This research was directed toward quantitatively characterizing the effects of arterial mechanical treatment procedures on the stress and strain energy states of the artery wall. Finite element simulations of percutaneous transluminal angioplasty (PTA) and orbital atherectomy (OA) were performed on arterial lesion models with various extents and types of plaque. Stress fields in the artery were calculated and strain energy density was used as an explicit description of potential damage to the artery. The research also included numerical simulations of changes in arterial compliance due to orbital atherectomy. The angioplasty simulations show that the damage energy fields in the media and adventitia are predominant in regions of the lesion that are not protected by a layer of calcification. In addition, it was observed that softening the plaque components leads to a lower peak stress and therefore lesser damage energy in the media and adventitia under the action of a semicompliant balloon. Orbital atherectomy simulations revealed that the major portion of strain energy dissipated is concentrated in the plaque components in contact with the spinning tool. The damage and peak stress fields in the media and adventitia components of the vessel were significantly less. This observation suggests less mechanically induced trauma during a localized procedure like orbital atherectomy. Artery compliance was calculated pre- and post-treatment and an increase was observed after the orbital atherectomy procedure. The localized plaque disruption produced in atherectomy suggests that the undesirable stress states in angioplasty can be mitigated by a combination of procedures such as atherectomy followed by angioplasty.

## Introduction

The intent of the research described here was to provide detailed characterization of the mechanical effects produced in arterial disease treatments. This will increase understanding of the procedures and their effects on artery mechanical properties and so provide extended bases for treatment system design, development, and use.

Atherosclerosis is a build-up of plaque in the arterial wall. Growth of plaque may lead to narrowing of the arterial lumen and coronary heart disease (CHD). Approximately, one in every six deaths in the U.S. in 2010 was caused by CHD [1]. Atherosclerosis of peripheral arteries (arteries that supply blood to the legs, arms, stomach, or kidneys) causes peripheral arterial disease, which affects about 8 million people in the U.S. and in severe cases may necessitate limb amputation.

PTA is commonly referred to as balloon angioplasty (BA). It is one of the most frequently used procedures to reduce plaque burden and increase lumen size in lesions with atherosclerotic plaque build-up. In PTA, a catheter is passed on a guide wire to the site of lesion to be treated and a cylindrical balloon is expanded using internal pressure. The balloon is held inflated for 30–120 s, deflated, and retracted from the treatment site. PTA is sometimes accompanied by stenting.

The primary factor determining PTA success rates is restenosis (reocclusion by plaque growth) of the treated vessel. Restenosis has been associated with excessive repair reaction of the artery due to mechanical damage produced during the treatment procedure. Mechanical damage has been associated with overstretching of the arterial wall and denudation (removal of surface layer cells) of the endothelium due to vessel–balloon interactions [2]. While restenosis rates associated with stenting are lower than those observed in the traditional PTA procedure, restenosis is a persistent complication [3]. Stents with a coating of a drug to control the vessel's response to injury have been developed. Although drug eluting stents can reduce restenosis, increased rates of late stent thrombosis were found in patients with acute myocardial infarction treated with drug-eluting stents [4].

The likelihood of post-treatment biological complications, suggest that plaque removal rather than lesion modification procedures, could be a viable option. Several atherectomy, plaque removal, systems have been developed. Mechanical atherectomy systems use a tool rotating in the artery to deform and remove plaque. The tool may be an abrasive burr or a single or multiple edge cutter. The tool is rotated by a drive wire and advanced and retracted through the lesion region a number of times. The removed material is either collected in the atherectomy system or, if the particles are small enough, allowed to remain in the blood circulatory system for filtering and removal. Two abrasive-type atherectomy devices are distinct with one having a symmetric circular cross section burr around the drive wire. The other, orbital atherectomy (OA), uses eccentric burr or crown to produce a relatively slower speed orbital motion around the rapidly spinning drive wire. Due to the localization of deformation in atherectomy, the possibility of overstretching the artery and the effects of applying internal pressure over the entire diseased and nondiseased lumen periphery are minimized. There is no direct control of tool motion and tool–artery contact. In addition to material removal complications and uncertainties due to no direct control of the process, distal embolization is the major disadvantage of atherectomy systems [5].

In order to effectively design and operate mechanical atherosclerosis treatment devices, the effects of the process on the artery and on postoperative developments must be understood. In the research described here, the stress and strain energy density states in the artery during angioplasty and orbital atherectomy were quantitatively characterized. Even for the established angioplasty procedure, open questions remain as will be described below. Many process-related mechanical effects are not well understood for the relatively new atherectomy processes. In addition, changes in compliance of lesions due to interventional procedures have been rarely considered in the literature. Mechanical procedure-induced artery compliance changes were included in the research described below.

## Problems Considered and Approach

The goals of this research were to quantitatively characterize the mechanical effects in percutaneous transluminal angioplasty and orbital atherectomy treatment systems to foster better understanding of the processes and provide firm bases for device and process design.

Specifically, the objectives of the work presented here were

• to characterize the stresses, strain energy density, and changes in material properties and behavior in the arterial wall with a variable plaque morphology during the balloon angioplasty procedure

• to describe the stress and strain energy density states produced by orbital atherectomy in lesions with calcified plaque

• to compare the mechanical states that arise over the entire lumen periphery in angioplasty to the mechanical state in the local tool–artery contact region in atherectomy

• to observe and describe the change in compliance that has been reported only in passing in previous research

Numerical techniques were used to study the full range of mechanical actions and interactions in the complicated loading of the complex artery material system. The finite element analysis method provided a means for process simulation but extension of available solution procedures and extensive artery material modeling were needed. These significant aspects of the mechanical action and complex material system were developed and used in a commercial finite element program package. Related experimental work by our collaborators (Nguyen et al. [6]) was conducted to investigate the change in lesion compliance during OA procedures, and some results are used here to validate computational results. A displacement-based approach for the crown dynamics during OA was used in this research based on the study by our collaborators, Zheng et al. [7]. Fluid dynamic effects were outside the scope of this new research aimed at describing basic solid mechanic aspects of arterial and plaque deformation. However, the experimental work of Zheng et al. [7] did include fluids and some of the fluid environment effects are indirectly included in the finite element model.

Plasticity and permanent set effects were accounted for in some of the simulations. However, effects of plaque removal in the form of extremely small size particulates through the abrasive process in orbital atherectomy [8] and tearing/fracture in the tissue were not included in this research. Tissue material property data were selected from various sources to formulate material models since to the writers' knowledge, no experimental results for cyclic loading of plaque tissues across tension and compression regimes are available in the accessible literature. Atherosclerotic plaque in the simulations was assumed to be isotropic, and realistic nonisotropic material models were used to describe the artery structure. Biological tissues are known to exhibit time and deformation rate-dependent behaviors. Such viscoelastic behavior was not taken into account in this work. Also, possible residual stresses and axial prestretch of the vessel were not included in this study. Frictionless interactions were used to model the contact between the PTA balloon and the OA tool and lesion due to the presence of a fluid in the contact region and the absence of established friction coefficients.

## Background

### Material Models of the Human Artery.

For the research reported here, artery material behavior models that describe large deformation and material structure and property changes were needed. Single layer, linear elastic, isotropic, or orthotropic material models as traditionally developed are unable to capture the response of the tissue at supra-physiological loading as imposed in mechanical treatment procedures. Further, the individual layers of the artery have experimentally been demonstrated to have separate responses that are highly nonlinear and directional, e.g., Holzapfel et al. [9]. Such considerations have led to the development and use of more complex material models.

Holzapfel et al. [9] performed cyclic, quasi-static, tensile tests on 13 human left anterior coronary arteries. Specimens from the circumferential and the axial direction were used for material characterization, and ultimate tensile stresses of individual arterial layers were determined. The experimental results were used to fit parameters to the strain energy-based Holzapfel–Gasser–Ogden (HGO) material model, which includes dependence of the strain energy density function (SEDF) on the orientation of fibers in the artery wall. Lally et al. [10] performed uniaxial and biaxial loading tests on porcine aorta and human femoral arteries. The resulting response of the tissues was fit to a five parameter Mooney–Rivlin hyperelastic model. This isotropic material model was used to describe stent–artery interaction.

In explicitly studying damage to arteries, Balzani et al. [14] included effects on the arterial fiber component since the assumption was that damage occurs due to rupture of cross-bridges between collagenous microfibrils. In further work, Balzani et al. [15] compared different damage functions with smooth and nonsmooth stress–strain tangent moduli models and pointed out the numerical difficulties in using these models. Maher et al. [16] performed cyclic compression tests on human atherosclerotic plaques and fit a damage evolution function to describe softening of the material. Weisbecker et al. [13] fit a pseudo-elastic damage model to cyclic loading results from experiments on human arterial tissue. A damage function was applied to the collagenous and matrix components. It was noted that damage in the matrix components is negligible in comparison to the collagenous material. Calvo et al. [17] presented a large strain damage model for fibrous tissue. The authors state that nonphysiological loads drive the soft tissue damage that arises from the tearing or plasticity of fibers or biochemical degradation of the extracellular matrix.

### Material Models of Plaque.

Loree et al. [18] performed tensile tests of plaques from human abdominal and thoracic aorta using specimens in the artery circumferential direction. Cellular, hypo-cellular, and calcified plaques were the three distinct classifications made, and a circumferential tangent modulus of elasticity was determined for each. In a one eighth symmetry finite element model of the vessel cross section, Pericevic et al. [19] used the material data from Loree et al. [18]. A third-order polynomial strain energy density function was fit to each of the three different plaque types. It was concluded that the calcified plaque could play a protective role by reducing the stresses in the arterial wall.

Maher et al. [20] measured the compressive and tensile response of fresh human carotid plaques and classified them as echolucent (soft), mixed, and calcified. A general second-order polynomial strain energy density function was used to fit parameters to the material response in tension and in compression but the tensile and compression models were not linked. Lawlor et al. [21] performed uniaxial tensile tests on fresh carotid plaques excised from 14 patients. The plaques were classified as hard, mixed, and soft. A Yeoh material strain energy density function was used to fit the material responses. Maher et al. [16] studied the inelasticity of human carotid plaques through cyclic loading of radial compressive specimens. A constitutive model was proposed that incorporated the inelastic effects of softening and permanent set on echolucent, mixed, and calcified plaques. Softening curves and a constitutive model were developed using measured material parameters. These curves were used in this work to describe the plaque's behavior in the compressive regime.

In review of artery material modeling for mechanical artery treatments, two major observations stand out. One is that testing and modeling have been carried out for tensile and compression loading separately. There is no description of continuous compression to tensile loading as is expected in atherectomy and to a lesser extent in angioplasty. The other observation is that arterial damage is typically characterized as a softening of the material due to cyclic loading.

### Computational Studies of Balloon Angioplasty.

Undesirable outcomes of angioplasty have been reported such as “higher inflation pressures and larger balloon sizes may also cause greater neointimal hyperplasia” [2224] and “acute luminal stretching during angioplasty is shown to be an accurate predictor of later luminal loss” [2528].

Laroche et al. [29] introduced the first folded balloon model, which has been commonly used in computation studies of angioplasty. The study included mathematical formulations of the initial and expanded configurations of the balloon. The model was adopted by several other researchers. For example, Martin and Boyle [2] and Conway et al. [30] studied the effects of a semicompliant angioplasty balloon on the deployment of stents. It was concluded that the balloon configuration has a substantial influence on the transient response of the stent and on its impact on the mechanical state of the coronary artery.

Lumen enlargement by angioplasty has been shown to be primarily due to the stretching of the artery [3133]. The plaque structure in the case of collagenous caps did not change during radial compression, whereas the nondiseased portion of the wall was stretched. Holzapfel et al. [34] conducted a finite element analysis (FEA) study of balloon angioplasty and stenting of human iliac arteries. The artery layers were distinguished as diseased and nondiseased media, adventitia, and nondiseased intima with a collagenous cap, lipid pool fibrotic plaque, and calcification. The geometric model was based on a magnetic resonance imaging scan. The lesion layers were modeled as HGO type material with fiber orientations and included plasticity. The balloon and stent expansions were simulated using displacement boundary conditions. Plane strain, isotropic material simulations with an axial prestretch reference condition and without axial prestretch were performed and compared. It was noted that major simplifications of material models could lead to significant errors in stress states and the need for more realistic models was emphasized.

With the motivation being to investigate the deformation, stress evolution and overstretch of the vessel due to a balloon, Gasser and Holzapfel [34] performed a finite element analysis of arterial response to balloon angioplasty. The artery was modeled as adventitia, media, and plaque with HGO type behavior and with fiber distribution and inelastic effects included in the media. Fibrous components of the plaque were neglected and the plaque was modeled as a rigid calcified layer. The relevant observations from the study were that the primary mechanical effect of angioplasty is overstretching the nondiseased arterial wall and that the dominating stresses in the media and adventitia were in the circumferential and axial directions. It was suggested that the balloon-induced mechanical changes in the arterial wall might be responsible for the development of smooth muscle cell proliferation, neointimal hyperplasia, and refractory restenosis.

Conway et al. [36] used the finite element method to study the effects of different types of stent designs on overall lumen size gain including tissue damage risk for individual layers of arteries. The artery wall was modeled as equally thick intima, media, and adventitia layers of HGO-type material. A tissue damage risk parameter was introduced as the percentage of elements experiencing stress levels above the ultimate tensile strength of the layer. It was concluded that increasing the percentage occlusion in the lesion increased the tissue damage risk value.

Gasser and Holzapfel [35] reported that the vascular injury seen during angioplasty was especially pronounced in the media, whereas inflation tests on the adventitial layer showed that the adventitia behaves nearly elastically with negligible material damage. Also, it was noted that balloon-induced wall overstretch seems to be mainly due to the structural changes in the medial layer. Therefore, the peak stresses induced in the artery, especially in the media, may be an important measure to predict vessel damage and injury.

Reviewing previous research on characterizing mechanical aspects of angioplasty shows the usefulness of hyperelastic material models for describing the behavior of a three-layer artery. Results show variations of the stress state over the internal artery circumference and among the wall layers. Artery damage is usually attributed to overstretch but little has been done to quantitatively describe the extent of damage over the arterial section.

### Previous Studies of Orbital Atherectomy.

There have been a very limited number of experimental studies of orbital atherectomy and apparently no computational studies. The experimental and clinical work was focused on patient outcomes and more recently, comparative assessments of orbital atherectomy when accompanied by balloon angioplasty.

Safian et al. [32] conducted a study on 124 patients who suffered from chronic infrapopliteal arterial occlusive disease (occlusion of leg arteries by plaque). The study concluded that OA provides a predictable and safe lumen enlargement option for infrapopliteal disease. Short-term data indicated infrequent need for revascularization or amputation.

Shammas et al. [33] studied results of OA and subsequent BA versus standalone BA in patients with critical limb ischemia. Fifty patients were enrolled in the study of which 25 were treated with only BA and 25 with OA followed by BA. In the OA + BA category, the procedural success was found to be 93.1% as opposed to 82.4% for BA alone. It was noted that, since OA + BA needed lower balloon pressures and resulted in fewer dissections as compared to standalone BA; inclusion of OA in treatment procedures, increases the chances of a desirable outcome. Dattilo et al. [37] conducted a similar study to test the hypothesis that OA improves lesion compliance. It was seen that the mean maximum balloon pressures in OA + BA was 4 atm as compared to 9.1 atm in BA alone. The study concluded that OA + BA yields better luminal gain by improving lesion compliance.

To the authors' knowledge, no analytical or numerical studies quantitatively describing the vessel structural response to the OA procedure are available in the research literature.

In summary, there are some experimental results showing that atherectomy alone and atherectomy prior to angioplasty may be favorable with respect to disrupting and removing plaque and increasing artery compliance. Detailed, quantitative descriptions of the mechanical effects of atherectomy or combined atherectomy–angioplasty procedures are not available.

## Development of Tissue Behavior Models for Process Simulations

### Material Property Changes, Damage.

The starting point in developing the material model used in the angioplasty and atherectomy simulations is the material stress softening behavior that has been observed in cyclic loading of arterial specimens. This type of mechanical softening behavior is often characterized by the Mullin's Effect as shown in Fig. 1(a) in which λ is the stretch ratio.

In the Mullin's effect model described by Ogden and Roxburgh [38], the energy dissipation aspect of material behavior was described by including a damage function, $ϕ$, that depends on the damage parameter, η, in the strain energy density, ψ
(1)

in which the λs are the principal stretch ratios and $ψ¯$ is the underlying strain energy density function of the material without damage. The damage variable $η$ is governed by the constant damage parameters, $r,m$, and $β$ as seen in Eq. (3). The term $ψm$ in Eq. (3) is the maximum value of strain energy density seen throughout the loading history of the material. On the primary (first) loading cycle, the value of $ψm$ is zero and therefore the value of $η$ is unity. This renders the damage function $ϕ(η)=0$, (Eq. (4)), that is, no energy has been dissipated in damaging the material yet. As the material is unloaded and reloaded back to $ψm$, the value of the damage variable $η$ changes from 0 to 1 through Eq. (3), and describes the softening in the material. On further loading, the damage variable $η$ stays constant at 1 until unloading is initiated. The parameter accounts for the extent of damage in the material with respect to the virgin state, $β≥0$ is a scaling parameter for the damage, and $m≥0$ gives the dependence of the damage on the extent of deformation. At low values of $m$, a significant amount of damage occurs at low levels of strain. In contrast, for large values of $m$, the damage is lesser at low strain values, but the material response is significantly altered within the small strain region of the next cycle of loading. Depending on the type of material response, the parameters , and $m$ are used for curve fitting [38].

The applications under consideration in this study are arterial deformations under internal loading, which is primarily biaxial loading, since the stresses in the circumferential and axial directions are significantly higher than those in the radial direction. Therefore, under the action of bi-axial loading, the strain energy density model used here is
(2)
where the damage variable $η$ for bi-axial loading is given by
$η=1−1rerf[ψm−ψ¯(λ1,λ2)m+βψm]$
(3)
where $ψm$ is the maximum value of the strain energy attained throughout the loading history. The corresponding form of damage function $ϕ(η)$ as seen in Ref. [38] is chosen such that
$−ϕ′(η)=m erf−1(r(η−1))+ψm$
(4)
On complete unloading, the material strain energy has a residual value known as the damage energy
$ψ(1,1,ηm)=ηmψ¯(1,1)+ϕ(ηm)=0+ϕ(ηm)=ϕ(ηm)$
(5)

where $ηm$ is the value of $η$ at the maximum stress encountered during the loading cycle of the material. The damage energy may be interpreted as a measure of the softening that takes place in the material. The more the material is stressed and unloaded, the greater the damage energy dissipated in the material. During the cyclic loading in atherectomy, damage energy is accumulated.

### Range of Deformation.

As summarized above, previous material models of arteries and plaques used either the tensile or compressive test data available. With the rotational and longitudinal motions of the tool in atherectomy, a given region of the artery is subjected to sequential compression and tensile deformation. In angioplasty, the plaque is subjected primarily to compressive deformation but there may be local regions with tensile stresses, e.g., at the plaque–artery interface. Any differences in material behavior with tensile or compressive deformation must be included in simulation material models.

### Material Models of Plaque.

One of the crucial parts of this work is modeling the plaque components of the lesion. Therefore, the material model for various types of plaque as constructed by Maher et al. [16] was investigated. The study by Maher et al. [16] involved performing cyclic compression tests on plaque samples of calcified, echolucent, and mixed type. The material models and parameter definitions as introduced by Maher et al. [16] are in the Appendix. The average values for the different plaque samples in Ref. [16] are listed in Table 1 and were used in this study, to construct the normal stress–normal strain curves as shown in Fig. 2.

A third-order Ogden primary hyperelastic model with Mullin's effect was fit to the plots in Fig. 2 and gave the Mullin's damage parameters, r, m, and β. Different material parameters were obtained for different plaque types. The resulting normal stress-normal strain material behaviors for calcified and echolucent plaques are shown in Fig. 3.

In order to simplify the data fitting of Mullin's damage parameters r, m, and β, permanent set was not included in the models. However, to test its effect during atherectomy, permanent set behavior was included directly in the simulation discussed in Sec. 6.3.2.

The data and models used to combine compression and tensile plaque material behavior models for soft and calcified plaques were chosen after investigating several possibilities in the literature. The selected data resulted in a continuous transition from compressive to tensile stress regions, and satisfied the Drucker stability criterion [39] of positive energy dissipation. For soft plaque, a stress–strain model was developed from the average data presented in Lawlor et al. [21], who conducted uniaxial tensile loading tests on fresh human carotid plaques and fit the data to the standard Yeoh strain energy density function
(6)

The material parameters , and $C30$ were reported for the different plaque samples along with average values for the three plaque types (hard, soft, and mixed plaques). The plots for the three different plaque types are shown in Fig. 4.

The combined compression–tension plot for echolucent plaque generated from Refs. [16] and [21] is shown in Fig. 5(a).

For calcified plaque in the tensile region, the results from Maher et al. [20] (sample 11) were used. As stated earlier, the selection of data was based on whether it provided an accurate, continuous fit at the origin along with satisfying the Drucker stability criterion. In the study by Maher et al. [20], a second-order polynomial hyperelastic strain energy density function was fit to tensile test results on carotid plaques. The plaque response for calcified plaque in tension and compression is shown in Fig. 5(b).

The combined compression–tension third-order Ogden function models with Mullin's effect, for the two kinds of plaque were extended over strain ranges useful for angioplasty and atherectomy simulations and are shown if Figs. 6 and 7.

### Material Models of Arterial Wall.

In contrast to the compressive and tensile strains produced in the plaque in atherectomy and to a lesser extent in angioplasty, in physiological loading the artery wall is subjected almost exclusively to tensile deformation. For realistic material modeling, the fibrous layer in the arterial wall has to be included in an artery material model since large deformations are expected.

The experimental results of Weisbecker et al. [13] were used to develop a Holzapfel–Gasser–Ogden (HGO) hyperelastic material model with Mullin's effect. The Cauchy stress–stretch curves were used to determine the Mullins effect parameters r, m, and β. The strain energy density of the HGO type as used in Ref. [13] is given by
$ψ=ψmatrix+ψfibers$
(7)

$ψmatrix=μ2(I1−3)$
(8)

$ψfibers=k12k2[ek2(I*i−1)2−1], Ii*=κI1+(1−3κ)Ii;for i=4,6$
(9)

where are material properties and $Ii$ are corresponding invariants of the right Cauchy Green deformation tensor. An important characteristic of the HGO model is the presence of an orientation effect that allows for including the fibrous component of the arterial structure into material behavior models. To completely characterize the HGO model used in simulations, finite element analyses with incompressible and compressible (almost incompressible) material models of tensile specimens were performed with fiber direction vectors assigned to elements. Results from the analyses were compared with the experimental results of Weisbecker et al. [13] and the final material model was formulated. The final arterial material models and the correspondence of them to experimental data are shown in Fig. 8. The ratio of the difference between the measured response and that predicted by the model with respect to the measured response reaches a maximum deviation of 6.1%.

The Mullin's effect parameters for the different types of plaques and the layers of the artery are reported in Table 2.

## Percutaneous Transluminal Angioplasty Simulations

Angioplasty process simulations were used to characterize the effects of the extent of the calcified region on the variation of stress and strain energy density around the internal surface of the artery.

### Geometry, Mesh, and Mesh Properties.

The vessel geometry used in this study was a slightly simplified version of a class of diseased arteries. The basic physical structure considered was a 3 cm long segment of a superficial femoral artery (SFA) with a cross section such as that in the histology image in Fig. 9. A sector of calcification can be seen in a layer of soft plaque which is fused with the intima.

For process simulations, the artery wall was modeled as media and adventitia surrounded by a layer of tissue to mimic the external surrounding tissue for the vessel. As shown in Fig. 10(a), the plaque was modeled as a sector of calcified plaque embedded in a layer of echolucent plaque fused with the intima. The vessel geometry was modeled to be constant over a length of 3 cm. The plaque layers extend over the central portion of 2 cm length. An idealized meshed geometry was created with dimensions derived as average values from the SFA. The effects of varying degrees of plaque burden were determined.

The artery was modeled as anisotropic HGO type, with fiber distribution and Mullins effect as described above. A brick-type hexahedral element with reduced-order integration and enhanced hourglass and distortion control was used. Mass proportional damping was included in the model. The muscle tissue surrounding the artery of 0.1 mm thickness and linear elastic material behavior with 50 kPa elastic modulus and 0.3 Poison ratio was modeled with shell elements with reduced order integration. Taking advantage of symmetry, half the length of the model was used in process simulations. The total element count was 229,048 elements, which was based on a mesh convergence study.

### Balloon Model.

The angioplasty process simulation was performed using a tri-folded balloon described by Laroche et al. [29] with an expanded diameter of 5.5 mm which is the diameter of the vessel. The tri-folded balloon model used in the simulations is shown in Fig. 11.

To describe the balloon inflation process, the geometry of the folded balloon surface was constructed by mapping the points of the deployed balloon of radius c in Fig. 11(c), onto the folded state shown in Fig. 11(a) described by inner radius a and outer radius b. Figure 11(b) represents an intermediate shape of a freely expanding balloon. The thickness of the balloon was 0.02 mm and its initial internal and external radii were 0.75 mm and 1 mm, respectively. Balloon expansion in the simulations was described by the radial and angular positions of balloon elements.

The finite element mesh for the balloon was composed of four node membrane elements with reduced integration. The elastic modulus and Poisson ratio of the balloon were 920 MPa and 0.4. The balloon length was 30 mm with a constant folded cross section for the majority of the length (26 mm). The loading conditions on the balloon have been mentioned in Sec. 5.3. A surface blend was created at the ends of the balloon (2 mm on each end) to an unfolded radial ring of 1 mm radius which was kept fixed to emulate the attachment to the catheter. The surface blend was meshed with a combination of triangular and quadrilateral membrane elements.

The artery was assumed to have a constant length during the process since its length in vivo is much greater the balloon length. Therefore, the distal and proximal ends of the artery model were fixed in the axial direction and free in the radial and circumferential directions. The model was simulated using a central plane of symmetry across the lesion. Appropriate symmetry boundary conditions were applied on the face. The balloon was given an internal pressure load of 8 atm, which lies within the typical pressure range for angioplasty of calcified lesions. The angioplasty procedure was simulated in the two steps of balloon unfolding followed by radial expansion to full internal pressure. The balloon was held inflated for 5 s since the peak stresses in the lesion were seen to plateau by this time. Due to the presence of a fluid environment in the vessel and lack of data regarding the friction coefficient, the interaction of the balloon with the internal wall of the vessel was assumed to be frictionless.

### Results of Angioplasty Simulations.

The effects of plaque extent on artery deformation are described here for plaques over 90 deg, 180 deg, and 270 deg of the artery internal periphery. The damage energy represents the amount of strain energy dissipated per unit volume in the element.

#### Results for 90 deg Case.

The von Mises stress and damage energy density are shown over the artery–plaque cross section in Fig. 12.

As can be observed in Fig. 12(a) while there is a high stress region in the calcified plaque, the largest stress in the media is in the noncalcified region of the lesion. The higher the stresses induced, the higher the damage energy dissipation since it is associated with the peak stresses developed in the material. Thus, the damage energy density shown in Fig. 12(b) is greatest on the noncalcified side of the artery in the media and adventitia. These findings are emphasized and the stress and damage energy effects shown clearly in the longitudinal sections in Fig. 13.

In the media and adventitia, the high stress regions extend around the artery and decrease as the calcified region is approached, Fig. 12(a), and are minimum behind the calcified region, Figs. 13(a) and 13(c). The peak von Mises stress in the media is much larger than in the adventitia, 303 MPa compared to 84 MPa in this particular case. In the calcified plaque, there is a central region of high stress shown in Fig. 13(c). The soft plaque component of the artery structure is subjected to high stresses but with much less variation than in the media.

Of interest with respect to treatment effects is that the region of the medial and adventitial tissue located behind the calcified layer experiences lower stress than other regions of the artery. It is in a sense protected by the calcified material. This implies that the possibility of vascular injury may be more pronounced in the noncalcified portion of the vessel wall. This result is consistent with the observations of Casserly [40], who mentioned that in the case of calcified lesions, high balloon pressures often result in increasing the risk of dissections to the compliant noncalcified portion of the media.

#### Results for 180 deg and 270 deg Cases.

Cross-sectional views of the von Mises stress for calcifications subtending 180 deg and 270 deg are shown in Fig. 14. A similar trend as seen in the 90 deg case is observed for the 180 and 270 deg cases. It is observed once again, that the section of the media and adventitia which have calcifications face low stresses, therefore exposing the opposite side to higher localized stresses.

The von Mises stresses in the media and adventitia with plaques extending over 90 deg, 180 deg, and 270 deg are shown in Fig. 15. Since the damage energy depends on the peak stress, the qualitative behavior of the strain energy density distribution will be similar to the von Mises stress.

As the region of calcified plaque increases, the high stress region in the media away from the plaque becomes more localized and the level of stress increases as shown in Figs. 15(a), 15(c), and 15(e). Accepting that the plaque offers some stress shielding to the media behind it, this effect increases with increasing plaque coverage Another observation is that the same kind of increasing high stress region localization occurs in the soft plaque stress state with increasing plaque coverage, Figs. 15(b), 15(d), and 15(f). The stresses observed in the soft plaque are significant; however, this layer has been shown to have an average ultimate tensile strength value of 231.5 kPa [5] that is very much larger than the stress acting. To further investigate the stress states, simulations with perfect plasticity included were carried out (detailed results not included here and are in Ref. [41]). The results from the plasticity simulations show similar trends for the media and adventitia to the ones reported above. Plasticity and damage simulations were also carried out for the original SFA geometry shown in Fig. 10(a). The results are described in Ref. [41].

#### Effect of Plaque Hardness.

It is expected that the magnitude of the stress in the media will depend on plaque stiffness or hardness, and hence affect the extent of changes in artery structure and behavior.

Angioplasty simulations were carried out with artery models having plaque components with stiffness of 50% and 20% of the baseline stiffness used in the previous simulations. The von Mises stress in the media for the three plaque hardness cases for 180 deg plaque extent is shown in Fig. 16.

As the plaque stiffness decreases, the high stress region becomes more diffuse with the maximum stress decreasing to approximately 85% of the initial value for the 50% stiffness case and to 60% for the 20% stiffness case. This demonstrates the value of softening the plaque components and this concept is pursued in the atherectomy process simulations.

### Summary.

The effects of angioplasty, and other mechanical processes, on modifying artery properties can be quantitatively described using the dissipated strain energy or damage energy density, which is expected to change the structure and behavior of the artery postprocedure.

The primary results of the angioplasty process simulations are summarized in Table 3.

The general results show that the major part of the strain energy, 60–70%, is dissipated in the calcified region of the arterial structure. As the extent of the calcified region around the artery internal surface increases from 90 deg to 270 deg, the amount of damage energy increases. A much smaller damage energy content is seen in the echolucent or soft plaque. A still smaller energy dissipation occurs in the media. But these smaller energy density levels do not imply a lack of concern for this effect on the artery wall. While disruption of the plaque may be beneficial, media layer damage may subsequently affect the extent and rate of restenosis.

In considering these results, two thoughts arise regarding arterial treatment processes. The first being that plaque disruption and mechanical effects in the media may change the overall artery compliance and this possibility should be examined. A second thought is that locally imposed deformation concentrated at the plaque region can be desirable. If relatively larger local deformation is imposed on the plaque compared to little or no deformation of the media, the process effectiveness may be increased. These issues were considered in the part of the research described below. Simulations of orbital atherectomy were aimed at characterizing effects of local, repeated deformation of lesions.

## Orbital Atherectomy Simulations

The initial aim of the research was to characterize process local effects on plaque disruption. In the course of the work, significant changes in artery compliance were seen. This previously overlooked effect was also quantitatively described.

### Process, Artery Geometry, Mesh, and Material Model.

The tool or crown model and process parameters were based on experimental observations of OA operation in a phantom artery by our collaborators, Zheng et al. [7]. The model 1.5 mm crown is shown in Fig. 17(a) and shell type rigid elements were used in the model. The artery model used in atherectomy simulations was the same as the artery model in the angioplasty simulations with respect to material layers, artery and muscle tissues, and element types. No symmetry approximation was made since for the crown dynamics no plane of symmetry exists. The artery models had 2 cm long diseased, calcified, regions centered along 3 cm long arteries as shown in Fig. 17(b). The cross section of the artery is shown in Fig. 17(c) and cases with the calcified region extending over 90 deg and 180 deg of the section were considered. The mesh was based on results of a mesh convergence analysis and was composed of 775,710 elements.

In process modeling, an initial crown radial displacement of 0.5 mm into the artery wall was imposed and held fixed and then the crown was given an orbiting frequency of 40 Hz. The crown was moved axially over the 2 cm long calcified section of the vessel at a rate of 1 cm/s. One complete forward and return pass was simulated. The interaction of the crown and the lesion was modeled as frictionless with kinematic contact control. The 3 cm long vessel was constrained at both the ends in the axial direction to emulate a semi-infinite vessel and in the circumferential direction to describe surrounding tissue and to negate rigid body rotation.

### Atherectomy Simulations: Results

#### Arterial Stresses and Effect of Varying Degree of Calcification.

The von Mises equivalent stress under the crown in the calcified region for the 90 deg calcification case at processing time of 2 s over the longitudinal and cross sections of the artery are displayed in Fig. 18. For the relatively long crown, this location includes both overall and end effects.

As expected, the high stress regions are localized near the crown–artery contact region. Of particular interest with regard to mechanical effects of the procedure is that the highest stress regions are in the calcification and soft plaque regions, and do not extend into the arterial wall. For this 90 deg calcification case, the maximum stress in the echolucent or soft plaque is 757 kPa and 197 kPa in the calcified plaque with the stress decreasing rapidly to maximum of 39 kPa in the media and 15 kPa in the adventitia.

Since the damage energy density is directly related to peak stress, this implies localization of material changes in the plaque and very little effect in the arterial wall. Several conclusions arise. In contrast to the widespread high stress fields in angioplasty, atherectomy effects are localized in the area requiring treatment, the plaque. With this localization of the high stress area, arterial disruption is small and so fewer undesirable complications are expected. Further, increases in artery compliance with atherectomy mentioned in a few previous works can be explained by local breakdown of the plaque.

Since the effects of the crown–plaque interaction are local with respect to crown position, similar stress fields were seen for the 180 deg plaque coverage case. The maximum stresses calculated were 802 kPa in the soft plaque, 282 kPa in the calcified plaque, 29 kPa in the media, and 17 kPa in the adventitia. In contrast to the small changes in stress in the other structure components, the large increase in calcified plaque stress from 197 kPa for the 90 deg calcification to 282 kPa in the 180 deg case suggests an increase in constraint imposed by surrounding stiff calcified material on the local crown–plaque deformation region.

#### Inclusion of Permanent Set in Material Model.

In addition to the strain softening observed in experimental characterizations of artery materials in cyclic loading, a permanent strain or set on unloading often was seen. Depending on the size of this effect, it may need to be included in describing the repeated loading imposed in orbital atherectomy. The strain softening Mullins effect model can be extended to include permanent set, and this was done in process simulations. A typical result is shown in Fig. 19 for a 90 deg calcification case.

The general characteristics of the stress fields are similar to a material model with no permanent set, Fig. 18, but the stress levels are different. They are higher for the material model with permanent set, for example 51 kPa versus 39 kPa in the media.

While the use of a more inclusive material model does have clear effects on the level of stress, what may be most relevant are the similar patterns of stress and energy dissipation for material models that do not include permanent set components. Noteworthy is the localization of high stress regions in the plaque and so very little arterial wall disruption is expected.

### Damage Energy Analysis.

The stresses in simulations are directly related to strain energy density, or damage energy density, and so the energy density fields are similar to the stress patterns shown above. The mean damage energy density in the various components of the arterial structure was calculated. The results are in Table 4.

The strain energy density results show that the major portion of energy dissipation, 50–70% is in the soft, or echolucent, plaque followed by calcified plaque. This is due to the very localized mechanical effects of the orbital atherectomy crown. The plaque regions result in little stress transfer to the media and adventitia and so very small amounts of energy are dissipated in these components of the artery–plaque structure. When the material models include permanent set effects, stress in the crown–soft plaque contact area is increased as shown above in Fig. 18 and this raises the stress level in the calcified plaque. Again, given the very rapid decrease in stress away from the crown, only very small damage energy values are seen in the media and adventitia.

### Change in Artery Compliance.

In overview, the high stress levels and concomitant energy dissipation in the plaque region of the artery can lead to degradation of this relatively stiff material and result in increased lesion compliance. To describe this aspect of the OA procedure more fully, changes in artery lumen area with internal pressurization were determined in simulations on artery models in an initial condition and after OA.

Two initial state models with 90 deg plaque arcs were used. One artery model was the same as used for angioplasty and atherectomy simulation and is shown in Fig. 10. The other initial condition artery model was the final state artery from the OA simulation. Each of these models was loaded by internal pressures of 40 mmHg, 80 mmHg, 120 mmHg, and 160 mmHg and the lumen areas calculated. Results are shown in Fig. 20. Compliance was defined as the change in area with change in pressure and determined from plots of area versus pressure such as those in Fig. 20.

Using the slopes of the lines in Fig. 20, the compliance of the initial artery is (13.7 − 9.5) mm2/120 mmHg = 0.035 mm2/mmHg and after OA it is (17.4 − 10.1) mm2/120 mmHg = 0.061 mm2/mmHg. The change in compliance with OA for this lesion is (0.061 − 0.035)/0.035 = 75%.

Experimental measurements of compliance change with OA reported by Nguyen et al. [6] also show an increase in compliance. Given the uncertainties of the variation of actual arterial mechanical properties, lesion cross section shapes and unknown degrees of calcification at specific artery cross sections and along the artery length, the numerical degree of change in compliance is expected to vary. However, the conclusion that emerges is that OA may increase artery compliance and as in other research results, the extent of the effect will vary greatly with the individual situation.

### Summary.

Simulations of orbital atherectomy showed that the very localized crown–artery contact region resulted in high stresses in the contact region but that the stress level decreased rapidly with increasing distance into the arterial structure. An implication is that plaque disruption in atherectomy will produce only small amounts of arterial wall damage. In addition, orbital atherectomy can increase artery compliance.

## Summary of Key Findings

Stress and strain energy density fields in balloon angioplasty and orbital atherectomy processes were simulated in artery models with various degrees and types of plaque. The research also investigated the change in compliance after orbital atherectomy.

In OA, the artery is subjected to a variation from compressive to tensile loading. There is very little information available about arterial material properties and behavior over a continuous variation from compressive to tensile loading regimes. Material models were constructed from data accumulated from tensile and compressive experiments from various sources and a continuous compression to tensile material behavior model developed. Appropriate material models were used for angioplasty and atherectomy simulations.

Angioplasty simulations showed that the mechanical loading over the entire lumen periphery produced high stress and larger damage energy density regions in the noncalcified region of the lesion. The calcification can be seen to serve as a protective layer for the arterial wall situated behind it. This result was consistently observed with the different degrees of calcification that were simulated.

In OA simulations, the crown–artery contact produces localized deformation in the soft and calcified plaque regions. There is little stress progression from the plaque into the arterial wall. This suggests that OA is a highly localized process that results in change in the plaque material properties through Mullin's effect, without excessive mechanical trauma to the artery.

One of the localized deformation effects shown in OA simulations is plaque disruption which can result in increased artery compliance.

## Acknowledgment

This research was funded by the National Science Foundation, under Award No. 1232655. Additional support was provided by Cardiovascular Systems Inc. St Paul, MN. The authors acknowledge the Minnesota Supercomputing Institute (MSI) at the University of Minnesota for providing resources that contributed to the research results reported in this paper.

### Appendix

The material model, as described in Ref. [16], was given as
$σ=(1−D)(σIL−σIN)$
(A1)

$σIL=2J−1F∂ψIL∂CFT and$
(A2)

$σIN=2J*−1F*∂N∂C*F*T$
(A3)
are the deformation gradient tensor, the Jacobian determinant, and the right Cauchy Green deformation tensor, respectively. are the values of the deformation gradient tensor, the Jacobian determinant, and the right Cauchy Green deformation tensor, respectively, calculated at the maximum loading during the deformation history. The functions $ψIL$ and $N$ were defined as the elastic and inelastic strain energy densities, and $σIL$ and $σIN$ are the corresponding Cauchy stresses. The elastic strain energy density function (SEDF) $ψIL$ was reported as a function of parameters $a$ (kPa) and $b$, and is given as follows:
(A4)
The inelastic SEDF $N$ was only a function of the parameter $c*(kPa)$ as follows:
(A5)
where the invariant $I1*$ is updated only on unloading from the maximum deformation state. The scalar parameter $D(ζ∞,a,b)$ seen in Eq. (A1), known as a softening parameter, is a function of the maximum value attained by the strain energy density during the deformation of the material and is given as
(A6)

where $ζ∞$ and $i$ are material parameters as seen in Table 1.

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