The diagnosis of many neurologic diseases benefits from the ability to quantitatively assess iron in the brain. Paramagnetic iron modifies the magnetic susceptibility causing magnetic field inhomogeneity in MRI. The local field can be mapped using the MR signal phase, which is discarded in a typical image reconstruction. The calculation of the susceptibility from the measured magnetic field is an ill-posed inverse problem. In this work, a bayesian regularization approach that adds spatial priors from the MR magnitude image is formulated for susceptibility imaging. Priors include background regions of known zero susceptibility and edge information from the magnitude image. There is a growing scientific and clinical interest in quantitatively mapping magnetic biomaterials by measuring their susceptibilities using MRI. Quantifying endogenous paramagnetic iron would be useful for assessing blood oxygenation (1-3) and iron overloading in organs such as the liver (4) and the heart (5). The diagnosis and monitoring of vascular and neurodegenerative diseases in the brain would benefit directly from iron quantification (6). Susceptibility quantification may allow exploiting the strong diamagnetism of calciumbased structures to characterize osteoporosis (7,8) or calcifications in the breast and brain. Furthermore, quantitative susceptibility mapping (QSM) would allow robust quantification of paramagnetic and superparamagnetic contrast agents essential to molecular and cellular imaging (9-11) and also be valuable to the characterization of cardiovascular function (12)(13)(14). Recently, an MR reporter gene enabling iron accumulation within the cell was demonstrated (15), and quantifying the induced iron would be very important for investigating in vivo molecular biology.Quantifying the susceptibility from the magnetic field is an inverse problem similar to magnetoencephalography, in which magnetic sources inside the brain must be located and quantified from limited measurements of the field outside the head (16). While quantification based on geometrical models has long been used for specific applications (1,2,4,8,14,(17)(18)(19)(20), the reconstruction of susceptibility maps in which each voxel has an unknown susceptibility is a much more complex problem. While some approaches have been theoretically proposed (6,21,22), the ill-posedness due to limited measurements was dealt with recently by using regularization approaches (8,23) or acquisition strategies (24). Here, a bayesian regularized approach is presented that introduces priors derived from the MR magnitude image. It is shown that imposing values at given locations together with seeking a solution that shares edges with the MR magnitude image is more robust that the previously proposed methods (23). The technique is validated using simulations and phantom experiments. Additionally, in vivo brain susceptibility maps are obtained, introducing a new quantitative contrast that is directly linked to the amount of iron in the brain. MATERIALS AND METHODS Susceptibility an...

The magnetic susceptibility of tissue can be determined in gradient echo MRI by deconvolving the local magnetic field with the magnetic field generated by a unit dipole. This Quantitative Susceptibility Mapping (QSM) problem is unfortunately ill-posed. By transforming the problem to the Fourier domain, the susceptibility appears to be undersampled only at points where the dipole kernel is zero, suggesting that a modest amount of additional information may be sufficient for uniquely resolving susceptibility. A Morphology Enabled Dipole Inversion (MEDI) approach is developed that exploits the structural consistency between the susceptibility map and the magnitude image reconstructed from the same gradient echo MRI. Specifically, voxels that are part of edges in the susceptibility map but not in the edges of the magnitude image are considered to be sparse. In this approach an L1 norm minimization is used to express this sparsity property. Numerical simulations and phantom experiments are performed to demonstrate the superiority of this L1 minimization approach over the previous L2 minimization method. Preliminary brain imaging results in healthy subjects and in patients with intracerebral hemorrhages illustrate that QSM is feasible in practice.

Quantitative susceptibility mapping (QSM) opens the door for measuring tissue magnetic susceptibility properties that may be important biomarkers, and QSM is becoming an increasingly active area of scientific and clinical investigations. In practical applications, there are sources of errors for QSM including noise, phase unwrapping failures, and signal model inaccuracy. To improve the robustness of QSM quality, we propose a nonlinear data fidelity term for frequency map estimation and dipole inversion to reduce noise and effects of phase unwrapping failures, and a method for model error reduction through iterative tuning. Compared with the previous phase based linear QSM method, this nonlinear QSM method reduced salt and pepper noise or checkerboard pattern in high susceptibility regions in healthy subjects and markedly reduced artifacts in patients with intracerebral hemorrhages.

With the recent development of iron-based contrast agents and biomarkers for drug delivery (1) and molecular imaging (2), a robust technique to quantify iron content has become an increasingly important need. Iron oxides as well as other magnetic biomarkers may be mapped in MRI by identifying the corresponding susceptibility distributions that modify the MR signal. Indeed, susceptibility has been investigated to reveal information about oxygen saturation level in blood, and to measure calcium or iron concentration in tissue, especially in the brain and bone (3-8). Therefore, there has been a major interest in quantifying susceptibility in MRI in general, as it could lead to a unique quantitative tool and provide a novel contrast mechanism.Quantifying arbitrary susceptibility distributions by inverting the measured magnetic field remains challenging because susceptibility inversion is intrinsically ill-posed (9). To circumvent this issue, several techniques have been proposed. Some of these techniques assume a uniform susceptibility distribution, or further require a well-defined geometric shape (3-5,10 -12). A voxel-based inversion has been proposed assuming there are sufficient measurement points (13), but it is computationally intensive and no experimental work applying this technique has been published. The numerical difficulty may be sidestepped by recasting the inverse problem as an iterative model fitting problem, but such a solution underestimates susceptibility by 50% (14). The inverse problem is further complicated by the nonuniform noise in the field measurement and by the high phase noise in regions with strong susceptibility due to signal voids caused by T* 2 effects. An experimentally robust voxel-based susceptibility quantification of arbitrary distribution remains to be developed.Here the ill-posed nature of this field to the source inverse problem is analyzed and a novel method to stabilize the inversion by imaging the object at multiple orientations with respect to B 0 is presented. Theoretical considerations and experimental validations on various objects are shown to examine the robustness of this technique. THEORY Relationship Between Susceptibility and Magnetic FieldIn the following, susceptibility refers to volume susceptibility. The spatially varying susceptibility distribution in an applied external uniform magnetic field changes the local field experienced by a spin in MRI. It can be shown from Maxwell magnetostatic equations and the Lorentz correction for media effects that the susceptibility distribution affects the local field component along the main magnetic field according to ␦ B ͑r ៝͒ ϭ 1 4 ͵ ͑r ៝Ј͒ 3 cos 2 ␣ Ϫ 1 ͉r ៝Ј Ϫ r ͉៝ 3 d 3 r ៝Ј,where r ៝ is the spatial coordinate vector, ␣ is the angle between r ៝Ј Ϫ r ៝ and the applied field, and ␦ B is the relative difference field given by

For optimal image quality in susceptibility-weighted imaging and accurate quantification of susceptibility, it is necessary to isolate the local field generated by local magnetic sources (such as iron) from the background field that arises from imperfect shimming and variations in magnetic susceptibility of surrounding tissues (including air). Previous background removal techniques have limited effectiveness depending on the accuracy of model assumptions or information input. In this article, we report an observation that the magnetic field for a dipole outside a given region of interest (ROI) is approximately orthogonal to the magnetic field of a dipole inside the ROI. Accordingly, we propose a nonparametric background field removal technique based on projection onto dipole fields (PDF). In this PDF technique, the background field inside an ROI is decomposed into a field originating from dipoles outside the ROI using the projection theorem in Hilbert space. This novel PDF background removal technique was validated on a numerical simulation and a phantom experiment and was applied in human brain imaging, demonstrating substantial improvement in background field removal compared with the commonly used high-pass filtering method.