Mathematical modelling of the arterial cellular communication.
Degree GrantorUniversity of Canterbury
Degree NameDoctor of Philosophy
A study of the arterial cellular communication may be regarded as of greatest importance for understanding cellular physiology and its consequences leading to pathological states of the human arterial diseases. However, it is complex and not an easy task, as biological structures are heterogeneous on many scales such as micro, macro, etc. In particular, a small arterial segment consists of billions of cells among which not only a certain types of cells are connected through cellular gap junctions, but also thousands of nonlinear ion pathways within each cell, which can be regarded to be in nano-scale structures. Mathematical models of these communities of cells typically involve thousands of billions of ordinary differential equations (ODEs), which require simulation on distributed high performance computer. As a result, an attempt to incorporate all of these possible structures leads the mathematical modeler to encounter difficulties due to the computational size of the models. However, multiscale analysis and the theory of mathematical homogenisation provide the mathematical modeler with a way of avoiding vast-scale calculations and are useful in obtaining a small number of partial differential equations for the study of such complex system. One of the main goals of this project is to further develop the homogenised partial differential equations (HPDEs) derived so far into a more comprehensive and cohesive model, so as to predict communication protocols between cells in the macroscopic arterial domain. In this interest we developed one- and two-dimensional HPDEs for the homocellular (same type of cells) membrane potential of a population of discrete and continuous cells around an arterial ring and along the arterial axis, respectively. For this, each homogenic cell is treated to be an identical and electrical equivalents of a single FitzHuge-Nagumo (FHN) analogue. Moreover, cells are assumed to as human arterial smooth muscle cells (SMCs). An argument of using FHN equations in the modelling of SMCs is given in relation to the local topological (dynamical) similarities between our FHN equations and the SMC models of koenigsberger et al.. Furthermore, as an application of the homogenised models the effect of macroscopic potential variation coefficient from the microscopic variation was illustrated by constructing numerical examples. As the next main goal of this project, we carried out a numerical dynamical study for a ring of three discrete cells. As a result we showed the dynamics of the cellular membrane potential of SMCs as oscillations when the system undergoes Hopf bifurcation. In the subsequent numerical simulations a spatial patterning of the membrane potential in one-dimension revealed a propagation of forward and backward plane waves under the application of a stimulus in the form of a second-order spatiallyvarying spline with respect to the Hopf points identified. Numerical simulations of the HPDEs in two-dimension discovered the membrane potential to be a propagation of curved wave-fronts and -backs, circular wave-fronts (moving in the direction of the outward normal) and -backs (moving opposite to the outward normal) and finally, rigidly rotating spiral waves under a physiological symmetry breaking condition (zero-state depolarization). Dynamics of these membrane potential waves in one and two dimensions were explained in terms of multiple steady states based on new notions, ‘moving point of initiation of waves’ and interference of waves, and drifts, meanders and wave-breaks, respectively. Finally, mathematical models for the membrane potential heterocellular communication between SMCs and endothelial cells were derived in one and two dimensions. However, the modelling of HPDEs of these models and their related dynamics have been left for future work.