BB6: Developing a novel mathematical model of coronary blood flow in the heart

Background

Motivated by the study of micro-vascular disease, the goal of our work is to develop continuum models of blood flow through the capillaries and link this to a discrete model of flow through branching arteriolar trees. Our focus is on the coronary microcirculation using data from imaging vessels in the muscular walls of the rat heart [1]. With the data, we explore the relationship between the underlying structure of capillary networks and the resulting averaged flow properties.

Techniques and Challenges

Our challenge is to determine an appropriate cut-off point from root-like branching arterioles to mesh-like capillaries based only on geometrical information. Considering the capillaries in isolation, we aim to capture the slowly-varying change in vessel orientation across the block using principal component analysis. Then we use homogenisation to derive effective fluid transport equations [2]. We develop an algorithm to automatically construct synthetic, periodic networks on which to numerically solve our capillary-scale equations sampled from distributions using geometric information from our data set.

Results

So far, we have developed an algorithm to distinguish arterioles from capillaries. Testing this on small mesentery networks, we found that the flow directions in the arterioles and venules were correct. Using our homogenisation methods, we have shown that the tissue-scale flow can be modelled using Darcy's Law and explicitly calculated using the permeability tensor in this expression by averaging the solution of the capillary-scale equations. By validating against an explicit solution, we have seen that our homogenisation method is able to efficiently capture the averaged flow properties.

The Future

Future work will involve improving our synthetic networks so that they capture the three-dimensional capillary network structure found in the data. To validate our model, we aim to extract sub-networks from the data that have minimal boundary conditions and estimate the explicit flow solution. Finally, our goal is to incorporate point sources representative of inflow from the arterioles, thus linking our homogenisation model to discrete flow in the arterioles.

References

[1] Lee, J.: Computational Modelling of Coronary Structure and Flow Mechanics, DPhil thesis, University of Oxford, 2009

[2] Shipley R., Chapman S.J.: Multiscale Modelling of Fluid and Drug Transport in Vascular Tumours, Bulletin of Mathematical Biology, 2010