Medical Engineering & Physics 36 (2014) 1480–1486
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The effect of stent graft oversizing on radial forces considering nitinol wire behavior and vessel characteristics B. Senf a,∗ , S. von Sachsen b , R. Neugebauer a , W.-G. Drossel a , H.-J. Florek c , F.W. Mohr b,d , C.D. Etz d,e a
Fraunhofer Institute for Machine Tools and Forming Technologies, Dresden, Germany Innovation Center Computer Assisted Surgery, Leipzig, Germany c Weisseritztal-Kliniken GmbH, Freital, Germany d Herzzentrum Leipzig GmbH, University Hospital, Leipzig, Germany e Mount Sinai Hospital, Cardiothoracic Surgery, New York, USA b
a r t i c l e
i n f o
Article history: Received 28 June 2012 Received in revised form 2 May 2014 Accepted 31 July 2014 Keywords: Surgical planning Finite element analysis Stent graft oversizing Endovascular surgery
a b s t r a c t Stent graft fixation in the vessel affects the success of endovascular aneurysm repair. Thereby the radial forces of the stent, which are dependent on several factors, play a significant role. In the presented work, a finite element sensitivity study was performed. The radial forces are 29% lower when using the hyperelastic approach for the vessel compared with linear elastic assumptions. Without the linear elastic modeled plaque, the difference increases to 35%. Modeling plaque with linear elastic material approach results in 8% higher forces than with a hyperelastic characteristic. The significant differences resulting from the investigated simplifications of the material lead to the conclusion that it is important to apply an anisotropic nonlinear approach for the vessel. The oversizing study shows that radial forces increase by 64% (0.54 N) when raising the oversize from 10 to 22%, and no further increase in force can be observed beyond these values (vessel diameter D = 12 mm). Starting from an oversize of 24%, the radial force steadily decreases. The findings of the investigation show that besides the oversizing the material properties, the ring design and the vessel characteristics have an influence on radial forces. © 2014 IPEM. Published by Elsevier Ltd. All rights reserved.
1. Introduction In the treatment of an abdominal aortic aneurysm with a stent graft, the choice of the oversize affects the therapeutic success. Research so far shows a link between stent graft oversizing and post-surgery complications such as prosthesis migration and endoleak type 1, which frequently occur. An endoleak type 1 exists if there is not an overall contact between stent graft rings and vessel wall components. Due to leaky areas, the blood can still flow into the aneurysm sac and therefore the therapy has failed. According to Mohan et al. [1], an increasing risk of endoleak type 1 exists for a stent graft oversize of 5 mm). Nowadays, there exist stent
∗ Corresponding author. E-mail address: [email protected] (B. Senf). http://dx.doi.org/10.1016/j.medengphy.2014.07.020 1350-4533/© 2014 IPEM. Published by Elsevier Ltd. All rights reserved.
grafts with and without additional proximal anchorage. Regarding the devices with hooks or bare springs, the radial force is not the most important factor. However, determination of the radial force through calculation of the stent graft vessel interaction can assist the surgeon to determine an oversizing for the stent graft. Thus, the question arises of which stent graft dimensioning is optimal for the specific patient, considering calcification, and if this task performed by the vascular surgeon can be supported by a simulation model. A stent graft has to have a sufficient fixation in all three landing zones to provide a good long-term outcome [4,5]. The fixation in the proximal landing zone is important to prevent migration and endoleak type 1a. In the distal landing zone, the fixation force is important to reduce endoleak type 1b. The focus of the presented work is the identification of the influence of material parameters and plaque on the simulation result value radial force. Against this background, it makes no difference if the parameter study is performed for the proximal or distal part of the stent graft. The reason for focusing the distal zone in the presented work is in the easier modelling of the geometry of the distal part compared with the proximal body which can have hooks or bare springs for additional anchorage.
B. Senf et al. / Medical Engineering & Physics 36 (2014) 1480–1486
The availability of material parameter for a diseased iliac artery by Holzapfel enables a good approximation of the vessel wall behavior. Usually the finite element modeling process starts with simplified approaches and later increases the level of detail. Therefore, the stent graft covering was first neglected and only one stent ring was modeled. One challenge for the surgeon in EVAR planning is to choose the right limb length and distal diameter which enables a good sealing and fixation. This can be handicapped by thrombus and plaque. Modeling the iliac limb in the distal landing zone is therefore of more interest for the surgeon than modeling the attachment point of bifurcation body and limb which boundary conditions like diameter and length of overlapping zone is usually determined by the manufacturer. While simulation-based analysis on the interaction of the wire stent (for the treatment of stenosis) and the vessel wall had been carried out by several research teams [6–10], only two relevant papers on the numerical description of the mechanical behavior of a stent graft with Nitinol rings have been detected so far [11,12]. By means of a simplified model, Kleinstreuer et al. analyzed the effects of different material combinations of the stent graft regarding fatigue resistance, radial forces and flexibility. The main focus was on assessing various Nitinol alloys for stent rings and the comparison of PET and ePTFE for the covering of the stent graft. Thereby, only the stent graft had been modeled and vessel wall properties were not taken into consideration. For the sealing section, Kleinstreuer et al. found that the graft material has no significant influence on the mechanical response of the stent wires [8]. Consequently, the limb covering of the stent graft is not considered within this study. Within the first simulation study by Scherer et al., the support of the vascular surgeon during the planning of a stent graft intervention was discussed. Blood pressure, consistency of intima and the existence of plaque were found to be influential parameters for the representation of the interaction between stent graft and vessel. Furthermore, the choice of the respective material models was presented [12]. The fixing force of the stent graft, as well as the contact pressure of the stent ring towards vessel wall and plaque have also been detected as helpful output variables for the optimization of implant selection. In order to represent vessel wall properties as realistically as possible, Scherer et al. used the commercially available materials approach AHYPER with the input variables taken from the material model by Holzapfel [13]. A model which is relevant for planning should contain both a realistic vessel wall and plaque, and the calculation time should be acceptable. Based on these considerations, it must be assessed which simplifications in the model may be carried out without significantly influencing the results. In this context, the requirement by Viceconti et al. concerning modeling in biomedical research must be mentioned, which considers sensitivity studies for the quantification of model uncertainty (inaccuracies in the model) an important task [14]. This model uncertainty, caused by making assumptions and using highly variable input parameters, must be taken into consideration when analyzing simulation results. This article discusses modeling variants with the aim of quantifying the effects of different model approaches on the output variable radial force. Moreover, analysis of possible correlation of radial force and oversize considering superelastic material properties of the Nitinol rings used in the stent graft are presented.
Fixing force = · (stent force + compression force)radial
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(1)
The radial force between the stent and the vessel is influenced by the oversize of the stent graft (stento ), the blood pressure (Bloodp ) and the material properties of vessel, plaque and stent (Vesselm , Plaquem , Stentm ), which will be analyzed in the following discussion: Stent force = f (Stento , Bloodp , Vesselm , Plaquem , Stentm )
(2)
The calculated radial force depends on the input variables. In this context, a sensitivity analysis was performed to determine the sensitivity of the output variable radial force with regard to the input variables. The above-mentioned requirement towards modeling in a biomedical context by Viceconti et al. was considered, and first results in the identification of model uncertainty were achieved. Within the analysis carried out, the radial force has been calculated for several model variants in order to investigate the influence of material model and material variables. 2.1. Materials approach in the finite element model For the description of the interaction between the stent and the vessel, the finite element method (FEM) is applied, enabling the representation of the complete nonlinear behavior. Besides the nonlinearity of the material of the stent ring and the vessel, large deformations and the difficulty of contact are considerable challenges for the calculation. The quality of the finite element model is therefore crucial [16]. Hexahedron elements with quadratic displacement behavior are applied, providing a homogenous meshing with the underlying topology. Tetrahedron elements, on the other hand, lead to increased system stiffness. Therefore, hexahedron elements should be applied as long as a representative meshing of topology can be carried out [17]. For more complex geometries (e.g. real vessel from CT scan), the use of four-sided elements may be required. 2.1.1. Stent graft For the development of the calculation model, the company Vascutek GmbH provided construction and material data of the Anaconda stent graft (see Fig. 1). In the simulation carried out, a model was applied in which the stent ring diameter matches the frequently used iliac limb L13 × 120 (Dring = 13 mm, length l = 120 mm). The stent ring consists of multiple winding Nitinol wires, which is described by means of a stress–strain diagram. The wire windings with its diameter of dwire = 0.14 mm do not go into the model separately in the form of a bundle, but they are modeled with a wire diameter resulting from the sum of the individual wires. Every wire winding experiences
2. Material and method According to Gebert de Uhlenbrock, the fixing force of the stent graft is made up of the radial forces and the systemic compression force (blood pressure) and takes into consideration the friction couple stent graft/vessel by using a friction coefficient with a value of 0.4 [15]:
Fig. 1. Anaconda stent graft (bifurcate body with two iliac limbs), Vascutek GmbH.
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Fig. 2. Comparison of a real stent ring “Anaconda iliac limb” (Vascutek GmbH) and FE stent detail.
Fig. 3. SMA SSD from ANSYS simulation of Anaconda stent ring.
a maximum strain of ε = 0.01 at the minor outer cross-sectional regions due to its circular bending during manufacturing process that is low in relation to the strain in use and will be neglected (Fig. 2). ε=
dwire Dring
(3)
Suitable material models are available for the numerical simulation of Nitinol properties. The commercial finite element software ANSYS features the nonlinear material model “Shape Memory Alloy” (SMA) according to Petrini/Auricchio for the characterization of superelastic material behavior [18]. The applied SMA material law in ANSYS by Aurrichio offers the consideration of the asymmetrical tension/compression behavior of nickel–titanium [19]. The stress–strain diagram (SSD) of the superelastic NiTi provided by the manufacturer Euroflex GmbH delivers important input variables but without distinguishing between tension and compression. This simplification should be avoided but may be accepted for the herewith presented finite element model development and oversizing investigation. From a cyclical-aged material test, relevant information on the Young’s modulus, Poisson ratio, limit of elastic strain and four strain values of hysteresis were attained as shown in Fig. 3. The characteristic key quantities for the superelastic material of the stent graft are listed in Table 1 and visualized in Fig. 3 as SSD. The radial forces of the stent graft are not depending on the very low bending stiffness of the covering. Thus, the covering is neglected in the simulation model, as it is assumed of subordinate importance for the forces between the implant and the vessel. This is a proper assumption in a structural, mechanical consideration but may be reconsidered in a fluid mechanics simulation [8].
2.1.2. Vessel with plaque Currently, there are several approaches for modeling of blood vessels, differing in the number of vessel wall layers and material models of the vessel wall. Linear elastic [20,21] and nonlinear elastic material models have been applied so far [6–9,22]. According to an analysis performed by Holzapfel, the nonlinear elastic approach, referred to as hyperelastic, is closest to reality [13]. For characterizing this behavior, the Mooney–Rivlin model is widely used, though. This approach can only represent hyperelastic behavior with isotropic properties. Histological analysis of vessel material by Holzapfel shows that the fibers of the vessel wall have different orientations and therefore the material is anisotropic [13]. The material model developed by Holzapfel for the characterization of hyperelastic, anisotropic properties of vessels is recently available for the FE simulation software ANSYS Version 14. A means of description which features both the nonlinear elastic and the direction-sensitive characteristics is available as standard with AHYPER in the commercial simulation software ANSYS Version 13 that was available within the research period and therefore AHYPER has been used during the analysis [11]. For describing the plaque properties, a hyperelastic isotropic material approach (Mooney–Rivlin) is mostly selected in the simulation models currently available [6,8,9,22]. Exceptions can be found in Liang et al. who use a viscoelastic model and in Holzapfel et al. using a linear elastic model (Young’s modulus of 12.6 ± 4.7 MPa). Holzapfel refers to his own experimental data about calcified plaque [23]. For model design, the use of a quarter segment of the vessel instead of a real CT scan is suitable. Basic results can be gathered from this stent–vessel interaction simulation model with minimized calculation effort. It is desirable in future to extend the simulation model to a real vessel topology. The geometrical parameters for vessel and plaque thicknesses which were used are listed in Table 2. For the unloaded vessel, a diameter of D = 12 mm was set. In contrast to the polymorphous shape in nature, an idealized sine-shaped cross-section is chosen to represent the plaque as shown in Fig. 4. The material parameters for the description of vessel and plaque, both with linear elastic and hyperelastic properties [24], are listed in Table 3. The applied constitutive strain–energy density formula consisting of a volumetric Wv and a deviatoric Wd part of the strain energy is stated in the following function. The approximation of the Holzapfel material law due to AHYPER is described in [12].
W = Wv (J) + Wd C, A ⊗ A, B ⊗ B Wv (J) =
1 (J − 1)2 d
Wd (I4 , I6 ) =
6
k=2
(4) (5)
ck (I4 − 1)k +
6 m=2
em (I6 − 1)m
Table 2 Geometrical parameters for vessel and plaque [13].
Table 1 Material variables for superelastic behavior of Anaconda stent ring. Young’s Modulus (MPa)
Poisson Ratio [1]
Elastic Strain [1]
P1
P2
P3
P4
(MPa)
(MPa)
(MPa)
(MPa)
53900
0.33
0.067
600
608
368
360
Model item
Thickness (mm)
Intima Media Adventitia Plaque
0.33 1.32 0.96 0.50
(6)
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Fig. 4. FE model consisting of three layers of vessel wall, plaque and stent ring (rotational symmetric multiplication of the ¼ detail).
Table 3 Vessel and plaque material parameters. Linear elastic
Young’s modulus (MPa)
Adventitia Media Intima Plaque
2.98 [21] 8.95 [21] 2.98 [21] 12.6 [25]
Anisotropic hyperelastic tissue types for an external iliac artery for use in the AHYPER model [11,13]
c (kPa)
c2 = e2 (kPa)
c4 = e4 (kPa)
c6 = e6 (kPa)
ϕ (◦ )
Adventitia (A) Non-diseased media (M-nos) Fibrous (diseased) intima (I-fm)
3.5 30.0 61.6
32.8 2.0 22.5
1013.5 2.3 110.3
20,878.5 1.8 360.2
49 7 0
Isotropic hyperelastic (Mooney-Rivlin)
c10 (kPa)
c01 (kPa)
c20 (kPa)
c11 (kPa)
c02 (kPa)
c30 (kPa)
Plaque
−0.496
0.507
3.638
1.194
0.000
4.737
The constitutive equation for the strain energy potential of the isotropic hyperelasticity according to Mooney–Rivlin [19] is listed below.
2
W = c10 I 1 − 3 + c01 I 2 − 3 + c20 I 1 − 3
2
+c21 I 1 − 3
+ c11 I 1 − 3
I 2 − 3 + c12 I 1 − 3
2
I2 − 3
2
I 2 − 3 + c02 I 2 − 3
3
+ c03 I 2 − 3
Due to the use of non ECG-triggered CT images, which are usually produced during diagnostics, it is not possible to determine whether the recorded vessel diameter represents the diastolic or the systolic state. In order to consider the maximum radial prestress in the vessel, the systolic blood pressure is used for the creation of the initial state within the calculation model. 2.2. Boundary conditions and load steps in the finite element model According to the values of the axial extension of the vessel determined experimentally by Holzapfel, the extension is considered as preload with a strain of 10% of the initial state in z-direction [26]. The axial section of the circular vessel is fixed in positive direction at the upper cross-sectional area and negative direction at the lower area with node displacements. The rotational symmetric model is simplified into a quarter cutout. Both peripheral cross-sectional areas are cyclical restrained by a multiple set of displacement constraint equations distinguishing the nodes on the edges and the nodes within the upper and lower half of the areas. In order to determine the interaction between the stent and the vessel, five steps are planned, which are shown in the flowchart (Fig. 5). The vessel model in unstressed initial state is the basis for the calculation. For all three layers of the vessel and plaque, the material model and the respective parameters are defined. The first calculation step includes the axial extension. In the following step, the
+
3
+ c30 I 1 − 3
(7)
1 (J − 1)2 d
1. step: Vessel modeling in initial state
2. step: Axial extension of vessel in z-direction
3. step: Radial vessel prestress by blood pressure
4. step: Stent modeling with oversizing
5. step: Interaction between stent and vessel Fig. 5. Flowchart of finite element analysis.
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blood pressure is applied. Based on this state, the stent ring diameter can be detected using a defined oversize. After the modeling of the stent, contact elements on the inner surface of the vessel and the outer surface of the stent are used for detecting the interaction between both contact partners. 2.3. Sensitivity analysis based on finite element simulation The advantage of the simulation inter alia is the possibility of varying the parameters to examine their influence on the overall system or selected output variables. For the detection of the optimal degree of detail suitable for the calculation of output variables relevant for planning, several model variants have been created. They differ in the number of nodes and the material approaches for the description of the vessel wall and plaque. A variation of the stent oversize provides insights into its influence on the radial forces. For all calculations, a constant internal pressure of 180 mmHg is applied. For the oversizing study an inner diameter of D = 12 mm with an oversize of 15% was assumed. 2.3.1. Discretization study To determine the optimal degree of detail, the required number of nodes of the numerical analogous model was examined regarding its result-quality within a discretization study. For this purpose, a coarse mesh was generated as initial basis, increasing the degree of meshing by means of multiple iterations. During the solution of the equation system, the radial force of the stent ring in correlation with the number of nodes has been considered as the decisive output parameter. 2.3.2. Influence of the material approach Finite element calculations of nonlinear tasks are highly complex. Therefore, the linear elastic material approach, which is simpler from a numerical point of view, is compared to the more complex anisotropic hyperelastic material law. Besides the calculation of the stent radial forces for a vessel without plaque, calculations for a vessel with linear elastic plaque and hyperelastic plaque using a vessel with linear elastic and anisotropic hyperelastic material behavior have been examined (model variants 1–5). 2.3.3. Correlation of radial forces and oversizing By means of a parameter study, the dependence of the radial forces of a Nitinol stent graft on the selected oversize is examined (model variant 6). Thereby, an anisotropic hyperelastic approach is applied for the vessel and a linear elastic approach is applied to the plaque. In several steps, the variation of the oversize is performed, starting at a minimum of 10% up to a maximum of 28%. Depending on the oversize selected, a radial force emerges between the stent ring and the vessel which is significant for the calculations. 3. Results
Fig. 6. Radial forces between stent and vessel in correlation with the number of nodes. Table 4 Calculation time of model variants. FE model
Calculation time (h)
Variant 1 Variant 2 Variant 3 Variant 4 Variant 5 Variant 6
0.7 4.5 0.8 4.6 8.3 4.6
The application of a simulation model in everyday clinical practice does not only require a suitable degree of detail, but should also take as little calculation time as possible. This aspect must be taken into consideration when the optimal degree of detail for the support of stent graft planning is determined. The calculation times, which have been detected within the model study, are listed in Table 4. For calculation, a 64-bit system with Windows server 2008 was used, which could process all data in the central memory without swapping. The system uses four processor cores with a frequency of 3 GHz for solving the calculation task. 3.2. Influence of the material approach The analysis of different material models for the vessel (cf. Tables 5 and 6) shows that the radial forces with linear elastic plaque and a hyperelastically modeled vessel are approx. 29% lower than with a linear elastic vessel (when using an stent graft oversize of 15%). The same tendency was detected for the calculation model without plaque. For this comparison, the force with the hyperelastic material approach was 35% lower than with the linear approach. Examining potential alterations of radial forces when using different material models for the description of plaque properties, the radial force was found to be ca. 8% higher with a hyperelastic vessel, a stent graft oversize of 15% and linear elastic plaque, compared to hyperelastic plaque (cf. Table 7).
3.1. Discretization study–application-specific degree of detail for the calculation of the radial forces of the stent graft
3.3. Correlation of radial forces and ring diameter
Different degrees of meshing resulted in the finding that no significant alteration of results can be detected whether 20,000 nodes or more are used. Fig. 6 shows the behavior of radial forces in correlation with the number of nodes. For the quarter model, 15,000 elements were used for the meshing of the vessel and 1000 elements for the Nitinol stent ring. Thus, the model altogether consists of just fewer than 16,000 solid elements and approximately 3000 contact elements. The cross-sectional area of the stent is meshed with 32 quadratic elements with a length of 0.32 mm along the wire.
The examination of the optimal stent ring oversize was performed for a model with vessel wall thickness of 2.61 mm (cf. Table 2) and an internal diameter variation between 12 and 20 mm (hyperelastic material approach). The plaque has a thickness of 0.5 mm (linear elastic material approach). For plaque and vessel, a blood pressure of 180 mmHg is applied. Thereby, the oversize was steadily raised in several steps from 10 to 28% which is indicated as variant 6. The results in Fig. 7 give information on the effects of oversize variation on radial forces. Based on the assumed boundary conditions, the maximum radial
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Table 5 Influence of material approach without plaque. Variant
Vessel
Oversizing (%)
Blood pressure (mmHg)
Plaque
Radial force (N)
1 2
Linear elastic Anisotropic hyperelastic
15 15
180 180
None None
1.75 1.25
Table 6 Influence of material approach with plaque. Variant
Vessel
Oversizing (%)
Blood pressure (mmHg)
Plaque
Radial force (N)
3 4
Linear elastic Anisotropic hyperelastic
15 15
180 180
Linear elastic Linear elastic
1.80 1.17
Table 7 Material approaches for plaque–hyperelastic vessel. Variant
Vessel
Oversizing (%)
Blood pressure (mmHg)
Plaque
Radial force (N)
4 5
Anisotropic hyperelastic Anisotropic hyperelastic
15 15
180 180
Linear elastic Hyperelastic
1.17 1.08
bending is at the maximum. The stent ring is stretched on the one side and compressed on the opposite side. The stretching is associated with a strain of material, causing the stent to rotate in the direction of compression. The compression is associated with a material compression, which is also meant to balance through a tendency towards skew in the direction of expansion. The compression of the sinus-shaped Anaconda stent ring results in torsion at the reversal point (cf. Fig. 8). A torsion bar shows the same effect when the relation of length and radius increases and thereby the torsional moment decreases with the same skew. 4. Discussion
Fig. 7. Development of radial forces with increasing stent oversize.
force was detected with an oversize of 18 to 22% depending on the vessel diameter. With a diameter of D = 12 mm, the radial force increases by 0.54 N (approx. 64%) when raising the oversize from 10 to 22%. No further increase in force can be observed beyond these values when raising the oversize further. Starting from an oversize of 24%, the radial force decreases steadily. The radius of the examined sinus-shaped Anaconda stent graft decreases by compression and thereby, the amplitude of its sinus shape increases. At the (upper and lower) dead centers, the
Based on the results of the calculated model variants with different degrees of meshing, assumptions can be made about the application-specific degree of detail. In the applied quarter model, the optimal number of nodes is 20,000, as finer meshing did not show considerable variations regarding radial forces in the results. The determined element size can be used as an indicator for the intended patient-specific model. The simulation study shows a correlation of radial forces and oversizing. The material properties of the Nitinol ring and its design are of significant importance in this regard. While retaining the oversize, the extension of the ring diameter results in a decreased sum of radial forces. This occurrence can be explained with the relation of the wire ring diameter and the wire cross-section. The cross-section remains constant, whereas the wire ring diameter
Fig. 8. Radial forces of Nitinol ring in N.
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depends on the internal pressure or the vessel diameter, respectively. Furthermore, the results show that the highest possible radial force also depends on patient-specific boundary conditions, such as vessel wall properties and calcifications. One limitation of the study is the use of an idealized model part which cannot be used in clinical practice. Based on the results in this work providing important model parameters, a patient-specific model is achievable. This should also consider stent graft covering which can assist the surgeon in determining an optimal length of the prosthesis (leg length). Another important aspect which is required by vascular surgeons is the development of radial forces under pulsation. Within a patient study on the influence of the cardiac cycle on the alteration of the vessel diameter, Pol et al. [27] found that this area of the iliac artery can cause a change of the diameter by up to 2.3 mm. Therefore, further work has to simulate stent graft behavior for systole and diastole state to quantify the influence of changes in blood pressure on radial forces. Because this study only deals with the numerical consideration, further work should aim an experimental validation of the calculation results. The parameter study shows that the radial force is influenced by the vessel wall properties. Therefore, future work has to aim on the integration of individual material parameters for attaining a more realistic simulation model. Echocardiogram-based MRI data can be examined for determination of the vessel extension dependent on blood pressure. Hoskins et al. found out that the change in vessel extension can be up to 10% of the diameter. According to Peterson et al. [28], the measured change in diameter can be used for calculation of the pressure–strain elastic modulus. The pressure–strain elastic modulus does not consider wall thickness and therefore it can usually be used as an index for better estimation of vessel wall stiffness. According to Hayashi et al., it is possible to calculate the Young’s modulus of the vessel wall based on the pressure–strain elastic modulus and the wall thickness which can used as input parameter for a finite element analysis (FEA) [29]. However, this FEA is then based on a 1-layer vessel wall. Ethical approval Concerning the present study, no ethical approval is required. Acknowledgements This work was sponsored by funds of the European Regional Development Fund (ERDF) and the state of Saxony within the framework of measures supporting the technology sector. Conflict of interest The authors have no conflict of interest in relation to the present study. References [1] Mohan IV, Laheij JP, Harris PL. Risk factors for endoleak and the evidence for stent-graft oversizing in patients undergoing endovascular aneurysm repair. Eur J Vasc Endovasc Surg 2001;21:344–9. [2] Schurink GWH, Aarts NJM, van Baalen JM, Kool LJS, van Bockel JH. Stent attachment site-related endoleakage after stent graft treatment: an in vitro study of the effects of graft size, stent type, and atherosclerotic wall changes. J Vasc Surg 1999;30:658–67.
[3] Sternbergh WC, Money SR, Greenberg RK, Chuter TAM. Influence of endograft oversizing on device migration, endoleak, aneurysm shrinkage, and aortic neck dilation: results from the Zenith multicenter trial. J Vasc Surg 2004;39(1): 20–6. [4] Waasdorp EJ, de Vries J-PPM, Sterkenburg A, Vos J-A, Kelder HJC, Moll FL, et al. The association between iliac fixation and proximal stent-graft migration during EVAR follow-up: mid-term results of 154 talent devices. Eur J Vasc Endovasc Surg 2009;37:681–7. [5] Figueroa CA, Zarins CK. Computational analysis of displacement forces acting on endografts used to treat aortic aneurysms. Stud Mechanobiol Tissue Eng Biomater 2011;7:221–46. [6] Auricchio F, Di Loreto M, Sacco E. Finite-element analysis of a stenotic artery revascularization through a stent insertion. Comput Methods Biomech Biomed Eng 2001;4:249–63, http://dx.doi.org/10.1080/10255840108908007. [7] Liang DK, Yang DZ, Qi M, Wang WQ. Finite element analysis of the implantation of a balloon-expandable stent in a stenosed artery. Int J Cardiol 2005;104(3):314–8. [8] Migliavacca F, Petrini L, Massarotti P, Schievano S, Auricchio F, Dubini G. Stainless and shape memory alloy coronary stents: a computational study on the interaction with the vascular wall. Biomech Model Mechanobiol 2004;2(4):205–17. [9] Pericevic I, Lally C, Toner D, Kelly D. The influence of plaque composition on underlying arterial wall stress during stent expansion: the case for lesionspecific stents. Med Eng Phys 2009;31:428–33. [10] Wu W, Qi M, Liu XP, Yang DZ, Wang WQ. Delivery and release of Nitinol stent in carotid artery and their interactions: a finite element analysis. J Biomech 2007;40(13):3034–40. [11] Kleinstreuer C, Li Z, Basciano CA, Seelecke S, Farber MA. Computational mechanics of Nitinol stent grafts. J Biomech 2008;41:2370–8. [12] Scherer S, Treichel T, Ritter N, Triebel G, Drossel WG, Burgert O. Surgical stent planning—simulation parameter study for models based on DICOM standards. Int J Comput Assisted Radiol Surg 2011;6(3):319–27. [13] Holzapfel GA. Biomechanics of soft tissues with application to arterial walls. Math Comput Model Biol Syst 2002;1:1–37. [14] Viceconti M, Olsen S, Nolte LP. Extracting clinically relevant data from finite element simulations. Clin Biomech 2005;20:451–4. [15] Gebert de Uhlenbrock A. Designanalyse von endovaskulären Aortenprothesen. Dissertation Technische Universität Hamburg-Harburg; 2009. [16] Rust W. Nichtlineare Finite-Elemente-Berechnungen—Kontakt, Geometrie, Material, vol. 1. Wiesbaden: Vieweg + Teubner; 2009. p. 254–60. [17] Fröhlich P. FEM-Anwendungspraxis. Einstieg in die Finite Elemente Analyse, vol. 1. Wiesbaden: Vieweg; 2005. p. 40–1. [18] Petrini L, Migliavacca F, Massarotti P, Schievano S, Dubini G, Auricchio F. Computational studies of shape memory alloy behavior in biomedical applications. J Biomech Eng 2005;127(4):716–25. [19] ANSYS, Inc. Material Reference. ANSYS Manual 2011; Version 13. [20] Giannoglu G, Giannakoulas G, Soulis J, Chatzizisis Y, Perdikides T, Melas N, et al. Predicting the risk of rupture of abdominal aortic aneurysms by utilizing various geometrical parameters: revisiting the diameter criterion. Angiology 2006;57(4):487–94. [21] Gao F, Watanabem M, Matsuzawa T. Stress analysis in a layered aortic arch model under pulsatile blood flow. Biomed Eng Online 2006, http://dx.doi.org/10.1186/1475-925X-5-25. [22] Wang WQ, Liang DK, Yang DZ, Qi M. Analysis of the transient expansion behavior and design optimization of coronary stents by finite element method. J Biomech 2006;39:21. [23] Gasser C, Holzapfel GA. Modelling plaque fissuring and dissection during balloon angioplasty intervention. Ann Biomed Eng 2007;35:711–23. [24] Fritsch A. ANSYS user defined materials: implementation of an anisotropic, hyperelastic material law for soft biological tissues. Infoplaner 2008:34–7, 02/2008. [25] Holzapfel GA, Sommer G, Regitnig P. Anisotropic mechanical properties of tissue components in human atherosclerotic plaques. J Biomech Eng 2004;126:657–65. [26] Holzapfel GA, Gasser TC. A new constitutive framework for arterial wall mechanics and a comparative study of material models. J Elasticity 2000: 1–48. [27] Pol JA, Truijers M, van der Vliet JA, Fillinger MF, Marra SP, Renema WKJ, et al. Impact of dynamic computed tomographic angiography on endograft sizing for endovascular aneurysm repair. J Endovasc Therapy 2009;16:546–51. [28] Peterson LH, Jensen RE, Parnell J. Mechanical properties of arteries in vivo. Circ Res 1960;8:622–39. [29] Hayashi K. Experimental approaches on measuring the mechanical-properties and constitutive laws of arterial-walls. J Biomech Eng 1993;115:481–8.
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Medical Engineering & Physics journal homepage: www.elsevier.com/locate/medengphy
The effect of stent graft oversizing on radial forces considering nitinol wire behavior and vessel characteristics B. Senf a,∗ , S. von Sachsen b , R. Neugebauer a , W.-G. Drossel a , H.-J. Florek c , F.W. Mohr b,d , C.D. Etz d,e a
Fraunhofer Institute for Machine Tools and Forming Technologies, Dresden, Germany Innovation Center Computer Assisted Surgery, Leipzig, Germany c Weisseritztal-Kliniken GmbH, Freital, Germany d Herzzentrum Leipzig GmbH, University Hospital, Leipzig, Germany e Mount Sinai Hospital, Cardiothoracic Surgery, New York, USA b
a r t i c l e
i n f o
Article history: Received 28 June 2012 Received in revised form 2 May 2014 Accepted 31 July 2014 Keywords: Surgical planning Finite element analysis Stent graft oversizing Endovascular surgery
a b s t r a c t Stent graft fixation in the vessel affects the success of endovascular aneurysm repair. Thereby the radial forces of the stent, which are dependent on several factors, play a significant role. In the presented work, a finite element sensitivity study was performed. The radial forces are 29% lower when using the hyperelastic approach for the vessel compared with linear elastic assumptions. Without the linear elastic modeled plaque, the difference increases to 35%. Modeling plaque with linear elastic material approach results in 8% higher forces than with a hyperelastic characteristic. The significant differences resulting from the investigated simplifications of the material lead to the conclusion that it is important to apply an anisotropic nonlinear approach for the vessel. The oversizing study shows that radial forces increase by 64% (0.54 N) when raising the oversize from 10 to 22%, and no further increase in force can be observed beyond these values (vessel diameter D = 12 mm). Starting from an oversize of 24%, the radial force steadily decreases. The findings of the investigation show that besides the oversizing the material properties, the ring design and the vessel characteristics have an influence on radial forces. © 2014 IPEM. Published by Elsevier Ltd. All rights reserved.
1. Introduction In the treatment of an abdominal aortic aneurysm with a stent graft, the choice of the oversize affects the therapeutic success. Research so far shows a link between stent graft oversizing and post-surgery complications such as prosthesis migration and endoleak type 1, which frequently occur. An endoleak type 1 exists if there is not an overall contact between stent graft rings and vessel wall components. Due to leaky areas, the blood can still flow into the aneurysm sac and therefore the therapy has failed. According to Mohan et al. [1], an increasing risk of endoleak type 1 exists for a stent graft oversize of 5 mm). Nowadays, there exist stent
∗ Corresponding author. E-mail address: [email protected] (B. Senf). http://dx.doi.org/10.1016/j.medengphy.2014.07.020 1350-4533/© 2014 IPEM. Published by Elsevier Ltd. All rights reserved.
grafts with and without additional proximal anchorage. Regarding the devices with hooks or bare springs, the radial force is not the most important factor. However, determination of the radial force through calculation of the stent graft vessel interaction can assist the surgeon to determine an oversizing for the stent graft. Thus, the question arises of which stent graft dimensioning is optimal for the specific patient, considering calcification, and if this task performed by the vascular surgeon can be supported by a simulation model. A stent graft has to have a sufficient fixation in all three landing zones to provide a good long-term outcome [4,5]. The fixation in the proximal landing zone is important to prevent migration and endoleak type 1a. In the distal landing zone, the fixation force is important to reduce endoleak type 1b. The focus of the presented work is the identification of the influence of material parameters and plaque on the simulation result value radial force. Against this background, it makes no difference if the parameter study is performed for the proximal or distal part of the stent graft. The reason for focusing the distal zone in the presented work is in the easier modelling of the geometry of the distal part compared with the proximal body which can have hooks or bare springs for additional anchorage.
B. Senf et al. / Medical Engineering & Physics 36 (2014) 1480–1486
The availability of material parameter for a diseased iliac artery by Holzapfel enables a good approximation of the vessel wall behavior. Usually the finite element modeling process starts with simplified approaches and later increases the level of detail. Therefore, the stent graft covering was first neglected and only one stent ring was modeled. One challenge for the surgeon in EVAR planning is to choose the right limb length and distal diameter which enables a good sealing and fixation. This can be handicapped by thrombus and plaque. Modeling the iliac limb in the distal landing zone is therefore of more interest for the surgeon than modeling the attachment point of bifurcation body and limb which boundary conditions like diameter and length of overlapping zone is usually determined by the manufacturer. While simulation-based analysis on the interaction of the wire stent (for the treatment of stenosis) and the vessel wall had been carried out by several research teams [6–10], only two relevant papers on the numerical description of the mechanical behavior of a stent graft with Nitinol rings have been detected so far [11,12]. By means of a simplified model, Kleinstreuer et al. analyzed the effects of different material combinations of the stent graft regarding fatigue resistance, radial forces and flexibility. The main focus was on assessing various Nitinol alloys for stent rings and the comparison of PET and ePTFE for the covering of the stent graft. Thereby, only the stent graft had been modeled and vessel wall properties were not taken into consideration. For the sealing section, Kleinstreuer et al. found that the graft material has no significant influence on the mechanical response of the stent wires [8]. Consequently, the limb covering of the stent graft is not considered within this study. Within the first simulation study by Scherer et al., the support of the vascular surgeon during the planning of a stent graft intervention was discussed. Blood pressure, consistency of intima and the existence of plaque were found to be influential parameters for the representation of the interaction between stent graft and vessel. Furthermore, the choice of the respective material models was presented [12]. The fixing force of the stent graft, as well as the contact pressure of the stent ring towards vessel wall and plaque have also been detected as helpful output variables for the optimization of implant selection. In order to represent vessel wall properties as realistically as possible, Scherer et al. used the commercially available materials approach AHYPER with the input variables taken from the material model by Holzapfel [13]. A model which is relevant for planning should contain both a realistic vessel wall and plaque, and the calculation time should be acceptable. Based on these considerations, it must be assessed which simplifications in the model may be carried out without significantly influencing the results. In this context, the requirement by Viceconti et al. concerning modeling in biomedical research must be mentioned, which considers sensitivity studies for the quantification of model uncertainty (inaccuracies in the model) an important task [14]. This model uncertainty, caused by making assumptions and using highly variable input parameters, must be taken into consideration when analyzing simulation results. This article discusses modeling variants with the aim of quantifying the effects of different model approaches on the output variable radial force. Moreover, analysis of possible correlation of radial force and oversize considering superelastic material properties of the Nitinol rings used in the stent graft are presented.
Fixing force = · (stent force + compression force)radial
1481
(1)
The radial force between the stent and the vessel is influenced by the oversize of the stent graft (stento ), the blood pressure (Bloodp ) and the material properties of vessel, plaque and stent (Vesselm , Plaquem , Stentm ), which will be analyzed in the following discussion: Stent force = f (Stento , Bloodp , Vesselm , Plaquem , Stentm )
(2)
The calculated radial force depends on the input variables. In this context, a sensitivity analysis was performed to determine the sensitivity of the output variable radial force with regard to the input variables. The above-mentioned requirement towards modeling in a biomedical context by Viceconti et al. was considered, and first results in the identification of model uncertainty were achieved. Within the analysis carried out, the radial force has been calculated for several model variants in order to investigate the influence of material model and material variables. 2.1. Materials approach in the finite element model For the description of the interaction between the stent and the vessel, the finite element method (FEM) is applied, enabling the representation of the complete nonlinear behavior. Besides the nonlinearity of the material of the stent ring and the vessel, large deformations and the difficulty of contact are considerable challenges for the calculation. The quality of the finite element model is therefore crucial [16]. Hexahedron elements with quadratic displacement behavior are applied, providing a homogenous meshing with the underlying topology. Tetrahedron elements, on the other hand, lead to increased system stiffness. Therefore, hexahedron elements should be applied as long as a representative meshing of topology can be carried out [17]. For more complex geometries (e.g. real vessel from CT scan), the use of four-sided elements may be required. 2.1.1. Stent graft For the development of the calculation model, the company Vascutek GmbH provided construction and material data of the Anaconda stent graft (see Fig. 1). In the simulation carried out, a model was applied in which the stent ring diameter matches the frequently used iliac limb L13 × 120 (Dring = 13 mm, length l = 120 mm). The stent ring consists of multiple winding Nitinol wires, which is described by means of a stress–strain diagram. The wire windings with its diameter of dwire = 0.14 mm do not go into the model separately in the form of a bundle, but they are modeled with a wire diameter resulting from the sum of the individual wires. Every wire winding experiences
2. Material and method According to Gebert de Uhlenbrock, the fixing force of the stent graft is made up of the radial forces and the systemic compression force (blood pressure) and takes into consideration the friction couple stent graft/vessel by using a friction coefficient with a value of 0.4 [15]:
Fig. 1. Anaconda stent graft (bifurcate body with two iliac limbs), Vascutek GmbH.
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Fig. 2. Comparison of a real stent ring “Anaconda iliac limb” (Vascutek GmbH) and FE stent detail.
Fig. 3. SMA SSD from ANSYS simulation of Anaconda stent ring.
a maximum strain of ε = 0.01 at the minor outer cross-sectional regions due to its circular bending during manufacturing process that is low in relation to the strain in use and will be neglected (Fig. 2). ε=
dwire Dring
(3)
Suitable material models are available for the numerical simulation of Nitinol properties. The commercial finite element software ANSYS features the nonlinear material model “Shape Memory Alloy” (SMA) according to Petrini/Auricchio for the characterization of superelastic material behavior [18]. The applied SMA material law in ANSYS by Aurrichio offers the consideration of the asymmetrical tension/compression behavior of nickel–titanium [19]. The stress–strain diagram (SSD) of the superelastic NiTi provided by the manufacturer Euroflex GmbH delivers important input variables but without distinguishing between tension and compression. This simplification should be avoided but may be accepted for the herewith presented finite element model development and oversizing investigation. From a cyclical-aged material test, relevant information on the Young’s modulus, Poisson ratio, limit of elastic strain and four strain values of hysteresis were attained as shown in Fig. 3. The characteristic key quantities for the superelastic material of the stent graft are listed in Table 1 and visualized in Fig. 3 as SSD. The radial forces of the stent graft are not depending on the very low bending stiffness of the covering. Thus, the covering is neglected in the simulation model, as it is assumed of subordinate importance for the forces between the implant and the vessel. This is a proper assumption in a structural, mechanical consideration but may be reconsidered in a fluid mechanics simulation [8].
2.1.2. Vessel with plaque Currently, there are several approaches for modeling of blood vessels, differing in the number of vessel wall layers and material models of the vessel wall. Linear elastic [20,21] and nonlinear elastic material models have been applied so far [6–9,22]. According to an analysis performed by Holzapfel, the nonlinear elastic approach, referred to as hyperelastic, is closest to reality [13]. For characterizing this behavior, the Mooney–Rivlin model is widely used, though. This approach can only represent hyperelastic behavior with isotropic properties. Histological analysis of vessel material by Holzapfel shows that the fibers of the vessel wall have different orientations and therefore the material is anisotropic [13]. The material model developed by Holzapfel for the characterization of hyperelastic, anisotropic properties of vessels is recently available for the FE simulation software ANSYS Version 14. A means of description which features both the nonlinear elastic and the direction-sensitive characteristics is available as standard with AHYPER in the commercial simulation software ANSYS Version 13 that was available within the research period and therefore AHYPER has been used during the analysis [11]. For describing the plaque properties, a hyperelastic isotropic material approach (Mooney–Rivlin) is mostly selected in the simulation models currently available [6,8,9,22]. Exceptions can be found in Liang et al. who use a viscoelastic model and in Holzapfel et al. using a linear elastic model (Young’s modulus of 12.6 ± 4.7 MPa). Holzapfel refers to his own experimental data about calcified plaque [23]. For model design, the use of a quarter segment of the vessel instead of a real CT scan is suitable. Basic results can be gathered from this stent–vessel interaction simulation model with minimized calculation effort. It is desirable in future to extend the simulation model to a real vessel topology. The geometrical parameters for vessel and plaque thicknesses which were used are listed in Table 2. For the unloaded vessel, a diameter of D = 12 mm was set. In contrast to the polymorphous shape in nature, an idealized sine-shaped cross-section is chosen to represent the plaque as shown in Fig. 4. The material parameters for the description of vessel and plaque, both with linear elastic and hyperelastic properties [24], are listed in Table 3. The applied constitutive strain–energy density formula consisting of a volumetric Wv and a deviatoric Wd part of the strain energy is stated in the following function. The approximation of the Holzapfel material law due to AHYPER is described in [12].
W = Wv (J) + Wd C, A ⊗ A, B ⊗ B Wv (J) =
1 (J − 1)2 d
Wd (I4 , I6 ) =
6
k=2
(4) (5)
ck (I4 − 1)k +
6 m=2
em (I6 − 1)m
Table 2 Geometrical parameters for vessel and plaque [13].
Table 1 Material variables for superelastic behavior of Anaconda stent ring. Young’s Modulus (MPa)
Poisson Ratio [1]
Elastic Strain [1]
P1
P2
P3
P4
(MPa)
(MPa)
(MPa)
(MPa)
53900
0.33
0.067
600
608
368
360
Model item
Thickness (mm)
Intima Media Adventitia Plaque
0.33 1.32 0.96 0.50
(6)
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Fig. 4. FE model consisting of three layers of vessel wall, plaque and stent ring (rotational symmetric multiplication of the ¼ detail).
Table 3 Vessel and plaque material parameters. Linear elastic
Young’s modulus (MPa)
Adventitia Media Intima Plaque
2.98 [21] 8.95 [21] 2.98 [21] 12.6 [25]
Anisotropic hyperelastic tissue types for an external iliac artery for use in the AHYPER model [11,13]
c (kPa)
c2 = e2 (kPa)
c4 = e4 (kPa)
c6 = e6 (kPa)
ϕ (◦ )
Adventitia (A) Non-diseased media (M-nos) Fibrous (diseased) intima (I-fm)
3.5 30.0 61.6
32.8 2.0 22.5
1013.5 2.3 110.3
20,878.5 1.8 360.2
49 7 0
Isotropic hyperelastic (Mooney-Rivlin)
c10 (kPa)
c01 (kPa)
c20 (kPa)
c11 (kPa)
c02 (kPa)
c30 (kPa)
Plaque
−0.496
0.507
3.638
1.194
0.000
4.737
The constitutive equation for the strain energy potential of the isotropic hyperelasticity according to Mooney–Rivlin [19] is listed below.
2
W = c10 I 1 − 3 + c01 I 2 − 3 + c20 I 1 − 3
2
+c21 I 1 − 3
+ c11 I 1 − 3
I 2 − 3 + c12 I 1 − 3
2
I2 − 3
2
I 2 − 3 + c02 I 2 − 3
3
+ c03 I 2 − 3
Due to the use of non ECG-triggered CT images, which are usually produced during diagnostics, it is not possible to determine whether the recorded vessel diameter represents the diastolic or the systolic state. In order to consider the maximum radial prestress in the vessel, the systolic blood pressure is used for the creation of the initial state within the calculation model. 2.2. Boundary conditions and load steps in the finite element model According to the values of the axial extension of the vessel determined experimentally by Holzapfel, the extension is considered as preload with a strain of 10% of the initial state in z-direction [26]. The axial section of the circular vessel is fixed in positive direction at the upper cross-sectional area and negative direction at the lower area with node displacements. The rotational symmetric model is simplified into a quarter cutout. Both peripheral cross-sectional areas are cyclical restrained by a multiple set of displacement constraint equations distinguishing the nodes on the edges and the nodes within the upper and lower half of the areas. In order to determine the interaction between the stent and the vessel, five steps are planned, which are shown in the flowchart (Fig. 5). The vessel model in unstressed initial state is the basis for the calculation. For all three layers of the vessel and plaque, the material model and the respective parameters are defined. The first calculation step includes the axial extension. In the following step, the
+
3
+ c30 I 1 − 3
(7)
1 (J − 1)2 d
1. step: Vessel modeling in initial state
2. step: Axial extension of vessel in z-direction
3. step: Radial vessel prestress by blood pressure
4. step: Stent modeling with oversizing
5. step: Interaction between stent and vessel Fig. 5. Flowchart of finite element analysis.
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blood pressure is applied. Based on this state, the stent ring diameter can be detected using a defined oversize. After the modeling of the stent, contact elements on the inner surface of the vessel and the outer surface of the stent are used for detecting the interaction between both contact partners. 2.3. Sensitivity analysis based on finite element simulation The advantage of the simulation inter alia is the possibility of varying the parameters to examine their influence on the overall system or selected output variables. For the detection of the optimal degree of detail suitable for the calculation of output variables relevant for planning, several model variants have been created. They differ in the number of nodes and the material approaches for the description of the vessel wall and plaque. A variation of the stent oversize provides insights into its influence on the radial forces. For all calculations, a constant internal pressure of 180 mmHg is applied. For the oversizing study an inner diameter of D = 12 mm with an oversize of 15% was assumed. 2.3.1. Discretization study To determine the optimal degree of detail, the required number of nodes of the numerical analogous model was examined regarding its result-quality within a discretization study. For this purpose, a coarse mesh was generated as initial basis, increasing the degree of meshing by means of multiple iterations. During the solution of the equation system, the radial force of the stent ring in correlation with the number of nodes has been considered as the decisive output parameter. 2.3.2. Influence of the material approach Finite element calculations of nonlinear tasks are highly complex. Therefore, the linear elastic material approach, which is simpler from a numerical point of view, is compared to the more complex anisotropic hyperelastic material law. Besides the calculation of the stent radial forces for a vessel without plaque, calculations for a vessel with linear elastic plaque and hyperelastic plaque using a vessel with linear elastic and anisotropic hyperelastic material behavior have been examined (model variants 1–5). 2.3.3. Correlation of radial forces and oversizing By means of a parameter study, the dependence of the radial forces of a Nitinol stent graft on the selected oversize is examined (model variant 6). Thereby, an anisotropic hyperelastic approach is applied for the vessel and a linear elastic approach is applied to the plaque. In several steps, the variation of the oversize is performed, starting at a minimum of 10% up to a maximum of 28%. Depending on the oversize selected, a radial force emerges between the stent ring and the vessel which is significant for the calculations. 3. Results
Fig. 6. Radial forces between stent and vessel in correlation with the number of nodes. Table 4 Calculation time of model variants. FE model
Calculation time (h)
Variant 1 Variant 2 Variant 3 Variant 4 Variant 5 Variant 6
0.7 4.5 0.8 4.6 8.3 4.6
The application of a simulation model in everyday clinical practice does not only require a suitable degree of detail, but should also take as little calculation time as possible. This aspect must be taken into consideration when the optimal degree of detail for the support of stent graft planning is determined. The calculation times, which have been detected within the model study, are listed in Table 4. For calculation, a 64-bit system with Windows server 2008 was used, which could process all data in the central memory without swapping. The system uses four processor cores with a frequency of 3 GHz for solving the calculation task. 3.2. Influence of the material approach The analysis of different material models for the vessel (cf. Tables 5 and 6) shows that the radial forces with linear elastic plaque and a hyperelastically modeled vessel are approx. 29% lower than with a linear elastic vessel (when using an stent graft oversize of 15%). The same tendency was detected for the calculation model without plaque. For this comparison, the force with the hyperelastic material approach was 35% lower than with the linear approach. Examining potential alterations of radial forces when using different material models for the description of plaque properties, the radial force was found to be ca. 8% higher with a hyperelastic vessel, a stent graft oversize of 15% and linear elastic plaque, compared to hyperelastic plaque (cf. Table 7).
3.1. Discretization study–application-specific degree of detail for the calculation of the radial forces of the stent graft
3.3. Correlation of radial forces and ring diameter
Different degrees of meshing resulted in the finding that no significant alteration of results can be detected whether 20,000 nodes or more are used. Fig. 6 shows the behavior of radial forces in correlation with the number of nodes. For the quarter model, 15,000 elements were used for the meshing of the vessel and 1000 elements for the Nitinol stent ring. Thus, the model altogether consists of just fewer than 16,000 solid elements and approximately 3000 contact elements. The cross-sectional area of the stent is meshed with 32 quadratic elements with a length of 0.32 mm along the wire.
The examination of the optimal stent ring oversize was performed for a model with vessel wall thickness of 2.61 mm (cf. Table 2) and an internal diameter variation between 12 and 20 mm (hyperelastic material approach). The plaque has a thickness of 0.5 mm (linear elastic material approach). For plaque and vessel, a blood pressure of 180 mmHg is applied. Thereby, the oversize was steadily raised in several steps from 10 to 28% which is indicated as variant 6. The results in Fig. 7 give information on the effects of oversize variation on radial forces. Based on the assumed boundary conditions, the maximum radial
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Table 5 Influence of material approach without plaque. Variant
Vessel
Oversizing (%)
Blood pressure (mmHg)
Plaque
Radial force (N)
1 2
Linear elastic Anisotropic hyperelastic
15 15
180 180
None None
1.75 1.25
Table 6 Influence of material approach with plaque. Variant
Vessel
Oversizing (%)
Blood pressure (mmHg)
Plaque
Radial force (N)
3 4
Linear elastic Anisotropic hyperelastic
15 15
180 180
Linear elastic Linear elastic
1.80 1.17
Table 7 Material approaches for plaque–hyperelastic vessel. Variant
Vessel
Oversizing (%)
Blood pressure (mmHg)
Plaque
Radial force (N)
4 5
Anisotropic hyperelastic Anisotropic hyperelastic
15 15
180 180
Linear elastic Hyperelastic
1.17 1.08
bending is at the maximum. The stent ring is stretched on the one side and compressed on the opposite side. The stretching is associated with a strain of material, causing the stent to rotate in the direction of compression. The compression is associated with a material compression, which is also meant to balance through a tendency towards skew in the direction of expansion. The compression of the sinus-shaped Anaconda stent ring results in torsion at the reversal point (cf. Fig. 8). A torsion bar shows the same effect when the relation of length and radius increases and thereby the torsional moment decreases with the same skew. 4. Discussion
Fig. 7. Development of radial forces with increasing stent oversize.
force was detected with an oversize of 18 to 22% depending on the vessel diameter. With a diameter of D = 12 mm, the radial force increases by 0.54 N (approx. 64%) when raising the oversize from 10 to 22%. No further increase in force can be observed beyond these values when raising the oversize further. Starting from an oversize of 24%, the radial force decreases steadily. The radius of the examined sinus-shaped Anaconda stent graft decreases by compression and thereby, the amplitude of its sinus shape increases. At the (upper and lower) dead centers, the
Based on the results of the calculated model variants with different degrees of meshing, assumptions can be made about the application-specific degree of detail. In the applied quarter model, the optimal number of nodes is 20,000, as finer meshing did not show considerable variations regarding radial forces in the results. The determined element size can be used as an indicator for the intended patient-specific model. The simulation study shows a correlation of radial forces and oversizing. The material properties of the Nitinol ring and its design are of significant importance in this regard. While retaining the oversize, the extension of the ring diameter results in a decreased sum of radial forces. This occurrence can be explained with the relation of the wire ring diameter and the wire cross-section. The cross-section remains constant, whereas the wire ring diameter
Fig. 8. Radial forces of Nitinol ring in N.
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depends on the internal pressure or the vessel diameter, respectively. Furthermore, the results show that the highest possible radial force also depends on patient-specific boundary conditions, such as vessel wall properties and calcifications. One limitation of the study is the use of an idealized model part which cannot be used in clinical practice. Based on the results in this work providing important model parameters, a patient-specific model is achievable. This should also consider stent graft covering which can assist the surgeon in determining an optimal length of the prosthesis (leg length). Another important aspect which is required by vascular surgeons is the development of radial forces under pulsation. Within a patient study on the influence of the cardiac cycle on the alteration of the vessel diameter, Pol et al. [27] found that this area of the iliac artery can cause a change of the diameter by up to 2.3 mm. Therefore, further work has to simulate stent graft behavior for systole and diastole state to quantify the influence of changes in blood pressure on radial forces. Because this study only deals with the numerical consideration, further work should aim an experimental validation of the calculation results. The parameter study shows that the radial force is influenced by the vessel wall properties. Therefore, future work has to aim on the integration of individual material parameters for attaining a more realistic simulation model. Echocardiogram-based MRI data can be examined for determination of the vessel extension dependent on blood pressure. Hoskins et al. found out that the change in vessel extension can be up to 10% of the diameter. According to Peterson et al. [28], the measured change in diameter can be used for calculation of the pressure–strain elastic modulus. The pressure–strain elastic modulus does not consider wall thickness and therefore it can usually be used as an index for better estimation of vessel wall stiffness. According to Hayashi et al., it is possible to calculate the Young’s modulus of the vessel wall based on the pressure–strain elastic modulus and the wall thickness which can used as input parameter for a finite element analysis (FEA) [29]. However, this FEA is then based on a 1-layer vessel wall. Ethical approval Concerning the present study, no ethical approval is required. Acknowledgements This work was sponsored by funds of the European Regional Development Fund (ERDF) and the state of Saxony within the framework of measures supporting the technology sector. Conflict of interest The authors have no conflict of interest in relation to the present study. References [1] Mohan IV, Laheij JP, Harris PL. Risk factors for endoleak and the evidence for stent-graft oversizing in patients undergoing endovascular aneurysm repair. Eur J Vasc Endovasc Surg 2001;21:344–9. [2] Schurink GWH, Aarts NJM, van Baalen JM, Kool LJS, van Bockel JH. Stent attachment site-related endoleakage after stent graft treatment: an in vitro study of the effects of graft size, stent type, and atherosclerotic wall changes. J Vasc Surg 1999;30:658–67.
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