Numerical simulation of the blood oxygenation level-dependent functional magnetic resonance signal using finite element method

Since the introduction of functional magnetic resonance imaging (fMRI), several computational approaches have been developed to examine the effect of the morphology and arrangement of blood vessels on the blood oxygenation-level dependent (BOLD) signal in the brain. In the present work, we implemented the original Ogawa's model using a numerical simulation based on the finite element method (FEM) instead of the analytical models. In literature, there are different works using analytical methods to analyse the transverse relaxation rate ( $R_2^{\ast }$ ), which BOLD signal is related to, modelling the vascular system with simple and canonical geometries such as an infinite cylinder model (ICM) or a set of cylinders. We applied the numerical simulation to the extravascular BOLD signal as a function of angular vessel distribution (perpendicular vs parallel to the static magnetic field) relevant for anatomical districts characterized by geometrical symmetries, such as spinal cord. Numerical simulations confirmed analytical results for the canonical ICM. Moreover, the perturbation to the magnetic field induced by blood deoxyhaemoglobin, as quantified assuming Brownian diffusion of water molecules around the vessel, revealed that vessels contribute the most to the variation of the $R_2^{\ast }$ when they are preferentially perpendicular to the external magnetic field, regardless of their size. Our results indicate that the numerical simulation method is sensitive to the effects of different vascular geometry. This work highlights the opportunity to extend $R_2^{\ast }$ simulations to realistic models of vasculature based on high-resolution anatomical images.