Quantitative MRI by nonlinear inversion of the Bloch equations

Abstract

Purpose: Development of a generic model-based reconstruction framework for multiparametric quantitative MRI that can be used with data from different pulse sequences.

Methods: Generic nonlinear model-based reconstruction for quantitative MRI estimates parametric maps directly from the acquired k-space by numerical optimization. This requires numerically accurate and efficient methods to solve the Bloch equations and their partial derivatives. In this work, we combine direct sensitivity analysis and pre-computed state-transition matrices into a generic framework for calibrationless model-based reconstruction that can be applied to different pulse sequences. As a proof-of-concept, the method is implemented and validated for quantitative $T_1$ and $T_2$ mapping with single-shot inversion-recovery (IR) FLASH and IR bSSFP sequences in simulations, phantoms, and the human brain.

Results: The direct sensitivity analysis enables a highly accurate and numerically stable calculation of the derivatives. The state-transition matrices efficiently exploit repeating patterns in pulse sequences, speeding up the calculation by a factor of 10 for the examples considered in this work, while preserving the accuracy of native ordinary differential equations solvers. The generic model-based method reproduces quantitative results of previous model-based reconstructions based on the known analytical solutions for radial IR FLASH. For IR bSFFP it produces accurate $T_1$ and $T_2$ maps for the National Insitute of Standards and Technology (NIST) phantom in numerical simulations and experiments. Feasibility is also shown for human brain, although results are affected by magnetization transfer effects.

Conclusion: By developing efficient tools for numerical optimizations using the Bloch equations as forward model, this work enables generic model-based reconstruction for quantitative MRI.

Keywords: Bloch equations; model-based reconstruction; nonlinear inversion; quantitative MRI; sensitivity analysis; state-transition matrix.