On uniqueness of coefficient identification in the Bloch-Torrey equation for magnetic resonance imaging

TL;DR

This paper investigates the uniqueness of reconstructing spatially varying parameters in the Bloch-Torrey equation for MRI, using sampling, explicit formulas, and diffusion properties to establish conditions for unique coefficient identification.

Contribution

It provides new uniqueness results for coefficient reconstruction in the Bloch-Torrey equation, combining sampling-based and diffusion-based approaches.

Findings

Uniqueness results for coefficient identification in MRI models.

Lipschitz continuous differentiability of the coefficient-to-state map.

Potential applications in MRI reconstruction and experimental design.

Abstract

In this paper we provide some uniqueness results for the (multi-)coefficient identification problem of reconstructing the spatially varying spin density as well as the spin-lattice and spin-spin relaxation times and the local field inhomogeneity in the Bloch-Torrey equation, as relevant in magnetic resonance imaging MRI. To this end, we follow two approaches: (a) Relying on sampling of the k-space and (approximately) explicit reconstruction formulas in the simplified (Bloch) ODE setting, along with perturbation estimates; (b) Relying on infinite speed of propagation due to diffusion. The results on well-posendess and Lipschitz continuous differentiability of the coefficient-to-state map derived for this purpose, are expected to be useful also in the convergence analysis of reconstruction schemes as well in mathematical optimization of the experimental design in MRI.