In the last decade, numerical models have been an increasingly important tool in biological and medical science both for the fundamental understanding of physiology as well as the potential for novel diagnostics and treatments tools in clinical application. In this paper, a nonlinear multi-scale model framework is developed for blood flow distribution in the full vascular system of an organ. We couple a quasi 1D vascular graph model to represent blood flow in larger vessels and a porous media model to describe flow in smaller vessels and capillary bed. The vascular model is based on Poiseuille’s law, with pressure correction by elasticity and pressure drop estimation at vessels junctions. The porous capillary bed is modelled as a two-compartment domain (artery and venous) using Darcy’s law. The fluid exchange between the artery and venous capillary bed compartments are defined as blood perfusion. 

The numerical experiments show that the proposed model for blood circulation: 1) is closely dependent on the structure and parameters of both the vascular vessels and of the capillary bed, and 2) it provides a realistic blood circulation in the organ. The advantage of the proposed model is that it is complex enough to reliably capture the main underlying physiological function, yet highly flexible as it offers the possibility of incorporating various local effects. Furthermore, the numerical implementation of the model is straightforward and allows for simulations on a regular desktop computer. 

The vascular system was segmented from an image in a classical medical textbook, Cohnheim JF. 1872 Untersuchungen über die embolischen Prozesse. Berlin: Hirschwald.

Segmentation based on Hanson EA, Lundervold A. 2013 Local/non-local regularized image segmentation using graph-cuts. International Journal of Computer Assisted Radiology and Surgery 8, 1073–1084.