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

Barbara Kaltenbacher

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.

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

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.

Paper Structure

This paper contains 19 sections, 12 theorems, 192 equations.

Table of Contents

  1. Introduction
  2. Problem setting
  3. The forward problem
  4. Well-posedness of the PDE for given parameters
  5. Explicit solution formulas with rectangular pulse RF fields
  6. Perturbation results
  7. Error estimates in the explicit Bloch ODE and k-space Bloch-Torrey solution formulas
  8. Smoothness of the Bloch-Torrey parameter-to-state map
  9. Uniqueness
  10. Uniqueness and stability via perturbation of Bloch equation
  11. The MRI isomorphism
  12. Recovery of $M^{eq}$, $R_1$ and $R_2^*$ from Bloch ODE explicit solution formulas
  13. Recovery of $(M^{eq},R_1)$ from Bloch-Torrey and $(M^{eq},R_1,R_2^*)$ from Bloch parameter-to state map
  14. Linearized uniqueness by diffusion
  15. Recovery of $M^{eq}$
  16. ...and 4 more sections

Key Result

Proposition 2.1

Let $T\in(0,\infty]$, $\gamma\in\mathbb{R}$, $R_1\in L^\infty(\Omega;\mathbb{R})$, $R_2^*,\, c^+\in L^\infty(\Omega;\mathbb{C})$, $\vec{v}\in L^1(0,T;W^{1,\infty}(\Omega;\mathbb{R}^3))$, $D\in L^\infty(\Omega;\mathbb{R}^{3\times 3})$, satisfying divv, definiteness, $\vec{G}\in L^1(0,T;\mathbb{R}^3)$ and the estimate holds for any $q\in[2,\infty]$ with a constant $C$ depending only on $q$ and $1-\

Theorems & Definitions (18)

  • Proposition 2.1
  • Remark 2.1
  • Remark 2.2
  • Lemma 2.1
  • Proposition 2.2
  • Proposition 2.3
  • Remark 2.3
  • Proposition 2.4
  • Proposition 2.5
  • Proposition 3.1
  • ...and 8 more