## Abstract

A nonlinear elastic, anisotropic and axisymmetric balloon angioplasty model, consisting of the balloon, the atherosclerotic plaque and the artery wall, has been developed and analysed in this paper. The deformation of the angioplasty compound, i.e. the balloon, the plaque and the artery, for slowly increasing dilation pressure within the balloon, is investigated. The plaque has been considered as a circular cylindrical tube extending all around the artery circumference. Normal radial stress and radial displacement boundary conditions have been imposed along the balloon–plaque and the plaque–artery interfaces. Large elastic deformations have been accommodated in the model. The balloon material has been considered as isotropic, the plaque material as transversely isotropic and the artery wall material as orthotropic. A strain energy density function, that satisfies existing incomplete experimental data, has been constructed for the plaque material. For a given dilation pressure within the balloon, the deformation of the angioplasty compound is determined via the numerical solution of a system of four nonlinear equations. For a medium hard balloon and a medium stiff plaque, a dilation pressure 7.5 times bigger than the blood pressure is required for an almost full reopening of the narrowed lumen. The percentage increase in length of the inner radius of the plaque is twice bigger than that of the radius of the artery. Results also indicate that the process of full reopening may be achievable, without stressing the artery wall dramatically.

## 1. Introduction

The general theory of large elastic deformations is presented by Green & Adkins (1970). The theory relevant to the extension, inflation and torsion of an incompressible circular cylindrical tube, the theory of curvilinear aeolotropy and the theory of elastic membranes, are included in that book. Spencer (1970) has written a review article on the equilibrium theory of finite deformations of elastic solids. Pipkin (1968) has published a note on the integration of an equation in membrane theory. In this brief communication, he pointed out that the equilibrium equation is valid for anisotropic materials, if the azimuthal and meridional directions of the membrane are symmetry axes of the membrane material. Kydoniefs (1969) and Kydoniefs & Spencer (1969) investigated the axisymmetric deformation of an initially cylindrical elastic membrane. Hart & Shi have used Kydonief's (1969) paper to study the deformation of the arteries, under static or dynamic blood pressure and considering the arterial wall either as isotropic or orthotropic. In Hart & Shi (1991, 1992), the deformation of joined dissimilar isotropic elastic membranes, under internal pressure and longitudinal tension has been investigated. In Hart & Shi (1993), the deformation of joined dissimilar orthotropic membranes has been studied. In Hart & Shi (1995*a*,*b*) the reflection and transmission of deformation and stress waves in two dissimilar orthotropic tubes, has been dealt with. The wrinkling of one of the two cylindrical tubes joined together is studied by Haughton & McKay (1997).

The development of stresses and displacements within the artery wall and the atherosclerotic plaque during the process of balloon angioplasty has been the subject of at least four papers. Holzapfel *et al*. (2002) have developed a 3D model for balloon angioplasty, consisting of eight distinct arterial components associated with specific mechanical responses, including inelastic ones. Their geometric model is based on *in vitro* magnetic resonance imaging of a human stenotic postmortem artery. A stent has also been incorporated in their model. Their 3D finite element realization considers the balloon–artery interaction and accounts for axial *in situ* prestretches. Gourisankaran & Sharma (2000) have developed finite-element models of diseased arteries, subject to balloon dilation, to study the stresses in the arterial wall and in the atherosclerotic plaque at peak balloon pressures. Oh *et al*. (1994) have also used finite element modelling to simulate the response of atherosclerotic arteries to balloon angioplasty procedures. Plain strain models and isotropic material properties have been used in both the above papers. The elastic pre- and post-angioplasty response of human common-iliac arteries *in vitro*, was studied by Medynsky *et al*. (1998). They related the lumen circumference with the lumen pressure via a nonlinear relationship.

Atkinson & Peltier (1993) have studied the shape of an inflatable packer, fixed against an oil wellbore. The shape of the packer is given as a function of the internal pressure within it. Using the large deformation theory for membranes, the tensile stresses on the packer surface have been calculated. Atkinson *et al*. (2001) studied the wellbore stresses induced by the nonlinear deformation of an inflatable packer. In both the papers above, the theory of finite axisymmetric deformations of cylindrical elastic membranes, reinforced with inextensible cords, developed by Kydoniefs (1970), was used. The knowledge of the wellbore stresses is of particular interest in oil recovery.

A nonlinear, anisotropic and axisymmetric balloon angioplasty model, consisting of the balloon, the atherosclerotic plaque and the artery wall, has been developed and analysed in this paper. The deformation of the angioplasty compound, i.e. the balloon, the plaque and the artery, for slowly increasing dilation pressure within the balloon, is dealt with. The plaque has been assumed to extend all around the artery circumference. This may be the case for hypocellular or fibrous plaques, but calcified plaques usually span over a part of the artery circumference. The balloon and the artery have been considered as elastic membranes under axisymmetric deformations and are dealt with by using the theory of Kydoniefs (1969). The plaque has been considered as a circular cylindrical tube under uniform extension and inflation and has been tackled by using the relevant theory of Green & Adkins (1970).

Nonlinear, anisotropic material properties and large elastic deformations have been accommodated in the model. The balloon material has been considered as isotropic of the Mooney–Rivlin type, the plaque material has been dealt with as transversely isotropic and the artery wall material has been considered as orthotropic. Regarding the plaque material, the curvilinear aeolotropy theory from Green & Adkins (1970) has been implemented, since the axis of transverse isotropy of the plaque is along the radial direction, i.e. it is an axis that is different from point-to-point within the plaque material. Experimental data from Loree *et al*. (1994) and Topoleski & Salunke (2000), have been used for the construction of a strain energy density function for the plaque.

Radial normal stress and radial displacement boundary conditions are satisfied along the balloon–plaque and the plaque–artery interfaces, for any particular dilation pressure within the balloon. Analytical integrations involving partial fractions and integration by parts are used for the evaluation of the pressure exerted on the balloon–plaque and the plaque–artery interfaces. A system of four nonlinear equations with four unknowns is eventually formed, which is solved numerically via Broyden's method (see Press *et al*. 1994), which does not require the knowledge of the Jacobian of the functions to be zeroed.

Results indicate that for a medium hard balloon and a medium stiff plaque, an inflation pressure of around 7.5 times the blood pressure is required for an almost full reopening of the narrowed lumen of the artery. At that dilation pressure, while the radius of the artery is increased by around 19%, the inner radius of the plaque is increased by around 42%. Hence the process of unclogging the narrowed artery gives rise to a radial displacement of the artery wall, which is half the radial displacement of the inner wall of the plaque. This is true for intermediate balloon pressures as well. The plaque is more complaint (i.e. it is squeezed more easily), at lower balloon pressures than at higher balloon pressures. This observation may be attributed to the nonlinear stress–stretch relation of the plaque material. For small inflation pressures close to blood pressure, the plaque is slightly compressed along the axial direction. For an almost full reopening of the narrowed artery lumen, the pressure exerted on the artery wall is four times less than the pressure exerted on the plaque by the balloon. Furthermore, at nearly full reopening, the pressure exerted on the artery wall is 1.5 times bigger than the blood pressure, a value which is comparable to physiological blood pressure of hypertasic patients. Hence the process of full reopening may be achievable, without stressing the artery wall dramatically.

## 2. The geometry and the external loadings of the model

Following Green & Adkins (1970), we use two curvilinear, orthogonal coordinate systems, namely (*θ*′_{1}, *θ*′_{2}, *θ*′_{3}) for the undeformed configuration and (*θ*_{1}, *θ*_{2}, *θ*_{3}) for the deformed configuration. The system (*θ*′_{1}, *θ*′_{2}, *θ*′_{3}) coincides with the cylindrical polar coordinate system (*ρ*, *ϑ*, *η*) of the undeformed configuration, i.e. we have the correspondence(2.1)while the system (*θ*_{1}, *θ*_{2}, *θ*_{3}) coincides with the cylindrical polar coordinate system (*r*, *θ*, *z*) of the deformed configuration, i.e.(2.2)Both coordinates *ϑ* and *θ*, of the reference and the current configurations, respectively, do not appear in figures 1–3 since the problem considered here is axisymmetric, regarding the geometry and the loadings, with respect to the axes *η* or *z*. Note that axes *η* and *z* coincide, while generally coordinates *η* and *z* are not equal when measured along them. Further, in our model all quantities involved are independent from *ϑ* and(2.3)Further, since in our model we also have symmetry with respect to the plane *η*=0 or *z*=0 (these two planes coincide), only the upper right quarter of the angioplasty configuration, i.e. for *η*>0 or *z*>0 and *ρ*>0 or *r*>0, is considered in our analysis. It is also only this part of the geometry that is shown in figures 1–3.

Both the above coordinate systems appear in figure 1 which shows the undeformed and the deformed shape of the artery wall alone, with respect to the action of internal blood pressure only. The undeformed artery wall in figure 1*a* has radius *ρ*=*ρ*_{0} when zero blood pressure acts on it. When the blood pressure acts on the artery wall, the radius of the deformed artery becomes *r*=*ρ*_{1}, as shown in figure 1*b*. We use the symbol *ρ*_{1} for the deformed radius of the artery in the configuration of figure 1*b*, because it is this configuration that will be introduced as initial configuration of the artery in the angioplasty model later on.

Figure 2 shows the geometry and the loadings of the angioplasty model. Part (*a*) of figure 2 refers to the undeformed configuration while part (*b*) of figure 2 refers to the deformed configuration.

The balloon has the shape of a circular cylinder. Line H′M′D′C′ represents a cross-section of the upper right part of the balloon membrane in the undeformed configuration. The straight line segment M′D′C′ corresponds to the right upper half of the meridian of the balloon in the reference configuration and (H′M′)=*ρ*_{2} is the length of the undeformed radius of the balloon. The balloon axial length is *L*_{b} and the initial thickness of its membrane is .

The atherosclerotic plaque has the shape of a circular cylindrical tube in the reference configuration. The rectangle A′B′C′D′ represents the upper right part of the cross-section of the plaque with a vertical plane passing through the axis *Oη*. The internal radius of the plaque tube is *ρ*_{2}, while its external radius is *ρ*_{1}. The plaque axial length is *L*_{p} and its radial thickness is *ρ*_{1}−*ρ*_{2}.

The upper right part, of the cross-section of the artery wall with a vertical plane passing through the axis *Oη*, is shown in figure 2*a* by the straight line J′A′B′. In this reference configuration the artery wall is acted upon by the blood pressure *p*_{0} and the radius of the artery is *ρ*_{1}. In the article, the blood pressure *p*_{0} is considered spatially and temporally constant and no flow of blood in the artery is considered. The initial thickness of the membrane of the artery wall, with zero pressure acting on it, is .

Note that the dimensions *L*_{b}, , *ρ*_{2}, *L*_{p}, *ρ*_{1}−*ρ*_{2} of the balloon and the plaque refer to their undeformed shape before the action of the blood pressure *p*_{0} on them. On the contrary, the radius *ρ*_{1} of the artery in the undeformed configuration of figure 2*a*, has emerged after the action of the blood pressure *p*_{0} on the undeformed, with radius *ρ*_{0}, artery wall of figure 1*a*.

The deformed configuration of the balloon–plaque–artery system is shown in figure 2*b*. The balloon meridian H′M′D′C′ maps to the line HMGDC, the plaque meridian A′B′C′D′ maps to the line ABCD and the artery meridian J′A′B′ maps to the line JAB. The plaque undergoes uniform extension along the axial *z* direction and uniform inflation, thus keeping the rectangular meridional perimeter that it had in the reference configuration. It should be mentioned that while the points H′, M′, C′ and B′ map to the points H, M, C and B, respectively, the points D′ of the balloon membrane and A′ of the artery wall do not map to D and A, respectively. On the other hand, the points D′ and A′ of the undeformed plaque, do map to points D and A of the deformed plaque. This happens because no condition of equal axial displacement along the balloon–plaque and the plaque–artery interfaces has been imposed. Only radial normal stress and radial displacement boundary conditions have been imposed along these surfaces.

Once the balloon is inflated with pressure *p*_{b}, greater than the blood pressure *p*_{0}, the balloon membrane deforms to the shape HMGDC. Note that the portion HM remains straight and horizontal and (H′M′)=(HM)=*ρ*_{2}, since it is assumed that the end edges of the balloon are clamped on rigid circular rings of radius *ρ*_{2}. It is also assumed that the plaque resists the balloon inflation, i.e. that the plaque radial compressive stiffness is sufficiently large, and a portion DC of the inflated balloon membrane is in contact with the whole of the plaque meridian DC. The pressure that the plaque meridian DC feels, owing to the balloon inflation is denoted by *p*_{2}. Finally, the part MGD of the balloon meridian, inflates into the blood, acted upon by pressure *p*_{0} from the external side of the balloon. The balloon axial length in the deformed configuration remains *L*_{b}, i.e. the balloon tips (e.g. at H′≡H) are fixed.

The plaque meridional perimeter A′B′C′D′ deforms to the rectangle ABCD. The deformed plaque internal and external radii are *r*_{2} and *r*_{1}, respectively. The plaque deformed internal and external surfaces are acted upon by pressures *p*_{2} and *p*_{1}, respectively, owing to the contact of the plaque with the balloon and the artery along the line portions CD and AB, respectively. The deformed axial length of the plaque is equal to *λL*_{p}, where *λ* is the uniform stretch ratio of the plaque tube along the axial *z* direction.

The meridian J′A′B′ of the undeformed artery wall deforms to the line JAB. Along the portion AB, the radius of the deformed artery is *r*=*r*_{1}, while at *z*→∞, the radius of the deformed artery is *r*=*ρ*_{1}, since the effect of the balloon inflation is not felt by the artery, far away from the balloon and the plaque.

## 3. The material properties for the balloon, the artery and the plaque

All materials involved in our model are considered as incompressible and hyperelastic; i.e. a strain energy density function, expressed in terms of strain components, exists for each one of the balloon, artery and plaque materials. In all cases the stress–strain relations are nonlinear and large deformations are also included.

Following Gourisankaran & Sharma (2000), an isotropic strain energy density function *W*^{(b)} for a Mooney–Rivlin material, of the form(3.1)has been selected for the balloon. and are constants, whose values are given in table 1 for a medium grade balloon. A medium grade balloon was selected, such that the whole axial length (CD)=*λL*_{p} (see figure 2) of the plaque is in contact with the balloon, at all stages of inflation.

As presented by Kydoniefs (1969), *I*_{1} and *I*_{2} are the strain invariants given by(3.2)(3.3)where *λ*_{1}, *λ*_{2} and *λ*_{3} are the principal stretch ratios along the principal directions of strain. Owing to the axially symmetric deformation, these principal directions of strain coincide with the normal to the deformed surface, the lines of latitude and the tangents to the meridian. Hence *λ*_{1}, *λ*_{2} and *λ*_{3} are given by the relations(3.4)(3.5)(3.6)where *h*^{(b)} is the thickness of the balloon membrane in the deformed configuration. The arc length along the meridian C of the balloon membrane is denoted by *ξ* and the angle between the tangent to C and the *z*-axis is denoted by *ω* (see figure 3). An orthotropic strain energy density function *W*^{(a)} has been selected for the artery wall, according to Vaishnav (1980) and Hart & Shi (1993, 1995*a*). The form of *W*^{(a)} is given by the equation(3.7)where *γ*′_{(22)} and *γ*′_{(33)} are physical components of strain along the circumferential *θ*′_{2}=*ϑ* and axial *θ*′_{3}=*η* directions, referred to the system (*θ*′_{1}, *θ*′_{2}, *θ*′_{3})=(*ρ*, *ϑ*, *η*) of the undeformed material (see Green & Adkins 1970, pp. 30–31). Note that *γ*′_{(22)} and *γ*′_{(33)} are not components of a tensor and therefore their suffices are bracketed. *γ*′_{(22)} and *γ*′_{(33)} are given by the formulae(3.8)(3.9)All quantities through are constants. Their values are given in Hart & Shi (1993) and are listed here in table 2.

As long as the atherosclerotic plaque is concerned, Lee (2000) reports that the atherosclerotic lesion should be considered as transversely isotropic material, with the radial direction *θ*′_{1}=*ρ* of the undeformed material, as the axis of elastic symmetry. Also Loree *et al*. (1994) have reported strongly anisotropic and nonlinear behaviour for cellular and hypocellular plaques along the circumferential and radial directions. Hayashi & Imai (1997) have used a strain energy density function for an isotropic material in their analysis of stresses in the atherosclerotic wall. In the literature, no transversely isotropic strain energy density function for the plaque was found. Subsequently in this article, we formed a strain energy density function *W*^{(p)} for the plaque, such that the existing experimental data are satisfied.

From Green & Adkins (1970, p. 26), a strain energy density function *W* for a transversely isotropic material, with respect to a rectangular Cartesian system of coordinates (*x*_{1}, *x*_{2}, *x*_{3}) which define points in the undeformed configuration and having the axis *x*_{1} as axis of elastic symmetry, is a polynomial function of the two strain invariants *I*_{1} and *I*_{2} ( due to incompressibility) and of the quantities *K*_{1} and *K*_{2} given by the equations(3.10)(3.11)where *e*_{ij} are the components of the Lagrangian strain tensor referred to the coordinate system (*x*_{1}, *x*_{2}, *x*_{3}) of the undeformed material. Hence, the strain energy density function has the form(3.12)where *I*_{1} and *I*_{2} are given by equations (3.2) and (3.3) in terms of the principal stretch ratios *λ*_{1} and *λ*_{2}, or by the equations (see Green & Adkins 1970, p. 4)(3.13)(3.14)in terms of the Lagrangian strain components. In equations (3.13) and (3.14), the summation convention over repeated Latin suffices has been adopted.

However, regarding the plaque transversely isotropic material, we have the axis of material symmetry along the radial direction *θ*′_{1}=*ρ* of the undeformed configuration. This means that for different material points within the plaque tube, the axis of material symmetry is rotated and we have curvilinear transverse isotropy (see Green & Adkins 1970, pp. 29–33). Then the strain energy density *W*, with respect to the system (*θ*′_{1}, *θ*′_{2}, *θ*′_{3})=(*ρ*, *ϑ*, *η*) of the undeformed configuration, is expressed as a polynomial function(3.15)where(3.16)(3.17)(3.18)(3.19)where *γ*′_{(rs)} are physical strain components.

For our problem, the strain energy density function *W*^{(p)} for the plaque was assumed to have the form(3.20)where *f*(*K*′_{2}) is a function of *K*′_{2}, which in turn is given by equation (3.19), and are constants to be determined. The function *f*(*K*′_{2}) will not be involved in our calculations and it will not be determined. Also, according to our knowledge, experimental data for the plaque material under shear deformation do not exist, hence the determination of *f*(*K*′_{2}) was not possible. Eventually we will have a partially determined strain energy density function for the plaque which will be sufficient for this problem, but may not be sufficient for other problems involving the plaque material under different boundary conditions. In other words, with *f*(*K*′_{2}) undetermined, the form of *W*^{(p)} given by equation (3.20) will be adequate for the solution of the problem presented in this article.

For the determination of constants of equation (3.20), we used the experimental data of Loree *et al*. (1994) and Topoleski & Salunke (2000), regarding uniaxial tension along the circumferential direction *θ*′_{2}=*ϑ* and uniaxial compression along the radial direction *θ*′_{1}=*ρ*_{2} of the undeformed material, respectively. In both the above papers, plaque rectangular specimens, taken from arteries of humans after death, were used. Hence it was more appropriate to use the Cartesian coordinates (*x*_{1}, *x*_{2}, *x*_{3}), which define points in the undeformed configuration, and to have the axis *x*_{1} as the axis of elastic symmetry. The Cartesian coordinates (*x*_{1}, *x*_{2}, *x*_{3}) correspond to the cylindrical polar coordinator (*ρ*, *ϑ*, *η*), respectively.

From both papers we selected data referring to fibrous plaques. From Loree *et al*. (1994) we selected for our data the right most curve in their fig. 3(B) (hypocellular specimens). From Topoleski & Salunke (2000), we selected for our data the right most curve given in their fig. 2(b). In both these papers, the stresses were measured as force per unit area in the undeformed configuration, i.e. the nominal stress or the first Piola–Kirchhoff stress components *Π*_{22} and *Π*_{11} (see Spencer 1980, pp. 132–134) were measured. Regarding the strain, the engineering tensile strain *ϵ* with(3.21)where *L* is the original length of the specimen along the circumferential direction and Δ*L* is the positive change in length along the circumferential direction, after the deformation was measured in Loree *et al*. (1994); see figure 4. Then the tensile stretch ratio *λ*_{2} along the circumferential direction, is given by(3.22)In Topoleski & Salunke (2000), the compressive stretch ratio *λ*_{1} along the radial direction, i.e.(3.23)was measured, *w* was the thickness of the specimen (see figure 4) and Δ*w* was the positive change in thickness.

From Spencer (1980, p. 139), the first Piola–Kirchhoff stress tensor *Π*_{ij} is related to the deformation gradient tensor *F*_{ij} via the relation(3.24)where *W* is the strain energy density function per unit volume of the material in the undeformed configuration. Regarding the uniaxial tension and compression experiments, on rectangular bars of atherosclerotic plaque material, and along the axes *x*_{2} and *x*_{1}, respectively, equation (3.24) yields(3.25)(3.26)where(3.27)(3.28)for each particular experiment. Equations (3.25) and (3.27) refer to the tensile experiment done by Loree *et al*. (1994) along the circumferential direction of the plaque, while equations (3.26) and (3.28) refer to the compression experiment done by Topoleski & Salunke (2000). In equation (3.25), the strain invariants *I*_{1} and *I*_{2} have been expressed in terms of the principal stretch ratios *λ*_{2} and *λ*_{3} only, since *λ*_{1}=(*λ*_{2}*λ*_{3})^{−1} owing to incompressibility. In equation (3.26), *I*_{1} and *I*_{2} have been expressed in terms of the principal stretch ratio *λ*_{1} only, since for that particular experiment(3.29)because the axis of material symmetry is the *x*_{1} and the extensions in the *x*_{2} and *x*_{3} directions are equal. Note that in equations (3.25) and (3.26), *f*(*K*_{2}) is not involved in the result while *K*_{1} is, since the experiments were performed along the principal directions of strain.

Regarding equation (3.25), from the experiments of Loree *et al*. (1994) we can extract data curves for *Π*_{22} versus *λ*_{2}. *λ*_{3} was not measured in these experiments. Also, *λ*_{3} cannot be obtained in terms of *λ*_{2} since the material of the plaque is transversely isotropic and equation (3.29) does not hold when the material is stretched in the *x*_{2} direction. Hence we used the function(3.30)in order to relate *λ*_{3} with *λ*_{2}. *c*_{1}, *c*_{2} and *c*_{3} are unknown constants. We require that the curves given by equation (3.30), pass from the point (1, 1), which reflects the situation in the unstretched configuration of the plaque rectangular specimen. We also impose the condition that at *λ*_{2}=1, these curves have a horizontal slope. By implementing the two conditions mentioned above, we reduce the number of constants to one and equation (3.30) becomes(3.31)where *m*=−*c*_{3} is a parameter, Regardless of the value of the parameter *m*, the family of curves given by equation (3.31), pass from the point (1, 1), and at *λ*_{2}=1, these curves have a horizontal slope. They also have a negative curvature equal to −2(*m*+1), if *m* is positive. This curvature becomes increasingly negative if *m* is allowed to take positive increasing values. We also require that(3.32)(3.33)and that *λ*_{3} and *λ*_{1} are monotonically decreasing for increasing *λ*_{2}. By inspection, we concluded that we must have 0≤*m*≤0.6 for the above conditions to be satisfied for 0≤*λ*_{2}≤5. A plot of *λ*_{3} and *λ*_{1} versus *λ*_{2}, obtained by using relations (3.31) and (3.33), is shown in figure 5, for *m*=0.3.

Since we now know *λ*_{3} as a function of *λ*_{2} for a uniaxial tension loading along the *x*_{2} direction, we can use the experimental data of Loree *et al*. (1994) and Topoleski & Salunke (2000), to find the constants of equation (3.20), by using equations (3.25) and (3.26), via the implementation of the linear least square fitting routine lfit of Press *et al*. (1994). The outcome of the fitting is presented in figures 6 and 7, when the parameter *m*, used in relation (3.31), is equal to 0.3. The numerical values of the constants are given in table 3. Note that the function *f*(*K*′_{2}), involved in equation (3.20) was not determined.

## 4. The underlying theory regarding our angioplasty model

The underlying theory for the analysis of our angioplasty problem is presented in this section. First, the theory for finite axisymmetric deformations of an initially cylindrical elastic membrane, related to both the balloon membrane and the artery wall, is presented in §4a. Next, the theory for the extension and inflation of an incompressible circular cylindrical tube is outlined in §4b.

### (a) Finite axisymmetric deformations of an initially cylindrical elastic membrane

Kydoniefs (1969) has presented the theory for finite axisymmetric deformations of an initially cylindrical isotropic elastic membranes. Pipkin (1968) has pointed out that the theory of Kydoniefs (1969) holds for anisotropic materials, if the azimuthal and meridional directions of the membrane are symmetry axes of the membrane material.

The strain energy density function of the membrane material is denoted by *W* and its initial thickness in the undeformed configuration is denoted by *h*_{0}. Referring to figures 2 and 3, the stress resultants *T*_{3} along the meridional (*ξ* measures arc length along that direction) and *T*_{2} along the azimuthal (*θ*_{2}=*θ*) directions in the deformed configuration, are given by the equations(4.1)(4.2)If *κ*_{3} and *κ*_{2} are the principal curvatures of the surface of the membrane, *κ*_{3} being the curvature of the meridian curve (along *ξ*) and *κ*_{2} the curvature of the curve of latitude (along *θ*_{2}=*θ*), the equilibrium equations can be written in the form(4.3)(4.4)where(4.5)(4.6)A principal curvature is positive when the corresponding centre of curvature is on the same side of the surface as the inward pointing normal. −*p* is the internal pressure acting on the deformed membrane (*p*<0).

By using equations (4.1), (4.2), (4.5) and (4.6) in equations (4.3) and (4.4), Kydoniefs has shown that the equilibrium equations take the form(4.7)(4.8)where *ρ* is the radius of the undeformed circular cylindrical membrane. Pipkin (1968) has integrated equation (4.7) and the result is(4.9)where *A*_{1} is an integration constant. The integration of equation (4.8), for constant pressure *p*=*p*_{0} applied on the membrane, has given(4.10)where *B*_{1} is another integration constant. From figure 3, the axial length *z* and the arc length *ξ* along a meridian, in the deformed configuration, are given by(4.11)(4.12)where *Λ*_{2} is the value of *λ*_{2} at *z*=0. The positive or negative sign in equations (4.11) and (4.12) is chosen for increasing or decreasing *λ*_{2} along *ξ* and *ξ*=0 for *z*=0. Also,(4.13)where *η* is measured from the cross-section of the undeformed body which corresponds to the cross-section *z*=0 of the deformed body.

### (b) Axisymmetric extension and inflation of an initially circular cylindrical tube

The atherosclerotic plaque, shown by the rectangle A′B′C′D′ in the undeformed configuration and by the rectangle ABCD in the deformed configuration in figure 2, is considered as a circular cylindrical tube, which undergoes extension and inflation. The relevant theory for this problem is presented by Green & Adkins (1970). Following Green & Adkins (1970, p. 39), and with reference to figure 2, we consider the deformation defined by the mapping(4.14)(4.15)(4.16)between the current and the reference configurations. *λ* is the constant, along *z*, axial extension ratio of the plaque tube. The principal directions of strain are along the radial *θ*_{1}=*r*, the circumferential *θ*_{2}=*θ* and the axial *θ*_{3}=*z* directions of the deformed configuration. The principal stretch ratios along these directions are(4.17)(4.18)(4.19)and from equations (4.17) to (4.19), owing to incompressibility, we immediately get(4.20)or(4.21)after integrating equation (4.20). *K* is an integration constant. The physical strain components *γ*′_{(ij)} with respect to the undeformed system (*θ*′_{1}, *θ*′_{2}, *θ*′_{3})=(*ρ*, *ϑ*, *η*) are given (see Green & Adkins 1970, p. 60) by the relations(4.22)(4.23)where *i* is not summed in equation (4.22). For incompressible materials for which the condition (4.20) holds, and for which the strain energy density function *W* contains the components *γ*′_{(13)} and *γ*′_{(23)} in powers higher than the first or as products with each other, the stress tensor *τ*^{ij}, measured per unit area of the deformed body and with the stress vector associated with the *θ*_{i} surfaces in the deformed body, is given by the relations(4.24)(4.25)(4.26)We recall that *W*^{(p)} is the strain energy density function for the plaque. From equations (4.22) and (4.23) we conclude that the strain components *γ*′_{(ij)} are independent of *θ* and *z* and depend only on *r* (or *ρ*). Also, from equations (3.16)–(3.20) we conclude that *W*^{(p)} is a function of *r* (or *ρ*) only as well. Hence, equations (4.24) and (4.25) indicate that the stresses *τ*^{ij} are functions of *r* (or *ρ*). Note that the pressure constant that appears in the stress–strain relations for nonlinear incompressible materials, is incorporated in the stress *τ*^{11}.

The equilibrium equations, in the absence of body forces, are expressed via the equations(4.27)where *i* is summed now. The symbol ‖_{i} denotes covariant differentiation with respect to *θ*_{i}. Using the metric tensors *G*_{ij}, *G*^{ij} where(4.28)and(4.29)(4.30)(4.31)and also using equations (4.17)–(4.19) and (4.22)–(4.26), and the incompressibility condition (4.20), Green & Adkins (1970, p. 61) have eventually shown that the equilibrium equations yield(4.32)which after integration gives(4.33)

## 5. The interaction between the balloon, the artery and the plaque

First, the artery alone, pressurized by the blood pressure, is considered in §5a. Next, the interaction between the balloon, the artery and the plaque is analysed in §5b.

### (a) The artery pressurized by the blood pressure alone

Referring to figure 1, we note that the artery is uniformly inflated, i.e. its radius is changed from *ρ*_{0} to *ρ*_{1} all along its axial length *z*. Following Kydoniefs (1969), we write the relation between the axial force *F* applied on the artery at *z*→∞ and the meridional stress resultant *T*_{3} as(5.1)From equations (4.1) and (5.1) we get(5.2)where is the thickness of the artery membrane when zero blood pressure acts on it, i.e. in the undeformed configuration shown in figure 1*a*. Also, *W*^{(a)} is the strain energy density function for the artery wall material. For such a deformation of the artery, with constant radius *ρ*_{1} all along its axial length, we have that(5.3)where *κ*_{3} is the principal curvature along the meridional direction. From equations (4.2), (4.4) and (4.5) we have for *z*→∞(5.4)We can solve equations (5.2) and (5.4) for *λ*_{2} and *λ*_{3}. Inserting these values of *λ*_{2} and *λ*_{3} in the equation of equilibrium (4.9), i.e. in(5.5)we can find the constant for the artery.

### (b) The interaction between the balloon, the artery and the plaque

With reference to figure 2, we consider the interaction of the balloon, the atherosclerotic plaque and the artery. For *z*→∞, i.e. far from the plaque along the axial *z* direction, the deformation of the artery wall is identical with the one described in §5a. Hence for a given blood pressure *p*_{0}, the stretch ratios and at *z*→∞ along the artery can be determined via the solution of the system of equations (5.2) and (5.4). The integration constant can then be determined from the solution of equation (5.5) for *z*→∞. Note that has constant value along the axial length of the artery.

When the balloon, is inflated with a pressure *p*_{b} bigger than the blood pressure, the system balloon–plaque–artery is displaced as shown in figure 2. Using the equilibrium equation (4.33), we can write for the equilibrium of the plaque along the radial *r* direction, the relation(5.6)where *p*_{1} and *p*_{2} are the normal stresses per unit area of the deformed configuration, applied on the outer AB an inner CD side of the plaque cylinder, respectively. *p*_{1} and *p*_{2} are unknown as yet. The resultant vertical force *N* applied on the top part AD of the plaque cylinder is(5.7)In terms of the undeformed inner and outer radii *ρ*_{2} and *ρ*_{1}, respectively, of the plaque and using the incompressibility relations (see equation (4.21))(5.8)(5.9)equation (5.7) yields(5.10)The vertical force *N* is also given in terms of the stress *τ*^{33} via the integral(5.11)or alternatively, by using equations (4.21), (5.8) and (5.9)(5.12)Note that the stress *τ*^{33} on the top edge AD of the plaque cylinder is generally a function of *r*, i.e. it is variable along the radial direction and not constant and equal to *p*_{0}. However, here we use a boundary condition for the axial force *N* in weak form. Note that in equation (5.11), *τ*^{33} is a function of *r*, as it is given by equation (4.25). In equation (5.12), *τ*^{33} is a function of *ρ* different from that in equation (5.11), but obtained from equation (5.11) if equations (5.8) and (5.9) are taken into account. Equating the right-hand sides of equations (5.10) and (5.12), we get(5.13)which is an expression of the stress boundary condition on the edge AD in weak form.

As far as the compressive *p*_{1} applied on the segment AB of the artery is concerned, taking it account equations (4.2) and (4.4), we can write(5.14)where and are the principal stretch ratios at the point A of the artery wall (see figure 2). Since the constant has been determined from equation (5.5), for a given value of we can find as a function of alone, by solving the equation(5.15)for . Recalling that(5.16)and taking equation (5.8) into account, we get(5.17)which means that is a function of *K* and *λ*. From equations (5.14)–(5.17) it is also concluded that *p*_{1} is a function of *K* and *λ* as well.

Regarding the balloon and with reference to figure 2, we have for the pressure *p*_{2} (*p*_{2}<0) acting externally, along the meridian surface CD of the balloon, the relation(5.18)analogous to equation (5.14) for the artery and(5.19)or(5.20)analogous to equations (5.16) and (5.17), respectively. and are the principal stretch ratios at point D, along the meridional and azimuthal directions of the balloon membrane, respectively. *W*^{(b)} is the strain energy density function of the balloon. From equations (5.18)–(5.20) we can see that *p*_{2} can be determined as a function of *K*, *λ* and . From equation (4.9), applied at point D of the balloon membrane, we obtain(5.21)which gives the constant for the balloon membrane in terms of *K*, *λ* and .

At point G of the balloon, the tangent to the balloon meridian is parallel to the *z* axis and thus cos *ω*=1 at that point. Also along the portion DGMH of the balloon, the pressure acted externally upon the balloon is *p*_{0} (*p*_{0}<0), i.e. the blood pressure. Implementing equation (4.10) for the deformation at point G, we get(5.22)which gives *B*_{2} in terms of , , *K* and *λ*. , are again the two principal stretch ratios along the meridional and azimuthal directions at point G of the balloon. But equation (4.9) for point G takes the form(5.23)Equation (5.23) can be solved to give in terms of , *K* and *λ* if equation (5.21) is taken into account. Hence from equations (5.22) and (5.23), *B*_{2} can be found in terms of , , *K* and *λ*. We can now formulate the deformation of the balloon in terms of , , *K* and *λ*.

Recalling equations (4.11) and (4.13) and regarding the axial length between two points along a meridian of the balloon, and since the tips of the balloon are considered fixed (i.e. H′≡H), we can write(5.24)(5.25)Now, for a given inflation pressure *p*_{b} in the balloon, we can solve the system of the four nonlinear equations (5.6), (5.13), (5.24) and (5.25) for the four unknowns , , *K* and *λ*. Note that equations (5.24) and (5.25) are nonlinear integral equations. Then the deformation of the angioplasty model (i.e. of the compound consisting of the balloon), the atherosclerotic plaque and the artery can be fully determined.

## 6. Results

The sources for the data regarding our model were Hart & Shi (1993), Gourisankaran & Sharma (2000) and Plastics Engineering (2000). Regarding figures 8–18, the material constants shown in table 3, i.e. the ones obtained with the parameter *m* of equation (3.31) equal to 0.3, were used. In figure 19, the parameter *m* varied from *m*=0 to *m*=0.6. Numerical results were obtained for a balloon–atherosclerotic plaque–artery configuration, with the following geometrical and load data; the undeformed radius and wall thickness (see figure 1) of the artery alone, i.e. with zero blood pressure (*p*_{0}=0) and zero axial force (*F*=0) acting on the artery at *z*→∞, are *ρ*_{0}=2.5 mm and , respectively, and were found in Hart & Shi (1993). The blood pressure was taken as *p*_{0}=−15 kPa and was found in Hart & Shi (1993) also. The axial force applied on the artery at *z*→∞ was taken as *F*=1.314*N* from Hart & Shi (1993). The radial thickness and the axial length of the plaque, in the undeformed configuration were taken as *t*=ρ_{1}−*ρ*_{2}=1.41 mm and *L*_{p}=5 mm, respectively. The thickness of the balloon was taken as , and that value was found in Plastics Engineering (2000). The axial length of the balloon was taken as *L*_{b}=10 mm.

Results were obtained by solving the system of the four nonlinear equations (5.6), (5.13), (5.24) and (5.25) for the four unknowns , , *K* and *λ*. The routine broydn from Press *et al*. (1994), based on Broyden's method which does not require the analytical evaluation of the Jacobian of the functions to be zeroed, was used for the solution of this system of the four nonlinear equations. The integral in the right-hand side of equation (5.6) was evaluated analytically by using integration by parts and partial fractions.

In figure 8, the variation of the stretch ratios and of the plaque at points D and A, respectively, in the deformed configuration (see figure 2), versus the normalized inflation pressure is shown. Essentially, in this graph, the increase of the radius of the narrowed artery lumen *ρ*_{2} and of the radius of the artery wall *ρ*_{1}, versus the inflation pressure *p*_{b} within the balloon, is shown. The variation is nonlinear but close to linear. An inflation pressure around 7.5 times bigger than the blood pressure is required for an almost full reopening, i.e. for a value of *r*_{2} close to *ρ*_{1} (see figure 2). While the outer meridian A′B′ of the plaque is displaced by around 19% further to the right to AB for an inflation pressure *p*_{b} around 7.5 times the blood pressure *p*_{0}, the inner meridian C′D′ is moved by around 42% to CD. Hence the process of unclogging the narrowed artery gives rise to a radial displacement of the artery wall, which is half the radial displacement of the inner wall of the plaque. As it is evident from figure 8, this is true for intermediate balloon pressures as well.

The variation of the average radial stretch ratio *r*_{1}−*r*_{2}/*ρ*_{1}−*ρ*_{2} in the plaque versus the normalized inflation pressure is depicted in figure 9. The plaque is more compliant (i.e. it is squeezed more easily), at lower balloon pressures than at higher balloon pressures. This observation may be attributed to the nonlinear stress–stretch relation of the plaque material along the radial direction (see figure 7).

The variation of the axial stretch ratio *λ* in the plaque, versus the normalized inflation pressure, is presented in figure 10. For small inflation pressures, i.e. for values of *p*_{b} close to *p*_{0}, the plaque is slightly compressed along the axial direction. For balloon pressures up to 3.5*p*_{0}, the increase of the axial length is more rapid than at higher pressures.

The variation of the normalized pressure and on the inner and outer side of the plaque, respectively, are portrayed in figure 11. For an almost full reopening of the narrowed artery lumen, the pressure *p*_{1} exerted on the artery wall is four times less than the pressure *p*_{2} exerted on the plaque by the balloon. Furthermore, *p*_{1}=1.5*p*_{0}, a value which is comparable to a physiological blood pressure of hypertasic patients. Hence the process of full reopening may be achievable without stressing the artery wall dramatically.

In figure 12, the dependence of the stretch ratios and at point A of the artery, on the balloon pressure *p*_{b}, is presented. While increases, decreases since the pressure *p*_{1} on the part AB increases while the axial force *F* on the artery at *z*→∞ and the blood pressure *p*_{0} on the part AJ of the artery remain constant. Note that for the artery wall is different for for the plaque since the radii for these two constituents of the model in the undeformed configuration are different, i.e. they are *ρ*_{0} and *ρ*_{1}, respectively. Also, for a given value of , is found via the solution of equation (4.9) for _{,} since is already known from the solution of equation (5.5) for *z*→∞.

The variation of the normalized stress resultants and , versus the normalized inflation pressure is shown in figure 13. These stress resultants vary almost linearly, with *p*_{b}. From figures 12 and 13 it is apparent that while decreases, increases. This is due to the action of the lateral pressure *p*_{1} along the portion AB of the artery.

The dependence of the meridional stretch ratio at point D of the balloon, on , is nonlinear as it is evident in figure 14. Close to inflation pressure that causes almost full reopening of the artery, we have , while for smaller pressures we have .

Figure 15 indicates that the meridional stretch ratio at point G in the balloon, is smaller than the azimuthal stretch ratio for all inflation pressures. However, for pressures close to full reopening, these values become almost equal to each other.

The variation of the normalized stress resultants and , versus the normalized inflation pressure is shown in figure 16. The azimuthal stress resultant is generally bigger than its meridional counterpart . However, close to full reopening, the two stress resultants converge to the same value. This means that, at point G in the balloon where the azimuthal stretch ratio *λ*_{2} is maximum, the balloon is stressed equally along the azimuthal and the meridional directions, at nearly full reopening.

The variation of the radial stretch ratios and at points A and G on the artery and the balloon, respectively, versus the normalized inflation pressure is presented in figure 17. Essentially, and measure the thinning of the artery and the balloon wall, respectively. Note that the values of are associated with the initial thickness of the artery, with zero internal pressure and zero axial force at *z*→∞. When the blood pressure *p*_{0}=−15 kPa and the axial force *F*=1.314*N* act on the artery, the artery wall thickness is already about 40%. At almost full reopening, the artery wall thickness is reduced by 12.5% with respect to the wall thickness at blood pressure *p*_{0}. On the other hand, the balloon wall thickness at nearly full reopening is reduced by about 60% of the balloon thickness in the undeformed configuration.

The dimensions (in mm) of the initial configuration of the angioplasty model, for balloon inflation pressure *p*_{b}=0 and the current configuration of the angioplasty model for balloon inflation pressure *p*_{b}=−110 kPa, are presented in figure 18. The narrowed lumen of the artery has been reopened at about 96% (almost full reopening). The artery wall has been displaced by about 19% further to the right, while the inner side of the plaque is displaced by about 42% to the right. Thus the angioplasty method seems to be quite effective since during its process, the narrowed lumen opens more dramatically than the artery wall is displaced.

In figure 19, the variation of the balloon pressure *p*_{b} required for the almost full reopening of the artery, is shown along with the varying parameter *m* of relation (3.31). The balloon pressure required for nearly full reopening seems to be independent from the value of the parameter *m* used in equation (3.31), which relates *λ*_{2} and *λ*_{3} during a uniaxial tension experiment on a plaque rectangular specimen along the circumferential direction.

## 7. Conclusions

A balloon angioplasty model, consisting of the balloon, the atherosclerotic plaque and the artery wall, has been developed and implemented in this paper. Normal radial stress and radial displacement boundary conditions have been imposed along the balloon–plaque and plaque–artery interfaces. The balloon and the artery have been considered as elastic membranes and the plaque has been considered as a circular cylindrical tube under extension and inflation. Nonlinear material properties and large elastic deformations have been accommodated in the model. The balloon material has been considered as isotropic of the Mooney–Rivlin type, the plaque material has been dealt with as transversely isotropic and the artery wall material has been taken as orthotropic. A medium grade balloon and a medium stiff plaque have been used as data for our numerical results. Our main conclusions are:

An inflation pressure around 7.5 times bigger than the blood pressure is required for an almost full reopening, i.e. for a value of

*r*_{2}close to*ρ*_{1}(see figure 2). While the outer meridian A′B′ of the plaque is displaced by around 19% further to the right to AB for an inflation pressure*p*_{b}around 7.5 times the blood pressure*p*_{0}, the inner meridian C′D′ is moved by around 42% to CD. Hence the process of unclogging the narrowed artery gives rise to a radial displacement of the artery wall, which is half the radial displacement of the inner wall of the plaque. This is true for intermediate balloon pressures as well.The plaque is more compliant, i.e. it is squeezed more easily, at lower balloon pressures than at higher balloon pressures. This observation may be attributed to the nonlinear stress–stretch relation of the plaque material.

For small inflation pressures, i.e. for values of

*p*_{b}close to*p*_{0}, the plaque is slightly compressed along the axial direction. For balloon pressure up to 3.5*p*_{0}, the increase of the axial length is more rapid than for higher balloon inflation pressures.For an almost full reopening of the narrowed artery lumen, the pressure

*p*_{1}exerted on the artery wall is four times less than the pressure*p*_{2}exerted on the plaque by the balloon. Furthermore,*p*_{1}=1.5*p*_{0}, a value which is comparable to physiological blood pressure of hypertasic patients. Hence the process of full reopening may be achievable without stressing the artery wall dramatically.The meridional stretch ratio at point G in the balloon is smaller than the azimuthal stretch ratio for all inflation pressures. However, for pressures close to full reopening, these values become almost equal to each other.

At almost full reopening, the artery wall thickness has been reduced by 12.5% with respect to the wall thickness at blood pressure

*p*_{0}. On the other hand, the balloon wall thickness at almost full reopening is reduced by about 60% of the balloon thickness in the undeformed configuration.The balloon pressure at almost full reopening seems to be independent from the value of the parameter

*m*used in equation (3.31), which relates*λ*_{2}and*λ*_{3}during a uniaxial tension experiment on a plaque rectangular specimen along the circumferential direction.

## 8. Discussion—limitations

Obviously, the angioplasty model that is analysed in this paper, has several limitations. First, the narrowed artery lumen has been considered as circular and concentric with the artery wall, and subsequently, the atherosclerotic plaque annulus has the same thickness all around its circumference. Clearly, such a configuration is only a particular one. Atherosclerotic plaques can also be eccentric and circular and eccentric and noncircular (see Oh *et al*. 1994). It is also possible that calcified plaques may not extend all along the artery wall circumference (see Hayashi & Imai 1997). Further, even if a plaque having the shape of a circular cylindrical tube is considered, with the profile A′B′C′D′ of figure 2*a*, it is not certain that after the deformation the plaque will keep the rectangular profile ABCD of figure 2*b*. For instance, the length of the straight line segment AB may not be equal to the length of the straight line segment CD. Also, in some cases the plaque may be in partial contact with the artery wall, e.g. when a pre-operative dissection has been done or when the plaque is torn apart from the artery wall during angioplasty (see Medynsky *et al*. 1998).

The experimental data of Loree *et al*. (1994), concerning uniaxial tension of rectangular plaque specimens along the circumferential direction, do not include any measurements of the change of the lateral width and thickness of the specimens. Since the plaque material is transversely isotropic with the axis of anisotropy parallel to the radial direction, the change in length of the lateral width and thickness of the specimens during the tensile experiment along the circumferential direction cannot be determined. Hence we have assigned arbitrary discrete values to *λ*_{3} corresponding to the measured discrete values of *λ*_{2} during the uniaxial tension experiment. The corresponding *λ*_{1} values could easily be obtained from the incompressibility condition. However, since during this experiment *λ*_{2}>1, we have taken care that *λ*_{3}<1 and *λ*_{1}<1. In this way we have made sure that during uniaxial tension along the circumferential direction of the plaque, we have lateral contractions along the axial and radial directions, which is a physically acceptable situation for a nonlinear, elastic and incompressible material.

Medynsky *et al*. (1998) and Gourisankaran & Sharma (2000) have separately reported balloon dilation pressures from 300 to 450 kPa during balloon angioplasty. In this paper we have found that a dilation balloon pressure of 110 kPa is adequate for an almost full reopening of the narrowed artery lumen. However, the plaque elastic properties may be different in these two papers than in our paper. Also, the plaque in these papers does not extend all around the artery circumference, while our paper, the plaque does extend all around the artery circumference. In this study, we have used radial normal stress and radial displacement boundary conditions between the plaque, the artery and the balloon, while Gourisankaran & Sharma (2000) have used pure bond boundary conditions in their plain strain angioplasty model.

## Acknowledgements

We would like to thank Dr. A. Tsoukas, of the Department of Medicine of the University of Athens for his valuable advice regarding the model of balloon angioplasty that we have used in this paper.

## Footnotes

- Received August 21, 2003.
- Accepted October 4, 2004.

- © 2005 The Royal Society