Analytical methods for braided stents design and comparison with FEA

Abstract

Braiding technology is nowadays commonly adopted to build stent-like devices. Indeed, these endoprostheses, thanks to their typical great flexibility and kinking resistance, find several applications in mini-invasive treatments, involving but not limiting to the cardiovascular field. The design process usually involves many efforts and long trial and error processes before identifying the best combination of manufacturing parameters. This paper aims to provide analytical tools to support the design and optimization phases: the developed equations, based on few geometrical parameters commonly used for describing braided stents and material stiffness, are easily implementable in a worksheet and allow predicting the radial rigidity of braided stents, also involving complex features such as multiple twists and looped ends, and the diameter variation range. Finite element simulations, previously validated with respect to experimental tests, were used as a comparator to prove the reliability of the analytical results. The illustrated tools can assess the impact of each selected parameter modification and are intended to guide the optimal selection of geometrical and mechanical stent proprieties to obtain the desired radial rigidity, deliverability (minimum diameter), and, if forming processes are planned to modify the shape of the stent, the required diameter variations (maximum and minimum diameters).

Full text

journal of the mechanical behavior of biomedical materials 119 (2021) 104560
Available online 24 April 2021
1751-6161/© 2021 Elsevier Ltd. All rights reserved.
Analytical methods for braided stents design and comparison with FEA
Alissa Zaccaria
a
,
b
, Giancarlo Pennati
a
, Lorenza Petrini
c
,
*
a
LaBS, Dept. of Chemistry, Materials and Chemical Engineering, Politecnico di Milano, Milan, Italy
b
Consorzio Intellimech, Bergamo, Italy
c
Dept. of Civil and Environmental Engineering, Politecnico di Milano, Milan, Italy
ARTICLE INFO
Keywords:
Radial rigidity
Self-expandable stent
In-silico modeling
Multiple twist
Looped end
ABSTRACT
Braiding technology is nowadays commonly adopted to build stent-like devices. Indeed, these endoprostheses,
thanks to their typical great exibility and kinking resistance, nd several applications in mini-invasive treat-
ments, involving but not limiting to the cardiovascular eld. The design process usually involves many efforts
and long trial and error processes before identifying the best combination of manufacturing parameters. This
paper aims to provide analytical tools to support the design and optimization phases: the developed equations,
based on few geometrical parameters commonly used for describing braided stents and material stiffness, are
easily implementable in a worksheet and allow predicting the radial rigidity of braided stents, also involving
complex features such as multiple twists and looped ends, and the diameter variation range. Finite element
simulations, previously validated with respect to experimental tests, were used as a comparator to prove the
reliability of the analytical results. The illustrated tools can assess the impact of each selected parameter
modication and are intended to guide the optimal selection of geometrical and mechanical stent proprieties to
obtain the desired radial rigidity, deliverability (minimum diameter), and, if forming processes are planned to
modify the shape of the stent, the required diameter variations (maximum and minimum diameters).
1. Introduction
Mini-invasive procedures involving implant of stent-like devices are
playing an increasingly important role in the treatment of several dis-
eases (White et al., 2016). Cardiovascular prosthesis are nowadays
commonly exploited in different anatomical areas: to restore the phys-
iological ow in coronaries (Schmidt and Abbott, 2018), peripheral
arteries (Kokkinidis and Armstrong, 2020), or veins (Murphy, 2019); in
the presence of aortic aneurysms and dissections (Maeda et al., 2018); as
ow diverter to promote thrombus formation in intracranial aneurysms
(Cagnazzo et al., 2017); to occlude the atrial septal defect (Yang and Wu,
2018) or the left atrial appendage in patients affected by non-valvular
atrial brillation (Pacha et al., 2019); to replace the mitral (Flynn
et al., 2018) or aortic native valve (Mahmaljy et al., 2020). Moreover,
stents are used to treat pathology affecting the esophagus (Vermeulen
and Siersema, 2018), urethra (De Grazia et al., 2019) and tracheo-
bronchial conduit (Grewal et al., 2019).
Based on the specic target, the device must comply with multiple
contrasting requirements (Chichareon et al., 2019; Watson et al., 2017),
including position stability and appropriate radial stiffness, exibility,
and permeability. In general, it should be possible to implant the stent
through a mini-invasive procedure involving the crimping of the device,
delivery at the region of interest, and deployment. Based on the prin-
ciple behind the last phase, the stents are subdivided into
balloon-expandable and self-expanding (Schmidt and Abbott, 2018). In
the former category, the deployment is forced by a pressurized balloon,
for the latter one, featuring large recoverable deformations, the expan-
sion is driven by the elastic recoil, resulting in a lower impact on the
vessel. Once in-situ, the stent should be exible to comply with vessel
anatomy and movements, should adequately adhere to the wall, and, in
case of stenotic pathologies, should also provide the radial strength
needed to restore the physiological ow (Watson et al., 2017).
The design process aims to optimize the geometrical and material
parameters providing the best compromise among all the requirements
at issue (Bressloff et al., 2016). In-silico simulations have proved to be a
valuable support tool during the design stage, investigating the impact
of the involved parameters with a signicant saving of money and time
with respect to a similar experimental procedure (Kumar et al., 2019),
providing additional levels of detail, and expanding the analyses to
scenarios resembling in-vivo conditions (Karanasiou et al., 2017;
* Corresponding author. Department of Civil and Environmental Engineering Politecnico di Milano Piazza Leonardo da Vinci, 32 20133, Milano, Italy.
E-mail addresses: alissa.zaccaria@polimi.it (A. Zaccaria), giancarlo.pennati@polimi.it (G. Pennati), lorenza.petrini@polimi.it (L. Petrini).
Contents lists available at ScienceDirect
Journal of the Mechanical Behavior of Biomedical Materials
journal homepage: http://www.elsevier.com/locate/jmbbm
https://doi.org/10.1016/j.jmbbm.2021.104560
Received 7 January 2021; Received in revised form 13 April 2021; Accepted 19 April 2021

Journal of the Mechanical Behavior of Biomedical Materials 119 (2021) 104560
2
Morlacchi and Migliavacca, 2013; Roy et al., 2012). However, con-
cerning innovative design, the efforts needed to implement and validate
the numerical model may not be negligible, and the analyses of each
variable impact may be quite expensive. In this light, the advantage
deriving from applying simplied analytical models at an early devel-
opment stage, aiming to identify a suitable range of variation for each
parameter, becomes evident.
This work focuses on braided stents, self-expandable endoprosthesis
consisting of interlaced wires, and featuring great exibility and kinking
resistance, which make them suitable for several applications (Bishu and
Armstrong, 2015; Cremonesi et al., 2015; Madhkour et al., 2019; Sei-
german et al., 2019; Subramaniam et al., 2019). The numerical or
experimental studies available in the literature on braided stent
behavior are very few and very recent (Zaccaria et al., 2020a; Shanahan
et al., 2017; McKenna and Vaughan, 2020, 2021). In this context, the
prediction of mechanical properties through analytical models is not
trivial. Jedwab and Clerc (1993) illustrated how to evaluate the longi-
tudinal and radial properties of a cylindrical basic braided structure and
validated the results with respect to experimental tests. Moreover, the
equations proposed were subsequently used as a comparator to validate
nite element simulations (Shanahan et al., 2017). However, so far, no
indications have been presented to predict the radial strength when
more complicated features are introduced, like multiple twists and
looped ends (Fig. 1a). Moreover, a crucial aspect in the design of braided
stents is the denition of the diameter variation range given the initial
conguration. On one side, the minimum diameter that can be reached
during crimping impacts the device’s deliverability. On the other side,
complex shapes are usually obtained by deforming an original cylin-
drical braided texture. Thus, a priori knowledge of the diameter
variation range enables estimating (on the basis of predened geomet-
rical parameters) if a specic conguration is achievable, reducing the
trial-and-error efforts.
In this paper, approximated analytical formulae for the prediction of
the radial strength of different braided structures are presented and
compared with a numerical model validated in a previous work (Zac-
caria et al., 2020a). Moreover, equations based on the braided
geometrical parameters to obtain the minimum and maximum diameter
are illustrated and validated comparing with numerical simulations on a
repetitive braided unit.
2. Materials and methods
In the following paragraphs, the analytical formulae developed to
predict the radial rigidity of a braided stent are outlined, as well as the
equations to derive the minimum and maximum diameter. Finally, the
nite element models used as comparators and the numerical analyses
performed for validation purposes are presented.
2.1. Radial pressure
2.1.1. Cylindrical braided stent
First, the formulae proposed by Jedwab and Clerc (1993) are sum-
marized. These equations allow for calculating longitudinal and radial
properties of a cylindrical braided stent, providing the material elastic
modulus (E) and Poisson coefcient (
ν
), and the following geometrical
parameters (Fig. 1b):
- single wire diameter (d);
Fig. 1. (a) Braided features exemplication: multiple twist (double and triple), looped end. (b) Braided geometrical parameters: average stent diameter (D), stent
length (L), pitch angle (β), wire diameter (d). The initial values are marked with the subscript ‘0’. The stent longitudinal axis is indicated by the dashed line. On the
right, the single unwound wire is represented in cyan (constant length =L
wire
) to clarify the relationships among the geometrical parameters.
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Journal of the Mechanical Behavior of Biomedical Materials 119 (2021) 104560
3
- initial average stent diameter (D
0
): easily obtainable by summing 2d
to the initial internal diameter, dened by the mandrel used during
the braiding procedure;
- initial pitch/braid angle (β
0
): dening, in this study, the slope of the
wires with respect to the circumferential axis;
- initial length (L
0
): longitudinal stent length;
- total number of wires (N): involving both clock-wise and counter
clock-wise wires.
Given these values, the initial pitch and the number of coils can be
calculated as:
p0=
π
D0tan(β0)(1)
c=L0/p0(2)
Assuming that the wire length (L
wire
) remains constant, and so the
sides length of the braided cells (Fig. 1b), and that the wire rotation
around the longitudinal axis is prevented by wires intertwining, the
deformed stent diameter and length (D, L) are related to the pitch angle
(β) through the following equations (Fig. 1b):
D=D0cos(β)
cos(β0)(3)
L=L0sin(β)
sin(β0)(4)
Thus, a longitudinal elongation is necessarily associated with a
diameter reduction, and it is possible to obtain the same deformed
conguration applying a longitudinal force or radial pressure. Consid-
ering the stent undergoing a tension test and focusing on the single wire,
it is possible to represent it as a helical spring with xed ends (Fig. 2a)
subject to a longitudinal force (F
wire
) and a moment (M
wire
), preventing
the rotation around the longitudinal axis. Thus, the bending and twisting
moments on the wire (m
B
, m
T
) are calculated as:
mB=Mwire cos(β) − FwireRsin(β)(5)
mT=Mwire sin(β) + FwireRcos(β)(6)
where R is the average radius of the stent (D/2). Moreover, it is possible
to connect the bending and twisting moment to the change in curvature
(Δκ) and the twist (Δθ) of the wire.
Fig. 2. (a) Open-coiled helical spring with end xed against rotation: external load and corresponding internal actions acting on a generic element along the helical
spring. (b) Double twist geometrical parameters: standard braided length (
h0), twist length (htwist), twisted cell length (ht0), circumferential dimension (bt0) and twist
pitch angle (βtwist0). (c) Quadruple twist geometrical parameters: twist length (3htwist ), twisted cell length (ht0), circumferential dimension (bt0). (d) Looped end
geometrical parameters: length (hle0), circumferential dimension (ble0) and looped end pitch angle (βle0). (e) Double twist: comparison between 1 and 2 standard
braided cells between adjacent multiple twist layers.
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Journal of the Mechanical Behavior of Biomedical Materials 119 (2021) 104560
4
mB
EI =Δκ =cos2(β)
R−cos2(β0)
R0
(7)
mT
GIP
=Δθ =cos(β)sin(β)
R−cos(β0)sin(β0)
R0
(8)
To obtain the curvature and twist denition reported above, the
natural parametrization of the helix γ(t)should be considered γ(t) =
[R ⋅cos(t⋅cos(β)/R);R ⋅sin(t⋅cos(β)/R);t⋅cos(β)⋅tan(β)]. Then, the cur-
vature κ is calculated as κ=γ’’(t) = cos2(β)/R, while the twist θ can be
obtained as θ=b’(t)where b(t) = γ′(t) × γ′′(t)/γ’’(t).
Thus, the force and moment applied on the wire extremities are
related to the pitch angle based on the following equations:
Fwire =GIPcos(β)
Rcos(β)sin(β)
R−cos(β0)sin(β0)
R0
−EIsin(β)
Rcos2(β)
R−cos2(β0)
R0(9)
Mwire =GIPsin(β)cos(β)sin(β)
R−cos(β0)sin(β0)
R0
+EIcos(β)cos2(β)
R−cos2(β0)
R0(10)
In the whole stent, the moment reects the internal action exerted by
the intertwined structure, while the total force F =F
wire
⋅N can be rep-
resented by substituting the terms involving initial geometrical param-
eters with three constants:
K1=sin(2β0)
D0
K2=2cos2(β0)
D0
K3=D0
cos(β0)(11)
F=2NGIP
K32sin(β)
K3
−K1−EI⋅tan(β)
K32cos(β)
K3
−K2 (12)
where I and Ip are the moment of inertia and the polar moment of
inertia, and G is the shear modulus. To extend or compress the stent, the
work done by the external force is equal to dW =F⋅ (L−L0). The same
deformation can be obtained applying a radial pressure P over the lateral
surface
π
DL, corresponding to an energy equal to dW =P⋅
π
DL⋅ (D−
D0)/2. Thus, the pressure needed to vary the average stent diameter can
be calculated as follows:
P=2F
π
DL
(L−L0)
(D−D0)(13)
And given the relation among length, diameter and pitch angle:
P=2Fc
DLtan(β)(14)
Or, if the length is unknown, considering Eqs. (1,2)
P=2F
π
D2tan2(β)(15)
2.1.2. Double twist
The double twist geometry is illustrated in Figs. 1a and 2b and is
obtained by inverting once the braiding direction. The geometrical pa-
rameters dening this feature are: bt0, ht0, htwist and βtwist0. While the
rst variable is directly deducible from the braiding parameters as bt0=
2
π
D0
N, the value ht0 was measured on real devices. In the specic case,
this entity was observed to be related to the length of a standard braided
texture as ht0=1.24
h0=1.24 bt0tan(β0)(Fig. 2b). Similarly, htwist was
measured, and, in the specic case, this entity was observed to be related
to the wire diameter as htwist =1.84 d (Fig. 2b). Finally, βtwist0 was
evaluated starting from the previous parameters as βtwist0=
arccos(bt0/Ldt)where Ldt =
bt0
2+ (ht0+htwist)2
.
During the crimping, the pitch angle should change according to Eq.
(3). However, at the double twists, the wires are not free to rotate
(Fig. 3a). It is possible to compare this segment, involving two crossing
points, to a beam subject to a concentrated load whose length is
approximated with Ldt . Then, the load can be evaluated assuming that
the pitch angle variation in correspondence of the extremity (red dot in
Fig. 3a) should equalize to the standard braided texture (β−β0). For the
displacement, no conditions were imposed since the double twist wires
can separate and, thus, the length of the cell sides (Fig. 1b) is not
required to remain constant. Thus, given a diameter variation, it is
possible to deduce the deformed pitch angle (β) from Eq. (3) and, based
on the linear theory of elasticity (Fig. 3d), the concentrated force acting
perpendicular on the wire extremity can be calculated as:
Fwire =(β−β0)⋅2EI
L2
dt
(16)
The total force acting on the stent in the longitudinal direction
should be equal to
F=NFwire
cos(β)(17)
and, starting from the energy equivalence presented by Jedwab and
Clerc (1993), Eq. (13), the average radial pressure can be calculated as
follows.
P=2F
Dhdt
⋅hdt − (ht0+htwist)
D−D0
(18)
where hdt can be deduced based on the linear theory of elasticity
(Fig. 3d, Δl=hdt − (ht0+htwist)) as a projection of the displacement
connected with the angle variation.
hdt = (ht0+htwist) + 2
3Ldt(β−β0)cos(βtwist )(19)
Note that the displacement was projected with respect to the twist
angle (Fig. 2b) evaluated as
βtwist =βtwist0+ (β−β0)(20)
2.1.3. Triple/quadruple twist
The same logic could be applied to more complex geometries
involving additional crossings. Note that the triple twist (Fig. 1a) does
not allow to increase the radial rigidity of the standard braided stent
since its design does not oppose the wire’s rotation due to radial
compression, allowing the pitch angle variation (Fig. 3b1). Concerning
the quadruple twist, the geometrical parameters considered are the
same: bt0, ht0, htwist and βtwist0 (Fig. 2c). However, the deformable length
is calculated as Lqt =
bt0
2+ (ht0+3htwist)2
. The rationale is similar to
the previous case, except for the condition imposed on the cantilevered
beam extremity. Indeed, in this case, the wires are not free to separate
(Fig. 3b2), and the sides of the cells preserve their original length. Thus,
besides the pitch angle, the displacement should also comply with the
geometry constraints. It is possible to assume the beam subject to a
concentrated moment and force (Fig. 3b) and, through the superposition
principle, deduce their value by imposing at the wire extremity the angle
variation, equal to (β−β0), and the displacement obtained considering
the twist hindrance. Indeed, at a given diameter D, while the circum-
ferential dimension for a standard braided cell is equal to 2
π
D/N, for the
multiple twist geometry, the same dimension would vary according to
the formula:
bqt =bt0
cos(βtwist)
cosβtwist0(21)
where βtwist is calculated as in Eq. (20), with β depending on the current
D value (Eq. (3)). Thus, the displacement of the extremities was
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Journal of the Mechanical Behavior of Biomedical Materials 119 (2021) 104560
5
calculated based on the difference between the circumferential dimen-
sion calculated through Eq. (21) and the braided texture constraint.
Fwire =6EI
L3
qt 2
sin(βtwist)bqt −2
π
ND+Lqt(β−β0)(22)
Mwire =EI
Lqt
(β−β0) + 1
2FwireLqt (23)
The boundary conditions were applied in the second overlapping
point, corresponding to a length estimated with Lqt , assuming that the
wires may be able to slightly adjust their position in the correspondence
of the rst intersection (Fig. 3b2):
The total force acting on the stent in the longitudinal direction is
obtained as previously based on Eq. (17), and the average radial pres-
sure may be calculated as:
P=2F
π
Dhqt +2htwist⋅
hqt −(ht0+htwist)sin(βtwist )
sin(βtwist0)
D−D0
(24)
hqt can be deduced projecting the displacement imposed at the wire
extremity and summing the elongation derived from the rotation in the
cylindrical plane. Note that the length variation in the second factor was
calculated with respect to the length resulting from the rigid rotation in
the cylindrical plane.
hqt = (ht0+htwist)sin(βtwist )
sinβtwist0+1
tan(βtwist)bqt −2
π
ND(25)
2.1.4. Multiple twist: two separation layers
While in the previous sections only one layer of standard braided
cells was assumed to separate the subsequent multiple twists, this sec-
tion aims to extend the formulae to stent designs featuring an additional
separation layer (Fig. 2e). In this case, the twist is assumed to behave as
previously, considering the same geometrical parameters, while the
surface on which the average pressure is evaluated is modied by
including the length of the additional braided layer
((
π
D/N)tan(β)(sin(β)/sin(β0))). Thus, only Eqs. (18, 24) were adjusted
as follows:
P=2F
π
Dhdt +
π
D
Ntan(β)sin(β)
sin(β0)⋅hdt −hdt0
D−D0
(26)
P=2F
π
Dhqt +2htwist +
π
D
Ntanβsin(β)
sin(β0)⋅
hqt −hqt0
sin(βtwist)
sin(βtwist0)
D−D0
(27)
2.1.5. Looped end
The looped end geometry is illustrated in Figs. 1a and 2c. The
geometrical parameters dening this feature are: ble0, hle0 and βle0. As
Fig. 3. (a-b-c) Beam approximation for double twist, multiple twists (triple and quadruple) and looped end (black). In yellow the length involved in the radial
pressure calculation. (d) Linear theory of elasticity: displacement and rotation for a beam subject to an external concentrated force or a moment.
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Journal of the Mechanical Behavior of Biomedical Materials 119 (2021) 104560
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previously, the rst variable was calculated as ble0=2
π
D0/N, while hle0
was measured on real devices. In the specic case, this entity was
observed to be related to the length of a standard braided texture as
hle0=0.714
h0=0.714 ble0tan(β0)(Fig. 2c). Finally, βle0 was evaluated
starting from the previous parameters as βle0=arccos(ble0/Lle )where
Lle =
ble0
2+hle0
2
.
Like the quadruple twist feature, the looped end was approximated
with a beam subject to a concentrated moment and force, since the
boundary conditions involve both angle variation and displacement
(Fig. 3c). As previously, the beam was assumed to cover two overlapping
points (Fig. 3c), corresponding to the deformable length Lle . The angle
variation at its extremity is equal to (β−β0). Concerning the displace-
ment, the same process applied for the quadruple twist was followed.
First, the pitch angle characteristic of the looped end beam, βle0, was
calculated. As previously, the circumferential dimension due to the pitch
angle variation would vary according to the formula:
ble =ble0
cos(βle)
cosβle0(28)
with βle =βle0+ (β−β0). Thus, the force and moment acting at the
extremity was calculated as:
Fwire =6EI
L3
le 2
sin(βle)ble −2
π
ND+Lle(β−β0)(29)
Mwire =EI
Lle
(β−β0) + 1
2FwireLle (30)
It can be observed that a lower looped end length (↓hle0)corresponds
to a lower looped end pitch angle (↓βle0) and a higher circumferential
dimension following the same rigid rotation (↑ble at the same β),
requiring higher displacement and force (↑Fwire) to comply with the
geometry restriction.
As previously the total longitudinal force (F) is calculated through
Eq. (17) and, nally, the radial pressure can be obtained as:
P=2F
π
Dhle +2
π
D
Ntanβsin(β)
sin(β0)⋅
hle −hle0
sin(βle)
sin(βle0)
D−D0
(31)
where:
hle =hle0
sin(βle)
sinβle0+1
tan(βle)ble −2
π
ND(32)
The pressure was calculated on a larger surface involving two more
crossing layers (Fig. 3c) to facilitate the comparison with the FE model.
2.2. Minimum and maximum diameter
The intertwined structure of the braided allows the wires to rotate
and relatively easily adjust their position, elongating or compressing the
stent, under external radial loads. However, each device presents a
specic limit, both considering crimping and expansion deformations,
after which, to further deform it, signicant forces would be required.
This stiffening is due to the activation of contacts among the wires in
points distant from the overlapping area (Fig. 4b). If higher forces are
applied, the stent may deform further but never exceed the physical
limit dened by the wires’ diameter (Fig. 4c).
The pitch angle related to the maximum compressed conguration
(βmin in Fig. 4c) is obtained assuming that between two crossing points
(circumferential distance ≈
π
D/N), the longitudinal displacement
should be equal to the diameter of the wires.
βmin =arctandN
π
D(33)
As regards the pitch angle at which the rst contact among the wires
Fig. 4. (a-b-c) Braided texture in the longitudinal-radial plane (zr, top) and in the longitudinal-circumferential plane (zθ, bottom): (a) undeformed conguration; (b)
deformation at which the lateral contact among the wires is rst recorded; (c) maximum deformation. (d) Detection of the pitch angle corresponding with the rst
contact deformation (β
1c
), based on the radial coordinate trend.
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Journal of the Mechanical Behavior of Biomedical Materials 119 (2021) 104560
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appeared (β1c in Fig. 4b), the intertwined geometry plays the main role.
It is possible to assume that the contact occurs on the mean plane (cy-
lindrical surface with diameter equal to the average stent diameter:
orange surface in Fig. 4d). The width of the wire footprint on this surface
(FP) is determined by the radial coordinate oscillation trend (ΔR) based
on the following equation
FP =d
2cosasin2ΔR
d (34)
Thus, β1c corresponds to the arctangent of FP rst derivative with
respect to the wire length (l
1
/l
2
coordinate in Fig. 4d) at the origin.
Considering a sinusoidal intertwining, meaning a radial coordinate
oscillation equal to
ΔR =d
2
cos(θN)θ∈0,
π
N(35)
And observing that the wire length is related to the circumferential
coordinate θ through the initial stent diameter and pitch angle
l=θD0
cos(β0)θ∈0,
π
N(36)
It is possible to obtain the rst derivative of FP with respect to l from
Eqs. (34–36)
δFP
δΔR = − sinasin2ΔR
d 1

1−2ΔR
d2
= − cos(θN)1
sin(θN)(37)
δΔR
δθ = − d
2Nsin(θN)(38)
δθ
δl =cos(β0)
D0
(39)
δFP
δl =δFP
δΔR
δΔR
δθ
δθ
δl =dN cos(β0)
2D0
cos(θN)(40)
δFP
δl (l=0) = dN cos(β0)
2D0
(41)
Thus, β1c is equal to
β1c =atandN cos(β0)
2D0(42)
Finally, it is possible to obtain the higher and lower diameter at
which the rst contact is recorded as
Dmin1c =D0
cos
π
2−β1c
cos(β0)(43)
Dmax1c =D0
cos(β1c)
cos(β0)(44)
2.3. Numerical simulations
2.3.1. Radial pressure
The validation was performed comparing the results of the analytical
formulae with output of numerical simulations performed in Abaqus/
Explicit 2019 (Dassault Systemes Simulia, Providence, RI, USA). Four
braided models were built, using 3D parametric equations (Zaccaria
et al., 2020b) and starting from the geometrical and material properties
of the braided component of the ID Venous System (ID NEST MEDICAL,
Strasbourg, FR), named ID Branch, that was deeply analyzed and vali-
dated with respect to experimental tests in previous works (Zaccaria
et al., 2020a): a cylindrical braided stent, a braided stent with one
looped end, the ID Branch design (involving one looped end and ve
double twist layers), the ID Branch design with triple and quadruple
twists instead of double twists.
The standard braided design was compared with the analytical
formulae already proposed and validated by Jedwab and Clerc (1993) to
assess the nite element model reliability in predicting the average
contact pressure on the wall, similar to Shanahan et al. (2017). The
remaining were exploited to validate the newly proposed equations.
Moreover, the predictions on multiple twists and looped ends were also
validated considering design variations: lower stent diameter (internal
diameter =10 mm), variable wire diameter, increased pitch angle
(length x1.5), two standard braided cells between multiple twist layers.
The comparison was based on crimping simulations in which the
stent was surrounded by a cylindrical surface whose diameter is varied
to reduce the initial average stent diameter down to 20%. The free open
extremity was constrained to stabilize the stent and minimize the
boundary effect by preventing the displacement along the circumfer-
ential and longitudinal axis, as well as the rotation around the same axis.
Differences due to this constraint are highlighted in the results and
discussion sections.
The general contact algorithm was exploited to describe the inter-
action among the wires (Zaccaria et al., 2020b), imposing a friction
coefcient of 0.2 (Kelly et al., 2019; Ma et al., 2012). Concerning the
material model, the original stent is made of Nitinol, a super-elastic
material whose properties were extracted from experimental tests on
wire samples at both 25 ◦C and 37 ◦C (Zaccaria et al., 2020a). However,
for the purpose of this study, only the austenite behavior, dening the
stress-strain relationship for low deformations, was considered. Indeed,
since the interest is to predict the radial pressure on the wall when the
stent is fully deployed (≈20% oversizing), it is fair to assume that the
local stress remains in the linear elastic region, dened by an elastic
modulus of 45000 MPa. In the results and discussion sections, attention
was paid to verify the reliability of this approximation.
2.3.2. Minimum and maximum diameter
A repetitive unit involving two crossing points was built to validate
the formulae for the maximum and minimum diameter (Fig. 5). Since
the equations proposed do not distinguish between a planar or a cylin-
drical braided structure, the unit was drawn considering both a planar
(xyz) and a cylindrical (rϑz) average surface in order to identify po-
tential unpredicted discrepancies. For the planar conguration, the
drawing equations (Zaccaria et al., 2020b) were modied as follows:

















x(ϑ) = d
2⋅cos(ϑN)
y(ϑ) = D0ϑ
2
z(ϑ) = D0ϑ
2⋅tan(
α
)
with ϑ∈0,2
π
n(45)
To simulate an elongation and compression deformation, boundary
conditions and equation constraints, accounting for the effect of the
adjacent hidden braided structure, were introduced at reference points
placed in the center of the wires’ extremities and rigidly connected with
these. Refer to Fig. 5 for the node labels and coordinate systems
conventions.
For the cylindrical structure, the rotation around the ϑ/z-axis and the
translation along the ϑ-axis were prevented. The longitudinal displace-
ment was imposed at nodes 1 and 2, while nodes 5 and 6 were xed in
the same direction. Finally, to account for the repetitive braided texture,
the equation constraints have been introduced to ensure that the radial
displacement at nodes 1, 2 and 3 is equal to the one at nodes 6, 5 and 4,
respectively, and to guarantee the same longitudinal translation at nodes
3 and 4.
For the planar structure, the rotation around the y/z-axis and the
translation along the x-axis were prevented. Moreover, nodes 2 and 5
were xed in the y-direction to avoid unconned motions. The
displacement was imposed at nodes 1 and 2 in the z-direction, while
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Journal of the Mechanical Behavior of Biomedical Materials 119 (2021) 104560
8
Fig. 5. Numerical boundary conditions to identify the maximum and minimum diameter for a standard braided unit.
Fig. 6. Crimping of a standard braided sample. From left to right: undeformed conguration; stent deformed conguration with von Mises stress colored map (note
that on one end only radial displacement and rotation around the radial axis are allowed); crimping surface with contact pressure colored map; comparison between
FE results and analytical prediction in terms of average contact pressure (calculated for the model on the section enclosed by the black rectangle) and cell length. Lo
indicate the cell length in the undeformed conguration.
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Journal of the Mechanical Behavior of Biomedical Materials 119 (2021) 104560
9
nodes 5 and 6 were xed in the same direction. Finally, to account for
the repetitive braided texture, the equation constraints have been
introduced to ensure that the displacement along y at nodes 1 and 3 is
equal and opposite to the one recorded at nodes 6 and 4, respectively,
and to guarantee the same translation along z at nodes 3 and 4.
3. Results
3.1. Radial pressure
3.1.1. Cylindrical braided stent
First, the FE model accuracy in predicting the contact pressure was
assessed. More specically, a crimping simulation on a standard braided
texture was performed and compared with the analytical formulae
proposed by Jedwab and Clerc (1993) and already validated with
respect to experimental tests. The ID Branch geometrical parameters
were used: D0 =13.77 mm; d =0.23 mm; β0 =32◦; N =26 (Fig. 6). The
numerical average contact pressure at 20% oversizing differs from the
analytical prediction for less than 1.39% in the central portion, proving
the reliability of the simulation. Note that the FE model is also able to
assess the extremity weakness. Indeed, the average pressure on the last
three cells in proximity of the free open end is decreased by 21.95%.
Moreover, the length variation in the model is equal to 38.73%, close to
the 38.50% variation predicted by the formulae.
3.1.2. Double twist
Concerning the double twist feature, the ID Branch original design,
previously validated through the comparison with experimental tests
(Zaccaria et al., 2020a), was used to assess the accuracy of the analytical
predictions (Fig. 7 “Double twist”). The geometrical parameters of the
braided texture are reported above, while the dimensions dening the
double twist geometry are: hdt0 =2.563 mm, htwist =0.423 mm, bdt0 =
3.327 mm. The analytical and FE results show differences of 0.94% and
2.44% in terms of contact pressure and deformed length, respectively
(Fig. 8 “ID Branch”).
To further validate the formulae, some design modications were
investigated. First, the stent diameter was modied keeping the same
pitch angle (10 mm internal diameter), and three different wires
diameter were evaluated (0.16/0.17/0.18 mm). Moreover, the pitch
angle impact was assessed by scaling the length of the original model by
3/2. The analytical radial pressure and the nal length differ from FE
results for lower than 3.35% and 2.15%, respectively (Fig. 8 “D0 10.36
mm”, “D0 10.34 mm”, “D0 10.32 mm”, “L0 x 1.5”).
3.1.3. Triple/quadruple twist
Concerning the triple twist, no signicant variations were recorded
with respect to the standard braided design (average contact pressure =
0.00322 MPa). For the quadruple twist (Fig. 7 “Quadruple twist”), the
formulae proposed correctly predict the contact pressure and the nal
length of the designs investigated, showing differences with respect to
the FE results below 6.2% (Fig. 8). Note that the average contact pres-
sure is close to the double twist results (difference <6.7%).
3.1.4. Multiple twist: two separation layers
As an additional validation, a second separation layer was intro-
duced between two sequential multiple twist traits (Fig. 2d) and the
equations were updated as outlined in the materials and methods sec-
tion. The analytical formulae reliability is demonstrated by the low
difference with respect to the FE model, below 5.1% and 2.9% in terms
of radial pressure and elongation (Fig. 8, “2 cells”).
3.1.5. Looped end
The Looped end prediction were validated based on the original ID
Branch geometrical parameters (Fig. 9 “ID Branch”), the reduced stent
diameter design with variable wires diameter (Fig. 9 “D0 10.36 mm”,
“D0 10.34 mm”, “D0 10.32 mm”), and the elongated conguration
(Fig. 9 “L0 x 1.5”). The differences in terms of average contact pressure
and nal length do not exceed 6.7%. Note that, in this case, the pressure
Fig. 7. Crimping of the ID Branch model with double or quadruple twists. From left to right: undeformed conguration; stent deformed conguration with von Mises
stress colored map (note that on the open end only radial displacement and rotation around the radial axis are allowed to minimize boundary effects); crimping
surface with contact pressure colored map.
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Journal of the Mechanical Behavior of Biomedical Materials 119 (2021) 104560
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was evaluated on the last two cell lines (Fig. 9 “Unstressed”).
3.2. Minimum and maximum diameter
Both a tensile and compression simulation were performed on a
planar and cylindrical braided unit involving two crossing points.
Generic geometrical parameters were used: D0 =10 mm; d =0.2 mm;
β0 =45◦; N =40.
Fig. 10 reports the total longitudinal force-diameter curves. The total
longitudinal force was calculated as
Fl=N
4((Fl1+Fl2) − (Fl5+Fl6)) (46)
while the diameter in the planar conguration was calculated as
D=D0−NΔy
2
π
(47)
where Δy is half of the unit length variation in this direction (Fig. 10b).
Observing the longitudinal force–diameter curves in all the simula-
tions it can be identied a stiffening point, associated with the rst
contact diameter. The planar and cylindrical congurations differ due to
the absence of force in the rst case before the rst contact appear.
However, the point in which the slope variation appear does not vary,
proving the reliability of the approximation considered. The rst contact
points are reported below and were calculated as the mean between the
rst point associated with a signicant force increase and the point
before:
- Dmin
1c
cylindrical =3.77 ±0.24 mm;
- Dmin
1c
planar =3.77 ±0.23 mm;
- Dmax
1c
cylindrical =13.62 ±0.02 mm;
- Dmax
1c
planar =13.62 ±0.02 mm.
The prediction based on the formulae proposed are within the
identied range: Dmin1c =3.85 mm and Dmax1c =13.61 mm.
Fig. 8. Comparison for the double and quadruple twist designs between FE results and analytical prediction in terms of average contact pressure (calculated for the
model on the sections enclosed by the black rectangles in Fig. 7) and unit length.
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Journal of the Mechanical Behavior of Biomedical Materials 119 (2021) 104560
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4. Discussion
This work focuses on braided stent design and presents approximated
formulae to predict the radial pressure and the diameter variation range.
Jedwab and Clerc (1993) have already outlined and validated with
respect to experimental data the equations that allow to connect the
longitudinal and the radial deformation and predict the radial stiffness
at a given oversizing. These formulae were illustrated in this work and
applied to derive the average radial pressure at 20% oversizing for a
standard braided stent using as reference the geometrical parameters of
the ID Branch (the braided component of the ID Venous System, ID Nest
Medica). Then, a FE simulation was performed to assess the numerical
model capability in evaluating the contact pressure on the conned wall.
Since the errors in terms of average contact pressure and length varia-
tion do not exceed 1.4%, the FE model is deemed reliable. Moreover, the
simulation highlighted the weakness associated with open ends. In
general, the free ends of the wires tend to collapse, decreasing the
pressure recorded at this level. Thus, in the subsequent simulations, the
open ends were constrained, as outlined in the material and methods
section. Indeed, as visible in Fig. 6 for the standard braided model, the
boundary effects at the constrained open end are signicantly reduced
with respect to the free extremity.
Following, the FE simulations were used to validate the analytical
predictions concerning multiple twists and looped ends with variable
design parameters. Given the comparability of the results both in terms
of average radial pressure and length variation, differing for less than
10%, the analytical formulae can be judged trustworthy. Note that, in
the diameter variation range considered, the stress distribution within
the wires do not exceed 450 MPa, complying with the assumption of
linearity for the Nitinol stress-strain relationship. Nevertheless, this
hypothesis would not be valid at higher deformation, for example if the
radial pressure on the delivery catheter has to be investigated.
Concerning the multiple twist feature, it could also be interesting to
observe the low variation in terms of radial pressure for double and
quadruple twists. Indeed, even if the longitudinal force is higher for the
quadruple twists, given the displacement constraint, this effect is
balanced by the lower axial deformation of the cantilevered beam and
the larger surface. Instead, as explained in the materials and methods
section, the triple twist does not allow to increase the radial rigidity.
Indeed, a similar simulation performed with a triple twist design showed
an average radial pressure of 0.00322 MPa, 5.3% lower than the stan-
dard braided model result. This is due to the fact that at the triple twist,
the wires are able to adjust their position, minimizing the bending and
torsion moments that would arise in a standard helicoidal wire due to
the curvature variation. For the looped end feature, observing the con-
tact pressure distribution in Fig. 9, it becomes clear the need to evaluate
the radial pressure on the last two cells layers instead of considering only
the last one. Indeed, the contact pressure is localized in correspondence
of the second and third crossing points.
Note that, even if the formulae were presented starting from the ID
Branch device geometry (Fig. 2b and c), specically, on the cylindrical
segment of the stent, involving one looped end and ve double twist
traits separated from each other by one braided layer (Fig. 2d “1 cell”),
their extendibility to different design has been demonstrated. The most
critical geometrical parameters to be dened are hdt0/hqt0,htwist and hle0.
If no data are available, the relations illustrated between these values
and the braiding parameters could provide preliminary information.
However, as soon as one sample is manufactured, these values should be
measured and updated.
Concerning the diameter variation range, the formulae illustrated for
a sinusoidal intertwining correctly identify the maximum and minimum
diameter at which the rst contact among the wires occurs. Further
deformations beyond these limits are possible but require a signicant
compressive or tensile force increase. Note that the planar conguration
does not record any force until that point. Indeed, without contacts
among the wires, a longitudinal elongation reects in a rigid rotation, in
contrast with the cylindrical conguration where the z-displacement
modify the curvature of the beam, generating bending and twisting
Fig. 9. Crimping of a standard braided sample with a looped end. From left to right: undeformed conguration; stent deformed conguration with von Mises stress
colored map (note that on the open end only radial displacement and rotation around the radial axis are allowed); crimping surface with contact pressure colored
map; comparison between FE results and analytical prediction in terms of average contact pressure (calculated for the model on the section enclosed by the black
rectangle) and cell length.
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Journal of the Mechanical Behavior of Biomedical Materials 119 (2021) 104560
12
moments. The values identied are good estimators of the maximum
and minimum diameter, the latter of which affect the system deliver-
ability. However, during the braiding procedure, a tensile load is usually
applied at the wire extremities, resulting in a more linear trend with
respect to the sinusoidal oscillation and, subsequently, in a narrower
diameter variation range (↑β1c , Fig. 4). Thus, the β1c obtained with the
sinusoidal approximation may be considered as the lower limit to which
the actual device tends with decreasing load applied during the
manufacturing. A more accurate result could be obtained by modifying
the wire’s radial oscillation (ΔR, Fig. 4, Eq. (35)) based on the applied
load.
5. Conclusion
In this paper, analytical formulae, based on geometrical and me-
chanical parameters, have been developed for the prediction of the
radial strength of different braided structures and the evaluation of
minimum and maximum diameter, obtainable once dened the geom-
etry. The efcacy of the proposed approach has been checked by
comparing the analytical results with computational data obtained from
a nite element model validated in previous work (Zaccaria et al.,
2020a).
The validity of the illustrated model, as a simplistic strategy, is
limited. Indeed, non-linearities and manufacturing process specicity
are not considered. Nevertheless, it can be deemed a helpful tool for the
design and optimization phases, easy to be implemented, and associated
with a signicant reduction in terms of time and production cost thanks
to the possibility to preliminary identify a suitable range for design
parameters such as stent diameter, pitch angle, the number of wires and
their diameter, as well as material stiffness and the introduction of
multiple twists and looped end features.
CRediT authorship contribution statement
Alissa Zaccaria: Conceptualization, Software, Validation, Data
curation, Formal analysis, Investigation, MethodologyMethodology,
Writing – original draft, VisualizationVisualization. Giancarlo Pennati:
Supervision, Resources, Writing – review & editing. Lorenza Petrini:
Fig. 10. Maximum and minimum diameter for a standard braided unit: (a) force-diameter curves for a planar and a cylindrical unit undergoing a traction and a
compression test; (b) undeformed (left) and deformed conguration under compression (center) and tension (right) for the planar unit at the diameter corresponding
with the rst contact deformation (1c) and at the maximum compression and traction, and illustration of the Δy term used in Eq. (47). The black dashed lines in (c)
indicates the longitudinal axis.
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Journal of the Mechanical Behavior of Biomedical Materials 119 (2021) 104560
13
Supervision, Project administrationProject administration, Writing –
review & editing.
Declaration of competing interest
The authors declare that they have no known competing nancial
interests or personal relationships that could have appeared to inuence
the work reported in this paper.
Acknowledgements
The authors acknowledge David Contassot of ID Nest Medical and
Florent Budillon of ADMEDES GmbH for discussing the issues with the
authors and provide useful insights.
References
Bishu, K., Armstrong, E.J., 2015. Supera self-expanding stents for endovascular
treatment of femoropopliteal disease: a review of the clinical evidence. Vasc. Health
Risk Manag. 11, 387–395. https://doi.org/10.2147/VHRM.S70229.
Bressloff, N.W., Ragkousis, G., Curzen, N., 2016. Design optimisation of coronary artery
stent systems. Ann. Biomed. Eng. 44, 357–367. https://doi.org/10.1007/s10439-
015-1373-9.
Cagnazzo, F., Mantilla, D., Lefevre, P.H., Dargazanli, C., Gascou, G., Costalat, V., 2017.
Treatment of middle cerebral artery aneurysms with ow- diverter stents: a
systematic review and meta-Analysis. Am. J. Neuroradiol. 38, 2289–2294. https://
doi.org/10.3174/ajnr.A5388.
Chichareon, P., Katagiri, Y., Asano, T., Takahashi, K., Kogame, N., Modolo, R.,
Tenekecioglu, E., Chang, C.C., Tomaniak, M., Kukreja, N., Wykrzykowska, J.J.,
Piek, J.J., Serruys, P.W., Onuma, Y., 2019. Mechanical properties and performances
of contemporary drug-eluting stent: focus on the metallic backbone. Expet Rev. Med.
Dev. 16, 211–228. https://doi.org/10.1080/17434440.2019.1573142.
Cremonesi, A., Castriota, F., Secco, G.G., Macdonald, S., Rof, M., 2015. Carotid artery
stenting: an update. Eur. Heart J. 36, 13–21. https://doi.org/10.1093/eurheartj/
ehu446.
De Grazia, A., Somani, B.K., Soria, F., Carugo, D., Mosayyebi, A., 2019. Latest
advancements in ureteral stent technology. Transl. Androl. Urol. 8, S436–S441.
https://doi.org/10.21037/tau.2019.08.16.
Flynn, C.D., Wilson-Smith, A.R., Yan, T.D., 2018. Novel mitral valve technologies-
transcatheter mitral valve implantation: a systematic review. Ann. Cardiothorac.
Surg. 7, 716–723. https://doi.org/10.21037/acs.2018.11.01.
Grewal, H.S., Dangayach, N.S., Ahmad, U., Ghosh, S., Gildea, T., Mehta, A.C., 2019.
Treatment of tracheobronchial injuries: a contemporary review. Chest 155, 595–604.
https://doi.org/10.1016/j.chest.2018.07.018.
Jedwab, M.R., Clerc, C.O., 1993. A study of the geometrical and mechanical properties of
a self-expanding metallic stent—theory and experiment. J. Appl. Biomater. 4, 77–85.
Karanasiou, G.S., Papafaklis, M.I., Conway, C., Michalis, L.K., Tzafriri, R., Edelman, E.R.,
Fotiadis, D.I., 2017. Stents: biomechanics, biomaterials, and insights from
computational modeling. Ann. Biomed. Eng. 45, 853–872. https://doi.org/10.1007/
s10439-017-1806-8.
Kelly, N., McGrath, D.J., Sweeney, C.A., Kurtenbach, K., Grogan, J.A., Jockenhoevel, S.,
O’Brien, B.J., Bruzzi, M., McHugh, P.E., 2019. Comparison of computational
modelling techniques for braided stent analysis. Comput. Methods Biomech. Biomed.
Eng. 22, 1334–1344. https://doi.org/10.1080/10255842.2019.1663414.
Kokkinidis, D.G., Armstrong, E.J., 2020. Current developments in endovascular therapy
of peripheral vascular disease. J. Thorac. Dis. 12, 1681–1694. https://doi.org/
10.21037/jtd.2019.12.130.
Kumar, A., Ahuja, R., Bhati, P., Vashisth, P., Bhatnagar, N., 2019. Design methodology of
a balloon expandable polymeric stent. J. Biomed. Eng. Med. Devices 4. https://doi.
org/10.35248/2475-7586.19.4.139.
Ma, D., Dargush, G.F., Natarajan, S.K., Levy, E.I., Siddiqui, A.H., Meng, H., 2012.
Computer modeling of deployment and mechanical expansion of neurovascular ow
diverter in patient-specic intracranial aneurysms. J. Biomech. 45, 2256–2263.
https://doi.org/10.1016/j.jbiomech.2012.06.013.
Madhkour, R., Wahl, A., Praz, F., Meier, B., 2019. Amplatzer patent foramen ovale
occluder: safety and efcacy. Expet Rev. Med. Dev. 16, 173–182. https://doi.org/
10.1080/17434440.2019.1581060.
Maeda, K., Ohki, T., Kanaoka, Y., 2018. Endovascular treatment of various aortic
pathologies: review of the latest data and technologies. Int. J. Angiol. 27, 81–91.
https://doi.org/10.1055/s-0038-1645881.
Mahmaljy, H., Tawney, A., Young, M., 2020. Transcatheter aortic valve replacement
(TAVR/TAVI , percutaneous replacement) contraindications. Clinical Signi cance
Enhancing Healthcare Team Outcomes 9–12.
McKenna, C.G., Vaughan, T.J., 2020. An experimental evaluation of the mechanics of
bare and polymer-covered self-expanding wire braided stents. J. Mech. Behav.
Biomed. Mater. 103, 103549. https://doi.org/10.1016/j.jmbbm.2019.103549.
McKenna, C.G., Vaughan, T.J., 2021. A nite element investigation on design parameters
of bare and polymer-covered self-expanding wire braided stents. J. Mech. Behav.
Biomed. Mater. 115, 104305. https://doi.org/10.1016/j.jmbbm.2020.104305.
Morlacchi, S., Migliavacca, F., 2013. Modeling stented coronary arteries: where we are,
where to go. Ann. Biomed. Eng. 41, 1428–1444. https://doi.org/10.1007/s10439-
012-0681-6.
Murphy, E., 2019. Surveying the 2019 venous stent landscape. Endovasc. Today 18,
53–64.
Pacha, H.M., Al-khadra, Y., Soud, M., Darmoch, F., Pacha, A.M., Alraies, M.C., 2019.
Percutaneous devices for left atrial appendage occlusion: a contemporary review.
World J. Cardiol. 11, 57–70. https://doi.org/10.4330/wjc.v11.i2.57.
Roy, D., Kauffmann, C., Delorme, S., Lerouge, S., Cloutier, G., Soulez, G., 2012.
A literature review of the numerical analysis of abdominal aortic aneurysms treated
with endovascular stent grafts. Comput. Math. Methods Med. https://doi.org/
10.1155/2012/820389.
Schmidt, T., Abbott, J., 2018. Coronary stents: history, design, and construction. J. Clin.
Med. 7, 126. https://doi.org/10.3390/jcm7060126.
Seigerman, M.E., Nathan, A., Anwaruddin, S., 2019. The Lotus valve system: an in-depth
review of the technology. Curr. Cardiol. Rep. 21, 157. https://doi.org/10.1007/
s11886-019-1234-5.
Shanahan, C., Tiernan, P., Tofail, S.A.M., 2017. Looped ends versus open ends braided
stent: a comparison of the mechanical behaviour using analytical and numerical
methods. J. Mech. Behav. Biomed. Mater. 75, 581–591. https://doi.org/10.1016/j.
jmbbm.2017.08.025.
Subramaniam, K., Ibarra, A., Boisen, M.L., 2019. Echocardiographic guidance of
AMPLATZER amulet left atrial appendage occlusion device placement. Semin.
CardioThorac. Vasc. Anesth. 23, 248–255. https://doi.org/10.1177/
1089253218758463.
Vermeulen, B.D., Siersema, P.D., 2018. Esophageal stenting in clinical practice: an
overview. Curr. Treat. Options Gastroenterol. 16, 260–273. https://doi.org/
10.1007/s11938-018-0181-3.
Watson, T., Webster, M.W.I., Ormiston, J.A., Ruygrok, P.N., Stewart, J.T., 2017. Long
and short of optimal stent design. Open Hear 4. https://doi.org/10.1136/openhrt-
2017-000680.
White, D.C., Elder, M., Mohamad, T., Kaki, A., Schreiber, T.L., 2016. Endovascular
interventional cardiology: 2015 in review. J. Intervent. Cardiol. 29, 5–10. https://
doi.org/10.1111/joic.12273.
Yang, M.C., Wu, J.R., 2018. Recent review of transcatheter closure of atrial septal defect.
Kaohsiung J. Med. Sci. 34, 363–369. https://doi.org/10.1016/j.kjms.2018.05.001.
Zaccaria, A., Migliavacca, F., Contassot, D., Heim, F., Chakfe, N., Pennati, G., Petrini, L.,
2020a. Finite element simulations of the ID venous system to treat venous
compression Disorders : from model validation to realistic implant prediction. Ann.
Biomed. Eng. https://doi.org/10.1007/s10439-020-02694-8.
Zaccaria, A., Migliavacca, F., Pennati, G., Petrini, L., 2020b. Modeling of braided stents:
comparison of geometry reconstruction and contact strategies. J. Biomech. 107,
109841. https://doi.org/10.1016/j.jbiomech.2020.109841.
A. Zaccaria et al.