Piecewise-Linear Mechano-Hydraulic Models of Intracranial Pressure Dynamics
Open Access
- Author:
- Evans, Davis James
- Graduate Program:
- Engineering Science and Mechanics
- Degree:
- Master of Science
- Document Type:
- Master Thesis
- Date of Defense:
- October 31, 2016
- Committee Members:
- Joseph Paul Cusumano, Thesis Advisor/Co-Advisor
- Keywords:
- intracranial pressure
dynamics
dynamical systems
brain
mathematical model
icp
piecewise
limit cycle
bifurcation - Abstract:
- Elevated intracranial pressure (ICP) is an extremely dangerous condition for patients who are suffering from traumatic brain injury, hydrocephalus, or related neurological disorders. To make informed decisions when treating patients, clinicians must understand how the body regulates ICP. Mathematical models can aid in this task. In addition to making quantitative predictions, accurate mathematical models can also describe the transition between different qualitative behaviors of ICP. This is crucial, because, for instance, in a pathological case known as Lundberg A-Waves, ICP oscillates between high and low pressures over a long period. This is in contrast to a healthy state, where ICP remains mostly constant with respect to time. In this thesis, we develop mathematical models of ICP dynamics with the goal of explaining how a healthy state of constant ICP can transition to the oscillatory case of Lundberg A-Waves. These low-dimensional models are built by coupling a lumped mechanical system with a hydraulic system, and applying the first-principles of the balance of mass and the balance of linear momentum to obtain the governing equations. The first of these models is a single-compartment model of the coupled mechanics and hydraulics of the ventricular cerebrospinal fluid (CSF) and the surrounding brain tissue. The governing equations are two coupled linear ordinary differential equations. Once we develop a nondimensional form of the governing equations, we postulate several different piecewise laws for the CSF formation rate that depend on either pressure or volume. Then, we examine the resulting piecewise-linear dynamical systems for the existence of a limit cycle. To do so, we look for fixed points in an analytic expression of the Poincar\'{e} map, which we construct as a composition of other maps represented as parametric curves. Once we determine which of these postulated CSF formation laws result in model equations that can exhibit limit cycles, we draw possible clinical conclusions based on this information. The second model is a dual-compartment model that couples the mechanics and hydraulics of the ventricular CSF, arterial blood, and brain tissue. The governing equations are four coupled linear ordinary differential equations. We numerically examine the stability characteristics of this linear system, once we specify values for the model parameters. In addition, we show that the steady-state ICP that the model predicts for these values agrees well with what is observed clinically.