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We propose a new technique to characterize a reflectionless arterial prosthesis. The corresponding transmission and reflection coefficients are determined from the geometric and the elastic properties of the arterial wall, and the interaction between the latter and the prosthesis are studied accordingly.
Elastic tubes are widely used nowadays in fluid flow, and their importance no longer needs to be demonstrated. From the biological field up to the industrial domain, different types of elastic tubes are encountered and while many studies have been done, there still exist unresolved questions.
The first studies on elastic tubes presented a linear approach of the fluid flow therein with differences arising only in the consideration of the tube wall contribution [1, 2]. The nonlinear studies that came after brought out the importance of the fluid convective term and particularly took into account the dynamical properties of the tube wall where the nonlinear stress–strain relationship is considered [3–6]. Most of the works that followed contributed to improve the modeling, in a more realistic way, of elastic tubes [7–9] and took into account viscosity, tapering aspect of the tubes [5, 10], pathological cases of vessels and prostheses .
Some studies on blood waves focused on the study of the behaviour of waves in the regions of discontinuities [12, 13]. In general, any discontinuity zone present in an artery gives rise to the phenomena of wave transmission and reflection. These discontinuities are in general due to the change in the elastic properties of the arterial wall, diseases, bifurcations and prostheses.
The objective of this paper is to determine the expression of the transmission and reflection coefficients in the prosthesis using a nonlinear model of blood flow. This theory is based on the separation of pressure waves into their forward and backward running components proximal and distal to the prosthesis. We then use the same approach to investigate the characteristics of a reflectionless prosthesis.
In the study, we consider that the fluid in the vessel is Newtonian, viscous, homogeneous and incompressible. Furthermore, we take into account the variation of the radius and elasticity of the wall. The rest of the paper is therefore organized as follows: in Section 2, the physical and mathematical problems are formulated. In Section 3, the nonlinear model of the flow is used to determine the transmission and reflection coefficients of the prosthesis, and the conditions for a reflectionless prosthesis are derived accordingly. The last section is devoted to concluding remarks.
The artery is assumed to be conical, with a nonlinear stress–strain relation. The geometry of the artery with the presence of a prosthesis is illustrated in Fig. 1. The index i (i=1,2 or 3) represents each portion, where i=2 stands for the prosthesis and i=1 and i=3 represent the leftmost and rightmost portions of the vessel, respectively. For the present discrete case, we apply the averaging procedure, used in the literature, to the Navier–Stokes equations  and we derive the following one-dimensional model
where z is the axial coordinate, t is time, Pi is the pressure inside the vessel, Wi is the axial component of the fluid velocity, Ri is the radius of the vessel, ρ is the fluid density and νi is kinematic viscosity.
For the wall dynamics, we use the second law of Newton on a portion of the vessel wall and we obtain the relation :
where is the pressure outside the vessel, δRi denotes the vessel deformation, ρ0i is the wall density, and Hi and hi are the effective inertial thickness and the thickness of the wall, respectively. γi is the viscoelasticity coefficient due to the absorption effects of the material. βi is the shear modulus, and i is another coefficient of viscoelasticity. σti is the approximated exponential function for the stress–strain relationship given in [15, 16]. This approximated relation is widely known as [3, 4, 6, 7]
where is the stationary radius of the vessel, Ei is the Young modulus and ai represents the nonlinear coefficient of elasticity in the portion i .
The vessel radius Ri(z,t) in the portion i is described as
As vessel segments are tapered elastic tubes with increasing rigidity away from the heart, they represent general cases of thin-walled elastic tubes with variable Young modulus. Many authors [4, 7, 8] showed that the stationary radius and the variation of elasticity along the wall will obey the relations
with f (z)=−mi(z−z0i), g(z)=ni(z−z0i). Here, f (z) denotes the decrease of the tube radius and g(z) the increase of the tube wall rigidity along the tube originating from the heart. R0i is the radius at the entrance of the portion i and z0i the abscissa at the entrance of the vessel segment, in the same portion. The positive coefficients mi and ni characterize the rate of decrease and increase of the radius and wall rigidity in the portion i, respectively.
As we consider weak wall displacement, we introduce the approximation:
where and are the normalized cross sectional area of the tube and the cross sectional area at the entrance of the ith portions, respectively. R0, w0 and E0 are, respectively, the reference values of radius, velocity and Young modulus at the entrance of the vessel. We define the viscosity coefficients and as:
and the dimensionless quantities
and we also include the following coefficients
We assume that the first and last portions (i=1 and i=3) are made up of the same material properties.
where ϵ is a small parameter measuring the weakness of dispersion and nonlinearity. Also weak are the parameters modelling the taper effect of the wall, the compliance variation and the blood viscosity which become
Next, the coordinate system is changed to slowly varying coordinates
where ci represents the wave group velocity and is assumed to be the velocity of the linear wave as it enters the medium. ξi is an independent variable representing the forward wave whereas ηi stands for the backward wave.
The approximation of the flow motion at the first order, ϵ1, gives:
In the present work, the study will be limited to this first order of perturbation. The second order, useful to obtain the analytic expression of the wave, is not used here. So, we consider the waves to be separated explicitly as forward and backward elements (where f and b are, respectively, used as superscripts) [18, 19].
In this part, we propose a new technique for the estimation of the transmission, T, and the reflection, B, coefficients in the prosthesis proportional to the local transmission, Ti, and reflection, Bi, coefficients at the interface of the prosthesis.
The local transmission, Ti, and reflection, Bi, coefficients are defined as
and are expressed in terms of the flows. Of course, in the literature, this is not usual, but from the conservation law the relation Ti+Bi=1 is not verified when the pressures are mainly used. This also brings out the difference between the method used in this paper and the one used by Stergiopulos et al. .
If we consider at the interfaces the conservation of mass and energy, it will appear that, at the first interface,
while at the second interface
Inserting Eqs. 25–27 into Eqs. 30 and 31 helps to obtain the local transmission coefficient T1 (resp. T2) and the local reflection coefficient B1 (resp. B2), respectively, at the first and second interfaces
with Lv as the length of the left portion of the vessel and Lp as the length of the prosthesis.
In this work, we suppose that the distal forward wave, , is the sum of two waves: the part of the proximal forward wave that became transmitted through the prosthesis and the part of the distal backward wave that became reflected by the prosthesis. And we obtain a similar equation for the proximal backward running wave, . These equations are given by
where T and B are the global transmission and reflection coefficients due to the overall prosthesis. Using Eq. 34, we obtain the expressions of the transmission and reflection coefficients, as
From Eq. 37, we get the conditions for a reflectionless arterial prostheses. The reflectionless arterial prosthesis is obtained when the reflection coefficient is equal to 0 and the transmission coefficient equal to 1. This gives us the following condition (with B1≠0):
Using this condition (see relation 38), the optimal thickness of the prosthesis is derived as
The manufacturing of reflectionless arterial prostheses being one of the primary objectives of research in this field, our result gives a new mathematical condition for the reflectionless arterial prosthesis. This condition takes into account the geometric and elastic properties of the vessel wall.
For numerical calculation, we have used the characteristic parameters for the femoral artery of a dog : Lv=4.5 cm, dyn/cm2, n1=0.067 cm−1, m1=0.080 cm−1 and the thickness of the vessel is 0.018 cm. We consider the prosthesis parameters to be Lp=4.0 cm, n2=0.069 cm−1 and m2=0.089 cm−1.
Figure 2 shows that the transmission coefficient increases to the value 1 on the positive side of the ordinate and the reflection coefficient increases to the value 0 on the negative side, as the ratio of the product of the Young modulus and the wall thickness is increased and for R01/R02=0.86. This behaviour is comparable with the effects of the severity of an extended stenosis as obtained by Stergiopulos et al. . The negative values obtained in the case of the reflection coefficient are due to the fact that the reflected waves velocity is negative (see Eq. 26).
In Fig. 3, the ratio of thickness (Eq. 41) is plotted as the ratio of the radius varies. When the ratio of the radii increases, the thickness ratio decreases. This figure illustrates a reflectionless arterial prosthesis. For example, in the case of a Palmaz stent with Young modulus dyn/cm2 , inserted in the femoral artery, the ratio of the thickness of the stent to that of the vessel is . Using this condition, the value of the transmission coefficient is approximately equal to 1 while the reflection coefficient is equal to 0, when the ratio of the radius is R01/R02=0.94. Taking into account the thickness of the femoral artery (h01=0.018 cm), the thickness of the stent is obtained as h02= 0.173 cm.
In Fig. 4, we show that the variation of the length of the left portion of the vessel brings about modifications in the characteristics of the perfect prosthesis. When the length of the vessel increases, the characteristics of the reflectionless prosthesis decrease. As a whole, these results present the interactions between the vessel wall and the prosthesis. Such a behaviour would also be observed for the variation of the elasticity of the vessel. These results are further confirmed by the case of Fig. 5, where the variation of the prosthesis length is taken into account. So, when the length of the prosthesis increases, the characteristics of the reflectionless prosthesis increase. In fact, if there is a perturbation at the entrance of the prothesis, that prothesis, due to its characteristics will overcome the perturbation for a better blood flow. That is why increasing the length of the prothesis deeply modifies the condition for a reflectionless prothesis as depicted in Fig. 5.
In this study, the concept of wave separation has been used to derive the reflection and transmission coefficients of blood waves created by a prosthesis inserted in an artery. This has led to the derivation of conditions for a reflectionless prosthesis. These conditions are useful for the design of a reflectionless arterial prosthesis. In our methodology, we take explicitly into account the interactions between the vessel wall and the prosthesis at the entrance and the exit of the prosthesis.