Pulse wave propagation in a deformable artery filled with blood: an analysis of the fifth-order mKdV equation with variable coefficients

Abstract

In this paper, the propagation of pulse wave in a deformable elastic vessel filled with inviscid blood is studied. Starting from the stress–strain relationship of blood vessel wall, momentum conservation equation and the Naiver–Stokes equation, the basic equations describing the wall motion and blood flow are established. By utilizing reductive perturbation technique and long wave approximation theory, the basic equations are simplified into a classical third-order mKdV equation with variable coefficients. In order to describe the propagation characteristics of pulse wave more accurately, a fifth-order variable-coefficient mKdV equation is derived. Then, the tanh-function method is applied to find the localized traveling wave solutions of these equations. Based on these localized traveling wave solutions, we further investigate the effects of higher order terms and initial vessel deformation on the characteristics of pulse wave propagation, blood flow velocity and the volume of blood flow. The results show that the higher-order nonlinear and dispersion terms lead to the distortion of the wave, while the initial deformation of the tube wall will influence the wave amplitude and wave width.

Keywords:

• Pulse wave
• variable-coefficient fifth-order mKdV equation
• deformable arterial vessel
• reductive perturbation technique

Acknowledgments

The first author was partially supported by the National Natural Science Foundation of China (Grant No.11975143), Nature Science Foundation of Shandong Province of China (Grant No.ZR2018MA017), the Taishan Scholars Program of Shandong Province (Grant No.ts20190936), and the Shandong University of Science and Technology Research Fund (Grant No.2015TDJH102).

Disclosure statement

No potential conflict of interest was reported by the author(s).

Additional information

Funding

The first author was partially supported by the National Natural Science Foundation of China (Grant No. 11975143), Nature Science Foundation of Shandong Province of China (Grant No. ZR2018MA017), the Taishan Scholars Program of Shandong Province (Grant No. ts20190936), and the Shandong University of Science and Technology Research Fund (Grant No. 2015TDJH102).