A Hybrid Asymptotic-Numerical Study of a Model for Intracranial Pressure Dynamics

Abstract

Lumped parameter, compartmental models of the human intracranial system are studied through development of a hybrid asymptotic-numerical technique. Dimensionless variables are introduced so that disparate time scales can be identified, and analysis shows that the system of model equations varies over both a fast and a slow time scale. On the fast time scale, the 5 × 5 system of equations may be decoupled to give a reduced 3 × 3 system combined with two conservation laws for the cerebrospinal fluid and brain compartmental volumes, respectively. The stiffness condition of the reduced system is shown to be considerably improved over that of the original system. For the general nonlinear problem, a uniformly valid asymptotic approximation for large time is derived by a hybrid asymptotic-numerical technique. In the special case of the linear problem, where compliances and resistances are assumed to be constants, the uniform approximation for large time is obtained analytically. To verify accuracy, both asymptotic and hybrid asymptotic-numerical results are compared with direct numerical integration of the full system. Physiological interpretations of the results are also given.