Efficient discretization scheme for semi-analytical solutions of the Bloch-Torrey equation

Abstract

Calculation of the NMR signal behaviour inside physiological muscle tissue models is of high interest for human and biological studies. Both diffusion and susceptibility effects have to be considered to characterize the time evolution of the transverse magnetization. Herein, we demonstrate a novel discretization scheme which allows to semi-analytically solve the Bloch-Torrey equation efficiently using a Heisenberg-style Hamiltonian matrix. Angular eigenfunctions are calculated as Chebyshev polynomials which can be implemented without needing special functions. The method allows a correct calculation of both, the gradient echo free induction decay and the inhomogeneous relaxation rate R2', over the whole diffusion range. We further provide an efficient implementation and demonstrate its use.

Keywords

• Diffusion magnetic resonance imaging
• Bloch-Torrey equation
• Krogh model
• Discretization scheme