Available online at www.sciencedirect.com

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j o u r n a l o f t h e m e c h a n i c a l b e h a v i o r o f b i o m e d i c a l m a t e r i a l s 4 9 ( 2 0 1 5 ) 4 3 – 6 0

http://dx.doi.org/101751-6161/& 2015 El

nCorresponding autE-mail address:

Research Paper

On the role of SMA modeling in simulating NiTinolself-expanding stenting surgeries to assessthe performance characteristics of mechanicaland thermal activation schemes

A.F. Saleebn, B. Dhakal, J.S. Owusu-Danquah

Department of Civil Engineering, The University of Akron, Akron, OH 44325-3905, USA

a r t i c l e i n f o

Article history:

Received 9 December 2014

Received in revised form

10 April 2015

Accepted 12 April 2015

Available online 22 April 2015

Keywords:

NiTinol

Self-expanding stent

Superelasticity

Shape memory effect

Pressure cycles

.1016/j.jmbbm.2015.04.012sevier Ltd. All rights rese

hor. Tel.: þ1 330 972 7692saleeb@uakron.edu (A.F.

a b s t r a c t

The work is focused on a detailed simulation of the key stages involved in the NiTinol self-

expanding stenting surgical procedure; i.e., crimping, deployment, SMA activation, as well

as post-surgery steady-state cyclic behavior mimicking the systolic-to-diastolic pressure

oscillations. To this end, a general multi-mechanism SMA model was utilized, whose

calibration was completed using the test data from simple isothermal uniaxial tension

experiments. The emphasis in the study was placed on the comparison of two alternative

SMA activation protocols, in terms of both the immediate and long-term (post-surgery)

performance characteristics. The first is ‘hard’ mechanical activation utilizing super-

elasticity, and the second is ‘soft’ thermal activation relying upon the combined one-

way shape memory effect and constrained-recovery characteristics of the NiTinol material.

The important findings are (1) the thermal activation protocol is far superior compared

to the mechanical counterpart, from the point of view of lower magnitudes of the induced

outward chronic forces, lesser developed stresses in the host tissue, as well as higher

compression ratio with lesser crimping force for the same geometry of initial stent

memory configuration, (2) the thermal activation protocol completely bypassed the

complications of maintaining the high restraining force during deployment of the stent,

and (3) there is no indication of any detrimental functional fatigue/degradation in the

cured stenotic artery during cyclic pressure oscillations.

& 2015 Elsevier Ltd. All rights reserved.

rved.

; fax: þ1 330 972 6020.Saleeb).

1. Introduction

NiTi shape memory alloys (SMA) are widely used in manybiomedical devices due to their biocompatibility, superelasticcapability and shape memory effect. These beneficial proper-ties of NiTi, such as superelasticity and shape memory

effects, are known to be highly nonlinear in character and

are acutely dependent on the composition, processing, as

well as temperature and/or stress states to which the mate-rial is subjected. Particularly, slight changes in the composi-tion, hot/cold processing, and heat treatment of NiTi SMA candrastically modify the key characteristics of the material,

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allowing engineers and designers a plethora of ways to adjustand manipulate the alloy to better suit the intended applica-tion. For example, a small increase in the percentage of nickelin NiTi SMA will dramatically lower the austenite finishtemperature below body temperature, an important criteriafor biomedical applications (Gall et al., 2008). These smallchanges produce large effects on the material behaviormaking traditional prototype experimentation impractical,while generating a rapidly growing interest in the use ofcomputer simulations to assess the effectiveness and opti-mize the design of NiTi SMA devices for use in complexsurgical procedures. Hence, for such complex class of mate-rial, its development hinges on the availability of accuratemathematical models accounting for the important andrelevant effects of such material’s unique phenomena.

World-wide, the coronary artery disease has become a lead-ing cause of death (Murphy et al., 2010; Lewis, 2008). Thus, thecardiovascular treatments (surgical stenting, angioplasty, ablat-ing the plaque by pulverization, etc.) have gained a growinginterest within the medical community in the recent years(O'Brien and Bruzzi, 2011). In an effort to find less invasivetreatments, SMA self-expandable stents offer many benefits overconventional balloon expanding devices. Furthermore, pre-viously problematic areas, such as peripheral arteries and intercranial locations where severe complications are prone todevelop, are now being treated successfully with self-expan-ding stents (Liebig et al., 2006).

In this case of self-expanding stent, SMA materials utilizedare crucial due to their ability to accommodate large strains up to8% (O'Brien and Bruzzi, 2011). This enables stents made fromSMAs to be created at an initial fully expanded diameter, com-pressed (crimped/crushed) down to fit into a catheter, and thenre-expanded once it is placed in the vessel (as in the schematicshown in Fig. 1). The importance of effective SMA modelingbecomes even more critical for biomedical applications inview of a number of other complicating factors involved in thesecases. In particular, this includes handling the challenging issuesencountered in biomedical simulations such as complex geome-

Fig. 1 – General schematic showing two alternative schemes of thmemory effect of the self-expanding SMA stent. In this, the followshape memory effect, ‘T’ for Temperature, ‘Af’ for austenite finish

tries, coupled extensional/flexural/twisting deformation modes,large spatial rotations, as well as elaborate procedures formechanical/thermal load control and contact conditions.

Depending on the application, location, and condition of theartery, stents come in a variety of configurations. Stents can bedescribed and categorized by their geometry, such as coil,helical spiral, woven, honeycomb, sequential rings, closed cell,open cell, slotted, modular, etc. (Stoeckel et al., 2002; Palmaz,2004; Bonsignore, 2003), or on the basis of the method offorming the stent, e.g., wire-based, sheet-based, and tube-based (Stoeckel et al., 2004). The stent geometry may alsodictate the extent of the mechanical work required to compressthe stent into the catheter during crimping stage of the stentingprocedure. There are different approaches utilized in theliterature to crush the stent. In Migliavacca et al. (2002 and2004), the uniform radial pressure and/or radial displacementwere applied to reduce the internal diameter of the stent. Amovable rigid surface subroutine was utilized to crimp the stentin Kleinstreuer et al. (2008). In addition to this, in the Reese andChrist (2008), a radial displacement was applied by an analyticalfunction on the outer surface of the stent to crimp it. Any of theabove methods for crimping the stent can be utilized in eitherone of the two states of the SMA material; i.e., soft (martensite)or hard (austenite) states. This in turn will also affect the degreeof mechanical work required to crush the stent.

More specifically, there are two possible alternative schemesfor the self-expanding stenting surgical procedure (see schematicin Fig. 1). The first one is related to the crushing/crimping of thememory stent configuration at the superelastic (SE) regime of theSMA material, i.e., at temperature higher than the characteristicaustenite finish (Af) temperature. This is then followed by thedeployment of the crushed stent inside the artery while main-taining the crimping force. Afterwards, the crimped stent isactivated/released inside the diseased artery (host tissue) to re-expand against the impediment of the vessel wall (thus estab-lishing contact interaction between the stent and host tissue).The final state of self-expansion of the stent will result in theincrease of the clear opening of the clogged/blocked artery.

e stenting surgical procedure utilizing superelasticity or shapeing abbreviations are used: ‘SE’ for superelasticity, ‘SME’ fortemperature, and ‘Mf’ for martensite finish temperature.

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In contrast, the second scheme for the stenting procedureutilizes the shape memory effect (SME) of the SMA material. Inthis case, the initial memory configuration of the stent (to whichthe material will try to return upon activation) exists at atemperature lower than the characteristic martensite finish(Mf) temperature. The crushing of the stent at this low tempera-ture is achieved mechanically and the crimping force is imme-diately released (thus, causing small “elastic” rebound/spring-back/recoil) at this same low temperature. The crushed stent is thendelivered into the diseased artery region and its self-expansionstage is activated by the shape memory effect occurring in theSMA material due to the gradual increase in temperature of thestent to attain the body temperature.

The long-term performance of these stents inside the dis-eased artery is dependent on two major factors: vessel patency/restenosis and the mechanical characteristics of the device (Tanet al., 2001). Considering that a stent is simply a scaffolding forthe vessel, providing a mechanical support to a biological tissue(Sigwart, 1997), the mechanical performance of the material atbody temperature is critical to the proper design of the device(Xia et al., 2007). Understanding how stresses will form in thestent structure, as well as the host tissue, under the effect of theoscillatory (systolic/diastolic) pressure cycles, and how to mitigatethem becomes the key to optimizing the new SMA stent designs.

In this study, the SMA model (Saleeb et al., 2011) is imple-mented for the calibration of the commercially available nickel-rich NiTinol (Pelton et al., 2000; Duerig et al., 2002), and thiscalibrated SMA model is utilized in the numerical simulation ofthe stenting procedure. More specifically, the overall simulationof the stenting procedure involves the following components: (i) ahoneycomb-pattern geometric configuration for the stent, (ii) acylindrical artery, (iii) a layer of a plaque, representing a diseasedblood vessel, and (iv) contact modeling to account for the mutualstent–artery–plaque interaction effects. The emphasis is placedon investigating the specific details affecting the various stages ofthe stenting procedure, i.e., crimping of stent, its deployment tothe targeted artery region, stent re-expansion to reduce thestenosis, and its long-term performance inside the artery whenit is subjected to oscillatory (systolic/diastolic) pressure cycles. Itis important to note that the presently available literatures on theSMA stent applications have mostly focused on the geometricdetails of the memory stent configuration and its crushing,whereas our present work attempts to quantify the effect ofthe SMA material behavior on the finer details of the surgicalprocedure.

2. Overview of SMA modeling framework andits calibration for NiTinol

2.1. SMA modeling framework

There are a variety of constitutive, mathematical models devel-oped to capture the high degrees of nonlinearity (superelasticity,pseudoplasticity, one/two-way shape memory effect) displayedby shape memory alloys. The reader is referred to the availableliterature (Patoor et al. 2006; Lagoudas et al., 2006; Lagoudas,2008; Lexcellent, 2013) for the background on such materialmodeling. The existing and emerging applications of SMAmaterials call for the ability of these models to predict the actual

multiaxial deformation modes, which are normally experiencedby the “real life” SMA-based devices during their operationperiods involving a complex initial/boundary/value problem. Inparticular, when targeting the utilization of these SMA devices inbiomedical fields, the model’s ability to efficiently analyze theSMA device’s intricate contact interaction with other internalorgans/tissues is of high necessity. The specific simulationsdiscussed here make use of the recent, multi-mechanism based,three-dimensional, SMA modeling framework developed bySaleeb et al. (2011, 2013a,b and 2014).

The SMAmodeling framework was formulated as interplay ofmultiple inelastic mechanisms to capture the nonlinear, hys-teretic, response of different ordinary and high temperature SMAmaterials under different thermomechanical loading. The gen-erality of this model allows its application in geometricallycomplex SMA-based devices, which are subjected to variabledegrees of displacements and rotations. A brief outline of thisformulation is given in the sequel, but further details can befound in the cited references.

In order to minimize the clutter in writing the governingequations of the SMA material model, the terminology of small-deformation analysis is used. For the actual use of the SMAmodel in the large deformation numerical simulation of thestenting procedure (to be described below in Sections 4–6), all thetensoral quantities mentioned here (stress, strains, inelasticstrains, internal state variables, etc.) are simply formulated inthe rotated configuration, e.g. for the Cauchy stress tensor, onehas Rmi�σmn�Rnj, where Rij is the rigid-rotation tensor obtainedfrom the polar decomposition of the deformation gradient into apure rotation part (orthogonal tensor) and a pure stretching part,i.e.,Fij ¼ RimUmj, where Fij is the deformation gradient tensor, Umj

is the right stretch tensor, and Rim is the pure rotation tensor(Saleeb et al., 1990).

In the present model, there are two different sets of statevariables, i.e., controllable state variables and internal state vari-ables. The controllable state variables include the temperature(scalar) and the second-order tensor components of the stress,σij, and/or total strain, εij. On the other hand, the internal statevariables are all second-order tensors, and they are grouped intosets of internal stress-like and their conjugate strain-like quan-tities, α bð Þ

ij and γ bð Þij , respectively, for each of the six different

inelastic mechanisms (signified here by the superscript counter-index ‘b’, where b¼1–6), together with the inelastic (transforma-tion) strain tensor, εIij.

Each of these internal state tensors is governed by an evolutionequation, i.e., first-order time-differential equation for the ratesof the tensoral quantity, where time rates are indicated below bya dot over the symbol. The development of these evolutionequations was carefully designed to enable the proper partition-ing of the energy storage versus energy dissipation for all thenonlinear hysteresis loops of the stress–strain–temperatureresponses in both the superelastic and the pseudoplasticregimes (Saleeb et al., 2011). This is achieved by the selectedfunctional forms (described below) together with the sets ofmaterial parameters entering each of these functional forms. Morespecifically, the material parameters were classified into twomain groups, i.e., fixed parameters as well as dependent para-meters (which can generally be dependent on the temperatureand/or the stress-state). Furthermore, in its most general form(Saleeb et al., 2014) the model contains specific sets of material

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parameters responsible for capturing any asymmetry in tension–compression–shear response (ATC). For the NiTinol materialused here (Sections 2.2 and 3.1), no test data were availableunder compression- or shear-type loading, therefore all aspectsrelated to ATC were deactivated here. Furthermore, in calibratingthe model parameters from the simple isothermal experimentsin Section 2.2, it was determined that no stress-dependency isneeded.

In the formulation, the total strain, εij, occurring during thematerial’s deformation process, is decomposed into an elastic(reversible) strain, εeij, and viscoplastic (irreversible) strain, εIij.

εij ¼ εeij þ εIij ð1Þ

Here, all deformation-inducing transformations, i.e., directtransformation of Austenite-Martensite, detwinning of Mar-tensite variants, and reorientation of Martensite variants, areaccounted for implicitly in the εIij tensor.

The SMA model utilizes two energy functions, i.e., the Gibbscomplementary function, Φ, and the dissipation-energy poten-tial, Ω in formulating the following evolution equations.

_εij� _εIij ¼ddt

∂ΦR

∂σij

!¼ E�1

ijkl _σkl ð2Þ

_α bð Þkl ¼ ∂2ΦIR

∂α bð Þij ∂α bð Þ

kl

24

35

�1

_γ bð Þij ð3Þ

_εIij ¼∂Ω∂σij

and _γ bð Þij ¼ � ∂Ω

∂α bð Þij

ð4Þ

In the prior equations (Eqs. (2)–(4)), the following forms forthe stored energy and the dissipation potentials are used:

Φ¼ΦR σij� �þ ΦIR σij; α

bð Þij

� �ð5Þ

ΦR σij� �¼ 1

2σijE�1

ijkl σkl ð6Þ

ΦIR σij; αbð Þij

� �¼ σijε

Iij þ

XNb ¼ 1

H bð Þ ð7Þ

Ω σij�αij� �

; α bð Þij

� �¼Z

κ2Fn

2μdF ð8Þ

In Eq. (6), Eijkl is the isotropic fourth order tensor of elasticmoduli and can be evaluated using E¼elastic modulus andν¼Poisson’s ratio as

Eijkl ¼νE

ð1þ νÞð1�2νÞ δijδkl

þ E2ð1þ νÞ δikδjl þ δilδjk

� �with δij ¼ Kronecker delta ð9Þ

In Eq. (7) above, H bð Þ is the nonlinear “power-type” visco-plastic function, dependent on the internal state tensor α bð Þ

ij

(or its conjugate internal strain γ bð Þij ), and h g bð Þ� �

is the hard-ening function and h G bð Þ

� �defines the dissipative function.

H bð Þ ¼κ2bð ÞR

1h g bð Þð ÞdG

bð Þ; for b¼ 1; 2; 3

κ2bð ÞR

1h G bð Þð ÞdG

bð Þ; for bZ4

8><>: ð10Þ

h g bð Þ� �

¼

κ bð ÞH bð Þffiffiffiffiffiffig bð Þ

p� � β bð Þ � 1ð Þκ bð ÞþH bð Þ

ffiffiffiffiffiffig bð Þ

p� �β bð Þ ; for b¼ 1; 2

H bð Þ 1þffiffiffiffiffiffig bð Þ

pκ bð Þ=H bð Þ

� �β bð Þ" #

; for b¼ 3

8>>>>><>>>>>:

ð11Þ

h G bð Þ� �

¼H bð Þ 1�ffiffiffiffiffiffiffiffiG bð Þ

p� �β bð Þh Lð Þ

; for bZ4 ð12Þ

G bð Þ α bð Þij

� �¼ 1

2κ2bð Þα bð Þij Mijklα

bð Þkl

� �ð13Þ

g bð Þ γ bð Þij

� �¼ γ bð Þ

ij γ bð Þij ð14Þ

Here, the h Lð Þ¼the Heaviside function with argumentbeing the loading index;L¼ α bð Þ

ij Γij , where Γij ¼ ∂F=∂ σij�αij� �

.In addition, HðbÞ, βðbÞ, and κðbÞ are material parameters for theindividual hardening mechanisms for b¼1–6.

In Eq. (8), κ, n and μ are the inelastic flow materialconstants. In particular, κ is the threshold for inelastic flow,n and μ are the rate dependency factors. Furthermore, thetransformation function F is given as

F σij�αij� �¼ 1

κ212

σij�αij� �

Mijkl σkl�αklð Þ

ð15Þ

Mijkl ¼12

δikδjl þ δilδjk� �� 1

3δijδkl ð16Þ

In the above model formulation, there are a total of 23material parameters required to be estimated. For the specificNiTinol material used here, this task will be accomplished inSection 2.2.

Among the material parameters, there are the elastic mod-ulus, ‘E’ and Poisson’s ratio, ‘υ’ to account for the elastic/reve-rsible part of the model. The remaining 21 parameters areresponsible for the inelastic, non-linear part of the model. Theseinclude two sets (1) parameters accounting for the inelastic/transformation strain (rate-dependency factors, ‘n’ and ‘μ’; andthreshold ‘κ’ in the transformation function), and (2) thresholdvalues κðbÞ, and respective hardening-rate parameters, i.e., mod-ulus HðbÞ and exponent βðbÞ for mechanisms b¼1–6 in theevolution equations of the internal state variables. For thepurpose of the model calibration as well as the numericalsimulations, this SMA modeling framework was implementedas a UMAT subroutine in ABAQUSs Standard (ABAQUS, 2012).

2.2. Calibration of NiTinol

The above SMA model is utilized to characterize a commerciallyavailable nickel-rich NiTinol (Ni50.8Ti49.2 (at%)), having the char-acteristic transformation temperatures; austenite start As¼�22 1

C and austenite finish Af¼11 1C ( Pelton et al., 2000; Duerig et al.,2002) based upon the available tensile isothermal experiments.Unfortunately these two references did not provide all the fourtransformation temperatures of the NiTinol SMA material.However, the DSC studies conducted by McNaney et al. (2003)on the same composition of NiTi, i.e., for a Ni50.8Ti49.2 (at%),revealed the following set of the phase transformation tempera-tures: martensite finish Mf¼�87.43 1C, martensite startMs¼�51.55 1C, austenite start As¼�6.36 1C, and austenite finishAf¼18.13 1C. For completeness, these values from the twodifferent sources, i.e., estimates for As and Af from Pelton et al.

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(2000) and Ms and Mf from McNaney et al. (2003) will be used,together with the several isothermal tensile stress–strain testdata from Pelton et al. (2000) in order to complete our calibrationof the model for the specific NiTinol material used here (seeTable 1).

More specifically, the available uniaxial, tensile, isother-mal experiments in Pelton et al. (2000) were performed on theNiTinol wire specimen at different temperatures of �100 1C,�50 1C, 0 1C, 10 1C, 22 1C and 40 1C as shown in Fig. 2(a)(Pelton et al., 2000; Duerig et al., 2002). Each of the abovetests involves two steps, i.e., a displacement-controlled

Table 1 – Calibrated SMA model material parameters for NiTin

Fixed material parameters

Parameters Units Value

E MPa 50,000ν Dimensionless 0.3n Dimensionless 5μ MPa s 105

κ MPa 60κ(b), b¼3 MPa 8κ(b), b¼5 MPa 20κ(b), b¼6 MPa 10H(b) for b¼1–6 MPa 300�103 30β(b) for b¼1–6 – 1 1

Temperature-dependency of material parameters κ and κ(b) for b¼1,2, an

Temperature (1C) Parameters (in MPa)

κ(b), b¼1,2

�100 1C 10�4

�50 1C 150 1C 8810 1C 10522 1C 14540 1C 175

Note: The intermediate values in the above table are interpolated linearl

Fig. 2 – Stress-vs.-strain response for the NiTinol material underat �100 1C, �50 1C, 0 1C, 10 1C, 22 1C and 40 1C. Each test involvefollowed by a force-controlled unloading step (to zero force). Part2000; Duerig et al., 2002), and part (b) gives the counterpart resultwo additional SMA model predictions at 24 1C and 37 1C.

loading step (till 6% strain), followed by a force-controlledunloading step (to zero force). These six tests were all utilizedin the calibration of the SMA model parameters.

The calibration was achieved by grouping the parametersinto two categories: (1) 20 fixed material parameters, i.e., E, υ,n, μ, HðbÞ and βðbÞ for b¼1–6, κ, and κðbÞ for b¼3, 5, 6 (see upperpart of Table 1); and (2) 3 temperature-dependent parametersκðbÞ for b¼1, 2 and 4 (see lower part of Table 1).

Utilizing the calibration procedure explained in Saleebet al. (2014), the above material parameters were determinedfor NiTinol SMA material (see Table 1). In particular, ‘E’ and ‘υ’

ol.

0� 103 100 103 15�103 11505 1 1 1

d 4:

κ(b), b¼4

2010�4

10�4

10�4

10�4

10�4

y between the tabulated values.

the conditions of simple, isothermal, tensile tests conducteds 2 steps, i.e., a displacement-controlled loading step,(a) of the figure shows the experimental results (Pelton et al.,ts from the calibrated SMA model. Note that, part (b) includes

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are estimated from the typical data available in the literature.The rate-dependency parameters ‘n’ and ‘μ’ are elected tohave the “large” value for ‘n’ (n¼5 here) and relatively “smallvalue” ‘μ’ (μ¼105 MPa s here), respectively, thus leading toweak rate-dependency (see Saleeb et al. (2011) for furtherdiscussion and justification of this choice of weak ratedependency) in the temperature ranges considered for thetest data used here. The key reference thresholds, κ in thetransformation function, and κðbÞ for mechanisms b¼1–6 andtheir respective hardening rate parameters HðbÞ and βðbÞ weredetermined from the isothermal test at 22 1C. For the calibra-tion of the material responses at other temperatures, only 3material parameters κðbÞ for b¼1, 2 and 4 were taken to varywith temperature in a piecewise-linear fashion. Furthermore,the values of κðbÞ for b¼1 and 2 were taken to be identical.

Note that the parameter κðbÞ for b¼4 (which is responsiblefor the pseudoplasticity behavior at the extreme cold tempera-ture) is assumed to vary linearly between �100 1C and �50 1Cwhere it will reach a negligible value at �50 1C that willcontinue till the end of the considered temperature range(40 1C here). These temperature variations at the three thresh-olds enabled us to capture the transition of the behavior fromsuperelastic (at high temperature) to pre-dominantly pseudo-plastic (at low temperature) as observed in the experiments.

The calibrated SMA model is utilized to simulate eightisothermal responses at �100 1C, �50 1C, 0 1C, 10 1C, 22 1C,24 1C, 37 1C and 40 1C. Note that these include the six testsutilized in determining the material parameters of the model,as well as predictions for two additional tests not employedin model calibration (at 24 1C and 37 1C).

3. Materials

The study of the biomedical surgical stenting procedureemploys different classes of engineering materials. In parti-cular, the stent exploits the superelastic and shape memorybehavior of equiatomic NiTinol (Pelton et al., 2000; Dueriget al., 2002) for the expansion to its original memory config-uration inside the diseased artery to reduce the plaque

Fig. 3 – SMA model prediction for the stress relaxation and inveholdings (at 1%, 2%, 3%, 4% and 5%) in the loading and unloadinPart (a) shows stress-vs.-strain diagram, and part (b) gives the s

occluded region. The artery and the plaque are modeled hereas hyperelastic materials.

3.1. Superelastic NiTinol for self-expanding stent

A commercially available nickel-rich NiTinol (Ni50.8Ti49.2 (at%)),having the characteristic transformation temperatures,As¼�22 1C and Af¼11 1C (Pelton et al., 2000), was selectedfor stent material and was calibrated using SMA model (seeSection 2.2).The calibrated SMA model predictions are plottedin Fig. 2(b) for isothermal, tensile tests at �100 1C, �50 1C, 0 1C,10 1C, 22 1C and 40 1C. Qualitatively and quantitatively, there isa good agreement between the experimental result (Fig. 2a)and the model counterpart (Fig. 2b). In particular, the modelsuccessfully captured the pseudoplastic response at �50 1Cand �100 1C, as well as superelastic response at 10 1C, 22 1Cand 40 1C. Moreover, two additional SMA model responses atroom temperature (24 1C) and body temperature (37 1C) arepredicted and given in Fig. 2(b). Note that these additional testsare relevant to the specific stages of the stenting procedureconsidered in the simulations of Sections 4–6.

Several other simulated responses from the SMA modelwere made to investigate the important aspects of rate-dependency of the material. More specifically, a strain-controlled (strain rate¼1E–4/s) uniaxial, isothermal test wassimulated at 37 1C with different hold periods, i.e., the strainsheld constant for 6 h at 1%, 2%, 3%, 4%, and 5% during loadingand unloading (see Fig. 3a and b). Fig. 3(a) shows the stress-vs.-strain response and Fig. 3(b) demonstrates the stressvariation with time at the respective strain holds. In theloading steps, the material demonstrates the “conventional”phenomena of stress relaxation where the magnitude oftensile stress decreases during the periods of constant strain,while the strain-holds in the unloading step shows signifi-cantly different character in variation of the stress, i.e., itexhibits an inverse-relaxation phenomena resulting in anincrease in the magnitude of the tensile stress at the periodsof strain-holds.

These phenomena of relaxation and inverse-relaxationhave also been observed in the isothermal (constant-tem-perature) experimental investigations (Fig. 18 in Grabe and

rse-relaxation phenomena during multiple periods of straing branches, respectively, at the body temperature (37 1C4Af).tress variation in time during the strain-holding periods.

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Bruhns (2008) and Fig. 2.10 in Helm (2001)) for the NiTi-basedSMA material at temperatures higher than Af, where strain isheld constant for a period of time. Such time-dependency ofthe stresses in the material becomes an important factor forthe long-term performance of the designed application, e.g.,self-expanding stent here.

3.2. Hyperelastic material for plaque and vessel

The simulation of the stenting surgical procedure involvesthe interaction of the stent with the diseased (plaqued) hosttissue. For simplicity, both the vessel and the plaque areassumed to be isotropic, hyperelastic, incompressible materi-als. To this end, a reduced polynomial hyperelastic modelapplicable to large-deformation analysis was selected in theABAQUSs Standard program (Abaqus, 2012).

The equation of the strain energy potential is given below,where W is the strain energy per unit volume, Ci0 and Di arematerial parameters, I1 is the first deviatoric strain tensor, J is

Table 2 – A reduced polynomial “incompressible” hyperelastic

Materials Parameters

C1 C2

Artery 0.000149 0.004529Plaque 0.000427 0.008673

Fig. 4 – Geometric details of the memory configuration of the barethe joint and the mid of the strut near the centerline of the stentin the simulation.

the total volume ratio (determinant of the deformation grad-ient tensor), and λi are the principle stretches.

W¼X3i ¼ 1

Ci0 I1 �3� �i þX3

i ¼ 1

1Di

J�1ð Þ2i ð17Þ

I1 ¼X3i ¼ 1

1Di

J�13λi

� �2ð18Þ

The parameters utilized for the vessel and plaque materialin this reduced polynomial hyperelastic model are shown inTable 2. Note that, for the present application, the value of Di

is taken to be zero, signifying a strictly incompressiblematerial, for which the special hybrid brick elements(C3D8H) are used. The stress–strain response for the equi-biaxial, tensile, load–unload tests for the vessel and theplaque materials is presented later in Fig. 5(b). In the presenttreatment, the plaque material was taken to be stiffer thanthe vessel.

model material parameters for the artery and plaque.

C3 D1 D2 D3

0.002552 0 0 00.450569 0 0 0

-metal, honeycomb stent. Elements ‘A’ and ‘B’ are located atand were chosen to highlight the key stress–strain responses

Fig. 5 – (a) The geometric details, and (b) the hyperelastic material behavior (in equi-biaxial, tensile, load–unload tests) of thediseased host tissue.

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4. Geometry and structure

The selected geometry of the stent investigated here is acommonly-used, bare-metal, honeycomb pattern (Stoeckelet al., 2002, 2004) in a tubular configuration as seen in Fig. 4.The initial, stress-free, memory configuration of the stent is28.3 mm long with an expanded diameter of 6.0 mm. Thethickness of the stent is 0.07 mm with a strut width of0.1 mm. The stent’s geometrical information utilized here isbased upon the technical specifications provided in Henkeset al. (2003). In Fig. 4, the specific locations of element ‘A’ (inthe joint region) and element ‘B’ (in the middle of the strut)are marked, as they are the critical locations where max-imum stresses are likely to occur and thus the response ofthe state of the maximum shear stress and strain will beextracted and reported at these two elements for the futurediscussion of the results. In particular, we note that the stentgeometry keeps changing at the different stages of thesimulated procedure, thus rendering the global stress andstrain components (which typically constitute the outputquantities in three-dimensional solid modeling) difficult tointerpret physically. This justifies the use of maximum shearstress and maximum shear strain in the subsequent sectionsand figures. More specifically, the maximum shear stress isdefined as one-half the difference between the maximumand minimum principal stresses, and the counterpart max-imum engineering shear strain is defined as the differencebetween the maximum and minimum principal logarithmicstrains.

The diseased blood vessel (Fig. 5a), to which the stent willbe inserted, is modeled as a simple, straight tubular part withan inner diameter of 5.4 mm (smaller than the diameter ofthe memory configuration of the stent), a thickness of 0.3 mmand a very long length (100 mm) so as to minimize theinfluence of end boundary conditions during the simulation.The layer of plaque covers 60 mm of the artery (in length) andis 1.0 mm thick at the center, tapering at each end to mergewith the artery wall. At the narrowest part the inner diameter

of the plaqued vessel becomes 3.4 mm wide, with the plaqueoccluding 60% of the original area of the artery.

5. Simulation methods and details

Making use of the symmetry conditions, only 1/8th of thestent, artery, and plaque were simulated. For the symmetricboundary conditions, the cut edges were fixed in the directionnormal to the cut surface to fix the rigid-body motion of theFEA models. A mesh convergence study was performed forthe stent with respect to response measures, i.e., the radialdisplacement at the plane of symmetry, as well as thedistribution of the effective stress at the elements ‘A’ and‘B’ in the mesh. This resulted in the identification of asufficiently-refined mesh size having 660 linear, 8-noded,three-dimensional (3D), hybrid brick elements (C3D8H) forthe stent. Correspondingly, 500 and 590 C3D8H elementswere used for the artery and plaque, respectively.

During the activation stage (see Fig. 6), the self-expandingstent expands and comes in contact with the wall of theplaqued artery. In order to account for these contact interac-tions in the numerical simulation, a soft contact modelinginvolving a frictionless, surface-to-surface and exponentialpressure-overclosure behavior is implemented. The diseasedartery wall is treated as a master surface and the outer surfaceof the stent is considered as a slave surface in the contactmodeling.

In this work, the two alternative schemes (see Fig. 6 and alsoalluded to above) are considered in completing the stentingsurgery, i.e., whether the initial, unstressed, fully-expanded,bare-metal stent exists in the hard (i.e., superelastic austenite)or soft (pre-dominantly pseudoplastic martensite) state. Forconvenience, these two alternative procedures will be referredto as Case 1 (hard) and Case 2 (soft), respectively. In the firstscheme (Case 1), all the steps of the stenting procedure areperformed at the constant temperature greater than the char-acteristic Af temperature of the SMA material (11 1C here). The

Fig. 6 – Specifics of the thermomechanical controls for the two alternative stenting schemes (Cases 1 and 2). Note that in bothschemes, the same compression ratio of ‘8’ was achieved in the final crushed configuration.

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stent is crimped at the room temperature (24 1C) into a deliverysystem, and then it is deployed to the plaque region in the bloodvessel maintaining the crimping force. Once the crimping force isreleased, the stent will be activated to undergo self-expansiondue to the superelastic nature of the NiTinol material at thistemperature (24 1C). Subsequently, the self-expanded stent gra-dually gets heated to the body temperature (37 1C).

In the second scheme (Case 2), the stent will be crimped andsubsequently the crimpling force will be released at the tempera-ture of �50 1C, i.e., much lower than the characteristic AS

temperature of the SMA material (here �22 1C) and closer toits MS temperature (here �52 1C). Due to the pseudoplasticbehavior of the NiTi material at this low temperature of�50 1C, only very small “elastic” rebound will occur upon theforce release, leaving the stent in its fully crushed, unstressed,deformed state. The crushed stent is then deployed to thediseased artery region where it is thermally-activated to expandexhibiting One-Way Shape Memory Effect (OWSME) against theresisting plaqued artery wall. This thermal activation end whenthe stent reaches the body temperature (37 1C) which is higherthan the characteristic Af temperature of the NiTinol material.

In both Cases 1 and 2 above, the stent crushing was achievedby imposing a simple, prescribed, axial extension of 8.5mm at arate of 0.017mm/s. This signifies over a 50% increase in the stentlength. At its final compressed state, the stent reaches acrushing ratio of nearly 8 (outer diameter measures 0.75mmcompared to the initial diameter of 6.0mm) in Case 1, and acrushing ratio of approximately 12 (outer diameter measures

0.5 mm compared to the initial diameter of 6.0 mm) in Case 2.This crushing ratio is sufficiently small to use in the catheter forperipheral and intracranial applications. When needed, all thereleases of the crimping forces (the axial reaction force devel-oped due to the prescribed axial extension) were conducted at aselected rate of 2 N/s. Also, any heating steps needed wereperformed at the rate of 0.85 1C/s.

Furthermore, the long-term performance of the stentinside the diseased artery is known to depend upon thebehavior of the stent in the presence of the cyclic arterialpressure (Kleinstreuer et al., 2008) from 50 mmHg (diastolic)to 150 mmHg (systolic). To account for the effect of thispressure oscillation in our present study, a simulation ofthe pressure-loading cycles (10 here) were performed at theend of the stenting surgery. For this purpose, internal pres-sure amplitude of 50 mmHg (6.67 kPa) was applied to theinternal wall of the stented artery with a time periodequivalent to 2 Hz (Duerig et al., 1999).

In addition to the above controls for the “real” basicstenting procedure, we have also established a number ofreference solutions to gain further insight into the factorsaffecting the complex stent–plaque–vessel interaction. Inparticular, the following reference solutions were established:(a) for Case 1, the crimping force was released with varyingrelease rates (in comparison to the basic one of 2 N/s) whileexcluding the interaction with the host tissue, and (b) forCase 2, the unstressed stent was heated to the body tem-perature without any contact constraints from the artery.

Fig. 8 – Successive deformation states of Case 2 showing theeffective stress distribution contours. State (i): the initialmemory configuration at �50 1C. State (ii): crushed state atthe end of applied displacement. State (iii): final crushedconfiguration after a small elastic rebound at �50 1C. State(iv): partially re-expanded stent just coming into contactwith the diseased vessel at �18 1C. State (v): the final-

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6. Simulation results

The results from the simulation of the two alternativestenting procedures Case 1 and Case 2 (see Fig. 6) will bediscussed in this section. To this end, all the importantresponses from these two cases will be presented in parallel.Figs. 7 and 8 show the key states (i–v) in Cases 1 and 2 of thesimulations. Here, only one-half of the models are shown forthe clear portray of the stent–plaque–vessel interaction (seeupper parts of Figs. 7 and 8). For the reference analysis, thefull model of the stent is presented and corresponding statesare identified by the same letters with apostrophe (i0–v0) (seelower parts of Figs. 7 and 8). Note that the reference analysisdiffers from the actual surgical procedure only when thestent comes in contact with the artery. The results from thisreference analysis are plotted as a red (either by full orbroken) line in subsequent Figs. 9 and 12 for Case 1 andFigs. 10 and 13 for Case 2. Furthermore, Figs. 9 and 10demonstrates the variation of the applied axial displacement,stent diameter and axial ‘crimping’ force during the stentingprocedure in Cases 1 and 2, respectively. The solution of Case2 requires controlling the temperature of the stent in thevarious stages of the simulation; hence, the results from Case2 will be depicted in 3D space with temperature representedby one of the axes.

The results in Fig. 11 show the comparison of the crimpingforce required to crush the stent in Cases 1 and 2. Similarly,

Fig. 7 – Successive deformation states of Case 1 showing theeffective stress distribution contours. State (i): the initialmemory configuration at 24 1C. State (ii): final crushed state.State (iii): partially re-expanded stent just coming intocontact with the diseased vessel. State (iv): the final-expanded stent after the complete release of the crimpingforce at 24 1C. State (v): final state upon the completion ofself-heating to body temperature (37 1C). For comparison, thelower row (states (i0)–(v0)) in the figure shows the counterpartdeformation states for the reference solution of the freelyexpanding stent.

expanded stent after the completion of self-heating to bodytemperature (37 1C). For comparison, the lower row (states(i0)–(v0)) in the figure shows the counterpart deformationstates for the reference solution of the freelyexpanding stent.

Figs. 12 and 13 present the development of the maximumshear stress (calculated as maximum principal stress minusminimum principal stress divided by 2)and the maximumshear strain (calculated as maximum principal logarithmicstrain minus minimum principal logarithmic strain) at ele-ment ‘A’ (at joint of the stent) and element ‘B’ (at mid of thestrut of the stent) in Cases 1 and 2, respectively. Fig. 14 showsthe comparison of the developed shear stress in the hosttissue with the increase in the internal vessel diameterduring the activation stages of the stenting procedure, i.e.,by releasing the crimping force in Case 1, and by thermalactivation in Case 2. More specific discussions pertinent tothe individual responses mentioned in the above figures aregiven below. For convenience, these are grouped into therespective stages of the stenting procedure.

6.1. Crimping stage

Crimping of the initial, memory configuration of the honey-comb stent constitutes the first stage in the stenting proce-dure. The memory configurations of the stents are as shownin states (i and i0) in Figs. 7 and 8, and they are taken at 24 1Cin Case 1 and �50 1C in Case 2. States (ii and ii0) in Figs. 7 and8 show the fully crushed/crimped stent configuration at 24 1Cin Case 1 and �50 1C in Case 2. The stent diameter was

Fig. 9 – Case 1 stenting procedure: (a) axial ‘crimping’ force-vs.-axial displacement and (b) axial ‘crimping’ force-vs.-stentdiameter at 24 1C. (For interpretation of the references to color in this figure, the reader is referred to the web version of thisarticle.)

Fig. 10 – Case 2 stenting procedure: (a) the variations in the axial ‘crimping’ force and the axial displacement withtemperature, and (b) the variations in the axial ‘crimping’ force and the stent diameter with temperature. (For interpretation ofthe references to color in this figure, the reader is referred to the web version of this article.)

Fig. 11 – Comparison of the axial ‘crimping’ force in Case 1 and Case 2: (a) axial force-vs.-applied axial displacement, and(b) axial force-vs.-stent diameter.

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Fig. 12 – Maximum shear stress-vs.-maximum shear strain in Case 1: (a) at element ‘A’ and (b) at element ‘B’. Here, path(iv)–(v) corresponds to the stage of heating the stent from 24 1C to 37 1C while in contact with the host artery. (Forinterpretation of the references to color in this figure, the reader is referred to the web version of this article.)

Fig. 13 – The variation of the maximum shear stress and the maximum shear strain with temperature in Case 2: (a) at element‘A’ and (b) at element ‘B’. (For interpretation of the references to color in this figure, the reader is referred to the web version ofthis article.)

Fig. 14 – Comparison of the variation in the developedmaximum shear stress in the host tissue with the increasein the internal vessel diameter in Cases 1 and 2.

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crimped from 6mm to �0.75 mm (hence, a compression ratioof �8) in Case 1 and to �0.5 mm (hence, a compression ratioof �12) in Case 2 at the end of the applied axial displacementof 8.5 mm.

Considering the crimping phase of the stenting procedure,i.e., along the path i–ii (also same for the i0–ii0 of the referencesolution) in Figs. 9 and 10, at the specific value of 6 mm axialdisplacement, the stent diameter was already reduced to3.4 mm (�57% of initial diameter) in Case 1 and to 3.2 mm(�53% of initial diameter) in Case 2. The corresponding axialreaction ‘crimping’ force at this intermediate state of stentcrushing was found to be 3 N in Case 1 and 0.87 N in Case 2.Considering next the remaining phase of the stent crimping,i.e., from axial displacement of value 6 mm (intermediatestent crushing) to 8.5 mm (final stent crushing), the resultingcrimping force increased rapidly to 44 N in Case 1 and 14 N inCase 2. This rapid increase in the crimping force is the effect

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of the geometric constraint of the complex, honeycomb, stentconfiguration. The crimping force required to crush the stentin Case 2 is only 32% of that in Case 1, thus demonstrating thetwo different characters of the soft (pre-dominantly marten-site) behavior, and hard (austenite) behavior of the NiTinolmaterial at �50 1C (Case 2) and 24 1C (Case 1), respectively.Corresponding to the above phases of stent crushing, thechanged diameters of the stents are as follows: from 6mm to�0.75 mm in Case 1, and from 6mm to �0.5 mm in Case 2.

The maximum shear stress-vs.-maximum shear strainresponse along the path i–ii (also same for the i0–ii0 of thereference solution) in Figs. 12 and 13 shows higher level of thetransformation stresses, i.e., 250 MPa in Case 1 and much lowerstress value, i.e., 80 MPa in Case 2 at the end of the crimpingstage. This is the expected manifestation of the differences inthe uniaxial tensile stress observed in the experimental resultsand their corresponding calibrated model predictions at the twodifferent temperatures (24 1C and �50 1C) as can be seen inFig. 2. During the applied deformation, the material exhibitedthe characteristic forward transformation from austenite tostress-induced martensite state at 24 1C in Case 1 and fromself-accommodatedmartensite to oriented martensite at �50 1Cin Case 2. Similar magnitudes of the maximum shear stresseswere reached at the end of this crimping stage at each of thetwo elements ‘A’ and ‘B”. Despite this, differing magnitudes ofthe maximum shear strain were accumulated at the end ofcrimping, i.e., nearly 7% for element ‘A’ and 10.5% for element ‘B’in Case 1. Similarly, maximum shear strain of 3.3% at element‘A’ and 1.5% in element ‘B’was developed in Case 2 at the end ofcrimping stage.

6.2. Deployment stage

This is the second stage in the surgical stenting procedure. InCase 1, the crimping force developed at the end of crushingstage (state (ii) in Fig. 7), i.e., 44 N is maintained constantinside the catheter at 24 1C during the deployment in order toprevent the premature expansion of the stent.

On the other hand, in Case 2, the crimping force (of 14 N) isreleased at �50 1C. Due to pseudoplastic behavior of NiTinol at�50 1C, a small elastic rebound (also known as spring-back/recoil) as seen along path ii–iii(also same for the ii0–iii0 of thereference solution) in Figs. 10 and 13 has occurred. This elasticrebound results in a small increase in the stent diameter(compared to initial diameter of 6mm) from 0.5mm after crim-ping to 0.75mm after rebound (state iii/iii0 in Fig. 8). Here, inCase 2, the stent is inserted in the catheter in a force-freecondition, hence providing the significant advantage of com-pletely eliminating any need for maintaining a force (as wasrequired in the simulation of Case 1).

The final stent diameters before its insertion into thecatheter for deployment are 0.75mm in both Cases 1 and 2.An appropriate catheter size, i.e., here selected to be 1mm indiameter (so that it is slightly greater than the 0.75mm men-tioned above, but simultaneously far lesser than the dimension3.4 mm representing the inner diameter of the narrowest part ofthe plaqued vessel). The crushed stents with diameter 0.75mmare then inserted inside the catheter, and together, they aredeployed to the targeted region of the diseased artery. Once thisassembly reaches the plaque–artery site, the catheter is drawn

back, leaving the stent to activate for expansion, i.e., by releasingthe crimping force in Case 1, and by heating in Case 2.

6.3. Activation stage

This activation stage involves the invoking of the super-elasticity and shape memory effect of the material in Cases1 and 2, respectively, for the expansion of the stent inside thediseased artery. In Case 1, the bare-metal, NiTinol stent,expands to its initial memory configuration by the gradualrelease of the crimping force at 24 1C.

Similarly, in Case 2, the stent attempts to expand to returnto its initial memory configuration by gradual self-heating to37 1C. During the gradual expansion, the stent comes incontact with the diseased wall of the artery and pushes theartery wall radially outward reducing the stenosis. Theamount of this outward radial motion depends on themethod of activation and will be discussed in details in thesequel.

6.3.1. Deformation statesStates (iii/iii0) in Fig. 7 for Case 1 and (iv/iv0) in Fig. 8 for Case 2represent the state at which the self-expanding stent will firstcome into contact with the diseased wall of the blood vessel.For Case 1, states (vi) and (v) in Fig. 7 correspond to the stateof the stent at the end of the release of the crimping forceduring the contact with the wall of plaqued vessel at 24 1C,and the final deformation state of the stent–plaque–vesselmodel at the end of self-heating of the stent from 24 1C to37 1C. For Case 2, state (v) in Fig. 8 represents the re-expandedstent in contact with the diseased wall of the blood vessel atthe end of the gradual self-heating from �50 1C to 37 1C.

In Case 1, referring to Fig. 7, the states (i0), (ii0) and (iii0) forreference analysis are identical to the states (i), (ii) and (iii), as nocontact was occurring during these stages. On the other hand,once the contact begins at state (iii), the reference solution willstart deviating from the actual simulated stenting procedure.More specifically, for the reference solution of Case 1, states (vi0)and (v0) in Fig. 7 show the recovery of the memory configurationupon release of the crimping forces and the final shape uponheating from 24 1C to 37 1C, respectively. In comparison to thisreference solution, the results of the actual simulation of Case 1indicated that the stent was not able to completely recover itsmemory configuration due to the contact interaction (comparestates (iv) and (iv0) in Fig. 7).

Similarly, in Case 2, in conjunction with Fig. 8, the states (i0),(ii0), (iii0) and (iv0) for the reference analysis are identical to thestates (i), (ii), (iii) and (iv), before the activation of contactinteraction. Starting with states (iv/iv0), the reference solutionalong path (iv0–v0) deviated from the actual simulation alongpath iv–v. State (v0) in Fig. 8 shows the final recovery of thememory configuration upon the gradual heating of the stent to37 1C in the reference solution of Case 2. Here also, the stent inthe actual simulation of Case 2 was not able to fully recover itsmemory configuration due to the impediment presented by thecontact with the inner wall of the plaqued vessel (comparestates (v) and (v0) in Fig. 8) for Case 2.

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6.3.2. Global force-deformation responseIn Case 1, the expansion of the bare-metal, NiTinol stent, inthe diseased artery location was activated by releasing thecrimping force of 44 N at the rate of 2 N/s at 24 1C. As the stentdiameter expands upon the gradual release of crimping force,the self-expanding stent comes in contact with the diseasedwall of the blood vessel in �20 s (out of 22 s of total time ofrelease at state-iii in Fig. 7 and location “iii” in path ii–iii–iv/iv0

in Fig. 9). Upon the further release of remaining ‘4 N’ crimpingforce in the additional period of 2 s, the stent in the actualsimulation expands against the effect of ‘obstacle’ force prov-ided by the contact, enlarging the stenotic vessel internaldiameter from 3.4 mm initially to 4.92 mm (path iii‐iv inFig. 9).

On the other hand, in Case 2, the stent was expanding due togradual heating from �50 1C to 37 1C at the rate of 0.85 1C/s. Theself-expanding stent comes in contact with the plaqued wall ofthe artery in �37 s (out of 102 s of total time of heating at state-iv in Fig. 8 and location “iv” along path iii–iv–v/v0 in Fig. 10). Thestent ambient temperature at this state is approximately�18 1C. Upon the further heating from �18 1C to 37 1C (i.e.,the duration of 65 s in this heating/activation step), the stentexpands to attain its memory configuration against the diseasedwall of the artery in contact reducing the blockage by increasingthe opening of the diseased artery from initial 3.4 mm to5.11mm (path iv–v in Fig. 10).

In the both Cases 1 and 2, the results from the referenceanalyses overlap with those from the actual stenting proce-dures before the contact, and only deviates from the actualsimulation after the contact is established (see red lines orpath ii–iii–iv0 in Fig. 9 for Case 1, and path iii–iv–v0 in Fig. 10for Case 2). The reference analysis shows the full recovery ofthe overall (global) initial memory configuration of the stentin both Cases 1 and 2.

To summarize, at the beginning of the simulation, stenoticvessel is 60% obstructed by the plaque, which is above thethreshold for surgical intervention. At the end of the stentingprocedure, the vessel blockage is reduced to �18% in Case 1 and�15% in Case 2, which is below the blockage levels normallydetected on an exercise electrocardiogram (Grant et al., 2003).

6.3.3. Maximum shear stress–maximum shear strainresponseIn Case 1, the stent utilizes the superelastic behavior of theNiTinol material where the material undergoes reverse trans-formation (i.e., from stress-induced martensite to the austenitephase) during the two-phase activation stage in our Case 1, i.e.,the first phase involving the release of the crimping force at24 1C, followed by the second phase in which the stent is heatedfrom 24 1C to body temperature. As a result of this reversetransformation, the stent will essentially attempt to recover thestress and strain accumulated in the stent elements upon therelease of the crimping forces. During the first phase of activa-tion, the first contact between the self-expanding stent and theplaqued artery (initial opening of 3.4 mm) is established at state(iii) along path (ii–iii–iv) in Fig. 12, where the correspondingmaximum shear stresses are 132 MPa and 22.5 MPa, and themaximum shear strains are1.24% and 0.46% at elements ‘A’ and‘B’, respectively.

Considering now state (iv) at the end of first activation phase,i.e., upon the completed release of the crimping force (alongpath ii–iii–iv in Fig. 12a), one observes a state of residualmaximum shear strain of magnitude 0.55% and a state ofresidual maximum shear stress of value 91MPa at the centroidof element ‘A’. This is to be contrasted with the counterpartvalues of 0.45% and 7MPa for the residual states of maximumshear strain and maximum shear stress, respectively, at thecentroid of element ‘B’(location ‘iv’ in path iii–iv in Fig. 12b).Considering the conditions upon the completion of the secondphase of the activation, i.e., heating the unloaded stent from24 1C to body temperature, no significant change in the residualstates of stress and strains were observed in element ‘B’,whereas at element ‘A’ the stress increased to 92MPa and thestrain decreased to 0.52% (state “v” along path iv–v in Fig. 12a).

Proceeding now to the consideration of the results of Case2, the stent utilizes the shape memory effect of the NiTinolduring the thermal activation in which the stent undergoestransformation from (pre-dominantly) detwinned martensiteto austenite state, attempting to recover the memory config-uration. The results for the local stress and strain statesinside elements ‘A’ and ‘B’ have indicated marked differencesin the changes with temperature during two distinct regions,i.e., �50 1C to �18 1C (path iii–iv in Fig. 13) and �18 1C to37 1C (path iv–v in Fig. 13), noting that state iv corresponds tothe concurrence of first contact between the stent andplaqued artery. In the first region, the maximum shear strainreduced significantly with temperature (from 2.9% to 1.6% inelement ‘A’ and from 1.13% to 0.46% in element ‘B’), but thestress remained essentially zero during this whole period. Atthe end of the second region of thermal activation, i.e., from�18 1C to 37 1C, significant changes occurred in both the stateof stress and strain. In particular, the maximum shear stressand strain developed are �6.6 MPa and 0.26% at the element‘B’ (path iv–v in Fig. 13b) as well as �57 MPa and 0.54% atelement ‘A’ (path iv–v in Fig. 13a).

To better access the differences in results obtained in Case1 vs. Case 2, we also give in Fig. 14, a detailed comparison ofthe maximum shear stress in the host tissue plotted againstthe change in the internal vessel diameter. More specifically,a significantly lesser value of the maximum shear stress (i.e.,1.18 kPa) was developed in the host (artery) tissue in Case 2 incomparison to 1.6 kPa in Case 1. This directly translates intoan important performance index (i.e., patient comfort) of thesurgical stenting procedure.

In the both Cases 1 and 2, as noticed in Section 6.3.2, thepresent results for the local stress/strain states at elements ‘A’and ‘B’ from the reference analyses overlap with those from theactual stenting procedures before the contact, and only deviatefrom the actual simulation after the contact is established (seered lines or path ii–iii–iv0 in Fig. 12 for Case 1, and path iii–iv–v0 inFig. 13 for Case 2). The reference analyses show the full recoveryof the local stress components (to zero values) at both elements‘A’ and ‘B’ for both Cases 1 and 2. However, there is difference inthe residual strains remaining in the two elements, i.e., residualmaximum shear strains of values 0.08% and 0.5% for elements‘A’ and ‘B’, respectively, in Case 1, and counterpart values of0.03% and 0.3% for elements ‘A’ and ‘B’ in Case 2.

To summarize, the present results of the local stresses andstrains developed in both the SMA stent as well as the host

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tissue, in each of the two alternative stenting procedures ofCases 1 and 2, indicate the following clinically-important con-clusions regarding three performance indices:

(1)

The magnitude of the stress developed at the joint region(element ‘A’) of the stent is far more significant inproviding the desirable, steady values of the “low chronicoutward” force (Duerig et al. 2002).

(2)

Collectively, stenting scenario of Case 1 (‘hard’ mechanicalactivation) produced a stress change in the SMA element ‘A’of 40MPa to achieve an increase in the stent diameter from3.4mm to 4.92mm, and this resulted in stress change of1.6 kPa in the host tissue. In contrast, the stenting scenario ofCase 2 (‘soft’ thermal activation) produced a stress change of57MPa in the SMA element ‘A’ to achieve a bigger increase inthe stent diameter from 3.4mm to 5.11mm, and thisresulted in a much smaller (thus clinically more favorable)stress change of 1.18 kPa in the host tissue.

(3)

As a corollary to (2) above, the soft thermal activation of SMAprovides a far more effective scheme for stenting comparedto the hard mechanical counterpart. In particular, this is truewith regard to (a) a lesser magnitude of the residual stressesdeveloping in SMA element ‘A’ (57 MPa in Case 2 vs. 92 MPain Case 1), and (b) simultaneously providing larger reductionin the vessel blockage area, i.e., from 60% to 15% in Case 2compared to from 60% to 18% in Case 1.

6.3.4. Additional results: effect of rate in releasing thecrimping forceRecalling the results in Section 6.3.3 above, a pattern can beestablished for the hard mechanical activation scheme withregard to the significance (or otherwise) of changes in theconstraining force responsible for the mechanical activationin Case 1. In particular, considering the numerical resultscited in item (2), i.e., 40 MPa stress change producing internalvessel diameter increase of 1.52 mm, a change of order�3 MPa or less in the SMA element ‘A’ is negligible. On thisbasis, it becomes of interest now to explore the variousfactors that may affect this mechanical force during theprocedural controls of Case 1. This has motivated the addi-tional results presented in this section.

The activation stage for Case 1 depends heavily on themanner by which we eliminate (release) the restraining force.Thus, additional simulations were made to study the effect ofrelease rate of the constraining force. To this end, only thereference analysis at 24 1C is considered with the focus beingplaced on the re-expansion stage of the stent (from the fullycrushed configuration). In the base case, i.e., 2 N/s force-releaserate was utilized in the simulation of Sections 6.3.2 and 6.3.3. Inparticular, when the expanding stent diameter reached 3.4mmit signified the beginning of the treatment for the consideredstenosis of 60% blockage, which was especially important indictating the subsequent changes resulting in the stress of theSMA elements in Case 1. Correspondingly, when the stentdiameter reaches the different values of 2.4mm and 1.4mm,one can speak of treating a more severe stenosis condition with80% and 93% blockage, respectively. These three specific dia-meters, and their associates stress values in the SMA elements

will be emphasized in the results of the present section dealingwith force-release rate effects.

In addition to the base rate of force release utilized above, i.e.,2 N/s (equivalently, the total time required to release the force of44 N to zero is 22 s), two additional slower force release rates areconsidered, i.e., 0.4 N/s (total time¼110 s) and 0.08 N/s (totaltime¼550 s). Fig. 15(a) and (b) demonstrates the effect of thedifferent rates of force-release on the state of maximum shearstress and maximum shear strain in the elements ‘A’ and ‘B’ ofthe stent. The state of stress and corresponding strain at whichthe stent diameter reaches values of 3.4 mm (base condition),2.4 mm, and 1.4mm are marked on the respective stress–straincurves for each of the rates of force-release. Fig. 15(c) is theenlargement of the stress–strain region that covers the stress–strain states at the above three stent diameters.

The decrease in the rate of force-releases increased themagnitude of maximum shear strains in the early unloadingregion and subsequently increased themaximum shear stress inthe centroids of elements ‘A’ and ‘B’ during the “plateau” regionof the reverse transformation (along the unloading path from10% to 2% in Fig. 15(a) and from 4.8% to 0.9% in Fig. 15b). At allthe stent diameters, the maximum shear stress in element ‘A’ ishigher at the lower rate of force-release as can be seen in Fig. 15(c). Also, for the smaller stent diameter, the stresses are higher.In particular, in the case of 2 N/s force-release rate, stress valuesat the stent diameters of 3.4 mm, 2.4 mm and 1.4mm are132.18MPa, 161.19 MPa, and 165.45 MPa, respectively. Similarly, in thecase of 0.4 N/s force-release rate, stress values at the stentdiameters of 3.4mm, 2.4mm and 1.4mm are 136.61 MPa,164.87 MPa, and 173.86 MPa, respectively. Furthermore, the stressvalues in the case of 0.08 N/s force-release rate are 138.22 MPa,167.11 MPa, and 178.64 MPa at the stent diameters of 3.4 mm,2.4 mm and 1.4mm, respectively.

In summary, together with the arguments made earlier in thefirst paragraph of this section regarding the order of significanceof the changes in the SMA stress, the above results (for the rangeof release rates considered) indicate that the effect of the rate ofrelease of the activation mechanical force of Case 1 will be mostsignificant in the case of severe stenosis of 93% blockage(corresponding to 1.4mm stent diameter) and almost negligiblefor the base case of 60% blockage (3.4mm stent diameter).Moreover, for the critical stenosis of 93% blockage, a slowerrelease rate seems to be favorable clinically.

6.4. Long-term performance

Additional simulations were made to assess the long-term perfor-mance of the two surgical stenting protocols of Case 1 and 2considered in Sections 6.1-6.3. This involves consideration of theactual scenario of cyclic arterial pressure oscillations between thesystolic and diastolic states. The considered case corresponds topressure amplitude of 50mmHg (i.e., 6.67 kPa) and frequency of2 Hz (cyclic time period of 0.5 s) at the body temperature (37 1C).

The results for the initial 10 cycles are presented here in Fig. 16(a) and (b). In particular, Fig. 16(a) shows the variation of theinternal vessel diameter in time and Fig. 16(b) depicts the timehistory of the maximum shear stress in the host tissue. During 10pressure cycles, the diameters of the inner wall of the diseasedvessel at the end of systolic half of the cycles (peaks) are observedto be increased by 0.229% (from 5.13655mm to 5.14832mm) in

Fig. 15 – Parts (a) and (b) show the effect of the rate of release of the axial ‘crimping’ force at three different rates of 2 N/s (base),0.4 N/s and 0.08 N/s on the maximum shear stress-vs.-maximum shear strain response for the reference solution of Case 1(at 24 1C) measured at the centroids of elements ‘A’ and ‘B’, respectively. Part (c) is the enlargement of the box region of part(a) with the maximum shear stress (in kPa) to clearly show the respective stress values corresponding to each rate of force-release at the stent diameters of 1.4 mm, 2.4 mm and 3.4 mm.

j o u r n a l o f t h e m e c h a n i c a l b e h a v i o r o f b i o m e d i c a l m a t e r i a l s 4 9 ( 2 0 1 5 ) 4 3 – 6 058

Case 2, whereas it was observed to be decreasing by small value of0.0199% (from 4.97843mm to 4.97744mm) in Case 1. Furthermore,the maximum shear stresses in the host tissue decreased by0.138% (from 2.48977 kPa to 2.48633 kPa) during the 10 cycles inCase1. In addition, in Case 2, the maximum shear stresses in thehost tissue decreased by 1.174% (from 1.87862 kPa to 1.85660 kPa)in the 10 pressure cycles. It is especially important to note theparticular pattern of change in internal vessel diameter during thelatter part of the pressure cycles, i.e., the tendency for the diameterto increase with cycles (hence, favorable avoiding any potentialrestenosis) in Fig. 16(a). This is amanifestation of the rather uniqueset of intrinsic SMA response characteristics referred to as theinverse-relaxation phenomena in Fig. 3 of Section 3.1 (i.e., tendencyto approach the “limit-equilibrium” hysteresis).

In summary, the stresses in the host tissue are higher in Case1 than in Case 2 during the 10 pressure cycles. Note that thedifferences between the systolic (peaks) and diastolic (valley)stress values are very similar in Cases 1 and 2. On the other hand,the internal openings of the stenotic vessel (i.e., internal vesseldiameter) at the systolic states of pressure cycles are higher inCase 2 than in Case 1, while change in opening diameter duringthe pressure cycles in Case 2 is nearly double of that in Case 1.

These results suggest that the stent performance will beimproving during the extended pressure cycles, as no sign ofdetrimental functional fatigue/degradation in the opening of theplaqued vessel was observed (particularly in Case 2 compared toCase 1). The present results are further supported by similarexperimental observations in Pelton et al. (2008) regarding theenhanced constant fatigue life of NiTinol stents. In addition, thestress in the host artery is decreasing with cycles in both Cases 1and 2 (decrease by1.174% in Case 2 versus 0.138% in Case 1 during10 pressure cycles). This provides the additional benefit of lessen-ing any pain felt by the patient from the sent–artery interaction.

7. Conclusions

The simulations of the key stages (crimping, deployment, activa-tion, and steady-state cyclic behavior) in the self-expanding,NiTinol stenting procedure were successfully performed utilizingthe general, multi-mechanism SMA model. The SMA model wascalibrated from test data of six isothermal tensile experiments.These calibration experiments covered a temperature rangebetween �100 1CoMf and 40 1C and 4Af. The calibrated SMA

Fig. 16 – Comparison of the time history responses for (a) internal vessel diameter and (b) maximum shear stress in the hosttissue, during 10 pressure cycles for Cases 1 and 2.

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model responses were found to be in very good agreement withthe experimental results. This calibratedmodel was subsequentlyapplied to the complex thermomechanical loading controlsinvolved in the stenting surgical procedure.

In particular, two different alternative schemes of activatingthe self-expanding SMA stent were considered: Case 1 and Case 2.These correspond to “hard” mechanical activation utilizing super-elasticity in Case 1, and “soft” thermal activation relying upon one-way SME and constrained-recovery characteristics in Case 2.

Based on the results of the present investigation, thefollowing conclusions were drawn:

1)

The SMA model calibrated from simple, unaxial, tensileisothermal tests was successfully implemented in thelarge-scale boundary value problem of the surgical stentingprocedure involving complex stent geometry, large displace-ments and rotations, contact interactions, cyclic thermome-chanical loadings, etc.

2)

As a corollary to item (1) above, the global force-deformationresponse shows significant geometric effect of the honey-comb stent, while the local stress and strain states in thestent exhibit genuine NiTinol SMAmaterial characteristics ofthe respective austenite or martensite states of the material.

3)

The stent programmed memory configuration was more-favorably crimped with lesser force to a higher crushingratio in the thermal activation scheme compared to itsmechanical counterpart.

4)

The mechanical activation procedure requires the maintain-ing of a high restraining force during the deployment toprevent any premature superelastic expansion. This iscompletely bypassed in the thermal activation protocol.

5)

At the end of the activation stage in the stenting proce-dure, for an initial stenosis of 60% blockage, the vesselblockage is reduced to �18% in the mechanical activationcase and �15% in the thermal counterpart.

6)

The magnitude of the SMA stresses developed at the jointregions of the stent are far more significant (compared tothe strut region) in providing the desirable, steady valuesof the “chronic outward” forces.

7)

From the standpoint of comparing the chronic outwardforces in the two alternative activation schemes, thethermal protocol provided a far more clinically favorablecondition. More specifically, this is reflected in lower SMAstresses of 57 MPa in the thermal protocol compared to92 MPa in the mechanical counterpart.

8)

The thermal activation protocol has also shown a morefavorable stress distribution in the host tissue, whichdirectly translates into an important clinical performanceindex (i.e., patient comfort) in the stenting surgery.

9)

For the considered geometry of the stent, the rate offorce-release in the mechanical activation protocol wasfound to be significant, especially in case of severe initialstenosis. In this latter case, lower force-release rates werefound to be more favorable.

10)

No sign of detrimental functional fatigue/degradation inthe “cured” stenotic vessel was observed (particularly inthe thermal activation case) when the effect of 10 cyclesof 50 mmHg of pressure oscillations were considered.Furthermore, the stress in the host artery was found tobe decreasing during the pressure cycles (decrease by1.174% in the thermal case vs. 0.138% in themechanical case).

11)

The use of the present SMA model in simulating NiTinolself-expanding stenting surgeries has provided importantinformation that are extremely difficult to obtain fromthe physical experiments. This indicates the great poten-tial of utilizing such material models as a tool to ulti-mately optimize the procedural controls and clinicalforces in other SMA medical devices.

Acknowledgment

This work was supported by NASA GRC, the FundamentalAeronautics Program, Subsonic, Fixed-Wing, Project no.NNH10ZEA001N-SFW1 and Grant no. NNX11AI57A to theUniversity of Akron. The authors would like to acknowledgethe SMA team at NASA GLENN Research center for their

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guidance and the many discussions regarding the character-istics and nature of these shape memory materials during thedifferent phases of the project.

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