On nonlinear waves of the blood flow through arteries

Abstract

We discuss propagation of traveling waves in a blood filled elastic artery with an axially symmetric dilatation (an idealized aneurysm). The processes in the injured artery are modelled by equations for the motion of the wall of the artery and by equation for the motion of the fluid (the blood). For the case when long-wave approximation holds the model equations are reduced to a version of the perturbed Korteweg-deVries equation. Exact travelling-wave solutions of this equation are obtained by the modified method of simplest equation where the differential equation of Abel is used as a simplest equation. A particular case of the obtained exact solution is discussed from the point of view of arterial mechanics.

Simulation studies on hemodynamic models for blood flow

“…The characteristics of two structures are important here; blood and blood vessels/arteries [2,3]. Blood is a special suspension of various blood cells and plasma, and its rheological properties are non-Newtonian fluid, but it is considered an incompressible inviscid fluid [4,5]. Blood flow submits the principles of universal conservation of mass, momentum and energy [2,6].…”

“…Blood flow submits the principles of universal conservation of mass, momentum and energy [2,6]. The forces that direct the blood flow are gravity and pressure gradient force, while the hindering forces are shear forces resulting from viscosity and turbulence [5,7]. The wall thickness to vessel radius (diameter) ratio due to arteries, are considered flat incompressible, thick viscoelastic [5,8,9] or elastic tubes [10], as well as thin elastic [5,[11][12][13] or viscoelastic tubes [5,14,15].…”

“…Therefore, these features show obvious nonlinearity. Usually, for simplicity, arteries are modeled as circular cylindrical elongated tubes with a constant cross-section [1,5]. Under these assumptions, many models were proposed by well-known equations in the literature [3,[16][17][18][19].…”

“…Firstly, the Korteweg-de Vries (KdV) equation was determined by combining nonlinear science with hemodynamics for simplest assumptions [4,20]. Later, considering more complex situations, the variable/constant coefficient forced KdV equation was determined [5,[21][22][23]. Boussinesq-type equations derived from the flow equations in elastic tubes and analyzed the effects of a local increase of radius followed by local variation of the thickness or rigidity of an elastic tube on the behavior of solitary waves [1,4,24].…”