Pressure–Poisson equation in numerical simulation of cerebral arterial circulation and its effect on the electrical conductivity of the brain

Abstract

Background and ObjectiveThis study considers dynamic modeling of the cerebral arterial circulation and reconstructing an atlas for the electrical conductivity of the brain. Electrical conductivity is a governing parameter in several electrophysiological modalities applied in neuroscience, such as electroencephalography (EEG), transcranial electrical stimulation (tES), and electrical impedance tomography (EIT). While high-resolution 7-Tesla (T) Magnetic Resonance Imaging (MRI) data allow for reconstructing the cerebral arteries with a cross-sectional diameter larger than the voxel size, electrical conductivity cannot be directly inferred from MRI data. Brain models of electrophysiology typically associate each brain tissue compartment with a constant electrical conductivity, omitting any dynamic effects of cerebral blood circulation. Incorporating those effects poses the challenge of solving a system of incompressible Navier–Stokes equations (NSEs) in a realistic multi-compartment head model. However, using a simplified circulation model is well-motivated since, on the one hand, the complete system does not always have a numerically stable solution and, on the other hand, the full set of arteries cannot be perfectly reconstructed from the MRI data, meaning that any solution will be approximative. MethodsWe postulate that circulation in the distinguishable arteries can be estimated via the pressure–Poisson equation (PPE), which is coupled with Fick's law of diffusion for microcirculation. To establish a fluid exchange model between arteries and microarteries, a boundary condition derived from the Hagen–Poisseuille model is applied. The relationship between the estimated volumetric blood concentration and the electrical conductivity of the brain tissue is approximated through Archie's law for fluid flow in porous media. ResultsThrough the formulation of the PPE and a set of boundary conditions (BCs) based on the Hagen–Poisseuille model, we obtained an equivalent formulation of the incompressible Stokes equation (SE). Thus, allowing effective blood pressure estimation in cerebral arteries segmented from open 7T MRI data. ConclusionsAs a result of this research, we developed and built a useful modeling framework that accounts for the effects of dynamic blood flow on a novel MRI-based electrical conductivity atlas. The electrical conductivity perturbation obtained in numerical experiments has an appropriate overall match with previous studies on this subject. Further research to validate these results will be necessary.