Durrleman, S. , Gilmore, J.H. , Gerig, G. Longitudinal growth modeling of discrete-time functions with application to DTI tract evolution in early neurodevelopment. Proc. of 9th IEEE International Symposium on Biomedical Imaging (ISBI May'12), p.1397-1400.
* Corouge, I., Fletcher, P.T., Joshi, S., Gouttard, S., Gerig, G., 2006. Fiber tract-oriented statistics for quantitative diffusion tensor MRI analysis. Med Image Anal, pp. 786-798.
* Goodlett, C.B., Fletcher, P.T., Gilmore, J.H., Gerig, G., 2009. Group analysis of DTI fiber tract statistics with application to neurodevelopment. Neuroimage, pp. S133-142.

==Key Investigators==
* Utah: Anuja Sharma, Stanley Durrleman, Guido Gerig
* INRIA/ICM, Pitie Salpetriere Hospital: Stanley Durrleman
[http://www.na-mic.org/Wiki/index.php/Algorithm:Utah2 Back to Utah 2 Algorithm Core]

== Regression of probability distributions obtained along white matter tracts to create spatio-temporal growth models ==

Temporal modeling frameworks often operate on scalar variables by
summarizing data at initial stages as statistical summaries of the underlying
distributions. For instance, DTI analysis often employs
summary statistics, like mean, for regions of interest and properties
along fiber tracts for population studies and hypothesis testing.
This reduction via discarding of variability information may introduce
significant errors which propagate through the procedures. Moreover, the data variability information is lost at
all the intermediate interpolated times.

{| border="0" style="background:transparent;"
|[[Image:DistributionRegression_Synthetic.png|thumb|600px|center|Left:Two synthetically generated distributions along time
(blue:t0, orange:t1) with translated locations and opposite skews.
Right: Simple linear interpolation using classic single-valued summary
statistics (means: red, medians: blue) and ignoring data variability
(shown as error bars) at t0 and t1. The decision to choose mean versus median can heavily influence the final regression estimates. Additionally, once the inherent variability is discarded for an early decision regarding a summary statistics, it cannot be recovered for the interpolated time points.]]
|-
|}

We propose a novel framework which uses distribution-valued variables
to retain and utilize the local variability information. We propose two different regression approaches to achieve our goals. The first is a non-parametric moving average method where minimal assumptions are involved in terms of the growth trajectory evolution (Sharma ISBI 2013). The second is a parametric regression scheme where the expected temporal trend is linear (Sharma ISBI 2014). It provides a compact representation in terms of model parameters for further statistical inference and has a closed form solution.

Our driving application is the modeling of age-related changes along DTI
white matter tracts. Results are shown for the spatiotemporal population
trajectory of genu tract estimated from a group of healthy infants and compared with an infant with Krabbe's disease.

{| border="0" style="background:transparent;"
|[[Image:DWIalongTime.png|thumb|400px|center|Model age related DTI changes in early neurodevelopment.]]
|[[Image:genu_distributions_along_time.png|thumb|600px|center|Distributions of DTI-derived scalar diffusion parameters like FA are obtained along the length of the genu tract. The distributions correspond to the cross-sections of the 3D tract geometry and are a function of the arc-length along the tract's total length as we move from one end of the tract to the other. This provides a functional curve which is parametrized by a feature of the local tract geometry (arc-length along the tract in this case) and is attributed by probability distributions (instead of scalar values like the 'mean' FA obtained from these local distributions).]]
|-
|}

=== Non-parametric regression for distribution-valued data ===

The proposed method
presents a completely nonparametric regression framework for spatial-temporal evolution of
statistical distributions. This avoids any parametric assumptions regarding the nature of the
underlying noise model in the DTI data and retains and utilizes the variability information to
estimate the regression trend. The estimated trajectory is completely characterized by statistical
distributions in space and time enabling a very powerful statistical inference framework. Both
population and individual trajectories can be estimated. The concept of ’distance’ between distributions and an ’average’ of distributions is employed. The dissimilarity metric employed is the Mallow's distance between distributions. It is the L2 norm of the generic Wasserstein metric and captures translation, changing width and shape of the participating distributions.

{| border="0" style="background:transparent;"
|[[Image:dist2_3d_bary_300_revrev.png|thumb|400px|center|Temporal evolution of the synthetic distributions between two timepoints as estimated by the proposed method.]]
|[[Image:bary_stat_300.png|thumb|400px|center|Since the regression estimates the complete probability distribution continuously along time, we now have access to the complete variability information at the interpolated time points. This can enable statistical inference using statistical summary measures as well as distribution differences at time points where the original image data is not available.]]
|-
|}

{| border="0" style="background:transparent;"
|[[Image:conte_krabbe_median_3d_300_2.png|thumb|400px|center|The medians of the estimated probability distributions are shown here for the healthy population (in red/pink) versus a single infant with Krabbe's disease (blue/black). Note that the method can applied to estimate normative population trajectories as well as subject-specific evolutions. Each point on these statistical manifolds are characterized by the complete distribution information (instead of just a scalar value).]]
|[[Image:conte_vs_krabbe_quartiles.png|thumb|600px|center|Highlighting the quantiles obtained at one of the arc length locations in the left plot. The Krabbe's subject seems to lag behind the normative population and the difference can be quantified in terms of distribution-based inferences.]]
|-
|}

=== Parametric linear regression for distribution-valued data ===

We build on the motivation as highlighted in the previous distribution-based regression. However,
the previous method is completely nonparametric leading to an intensive and complex statistical
inference framework based on comparison of statistical manifolds. Therefore, we simplify the
analysis for situations where a linear time-course trajectory can be assumed. We apply the classic
linear regression technique, extended to work with distribution-valued data. The advantage lies in
the compact description of the spatial-temporal trajectory in terms of the estimated regression
coefficients which can now be conveniently used for statistical inference. A synthetic validation
experiment is conducted to show the improved reliability and robustness of this method when
compared with methods working with statistical summaries of the underlying data.

= References =

* A. Sharma, P.T. Fletcher, J.H. Gilmore, M.L. Escolar, A. Gupta, M. Styner, G. Gerig. “ Parametric Regression Scheme For Distributions: Analysis of DTI Fiber Tract Diffusion Changes In Early Brain Development,” In Proceedings of the 2014 IEEE 11th International Symposium on Biomedical Imaging (ISBI), pp. 559-562. 2014.
* A. Sharma, P.T. Fletcher, J.H. Gilmore, M.L. Escolar, A. Gupta, M. Styner, G. Gerig. “Spatiotemporal Modeling of Discrete-Time Distribution-Valued Data Applied to DTI Tract Evolution in Infant Neurodevelopment,” In Proceedings of the 2013 IEEE 10th International Symposium on Biomedical Imaging (ISBI), pp. 684--687. 2013.
* A. Sharma, S. Durrleman, J.H. Gilmore, G. Gerig. “Longitudinal Growth Modeling of Discrete-Time Functions with Application to DTI Tract Evolution in Early Neurodevelopment,” In Proceedings of IEEE ISBI 2012, pp. 1397--1400. 2012.
* Corouge, I., Fletcher, P.T., Joshi, S., Gouttard, S., Gerig, G., 2006. Fiber tract-oriented statistics for quantitative diffusion tensor MRI analysis. Med Image Anal, pp. 786-798.
* Goodlett, C.B., Fletcher, P.T., Gilmore, J.H., Gerig, G., 2009. Group analysis of DTI fiber tract statistics with application to neurodevelopment. Neuroimage, pp. S133-142.

Projects:TractLongitudinalDTI

2014-10-15T22:45:10Z

Anuja: /* Non-parametric regression for distribution-valued data */

Back to [[Algorithm:Utah2|Utah 2 Algorithms]]
__NOTOC__

== Tract-based longitudinal modeling of DTI data ==

This project develops a methodology to explore subject-specific, DTI data obtained from brain's white matter tracts, available at multiple but often sparsely present timepoints. The challenge is to develop a continuous spatio-temporal growth model, given discrete 4D DWI images. This would enable comparison of growth trajectories across subjects and along tracts which are biologically of interest in developmental and pathological changes.

== Background ==

We use the arc length parametrization scheme initially proposed by Corouge et al. It represents white matter fiber tracts obtained via streamline tractography in the brain's atlas tensor image as a function of arc length. (Atlas construction uses unbiased atlas building schemes followed by back transformations to subjects' DTI images to obtain identical fiber tract geometry across subjects, populated with subject specific diffusion data). We use the mean diffusion scalar invariants derived from these individual fiber bundle cross sections, as our input longitudinal diffusion profiles.

{| border="0" style="background:transparent;"
|[[Image:Tract-stats.png|thumb|700px|White matter diffusion properties along fiber tract: Left: A fiber tract with an origin plane defined for arc length parametrization, Right: FA mean and standard deviation as function of arc-length. Dots mark location of coordinate origin.]]
|-
|}

== Subject-specific spatiotemporal continuous growth model ==

We propose the use of Verhulst-Pearl logistic equation to capture temporal changes, while using a non-parametric kernel along arc length to account for biologically motivated functional along-tract relationship in the diffusion data.

{| border="0" style="background:transparent;"
|[[Image:logistic_eq.png|thumb|600px|Logistic equation with P as the population value at a given time, r being the growth rate of the population and K being the carrying capacity (also the limiting asymptote value).]]
|[[Image:logistic_simulation_growthrate_r.png|thumb|400px|Effect of growth rate r on the logistic curve.]]
|[[Image:logistic_simulation_asymptote_K.png|thumb|400px|Effect of carrying capacity K on the logistic curve.]]
|-
|}

From the definition of the logistic function, the parameter r represents the growth rate of the diffusion invariant and the parameter K represents the asymptote value. The overall function shape intuitively follows the growth pattern we expect to see during brain maturation. The diffusion invariants start with an initial diffusion profile along tract, and have a non linear temporal growth trajectory showing maximum changes in early childhood and then slowing down or almost saturating at a certain age. (For instance, observed diffusion changes are much more in neonates than in an adult brain). The temporal trajectories may also differ along the tract's length giving localized changes. Since our method gives us continuous along-tract, growth trajectories all along time, we can compare subjects with respect to differences in diffusion profiles at birth, growth rates at any given age as well as the asymptote saturation values. This gives us important information to understand delayed or abnormal brain maturation by comparing a normative growth surface with an individual's or by comparing the model's parameters across subjects.

== Results ==

Below are some results using synthetic data. For more validation results and extension of the framework to jointly estimate individual subject trajectories together with a normative trajectory, refer to [http://ieeexplore.ieee.org/xpl/login.jsp?tp=&arnumber=6235829&url=http%3A%2F%2Fieeexplore.ieee.org%2Fxpls%2Fabs_all.jsp%3Farnumber%3D6235829 Sharma et al. (ISBI '12)].

{| border="0" style="background:transparent;"
|[[Image:namic_tract_longitudinal_main.png|thumb|400px|Longitudinal modeling of synthetic data. The data is generated from a template neonate FA mean curve and uses known logistic parameters to generate the three timepoint curves (red). Our method faithfully recovers the original curves (blue) while simultaneously creating a continuous longitudinal growth profile (green) without having any prior knowledge of the logistic parameters used for data generation. It also estimates an unbiased time=0 template curve (black) along with parameters r and K to avoid biasing the estimation with any assumed initialization at time=0.]]
|[[Image:namic_tract_longitudinal_3d.png|thumb|400px|Results on real data (mean FA curves for a subject at three timepoints:neonate, 1 year and 2 year).]]
|-
|}
{| border="0" style="background:transparent;"
|[[Image:namic_tract_longitudinal_p1p2only.png|thumb|400px|For the above real data (right), these are the final along-tract initial values and the final estimates for parameters-growth rate r(mentioned as p1 here) and asymptote K(mentioned as p2 here).]]
|[[Image:namic_tract_longitudinal_alphaonly.png|thumb|400px|The complete continuous growth grid as estimated for the real data above.]]

|-
|}

The below images show jointly estimated personalized trajectories for 15 control subjects along with the average growth trajectory. The estimated model parameters for 15 subjects as well as the average trajectory are also shown. The normative trajectory is colored by the local standard deviation. It points to the fact that despite individual variability in the maturation process, there is a strong agreement in the asymptote FA values seen around a gestational age of 2.5 years indicating a relative stabilization of the white matter changes across subjects. The framework thus quantifies patient-specific changes in serial diffusion data given discrete-time diffusion curves.

{| border="0" style="background:transparent;"
|[[Image:CONTE_Final_3d.png|thumb|400px|The jointly estimated individual growth trajectories for 4 of the 15 control subjects.]]
|[[Image:CONTE_Norm_Params.png|thumb|400px|Estimated model parameters for 15 subjects together with the normative growth trajectory colored by the local variability and the estimated normative model parameters.]]

|-
|}

== Experiments with Huntington's data ==

During the NAMIC Winter project week, we worked on registering subjects with Huntington's disease in the same coordinate space as control subjects. We then applied the above framework [http://ieeexplore.ieee.org/xpl/login.jsp?tp=&arnumber=6235829&url=http%3A%2F%2Fieeexplore.ieee.org%2Fxpls%2Fabs_all.jsp%3Farnumber%3D6235829 Sharma et al. ISBI'12] to quantify the differences in normal expected aging versus the accelerated white matter changes expected in HD. Details of the data pre-processing steps and image registration are available on the [http://www.na-mic.org/Wiki/index.php/2012_Winter_Project_Week:_DTI_Change_Modeling Winter AHM page]. Some results are summarized here for FA value along the genu tract.

Two subjects- one being a HD patient (10027) with a high burden factor (higher factor value (of factor 12) implying that the subjects is closer to the onset) and the other being a control subject (10004) are chosen for concept demonstration. Both have a similar baseline scan age (42 and 43 years respectively) as well as a similar time separation between follow up scans. This allows an intuitive comparison of the estimated growth trajectories.

{| border="0" style="background:transparent;"
|[[Image:ORIG_hd_cn.png|thumb|300px|center|FA curves corresponding to the genu tract (middle clipped portions) for one HD case (10027) and one control case (10004). Visual inspection clearly shows the huge decrease relative to the Control in the HD subject. On the other hand, the control shows normal expected decrease due to aging. Red-timepoint 1, Green-timepoint 2, Blue-timepoint 3. The left is HD case and the right plot is Control.]]
|[[Image:HD_vs_CN_3d.png|thumb|300px|center|The red trajectory corresponds the the HD subjects while the blue is a control subject with healthy aging. HD patient clearly shows a much sharper FA decrease along time than the Control.]]
|[[Image:HD_vs_CN_params.png|thumb|300px|center|The estimated model parameters (Red-HD case, Blue-Control case) for the two subjects. The top plot is the rate of change per unit time, per unit FA value. The middle plot is the function asymptote and the last plot is the common baseline curve estimation for providing a common reference frame for model estimation and comparison. The asymptote clearly shows a much lower FA range for HD when compared to the Control which already shows promising early prediction capabilities for HD cases where white matter decay is much faster than healthy patients. Also, a higher per unit rate of change quantifies the sharper FA decrease compared to control.]]
|-
|}

== Experiments with Krabbe's data ==

We have also applied the framework to study white matter changes in infants with Krabbe's disease. Here we show some preliminary results for the FA tract profile along the genu tract for a single subject with Krabbe's disease. In theory, FA non-linearly increases with time along genu owing to a healthy brain maturation process. However, in Krabbe's case, the FA does not follow a monotonic change pattern.

{| border="0" style="background:transparent;"
|[[Image:Genu_fa_3d_conte_vs_krabbe.png|thumb|400px|center|FA curves corresponding to the genu tract for the Krabbe's patient are shown (Neonate:black, 6months:cyan, 1year:magenta). The other FA profiles are for the 15 healthy control subjects as discussed previously (neonate:red, 1year:green, 2year:blue). This also highlights the fact that the framework does not assume uniformity or correspondence in the scan timings across subjects.]]
|[[Image:Genu_fa_3dnorm_conte_vs_krabbe.png|thumb|400px|center|The black trajectory is the normative FA growth estimated using 15 controls as discussed previously. The three FA curves for the Krabbe case are also plotted and clearly show how the patient is lagging behind on expected brain maturation.]]
|-
|}

== Future work ==

As we see in the Krabbe experiment, the model assumes a monotonic change along time. Neighboring locations along the tract can have different, yet correlated time changes. For eg. we can see localized increases or decreases along the tract length with respect to the time evolution. However, owing to the definition of the logistic function, any non-monotonic behavior along time at any given tract location is not modeled accurately. Therefore, in cases like the Krabbe's subject, we need a more relaxed and flexible semi-parametric or non-parametric model to remove the constraints coming in from the parametric nature of the current model. This is currently work under progress.
To utilize the quantification of growth trajectories enabled by this method, we also require a systematic hypothesis testing scheme to draw conclusive results of individuals differing from normative patterns or populations differing in their evolution behaviors. This too is currently being worked upon.

= References =

* Sharma, A., Durrleman, S. , Gilmore, J.H. , Gerig, G. Longitudinal growth modeling of discrete-time functions with application to DTI tract evolution in early neurodevelopment. Proc. of 9th IEEE International Symposium on Biomedical Imaging (ISBI May'12), p.1397-1400.
* Corouge, I., Fletcher, P.T., Joshi, S., Gouttard, S., Gerig, G., 2006. Fiber tract-oriented statistics for quantitative diffusion tensor MRI analysis. Med Image Anal, pp. 786-798.
* Goodlett, C.B., Fletcher, P.T., Gilmore, J.H., Gerig, G., 2009. Group analysis of DTI fiber tract statistics with application to neurodevelopment. Neuroimage, pp. S133-142.

==Key Investigators==
* Utah: Anuja Sharma, Stanley Durrleman, Guido Gerig
* INRIA/ICM, Pitie Salpetriere Hospital: Stanley Durrleman
[http://www.na-mic.org/Wiki/index.php/Algorithm:Utah2 Back to Utah 2 Algorithm Core]

== Regression of probability distributions obtained along white matter tracts to create spatio-temporal growth models ==

Temporal modeling frameworks often operate on scalar variables by
summarizing data at initial stages as statistical summaries of the underlying
distributions. For instance, DTI analysis often employs
summary statistics, like mean, for regions of interest and properties
along fiber tracts for population studies and hypothesis testing.
This reduction via discarding of variability information may introduce
significant errors which propagate through the procedures. Moreover, the data variability information is lost at
all the intermediate interpolated times.

{| border="0" style="background:transparent;"
|[[Image:DistributionRegression_Synthetic.png|thumb|600px|center|Left:Two synthetically generated distributions along time
(blue:t0, orange:t1) with translated locations and opposite skews.
Right: Simple linear interpolation using classic single-valued summary
statistics (means: red, medians: blue) and ignoring data variability
(shown as error bars) at t0 and t1. The decision to choose mean versus median can heavily influence the final regression estimates. Additionally, once the inherent variability is discarded for an early decision regarding a summary statistics, it cannot be recovered for the interpolated time points.]]
|-
|}

We propose a novel framework which uses distribution-valued variables
to retain and utilize the local variability information. We propose two different regression approaches to achieve our goals. The first is a non-parametric moving average method where minimal assumptions are involved in terms of the growth trajectory evolution (Sharma ISBI 2013). The second is a parametric regression scheme where the expected temporal trend is linear (Sharma ISBI 2014). It provides a compact representation in terms of model parameters for further statistical inference and has a closed form solution.

Our driving application is the modeling of age-related changes along DTI
white matter tracts. Results are shown for the spatiotemporal population
trajectory of genu tract estimated from a group of healthy infants and compared with an infant with Krabbe's disease.

{| border="0" style="background:transparent;"
|[[Image:DWIalongTime.png|thumb|400px|center|Model age related DTI changes in early neurodevelopment.]]
|[[Image:genu_distributions_along_time.png|thumb|600px|center|Distributions of DTI-derived scalar diffusion parameters like FA are obtained along the length of the genu tract. The distributions correspond to the cross-sections of the 3D tract geometry and are a function of the arc-length along the tract's total length as we move from one end of the tract to the other. This provides a functional curve which is parametrized by a feature of the local tract geometry (arc-length along the tract in this case) and is attributed by probability distributions (instead of scalar values like the 'mean' FA obtained from these local distributions).]]
|-
|}

=== Non-parametric regression for distribution-valued data ===

The proposed method
presents a completely nonparametric regression framework for spatial-temporal evolution of
statistical distributions. This avoids any parametric assumptions regarding the nature of the
underlying noise model in the DTI data and retains and utilizes the variability information to
estimate the regression trend. The estimated trajectory is completely characterized by statistical
distributions in space and time enabling a very powerful statistical inference framework. Both
population and individual trajectories can be estimated. The concept of ’distance’ between distributions and an ’average’ of distributions is employed. The dissimilarity metric employed is the Mallow's distance between distributions. It is the L2 norm of the generic Wasserstein metric and captures translation, changing width and shape of the participating distributions.

{| border="0" style="background:transparent;"
|[[Image:dist2_3d_bary_300_revrev.png|thumb|400px|center|Temporal evolution of the synthetic distributions between two timepoints as estimated by the proposed method.]]
|[[Image:bary_stat_300.png|thumb|400px|center|Since the regression estimates the complete probability distribution continuously along time, we now have access to the complete variability information at the interpolated time points. This can enable statistical inference using statistical summary measures as well as distribution differences at time points where the original image data is not available.]]
|-
|}

{| border="0" style="background:transparent;"
|[[Image:conte_krabbe_median_3d_300_2.png|thumb|400px|center|The medians of the estimated probability distributions are shown here for the healthy population (in red/pink) versus a single infant with Krabbe's disease (blue/black). Note that the method can applied to estimate normative population trajectories as well as subject-specific evolutions. Each point on these statistical manifolds are characterized by the complete distribution information (instead of just a scalar value).]]
|[[Image:conte_vs_krabbe_quartiles.png|thumb|600px|center|Highlighting the quantiles obtained at one of the arc length locations in the left plot. The Krabbe's subject seems to lag behind the normative population and the difference can be quantified in terms of distribution-based inferences.]]
|-
|}

=== Parametric linear regression for distribution-valued data ===

Classic linear regression is adapted to employ distribution-valued variables for model estimation.

= References =

* A. Sharma, P.T. Fletcher, J.H. Gilmore, M.L. Escolar, A. Gupta, M. Styner, G. Gerig. “ Parametric Regression Scheme For Distributions: Analysis of DTI Fiber Tract Diffusion Changes In Early Brain Development,” In Proceedings of the 2014 IEEE 11th International Symposium on Biomedical Imaging (ISBI), pp. 559-562. 2014.
* A. Sharma, P.T. Fletcher, J.H. Gilmore, M.L. Escolar, A. Gupta, M. Styner, G. Gerig. “Spatiotemporal Modeling of Discrete-Time Distribution-Valued Data Applied to DTI Tract Evolution in Infant Neurodevelopment,” In Proceedings of the 2013 IEEE 10th International Symposium on Biomedical Imaging (ISBI), pp. 684--687. 2013.
* A. Sharma, S. Durrleman, J.H. Gilmore, G. Gerig. “Longitudinal Growth Modeling of Discrete-Time Functions with Application to DTI Tract Evolution in Early Neurodevelopment,” In Proceedings of IEEE ISBI 2012, pp. 1397--1400. 2012.
* Corouge, I., Fletcher, P.T., Joshi, S., Gouttard, S., Gerig, G., 2006. Fiber tract-oriented statistics for quantitative diffusion tensor MRI analysis. Med Image Anal, pp. 786-798.
* Goodlett, C.B., Fletcher, P.T., Gilmore, J.H., Gerig, G., 2009. Group analysis of DTI fiber tract statistics with application to neurodevelopment. Neuroimage, pp. S133-142.

Projects:TractLongitudinalDTI

2014-10-15T22:44:18Z

Anuja: /* Non-parametric regression for distribution-valued data */

Back to [[Algorithm:Utah2|Utah 2 Algorithms]]
__NOTOC__

== Tract-based longitudinal modeling of DTI data ==

This project develops a methodology to explore subject-specific, DTI data obtained from brain's white matter tracts, available at multiple but often sparsely present timepoints. The challenge is to develop a continuous spatio-temporal growth model, given discrete 4D DWI images. This would enable comparison of growth trajectories across subjects and along tracts which are biologically of interest in developmental and pathological changes.

== Background ==

We use the arc length parametrization scheme initially proposed by Corouge et al. It represents white matter fiber tracts obtained via streamline tractography in the brain's atlas tensor image as a function of arc length. (Atlas construction uses unbiased atlas building schemes followed by back transformations to subjects' DTI images to obtain identical fiber tract geometry across subjects, populated with subject specific diffusion data). We use the mean diffusion scalar invariants derived from these individual fiber bundle cross sections, as our input longitudinal diffusion profiles.

{| border="0" style="background:transparent;"
|[[Image:Tract-stats.png|thumb|700px|White matter diffusion properties along fiber tract: Left: A fiber tract with an origin plane defined for arc length parametrization, Right: FA mean and standard deviation as function of arc-length. Dots mark location of coordinate origin.]]
|-
|}

== Subject-specific spatiotemporal continuous growth model ==

We propose the use of Verhulst-Pearl logistic equation to capture temporal changes, while using a non-parametric kernel along arc length to account for biologically motivated functional along-tract relationship in the diffusion data.

{| border="0" style="background:transparent;"
|[[Image:logistic_eq.png|thumb|600px|Logistic equation with P as the population value at a given time, r being the growth rate of the population and K being the carrying capacity (also the limiting asymptote value).]]
|[[Image:logistic_simulation_growthrate_r.png|thumb|400px|Effect of growth rate r on the logistic curve.]]
|[[Image:logistic_simulation_asymptote_K.png|thumb|400px|Effect of carrying capacity K on the logistic curve.]]
|-
|}

From the definition of the logistic function, the parameter r represents the growth rate of the diffusion invariant and the parameter K represents the asymptote value. The overall function shape intuitively follows the growth pattern we expect to see during brain maturation. The diffusion invariants start with an initial diffusion profile along tract, and have a non linear temporal growth trajectory showing maximum changes in early childhood and then slowing down or almost saturating at a certain age. (For instance, observed diffusion changes are much more in neonates than in an adult brain). The temporal trajectories may also differ along the tract's length giving localized changes. Since our method gives us continuous along-tract, growth trajectories all along time, we can compare subjects with respect to differences in diffusion profiles at birth, growth rates at any given age as well as the asymptote saturation values. This gives us important information to understand delayed or abnormal brain maturation by comparing a normative growth surface with an individual's or by comparing the model's parameters across subjects.

== Results ==

Below are some results using synthetic data. For more validation results and extension of the framework to jointly estimate individual subject trajectories together with a normative trajectory, refer to [http://ieeexplore.ieee.org/xpl/login.jsp?tp=&arnumber=6235829&url=http%3A%2F%2Fieeexplore.ieee.org%2Fxpls%2Fabs_all.jsp%3Farnumber%3D6235829 Sharma et al. (ISBI '12)].

{| border="0" style="background:transparent;"
|[[Image:namic_tract_longitudinal_main.png|thumb|400px|Longitudinal modeling of synthetic data. The data is generated from a template neonate FA mean curve and uses known logistic parameters to generate the three timepoint curves (red). Our method faithfully recovers the original curves (blue) while simultaneously creating a continuous longitudinal growth profile (green) without having any prior knowledge of the logistic parameters used for data generation. It also estimates an unbiased time=0 template curve (black) along with parameters r and K to avoid biasing the estimation with any assumed initialization at time=0.]]
|[[Image:namic_tract_longitudinal_3d.png|thumb|400px|Results on real data (mean FA curves for a subject at three timepoints:neonate, 1 year and 2 year).]]
|-
|}
{| border="0" style="background:transparent;"
|[[Image:namic_tract_longitudinal_p1p2only.png|thumb|400px|For the above real data (right), these are the final along-tract initial values and the final estimates for parameters-growth rate r(mentioned as p1 here) and asymptote K(mentioned as p2 here).]]
|[[Image:namic_tract_longitudinal_alphaonly.png|thumb|400px|The complete continuous growth grid as estimated for the real data above.]]

|-
|}

The below images show jointly estimated personalized trajectories for 15 control subjects along with the average growth trajectory. The estimated model parameters for 15 subjects as well as the average trajectory are also shown. The normative trajectory is colored by the local standard deviation. It points to the fact that despite individual variability in the maturation process, there is a strong agreement in the asymptote FA values seen around a gestational age of 2.5 years indicating a relative stabilization of the white matter changes across subjects. The framework thus quantifies patient-specific changes in serial diffusion data given discrete-time diffusion curves.

{| border="0" style="background:transparent;"
|[[Image:CONTE_Final_3d.png|thumb|400px|The jointly estimated individual growth trajectories for 4 of the 15 control subjects.]]
|[[Image:CONTE_Norm_Params.png|thumb|400px|Estimated model parameters for 15 subjects together with the normative growth trajectory colored by the local variability and the estimated normative model parameters.]]

|-
|}

== Experiments with Huntington's data ==

During the NAMIC Winter project week, we worked on registering subjects with Huntington's disease in the same coordinate space as control subjects. We then applied the above framework [http://ieeexplore.ieee.org/xpl/login.jsp?tp=&arnumber=6235829&url=http%3A%2F%2Fieeexplore.ieee.org%2Fxpls%2Fabs_all.jsp%3Farnumber%3D6235829 Sharma et al. ISBI'12] to quantify the differences in normal expected aging versus the accelerated white matter changes expected in HD. Details of the data pre-processing steps and image registration are available on the [http://www.na-mic.org/Wiki/index.php/2012_Winter_Project_Week:_DTI_Change_Modeling Winter AHM page]. Some results are summarized here for FA value along the genu tract.

Two subjects- one being a HD patient (10027) with a high burden factor (higher factor value (of factor 12) implying that the subjects is closer to the onset) and the other being a control subject (10004) are chosen for concept demonstration. Both have a similar baseline scan age (42 and 43 years respectively) as well as a similar time separation between follow up scans. This allows an intuitive comparison of the estimated growth trajectories.

{| border="0" style="background:transparent;"
|[[Image:ORIG_hd_cn.png|thumb|300px|center|FA curves corresponding to the genu tract (middle clipped portions) for one HD case (10027) and one control case (10004). Visual inspection clearly shows the huge decrease relative to the Control in the HD subject. On the other hand, the control shows normal expected decrease due to aging. Red-timepoint 1, Green-timepoint 2, Blue-timepoint 3. The left is HD case and the right plot is Control.]]
|[[Image:HD_vs_CN_3d.png|thumb|300px|center|The red trajectory corresponds the the HD subjects while the blue is a control subject with healthy aging. HD patient clearly shows a much sharper FA decrease along time than the Control.]]
|[[Image:HD_vs_CN_params.png|thumb|300px|center|The estimated model parameters (Red-HD case, Blue-Control case) for the two subjects. The top plot is the rate of change per unit time, per unit FA value. The middle plot is the function asymptote and the last plot is the common baseline curve estimation for providing a common reference frame for model estimation and comparison. The asymptote clearly shows a much lower FA range for HD when compared to the Control which already shows promising early prediction capabilities for HD cases where white matter decay is much faster than healthy patients. Also, a higher per unit rate of change quantifies the sharper FA decrease compared to control.]]
|-
|}

== Experiments with Krabbe's data ==

We have also applied the framework to study white matter changes in infants with Krabbe's disease. Here we show some preliminary results for the FA tract profile along the genu tract for a single subject with Krabbe's disease. In theory, FA non-linearly increases with time along genu owing to a healthy brain maturation process. However, in Krabbe's case, the FA does not follow a monotonic change pattern.

{| border="0" style="background:transparent;"
|[[Image:Genu_fa_3d_conte_vs_krabbe.png|thumb|400px|center|FA curves corresponding to the genu tract for the Krabbe's patient are shown (Neonate:black, 6months:cyan, 1year:magenta). The other FA profiles are for the 15 healthy control subjects as discussed previously (neonate:red, 1year:green, 2year:blue). This also highlights the fact that the framework does not assume uniformity or correspondence in the scan timings across subjects.]]
|[[Image:Genu_fa_3dnorm_conte_vs_krabbe.png|thumb|400px|center|The black trajectory is the normative FA growth estimated using 15 controls as discussed previously. The three FA curves for the Krabbe case are also plotted and clearly show how the patient is lagging behind on expected brain maturation.]]
|-
|}

== Future work ==

As we see in the Krabbe experiment, the model assumes a monotonic change along time. Neighboring locations along the tract can have different, yet correlated time changes. For eg. we can see localized increases or decreases along the tract length with respect to the time evolution. However, owing to the definition of the logistic function, any non-monotonic behavior along time at any given tract location is not modeled accurately. Therefore, in cases like the Krabbe's subject, we need a more relaxed and flexible semi-parametric or non-parametric model to remove the constraints coming in from the parametric nature of the current model. This is currently work under progress.
To utilize the quantification of growth trajectories enabled by this method, we also require a systematic hypothesis testing scheme to draw conclusive results of individuals differing from normative patterns or populations differing in their evolution behaviors. This too is currently being worked upon.

= References =

* Sharma, A., Durrleman, S. , Gilmore, J.H. , Gerig, G. Longitudinal growth modeling of discrete-time functions with application to DTI tract evolution in early neurodevelopment. Proc. of 9th IEEE International Symposium on Biomedical Imaging (ISBI May'12), p.1397-1400.
* Corouge, I., Fletcher, P.T., Joshi, S., Gouttard, S., Gerig, G., 2006. Fiber tract-oriented statistics for quantitative diffusion tensor MRI analysis. Med Image Anal, pp. 786-798.
* Goodlett, C.B., Fletcher, P.T., Gilmore, J.H., Gerig, G., 2009. Group analysis of DTI fiber tract statistics with application to neurodevelopment. Neuroimage, pp. S133-142.

==Key Investigators==
* Utah: Anuja Sharma, Stanley Durrleman, Guido Gerig
* INRIA/ICM, Pitie Salpetriere Hospital: Stanley Durrleman
[http://www.na-mic.org/Wiki/index.php/Algorithm:Utah2 Back to Utah 2 Algorithm Core]

== Regression of probability distributions obtained along white matter tracts to create spatio-temporal growth models ==

Temporal modeling frameworks often operate on scalar variables by
summarizing data at initial stages as statistical summaries of the underlying
distributions. For instance, DTI analysis often employs
summary statistics, like mean, for regions of interest and properties
along fiber tracts for population studies and hypothesis testing.
This reduction via discarding of variability information may introduce
significant errors which propagate through the procedures. Moreover, the data variability information is lost at
all the intermediate interpolated times.

{| border="0" style="background:transparent;"
|[[Image:DistributionRegression_Synthetic.png|thumb|600px|center|Left:Two synthetically generated distributions along time
(blue:t0, orange:t1) with translated locations and opposite skews.
Right: Simple linear interpolation using classic single-valued summary
statistics (means: red, medians: blue) and ignoring data variability
(shown as error bars) at t0 and t1. The decision to choose mean versus median can heavily influence the final regression estimates. Additionally, once the inherent variability is discarded for an early decision regarding a summary statistics, it cannot be recovered for the interpolated time points.]]
|-
|}

We propose a novel framework which uses distribution-valued variables
to retain and utilize the local variability information. We propose two different regression approaches to achieve our goals. The first is a non-parametric moving average method where minimal assumptions are involved in terms of the growth trajectory evolution (Sharma ISBI 2013). The second is a parametric regression scheme where the expected temporal trend is linear (Sharma ISBI 2014). It provides a compact representation in terms of model parameters for further statistical inference and has a closed form solution.

Our driving application is the modeling of age-related changes along DTI
white matter tracts. Results are shown for the spatiotemporal population
trajectory of genu tract estimated from a group of healthy infants and compared with an infant with Krabbe's disease.

{| border="0" style="background:transparent;"
|[[Image:DWIalongTime.png|thumb|400px|center|Model age related DTI changes in early neurodevelopment.]]
|[[Image:genu_distributions_along_time.png|thumb|600px|center|Distributions of DTI-derived scalar diffusion parameters like FA are obtained along the length of the genu tract. The distributions correspond to the cross-sections of the 3D tract geometry and are a function of the arc-length along the tract's total length as we move from one end of the tract to the other. This provides a functional curve which is parametrized by a feature of the local tract geometry (arc-length along the tract in this case) and is attributed by probability distributions (instead of scalar values like the 'mean' FA obtained from these local distributions).]]
|-
|}

=== Non-parametric regression for distribution-valued data ===

The proposed method
presents a completely nonparametric regression framework for spatial-temporal evolution of
statistical distributions. This avoids any parametric assumptions regarding the nature of the
underlying noise model in the DTI data and retains and utilizes the variability information to
estimate the regression trend. The estimated trajectory is completely characterized by statistical
distributions in space and time enabling a very powerful statistical inference framework. Both
population and individual trajectories can be estimated. The concept of ’distance’ between distributions and an ’average’ of distributions is employed. The dissimilarity metric employed is the Mallow's distance between distributions. It is the L2 norm of the generic Wasserstein metric and captures translation, changing width and shape of the participating distributions.

{| border="0" style="background:transparent;"
|[[Image:synbarypdf_barycenterhist.png|thumb|400px|center|An 'average' histogram (red) created from the participating histograms (grey) using the Mallow's distance metric.]]
|[[Image:dist2_3d_bary_300_revrev.png|thumb|400px|center|Temporal evolution of the synthetic distributions between two timepoints as estimated by the proposed method.]]
|[[Image:bary_stat_300.png|thumb|400px|center|Since the regression estimates the complete probability distribution continuously along time, we now have access to the complete variability information at the interpolated time points. This can enable statistical inference using statistical summary measures as well as distribution differences at time points where the original image data is not available.]]
|-
|}

{| border="0" style="background:transparent;"
|[[Image:conte_krabbe_median_3d_300_2.png|thumb|400px|center|The medians of the estimated probability distributions are shown here for the healthy population (in red/pink) versus a single infant with Krabbe's disease (blue/black). Note that the method can applied to estimate normative population trajectories as well as subject-specific evolutions. Each point on these statistical manifolds are characterized by the complete distribution information (instead of just a scalar value).]]
|[[Image:conte_vs_krabbe_quartiles.png|thumb|600px|center|Highlighting the quantiles obtained at one of the arc length locations in the left plot. The Krabbe's subject seems to lag behind the normative population and the difference can be quantified in terms of distribution-based inferences.]]
|-
|}

=== Parametric linear regression for distribution-valued data ===

Classic linear regression is adapted to employ distribution-valued variables for model estimation.

= References =

* A. Sharma, P.T. Fletcher, J.H. Gilmore, M.L. Escolar, A. Gupta, M. Styner, G. Gerig. “ Parametric Regression Scheme For Distributions: Analysis of DTI Fiber Tract Diffusion Changes In Early Brain Development,” In Proceedings of the 2014 IEEE 11th International Symposium on Biomedical Imaging (ISBI), pp. 559-562. 2014.
* A. Sharma, P.T. Fletcher, J.H. Gilmore, M.L. Escolar, A. Gupta, M. Styner, G. Gerig. “Spatiotemporal Modeling of Discrete-Time Distribution-Valued Data Applied to DTI Tract Evolution in Infant Neurodevelopment,” In Proceedings of the 2013 IEEE 10th International Symposium on Biomedical Imaging (ISBI), pp. 684--687. 2013.
* A. Sharma, S. Durrleman, J.H. Gilmore, G. Gerig. “Longitudinal Growth Modeling of Discrete-Time Functions with Application to DTI Tract Evolution in Early Neurodevelopment,” In Proceedings of IEEE ISBI 2012, pp. 1397--1400. 2012.
* Corouge, I., Fletcher, P.T., Joshi, S., Gouttard, S., Gerig, G., 2006. Fiber tract-oriented statistics for quantitative diffusion tensor MRI analysis. Med Image Anal, pp. 786-798.
* Goodlett, C.B., Fletcher, P.T., Gilmore, J.H., Gerig, G., 2009. Group analysis of DTI fiber tract statistics with application to neurodevelopment. Neuroimage, pp. S133-142.

File:Synbarypdf barycenter.png

2014-10-15T22:43:39Z

Anuja:

File:Synbarypdf.png

2014-10-15T22:43:03Z

Anuja:

Projects:TractLongitudinalDTI

2014-10-15T22:42:50Z

Anuja: /* Non-parametric regression for distribution-valued data */

Back to [[Algorithm:Utah2|Utah 2 Algorithms]]
__NOTOC__

== Tract-based longitudinal modeling of DTI data ==

This project develops a methodology to explore subject-specific, DTI data obtained from brain's white matter tracts, available at multiple but often sparsely present timepoints. The challenge is to develop a continuous spatio-temporal growth model, given discrete 4D DWI images. This would enable comparison of growth trajectories across subjects and along tracts which are biologically of interest in developmental and pathological changes.

== Background ==

We use the arc length parametrization scheme initially proposed by Corouge et al. It represents white matter fiber tracts obtained via streamline tractography in the brain's atlas tensor image as a function of arc length. (Atlas construction uses unbiased atlas building schemes followed by back transformations to subjects' DTI images to obtain identical fiber tract geometry across subjects, populated with subject specific diffusion data). We use the mean diffusion scalar invariants derived from these individual fiber bundle cross sections, as our input longitudinal diffusion profiles.

{| border="0" style="background:transparent;"
|[[Image:Tract-stats.png|thumb|700px|White matter diffusion properties along fiber tract: Left: A fiber tract with an origin plane defined for arc length parametrization, Right: FA mean and standard deviation as function of arc-length. Dots mark location of coordinate origin.]]
|-
|}

== Subject-specific spatiotemporal continuous growth model ==

We propose the use of Verhulst-Pearl logistic equation to capture temporal changes, while using a non-parametric kernel along arc length to account for biologically motivated functional along-tract relationship in the diffusion data.

{| border="0" style="background:transparent;"
|[[Image:logistic_eq.png|thumb|600px|Logistic equation with P as the population value at a given time, r being the growth rate of the population and K being the carrying capacity (also the limiting asymptote value).]]
|[[Image:logistic_simulation_growthrate_r.png|thumb|400px|Effect of growth rate r on the logistic curve.]]
|[[Image:logistic_simulation_asymptote_K.png|thumb|400px|Effect of carrying capacity K on the logistic curve.]]
|-
|}

From the definition of the logistic function, the parameter r represents the growth rate of the diffusion invariant and the parameter K represents the asymptote value. The overall function shape intuitively follows the growth pattern we expect to see during brain maturation. The diffusion invariants start with an initial diffusion profile along tract, and have a non linear temporal growth trajectory showing maximum changes in early childhood and then slowing down or almost saturating at a certain age. (For instance, observed diffusion changes are much more in neonates than in an adult brain). The temporal trajectories may also differ along the tract's length giving localized changes. Since our method gives us continuous along-tract, growth trajectories all along time, we can compare subjects with respect to differences in diffusion profiles at birth, growth rates at any given age as well as the asymptote saturation values. This gives us important information to understand delayed or abnormal brain maturation by comparing a normative growth surface with an individual's or by comparing the model's parameters across subjects.

== Results ==

Below are some results using synthetic data. For more validation results and extension of the framework to jointly estimate individual subject trajectories together with a normative trajectory, refer to [http://ieeexplore.ieee.org/xpl/login.jsp?tp=&arnumber=6235829&url=http%3A%2F%2Fieeexplore.ieee.org%2Fxpls%2Fabs_all.jsp%3Farnumber%3D6235829 Sharma et al. (ISBI '12)].

{| border="0" style="background:transparent;"
|[[Image:namic_tract_longitudinal_main.png|thumb|400px|Longitudinal modeling of synthetic data. The data is generated from a template neonate FA mean curve and uses known logistic parameters to generate the three timepoint curves (red). Our method faithfully recovers the original curves (blue) while simultaneously creating a continuous longitudinal growth profile (green) without having any prior knowledge of the logistic parameters used for data generation. It also estimates an unbiased time=0 template curve (black) along with parameters r and K to avoid biasing the estimation with any assumed initialization at time=0.]]
|[[Image:namic_tract_longitudinal_3d.png|thumb|400px|Results on real data (mean FA curves for a subject at three timepoints:neonate, 1 year and 2 year).]]
|-
|}
{| border="0" style="background:transparent;"
|[[Image:namic_tract_longitudinal_p1p2only.png|thumb|400px|For the above real data (right), these are the final along-tract initial values and the final estimates for parameters-growth rate r(mentioned as p1 here) and asymptote K(mentioned as p2 here).]]
|[[Image:namic_tract_longitudinal_alphaonly.png|thumb|400px|The complete continuous growth grid as estimated for the real data above.]]

|-
|}

The below images show jointly estimated personalized trajectories for 15 control subjects along with the average growth trajectory. The estimated model parameters for 15 subjects as well as the average trajectory are also shown. The normative trajectory is colored by the local standard deviation. It points to the fact that despite individual variability in the maturation process, there is a strong agreement in the asymptote FA values seen around a gestational age of 2.5 years indicating a relative stabilization of the white matter changes across subjects. The framework thus quantifies patient-specific changes in serial diffusion data given discrete-time diffusion curves.

{| border="0" style="background:transparent;"
|[[Image:CONTE_Final_3d.png|thumb|400px|The jointly estimated individual growth trajectories for 4 of the 15 control subjects.]]
|[[Image:CONTE_Norm_Params.png|thumb|400px|Estimated model parameters for 15 subjects together with the normative growth trajectory colored by the local variability and the estimated normative model parameters.]]

|-
|}

== Experiments with Huntington's data ==

During the NAMIC Winter project week, we worked on registering subjects with Huntington's disease in the same coordinate space as control subjects. We then applied the above framework [http://ieeexplore.ieee.org/xpl/login.jsp?tp=&arnumber=6235829&url=http%3A%2F%2Fieeexplore.ieee.org%2Fxpls%2Fabs_all.jsp%3Farnumber%3D6235829 Sharma et al. ISBI'12] to quantify the differences in normal expected aging versus the accelerated white matter changes expected in HD. Details of the data pre-processing steps and image registration are available on the [http://www.na-mic.org/Wiki/index.php/2012_Winter_Project_Week:_DTI_Change_Modeling Winter AHM page]. Some results are summarized here for FA value along the genu tract.

Two subjects- one being a HD patient (10027) with a high burden factor (higher factor value (of factor 12) implying that the subjects is closer to the onset) and the other being a control subject (10004) are chosen for concept demonstration. Both have a similar baseline scan age (42 and 43 years respectively) as well as a similar time separation between follow up scans. This allows an intuitive comparison of the estimated growth trajectories.

{| border="0" style="background:transparent;"
|[[Image:ORIG_hd_cn.png|thumb|300px|center|FA curves corresponding to the genu tract (middle clipped portions) for one HD case (10027) and one control case (10004). Visual inspection clearly shows the huge decrease relative to the Control in the HD subject. On the other hand, the control shows normal expected decrease due to aging. Red-timepoint 1, Green-timepoint 2, Blue-timepoint 3. The left is HD case and the right plot is Control.]]
|[[Image:HD_vs_CN_3d.png|thumb|300px|center|The red trajectory corresponds the the HD subjects while the blue is a control subject with healthy aging. HD patient clearly shows a much sharper FA decrease along time than the Control.]]
|[[Image:HD_vs_CN_params.png|thumb|300px|center|The estimated model parameters (Red-HD case, Blue-Control case) for the two subjects. The top plot is the rate of change per unit time, per unit FA value. The middle plot is the function asymptote and the last plot is the common baseline curve estimation for providing a common reference frame for model estimation and comparison. The asymptote clearly shows a much lower FA range for HD when compared to the Control which already shows promising early prediction capabilities for HD cases where white matter decay is much faster than healthy patients. Also, a higher per unit rate of change quantifies the sharper FA decrease compared to control.]]
|-
|}

== Experiments with Krabbe's data ==

We have also applied the framework to study white matter changes in infants with Krabbe's disease. Here we show some preliminary results for the FA tract profile along the genu tract for a single subject with Krabbe's disease. In theory, FA non-linearly increases with time along genu owing to a healthy brain maturation process. However, in Krabbe's case, the FA does not follow a monotonic change pattern.

{| border="0" style="background:transparent;"
|[[Image:Genu_fa_3d_conte_vs_krabbe.png|thumb|400px|center|FA curves corresponding to the genu tract for the Krabbe's patient are shown (Neonate:black, 6months:cyan, 1year:magenta). The other FA profiles are for the 15 healthy control subjects as discussed previously (neonate:red, 1year:green, 2year:blue). This also highlights the fact that the framework does not assume uniformity or correspondence in the scan timings across subjects.]]
|[[Image:Genu_fa_3dnorm_conte_vs_krabbe.png|thumb|400px|center|The black trajectory is the normative FA growth estimated using 15 controls as discussed previously. The three FA curves for the Krabbe case are also plotted and clearly show how the patient is lagging behind on expected brain maturation.]]
|-
|}

== Future work ==

As we see in the Krabbe experiment, the model assumes a monotonic change along time. Neighboring locations along the tract can have different, yet correlated time changes. For eg. we can see localized increases or decreases along the tract length with respect to the time evolution. However, owing to the definition of the logistic function, any non-monotonic behavior along time at any given tract location is not modeled accurately. Therefore, in cases like the Krabbe's subject, we need a more relaxed and flexible semi-parametric or non-parametric model to remove the constraints coming in from the parametric nature of the current model. This is currently work under progress.
To utilize the quantification of growth trajectories enabled by this method, we also require a systematic hypothesis testing scheme to draw conclusive results of individuals differing from normative patterns or populations differing in their evolution behaviors. This too is currently being worked upon.

= References =

* Sharma, A., Durrleman, S. , Gilmore, J.H. , Gerig, G. Longitudinal growth modeling of discrete-time functions with application to DTI tract evolution in early neurodevelopment. Proc. of 9th IEEE International Symposium on Biomedical Imaging (ISBI May'12), p.1397-1400.
* Corouge, I., Fletcher, P.T., Joshi, S., Gouttard, S., Gerig, G., 2006. Fiber tract-oriented statistics for quantitative diffusion tensor MRI analysis. Med Image Anal, pp. 786-798.
* Goodlett, C.B., Fletcher, P.T., Gilmore, J.H., Gerig, G., 2009. Group analysis of DTI fiber tract statistics with application to neurodevelopment. Neuroimage, pp. S133-142.

==Key Investigators==
* Utah: Anuja Sharma, Stanley Durrleman, Guido Gerig
* INRIA/ICM, Pitie Salpetriere Hospital: Stanley Durrleman
[http://www.na-mic.org/Wiki/index.php/Algorithm:Utah2 Back to Utah 2 Algorithm Core]

== Regression of probability distributions obtained along white matter tracts to create spatio-temporal growth models ==

Temporal modeling frameworks often operate on scalar variables by
summarizing data at initial stages as statistical summaries of the underlying
distributions. For instance, DTI analysis often employs
summary statistics, like mean, for regions of interest and properties
along fiber tracts for population studies and hypothesis testing.
This reduction via discarding of variability information may introduce
significant errors which propagate through the procedures. Moreover, the data variability information is lost at
all the intermediate interpolated times.

{| border="0" style="background:transparent;"
|[[Image:DistributionRegression_Synthetic.png|thumb|600px|center|Left:Two synthetically generated distributions along time
(blue:t0, orange:t1) with translated locations and opposite skews.
Right: Simple linear interpolation using classic single-valued summary
statistics (means: red, medians: blue) and ignoring data variability
(shown as error bars) at t0 and t1. The decision to choose mean versus median can heavily influence the final regression estimates. Additionally, once the inherent variability is discarded for an early decision regarding a summary statistics, it cannot be recovered for the interpolated time points.]]
|-
|}

We propose a novel framework which uses distribution-valued variables
to retain and utilize the local variability information. We propose two different regression approaches to achieve our goals. The first is a non-parametric moving average method where minimal assumptions are involved in terms of the growth trajectory evolution (Sharma ISBI 2013). The second is a parametric regression scheme where the expected temporal trend is linear (Sharma ISBI 2014). It provides a compact representation in terms of model parameters for further statistical inference and has a closed form solution.

Our driving application is the modeling of age-related changes along DTI
white matter tracts. Results are shown for the spatiotemporal population
trajectory of genu tract estimated from a group of healthy infants and compared with an infant with Krabbe's disease.

{| border="0" style="background:transparent;"
|[[Image:DWIalongTime.png|thumb|400px|center|Model age related DTI changes in early neurodevelopment.]]
|[[Image:genu_distributions_along_time.png|thumb|600px|center|Distributions of DTI-derived scalar diffusion parameters like FA are obtained along the length of the genu tract. The distributions correspond to the cross-sections of the 3D tract geometry and are a function of the arc-length along the tract's total length as we move from one end of the tract to the other. This provides a functional curve which is parametrized by a feature of the local tract geometry (arc-length along the tract in this case) and is attributed by probability distributions (instead of scalar values like the 'mean' FA obtained from these local distributions).]]
|-
|}

=== Non-parametric regression for distribution-valued data ===

The proposed method
presents a completely nonparametric regression framework for spatial-temporal evolution of
statistical distributions. This avoids any parametric assumptions regarding the nature of the
underlying noise model in the DTI data and retains and utilizes the variability information to
estimate the regression trend. The estimated trajectory is completely characterized by statistical
distributions in space and time enabling a very powerful statistical inference framework. Both
population and individual trajectories can be estimated. The concept of ’distance’ between distributions and an ’average’ of distributions is employed. The dissimilarity metric employed is the Mallow's distance between distributions. It is the L2 norm of the generic Wasserstein metric and captures translation, changing width and shape of the participating distributions.

{| border="0" style="background:transparent;"
|[[Image:synbarypdf.png|thumb|400px|center|An 'average' histogram (red) created from the participating histograms (grey) using the Mallow's distance metric.]]
|[[Image:dist2_3d_bary_300_revrev.png|thumb|400px|center|Temporal evolution of the synthetic distributions between two timepoints as estimated by the proposed method.]]
|[[Image:bary_stat_300.png|thumb|400px|center|Since the regression estimates the complete probability distribution continuously along time, we now have access to the complete variability information at the interpolated time points. This can enable statistical inference using statistical summary measures as well as distribution differences at time points where the original image data is not available.]]
|-
|}

{| border="0" style="background:transparent;"
|[[Image:conte_krabbe_median_3d_300_2.png|thumb|400px|center|The medians of the estimated probability distributions are shown here for the healthy population (in red/pink) versus a single infant with Krabbe's disease (blue/black). Note that the method can applied to estimate normative population trajectories as well as subject-specific evolutions. Each point on these statistical manifolds are characterized by the complete distribution information (instead of just a scalar value).]]
|[[Image:conte_vs_krabbe_quartiles.png|thumb|600px|center|Highlighting the quantiles obtained at one of the arc length locations in the left plot. The Krabbe's subject seems to lag behind the normative population and the difference can be quantified in terms of distribution-based inferences.]]
|-
|}

=== Parametric linear regression for distribution-valued data ===

Classic linear regression is adapted to employ distribution-valued variables for model estimation.

= References =

* A. Sharma, P.T. Fletcher, J.H. Gilmore, M.L. Escolar, A. Gupta, M. Styner, G. Gerig. “ Parametric Regression Scheme For Distributions: Analysis of DTI Fiber Tract Diffusion Changes In Early Brain Development,” In Proceedings of the 2014 IEEE 11th International Symposium on Biomedical Imaging (ISBI), pp. 559-562. 2014.
* A. Sharma, P.T. Fletcher, J.H. Gilmore, M.L. Escolar, A. Gupta, M. Styner, G. Gerig. “Spatiotemporal Modeling of Discrete-Time Distribution-Valued Data Applied to DTI Tract Evolution in Infant Neurodevelopment,” In Proceedings of the 2013 IEEE 10th International Symposium on Biomedical Imaging (ISBI), pp. 684--687. 2013.
* A. Sharma, S. Durrleman, J.H. Gilmore, G. Gerig. “Longitudinal Growth Modeling of Discrete-Time Functions with Application to DTI Tract Evolution in Early Neurodevelopment,” In Proceedings of IEEE ISBI 2012, pp. 1397--1400. 2012.
* Corouge, I., Fletcher, P.T., Joshi, S., Gouttard, S., Gerig, G., 2006. Fiber tract-oriented statistics for quantitative diffusion tensor MRI analysis. Med Image Anal, pp. 786-798.
* Goodlett, C.B., Fletcher, P.T., Gilmore, J.H., Gerig, G., 2009. Group analysis of DTI fiber tract statistics with application to neurodevelopment. Neuroimage, pp. S133-142.

File:Conte vs krabbe quartiles.png

2014-10-15T22:38:01Z

Anuja:

File:Conte krabbe median 3d 300 2.png

2014-10-15T22:37:48Z

Anuja:

File:Bary stat 300.png

2014-10-15T22:37:27Z

Anuja:

File:Dist2 3d bary 300 revrev.png

2014-10-15T22:37:15Z

Anuja:

Projects:TractLongitudinalDTI

2014-10-15T22:36:55Z

Anuja: /* Regression of probability distributions obtained along white matter tracts to create spatio-temporal growth models */

Back to [[Algorithm:Utah2|Utah 2 Algorithms]]
__NOTOC__

== Tract-based longitudinal modeling of DTI data ==

This project develops a methodology to explore subject-specific, DTI data obtained from brain's white matter tracts, available at multiple but often sparsely present timepoints. The challenge is to develop a continuous spatio-temporal growth model, given discrete 4D DWI images. This would enable comparison of growth trajectories across subjects and along tracts which are biologically of interest in developmental and pathological changes.

== Background ==

We use the arc length parametrization scheme initially proposed by Corouge et al. It represents white matter fiber tracts obtained via streamline tractography in the brain's atlas tensor image as a function of arc length. (Atlas construction uses unbiased atlas building schemes followed by back transformations to subjects' DTI images to obtain identical fiber tract geometry across subjects, populated with subject specific diffusion data). We use the mean diffusion scalar invariants derived from these individual fiber bundle cross sections, as our input longitudinal diffusion profiles.

{| border="0" style="background:transparent;"
|[[Image:Tract-stats.png|thumb|700px|White matter diffusion properties along fiber tract: Left: A fiber tract with an origin plane defined for arc length parametrization, Right: FA mean and standard deviation as function of arc-length. Dots mark location of coordinate origin.]]
|-
|}

== Subject-specific spatiotemporal continuous growth model ==

We propose the use of Verhulst-Pearl logistic equation to capture temporal changes, while using a non-parametric kernel along arc length to account for biologically motivated functional along-tract relationship in the diffusion data.

{| border="0" style="background:transparent;"
|[[Image:logistic_eq.png|thumb|600px|Logistic equation with P as the population value at a given time, r being the growth rate of the population and K being the carrying capacity (also the limiting asymptote value).]]
|[[Image:logistic_simulation_growthrate_r.png|thumb|400px|Effect of growth rate r on the logistic curve.]]
|[[Image:logistic_simulation_asymptote_K.png|thumb|400px|Effect of carrying capacity K on the logistic curve.]]
|-
|}

From the definition of the logistic function, the parameter r represents the growth rate of the diffusion invariant and the parameter K represents the asymptote value. The overall function shape intuitively follows the growth pattern we expect to see during brain maturation. The diffusion invariants start with an initial diffusion profile along tract, and have a non linear temporal growth trajectory showing maximum changes in early childhood and then slowing down or almost saturating at a certain age. (For instance, observed diffusion changes are much more in neonates than in an adult brain). The temporal trajectories may also differ along the tract's length giving localized changes. Since our method gives us continuous along-tract, growth trajectories all along time, we can compare subjects with respect to differences in diffusion profiles at birth, growth rates at any given age as well as the asymptote saturation values. This gives us important information to understand delayed or abnormal brain maturation by comparing a normative growth surface with an individual's or by comparing the model's parameters across subjects.

== Results ==

Below are some results using synthetic data. For more validation results and extension of the framework to jointly estimate individual subject trajectories together with a normative trajectory, refer to [http://ieeexplore.ieee.org/xpl/login.jsp?tp=&arnumber=6235829&url=http%3A%2F%2Fieeexplore.ieee.org%2Fxpls%2Fabs_all.jsp%3Farnumber%3D6235829 Sharma et al. (ISBI '12)].

{| border="0" style="background:transparent;"
|[[Image:namic_tract_longitudinal_main.png|thumb|400px|Longitudinal modeling of synthetic data. The data is generated from a template neonate FA mean curve and uses known logistic parameters to generate the three timepoint curves (red). Our method faithfully recovers the original curves (blue) while simultaneously creating a continuous longitudinal growth profile (green) without having any prior knowledge of the logistic parameters used for data generation. It also estimates an unbiased time=0 template curve (black) along with parameters r and K to avoid biasing the estimation with any assumed initialization at time=0.]]
|[[Image:namic_tract_longitudinal_3d.png|thumb|400px|Results on real data (mean FA curves for a subject at three timepoints:neonate, 1 year and 2 year).]]
|-
|}
{| border="0" style="background:transparent;"
|[[Image:namic_tract_longitudinal_p1p2only.png|thumb|400px|For the above real data (right), these are the final along-tract initial values and the final estimates for parameters-growth rate r(mentioned as p1 here) and asymptote K(mentioned as p2 here).]]
|[[Image:namic_tract_longitudinal_alphaonly.png|thumb|400px|The complete continuous growth grid as estimated for the real data above.]]

|-
|}

The below images show jointly estimated personalized trajectories for 15 control subjects along with the average growth trajectory. The estimated model parameters for 15 subjects as well as the average trajectory are also shown. The normative trajectory is colored by the local standard deviation. It points to the fact that despite individual variability in the maturation process, there is a strong agreement in the asymptote FA values seen around a gestational age of 2.5 years indicating a relative stabilization of the white matter changes across subjects. The framework thus quantifies patient-specific changes in serial diffusion data given discrete-time diffusion curves.

{| border="0" style="background:transparent;"
|[[Image:CONTE_Final_3d.png|thumb|400px|The jointly estimated individual growth trajectories for 4 of the 15 control subjects.]]
|[[Image:CONTE_Norm_Params.png|thumb|400px|Estimated model parameters for 15 subjects together with the normative growth trajectory colored by the local variability and the estimated normative model parameters.]]

|-
|}

== Experiments with Huntington's data ==

During the NAMIC Winter project week, we worked on registering subjects with Huntington's disease in the same coordinate space as control subjects. We then applied the above framework [http://ieeexplore.ieee.org/xpl/login.jsp?tp=&arnumber=6235829&url=http%3A%2F%2Fieeexplore.ieee.org%2Fxpls%2Fabs_all.jsp%3Farnumber%3D6235829 Sharma et al. ISBI'12] to quantify the differences in normal expected aging versus the accelerated white matter changes expected in HD. Details of the data pre-processing steps and image registration are available on the [http://www.na-mic.org/Wiki/index.php/2012_Winter_Project_Week:_DTI_Change_Modeling Winter AHM page]. Some results are summarized here for FA value along the genu tract.

Two subjects- one being a HD patient (10027) with a high burden factor (higher factor value (of factor 12) implying that the subjects is closer to the onset) and the other being a control subject (10004) are chosen for concept demonstration. Both have a similar baseline scan age (42 and 43 years respectively) as well as a similar time separation between follow up scans. This allows an intuitive comparison of the estimated growth trajectories.

{| border="0" style="background:transparent;"
|[[Image:ORIG_hd_cn.png|thumb|300px|center|FA curves corresponding to the genu tract (middle clipped portions) for one HD case (10027) and one control case (10004). Visual inspection clearly shows the huge decrease relative to the Control in the HD subject. On the other hand, the control shows normal expected decrease due to aging. Red-timepoint 1, Green-timepoint 2, Blue-timepoint 3. The left is HD case and the right plot is Control.]]
|[[Image:HD_vs_CN_3d.png|thumb|300px|center|The red trajectory corresponds the the HD subjects while the blue is a control subject with healthy aging. HD patient clearly shows a much sharper FA decrease along time than the Control.]]
|[[Image:HD_vs_CN_params.png|thumb|300px|center|The estimated model parameters (Red-HD case, Blue-Control case) for the two subjects. The top plot is the rate of change per unit time, per unit FA value. The middle plot is the function asymptote and the last plot is the common baseline curve estimation for providing a common reference frame for model estimation and comparison. The asymptote clearly shows a much lower FA range for HD when compared to the Control which already shows promising early prediction capabilities for HD cases where white matter decay is much faster than healthy patients. Also, a higher per unit rate of change quantifies the sharper FA decrease compared to control.]]
|-
|}

== Experiments with Krabbe's data ==

We have also applied the framework to study white matter changes in infants with Krabbe's disease. Here we show some preliminary results for the FA tract profile along the genu tract for a single subject with Krabbe's disease. In theory, FA non-linearly increases with time along genu owing to a healthy brain maturation process. However, in Krabbe's case, the FA does not follow a monotonic change pattern.

{| border="0" style="background:transparent;"
|[[Image:Genu_fa_3d_conte_vs_krabbe.png|thumb|400px|center|FA curves corresponding to the genu tract for the Krabbe's patient are shown (Neonate:black, 6months:cyan, 1year:magenta). The other FA profiles are for the 15 healthy control subjects as discussed previously (neonate:red, 1year:green, 2year:blue). This also highlights the fact that the framework does not assume uniformity or correspondence in the scan timings across subjects.]]
|[[Image:Genu_fa_3dnorm_conte_vs_krabbe.png|thumb|400px|center|The black trajectory is the normative FA growth estimated using 15 controls as discussed previously. The three FA curves for the Krabbe case are also plotted and clearly show how the patient is lagging behind on expected brain maturation.]]
|-
|}

== Future work ==

As we see in the Krabbe experiment, the model assumes a monotonic change along time. Neighboring locations along the tract can have different, yet correlated time changes. For eg. we can see localized increases or decreases along the tract length with respect to the time evolution. However, owing to the definition of the logistic function, any non-monotonic behavior along time at any given tract location is not modeled accurately. Therefore, in cases like the Krabbe's subject, we need a more relaxed and flexible semi-parametric or non-parametric model to remove the constraints coming in from the parametric nature of the current model. This is currently work under progress.
To utilize the quantification of growth trajectories enabled by this method, we also require a systematic hypothesis testing scheme to draw conclusive results of individuals differing from normative patterns or populations differing in their evolution behaviors. This too is currently being worked upon.

= References =

* Sharma, A., Durrleman, S. , Gilmore, J.H. , Gerig, G. Longitudinal growth modeling of discrete-time functions with application to DTI tract evolution in early neurodevelopment. Proc. of 9th IEEE International Symposium on Biomedical Imaging (ISBI May'12), p.1397-1400.
* Corouge, I., Fletcher, P.T., Joshi, S., Gouttard, S., Gerig, G., 2006. Fiber tract-oriented statistics for quantitative diffusion tensor MRI analysis. Med Image Anal, pp. 786-798.
* Goodlett, C.B., Fletcher, P.T., Gilmore, J.H., Gerig, G., 2009. Group analysis of DTI fiber tract statistics with application to neurodevelopment. Neuroimage, pp. S133-142.

==Key Investigators==
* Utah: Anuja Sharma, Stanley Durrleman, Guido Gerig
* INRIA/ICM, Pitie Salpetriere Hospital: Stanley Durrleman
[http://www.na-mic.org/Wiki/index.php/Algorithm:Utah2 Back to Utah 2 Algorithm Core]

== Regression of probability distributions obtained along white matter tracts to create spatio-temporal growth models ==

Temporal modeling frameworks often operate on scalar variables by
summarizing data at initial stages as statistical summaries of the underlying
distributions. For instance, DTI analysis often employs
summary statistics, like mean, for regions of interest and properties
along fiber tracts for population studies and hypothesis testing.
This reduction via discarding of variability information may introduce
significant errors which propagate through the procedures. Moreover, the data variability information is lost at
all the intermediate interpolated times.

{| border="0" style="background:transparent;"
|[[Image:DistributionRegression_Synthetic.png|thumb|600px|center|Left:Two synthetically generated distributions along time
(blue:t0, orange:t1) with translated locations and opposite skews.
Right: Simple linear interpolation using classic single-valued summary
statistics (means: red, medians: blue) and ignoring data variability
(shown as error bars) at t0 and t1. The decision to choose mean versus median can heavily influence the final regression estimates. Additionally, once the inherent variability is discarded for an early decision regarding a summary statistics, it cannot be recovered for the interpolated time points.]]
|-
|}

We propose a novel framework which uses distribution-valued variables
to retain and utilize the local variability information. We propose two different regression approaches to achieve our goals. The first is a non-parametric moving average method where minimal assumptions are involved in terms of the growth trajectory evolution (Sharma ISBI 2013). The second is a parametric regression scheme where the expected temporal trend is linear (Sharma ISBI 2014). It provides a compact representation in terms of model parameters for further statistical inference and has a closed form solution.

Our driving application is the modeling of age-related changes along DTI
white matter tracts. Results are shown for the spatiotemporal population
trajectory of genu tract estimated from a group of healthy infants and compared with an infant with Krabbe's disease.

{| border="0" style="background:transparent;"
|[[Image:DWIalongTime.png|thumb|400px|center|Model age related DTI changes in early neurodevelopment.]]
|[[Image:genu_distributions_along_time.png|thumb|600px|center|Distributions of DTI-derived scalar diffusion parameters like FA are obtained along the length of the genu tract. The distributions correspond to the cross-sections of the 3D tract geometry and are a function of the arc-length along the tract's total length as we move from one end of the tract to the other. This provides a functional curve which is parametrized by a feature of the local tract geometry (arc-length along the tract in this case) and is attributed by probability distributions (instead of scalar values like the 'mean' FA obtained from these local distributions).]]
|-
|}

=== Non-parametric regression for distribution-valued data ===

The proposed method
presents a completely nonparametric regression framework for spatial-temporal evolution of
statistical distributions. This avoids any parametric assumptions regarding the nature of the
underlying noise model in the DTI data and retains and utilizes the variability information to
estimate the regression trend. The estimated trajectory is completely characterized by statistical
distributions in space and time enabling a very powerful statistical inference framework. Both
population and individual trajectories can be estimated. The concept of ’distance’ between distributions and an ’average’ of distributions is employed. The dissimilarity metric employed is the Mallow's distance between distributions. It is the L2 norm of the generic Wasserstein metric and captures translation, changing width and shape of the participating distributions.

{| border="0" style="background:transparent;"
|[[Image:dist2_3d_bary_300_revrev.png|thumb|400px|center|Temporal evolution of the synthetic distributions between two timepoints as estimated by the proposed method.]]
|[[Image:bary_stat_300.png|thumb|400px|center|Since the regression estimates the complete probability distribution continuously along time, we now have access to the complete variability information at the interpolated time points. This can enable statistical inference using statistical summary measures as well as distribution differences at time points where the original image data is not available.]]
|-
|}

{| border="0" style="background:transparent;"
|[[Image:conte_krabbe_median_3d_300_2.png|thumb|400px|center|The medians of the estimated probability distributions are shown here for the healthy population (in red/pink) versus a single infant with Krabbe's disease (blue/black). Note that the method can applied to estimate normative population trajectories as well as subject-specific evolutions. Each point on these statistical manifolds are characterized by the complete distribution information (instead of just a scalar value).]]
|[[Image:conte_vs_krabbe_quartiles.png|thumb|600px|center|Highlighting the quantiles obtained at one of the arc length locations in the left plot. The Krabbe's subject seems to lag behind the normative population and the difference can be quantified in terms of distribution-based inferences.]]
|-
|}

=== Parametric linear regression for distribution-valued data ===

Classic linear regression is adapted to employ distribution-valued variables for model estimation.

= References =

* A. Sharma, P.T. Fletcher, J.H. Gilmore, M.L. Escolar, A. Gupta, M. Styner, G. Gerig. “ Parametric Regression Scheme For Distributions: Analysis of DTI Fiber Tract Diffusion Changes In Early Brain Development,” In Proceedings of the 2014 IEEE 11th International Symposium on Biomedical Imaging (ISBI), pp. 559-562. 2014.
* A. Sharma, P.T. Fletcher, J.H. Gilmore, M.L. Escolar, A. Gupta, M. Styner, G. Gerig. “Spatiotemporal Modeling of Discrete-Time Distribution-Valued Data Applied to DTI Tract Evolution in Infant Neurodevelopment,” In Proceedings of the 2013 IEEE 10th International Symposium on Biomedical Imaging (ISBI), pp. 684--687. 2013.
* A. Sharma, S. Durrleman, J.H. Gilmore, G. Gerig. “Longitudinal Growth Modeling of Discrete-Time Functions with Application to DTI Tract Evolution in Early Neurodevelopment,” In Proceedings of IEEE ISBI 2012, pp. 1397--1400. 2012.
* Corouge, I., Fletcher, P.T., Joshi, S., Gouttard, S., Gerig, G., 2006. Fiber tract-oriented statistics for quantitative diffusion tensor MRI analysis. Med Image Anal, pp. 786-798.
* Goodlett, C.B., Fletcher, P.T., Gilmore, J.H., Gerig, G., 2009. Group analysis of DTI fiber tract statistics with application to neurodevelopment. Neuroimage, pp. S133-142.

Projects:TractLongitudinalDTI

2014-10-15T21:07:29Z

Anuja: /* Regression of probability distributions obtained along white matter tracts to create spatio-temporal growth models */

Back to [[Algorithm:Utah2|Utah 2 Algorithms]]
__NOTOC__

== Tract-based longitudinal modeling of DTI data ==

This project develops a methodology to explore subject-specific, DTI data obtained from brain's white matter tracts, available at multiple but often sparsely present timepoints. The challenge is to develop a continuous spatio-temporal growth model, given discrete 4D DWI images. This would enable comparison of growth trajectories across subjects and along tracts which are biologically of interest in developmental and pathological changes.

== Background ==

We use the arc length parametrization scheme initially proposed by Corouge et al. It represents white matter fiber tracts obtained via streamline tractography in the brain's atlas tensor image as a function of arc length. (Atlas construction uses unbiased atlas building schemes followed by back transformations to subjects' DTI images to obtain identical fiber tract geometry across subjects, populated with subject specific diffusion data). We use the mean diffusion scalar invariants derived from these individual fiber bundle cross sections, as our input longitudinal diffusion profiles.

{| border="0" style="background:transparent;"
|[[Image:Tract-stats.png|thumb|700px|White matter diffusion properties along fiber tract: Left: A fiber tract with an origin plane defined for arc length parametrization, Right: FA mean and standard deviation as function of arc-length. Dots mark location of coordinate origin.]]
|-
|}

== Subject-specific spatiotemporal continuous growth model ==

We propose the use of Verhulst-Pearl logistic equation to capture temporal changes, while using a non-parametric kernel along arc length to account for biologically motivated functional along-tract relationship in the diffusion data.

{| border="0" style="background:transparent;"
|[[Image:logistic_eq.png|thumb|600px|Logistic equation with P as the population value at a given time, r being the growth rate of the population and K being the carrying capacity (also the limiting asymptote value).]]
|[[Image:logistic_simulation_growthrate_r.png|thumb|400px|Effect of growth rate r on the logistic curve.]]
|[[Image:logistic_simulation_asymptote_K.png|thumb|400px|Effect of carrying capacity K on the logistic curve.]]
|-
|}

From the definition of the logistic function, the parameter r represents the growth rate of the diffusion invariant and the parameter K represents the asymptote value. The overall function shape intuitively follows the growth pattern we expect to see during brain maturation. The diffusion invariants start with an initial diffusion profile along tract, and have a non linear temporal growth trajectory showing maximum changes in early childhood and then slowing down or almost saturating at a certain age. (For instance, observed diffusion changes are much more in neonates than in an adult brain). The temporal trajectories may also differ along the tract's length giving localized changes. Since our method gives us continuous along-tract, growth trajectories all along time, we can compare subjects with respect to differences in diffusion profiles at birth, growth rates at any given age as well as the asymptote saturation values. This gives us important information to understand delayed or abnormal brain maturation by comparing a normative growth surface with an individual's or by comparing the model's parameters across subjects.

== Results ==

Below are some results using synthetic data. For more validation results and extension of the framework to jointly estimate individual subject trajectories together with a normative trajectory, refer to [http://ieeexplore.ieee.org/xpl/login.jsp?tp=&arnumber=6235829&url=http%3A%2F%2Fieeexplore.ieee.org%2Fxpls%2Fabs_all.jsp%3Farnumber%3D6235829 Sharma et al. (ISBI '12)].

{| border="0" style="background:transparent;"
|[[Image:namic_tract_longitudinal_main.png|thumb|400px|Longitudinal modeling of synthetic data. The data is generated from a template neonate FA mean curve and uses known logistic parameters to generate the three timepoint curves (red). Our method faithfully recovers the original curves (blue) while simultaneously creating a continuous longitudinal growth profile (green) without having any prior knowledge of the logistic parameters used for data generation. It also estimates an unbiased time=0 template curve (black) along with parameters r and K to avoid biasing the estimation with any assumed initialization at time=0.]]
|[[Image:namic_tract_longitudinal_3d.png|thumb|400px|Results on real data (mean FA curves for a subject at three timepoints:neonate, 1 year and 2 year).]]
|-
|}
{| border="0" style="background:transparent;"
|[[Image:namic_tract_longitudinal_p1p2only.png|thumb|400px|For the above real data (right), these are the final along-tract initial values and the final estimates for parameters-growth rate r(mentioned as p1 here) and asymptote K(mentioned as p2 here).]]
|[[Image:namic_tract_longitudinal_alphaonly.png|thumb|400px|The complete continuous growth grid as estimated for the real data above.]]

|-
|}

The below images show jointly estimated personalized trajectories for 15 control subjects along with the average growth trajectory. The estimated model parameters for 15 subjects as well as the average trajectory are also shown. The normative trajectory is colored by the local standard deviation. It points to the fact that despite individual variability in the maturation process, there is a strong agreement in the asymptote FA values seen around a gestational age of 2.5 years indicating a relative stabilization of the white matter changes across subjects. The framework thus quantifies patient-specific changes in serial diffusion data given discrete-time diffusion curves.

{| border="0" style="background:transparent;"
|[[Image:CONTE_Final_3d.png|thumb|400px|The jointly estimated individual growth trajectories for 4 of the 15 control subjects.]]
|[[Image:CONTE_Norm_Params.png|thumb|400px|Estimated model parameters for 15 subjects together with the normative growth trajectory colored by the local variability and the estimated normative model parameters.]]

|-
|}

== Experiments with Huntington's data ==

During the NAMIC Winter project week, we worked on registering subjects with Huntington's disease in the same coordinate space as control subjects. We then applied the above framework [http://ieeexplore.ieee.org/xpl/login.jsp?tp=&arnumber=6235829&url=http%3A%2F%2Fieeexplore.ieee.org%2Fxpls%2Fabs_all.jsp%3Farnumber%3D6235829 Sharma et al. ISBI'12] to quantify the differences in normal expected aging versus the accelerated white matter changes expected in HD. Details of the data pre-processing steps and image registration are available on the [http://www.na-mic.org/Wiki/index.php/2012_Winter_Project_Week:_DTI_Change_Modeling Winter AHM page]. Some results are summarized here for FA value along the genu tract.

Two subjects- one being a HD patient (10027) with a high burden factor (higher factor value (of factor 12) implying that the subjects is closer to the onset) and the other being a control subject (10004) are chosen for concept demonstration. Both have a similar baseline scan age (42 and 43 years respectively) as well as a similar time separation between follow up scans. This allows an intuitive comparison of the estimated growth trajectories.

{| border="0" style="background:transparent;"
|[[Image:ORIG_hd_cn.png|thumb|300px|center|FA curves corresponding to the genu tract (middle clipped portions) for one HD case (10027) and one control case (10004). Visual inspection clearly shows the huge decrease relative to the Control in the HD subject. On the other hand, the control shows normal expected decrease due to aging. Red-timepoint 1, Green-timepoint 2, Blue-timepoint 3. The left is HD case and the right plot is Control.]]
|[[Image:HD_vs_CN_3d.png|thumb|300px|center|The red trajectory corresponds the the HD subjects while the blue is a control subject with healthy aging. HD patient clearly shows a much sharper FA decrease along time than the Control.]]
|[[Image:HD_vs_CN_params.png|thumb|300px|center|The estimated model parameters (Red-HD case, Blue-Control case) for the two subjects. The top plot is the rate of change per unit time, per unit FA value. The middle plot is the function asymptote and the last plot is the common baseline curve estimation for providing a common reference frame for model estimation and comparison. The asymptote clearly shows a much lower FA range for HD when compared to the Control which already shows promising early prediction capabilities for HD cases where white matter decay is much faster than healthy patients. Also, a higher per unit rate of change quantifies the sharper FA decrease compared to control.]]
|-
|}

== Experiments with Krabbe's data ==

We have also applied the framework to study white matter changes in infants with Krabbe's disease. Here we show some preliminary results for the FA tract profile along the genu tract for a single subject with Krabbe's disease. In theory, FA non-linearly increases with time along genu owing to a healthy brain maturation process. However, in Krabbe's case, the FA does not follow a monotonic change pattern.

{| border="0" style="background:transparent;"
|[[Image:Genu_fa_3d_conte_vs_krabbe.png|thumb|400px|center|FA curves corresponding to the genu tract for the Krabbe's patient are shown (Neonate:black, 6months:cyan, 1year:magenta). The other FA profiles are for the 15 healthy control subjects as discussed previously (neonate:red, 1year:green, 2year:blue). This also highlights the fact that the framework does not assume uniformity or correspondence in the scan timings across subjects.]]
|[[Image:Genu_fa_3dnorm_conte_vs_krabbe.png|thumb|400px|center|The black trajectory is the normative FA growth estimated using 15 controls as discussed previously. The three FA curves for the Krabbe case are also plotted and clearly show how the patient is lagging behind on expected brain maturation.]]
|-
|}

== Future work ==

As we see in the Krabbe experiment, the model assumes a monotonic change along time. Neighboring locations along the tract can have different, yet correlated time changes. For eg. we can see localized increases or decreases along the tract length with respect to the time evolution. However, owing to the definition of the logistic function, any non-monotonic behavior along time at any given tract location is not modeled accurately. Therefore, in cases like the Krabbe's subject, we need a more relaxed and flexible semi-parametric or non-parametric model to remove the constraints coming in from the parametric nature of the current model. This is currently work under progress.
To utilize the quantification of growth trajectories enabled by this method, we also require a systematic hypothesis testing scheme to draw conclusive results of individuals differing from normative patterns or populations differing in their evolution behaviors. This too is currently being worked upon.

= References =

* Sharma, A., Durrleman, S. , Gilmore, J.H. , Gerig, G. Longitudinal growth modeling of discrete-time functions with application to DTI tract evolution in early neurodevelopment. Proc. of 9th IEEE International Symposium on Biomedical Imaging (ISBI May'12), p.1397-1400.
* Corouge, I., Fletcher, P.T., Joshi, S., Gouttard, S., Gerig, G., 2006. Fiber tract-oriented statistics for quantitative diffusion tensor MRI analysis. Med Image Anal, pp. 786-798.
* Goodlett, C.B., Fletcher, P.T., Gilmore, J.H., Gerig, G., 2009. Group analysis of DTI fiber tract statistics with application to neurodevelopment. Neuroimage, pp. S133-142.

==Key Investigators==
* Utah: Anuja Sharma, Stanley Durrleman, Guido Gerig
* INRIA/ICM, Pitie Salpetriere Hospital: Stanley Durrleman
[http://www.na-mic.org/Wiki/index.php/Algorithm:Utah2 Back to Utah 2 Algorithm Core]

== Regression of probability distributions obtained along white matter tracts to create spatio-temporal growth models ==

Temporal modeling frameworks often operate on scalar variables by
summarizing data at initial stages as statistical summaries of the underlying
distributions. For instance, DTI analysis often employs
summary statistics, like mean, for regions of interest and properties
along fiber tracts for population studies and hypothesis testing.
This reduction via discarding of variability information may introduce
significant errors which propagate through the procedures. Moreover, the data variability information is lost at
all the intermediate interpolated times.

{| border="0" style="background:transparent;"
|[[Image:DistributionRegression_Synthetic.png|thumb|600px|center|Left:Two synthetically generated distributions along time
(blue:t0, orange:t1) with translated locations and opposite skews.
Right: Simple linear interpolation using classic single-valued summary
statistics (means: red, medians: blue) and ignoring data variability
(shown as error bars) at t0 and t1. The decision to choose mean versus median can heavily influence the final regression estimates. Additionally, once the inherent variability is discarded for an early decision regarding a summary statistics, it cannot be recovered for the interpolated time points.]]
|-
|}

We propose a novel framework which uses distribution-valued variables
to retain and utilize the local variability information. We propose two different regression approaches to achieve our goals. The first is a parametric regression scheme where the expected temporal trend is linear (Sharma ISBI 2014). The second is a non-parametric method where minimal assumptions are involved in terms of the growth trajectory evolution (Sharma ISBI 2013).

Our driving application is the modeling of age-related changes along DTI
white matter tracts. Results are shown for the spatiotemporal population
trajectory of genu tract estimated from a group of healthy infants and compared with an infant with Krabbe's disease.

{| border="0" style="background:transparent;"
|[[Image:DWIalongTime.png|thumb|400px|center|Model age related DTI changes in early neurodevelopment.]]
|[[Image:genu_distributions_along_time.png|thumb|600px|center|Distributions of DTI-derived scalar diffusion parameters like FA are obtained along the length of the genu tract. The distributions correspond to the cross-sections of the 3D tract geometry and are a function of the arc-length along the tract's total length as we move from one end of the tract to the other. This provides a functional curve which is parametrized by a feature of the local tract geometry (arc-length along the tract in this case) and is attributed by probability distributions (instead of scalar values like the 'mean' FA obtained from these local distributions).]]
|-
|}

=== Parametric linear regression for distribution-valued data ===

Classic linear regression is adapted to employ distribution-valued variables for model estimation.

=== Non-parametric regression for distribution-valued data ===

The concept of ’distance’
between distributions and an ’average’ of distributions is employed.
The framework quantifies growth trajectories for individuals
and populations in terms of the complete data variability estimated
along time and space. Concept is demonstrated in the context of our
driving application which is modeling of age-related changes along
white matter tracts in early neurodevelopment. Results are shown
for a single subject with Krabbe’s disease in comparison with a normative
trend estimated from 15 healthy controls.

= References =

* A. Sharma, P.T. Fletcher, J.H. Gilmore, M.L. Escolar, A. Gupta, M. Styner, G. Gerig. “ Parametric Regression Scheme For Distributions: Analysis of DTI Fiber Tract Diffusion Changes In Early Brain Development,” In Proceedings of the 2014 IEEE 11th International Symposium on Biomedical Imaging (ISBI), pp. 559-562. 2014.
* A. Sharma, P.T. Fletcher, J.H. Gilmore, M.L. Escolar, A. Gupta, M. Styner, G. Gerig. “Spatiotemporal Modeling of Discrete-Time Distribution-Valued Data Applied to DTI Tract Evolution in Infant Neurodevelopment,” In Proceedings of the 2013 IEEE 10th International Symposium on Biomedical Imaging (ISBI), pp. 684--687. 2013.
* A. Sharma, S. Durrleman, J.H. Gilmore, G. Gerig. “Longitudinal Growth Modeling of Discrete-Time Functions with Application to DTI Tract Evolution in Early Neurodevelopment,” In Proceedings of IEEE ISBI 2012, pp. 1397--1400. 2012.
* Corouge, I., Fletcher, P.T., Joshi, S., Gouttard, S., Gerig, G., 2006. Fiber tract-oriented statistics for quantitative diffusion tensor MRI analysis. Med Image Anal, pp. 786-798.
* Goodlett, C.B., Fletcher, P.T., Gilmore, J.H., Gerig, G., 2009. Group analysis of DTI fiber tract statistics with application to neurodevelopment. Neuroimage, pp. S133-142.

File:Genu distributions along time.png

2014-10-15T21:06:08Z

Anuja:

File:DWIalongTime.png

2014-10-15T21:05:53Z

Anuja:

Projects:TractLongitudinalDTI

2014-10-15T21:05:22Z

Anuja: /* Regression of probability distributions obtained along white matter tracts to create spatio-temporal growth models */

Back to [[Algorithm:Utah2|Utah 2 Algorithms]]
__NOTOC__

== Tract-based longitudinal modeling of DTI data ==

This project develops a methodology to explore subject-specific, DTI data obtained from brain's white matter tracts, available at multiple but often sparsely present timepoints. The challenge is to develop a continuous spatio-temporal growth model, given discrete 4D DWI images. This would enable comparison of growth trajectories across subjects and along tracts which are biologically of interest in developmental and pathological changes.

== Background ==

We use the arc length parametrization scheme initially proposed by Corouge et al. It represents white matter fiber tracts obtained via streamline tractography in the brain's atlas tensor image as a function of arc length. (Atlas construction uses unbiased atlas building schemes followed by back transformations to subjects' DTI images to obtain identical fiber tract geometry across subjects, populated with subject specific diffusion data). We use the mean diffusion scalar invariants derived from these individual fiber bundle cross sections, as our input longitudinal diffusion profiles.

{| border="0" style="background:transparent;"
|[[Image:Tract-stats.png|thumb|700px|White matter diffusion properties along fiber tract: Left: A fiber tract with an origin plane defined for arc length parametrization, Right: FA mean and standard deviation as function of arc-length. Dots mark location of coordinate origin.]]
|-
|}

== Subject-specific spatiotemporal continuous growth model ==

We propose the use of Verhulst-Pearl logistic equation to capture temporal changes, while using a non-parametric kernel along arc length to account for biologically motivated functional along-tract relationship in the diffusion data.

{| border="0" style="background:transparent;"
|[[Image:logistic_eq.png|thumb|600px|Logistic equation with P as the population value at a given time, r being the growth rate of the population and K being the carrying capacity (also the limiting asymptote value).]]
|[[Image:logistic_simulation_growthrate_r.png|thumb|400px|Effect of growth rate r on the logistic curve.]]
|[[Image:logistic_simulation_asymptote_K.png|thumb|400px|Effect of carrying capacity K on the logistic curve.]]
|-
|}

From the definition of the logistic function, the parameter r represents the growth rate of the diffusion invariant and the parameter K represents the asymptote value. The overall function shape intuitively follows the growth pattern we expect to see during brain maturation. The diffusion invariants start with an initial diffusion profile along tract, and have a non linear temporal growth trajectory showing maximum changes in early childhood and then slowing down or almost saturating at a certain age. (For instance, observed diffusion changes are much more in neonates than in an adult brain). The temporal trajectories may also differ along the tract's length giving localized changes. Since our method gives us continuous along-tract, growth trajectories all along time, we can compare subjects with respect to differences in diffusion profiles at birth, growth rates at any given age as well as the asymptote saturation values. This gives us important information to understand delayed or abnormal brain maturation by comparing a normative growth surface with an individual's or by comparing the model's parameters across subjects.

== Results ==

Below are some results using synthetic data. For more validation results and extension of the framework to jointly estimate individual subject trajectories together with a normative trajectory, refer to [http://ieeexplore.ieee.org/xpl/login.jsp?tp=&arnumber=6235829&url=http%3A%2F%2Fieeexplore.ieee.org%2Fxpls%2Fabs_all.jsp%3Farnumber%3D6235829 Sharma et al. (ISBI '12)].

{| border="0" style="background:transparent;"
|[[Image:namic_tract_longitudinal_main.png|thumb|400px|Longitudinal modeling of synthetic data. The data is generated from a template neonate FA mean curve and uses known logistic parameters to generate the three timepoint curves (red). Our method faithfully recovers the original curves (blue) while simultaneously creating a continuous longitudinal growth profile (green) without having any prior knowledge of the logistic parameters used for data generation. It also estimates an unbiased time=0 template curve (black) along with parameters r and K to avoid biasing the estimation with any assumed initialization at time=0.]]
|[[Image:namic_tract_longitudinal_3d.png|thumb|400px|Results on real data (mean FA curves for a subject at three timepoints:neonate, 1 year and 2 year).]]
|-
|}
{| border="0" style="background:transparent;"
|[[Image:namic_tract_longitudinal_p1p2only.png|thumb|400px|For the above real data (right), these are the final along-tract initial values and the final estimates for parameters-growth rate r(mentioned as p1 here) and asymptote K(mentioned as p2 here).]]
|[[Image:namic_tract_longitudinal_alphaonly.png|thumb|400px|The complete continuous growth grid as estimated for the real data above.]]

|-
|}

The below images show jointly estimated personalized trajectories for 15 control subjects along with the average growth trajectory. The estimated model parameters for 15 subjects as well as the average trajectory are also shown. The normative trajectory is colored by the local standard deviation. It points to the fact that despite individual variability in the maturation process, there is a strong agreement in the asymptote FA values seen around a gestational age of 2.5 years indicating a relative stabilization of the white matter changes across subjects. The framework thus quantifies patient-specific changes in serial diffusion data given discrete-time diffusion curves.

{| border="0" style="background:transparent;"
|[[Image:CONTE_Final_3d.png|thumb|400px|The jointly estimated individual growth trajectories for 4 of the 15 control subjects.]]
|[[Image:CONTE_Norm_Params.png|thumb|400px|Estimated model parameters for 15 subjects together with the normative growth trajectory colored by the local variability and the estimated normative model parameters.]]

|-
|}

== Experiments with Huntington's data ==

During the NAMIC Winter project week, we worked on registering subjects with Huntington's disease in the same coordinate space as control subjects. We then applied the above framework [http://ieeexplore.ieee.org/xpl/login.jsp?tp=&arnumber=6235829&url=http%3A%2F%2Fieeexplore.ieee.org%2Fxpls%2Fabs_all.jsp%3Farnumber%3D6235829 Sharma et al. ISBI'12] to quantify the differences in normal expected aging versus the accelerated white matter changes expected in HD. Details of the data pre-processing steps and image registration are available on the [http://www.na-mic.org/Wiki/index.php/2012_Winter_Project_Week:_DTI_Change_Modeling Winter AHM page]. Some results are summarized here for FA value along the genu tract.

Two subjects- one being a HD patient (10027) with a high burden factor (higher factor value (of factor 12) implying that the subjects is closer to the onset) and the other being a control subject (10004) are chosen for concept demonstration. Both have a similar baseline scan age (42 and 43 years respectively) as well as a similar time separation between follow up scans. This allows an intuitive comparison of the estimated growth trajectories.

{| border="0" style="background:transparent;"
|[[Image:ORIG_hd_cn.png|thumb|300px|center|FA curves corresponding to the genu tract (middle clipped portions) for one HD case (10027) and one control case (10004). Visual inspection clearly shows the huge decrease relative to the Control in the HD subject. On the other hand, the control shows normal expected decrease due to aging. Red-timepoint 1, Green-timepoint 2, Blue-timepoint 3. The left is HD case and the right plot is Control.]]
|[[Image:HD_vs_CN_3d.png|thumb|300px|center|The red trajectory corresponds the the HD subjects while the blue is a control subject with healthy aging. HD patient clearly shows a much sharper FA decrease along time than the Control.]]
|[[Image:HD_vs_CN_params.png|thumb|300px|center|The estimated model parameters (Red-HD case, Blue-Control case) for the two subjects. The top plot is the rate of change per unit time, per unit FA value. The middle plot is the function asymptote and the last plot is the common baseline curve estimation for providing a common reference frame for model estimation and comparison. The asymptote clearly shows a much lower FA range for HD when compared to the Control which already shows promising early prediction capabilities for HD cases where white matter decay is much faster than healthy patients. Also, a higher per unit rate of change quantifies the sharper FA decrease compared to control.]]
|-
|}

== Experiments with Krabbe's data ==

We have also applied the framework to study white matter changes in infants with Krabbe's disease. Here we show some preliminary results for the FA tract profile along the genu tract for a single subject with Krabbe's disease. In theory, FA non-linearly increases with time along genu owing to a healthy brain maturation process. However, in Krabbe's case, the FA does not follow a monotonic change pattern.

{| border="0" style="background:transparent;"
|[[Image:Genu_fa_3d_conte_vs_krabbe.png|thumb|400px|center|FA curves corresponding to the genu tract for the Krabbe's patient are shown (Neonate:black, 6months:cyan, 1year:magenta). The other FA profiles are for the 15 healthy control subjects as discussed previously (neonate:red, 1year:green, 2year:blue). This also highlights the fact that the framework does not assume uniformity or correspondence in the scan timings across subjects.]]
|[[Image:Genu_fa_3dnorm_conte_vs_krabbe.png|thumb|400px|center|The black trajectory is the normative FA growth estimated using 15 controls as discussed previously. The three FA curves for the Krabbe case are also plotted and clearly show how the patient is lagging behind on expected brain maturation.]]
|-
|}

== Future work ==

As we see in the Krabbe experiment, the model assumes a monotonic change along time. Neighboring locations along the tract can have different, yet correlated time changes. For eg. we can see localized increases or decreases along the tract length with respect to the time evolution. However, owing to the definition of the logistic function, any non-monotonic behavior along time at any given tract location is not modeled accurately. Therefore, in cases like the Krabbe's subject, we need a more relaxed and flexible semi-parametric or non-parametric model to remove the constraints coming in from the parametric nature of the current model. This is currently work under progress.
To utilize the quantification of growth trajectories enabled by this method, we also require a systematic hypothesis testing scheme to draw conclusive results of individuals differing from normative patterns or populations differing in their evolution behaviors. This too is currently being worked upon.

= References =

* Sharma, A., Durrleman, S. , Gilmore, J.H. , Gerig, G. Longitudinal growth modeling of discrete-time functions with application to DTI tract evolution in early neurodevelopment. Proc. of 9th IEEE International Symposium on Biomedical Imaging (ISBI May'12), p.1397-1400.
* Corouge, I., Fletcher, P.T., Joshi, S., Gouttard, S., Gerig, G., 2006. Fiber tract-oriented statistics for quantitative diffusion tensor MRI analysis. Med Image Anal, pp. 786-798.
* Goodlett, C.B., Fletcher, P.T., Gilmore, J.H., Gerig, G., 2009. Group analysis of DTI fiber tract statistics with application to neurodevelopment. Neuroimage, pp. S133-142.

==Key Investigators==
* Utah: Anuja Sharma, Stanley Durrleman, Guido Gerig
* INRIA/ICM, Pitie Salpetriere Hospital: Stanley Durrleman
[http://www.na-mic.org/Wiki/index.php/Algorithm:Utah2 Back to Utah 2 Algorithm Core]

== Regression of probability distributions obtained along white matter tracts to create spatio-temporal growth models ==

Temporal modeling frameworks often operate on scalar variables by
summarizing data at initial stages as statistical summaries of the underlying
distributions. For instance, DTI analysis often employs
summary statistics, like mean, for regions of interest and properties
along fiber tracts for population studies and hypothesis testing.
This reduction via discarding of variability information may introduce
significant errors which propagate through the procedures. Moreover, the data variability information is lost at
all the intermediate interpolated times.

{| border="0" style="background:transparent;"
|[[Image:DistributionRegression_Synthetic.png|thumb|800px|center|Left:Two synthetically generated distributions along time
(blue:t0, orange:t1) with translated locations and opposite skews.
Right: Simple linear interpolation using classic single-valued summary
statistics (means: red, medians: blue) and ignoring data variability
(shown as error bars) at t0 and t1. The decision to choose mean versus median can heavily influence the final regression estimates. Additionally, once the inherent variability is discarded for an early decision regarding a summary statistics, it cannot be recovered for the interpolated time points.]]
|-
|}

We propose a novel framework which uses distribution-valued variables
to retain and utilize the local variability information. We propose two different regression approaches to achieve our goals. The first is a parametric regression scheme where the expected temporal trend is linear (Sharma ISBI 2014). The second is a non-parametric method where minimal assumptions are involved in terms of the growth trajectory evolution (Sharma ISBI 2013).

Our driving application is the modeling of age-related changes along DTI
white matter tracts. Results are shown for the spatiotemporal population
trajectory of genu tract estimated from a group of healthy infants and compared with an infant with Krabbe's disease.

{| border="0" style="background:transparent;"
|[[Image:DWIalongTime.png|thumb|600px|center|Model age related DTI changes in early neurodevelopment.]]
|[[Image:genu_distributions_along_time.png|thumb|600px|center|Distributions of DTI-derived scalar diffusion parameters like FA are obtained along the length of the genu tract. The distributions correspond to the cross-sections of the 3D tract geometry and are a function of the arc-length along the tract's total length as we move from one end of the tract to the other. This provides a functional curve which is parametrized by a feature of the local tract geometry (arc-length along the tract in this case) and is attributed by probability distributions (instead of scalar values like the 'mean' FA obtained from these local distributions).]]
|-
|}

=== Parametric linear regression for distribution-valued data ===

Classic linear regression is adapted to employ distribution-valued variables for model estimation.

=== Non-parametric regression for distribution-valued data ===

The concept of ’distance’
between distributions and an ’average’ of distributions is employed.
The framework quantifies growth trajectories for individuals
and populations in terms of the complete data variability estimated
along time and space. Concept is demonstrated in the context of our
driving application which is modeling of age-related changes along
white matter tracts in early neurodevelopment. Results are shown
for a single subject with Krabbe’s disease in comparison with a normative
trend estimated from 15 healthy controls.

= References =

* A. Sharma, P.T. Fletcher, J.H. Gilmore, M.L. Escolar, A. Gupta, M. Styner, G. Gerig. “ Parametric Regression Scheme For Distributions: Analysis of DTI Fiber Tract Diffusion Changes In Early Brain Development,” In Proceedings of the 2014 IEEE 11th International Symposium on Biomedical Imaging (ISBI), pp. 559-562. 2014.
* A. Sharma, P.T. Fletcher, J.H. Gilmore, M.L. Escolar, A. Gupta, M. Styner, G. Gerig. “Spatiotemporal Modeling of Discrete-Time Distribution-Valued Data Applied to DTI Tract Evolution in Infant Neurodevelopment,” In Proceedings of the 2013 IEEE 10th International Symposium on Biomedical Imaging (ISBI), pp. 684--687. 2013.
* A. Sharma, S. Durrleman, J.H. Gilmore, G. Gerig. “Longitudinal Growth Modeling of Discrete-Time Functions with Application to DTI Tract Evolution in Early Neurodevelopment,” In Proceedings of IEEE ISBI 2012, pp. 1397--1400. 2012.
* Corouge, I., Fletcher, P.T., Joshi, S., Gouttard, S., Gerig, G., 2006. Fiber tract-oriented statistics for quantitative diffusion tensor MRI analysis. Med Image Anal, pp. 786-798.
* Goodlett, C.B., Fletcher, P.T., Gilmore, J.H., Gerig, G., 2009. Group analysis of DTI fiber tract statistics with application to neurodevelopment. Neuroimage, pp. S133-142.

File:DistributionRegression Synthetic.png

2014-10-15T20:45:11Z

Anuja:

Projects:TractLongitudinalDTI

2014-10-15T20:44:48Z

Anuja:

Back to [[Algorithm:Utah2|Utah 2 Algorithms]]
__NOTOC__

== Tract-based longitudinal modeling of DTI data ==

This project develops a methodology to explore subject-specific, DTI data obtained from brain's white matter tracts, available at multiple but often sparsely present timepoints. The challenge is to develop a continuous spatio-temporal growth model, given discrete 4D DWI images. This would enable comparison of growth trajectories across subjects and along tracts which are biologically of interest in developmental and pathological changes.

== Background ==

We use the arc length parametrization scheme initially proposed by Corouge et al. It represents white matter fiber tracts obtained via streamline tractography in the brain's atlas tensor image as a function of arc length. (Atlas construction uses unbiased atlas building schemes followed by back transformations to subjects' DTI images to obtain identical fiber tract geometry across subjects, populated with subject specific diffusion data). We use the mean diffusion scalar invariants derived from these individual fiber bundle cross sections, as our input longitudinal diffusion profiles.

{| border="0" style="background:transparent;"
|[[Image:Tract-stats.png|thumb|700px|White matter diffusion properties along fiber tract: Left: A fiber tract with an origin plane defined for arc length parametrization, Right: FA mean and standard deviation as function of arc-length. Dots mark location of coordinate origin.]]
|-
|}

== Subject-specific spatiotemporal continuous growth model ==

We propose the use of Verhulst-Pearl logistic equation to capture temporal changes, while using a non-parametric kernel along arc length to account for biologically motivated functional along-tract relationship in the diffusion data.

{| border="0" style="background:transparent;"
|[[Image:logistic_eq.png|thumb|600px|Logistic equation with P as the population value at a given time, r being the growth rate of the population and K being the carrying capacity (also the limiting asymptote value).]]
|[[Image:logistic_simulation_growthrate_r.png|thumb|400px|Effect of growth rate r on the logistic curve.]]
|[[Image:logistic_simulation_asymptote_K.png|thumb|400px|Effect of carrying capacity K on the logistic curve.]]
|-
|}

From the definition of the logistic function, the parameter r represents the growth rate of the diffusion invariant and the parameter K represents the asymptote value. The overall function shape intuitively follows the growth pattern we expect to see during brain maturation. The diffusion invariants start with an initial diffusion profile along tract, and have a non linear temporal growth trajectory showing maximum changes in early childhood and then slowing down or almost saturating at a certain age. (For instance, observed diffusion changes are much more in neonates than in an adult brain). The temporal trajectories may also differ along the tract's length giving localized changes. Since our method gives us continuous along-tract, growth trajectories all along time, we can compare subjects with respect to differences in diffusion profiles at birth, growth rates at any given age as well as the asymptote saturation values. This gives us important information to understand delayed or abnormal brain maturation by comparing a normative growth surface with an individual's or by comparing the model's parameters across subjects.

== Results ==

Below are some results using synthetic data. For more validation results and extension of the framework to jointly estimate individual subject trajectories together with a normative trajectory, refer to [http://ieeexplore.ieee.org/xpl/login.jsp?tp=&arnumber=6235829&url=http%3A%2F%2Fieeexplore.ieee.org%2Fxpls%2Fabs_all.jsp%3Farnumber%3D6235829 Sharma et al. (ISBI '12)].

{| border="0" style="background:transparent;"
|[[Image:namic_tract_longitudinal_main.png|thumb|400px|Longitudinal modeling of synthetic data. The data is generated from a template neonate FA mean curve and uses known logistic parameters to generate the three timepoint curves (red). Our method faithfully recovers the original curves (blue) while simultaneously creating a continuous longitudinal growth profile (green) without having any prior knowledge of the logistic parameters used for data generation. It also estimates an unbiased time=0 template curve (black) along with parameters r and K to avoid biasing the estimation with any assumed initialization at time=0.]]
|[[Image:namic_tract_longitudinal_3d.png|thumb|400px|Results on real data (mean FA curves for a subject at three timepoints:neonate, 1 year and 2 year).]]
|-
|}
{| border="0" style="background:transparent;"
|[[Image:namic_tract_longitudinal_p1p2only.png|thumb|400px|For the above real data (right), these are the final along-tract initial values and the final estimates for parameters-growth rate r(mentioned as p1 here) and asymptote K(mentioned as p2 here).]]
|[[Image:namic_tract_longitudinal_alphaonly.png|thumb|400px|The complete continuous growth grid as estimated for the real data above.]]

|-
|}

The below images show jointly estimated personalized trajectories for 15 control subjects along with the average growth trajectory. The estimated model parameters for 15 subjects as well as the average trajectory are also shown. The normative trajectory is colored by the local standard deviation. It points to the fact that despite individual variability in the maturation process, there is a strong agreement in the asymptote FA values seen around a gestational age of 2.5 years indicating a relative stabilization of the white matter changes across subjects. The framework thus quantifies patient-specific changes in serial diffusion data given discrete-time diffusion curves.

{| border="0" style="background:transparent;"
|[[Image:CONTE_Final_3d.png|thumb|400px|The jointly estimated individual growth trajectories for 4 of the 15 control subjects.]]
|[[Image:CONTE_Norm_Params.png|thumb|400px|Estimated model parameters for 15 subjects together with the normative growth trajectory colored by the local variability and the estimated normative model parameters.]]

|-
|}

== Experiments with Huntington's data ==

During the NAMIC Winter project week, we worked on registering subjects with Huntington's disease in the same coordinate space as control subjects. We then applied the above framework [http://ieeexplore.ieee.org/xpl/login.jsp?tp=&arnumber=6235829&url=http%3A%2F%2Fieeexplore.ieee.org%2Fxpls%2Fabs_all.jsp%3Farnumber%3D6235829 Sharma et al. ISBI'12] to quantify the differences in normal expected aging versus the accelerated white matter changes expected in HD. Details of the data pre-processing steps and image registration are available on the [http://www.na-mic.org/Wiki/index.php/2012_Winter_Project_Week:_DTI_Change_Modeling Winter AHM page]. Some results are summarized here for FA value along the genu tract.

Two subjects- one being a HD patient (10027) with a high burden factor (higher factor value (of factor 12) implying that the subjects is closer to the onset) and the other being a control subject (10004) are chosen for concept demonstration. Both have a similar baseline scan age (42 and 43 years respectively) as well as a similar time separation between follow up scans. This allows an intuitive comparison of the estimated growth trajectories.

{| border="0" style="background:transparent;"
|[[Image:ORIG_hd_cn.png|thumb|300px|center|FA curves corresponding to the genu tract (middle clipped portions) for one HD case (10027) and one control case (10004). Visual inspection clearly shows the huge decrease relative to the Control in the HD subject. On the other hand, the control shows normal expected decrease due to aging. Red-timepoint 1, Green-timepoint 2, Blue-timepoint 3. The left is HD case and the right plot is Control.]]
|[[Image:HD_vs_CN_3d.png|thumb|300px|center|The red trajectory corresponds the the HD subjects while the blue is a control subject with healthy aging. HD patient clearly shows a much sharper FA decrease along time than the Control.]]
|[[Image:HD_vs_CN_params.png|thumb|300px|center|The estimated model parameters (Red-HD case, Blue-Control case) for the two subjects. The top plot is the rate of change per unit time, per unit FA value. The middle plot is the function asymptote and the last plot is the common baseline curve estimation for providing a common reference frame for model estimation and comparison. The asymptote clearly shows a much lower FA range for HD when compared to the Control which already shows promising early prediction capabilities for HD cases where white matter decay is much faster than healthy patients. Also, a higher per unit rate of change quantifies the sharper FA decrease compared to control.]]
|-
|}

== Experiments with Krabbe's data ==

We have also applied the framework to study white matter changes in infants with Krabbe's disease. Here we show some preliminary results for the FA tract profile along the genu tract for a single subject with Krabbe's disease. In theory, FA non-linearly increases with time along genu owing to a healthy brain maturation process. However, in Krabbe's case, the FA does not follow a monotonic change pattern.

{| border="0" style="background:transparent;"
|[[Image:Genu_fa_3d_conte_vs_krabbe.png|thumb|400px|center|FA curves corresponding to the genu tract for the Krabbe's patient are shown (Neonate:black, 6months:cyan, 1year:magenta). The other FA profiles are for the 15 healthy control subjects as discussed previously (neonate:red, 1year:green, 2year:blue). This also highlights the fact that the framework does not assume uniformity or correspondence in the scan timings across subjects.]]
|[[Image:Genu_fa_3dnorm_conte_vs_krabbe.png|thumb|400px|center|The black trajectory is the normative FA growth estimated using 15 controls as discussed previously. The three FA curves for the Krabbe case are also plotted and clearly show how the patient is lagging behind on expected brain maturation.]]
|-
|}

== Future work ==

As we see in the Krabbe experiment, the model assumes a monotonic change along time. Neighboring locations along the tract can have different, yet correlated time changes. For eg. we can see localized increases or decreases along the tract length with respect to the time evolution. However, owing to the definition of the logistic function, any non-monotonic behavior along time at any given tract location is not modeled accurately. Therefore, in cases like the Krabbe's subject, we need a more relaxed and flexible semi-parametric or non-parametric model to remove the constraints coming in from the parametric nature of the current model. This is currently work under progress.
To utilize the quantification of growth trajectories enabled by this method, we also require a systematic hypothesis testing scheme to draw conclusive results of individuals differing from normative patterns or populations differing in their evolution behaviors. This too is currently being worked upon.

= References =

* Sharma, A., Durrleman, S. , Gilmore, J.H. , Gerig, G. Longitudinal growth modeling of discrete-time functions with application to DTI tract evolution in early neurodevelopment. Proc. of 9th IEEE International Symposium on Biomedical Imaging (ISBI May'12), p.1397-1400.
* Corouge, I., Fletcher, P.T., Joshi, S., Gouttard, S., Gerig, G., 2006. Fiber tract-oriented statistics for quantitative diffusion tensor MRI analysis. Med Image Anal, pp. 786-798.
* Goodlett, C.B., Fletcher, P.T., Gilmore, J.H., Gerig, G., 2009. Group analysis of DTI fiber tract statistics with application to neurodevelopment. Neuroimage, pp. S133-142.

==Key Investigators==
* Utah: Anuja Sharma, Stanley Durrleman, Guido Gerig
* INRIA/ICM, Pitie Salpetriere Hospital: Stanley Durrleman
[http://www.na-mic.org/Wiki/index.php/Algorithm:Utah2 Back to Utah 2 Algorithm Core]

== Regression of probability distributions obtained along white matter tracts to create spatio-temporal growth models ==

Temporal modeling frameworks often operate on scalar variables by
summarizing data at initial stages as statistical summaries of the underlying
distributions. For instance, DTI analysis often employs
summary statistics, like mean, for regions of interest and properties
along fiber tracts for population studies and hypothesis testing.
This reduction via discarding of variability information may introduce
significant errors which propagate through the procedures. Moreover, the data variability information is lost at
all the intermediate interpolated times.

{| border="0" style="background:transparent;"
|[[Image:DistributionRegression_Synthetic.png|thumb|800px|center|Left:Two synthetically generated distributions along time
(blue:t0, orange:t1) with translated locations and opposite skews.
Right: Simple linear interpolation using classic single-valued summary
statistics (means: red, medians: blue) and ignoring data variability
(shown as error bars) at t0 and t1. The decision to choose mean versus median can heavily influence the final regression estimates. Additionally, once the inherent variability is discarded for an early decision regarding a summary statistics, it cannot be recovered for the interpolated time points.]]
|-
|}

We
propose a novel framework which uses distribution-valued variables
to retain and utilize the local variability information. We propose two different regression approaches to achieve our goals. The first is a parametric regression scheme where the expected temporal trend is linear (Sharma ISBI 2014). The second is a non-parametric method where minimal assumptions are involved in terms of the growth trajectory evolution (Sharma ISBI 2013).

Our driving application is the modeling of age-related changes along DTI
white matter tracts. Results are shown for the spatiotemporal population
trajectory of genu tract estimated from 45 healthy infants and
compared with a Krabbe’s patient.

=== Parametric linear regression for distribution-valued data ===

Classic linear regression is adapted to employ distribution-valued variables for model estimation.

=== Non-parametric regression for distribution-valued data ===

The concept of ’distance’
between distributions and an ’average’ of distributions is employed.
The framework quantifies growth trajectories for individuals
and populations in terms of the complete data variability estimated
along time and space. Concept is demonstrated in the context of our
driving application which is modeling of age-related changes along
white matter tracts in early neurodevelopment. Results are shown
for a single subject with Krabbe’s disease in comparison with a normative
trend estimated from 15 healthy controls.

= References =

* A. Sharma, P.T. Fletcher, J.H. Gilmore, M.L. Escolar, A. Gupta, M. Styner, G. Gerig. “ Parametric Regression Scheme For Distributions: Analysis of DTI Fiber Tract Diffusion Changes In Early Brain Development,” In Proceedings of the 2014 IEEE 11th International Symposium on Biomedical Imaging (ISBI), pp. 559-562. 2014.
* A. Sharma, P.T. Fletcher, J.H. Gilmore, M.L. Escolar, A. Gupta, M. Styner, G. Gerig. “Spatiotemporal Modeling of Discrete-Time Distribution-Valued Data Applied to DTI Tract Evolution in Infant Neurodevelopment,” In Proceedings of the 2013 IEEE 10th International Symposium on Biomedical Imaging (ISBI), pp. 684--687. 2013.
* A. Sharma, S. Durrleman, J.H. Gilmore, G. Gerig. “Longitudinal Growth Modeling of Discrete-Time Functions with Application to DTI Tract Evolution in Early Neurodevelopment,” In Proceedings of IEEE ISBI 2012, pp. 1397--1400. 2012.
* Corouge, I., Fletcher, P.T., Joshi, S., Gouttard, S., Gerig, G., 2006. Fiber tract-oriented statistics for quantitative diffusion tensor MRI analysis. Med Image Anal, pp. 786-798.
* Goodlett, C.B., Fletcher, P.T., Gilmore, J.H., Gerig, G., 2009. Group analysis of DTI fiber tract statistics with application to neurodevelopment. Neuroimage, pp. S133-142.

Projects:TractLongitudinalDTI

2014-10-15T11:04:53Z

Anuja: /* Key Investigators */

File:DTIFiberTractStatistics.png

2014-10-13T09:13:08Z

Anuja: uploaded a new version of "File:DTIFiberTractStatistics.png": Reverted to version as of 18:39, 17 August 2010

File:DTIFiberTractStatistics.png

2014-10-13T09:12:19Z

Anuja: uploaded a new version of "File:DTIFiberTractStatistics.png"

Algorithm:Utah2

2014-10-13T08:52:03Z

Anuja: /* Tract-based longitudinal modeling of DTI data */

Back to [[Algorithm:Main|NA-MIC Algorithms]]
__NOTOC__
= Overview of Utah 2 Algorithms (PI: Guido Gerig) =

The Utah II group, guided by Guido Gerig and Marcel Prastawa, is interested in providing new methodology for analysis of DTI (including group analysis, fiber tract parametrization and quantification, longitudinal analysis), of image data including pathology (lesions (MS, lupus), tumor, trauma etc.), and methodology for registration and segmentation of serial/longitudinal image data. Methodology development will focus on subject-specific analysis in personalized medicine applications, and on providing normative data in the form of spatial altlases and descriptive information on geometric and appearance change in the presence of pathology.

= Utah 2 Projects =

{| cellpadding="10" style="text-align:left;"

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== [[Projects:LongitudinalShapeAnalysis|Analysis of Longitudinal Shape Variability]] ==

<font color="red">'''Ongoing (Updated 08/2012): '''</font> Statistical analysis of longitudinal imaging data is crucial for understanding normal anatomical development as well as disease progression. This fundamental task is challenging due to the difficulty in modeling longitudinal changes, such as growth, and comparing changes across different populations. In this project, we develop a new method for the statistical analysis of longitudinal shape variability.
[[Projects:LongitudinalShapeAnalysis|More...]]

''Fishbaugh, J., Prastawa, M., Durrleman, S., Piven, J., Gerig, G. Analysis of Longitudinal Shape Variability via Subject Specific Growth Modeling. In N. Ayache et al. (Eds.): MICCAI 2012, Part I, LNCS 7510, pp. 730--737. (2012)''

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== [[Projects:PathologyAnalysis|Analysis of Brain Images with Pathological Changes]] ==

<font color="red">'''Ongoing: '''</font> Quantification, analysis and display of brain pathology as observed in MRI is important for diagnosis, monitoring of disease progression, improved understanding of pathological processes and for studying treatment strategies.
We conduct research on developing novel methodologies that covers registration, segmentation, visualization of pathological structures
with the aim of quantifying changes due to lesions, bleeding, and deformations over time.
[[Projects:PathologyAnalysis|More...]]

''Bo Wang, Wei Liu, Marcel Prastawa, Andrei Irimia, Paul M. Vespa, John D. Van Horn, P. Thomas Fletcher, and Guido Gerig . 4D Active Cut: An Interactive Tool for Pathological Anatomy Modeling, In Biomedical Imaging (ISBI), 2014 IEEE 11th International Symposium on, pp. 529-532. IEEE, 2014.''

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== [[Projects:TractLongitudinalDTI|Tract-based longitudinal modeling of DTI data]] ==

This project develops a methodology to explore subject-specific, DTI data obtained from brain's white matter tracts, available at multiple but often sparsely present timepoints. The aim is to have a continuous representation of the longitudinal diffusion changes along tract definitions.
[[Projects:TractLongitudinalDTI|More...]]

''Sharma, A., Durrleman, S. , Gilmore, J.H. , Gerig, G. Longitudinal growth modeling of discrete-time functions with application to DTI tract evolution in early neurodevelopment. Proc. of 9th IEEE International Symposium on Biomedical Imaging (ISBI May'12), p.1397-1400.''

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== [[Projects:longitudinaldwi|Longitudinal analysis of DWI]] ==

This project develops methodology to analyze serial/longitudinal DWI data, e.g. as baseline and follow-up in trauma, serial follow-up scans as acquired in the Huntington PREDICT study, or subject-specific white matter maturation in early brain development (see picture). [[Projects:longitudinaldwi|More...]]

''Sharma, A., Durrleman, S. , Gilmore, J.H. , Gerig, G. Longitudinal growth modeling of discrete-time functions with application to DTI tract evolution in early neurodevelopment. Proc. of 9th IEEE International Symposium on Biomedical Imaging (ISBI May'12), p.1397-1400.''
''N. Sadeghi, M. Prastawa, J. H. Gilmore, W. Lin, and G. Gerig, "Spatio-Temporal Analysis of Early Brain Development," in Proceedings IEEE Asilomar Conference on Signals, Systems & Computers, Nov. 2010, in print''

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== [[Projects:dtistatisticsfibers|Quantitative Description of White Matter Fiber Tracts]] ==

As part of this project, we are processing DTI data and computing statistics along white matter fiber tracts. For this, we are building a command line tool which allows the user to study the behavior of water diffusion along the length of the tracts.
[[Projects:dtistatisticsfibers|More...]]

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== [[Projects:Utah2ShapeRegression|Smooth Growth Trajectories from Time Series Shape Data]] ==

<font color="red">'''Ongoing (Updated 08/16/12):'''</font> Clinical research is interested in the spatiotemporal analysis of changes of anatomical shapes and structures, which potentially leads to improved understanding of the rate of change, locality and growth trajectory of structures of interest. This project develops a new methodology for the generation of a continuous growth model generated from a sparse set of shapes.
[[Projects:Utah2ShapeRegression|More...]]

''Fishbaugh, J., Durrleman, S., Gerig, G. Estimation of Smooth Growth Trajectories with Controlled Acceleration from Time Series Shape Data. Proc. of Medical Image Computing and Computer Assisted Intervention (MICCAI '11). September 2011.''

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== [[Projects:UtahAtlasSegmentation|Atlas Based Brain Segmentation]] ==

<font color="red">'''Ongoing: '''</font> Automatic segmentation can be performed reliably using priors from brain atlases and an image generative model. We have developed a tool that provides an automatic segmentation pipeline in a modular framework. Input are arbitrary number of image channels and a normative statistical brain atlas representing the population.
[[Projects:UtahAtlasSegmentation|More...]]

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== [[Projects:DTIPopulationAnalysis|Group Analysis of DTI Fiber Tracts]] ==

Analysis of populations of diffusion images typically requires time-consuming manual segmentation of structures of interest to obtain correspondance for statistics. This project uses non-rigid registration of DTI images to produce a common coordinate system for hypothesis testing of diffusion properties. [[Projects:DTIPopulationAnalysis|More...]]

''Casey B. Goodlett, P. Thomas Fletcher, John H. Gilmore, Guido Gerig. Group Analysis of DTI Fiber Tract Statistics with Application to Neurodevelopment. NeuroImage 45 (1) Supp. 1, 2009. p. S133-S142.''

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== [[Projects:RegistrationEvaluation|Evaluation of Registration Algorithms]] ==

We are interested in comparing existing registration packages to determine how registration in Slicer3 can be improved. This work focuses on examining various packages researchers are currently using for registration and comparing results on a set of examples representative of common registration tasks.
[[Projects:RegistrationEvaluation|More...]]

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== [[Projects:RegistrationTestbed|A Framework for Registration Evaluation and Exploration]] ==
In response to the difficulties of image registration, we propose an environment where registration applications can be explored, tested, and compared.
[[Projects:RegistrationTestbed|More...]]

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== [[Projects:LesionSegmentation|Lesion Segmentation]] ==

Quantification, analysis and display of brain pathology such as white matter lesions as observed in MRI is important for diagnosis, monitoring of disease progression, improved understanding of pathological processes and for developing new therapies.
[[Projects:LesionSegmentation|More...]]

Marcel Prastawa and Guido Gerig. Automatic MS Lesion Segmentation by Outlier Detection and Information Theoretic Region Partitioning. 3D Segmentation in the Clinic: A Grand Challenge II Workshop at Medical Image Computing and Computer Assisted Intervention (MICCAI) 2008.

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== [[Projects:DTINoiseStatistics|Influence of Imaging Noise on DTI Statistics]] ==

Clinical acquisition of diffusion weighted images with high signal to noise ratio remains a challenge. The goal of this project is to understand the impact of MR noise on quantiative statistics of diffusion properties such as anisotropy measures, trace, etc. [[Projects:DTINoiseStatistics|More...]]

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== [[Projects:UtahTumorSimulation|Tumor Simulation for Validating Change Tracking Applications]] ==

Determining extent of pathology as it changes over time is an important clinical task.
However, there is a lack of a reliable, objective ground truth for evaluating automatic tracking methods. We have developed a simulation tool that can generate MR images with known
tumor and edema.
[[Projects:UtahTumorSimulation|More...]]

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