A second order control-volume finite-element least-squares strategy for simulating diffusion in strongly anisotropic media

An unstructured mesh �nite volume discretisation method for simulating di�usion in anisotropic media in two-dimensional space is discussed. This technique is considered as an extension of the fully implicit hybrid control-volume �nite-element method and it retains the local continuity of the ux at the control volume faces. A least squares function recon- struction technique together with a new ux decomposition strategy is used to obtain an accurate ux approximation at the control volume face, ensuring that the overall accuracy of the spatial discretisation maintains second order. This paper highlights that the new technique coincides with the traditional shape function technique when the correction term is neglected and that it signi�cantly increases the accuracy of the previous linear scheme on coarse meshes when applied to media that exhibit very strong to extreme anisotropy ratios. It is concluded that the method can be used on both regular and irregular meshes, and appears independent of the mesh quality.

Diffusion tensor of water in model articular cartilage

We used Monte Carlo simulations of Brownian dynamics of water to study anisotropic water diffusion in an idealised model of articular cartilage. The main aim was to use the simulations as a tool for translation of the fractional anisotropy of the water diffusion tensor in cartilage into quantitative characteristics of its collagen fibre network. The key finding was a linear empirical relationship between the collagen volume fraction and the fractional anisotropy of the diffusion tensor. Fractional anisotropy of the diffusion tensor is potentially a robust indicator of the microstructure of the tissue because, in the first approximation, it is invariant to the inclusion of proteoglycans or chemical exchange between free and collagen-bound water in the model. We discuss potential applications of Monte Carlo diffusion-tensor simulations for quantitative biophysical interpretation of MRI diffusion-tensor images of cartilage. Extension of the model to include collagen fibre disorder is also discussed.

Computer modeling of diffusion in biological tissues

Molecular-level computer simulations of restricted water diffusion can be used to develop models for relating diffusion tensor imaging measurements of anisotropic tissue to microstructural tissue characteristics. The diffusion tensors resulting from these simulations can then be analyzed in terms of their relationship to the structural anisotropy of the model used. As the translational motion of water molecules is essentially random, their dynamics can be effectively simulated using computers. In addition to modeling water dynamics and water-tissue interactions, the simulation software of the present study was developed to automatically generate collagen fiber networks from user-defined parameters. This flexibility provides the opportunity for further investigations of the relationship between the diffusion tensor of water and morphologically different models representing different anisotropic tissues.

Characterisation of the micro-architecture of direct writing melt electrospun tissue engineering scaffolds using diffusion tensor and computed tomography microimaging

This article describes the first steps toward comprehensive characterization of molecular transport within scaffolds for tissue engineering. The scaffolds were fabricated using a novel melt electrospinning technique capable of constructing 3D lattices of layered polymer fibers with well - defined internal microarchitectures. The general morphology and structure order was then determined using T 2 - weighted magnetic resonance imaging and X - ray microcomputed tomography. Diffusion tensor microimaging was used to measure the time - dependent diffusivity and diffusion anisotropy within the scaffolds. The measured diffusion tensors were anisotropic and consistent with the cross - hatched geometry of the scaffolds: diffusion was least restricted in the direction perpendicular to the fiber layers. The results demonstrate that the cross - hatched scaffold structure preferentially promotes molecular transport vertically through the layers ( z - axis), with more restricted diffusion in the directions of the fiber layers ( x – y plane). Diffusivity in the x – y plane was observed to be invariant to the fiber thickness. The characteristic pore size of the fiber scaffolds can be probed by sampling the diffusion tensor at multiple diffusion times. Prospective application of diffusion tensor imaging for the real - time monitoring of tissue maturation and nutrient transport pathways within tissue engineering scaffolds is discussed.

Brain fiber architecture, genetics, and intelligence: a high angular resolution diffusion imaging (HARDI) study

We developed an analysis pipeline enabling population studies of HARDI data, and applied it to map genetic influences on fiber architecture in 90 twin subjects. We applied tensor-driven 3D fluid registration to HARDI, resampling the spherical fiber orientation distribution functions (ODFs) in appropriate Riemannian manifolds, after ODF regularization and sharpening. Fitting structural equation models (SEM) from quantitative genetics, we evaluated genetic influences on the Jensen-Shannon divergence (JSD), a novel measure of fiber spatial coherence, and on the generalized fiber anisotropy (GFA) a measure of fiber integrity. With random-effects regression, we mapped regions where diffusion profiles were highly correlated with subjects' intelligence quotient (IQ). Fiber complexity was predominantly under genetic control, and higher in more highly anisotropic regions; the proportion of genetic versus environmental control varied spatially. Our methods show promise for discovering genes affecting fiber connectivity in the brain.

Mapping genetic influences on brain fiber architecture with high angular resolution diffusion imaging (HARDI)

We report the first 3D maps of genetic effects on brain fiber complexity. We analyzed HARDI brain imaging data from 90 young adult twins using an information-theoretic measure, the Jensen-Shannon divergence (JSD), to gauge the regional complexity of the white matter fiber orientation distribution functions (ODF). HARDI data were fluidly registered using Karcher means and ODF square-roots for interpol ation; each subject's JSD map was computed from the spatial coherence of the ODFs in each voxel's neighborhood. We evaluated the genetic influences on generalized fiber anisotropy (GFA) and complexity (JSD) using structural equation models (SEM). At each voxel, genetic and environmental components of data variation were estimated, and their goodness of fit tested by permutation. Color-coded maps revealed that the optimal models varied for different brain regions. Fiber complexity was predominantly under genetic control, and was higher in more highly anisotropic regions. These methods show promise for discovering factors affecting fiber connectivity in the brain.

Detailed analysis of a conservative difference approximation for the time fractional diffusion equation

Diffusion equations that use time fractional derivatives are attractive because they describe a wealth of problems involving non-Markovian Random walks. The time fractional diffusion equation (TFDE) is obtained from the standard diffusion equation by replacing the first-order time derivative with a fractional derivative of order α ∈ (0, 1). Developing numerical methods for solving fractional partial differential equations is a new research field and the theoretical analysis of the numerical methods associated with them is not fully developed. In this paper an explicit conservative difference approximation (ECDA) for TFDE is proposed. We give a detailed analysis for this ECDA and generate discrete models of random walk suitable for simulating random variables whose spatial probability density evolves in time according to this fractional diffusion equation. The stability and convergence of the ECDA for TFDE in a bounded domain are discussed. Finally, some numerical examples are presented to show the application of the present technique.