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A general finite element analysis method for balloon expandable stents based on repeated unit cell (RUC) model
Finite Elements in Analysis and Design 43 (2007) 649 – 658 www.elsevier.com/locate/finel
A general finite element analysis method for balloon expandable stents based on repeated unit cell (RUC) model Zihui Xia a,∗ , Feng Ju a , Katsuhiko Sasaki b a Department of Mechanical Engineering, University of Alberta, Edmonton, Alta., Canada T6G 2G8 b Division of Mechanical Science, Hokkaido University, N13, W8, Kita-ku, Sapporo 060-8628, Japan
Received 9 May 2006; received in revised form 12 January 2007; accepted 12 January 2007 Available online 28 February 2007
Abstract A successful deployment of the stent is dependent on the good understanding of its mechanical properties. This paper presents a general RUC (repeated unit cell) approach for the analysis of balloon expandable stents with various types of closed cells. The application of the unified periodic boundary conditions on the RUC model of the stent is formulated. Numerical analysis is performed by the ABAQUS FEM package. Stents of the Palmaz-Schatz type and NIR䉸 type with V- or S-shaped links are analyzed with the RUC models and the numerical results agree well with those obtained by much larger models in available publications. The global deformation pattern and the stress distribution of the entire stent can also be obtained by a geometrical tessellation or assembling of the deformed RUCs. It is found from the numerical analysis that the diameter of the stent during deployment changes slowly at the start and increases drastically when the internal pressure reaches a critical value where yielding hinges are formed. V- and S-stents are much easier to expand than the Palmaz-Schatz stent. However, the foreshortenings of the V-stent and S-stent are noticeable. The advantages of the RUC approach are that the existing periodicities in the stent structure are better and directly represented by applying the periodic boundary conditions and a much smaller FEM model and less computational efforts are required to achieve the same analysis accuracy as that by larger models adopted in previous publications. 䉷 2007 Elsevier B.V. All rights reserved. Keywords: Stent; Repeated unit cell model; Finite element method; Unified periodic boundary conditions; Rotational symmetry; Nonlinear analysis
1. Introduction The use of intravascular stents has become a common procedure in the treatment of vascular disease. There are currently over 100 different types of stents in the market and laboratories in the world [1]. Stents can be classified as slotted tube, coil and mesh types based on their original cell patterns. The geometric cells can be in closed or open patterns to balance the strength and flexibility requirements. Before deployment, the stent is collapsed to a small diameter and put over a balloon catheter. It is then moved into the area of the blockage in a blood vessel and expanded by the inflation of the balloon. The expanded stent permanently locks in the place of stenosis and forms a scaffold that holds the artery open so that blood flow is improved. From the mechanics point of view, the ∗ Corresponding author. Tel.: 1 780 492 3870; fax: 1 780 492 2200.
E-mail address: [email protected] (Z. Xia). 0168-874X/$ - see front matter 䉷 2007 Elsevier B.V. All rights reserved. doi:10.1016/j.finel.2007.01.001
deployment of stents involves large plastic deformation and nonlinear contacts. Therefore, understanding of stresses and strains experienced by the stents during deployment is important for successful application of the stenting technology. The finite element method (FEM) has been extensively used in the numerical analysis of the mechanical properties (strains, stresses, deformation, stiffness and flexibility, etc.) in the stent deployment. Starting from the simulation of the expansion of the stent itself, Chua et al. [2–5] analyzed the interaction between the balloon and the slotted tube stent during the stent delivery as well as the stresses in the plaque and artery due to the interference of the slotted tube stent. In their studies, half and quarter models are used in the finite element analysis with LS-DYNA and ANSYS codes. Etave et al. [6] studied the mechanical properties of both tubular and coil stents by using ABAQUS code. Dumoulin and Cochelin [7] studied the shortening percentage, radial and longitudinal recoils and the weakness of the structure of balloon-expandable stents, where
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for a special type of stent a generic part of the structure with periodic boundary conditions was used. McGarry et al. [8] investigated the mechanical behaviour of a balloon-expandable stent using computational micromechanics in the context of the FEM, where a 2D unit cell is used for the numerical analysis. Unfortunately, the unit cell proposed in their study is not a true repeated unit cell (RUC) and cannot be used to analyze the complete mechanical properties of the stent. The research on the mechanical behavior of stents was also conducted by Migliavacca et al. [9] through both numerical and experimental studies. Wang et al. [10] simulated the transient expansion process of stent/balloon system with different stent structure and balloon length under the internal pressure with a full 3D finite element model, where the transitory non-uniform expansion or the so-called dogboning phenomenon is highlighted. Walke et al. [11] presented their results of the experimental and numerical research of a vascular stent used in the treatment of blood vessel stenosis including the biomechanical characteristic of the stent. It can be seen that finite element analysis appears to be a quick and cost-effective method to evaluate the deployment behavior and mechanical properties of stents, including the effective yielding limit of the structure, the expanded geometry, the stress and strain fields within the stent and interactions between balloon–stent, stent–plaque–blood vessel system under various loading conditions. Geometrically, a closed-cell stent is an assembly of a number of RUCs and exhibits the periodicity in both longitudinal and circumferential directions. The periodical pattern along the circumferential direction is called rotationally or cyclically periodicity, which has been extensively exploited in the analysis of turbomachinery structures in order to reduce the size of numerical models [12–14]. This periodicity reduces a numerical model of the whole structure to a small panel or a pie portion without losing the accuracy of analysis. The periodicity along the longitudinal direction also warrants the use of a repeated representative unit cell in lieu of the analysis of the whole stent structure. The concept has been well adopted in the micromechanical analysis of composites where the RUC consists of both fiber and matrix with various configurations [15–18]. The RUC based approach is essentially stemmed from the earlier mathematical work of asymptotic homogenization theory by Benssousan et al. [19] and Suquet [20]. The strong material and geometrical nonlinearities and complex multiple contact behaviors in the stent analysis result in a large and complicated finite element model that is computationally intensive and challenging. With the RUC approach, the size of the numerical model is largely reduced so that a quick solution is possible. This is especially significant in the preliminary design stage of stents when many cycles of iteration between the numerical analysis and parameter selection are needed. The objective of this study is to apply the RUC method in the analysis of the mechanical properties of the balloon expandable stents with different closed cells. The unified periodic boundary conditions in the longitudinal and circumferential directions of the stent under general loading conditions are first formulated. The advantages of the RUC approach are that the
existing periodicities in the stent structure are better and directly represented by applying the periodic boundary conditions and a much smaller FEM model and even less computational effort are needed to achieve the same analysis accuracy as that by larger models adopted in previous publications. As examples, stents of the Palmaz-Schatz type and with V- or S-shaped links are analyzed with the RUC models and the numerical results are compared with those by using larger models. The global picture of the deformation pattern and the stress/strain distributions of the entire stent can also be obtained by a geometrical tessellation, or assembling, of the deformed RUCs. 2. Unified periodic boundary conditions for RUC models Consider a cylindrical structure with both longitudinal and circumferential periodicities as show in Fig. 1(a), where the applied global loads include the internal pressure p and axial force Fz . The cylindrical structure can be seen as a periodic array of a chosen RUC. Under the above global load conditions the deformation mode and stress/strain distributions should also have periodicities except in the limited area near the free ends of the cylinder. Therefore, instead of modeling the whole structure the numerical analysis only needs to be performed on a single RUC with application of appropriate periodic boundary conditions. A RUC in a cylindrical coordinate system is shown in Fig. 1(b), which is cut from the cylindrical periodic structure, Fig. 1(a). Planes k − and k + are two rotationally periodical boundary planes along the -direction, while planes j − and j + are two periodical boundary planes along the z-axis direction. Each pair of the above boundary planes of the RUC has the same shape. Therefore, for each point on the plane k − , there is a corresponding point on the opposite plane k + , where the coordinates r and z are the same for this pair of points. Similarly, the corresponding two points on the two boundary surfaces, planes j − and j + , have the same r and coordinates. For the boundary planes k − and k + , the rotational periodic boundary conditions can be written as k− uk+ r = ur ,
(1a)
k− uk+ = u ,
(1b)
k− uk+ z = uz ,
(1c)
where ur , u and uz are the displacements along r, and z directions, respectively. Eqs. (1a)–(1c) indicate that a pair of corresponding points on planes k − and k + has the same translational displacements. The periodic boundary conditions for the boundary planes j − and j + can be written as j+
j−
j+
j−
j+
j−
ur = ur , u = u , uz − uz = e¯z (zj + − zj − ) = e¯z z = Cz .
(2a) (2b) (2c)
In the above, e¯z is the elongation per unit length in z-direction averaged over the volume of the RUC. Since the two planes j −
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One would of course select the one with simplest boundaries for convenience of meshing and applying boundary conditions such as RUC 1, which is preferred to RUCs 2 and 3 in Fig. 1. Interested readers can refer to Ref. [18] for more details. 3. FEM implementation
Fig. 1. An illustrative cylindrical structure with periodicities at both longitudinal and circumferential directions: (a) the periodic structure and different repeated unit cells (RUCs); (b) a typical RUC.
and j + are parallel to each other, z = zj + − zj − is constant for any pair of corresponding points on the two planes and thus the difference of displacement component in z-direction is a constant. Eqs. (1) and (2) consist of the unified periodic boundary conditions for the RUC under any combination of global loads of internal pressure p and axial force Fz . Instead of giving known displacements on the boundary surfaces, Eqs. (1) and (2) specify differences of displacements on the two opposite boundary surfaces. In FEM analysis, they can be applied as the nodal displacement constraint equations. Since the RUCs should be assembled as a continuous physical body, two continuity conditions must be satisfied at the neighboring RUCs: (1) displacement continuity, i.e. the two neighboring boundaries cannot be separated from or encroach into each other; (2) traction continuity, i.e. the traction distribution must be the same at the neighboring boundary surfaces. It has proved in Ref. [18] that in a displacement-based FEM analysis, applying the above type of displacement-difference periodic boundary conditions on a RUC, a unique solution can be obtained and the solution also meets both the displacement and traction continuities. It is also noted that selection of the RUC is not unique as shown in Fig. 1(a), where three different RUCs are highlighted. RUC 1 has the symmetry in both longitudinal and circumferential directions with four flat boundary surfaces, while RUC 2 has no symmetry in the above two directions and RUC 3 is with curved boundary surfaces along the longitudinal direction. It has also been proven in [18] that for a fixed periodic structure, the solutions of different RUCs are the same if the unified periodic boundary conditions are applied.
For the convenience of the implementation of these periodical boundary conditions, Eqs. (1) and (2), planes k − and k + and planes j − and j + should have the same meshing pattern of nodes and elements so that each pair of nodes in the opposite surfaces having two same coordinates in the specified cylindrical coordinate system as shown in Fig. 1(a). Assuming L nodes each on the boundary surfaces along the circumferential direction, planes k − and k + , Eq. (1) will contain 3 × L nodal displacement constraint equations in the FEM formulation. Assuming M nodes each on the boundary surfaces along the longitudinal direction, planes j − and j + , Eq. (2) is equivalent to 3 × M nodal displacement constraint equations. Applications of Eqs. (1a)–(1c) and (2a) and (2b) are straightforward, while the application of Eq. (2c) may need further explanation since the constant Cz is generally unknown for a given global axial force Fz . As shown in Fig. 1(b) the nodes on the plane j − and plane j + are assumed to range, respectively, from 1 to M and N + 1 to N + M, where N can be considered as a shift of node numbers from plane j − to plane j + . Nodes i and N +i are therefore a pair of nodes having the same r- and -coordinates. Eq. (2c) can then be written as uz,N+i − uz,i = Cz
(i = 1, M).
(3)
To eliminate rigid body displacements at the z-direction, without losing generality, let uz be zero at node 1. That is uz,1 = 0.
(4)
Applying Eq. (4) to Eq. (3) with i = 1 gives uz,N+1 = Cz .
(5)
Replacing Cz by uz,N+1 , the rest of the constraint equations from the Eq. (3) can be then written as uz,N+i − uz,i = uz,N+1
(i = 2, M).
(6)
The unified periodic boundary conditions along the z-direction are now represented by Eq. (6) instead of Eq. (3). If the periodic cylindrical structure is subjected to a resultant axial force Fz , in addition to apply Eq. (6), one should apply a nodal force at the node N + 1, fz,N+1 =
· Fz , 2
(7)
where is the angle between plane k − and plane k + , Fig. 1(b). If only internal pressure p is applied on the cylindrical periodic structure, as the nature of the displacement-based FEM, applying Eq. (6) only will automatically yield the resultant force Fz = 0.
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The above periodic displacement boundary conditions can be further simplified if the RUC structure and the applied load is symmetrical at the z-direction as this shown in Fig. 1(b) where plane A–B–C–D is the symmetrical plane. In this case, the planes j − and j + must remain planes during deformation since the RUCs will otherwise be impossible to be assembled into a continuous structure in the z-direction. This conclusion can also be derived from Eq. (3). Applying symmetrical condition uz,N +i = −uz,i to Eq. (3) gives uz,i = Cz /2
(i = 1, M).
(8)
Eq. (8) indicates all the nodes on plane j − have the same displacement at the z-direction and plane j − therefore remains plane. Similarly, one can prove that plane j + also remains plane. The periodic displacement boundary conditions for symmetrical RUC can thus be specified as uz,1 = uz,2 = · · · = uz,M = 0, uz,N +1 = uz,N+2 = · · · = uz,N+M .
(9)
4. Application examples 4.1. Stent of Palmaz-Schatz type [1,3] Fig. 2(a) shows a quarter of the geometry of the PalmazSchatz stent. The advantages of these type of stents include high vessel surface area coverage, high radial strength and consistent circumferential deployment pattern [1,3]. It can be seen that the closed unit cell is repeated to form the whole stent. The natural and convenient representation of stent geometry would be in a cylindrical coordinate system. However, it would be tedious and difficult to model the 3D stent geometry directly in cylindrical coordinates, especially for some complex stent with fillets and round corners. In this study, stent is first modeled and meshed in a rectangular coordinate system based on the roll-out geometry shown in Fig. 2(a), and then mapped onto a cylindrical surface by a simple geometrical transformation of nodal coordinates as shown in Fig. 2(b). The RUC model selected for this stent is shown in Fig. 2(c). As discussed in the previous section, planes j − and j + are the periodic planes where constraint equations, Eqs. (2a), (2b) and (6) should be applied, and the rotational symmetric conditions, Eqs. (1a)–(1c), should be imposed for the nodes on planes k − and k + . (Only internal pressure load is considered in all examples in this paper.) Fig. 3(a) shows the finite element mesh of the RUC of a Palmaz-Schatz stent with the balloon, which has been studied by Chua et al. [3]. In the present calculation same material models and parameters as in [3] were used in order to make a comparison. The material of the stent is stainless steel. A bi-linear elasto-plastic material model is assumed with Young’s modulus E = 193 GPa, tangent modulus ET = 692 MPa, yield strength y = 0.207 GPa and Poisson’s ratio = 0.27. The balloon material is considered to be hyperelastic and represented by the two-parameter Mooney–Rivlin model with C10 =1.06881 MPa, C01 =0.710918 MPa and density=1070 kg/m3 . This hyperelastic model is one of the most commonly used in balloon model-
Fig. 2. A quarter of a Palmaz-Schatz stent and a repeated unit cell (RUC) model: (a) the roll-out geometry of stent in rectangular coordinates; (b) real 3D geometry in cylindrical coordinates; (c) a typical RUC of this stent.
Fig. 3. RUC model for the Palmaz-Schatz stent: (a) RUC before the expansion of balloon; (b) RUC after the expansion of the balloon with the contour of von Mises strain distribution at p = 0.409 MPa.
ing. In the simulation process, the inner surface of the balloon is subjected to a uniform internal pressure which gradually increased from 0 to 0.409 MPa as used in Ref. [3]. Numerical analysis is conducted by ABAQUS Explicit codes and the total kinetic energy is checked to be less than 1.0% to ensure the quasi-static nature of the stent deployment in numerical simulation. Both the stent and the balloon are modeled by 3D solid elements. A surface to surface contact algorithm is selected to model the nonlinear contact between the balloon and the stent. When surfaces are in contact they usually transmit shear as well as
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Fig. 4. A complete round of stent tessellated by RUC before and after balloon expansion.
normal compressive forces across their interface. Coulomb friction model is used to model the frictional contacts between the balloon and the stent. The basic concept of this friction model is that the contacting surfaces can only carry a certain magnitude of shear force before they start to slide from each other. This maximum shear force is determined by the coefficient of friction. In this study, the coefficient of friction is chosen to be 0.2. Different values of the coefficient ranging from 0.05 to 0.25 are also tested and the difference in the computational results is found insignificant since the balloon surface would crimple into the vanity parts of the stent, making the contacting surfaces hard to slip from each other. 4.1.1. Deformation and expansion–pressure relation Fig. 3(b) shows the deformed shape and the distributions of von Mises strain (equivalent plastic strain) in the stent RUC when the pressure on the balloon, p, reaches 0.409 MPa. Fig. 4 shows a full round of the stent before and after the balloon expansion that are tessellated from the undeformed and deformed RUC models, respectively. A clearer global deformation pattern of the stent can be observed from the figure. In the stent deployment, internal pressure inflates the balloon and further makes the stent expand, and the relationship between the internal pressure and the extent of stent expansion is therefore of importance for the stent design and clinical application. For the current stent/balloon system the stent diameter–pressure relation is shown in Fig. 5 (the curve with square symbols). It can be seen from the curve that the dilation of stent diameter from the start increases very slowly. Numerical results show that initial yielding starts from the corners of the bridges and the struts when the pressure reached 0.12 MPa (as point A shown in Fig. 3). Yielding areas further increase with the increase of the pressure and become large enough to form yielding hinges when the pressure is about 0.40 MPa. At this point the diameter of the stent increases drastically, meaning that a little bit of increment in pressure after this critical value causes a big increase in the stent diameter. For clinical deployment of stents, these critical pressure values have practical significance.
Fig. 5. Stent diameter vs. internal pressure.
In some analyses of stents, the balloon was excluded from the computational model [2,6,9,10], i.e. the internal pressure is directly applied onto the inner surface of the stents. This is the case when the relationship between the stress status and the expanded stent diameter is of interest. To make a comparison, the stent expansion curve without balloon and the expansion curve of balloon itself are computed separately by applying pressure directly into the inner surface of the balloon or the stent. The results are shown in Fig. 5 by the curves with triangular and circular symbols, respectively. It can be seen from the three curves in the figure that a significant part of work done by the internal pressure is transferred into the strain energy of the balloon. In addition, the pressure needed to expand the balloon and stent together as a system is not a simple summation of the pressures to expand the balloon and the stent individually. Therefore, the stent/balloon catheter model should be used in order to obtain correct stent expansion–pressure relation. 4.1.2. Stress distribution Fig. 6 shows the distributions of the von Mises equivalent stress, longitudinal and hoop stresses when the internal
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Fig. 6. Distribution of stresses e (von Mises), z and at p = 0.409 MPa calculated by RUC model.
pressure is 0.409 MPa. The stress at the radial direction, r , is found much smaller than z or . It can be seen that the longitudinal stress, z , is a primary stress component, instead of the hoop stress component, , although it seems that the largest deformation is in the hoop direction. This can be explained as that the large expansion in diameter is introduced by the large rotational deformation of the strut around the bridge (see Fig. 6). This is caused by the bending of the stent strut in the –z plane and the longitudinal bending stress is thus the dominant component. The distribution of z across the strut’s cross section in Fig. 6 clearly shows that the stress ranges from maximum tensile to maximum compressive stresses, which indicates the bending phenomenen of the struts. From Fig. 3(b) one can also see that the maximum von Mises strain is about 0.168, which is not at the same order as the expansion rate of the diameter (from 3.0 to 4.8 mm). From the mechanics point of view, this is a large displacement (or rotation) and medium strain problem in nature. Note that two layers of elements are used in the thickness direction of the RUC model. The FEM results show that z and , are almost uniformly distributed along the thickness direction. This is true since the thickness of the stent is much smaller than its other dimensions and the stent can be considered as a thin-walled structure. 4.1.3. Comparison of results of RUC model with quarter model To make a comparison, a quarter model of the stent, which was employed by Chua et al. [3], is also re-analyzed as shown in Fig. 7. The size (nodes and elements) and CPU time of the quarter model are, respectively, 12 and 28 times those of the RUC
Fig. 7. Quarter model for the Palmaz-Schatz stent with deformation at p = 0.409.
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Table 1 Comparison of the expanded diameter and von Mises stress at a corner node between RUC and other models Pressure (MPa)
0.1
Expanded diameter (mm) RUC Model 3.05 Quarter Model 3.05 3.1 Chua et al. [3]a Von Mises Stress (MPa) RUC Model 25.3 Quarter Model 25.0 – Chua et al. [3]a a Numbers
0.2
3.11 3.11 3.2 214 212 –
0.3
3.46 3.47 3.4 225 220 –
0.409
4.79 4.83 4.6 283 281 286
are directly measured from the figures from [3].
model. The corner node at the RUC model (Node A, Fig. 3) and a corresponding node in the quarter model (Node A’ in Fig. 7) are identified to compare the computational accuracy of the two models. Table 1 gives the comparisons of the expanded stent diameters and von Mises stresses between the RUC model at Node A and quarter model at Node A’ as well as the results from Ref. [3]. It can be seen that the differences in the displacement and stress between RUC model and the quarter model are well below 1.0%. The table also shows that results of the RUC and our re-calculated quarter model are also close to those in Ref. [3], which are directly measured from the figures in the reference since precise numerical data were not given in the paper. One can therefore conclude that the RUC approach with a much smaller model size can produce same accurate stress and deformation results of the stent as much bigger models. 4.2. Closed-cell stents with V- or S-links Stents of the Palmaz-Schatz type are strong but too rigid. To improve the flexibility of such stents, V-shaped links were first added in the NIR䉸 stents to accommodate the change in shape during plastic deformation. Based on this concept, stents with U-, S-, N- or W-shaped links have also been designed and analyzed [21,22,10]. Figs. 8(a) and 10(a) show the closed-cell stents with V- and S- links, which are, respectively, termed as the V-stent and S-stent in this study. The percentages of the metal surface over that of artery are 17.6% and 19.1% for the V-stent and the S-stent, respectively. From the roll-out geometries of the V- and S-stent shown in the figures, one can clearly see the existence of RUCs in these stent structures. Figs. 8(b) and 10(b) show RUCs selected for these stents, respectively. As indicated in the previous section, the selection of RUCs is not unique and an appropriate RUC should facilitate the application of the periodic boundary conditions and save the labor in data preparations. It can be seen that there is only one symmetrical plane in the RUC of the V-stent and no symmetry for that of the S-stent. The applications of periodic boundary conditions are the same as that for the Palmaz-Schatz stent. Also the same material model and constants are used in the calculations.
Fig. 8. RUC model for the V-stent; (a) the roll-out geometry of stent in rectangular coordinates; (b) RUC before stent expansion; (c) deformed RUC and Mises stress when stent diameter is expanded from 3 to 6.4 mm.
4.2.1. Deformation and stress distribution Numerical analyses are conducted for stents with balloons. The maximum pressure applied on the inner surface of the balloon is 0.16 MPa. Figs. 8(c) and 10(c) give the deformed shapes and the distribution of von Mises stresses when the stents are expanded to 6.4 mm in diameter for V-stent and 9.9 mm for S-stent, respectively. To have a clear observation of the deformation and stress distribution patterns of the whole stent, parts of stent geometries before and after expansion are also tessellated from the corresponding undeformed and deformed RUCs and shown in Figs. 9 and 11 for V-stent and S-stent, respectively. It is obvious that multiple yielding hinges are developed with the increasing internal pressure and they play a crucial role to the stent expansion. Also, critical areas of stress concentration and deformation can be identified from the numerical results, which may be used to guide the stent design and clinical application (see Figs. 9–11). The numerical results show that the boundary planes in z-direction of the RUC of the V-stent still remains plane, while the boundary planes of the S-stent no longer remains plane after deformation as shown in Fig. 12. The reason for this phenomenon is that the RUC of the V-stent is symmetrical with respect to A–A plane shown in Figs. 8(b) and (c), while there is no such symmetrical plane in the RUC of S-stent. As we have discussed in Section 3 that if a RUC is symmetrical in the z-direction, its boundary planes at the two ends will remain planes during the deformation. It is worth to note that, for asymmetrical RUCs such as the RUC of the S-stent, applying “plane remains plane” condition at the two boundary planes in z-direction of the RUC will induce invalid results. It can be further concluded that if the RUC and the applied load have the symmetry in both - and z-directions, such
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Fig. 11. The tessellated geometries of the S-stent before and after expansion based on RUC models. Fig. 9. The tessellated geometries of the V-stent before and after expansion based on RUC model.
Fig. 12. Deformation of the end plane of RUC of the S-stent in the z-direction. Fig. 10. RUC model for the S-stent; (a) the roll-out geometry of stent in rectangular coordinates; (b) RUC before stent expansion; (c) deformed RUC and Mises stress when stent diameter is expanded from 3 to 9.9 mm.
as in the case of the stent of the Palmaz-Schatz type, a further smaller model of a quarter of the RUC can be used in the FEM analysis. If the RUC and applied load are symmetrical only in z-direction, such as the V-stent, half of the RUC model is enough for the analysis. However, for the asymmetrical RUCs, such as the S-stent, a full RUC model should be used and
the general periodic boundary conditions, Eqs. (1a)–(1c) and (2a)–(2c) must be applied. In contrast, if the proposed RUC model and corresponding periodic boundary conditions are not applied for the analysis of the above three types of stents with an odd number of cells along the circumferential direction, a quarter model should be used for the Palmaz-Schatz stents, a half model for V-stents and a whole model for S-stents. Therefore the advantages of the RUC model become more evident for the stents with less symmetries.
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the foreshortening of both the V-stent and S-stent is significant and this is the disadvantage of the flexible stents. Also the foreshortening of the V-stent is more than that of the S-stent if both stents are expanded to the same diameter. This phenomenon is probably due to the weaker longitudinal stiffness of V shaped link comparing to that of the S shaped link with the dimensions used in this study. The superior flexibility is the advantage of the S-stent and V-stent, the noticeable foreshortening should be, however, considered in the design and deployment of such type of stents. 5. Conclusions Fig. 13. Expanded stent diameter with respect to the inner pressure.
Fig. 14. Foreshortening of stent with respect to expanded stent diameter.
Fig. 13 presents the relationship between the expanded diameter and pressure applied on the inner surface of the balloon. It can be seen from the figure that, similar to the case of the Palmaz-Schatz stent, the diameter of the stent changes slowly at the start of the deployment and increases drastically when the pressure reaches a critical value where the yielding hinges are formed. By comparing the stent expansion–pressure curves in Figs. 5 and 13 (all the three stents have the same initial diameter of 3 mm) one can see that the V- and S-stents are much easier to expand than the Palmaz-Schatz stent. The S-stent is more flexible than the V-stent when they have similar percentage of the metal surface over that of artery. 4.2.2. Foreshortening of stent When a stent is expanded, its dimension in the longitudinal direction will be reduced. This behavior is termed as “foreshortening” in the stent design and defined as Foreshortening =
L0 − L × 100%, L0
(10)
where L0 is original length of the stent and L is the length of the stent after expansion, as indicated in Figs. 8(b) and 10(b). Fig. 14 shows the foreshortening of V- and S-stents with respect to the expanded diameter of the stent. It can be seen that
The repeated unit cell (RUC) model can be efficiently used in the analysis of the mechanical properties of stents with closed cells. A unified form of periodic boundary conditions for FEM analysis of stents has been presented. Stents of the PalmazSchatz type and with V- or S-shaped links are analyzed with the RUC approach. The computed deformation and stresses agree well with results that are obtained with much larger stent models and with much more computational efforts comparing to the present approach. The global picture of the deformation pattern and the stress distribution can be obtained by a simple geometrical tessellation of the deformed RUCs. With the application of the internal pressure, the diameter of the stent changes slowly at the start of the deployment and increases drastically when the pressure reaches a critical value where the yielding hinges are formed. Numerical results show that V- and S-stents are easier to expand than the Palmaz-Schatz stent. However, the foreshortening of the S-stent and V-stent is noticeable and this is the shortcoming of such flexible stents. The present RUC model with application of the unified periodic boundary conditions can accurately predict the behavior of most part of the whole stent except for the limited areas at the two ends. The main advantage of the RUC approach is that a much smaller numerical model with less computational effort is needed to achieve the same analysis accuracy as that by much larger models. The limitation of the current RUC approach is that it is incapable to provide stress and deformation information where the effect of free edge exists. A comprehensive numerical analysis should also include the artery and plaque, which are neglected in this study. References [1] D. Stoeckel, C. Bonsignore, S. Duda, A survey of stent designs, Minimally Invasive Ther. Allied Technol. 11 (4) (2002) 137–147. [2] S.N.D. Chua, B.J. Mac Donald, M.S.J. Hashmi, Finite-element simulation of stent expansion, J. Mater. Process. Technol. 120 (2002) 335–340. [3] S.N.D. Chua, B.J. Mac Donald, M.S.J. Hashmi, Finite element simulation of stent and balloon interaction, J. Mater. Process. Technol. 143–144 (2003) 591–597. [4] S.N.D. Chua, B.J. Mac Donald, M.S.J. Hashmi, Effects of varying slotted tube (stent) geometry on its expansion behaviour using finite element method, J. Mater. Process. Technol. 155–156 (2004) 1764–1771. [5] S.N.D. Chua, B.J. Mac Donald, M.S.J. Hashmi, Finite element simulation of slotted tube (stent) with the presence of plaque and artery by balloon expansion, J. Mater. Process. Technol. 155–156 (2004) 1772–1779.
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[6] F. Etave, G. Finet, M. Boivin, J. Boyer, G. Rioufol, G. Thollet, Mechanical properties of coronary stents determined by using finite element analysis, J. Biomechanics 34 (8) (2001) 1065–1075. [7] C. Dumoulin, B. Cochelin, Mechanical behaviour modelling of balloonexpandable stents, J. Biomechanics 33 (11) (2000) 1461–1470. [8] J.P. McGarry, B.P. O’Donnell, P.E. McHugh, J.G. McGarry, Analysis of the mechanical performance of a cardiovascular stent design based on micromechanical modelling, Comput. Mater. Sci. 31 (3–4) (2004) 421–438. [9] F. Migliavacca, L. Petrini, V. Montanari, I. Quagliana, F. Auricchio, G. Dubini, A predictive study of the mechanical behaviour of coronary stents by computer modeling, Med. Eng. Phys. 27 (1) (2005) 13–18. [10] W.-Q. Wang, D.-K. Liang, D.-Z. Yang, M. Qi, Analysis of the transient expansion behavior and design optimization of coronary stents by finite element method, J. Biomechanics 39 (1) (2006) 21–32. [11] W. Walke, Z. Paszenda, J. Filipiak, Experimental and numerical biomechanical analysis of vascular stent, J. Mater. Process. Technol. 164–165 (2005) 1263–1268. [12] A.J. Fricker, S. Potter, Transient forced vibration of rotationally periodic structures, Int. J. Numer. Methods Eng. 17 (1981) 957–974. [13] D.L. Thomas, Dynamics of rotationally periodic structures, Int. J. Numer. Methods Eng. 14 (1979) 81–102. [14] I.Y. Shen, Vibration of rotationally periodic structures, J. Sound Vib. 172 (4) (1994) 459–470.
[15] J. Aboudi, Mechanics of Composite Materials, A Unified Micromechanical Approach, Elsevier Science Publishers, Amsterdam, 1991. [16] Z. Xia, Y. Zhang, F. Ellyin, A unified periodical boundary condition for representative volume elements of composites and applications, Int. J. Solids Struct. 40 (2003) 1907–1921. [17] Y. Zhang, Z. Xia, F. Ellyin, Nonlinear viscoelastic micromechanical analysis of fibre-reinforced polymer laminates with damage evolution, Int. J. Solids Struct. 42 (2004) 591–604. [18] Z. Xia, C. Zhou, Q. Yong, X. Wang, On selection of repeated unit cell and application of unified periodic boundary conditions in micromechanical analysis of composites, Int. J. Solids Struct. 43 (2006) 266–278. [19] A. Benssousan, J.L. Lions, G. Papanicoulau, Asymptotic Analysis for Periodic Structures, North-Holland, Amsterdam, 1978. [20] P. Suquet, Elements of homogenization theory for inelastic solid mechanics, in: E. Sanchez-Palencia, A. Zaoui (Eds.), Homogenization Techniques for Composite Media, Springer, Berlin, 1987, pp. 194–275. [21] D.V. Kandzari, J.E. Tcheng, J. Zidar, Coronary artery stents: evaluating new designs for contemporary percutaneous intervention, Catheterization Cardiovasc. Interventions 56 (2002) 562–576. [22] K. Mori, T. Saito, Effects of stent structure on stent flexibility measurements, Ann. Biomed. Eng. 33 (6) (2005) 733–742.
A general finite element analysis method for balloon expandable stents based on repeated unit cell (RUC) model Zihui Xia a,∗ , Feng Ju a , Katsuhiko Sasaki b a Department of Mechanical Engineering, University of Alberta, Edmonton, Alta., Canada T6G 2G8 b Division of Mechanical Science, Hokkaido University, N13, W8, Kita-ku, Sapporo 060-8628, Japan
Received 9 May 2006; received in revised form 12 January 2007; accepted 12 January 2007 Available online 28 February 2007
Abstract A successful deployment of the stent is dependent on the good understanding of its mechanical properties. This paper presents a general RUC (repeated unit cell) approach for the analysis of balloon expandable stents with various types of closed cells. The application of the unified periodic boundary conditions on the RUC model of the stent is formulated. Numerical analysis is performed by the ABAQUS FEM package. Stents of the Palmaz-Schatz type and NIR䉸 type with V- or S-shaped links are analyzed with the RUC models and the numerical results agree well with those obtained by much larger models in available publications. The global deformation pattern and the stress distribution of the entire stent can also be obtained by a geometrical tessellation or assembling of the deformed RUCs. It is found from the numerical analysis that the diameter of the stent during deployment changes slowly at the start and increases drastically when the internal pressure reaches a critical value where yielding hinges are formed. V- and S-stents are much easier to expand than the Palmaz-Schatz stent. However, the foreshortenings of the V-stent and S-stent are noticeable. The advantages of the RUC approach are that the existing periodicities in the stent structure are better and directly represented by applying the periodic boundary conditions and a much smaller FEM model and less computational efforts are required to achieve the same analysis accuracy as that by larger models adopted in previous publications. 䉷 2007 Elsevier B.V. All rights reserved. Keywords: Stent; Repeated unit cell model; Finite element method; Unified periodic boundary conditions; Rotational symmetry; Nonlinear analysis
1. Introduction The use of intravascular stents has become a common procedure in the treatment of vascular disease. There are currently over 100 different types of stents in the market and laboratories in the world [1]. Stents can be classified as slotted tube, coil and mesh types based on their original cell patterns. The geometric cells can be in closed or open patterns to balance the strength and flexibility requirements. Before deployment, the stent is collapsed to a small diameter and put over a balloon catheter. It is then moved into the area of the blockage in a blood vessel and expanded by the inflation of the balloon. The expanded stent permanently locks in the place of stenosis and forms a scaffold that holds the artery open so that blood flow is improved. From the mechanics point of view, the ∗ Corresponding author. Tel.: 1 780 492 3870; fax: 1 780 492 2200.
E-mail address: [email protected] (Z. Xia). 0168-874X/$ - see front matter 䉷 2007 Elsevier B.V. All rights reserved. doi:10.1016/j.finel.2007.01.001
deployment of stents involves large plastic deformation and nonlinear contacts. Therefore, understanding of stresses and strains experienced by the stents during deployment is important for successful application of the stenting technology. The finite element method (FEM) has been extensively used in the numerical analysis of the mechanical properties (strains, stresses, deformation, stiffness and flexibility, etc.) in the stent deployment. Starting from the simulation of the expansion of the stent itself, Chua et al. [2–5] analyzed the interaction between the balloon and the slotted tube stent during the stent delivery as well as the stresses in the plaque and artery due to the interference of the slotted tube stent. In their studies, half and quarter models are used in the finite element analysis with LS-DYNA and ANSYS codes. Etave et al. [6] studied the mechanical properties of both tubular and coil stents by using ABAQUS code. Dumoulin and Cochelin [7] studied the shortening percentage, radial and longitudinal recoils and the weakness of the structure of balloon-expandable stents, where
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for a special type of stent a generic part of the structure with periodic boundary conditions was used. McGarry et al. [8] investigated the mechanical behaviour of a balloon-expandable stent using computational micromechanics in the context of the FEM, where a 2D unit cell is used for the numerical analysis. Unfortunately, the unit cell proposed in their study is not a true repeated unit cell (RUC) and cannot be used to analyze the complete mechanical properties of the stent. The research on the mechanical behavior of stents was also conducted by Migliavacca et al. [9] through both numerical and experimental studies. Wang et al. [10] simulated the transient expansion process of stent/balloon system with different stent structure and balloon length under the internal pressure with a full 3D finite element model, where the transitory non-uniform expansion or the so-called dogboning phenomenon is highlighted. Walke et al. [11] presented their results of the experimental and numerical research of a vascular stent used in the treatment of blood vessel stenosis including the biomechanical characteristic of the stent. It can be seen that finite element analysis appears to be a quick and cost-effective method to evaluate the deployment behavior and mechanical properties of stents, including the effective yielding limit of the structure, the expanded geometry, the stress and strain fields within the stent and interactions between balloon–stent, stent–plaque–blood vessel system under various loading conditions. Geometrically, a closed-cell stent is an assembly of a number of RUCs and exhibits the periodicity in both longitudinal and circumferential directions. The periodical pattern along the circumferential direction is called rotationally or cyclically periodicity, which has been extensively exploited in the analysis of turbomachinery structures in order to reduce the size of numerical models [12–14]. This periodicity reduces a numerical model of the whole structure to a small panel or a pie portion without losing the accuracy of analysis. The periodicity along the longitudinal direction also warrants the use of a repeated representative unit cell in lieu of the analysis of the whole stent structure. The concept has been well adopted in the micromechanical analysis of composites where the RUC consists of both fiber and matrix with various configurations [15–18]. The RUC based approach is essentially stemmed from the earlier mathematical work of asymptotic homogenization theory by Benssousan et al. [19] and Suquet [20]. The strong material and geometrical nonlinearities and complex multiple contact behaviors in the stent analysis result in a large and complicated finite element model that is computationally intensive and challenging. With the RUC approach, the size of the numerical model is largely reduced so that a quick solution is possible. This is especially significant in the preliminary design stage of stents when many cycles of iteration between the numerical analysis and parameter selection are needed. The objective of this study is to apply the RUC method in the analysis of the mechanical properties of the balloon expandable stents with different closed cells. The unified periodic boundary conditions in the longitudinal and circumferential directions of the stent under general loading conditions are first formulated. The advantages of the RUC approach are that the
existing periodicities in the stent structure are better and directly represented by applying the periodic boundary conditions and a much smaller FEM model and even less computational effort are needed to achieve the same analysis accuracy as that by larger models adopted in previous publications. As examples, stents of the Palmaz-Schatz type and with V- or S-shaped links are analyzed with the RUC models and the numerical results are compared with those by using larger models. The global picture of the deformation pattern and the stress/strain distributions of the entire stent can also be obtained by a geometrical tessellation, or assembling, of the deformed RUCs. 2. Unified periodic boundary conditions for RUC models Consider a cylindrical structure with both longitudinal and circumferential periodicities as show in Fig. 1(a), where the applied global loads include the internal pressure p and axial force Fz . The cylindrical structure can be seen as a periodic array of a chosen RUC. Under the above global load conditions the deformation mode and stress/strain distributions should also have periodicities except in the limited area near the free ends of the cylinder. Therefore, instead of modeling the whole structure the numerical analysis only needs to be performed on a single RUC with application of appropriate periodic boundary conditions. A RUC in a cylindrical coordinate system is shown in Fig. 1(b), which is cut from the cylindrical periodic structure, Fig. 1(a). Planes k − and k + are two rotationally periodical boundary planes along the -direction, while planes j − and j + are two periodical boundary planes along the z-axis direction. Each pair of the above boundary planes of the RUC has the same shape. Therefore, for each point on the plane k − , there is a corresponding point on the opposite plane k + , where the coordinates r and z are the same for this pair of points. Similarly, the corresponding two points on the two boundary surfaces, planes j − and j + , have the same r and coordinates. For the boundary planes k − and k + , the rotational periodic boundary conditions can be written as k− uk+ r = ur ,
(1a)
k− uk+ = u ,
(1b)
k− uk+ z = uz ,
(1c)
where ur , u and uz are the displacements along r, and z directions, respectively. Eqs. (1a)–(1c) indicate that a pair of corresponding points on planes k − and k + has the same translational displacements. The periodic boundary conditions for the boundary planes j − and j + can be written as j+
j−
j+
j−
j+
j−
ur = ur , u = u , uz − uz = e¯z (zj + − zj − ) = e¯z z = Cz .
(2a) (2b) (2c)
In the above, e¯z is the elongation per unit length in z-direction averaged over the volume of the RUC. Since the two planes j −
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One would of course select the one with simplest boundaries for convenience of meshing and applying boundary conditions such as RUC 1, which is preferred to RUCs 2 and 3 in Fig. 1. Interested readers can refer to Ref. [18] for more details. 3. FEM implementation
Fig. 1. An illustrative cylindrical structure with periodicities at both longitudinal and circumferential directions: (a) the periodic structure and different repeated unit cells (RUCs); (b) a typical RUC.
and j + are parallel to each other, z = zj + − zj − is constant for any pair of corresponding points on the two planes and thus the difference of displacement component in z-direction is a constant. Eqs. (1) and (2) consist of the unified periodic boundary conditions for the RUC under any combination of global loads of internal pressure p and axial force Fz . Instead of giving known displacements on the boundary surfaces, Eqs. (1) and (2) specify differences of displacements on the two opposite boundary surfaces. In FEM analysis, they can be applied as the nodal displacement constraint equations. Since the RUCs should be assembled as a continuous physical body, two continuity conditions must be satisfied at the neighboring RUCs: (1) displacement continuity, i.e. the two neighboring boundaries cannot be separated from or encroach into each other; (2) traction continuity, i.e. the traction distribution must be the same at the neighboring boundary surfaces. It has proved in Ref. [18] that in a displacement-based FEM analysis, applying the above type of displacement-difference periodic boundary conditions on a RUC, a unique solution can be obtained and the solution also meets both the displacement and traction continuities. It is also noted that selection of the RUC is not unique as shown in Fig. 1(a), where three different RUCs are highlighted. RUC 1 has the symmetry in both longitudinal and circumferential directions with four flat boundary surfaces, while RUC 2 has no symmetry in the above two directions and RUC 3 is with curved boundary surfaces along the longitudinal direction. It has also been proven in [18] that for a fixed periodic structure, the solutions of different RUCs are the same if the unified periodic boundary conditions are applied.
For the convenience of the implementation of these periodical boundary conditions, Eqs. (1) and (2), planes k − and k + and planes j − and j + should have the same meshing pattern of nodes and elements so that each pair of nodes in the opposite surfaces having two same coordinates in the specified cylindrical coordinate system as shown in Fig. 1(a). Assuming L nodes each on the boundary surfaces along the circumferential direction, planes k − and k + , Eq. (1) will contain 3 × L nodal displacement constraint equations in the FEM formulation. Assuming M nodes each on the boundary surfaces along the longitudinal direction, planes j − and j + , Eq. (2) is equivalent to 3 × M nodal displacement constraint equations. Applications of Eqs. (1a)–(1c) and (2a) and (2b) are straightforward, while the application of Eq. (2c) may need further explanation since the constant Cz is generally unknown for a given global axial force Fz . As shown in Fig. 1(b) the nodes on the plane j − and plane j + are assumed to range, respectively, from 1 to M and N + 1 to N + M, where N can be considered as a shift of node numbers from plane j − to plane j + . Nodes i and N +i are therefore a pair of nodes having the same r- and -coordinates. Eq. (2c) can then be written as uz,N+i − uz,i = Cz
(i = 1, M).
(3)
To eliminate rigid body displacements at the z-direction, without losing generality, let uz be zero at node 1. That is uz,1 = 0.
(4)
Applying Eq. (4) to Eq. (3) with i = 1 gives uz,N+1 = Cz .
(5)
Replacing Cz by uz,N+1 , the rest of the constraint equations from the Eq. (3) can be then written as uz,N+i − uz,i = uz,N+1
(i = 2, M).
(6)
The unified periodic boundary conditions along the z-direction are now represented by Eq. (6) instead of Eq. (3). If the periodic cylindrical structure is subjected to a resultant axial force Fz , in addition to apply Eq. (6), one should apply a nodal force at the node N + 1, fz,N+1 =
· Fz , 2
(7)
where is the angle between plane k − and plane k + , Fig. 1(b). If only internal pressure p is applied on the cylindrical periodic structure, as the nature of the displacement-based FEM, applying Eq. (6) only will automatically yield the resultant force Fz = 0.
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The above periodic displacement boundary conditions can be further simplified if the RUC structure and the applied load is symmetrical at the z-direction as this shown in Fig. 1(b) where plane A–B–C–D is the symmetrical plane. In this case, the planes j − and j + must remain planes during deformation since the RUCs will otherwise be impossible to be assembled into a continuous structure in the z-direction. This conclusion can also be derived from Eq. (3). Applying symmetrical condition uz,N +i = −uz,i to Eq. (3) gives uz,i = Cz /2
(i = 1, M).
(8)
Eq. (8) indicates all the nodes on plane j − have the same displacement at the z-direction and plane j − therefore remains plane. Similarly, one can prove that plane j + also remains plane. The periodic displacement boundary conditions for symmetrical RUC can thus be specified as uz,1 = uz,2 = · · · = uz,M = 0, uz,N +1 = uz,N+2 = · · · = uz,N+M .
(9)
4. Application examples 4.1. Stent of Palmaz-Schatz type [1,3] Fig. 2(a) shows a quarter of the geometry of the PalmazSchatz stent. The advantages of these type of stents include high vessel surface area coverage, high radial strength and consistent circumferential deployment pattern [1,3]. It can be seen that the closed unit cell is repeated to form the whole stent. The natural and convenient representation of stent geometry would be in a cylindrical coordinate system. However, it would be tedious and difficult to model the 3D stent geometry directly in cylindrical coordinates, especially for some complex stent with fillets and round corners. In this study, stent is first modeled and meshed in a rectangular coordinate system based on the roll-out geometry shown in Fig. 2(a), and then mapped onto a cylindrical surface by a simple geometrical transformation of nodal coordinates as shown in Fig. 2(b). The RUC model selected for this stent is shown in Fig. 2(c). As discussed in the previous section, planes j − and j + are the periodic planes where constraint equations, Eqs. (2a), (2b) and (6) should be applied, and the rotational symmetric conditions, Eqs. (1a)–(1c), should be imposed for the nodes on planes k − and k + . (Only internal pressure load is considered in all examples in this paper.) Fig. 3(a) shows the finite element mesh of the RUC of a Palmaz-Schatz stent with the balloon, which has been studied by Chua et al. [3]. In the present calculation same material models and parameters as in [3] were used in order to make a comparison. The material of the stent is stainless steel. A bi-linear elasto-plastic material model is assumed with Young’s modulus E = 193 GPa, tangent modulus ET = 692 MPa, yield strength y = 0.207 GPa and Poisson’s ratio = 0.27. The balloon material is considered to be hyperelastic and represented by the two-parameter Mooney–Rivlin model with C10 =1.06881 MPa, C01 =0.710918 MPa and density=1070 kg/m3 . This hyperelastic model is one of the most commonly used in balloon model-
Fig. 2. A quarter of a Palmaz-Schatz stent and a repeated unit cell (RUC) model: (a) the roll-out geometry of stent in rectangular coordinates; (b) real 3D geometry in cylindrical coordinates; (c) a typical RUC of this stent.
Fig. 3. RUC model for the Palmaz-Schatz stent: (a) RUC before the expansion of balloon; (b) RUC after the expansion of the balloon with the contour of von Mises strain distribution at p = 0.409 MPa.
ing. In the simulation process, the inner surface of the balloon is subjected to a uniform internal pressure which gradually increased from 0 to 0.409 MPa as used in Ref. [3]. Numerical analysis is conducted by ABAQUS Explicit codes and the total kinetic energy is checked to be less than 1.0% to ensure the quasi-static nature of the stent deployment in numerical simulation. Both the stent and the balloon are modeled by 3D solid elements. A surface to surface contact algorithm is selected to model the nonlinear contact between the balloon and the stent. When surfaces are in contact they usually transmit shear as well as
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Fig. 4. A complete round of stent tessellated by RUC before and after balloon expansion.
normal compressive forces across their interface. Coulomb friction model is used to model the frictional contacts between the balloon and the stent. The basic concept of this friction model is that the contacting surfaces can only carry a certain magnitude of shear force before they start to slide from each other. This maximum shear force is determined by the coefficient of friction. In this study, the coefficient of friction is chosen to be 0.2. Different values of the coefficient ranging from 0.05 to 0.25 are also tested and the difference in the computational results is found insignificant since the balloon surface would crimple into the vanity parts of the stent, making the contacting surfaces hard to slip from each other. 4.1.1. Deformation and expansion–pressure relation Fig. 3(b) shows the deformed shape and the distributions of von Mises strain (equivalent plastic strain) in the stent RUC when the pressure on the balloon, p, reaches 0.409 MPa. Fig. 4 shows a full round of the stent before and after the balloon expansion that are tessellated from the undeformed and deformed RUC models, respectively. A clearer global deformation pattern of the stent can be observed from the figure. In the stent deployment, internal pressure inflates the balloon and further makes the stent expand, and the relationship between the internal pressure and the extent of stent expansion is therefore of importance for the stent design and clinical application. For the current stent/balloon system the stent diameter–pressure relation is shown in Fig. 5 (the curve with square symbols). It can be seen from the curve that the dilation of stent diameter from the start increases very slowly. Numerical results show that initial yielding starts from the corners of the bridges and the struts when the pressure reached 0.12 MPa (as point A shown in Fig. 3). Yielding areas further increase with the increase of the pressure and become large enough to form yielding hinges when the pressure is about 0.40 MPa. At this point the diameter of the stent increases drastically, meaning that a little bit of increment in pressure after this critical value causes a big increase in the stent diameter. For clinical deployment of stents, these critical pressure values have practical significance.
Fig. 5. Stent diameter vs. internal pressure.
In some analyses of stents, the balloon was excluded from the computational model [2,6,9,10], i.e. the internal pressure is directly applied onto the inner surface of the stents. This is the case when the relationship between the stress status and the expanded stent diameter is of interest. To make a comparison, the stent expansion curve without balloon and the expansion curve of balloon itself are computed separately by applying pressure directly into the inner surface of the balloon or the stent. The results are shown in Fig. 5 by the curves with triangular and circular symbols, respectively. It can be seen from the three curves in the figure that a significant part of work done by the internal pressure is transferred into the strain energy of the balloon. In addition, the pressure needed to expand the balloon and stent together as a system is not a simple summation of the pressures to expand the balloon and the stent individually. Therefore, the stent/balloon catheter model should be used in order to obtain correct stent expansion–pressure relation. 4.1.2. Stress distribution Fig. 6 shows the distributions of the von Mises equivalent stress, longitudinal and hoop stresses when the internal
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Fig. 6. Distribution of stresses e (von Mises), z and at p = 0.409 MPa calculated by RUC model.
pressure is 0.409 MPa. The stress at the radial direction, r , is found much smaller than z or . It can be seen that the longitudinal stress, z , is a primary stress component, instead of the hoop stress component, , although it seems that the largest deformation is in the hoop direction. This can be explained as that the large expansion in diameter is introduced by the large rotational deformation of the strut around the bridge (see Fig. 6). This is caused by the bending of the stent strut in the –z plane and the longitudinal bending stress is thus the dominant component. The distribution of z across the strut’s cross section in Fig. 6 clearly shows that the stress ranges from maximum tensile to maximum compressive stresses, which indicates the bending phenomenen of the struts. From Fig. 3(b) one can also see that the maximum von Mises strain is about 0.168, which is not at the same order as the expansion rate of the diameter (from 3.0 to 4.8 mm). From the mechanics point of view, this is a large displacement (or rotation) and medium strain problem in nature. Note that two layers of elements are used in the thickness direction of the RUC model. The FEM results show that z and , are almost uniformly distributed along the thickness direction. This is true since the thickness of the stent is much smaller than its other dimensions and the stent can be considered as a thin-walled structure. 4.1.3. Comparison of results of RUC model with quarter model To make a comparison, a quarter model of the stent, which was employed by Chua et al. [3], is also re-analyzed as shown in Fig. 7. The size (nodes and elements) and CPU time of the quarter model are, respectively, 12 and 28 times those of the RUC
Fig. 7. Quarter model for the Palmaz-Schatz stent with deformation at p = 0.409.
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Table 1 Comparison of the expanded diameter and von Mises stress at a corner node between RUC and other models Pressure (MPa)
0.1
Expanded diameter (mm) RUC Model 3.05 Quarter Model 3.05 3.1 Chua et al. [3]a Von Mises Stress (MPa) RUC Model 25.3 Quarter Model 25.0 – Chua et al. [3]a a Numbers
0.2
3.11 3.11 3.2 214 212 –
0.3
3.46 3.47 3.4 225 220 –
0.409
4.79 4.83 4.6 283 281 286
are directly measured from the figures from [3].
model. The corner node at the RUC model (Node A, Fig. 3) and a corresponding node in the quarter model (Node A’ in Fig. 7) are identified to compare the computational accuracy of the two models. Table 1 gives the comparisons of the expanded stent diameters and von Mises stresses between the RUC model at Node A and quarter model at Node A’ as well as the results from Ref. [3]. It can be seen that the differences in the displacement and stress between RUC model and the quarter model are well below 1.0%. The table also shows that results of the RUC and our re-calculated quarter model are also close to those in Ref. [3], which are directly measured from the figures in the reference since precise numerical data were not given in the paper. One can therefore conclude that the RUC approach with a much smaller model size can produce same accurate stress and deformation results of the stent as much bigger models. 4.2. Closed-cell stents with V- or S-links Stents of the Palmaz-Schatz type are strong but too rigid. To improve the flexibility of such stents, V-shaped links were first added in the NIR䉸 stents to accommodate the change in shape during plastic deformation. Based on this concept, stents with U-, S-, N- or W-shaped links have also been designed and analyzed [21,22,10]. Figs. 8(a) and 10(a) show the closed-cell stents with V- and S- links, which are, respectively, termed as the V-stent and S-stent in this study. The percentages of the metal surface over that of artery are 17.6% and 19.1% for the V-stent and the S-stent, respectively. From the roll-out geometries of the V- and S-stent shown in the figures, one can clearly see the existence of RUCs in these stent structures. Figs. 8(b) and 10(b) show RUCs selected for these stents, respectively. As indicated in the previous section, the selection of RUCs is not unique and an appropriate RUC should facilitate the application of the periodic boundary conditions and save the labor in data preparations. It can be seen that there is only one symmetrical plane in the RUC of the V-stent and no symmetry for that of the S-stent. The applications of periodic boundary conditions are the same as that for the Palmaz-Schatz stent. Also the same material model and constants are used in the calculations.
Fig. 8. RUC model for the V-stent; (a) the roll-out geometry of stent in rectangular coordinates; (b) RUC before stent expansion; (c) deformed RUC and Mises stress when stent diameter is expanded from 3 to 6.4 mm.
4.2.1. Deformation and stress distribution Numerical analyses are conducted for stents with balloons. The maximum pressure applied on the inner surface of the balloon is 0.16 MPa. Figs. 8(c) and 10(c) give the deformed shapes and the distribution of von Mises stresses when the stents are expanded to 6.4 mm in diameter for V-stent and 9.9 mm for S-stent, respectively. To have a clear observation of the deformation and stress distribution patterns of the whole stent, parts of stent geometries before and after expansion are also tessellated from the corresponding undeformed and deformed RUCs and shown in Figs. 9 and 11 for V-stent and S-stent, respectively. It is obvious that multiple yielding hinges are developed with the increasing internal pressure and they play a crucial role to the stent expansion. Also, critical areas of stress concentration and deformation can be identified from the numerical results, which may be used to guide the stent design and clinical application (see Figs. 9–11). The numerical results show that the boundary planes in z-direction of the RUC of the V-stent still remains plane, while the boundary planes of the S-stent no longer remains plane after deformation as shown in Fig. 12. The reason for this phenomenon is that the RUC of the V-stent is symmetrical with respect to A–A plane shown in Figs. 8(b) and (c), while there is no such symmetrical plane in the RUC of S-stent. As we have discussed in Section 3 that if a RUC is symmetrical in the z-direction, its boundary planes at the two ends will remain planes during the deformation. It is worth to note that, for asymmetrical RUCs such as the RUC of the S-stent, applying “plane remains plane” condition at the two boundary planes in z-direction of the RUC will induce invalid results. It can be further concluded that if the RUC and the applied load have the symmetry in both - and z-directions, such
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Fig. 11. The tessellated geometries of the S-stent before and after expansion based on RUC models. Fig. 9. The tessellated geometries of the V-stent before and after expansion based on RUC model.
Fig. 12. Deformation of the end plane of RUC of the S-stent in the z-direction. Fig. 10. RUC model for the S-stent; (a) the roll-out geometry of stent in rectangular coordinates; (b) RUC before stent expansion; (c) deformed RUC and Mises stress when stent diameter is expanded from 3 to 9.9 mm.
as in the case of the stent of the Palmaz-Schatz type, a further smaller model of a quarter of the RUC can be used in the FEM analysis. If the RUC and applied load are symmetrical only in z-direction, such as the V-stent, half of the RUC model is enough for the analysis. However, for the asymmetrical RUCs, such as the S-stent, a full RUC model should be used and
the general periodic boundary conditions, Eqs. (1a)–(1c) and (2a)–(2c) must be applied. In contrast, if the proposed RUC model and corresponding periodic boundary conditions are not applied for the analysis of the above three types of stents with an odd number of cells along the circumferential direction, a quarter model should be used for the Palmaz-Schatz stents, a half model for V-stents and a whole model for S-stents. Therefore the advantages of the RUC model become more evident for the stents with less symmetries.
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the foreshortening of both the V-stent and S-stent is significant and this is the disadvantage of the flexible stents. Also the foreshortening of the V-stent is more than that of the S-stent if both stents are expanded to the same diameter. This phenomenon is probably due to the weaker longitudinal stiffness of V shaped link comparing to that of the S shaped link with the dimensions used in this study. The superior flexibility is the advantage of the S-stent and V-stent, the noticeable foreshortening should be, however, considered in the design and deployment of such type of stents. 5. Conclusions Fig. 13. Expanded stent diameter with respect to the inner pressure.
Fig. 14. Foreshortening of stent with respect to expanded stent diameter.
Fig. 13 presents the relationship between the expanded diameter and pressure applied on the inner surface of the balloon. It can be seen from the figure that, similar to the case of the Palmaz-Schatz stent, the diameter of the stent changes slowly at the start of the deployment and increases drastically when the pressure reaches a critical value where the yielding hinges are formed. By comparing the stent expansion–pressure curves in Figs. 5 and 13 (all the three stents have the same initial diameter of 3 mm) one can see that the V- and S-stents are much easier to expand than the Palmaz-Schatz stent. The S-stent is more flexible than the V-stent when they have similar percentage of the metal surface over that of artery. 4.2.2. Foreshortening of stent When a stent is expanded, its dimension in the longitudinal direction will be reduced. This behavior is termed as “foreshortening” in the stent design and defined as Foreshortening =
L0 − L × 100%, L0
(10)
where L0 is original length of the stent and L is the length of the stent after expansion, as indicated in Figs. 8(b) and 10(b). Fig. 14 shows the foreshortening of V- and S-stents with respect to the expanded diameter of the stent. It can be seen that
The repeated unit cell (RUC) model can be efficiently used in the analysis of the mechanical properties of stents with closed cells. A unified form of periodic boundary conditions for FEM analysis of stents has been presented. Stents of the PalmazSchatz type and with V- or S-shaped links are analyzed with the RUC approach. The computed deformation and stresses agree well with results that are obtained with much larger stent models and with much more computational efforts comparing to the present approach. The global picture of the deformation pattern and the stress distribution can be obtained by a simple geometrical tessellation of the deformed RUCs. With the application of the internal pressure, the diameter of the stent changes slowly at the start of the deployment and increases drastically when the pressure reaches a critical value where the yielding hinges are formed. Numerical results show that V- and S-stents are easier to expand than the Palmaz-Schatz stent. However, the foreshortening of the S-stent and V-stent is noticeable and this is the shortcoming of such flexible stents. The present RUC model with application of the unified periodic boundary conditions can accurately predict the behavior of most part of the whole stent except for the limited areas at the two ends. The main advantage of the RUC approach is that a much smaller numerical model with less computational effort is needed to achieve the same analysis accuracy as that by much larger models. The limitation of the current RUC approach is that it is incapable to provide stress and deformation information where the effect of free edge exists. A comprehensive numerical analysis should also include the artery and plaque, which are neglected in this study. References [1] D. Stoeckel, C. Bonsignore, S. Duda, A survey of stent designs, Minimally Invasive Ther. Allied Technol. 11 (4) (2002) 137–147. [2] S.N.D. Chua, B.J. Mac Donald, M.S.J. Hashmi, Finite-element simulation of stent expansion, J. Mater. Process. Technol. 120 (2002) 335–340. [3] S.N.D. Chua, B.J. Mac Donald, M.S.J. Hashmi, Finite element simulation of stent and balloon interaction, J. Mater. Process. Technol. 143–144 (2003) 591–597. [4] S.N.D. Chua, B.J. Mac Donald, M.S.J. Hashmi, Effects of varying slotted tube (stent) geometry on its expansion behaviour using finite element method, J. Mater. Process. Technol. 155–156 (2004) 1764–1771. [5] S.N.D. Chua, B.J. Mac Donald, M.S.J. Hashmi, Finite element simulation of slotted tube (stent) with the presence of plaque and artery by balloon expansion, J. Mater. Process. Technol. 155–156 (2004) 1772–1779.
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