On the representation of magnetic particle imaging in fourier space

Magnetic particle imaging is a tracer-based medical imaging modality. Although various reconstruction methods are known, such as the ones based on a measured system matrix, the mathematical formulation of physical models of magnetic particle imaging is still lacking in several ways. Even for fairly simplified models, such as the Langevin model of paramagnetism, many properties are unproven. Only when one-dimensional excitation is used, the existing models are sufficient to derive simple and fast reconstruction techniques, like the so-called x-space and Chebyshev reconstruction approaches. Recently, an accurate formulation of the one-dimensional Fourier transform of the Langevin function and related functions has been provided. The present article extends the theory to multidimensional magnetic particle imaging. The derived formulations help us to calculate the exact relationship between the system function of Lissajous field-free-point trajectory based magnetic particle imaging and tensor products of Chebyshev polynomials and also uncover a direct relationship to tensor products of Bessel functions of first kind in the spatio-temporal Fourier domain. Moreover, the developed formulation consolidates the mathematical description of magnetic particle imaging and lays the basis for the investigation of different trajectories.