Nonlinear dynamic behavior and driven chaos from in vitro and computer models of a collapsible vascular segment

Abstract

Previous studies of vascular segments treated the vascular wall as a rigid or deformable structure with linear elasticity. The present study relieves this assumption, with the goal of modeling a vascular segment more completely. Because the vascular segment is deformable the three dynamic elements are lumen area dependent, i.e. the fluid resistance, inertance, and compliance are all functions of the lumen area and are thus also nonlinear. Due to the nonlinearities inherent in the vascular segment complex dynamic phenomenon, such as chaos, were investigated. Two lumped parameter three element nonlinear models were developed. A mathematical model was formulated from a system of nonlinear ordinary differential equations. These equations were numerically solved on a computer through a two element fourth order Runge-Kutta algorithm. An in vitro model was also employed, using a segment of latex tubing as a model for the collapsible vascular segment. Both models were investigated by applying a sinusoidal input pressure with a variable amplitude, offset, and frequency. Complex dynamic behavior was observed in both models for several parameter sets. In addition, the frequency response for the system was found to be dependent on transmural pressure and often demonstrated the presence of multiple resonances unlike a linear vessel model. Finally, these models can be applied to modeling flexible dynamic stenosis.