Computational models provide a powerful tool for pre -clinical assessment of medical devices and early evaluation of potential risks to the patient in terms of plaque fragmentation and in -stent restenosis (ISR). Using a suitable constitutive model for arterial tissue is key for the development of a reliable computational model. Although some inelastic pheno mena such as stress softening and permanent deformation likely occur due to the supra -physiological loading of arterial tissue during the stenting procedure, hyperelastic constitutive models have been employed in most of the previously developed computatio nal models. This study presents a finite element model for stent deployment into a patient -specific stenosed artery while inelastic arterial behaviors due to supra -physiological loading of the tissue have been considered. Specifically, the maximum stress i n the plaque and the arterial layers which is the main cause of plaque fracture during stent deployment and the surgically -induced injury (damage) in the arterial wall, as the main cause of ISR, are presented. The results are compared with the commonly -used hyperelastic behavior for arterial layers. Furthermore, the effects of arterial material parameter variation, analogues to different patients , are investigated. A higher amount of damage is predicted for the artery which shows a higher stress in a specif ic strain. Keywords : Stent, artery, inelastic constitutive model, damage, patient -specific, finite element . 1. Introduction Atherosclerosis is one of the most common types of cardiovascular disease in which fibrous and fatty materials, called plaque, build up inside the artery and cause a partial or total occlusion of the artery. One way to restore physiological blood flow to an occluded artery involves the deformation of plaque using intravascular balloon angioplasty with or without stenting. This clinical intervention is accompanied with supra -physiological loading of the arterial wall causing tissue damage (injury) (see (Holzapfel and Gasser, 2007) for more details), and that may lea d to tissue in -growth and a re -blockage, termed in -stent restenosis (ISR) (Boyle and Prendergast, 2011 ; Fereidoonnezhad et al. , 2017). Computational modeling of the stenting procedure provides a powerful tool for an early evaluation of the potential risk of a specific sten t for the patient in terms of (i) vessel wall damage, as the main cause of ISR, and (ii) plaque fragmentation which can cause a downstream occlusion of tiny blood vessels. Although the literature concerning the modeling of the stent deployment into a sten osed artery is rich in terms of stent design and deformation (Gu et al, 2012; Wang et al. , 2012 ; Liu et al. , 2020 ), material of the stent and its constitutive model (Liu et al. , 2019 ), the majority of the studies employed a rather simple cylindrical shape for the artery (Rivlin and A, 2006; Cui et al. , 2010 , Kai et al. , 2020 ). Some studies focused * Corresponding authors: baghani@ut.ac.ir , behrooz.fereidoonnezhad@nuigalway.ie 2 on more realistic geometries of the stenosed artery (Holzapfel, 2005; Voß et al. , 2019) and investigated the effects of stent insertion or blood flow on the artery . However, these studies employed hyperelastic constitutive models for the arterial tissues which are not enough for supra -physiological loading that occurs during stent deployment (Gu, Zhao and Froemming , 2012; Mozafari et al. , 2018; Ruan et al. , 2018) . In supra-physiological loading inelastic phenomena such as stress softening and permanent deformation may occur in arterial tissues (Maher et al. , 2012; Fereidoonnezhad, Naghdabadi and Holzapfel, 2016) . Although some approaches aim to model the inelastic phenomena able to investigate the injury in arte rial tissu es (Holzapfel and Fereidoonnezhad, 2017) , several inelastic models are still not considered in the computational modeling of stent deployment in a pati ent-specific lesion morphology. It is then critical that the constitutive models for arterial tissue account for these inelastic phenomena so that computational models can properly predict the outcome of the stenting procedure. Therefore, the goal of the c urrent study is to incorporate the aforementioned inelastic phenomena into the computational modeling of the stenting procedure in a patient specific -artery, which have not been considered in previous research studies. To accomplish that, we develop a fini te element (FE) model for the expansion of a stent into a patient -specific stenotic artery by accounting for the damage -induced inelasticity in the tissues based on the constitutive model proposed in et al. , 2016) and compared the results of the inelastic model to the commonly used hyperelastic model . It is noted that the word 'damage' in this work points to the changes in the microstructure of the tissue such as breaking of the bonds between collagen fibers which consequently results in some inelastic phenomena such as stress softening and permanent deformation in the tissue scale. The stress distribution in the plaque and the arterial tissue, which may be used for the assessment of plaque rupture risk in future studies, and the induced damage (injury) in the arterial wall, as the main cause of ISR, are presented. Moreover, to investigate the effect of the stent design on the in duced injury in the arterial wall we perform the simulations for two different stents geometry. We also examine the effect of arterial properties variation on the final outcome of the procedure. The paper is structured as follow: In Section 2 the geometry and mechanical behavior of the stent and the artery are introduced, and the simulation of the process of the analysis, i.e., boundary conditions, meshing and loading are described in detail. In Section 3 the related results are p resented, and the effects of different parameters on the results are discussed. Finally, we draw some conclusions in Section 4. 2. Materials and methods A finite element simulation of the stent expansion into a (patient -specific) stenotic artery requires the geometry and material properties of the stent and artery, the stent -artery interaction, the boundary and loading conditions that are described in the following. 2.1. Material model s Proper constitutive models for the plaque materials and the arterial tissues are requi red for a reliable computational model of the stenting procedure. In this section, the constitutive models, employed in computational modelling of the procedure, are discussed. 2.1.1 . Artery and plaque In this study we use two different constitutive models for the arterial layers and the results are compared. The first m odel is the hyperelastic model (Gasser, Ogden and Holzapfel, 2006) , the so -called HGO model, which extends the constitutiv e framework for arterial walls (Holzapfel , Gasser and Ogden 2000) , by considering distributed collagen fiber orientations. The seco nd model is the inelastic model 3 (Fereidoonnezhad, Naghdabadi and Holzapfel, 2016) , which considers the stress softening and permanent deformation in arterial tissues. The latter is also able to evaluate the damage in arter ial tissue after stent deployment. It is also noted that in all simulations the plaque are assumed to be a hyperelastic material. Both models are briefly reviewed in the following. The HGO model The anisotropic h yperelastic constitutive model (Gasser, Ogden and Holzapfel, 2006) was initially developed to describe the elastic properties of arterial tissue, but now is used extensively for modeling a variety of soft fibrous tissues, and it has been implemented in several commercially available FE progr ams to simulate soft tissue elasticity. In the following a brief review of this model is provided. The kinematics is described in terms of the deformation gradient F . Hyperelastic constitutive models often split the local deformati on into volumetric and isochoric parts. Accordingly a multiplicative decomposition of the deformation gradient 1 3J=FF is partly considered, where detJ= F is the volume ratio, and F is the modif ied deformation gradient with det 1=F . Consequently, T=C F F denotes the right Cauchy -Green tensor , and T=C F F is the modified right Ca uchy - Green tensor . Arterial tissues can be regarded as materials that consist of an isotropic matrix material within which (at least) two families of fibers are embedded (Schriefl et al. , 2012) . Here, the arterial layers in the initial configuration are characterized by two collagen fiber families, symmetrically distributed by an angle  with respect to the circumferential direction of the artery. Thus, the arterial laye rs respond orthotropically, and the families of collagen fibers are characterized by the two direction vectors 0, 4,6ii= a with 01i=a . The following invariants, required for the model, can then be defined. 1I tr=C , 4 04 04 . I=a Ca , 6 06 06 . I=a Ca (1) 2/3 11I J I−= , 2/3 44I J I−= , 2/3 66I J I−= (2) The strain -energy function  is then given in this form ()()()()04 06 04 06, , , ,vol iso fJ  = + +C a a C C a a (3) Where vol is the volumetric, iso the isochoric, isotropic and f the fiber contribution. Recent investigations demonstrate that using deviatoric invariants for fiber -reinforced materials may lead to some non-physical results when assuming some degree of compressibility for tissue (Nolan et al. , 2014; Gültekin, Dal and Holzapfel, 2018) . To avoid th ese non-physical results, henceforth, we use the right Cauchy - Green tensor C (instead of C ) in the last term of eq. (3), as suggested in (Nolan et al. , 2014) . Thus, the HGO strain -energy function is reformulated as ()()()()04 06 1 , 1 4,6, , ,vol iso f i i iJ I I I = = + +   C a a (4) where ()21ln22volKJJJ− = −  , ()() 11 3isoII  = − (5) and ()(),02 1 12 2, exp 12fi iikI I k Ek  = − , ()11 3 1iiE I I= + − − (6) 4 in which K is the initial bulk modulus ; , 1k, and 2k are material parameters , and  0,1 3 denotes the dispersion parameter. We enforced the material incompressibility by using a high value of 𝐾/𝜇. For further details on the material compressibility the reader is refer to the recent work by Mo erman et al., (2020 ). The inelastic model s Fereidoonnezhad et al. (Fereidoonnezhad, Naghdabadi and Holzapfel, 2016) proposed a constitutive model which accounts for the damage -induced inelastic phenomena, t he stress softening and permanent deformation, in arterial tissue under supra -physiological loading conditions which occur during balloon angioplasty. The strain -energy function  is presented as (Fereidoonnezhad, Naghdabadi and Holzapfel, 2016) ( )()04 06 ,4 ,6, , , , ,s s in vol J     = +C a a (7) () ()()()()() ,* 1 , 1 , 4,6,1 fi iso s i i i s i in in in iI I I N      =  = +  + − − +  C (8) where ,, 4,6sii= and in are damage variables introduced to capture the stress softening and permanent deformation, respectively. The functions i and in are dissipation functions and describe the inelastic energy dissipation. Moreover, the inelastic energy N is presented as (Fereidoonnezhad, Naghdabadi and Holzapfel, 2016) ()() ()* * * * * *2 1 12 * 4,6 23 exp 12i ikN I k Ek = = − + − C (9) ()* * * * * 11 3 1iiE I I= + − − (10) Where * , * 1k, * 2k , and * are material parameters describing the anisotropic damage in collagen fibers. The damage variables are defined as (Fereidoonnezhad, Naghdabadi and Holzapfel, 2016) ()max 0 , , , 11111s i f i f i erfrm= −  −  (11) ()2 0 maxtanh / tanh 1m in= (12) in which 1r , 1m and 2m are material parameters. max ,fi and max are maximum strain energy of the fiber family i and tissue, respectively, which obtained by the deformation history while ** 1I tr=C , ** 4 04 04 . I=a C a , and ** 6 06 06 . I=a C a . Here *C is a function of the modified right Cauchy -Green tensor at the peak deformation of the loading history. The damage induced in each fiber family can be characterized by the minimum value min ,si of the damage variable according to (Fereidoonnezhad, Naghdabadi and Holzapfel, 2016) ()min max ,, 11111s i f i erfrm= −   (13) 5 Stress -strain relationships are readily derived from (1) and (2) above, as described by Fereidoonnezhad et al., (Fereidoonnezhad , Naghdabadi and Holzapfel, 2016 ). ()0 0 , 1 , 4,62 2 2 1fi vol m s i in id NJdJ− =   = + + − −    SCC C C (14) This constitutive model is implicitly discretized and implemented in a user material subroutine (UMAT) by Fereidoonnezhad et al. (Fereidoonnezhad, Naghdabadi and Holzapfel, 2016 ) to be used in ABAQUS/Standar d. This subroutine has been used for all simulat ions in this work. Material parameters Different human tissues have different material properties. In Table 1 elastic and inelastic parameters for the three layers (intima, media, and adventitia) of two arteries and plaques are reported. The material parameters of the arteries are taken from (Fereidoonnezhad, Naghdabadi and Holzapfel, 2016) . Material parameters of atherosclerotic plaque are obtained by calibrating the constitutive model (equation (14)) to the e xperimental data of sample I documented in (Holzapfel, Sommer and Regitnig, 2004) , using a least- square optimization algorithm in MATLAB. It is noted that the plaque comprises of different components and an average mechanical response of all constituents are considered here, as shown in Figure 1. Table 1: Elastic and in elastic propertie s of two human arterial layers (Fereidoonnezhad, Naghdabadi and Holzapfel, 2016) , and elastic properties of plaque according to the e xperimental d ata documented in (Holzapfel, Sommer and Regitnig, 2004) .  (MPa) 1k (MPa) 2k   * (MPa) * 1k (MPa) * 2k * 1r 1m (MPa) 2m Artery I Intima 0.017 0.263 41.191 0.3 90 0.001292 0.014591 0 0 1.49 0.0192 3 Media 0.012 0.059 3.871 0.02 45.3 0.000098 0.000611 0.665 0.29 1.44 0.0077 3.32 Adventitia 0.008 0.222 9.948 0.23 53.1 0.001206 0.000001 0.011 0.33 4 0.001 4 Artery II Intima 0.003 1.073 41.7 0.33 69 1.18 0.0064 0.0003 0.27 1.43 0.0101 2.61 Media 0.02 0.078 19.4 0.32 0.9 1.26 0.00177 0.017 0 2.83 0.0084 1.87 Adventitia 0.008 0.212 4.63 0.28 34.4 1.45 0.001239 0.734 0.33 3.72 0.047 1.5 Plaque 0.0075 3.5806 12 0.225 58 - - - - - - - 6 Figure 1: Conformability of the model to the uniaxial experimental data of atherosclerotic plaque (Holzapfel, Sommer and Regitnig, 2004) . 2.1.2. Stent Numerous balloon expandable stents are made from SS316LN stainles s steel. Different values of material parameters are reported for SS316LN in the literature, which are in close agreement. Here, the mate rial parameters are taken from (Auricchio et al. , 2001) , where plastic de forma tion was considered (see Table 2). Table 2: Material parameters for the stent Young modulus (MPa) 196000 Poisson ratio 0.3 Yield stress (MPa) 205 Limit stress (MPa) 515 Limit nominal strain 60% 2.2. Model geometry Clinical studies have identified factors such as the stent design and the deployment technique that are one cause for the success or failure of a stent treatment. In addition, the success rate may also depend on the stenosis type. Hence , for a particular stenotic artery, the optimal intervention can be identified by studying the influence of factors such as stent type, strut thickness and geometry of the stent cell (Holzapfel, 2005) . In the present study we investigate the individual h uman stenosis, as presented in (Holzapfel, 2005) . Therein, the geometrical model of the considered stenosis was composed of different tissue components, which were specified from high resolution magnetic resonance images and reconstructed by means of non - uniform rational B -splines. As ill ustrated in Figure 2, a high degree of non -symmetric stenosis is observed in the artery, in contrary to the symmetric assumption used in the majority of studies , also the dimensions of the stent, and the pattern w hich has been used for making the stent geometry in the CAD software Autodesk Inventor, are shown in Figure 3. 7 Figure 2: The geometry of the stenosis artery. The tissue components are adventiti a , media , int ima and plaque . Figure 3: stent geometry . All dimensions are in mm . The inner and outer diameter of the stent are 2mm and 2.1mm, respectively and the strut width is 0. 1mm, and t he initial lengths of the stent was 31mm. 2.3. Loading, boundary condition s, and meshing The stent expansion is performed by imposition of a radial displacement to the stent. More sophisticated methods such as using an inflatable balloon has also been previously employed for this purpose (Cui et al., 2010; De Beule et al. 2008). The radial disp lacement is applied on the inner surface of the stent struts to expand the stent from the crimped state, with the inner diameter of 2mm, to the fully -expanded state of 5.4mm diameter. Finally, the load is removed and the elastic recoil of stent is occurred . The general contact in ABAQUS together with a frictionless tangential behavior are employed to simulate the contact between the outer surface of the stent and the inner surface of the plaque and the artery. Both ends of the artery are constrained in the longitudinal and circumferential directions, while a radial displacement is allowed. The stent is constrained in the longitudinal direction at one end; therefore, the stent length can change in the expansion process, and both ends of the stent are constrai ned in the circumferential direction. Three -dimensional (3D) eight node linear, hybrid elements (C3D8H) are assigned to the mesh of the artery and the plaque and 3D eight node linear elements (C3D8) are employed for the stent. To check the mesh convergence, the analysis is performed with a ref ined mesh for the artery I. The results are shown in Table 4.45 1.04 0.1 0.28 8 3, which indicates the mesh independent result. After a mesh study, the total number of 4032 and 11825 elements have been used for the stent and the stenosed artery, respectively. Table 3: The result of mesh analysis, which shows Max. principal stress and Max. principal strain distribution in simulations with different number of elements for the artery. NO. of elements (artery+stent) Max. principal stress (MPa) Max. Principal strain Max principal stress of the stent (MPa) 14487 0.33 0.41 363 15857 0.32 0.40 362.3 18602 0.32 0.39 362.1 3. Results and discussio n In this section, the results are presented and the implications of stenting on the arterial tissue is discussed. The cross sectional area of stenosed artery and the distribution of maximum principal stress after stent deployment is demonstr ated in Figure 4. The inelastic constitutive model was used for arterial tissue. The results clearly show the lumen gain after stent deployment. Figure 4: The distribution of the max. principal stress in arterial tissue a) before and b) after stent expansion , using the inelastic constitutive model. As mentioned in the previous section, the diameter of the stainless steel stent decrease s duo to its recoils. In the Figure 5 the effect of this recoil is shown. After the stent recoil, the stress of the artery will decline, but the important point is that its stress -strain diagram , which is presented in Figure 6, is not compatible on the stress -strain diagram in the first step, which is the expansion step. This diagram is plotted for the point with the maximum principal stress in the intima layer. Max. principal stress (MPa) b a 9 Figure 5: Max. principal stress of the artery and plaque after the first (in the left) and second step (in the right) , which are the expansion and recoil of the stent, respectively. Figure 6: The variation of s tress versus s train in the artery during expansion and recoil steps . Given this result, in the next section results of the inelastic model and HGO model are compared. We examine how constitutive behavior of arterial tissue affect the stress distribution in arterial tissue and show the importance of considering damage on the correct prediction of the outcomes. In the next step, the effects of variation in arterial mater ial parameters, analogous to different patients, on the final outcomes o f the procedure are investigated. 3.1 Effects of damage -induced arterial Using the same stent geometry, two constitutive models i.e., the hyperelastic model and the inelastic model, as discussed in Section 2, are considered for the arterial wall. As depicted in Figure 7, the distributions of the maximum principal stress during expansion of the stent are generally similar for both the hyperelastic and inelastic models. However, as shown in Figure 7 the maximum principal stresses are different for hyperelastic and inelasti c models after stent recoil . In the following, we do not show the plaque since we consider the inelastic model only for arterial layer, which are intima, media, and adventitia. Max. principal stress (MPa) Max. principal stress (MPa) 10 Max. principal stress (MPa) Max. principal stress (MPa) Max. principal strain Max. principal strain Max. principal strain Max. principal stress (MPa) 11 Figure 7: Distribution of maximum principal stress after (a) fully expansion of stent and (b) elastic recoil, and maximum principal strain after (c) fully expansion of stent, (d) elastic recoil of the stent. Left figures show the results for the inelastic model and the results of the hyperelastic model are shown in the right column. Figure 8: Maximum principal stress versus radial displacement for the hyperelastic and inelastic models during stent expansion and elastic recoil (This diagram is plotted for the point with the maximum principal stress in the intima layer) . It is noted that although the global stress distribution is not affected significantly by ignoring the inelastic behavior of arterial tissue, the hyperelastic models, as used in most of the previous computational models, are not able to predict the surgically -induced damage after procedure, contrary to the inelastic model considered in the present study. The importance of this behavior is clear when one experience s more than one stenting, as a result, using the inelastic model help s us to predict and analyze this proc edure accurately . 3.2 Effects of arterial properties To investigate the effect of variation in material properties of arterial tissue on the final outcomes of the procedure, the simulation is conducted for two sets of arterial parameters belonging to two different patients (see table 1). The distribution of the maximum principal stress, maximum principal strain and radial displacement of the artery are presented in Figure 9. Max. principal stress (MPa) 12 Figure 9: Distribution of (a) maximum principal stress, (b) maximum principal strain and (c) radial displacement of artery with parameters set I ( left) and parameters set II ( right). In view of Figure 9, one may also observe that the location of the maximum stress are not coincident with the location of maximum strain and displacement. This is strongly dependent on the non-symmetric geometry of the stenosed artery and heterogeneous properties of plaque and artery. It is noteworthy to mention that this attribute could not be predicted where a symmetric geometry is assumed for the artery. In Figure 10 , the maximum principa l stress and damage evolution (equation (13)) versus maximum principal strain in the arterial tissue are demonstrated for two inelastic parameter sets , at the point with highest value of maximum principal stress . A higher amount of damage is predicted for artery I which also shows a higher stress in a specific strain compared to artery II . Max. principal strain Max. principal strain Radial Disp. (mm) Radial Disp. (mm) 13 Figure 10: a) Damage evolution and b) stress -strain curve in arterial tissue for two different sets of inelastic properties. 4. Concluding Remarks Balloon angioplasty with stent deployment in one of the most common treatments for atherosclerosis. Despite the significant benefits of this procedure, there are some short -term and long -term issues for the patients; such as plaque fragmentation and ISR. C omputational models as a powerful tool for pre -clinical assessment of medical devices such as stent have gained a lot of momentum in the last years. However, most of those models are based on the simplified (idealized) geometry of stenosed artery. On the o ther hand, those models which consider the patient -specific geometry of stenosed artery have employed a hyperelastic constitutive model for arterial tissue, which is not enough for supra -physiological loading during clinic al treatments, as discussed in (Fereidoonnezhad, Naghdabadi and Holzapfel, 2016) . Knowing that both patient -specific geometry of stenosed artery and suitable constitutive modeling of arterial tiss ue are critical for developing a precise computational model, a finite element model for stent deployment into a patient -specific stenosed artery was investigated in this paper, while inelastic arterial behavior due to the supra -physiological loading has b een considered. Specially, the surgically induced injury (damage) in arterial wall, as the main cause of ISR, has been presented and the results are compared with the commonly -used hyperelastic behavior of arterial layers. Furthermore, the effects o f arter ial properties variation and analogues to different patients are investigated by doing the simulations for two different parameter sets. In fact, the developed FE model provides a powerful tool and may serve as a ‘virtual workbench’ to investigate the ste nting procedure computationally, allowing early evaluation of potential risks to the patient in terms of vessel wall damage, as the main cause of ISR (Fereidoonnezhad et al., 2017) , for a specific stent. The developed computational model is clinically relevant in two ways: (i) the proposed framework can be used in future studies to optimize stent design, e.g., to minimize tissue damage while lumen gain is maximum; (ii) the proposed framework could enable t he prediction of surgically -induced damage evolution allowing a better prediction of the long -term adaptive behavior of an artery such as mechanically -induced growth which is the main cause of in -stent restenosis. (a) (b) 14 Acknowledgement The authors would like to acknowledge Professor Gerhard A. Holzapfel for his valuable discussion and comments. We also wish to thank Dr. Michael Stadler for providing the geometric al data of the stenotic artery.