Parsimonious continuous time random walk models and kurtosis for diffusion in magnetic resonance of biological tissue.

Carson Ingo,Yi Sui,Andrew G. Webb,Yufen Chen,Todd B. Parrish,Itamar Ronen

doi:10.3389/fphy.2015.00011

Abstract

In this paper, we provide a context for the modeling approaches that have been developed to describe non-Gaussian diffusion behavior, which is ubiquitous in diffusion weighted magnetic resonance imaging of water in biological tissue. Subsequently, we focus on the formalism of the continuous time random walk theory to extract properties of subdiffusion and superdiffusion through novel simplifications of the Mittag-Leffler function. For the case of time-fractional subdiffusion, we compute the kurtosis for the Mittag-Leffler function, which provides both a connection and physical context to the much-used approach of diffusional kurtosis imaging. We provide Monte Carlo simulations to illustrate the concepts of anomalous diffusion as stochastic processes of the random walk. Finally, we demonstrate the clinical utility of the Mittag-Leffler function as a model to describe tissue microstructure through estimations of subdiffusion and kurtosis with diffusion MRI measurements in the brain of a chronic ischemic stroke patient.

Highlights

In the first measurements of water diffusion in biological tissue using magnetic resonance imaging (MRI) systems, the term “apparent diffusion coefficient” (ADC) was chosen to highlight the fact that, the free diffusion coefficient of water at body temperature is ∼ 3 × 10−3 mm2/s, typical values in white matter (WM) and gray matter (GM) regions of interest in the human brain were found to be an order of magnitude smaller ∼ 0.6 − 1.0 × 10−3 mm2/s due to hindrances imposed on water self-diffusion by the tissue microstructure [1,2,3]
We have presented new, simplified fitting forms for the Mittag-Leffler function (MLF) as a three-parameter model in Equation (21) and a two-parameter model in Equation (22)
The concepts of subdiffusion, superdiffusion, and Brownian motion have been simulated to illustrate the physical consequences of the movement of a particle in the statistical context of the continuous time random walk (CTRW) theory, which potentially can have biological correlates in diffusion MRI

Summary

Introduction

In the first measurements of water diffusion in biological tissue using magnetic resonance imaging (MRI) systems, the term “apparent diffusion coefficient” (ADC) was chosen to highlight the fact that, the free diffusion coefficient of water at body temperature is ∼ 3 × 10−3 mm2/s, typical values in white matter (WM) and gray matter (GM) regions of interest in the human brain were found to be an order of magnitude smaller ∼ 0.6 − 1.0 × 10−3 mm2/s due to hindrances imposed on water self-diffusion by the tissue microstructure [1,2,3]. There has been an attempt to characterize fixed biological tissue with higher moment analysis of the diffusion propagator in order to extract fractal-dimension measures [18, 19] Another way to identify non-Gaussian diffusion, known as diffusional kurtosis imaging (DKI), uses a Taylor-series expansion of the argument in the exponential function in order to estimate excess kurtosis of the measured signal vs the Gaussian case of a monoexponential decay [20]. We formulate simplified fitting forms of the MLF, connect subdiffusion to kurtosis, provide simulations of random walk conditions to illustrate the diffusion physics, and demonstrate measurements of non-Gaussian diffusion in the brain of a chronic stroke patient

Methods

Results

Discussion

Conclusion