Mathematical modeling of blood flow through a deformable thin-walled vessel:an analytical and quantitative investigation of the advection and diffusion contribution in the global and local equations of motion

In hemodynamics, the interaction of the blood flow through an arterial vessel and the geometrical and mechanical properties of the vessel is studied. Very often, numerical methods are used to determine important flow characteristics, such as the pressure distribution, in estimating the behavior of blood flow. This is a very useful method, often based on measurements of patient specific parameters, in predicting and preventing complications in cardiovascular surgery or in the application of cardiovascular equipment. Mathematical investigation of the order of magnitude of the flow characteristics can help to reduce the computational costs in the numerical analysis of the blood flow. In this project, the flow of blood through a thin-walled, linearly elastic deformable vessel is modeled. The assumption of linearly elastic behavior of the vessel wall gives rise to an affine constitutive relationship between the crosssectional surface of the vessel and the pressure distribution. The blood, as a fluid, is regarded as an incompressible Newtonian fluid. We distinguish between the global one-dimensional equations of motion and the local three-dimensional equations of motion, describing the relation between flow characteristics like the volumetric flow rate, the pressure distribution and the axial velocity. Of special interest is the order of relevance of the contributions of radial diffusion, i.e. the shear stress at the wall of the vessel, and nonlinear advection, which for the global equations of motion are assumed to be of the forms as proposed by Bessems in [1]. In contrast with most studies in this area, where periodic solutions are studied, we focus on the transient solutions by imposing a single-pulse inlet flow. Moreover, the solutions obtained from the global and local equations of motion are as far as possible analytical and they are compared which each other, based on plots of the solutions for a set of representative arteries, both on quantitative and qualitative features. A parameter of great relevance in this study is the dimensionless compliance (in this report denoted by δ), which equals the ratio of the characteristic speed of the blood flow and the elastic wave speed of the wall (for flexural waves). This parameter is always less than one, and often much smaller than one. For arteries having a small compliance, i.e. for a compliance much smaller than one, we observe that radial diffusion is of important influence. For arteries with larger values of the compliance, close to but still less than one, we still observe the effect of the radial diffusion term, however the flow is now dominated by the effects of the spatial and time derivatives of the volumetric flow rate and the pressure distribution. We expect that the influence of the nonlinear advection term is almost always small compared to the effect of all other terms in the equations of motion. However, we can not draw any conclusions about this statement based on the results in this project, due to the asymptotic assumptions we make in deriving the solutions for the nonlinear advection contribution, which lead to large variances in the qualitative and quantitative aspects of the solution. This holds specifically if the compliance is not too small (i.e. close to one). Technische Universiteit Eindhoven University of Technology 2 Mathematical modeling of blood flow through a deformable thin-walled vessel Technische Universiteit Eindhoven University of Technology “Just as the body cannot exist without blood, so the soul needs the matchless and pure strength of faith” Mahatma Gandhi, 1869-1948 Mathematical modeling of blood flow through a deformable thin-walled vessel 3 Technische Universiteit Eindhoven University of Technology 4 Mathematical modeling of blood flow through a deformable thin-walled vessel Technische Universiteit Eindhoven University of Technology