A cut finite element method for spatially resolved energy metabolism models in complex neuro-cell morphologies with minimal remeshing

A thorough understanding of brain metabolism is essential to tackle neurodegenerative diseases. Astrocytes are glial cells which play an important metabolic role by supplying neurons with energy. In addition, astrocytes provide scaffolding and homeostatic functions to neighboring neurons and contribute to the blood–brain barrier. Recent investigations indicate that the complex morphology of astrocytes impacts upon their function and in particular the efficiency with which these cells metabolize nutrients and provide neurons with energy, but a systematic understanding is still elusive. Modelling and simulation represent an effective framework to address this challenge and to deepen our understanding of brain energy metabolism. This requires solving a set of metabolic partial differential equations on complex domains and remains a challenge. In this paper, we propose, test and verify a simple numerical method to solve a simplified model of metabolic pathways in astrocytes. The method can deal with arbitrarily complex cell morphologies and enables the rapid and simple modification of the model equations by users also without a deep knowledge in the numerical methods involved. The results obtained with the new method (CutFEM) are as accurate as the finite element method (FEM) whilst CutFEM disentangles the cell morphology from its discretisation, enabling us to deal with arbitrarily complex morphologies in two and three dimensions.