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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.


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.

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.

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).
Effect of growth rate r on the logistic curve.
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.


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 Sharma et al. (ISBI '12).

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.
Results on real data (mean FA curves for a subject at three timepoints:neonate, 1 year and 2 year).
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).
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.

The jointly estimated individual growth trajectories for 4 of the 15 control subjects.
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 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 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.

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.
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.
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.

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.
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.


  • 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

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.

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.

Model age related DTI changes in early neurodevelopment.
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.

Temporal evolution of the synthetic distributions between two timepoints as estimated by the proposed method.
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.
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).
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.

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.
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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).

Continuous spatiotemporal normative growth trajectory: Estimated mean FA (gray), Age range where control data is originally available (green, orange, purple).
Krabbe's subject's scans (solid lines): (14 days, 6 mo., 1 yr.) with respect to the estimated healthy trajectory.
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 days).


  • 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.