Weak solutions to the Bloch equations with distant dipolar field.

Area of Science:

• Physics, Computational Physics, Magnetic Resonance Imaging

Background:

• The distant dipolar field (DDF) significantly impacts spin dynamics in liquids, influencing multiple-quantum coherences and magnetic resonance imaging (MRI) contrast.
• Standard Fast Fourier Transform (FFT) methods for DDF calculations are limited to periodic or Cartesian grids, posing challenges for bounded, complex geometries.
• Accurate simulation of Bloch-DDF dynamics on bounded domains with realistic diffusion boundary conditions is crucial for advanced MRI applications.

Purpose of the Study:

• To develop and validate a robust computational framework for simulating Bloch-DDF dynamics on bounded domains with complex geometries and Neumann diffusion conditions.
• To provide an analyzable and reproducible method for understanding spin dynamics influenced by DDF in realistic sample environments.

Main Methods:

• Derived a conforming finite-element weak formulation for Bloch-DDF equations, incorporating spatially varying diffusion/relaxation and short-distance regularization of the DDF kernel.
• Proved operator boundedness and established an L2 energy balance, demonstrating the dissipative nature of diffusion and relaxation.
• Employed a matrix-free near/far scheme for real-space DDF evaluation and an implicit-explicit splitting method for time integration, with explicit precession treatment using Rodrigues rotations.

Main Results:

• Established local well-posedness and continuous dependence on data for the Bloch-DDF model under specific conditions.
• Validated the finite-element method against three closed-form benchmarks, confirming its accuracy in isolating model components.
• Quantified curved-boundary effects by comparing finite-element results with finite-difference methods on a spherical domain.

Conclusions:

• The developed finite-element method provides a stable, accurate, and reproducible approach for simulating Bloch-DDF dynamics on bounded domains with complex geometries.
• This framework supports multi-cycle, lab-frame calculations and offers a pathway for more precise modeling of MRI contrast in diverse sample types.