Quantitative susceptibility mapping aims to solve the magnetic dipolar inverse problem to reconstruct tissue magnetic susceptibility distributions from single- or multi-echo GRE phase data. Being an ill-conditioned inverse problem, computation of magnetic susceptibility is challenging and requires conditioning. Several approaches to solve this problem exist, including threshold-based masking or kernel modification, utilizing data redundancy achieved by multiple MRI measurements with different orientations of the object, or applying regularization techniques that incorporate prior information about the spatial distribution of susceptibility. Several of these approaches will be reviewed in this lecture.
Quantitative susceptibility mapping (QSM) aims to estimate the distribution of tissue magnetic susceptibility from local MRI phase measurements. Whereas the related forward problem, i.e., determining the resulting magnetic field arising from a known susceptibility distribution, is well-posed, reconstruction of the susceptibility distribution from such phase information unfortunately represents an ill-conditioned inverse problem due to values of the dipole field equal or close to zero at the magic angle in the Fourier space that cause non-uniqueness and severe noise amplification.
Several methods to overcome the ill-posed nature of this field-to-source inversion problem have been developed in recent years, which rely either on repeated measurements of the object after having been rotated with respect to the magnetic field and combining data or in case of single orientation imaging on the utilization of non-iterative k-space (e.g., simple k-space thresholding) or iterative image-space based regularization methods. Applying regularization one seeks to incorporate a priori knowledge into the solution process (i.e. knowledge like, e.g., amount or type of noise, smoothness or sparsity of the solution, or restrictions on the values the solution may take on), which introduce assumptions on the solution and prevent overfitting but also require choosing one or more regularization parameters.
One approach to compute magnetic susceptibility maps (although clinically not feasible) is to acquire multiple MRI measurements under different orientations of the object (e.g. human head) with respect to the main magnetic field. This method has been denoted as ‘calculation of susceptibility through multiple orientation sampling’ (COSMOS) (45,64,91,99,100) and exploits the fact that the orientation of the magic angle cone regions follows the object rotation relative to the magnetic field. With three or more independent object orientations, the zero cone surfaces of the Fourier domain kernels do not intercept and all Fourier coefficients can be determined (91,99), which enables the computation of the magnetic susceptibility distribution without priors and regularization by solving simultaneously the now over-determined field-to-susceptibility Problem.
More sophisticated algorithms extend the data fidelity term by one or more regularization terms that derive scalar measures from the susceptibility distribution, where the – usually non-linear – functions convert the susceptibility distribution into single scalars summarizing certain image properties, such as the image power, power of the gradient image, or the total variation. The scalar regularization parameters determine the trade-off between data fidelity and penalization of image features described by the corresponding function
Wharton SJ, Bowtell RW.Whole-brain susceptibility mapping at high field: a comparison of multiple- and single-orientation methods. NeuroImage 2010;53(2):515–25
Choi JK, Park HS, Wang S, Wang Y, Seo JK. Inverse Problem in Quantitative Susceptibility Mapping. SIAM J Imag Sci 2014;7(3):1669–89
Haacke EM, Liu S, Buch S, Zheng W, Wu D, Ye Y. Quantitative susceptibility mapping: current status and future directions. Magn Reson Imaging 2015;33(1):1–25.
Khabipova D,Wiaux Y, Gruetter R, Marques JP. A Modulated Closed Form solution for Quantitative Susceptibility Mapping - A thorough evaluation and comparison to iterative methods based on edge prior knowledge. NeuroImage 2014;107:163–74
Wang Y, Liu T. Quantitative susceptibility mapping (QSM): Decoding MRI data for a tissue magnetic biomarker. Magn Reson Med 2015;73(1):82–101