Date of Award
Program or Major
Doctor of Philosophy
Russell T Carr
It has been found that spontaneous oscillations of nodal pressures, hematocrit, and blood velocity can occur in microvascular networks in the absence of biological control. In this paper, both analytical and numerical methods have been used to investigate the nonlinear dynamics of microvascular blood flow in simple networks. First, the steady state solutions for the system are found. Then the governing coupled PDE's are transformed into state dependent time delay differential equations, DDE's. The DDE's are then linearized about a steady state and normalized.
The characteristic equation for the network is found by assuming the linearized DDE's have a nontrivial exponential solution. The solutions of the characteristic equation are also called the eigenvalues for the network dynamics at that steady state. It is known that the steady state is unstable when the real part of the rightmost eigenvalue is positive. Thus, a theoretical prediction of the stability of the blood flow in the network can be based on the rightmost eigenvalues. The analysis has been performed on the networks with two node topology and with three node topology.
Due to the nonlinearity of the characteristic equation, solutions are found numerically using a software package called "DDE-BIFTOOL". After the eigenvalues are found, predictions of the stability of steady states are compared to direct numerical simulations for blood flow in the networks. Effects of physical parameters and inlet conditions on hemodynamics are investigated in the two node microvascular networks and the three node microvascular networks (2 inlets).
For the two node networks, the region of instability in parameter space is quite narrow. This means that experimental verification of spontaneous blood flow oscillations will be very difficult for the two node topology. The numerical results for the three node networks showed the three node system has instabilities over a much wider parameter ranges than the two node network. However, one of the most critical parameters, inlet hematocrit, is still quite high. This means such experiments are still very challenging.
Future work may involve continuing the search in wider parameter ranges and testing more complicated topologies to find realized conditions. Then in vitro experiments may be conducted to verify results of the linear stability analysis.
Wu, Fan, "Nonlinear dynamics of blood flow in simple microvascular networks" (2005). Doctoral Dissertations. 312.