Efficient Chebyshev reconstruction for the anisotropic equilibrium model in magnetic particle imaging

Magnetic Particle Imaging (MPI) is a tomographic imaging modality capable of real-time, high-sensitivity mapping of superparamagnetic iron oxide nanoparticles. Model-based image reconstruction provides an alternative to conventional methods that rely on a measured system matrix, eliminating the need for laborious calibration measurements. Nevertheless, model-based approaches must account for the complexities of the imaging chain to maintain high image quality. This work investigates and accelerates an adapted direct Chebyshev reconstruction (DCR) method that accounts for the physical effect of magnetic nanoparticle anisotropy. Approach: The adapted DCR method employs weighted Chebyshev polynomials in the frequency domain and incorporates a spatially variant deconvolution step to compensate for magnetic anisotropy effects. The method is evaluated on five experimental phantoms and compared with model-based and measured system matrix reconstructions. In addition, an efficient approximation of the spatially variant deconvolution is introduced to reduce computational runtime and memory consumption while maintaining reconstruction accuracy. The proposed approach is further investigated on simulated three-dimensional data to assess its scalability to volumetric MPI. Main results: The adapted DCR method significantly improves reconstruction quality in real measurements and achieves image fidelity comparable to that of the measured system matrix reconstruction and superior to that of the model-based system matrix reconstruction. The proposed approximation of the spatially variant deconvolution preserves the reconstruction accuracy while reducing computational and memory requirements. Its computational complexity is reduced to $\mathcal{O}(N \log N)$. The method also successfully reconstructs simulated 3D data, demonstrating its applicability to volumetric MPI. Significance: The results demonstrate the potential of the adapted DCR approach to improve model-based MPI reconstruction while avoiding the need for a system matrix. The efficient deconvolution approximation makes the method particularly attractive for high-resolution and three-dimensional imaging by substantially improving its computational efficiency and scalability.