Hemodynamics in the microcirculation
Tet Chuan Lee, David Long, Richard Clarke, Department of Engineering Science
The microcirculation is the network of arterioles, capillaries and venules that deliver blood to the body’s tissues.
Lining the walls of these microvessels is a layer known as the Endothelial Glycocalyx Layer (EGL). The diameter of the vessels we consider (e.g. post-capillary venules) typically lie in the range of about ten microns and the EGL can extend half a micron or more into the blood flow. It plays an important role in the microcirculation as it modifies the velocity profile of blood flow and is believed to alleviate the fluid shear stresses exerted upon the vessel wall. It is also theorised to play a role in regulating the permeability of the vessel as well as being involved in mechanical signalling and inflammatory cell trafficking. Moreover, it is also believed that it plays an important function in disease states such as ischemia-reperfusion injury, diabetes and atherosclerosis.
However, the EGL does not behave the same in the laboratory as in the body. This issue is compounded by the fact that in-vivo measurements of the EGL are extremely challenging. For these reasons, it is hoped that models can help us to better understand the EGL and confirm some of the above-hypothesised behaviours.
Current models of the EGL, however, typically assume an idealised vessel geometry which may not necessarily represent biological reality. In order to model a more realistic microvessel, a Boundary Element computational model was developed that can simulate blood flow through a physiologically realistic microvessel based upon real biological data obtained using confocal microscopy.
Figure 1 shows a computational mesh for a post-capillary venule obtained from the Centre for Microvascular Research at Queen Mary University of London. Triangular elements were used to approximate the geometry with a barycentric coordinate system used on each element. Examples of the fluid shear stresses predicted by simulations for a particular choice of EGL and vessel geometry are shown in Figure 2.
Figure 1: Computational mesh of a post-capillary venule based on real biological data. The endothelial cells (exaggerated here) can be seen to protrude into the vessel.
Figure 2: Examples of Fluid Shear Stresses exerted upon the vessel wall (taken from Lee, T-C. Masters Thesis 2013, University of Auckland)
Constructing matrices
The Boundary Element Method, which was written in the C programming language, readily lends itself to parallelisation. Construction of the full matrices required by the Boundary Element Method was multithreaded using OpenMP, which enabled the multi-core environment on the University of Auckland’s high performance computing to be exploited. For the calculations shown in Figure 2, the simulations were run on a single node across sixteen cores, although larger computations (i.e. relating to a greater number of partitions, necessary for finer meshes) could easily be run across multiple nodes using MPI. Once built, the matrices were assembled into a linear system which was then solved iteratively using Generalised Minimal Residual (GMRES) again multithreaded across multiple cores using OpenMP. Figure 3 provides details of computational time as a function of the number of cores and memory usage as a function of mesh size.
There are numerous physiological effects which are still to be incorporated, including the elastic behaviour of the EGL, as well as possible charge and osmotic effects, all of which will increase the size of the numerical scheme and the computational burden.
