Speaker
Description
We present a computational framework for model-based quantitative imaging through the identification of spatially varying relaxation parameters in the Bloch-Torrey equation, a time dependent PDE governing magnetization dynamics in magnetic resonance imaging. The equation couples RF induced precession, longitudinal and transverse relaxation, and diffusion advection transport, a structure that poses particular challenges for both discretization and adjoint derivation due to the interplay of rotational, dissipative, and elliptic operators acting on different time scales. The forward problem is discretized using a first order Lie operator splitting that preserves the physical structure of each sub process: precession is handled using the exact Rodrigues formula, relaxation using pointwise exponentials, and diffusion advection using an implicit P1 finite element scheme. The order of accuracy is rigorously verified using particular known solutions, confirming the expected temporal and spatial convergence rates. A key contribution is the derivation of a discrete adjoint consistent with the splitting scheme. In particular, the adjoint of the Rodrigues sub-step is its exact inverse rotation, a nontrivial requirement that ensures gradient consistency without resorting to continuous adjoints or automatic differentiation. Gradients of the least-squares cost functional are used within an L-BFGS method formulated in the L²(Ω) inner product, enabling distributed parameter identification for three spatially varying fields: the longitudinal rate R₁, the transverse rate R₂, and the equilibrium magnetization M_eq. Numerical experiments on a synthetic phantom demonstrate accurate simultaneous reconstruction of all three parameters from CPMG multi echo measurements, across a hierarchy of mesh refinements with warm-started interpolation between levels.