Math 2000

Biofluid Dynamics & Medical Science / Dynamique des biofluides et sciences médicales
(Siv Sivaloganathan, Organizer)

JAMES DRAKE, University of Toronto, Hospital for Sick Children, 555 University Avenue, Toronto, Ontario  M5G 1X8
Requirement for new mathematical models of the brain for hydrocephalus treatment

Hydrocephalus is a common condition in which spinal fluid (essentially water) builds up inside the brain, compressing the brain tissue and producing neurological dysfunction. Hydrocephalus is usually treated with an indwelling valved tube called a shunt. Most shunts over drain the brain, leading to collapse of the fluid cavities. The average patient develops a shunt complication within 2 years, often related to collapse of the fluid cavities (brain ventricles) around the shunt drainage holes. Patients often endure many repeat surgeries for complications.
A number of mathematical models of the brain and shunt systems have been developed to reduce shunt complication rates. Shunt valves with more sophisticated fluid flow characteristics, based on simplified pressure-volume models of the brain, have been developed to reduce this over draining tendency. Unfortunately they have had no effect on the size of the ventricles, or the complication rate. Neurosurgeons would prefer to place the shunt in what will ultimately be a slightly dilated portion of the ventricle but currently have no strategy for doing so.
More sophisticated models of the brain, which span a longer time frame, and address the shape changes in the hydrocephalic brain, are needed to develop improved shunt designs, and placement strategies.

CORINA DRAPACA, Waterloo
Transient ventricular wall dynamics in shunted hydrocephalus

Hydrocephalus is still an endemic condition in the pediatric population with a prevalence of 1-1.5%. Within limits, the dilatation of the ventricles can be reversed by shunting procedures. Unfortunately, the rate of shunt failure is unacceptably high and an important risk factor in this regard is the positioning of the catheter's tip. Finding an optimal solution to this problem requires the creation of a good mathematical model capable of predicting the evolution of the ventricular walls configuration in shunted hydrocephalus. In this paper we report on some progress recently made in the mathematical description of the short-time ventricular wall dynamics in a simplified (cylindrical) geometry. In particular, we shall focus on the free boundary value problem resulting from an application of the theory of consolidation in porous media.

TONY HEENAN, Department of Mechanical Engineering, Queen's University, Kingston, Ontario  K7L 3N6
Visualization of the flow through the human oro-pharynx

The overall structure and dynamics of breathing flow through the human oro-pharynx (OP) is poorly understood. This is due to a lack of good quality data on the flow regimes. The complex and variable geometry of the OP makes traditional point-based measurements impractical, while the combination of complex geometry and sufficiently high Reynolds number makes accurate numerical simulation expensive.
This paper presents preliminary, qualitative results from an experimental, field-measurement (PIV) based study of OP fluid dynamics. A series of high-speed digital video recordings were made of the flow in an idealized model of the OP, and these are presented. The flow was marked using the smoke-wire technique, and illuminated with an Argon Ion laser. Visualizations for constant flow rates of 30L/min and 90L/min are presented. The features and behaviour revealed in the recordings are discussed.
(joint work with A. Pollard and W. Finlay)

E. KRAUSE, Aerodynamisches Institut, Wuellnerstr., 52062  Aachen, Germany
Flow-modeling for the human circulatory system

The blood flow in the human circulatory system is modeled with simplified fluid mechanical relations. The system is decomposed into four active elements, which are used to simulate the filling and the emptying of the left and right atrium and ventricle. The contraction and relaxation of the heart muscle is described with Hill's model extended to the right ventricle. The opening and closing of the four heart valves are simulated with time-dependent resistance coefficients, approaching infinity when the valves close. Each of the other eleven passive elements is assumed to consist out of a reservoir with elastic walls and a rigid tube with variable cross-section, connecting the element to its neighbor. The simplified momentum and continuity equation are used to describe the time-dependent one-dimensional flow in the elements, with the viscous forces represented by resistance coefficients. The model results in 32 nonlinear ordinary differential equations of first order, which are solved with the Runge-Kutta method. Blood pressures and volumes in the atria, the ventricles and in the aorta, simulated so far, agree well with measured physiological data. Long-time simulations were carried out, also for physiologically high and varying heart rates, as well as for pathological changes, as for example valvular abnormalities.

KATRIN ROHLF, Department of Applied Mathematics, University of Waterloo, Waterloo Ontario  N2L 3G1
Stochastic theory of blood flow in small vessels

It is well-known that blood is a suspension of cells in plasma. Since the red blood cells (RBCs for short) are the largest of these, and by far the most in number, they are believed to be the main contributors to the many peculiarities of flowing blood.
One of the key attributes of RBCs is that in the absence of shear, they begin to stick together-a process called aggregation-and form chain-like aggregates called rouleaux. In fact, if the shear rates are not too large, aggregation still occurs even in flowing blood. Thus, if one is interested in a detailed mathematical description of blood flow in small vessels (e.g. arterioles and large capillaries where the shear forces are known to be small), one cannot ignore the effects of the aggregates, which in general, surface through non-Newtonian properties of the fluid.
In this talk, a stochastic description of an aggregation process will be presented. Multi-particle collisions will be taken into account, as well as the formation and break-up of the aggregates, which occur if shear forces are taken into account. The results will be discussed with specific application to RBC aggregation and their resulting connection to the macroscopic non-Newtonian flow properties of blood.

MAREK STASTNA, University of Waterloo, Waterloo, Ontario  N2L 3G1
Some Problems in the application of consolidation theory to hydrocephalus

This talk will outline several problems in the application of consolidation theory to the clinical condition of hydrocephalus. One set of problems relates to the fact that the equations of consolidation theory have several odd properties from a computational standpoint. Another relates to the phenomenological nature of the applications of consolidation theory in brain biomechanics (as well as other fields in which non-intrusive experimental data is difficult to obtain). This second set of problems relates to issues such as space dependent material parameters, deformation dependent permeability, and free boundaries. The talk will contain a mixture of analytic and numerical work. All material will be presented at an elementary level.

JOHN STOCKIE, Simon Fraser University, Burnaby, British Columbia  V5A 1S6
Simulating flexible fibres immersed in fluid

The motion of flexible, elastic fibres immersed in a surrounding fluid arises in many applications, including polymer composite materials, swimming microorganisms, and wood pulp processing. We develop a model of a single fibre using the ``immersed boundary'' framework, which has proven effective in modeling elastic boundaries such as the heart wall. The primary advantage in using the immersed boundary method is its ability capture the complex, hydrodynamic interaction between fibre and fluid that is typically neglected in other models of fibre motion. We perform a series of 2D and 3D numerical simulations of a single fibre subjected to planar shear, and draw comparisons with experimental observations. Our results reproduce the observed orbital motions of actual fibres, and exhibit behaviour not captured by other models.