Advances in Computational and Statistical Diffusion MRI

Abstract

Computational methods are crucial for the analysis of diffusion magnetic resonance imaging of the brain. Computational diffusion MRI can provide rich information at many size scales, including local microstructure measures such as diffusion anisotropies or apparent axon diameters, whole-brain connectivity information that describes the brain’s wiring diagram, and population-based studies in health and disease. Many of the diffusion MRI analyses performed today were not possible five, ten or twenty years ago due to requirements for large amounts of computer memory or processor time. In addition, mathematical frameworks had to be developed or adapted from other fields to create new ways to analyze diffusion MRI data. The purpose of this review is to highlight recent computational and statistical advances in diffusion MRI and to put these advances into context by comparison to the more traditional computational methods that are in popular clinical and scientific use. We aim to provide a high-level overview of interest to diffusion MRI researchers, with a more in-depth treatment to illustrate selected computational advances.

Graphical Abstract

Computational analyses of diffusion MRI can provide rich data about the brain, including tissue microstructure measures, whole-brain connectivity information, and population-based statistical analyses. In this review, we highlight recent computational and statistical advances in diffusion MRI and put these advances into context by comparison to the more traditional computational methods that are in popular clinical and scientific use.

1 Introduction

By measuring the diffusion of water molecules that probe the brain’s microstructure at the cellular scale, diffusion magnetic resonance imaging (dMRI) provides unique information about the living human brain in health and disease [1, 2]. Unlike traditional structural MRI acquisitions that directly provide scalar maps (such as T1-, T2-, or susceptibility-weighted images), dMRI acquisitions require multiple measurements to assess brain microstructure and connectivity. Current dMRI protocols produce from seven to hundreds of diffusion-weighted measurements, which are acquired at different orientations and, often, using different amounts of diffusion weighting. Due to the complex nature of this data, computational analysis methods are essential to produce advanced visualization or quantitation (Figure 1). In the application of dMRI to the study of brain microstructure and connectivity, computational analyses can be performed at multiple neuroanatomical scales ranging from the microstructure scale of diffusion compartments, to the voxel scale of local white or gray matter anatomy, to the whole-brain scale of connections estimated using tractography, and finally to the population scale for neuroscientific studies [3, 4, 5, 6, 7, 8]. Note, though, that there is an important mismatch of size scales between white matter neurons and much larger dMRI measurement voxels [9]. Many mathematical modeling approaches have tried to bridge this gap.

Figure 1.

Computational methods for diffusion MRI are needed to analyze input diffusion-weighted image (DWI) data to enable visualization or quantitation. At left, coronal images show DWIs acquired using several gradient directions. The yellow box indicates the zoomed region shown in the images at the right, which result from computational analysis of the DWIs. Colors encode fiber orientation such that left-right is red, inferior-superior is blue, and anterior-posterior is green. The traditional single diffusion tensor model (DTI, top right) is shown using ellipsoids for tensor visualization. DTI analysis has led to important findings, but it cannot represent anatomical fiber tract crossings (bottom right image, arrows) that can be seen with a multi-fiber model such as the two-tensor (2-T) [10, 11] tractography shown here. These images were created using the first dataset from the Human Connectome Project [12] using the SlicerDMRI package in 3D Slicer [13, 14].

Since the introduction of the diffusion tensor model in 1994 [15], computational methods for dMRI have become an increasingly active area of research. Techniques initially developed for scalar medical images, such as registration, segmentation, and statistical analyses, have been extended and adapted to various forms of dMRI data, including voxel-level and tract-level data representations. dMRI has enabled one unique computational analysis method, tractography, which is the process of non-invasively tracing the brain’s white matter connections (fiber tracts). Many of the computational dMRI studies performed today were not possible five, ten or twenty years ago: many analyses require large amounts of memory and long processor times, or had high computational complexity and required optimization to become feasible. In some cases (see Section 3), mathematical frameworks had to be developed or adapted from other fields (e.g. statistics or geometry) to create new ways to analyze dMRI data.

In this review, we will focus on the historical evolution and recent advances in three selected computational analyses of dMRI: tractography, statistical analysis, and registration. (dMRI tract-based segmentation was recently reviewed [16].) We will begin this review at the whole-brain scale, with tractography (Section 2), which is commonly performed in individual subject data. Next we will give an overview of the mathematics of statistical analysis of local diffusion models (Section 3), which are generally analyzed in groups of subjects. Finally we will discuss how different representations of dMRI data, both tractography and voxel-level diffusion information, can be used for registration (Section 4), which is most often employed across subjects. Where possible, we will contrast standard dMRI computational methods with more advanced ones to show when the latter might be useful, with consideration of the potential differences in computational cost. Finally, we will conclude with an overall look at future directions of computational diffusion MRI.

2 White matter tractography: a computational perspective

Tractography, the process of inferring white matter pathways from dMRI data, is an invaluable tool for noninvasively studying the brain’s structural connectivity [17, 18]. Despite the large number of available methods (see [19] for a comprehensive review), the reconstructed pathways are not truly quantitative and only indirectly reflect the actual white matter structure [20, 21]. In this section, we describe selected computational advances that represented major steps forward in terms of accuracy and interpretation of connectivity estimates (Fig. 2). All reported benchmarks are intended as indicative values of the algorithms (as described in the original publications) on a typical dataset size and a standard workstation, without using multi-threading or parallel computing, unless otherwise specified.

Figure 2.

Computational demands and benefits for quantifying the connectivity of notable formulations proposed to date for tractography: line propagation ( a, deterministic; b, probabilistic), global inverse problem ( c, bottom-up), microstructure informed tractography ( d, top-down; e, dictionary-based).

2.1 Tractography as line propagation

Tractography was originally formulated as a line-propagation problem:

$$\frac{\mathrm{d}\mathbf{f}(s)}{\mathrm{d}s}=\mathbf{v}\left(\mathbf{f}(s)\right)\text{with}\{\begin{array}{l}0\le s\le 1\hfill \\ \mathbf{f}(0)={\mathbf{f}}_0\hfill \end{array}$$ (1)

where a streamline f, parameterized by arc-length s, is constructed iteratively from a starting point f⁰ ∈ ℝ³ following the principal diffusion direction v estimated along the path with DTI [22, 23]. This deterministic procedure (Fig. 2(a)) is very fast (few minutes/subject) but also extremely sensitive to local inaccuracies in estimating the propagation direction; moreover, the reconstructed streamlines do not provide a direct means of quantifying the underlying connectivity [20, 24]. In probabilistic methods [25, 26], the propagation direction is sampled from a distribution estimated from advanced diffusion models (Fig. 2(b)), and probability maps are created by generating a high number of streamlines from each seed location. This tracking strategy is computationally expensive (≈ 1 day/subject) and may not substantially improve the estimation of connectivity, as these probabilities are difficult to interpret and have often been mistakenly considered as anatomical connection strengths [20].

2.2 Tractography as an inverse problem

Recent approaches have directly acknowledged the ill-posed nature of tractography and tackled the reconstruction as a global inverse problem (Fig. 2(c,d,e)):

$$\mathcal{F}=\text{argmax}\mathcal{E}\left(S,\widehat{S}(\mathcal{F})\right)$$ (2)

in which the set of streamlines ℱ, also known as tractogram, that maximizes the similarity ℰ(·, ·) between the acquired diffusion MR images, S, and those predicted from ℱ, Ŝ(ℱ), are estimated simultaneously using global optimization.

Bottom-up methods [27, 28] construct the tractogram starting from an initial set of fixed-length segments, where each segment contributes to the predicted signal according to a tensor-based generative model and the optimization encourages them to concatenate into long chains while fitting the measured data (Fig. 2(c)). While reconstructions showed an unparalleled level of detail, the tracking process became computationally impractical (≈ 1 month/subject). The model was later simplified by Reisert et al. [29], who significantly decreased the processing time to few hours/subject. Sherbondy et al. [30] approached the reconstruction problem in a top-down fashion (Fig. 2(d)). The BlueMatter algorithm built a massive collection of candidate streamlines (≈ 180 billion) combining the output from different tractography algorithms and used a stochastic search to determine the optimal subset that best explained the acquired dMRI data. Although it required 9 days on a 2048-core supercomputer with 500 GB of memory, BlueMatter offered a novel solution to tackle tractography from a different angle, opening new perspectives for the quantification of connectivity.

The Spherical-deconvolution Informed Filtering of Tractograms (SIFT) [31] followed the same pruning strategy but cut computational demands down to ≈ 15–20 hours/subject on standard workstations using an efficient heuristic to approximate the similarity ℰ. SIFT selectively removes streamlines guided by the fiber ODF (fODF) estimated in each voxel with Constrained Spherical Deconvolution (CSD) [32]; however, the procedure requires a very large number of initial candidates (≈ 100 million) and a considerable amount of memory (≈ 26 GB/subject). In SIFT2 [33], rather than discarding streamlines, the algorithm attempts to estimate their “effective volume” using a CSD-based generative model and assigning a cross-sectional area to each. This approach needs a lower number of candidates (≈ 10 million) and improves both computation time (few hours/subject) and memory consumption (few GB/subject). Recent studies showed that tractograms optimized with these pruning techniques provide more robust and biologically meaningful estimates of the underlying connectivity [34, 35]. Nevertheless, SIFT/SIFT2 use a similarity function based on the reconstructed fODFs rather than the full dMRI data, and thus cannot exploit possible additional information about the microstructural composition of the fascicles, e.g. apparent axon caliber, which may further improve quantification of connectivity.

2.3 Tractography with microstructure information

The integration of microstructure information into tractography represents the latest frontier in dMRI tractography. The first microstructure-informed tractography method was proposed by Sherbondy et al. [36], who extended the pruning strategy of BlueMatter (Fig. 2(d)) to allow simultaneous estimation of the optimal subset of fascicles as well as their biophysical parameters, invoking the assumption that these latter remain consistent along the fibers¹. MicroTrack uses a biophysical model derived from [37] to predict the diffusion MR images, Ŝ (ℱ), and infer the effective volume and apparent average axon caliber of each fiber f ∈ ℱ; specifically, the signal is expected to arise from water restricted inside the axons and hindered in the space around them, besides possible isotropic contributions from gray matter or cerebrospinal fluid. An efficient genetic algorithm enabled computation on standard workstations, but the processing time remained an issue (20 days/subject).

Malcolm et al. [10] approached the problem bottom-up and extended the line-propagation formulation of Eq. (1) to take into account microstructure information in the integration process. At each tracking position, an unscented Kalman filter is used to simultaneously fit a multi-tensor model and determine the most consistent propagation direction using the information estimated at previous positions. Recent methods extended this idea by incorporating more advanced tissue microstructural models; Reddy and Rathi [38] built on the previous approach and used an enhanced version of the NODDI model [39] to deal with crossing fibers, whereas Girard et al. [40] used the AMICO framework [41] to estimate, at each tracking position, the apparent mean axon caliber along the current tracking orientation. Being based on line propagation, these methods are considerably faster than MicroTrack (2 days/subject) but, at the same time, they also inherit all drawbacks of local formulations described earlier.

These pioneering approaches served as a proof of concept for the great potential of tractography to benefit from microstructure information; however, despite encouraging preliminary results, the complexity of these initial solutions has hampered their widespread application in clinical research. A dramatic reduction in the computational burden was achieved with the dictionary-based formulation introduced by Daducci et al. [42, 43] (Fig. 2(e)), who observed that the predicted signal from a tractogram ℱ, Ŝ (ℱ), can be conveniently expressed in matrix form as:

$$\widehat{S}(\mathcal{F})=\mathbf{Ax}$$ (3)

where A = {a_ij} ∈ ℝ^n_vn_d×n_c is a linear operator mapping a multicompartment model analogous to MicroTrack to the n_d diffusion measurements in all n_v voxels. The contributions $\mathbf{x}\in {ℝ}_{+}^{n_c}$ of the corresponding n_c compartments, e.g. the fraction of restricted water inside the axons corresponding to fibers in ℱ, can be efficiently estimated using convex optimization:

$$\underset{\mathbf{x}\ge 0}{\text{argmin}}{\Vert \mathbf{Ax}-\mathbf{y}\Vert }_2^2$$ (4)

where || · ||₂ is the usual ℓ₂-norm in ℝⁿ and the cost function is the distance between the acquired diffusion MR images S, contained in the vector $\mathbf{y}\in {ℝ}_{+}^{n_d{n}_v}$, and those predicted from the tractogram ℱ via the relation Ŝ (ℱ) = Ax.

This approach is named Convex Optimization Modeling for Microstructure Informed Tractography. COMMIT allows combination of tractography with virtually any diffusion model [44], and its convex formulation enables fast estimation (few minutes/subject) that is guaranteed to converge to the optimal solution. Its memory usage is quite efficient (few GB/subject): although A is extremely large, it is never stored explicitly and its columns are assembled at runtime, one at a time, using the Kronecker product and precomputed lookup tables. A similar mathematical formulation was recently implemented in [45], but the Linear Fascicle Evaluation (LiFE) method was designed for validating the existence of fascicles rather than assessing their microstructure properties.

Microstructure-informed tractography is a relatively young but very promising area of research and these recent advances offer new and exciting possibilities for connectivity mapping, potentially opening the door to truly quantitative and biologically meaningful analyses of the brain’s connectivity. Nonetheless, like any new technology, they also come with challenges and open questions (see [46] for an extensive discussion), which are expected to further stimulate research.

3 Statistics of local diffusion models

Statistical analyses of neuroimaging data enable the quantitative study of the human brain in health and disease [47, 48]. The mathematical characteristics of non-scalar entities derived from dMRI data, such as the DT or ODF, have led to exciting theoretical and computational developments. As mentioned in Section 4, defining a metric between these entities constitutes a key step to assess similarity, estimate distributions and test hypotheses which are critical for neuroscience [2] and clinical studies [49]. At the core of this line of research is the notion that the DT [1] and ODF [50, 51, 52] do not live in linear (Euclidean) spaces and therefore require appropriate theories to enforce physically meaningful properties, such as symmetry, positivity or integrability to 1, while avoiding potential statistical bias and significant increase in computational requirements. Compared to the computational costs of dMRI tractography or registration, the cost of statistical analyses often is lower and relates to the efficiency of metric computation.

The definition of differential geometrical structures for statistical models originated in 1936 with the work of Mahalanobis [53] for multivariate normal distributions with fixed covariance matrices. It was followed by Rao [54] who showed that the Fisher information matrix could be used to define a Riemannian metric, therefore called the Fisher-Rao metric [55] metric, between parameterized probability density functions. Since then, the information geometry field [56, 57] has provided a better understanding of those geometries, and several papers have leveraged these concepts to develop statistical analysis methods for dMRI data. We hereby review some of the works that have addressed these questions.

3.1 Diffusion tensors

Following a prior work where DTs were identified as elements of ℝ⁶ [58], which ignores the algebraic properties of symmetric and positive-definite matrices, the concept of a tensor-variate normal distribution [59] was introduced to capture the variability of DTs via a symmetric positive-definite fourth-order tensor. Such approaches provided a convenient framework to estimate DT distributions and perform hypotheses testing, albeit not taking into account the geometry of DTs. This was achieved shortly after by modeling the space of DTs as a Riemannian symmetric space GL⁺(3)/SO(3), where GL⁺(3) is the set of real 3 × 3 matrices with positive determinant and SO(3) is the set of real 3 × 3 orthogonal matrices with determinant 1 [60]. In this pioneering work, the authors established methods and provided algorithms to compute statistics and modes of variations for DTs, using a metric which is naturally invariant under transformations from GL⁺(3). Subsequently, Batchelor et al. [61], expanding on previous work by Moakher [62], used similar theoretical results to define a distance between tensors, obtain their mean and interpolate them. In addition, a new definition of anisotropy called Geodesic Anisotropy (GA) was introduced, which appeared to be more sensitive to small eigenvalues. Pennec et al. also leveraged this Lie group perspective and developed a computational framework to DT processing with a particular emphasis on interpolation, regularization and restoration of noisy tensor fields [63]. Finally, Lenglet et al. [64] relied on the information geometry of the space of multivariate normal distributions with zero mean to tackle similar theoretical questions, while focusing on the estimation and level-set based segmentation [65] of DTI data. A related and recent work, which focuses on the DT estimation problem [66], formulates a Bayesian framework for the simultaneous estimation and regularization of DTI data, while accounting for Rician noise in diffusion-weighted data and providing error bounds. The geodesic distance between two DTs, common to all these studies, is defined as

$$d(D_1,D_2)=\sqrt{tr\left({log}^2\left(D_1^{-1}{D}_2\right)\right)}$$ (5)

This affine-invariant metric prevents the generation of non-positive DTs and limits the “swelling” effect known to disproportionally increase eigenvalues and affect diffusivity and anisotropy estimates. However, one of the drawbacks of these approaches is their relatively high computational requirement due to matrix operations and the need for optimization methods, since closed form expressions do not exist in many cases. To address this situation, Arsigny et al. introduced the log-Euclidean framework [67, 68] and applied it to data interpolation and regularization. Under this framework, Riemmanian operations are converted to Euclidean ones by transforming DTs into their matrix logarithms (which are symmetric matrices). The key finding of this work was that, thanks to one-to-one correspondences via the matrix logarithm and exponential, tensors can be treated as vectors, thereby simplifying and speeding up computations. Equation 5 becomes

$$d(D_1,D_2)=\Vert log\left(D_1\right)-log\left(D_2\right)\Vert ,$$

but it should be noted that affine invariance is lost and replaced by similarity invariance. A comparison of these metrics, along with the introduction of a new Procrustes size-and-shape metric, with origin in shape statistics, was recently proposed by Dryden et al. [69].

Although scalar-based (fractional anisotropy, diffusivity, etc.) analyses still largely dominate when it comes to clinical or neuroscience applications, investigators have started to adopt these new developments to e.g. process clinical data with low SNR [70], and perform statistical analyses [71] to detect white matter lesions in patients with multiple sclerosis. However, despite their ability to enforce physically meaningful constraints, affine- or similarity-invariant metrics have been shown to potentially bias the analysis of normally distributed DT data [72], therefore requiring investigators to carefully select the metric in the light of their particular application.

Recent works have focused on multivariate parametric and nonparametric tests, as well as hypothesis testing procedures for DT data, and have investigated the relevance and impact of non-Euclidean metrics [73, 74, 75]. For instance, in Lee et al. [73], a Hotelling’s T² test applied to log-Euclidean transformed DT data was shown to outperform univariate analysis of DT-derived scalar measures. Again, careful considerations appear to be necessary for such parametric tests, which assume multivariate normality of DTs, and their interaction with non-Euclidean metrics [72, 76]. Using affine- or similarity-invariant metrics in combination with permutation-based [76, 77] or Cramer [76] tests also did not appear to improve the performance of these nonparametric tests. However, using the Cholesky decomposition of DTs, as introduced in [78], it was demonstrated that [79, 80], in addition to decreased computational time, a nonparametric two-sample test can detect differences in a specific brain region between young children with dyslexia and controls.

3.2 Orientation distribution functions

Generalizing the concepts, which we briefly outlined in the previous section, to orientation distribution functions derived from high angular resolution diffusion imaging is challenging, in part because of the multi-directional nature of ODFs. This can lead to the mixing or generation of erroneous fiber orientations when performing simple operations such as averaging [81]. ODFs are symmetric spherical probability density functions commonly approximated using the spherical harmonic basis [82], although other approximation methods using e.g. mixtures of von Mises-Fisher distributions [83] or spherical ridgelets [84] have also been proposed. The primary application of such approximation methods has been the accurate estimation and characterization of ODFs to extract white matter fiber orientations (e.g. for tractography). Defining metrics between ODFs for interpolation [85] or segmentation purposes [86] can easily be done by using the Euclidean distance between spherical harmonic coefficients [86], although this does not account for the intrinsic properties of the (Riemannian) space of ODFs. Chiang et al. [85] first introduced the idea of using the Fisher-Rao Riemannian metric to compare ODFs. In particular, they used this idea to perform spatial interpolation of ODFs and study genetic influences on white matter fiber organization. This metric had been introduced in the computer vision literature [87] and provides closed form expressions for geodesic distance, as well as exponential and logarithm maps. Considering the space of spherical probability density functions {p : 𝕊² ↦ ℝ⁺| ∫_𝕊² p(s)ds = 1}, the space of square-root representations $\psi (s)=\sqrt{p(s)}$ is the Hilbert unit sphere where the Fisher-Rao metric becomes the standard 𝕃² metric. It can be shown, for instance, that the geodesic distance is defined as

$$d({\psi }_1,{\psi }_2)={cos}^{-1}\left({\int }_{{\mathbb{S}}^2}{\psi }_1(s){\psi }_2(s)ds\right)$$

This was used by Cheng et al. [88] and Goh et al. [89, 90] to propose frameworks for computational and statistical analysis of ODF data. While Cheng et al. focused on the estimation and interpolation of ODFs, Goh et al. developed tools for the anisotropic filtering and statistical analysis of ODFs. They proposed applying the concept of principal geodesic analysis [91] to ODFs, to study the modes of variation of a set of ODFs, and they generalized the Hotelling’s T² statistic for performing multivariate hypothesis testing on two populations of ODFs. This was applied to detect asymmetries of the white matter, and led to the identification of significant clusters in the language and sensorimotor areas, for which the Riemannian metric appeared to be more sensitive than the Euclidean metric. Related work leveraging the properties of the Fisher-Rao metric in a sparse Riemannian spectral clustering framework also demonstrated the possibility to reliably extract fiber pathways in complex white matter areas [92].

Finally, recent contributions have aimed to design metrics producing more physically meaningful results, as they relate to the shape of ODFs. For instance, it has been demonstrated that interpolating single-fiber ODFs with peaks at different locations, using the Fisher-Rao metric, can result in two-fiber ODFs. To alleviate such issues, one can separate the orientation and shape information contained in an ODF, thereby defining a product space in which orientation and shape are independently compared and new statistics can be defined [93]. Similar ideas were pursued [81] by introducing a group action induced distance.

We have summarized some of the recent efforts to define metrics for DTs and ODFs, which respect their inherent mathematical properties, and with a particular emphasis on statistical analysis. It appears important to move beyond the scalar (e.g. fractional anisotropy, diffusivity) parametric or nonparametric population studies, to better leverage the great amount of information encoded in these local models of diffusion. Nonetheless, as we have outlined, care must be taken in doing so, as significant bias can be introduced when these metrics do not account for the proper distributions or noise models. However, it seems apparent that, at least for ODFs, recent works can enforce both mathematically and physically meaningful constraints which, ultimately, may prove useful in the statistical analysis of these data for neuroscience and clinical studies.

4 Registration methods for diffusion MRI

Registration is a key tool in medical imaging, with applications from population studies to surgical planning. Given two objects I and J, the registration problem is that of finding a transformation ϕ that, when applied to the object J, makes it “look” as close to I as possible. The registration problem is usually formulated as the minimization problem

$$arg\underset{\varphi \in \mathrm{\Phi }}{min}\mathcal{M}(I,{\varphi }^{\ast }J)+\mathcal{R}(\varphi ).$$ (6)

Equation (6) aims to find the transformation ϕ from the set of transformations Φ that minimizes the dissimilarity measure ℳ between I and ϕ^*J. Throughout this paper, ϕ^*J represents the J deformed according to the transformation ϕ. We prefer this notation to the less general composition rule J ∘ ϕ as these deformations have different mathematical interpretations depending on whether J is a scalar image, a diffusion-weighted image or a three-dimensional object such as a white matter tract. Furthermore, to reduce the possible space of solutions, hypotheses on ϕ such as the regularity are represented by the term ℛ(ϕ).

Finding the optimal deformation ϕ, which is usually either a linear transformation or a deformable one, by minimizing Equation (6), has three main points that must be specifically addressed in the dMRI case:

1. How to quantify the difference between two elements representing or computed from diffusivity, specifically the formulation of ℳ(·, ·)

2. How the diffusivity information or its derivatives such as tracts must be warped, specifically how ϕ^*J is formulated and implemented.

3. How to incorporate the previous two points into the minimization scheme in Equation (6).

In the remainder of this section, we focus on the different approaches to the aforementioned items and we refer the reader interested in the deformation space Φ, the regularization term ℛ(ϕ), and the overall optimization schemes to existing reviews on this topic [see e.g. 94].

4.1 Image-based registration

In the case where I and J are images, they are commonly viewed as spatial functions I, J : ℝ³ ↦ 𝕀. In this formulation, 𝕀 can denote a subset of real numbers 𝕀 ⊆ ℝ for scalar image registration, a subset of 3 × 3 positive definite matrices $\mathbb{I}\subseteq {\mathbb{S}}_{+}^3$ in the case of DTI, and more generally, a subset of spherical 𝕀 ⊆ (S² ↦ ℝ) for HARDI or three-dimensional functions 𝕀 ⊆ (ℝ³ ↦ ℝ) for multi-shell or DSI approaches. In the scalar case, the application of the transformation ϕ to the image J takes the form

$$({\varphi }^{\ast }J)(x)=J(\varphi (x)).$$ (7)

However, in cases where the images are vector fields, tensor fields, or more complex representations of the dMRI signal, an extra transformation is required. We have illustrated the need for this transformation in Figure 3. Then, applying the transformation ϕ to the image J can be divided into two steps:

Figure 3.

Two rotation transformations applied to the dMRI field on the left. The one on the top right does not include the warping of diffusivity information; the one on the bottom right does.

1. Obtain the diffusivity information N ∈ 𝕀 of the warped image ϕ^*J at x:

   $$N=J(\varphi (x))$$ (8)
2. Warp the diffusivity information N according to the deformation ϕ:

   $${\varphi }^{\ast }N=\text{warp}(N,D_x\varphi ).$$ (9)

In accordance to the formalization of Cao et al. [95], these two steps, represent the action of ϕ on tensor fields. Specifically, the first step is the change of coordinates, while second one, which depends on D_xϕ, the Jacobian of ϕ at x, accounts for reorientation and, if needed, scaling.

Much of the effort on dMRI registration is focused on the warp in Equation (9). This presents two important computational challenges: first, registration algorithms have to devise a warp operator for the diffusivity data; second, the algorithms must incorporate the warp in the optimization process.

The most common approach in statistical analyses of dMRI images is still to choose a scalar measure derived from a dMRI acquisition and perform registration on that scalar image [96]. However, these techniques drive registrations according to a sub-optimal subset of the information contained in a dMRI image. To address this problem, Alexander et al. [97] studied approaches for diffusion tensor (DT) registration. They compared using a combination of scalar measures vs. taking the Frobenius norm between tensors while testing different methodologies to warp the diffusion information. They showed that, for most DT applications, quantifying the image difference using full tensor information and warping by reorienting the tensors yielded an improved registration result. This first step in dMRI image registration created a lasting framework for diffusivity information warping and left two problems open: first, the DT warping was performed in an alternate optimization scheme, leading to instability in the solutions; second, the proposed framework was hard to extend to the novel trend of diffeomorphic optimization techniques, which was achieving more stable results in scalar registration and template estimation [98, 99].

The results produced by Alexander et al. [97] opened a new research trend: the exploration of similarity measures for DT images. Ruiz-Alzola et al. [100] proposed a generalization of the cross-correlation metric for scalar images to the tensor space. Their work was more robust to noise than that of Alexander et al. [97]. However, incorporating the DT rotation into the optimization framework proved to be a difficult task. To decrease the computational costs of deformable registration, Park et al. [101] proposed to frame the problem in terms of the multichannel Demons algorithm [102] using a vector with components including the T₂ image obtained by setting the b-value to 0s/mm² in the dMRI and scalar measures derived from DT images, as well as components of the DT. Although this made the registration much faster and reduced memory consumption, it added the non-trivial problem of how to weight the different channels. Additional multi-channel approaches explored the use of mutual information as a metric to perform affine and nonrigid registration using DTI or DWI images [103, 104], with tensor reorientation in the case of DTI images. Up to this point, registration techniques had begun to agree on the idea of using the Frobenius norm for tensors. Being able to perform deformable diffeomorphic registration and template estimation of dMRI images was, nonetheless, still an open problem.

Zhang et al. [105] proposed a diffeomorphic registration algorithm and deformable template estimation using the Euclidean norm between the deviatoric tensor of a DT as a metric and parameterizing the deformation field as a multi-scale patchwork of affine deformations. This parameterization is invertible, greatly improving the template estimation process. Within this framework, the gradient of the registration cost function, shown in Equation (6), can be computed analytically, enabling the use of a large number of efficient optimization strategies to register the images. Previous registration approaches had not produced an analytic gradient for the registration cost function, instead implementing it thorough numerical or heuristic approaches resulting in algorithms with reduced numerical stability. Soon after, Cao et al. [106] used the Frobenius norm and a clever formulation of the Preservation of Principal Direction (PPD) [97] warping technique to formulate the problem within the LDDMM framework [107], which most commonly uses the Euclidean metric between image elements. Both approaches were stable and suitable for generating dMRI templates, while the Zhang et al. [105] approach required less computation time. Later, Yeo et al. [108] took this a step further, deriving a general framework for incorporating the gradient of the DT warp, shown in Equation (9), into a larger set of registration techniques. This enabled them to produce a diffeomorphic Demons algorithm for DT registration and to compare different similarity metrics for tensor fields.

As more advanced acquisition techniques for dMRI became common [109, 110], registration approaches were extended and adapted. In these approaches, the diffusivity M at each point of the images was now a spherical function, such as the ODF, [see e.g. 109], or a 3-dimensional function, such as raw DSI information [see e.g. 110]. Goh et al. [111] addressed the ODF case by first devising a Riemmanian space to embed these functions and then deriving a diffeomorphic registration algorithm [112]. Hsu et al. [113] addressed the DSI case by registering the image in the full 6-dimensional space (3 spatial dimensions + 3 dimensions indicating a 3D vector in diffusion space). On the other hand, Zhang et al. [114] opted for a more general approach. Their work aimed to register the raw DWI images using the LDDMM framework while reorienting and resampling the spherical functions at each voxel to match the acquisition scheme [115].

4.2 White matter tract-based approaches

An alternative to image-based registration is to perform registration on the white matter tracts directly [19]. Note that image registration techniques act on the spatial domain of the diffusion image while tracts are usually defined as curves on the image’s co-domain. Hence, to register tracts while keeping the parameterization of deformations consistent with image-based registration, the deformation needs to be applied on the co-domain of the diffusion image. In practice, the tracts must be deformed in the opposite direction than the images (i.e. by applying the forward transformation to the tract points rather than the inverse or image resampling transformation). An example of this is the application of a deformation to a single point in the co-domain of an image: given an image consisting of a dimensionless point at coordinate c, J(x) = δ(c − x), applying the deformation ϕ to the domain of J is equivalent to applying ϕ⁻¹ to the coordinate c, which is defined on J’s co-domain. More formally,

$${\varphi }^{\ast }J=J(\varphi (x))=\delta (c-\varphi (x))=\delta ({\varphi }^{-1}(c)-x)$$ (10)

The first proposed tract registration approach performed rigid registration of whole-brain fiber tracts by first finding the best matched pairs of fibers across the two datasets, then estimating a local transformation using points from each fiber pair, and finally computing an overall global transform [116]. This process was repeated in an iterative multiscale fashion. To perform population statistics on tracts, Corouge et al. [117] proposed a framework to register groups of tracts using linear transformations. This technique represented tracts as sequences of N points and used the Euclidean distance between these matrices of ℝ^N×3. This representation imposes two artefacts on the tracts: an orientation and a fixed discrete length. To address this, Corouge et al’s approach required a careful reparameterization of the tracts to have the same length and orientation. Ziyan et al. [118, 119] produced a deformable registration for tracts. They achieved this by representing the tracts as images of point density in space and quantifying the similarity between tracts through well-known measures of density agreement. This representation enabled them to remove the requirement of a point-to-point correspondence and equivalent parameterization of white matter tracts. Durrleman et al. [120] took this a step further by representing the tracts as dense vector fields and implementing an LDDMM registration algorithm. Their approach used the currents mathematical framework to explicitly take into account the orientation of the tracts and enable sparse representation to reduce computation time. The main drawback of this approach was due to the formalization of tracts as oriented parametric curves. Consequently, their approach needs a coherent tract orientation: two otherwise identical tracts, but with opposite orientations, will be considered infinitely different. Then, Wassermann et al. [121] combined the two approaches based on density representations of tracts [120, 118] to produce a diffeomorphic tract registration and template estimation algorithm that did not require consistent orientation. Finally, two recent approaches produced efficient tract-based unbiased template estimations [122] and statistical analyses [123] based on explicit tract parameterizations.

4.3 Computational costs

For dMRI image registration algorithms, once the registration algorithm and the representation of the deformation are fixed, the computational cost increases with the dimensionality of the representation of diffusion information. This is due to the computation of the metric, which usually increases at most quadratically with the diffusion representation [108], but in general increases linearly with the number of components [e.g. 111, 113, 105]. Moreover, a more costly increment to the computational cost is the warp strategy, shown in eq. (9), which is at least quadratic with respect to the dimensionality of the diffusion information [e.g. 97], but it can be much higher when diffusion information is non-parametric [115]. Overall, the computational cost of dMRI registration depends on both the metric and warping operations needed for dMRI registration.

In the case of tract-based registration algorithms, the cost can be divided in two trends: techniques working directly on tracts [122, 123] whose computational cost depends directly on the number of tracts and the number of points representing each tract, and those representing tract bundles as spatial functions whose cost is equivalent to registering a scalar or vector image [e.g. 121, 120, 118]. In general, if the number of tracts registered is large, e.g., a full-brain tractography of a hemisphere, the run time is at least as long as that of full dMRI image registration. However, if it is not large, e.g., the registration is aimed at a specific bundle, the run time will be lower than dMRI image registration techniques.

5 Future directions in computational dMRI

Today, there are many possible dMRI acquisition and analysis strategies for neuroimaging research. The choice of methods may depend on the neuroscientific or clinical question of interest, the constraints for acquisition and computational processing time, and the researchers’ awareness of potential pitfalls. There are still many open challenges in dMRI, including non-specificity of traditional DTI-derived measurements [124, 125], uncertain anatomical accuracy of dMRI tractography [126, 127, 128], and a general lack of ground truth, which is partially addressed using synthetic or physical fiber phantoms [129, 130]. These challenges are expected to inspire future developments in computational dMRI.

Quantitative and anatomical validation is becoming an important aspect of computational dMRI. The dMRI analysis field is developing a strong track record of community validation challenges at MICCAI, ISMRM, and ISBI conferences², including challenges in tractography in phantoms [129, 131], tractography in clinical cases [132], and model reconstruction [133, 134]. The clinical community is becoming increasingly aware that multi-fiber models beyond the diffusion tensor can improve anatomical accuracy of tractography [135, 136, 137, 138, 139, 140, 11, 141, 142], and there is increasing development of open-source dMRI software that moves beyond the diffusion tensor [143, 10, 144, 14, 145, 146, 147, 148].

“Big data” is another important direction for future developments in computational dMRI. Large open datasets, such as the Human Connectome Project [12] and the UK Biobank [149], are being acquired and released to enable the study of multimodal imaging, genetics, and bioinformatics data. To handle big data, future computational dMRI algorithms are expected to require improvements in speed, memory usage, and the ability to correlate dMRI information with other multimodal data. Another research direction is to synthesize large dMRI datasets by statistically harmonizing multiple smaller datasets [150] to reduce across-sites variability of measures such as fractional anisotropy [151].

The advances described in this paper represent crucial steps forward for dMRI computational analysis. However, important basic research is underway that may change our assumptions about the brain’s microstructure and connectivity: current theories about the organization of axons into sheets [152, 153, 154, 155] and the possibility that axons take sharp turns or branch at 90 degrees throughout the brain [156] have important implications for the future of tractography methods and microstructure models. In fact, the true three-dimensional morphology of neurons is not yet known, and it is currently an active field of study [157]. In the future, more detailed microstructural and neuroanatomical information may become available using advanced “multidimensional” dMRI acquisitions that employ diffusion-sensitizing magnetic field gradients in sophisticated ways [158, 159]. Such novel dMRI acquisitions will require future advances in mathematical modeling and computational analysis.

6 Conclusion

Thus far, standard DTI-based computational methods, which have been available since the mid-nineties, have had a large impact in scientific research [160]. More recently, significant progress has been made with HARDI, multishell, q-ball, and DSI imaging [161]. However, there is still a need for methods that can measure and model the brain’s true underlying tissue geometry and biophysical parameters. Based on the current progress in validating advanced methods and the expected future increases in computational power, we look forward to an increasingly successful application of advances in computational and statistical dMRI to neuroimaging research.

Acronyms

MRI

magnetic resonance imaging

dMRI

diffusion magnetic resonance imaging

DWI

diffusion-weighted image

DT

diffusion tensor

DTI

diffusion tensor imaging

ODF

orientation distribution function

HARDI

high angular resolution diffusion imaging

DSI

diffusion spectrum imaging