Solution of the Bloch Equation Based on Spectral Diagonalization and Matrix Exponential Integration: Modeling of SSFP for SNMR

Tingting Lin, Qingyue Wang, Chuandong Jiang, Chunpeng Ren, Yunzhi Wang, Liang Wang

We present a spectral-diagonalization-based matrix exponential integration (SD-MEI) algorithm for efficient and stable solutions of fully coupled Bloch equations in surface nuclear magnetic resonance (SNMR). Conventional explicit numerical methods exhibit cumulative discretization errors and escalating computational costs due to step-size dependence and finite precision limitations. SD-MEI integrates spectral diagonalization with matrix exponential operations, replacing iterative computations with a single eigendecomposition of the system matrix. This approach achieves parameter-robust computational complexity while maintaining numerical stability across broad RF field strengths (10$^{-10}$ T to 10$^{-5}$ T) and relaxation times (10 ms to 1000 ms). Validated for steady-state free precession (SSFP) dynamics in heterogeneous geomagnetic environments, the method enables high-accuracy modeling of transient magnetization evolution with large time steps. The framework advances SNMR efficient forward modeling and inversion while optimizing protocols by resolving critical limitations in existing numerical and analytical approaches.