Dynamics and Bifurcations in Low-Dimensional Models of Intracranial Pressure

Abstract

Intracranial Pressure (ICP) is a physiological parameter of the brain which plays an important role in the diagnosis and treatment of pathologies such as hydrocephalus and traumatic brain injury. Currently, all reliable methods for ICP monitoring involve drilling through the skull to place a pressure probe inside the brain. As a result, ICP is only measured in the most critical cases which require neurosurgical intervention. Mathematical models that relate ICP to physiological parameters whose measurements are minimally invasive could contribute to better diagnostic and treatment protocols for brain disorders. Ideally, such mathematical models should have the capability to predict ICP in real time from non-invasive measurements of other clinically relevant parameters without the need for high-risk procedures. In this paper, we examine in detail the dynamics and stability of a mathematical model proposed by Ursino and Lodi in (J Appl Physiol 82(4):1256–1269, 1997) which predicts ICP from measurements of arterial blood pressure. We study how the equilibria vary with the model parameters and, aided by numerical simulations, we obtain bifurcation diagrams for the system of non-linear ordinary differential equations. Expanding upon the work of Ursino and Lodi, we show that the model exhibits not only one Hopf bifurcation but also a reverse Hopf bifurcation in certain parameter regimes. In addition, we present global phase portraits of the system in interesting parameter configurations.