A study of balloon type, system constraint and artery constitutive model used in finite element simulation of stent deployment
© Schiavone and Zhao; licensee Springer. 2015
Received: 17 September 2014
Accepted: 18 December 2014
Published: 5 May 2015
Finite element is an effective tool to simulate stent expansion inside stenotic arteries, which provides an insightful understanding of the biomechanical behaviour of the whole stent-artery system during the procedure. The choice of balloon type, system constraint and artery constitutive model plays an important role in finite element simulation of stent deployment.
Commercial finite element package ABAQUS was used to model the expansion of Xience stent inside a diseased artery with 40% stenosis. The arterial wall, consisting of intima, media and adventitia layers, and the stenotic plaque were described by different hyperelastic models. Both folded and rubber balloons were considered and inflated with a linearly increasing pressure of 1.4 MPa. Simulations were also carried out by considering free, partially and fully constrained arteries.
Folded balloon produces sustained stent expansion under a lower pressure when compared to rubber balloon, leading to increased stress level and enhanced final expansion for the system. Fully constrained artery reduces the stent expansion when compared to free and partially constrained arteries, due to the increased recoiling effect. Stress in the artery-plaque system has higher magnitude for stent expansion in a free artery due to more severe stretch. Calcified plaque limits stent expansion considerably when compared to hypocellular plaque. The negligence of the second stretch invariant in the strain energy potential leads to the disappearance of saturation behaviour during stent expansion. The use of anisotropic artery model reduces the system expansion at peak pressure when compared to the isotropic model, but with an increased final diameter due to reduced recoiling effect. The stress distribution in the artery-plaque system is also different for different combinations of artery and plaque constitutive models.
Folded balloon should be used in the simulation of stent deployment, with the artery partially constrained using spring elements with a proper stiffness constant. The blood vessel should be modelled as a three-layer structure using a hyperelastic potential that considers both the first and second stretch invariants as well as the anisotropy. The composition of the plaque also has to be considered due to its major effect on stent deployment.
Coronary stents are essentially scaffolds, made of metallic alloys or biopolymers, used to sustain the blood vessels once expanded inside the obstructed arteries. Stents are generally deployed inside the diseased artery by means of an angioplasty balloon (except for self-expandable stents). The scaffold is placed over the balloon and expands with the balloon when this is inflated by internal pressure. This surgery procedure has minimal invasive nature and provides fast and effective solutions to patients suffering from coronary stenosis, a major cause of heart attack.
Finite element is an effective tool to simulate the process of stent expansion inside stenotic arteries, which helps to understand the insight of the biomechanical behaviour of the whole stent-artery system during the procedure. The simulations provide essential information regarding the behaviour of stent expansion, recoiling, dogboning and residual stresses, which can be further utilised to guide stent design and surgery procedures (Chua et al., 2003; Lally et al., 2005; Pericevic et al., 2009; Zhao et al., 2012a; Morlacchi et al., 2013). One factor that has important influence in the simulation of stent expansion is the type of balloon used in the modelling. Rubber balloons, inflated by internal pressure, were generally used to simulate expansion of early generation stents (e.g. Palmaz-Schatz). Modelling of a rubber balloon is relatively easy, and produces reliable results including both stent expansion and stresses in the stent-artery system (Chua et al., 2003; Ju et al., 2008; Schiavone et al., 2014). While De Beule et al. (2008) assessed the importance of balloon folding in the expansion behaviour of a Cypher stent and proved that the folded balloon was the only model that produced results consistent with the data provided by the manufacturer, in terms of diameter change as a function of pressure. This is particularly the case for the recent generation of stents (e.g. Cypher, Cordis) which are designed to be expanded by folded balloons. A study by Gervaso et al. (2008) compared three different methods in modelling stent expansion, i.e., no balloon (pressure applied on the stent inner surface), free rubber cylinder inside the stent and folded balloon. The work showed that the folded balloon gave the expansion behaviour of Cordis BX-Velocity stent closer to the company data. Martin and Boyle (2013) also showed that considering the tri-folded shape of the balloon led to an expansion curve of Cordis BX-Velocity stent closer to the manufacturer data. Although the importance of balloon type has been recognised in recent studies, a full comparison of stent deployment using a rubber and a folded balloon has not been carried out yet, especially the essential expansion behaviour and the stress distributions for the whole stent-artery system.
In stent deployment simulation, different boundary conditions have been proposed to constrain the rigid body motion of artery. For instance, Gijsen et al. (2008) performed a simulation of stent deployment in a realistic coronary artery, which was obtained by three-dimensional reconstruction of a mildly stenosed coronary artery. In this simulation, the artery was fully constrained at the ends to avoid translations and rotations. Results showed the arterial walls were successfully enlarged by the stent, with peak stress values observed on the thinner portions of the artery wall. Zahedmanesh et al. (2010) simulated the stent expansion in a realistic artery using folded-balloon geometry. The artery was constrained at its ends using connector elements with a stiffness of 1kN/m. Expansion of stent inside the artery was simulated by either inflating the balloon or applying internal pressure directly on the stent. Application of direct pressure to the inner stent surface was found to be incapable of accurately predicting stress-strain field and the deformed configuration of both the stent and artery. However, application of pressure with restraining elements, which prevent expansion of the stent beyond the desired diameter, may be used as a computationally efficient method to accurately predict the stress-strain field in the vessel wall following full stent expansion and recoil. Currently, there is a lack of investigation of the effects of different artery constraints on stent deployment simulations. Most of existing work tended to fix both ends of the artery in order to remove the rigid body motion completely, which might impose an over constraint to the system and affect the simulation results. The use of connector elements seems to be a plausible method to overcome the issue of over constraint, but still needs to be assessed against other types of constraints (e.g. full constraint and free of constraint) generally used in modelling of stent deployment.
The artery constitutive model is another important factor that needs to be carefully thought about for simulation of stent deployment. Holzapfel and Ogden (2010) reviewed the mathematical modelling of the mechanical behaviour of the human artery wall. This review showed the state of the art of development of constitutive models in the last ten years to describe the mechanical behaviour of artery tissue. It highlighted the highly nonlinear and anisotropic behaviour of the blood vessel tissue. This work also suggested that in many cases the artery behaves purely elastically and can be characterized by hyperelastic strain energy functions, including the layer specific constitutive models used to describe mechanical behaviour of arteries. For instance, Dixon et al. (2003) developed a constitutive model to describe the layer-specific properties of porcine coronary arteries. This model consisted of a two-parameter logarithmic strain energy function, with parameter values determined from comprehensive experimental data. Results showed that the model was able to fit the experimental data, showing consistency in the model parameters obtained for different branches of coronary arteries. Holzapfel et al. (2005) published an extensive experimental work to determine the layer-specific mechanical properties of the human coronary arteries. Their results showed that the intima is the stiffest layer over the whole deformation domain whereas the media in the longitudinal direction is the softest. They also proposed a strain energy function to model human coronary arteries as a composite structure composed of three solid mechanically relevant layers with different mechanical properties. Although strain energy potentials have been dominantly employed to describe the constitutive behaviour of blood vessels, the influence of different hyperelastic potentials in simulation of stent expansion has not been studied yet. In addition, state of the art of arterial constitutive modelling suggests the use of hyperelastic anisotropic models (Holzapfel et al., 2004; Holzapfel et al., 2005; Zhang et al., 2007), which are expected to affect the expansion behaviour of stent-artery systems in computer simulations (Nolan et al., 2014). However, we hardly come across published papers which used anistropic hyperelastic models to simulate stent deployment in diseased arteries, and existing computational studies are dominantly limited to isotropic models.
In this paper, computational analyses have been carried out to evaluate the effect of different modelling practices used in the simulation of stent expansion inside a diseased artery. In particular, the simulations using folded balloon and polyurethane rubber balloon were compared against each other in terms of diameter change, recoiling, dogboning and stresses for the whole stent-artery system. The effects of boundary conditions were assessed by comparing the stent expansion in fully constrained, partially constrained and unconstrained arteries. Simulation of stent expansion was also studied by considering different strain energy potentials used to describe the constitutive behaviour of blood vessel and stenosis. This work is of help and guidance for reliable finite element analyses of stent deployment in-vivo.
Finite element model
The artery, plaque and rubber balloon were meshed into hexahedral elements with reduced integration, which are mostly used to increase the computing efficiency and numerical convergence associated with large deformation, especially for soft tissues (Mortier et al., 2009; Zahedmanesh et al., 2010; Gastaldi et al., 2010; Zhao et al., 2012b). Nevertheless, a comparison between full and reduced integration elements has been made by looking at the von Mises stress averaged along the middle ring of the plaque surface, which confirmed the consistency of the results obtained using the two types of elements. The stent was meshed into incompatible hexahedral elements (with full integration) in order to accommodate large bending deformation of the stent strut during expansion (Bower, 2008). This is also strongly recommended by Abaqus for stent deformation simulation (Simulia, 2010). The folded balloon was meshed using 4-node shell elements with reduced integration based on the consideration of computational efficiency. The number of elements is about 20,000 for the artery, with four layers of elements through the radial thickness, and about 50,000 for the stent, with four layers of elements through the strut thickness and two layers of elements across the strut width. The balloons were meshed into ~4000 elements for the rubber one and ~20,000 elements for the folded one.
Contacts between the stent, the artery and the balloon were defined as hard contact with a friction coefficient of 0.25 (Ju et al., 2008). Varied boundary conditions were applied to the ends of the artery to study the effect of constraints on stent expansion. The rubber balloon was fully constrained at the two ends to simulate the fixed connection of the balloon to catheter. The folded balloon was constrained only in the axial direction while allowing the ends to expand freely in the radial direction.
All analyses were carried out using Abaqus explicit solver. The simulations consisted of two steps: the inflation step (0.1 s) in which the applied pressure increased linearly to the peak value and the deflation step (0.1 s) in which the pressure dropped linearly to zero to allow the recoil of the artery and the stent. The pressure was applied on the inner surface of the balloon. A peak pressure of 1.4 MPa was used in this study to ensure that maximum expansion was achieved in all cases. The incremental time automatically chosen by Abaqus is on the order of 1.0E-8 s. The semi-automatic mass scaling (using a factor of 5) was used for both inflation and deflation steps, in order to reduce the computing time of the analysis. During the full process of analysis, the internal and kinetic energies have been monitored during the analyses, and the kinetic energy for whole the system was always less than 5% of the internal energy, which confirmed the validity of our quasi-static analyses (Gastaldi et al., 2010). Mesh sensitivity study has also been carried out, and confirmed the convergence of the results, in terms of stent diameter change, recoiling effect and residual stresses, for the mesh used in the present paper.
Material constitutive behaviour
Models for stent and balloon
Values of the Mooney-Rivlin model parameters for the polyurethane rubber balloon
ρ(kg/mm 3 )
The Ogden model
Values of the Ogden model parameters for the three vessel wall layers and the homogeneous artery
ρ(kg/mm 3 )
Values of the Ogden model parameters for hypocellular and calcified plaques
ρ (kg/mm 3 )
It should be clarified that the strain energy potential given in equation (1) is the third-order Ogden model and not a generic Ogden model. The Ogden model formulation used in this paper is exactly the same as that in Abaqus. Also, the parameter values taken from Zahedmanesh and Lally (2009) were fitted for the same model (i.e. the third-order Ogden model). This is slightly different from the standard (generic) form of Ogden model, which requires particular attention when fitting parameters are taken from published papers that refer to the standard Ogden formulation.
The polynomial model
Values of the polynomial model parameters for the three vessel wall layers and the hypocellular plaque
ρ (kg/mm 3 )
The Holzapfel-Gasser-Ogden model
Values of the Holzapfel-Gasser-Ogden model parameters for the three vessel layers
ρ(kg/mm 3 )
Results and discussion
Effect of balloon folding
The system expanded using a rubber balloon showed higher recoiling effect (22%) compared to that using a folded balloon (12%). These results can be explained by the different volume of plastic deformation achieved in the expanded stent, which was calculated to be 29% for the case of rubber balloon and 58% for the case of folded balloons. The volume of plastic deformation is closely linked with the recoiling of the system. The stent with larger volume of plastic deformation provides stronger resistance to the elastic recovery of the deformed artery, which explains the less recoiling effect for systems expanded using the folded balloons. However, the dogboning effect was lower for system expanded using rubber balloon (18%) than that using folded balloon (21%), which might be due to the lack of radial constraints for the folded balloons in our simulation.
The results suggest that balloon type has a significant effect on stent-artery expansion and their stress distributions. Stent deployment using folded balloon led to a sustained expansion until the balloon is unfolded. After this, a saturation stage is reached and further expansion became very slow and difficult. Furthermore, the recoiling appeared to be lower for a folded balloon than that for a rubber balloon, resulting in an overall larger expansion of the artery for folded balloon. This behaviour was mainly due to the fact that the balloon remained partially in contact with the stent during the whole deflation process, preventing the artery from recoiling completely. The dogboning was also affected by the balloon folding, but this may be due to different boundary conditions used for the folded balloon and rubber balloon. The ends of folded balloon are free of constraints in radial direction, and consequently less prohibitive for the recoiling at the distal sections, resulting in the increased dogboning effect.
It should be noted that there is a limitation for the simulations using the folded balloon. The folded balloon model adopted in this paper lacks the interaction with the catheter, which led to an overestimation of the dogboning effect during the expansion as also reported in Martin and Boyle (2013). However, this difference was noticed only at the beginning of the expansion, and became negligible (around 2%) at the final stage of expansion. For this reason, it is believed that the key results will not be affected by neglecting the interaction between folded balloon and angioplasty catheter.
Effect of artery constraints
Effect of artery constitutive models
Second stretch invariant in strain energy potential (Ogden model vs. Polynomial model)
Artery wall model (3-layer artery vs. Homogeneous artery)
Anisotropic behaviour of arterial layers
Stenosis composition (Hypocellular plaque vs. Calcified plaque)
From these results, it is plausible to say that the diameter change of the artery during deployment is highly determined by the composition of the stenotic plaque. The stent deployment simulated using calcified plaque model reached a saturation of expansion far earlier than that using hypocellular plaque model. The finally achieved expansion for calcified plaque model is also significantly lower than that for hypocellular plaque models. However, the recoiling effect was consistent with each other, regardless of the artery and plaque constitutive models.
The choice of balloon type, system constraints and artery constitutive models has a great influence on the simulation of stent expansion inside a stenotic artery. Simulations using folded and rubber balloons generate significantly different results in the stent expansion, such as sustained stent expansion behaviour, stress levels on the system and finally achieved diameter. The type of boundary conditions used for the artery highly affects the recoiling of the system, showing that over-constraints lead to higher recoiling effect but a decrease of stress magnitude on the artery wall due to reduced stretch. The use of different artery-plaque constitutive models also affects the simulated expansion behaviour of the system. For instance, calcified plaque model leads to a considerably lower expansion than hypocellular plaque model, with also higher stress levels. Stent expansion exhibits a saturation stage for simulation using Ogden model, which has not been observed for 6-parameter polynomial models due to the negligence of the second stretch invariant in the strain energy potential. Consideration of 3-layer structure of the artery wall showed the discontinuous stress distributions across the thickness of the blood vessel which cannot be captured by the homogeneous artery constitutive model. The anisotropy of the vessel layers leads to a lower maximum expansion and a higher stress level in the inner and outer arterial layers and on the surface of the stenotic plaque. These important factors need to be considered carefully in order to produce reliable and conclusive results in stent deployment simulations.
In conclusion, the hyperelastic strain energy potential that includes the first two invariants of the stretch tensor should be adopted to describe the constitutive behaviour of both the artery and the stenotic plaque. The constitutive model should also take into account the layered structure of the artery and the anisotropy of each layer. Over constraint of the artery ends can lead to an overestimation of recoiling effect of the system. From the results of this paper, it is more realistic to partially constrain the artery using spring elements with a proper stiffness constant. The plaque composition is also a major factor that affects the process of stent deployment and should be considered in computational modelling. For balloon-expandable stents, folded balloon should be modelled in order to obtain a more realistic diameter-pressure response of the system.
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