Magnetic susceptibility measurement has wide-ranging applications in MR technical development and medical applications. A general susceptibility quantitation method for objects of arbitrary shapes in inhomogeneous magnetic fields is presented in this study. Based on the mean value properties of magnetic fields, the polarizing magnetic field at the location of interest inside an object can be exactly obtained in situ from the field values on a spherical surface enclosing the object. With numerical computation of the self-demagnetizing field and correction of contact shifts, magnetic susceptibilities were quantitatively measured for CuSO 4 Magnetic susceptibility effects relate to wide-ranging MR applications, such as artifact corrections (1,2), blood/tumor oxygenation measurement (3,4), the BOLD (bloodoxygenation-level-dependent) signal contrast in functional MRI (5), medical diagnosis (6), and in vivo imaging of gene expression (7). Therefore, quantitative magnetic susceptibility measurement is important. A number of NMR methods exist to measure susceptibility or susceptibility related effects based on various mechanisms (8 -20). A common approach is to measure the MR frequency or phase shift due to the field deviation from the susceptibility effects (16 -20). However, for most of the existing methods magnetic susceptibility quantitation for macroscopically localized objects is limited to objects with regular shapes such as a sphere or a long cylinder (8,10,16 -18), and the static polarizing field is either assumed to be sufficiently homogeneous or accounted for with approximation methods such as an extra reference scan or polynomial fitting methods (17)(18)(19)(20). Wang et al. (20) reported a susceptibility quantitation method which has no presumption of sample shape and only needs measurement of the magnetic field difference across an interface of two regions of different susceptibilities. In practice, accuracy of the method is reduced because of the severe partial volume effects at the interface caused by abrupt field change and finite image resolution.The purpose of this article is to present a new susceptibility quantitation method that is theoretically rigorous in inhomogeneous polarizing fields and applicable to objects with arbitrary shapes. The susceptibility is quantified under a general condition: multiple localized distributions of materials with distinct susceptibilities are present in space permeated with a medium of known susceptibility o , and the inhomogeneous applied field H a . The goal is to determine the susceptibility value i of a localized region K (boundary denoted as ץK, with an arbitrary shape and uniform susceptibility inside) from the measurement of field distribution in space (Fig. 1).If one considers media in which magnetization M and magnetic field H are collinear and in proportion to each other, i.e., M ϭ H ( is the magnetic susceptibility), then the relationship between MR-measured magnetic induction field B nmr and magnetic field H at any point in space is given by the vect...