Developing Constitutive Laws for the Endothelial Glycocalyx Layer
In this work, we aim to develop new mathematical models for the Endothelial Glycocalyx Layer (EGL) in order to improve our understanding of the layer and its functions, particularly in the microcirculation. The EGL is a porous macromolecular layer that lines the insides of blood vessels. It is located on the important interface between the endothelium and flowing blood and as such is believed to play a number of important roles including transducing mechanical signals from flowing blood (mechanotransduction) and regulating vessel permeability. The use of mathematical modelling is motivated by the fact that the EGL is a delicate layer that is sensitive to environmental conditions, making it extremely challenging to study experimentally. In the first part of this work, we derive a low-permeability asymptotic model for the EGL based on biphasic mixture theory which has been used to model the EGL in the literature previously. In the physiological regime, the EGL has an extremely low permeability, making solution of the biphasic mixture theory equations computationally challenging. Using the low-permeability model, we create a boundary element scheme that is able to simulate a more physiologically realistic microvessel than has been done previously in the literature. We use this scheme to study the implications of a hypothesis from the literature that the EGL redistributes so that it is thickest at the cell-cell junctions. Next, in the second part of this work, we use homogenisation theory to derive a more sophisticated model for the EGL. Using homogenisation, we are able to obtain bulk EGL properties such as the permeability from the underlying microstructure of the EGL. In addition, we are also able to evaluate the torque experienced by the endothelium, which cannot be obtained directly when modelling the problem using biphasic mixture theory. This torque is important for mechanotransduction, as the majority of mechanical stress that is experienced by endothelial cells is believed to be of this form.
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