On well-posedness of a mathematical model for cerebrospinal fluid in the optic nerve sheath and the spinal subarachnoid space

Abstract

More often than not it is assumed a-priori that first-order, time-dependent systems of non-linear partial differential equations will be hyperbolic; such assumption may be incorrect and could produce misleading results, or no results at all, due to instabilities of the system itself. The present paper is concerned with the detailed theoretical and numerical study of a 4×4 non-linear, time-dependent first-order system that has been proposed for modelling the dynamics of cerebrospinal fluid in the spinal subarachnoid space [14,15,53]. The system has also been proposed as a model for the dynamics of the cerebrospinal fluid surrounding the optic nerve [45]; see also [39]; in this case however, there are physiological aspects that require clarification, before a realistic application of the present model to optic nerve physiology is considered. Here we address mathematical aspects and prove a well-posedness theorem that identifies necessary and sufficient conditions for the system to be strictly hyperbolic and conditions under which the system is of mixed elliptic-hyperbolic type. This result was anticipated in [53] and full details are given in [45]. We then address more generally the numerical implications for such type of systems. Solving numerically systems that contain a mixed elliptic-hyperbolic region is challenging. First, the use of upwind methods, so popular for hyperbolic first-order systems, is ruled out automatically here. One is then left with having to choose centred-type methods with great care. Moreover, the non-conservative character of the present system adds another challenge. Results for various representative examples are shown and analysed and the potential of the proposed model for simulating CSF dynamics in physiological and pathophysiological settings is discussed.