Theoretical Models of Blood Flow in the Vasculature: Combining Discrete and Continuum Approaches

The vasculature is a 3D multiscale network comprised of a hierarchy of vessels that is frequently categorized according to vessel size. Although the geometry and topology of the vasculature is organ-specific, blood flows into an organ from a feeding artery, through the arterioles into the microcirculation, and exits through the venules then veins. Gas exchange occurs primarily in the microcirculation and, indeed, the function of the vasculature is to bring oxygenated blood within a small distance of every tissue point in the body in order to meet metabolic demands. Understanding and predicting the flow of blood through these networks will play a crucial role in, for example, promoting angiogenesis and vascular remodelling to treat myocardial ischaemia.

Traditional modelling approaches have employed a discrete approach by solving equations for blood flow in each vessel of a network. However, recent advancements in imaging methods have led to a wealth of imaging data that describe vascular structure in a highly detailed way. As the resolution of this data increases, it is becoming too computationally intensive to simulate flow and mass transport in the complete vascular tree using a discrete approach. As such, continuum models must be developed that can be used alongside a discrete approach to capture the key functional properties of blood flow.

Continuum multiscale models that describe blood flow in the microcirculation, derived using the mathematical process of asymptotic homogenization, will be discussed. A strategy for combining discrete and continuum models to simulate blood flow in large networks will be presented, and results of this strategy for explicit examples of rat mesentery networks will be demonstrated.