# Propagation of Nonlinear Pressure Waves in Blood

Authors: A. Elgarayhi; E. K. El-Shewy; Abeer A. Mahmoud; Ali A. Elhakem
Journal: ISRN Computational Biology
 (2013)

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## Abstract

The propagation of weakly nonlinear pressure waves in a fluid-filled elastic tube has been investigated. The reductive perturbation method has been employed to derive the Korteweg-de Vries equation
for small but finite amplitude. The effect of the final inner radius of the tuberf on the basic properties of the soliton wave was discussed. Moreover, the conditions of stability and the soliton existence via the potential and the corresponding phase portrait were computed. The applicability of the present investigation to flow problems in arteries is discussed.


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## Body

## 1. Introduction

The intermittent ejection of blood from the left ventricle produces pressure waves that flow in the arterial tree. Experimental data found that the flow velocity in blood vessels largely depends on the elastic properties of the vessel wall and they propagate towards the periphery with a characteristic diagram [1]. In arterial mechanics, the propagation of pressure waves in fluid-filled distensible tubes has been theoretically studied by several researchers [2–4]. Experimental observation for the simultaneous changes in amplitude of the pressure waves at five sites from the ascending aorta to the saphenous artery in dogs showed that the pulsatile character of the blood is soliton waves [5]. Yomosa [6] investigated the nonlinear propagation of solitary waves in large blood vessels. He found that the pulse waves of pressure and flow propagating through the arteries can be described as solitary waves excited by cardiac ejections of blood, and the features of the pulse wave such as “peaking” and “steepening” are interpreted in the viewpoint of soliton. Later, R. M. Shoucri and M. M. Shoucri studied the application of the method of characteristics of shock waves in models of blood flow in the Aorta [7]. Recently, Gaik and Demiray [8] treated the arteries as an incompressible prestressed thin walled elastic tube with a stenosis and the blood as a Newtonian fluid with variable viscosity, which vanishes on the arterial wall, and it takes the maximum value at the center of the artery. They studied the propagation of weakly nonlinear waves in the long wave approximation by the use of the perturbation methods [9]. Many authors examined the stability and the soliton existence condition via the potential and the corresponding phase portrait [10–12]. Elwakil et al. [10] studied the propagation of solitary electron acoustic waves in unmagnetized collisionless plasma. They found that, there are saddle and two-center equilibrium state in the phase plane and there is two-finite separatrix going from a saddle returning to it in the phase portrait. The evolution of small but finite-amplitude propagating solitary structures was studied the  by means of Korteweg-de Vries equation (KdV). Many authors have been using the  KdV equation to study the properties of solitary waves [13, 14]. The major topic of this work is to study the propagation of pressure waves in weakly nonlinear waves in a fluid-filled elastic tube. This paper is organized as follows: in Section 2, we present the basic set of fluid equations governing our model. In Section 3, long wave approximation is used to drive KdV equation, and the solution for KdV equation is obtained. In Section 4, some discussions and conclusions are given.


## 2. Basic Equations

To drive the equation of motion of the tube, let us consider a circular cylindrical long tube of radiusR0with a uniform inner pressure P0, let the axial stretch ratioλzandr0be the radius of the cylindrical tube after such a finite static deformation. The position vector of a generic point of the tube may be described by the following:
(1)r=(r0+u*)er+z*ez,z*=λzZ,
whereu*(z*,t*),er,eθ,ez,z* andZare a finite time dependent radial displacement, unit base vectors in the cylindrical polar coordinates, the spatial coordinate in the intermediate configuration and the axial coordinate of a point in the undeformed configuration, respectively. The axial displacement is neglected in view of the external tethering. The unit tangent vectortto the meridional curve and the unit exterior normal vector n to the deformed membrane are given by
(2)t=1Λ(∂u*∂z*er+ez),n=1Λ(er-∂u*∂z*ez),
whereΛis defined by
(3)Λ=[1+(∂u*∂z*)2]1/2.The stretches in the axial and circumferential directions may be given as follows:(4)λ1=λzΛ,λ2=λθ+u*R0,
whereλθ=r0/R0is the stretch ratio in the circumferential direction after finite static deformation.LetF1 and F2 be the membrane forces acting along each unit length of the meridional and circumferential curves of the tube, respectively. The equation of motion of the tube in the radial direction is given by
(5)∂∂z*{μ[1+(∂u*/∂z*)2]1/2∂Σ∂λ1∂u*∂z*}-μλzR0∂Σ∂λ2+(λθ+u*R0)P*H=ρ0λz∂2u*∂t*2,
whereμis the shear modulus of the tube material,μΣis the strain energy density function,His the initial tube thickness,P*is the fluid pressure, andρ0is the mass density of the tube material. In order to complete the field equations one must know the value of the fluid pressureP*. Therefore, (5) is to be complemented with the equations governing the blood fluid. Blood is known to be an incompressible non-Newtonian fluid. The main factor for blood to behave like a non-Newtonian fluid is the deformability of red blood cells and the level of cell concentration (hematocrit ratio). When blood flows in arteries the red cells move to the central region of the artery and, thus, the hematocrit ratio is reduced near the arterial wall, where the shear rate is quite high, as can be seen from Poiseuille flow. In other words, experimental observations indicate that when the shear rate is high, blood behaves like a Newtonian fluid. The ratio of the viscous terms to the nonlinear term is(μv/ρf)(∂2υ*/∂z2)/υ*(∂υ*/∂z)≈μv/ρfV′T′υ*≈O(10-5), consideringμv=0.04pandρf=1.05g/cm3. Therefore, the viscous effect in comparison to the nonlinear effect can be neglected. Based on these observations, we assume that blood is an incompressible inviscid fluid whose equations of axially symmetrical motion in the cylindrical polar coordinates are given by
(6)∂Vr*∂r+Vr*r+∂Vz**∂z*=0,∂Vr*∂t*+Vr*∂Vr*∂r+Vz**∂Vr*∂z*+1ρf∂P¯∂r=0,∂Vz**∂t*+Vr*∂Vz**∂r+Vz**∂Vz**∂z*+1ρf∂P¯∂z*=0,
whereVr*andVz**are the fluid velocity components in the radial and axial directions, respectively,ρfis the mass density of the fluid, andP¯is the fluid pressure function. These field equations must satisfy the following boundary conditions:
(7)Vr*|r=rf=∂u*∂t*+∂u*∂z*Vz**|r=rf,P¯|r=rf=P*.HereP¯is the fluid pressure function,ρfis the fluid mass density, andrfis the final inner radius of the tube. whereλθ=(r0/R0)is the stretch ratio in the circumferential direction after the finite static deformation, where from (5) it is obtained that
(8)P*=ρ0R0Hλz(R0λθ+u*)∂2u*∂t*2+μHλz(R0λθ+u*)∂Σ∂λ2-R0H(R0λθ+u*)∂∂z*×{μ[1+(∂u*/∂z*)2]1/2∂Σ∂λ1∂u*∂z*}.In general, the strain energy densityΣis a function ofλ1andλ2. For our purposes, we shall assume thatΣis analytic inλ1andλ2and can be expanded into power series of the following form:
(9)P*=β1μHR02u*+ρ0Hλθλz∂2u*∂t*2-α0μH∂2u*∂z*2+β2μHR03u*2-ρ0Hλθ2λzR0u*∂2u*∂t*2-α1μHR0(∂u*∂z*)2+(α0μHλθR0-2α1μHR0)u*∂2u*∂z*2+μHR0P0,
where the coefficientsα0,…,β2are defined by
(10)α0=1λθλz∂Σ∂Λ|u*=0,β1=R0λθλz(R0∂2Σ∂u*2-1λθ∂Σ∂u*)|u*=0,β2=R032λθλz∂3Σ∂u*3|u*=0-β1λθ.Equations (6)–(9) give sufficient relations to determine the unknownsu*,Vr*,Vz**, and P¯.


## 3. Long Wave Approximation

The reductive perturbation method is used to study the propagation of small but finite amplitude. Let us introduce the following types of stretched coordinates [15]:
(11)ξ=ϵ1/2(z*-gt*),τ=ϵ3/2z*,
whereϵis a small parameter measuring the smallness of nonlinearity, dissipation, and dispersion,gis the phase velocity in the longwave approximation to be determined later. All physical quantities appearing in (6) and (7) are expanded as power series in ϵ about their equilibrium values as:
(12)Vr*=∑n=1∞‍ϵ1/2+nVr*(n)(ξ,τ,r),Vz**=∑n=1∞‍ϵnVz**(n)(ξ,τ,r),P¯=∑n=1∞‍ϵnP¯(n)(ξ,τ,r),u*=∑n=1∞‍ϵnun*(ξ,τ,r).We impose the boundary conditions as follows:(13)P¯|r=rf=ϵ(ρ0Hg2λθλz-α0μH)∂2u*∂ξ2+β1μHR02u*-ϵ3α0μH∂2u*∂τ2-2ϵ2α0μH∂2u*∂ξ∂τ+β2μHR03u*2-ϵρ0Hg2λθ2λzR0u*∂2u*∂ξ2-α1μHR0[ϵ(∂u*∂ξ)2+2ϵ2∂u*∂ξ∂u*∂τ+ϵ3(∂u*∂τ)2]+(α0μHλθR0-2α1μHR0)×u*(ϵ∂2u*∂ξ2+ϵ3∂2u*∂τ2+ϵ2∂2u*∂ξ∂τ)+μHR0P0,Vr*|r=rf=-ϵ1/2g∂u*∂ξ+ϵ1/2(∂u*∂ξ+ϵ∂u*∂τ)Vz**|r=rf.Substituting (11) and (12) into (6) and (9) and equating coefficients of like powers ofϵ. Then, from the lowest-order equations in ϵ the following results are obtained:
(14)1ρf∂P¯1∂r=0,-g∂Vz**(1)∂ξ+1ρf∂P¯1∂ξ=0,∂Vr*(1)∂r+Vr*(1)r+∂Vz**(1)∂ξ=0.And the boundary conditions can be written as follows:(15)P¯1|r=rf=β1μHR02u1*,Vr*(1)|r=rf=-g∂u1*∂ξ.From the solution of the sets (14) and (15) we obtain that
(16)u1*=U(ξ,τ),U(ξ,τ)=R02β1μHP¯1(ξ,τ),Vz**(1)=1gρfP¯1(ξ,τ),Vr*(1)=-12gρf∂P¯1(ξ,τ)∂ξr,g2=β1μHrf2ρfR02,
where U(ξ,τ) is an unknown function whose governing equation will be obtained later andgis the phase velocity in the longwave approximation.Considering now the coefficients ofO(ϵ2), we derive with the aid of (16) the following set of equations
(17)12ρf∂2P¯1(ξ,τ)∂ξ2r+1ρf∂P¯2∂r=0,-g∂Vz**(2)∂ξ+1g2ρf2P¯1∂P¯1∂ξ+1ρf∂P¯2∂ξ+1ρf∂P¯1∂τ=0,∂Vr*(2)∂r+Vr*(2)r+∂Vz**(2)∂ξ+1gρf∂P¯1∂τ=0.And the boundary conditions can be written as follows(18)P¯(2)|r=rf=(ρ0Hg2λθλz-α0μH)R02β1μH∂2P¯1∂ξ2+β1μHR02u2*+R0β2μHβ12(P¯1)2,Vr*(2)|r=rf=-g∂u2*∂ξ+R02gρfβ1μHP¯1∂P¯1∂ξ|r=rf.From (17-a) we have(19)P¯2=-14∂2P¯1∂ξ2r2.Eliminating the second order perturbed quantitiesP¯2andVz**(2), the desired KdV equation is  obtained as follows:
(20)∂P¯1∂τ+AP¯1∂P¯1∂ξ+B∂3P¯1∂ξ3=0,
where the coefficientsAandBare defined by
(21)A=R02β1μH(β1μH2g2ρfR02+2g2ρfβ2R0β12μHrf+1rf),B=(g2ρfR02rf4β1μH+ρ0ρfg4R04λθλzβ12μ2Hrf-rf216-α0g2R04ρfβ12μHrf).
Our system can support two kinds of potential structure depending on the sign of the coefficient of the nonlinear term (A). A stationary solitary wave solution of the KdV equation can be obtained by transforming the space variable to
(22)η=(ζ-vτ),
wherevis velocity of the wave. This has been done by imposing the boundary conditions for localized perturbations, namely, P¯1=0, dP¯1/dη=0, andd2P-1/dη2=0 for η→±∞. Thus, the steady state solution of (22) can be expressed as
(23)P-1=P¯0sech2[ηΔ],
where the soliton amplitudeP¯0and the soliton widthΔare given by
(24)P¯0=3vA,Δ=2Bv.


## 4. Numerical Results and Discussion

The weakly nonlinear pressure waves in a fluid-filled elastic tube have been investigated. To make our result physically relevant, numerical studies have been made using parameters close to those values corresponding to actual biologically relevant parameters for experimental data in dogs [6, 16]. The effect of the final inner radius of the tuberfon the basic properties of the amplitude and width of the soliton is shown in Figure 1. It is obvious that the magnitude of the soliton amplitude and width increases withrf. From (20), the nonlinear equation of motion can be obtained as
(25)12[dP¯1dη]2+V(P¯1)=0.Figure 1
The soliton profile for expression (23) for different values ofrfforβ1=296.105,β2=991.496,ρf=1.05gm/cm3,R0=0.38 cm,ρ0=1.03 gm/cm3,λz=λθ=1.6,α0=78.692,H=2×10-2cm,μ=0.4, andv=8cm/s.This equation can be regarded as an “energy integral” of an oscillating particle of unit mass, with a velocitydP¯1/dηand positionP¯1in a potential
(26)V(P¯1)=[P¯16B-v2B]P¯12.The kinetic energy of oscillating particle is shown in Figure2. A necessary condition for the existence of the solitary waves is thatd2V/dP¯12<0forP¯1=0. A value of d2V/dP¯12>0predicts the formation of a shock wave. The potential and the corresponding phase portrait are shown in Figure 3. Stable solitons will exist whenv/2B>0; otherwise stable solitons will not be present. The application of our model might be particularly interesting in the new observations for the biological experimental data.Figure 2
The kinetic energy forrf=0.75,β1=296.105,β2=991.496,ρf=1.05gm/cm3,R0=0.38 cm,ρ0=1.03 gm/cm3,λz=λθ=1.6,α0=78.692,H=2×10-2cm,μ=0.4, andv=8 cm/s.Figure 3
The potential and the corresponding phase portraitforrf=0.75,β1=296.105,β2=991.496,ρf=1.05gm/cm3,R0=0.38 cm,ρ0=1.03 gm/cm3,λz=λθ=1.6,α0=78.692,H=2×10-2cm,μ=0.4, andv=8 cm/s.



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*Source: 436267-2013-08-06.xml*