Deformation Mechanics of Self-expanding Venous Stents: Modelling and Experiments
Masoud Hejazi, Farrokh Sassani, Joel Gagnon, York Hsiang, Srikantha Phani
DDeformation Mechanics of Self-expanding VenousStents: Modelling and Experiments
Masoud Hejazi a , Farrokh Sassani a , J¨oel Gagnon b , York Hsiang b , A. SrikanthaPhani a, ∗ a Department of Mechanical Engineering, 6250 Applied Science Lane, University of BritishColumbia, Vancouver, B.C, Canada V6T 1Z4 b Division of Vascular Surgery, 4219-2775 Laurel Street, Vancouver General Hospital,Vancouver, B.C, Canada V5Z 1M9
Abstract
Deformation properties of venous stents based on braided design, chevron de-sign, Z design, and diamond design are compared using in vitro experimentscoupled with analytical and finite element modelling. Their suitability for de-ployment in different clinical contexts is assessed based on their deformationcharacteristics. Self-expanding stainless steel stents possess superior collapseresistance compared to Nitinol stents. Consequently, they may be more reli-able to treat diseases like May-Thurner syndrome in which resistance againsta concentrated (pinching) force applied on the stent is needed to prevent col-lapse. Braided design applies a larger radial pressure particularly for vesselsof diameter smaller than 75% of its nominal diameter, making it suitable for along lesion with high recoil. Z design has the least foreshortening, which aidsin accurate deployment. Nitinol stents are more compliant than their stainlesssteel counterparts, which indicates their suitability in veins. The semi-analyticalmethod presented can aid in rapid assessment of topology governed deformationcharacteristics of stents and their design optimization.
Keywords:
Venous Stents, Foreshortening, Radial Pressure, CollapseManuscript word count: 3500 ∗ Corresponding authorphone: +1 (604) 822-6998)fax: +1 (604) 822-2403email: [email protected]
Preprint submitted to Elsevier February 23, 2021 a r X i v : . [ phy s i c s . m e d - ph ] F e b . Introduction Venous obstruction is a common pathological condition of the lower ex-tremities, which reduces the vessel patency and hence the blood flow. Venousobstruction can be a result of a non-thrombotic syndrome (like May-Thurnerand venous insufficiency) or an acute/chronic venous thrombosis (Murphy et al.,2017; Beebe-Dimmer et al., 2005). The prevalence and incidence rates of thevenous obstruction vary depending on the underlying medical conditions. Forinstance, deep vein thrombosis (DVT) affects 300,000 people in North Americaeach year with an incidence rate of 0 . <
40% and < −
60% of acute DVT cases, the fibrotic thrombus is attached to the wallcausing vein thickening and post-thrombotic recoil (Razavi et al., 2015; Deatricket al., 2011). In summary, in designing a venous stent, one should consider thelocalized pinching forces (May-Thurner), high recoil (fibrotic veins), foreshort-ening (reduction of the length during expansion), and the large distensibility(high compliance) of the healthy wall proximal and distal to the pathologicalregion. We first review pertinent literature on the deformation properties ofstents.The effect of radial pressure on the hemodynamics of a stented vessel hasbeen investigated through experimental and computational studies (Freemanet al., 2010; Bedoya et al., 2006; Lally et al., 2005; Zahedmanesh and Lally,2009). It is shown that a stent with excessive radial pressure constrains thecyclic dilation of the vessel leading to post-deployment complications such asrestenosis (Morrow et al., 2005; Vernhet et al., 2001). Nevertheless, an optimalradial pressure is required to avoid post-deployment recoil and migration (Liand Kleinstreuer, 2006). The radial pressure is a function of stent structuralparameters and the material properties, which has been extensively studied forbraided stents (Zaccaria et al., 2020; Kim et al., 2008), Z stents (Snowhill et al.,3001), and closed-cell venous stents (Dabir et al., 2018). In venous stenting,oversizing the stent is a common technique to avoid recoil and migration. Herewe provide a semi-analytical method, in Section 3, to determine the radial pres-sure variation due to stent oversizing.Collapse is a common failure mode for stents that are placed in veins (Mur-phy et al., 2017). Three different buckling modes in the collapse of balloon-expandable stents are identified in (Dumoulin and Cochelin, 2000) identifiedand collapse in self-expanding stents has been reported in (Kim et al., 2008;Dabir et al., 2018; Schwein et al., 2018). They observed mechanical instabilityby applying a pinching force on the stent structure. It has been reported thatthe local collapse stiffness of a stent depends critically on its strut geometryand the elastic modulus of the stent material (Duerig et al., 2000). We evaluatethe performance of a venous stent against collapse experimentally and explainhow we can qualitatively compare a stent’s behavior under collapse through aunit-cell study.Compliance of a stent also plays a vital role in its clinical performance par-ticularly for the venous system with high distensibility. The stent is deployedto maintain adequate contact with the healthy wall proximally and distally inorder to increase the anchoring area and avoid migration. The compliance mis-match between the vessel and the stent magnifies the post-deployment complica-tions (Berry et al., 2002; Morris et al., 2016; Post et al., 2019). The complianceof a stent/vessel is defined as the variation of pressure over the variation ofdiameter. Here, the stent compliance is determined through experiment andanalysis.Despite the critical role of the foreshortening in precise stent placement, thereis little study on the foreshortening of venous stents. For balloon-expandablearterial stents, however, the significance of longitudinal strain and the foreshort-ening mechanism has been elaborated in the literature (Douglas et al., 2014; Tanet al., 2011). However, the method cannot be used here due to differences in theexpansion mechanism of self-expanding stents. Here, we measure the foreshort-ening during stent expansion and analytically study this parameter in Section4. The above studies confirm the importance of mechanical properties and de-formation characteristics on the clinical performance of venous stents and a needto develop rapid assessment tools to inform the choice of stent topologies. Defor-mation characteristics of venous stents based on braided design, chevron design,Z design, and diamond design are compared using in vitro experiments coupledwith analytical and finite element modelling. Their suitability for deployment indifferent clinical contexts is assessed based on their deformation characteristics.We start with an in vitro experiment to evaluate these parameters in Section2. Afterwards, we employ the unit-cell study in Section 3 to determine radialpressure, compliance, foreshortening and collapse of the candidate stents in thisstudy. We assess the validity of the unit-cell study by comparing the resultswith observations from the in vitro experiment and discuss clinical relevance inSections 4 and 5. Concluding remarks and future work are given in Section 6. In vitro experiments
Self-expanding stents are available in different structural design and materi-als. Here we chose two stainless steel stents (Z design and Braided design) andtwo Nitinol stents (Diamond design, Chevron design) shown in Fig. 1. Notethat these stents are commercial designs currently being used in practice forvenous stenting and here we use the design names instead of commercial namesprovided by companies. Among these, Braided design is the only one with anopen-cell structure and the rest are closed-cell designs. Diamond design andChevron design are the recent designs dedicated to venous stenting, Braideddesign is a common off-labeled design (used for both arteries and veins), and Zdesign is a trachea stent commonly used for venous stenting as a local reinforce-ment (Murphy et al., 2017). To study the effect of the material and structuraldesign, we start with a series of in vitro tests to determine the radial pressure,collapse resistance, and foreshortening. Compliance, defined as the diametervariation ratio to radial pressure can be calculated as well.5 tent Design Stent Structure Unit-cell Geometry Strut Thickness(mm) Strut Length ( L s ) (mm) MaterialVolume/ L ength ( mm /mm) C hevron design Diamond design
Z design
Braided design
NA 0.33 NA 1.03
Figure 1: Stent designs used in this study and their structural design parameters.
The experimental setup for the in vitro tests are shown in Fig. 2. An en-vironmental chamber was used to maintain the ambient temperature at 37 o centigrade to simulate the body temperature. The radial pressure (i.e., cir-cumferential resistive pressure) measured by the radial crimping test accordingto (Morris et al., 2016; Duda et al., 2000) shown in Fig. 2(a). An aluminumfabric of width equal to the undeployed length of the stent is wrapped aroundthe fully deployed stent and threaded through a narrow gap between two rollers(diameter of 3 mm ).The lower edge of the fabric is attached to the fixed jawwhile the upper edge is attached to the moving jaw and the load cell. As theupper jaw moves upward, the circumference of the aluminum wrap decreasesleading to reduction of the internal diameter and radial crimping of the stent.Note that in this case we only measure the circumferential resistive force, not6 oad cellEnvironmental chamberTransparent shieldHeaterFixed endMoving end FF RollerStentAluminum FabricSide view (b)
Localized collapse (c)
Global collapse (a) R adial pressure test
Figure 2:
In vitro tests. (a) Stent is wrapped in an aluminum fabric sheet that is attached tothe material testing machine (Instron 5965) grips at both ends. The rollers reduce the frictionwhile the upper grip pulls the fabric and reduces the internal diameter of the aluminum warp.(b) The anvil applies compression locally at the mid-section of the stent. (c) The stent isglobally compressed between two steel compression plates. the chronic outward force that is applied by the stent to the wall during de-ployment. This is due to the fact that increasing the patency is commonlyperformed by immediate angioplasty after venous stenting. Hence, even if thechronic outward pressure is not enough, the angioplasty using balloon expan-sion can assist during deployment. Accordingly, the circumferential resistiveforce, resisting the post-deployment recoil, is a more representative characteris-tic in terms of clinical durability. In a temperature control chamber, the global7ollapse and local collapse tests, based on the deformation modes suggestedby (Dumoulin and Cochelin, 2000; Bandyopadhyay and Bose, 2013), were per-formed by compressing the stent by rigid plates and an anvil (tip diameter of10 mm ), respectively (See Fig. 2(a) and b ). Results from the experiments willbe presented and compared with semi-analytical model in Sections 4 and 5.
3. Unit-cell study and finite element analysis
The deformation characteristics of a stent rely on its lattice expansion mech-anism, which can be investigated through a unit-cell study. This method canreduce the computation time and cost since the entire stent is not modelled.First, we start with defining the unit-cell for each design (see Fig. 3), where acylindrical polar co-ordinate system ( r − θ − z ) is introduced in Fig. 3(a). Theforces and moments associated with these co-ordinate axes are F r , F θ and M z , M θ , respectively. Each stent has n a number of unitcells along the axial ( z ) direc-tion and n c number of unitcells in the circumferential ( θ ) direction. Assumingperiodic boundary conditions and axisymmetry of the structure, we can identifya unit-cell and define the boundary conditions at the decoupled joints/links asshown in Fig. 3(c). For uniform expansion and axisymmetric boundary condi-tions, the force F r , and the moments M θ , and M z can be neglected (Hejazi,2018). Since Braided design is made through braiding, we cannot define aclosed-joint uni-cell. Consequently, the approach to study the expansion of thisstent will be different and discussed separately.The venous pressure distribution acting on the stent is assumed to be uni-form. Consequently, the resultant force F p due to pressure, which is applied byvessel on all struts, acts at the center of the unit-cell in a radial direction andit is related to F θ (the tangential force applied to the joints) as: F p = 2 F θ sin β, β = πn c . (1)We can define the length ( L ) and the diameter ( D ) of an expanding stent ateach stage of deployment by following the method introduced by (Douglas et al.,8 =0 Deformed Strut 𝐹 𝜃 𝑀 𝜃𝑧 𝑟𝜃𝑟𝑧 𝑟 𝑧 𝜃 𝛽𝛽𝑟𝜃𝑟
24 13 𝑀 𝑧 𝐹 𝜃 𝑀 𝜃 𝐹 𝑟 𝐹 𝜃 𝐹 𝜃 𝛽 (a) (e)(f ) 𝐹 𝑝 𝐹 𝑝 𝛽𝑟𝑧 (d)(b)(c) (g) Figure 3: Typical geometry and loading conditions of a unit-cell in a stent structure. (a) cylin-drical coordinate system; (b) top view of the isolated (highlighted) unit-cell in the structure;(c) four isolated joints of the unit-cell; (d) front view of the isolated unit-cell in the structure;(e) the reaction moments and forces applied to the isolated joints, F p (resultant force due tocontact pressure applied by the vessel to the stent), M z and M θ (the reaction moment along z axis and θ axis), F r and F θ (the reaction forces along r axis and θ axis); (f) the front view ofa unit-cell loading condition; (g) kinematic role of a strut in deformation of a stent unit-cell.Each stent has n a number of unitcells along the axial ( z ) direction and n c number of unitcellsin the circumferential ( θ ) direction. L = n a ( l − u ) , (2)9 = n c ( w + 2 v ) π , (3)where u, v, l , and w are respectively axial displacement, circumferential dis-placement, undeformed length, and width of a unit-cell (Fig. 3(g)). Note thatfor an unexpanded stent u = v = 0 so that L = n a l and D = n c w π define theinitial length and diameter, respectively.The main purpose of deploying a stent is to maintain the patency of thelumen. A vessel that tends to recoil applies a redial pressure to the stent,which governs post-deployment performance (Duerig et al., 2000; Morlacchi andMigliavacca, 2013). This pressure can be defined as: P = F p Dβ ( l − u ) , (4)Where P is the lumen radial pressure (circumferential resistive pressure), thenumerator is the total applied force, and the denominator is the circumferentialarea of a unit-cell. By substituting (1), and (3) into (4) we have: P = 2 F θ sin ( πn c )( w + 2 v )( l − u ) . (5)We introduce the foreshortening parameter ( f ) as f = l − ll = 2 ul , (6)where l = l − u is the length of the unit-cell at a given stage of deployment.Using (6), we can rewrite the radial pressure as a function of foreshortening asfollows: P = 2 F θ tan ( πn c )( w − v ) l (1 + f ) . (7)The above indicates that P and f are inversely related. This suggests thatwhile a large value of foreshortening is undesirable for precise deployment alarger radial pressure can be achieved. It is worth noting that this compromisein clinical performance can be avoided by choosing zero foreshortening stent-designs proposed in Douglas et al. (2014)10ompliance of a stent is defined according to C = D − D D ( P − P ) , (8)where D i and P i are the incremental stent diameter and pressure. By substi-tuting equations (2) and (3) into (8), we have C = v − v sin ( πn c ) (cid:16) ( w +2 v ) F θ ( w +2 v )( l − u ) − F θ l − u (cid:17) . (9)Equations (2) to (9) indicate that for calculating the deformation charac-teristics (radial pressure, compliance, and foreshortening) we need to correlatethe strut bending force ( F θ ) with displacements in the circumferential ( v ) andlongitudinal ( u ) directions. In Sections 3.1 and 3.2 we study the bending mech-anism of the candidate stents, which governs the deformation characteristics ofthe stent. Chevron design and Diamond stent designs used in this study are made ofNitinol alloy, which provides the desired mechanical properties such as super-elasticity. The Nitinol struts undergo phase transition depending on the mechan-ical strain which influences their deformation characteristics. A mathematicalmodel is introduced for the bending analysis of Nitinol beams in (Mirzaeifaret al., 2013) . We apply their model to find the deformation of the stent strutsubjected to bending described in Fig. 3(g). The bending analysis for a can-tilever beam is summarized in Appendix A1. The Chevron design has a morecomplex shape, which can be divided into curved and straight sections (Fig. 4).Hence, we have to modify (5), (6), (7), and (9) by substituting circumferentialdisplacement v by 3 v and longitudinal displacement u by 0 . u to account for thenumber of struts in the unit-cell and the different orientation of the joints. Wecan use equations (10) and (11) respectively, to calculate the bending momentin the curved and straight portion as: M c = 12 F l cos α − F r (1 − cos θ ) , (10)11 l = 12 F l cos α − F x cos α + F r. (11)where θ and x locate the section in the curved and the straight part, respectivelyin Fig. 4(b) and in Fig. 4(d). Using (10) and (11) we can calculate thebending moment throughout the strut of Chevron design as a function of θ for the curved portion and x for the straight part. In these equations, l isthe effective arm length of the strut (contributing in bending moment), r isthe radius of the curved portion, and F = F θ cos β based on Fig. 4. For theDiamond stent (Nitinol) since there is no curved region at the intersection ofjoints, we can directly use the analysis of a cantilever beam in Appendix A1 tocalculate the internal bending moments during deployment. Having thus foundthe internal forces we can calculate the deformation properties of these twodesigns by following the Appendix A1 to find the displacements and calculateradial pressure and compliance using (7) and (9) . Deformed Strut F 𝑀 (cid:3004) F 𝑀 (cid:3039) (a)(b) ( d ) LinkageCurved Deformed Strut F 𝑀 (cid:3004) F 𝑀 (cid:3039) ( (cid:68) )( (cid:69) ) ( (cid:70) )( (cid:71) )LinkageCurved C i r c u m f e r e n ti a l Axial (c ) Figure 4: (a) Chevron design unit-cell geometry; (b) a strut of the unit-cell including thecurved and linear parts; (c) the bending moment and shear force at a given cross section ofthe curve part; (d) the bending moment and shear force at a given cross section of the linkage. .2. Unit-cell deformation characteristics of Z design and Braided design Geometric design imparts self-expanding ability to Z design and Braideddesign stents. Two sets of steel links and a coil that connect these links arethe elements that define the Z design unit-cell. Accordingly, we can calculatethe displacement of the strut based on the coil angular twist and the elasticbending of the link. The displacement in a single strut is a combination ofthe coil angular twist and the link bending deflection. The torsional stiffness( k t ) of the coil and the torsion angle can be calculated through k t = Ed . Dn ,where E, d, D, n, and l are torsional stiffness, Young’s modulus, wire diameter,the diameter of the coil, number of coil body turns, the torsion angle and linklength (Budynas et al., 2008). Accordingly, the angular twist of the coil part,longitudinal and circumferential displacements can be determined through γ = k t F ( l + 2 d ) , (12) u = l (1 − cos γ ) , (13) v = F l EI + l sin γ. (14)In (12) we can determine the longitudinal displacement of the strut tip interms of the angular twist. Here, we ignore the contribution of the strut bendingas it is in the elastic region of the bending deformation and can be assumed smallin comparison to the effect of an angular twist. To calculate the circumferentialdisplacement, we can use (13), in which, the first term is the contribution ofbending displacement and the second term represents the effect of coil angulartwist.Braided design is fabricated by braided wires forming its entire structure.Consequently, it does not have any true geometrical unit-cell or joint and there-fore no real strut can be defined. The expansion of Braided design was inves-tigated to derive an equation that relates the pressure to the diameter of the13raided design structure based on slender bar theory (Wang and Ravi-Chandar,2004a,b) as: P = n cos α πr sin α EI sin αr (cid:18) cos αr − cos α r (cid:19) − GI p cos αr (cid:18) cos α sin αr − cos α sin α r (cid:19) , (15) f = (cid:112) λ + 4 π r − π r − λ λ , (16)Where n, E, G, I p , r, and r are respectively number of wires, Young’s modulus,shear modulus, the area moment of inertia, stent radius, and nominal radius.To calculate the foreshortening, we can use (16), in which λ and r are initialhelical wire pitch and diameter. Furthermore, to determine the compliance,instead of using (9), which has been used for other stents, we can directlyuse (8) associated with radial pressure values from (15). In this work, the simulation has been performed in ABAQUS/Standard com-mercial code linked with a user material subroutine (UMAT) based on (Lagoudas,2008). Here, the FE method was employed to evaluate two different problems.We studied the bending of the Chevron design and Diamond design struts tovalidate the method presented in (Mirzaeifar et al., 2013), which addresses thebending mechanics of a Nitinol beam. We used material properties of NITI-I forNitinol stents, and an elastic modulus of 193 GPa, Poisson’s ratio of 0.3 and uni-axial yield stress of 260 MPa for steel stents (Bandyopadhyay and Bose, 2013;Duerig et al., 2000). We employed UMAT (user material subroutine), which isa framework for ABAQUS users to implement a material (Nitinol model wasnot available in the software library at the time of this study). The foundationof the UMAT code is the thermo-mechanical constitutive model of Nitinol (Au-ricchio and Taylor, 1997; Bhattacharya, 2003; Lagoudas, 2008). We used fullyintegrated solid linear hexahedron element (C3D20) for other designs. Theglobal size for the meshing varies from 0.075 mm to 0.05 mm based on the mesh14ensitivity test. The numerical finite element results are used to validate thesemi-analytical method for bending of Nitinol struts, presented in Section 3.1.
4. Results
A comparison is drawn between FE simulation (Section 3.2) and the semi-analytical approach (Section 3.1) results for a strut bending of the Nitinol stentsin Fig. 5. The change of the slope reflects the onset of phase transition (austeniteto martensite). The FE solution here is stiffer for Chevron design. For Diamonddesign, however, the semi-analytical method demonstrated a stiffer response.This is because of the shear stress contribution to the phase transition, whichhas been ignored in Section 3.1. The change of the slope reflects the onset ofphase transition (austenite to martensite). If von Mises stress is more than 260MPa, the region has a pure martensite phase. A core of pure austenite alwaysexists for the Chevron design joints. For Diamond design, however, we observea core of martensite phase close to the joints.The foreshortening of the stents, illustrated in Fig. 6, has been calculatedthrough (6)) for joint based designs (Z design, Diamond design, and Chevrondesign) and (16) for Braided design. It should be noted that the foreshorteningis a dimensionless characteristic and the presented results in Fig. 6 are validfor different stent sizes. As shown, in all cases the experimental values of fore-shortening is smaller. This can be the result of longitudinal compressive forces(due to the friction) that is applied to the stent during the crimping test in thealuminum fabric.In clinical practice, stents are oversized to maintain a required radial pres-sure. Fig. 7 shows the radial pressure versus the over-sizing parameter ( D n /D v ),where D n and D v are stent nominal expanded diameter and vessel diameter,respectively. For a given D n = 16 mm , the experimental data points were mea-sured through the in vitro test setup shown in Section 2. In this case, wesimulate D v by adjusting the diameter of the aluminum wrap. For each stent,the joint analysis (Sections 3.2 and 3.3) yields the circumferential force ( F θ ) and15 / L s × -4 v/ L s × -4 FEAnal ys . F / E A A s F / E A A s FEAnal ys . vv (a) Diamond design(b) Chevron design Figure 5: Von Mises stress distribution over the stent strut at the delivery size (maximummagnitude of stress); (a) Diamond design joint; (b) Chevron design. Comparison betweenstrut bending force vs. circumferential displacement from the analytical method in Section 3and Finite Element calculations. longitudinal displacement ( u ), at a given circumferential displacement ( v ). Inthis case, circumferential displacement is adjusted to match the simulated vesseldiameter using (3). Accordingly, (5) was used to calculate the radial pressure.16 ( % ) Diamond design analysis
Diamond design e xp. C hevron design analysis C hevron design e xp. Z design analysis
Z design e xp. Braided design analysis
Braided design e xp. Radial strain
Figure 6: Analytical prediction of foreshortening compared with experimental measurements. P ( k P a ) (a) Diamond design P ( k P a ) (b) Braided design (d) Z design (c) Chevron design D n /D v D n /D v D n /D v D n /D v P ( k P a ) P ( k P a ) Figure 7: Radial pressure calculated based on (5), shown by solid line, compared with datapoints corresponding to in vitro experiment (Fig Fig. 2(a)). D n and D v are stent nominalexpanded diameter and vessel diameter, respectively. Usually the ratio D n D v does not exceed1.3 which corresponds to 30% oversizing. We can compare the radial pressure performance of the stents in Fig. 8(a),where we combine and compare all stents in Fig. 7 in a single plot for a bettercomparison. Note that the oversizing parameter is limited to D n /D v = 1 . D n /D v = 3, it loses 70%of its radial pressure at D n /D v = 1 .
5. To compare stent radial compliance,calculated using (8), we have Fig. 8(b). The steel stents (Braided design and Zdesign) are much more compliant at higher oversizing values. However, in therange of D n /D v < .
5, they are much stiffer. The compliance of Nitinol stents,Chevron design and Diamond design, does not change as much as steel stentswith changing the oversizing D n /D v . This can be due to the effect of the phasetransition, which is evident in the local maximum points in the compliancecurve.The results of the global and local collapse tests are shown in Fig. 9(a) and(b), respectively. Wall stent and Z design have the higher resistance in bothtests compared to Nitinol stents. Note that Z design has a higher resistance tolocalized collapse while Braided design performs better in global collapse test.It is worth mentioning that steel stents were also stiffer in the radial pressuretest when D n /D v < .
5. Discussion
The assumption on loading conditions and resultant deformation of the semi-analytical method is validated through comparison with experiments (Fig. 7).The experiments show a slightly higher pressure, especially at higher expansionratio ( D n D v > . ( k P a ) C ( % / mm H g ) D n /D v D n /D v (a) Radial pressure (b)
Compliance C hevron designZ designBraided designDiamond design C hevron designZ designBraided designDiamond design Figure 8: Experimental measurements of radial pressure and compliance of stents comparedin a single plot. the straight strut by another fillet curve. Since the connection between theseparts is assumed as a straight link for the purpose of simplification, the resultsdeviate from the experiments. This conclusion is also valid for the fillet thatconnects the curved and straight parts of the Z design (See Fig. 4).In Fig. 8(a), we observe a higher radial pressure (up to 30% oversizing, D n /D v =1 .
3) for Diamond design and Braided design (steel stents). They offer morescaffolding than Z design and Chevron design (Nitinol stents) due to highercoverage. The radial pressure is a function of bending force in each unit-cell,19 c ( N / c m ) F c ( N ) Z/DZ/D (b)
Global (a)
Local F c ZF c Z C hevron designZ designBraided designDiamond design C hevron designZ designBraided designDiamond design Figure 9: Experimental measurements of collapse force ( F c ) Vs. displacement ratio ( Z/D ). D is the internal diameter of the fully expanded stent. and the bending force itself is a function of the cross-section area, length of thestrut, the material of the stent, and foreshortening according to (7). Hence, withhigher foreshortening, the material volume per unit length increases and the to-tal contact surface decreases, which leads to higher radial pressure. Anotherremarkable observation is the relationship between compliance and collapse re-sistance. By comparing Fig. 8(b) and Fig. 9, we can conclude that steel stentsare less susceptible to collapse. Consequently, collapse resistance is inversely20orrelated to compliance. When the collapse mode has a spatially non-uniformdeformation as a result of structural instability the analytical method given hereis not applicable. However, given the relationship between collapse resistanceand radial compliance (calculated based on (9)), it is possible to qualitativelycompare the collapse behavior of stent designs.The limitation of the present study arises in both experimental and analyt-ical modelling. The in vitro experiment based on Section 2 only considers theuniform deployment and does not account for non-uniform lumen or a curvedanatomy of the vein. In general it is a challenge to excise fibrotic veins and thevein material properties are difficult to emulate using polymeric tubes. Thismeans that the vein-stent interaction is not accounted in this study. This isan area where further work is needed. However, for a given vein model therelative performance that we report in this study is expected to hold. Anotheralternative approach to measure the radial pressure is to use an aperture-type(crimper) machine (see (Dabir et al., 2018; McKenna and Vaughan, 2020). Thelimitations listed above are still unavoidable. Another consideration is the effectof foreshortening on the radial pressure test. In the setup used here, it is easyto use a wider fabric to account for stent elongation during the crimp test. Inthe aperture-type devices, however, the length of the stent is limited to the de-vice capacity. In the analytical approach (Section 3), we ignored the frictionalforces and the mechanical interaction between the vein wall and the stent wasmodelled as a uniform pressure distribution, which is not the case for curvedvessel geometries. This effect can be included in future studies by assuming theelasticity of the vessel wall. This study compares four different venous stent designs (Chevron design,Z design, Diamond design, and Braided design) based on their collapse, fore-shortening, radial pressure, and compliance. Usually, the stent deploymentfor abnormal/diseased vein is performed after the vein angioplasty under fluo-roscopy. Consequently, a surgeon can observe the length of occlusion, the regions21ith high forces (based on the shape of inflated balloon), and the stiffness ofthe occluded part by observing the balloon pressure. Based on the mechanicalproperties of the diseased vein, we can suggest the most suitable design for aspecific occlusion type. Based on our study, here we present a summary of someof the common vein occlusion scenarios. • Z design has superior performance against collapse deformation. Conse-quently, it may be more reliable to treat diseases like May-Thurner syn-drome, which tends to apply localized force. • Nitinol stents are more compliant. Thus they follow the joint movements.Accordingly, they are more suitable for deployment in proximity of ex-pected major vessel bending during limbs movements. Diamond designand Braided design apply a higher radial pressure. Thus, they may bechosen for the long lesion with high recoil. • Stents have a number of anchors at both ends, which attach them to thevessel wall to avoid migration. If the locations of the stent tips are criticalto be predicted (e.g., deployment close to branch orifice), Z design can be agood choice. Because of the small foreshortening of Z design in comparisonto other designs, we can predict the final location of the ends.
6. Conclusions
This study compared four different stents currently used in veins. Two de-signs (Z, braided stents) are off-label while the remaining two (Diamond andChevron) are specifically designed for venous stenting. Deformation characteris-tics of all four designs are compared under identical loading conditions through in -vtro testing and semi-analytical modelling. Particular attention is given toforeshortening, compliance, radial pressure, and collapse resistance. An inversecorrelation between radial compliance and collapse resistance, and foreshort-ening and radial pressure is found. A good agreement is found between thepredictions of the unit cell based semi-analytical modelling and experiments.22elative merits of each stent design for common vein occlusion scenarios areidentified. Venous stenting is a relatively new area of investigation comparedwith arterial stents. While we expect the relative comparision across the designsto hold, further work on vein-stent interaction is needed.
Conflict of interest
The authors claim no conflict of interest regarding the choice of candidatestents. No human or animal test was conducted for this study.
Acknowledgment
We sincerely acknowledge the Division of Vascular Surgery at VancouverGeneral Hospital for providing the stent models. Furthermore, the fundingthrough a Discovery Grant to Srikantha A. Phani from Natural Sciences andEngineering Research Council (NSERC) Canada is acknowledged.
References
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A1. Material properties and Elastic bending of Nitinol beams
The correlation between elastic curvature ( κ ) and the bending moment isgiven in (17), where y c and y c are respectively the distance from the neutralaxis to the boundaries of the transition, and martensite regions, which are undercompression (Fig. 10); the same terminology is true for y t and y t which are28epresenting the same variables for the tension portion. Thermo-mechanicalproperties for NITI-I are given in Table 1. To calculate I in (17), (Mirzaeifaret al., 2013) introduced four different functions for different loading conditions.Here, we used equation (24) in their paper. Accordingly, we determine thebending moment in each strut (a function of F θ ) and find the curvature of thestent strut based on the method presented by (Mirzaeifar et al., 2013). Oncewe have the curvature along the strut length, we can find the circumferentialand longitudinal displacements. M ( κ ) = − / E A κ ( y c − y t ) + ( I ( y c ) − I ( y c ))+ E M w [1 / κ (( h c ) / − y c ) − H c (( h c ) / − y c )] + ( I ( y t ) − I ( y t ))+ E M w [1 / κ ( y t − ( h t ) / − H c ( y t − ( h t ) / ] , (17)Where y c , y c , y t , and y t are given in equations (32) in (Mirzaeifar et al.,2013). Other parameters in the above expression are given in the Table 1. (a)(b) Bended Beam Center LineNeutral Axis h ℎ 𝑐 𝑦 𝑦 𝑦 𝑦 ℎ 𝑡 w Force
Figure 10: Phase distribution of a Nitinol beam under bending.; (a) phase distribution, blue(meshed), yellow (solid) and red (dashed) are respectively, Austenite, Transition and Marten-site Phases; (b) Cross section stress distribution. E A
72 GPa E M