**1. Introduction**

## **1.1. Numerical analysis of intravascular flow**

Numerical flow simulation is useful for understanding fluid phenomena such as blood flow or pulse wave propagation in the systemic arteries. For numerical analysis of intravascular flow, it is important to consider not only incompressible assumption and blood viscosity but also the viscoelasticity of the blood vessel wall; however, blood flow *in vivo* is complicated because of the unsteadiness of pulsatile flow and complex viscoelastic properties of the blood vessel wall. In order to conduct such numerical flow analysis in a viscoelastic blood vessel, it is effective to use the one-dimensional distributed parameter model, which can analyze flow along with the blood vessel axis. This distributed parameter model, pressure, flow volume and cross-section of the tube for every section element are defined and the time change is analyzed.

According to previous research, quantitative numerical simulation requires a model which take in both effects of unsteady viscous friction and viscoelasticity of the vessel wall in case flow unsteadiness is large (Reuderink et al., 1989). Conventional one-dimensional numerical simulation models can be classified into a linear distributed parameter model (Snyder et al., 1968; Avolio et al., 1980) and a nonlinear distributed parameter model (Anliker & Rockwall, 1971; Schaaf & Abbrecht, 1972; Porenta et al., 1986). The linear distributed parameter model has the feature that is easy to take in the influence of viscoelasticity and to conduct numerical analysis of the flow unsteadiness, since superposition of a periodic solution is possible; however, the influence of fluid nonlinearity cannot be disregarded. On the other hand, the conventional nonlinear distributed parameter model does not involve the effects of such flow unsteadiness and the viscoelastic behavior of the blood vessel wall concurrently with the difficulty of model construction. Hence, these models can be used only for qualitative discussion.

© 2012 Kitawaki, licensee InTech. This is an open access chapter distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. © 2012 The Author(s). Licensee InTech. This chapter is distributed under the terms of the Creative Commons Attribution License http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Consequently, in order to construct a numerical simulation model of intravascular flow for quantitative analysis with viscoelasticity of the blood vessel wall, it is necessary to use a nonlinear distributed parameter model to be able to include the viscoelasticity.

Numerical Simulation Model with Viscoelasticity of Arterial Wall 189

(1)

(2)

0 *<sup>t</sup>*

 

*i t*

(4)

(3)

calculated results using a one-dimensional numerical simulation model with terminal resistance treatment. Finally, we studied the effect of vessel wall viscoelasticity on the propagation of a periodic pulsatile wave by comparing numerical simulation results between the difference of viscoelastic models and viscoelastic parameters. The final **section** 

When constructing the numerical model, we neglect the effects of bends of vessels. We also assume that the tube does not leak and that the flow is axisymmetric and incompressible. Under these conditions, the equations of continuity and momentum conservation of the one-

> <sup>0</sup> *A Q t x*

where *A* is the cross-section of the tube, *Q* is the mean sectional flow volume, *p* is the mean pressure, *t* is time, *x* is distance along the vessel axis, *ρ* is the fluid density, and *Ft* is the

By assuming that we are dealing with a Newtonian fluid, and an oscillating flow velocity distribution in a cylindrical tube, using the Womersley model, the viscous resistance *Ft* is

<sup>0</sup> 42 () *<sup>t</sup>*

where *V* is the mean sectional velocity, *ν* is the kinematic viscosity, and *W(t)* is a weight

*J W t e d F i*

where 0 1 () () () *<sup>J</sup> F z zJ z J z* , and *J0*, *J1* are 0th and 1st order Bessel functions of the complex

For a long wavelength, the flow velocity distribution in a distensible tube is similar to that in a rigid tube. Therefore, the weight function in a distensible tube can be approximated by Eq. (4),

<sup>1</sup> ( ) lim

number *z*. *ω*(=2π*f*) is the angular frequency of the flow oscillating at frequency *f*, *R*

*t*

3 2

( )2

*<sup>V</sup> F V Wt d*

*t*

> 

2

*Q QA <sup>p</sup> <sup>F</sup> t xA x*

**5**, describes the conclusion of this chapter.

dimensional model are given by (Olufsen, 1999),

**2. Theory** 

**2.1. Basic equations** 

viscous resistance.

given by (Zielke, 1968),

function. In a rigid cylindrical tube,

is the Womersley number and *R* is the tube radius.
