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past events |
Cambridge University, England
delivered a set of lectures
Two major research themes have dominated the fluid dynamical study of blood flow in arteries: (a) propagation of the pressure pulse and (b) flow patterns and wall shear stress (WSS) distribution in complex arterial geometries. The former led to physiological understanding and permitted the interpretation of diagnostic measurements of the wave-forms of blood pressure and flow-rate, for example. The latter was driven by the need to understand the link between wall shear stress and the development of arterial disease, and the understanding gained is also used in the design of surgical interventions such as bypass grafts.
Pulse wave modelling has always been essentially mathematical, using one-dimensional linear or weakly nonlinear theory, and can therefore given significant understanding very simply, as will be demonstrated. The relatively new wave-intensity analysis of the pulse wave shows that the subject is still capable of giving new insight.
The study of time-dependent flow in complex three-dimensional geometry, even when the tubes are taken to be rigid and the fluid Newtonian, is much more difficult. Realistic simulation requires the computational solution of the full Navier-Stokes equations, in a geometry obtained from a particular subject by means of magnetic resonance imaging (say), using input flow or pressure data that are also obtained by non-invasive imaging. The combined computational procedure has not yet been developed to the point at which one can have confidence in its accuracy, but it soon will be. However, this is not mathematical modelling and does not clearly lead to new fluid dynamical understanding. For that one must go to idealised models such as uniform curved or helical tubes, or non-uniform two-dimensional channels, or small isolated three-dimensional protuberances on a smooth wall. Such problems lead to interesting fluid dynamics, but it is not clear how relevant they are to biomedical practice.
To show that mathematical modelling is not dead, the talk will conclude with a brief description of a recent model by S L Waters of the new process of transmyocardial laser revascularisation, developed to restore oxygen supply to heart muscle cut off by an infarct, for example.
Laboratory experiments designed to shed light on fluid flow through collapsible tubes, a problem with several physiological applications, invariably give rise to a wide variety of self-excited oscillations. The object of modelling is to provide scientific understanding of the complex dynamical system in question. This lecture outlines some of the models that have been developed to describe the standard experiment, of flow along a finite length of elastic tube mounted at its ends on rigid tubes and contained in a chamber whose pressure can be independently varied. Lumped and one-dimensional models have been developed for the study of steady flow and its instability, and a variety of oscillation types are indeed predicted. However, such models cannot be rationally derived from the full governing equations, relying as they do on several crude, ad hoc assumptions such as that concerning the energy loss associated with flow separation at the time-dependent constriction during large-amplitude oscillations. A complete scientific description can be given, however, for a related two-dimensional configuration, of flow in a parallel-sided channel with a segment of one wall replaced by a membrane under longitudinal tension T. The flow and membrane displacement have been calculated successively by lubrication theory, Stokes-flow computation, steady Navier-Stokes computation and unsteady Navier-Stokes computation. For a given Reynolds number, Re, steady flow becomes unstable when T falls below a critical value (equivalently, when Re exceeds a critical value for fixed T), and the consequent oscillations reveal at least one period-doubling bifurcation as T is further reduced. The effect of wall inertia has also been investigated: it is negligible if the flowing fluid is water, but leads to an independent, high frequency flutter when it is air. Most recently, rational asymptotic models have been developed for the 2-D problem, and considerable progress has been made towards full 3-D computations.