Multifield computational model for human brain development: Explicit numerical stabilization

Abstract

AbstractThe morphological surface of the human brain is still puzzling to scientists. During the last few decades, studies have addressed different aspects of this phenomenon. The underlying cellular processes occurring at the microscopic scale are certainly important for the folding process. However, there is also an essential contribution from mechanical forces generated at the macroscopic scale during these processes. Many previous studies confirmed this fact, but still, most of them have not considered the link between cellular processes and mechanical forces. Recently, we have explained how the proliferation in different zones of the brain, cell migration, and neuronal connectivity affect the folding through a multifield computational model coupling an advection‐diffusion model with the theory of finite growth. We mathematically describe the cellular behavior of cells to control mechanical growth in both radial and circumferential directions. To deal with issues regarding numerical stability of the advection‐diffusion equation, which lead to temporal and spatial numerical oscillations, reduced efficiency of the numerical solver, and inaccurate results, we introduce an artificial diffusivity, which we apply only when the actual cell density does not satisfy the balance equation. The proposed method stabilizes the numerical solution and significantly improves the simulation results without additional numerical cost. It allows us to study brain development in 2D and 3D.