The Effect of Flux Dysconnectivity Functions on Concentration Gradients Changes in a Multicomponent Model of Convectional Reaction-Diffusion by the Example of a Neurovascular Unit

Abstract

A convectional diffusion of nutrients around the blood vessels in brain occurs in well-structured neurovascular units (NVU) including neurons, glia and micro vessels. A common feature of the process is a combination of a relatively high-speed delivery solution stream inside the blood vessel and a low-speed convectional flow in parenchyma. The specific trait of NVU is the existence of a tight cover layer around the vessels which is formed by shoots (end-feet) of astrocytes. This layer forms so called blood-brain barrier (BBB). Under different pathological states the permeability of BBB is changed. The concentration gradient of a chemical compound in NVU has been modelled using a combination of mathematical description of a cerebral blood flow (CBF) and further 3D diffusion away from the blood vessels borders. The governing equation for the blood flow is the non-steady-state Navier–Stokes equation for an incompressible non-Newtonian fluid flow without buoyancy effects. BBB is modeled by the flux dysconnectivity functions. The velocity of fluid flow in the paravascular space was estimated using Darcy's law. Finally, the diffusion of the nutrient is considered as a convectional reaction-diffusion in a porous media. By the example of glucose, it was shown that increased permeability of BBB yields an increased level of the nutrient even under essential (on 70%) decrease of CBF. Contrarily, a low BBB permeability breeds a decreased concentration level under increased (on 50%) CBF. Such a phenomenon is explained by a smooth enlarge of the direct diffusion area for a blood-to-brain border glucose transport having three-level organization.