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Diffusion tensor imaging (DTI) provides a unique source of information about the underlying tissue structure of brain white matter in vivo including both the geometry of major fiber bundles as well as quantitative information about tissue properties represented by derived tensor measures. This paper presents a method for statistical comparison of fiber bundle diffusion properties between populations of diffusion tensor images. Unbiased diffeomorphic atlas building is used to compute a normalized coordinate system for populations of diffusion images. The diffeomorphic transformations between each subject and the atlas provide spatial normalization for the comparison of tract statistics. Diffusion properties, such as fractional anisotropy (FA) and tensor norm, along fiber tracts are modeled as multivariate functions of arc length. Hypothesis testing is performed non-parametrically using permutation testing based on the Hotelling T2 statistic. The linear discriminant embedded in the T2 metric provides an intuitive, localized interpretation of detected differences. The proposed methodology was tested on two clinical studies of neurodevelopment. In a study of one and two year old subjects, a significant increase in FA and a correlated decrease in Frobenius norm was found in several tracts. Significant differences in neonates were found in the splenium tract between controls and subjects with isolated mild ventriculomegaly (MVM) demonstrating the potential of this method for clinical studies.
Clinical neuroimaging studies increasingly rely on diffusion tensor imaging (DTI) for new insights into the tissue structure of brain white matter in vivo. Traditional structural magnetic resonance imaging (MRI) provides little contrast within white matter, which is displayed as a homogeneous volume without information about the underlying tissue orientation and microstructure. DTI, on the other hand, provides information about the axon bundles of brain white matter such as preferred orientation, myeli-nation, and density as reflected in measures of the diffusion tensor for each voxel (Basser and Pierpaoli, 1996). The diffusion tensor incorporates information about the preferred fiber orientation in the principal eigenvector as well as information about local tissue structure in measures of anisotropy and norm. This paper addresses the problem of normalizing geometric models of white matter bundles and making statistical inference about differences in diffusion properties.
Most approaches to group analysis in the clinical DTI literature have relied on voxel based analysis or manually drawn regions of interest (ROI). An overview of the differences between voxel based and ROI analysis in DTI population studies was described by Snook et al (2007). Voxel based analysis methods are characterized by alignment of images to a template followed by independent hypothesis tests per voxel which are smoothed and corrected for multiple comparisons. Voxelwise analysis has been applied in DTI studies including autism (Barnea-Goraly et al., 2004) and schizophrenia (Burns et al., 2003). The major challenge in voxel based analysis is the need for multiple comparison correction and smoothing which can make localization of changes challenging to interpret (Jones et al., 2005a). Other studies have used manually drawn ROIs for group comparison of DTI properties. Within the ROIs, diffusion properties such as FA or mean diffusivity (MD) are averaged to create a single statistic. Examples of studies using ROI methods can be found in normal development (Bonekamp et al., 2007; Hermoye et al., 2006), schizophrenia (Kubicki et al., 2005), and Krabbe disease (Guo et al., 2001). The major drawback of ROI analysis is the time consuming nature of manual identification of regions, especially the ability to identify the long curved structures common in DTI fiber tracts. Our method improves on previous methods in the ability to perform automatic processing through the use of high dimensional deformable registration as well as the ability to focus on testing specific hypotheses regarding tracts of interest using a novel method for joint analysis of multivariate tensor measures in a tract model.
Segmenting anatomically known fiber bundles remains an important challenge for DTI analysis. The most common approach, fiber tractography, integrates the field of tensor principal eigenvector to create streamlines which sample anatomical fiber bundles (Basser et al., 2000). Corouge et al. (2006), Jones et al. (2005b), Maddah et al. (2008), and Lin et al. (2006) proposed to analyze diffusion properties as a function sampled along arc length of fiber bundles. More recent work has focused on volumetric segmentation methods which also allow data within the tract to be reduced to a function of arc length (Fletcher et al., 2007; Melonakos et al., 2007). These methods emphasize the need to understand diffusion properties in the context of geometric models of fiber bundles.
The major challenge in implementing tract oriented statistics in population studies is finding a consistent spatial parametrization within and between populations. Defining anatomically equivalent ROIs to seed tractography for large population studies is time consuming, error prone, and often requires significant post-processing such as cleaning and clustering (Gilmore et al., 2007). Furthermore, even given tractography seeds for each image, the natural variability of brain size and shape prohibits a naturally consistent parametrization for arc length models of diffusion. To solve both the needs for tract segmentation in individual cases as well as shape normalization for fiber tracts, we apply a population based registration method. Jones et al. (2002) and Xu et al. (2003) described the advantages of spatial normalization for DTI population studies. Recent work has focused on the use of unbiased methods for mapping tensor images to a common coordinate system (Zhang et al., 2007; Peyrat et al., 2007). A reference atlas of fiber bundles visible in DTI was produced by Mori et al. (2005). Xu et al. (2008) highlighted the need for smooth invertible mappings in a registration framework. Other work on DTI atlas building has used the geometry information contained within tractography results rather than image registration to build a population model (O’Donnell and Westin, 2007). In our framework, atlas building for DTI creates a global spatial normalization which can be used to parametrize tract oriented measures across a population.
In work closely related to the proposed methodology, Yushkevich et al. (2008) propose a method for statistical analysis along the two-dimensional medial manifolds of fiber tracts for specific tracts of interest after unbiased group alignment. On the tract medial axis, permutation tests are applied to detect clusters of pointwise differences between MD of groups. Another approach proposed by Smith et al. (2006), tract-based spatial statistics, is a global approach for analysis of diffusion properties using non-linear registration to a template combined with a skeletonization of FA voxels. FA values are globally projected onto the skeleton followed by pointwise hypothesis tests on the skeleton. O’Donnell et al. (2007) used tracts obtained through clustering and performs pointwise statistics along tract-oriented functions. Our method differs from these primarily in the use of statistical analysis that incorporate multivariate tensor measure and tract-oriented statistics for a single hypothesis test per tract.
This paper presents a method for group comparison of DTI that combines a method for high-dimensional diffeomorphic registration with a statistical framework for detecting and understanding differences between the diffusion properties of fiber tracts. The emphasis of the DTI atlas building procedure is to model and normalize the geometry of fiber bundles to analyze differences of diffusion properties between groups. A schematic overview of the procedure is shown in Fig. 1. Atlas building is performed based on a feature which is sensitive to the medial location of brain white matter. The diffusion properties of fiber bundles are modeled as continuous functions of arc length, where the tract functions are multivariate functions which map arc length to orthogonal tensor invariants. Statistics appropriate to populations of continuous functions is applied for hypothesis testing and discrimination. The proposed methodology is applicable to general DTI population comparisons. For this paper, the methodology was evaluated on a large pediatric study of normal development and comparison of neonate controls to MVMs who are at higher risk for mental illness.
Comparison of diffusion properties in populations of diffusion tensor images requires a method for identifying corresponding regions of anatomy. We have extended a high-dimensional, unbiased registration procedure developed by Joshi et al. (2004) using a feature image that is sensitive to the geometry of brain white matter and is similar in spirit to methods proposed in the literature for modeling white matter by its medial sheet (Smith et al., 2006; Kindlmann et al., 2007). The goal of the atlas building procedure is to provide spatial normalization for analysis of diffusion values at corresponding locations. Further reference on the DTI atlas building procedure is described by Goodlett et al. (2006).
Diffusion tensors are estimated for each subject from the diffusion weighted images using weighted least squares tensor estimation (Salvador et al., 2005). Skull stripping is performed by applying a segmentation tool on the non-diffusion weighted image to create a mask of non-brain regions. After skull stripping, a feature image is computed for each subject. Given a tensor image I and the corresponding FA image FA, the feature image C is defined as the maximum eigenvalue of the Hessian of the FA image. The Hessian is computed by convolution of the FA image with a set of Gaussian second derivatives with a fixed aperture. The σ value for the kernel is chosen empirically to be proportional to the size of white matter structures in the brain. For example, a smaller value is used for neonates than for adults. Figure 2 shows the FA image of a tensor field and the corresponding structural image C. We apply the atlas building procedure of Joshi et al. (2004) to the set of feature images. The transformation for each image is initialized by affine registration of the non-diffusion weighted images of each subject to a T2 weighted atlas using normalized mutual information. The result of the atlas building procedure is a set of invertible transformations which map each subject tensor image from native space to the atlas coordinate system.
We choose the feature image C over alternative image match metrics for two main reasons. First, we observe that C is a good detector of major fiber bundles which occur as tubular or sheet-like structures. Callosal fibers form a thin swept U; the corona radiata is a thin fan; the cingulum is a tubular bundle, and C serves as a strong feature detector for all types of these thin structures. Consequently. C optimizes correspondence of fiber tracts better than the non-diffusion weighted image, because C has the strongest response at the center of major fiber bundles, while the non-diffusion weighted image has the strongest signal in the cerebro-spinal fluid (CSF). Secondly, we use C instead of a full tensor metric or FA itself to avoid using the same feature for registration that will be used for statistical comparison.
After nonlinear transformations have been computed for each feature image, they are applied to the corresponding tensor images1. Methods appropriate for tensor processing are used to resample the tensor fields in the atlas space. Tensors are reoriented using the finite strain approximation proposed by Alexander et al. (2001). We have chosen to use the finite strain over the preservation of principal diffusion direction model as we are not modeling mechanical transformations of the anatomy (Peyrat et al., 2007). During re-sampling, tensors are interpolated using Riemannian methods first proposed by Pennec et al. (2006) and Fletcher and Joshi (2007). For efficiency we employ the Log–Euclidean approximation of the Riemannian metric on the space of diffusion tensors (Arsigny et al., 2006). After all images are transformed into the atlas space, the transformed images are averaged using the Log-Euclidean method to produce a tensor atlas.
The tensor atlas provides an image with improved signal-to-noise ratio (SNR) that is used to create template fiber tracts. The diffusion tensors obtained from averaging across the population can be integrated in streamline tractography approaches with significantly less outliers than in noisy individual images. For the purpose of this study we use a simple streamline integration method based on fourth order Runge-Kutta integration of the principal eigenvector field. Manual seeding and clustering of resulting tracts is used to input prior anatomical knowledge into the segmentation of fiber bundles. After creation of the template fiber tracts, diffusion statistics from the individual cases are mapped to the atlas tracts. When mapping the diffusion properties from native space to the template tracts, scalar invariants are interpolated from the native space using tri-linear interpolation. As a result a fiber bundle is created for each subject using the geometry of the template atlas tract but replacing the diffusion properties with those mapped from the subject. In this work four tracts were chosen by the authors to illustrate the methodology, but future clinical studies should rely on clinical hypotheses and expert neuroanatomical knowledge for tract definition. The set of individual tracts with corresponding geometry but varying diffusion properties are then compared in a novel statistical framework.
After spatial normalization of tensor images, corresponding values of diffusion properties within a fiber tract can be compared. In previous work this has been accomplished primarily through voxel based tests that require sophisticated smoothing and multiple comparison correction. While this type of analysis is effective for hypothesis generation, the results are often challenging to interpret, and strong differences are necessary to overcome multiple comparison correction. We propose to use a semi-parametric b-spline model of diffusion statistics along fiber bundles as the basis for group analysis (Goodlett et al., 2008).
The diffusion properties of fiber tracts are modeled as smooth functions of arc length. In this model we reduce the diffusion data in a fiber bundle to a function of arc length for each tensor measure of interest. As illustrated in Fig. 3, tensor measures in a bundle are averaged at each cross-section along the bundle to produce a function of arc length. In our framework template fiber bundles are computed in the tensor atlas using tractography, and the improved signal-to-noise ratio of the atlas allows reliable extraction of fiber bundles. The template fiber is warped back into the individual subject images to collect the diffusion data as shown in Fig. 1. Figure 6 shows an example of the template genu and splenium fiber bundles from the study presented later in section 4.1. and the individual functions produced for this bundle. Because the geometry of the individual fiber bundles are identical in atlas space, the data from each subject is parametrized consistently.
We have chosen to focus our analysis of diffusion properties on measures of tensor shape for two primary reasons. First, tensor orientation is an unstable measurement due to approximations of tensor reorientation during deformation. More importantly, however, rotationally invariant measures such as anisotropy and MD have been linked to changes in tissue properties (Bihan, 2003). The most commonly used measurements of tensor shape are FA and MD. However, it has been shown by Ennis and Kindlmann (2006) that FA and MD are not orthogonal and therefore not appropriate for joint statistical analysis. The non-orthogonality implies that differences in FA have different meanings depending on the magnitude of the MD. In this framework we have chosen to use FA because of its common usage in the literature. As a measure orthogonal to FA, the Frobenius norm of the tensor D is used as a measure of tensor size and is defined as
The mean diffusivity of the tensor, by comparison, is given by the sum of the eigenvalues rather than the sum of squared eigenvalues.
The population of multivariate functions produced by the fiber tract model requires a new method for statistical inference. Image sampling as well as the fiber tract extraction process create a sampled representation of the fiber bundle diffusion properties. However, there exists a continuous underlying biology which generates these samples. Therefore, statistical analysis of the sampled diffusion functions must account for the underlying continuity and spatial correlation of the samples. We compute statistics of the diffusion curves as an infinite dimensional extension to multivariate statistics known as functional data analysis (Ramsay and Silverman, 2005). The simplest extensions of ordinary statistics to the functional setting is the sample mean function (t), parametrized by arc-length t for N samples, given by
and the sample variance-covariance function, which is the bivariate function parametrized by two arc-length functions s,t,
The diagonal of the function, v(t, t), is the pointwise variance of the population of functions. Hypothesis testing and discriminant analysis of the space of functions has an inherent high-dimension, low-sample-size problem because of the infinite-dimensional space of continuous functions. Regularization methods are, therefore, essential in the computation of functional statistics. To enforce regularity, B-spline fitting and functional principal components analysis (PCA) is used for data-driven smoothing, where the number of retained PGA modes acts as a smoothing parameter.
In order to make computations tractable, smooth basis functions are fit to the sampled diffusion curves. B-splines were selected as basis functions due to the nonperiodic nature of the data, the compact support of the B-spline basis, and the ability to enforce derivative continuity. A large number of B-spline bases are first fit to the sampled functions using a least squares approach. The number of basis functions is chosen empirically to maintain local features while providing some smoothing. Computation of the mean function is computed by the sample mean of the B-spline coefficients. Computation of the variance-covariance function is more complex and requires accounting for the mapping between basis coefficients and function values. Let fi(t) be the B-spline function fit to the samples from subject i. Following the notation of Ramsay and Silverman (2005), in matrix notation, we express all functions fi(t) as a matrix of coefficients C times the basis functions , so that
Assuming the functions are centered about the sample mean, the variance-covariance function of f(t) is
Note that the vector valued function f(t) gives a column vector for each value t, therefore taking the variance-covariance with respect to the arc length parameters s, t is transposed from the usual notation. PCA of the functions fi(t) decomposes v(s, t) into the orthogonal unit eigenfunctions ξ(t) which satisfy
Equation (6) can be solved numerically by rewriting in terms of the basis functions with coefficients b,
Let W be a matrix of basis function inner products with entries
equation (7) can be simplified in the matrix form,
The B-Spline basis is not orthonormal resulting in a non-symmetric eigenvalue problem to solve (9). This minimization can be solved by the symmetric eigenvalue problem for the basis coefficients b, with the change of variable W−1/2u = b:
In our analysis we consider a joint analysis of FA and Frobenius norm functions with basis coefficients C1 and C2 respectively. We therefore compute PCA from the eigenanalysis of Σ, where
Hypothesis testing and discriminant analysis is performed on the projection of the data onto the first K PCA modes, where K serves as a smoothing parameter. The PCA projection accounts for the different scaled of the FA and Frobenius norm value enabling a joint analysis despite the differences in scale of the two values. An example of the PCA modes for the genu tracts from one and two year old subjects is shown in Fig. 4. In this work, K is chosen to maintain 90% of the variability of the variance-covariance matrix. Let Xi and yi be the projection of the curves from two populations of functions onto the PCA space. In this space the basis mapping has already been incorporated and standard multivariate analysis can be applied. The normal parametric hypothesis test for mean differences is the Hotelling T2 statistic,
where S is the pooled covariance matrix. In order to relax the normality assumptions associated with the parametric test, we apply a permutation test based on the T2 statistic to compute p-values (Nichols and Holmes, 2002). The permutation test is run on the global multivariate T2 and does not require multiple comparison correction.
The T2 statistic is proportional to the difference between group means projected onto the subspace given by the Fisher linear discriminant (FLD),
The linear discriminant, therefore, provides a direction for interpreting the detected group differences of the hypothesis test. The coefficients of the discriminant can be expanded into the original function basis so that FLD(t) = (t)ω is a function whose inner product with the original data provides maximal separation between the groups. This function, FLD(t), describes changes of tensor parameters between the two groups as a function of arc length.
The registration framework of section 2 along with the analysis method of section 3 was applied to two clinical studies of neurodevelopment. We have evaluated the effect of normal development on fiber tract properties from age one to two in cross-sectional populations. We have also performed hypothesis tests for differences between neonate control subjects and neonates with isolated mild ventriculomegaly (MVM).
The proposed methodology was evaluated on a cross-sectional study of normal development including subjects at one and two years of age. Institutional review board (IRB) approval was obtained for the study, and informed consent was obtained for all subjects. This study was chosen as a test case because of the expected large differences in diffusion properties due to development. In this case the p-value of the hypothesis testing framework of section 3 is less relevant as large changes are expected, but the discriminant direction provides localized information about major changes in diffusion properties across age.
Subjects were imaged on a Siemens 3T Allegra scanner using a DTI protocol with 10 repetitions of a non-diffusion weighted image plus 6 diffusion weighted gradient directions using a b-value of 1000s/mm2 and a voxel size of 2x2x2mm3. After image acquisition each repetition of the sequence was corrected for head motion by registration to the non-diffusion weighted images using the DTIchecker software2. The atlas building procedure of section 2 was applied to a database of 49 healthy controls including 22 one year old subjects and 27 two year old subjects. The transformations were initialized to a template T2 atlas appropriate for pediatric images. The feature image for atlas building was computed with a Gaussian kernel width of σ = 2.0mm, and atlas building was performed in a multi-resolution framework.
After registration and averaging, the atlas tensor image was used to identify four tracts of interest: genu, splenium, left cortico-spinal tract, and right cortico-spinal tract. A streamlime tractography algorithm using Runge-Kutta integration of the principal eigenvector field was used to extract the fiber tracts 3. Fibers were tracked from manually drawn seed regions in the atlas image and constrained to pass through a manually drawn target region. As mentioned in section 2, the atlas image provides improved SNR which allows lower FA thresholds than typically used for processing of single images. Figure 5 shows a comparison of the genu tract for one subject mapped from the atlas and the corresponding tract generated by tractography in the native space. In the entire population, the mapped tracts and individually computed tracts for the genu were mapped to binary voxel images. Dice coefficients were computed for each individual. The median dice coefficient for the cohort was .69 which is a reasonable overlap given the instability of streamline tractography methods. Previous work on comparing histology to fiber tractography has shown the difficulty in using dice coefficients and their dependence on the various parameters for tractography (Dauguet et al., 2008).
After generation of the atlas template tracts, data from the individual subject images was mapped to these tracts using the transformations created during atlas building. Tract oriented functions were computed for each subject using an origin defined in the atlas. Thirty b-spline control points evenly spaced in the arc-length parameter t were used to fit each function, and the resulting functions for the genu, splenium, and the left and right cortico-spinal tracts are shown in Figs. 6 and and7.7. After evaluation of the variance-covariance matrix for each tract, the number of PGA modes for the tract was selected to maintain 90% of the variance resulting in between 6–10 PGA modes per tract. Permutation testing over the Hotelling T2 statistics was run for each tract with 100,000 permutations, and the FLD associated with the null permutation was computed for visualization. The resulting p-values as well as the maximum and average pointwise differences of diffusion measures between groups along the tract are summarized in Table 1. All the tracts indicate a general trend of increase in FA and a correlated decrease in Frobenius norm from one to two year old groups. Figure 8 shows a visualization of the discriminant function for the genu tract that indicates an increase in FA and a correlated decrease in Frobenius norm from the one year to two year old groups with the effect focused in the center of the tract and trailing off as the tract enters the grey matter regions of the cortex. In the cortico-spinal tracts there is some evidence of localized changes. Figure 9 shows the mean functions for the two groups and the discriminant direction. The discriminant indicates that FA increases from one to two years in regions of the tract inferior of the callosal fibers, while the FA decreases in regions at the callosal fibers and above. This localized change could indicate a possible increase in orientation complexity or crossing fibers during development.
Prenatal MVM is a condition characterized by enlargement of the lateral ventricles diagnosed by ultrasound and has been associated with increased risk of neuropsychatric disorders (Gilmore et al., 1998). Previous investigation of DTI quantities in MVM have found a significant decrease in FA from controls in manually identified regions of splenium as well as significant increase in MD in regions of the genu, splenium, and cortico-spinal tracts (Gilmore et al., 2008).
The atlas building method described in section 2 was applied to a database of 114 images including 85 controls, 13 MVMs, 12 offsprings of schizophrenics, and 4 offsprings of bi-polar. IRB approval was provided by the University of North Carolina School of Medicine, and informed consent was obtained from all subjects. Future work will study the different groups within this database, but for this study we focus on the comparison of MVMs to controls. Transformations were initialized to a neonate specific template with T2 weighting. The feature image was computed at σ = 1.5mm for each subject, and the atlas building procedure was applied in a multi-resolution framework. After diffeomorphic registration of each tensor image, an atlas tensor image was created by averaging the deformed images. In the atlas tensor image, tracts were computed for the genu, splenium. and left and right cortico-spinal tract as shown in Fig. 10.
Analysis of tracts was performed on the left and right cortico-spinal tracts, genu, and splenium. Statistically significant differences were found in the splenium tract but not the genu or cortico-spinal tracts. Figure 11 shows the discriminant direction for the splenium tract and indicates a decrease in FA and a correlated increase in Frobenius norm from control subjects to those with MVM. Results for all analyzed tracts are summarized in Table 2.
We have presented a method for making inferences about group differences in fiber tract diffusion properties. Our framework combines a method for spatial normalization of tensor images with a method for quantitative tract analysis. Within this framework we apply a novel method for joint analysis of tensor shape parameters using statistical inference of populations of multivariate continuous functions. The statistical framework provides a method for both hypothesis testing and localizing the differences determined by the hypothesis test.
There are several limitations to the proposed methodology. First, the atlas building methodology of section 2 assumes that the overall appearance of DTI images is sufficiently similar between the two groups that registration to a single coordinate system is feasible, as is common in brain mapping approaches. In studies where subjects with severe geometric distortions such as tumors are present, this approach is likely not feasible. Secondly, tracts which are small or inconsistent even among the same group will be challenging to identify in the atlas. For this reason, we have focused on large major fiber bundles where consistency is expected. Finally, the statistical analysis relies on tensor shape measures to make inference about potential changes in tissue structure. However, there are several other effects which could have an impact on the tensor shape besides tissue change. For example, varying degrees of partial voluming effects can cause differences in total diffusivity that do not necessarily reflect changes in axon density or myelination. This may be part of the underlying change found in the splenium of the MVM population due to partial voluming with the lateral ventricles. Further studies are necessary to investigate the underlying biological cause of detected differences in DTI measures.
In this work, we have focused on analysis of the diffusion tensor model rather than high angular resolution diffusion imaging (HARDI) models. This is done primarily because of the constraints of currently available clinical data. In the future we anticipate that some level of higher order modeling will be required to fully capture the geometry of anatomical fiber bundles and resolve fiber crossings. The statistical analysis presented in this paper can be readily extended to more complex information given appropriate measures of diffusion shape for more complex models. For example the generalized measures presented by Özarslan et al. (2005) could be used within the same framework. More advanced tractography methods incorporating HARDI data could also improve the generation of the template fiber tract.
In summary, we have presented a novel method for detecting and understanding differences between diffusion properties of DTI fiber bundles. The methodology is demonstrated on a clinical study of neurodevelopment. In this study changes between one and two year populations were assessed and localized in several key tracts and significant differences were found between neonate controls and MVMs.
This work is part of the National Alliance for Medical Image Computing (NA-MIC), funded by the National Institutes of Health through Grant U54 EB005149. Information on the National Centers for Biomedical Computing can be obtained from http://nihroadmap.nih.gov/bioinformatics. We also acknowledge support from the NIMH Silvio Conte Center for Neuroscience of Mental Disorders MH064065.
1Software for tensor estimation, and tensor resampling is available from http://www.sci.utah.edu/~gcasey/software
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