Quantitative Susceptibility Mapping (QSM) Algorithms: Mathematical Rationale and Computational Implementations

• Published in: IEEE transactions on biomedical engineering Vol. 64; no. 11; pp. 2531 - 2545
• Main Authors: Kee, Youngwook, Liu, Zhe, Zhou, Liangdong, Dimov, Alexey, Cho, Junghun, de Rochefort, Ludovic, Seo, Jin Keun, Wang, Yi
• Format: Journal Article
• Language: English
• Published: United States IEEE 01.11.2017; The Institute of Electrical and Electronics Engineers, Inc. (IEEE); Institute of Electrical and Electronics Engineers
• Subjects: Algorithms; Bayes methods; Bayes Theorem; Bayesian analysis; Bayesian image reconstruction; Bioengineering; Brain - diagnostic imaging; Computer applications; Computer Science; Computer Simulation; Convolution; Data acquisition; deconvolution; Dipoles; Forward problem; Humans; Ill posed problems; Image acquisition; Image Processing, Computer-Assisted - methods; Imaging; Integrals; Inverse problems; Kernel; Life Sciences; Magnetic fields; Magnetic permeability; Magnetic resonance imaging; Magnetic Resonance Imaging - methods; Magnetic susceptibility; Mapping; Medical Imaging; nonsmooth convex optimization; Partial differential equations; partial differential equations (PDEs); Preconditioning; quantitative susceptibility mapping (QSM); Resource description framework; Shadows; Inverse problems; Magnetic resonance imaging; Partial differential equations; Resource description framework; Bayes methods; Kernel; Magnetic susceptibility; Magnetic Resonance Imaging; snc
• ISSN: 0018-9294; 1558-2531; 1558-2531
• DOI: 10.1109/TBME.2017.2749298

Quantitative susceptibility mapping (QSM) solves the magnetic field-to-magnetization (tissue susceptibility) inverse problem under conditions of noisy and incomplete field data acquired using magnetic resonance imaging. Therefore, sophisticated algorithms are necessary to treat the ill-posed nature of the problem and are reviewed here. The forward problem is typically presented as an integral form, where the field is the convolution of the dipole kernel and tissue susceptibility distribution. This integral form can be equivalently written as a partial differential equation (PDE). Algorithmic challenges are to reduce streaking and shadow artifacts characterized by the fundamental solution of the PDE. Bayesian maximum a posteriori estimation can be employed to solve the inverse problem, where morphological and relevant biomedical knowledge (specific to the imaging situation) are used as priors. As the cost functions in Bayesian QSM framework are typically convex, solutions can be robustly computed using a gradient-based optimization algorithm. Moreover, one can not only accelerate Bayesian QSM, but also increase its effectiveness at reducing shadows using prior knowledge based preconditioners. Improving the efficiency of QSM is under active development, and a rigorous analysis of preconditioning needs to be carried out for further investigation.