Physical principles of venous hemodynamics and its mathematical modeling

Abstract

In physiological conditions, the cardiovascular system (CVS) is a closed circulatory system comprising a pump (the heart), a conduit system (vasculature), and a continuum media (blood) moving through the system. The heart is a major source of energy in this system. It pumps the blood through the two connected loops. From the mechanical point of view, CVS can be represented as a network of flexible tubes filled with a viscous incompressible fluid driven by a periodic energy source. The fluid dynamics are described by Navier-Stokes's equations, representing the fundamental physical principles of mass and momentum conservation. These equations allow computation of the blood velocity field and pressure depending on the forces exerted to the fluid's surface (surface forces) and to a unit portion of the fluid (mass forces). Equations of structural dynamics describe the motion of the vascular wall. The state-of-the-art models incorporate fluid and structure interaction (FSI). The blood flow in various parts of CVS has different features that must be considered during computational simulations. Elastic properties of the veins and arteries are different. The structural features of veins (valves) limit the backward flow. The geometry of venous cross-sections may be circular, elliptic, and dumbbell-shaped. It changes the flow characteristics. Blood rheology plays a significant role in venous flows. According to the mass conservation law, the work of the heart pump provides energy for the arterial flow and determines venous return to the heart atria. Venous hemodynamics comprises a lot of various processes with different physical and biological origins. Complex analysis of a patient requires computational simulations, which provide medical experts with a basis for prognosis and optimal surgical treatment. In this work, we review basic physical principles and modern mathematical models of venous hemodynamics. In conclusion, the blood flow in veins can be considered as a mechanical process. It obeys the fundamental physical principles and can be described by the well-known mathematical models of continuum mechanics. Thus, the flow characteristics can be simulated and predicted in various healthy and pathological conditions basing on the boundary conditions and material properties of the blood and veins.