A Group Distributional ICA Method for Decomposing Multi-subject Diffusion Tensor Imaging

Summary:

Diffusion tensor imaging (DTI) is a frequently used imaging modality to investigate white matter fiber connections of human brain. DTI provides an important tool for characterizing human brain structural organization. Common goals in DTI analysis include dimension reduction, denoising, and extraction of underlying structure networks. Blind source separation methods are often used to achieve these goals for other imaging modalities. However, there has been very limited work for multi-subject DTI data. Due to the special characteristics of the 3D diffusion tensor measured in DTI, existing methods such as standard independent component analysis (ICA) cannot be directly applied. We propose a Group Distributional ICA (G-DICA) method to fill this gap. G-DICA represents a fundamentally new blind source separation method that separates the parameters in the distribution function of the observed imaging data as a mixture of independent source signals. Decomposing multi-subject DTI using G-DICA uncovers structural networks corresponding to several major white matter fiber bundles in the brain. Through simulation studies and real data applications, the proposed G-DICA method demonstrates superior performance and improved reproducibility compared to the existing method.

1. Introduction

The field of neuroimaging has seen significant advancements in recent years, providing researchers with a range of techniques for investigating human brain function and organization. Among these, diffusion tensor imaging (DTI) has become particularly significant for studying brain structure and connectivity (Alexander et al., 2007). DTI maps the structure of white fiber tracts in the brain by measuring the three-dimensional diffusion of water, offering a unique view into the brain’s structural networks. DTI-derived structural connections aid in understanding and validating functional networks identified through fMRI. By complementing fMRI data, DTI accelerates advancements in brain network analysis and contributes critical insights for developing biologically plausible brain network models. Recent neuroimaging studies, such as the Philadelphia Neurodevelopmental Cohort (PNC) study that inspired our work, collect both fMRI and DTI to comprehensively investigate brain organization. The PNC cohort includes youths aged 8–21 years who visited the Children’s Hospital of Philadelphia for pediatric care. Functional and diffusion MRI scans were acquired for a subset of participants, providing a valuable multi-modal dataset to study brain functional and structural organization during childhood and adolescence. While well-established methods exist for analyzing functional networks, extracting meaningful structural networks from multi-subject DTI datasets remains analytically challenging, underscoring the need for robust and efficient methodologies.

A key challenge in analyzing DTI data arises from its unique data representation. At each brain location, DTI is expressed as a 3D tensor matrix that characterizes water diffusion patterns, which in turn enables the inference of white matter fiber structures. Specifically, each tensor matrix represents a 3D ellipsoid, with the eigenvectors defining the directions of the principal axes and the eigenvalues determining the ellipsoidal radii. The diffusion tensor matrix characterizes both the shape and orientation of the diffusion ellipsoid at each voxel where the eigenvectors of the tensor matrix represent the diffusion directions and the eigenvalues are associated with the speed of diffusion in these directions. Tractography techniques can be applied to the local diffusion data to map white matter fiber tracts in the brain. Compared to functional MRI (fMRI) which reveals brain functional networks by studying functional coherence between brain regions, DTI maps structural networks by inferring white matter connections from water diffusion patterns. A key objective in analyzing imaging data from these modalities is to extract brain networks, which has been successfully achieved for fMRI using independent component analysis (ICA). ICA is a widely applied blind source separation method that decomposes observed multivariate data into statistically independent additive components. This widely applied technique has shown great potential in a wide range of scientific fields, such as biomedical imaging and machine learning (Hyvärinen and Oja, 2000; Bartlett et al., 2002; Beckmann et al., 2005). The ICA model is typically described as follows. Let Y be the observed multivariate data matrix with dimensions K × V , where K represents the number of mixed signals and V represents the length of each signal. The classical noise-free ICA method aims to decompose Y into a linear combination of latent independent sources as,

$$Y_{K\times V}=A_{K\times L}{S}_{L\times V},$$

where A represents the mixing matrix, S represents the source signals, and L is the number of independent sources, which is smaller than K. These L source signals are statistically independent and follow non-Gaussian distributions. ICA has proven to be a highly effective tool for dimension reduction, denoising and extraction of latent source signals (Hyvärinen and Oja, 2000). Various ICA algorithms have been proposed including the widely used traditional ICA algorithms such as Infomax (Bell and Sejnowski, 1995) and FastICA (Hyvarinen, 1999), and recently developed advanced methods such as maximum likelihood estimations (e.g., Hastie and Tibshirani (2003)), rank-based estimations (e.g., Hallin and Mehta (2015)), and estimations using deep learning techniques (e.g., Ngiam et al. (2010); Le et al. (2011)). The optimization process of ICA has also been advanced in various ways to enhance its performance, such as incorporating sample dependence (Adali et al., 2014) and signal sparsity (Wang et al., 2024).

While ICA algorithms have been extensively applied to fMRI analysis in both individual level and group level studies (McKeown et al., 1998; Calhoun et al., 2001; Beckmann and Smith, 2005; Guo, 2011; Shi and Guo, 2016), its application to multi-subject DTI data has been scarce. This is due to the unique characteristics of the 3D diffusion tensors measured in DTI, which limit the direct application of standard ICA algorithms to group-level analysis. In a few cases where ICA was applied to DTI data (Li et al., 2012; Ouyang et al., 2015) or joint analysis of DTI and fMRI data (Sui et al., 2011), researchers have only considered decomposing multi-subject fractional anisotropy (FA), which is a scalar summary statistic defined based on the eigenvalues of diffusion tensor matrices. This FA-based ICA application has major limitations. The diffusion tensor matrix from DTI data characterizes both the shape and orientation of the diffusion ellipsoid at each voxel where the eigenvectors of the tensor matrix represent the diffusion directions and the eigenvalues are associated with the speed of diffusion in these directions. The FA is a scalar summary statistic derived from the differences among the eigenvalues of the diffusion tensor. While it captures the degree of anisotropy in the diffusion ellipsoid, FA does not utilize the full spectrum of eigenvalue information and, importantly, does not account for the eigenvectors. Consequently, the FA-based ICA overlooks crucial information in DTI data such as the orientation of the diffusion tensors.

Motivated by the novel distributional ICA (DICA) framework for single subject imaging analysis (Wu et al., 2022), we propose a group DICA (G-DICA) method that addresses the limitations of the existing ICA methods in analyzing multi-subject DTI data. Unlike traditional ICA methods, which perform ICA on the observed data, our G-DICA performs ICA on the distribution level by separating the parameters in the distribution function of the imaging data as independent source signals. Our method models multi-subject DTI data as a mixture of Wishart distributions, obtaining individual-level posterior weights that represent the probability loadings on each component. The weights are then registered to a common brain template, allowing for comparison across subjects. Finally, ICA is performed on the concatenated group posterior weights, uncovering population-level structural networks. In our method, the mixture of Wishart model in our approach considers information from both eigenvalues and eigenvectors of the diffusion tensors, enabling the analysis of both shape and orientation of the diffusion ellipsoids in the DTI data. Therefore, the proposed G-DICA method resolves the major difficulties in existing ICA analysis for decomposing multi-subject DTI data.

In this study, the G-DICA method was applied to decompose multi-subject DTI data from the Philadelphia Neurodevelopmental Cohort (PNC) study. The experimental results show that G-DICA effectively extracted independent components corresponding to structural networks aligned with major fiber bundles in the brain. Two simulation studies were conducted to compare the performance of G-DICA with an existing method, demonstrating that the proposed approach is more accurate and robust in recovering source signal regions. A comprehensive reliability and reproducibility analysis was conducted for G-DICA in comparison with individual DICA (Wu et al., 2022) and an FA-based group ICA. The results show that G-DICA consistently produced highly reliable outcomes across various factors, including parameter settings, initialization, and data resampling.

The paper is structured as follows: In Section 2, we present the methodological framework underlying the G-DICA approach. In Section 3, we demonstrate its practical application on real data from the PNC study, as well as a reliability and reproducibility analysis. The simulation studies are presented in Section 4. Finally, we conclude with a discussion in Section 5.

2. Methods

In this section, we present the G-DICA method for decomposing multi-subject DTI data to extract structural networks. A schematic plot of the G-DICA is illustrated in Figure 1. Let ${\mathit{Y}}_{iv}\in {𝒫𝒢}_3$ denotes the 3D diffusion tensor matrix for subject $i(i=1,\cdots ,N)$ at voxel $v\left(v=1,\cdots ,V_i\right)$, where $𝒫{𝒢}_3$ represents the space of 3 × 3 symmetric positive definite matrices, $V_i$ is the number of total voxels in the ith subject DTI image. ${\mathit{Y}}_{iv}$ characterizes the water diffusion pattern in the local white matter, which can be visualized as a 3D ellipsoid.

Figure 1:

Schematic plot for G-DICA

2.1. Step I: Distribution Modeling of the DTI tensor matrices

As Step I of G-DICA, we model the diffusion tensor ${\mathit{Y}}_{iv}$ using a mixture distribution of K Wishart components,

$$Y_{iv}\sim \sum _{k=1}^K{\pi }_k{𝓌}_3\left({\text{Θ}}_k\right),\text{for}v=1,\cdots ,V_i;i=1,\cdots ,N,$$ (1)

where ${𝓌}_3\left({\mathrm{\Theta }}_k\right)$ is the kth component in the mixture of 3-dimensional Wishart distributions and ${\pi }_k$ is the weight of the kth component in the mixture distribution. The parameters of the Wishart distribution are denoted as ${\mathrm{\Theta }}_k=\left({\mathrm{\Sigma }}_k,d_k\right)$, where ${\mathbf{\Sigma }}_k$ is the scale matrix which represents the expected tensor shape and $d_k$ is the degree of freedom. In accordance with the definition of the Wishart distribution, we have $d_k>2$ since the dimension of the diffusion tensors is three. The posterior probabilities for each of the K components based on the observed imaging data can be obtained as:

$$w_{ivk}=\frac{{\pi }_k{𝓌}_3\left({\mathit{Y}}_{iv};{\text{Θ}}_k\right)}{\sum _{K^{\prime }=1}^K{\pi }_{K^{\prime }}{𝓌}_3\left({\mathit{Y}}_{iv};{\text{Θ}}_{K^{\prime }}\right)},k=1,\cdots ,K,$$

where ${𝓌}_3\left({\mathit{Y}}_{iv};{\mathrm{\Theta }}_k\right)$ represents the probability density function of the 3-dimensional Wishart distribution evaluated at ${\mathit{Y}}_{iv}$ with parameter ${\mathrm{\Theta }}_k$ .

In the distribution model (1), the K Wishart component distributions capture the the 3D ellipsoids underlying the diffusion tensors measured across all brain locations. Given the Wishart component distributions, the probability ${\mathit{w}}_{iv}={\left(w_{iv1},\dots ,w_{ivK}\right)}^{\prime }$ provides a re-representation of the 3D diffusion tensor at voxel v and characterizes the 3D ellipsoid each local diffusion tensor is associated with. We fit the mixture of Wishart distribution model with the Expectation-Maximization (EM) algorithm, specifically utilizing the k-MLE method introduced by Nielsen (2012). This method can be viewed as a more efficient version of the EM algorithm with a hard membership clustering approach. The maximum likelihood estimates of each Wishart distribution are calculated using the techniques outlined in Saint-Jean and Nielsen (2014). The number of components in the mixture model is chosen based on the Bayesian information criteria (BIC).

2.2. Step II: Registration across subjects

The posterior probabilities from step I, ${\mathbf{w}}_{iv}$, at each voxel v are then transformed into a K – 1 dimensional vector ${\mathbf{m}}_{iv}$ using the mlogit function g,

$$m_{iv}=g\left(w_{iv}\right)={\left(\text{log}\left\{w_{iv1}/w_{ivK}\right\},\cdots ,\text{log}\left\{w_{ivK-1}/w_{ivK}\right\}\right)}^{\text{'}}.$$

This step maps the probabilities to the real line to facilitate the subsequent ICA decomposition. Across the brain of each subject i, we obtain a K – 1 channel logit map ${\mathbf{M}}_i=\left({\mathbf{m}}_{i1},\cdots ,{\mathbf{m}}_{i{V}_i}\right)$, a $(K-1)\times V_i$ matrix.

An important requirement of performing group analysis on the logit maps is that the logit posterior probability at each voxel v refers to the same spatial location in the brain across different subjects. To meet this requirement, we propose a procedure to register the logit maps ${\mathbf{M}}_i$ across subjects to a common template, which is the MNI152 standard space (Maintz and Viergever, 1998). Image registration is a critical preprocessing step that aligns images from different subjects into a standardized coordinate system, enabling group-level analyses. Various registration techniques exist, with affine transformation being the most commonly used. As a linear registration method, affine transformation adjusts for differences in brain size, shape, orientation, and position by applying scaling, translation, rotation, and shear transformations (Jenkinson and Smith, 2001). More advanced nonlinear registration techniques, such as diffeomorphic registration (Rueckert et al., 2006) and elastic transformations (Periaswamy and Farid, 2003), aim to capture finer anatomical variations by allowing localized deformations (Wang et al., 2017). However, these methods typically come with significantly higher computational costs and may introduce interpolation artifacts and numerical instability. In this paper, we employ affine transformation to map an original image ${\mathit{X}}_i$ in a subject’s native diffusion space to its corresponding image ${\mathit{X}}_i^{\mathrm{*}}$ in the MNI152 standard space. Affine transformation is chosen for its computational efficiency and effectiveness in aligning brain structures while preserving their geometric properties. The affine transformation comprises a linear transformation represented by a square matrix ${\mathbb{T}}_i$ and a translation component represented by a vector ${\mathbf{b}}_i$. ${\mathbb{T}}_i$ and ${\mathbf{b}}_i$ are estimated by maximizing the correlation ratio $\mathrm{V}\mathrm{a}\mathrm{r}\left[E\left({\mathit{X}}_i^{\mathrm{*}}\mid {\mathit{X}}_i\right)\right]/\mathrm{V}\mathrm{a}\mathrm{r}\left({\mathit{X}}_i^{\mathrm{*}}\right)$, ensuring optimal alignment between the subject’s image and the template.

In the registration, the coordinates of voxels in the standard space, ${\mathit{P}}_{i,X^{\mathrm{*}}}^{\mathrm{*}}=\left\{\left(x^{\mathrm{*}},y^{\mathrm{*}},z^{\mathrm{*}}\right)\right\}$, are linked to the coordinates of the diffusion space, ${\mathit{P}}_{i,X}=\left\{\right(x,y,z\left)\right\}$, by a linear transformation. Specifically, ${\mathit{P}}_{i,X^{\mathrm{*}}}^{{\mathrm{*}}^{\prime }}={\mathit{P}}_{i,X}^{\prime }{\mathbb{T}}_i+{\mathbf{b}}_i$, where ${\mathbb{T}}_i$ denotes a 3 × 3 transformation matrix that captures the affine transformation including rotation, shearing and scaling of the original image ${\mathit{X}}_i$, while ${\mathbf{b}}_i$ represents a 3-dimensional vector associated with the translation shift of ${\mathit{X}}_i$. To estimate these transformation parameters, we utilize FMRIB Software Library (FSL) (Jenkinson et al., 2012) to combine the transformation from a subject’s diffusion space to the subject’s structural space with the transformation from the subject’s structural space to the MNI152 standard space. We then apply the combined affine transformation to the original logit maps ${\mathbf{M}}_i$ to obtain the registered logit maps in the standard space, ${\mathbf{M}}_i^{\mathrm{*}}$. The mean(SD) computation time (in seconds) per subject is 20.5(2.3) for computing the affine transformation and 1.0(0.2) for applying the transformation. It is worth noting that both the transformation computation and application steps can be parallelized across subjects, ensuring computational efficiency. Each voxel in the registered map ${\mathbf{M}}_i^{\mathrm{*}}$ is derived from the original map ${\mathbf{M}}_i$ by transforming its coordinates as,

$$P_{i,M^{\text{*}}}^{{*}^{\prime }}={P^{\prime }}_{i,M}{\mathbb{T}}_i+b_i.$$

The transformation matrix ${\mathbb{T}}_i$ and vector ${\mathbf{b}}_i$ ensure that each voxel in the original logit maps is mapped accurately to its corresponding position in the standard space, thus creating the registered logit map ${\mathbf{M}}_i^{\mathrm{*}}$. Examples of logit maps before and after registration are presented in Web Figure 1 in Web Appendix A of Supplementary Materials. Finally, to facilitate the group analysis, a group white matter mask is generated and applied to the registered logit maps across subjects to obtain a common brain mask with a total of V voxels. The group mask is derived by thresholding the fractional anisotropy (FA) values to obtain common white matter regions across subjects (Karimi et al., 2023). Specifically, the FA maps from each subject are first registered to the MNI152 standard space using the affine transformation, and the group mask is generated by selecting only those voxels with average FA values exceeding a predefined threshold across all subjects. The FA threshold is usually set between 0.1-0.2 (Guevara et al., 2011) to select white matter regions. The subsequent group ICA analysis is then performed on the voxels within the group mask. This ensures the selected voxels reflect common and robust white matter regions across all subjects in the cohort.

2.3. Step III: Group ICA of the registered logit maps

At Step III, we apply Independent Component Analysis (ICA) to decompose the registered logit maps ${\mathbf{M}}^{\mathrm{*}}={\left({\mathbf{M}}_1^{{\mathrm{*}}^{\prime }},\dots ,{\mathbf{M}}_N^{{\mathrm{*}}^{\prime }}\right)}^{\prime }$, which takes the form of an $N(K-1)\times V$ matrix, into L independent latent source signals.

The group ICA model is expressed as:

$$M^{\text{*}}=AS,$$

where $\mathbf{A}={\left({\mathbf{A}}_1^{\prime },\dots ,{\mathbf{A}}_N^{\prime }\right)}^{\prime }\in {\mathbb{R}}^{N(K-1)\times L}$ contains the mixing coefficients. The matrix $\mathbf{S}={\left({\mathit{s}}_1,\dots ,{\mathit{s}}_L\right)}^{\prime }\in {\mathbb{R}}^{L\times V}$ consists of the L independent latent source signals, where each row represents a spatially independent component, and each column corresponds to a voxel. The decomposition is performed using the Infomax ICA algorithm (Bell and Sejnowski, 1995) on the concatenated logit map ${\mathbf{M}}^{\mathrm{*}}$. Each independent component $s_l(l=1,\dots ,L)$ represents a structural network comprising white matter regions with similar diffusion tensor properties.

Our ICA model assumes statistical independence in the spatial domain, a well-suited assumption for analyzing brain imaging data, where brain networks often exhibit sparse and spatially distributed patterns. This spatial ICA approach enables the extraction of shared brain structural networks, represented by S, from the logit maps across subjects. The mixing coefficient matrix, A, accounts for subject-specific variations, modulating how each subject’s logit map is constructed from the shared latent components. By leveraging the spatial independence assumption, we ensure that the extracted components are interpretable and correspond to biologically meaningful structures. The effectiveness of this method in detecting brain networks has been validated in prior research for both fMRI and DTI studies(Calhoun et al., 2001; Beckmann and Smith, 2005; Wu et al., 2022)

3. Real Data Application

3.1. Data Description

We applied the G-DICA model to the multi-subject DTI data obtained from the Philadelphia Neurodevelopmental Cohort (PNC) study. All images were acquired on a Siemens Tim Trio 3 Tesla scanner, and DTI data were generated based on diffusion weighted imaging (DWI) scans using a twice-refocused spin echo (TRSE) single-shot EPI sequence. The sequences consists of 64 diffusion-weighted directions with b = 1000s/mm² and 7 scans with b = 0s/mm². More details about image acquisition can be found in Satterthwaite et al. (2014).

The multi-subject DTI data from the PNC study were preprocessed using the standard pipeline in FSL. This involved removing non-brain regions through brain extraction, correcting phase reversal distortion, and aligning the diffusion-weighted images to the average non-diffusion weighted image through a rigid body affine transformation to remove motion artifacts and correct for eddy current distortions. The directional diffusion at each voxel was then estimated through a diffusion tensor model using the diffusion toolbox (FDT) of FSL. After quality control, DTI data from a total of 453 subjects were used in the G-DICA study. Step I of G-DICA modeled the diffusion tensors of the subjects using a mixture Wishart distribution. The logit maps obtained for each subject were registered to the MNI152 standard space, where the affine transformation matrices from subjects’ diffusion space to the MNI152 space were estimated using FSL. Latent structural networks were extracted via G-DICA decomposition of the registered logit maps across subjects.

3.2. Extracted structural networks

We applied a group white matter mask to our DTI images by thresholding the fractional anisotropy (FA) at 0.1. This threshold was chosen based on commonly used FA thresholds in DTI studies (Guevara et al., 2011), resulting in a white matter mask that retains most white matter regions. In Step I of the G-DICA analysis, we considered a mixture of K = 16 Wishart distributions and extracted L = 10 structural networks in Step III. Here, we provide practical recommendations on how to choose these parameters. For choosing K, since clustering analysis is used to initialize the EM algorithm in Step I, we recommend leveraging established techniques in clustering analysis for determining the number of clusters, such as the Silhouette score (Rousseeuw, 1987). For choosing L, we recommend using established approaches for selecting the number of components in ICA, such as the ICA-by-Blocks (Bouveresse et al., 2012). The details are presented in Web Appendix B of Supplementary Materials. To assess the impact of K and L on the results, we further conducted a sensitivity analysis in Section 3.3.2 by varying their values. The results demonstrate that the structural networks identified by G-DICA exhibited strong stability across a range of parameter specifications.

Among the extracted structural networks, several of them were associated with major white matter fiber pathways. The spatial distributions of the structural networks and the corresponding white matter fiber pathways are presented in Figure 2. We also present the major diffusion ellipsoid for each structural network which was inferred from the posterior weights within each network region. The first structural network represents the corpus callosum, a major fiber bundle that connects the left and right cerebral hemispheres and is distributed across the entire upper brain. The second structural network corresponds to the corticoponto-cerebellar tracts that establish communications between the cerebellum and the contralateral cerebral hemisphere. The third structural network encompasses the cingulum (Catani and De Schotten, 2008), a medial associative fiber bundle that runs an antero-posterior course within the cingulum gyrus surrounding the corpus callosum. The fourth structural network contains projection fiber tracts that pass through the internal capsule and corona radiate, consisting of both ascending and descending fibers, and are located symmetrically in both hemispheres of the brain. The fifth structural network is associated with a structural network close to the Fornix, with anterior fibers bending downward and connecting the medial temporal lobe to the mammillary bodies and hypothalamus. A portion of the sixth structural network corresponds to the inferior longitudinal fasciculus, a ventral associative bundle that connects the occipital and temporal lobes.

Figure 2:

Brain structural networks extracted from multi-subject DTI data from the PNC study using G-DICA (K = 16 and L = 10). Our analysis identified 6 structure networks that correspond to major white fiber pathways. The results are presented as (a): The spatial maps of the extracted structural network, (b): Tractography results of the white fiber tracts passing through the network, and (c): The ellipsoid of the major diffusion tensor associated with each structural network.

3.3. Stability and reliability analysis for G-DICA

In this Section, we conducted an extensive evaluation of the stability and reliability of the G-DICA results with regard to various factors including parameter specifications, initialization, and data resampling. The evaluation was conducted in comparison with the original individual DICA method (Wu et al., 2022) and the FA-based group ICA (Basser et al., 1994). We used the Dice coefficient as a similarity metric to quantify the consistency of the extracted structural networks with respect to the aforementioned factors.

3.3.1. Reliability w.r.t initial values selection

Initialization was identified as a key factor affecting the reproducibility of the individual DICA method (Keeratimahat and Nichols, 2022; Moerkerke and Seurinck, 2022). The analysis involves initializing weights and distributional bases for the Expectation-Maximization (EM) algorithm used to fit the mixture distribution in Step I, as well as setting an initial rotation matrix for the ICA algorithm in Step III. To assess the reliability of the G-DICA method, we evaluated the performance under both fixed and random initialization strategies in these steps. Fixed initialization refers to using the same starting values for the EM algorithm (Step I) and the group ICA (Step III) for each replication, ensuring consistent initial conditions across all runs. Random initialization, on the other hand, involves selecting different random starting values for either Step I and/or Step III for each replication, which evaluates the method’s robustness to varying initial conditions for either of the steps.

For these evaluations, we performed 50 replications to capture the variability in results and ensure a robust comparison between fixed and random initializations. Our results, shown in Figure 3(a), include assessing the reliability of random initialization for a single step while keeping the other step fixed, as well as evaluating the scenario where both steps are randomized simultaneously. We found that the initialization in Step I had a greater impact on the results, while the initialization in Step III had relatively little impact. This finding is reasonable because the distribution model determines the distributional bases and posterior weights. Based on these findings, we recommend using clustering methods to group the diffusion tensors across subjects into preliminary tensor clusters and use the cluster centroids as initial values for the EM algorithm in Step I of G-DICA. This approach generates more informative and representative initial values, leading to faster and more reliable convergence of the algorithm compared to random initialization. We further evaluate the reliability with respect to initial values between the proposed G-DICA and the original individual DICA method. Compared to the results obtained using the original DICA approach on each of the 453 subjects, our G-DICA method consistently demonstrated greater reliability for both fixed and random initialization types.

Figure 3:

Reliability analysis of G-DICA with respect to various factors using the PNC data. (a) Reliability with respect to different initial values. The Rand_both represents random initialization in step I and III, Rand_I represents random initialization for the EM algorithm in step I and fixed initialization for ICA in step III, and Rand_II represents fixed initialization for the EM algorithm in step I and random initialization for ICA in step III. (b) Stability of G-DICA for various selections of K and L compared to the original model (K = 16, L = 10) in the PNC study. (c) Reproducibility of results with different data samples. Data samples with different sample sizes, ranging from 22% to 77% of the full dataset, were subsampled from the PNC study. Dice coefficients were used to assess the reproducibility of the results.

3.3.2. Stability w.r.t parameter specifications

We evaluated the reliability of the G-DICA results with respect to the parameter specifications for K and L. Specifically, we treated our original specification (K = 16 and L = 10) as a reference, and compared the source signals estimated from the reference model to those estimated with different values of K and L, ranging from K = 12 to K = 20 and L = 6 to L = 14. In each case, we varied either K or L while keeping the other parameter fixed at its original value to study the effect of each parameter on the estimated source signals. For each parameter setting, we performed 50 replications. The estimated signals from each replication were matched to those from the reference model and the similarity between corresponding components was evaluated using Dice coefficients (Figure 3(b)). The Dice coefficients were higher than 0.75 in the majority of cases for the G-DICA, where higher consistency with the original results were observed with K and L closer to parameters specified in the original model. Furthermore, the G-DICA method consistently demonstrated higher stability than the individual DICA. The results show the robust stability of the proposed G-DICA with respect to parameter specifications.

3.3.3. Reproducibility w.r.t data resampling

We also evaluated the reproducibility of the G-DICA results with respect to variability in data samples. We generated various sizes of data samples by subsampling 22% to 77% of subjects from the PNC dataset 50 times and evaluated the reproducibility of the results from the original dataset in these different data samples. In comparison to G-DICA, we considered the FA-based group ICA method which is an existing group ICA method for DTI data. Since FA-based group ICA only uses the summary statistics of diffusion tensors, its results are expected to have more overlap across data samples. Figure 3(c) shows that the major fiber pathways uncovered by G-DICA exhibited good reproducibility across varying data samples, achieving a Dice coefficient above 0.5 with only 22% subsampling and generally exceeding 0.75 with 77% subsampling. Results show that the reproducibility of G-DICA is comparable to that of the FA-based group ICA.

3.4. Comparison between G-DICA and the FA-Based Group ICA

To further evaluate the performance of G-DICA, we compared our results with those derived from the FA-based Group ICA analysis of the PNC data (Web Figure 4 in Web Appendix C of Supplementary Materials). The results indicate structural networks extracted by FA-based group ICA exhibited substantial spatial overlap, encompassing highly similar white matter regions in the brain. This is due to the limited information captured by FA measures. As a simple scalar summary derived solely from the eigenvalues of the diffusion tensor matrix, FA only quantifies the degree of anisotropy but does not account for essential properties such as the orientation of diffusion. As a result, white matter regions with differing diffusion patterns can appear similar in FA, limiting its ability to distinguish between these regions. In contrast, the proposed G-DICA, which leverages the full information contained in the diffusion tensor matrices, identified more spatially distinct structural networks corresponding to different major white matter fiber bundles.

4. Simulation Studies

We conducted two simulation studies to evaluate the performance of the proposed G-DICA method, in comparison with the FA-based group ICA method. DTI data were generated from two distinct underlying models. In the first case, the data generative model aligned with the G-DICA model, where DTI data were linear mixtures of four underlying structural networks on the distribution level. In the second case, DTI data were generated from four clusters featuring different diffusion tensor ellipsoids, which deviated from the G-DICA model. For both simulation studies, we considered imaging data with dimensions 91×109×91, corresponding to the standard MNI space with a spatial resolution of 2×2×2 mm. The white matter mask was derived from PNC data, ensuring that only structurally relevant voxels are included. After applying the mask, a total of V = 97, 347 voxels were retained, providing comprehensive coverage of major structural networks. The procedure for generating diffusion tensors within each voxel is described in detail in the following sections. As ICA recovery is invariant to permutations, the estimated ICs were matched with the original source with which it had the highest spatial Dice coefficient.

4.1. Simulation I: Data from a model that aligns with G-DICA

In the first simulation, we generated DTI data for multiple subjects from four source signals S that correspond to major fiber bundles in human brain. First, to generate subjects’ logit maps ${\mathbf{M}}_i$, we mixed the source signals with subject-specific mixing matrices ${\mathbf{A}}_i$ that were sampled from estimates from real DTI data. Gaussian random noises ${\mathit{ϵ}}_{iv}\sim 𝒩\left(0,{\sigma }^2\mathbf{I}\right)$ were then added to the mixture to generate the logit vector ${\mathbf{m}}_{iv}$ at each voxel of the logit maps ${\mathbf{M}}_i$. The Softmax function was applied to the logit vector to generate a probability vector ${\mathbf{w}}_{iv}$. To generate the diffusion tensor for voxel v of subject i, we sampled random tensors ${\mathbf{T}}_{ivj}(j=1,\dots ,8)$ from eight Wishart distributions ${\mathbf{W}}_3\left({\mathrm{\Theta }}_j\right)$, with scaling matrices representing different ellipsoid shapes and orientations that mimic real diffusion tensors. These tensors were then weighted by the elements of the probability vector w_iv to obtain the diffusion tensor ${\mathbf{T}}_{iv}$ for each voxel.

We compared the performance of G-DICA and FA-based group ICA for uncovering the underlying source signals from DTI data generated with different noise levels ($\sigma$) and sample sizes (N). we specified K = 8 and L = 4 for G-DICA and 4 ICs for FA-based group ICA. Other specifications of K and L were also investigated for G-DICA and the results were found to be similar. The mean and standard deviation of the Dice coefficients between the true and estimated source signals for both methods were evaluated for 100 simulation runs and presented in Table 1. The results show that the proposed G-DICA method consistently demonstrated better accuracy in recovering the true source signals compared to the FA-based method. We also illustrate the z-scores of the true and recovered source signals by the two methods in randomly selected simulation runs in Figure 4. The z-scores were obtained by standardizing the estimated source signals using the mean and standard deviation of the entire source signal map, facilitating comparison between methods. Compared to the FA-based group ICA, G-DICA demonstrated much higher power in detecting brain regions with the true source signals and also fewer false positive findings across the brain.

Table 1:

Comparison of G-DICA and FA-based Group ICA in simulation I with 100 runs. The mean and standard deviation of the Dice coefficients between the true and estimated component regions are presented. The noise levels are Low (σ = 0.5), Medium (σ = 1.0), and High (σ = 2.0).

Noise	Dice coefficients of uncovering true signals Mean (SD)
	G-DICA	FA-based group ICA
Low
N = 20	0.657(0.102)	0.477(0.102)
N = 50	0.725(0.111)	0.668(0.116)
N = 100	0.738(0.114)	0.709(0.130)
Medium
N = 20	0.440(0.147)	0.249(0.133)
N = 50	0.602(0.164)	0.559(0.090)
N = 100	0.703(0.121)	0.665(0.122)
High
N = 20	0.226(0.166)	0.054(0.053)
N = 50	0.340(0.209)	0.133(0.099)
N = 100	0.473(0.184)	0.399(0.090)

Figure 4:

The z-scores of the estimated source signals from five random simulation runs in Simulation study I, using the proposed G-DICA and the existing FA-based group ICA.

4.2. Simulation II: Data generated from clusters of diffusion tensors

In the second simulation, we generated DTI data with four major fiber bundle regions characterized by the source signal maps as Simulation I. Each of the four fiber bundle regions were associated with a specific Wishart distribution. The diffusion tensors of voxels within each of the source signal regions were sampled from the corresponding Wishart distribution. Diffusion tensors in voxels in other regions in the brain were simulated by averaging random tensors generated from the four Wishart distributions. This scenario represents a deviation from the G-DICA model.

Figure 5 presents the true and uncovered sources from simulated data of N = 20 subjects using G-DICA and FA-based group ICA methods. For G-DICA, we specified K = 8 and L = 4. Other specifications of K and L were also investigated and the results were found to be similar. For the FA-based group ICA method, we also set the number of ICs to 4. The results showed that G-DICA performed very well in recovering the sources, while the FA-based method were unable to separate the true source signals. This result indicates that G-DICA provides a valuable tool for uncovering sources from DTI data derived from varying underlying mechanisms.

Figure 5:

The z-scores of the estimated source signals from five random simulation runs in Simulation study II, using the proposed G-DICA and the existing FA-based group ICA.

5. Discussion

In this paper, we introduce a group distributional ICA (G-DICA) approach for multi-subject DTI data analysis. To the best of our knowledge, it is the first ICA method that directly decomposes 3D diffusion tensors in multi-subject DTI studies. We conducted extensive reliability and reproducibility analyses for the proposed G-DICA model, showing highly consistent structural networks uncovered by G-DICA across different settings. For future work, additional analyses could be conducted to examine whether alternative registration methods, beyond the Affine transformation, result in meaningful differences in the structural networks.

G-DICA is a promising tool for performing blind source separation of multi-subject DTI data. It offers straightforward processing steps with the potential of being further simplified by incorporating predefined Wishart components or alternative distribution models to represent diffusion tensors. The derived structural networks offer an alternative to conventional voxel-wise mass-univariate approaches for investigating group differences in white matter microstructure and for linking white matter tracts to cognitive and behavioral performance in clinical diffusion MRI studies. Furthermore, the lower-dimensional representation of DTI data obtained in Step I of G-DICA holds great potential as a valuable resource for modeling DTI data. For future studies, we aim to develop a hierarchical modeling framework to incorporate subjects’ covariates in the decomposition of DTI data to investigate between-group heterogeneity in brain structural networks. Another direction for future research is to extend the application of our method to the joint decomposition of multi-subject imaging data from both fMRI and DTI modalities.

Supplementary Material

Web Appendices and Figures referenced in Sections 2.2, 3.2 and 3.4, example data, and R code implementing the proposed method are available with this paper at the Biometrics website on Oxford Academic.

Data Availability

The Philadelphia Neurodevelopmental Cohort study data (Satterthwaite et al., 2014) in this paper are publicly available for download from the Database of Genotypes and Phenotypes (dbGaP) via Authorized Access (https://www.ncbi.nlm.nih.gov/projects/gap/cgi-bin/study.cgi?study_id=phs000607.v3.p2).