CANADIAN APPLIED MATHEMATICS QUARTERLY Volume 7, Number 1, Spring 1999

BRAIN BIOMECHANICS: CONSOLIDATION THEORY

OF HYDROCEPHALUS. VARIABLE PERMEABILITY

AND TRANSIENT EFFECTS

M. STASTNA, G. TENTI, S. SIVALOGANATHAN AND J.M. DRAKE

ABSTRACT. The quasi-steady linear consolidation theory is applied to a model brain under hydrodynamic loading, due to a step increase of intracranial pressure, in order to gain insights into the onset and evolution of hydrocephalus and brain edema in children. Simple asymptotic formulas are derived that shed light on the transient behavior of the dynamics of filtration and of the deformation of the solid matrix, and it is shown that the results hold over time intervals much shorter than the consolidation time which is estimated to be of the order of several hours. Finally, the role of a variable permeability and of the complications introduced by it in the analysis of the transient response are discussed.

1. Introduction. The theory of consolidation, originally developed in soil mechanics [3],is one of the better established theories of flow through porous media [2]. Over the last several decades, the theory has found meaningful applications in the new disciple of biomechanics. Good examples of this are the works of Mow and collaborators, see, e.g., [15],in flow through cartilage, and of Kenyon [lo]in flow through arterial walls. More recently, consolidation theory has been applied to brain biomechanics with the specific objective of understanding the mechanisms of the onset and evolution of such disease states as hydrocephalus and brain edema 116, 19, 81.

Like most biological tissues, the brain consists of soft hydrated material. Under load, such as an increase of the cerebrospinal fluid (CSF) intraventricular pressure, it responds by deformation of the solid constituents and filtration of the CSF through the interstitial spaces of the parenchyma, in a manner analogous to the behavior of a sponge saturated with water [7].In such conditions, the dynamical behavior of filtration depends on the rate at which pore volume can change. This, in turn, depends on the brain's mechanical parameters, and in particular on its Poisson ration v and the shear modulus p,

Received by the editors in final form on April 8, 1998 Copyright @I999 Rocky Mountain Mathematics Consortium

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or G, as is more common in the engineering literature. Within the framework of the classical theory of consolidation, these important parameters must be determined experimentally. For some biological tissues this is possible by means of standard laboratory tests developed for engineering materials. Unfortunately, for brain tissue, because of its extreme fragility, alternative methods are required.

We reported progress in this direction in a recent paper [20], hence- forth referred to as TSD. Using a simplified geometry for the actual hydrocephalic brain, we were able to derive explicit formulas for all the relevant fields. It became possible to understand the critical role played by the Poisson ratio and to extract from previous data [14, 171 an ap- proximate estimate of its true value, which was considerably less than the theoretical upper limit, u = 112, used in some recent numerical simulations of hydrocephalus [16] and brain edema [19].

The analysis carried out in TSD was restricted to the steady-state limit of linear consolidation theory for a thick-walled, tethered fluid- filled porous tube, so that plane strain conditions would prevail. One may question the appropriateness of a linear theory to a phenomenon involving very large deformations, as in hydrocephalus. It may be argued, however, that, since the large expansion of the ventricular cavities occurs very slowly (on a time scale of the order of weeks or months), the deformation is relatively small on a much shorter time scale, with the mechanical properties remaining essentially constant. This point of view was assumed in TSD, but clearly required further analysis of two major points. First there was a need for a clarification of the role of the transient response to hydrodynamic loading of the brain parenchyma and of a proper identification of the time scale over which the system can be considered to be in steady state. And, second, there was a need to assess the validity of the assumption of constant permeability. the latter is a measure of the facility with which the fluid flows through the pores. Since pore deformation occurs during the consolidation process, constant permeability may not be a good assumption even for small deformation and strain.

The purpose of this paper is to address these issues in some detail. For completeness and to introduce the notation, we briefly review Biot's consolidation theory in Section 2, along with the model geometry and the boundary conditions appropriate to our case. In Section 3 we study the transient response of the model brain to a step increase