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Fiber orientation mapping using generalized diffusion tensor imaging
Abstract
Generalized diffusion tensor imaging uses tensors of arbitrary ranks to model the angular variations in the diffusivities measured by magnetic resonance imaging (MRI) methods. However, a diffusivity profile alone is not readily capable of producing distinct fiber orientations. In this work, we show how it is possible to get the displacement probability profile for water molecules from the higher rank diffusion tensors and validate the technique via simulations of one, two and three fiber systems. Finally, we present fiber orientation results for an image from an excised rat brain.
... We estimate 4 th order HOTs using the Riemannian and the "Ternary Quartic" (TQ) approaches and plot their ADCs. Further, since the maxima of the ADCs don't correspond to the fiber directions, we also compute the diffusion ensemble average propagators (EAPs) P(r) = (S i (q)/S 0 ) exp −2πiq T r dq, from the estimated 4 th order tensors [27,28]. ...
... In this experiment we estimate 4 th order GDTI diffusion tensors from the phantom dataset using both the Riemannian approach and the TQ approach. We then compute the EAPs from the tensors using the methods in [27,28] to validate the coherence of their geometry with the known layout of the phantom and to see if it is possible to infer the underlying fiber bundle directions. For the sake of comparison we also present the result of the orientation distribution function (ODF) computed from the analytical q-ball estimation technique in [30], which is an angular marginal distribution of the true and unknown EAP under a mono-exponential decay model that corresponds to the GDTI model. ...
... However, the increased estimation time due to the complexity of the positivity constraint is still tractable. Finally, we conclude the experiments, by computing the EAPs from tensors estimated using both the Riemannian method and the TQ method from the in vivo human dataset (using [27,28]). For comparison we include the EAPs computed from tensors estimated using the LS method (using [27,28]). ...
High Order Cartesian Tensors (HOTs) were introduced in Generalized DTI (GDTI) to overcome the limitations of DTI. HOTs can model the apparent diffusion coefficient (ADC) with greater accuracy than DTI in regions with fiber heterogeneity. Although GDTI HOTs were designed to model positive diffusion, the straightforward least square (LS) estimation of HOTs doesn’t guarantee positivity. In this chapter we address the problem of estimating 4th order tensors with positive diffusion profiles.Two known methods exist that broach this problem, namely a Riemannian approach based on the algebra of 4th order tensors, and a polynomial approach based on Hilbert’s theorem on non-negative ternary quartics. In this chapter, we review the technicalities of these two approaches, compare them theoretically to show their pros and cons, and compare them against the Euclidean LS estimation on synthetic, phantom and real data to motivate the relevance of the positive diffusion profile constraint.
... Generalized diffusion tensor imaging (GDTI) [1][2][3], was proposed to model the apparent diffusion coefficient (ADC) recovered by diffusion MRI (dMRI) when imaging the diffusion of water molecules in heterogeneous media like the cerebral white matter. Essentially, GDTI uses higher-order Cartesian tensors (HOTs) to model the spherical profile of the ADC. ...
... But computing the EAP from the HOT model of the ADC in GDTI is not an easy task. In [2], the authors proposed a numerical fast Fourier transform scheme-to emulate diffusion spectrum imaging (DSI) [15] from GDTI-to estimate the EAP and to recover the underlying fiber directions. However, this method is computationally expensive, and although the numerical Fourier transform can compute the values of the EAP at desired points, it cannot compute a continuous and differentiable function which has great advantages. ...
... However, for general k > 2, closed forms for the Cartesian Fourier transform of E(q) k are hard to compute, since in Cartesian coordinates, E(q) k is not separable in q 1 , q 2 , and q 3 , the components of q. In [2], where a method for recovering the EAP from GDTI is proposed, P(r) k is computed numerically by evaluating E(q) k more or less densely in q-space and by computing its fast Fourier transform. ...
Generalized diffusion tensor imaging (GDTI) was developed to model complex apparent diffusivity coefficient (ADC) using higher-order tensors (HOTs) and to overcome the inherent single-peak shortcoming of DTI. However, the geometry of a complex ADC profile does not correspond to the underlying structure of fibers. This tissue geometry can be inferred from the shape of the ensemble average propagator (EAP). Though interesting methods for estimating a positive ADC using 4th-order diffusion tensors were developed, GDTI in general was overtaken by other approaches, for example, the orientation distribution function (ODF), since it is considerably difficult to recuperate the EAP from a HOT model of the ADC in GDTI. In this paper, we present a novel closed-form approximation of the EAP using Hermite polynomials from a modified HOT model of the original GDTI-ADC. Since the solution is analytical, it is fast, differentiable, and the approximation converges well to the true EAP. This method also makes the effort of computing a positive ADC worthwhile, since now both the ADC and the EAP can be used and have closed forms. We demonstrate our approach with 4th-order tensors on synthetic data and in vivo human data.
... The tissue microstructure can be inferred from the shape of the EAP. However, to compute the EAP from the HOT model of the ADC in GDTI is no easy task [7]. That is perhaps the reason why GDTI approaches in general have been overtaken by other methods estimating the EAP or its characteristics directly , such as the ODF. ...
... 4,4 makes it computationally very efficient , especially since the expression for a fixed n can be hard coded and compiled. In the next experiment we compare our approach to [7] on again a synthetic dataset with b = 3000s/mm 2 and the profile for a single fiber coming from the diagonal diffusion tensor with values [1700, 300, 300] × 10 −6 mm 2 /s. For visualization and comparison we consider a slice with 30 × 30 voxels. ...
... For visualization and comparison we consider a slice with 30 × 30 voxels. For the implementation of [7] we evaluate E(q) 4,4 on a 21 × 21 × 21 Cartesian grid before computing the FFT. We evaluate the numerical EAP on a spherical mesh with 162 vertices. ...
Generalized Diffusion Tensor Imaging (GDTI) was developed to model complex Apparent Diffusivity Coefficient (ADC) using Higher Order Tensors (HOT) and to overcome the inherent single-peak shortcoming of DTI. However, the geometry of a complex ADC profile doesn't correspond to the underlying structure of fibers. This tissue geometry can be inferred from the shape of the Ensemble Average Propagator (EAP). Though interesting methods for estimating a positive ADC using 4th order diffusion tensors were developed, GDTI in general was overtaken by other approaches, e.g. the Orientation Distribution Function (ODF), since it is considerably difficult to recuperate the EAP from a HOT model of the ADC in GDTI. In this paper we present a novel closed-form approximation of the EAP using Hermite Polynomials from a modified HOT model of the original GDTI-ADC. Since the solution is analytical, it is fast, differentiable, and the approximation converges well to the true EAP. This method also makes the effort of computing a positive ADC worthwhile, since now both the ADC and the EAP can be used and have closed forms. We demonstrate on 4th order diffusion tensors.
... There are currently two classes of high order processing methods for these high resolution acquisition techniques. The first is based on apparent diffusion coefficient (ADC) modeling [2, 9, 12, 21, 22] and the other is based on the estimation of the probability density function of the average spin displacement of water molecules [1, 5, 7, 8, 20, 25, 26, 35, 38, 39]. ...
... In the first case, the natural generalization of classical DTI is to describe the ADC profile with a higher order diffusion tensor (HODT) [21, 24, 25]. This formulation has been shown ([10, 21]) to be equivalent to spherical harmonic series approximation techniques [2, 10, 12], where non-Gaussian diffusion can be characterized using high order anisotropy measures based on the HODT description of the spherical harmonic series coefficients of the ADC profile [10]. ...
... In the 2-fiber case, both fibers are identified only at high b-value of 4000 whereas in the 3-fiber case, the third fiber is never detected. While it is common to use min-max normalization ([38]) and minimum inscribed sphere (MIS) subtraction ([25]) to enhance visual angular contrast when visualizing ODFs, these methods do not enhance the different underlying fiber compartments and do not improve maxima extraction. One way to deal with this problem is to introduce ODF sharpening methods to enhance each underlying fiber distribution. ...
Due the well-known limitations of diffusion tensor imaging (DTI), high angular resolution diffusion imaging is currently of great interest to characterize voxels containing multiple fiber crossings. In particular, Q-ball imaging (QBI) is now a popular reconstruction method to obtain the orientation distribution function (ODF) of these multiple fiber distributions. The latter captures all important angular contrast by expressing the probability that a water molecule will diffuse into any given solid angle. However, QBI and other high order spin displacement estimation methods involve non-trivial numerical computations and lack a straightforward regularization process. In this paper, we propose a simple linear and regularized analytic solution for the Q-ball reconstruction of the ODF. First, the signal is modeled with a physically meaningful high order spherical harmonic series by incorporating the Laplace-Beltrami operator in the solution. This leads to an elegant mathematical simplification of the Funk-Radon transform using the Funk-Hecke formula. In doing so, we obtain a fast and robust model-free ODF approximation. We validate the accuracy of the ODF estimation quantitatively using the multi-tensor synthetic model where the exact ODF can be computed. We also demonstrate that the estimated ODF can recover known multiple fiber regions in a biological phantom and in the human brain. Another important contribution of the paper is the development of ODF sharpening methods. We show that sharpening the measured ODF enhances each underlying fiber compartment and considerably improves the extraction of fibers. The proposed techniques are simple linear transformations of the ODF and can easily be computed using our spherical harmonics machinery.
... However, in spite of the interests in HOTs to describe complex shaped ADCs, the tissue microstructure can only be inferred from the shape of the EAP. However, to compute the EAP from the HOT model of the ADC in GDTI1 is no easy task [84]. That is perhaps the reason why the GDTI1 approach has been overtaken by other methods that estimate the EAP or its characteristics directly from the signal, such as QBI, PAS-MRI, DOT, SD etc. [28,85,59,58,60,86,87]. ...
... However, for general k > 2, closed-forms for the Cartesian Fourier Transform of E(q) k are hard to compute, since in Cartesian coordinates E(q) k isn't separable in q 1 , q 2 , q 3 , the components of q. In [84], where a method for recovering the EAP from GDTI1 is proposed, P (r) k is computed numerically by evaluating E(q) k more or less densely in q-space and by computing its fast Fourier Transform. ...
... The synthetic data was generated by simulating the MR signal from a single fiber using the realistic diffusion MR simulation model in [15]. Then, we added different amounts of Riccian noise to the simulated dataset and we estimated the 4 th -order tensors from the noisy data by: a) minimizing without using the proposed parametrization to enforce PSD constraint, by employing the method in [11] and b) our method, which guarantees the PSD property of the tensors. (S i is the MR signal of the i th image and S 0 is the zero-gradient signal). ...
... First, we estimated a 4 th -order diffusion tensor field from this dataset by minimizing without using the proposed parametrization to enforce positivity [11]. As expected, some of the estimated tensors were not positive. ...
In Diffusion Weighted Magnetic Resonance Image (DW-MRI) processing a 2nd
order tensor has been commonly used to approximate the diffusivity function at each lattice point of the DW-MRI data. It is now well known that this 2nd
-order approximation fails to approximate complex local tissue structures, such as fibers crossings. In this paper we employ a 4th
order symmetric positive semi-definite (PSD) tensor approximation to represent the diffusivity function and present a novel technique to estimate these tensors from the DW-MRI data guaranteeing the PSD property. There have been several published articles in literature on higher order tensor approximations of the diffusivity function but none of them guarantee the positive semi-definite constraint, which is a fundamental constraint since negative values of the diffusivity coefficients are not meaningful. In our methods, we parameterize the 4th
order tensors as a sum of squares of quadratic forms by using the so called Gram matrix method from linear algebra and its relation to the Hilbert’s theorem on ternary quartics. This parametric representation is then used in a nonlinear-least squares formulation to estimate the PSD tensors of order 4 from the data. We define a metric for the higher-order tensors and employ it for regularization across the lattice. Finally, performance of this model is depicted on synthetic data as well as real DW-MRI from an isolated rat hippocampus.
... a common 'ad-hoc' enhancement principle already adopted in practice [31]. The ADC is typically expressed in terms of a fully symmetric higher order (Cartesian) tensor or in terms of real-valued spherical harmonic functions, both of which transform in a straightforward manner under the proposed enhancement. ...
In this short note we consider a method of enhancing diffusion MRI data based on analytically deblurring the ensemble average propagator. Because of the Fourier relationship between the normalized signal and the propagator, this technique can be applied in a straightforward manner to a large class of models. In the case of diffusion ten-sor imaging, a commonly used 'ad hoc' min-normalization sharpening method is shown to be closely related to this deblurring approach. The main goal of this manuscript is to give a formal description of the method for (generalized) diffusion tensor imaging and higher order apparent diffusion coefficient-based models. We also show how the method can be made adaptive to the data, and present the effect of our proposed enhancement on scalar maps and tractography output.
... Similar observation was also reported by Tuch et al. [152], Zhan and Yang [165] in vivo. Due to this fact, the diffusivity profile can not be used directly for extracting fiber orientations, and one might still need to investigate the average diffusion propagator by taking the Fourier transform of the signal attenuation implied by the diffusivity profile as done in [115,119]. ...
... Unlike in DTI, the analytical Fourier transform of the tensor model in Eq. (22) is unknown. In [91], a fast Fourier transform was performed on interpolated (and extrapolated) q-space data on a Cartesian grid generated from the tensor in Eq. (22) to numerically estimate the EAP. In [92], an analytical EAP on a single R 0 -shell, i.e., P(R 0 r ||r|| ), was proposed for this model. ...
Diffusion imaging is a noninvasive tool for probing the microstructure of fibrous nerve and muscle tissue. Higher-order tensors provide a powerful mathemat-ical language to model and analyze the large and complex data that is generated by its modern variants such as High Angular Resolution Diffusion Imaging (HARDI) or Diffusional Kurtosis Imaging. This survey gives a careful introduction to the foundations of higher-order tensor algebra, and explains how some concepts from linear algebra generalize to the higher-order case. From the application side, it re-views a variety of distinct higher-order tensor models that arise in the context of diffusion imaging, such as higher-order diffusion tensors, q-ball or fiber Orientation Distribution Functions (ODFs), and fourth-order covariance and kurtosis tensors. By bridging the gap between mathematical foundations and application, it provides an introduction that is suitable for practitioners and applied mathematicians alike, and propels the field by stimulating further exchange between the two.
... Hence, diffusion orientation can be measured through the Orientation Distribution Function (ODF) defined as the radial projection of the spherical diffusion function. Other higher order models based also on the Gaussian assumption of the diffusion have been proposed in the literature: multi-fiber Gaussian tensors which model the signal as a finite number of Gaussian fibers [25] and spherical deconvolution techniques which use Gaussian kernels to estimate the diffusion signal [17, 24]. These approaches are model-based methods, implying a strong a priori knowledge about the local fiber configuration. ...
Abstract We address the problem,of estimating complex,diffusion models from high angular resolution diffusion MRI images (also known,as HARDI datasets). Rather than considering a classical 2nd-order tensor to model the water molecule diffusion in tissues, we describe each voxel diffusion by a model-free orientation distribution function (ODF) expressed as a set of spherical harmonics,coefficients. We propose to estimate the ODFs volume,directly from the raw HARDI data by minimizing a nonlinear energy functional which considers the non-gaussianity of the MRI Rician noise as well as introduces a regularity constraint on the estimated field. The estimation is thus performed,by a set of multi-valued partial differential equations composed,of both robust estimation and discontinuity-preserving regularization terms. We show that fiber-tracking is more accurate when using this regularized estimation as opposed to non-regularized methods. We finally illustrate the importance of these constraints in the ODFs estimation process through both synthetic and real HARDI datasets.
... Second, we only compared three kernel smoothing methods and didn't consider entirely different DTI smoothing techniques that do not fit into the kernel smoothing framework. Third, our experiments were applied to diffusion tensor data, and it is not clear how they generalize to High Angular Resolution Diffusion Imaging (HARDI), a more advanced and increasingly widespread extension of DTI that allows a much richer representation of local water diffusion properties [9]. ...
Diffusion tensor magnetic resonance imaging (DTI), a method for measuring the integrity of axon fiber tracts in the brain, plays an important role in clarifying brain changes that accompany aging and aging-associated neurodegenerative disease. While DTI smoothing methods theoretically have the potential to enhance such studies by reducing noise, it is unclear whether DTI smoothing has any practical impact on computed associations between fiber tract integrity and scientific variables of interest. Therefore we smoothed DTI images from 154 older adults using three kernel smoothing methods hypothesized to have differing strengths (the affine and log-Euclidean smoothers were hypothesized to enhance highly organized tracts better than the Euclidean smoother). Smoothing increased the strengths of expected associations between DTI and age, cognitive function, and the diagnosis of dementia. However, no particular smoothing method was uniformly superior in strengthening these associations. This data suggests that DTI smoothing enhances the sensitivity of studies of brain aging, but further research is needed to determine which smoothing technique is optimal.
... The algorithm recovers the orientation of the capillaries consistently. In Chap. 10 and [38], Ozarslan et al. fit higher-order tensor models (see Sect. 5.3.2) to measurements from a spherical acquisition scheme. They assume that A(q) decays exponentially with increasing |q| and fixedˆqfixedˆ fixedˆq. ...
This chapter gives an introduction to the principles of diffusion magnetic resonance imaging (MRI) with emphasis on the computational
aspects. It introduces the philosophies underlying the technique and shows how to sensitize MRI measurements to the motion
of particles within a sample material. The main body of the chapter is a technical review of diffusion MRI reconstruction
algorithms, which determine features of the material microstructure from diffusion MRI measurements. The focus is on techniques
developed for biomedical diffusion MRI, but most of the methods discussed are applicable beyond this domain. The review begins
by showing how the standard reconstruction algorithms in biomedical diffusion MRI, diffusion-tensor MRI and diffusion spectrum
imaging, arise from the principles of the measurement process. The discussion highlights the weaknesses of the standard approaches
to motivate the development of a new generation of reconstruction algorithms and reviews the current state-of-the-art. The
chapter concludes with a brief discussion of diffusion MRI applications, in particular fibre tracking, followed by a summary
and a glimpse into the future of diffusion MRI acquisition and reconstruction.
... Both affine invariant and Log-Euclidean frameworks can be employed for processing fields of 2 nd order tensors. Use of higher order tensors was proposed in [5] to represent more complex diffusivity profiles which better approximate the diffusivity of the local tissue geometry. However to date, none of the reported methods in literature for the estimation of the coefficients of higher order tensors preserve the positive definiteness of the diffusivity function. ...
In Diffusion Tensor Magnetic Resonance Image (DT-MRI) processing a 2(nd) order tensor has been commonly used to approximate the diffusivity function at each lattice point of the 3D volume image. These tensors are symmetric positive definite matrices and the appropriate constraints required in algorithms for processing them makes these algorithms complex and significantly increases their computational complexity. In this paper we present a novel parameterization of the diffusivity function using which the positive definite property of the function is guaranteed without any increase in computation. This parameterization can be used for any order tensor approximations; we present Cartesian tensor approximations of order 2, 4, 6 and 8 respectively, of the diffusivity function all of which retain the positivity property in this parameterization without the need for any explicit enforcement. Furthermore, we present an efficient framework for computing distances and geodesics in the space of the coefficients of our proposed diffusivity function. Distances & geodesics are useful for performing interpolation, computation of statistics etc. on high rank positive definite tensors. We validate our model using simulated and real diffusion weighted MR data from excised, perfusion-fixed rat optic chiasm.
... Thus, measurements are generally taken on a ball in q-space, with fixed gradient magnitude g and duration δ. Assuming that the signal decays exponentially with g (Equation (2.20)), the expected intensity values on a Cartesian grid in q-space can be 2. Background computed numerically [151]. It is even possible to evaluate the radial part of the integral analytically, which leads to the diffusion orientation transformation [150]. ...
Diffusion Weighted Magnetic Resonance Imaging (DW-MRI) is a recent modality to investigate the major neuronal pathways of the human brain. However, the rich DW-MRI datasets cannot be interpreted without proper preprocessing. In order to achieve under- standable visualizations, this dissertation reduces the complex data to relevant features. The first part is inspired by topological features in flow data. Novel features reconstruct fuzzy fiber bundle geometry from probabilistic tractography results. The topological prop- erties of existing features that extract the skeleton of white matter tracts are clarified, and the core of regions with planar diffusion is visualized. The second part builds on methods from computer vision. Relevant boundaries in the data are identified via regularized eigenvalue derivatives, and boundary information is used to segment anisotropy isosurfaces into meaningful regions. A higher-order structure tensor is shown to be an accurate descriptor of local structure in diffusion data. The third part is concerned with fiber tracking. Streamline visualizations are improved by adding features from structural MRI in a way that emphasizes the relation between the two types of data, and the accuracy of streamlines in high angular resolution data is increased by modeling the estimation of crossing fiber bundles as a low-rank tensor approximation problem.
... However, 2 nd -order tensors are incapable of modeling complex geometry of the diffusivity function in practice for many cases (see [2,3], such as in the presence of fiber-crossings, and a higher-order approximation must be employed instead. Higher-order tensors have been used to model either the local diffusivity function [4,5], or the Kurtosis component of it [6]. However, in all cases the peaks of the estimated higher-order tensor do not necessarily yield the distinct orientations of the underlying distinct fiber bundles [2]. ...
Cartesian tensor basis have been widely used to approximate spherical functions. In Medical Imaging, tensors of various orders have been used to model the diffusivity function in Diffusion-weighted MRI data sets. However, it is known that the peaks of the diffusivity do not correspond to orientations of the underlying fibers and hence the displacement probability profiles should be employed instead. In this paper, we present a novel representation of the probability profile by a 4(th) order tensor, which is a smooth spherical function that can approximate single-fibers as well as multiple-fiber structures. We also present a method for efficiently estimating the unknown tensor coefficients of the probability profile directly from a given high-angular resolution diffusion-weighted (HARDI) data set. The accuracy of our model is validated by experiments on synthetic and real HARDI datasets from a fixed rat spinal cord.
... (ii) The tensor model cannot handle properly fibers crossings or kissings that may occur within a voxel. Emerging approaches using higher order models based on High Angular Resolution Diffusion Imaging (HARDI) [19, 54, 48, 3, 13, 14] may provide an answer to this issue. An intrinsic problem of the geometrical connectivity mapping approach used here comes from the absence of absolute threshold to confidently estimate fiber tracts from the connectivity maps [49]. ...
Various approaches have been introduced to infer the organization of white matter connectivity using Diffusion Tensor Imaging (DTI). In this study, we validate a recently introduced geometric tractography technique, Geodesic Connectivity Mapping (GCM), able to overcome the main limitations of geometrical approaches. Using the GCM technique, we could successfully characterize anatomical connections in the human low-level visual cortex. We reproduce previous findings regarding the topology of optic radiations linking the LGN to V1 and the regular organization of splenium fibers with respect to their origin in the visual cortex. Moreover, our study brings further insights regarding the connectivity of the human MT complex (hMT+) and the retinotopic areas.
... The diffusion tensor imaging model described earlier represents diffusion using a rank-2 tensor. Diffusion has been described more generally by Özarslan et al. [8] by considering Cartesian tensors of higher rank. A Cartesian tensor of rank n will, in general, have 3 n elements. ...
High angular resolution diffusion imaging (HARDI) permits the computation of water molecule displacement probabilities over the sphere. This probability is often referred to as the orientation distribution function (ODF). In this paper we present a novel model for representing this diffusion ODF namely, a mixture of von Mises-Fisher (vMF) distributions. Our model is compact in that it requires very few parameters to represent complicated ODF geometries which occur specifically in the presence of heterogeneous nerve fiber orientations. We present a Riemannian geometric framework for computing intrinsic distances (in closed-form) and for performing interpolation between ODFs represented by vMF mixtures. We also present closed-form equations for entropy and variance based anisotropy measures that are then computed and illustrated for real HARDI data from a rat brain.
... The synthetic data was generated by simulating the MR signal from a highly anisotropic single fiber (fractional anisotropy> 0.9) using the realistic diffusion MR simulation model in [35] (b-value= 1250s/ mm 2 , 21 gradient directions). Then, we added different amounts of Riccian noise to the simulated data set and estimated the 4 th -order tensors from the noisy data by: a) minimizing without using the proposed parametrization to enforce the SPD constraint, by employing the method in [29] and b) our method, which guarantees the SPD property of the tensors. (S i is the MR signal of the i th image and S 0 is the zero-gradient signal). ...
In Diffusion Weighted Magnetic Resonance Image (DW-MRI) processing, a 2nd order tensor has been commonly used to approximate the diffusivity function at each lattice point of the DW-MRI data. From this tensor approximation, one can compute useful scalar quantities (e.g. anisotropy, mean diffusivity) which have been clinically used for monitoring encephalopathy, sclerosis, ischemia and other brain disorders. It is now well known that this 2nd-order tensor approximation fails to capture complex local tissue structures, e.g. crossing fibers, and as a result, the scalar quantities derived from these tensors are grossly inaccurate at such locations. In this paper we employ a 4th order symmetric positive-definite (SPD) tensor approximation to represent the diffusivity function and present a novel technique to estimate these tensors from the DW-MRI data guaranteeing the SPD property. Several articles have been reported in literature on higher order tensor approximations of the diffusivity function but none of them guarantee the positivity of the estimates, which is a fundamental constraint since negative values of the diffusivity are not meaningful. In this paper we represent the 4th-order tensors as ternary quartics and then apply Hilbert's theorem on ternary quartics along with the Iwasawa parametrization to guarantee an SPD 4th-order tensor approximation from the DW-MRI data. The performance of this model is depicted on synthetic data as well as real DW-MRIs from a set of excised control and injured rat spinal cords, showing accurate estimation of scalar quantities such as generalized anisotropy and trace as well as fiber orientations.
... Ozarslan et al. 24 fit higher-order tensor models to measurements from a spherical acquisition scheme. They assume that A(q) decays exponentially with increasing ⏐q⏐ and fixed . ...
This chapter reviews multiple-fiber reconstruction algorithms for diffusion magnetic resonance imaging (MRI) and provides some initial comparative results for two such algorithms, q-ball imaging and PASMRI, on data from a typical clinical diffusion MRI acquisition. The chapter highlights the problems with standard approaches, such as diffusion-tensor MRI, to motivate a recent set of alternative approaches. The review concentrates on the software implementation of the new techniques. Results of the preliminary comparison show that PASMRI recovers the principal directions of simple test functions more consistently than q-ball imaging and produces qualitatively better results on the test data set. Further simulations suggest that a moderate increase in data quality allows q-ball, which is much faster to run, to recover directions with consistency comparable to that of PASMRI on the test data.
... The multi-tensor model has also been used for synthetic data experiments by other authors (e.g. Tuch [41]) and it is worth mentioning that there exists another popular synthetic data generation model used in other works [44, 14, 29]. In this formulation, Soderman and Jonsson [36] suppose that fibers are perfect cylinders and that water molecules are confined to diffuse within the walls of these cylinders. ...
High angular resolution diffusion imaging has recently been of great interest in characterizing non-Gaussian diffusion processes. One important goal is to obtain more accurate fits of the apparent diffusion processes in these non-Gaussian regions, thus overcoming the limitations of classical diffusion tensor imaging. This paper presents an extensive study of high-order models for apparent diffusion coefficient estimation and illustrates some of their applications. Using a meaningful modified spherical harmonics basis to capture the physical constraints of the problem, a new regularization algorithm is proposed. The new smoothing term is based on the Laplace-Beltrami operator and its closed form implementation is used in the fitting procedure. Next, the linear transformation between the coefficients of a spherical harmonic series of order l and independent elements of a rank-l high-order diffusion tensor is explicitly derived. This relation allows comparison of the state-of-the-art anisotropy measures computed from spherical harmonics and tensor coefficients. Published results are reproduced accurately and it is also possible to recover voxels with isotropic, single fiber anisotropic, and multiple fiber anisotropic diffusion. Validation is performed on apparent diffusion coefficients from synthetic data, from a biological phantom, and from a human brain dataset.
... It has been shown that the diffusivity function has a complicated structure in voxels with orientational heterogeneity (von dem Hagen and Henkelman, 2002; Tuch et al., 2002). Several studies proposed to represent the diffusivity function using the spherical harmonic expansion (Frank, 2002; Alexander et al., 2002) or higher order Cartesian tensors leading to a generalization of DTI (Özarslan and Mareci, 2003; Özarslan et al., 2004). A second class of approaches attempts to transform the limited number of multidirectional signals into a probability function, which is typically a compromised version of P(r) with presumably the same directional characteristics. ...
Diffusion MRI is a non-invasive imaging technique that allows the measurement of water molecule diffusion through tissue in vivo. The directional features of water diffusion allow one to infer the connectivity patterns prevalent in tissue and possibly track changes in this connectivity over time for various clinical applications. In this paper, we present a novel statistical model for diffusion-weighted MR signal attenuation which postulates that the water molecule diffusion can be characterized by a continuous mixture of diffusion tensors. An interesting observation is that this continuous mixture and the MR signal attenuation are related through the Laplace transform of a probability distribution over symmetric positive definite matrices. We then show that when the mixing distribution is a Wishart distribution, the resulting closed form of the Laplace transform leads to a Rigaut-type asymptotic fractal expression, which has been phenomenologically used in the past to explain the MR signal decay but never with a rigorous mathematical justification until now. Our model not only includes the traditional diffusion tensor model as a special instance in the limiting case, but also can be adjusted to describe complex tissue structure involving multiple fiber populations. Using this new model in conjunction with a spherical deconvolution approach, we present an efficient scheme for estimating the water molecule displacement probability functions on a voxel-by-voxel basis. Experimental results on both simulations and real data are presented to demonstrate the robustness and accuracy of the proposed algorithms.
In this paper, we review the state of the art in diffusion magnetic resonance imaging (dMRI) and we present current trends in modelling the brain's tissue microstructure and the human connectome. dMRI is today the only tool that can probe the brain's axonal architecture in vivo and non-invasively, and has grown in leaps and bounds in the last two decades since its conception. A plethora of models with increasing complexity and better accuracy have been proposed to characterise the integrity of the cerebral tissue, to understand its microstructure and to infer its connectivity. Here, we discuss a wide range of the most popular, important and well-established local microstructure models and biomarkers that have been proposed from these models. Finally, we briefly present the state of the art in tractography techniques that allow us to understand the architecture of the brain's connectivity.
By analyzing stochastic processes on a Riemannian manifold, in particular Brownian motion, one can deduce the metric structure of the space. This fact is implicitly used in diffusion tensor imaging of the brain when cast into a Riemannian framework. When modeling the brain white matter as a Riemannian manifold one finds (under some provisions) that the metric tensor is proportional to the inverse of the diffusion tensor, and this opens up a range of geometric analysis techniques. Unfortunately a number of these methods have limited applicability, as the Riemannian framework is not rich enough to capture key aspects of the tissue structure, such as fiber crossings. An extension of the Riemannian framework to the more general Finsler manifolds has been proposed in the literature as a possible alternative. The main contribution of this work is the conclusion that simply considering Brownianmotion on the Finsler base manifold does not reproduce the signal model proposed in the Finslerian framework, nor lead to a model that allows the extraction of the Finslerian metric structure from the signal.
Diffusion weighted imaging (DWI) techniques have been used to study hu-man brain white matter fiber structures in vivo. Commonly used standard diffusion tensor magnetic resonance imaging (DTI) tractography derived from the second order diffusion tensor model has limitations in its ability to re-solve complex fiber tracts. We propose a new fiber tracking method based on the generalized diffusion tensor (GDT) model. This new method better mod-els the anisotropic diffusion process in human brain by using the generalized diffusion simulation-based fiber tractography (GDST). Due to the additional information provided by GDT, the GDST method simulates the underlying physical diffusion process of the human brain more accurately than does the standard DTI method. The effectiveness of the new fiber tracking algorithm was demonstrated via analyses on real and synthetic DWI datasets. In addi-tion, the general analytic expression of high order b matrix is derived in the case of twice refocused spin-echo (TRSE) pulse sequence which is used in the DWI data acquisition. Based on our results, we discuss the benefits of GDT and the second order diffusion tensor on fiber tracking.
In this work we review how the diffusivity profiles obtained from diffusion MRI can be expressed in terms of Cartesian tensors
of ranks higher than 2. When the rank of the tensor being used is 2, one recovers traditional diffusion tensor imaging (DTI).
Therefore our approach can be seen as a generalization of DTI. The properties of generalized diffusion tensors are discussed.
The shortcomings of DTI experienced in the presence of orientational heterogeneity may cause inaccurate anisotropy values
and incorrect fiber orientations. Employment of higher rank tensors is helpful in overcoming these difficulties.
High angular resolution diffusion imaging (HARDI) permits the computation of water molecule displacement probabilities over a sphere of possible displacement directions. This probability is often referred to as the orientation distribution function (ODF). In this paper we present a novel model for the diffusion ODF namely, a mixture of von Mises-Fisher (vMF) distributions. Our model is compact in that it requires very few variables to model complicated ODF geometries which occur specifically in the presence of heterogeneous nerve fiber orientation. We also present a Riemannian geometric framework for computing intrinsic distances, in closed-form, and performing interpolation between ODFs represented by vMF mixtures. As an example, we apply the intrinsic distance within a hidden Markov measure field segmentation scheme. We present results of this segmentation for HARDI images of rat spinal cords - which show distinct regions within both the white and gray matter. It should be noted that such a fine level of parcellation of the gray and white matter cannot be obtained either from contrast MRI scans or Diffusion Tensor MRI scans. We validate the segmentation algorithm by applying it to synthetic data sets where the ground truth is known.
Cette thèse est consacrée au développement d'outils de traitement pour l'Imagerie par Résonance Magnétique du Tenseur de Diffusion (IRM-TD). Cette technique d'IRM récente est d'une grande importance pour comprendre le fonctionnement du cerveau ou pour améliorer le diagnostic de pathologies neurologiques. Nous proposons des méthodes de traitement basées sur la géométrie Riemannienne, les équations aux dérivées partielles et les techniques de propagation de front. La première partie de ce travail est théorique. Après des rappels sur le système nerveux humain, l'IRM et la géométrie différentielle, nous étudions l'espace des lois normales multivariées. L'introduction d'une structure Riemannienne sur cet espace nous permet de définir des statistiques et des schémas numériques intrinsèques qui sont à la base des algorithmes proposés dans la seconde partie. Les propriétés de cet espace sont importantes pour l'IRM-TD car les tenseurs de diffusion sont les matrices de covariance de lois normales modélisant la diffusion des molécules d'eau en chaque voxel du milieu imagé. La seconde partie est méthodologique. Nous y introduisons des approches originales pour l'estimation et la régularisation d'IRM-TD. Puis nous montrons comment évaluer le degré de connectivité entre aires corticales et introduisons un modèle statistique d'évolution de surface permettant de segmenter ces images. Finalement, nous proposons une méthode de recalage non-rigide. La dernière partie de cette thèse est consacrée à l'analyse des connexions entre le cortex cérébral et les noyaux gris centraux, impliquées dans des tâches motrices, et à l'étude du réseau anatomo-fonctionnel du cortex visuel humain..
This article describes an accurate and fast method for fiber orientation mapping using multidirectional diffusion-weighted magnetic resonance (MR) data. This novel approach utilizes the Fourier transform relationship between the water displacement probabilities and diffusion-attenuated MR signal expressed in spherical coordinates. The radial part of the Fourier integral is evaluated analytically under the assumption that MR signal attenuates exponentially. The values of the resulting functions are evaluated at a fixed distance away from the origin. The spherical harmonic transform of these functions yields the Laplace series coefficients of the probabilities on a sphere of fixed radius. Alternatively, probability values can be computed nonparametrically using Legendre polynomials. Orientation maps calculated from excised rat nervous tissue data demonstrate this technique's ability to accurately resolve crossing fibers in anatomical regions such as the optic chiasm. This proposed methodology has a trivial extension to multiexponential diffusion-weighted signal decay. The developed methods will improve the reliability of tractography schemes and may make it possible to correctly identify the neural connections between functionally connected regions of the nervous system.
This paper introduces a new, accurate and fast method for fiber orientation mapping using high angular resolution diffusion imaging (HARDI) data. The approach utilizes the Fourier relationship between the water displacement probabilities and diffusion attenuated magnetic resonance (MR) signal expressed in spherical coordinates. The Laplace series coefficients of the water displacement probabilities are evaluated at a fixed distance away from the origin. The computations take under one minute for most three-dimensional datasets. We present orientation maps computed from excised rat optic chiasm, brain and spinal cord images. The developed method will improve the reliability of tractography schemes and make it possible to correctly identify the neural connections between functionally connected regions of the nervous system.
This paper proposes a maximum entropy method for spherical deconvolution. Spherical deconvolution arises in various inverse problems. This paper uses the method to reconstruct the distribution of microstructural fibre orientations from diffusion MRI measurements. Analysis shows that the PASMRI algorithm, one of the most accurate diffusion MRI reconstruction algorithms in the literature, is a special case of the maximum entropy spherical deconvolution. Experiments compare the new method to linear spherical deconvolution, used previously in diffusion MRI, and to the PASMRI algorithm. The new method compares favourably both in simulation and on standard brain-scan data.
We propose a regularized, fast, and robust analytical solution for the Q-ball imaging (QBI) reconstruction of the orientation distribution function (ODF) together with its detailed validation and a discussion on its benefits over the state-of-the-art. Our analytical solution is achieved by modeling the raw high angular resolution diffusion imaging signal with a spherical harmonic basis that incorporates a regularization term based on the Laplace-Beltrami operator defined on the unit sphere. This leads to an elegant mathematical simplification of the Funk-Radon transform which approximates the ODF. We prove a new corollary of the Funk-Hecke theorem to obtain this simplification. Then, we show that the Laplace-Beltrami regularization is theoretically and practically better than Tikhonov regularization. At the cost of slightly reducing angular resolution, the Laplace-Beltrami regularization reduces ODF estimation errors and improves fiber detection while reducing angular error in the ODF maxima detected. Finally, a careful quantitative validation is performed against ground truth from synthetic data and against real data from a biological phantom and a human brain dataset. We show that our technique is also able to recover known fiber crossings in the human brain and provides the practical advantage of being up to 15 times faster than original numerical QBI method.
Functional imaging with positron emission tomography and functional MRI has revolutionized studies of the human brain. Understanding the organization of brain systems, especially those used for cognition, remains limited, however, because no methods currently exist for noninvasive tracking of neuronal connections between functional regions [Crick, F. & Jones, E. (1993) Nature (London) 361, 109-110]. Detailed connectivities have been studied in animals through invasive tracer techniques, but these invasive studies cannot be done in humans, and animal results cannot always be extrapolated to human systems. We have developed noninvasive neuronal fiber tracking for use in living humans, utilizing the unique ability of MRI to characterize water diffusion. We reconstructed fiber trajectories throughout the brain by tracking the direction of fastest diffusion (the fiber direction) from a grid of seed points, and then selected tracks that join anatomically or functionally (functional MRI) defined regions. We demonstrate diffusion tracking of fiber bundles in a variety of white matter classes with examples in the corpus callosum, geniculo-calcarine, and subcortical association pathways. Tracks covered long distances, navigated through divergences and tight curves, and manifested topological separations in the geniculo-calcarine tract consistent with tracer studies in animals and retinotopy studies in humans. Additionally, previously undescribed topologies were revealed in the other pathways. This approach enhances the power of modern imaging by enabling study of fiber connections among anatomically and functionally defined brain regions in individual human subjects.
To understand evolving pathology in the central nervous system (CNS) and develop effective treatments, it is essential to correlate the nerve fiber connectivity with the visualization of function. Diffusion tensor imaging (DTI) can provide the fundamental information required for viewing structural connectivity. We present a novel algorithm for automatic fiber tract mapping in the CNS specifically, the spinal cord. The automatic fiber tract mapping problem is solved in two phases, namely a data smoothing phase and a fiber tract mapping phase. In the former, smoothing is achieved via a new weighted total variation (TV)-norm minimization (for vector-valued data) which strives to smooth while retaining all relevant detail. For the fiber tract mapping, a smooth 3D vector field indicating the dominant anisotropic direction at each spatial location is computed from the smoothed data. Fiber tracts are then determined as the smooth integral curves of this vector field in a variational framework
The phenomenological Bloch equations in nuclear magnetic resonance are generalized by the addition of terms due to the transfer of magnetization by diffusion. The revised equations describe phenomena under conditions of inhomogeneity in magnetic field, relaxation rates, or initial magnetization. As an example the equations are solved in the case of the free precession of magnetic moment in the presence of an inhomogeneous magnetic field following the application of a 90° pulse with subsequent applications of a succession of 180° pulses. The spin-echo amplitudes agree with the results of Carr and Purcell from a random walk theory.
A derivation is given of the effect of a time-dependent magnetic field gradient on the spin-echo experiment, particularly in the presence of spin diffusion. There are several reasons for preferring certain kinds of time-dependent magnetic field gradients to the more usual steady gradient. If the gradient is reduced during the rf pulses, H1 need not be particularly large; if the gradient is small at the time of the echo, the echo will be broad and its amplitude easy to measure. Both of these relaxations of restrictions on the measurement of diffusion coefficients by the spin-echo technique serve to extend its range of applicability. Furthermore, a pulsed gradient can be recommended when it is critical to define the precise time period over which diffusion is being measured.
The theoretical expression derived has been verified experimentally for several choices of time dependent magnetic field gradient. An apparatus is described suitable for the production of pulsed gradients with amplitudes as large as 100 G cm−1. The diffusion coefficient of dry glycerol at 26°±1°C has been found to be (2.5±0.2)×10−8 cm2 sec−1, a value smaller than can ordinarily be measured by the steady gradient method.
Diffusion-weighted MR images were compared with T2-weighted MR images and correlated with 1H spin-echo and 31P MR spectroscopy for 6-8 h following a unilateral middle cerebral and bilateral carotid artery occlusion in eight cats. Diffusion-weighted images using strong gradient strengths (b values of 1413 s/mm2) displayed a significant relative hyperintensity in ischemic regions as early as 45 min after onset of ischemia whereas T2-weighted spin-echo images failed to clearly demonstrate brain injury up to 2-3 h postocclusion. Signal intensity ratios (SIR) of ischemic to normal tissues were greater in the diffusion-weighted images at all times than in either TE 80 or TE 160 ms T2-weighted MR images. Diffusion- and T2-weighted SIR did not correlate for the first 1-2 h postocclusion. Good correlation was found between diffusion-weighted SIR and ischemic disturbances of energy metabolism as detected by 31P and 1H MR spectroscopy. Diffusion-weighted hyperintensity in ischemic tissues may be temperature-related, due to rapid accumulation of diffusion-restricted water in the intracellular space (cytotoxic edema) resulting from the breakdown of the transmembrane pump and/or to microscopic brain pulsations.
The diagonal and off-diagonal elements of the effective self-diffusion tensor, Deff, are related to the echo intensity in an NMR spin-echo experiment. This relationship is used to design experiments from which Deff is estimated. This estimate is validated using isotropic and anisotropic media, i.e., water and skeletal muscle. It is shown that significant errors are made in diffusion NMR spectroscopy and imaging of anisotropic skeletal muscle when off-diagonal elements of Deff are ignored, most notably the loss of information needed to determine fiber orientation. Estimation of Deff provides the theoretical basis for a new MRI modality, diffusion tensor imaging, which provides information about tissue microstructure and its physiologic state not contained in scalar quantities such as T1, T2, proton density, or the scalar apparent diffusion constant.
We review several methods that have been developed to infer microstructural and physiological information about isotropic and anisotropic tissues from diffusion weighted images (DWIs). These include Diffusion Imaging (DI), Diffusion Tensor Imaging (DTI), isotropically weighted imaging, and q-space imaging. Just as DI provides useful information about molecular displacements in one dimension with which to characterize diffusion in isotropic tissues, DTI provides information about molecular displacements in three dimensions needed to characterize diffusion is anisotropic tissues. DTI also furnishes scalar parameters that behave like quantitative histological or physiological 'stains' for different features of diffusion. These include Trace(D), which is related to the mean diffusivity, and a family of parameters derived from the diffusion tensor, D, which characterize different features of anisotropic diffusion. Simple thought experiments and geometrical constructs, such as the diffusion ellipsoid, can be used to understand water diffusion in isotropic and anisotropic media, and the NMR experiments used to characterize it.
The methods of group theory are applied to the problem of characterizing the diffusion measured in high angular resolution MR experiments. This leads to a natural representation of the local diffusion in terms of spherical harmonics. In this representation, it is shown that isotropic diffusion, anisotropic diffusion from a single fiber, and anisotropic diffusion from multiple fiber directions fall into distinct and separable channels. This decomposition can be determined for any voxel without any prior information by a spherical harmonic transform, and for special cases the magnitude and orientation of the local diffusion may be determined. Moreover, non-diffusion-related asymmetries produced by experimental artifacts fall into channels distinct from the fiber channels, thereby allowing their separation and a subsequent reduction in noise from the reconstructed fibers. In the case of a single fiber, the method reduces identically to the standard diffusion tensor method. The method is applied to normal volunteer brain data collected with a stimulated echo spiral high angular resolution diffusion-weighted (HARD) acquisition.
Several new MR techniques have been introduced to infer direction through diffusion in multiple nerve fiber bundles within a voxel. To date, however, there has been no physical model reported to evaluate these methodologies and their ability to determine fiber orientation. In this article a model of diffusion analogous to nerve fibers is presented. Diffusion measurements at multiple closely spaced angles of 15 degrees in samples with different fiber orientations are compared with theoretical calculations for restricted diffusion in cylindrical geometry. Orientational diffusion measurements are shown to reflect fiber geometry and theoretical predictions to within 10%. Simulations of fiber crossings within a voxel suggest fiber orientation does not correspond to the direction of the largest measured diffusion coefficient, but theoretical knowledge of signal decay curves can predict the shape of these diffusion coefficient contours for given fiber orientation probabilities.
A new method for mapping diffusivity profiles in tissue is presented. The Bloch-Torrey equation is modified to include a diffusion term with an arbitrary rank Cartesian tensor. This equation is solved to give the expression for the generalized Stejskal-Tanner formula quantifying diffusive attenuation in complicated geometries. This makes it possible to calculate the components of higher-rank tensors without using the computationally-difficult spherical harmonic transform. General theoretical relations between the diffusion tensor (DT) components measured by traditional (rank-2) DT imaging (DTI) and 3D distribution of diffusivities, as measured by high angular resolution diffusion imaging (HARDI) methods, are derived. Also, the spherical tensor components from HARDI are related to the rank-2 DT. The relationships between higher- and lower-rank Cartesian DTs are also presented. The inadequacy of the traditional rank-2 tensor model is demonstrated with simulations, and the method is applied to excised rat brain data collected in a spin-echo HARDI experiment.