Mathematical Modelling of The Liver Microcirculation

The models of the microcirculation of blood and interstitial fluid in the human liver lobule are developed based on the classical hexagon model of Kiernan. Both blood and interstitial flows in the lobule are treated as flows in porous medium connected via the fenestrated membrane of sinusoids. Several important physiological components are developed and included in the models. The lobule with tissue elasticity shows that the pressure-flux relationship is non-linear and the poroelastic model has more compliance than the solid elastic model. Models of the interstitial flow in both a single lobule and the whole liver are also developed. The results show that our models can predict the amount of interstitial fluid drainage including the ascites. From the parameter studies, we find that the permeabilities of the sinusoids and the interstitial space, and the portal pressure are the most important factors on ascites production. We further investigate the oxygen transportation and uptake by liver cells using the advection–diffusion equations and Michaelis-Menten kinetics. The studies show that the main mechanism of oxygen transportation within the sinusoids is advection; however, the transportations within the interstitial space and across the fenestrated endothelial cells are mainly from diffusion process. The effect of the arrangement of the vessels and the geometry of the lobule on blood perfusion and oxygen distribution is also studied. The results show that the classical hexagonal lobule with the vascular septa provides the optimal perfusion compared to other geometries of the lobule. In summary, this thesis contributes to the development of mathematical models of several important features in the liver microcirculation such as the tissue elasticity, the interstitial flow, the oxygen distribution, and the arrangement of the vessels in the lobule