Superparamagnetic iron oxide (SPIO) nanoparticles generate a strong susceptibility gradient making them an ideal contrast agent in magnetic resonance imaging (MRI). Contrast agents based on SPIO nanoparticles are ideally suited for a wide range of applications in MRI. Applications of SPIO include: liver imaging (1, 2), MR angiography (3), and tracking of labeled cells in vivo (4–7). Quantification of the total amount of SPIO-based contrast agent allows researchers and clinicians to quantitatively evaluate the results from different treatments and studies (7).

Contrast agent quantification methods can be broken into two categories: relaxometry methods and model-based methods. The relaxometry methods use the magnitude information from multiple complex-valued MR images and quantify the concentration of SPIOs within a given area by mapping the relaxation rate of SPIOs within tissue. The model-based methods quantify the concentration of SPIOs using the information about magnetic field inhomogeneities from the phase map and then model the magnetic field inhomogeneities as simple geometries.

The relaxometry methods utilize signal enhancement or decay associated with areas containing SPIOs. The effective transverse relaxation rate, denoted *R*, of SPIO labeled cells satisfies the static dephasing regime theory (8, 9). Since the value of *R*_{2} for SPIOs is approximately two orders smaller than *R* (9), most relaxometry methods attempt to measure *R* over *R*_{2} (10, 11). However, some relaxometry methods quantify the concentration by measuring *R*_{1} (12). Recent advances in the relaxometry-based methods utilize multiple acquisition pulse sequences that significantly reduce the total number of scans (13). Relaxometry methods map *R* in a particular region and assume that *R* varies linearly with contrast agent concentration. The equation that governs how the relaxation rate changes with concentration is

where *R* denotes the intrinsic relaxation rate (e.g., no contrast agent in tissue), *r* denotes the relaxivity of the contrast agent, *c* denotes the concentration of the contrast agent.

The model-based quantification methods rely on analytic models of simple geometries and use the magnetic field inhomogeneities generated by areas containing SPIOs to quantify the concentration of the SPIOs. The magnetic field inhomogeneities created by the SPIOs are identified using phase maps. The concentration of SPIOs is quantified by, first, modeling the magnetic field inhomogeneities as simple geometries such as a sphere or infinite cylinder; then, fitting the magnetic field of the model to the magnetic field inhomogeneities acquired experimentally. The method presented in (14) used an infinite cylinder model and the method presented in (15) used a spherical dipole model to measure the magnetization of a solution of SPIO-based contrast agent suspended within a phantom. Recently published phase-based quantification methods (16, 17) model distributions of iron as spherical dipoles. However, the two methods differ on how the magnetic field inhomogeneities from the iron distributions are calculated.

One major limitation of the model-based quantification method is that the method breaks down when an object with a complex geometric shape is encountered (16). This limitation can be dealt with by applying different numerical techniques to take into account of complex geometry and then calculate the magnetic field inhomogeneities. The numerical techniques can be divided into four categories: methods based on the finite element or finite difference methods (18–20), methods that utilize boundary conditions (21, 22), approximation methods (23), and Fourier-based methods (23–27).

A new SPIO quantification algorithm is introduced in this work to deal with a complex geometric case. The proposed method utilizes a positive contrast technique, known as phase gradient mapping, to bypass the phase unwrapping step in the quantification process. A modified finite perturber method is used to model the magnetic field for complex geometries (23). The proposed method quantifies SPIOs in two steps: first, calculating the phase gradient map of the acquired MR data; and second, fitting the phase gradient map to the gradient generated by the theoretical model, a distribution of spherical dipoles occupying the geometry of the object under consideration. The phase gradient is considered because it does not require the phase unwrapping step present in previous model-based quantification methods. In this work, the proposed quantification method is validated on both phantom and in vivo mouse models with SPIO labeled C6 glioma cells injected in the flanks. The performance of the proposed quantification method in a low SNR environment is also evaluated using the phantom data set.