Nonlinear Inversion of the Bloch Equations
| Duration: | 2019 - 2024 |
| Technologies: | C, C++, IDEA, Shell Scripting, Python |
| Collaborators: | Xiaoqing Wang, Volkert Roeloffs, Sebastian Rosenzweig, Martin Uecker |
Illustration of the cost function and forward operator of the nonlinear inversion of the Bloch equations. It includes the pattern \(\mathcal{P}\), Fourier \(\mathcal{F}\), coil \(\mathcal{C}\), and signal model (Bloch) \(\mathcal{B}\) operators, the measured data \(\boldsymbol{y}\), the estimated parameter \(\boldsymbol{x_p}\) and coil maps \(\boldsymbol{x_c}\), as well as the added smoothness \(\mathcal{Q}\) and sparsity \(\mathcal{R}\) constraints with their scaling factors \(\alpha\) and \(\beta\).
In the work "Quantitative MRI by nonlinear inversion of the Bloch equations", we developed a generic framework for simultaneous quantification of multiple physical parameters for arbitrary sequences.
The technique has been developed at the University Medical Center in Göttingen and the Graz University of Technology.
The novelty of the proposed work is the incorporation of a full Bloch simulation as forward model into a nonlinear reconstruction algorithm.
This avoids over-simplified analytical models and a specialization of the reconstruction to individual sequences.
It renders the developed technique applicable to all MRI sequences that can be simulated and especially includes sequences with non-analytical signal dynamics.
The proposed work required two main innovations.
First, the generalization of a forward model from analytical representations to arbitrary dynamics adds much computational complexity.
Conventional simulations based on ordinary differential equation solvers and operator splitting techniques were found too slow to combine a high accuracy with a reasonable reconstruction time.
Therefore, we developed a new simulation technique that can efficiently exploit repeating patterns in a sequence.
The developed state-transition matrix simulation determines time-ordered exponentials, which model the dynamics of the Bloch equations accurately without the need for continuous fine discretization or analytical approximations.
The second innovation in this work is the introduced combination of the STM simulation with a direct sensitivity analysis of the Bloch equations.
This enabled a simultaneous estimation of the signal together with its partial derivatives. The method was found to be efficient and especially numerically stable and accurate. Its development allowed us to incorporate the generic forward model into a gradient-based nonlinear optimization solver for a calibrationless model-based reconstruction.
Together these components define a generic model-based reconstruction framework for multi-parametric quantitative MRI that can be used with data from different pulse sequences and that allows realistic modelling of non-analytical signal dynamics.
Resources
A main focus of this work has been to make it fully open-source and easily reproducible. The manuscript, all scripts and data to recreate the individual figures, the code of all developed methods, and a tutorial have been published under a free CC-BY 4.0 license and can be accessed online by everyone.| Resources | Location |
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References
The technique and parts of this page have been published in:- Scholand N, Wang X, Roeloffs V, Rosenzweig S, Uecker M. Quantitative MRI by nonlinear inversion of the Bloch equations. Magn Reson Med. 2023; 90: 520-538. doi: 10.1002/mrm.29664
- Scholand N. Quantitative Multi-Parameter Mapping in Magnetic Resonance Imaging. Göttingen Graduate Center for Neurosciences, Biophysics, and Molecular Biosciences, PhD Thesis, 2023. doi: 10.53846/goediss-10028