Theoretical studies on blood flow in small vessels

Detailed knowledge of the dynamics of blood flow has been recognized as very important in a number of applications of clinical interest. For instance, variations in the values of some of the blood's transport coefficients or mechanical parameters—such as its yield stress—have been directly connected to disease. The potential to use such knowledge for diagnostic purposes has been greatly limited for a number of reasons, the principal of which is intimately tied to the serious mathematical difficulties encountered in the attempt to develop a comprehensive theory of blood flow.; When a continuum mechanics description can be justified, most of the work done in this area involves ad hoc assumptions for the constitutive equation. Despite this lack of sophistication, even this type of engineering analysis leaves room for improvement, for the nonlinearities often prevent determination of explicit solutions. In Chapter 4, we discuss a new interpretation of the Womersley parameter, which has quite frequently been used as a perturbation parameter. With the mathematical problems set aside, the success of such approaches—as also the success for any clinical application—primarily rests on the availability of measured values for the transport and mechanical parameters. For the reasons explained in Chapters 2 to 4, the difficulties in measurement make more fundamental theoretical approaches highly desirable.; There appears to be agreement in the literature that the non-Newtonian properties of blood are due to the ability of red blood cells—its main constituents—to aggregate in clusters (often called rouleaux) at low shear rates. This interaction, as well as the complicated behaviour of individual red cells in general, has led to the belief that most of the flow anomalies associated with blood, and its non-Newtonian behaviour are reducible to the dynamics of these cells. Due to their extremely large number in blood, it seems clear that a fruitful approach should be based on some of the classical methods developed over the last century in non-equilibrium statistical mechanics.; The second part of this Thesis contains precisely such an approach. Since blood may be considered as a colloidal suspension, we start by reviewing in Chapter 5 the classical Smoluchowski approach to colloidal aggregation. The inability of this description to account for the inhomogeneities and flow conditions of our system, necessitates generalization. We propose in Chapter 7, a number of stochastic descriptions in terms of probability densities that depend on the phase-space coordinates of the particles, in much the same spirit as the kinetic theory of molecular gases at the level of the Wang Chang-Uhlenbeck equation. This means that we assume the stochastic process to be a Markov process, in order to facilitate the mathematical derivation of a generalized Master Equation; that is, an evolution equation for the conditional probability density in which the transition rates for aggregation and breakup can be calculated, in principle, from a study of the collisional events. In particular, we present a new approach in Chapter 8 from which we derive the Smoluchowski equation as a check of its promise.; The work however, is far from complete, and in Chapter 9 we elaborate on the challenging work that remains to be done, and explain why it is extremely worthwhile not only for blood flow, but for the theory of colloidal suspensions in general.