Independent Component Analysis Tractography Combined with a Ball-Stick Model to Isolate Intra-Voxel Crossing Fibers of the Corticospinal Tracts in Clinical Diffusion MRI

Abstract

The independent component analysis (ICA) tractography method has improved the ability to isolate intravoxel crossing fibers; however, the accuracy of ICA is limited in cases with voxels in local clusters lacking sufficient numbers of fibers with the same orientations. To overcome this limitation, the ICA was combined with a ball-stick model (BSM) [“ICA+BSM”]. An ICA approach is applied to identify crossing fiber components in voxels of small cluster, which are maximally independent in orientation. The eigenvectors of these components are numerically optimized via the subsequent BSM procedure. Simulation studies for two or three crossing fibers demonstrate that ICA+BSM overcomes the limitation of the original ICA method by refining regional ICA solutions in diffusion measurement of a single voxel. It shows 2°–5° of angular errors to isolate two or three fibers, providing a better recovery of simulated fibers compared to ICA alone. Human studies show that ICA+BSM achieves high anatomical correspondence of cortico-spinal tracts (CST) compared to post-mortem CST histology, yielding 92.2% true positive detection including both lateral and medial projections, compared to 84.1% for ICA alone. This study demonstrates that the intra-voxel crossing fiber problem in clinical diffusion MRI may be sorted out more efficiently by combining ICA with BSM.

Introduction

A common problem encountered in the streamline tractography of white matter fiber systems when relying on a rank-2 diffusion tensor model (1,2) is the existence of multiple fiber orientations within an imaging voxel. The simple diffusion tensor model used routinely in streamline tractography is inadequate in modeling the probability density function of water displacement when the model is affected by complex fiber arrangement (3). In such a situation, the primary eigenvector of a rank-2 tensor model of a voxel’s diffusion field is likely to point in an erroneous direction, biased toward the highest density fibers thus designating many tracts to seemingly propagate in the wrong direction (4). This intra-voxel problem limits current diffusion weighted imaging (DWI) technology in clinical applications.

The strategies to solve the intra-voxel problem have included various methods relying on a more complex model characterizing diffusion signals acquired by high angular resolution diffusion imaging, which samples data on a shell in the diffusion-encoding q-space (5). The candidate methods include the Gaussian mixture model (GMM) such as the ball-stick model (BSM) (5–8), the generalized diffusion tensor (9,10), spherical harmonic decomposition (11), and spherical deconvolution (SD) (12–15). Also, the model-free methods, known as q-space imaging (16,17), include diffusion spectrum imaging (DSI) (18) and generalized q-sampling imaging (GQI) (19). These methods are widely used to delineate crossing fiber patterns and estimate their tensors in high-angular-resolution DWI data (16–19).

Recently, a novel fast independent component analysis (ICA) tractography method (4) that estimates the orientations of up to three fibers per voxel was proposed to allow the isolation of multiple fibers in clinical DWI data. By assuming that the orientations of all crossing fibers are the same inside each voxel of the local cluster, the ICA method extracts up to three orientations by performing the F-test model selection to estimate an optimal number of crossing fibers in voxels of the cluster. However, the assumption that tracts have a uniform orientation inside the cluster is not necessarily correct in the real brain. Especially at the boundary of different white matter regions where multiple fibers likely cross only at some voxels of the cluster, the sensitivity of the ICA decomposition might be significantly reduced by partial volume effects. In fact, it was found that the ICA method alone does not allow entire delineation of the cortico-spinal tract (CST) since it fails to track the fibers of those projections in voxels near the precentral gyrus/sulcus where fibers of the arcuate fasciculus (AF), the superior longitudinal fasciculus (SLF), and the corpus callosum intermingle with the lateral projections of the CST (20).

The present study combines BSM with ICA in order to further resolve the orientations of multiple fibers within a voxel. The applicability of ICA combined with BSM, termed “ICA+BSM”, is demonstrated in computer simulations and clinical DWI data to isolate individual tensors of multiple crossing fibers such as CST, AF, and SLF fibers at voxels near the precentral gyrus and the central sulcus. Based on the hypothesis that multi-compartment Gaussian tensors correspond to multiple intersecting fiber bundles present within a voxel, two complementary approaches are combined to isolate multiple fiber bundles within a single voxel: the first is ICA to approximate fiber orientations of multiple tensors existing in a local cluster and the second is BSM to refine information across multiple ICA driven tensors mixed into a single voxel of a local cluster.

Theory

The BSM fitting approach (6) is used to supplement the ICA approach in order to better define fiber bundle orientations within a voxel. For every voxel of white matter, (x,y,z), a small neighborhood cluster of 11 voxels is defined around it by the next nearest voxel, i.e. 3×3 voxels in the x-y plane, and one voxel above and below the center voxel, (x,y,z), along the z-axis. The diffusion data within this cluster is used to construct the measurement matrix, U, whose i-th row vector, u_i, represents diffusion sensitizing measurements of all N gradient directions with zero-mean at the i-th voxel of the cluster (voxel index i=1,2,…,11); see Eq. [1]. Blind source decomposition using a fast fixed point ICA (21) is applied to estimate a generative model that describes how the observed diffusion weighted signal of u_i is generated by a process of mixing the statistically independent diffusion processes with zero-mean and unit-variance, s_ij (j=1,2,..,11) (7).

$${\mathbf{u}}_{\text{i}}=\left[\begin{array}{ccc}\frac{{\text{m}}_{\text{i}1}}{{\text{m}}_{\text{i}0}}& \cdots & \frac{{\text{m}}_{\text{iN}}}{{\text{m}}_{\text{i}0}}\end{array}\right]-\left[\begin{array}{ccc}\frac{1}{\text{N}}{\sum }_{\text{n}=1}^{\text{N}}\frac{{\text{m}}_{\text{in}}}{{\text{m}}_{\text{i}0}}& \dots & \frac{1}{\text{N}}{\sum }_{\text{n}=1}^{\text{N}}\frac{{\text{m}}_{\text{in}}}{{\text{m}}_{\text{i}0}}\end{array}\right]={\sum }_{\text{j}=1}^{11}{\text{a}}_{\text{ij}}{\mathbf{s}}_{\text{ij}}$$ [1]

where m_in indicates the signal intensity at the i-th voxel of the n-th diffusion gradient image, n=0 for the b0 image and n=1,…, N for the N-diffusion weighted images. a_ij defines a mixing weight for the j-th independent component vector, s_ij, in the i-th voxel measurement that will yield u_i. It is assumed that each of the j-th row vectors, s_ij represents either multiple diffusion components independently attenuated per diffusion sensitizing gradient or some unknown confounding artifact. The elements of each row vector, s_ij are latent variables that cannot be directly observed in the elements of the measurements in u_i. The weights of a_ij are assumed to be unknown and are therefore iteratively learned to maximize the statistical independence (or Non-Gaussianity) between s_ij. This configuration involving a neighborhood of 11 voxels allows a total of eleven s_ij vectors to be learned within the framework of an unsupervised neural network (21). In vector-matrix notation, U=AS or S = A⁻¹U where the weights of a_ij consist of the elements of A. u_i and s_ij constitute the i-th row vectors of U and S, respectively.

In the present study, principal component analysis (PCA) consisting of a whitening transformation with reduced dimensionality, K, is applied to the measurement matrix, U, in order to improve the ICA estimation,

$$\stackrel{\sim }{\mathbf{U}}=\left({\mathbf{Y}}_{\mathbf{K}}^{-\frac{1}{2}}{\mathbf{X}}^{\mathbf{T}}\right)\mathbf{U}=\stackrel{\sim }{\mathbf{A}}\widehat{\mathbf{S}}$$ [2]

where X and Y are the matrices of eigenvectors and eigenvalues in the covariance of U^T, respectively. Y is a diagonal matrix denoted by diag(y₁,….,y₁₁). Y_K^−1/2 represents a diagonal matrix to keep the first K-eigenvalues and disregard other small eigenvalues (i.e., diag(y₁^−1/2, … , y_K^−1/2, 0, …,0)), with K representing the assumed number of fiber directions in the measurement, u_i. The superscript T denotes the transpose of a matrix.

Eq. [2] reduces the number of sources, s_ij, to a scalar K, reduces noise, and prevents over-learning of s_ij. It yields a new representation of the diffusion measurement, ũ_i at the i-th voxel,

$${\stackrel{\sim }{\mathbf{u}}}_{\text{i}}={\sum }_{\text{j}=1}^{\text{K}}{\stackrel{\sim }{\text{a}}}_{\text{ij}}{\widehat{\mathbf{s}}}_{\text{ij}}$$ [3]

where ã_ij denotes the new mixing fraction of the j-th source, ŝ_ij, in ũ_i that represents a diffusion profile reconstructed by combining directionally independent K-diffusion processes, ŝ_ij without any noise or artifact.

Based on the ICA assumption in Eq. [2], the definition of u_i in Eq. [1] is substituted into the first term of Eq. [2] whose vector form is eventually equal to Eq. [3]. The first line of Eq. [4] shows that this equality results in the new definition of ṽ_i in the third line of Eq. [4], indicating that ṽ_i (the diffusion profile with reduced dimension) can be represented as a linear summation of “mean offsets of the diffusion signals reconstructed by whitening and dewhitening” and “uncorrelated-zero-mean-unit-variance diffusion signals of multiple K-fibers” that are directionally independent within voxels of the local cluster. ṽ_i establishes a formal connection between ICA and BSM, assuming that ṽ_i approximates a linear summation of the “isotropic diffusion tensor, D_i0” and “K-anisotropic tensors, {D_ij} of the K-fiber compartments” in the conventional BSM (6, 22).

$$\begin{array}{c}{\stackrel{\sim }{\mathbf{U}}}_{\text{i}}=\left({\mathbf{Y}}_{\text{K}}^{-\frac{1}{2}}{\mathbf{X}}^{\text{T}}\right)[\mathbf{M}-\overline{\mathbf{M}}],\stackrel{\sim }{\mathbf{V}}=\left({\mathbf{XY}}_{\text{K}}^{-\frac{1}{2}}\right){\stackrel{\sim }{\mathbf{U}}}_{\text{i}}\\ \stackrel{\sim }{\mathbf{V}}=\left({\mathbf{XY}}_{\text{K}}^{-\frac{1}{2}}\right)\left({\mathbf{Y}}_{\text{K}}^{-\frac{1}{2}}{\mathbf{X}}^{\text{T}}\right)\mathbf{M}=\left({\mathbf{XY}}_{\text{K}}^{-\frac{1}{2}}\right)\left({\mathbf{Y}}_{\text{K}}^{-\frac{1}{2}}{\mathbf{X}}^{\text{T}}\right)\overline{\mathbf{M}}+\stackrel{\sim }{\mathbf{A}}\widehat{\mathbf{S}}=\mathbf{TM}=\mathbf{T}\overline{\mathbf{M}}+\stackrel{\sim }{\mathbf{A}}\widehat{\mathbf{S}}\\ \iff {\stackrel{\sim }{\mathbf{v}}}_{\text{i}}=[\begin{array}{ccc}\cdots & {\mathbf{t}}_{\text{i}}·{\mathbf{m}}_{\text{n}}& \cdots \end{array}]=[\begin{array}{ccc}\cdots & {\mathbf{t}}_{\text{i}}·{\overline{\mathbf{m}}}_{\text{n}}& \cdots \end{array}]+{\sum }_{\text{j}=1}^{\text{K}}{\stackrel{\sim }{\text{a}}}_{\text{ij}}{\widehat{\mathbf{s}}}_{\text{ij}}\\ \cong \left[\begin{array}{ccc}\cdots & \left(1-{\sum }_{\text{j}=1}^{\text{K}}{\text{f}}_{\text{ij}}\right)exp(-\text{b}{\mathbf{r}}_{\text{n}}^{\text{T}}{\mathbf{D}}_{\text{i}0}{\mathbf{r}}_{\text{n}})& \cdots \end{array}\right]+\left[\begin{array}{ccc}\cdots & {\sum }_{\text{j}=1}^{\text{K}}{\text{f}}_{\text{ij}}exp(-\text{b}{\mathbf{r}}_{\text{n}}^{\text{T}}{\mathbf{D}}_{\text{i}0}{\mathbf{r}}_{\text{n}})& \cdots \end{array}\right]\end{array}$$ [4]

where T represents the product of XY_K^−1/2 and Y_K^−1/2X^T. M is a matrix whose i-th row and n-th column element consists of m_in/m_i0. M̄ is a matrix whose i-th row and n-th column element is $({\sum }_{\text{n}=1}^{\text{N}}{\text{m}}_{\text{in}}/{\text{m}}_{\text{i}0})/\text{N}$. t_i indicates the i-th row vector of T. m_n and m̄_n indicate the n-th column vector of M and M̄, respectively. f_ij is the volume fraction of the j-th BSM compartment, characterized by the diffusion tensor D_ij (= E_jV_jE_j^T, E_j and V_j denote eigenvector and eigenvalue matrices, diag (λ_j1, λ_j2, λ_j3), of tensor matrix D_ij, respectively). D_i0 represents the diffusion tensor of isotropic diffusion. b and r_n are the b-value and the unit vector, respectively, of the N-applied diffusion gradients; note: n=1,2,…, N.

The approximation of ṽ_i using the compartments of BSM can be supported by significant correlations between ICA-driven diffusion components and BSM compartments (Fig. 1). Two cylindrical fiber compartments were generated by BSM (6,22) through parts of Eq. [4] and their ICA components were then estimated by the fast fixed point ICA approach (21) of Eq. [3]. It is notable that “mean offsets of the diffusion signals reconstructed by whitening an dewhitening” are highly correlated with the weighted isotropic diffusion profile (Fig. 1 (a)). Weights of individual ICA sources apparently correspond to volume fractions of individual compartments (Fig. 1 (b)). Apparent diffusion coefficients and the first eigenvectors of individual ICA sources are significantly correlated with those of individual compartment signals (Fig. 1 (c)–(d)). Although ICA is unable to estimate both variance and order of the original sources (21), it is reasonable to assume that ICA-driven diffusion offset and sources can approximate the “isotropic diffusion tensor, D_i0” and “K-anisotropic tensors, {D_ij} of the K-fiber compartments” in BSM.

Figure 1.

Correlation between ICA-driven components and BSM compartments. (a) apparent diffusion coefficients (ADC) of whitened mean offsets of the diffusion signals vs. ADC of the weighted ball compartment. (b) weights of individual ICA sources vs. volume fractions of individual stick compartments. (c) ADC of the individual ICA source mixture vs. ADC of the individual stick compartment mixture. (d) the first eigenvectors of individual ICA sources vs. the first eigenvectors of individual stick compartments. An isotropic diffusion tensor (D_i0 with [λ₁,λ₂,λ₃] = [1.7, 1.7, 1.7]×10⁻³mm²/s) and two stick tensors (D_ij with [λ₁,λ₂,λ₃] = [1.7, 0, 0]×10⁻³mm²/s) were randomly mixed using the BSM parts of Eq. [4] with different values of fractions, f_ij, and eigenvectors, [e_j1, e_j2, e_j3]. For each random mixture of three BSM compartments, ICA-driven components were estimated by the fast ICA approach of Eq. [3]. For (a)–(d), Black dots represent corresponding values of ICA-driven components and BSM compartments in order to support the experimental feasibility of Eq. [4]. The red line indicates the fitted linear regression of the scattered dots. The square of the Pearson correlation coefficient (R²) was reported in order to show the correlation between ICA-driven components and BSM compartments.

The present study parameterizes BSM in Eq. [4] by adopting a single ball and K-stick model (6, 22) where all D_ij, including D_i0, have equal λ₁: the ball compartment of D_i0 is completely isotropic (λ₁=λ₂=λ₃) while the remaining “stick” compartments of all D_ij are perfectly linear (λ_j1=λ₁, λ_j2=λ_j3=0). For K-fiber terms, this model leads to K+1 compartments and 3×K+1 degrees of freedom. An optimal number of compartments (or fibers) in ṽ_i, K_opt is finally found by the Bayesian Information Criterion (BIC) (23).

$$\begin{array}{c}{\text{BIC}}_{\text{K}}=log\left(\frac{{\text{RMSE}}_{\text{K}}}{\text{N}}\right)+{\text{p}}_{\text{K}}\frac{log(\text{N})}{\text{N}}\\ {\text{RMSE}}_{\text{K}}=\sqrt{\frac{{\sum }_{\text{n}=1}^{\text{N}}{({\stackrel{\sim }{\mathbf{v}}}_{\text{i}}^{\text{K}}(\text{n})-{\mathbf{v}}_{\text{i}}^{\text{K}}(\text{n}))}^2}{\text{N}}}\\ {\mathbf{v}}_{\text{i}}^{\text{K}}=[\begin{array}{ccc}\cdots & (1-{\sum }_{\text{j}=1}^{\text{K}}{\text{f}}_{\text{ij}})exp(-\text{b}{\mathbf{r}}_{\text{n}}^{\text{T}}{\mathbf{D}}_{\text{i}0}{\mathbf{r}}_{\text{n}})+{\sum }_{\text{j}=1}^{\text{K}}{\text{f}}_{\text{ij}}exp(-\text{b}{\mathbf{r}}_{\text{n}}^{\text{T}}{\mathbf{D}}_{\text{i}0}{\mathbf{r}}_{\text{n}})& \cdots \end{array}]\end{array}$$ [5]

where p_K is the number of parameters at BSM with K-stick components. RMSE stands for the root mean square error between ${\stackrel{\sim }{\mathbf{v}}}_{\text{i}}^{\text{K}}$ and v_i^K. A key contribution of ICA+BSM is not to use a noisy diffusion measurement of u_i but to incorporate a diffusion profile, ṽ_i, as a target function for BSM of v_i. Since ṽ_i contains both K-anisotropic ICA components (existing in multiple voxels of the local cluster) and a mean diffusion component (existing only in a center voxel of the local cluster), it represents an explicit diffusion process without any confounding artifacts and directly models the diffusion field existing in a voxel of K-crossing fibers.

In our implementation of Eq. [5], the optimal parameters of f_ij, D_i0, and D_ij are iteratively refined to minimize the RMSE_K by using the Levenberg-Marquardt (LM) fitting algorithm (24) under the following constraints,

$$\begin{array}{c}{\sum }_{\text{j}=1}^{\text{K}}{\text{f}}_{\text{ij}}=1,0.1\le {\text{f}}_{\text{ij}}\le 0.9\\ {\mathbf{D}}_{\text{i}0}=\left[\begin{array}{ccc}{\lambda }_1& 0& 0\\ 0& {\lambda }_1& 0\\ 0& 0& {\lambda }_1\end{array}\right],0.001\le {\lambda }_1\le 0.002\\ {\mathbf{D}}_{\text{ij}}=\left[\begin{array}{ccc}{\text{e}}_{\text{j}1}& 0& 0\\ {\text{e}}_{\text{j}2}& 0& 0\\ {\text{e}}_{\text{j}3}& 0& 0\end{array}\right]\left[\begin{array}{ccc}{\lambda }_1& 0& 0\\ 0& 0& 0\\ 0& 0& 0\end{array}\right]{\left[\begin{array}{ccc}{\text{e}}_{\text{j}1}& 0& 0\\ {\text{e}}_{\text{j}2}& 0& 0\\ {\text{e}}_{\text{j}3}& 0& 0\end{array}\right]}^{\text{T}},-1\le {\text{e}}_{\text{j}1},{\text{e}}_{\text{j}2},{\text{e}}_{\text{j}3}\le 1\text{and}\Vert {\mathbf{e}}_{\text{j}}\Vert =1\end{array}$$ [6]

where f_ij is randomly initialized to have a floating number close to 1/(K+1). λ₁ of D_i0 is randomly initialized to have a floating number around 0.0017 mm²/s. [e_j1,e_j2,e_j3] of D_ij is initialized by the first eigenvector of the diffusion tensor matrix from the ICA source component, ŝ_ij, in Eq. [3]. It is assumed that there exists up to three fiber bundles, K=3, for the BIC selection (i.e., 0 ≤ K≤ 3). At K=0 (no fiber), only {λ₀₁} is optimized since there is no ICA component available for [e_j1,e_j2,e_j3] in BSM. For 1 ≤ K≤ 3, the parameter set of {f_ij, λ₁,e_j1,e_j2,e_j3} is optimized to provide the best fit of v_i to ṽ_i.

Until a global solution of the LM algorithm is achieved at each K (i.e. RMSE_K < 0.01), the overall fitting process is repeated up to 5 times with the same ICA driven e_j but different random values of f_ij and λ₁. A minimization of RMSE_K is selected as the global solution of the LM fitting algorithm and then converted into BIC_K of Eq. [5] that includes the cost of penalty caused by the model complexity (i.e., number of parameters). Finally, the index K providing the minimal BIC_K is selected as an optimal number of compartments, K_opt. The parameter sets {f_ij,λ₁,e_j1,e_j2,e_j3} providing the global solution at K = K_opt are used to configure K_opt-stick compartments existing at the i-th voxel of the cluster. That is, total K_opt fiber orientations are available at each voxel of white matter for the streamline tractography.

Methods

Subjects and MRI acquisition

The present study included twenty typically developing children (age: 15.0±3.3 years, 10.1–17.8 years, 10 boys) and two patients with a diagnosis of focal epilepsy. The Human Investigations Committee of Wayne State University granted permission for acquiring MRI scans of all children. All diffusion MRI scans were performed on a 3T GE Signa scanner (GE Healthcare, Milwaukee, WI) equipped with an 8-channel head coil at TR = 12,500 ms, TE = 88.7 ms, field of view = 240 cm, 128×128 acquisition matrix, contiguous 42 slices with 3 mm thickness using 55 isotropic gradient directions with b= 1000 s/mm², one b=0 acquisition, and number of excitations (NEX) = 1.

Computer Simulations

For simulating a single cylindrical compartment, a rank-2 tensor was assumed, D_ij =E_jV_jE_j^T that is completely described by its eigenvector matrix, E_j, and eigenvalue matrix, V_j=diag(λ₁,λ_2,λ₃). To simulate E_j, we utilized a random vector with unit norm as the first column vector of E_j where other elements of E_j were assumed to be zeros. Also, to mimic anisotropic diffusion of water molecule at three orthogonal directions, we assumed V_j of [λ₁,λ₂,λ₃] =[1.7, 0, 0]×10⁻³mm²/s or [1.8, 0, 0]×10⁻³mm²/s. An isotropic diffusion compartment, D_i0 was simulated with E₀ = diag(1,1,1) and V₀ = diag(λ₁,λ₁,λ₁). Other anisotropic diffusion compartments, D_ij were simulated by varying E_j with other random vectors with unit norm and a fixed V_j = diag(λ₁,λ₂,λ₃). The crossing angle between two D_ij (called “inter-fiber angle, α”) was ranged from 10° to 80°. For three fibers, the two inter-fiber angles were assumed to be equal.

The GMM compartments [5] were also used to assess how accurately ICA+BSM can work within a realistic configuration of a multiple fiber mixture. For simulating the mixture of two Gaussian compartments, D_i1 and D_i2 crossing at different inter-fiber angles, the eigenvalues of V_0, V_{1 ,} and V₂ were changed into [λ₁,λ₂,λ₃] =[1.7, 1.7, 1.7]×10⁻³mm²/s, [λ₁,λ₂,λ₃] =[1.7, 0.4, 0.2]×10⁻³mm²/s, and [1.8, 0.5, 0.3]×10⁻³mm²/s, respectively. E₀ was fixed to diag(1,1,1). A 3×3 identity matrix with 1’s on the diagonal and 0’s elsewhere was randomly rotated in the x-y-z plane to generate the eigenvectors of the first fiber, E₁. Discrete inter-fiber angle (α) was used to parameterize E₂ of the next fiber to rotate E₁ by α in the x-y-z plane. Each α was selected randomly to lie within a 10°–80° range and distributed equally within each 10°.

The volume fraction, f_ij, were randomly assigned in the range from 0.1 to 0.9 such that Σf_ij = 1 (i.e., as a function of K, different ranges of f_ij were applied to model the degree of any partial volume effect from an isotropic diffusion compartment, f_i0 = 1 for K=0, 0.1 ≤ f_i1 ≤ 0.9 for K=1, 0.2 ≤ f_i1,f_i2 ≤ 0.7 for K=2, 0.2 ≤ f_i1, f_i2, f_i3 ≤ 0.5 for K=3). A total of 1000 trials per each of the inter-fiber angles were repeated for each mixture of two and three fibers. For each random trial, multiple fiber compartments were selectively determined in the framework of the BIC model as shown in Eq. [5]. The accuracy of recovering fiber orientations was evaluated by the absolute angular error between the actual orientation (1st eigenvector of simulated D_ij) and the estimated tensor orientation (1st eigenvector of recovered D_ij).

To assess the effect of heterogeneous voxels on the accuracy of the original ICA method, the percentage of heterogeneous fiber orientations inside the cluster was adjusted in the range 0–0.7 where 0 represents all voxels inside the cluster have crossing fibers with the same orientations (no heterogeneous orientations: “theoretical ICA assumption”) and 0.5 represents 50% of voxels have crossing fibers with heterogeneous orientations. Randomly oriented tensors were replaced to simulate the heterogeneous fibers. At each heterogeneous ratio, the errors of ICA+BSM was compared to those of ICA with the F-test model selection (4), “BSM with the random initialization of {f_ij,λ₁,e_j1,e_j2,e_j3} (BSM)” (6, 22), “constrained spherical deconvolution (CSD)” (14,15), “BSM with the initialization of {e_j1,e_j2,e_j3} using the orientations of local peaks in the orientation distribution function (ODF) obtained from the CSD (CSD+BSM)”, DSI (18), and GQI (19). For standalone BSM, a set of {f_ij,λ₁,e_j1,e_j2,e_j3} was randomly initialized as described in Eq. [6]. For CSD+BSM, the orientations of local peaks in a CSD-driven ODF were used to initialize {e_j1,e_j2,e_j3} of individual stick compartments. Other parameters such as {f_ij,λ₁} were randomly initialized according to Eq. [6]. The above scheme of parameter initialization was also applied to human data.

Successful recovery of fiber orientation was assumed if absolute angular error was less than 5°. The rate of this successful recovery was used as another measure for the comparison. The ODF of CSD, DSI, and GQI were obtained using MRtrix package (available at www.nitrc.org) and DSI studio package (available at dsi-studio.labsolver.org) at ODF tessellation = 8 folds and number of peaks = 3. For all trials, Rician noise was randomly added under b0-SNR = 30 to simulate noisy MR diffusion signals.

ICA based Ball-Stick Model Tractography

For each voxel of fractional anisotropy (FA) > 0.2, an 11 neighborhood cluster was defined to create a diffusion data matrix, U, whose row vectors indicate the diffusion weighted signals at every voxel of a cluster. The first eigenvectors of K_opt-tensors resulting from the BIC selection (Eq. [5]) were utilized for tractography. The conventional streamline tracking algorithm was modified to accommodate multiple orientations in voxels (19). At the center of each seeding voxel, tracking was started in the orientation of one fiber that was randomly selected among K_opt- fibers. The propagation direction was calculated by applying trilinear interpolation on the fiber orientations provided from nearby voxels of the current point. For each nearby voxel, only the fiber orientation that had the smallest turning angle was considered for interpolation.

The lateral projection of CST was used to demonstrate how the ICA+BSM method performs compared to previously reported multi-fiber solution methods. Fiber orientation maps were generated from all subjects by using six different methods, ICA+BSM, ICA, CSD+BSM, CSD, BSM, and an in-house implementation of a conventional single diffusion tensor imaging method (DTI, 1,2). The same tractography algorithm was used to propagate streamlines from every voxel in native space. The CST pathway for each subject was sorted from different fiber orientation maps by applying a two-ROI approach, with one seed ROI placed in the precentral gyrus (PCG) and another ROI to select for the fibers in the posterior limb of internal capsule (PLIC). To minimize inter-operator variability, binary Montreal Neurological Institute (MNI) space atlases of both PCG and PLIC (WFU PickAtlas, www.fmri.wfubmc.edu/software) were placed into native space by applying the inverse of spatial deformation obtained between the subject’s b0 image and MNI b0 template (25).

To ensure group consistency in tracking the entire CST pathway obtained from each tractography method, a percentage overlap approach was utilized (26). A fiber visitation map was created by counting the number of fibers intersected per voxel (27). The resulting map was normalized to MNI space using the spatial deformation derived from the process of normalization of the b0 image, averaged across all subjects, and scaled by a value of maximum visitation in the map, which defines a percentage overlap map. The percentage overlap map of each tractography method was compared with that of a histological atlas of the CST as the gold standard (SPM anatomy toolbox, www.fil.ion.ac.uk). Receiver Operating Characteristic (ROC) curve analysis was performed to evaluate the performance of each method.

The ICA+BSM method was further validated by correlating the lateral projections of CST fibers and the primary motor face and hand areas determined by direct brain electrical stimulation in two patients with focal epilepsy. Functional motor mapping using electrical stimulation was performed during extraoperative electrocorticography recordings (28). A site with a contralateral movement induced by stimulation, without after-discharges, was defined as ‘the primary motor area’ for that given body part. Specifically, it was determined whether the CST connecting ‘hand motor cortex’ (seed), ‘PLIC’ (1st ROI), and ‘cerebellar peduncles’ (2nd ROI) corresponded to the primary hand CST pathway (29). Then, it was determined whether those connecting ‘face motor cortex’ (seed), ‘PLIC’ (1st ROI), and ‘facial colliculus’ (2nd ROI) corresponded to the primary face CST pathway, specially known as the corticobulbar tract (30).

Results

Simulation Studies

Simulation studies assessing the absolute error for two and three fibers per voxel (at the ratio of heterogeneous orientations = 0 %, 25%, and 50%) demonstrated that ICA+BSM is superior to all other methods using BSM, CSD, CSD+BSM, ICA, and PCA+BSM (Fig. 2). The ICA method showed high sensitivity to the ratio of heterogeneous orientations. At 0% (i.e., ICA assumption satisfied) the ICA method showed the smallest error (median value = 4.7°) for two fibers, but errors dramatically increase as the percentage of heterogeneous orientations increases (median value = 19.4°). Meanwhile the percentage of heterogeneous orientations had little effect on absolute errors in ICA+BSM. The ICA+BSM showed the smallest errors (< 2°–7°) in most inter-fiber angles for two and three fibers. Furthermore, it showed much smaller errors than BSM and CSD+BSM, suggesting that the parametric optimization using the ICA-driven initialization and target diffusion profile provides more accurate solutions in the BSM procedureUnlike standalone BSM and CSD+BSM that use noisy diffusion measurement, u_i, as a target diffusion profile for BSM optimization, ICA+BSM uses a diffusion profile, ṽ_i,as a target diffusion profile for BSM optimization that represents an explicit diffusion process without any confounding artifacts. The main improvement of ICA+BSM is to minimize error angles at K=2 and 3 resulting from dimensional reduction procedure using PCA, and a small additional improvement is provided by the ICA-based initialization, especially in K=3. This incorporation of ṽ_i to BSM dramatically reduces errors even in case where ICA alone fails to estimate the orientations of multiple compartments at high heterogeneity. Also it improves correct BIC selection of the model parameter, K_opt at significantly reduced updates by providing the best guesses of experimental parameters to the non-linear LM curve fitting optimization (Table 1).

Figure 2.

Error angle between actual and estimated fibers for two and three cylindrical fibers with [λ₁,λ₂,λ₃] =[1.7, 0, 0]×10⁻³mm²/s. 55 point shell sampling with b-value of 1000 s/mm² was utilized to generate the simulation data. Each box has lines at the values of the lower quartile (blue colored), median (red colored), and upper quartile (blue colored). The black colored whiskers are lines extending from each end of the quartiles to show the complete extent of the data. (a) A case of two fibers (K=2). (b) A case of three fibers (K=3). A b0-SNR =30 was assumed for (a) and (b). To address how ICA helps BSM to find the global optimum, a diffusion profile reconstructed by PCA including both whitening and dewhitening, ṽ_i in Eq. [4], was used as a target profile for BSM with five random initializations (PCA+BSM).

Table 1.

Percentage of trials and average number of updates in which the global solution of Levenberg-Marquardt curve fitting algorithm was achieved by the BIC model selection combined with the standalone BSM, CSD+BSM, and ICA+BSM.

True	Standalone BSM
	No fiber	1	2	3	Updates
No fiber	99.4	0.6	0	0	22
1	0	99.4	0.6	0	30
2	0	0.6	98.8	1.4	62
3	0	18.6	24.0	57.4	128

True	CSD + BSM
	No fiber	1	2	3	Updates
No fiber	99.4	0.6	0	0	22
1	0	99.6	0.4	0	32
2	0	0.2	98.6	1.2	53
3	0	18.2	21.0	60.8	92

True	ICA + BSM
	No fiber	1	2	3	Updates
No fiber	99.4	0.6	0	0	22
1	0	99.6	0.4	0	19
2	0	0.2	98.8	1.0	41
3	0	10.8	19.4	71.8	59

For all methods fractional increment for derivative was set at 10⁻⁵. The experimental parameters were iteratively updated until termination tolerance was reached at 10⁻⁶.

The performance of DSI, GQI, and ICA+BSM in resolving two Gaussian tensors with different FA values was also tested with two simulated data sets, ones with high and low angular resolutions, using 203 point grid sampling with multiple b-values of 307–4000 s/mm² and 55 shell sampling with a b-value of 1000 s/mm² (Fig. 3). In both angular sampling schemes, the ICA+BSM showed complete reconstructions of two fibers with the smallest errors and also produced the highest probability of successful recovery for most inter-fiber angles. This result is supported by the fact that high non-Gaussianity of the diffusion signals was properly generated at both angular sampling schemes (Fig. 4), demonstrating that a crucial assumption of ICA+BSM, “diffusion measurement u_i is a linear mixture of non-Gaussian diffusion components, s_ij” was met in simulated data.

Figure 3.

Error angle between actual and estimated orientation and probability of successful reconstruction for two fibers with different FA values, [λ₁,λ₂,λ₃] =[1.7, 0.4, 0.2]×10⁻³mm²/s for the first fiber and [λ₁,λ₂,λ₃] =[1.8, 0.5, 0.3]×10⁻³mm²/s for the second fiber. Blue and green ROIs were used to isolate the tracts of two crossing fibers seeded from red ROI (shown for 60° inter-fiber angle). Each box of the box plots has lines at the values of the lower quartile (blue colored), median (red colored), and upper quartile (blue colored). The black colored whiskers are lines extending from each end of the quartiles to show the complete extent the data. Red dots represent outliers beyond the ends of the whiskers. (a) high angular resolution set of 203 grid sampling with multiple b-values (307–4000 s/mm²). (b) low angular resolution set of 55 point shell sampling with a b-value of 1000 s/mm². A b0-SNR=30 was assumed for (a) and (b). Note that left boxplot of both panel (a) and (b) reports the result from fiber-crossings of 30°, 40°, 60°, and 80°.

Figure 4.

Examples of the probability density function, p(m_in/m_i0) of the diffusion signals, u_i = [m_i1/m_i0, m_i1/m_{i0, …,} m_i55/m_i0]generated by BSM (a ball D_i0 with [λ₁,λ₂,λ₃] = [1.7, 1.7, 1.7]×10⁻³mm²/s and a stick D_ij with [λ₁,λ₂,λ₃] = [1.7, 0, 0]×10⁻³mm²/s) using two different sampling schemes, (a) low angular resolution set of 55 point shell sampling with b-value of 1000 s/mm² and (b) high angular resolution set of 203 grid sampling with multiple b-values (307–4000 s/mm²). A b0-SNR=30 was assumed for (a) and (b). The p(m_in/m_i0) suggests significant non-Gaussianity of the diffusion signals (i.e., kurtosis > 0). For comparison, a corresponding Gaussian distribution with zero kurtosis is shown in red.

Assessment of Clinical Scans

Typical fiber orientations obtained from the voxels of PCG using ICA+BSM are presented in Fig. 5. In Fig. 5, the CST significantly intersects with AF and SLF in the two ROIs denoted by yellow boxes in slice₁ and slice₂. Compared to the ICA, the ICA+BSM images show more lateral components of CST fibers (left to right colored red) in every voxel of each ROI (yellow boxes), suggesting ICA+BSM can successfully identify lateral projections of CST fibers, which the ICA alone fails to realize.

Figure 5.

Independent component analysis tractography combined with a ball-stick model (ICA+BSM) to recover fiber orientations in lateral projections of the corticospinal tract (CST). Brown and yellow clusters indicate the voxels of the precentral gyrus (PCG) and ROIs with crossing fibers, respectively. (a) fiber orientations from the ICA method and (b) fiber orientations from the ICA+BSM method. The ICA+BSM method could isolate correct orientations of lateral projections (red bars in an ROI of slice₁ and slice₂) in voxels where the CST intersects with the superior longitudinal fasciculus and arcuate fasciculus.

Lateral CST projections were identified by whole brain tractography (Fig. 6) using single tensor-based DTI, CSD, ICA, BSM, CSD+BSM, and ICA+BSM with the same configuration of seed region (bilateral PCG), filtering ROIs (bilateral PIC and pons) and tractography parameters (fraction, f_ij > 0.1, step size = 0.2 voxel width, angular deviation < 45°, 100 seeds per voxel). The ICA+BSM produced the largest number of lateral projections connecting PIG to PCG.

Figure 6.

Comparison of lateral projections of the corticospinal tract obtained from two normal children using single tensor based DTI, CSD, ICA, BSM, CSD+BSM, and ICA+BSM. The same configuration of a seed region (bilateral precentral gyrus: PCG), filtering ROIs (bilateral posterior limb of internal capsule, PLIC: cyan color; pons: red color), and tractography parameters (fraction, f_ij > 0.1, step size = 20% of voxel width, angular deviation < 45°, 100 seeds per voxel) was applied for each method. Compared to the other methods, the ICA+BSM imaged more lateral projections connecting the PCG to the pons via the PLIC.

In order to determine whether the additional streamlines identified corresponded to the correct anatomical location of the CST, percentage overlap maps derived from ICA+BSM, ICA, CSD+BSM, CSD, BSM, and DTI were compared with the histological map of the CST atlas (Fig. 7). All methods show a good anatomical correspondence in medial but not lateral portions of the CST. The ICA+BSM achieves substantially larger overlap of lateral CST projections across subjects.

Figure 7.

Comparison between percentage maps of post-mortem CST histology and diffusion tractography obtained from single tensor-based DTI, BSM, CSD, CSD+BSM, ICA, and ICA+BSM. The CST tractography of each method was performed from diffusion data of 20 normal controls. The fiber visitation maps of individual subjects were then transformed to MNI space by applying the deformation field obtained between the b0 map of individual subjects and the b0 template of MNI space. The normalized maps of each method were averaged across subjects and then scaled by the maximum visitation in the map to create a percentage overlap map. The percentage overlap map of each method was finally compared to the post-mortem histological map.

To quantify this larger overlap of the ICA+BSM to identify the histological location of the CST pathway, the receiver operating characteristic (ROC) curve and accuracy plot were evaluated at five different thresholds of percentage overlap ranging from 2% to 10% (Fig. 8 (a)). The ICA+BSM achieved the highest accuracy at the threshold of 2% where sensitivity/specificity/accuracy= 92.2/99.8/96% for ICA+BSM, 82.3/99.9/91.1% for ICA, 78.3/99.9/89.1% for CSD, 84.1/99.9/91.5% for CSD+BSM, 84.3/99.9/92% for BSM, and 71.9/99.5/82.1% for DTI. For instance, as presented in Fig. 8 (b) showing regions of true positive (TP), false positive (FP), and false negative (FN) at the threshold of 10%, the ICA+BSM produced the largest region of true positive (green-colored voxels) and also the smallest region of false negative (red-colored voxels) compared to the other methods.

Figure 8.

Assessment of anatomical correspondence achieved by single-tensor based DTI, BSM, CSD, CSD+BSM, ICA, and ICA+BSM tractography. (a) Receiver operating characteristic (ROC) curve (left) and accuracy plot (right) were estimated at five different thresholds of percentage overlap, from 2% to 10%. The ICA+BSM achieved the highest accuracy at the threshold of 2% (i.e., sensitivity/specificity/accuracy= 92.2/99.8/96% for ICA+BSM, 82.3/99.9/91.1% for ICA, 78.3/99.9/89.1% for CSD, 84.1/99.9/91.5% for CSD+BSM, 84.3/99.9/92% for BSM, and 71.9/99.5/82.1% for DTI). (b) For comparison, regions of TP, FP, and FN at the threshold of 10% were obtained for each method.

Validation with the primary face and hand areas defined by electrical stimulation

The CST fibers of two patients with focal epilepsy were presented in Fig. 9. The ICA+BSM method successfully visualized the lateral projections of CST fibers originating from the primary face and hand areas in both patients, where localization of motor pathways is important to minimize the risk of a postoperative motor deficit. Compared to the ICA alone, the ICA+BSM method successfully found the hand CST pathway (red-colored fibers) and face CST pathway (blue-cyan fibers) in both patients.

Figure 9.

CST fibers in two patients with focal epilepsy.

Extra-operative electrical stimulation determined the primary hand motor areas (A: red) and face motor areas (B: mouth-tongue, blue and C: mouth-lip, cyan). The hand CST fibers connecting the primary hand areas to cerebellar peduncles (CP, green) via the posterior limb of internal capsule (PLIC) were obtained using ICA and ICA+BSM (red fibers). Similarly, the face CST fibers connecting the face motor areas to the facial colliculus (FC) via PLIC were obtained (blue- and cyan fibers). The seed ROIs (motor areas), Filtering ROIs (PLIC, CP, FC) and tractography parameters (fraction, f_ij > 0.1, step size = 20 % of voxel width, angular deviation < 40°, 100 seeds per voxel) were identically applied for all cases.

Discussion and Conclusions

Two major findings emerge from this work. First, the parametric optimization of BSM parameters by using ICA-driven eigenvectors, [e_j1,e_j2,e_j3], and target diffusion profile, ṽ_i, dramatically enhances the accuracy of BSM in recovering correct orientations of individual fibers. The simulation study showed that combining ICA with BSM reduces errors up to more than 10° in recovering the orientations of two or three fibers, especially when the inter-fiber angles are greater than 30°, leading to increased probability of accurate reconstruction up to 5–10%. Secondly, the ICA+BSM outperforms other voxel-wise BSM methods that parameterize the orientations of multiple fibers using either arbitrary random selection (standalone BSM) or local peaks in ODF (CSD and CSD+BSM). Up to now, most BSM approaches have estimated the likelihood of model parameters based on a raw diffusion profile from an “individual voxel”, rather than the diffusion profiles from “neighboring voxels” that have been utilized for the first time in the present study. In-vivo, the crossing of prominent fiber tracts such as AF and CST is most likely observable in the cluster of neighboring voxels. Thus, the independent diffusion processes existing in the cluster can be better estimates of multiple fibers compared to those of multiple fibers observed in a single voxel, that is feasibly suited to be being trapped at local minimum and also require computational complexity in overall fitting procedure. As demonstrated in human studies, the ICA+BSM provided the highest anatomical correspondence to the CST in clinical diffusion data. By successfully imaging the lateral portion of CST, it could increase true positive fibers up to 7–10% whereas it reduced true negative fibers up to 10–15%, compared to the original ICA, BSM, and CSD+BSM methods.

The present study applied dimensionality reduction and ICA to isolate multiple diffusion components that are independently attenuated in each direction of diffusion sensitizing gradients. Principal component analysis showed that there exists up to three principal components (K=3) in the covariance of our DWI signals, U, accounting for more than 97 % of total variance which implies three components are sufficient to explain overall independent diffusion processes. Thus, the diffusion signals at local neighborhood voxels were constructed to a 2-D matrix, U, sized by voxel (row)×gradients (column), cleaned by whitening via eigenvalue decomposition with dimensionality, K, and loaded column-wise to estimate an unmixing matrix in the framework of fast ICA. If subsequent model selection such as BIC properly identifies the correct number of components, K_opt, mixed in U, this configuration will result in K_opt-MR signal attenuation components that are maximally independent in each of the applied gradient directions. The present study clearly demonstrated that different patterns of these attenuated components feasibly represent independent diffusion processes along multiple axonal fibers crossing at a single voxel.

In the computer simulations, the angular resolution of DSI and GQI was relatively worse than reported in the previous literature, and the discrepancy may be due to the different diffusion sampling schemes used in the present study. Since the diffusion sampling scheme plays an important role in the angular resolution (31), the results of this study may not assert that DSI and GQI cannot resolve fibers at 60 degrees crossing. Although the ICA recovers directionally independent diffusion components with respect to the applied gradient directions, the present study has not fully examined what number of gradient directions and b-value optimizes the non-Gaussianity of a single fiber to determine the performance of ICA+BSM for clinical DWI study. Future work will thoroughly investigate the effect of the diffusion sampling scheme on the accuracy of ICA+BSM.

For comparison with ICA+BSM, the present study utilized a CSD+BSM that used local ODF peaks as initial estimates of the stick compartments in BSM. Since the computer simulations allowed a high volume fraction for an isotropic diffusion compartment in order to model conservative partial-volume effects from isotropic tissue such as gray matter or cerebrospinal fluid, it might be possible that the standard CSD method used in the present study (14,15) suffers from severe instability due to a high volume fraction of the isotropic diffusion compartment and noise robustness (32), providing subsequent BSM with inaccurate estimation of fiber orientation. The computer simulations of this study supported this possibility by presenting similar accuracy between CSD+BSM and BSM. To overcome this limitation of the standard CSD, a higher-order tensor approximation (22) might be useful. This approximation was designated to incorporate accurate fiber orientations and relative volume fractions that explain the overall ODF. It might be a better alternative to the standard CSD+BSM. The CSD+BSM with the high order tensor approximation (22) might provide an effective solution to many intra-voxel problems in a voxel level, perhaps comparable to the presented ICA+BSM.

In conclusion, the BSM procedure definitely helps to minimize potential errors in voxels where fiber configurations contain incompatible ICA assumptions: 1) multiple fibers are significantly changed in orientation within voxels of a neighborhood cluster, and 2) multiple fibers do not cross all voxels of a neighborhood cluster.