Étude numérique des origines hémodynamiques des oscillations dans des réseaux de capillaires
While simulating blood flow in a microvascular network (in the absence of biological control), it is possible to observe the presence of oscillations in certain parameters such as blood flow, nodal pressure and hematocrit (red blood cell concentration in blood). This behaviour seems consistent with certain in vivo experiments. Despite this agreement, one has to wonder if the fluctuations observed in simulations are physical in nature, numerical or an artefact of unrealistic models since there are always differences between modelling and in vivo experiments. To settle this question satisfactorily, we will study and analyze blood flow and the nature of the fluctuations in different microvascular networks using a convective model and a well-mixed model to depict the governing equations for conservation of red blood cell mass. These models take into account two important rheological effects : the Fåhraeus-Lindqvist effect describing the apparent viscosity of blood flow in a vessel and the plasma skimming effect which describes the separation of red blood cells at diverging nodes. To describe the latter effect, we will implement two plasma skimming models (Pries et al. and Fenton et al.). In this thesis, we will present a description of the physiological problem (blood rheology). We will introduce the mathematical models used (well-mixed and convective) as well as the plasma skimming models (Pries et al. and Fenton et al.) accompanied by a detailed analysis of the numerical methods implemented. For the well-mixed model, we use the traditional explicit Euler method as well as a new implicit scheme that allows us to solve the problem in an efficient manner. This is done using a Newton-Krylov method with a preconditioned conjugate gradient and GMRES method for the inner iterations as well as a quasi- Newton method (Broyden’s method). The implicit method will include the vi backward Euler and trapezoidal methods. For the convective model, the explicit method of Kiani et al. will be implemented as well as a new numerical implicit approach. The stability of these numerical schemes will be explored. Using three different topologies, we will compare the results of the two mathematical models as well as the two plasma skimming models and the various numerical methods in order to ascertain to what extent the oscillations that have been observed using the traditional schemes may be attributable to the choice of the mathematical models or the choice of the numerical methods.
oscillations Microvascular network oscillations blood flow hematocrit