The Mathematics of Quasi-Diffusion Magnetic Resonance Imaging

Abstract

Quasi-diffusion imaging (QDI) is a novel quantitative diffusion magnetic resonance imaging (dMRI) technique that enables high quality tissue microstructural imaging in a clinically feasible acquisition time. QDI is derived from a special case of the continuous time random walk (CTRW) model of diffusion dynamics and assumes water diffusion is locally Gaussian within tissue microstructure. By assuming a Gaussian scaling relationship between temporal () and spatial () fractional exponents, the dMRI signal attenuation is expressed according to a diffusion coefficient, (in mm2 s−1), and a fractional exponent, . Here we investigate the mathematical properties of the QDI signal and its interpretation within the quasi-diffusion model. Firstly, the QDI equation is derived and its power law behaviour described. Secondly, we derive a probability distribution of underlying Fickian diffusion coefficients via the inverse Laplace transform. We then describe the functional form of the quasi-diffusion propagator, and apply this to dMRI of the human brain to perform mean apparent propagator imaging. QDI is currently unique in tissue microstructural imaging as it provides a simple form for the inverse Laplace transform and diffusion propagator directly from its representation of the dMRI signal. This study shows the potential of QDI as a promising new model-based dMRI technique with significant scope for further development.

Optimization of quasi‐diffusion magnetic resonance imaging for quantitative accuracy and time‐efficient acquisition

“…Equation () describes Gaussian diffusion when $$$ \upalpha =1 $$$ and non‐Gaussian diffusion for $$$ 0<\upalpha <1 $$$. The fractional exponent is indicative of the inverse power law of the diffusion signal attenuation 1,5 . $$$ {D}_{1,2} $$$, and $$$ \upalpha $$$, are estimated by fitting Equation () to acquired dMRI data 1 .…”

“…The MLF can be considered to be a generalization of the exponential function and is completely monotone in the negative real axis for 0 < α ≤ 1. 5 The quasi-diffusion signal attenuation, S(b), at a given diffusion-sensitization, b, (in s mm −2 ) is given by,…”

“…5 This mathematical formulation is currently unique among dMRI signal representation and modeling techniques -QDI is the only approach in the literature, which has closed forms for its propagator and both the Fourier and Laplace transforms. The form of the propagator also allows derivation of quasi-diffusion mean apparent propagator (qMAP) imaging, 5 which is a model based alternative to standard MAP imaging. 9 The mathematical description of QDI provides insight into the heterogeneity of the diffusion environment, therefore, D 1,2 and α will be similarly sensitive to pathology as DKI and MAP techniques.…”

“…It provides feasible QDI parameter estimates in voxels, for which DKI assigns infeasible negative excess kurtosis, without requiring regularization 1,4 and represents the dMRI signal attenuation by a completely monotonic decreasing function unlike DKI. 5 The continuous time random walk (CTRW) model of diffusion 6 provides a mathematical description of dMRI signal decay with a physical interpretation of diffusion dynamics, which makes no assumptions about tissue geometry. The spin's random walk is described by 2 discrete stochastic processes, with fractional exponents relating to step lengths, β, over time intervals α. Quasi-diffusion is a special case of the CTRW model, which assumes a Gaussian scaling relationship between space and time fractional exponents, β = 2α, which leads to mean squared displacement that is linearly proportional to diffusion time.…”

“…QDI is mathematically robust. It provides feasible QDI parameter estimates in voxels, for which DKI assigns infeasible negative excess kurtosis, without requiring regularization 1,4 and represents the dMRI signal attenuation by a completely monotonic decreasing function unlike DKI 5 …”