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...

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

Magnetic susceptibility is an important physical property of tissues, and can be used as a contrast mechanism in magnetic resonance imaging. Recently, targeting contrast agents by conjugation with signaling molecules and labeling stem cells with contrast agents have become feasible. These contrast agents are strongly paramagnetic, and the ability to quantify magnetic susceptibility could allow accurate measurement of signaling and cell localization. Presented here is a technique to estimate arbitrary magnetic susceptibility distributions by solving an ill-posed inversion problem from field maps obtained in an MRI scanner. Two regularization strategies are considered, conventional Tikhonov regularization, and a sparsity promoting nonlinear regularization using the ℓ1 norm. Proof of concept is demonstrated using numerical simulations, phantoms, and in a stroke model mouse. Initial experience indicates that the nonlinear regularization better suppresses noise and streaking artifacts common in susceptibility estimation.

The pulmonary nodule is the most common manifestation of lung cancer, the most deadly of all cancers. Most small pulmonary nodules are benign, however, and currently the growth rate of the nodule provides for one of the most accurate noninvasive methods of determining malignancy. In this paper, we present methods for measuring the change in nodule size from two computed tomography image scans recorded at different times; from this size change the growth rate may be established. The impact of partial voxels for small nodules is evaluated and isotropic resampling is shown to improve measurement accuracy. Methods for nodule location and sizing, pleural segmentation, adaptive thresholding, image registration, and knowledge-based shape matching are presented. The latter three techniques provide for a significant improvement in volume change measurement accuracy by considering both image scans simultaneously. Improvements in segmentation are evaluated by measuring volume changes in benign or slow growing nodules. In the analysis of 50 nodules, the variance in percent volume change was reduced from 11.54% to 9.35% (p = 0.03) through the use of registration, adaptive thresholding, and knowledge-based shape matching.

Existing parallel MRI methods are limited by a fundamental trade-off in that suppressing noise introduces aliasing artifacts. Bayesian methods with an appropriately chosen image prior offer a promising alternative; however, previous methods with spatial priors assume that intensities vary smoothly over the entire image, resulting in blurred edges. Here we introduce an edge-preserving prior (EPP) that instead assumes that intensities are piecewise smooth, and propose a new approach to efficiently compute its Bayesian estimate. The estimation task is formulated as an optimization problem that requires a nonconvex objective function to be minimized in a space with thousands of dimensions. As a result, traditional continuous minimization methods cannot be applied. This optimization task is closely related to some problems in the field of computer vision for which discrete optimization methods have been developed in the last few years. We adapt these algorithms, which are based on graph cuts, to address our optimization problem