Peter Savadjiev

Mathematical methods for diffusion MRI processing

In this article, we review recent mathematical models and computational methods for the processing of diffusion Magnetic Resonance Images, including state-of-the-art reconstruction of diffusion models, cerebral white matter connectivity analysis, and segmentation techniques. We focus on Diffusion Tensor Images (DTI) and Q-Ball Images (QBI).

Labeling of ambiguous subvoxel fibre bundle configurations in high angular resolution diffusion MRI

Whereas high angular resolution reconstruction methods for diffusion MRI can estimate multiple dominant fibre orientations within a single imaging voxel, they are fundamentally limited in certain cases of complex subvoxel fibre structures, resulting in ambiguous local orientation distribution functions. In this article we address the important problem of disambiguating such complex subvoxel fibre tract configurations, with the purpose of improving the performance of fibre tractography. We do so by extending a curve inference method to distinguish between the cases of curving and fanning fibre bundles using differential geometric estimates in a local neighbourhood. The key benefit of this method is the inference of curves, instead of only fibre orientations, to model the underlying fibre bundles. This in turn allows distinct fibre geometries that contain nearly identical sets of fibre orientations at a voxel, to be distinguished from one another. Experimental results demonstrate the ability of the method to successfully label voxels into one of the above categories and improve the performance of a fibre-tracking algorithm.

Heart wall myofibers are arranged in minimal surfaces to optimize organ function

Heart wall myofibers wind as helices around the ventricles, strengthening them in a manner analogous to the reinforcement of concrete cylindrical columns by spiral steel cables [Richart FE, et al. (1929) Univ of Illinois, Eng Exp Stn Bull 190]. A multitude of such fibers, arranged smoothly and regularly, contract and relax as an integrated functional unit as the heart beats. To orchestrate this motion, fiber tangling must be avoided and pumping should be efficient. Current models of myofiber orientation across the heart wall suggest groupings into sheets or bands, but the precise geometry of bundles of myofibers is unknown. Here we show that this arrangement takes the form of a special minimal surface, the generalized helicoid [Blair DE, Vanstone JR (1978) Minimal Submanifolds and Geodesics 13–16], closing the gap between individual myofibers and their collective wall structure. The model holds across species, with a smooth variation in its three curvature parameters within the myocardial wall providing tight fits to diffusion magnetic resonance images from the rat, the dog, and the human. Mathematically it explains how myofibers are bundled in the heart wall while economizing fiber length and optimizing ventricular ejection volume as they contract. The generalized helicoid provides a unique foundation for analyzing the fibrous composite of the heart wall and should therefore find applications in heart tissue engineering and in the study of heart muscle diseases.

Targeting of White Matter Tracts with Transcranial Magnetic Stimulation

BACKGROUND TMS activations of white matter depend not only on the distance from the coil, but also on the orientation of the axons relative to the TMS-induced electric field, and especially on axonal bends that create strong local field gradient maxima. Therefore, tractography contains potentially useful information for TMS targeting. OBJECTIVE/METHODS Here, we utilized 1-mm resolution diffusion and structural T1-weighted MRI to construct large-scale tractography models, and localized TMS white matter activations in motor cortex using electromagnetic forward modeling in a boundary element model (BEM). RESULTS As expected, in sulcal walls, pyramidal cell axonal bends created preferred sites of activation that were not found in gyral crowns. The model agreed with the well-known coil orientation sensitivity of motor cortex, and also suggested unexpected activation distributions emerging from the E-field and tract configurations. We further propose a novel method for computing the optimal coil location and orientation to maximally stimulate a pre-determined axonal bundle. CONCLUSIONS Diffusion MRI tractography with electromagnetic modeling may improve spatial specificity and efficacy of TMS.

Inter-site and inter-scanner diffusion MRI data harmonization

We propose a novel method to harmonize diffusion MRI data acquired from multiple sites and scanners, which is imperative for joint analysis of the data to significantly increase sample size and statistical power of neuroimaging studies. Our method incorporates the following main novelties: i) we take into account the scanner-dependent spatial variability of the diffusion signal in different parts of the brain; ii) our method is independent of compartmental modeling of diffusion (e.g., tensor, and intra/extra cellular compartments) and the acquired signal itself is corrected for scanner related differences; and iii) inter-subject variability as measured by the coefficient of variation is maintained at each site. We represent the signal in a basis of spherical harmonics and compute several rotation invariant spherical harmonic features to estimate a region and tissue specific linear mapping between the signal from different sites (and scanners). We validate our method on diffusion data acquired from seven different sites (including two GE, three Philips, and two Siemens scanners) on a group of age-matched healthy subjects. Since the extracted rotation invariant spherical harmonic features depend on the accuracy of the brain parcellation provided by Freesurfer, we propose a feature based refinement of the original parcellation such that it better characterizes the anatomy and provides robust linear mappings to harmonize the dMRI data. We demonstrate the efficacy of our method by statistically comparing diffusion measures such as fractional anisotropy, mean diffusivity and generalized fractional anisotropy across multiple sites before and after data harmonization. We also show results using tract-based spatial statistics before and after harmonization for independent validation of the proposed methodology. Our experimental results demonstrate that, for nearly identical acquisition protocol across sites, scanner-specific differences can be accurately removed using the proposed method.

Local white matter geometry from diffusion tensor gradients

We introduce a mathematical framework for computing geometrical properties of white matter fibers directly from diffusion tensor fields. The key idea is to isolate the portion of the gradient of the tensor field corresponding to local variation in tensor orientation, and to project it onto a coordinate frame of tensor eigenvectors. The resulting eigenframe-centered representation then makes it possible to define scalar indices (or measures) that describe the local white matter geometry directly from the diffusion tensor field and its gradient, without requiring prior tractography. We derive new scalar indices of (1) fiber dispersion and (2) fiber curving, and we demonstrate them on synthetic and in vivo data. Finally, we illustrate their applicability to a group study on schizophrenia.

Fiber geometry in the corpus callosum in schizophrenia: evidence for transcallosal misconnection.

BACKGROUND Structural abnormalities in the callosal fibers connecting the heteromodal association areas of the prefrontal and temporoparietal cortices bilaterally have been suggested to play a role in the etiology of schizophrenia. AIMS To investigate for geometric abnormalities in these callosal fibers in schizophrenia patients by using a novel Diffusion-Tensor Imaging (DTI) metric of fiber geometry named Shape-Normalized Dispersion (SHD). METHODS DTIs (3T, 51 gradient directions, 1.7mm isotropic voxels) were acquired from 26 schizophrenia patients and 23 matched healthy controls. The prefrontal and temporoparietal fibers of the corpus callosum were extracted by means of whole-brain tractography, and their mean SHD calculated. RESULTS The schizophrenia patients exhibited subnormal levels of SHD in the prefrontal callosal fibers when controlling for between-group differences in Fractional Anisotropy. Reduced SHD could reflect either irregularly turbulent or inhomogeneously distributed fiber trajectories in the corpus callosum. CONCLUSIONS The results suggest that the transcallosal misconnectivity thought to be associated with schizophrenia could reflect abnormalities in fiber geometry. These abnormalities in fiber geometry could potentially be underpinned by neurodevelopmental irregularities.

Validation and regularization in diffusion MRI tractography

We present a physical phantom designed for fibre tractography validation and use it to evaluate tracking algorithms that employ (a) the classic diffusion tensor model of diffusion, (b) high angular resolution reconstruction of the diffusion orientation distribution function (ODF), and (c) a regularization algorithm capable of inferring complex subvoxel fibre configurations. This work addresses four issues in diffusion MRI tractography: validation of the tracking process using ground truth, evaluation of different approaches for diffusion ODF reconstruction, coping with imaging noise, and coping with confounding subvoxel fibre configurations

Gray Matter Alterations in Early Aging: A Diffusion Magnetic Resonance Imaging Study

Many studies have observed altered neurofunctional and structural organization in the aging brain. These observations from functional neuroimaging studies show a shift in brain activity from the posterior to the anterior regions with aging (PASA model), as well as a decrease in cortical thickness, which is more pronounced in the frontal lobe followed by the parietal, occipital, and temporal lobes (retrogenesis model). However, very little work has been done using diffusion MRI (dMRI) with respect to examining the structural tissue alterations underlying these neurofunctional changes in the gray matter. Thus, for the first time, we propose to examine gray matter changes using diffusion MRI in the context of aging. In this work, we propose a novel dMRI based measure of gray matter “heterogeneity” that elucidates these functional and structural models (PASA and retrogenesis) of aging from the viewpoint of diffusion MRI. In a cohort of 85 subjects (all males, ages 15–55 years), we show very high correlation between age and “heterogeneity” (a measure of structural layout of tissue in a region‐of‐interest) in specific brain regions. We examine gray matter alterations by grouping brain regions into anatomical lobes as well as functional zones. Our findings from dMRI data connects the functional and structural domains and confirms the “retrogenesis” hypothesis of gray matter alterations while lending support to the neurofunctional PASA model of aging in addition to showing the preservation of paralimbic areas during healthy aging. Hum Brain Mapp 35:3841–3856, 2014.

Streamline Flows for White Matter Fibre Pathway Segmentation in Diffusion MRI

We introduce a fibre tract segmentation algorithm based on the geometric coherence of fibre orientations as indicated by a streamline flow model. The inference of local flow approximations motivates a pairwise consistency measure between fibre ODF maxima. We use this measure in a recursive algorithm to cluster consistent ODF maxima, leading to the segmentation of white matter pathways. The method requires minimal seeding compared to streamline tractography-based methods, and allows multiple tracts to pass through the same voxels. We illustrate the approach with a segmentation of the corpus callosum and one of the cortico-spinal tract, with each example seeded at a single voxel.