| The work presented in this thesis is meant as a contribution to the development of Brain Biomechanics—that is, to the development of a mathematical theory capable of explaining and predicting the response of the brain to certain forms of mechanical loading. Although such a theory is of interest in several different phenomena of clinical relevance, we focus here on a detailed application to hydrocephalus, and leave other applications to future work. Hydrocephalus is a clinical condition which occurs when normal cerebrospinal fluid (CSF) circulation is impeded within the cranial cavity. As a result, the brain ventricles enlarge and the brain tissue is compressed. Within limits, the dilatation of the ventricles can be reversed by shunting procedures, but, unfortunately, the rate of shunt failure is unacceptably high in most of the reported series. One of the most common factors causing shunt failure is misplacement, especially of the proximal catheters. Precise and strict indications for the CSF-shunt position must be based on a viable model around which to design experiments and interpret readings to promise the highest possible success rate. Using only the shape of the enlarged ventricles as it appears in a CT or MRI scan and Sethian's level set method, we show how to predict the form of the ventricles at later times after the shunt implantation for a given speed function. The theoretical prediction of the latter is a very challenging task and takes up a considerable portion of the thesis, through the development of a quasi-linear viscoelastic model of the brain tissue as well as the determination of the material parameters. This is particularly arduous in view of the fragility of the brain and the resulting difficulty of performing standard engineering tests on it. A second difficulty arises from the complex geometry of the system. In a horizontal scan of a hydrocephalic brain the interface between the CSF and the parenchyma is a non-convex curve in the plane. Under the action of the pressure gradient set up by the shunt, each point on this curve moves inwardly at its own speed, and one would expect such a speed to depend on the local curvature. Taking this into account, however, would introduce many model parameters, while the only two sets of data available in the literature would make it impossible to determine them. Therefore, in order to make progress, and only for the purpose of calculating the speed function, we approximate a fully-developed hydrocephalic brain by a circular cylinder made of an incompressible, isotropic, homogeneous quasi-linear viscoelastic material with a closing inner surface and an outer surface reinforced by an elastic casing, and calculate the speed function of the healing inner surface. In the end we compare the theoretically obtained velocity with the velocities at which the ventricular walls shrink in reality after the insertion of different types of shunts. Taking into account the agreement noticed among these velocities we conclude that the velocity at which the ventricular wall shrinks after the shunt implantation does not appear to be very dependent on the original shape of the ventricular wall. These results are finally used to develop a software package of interest to neurosurgeons in their quest for the optimal ventricular catheter location. |
