Multi-scale homogenization of blood flow in 3-dimensional human cerebral microvascular networks

The microvasculature plays a crucial role in the perfusion of blood through cerebral tissue. Current models of the cerebral microvasculature are discrete, and hence only able to model the perfusion over small voxel sizes before becoming computationally prohibitive. Larger models are required to provide comparisons and validation against imaging data. In this work, multi-scale homogenization methods were employed to develop continuum models of blood flow in a capillary network model of the human cortex. Homogenization of the local scale blood flow equations produced an averaged form of Darcy׳s law, with the permeability tensor encapsulating the capillary bed topology. A statistically accurate network model of the human cortex microvasculature was adapted to impose periodicity, and the elements of the permeability tensor calculated over a range of voxel sizes. The permeability tensor was found to converge to an effective permeability as voxel size increased. This converged permeability tensor was isotropic, reflecting the mesh-like structure of the cerebral microvasculature, with off-diagonal terms normally distributed about zero. A representative elementary volume of 375 µm, with a standard deviation of 4.5% from the effective permeability, was determined. Using the converged permeability values, the cerebral blood flow was calculated to be around 55 mL min−1 100 g−1, which is in very close agreement with experimental values. These results open up the possibility of future multi-scale modeling of the cerebral vascular network.