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Simulating the Time Evolving Geometry, Mechanical Properties, and
Fibrous Structure of Bioprosthetic Heart Valve Leaflets Under Cyclic
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Will Zhanga,c, Shruti Motiwalea,c, Ming-Chen Hsub, Michael S. Sacksa,⇤
aJames T. Willerson Center for Cardiovascular Modeling and Simulation
Oden Institute for Computational Engineering and Sciences and the Department of Biomedical Engineering
The University of Texas at Austin, Austin, TX 78712-0027 U.S.A.bComputational Fluid–Structure Interaction Laboratory
Department of Mechanical Engineering
Iowa State University, Ames, IA 50011-2030, U.S.A.cBoth authors contributed equally to this work.
Keywords: soft tissue mechanics, bioprosthetic heart valve, simulation, time evolving properties
⇤Principal corresponding author:Michael S. Sacks, Ph.D.W. A. “Tex” Moncrief, Jr. Simulation-Based Engineering Science Chair and Professor of Biomedical EngineeringOden Institute for Computational Engineering and SciencesThe University of Texas at Austin201 East 24th Street, 1 University Station, C0200Austin, TX 78712-0027, U.S.A.
Email address: [email protected] (Michael S. Sacks)
Preprint submitted to Journal of the Mechanical Behavior of Biomedical Materials July 15, 2021
ABSTRACT
Currently, the most common replacement heart valve design is the ‘bioprosthetic’ heart valve(BHV), which has important advantages in that it does not require permanent anti-coagulationtherapy, operates noiselessly, and has blood flow characteristics similar to the native valve. BHVs aretypically fabricated from glutaraldehyde-crosslinked pericardial xenograft tissue biomaterials (XTBs)attached to a rigid, semi-flexible, or fully collapsible stent in the case of the increasingly populartranscutaneous aortic valve replacement (TAVR). While current TAVR assessments are positive,clinical results to date are generally limited to <2 years. Since TAVR leaflets are constructed usingthinner XTBs, their mechanical demands are substantially greater than surgical BHV due to theincreased stresses during in vivo operation, potentially resulting in decreased durability. Given thefunctional complexity of heart valve operation, in-silico predictive simulations clearly have potentialto greatly improve the TAVR development process. As such simulations must start with accuratematerial models, we have developed a novel time-evolving constitutive model for pericardial xenografttissue biomaterials (XTB) utilized in BHV (doi: 10.1016/j.jmbbm.2017.07.013). This model wasable to simulate the observed tissue plasticity e↵ects that occur in approximately in the first twoyears of in vivo function (50 million cycles). In the present work, we implemented this model into acomplete simulation pipeline to predict the BHV time evolving geometry to 50 million cycles. Thepipeline was implemented within an isogeometric finite element formulation that directly integratedour established BHV NURBS-based geometry (doi: 10.1007/s00466-015-1166-x). Simulations ofsuccessive loading cycles indicated continual changes in leaflet shape, as indicated by spatiallyvarying increases in leaflet curvature. While the simulation model assumed an initial uniform fiberorientation distribution, anisotropic regional changes in leaflet tissue plastic strain induced a complexchanges in regional fiber orientation. We have previously noted in our time-evolving constitutivemodel that the increases in collagen fiber recruitment with cyclic loading placed an upper bound onplastic strain levels. This e↵ect was manifested by restricting further changes in leaflet geometrypast 50 million cycles. Such phenomena was accurately captured in the valve-level simulations dueto the use of a tissue-level structural-based modeling approach. Changes in basic leaflet dimensionsagreed well with extant experimental studies. As a whole, the results of the present study indicatethe complexity of BHV responses to cyclic loading, including changes in leaflet shape and internalfibrous structure. It should be noted that the later e↵ect also influences changes in local mechanicalbehavior (i.e. changes in leaflet anisotropic tissue stress-strain relationship) due to internal fibrousstructure resulting from plastic strains. Such mechanism-based simulations can help pave the waytowards the application of sophisticated simulation technologies in the development of replacementheart valve technology.
2
1. INTRODUCTION
1.1. Background
The most popular replacement heart valves continue to be the ‘bioprosthetic’ heart valves (BHV),
and are typically fabricated from glutaraldehyde-crosslinked pericardial xenograft tissue biomaterials
(XTBs) sutured to a rigid or semi-flexible stent [1, 2]. While these devices continue to benefit many
patients in the short term, failure due to fatigue induced structural deterioration along with tissue
mineralization continue to be the central issues limiting their durability [3, 4]. Currently, the BHV
durability is assessed through costly and time-consuming in-vitro accelerated wear testing (AWT)
and pre-clinical animal model-based evaluations. AWT, while required by regulatory agencies, is
an empirically developed methodology that provides only limited information on durability using
accelerated frequencies (10-15 Hz), and utilizes aphysiologic loading patterns that do not realistically
load the valve. Thus, the relation between AWT evaluations and long-term durability remains
unclear at best. In contrast, large animal models are primarily focused on limited-term (¡6 months)
implant responses (mainly calcification) and thus cannot capture long-term mechanical fatigue e↵ects.
While biological interactions with blood elements and immunological processes are evident in BHV
[1, 2], basic functional durability issues remain largely unaddressed.
Increasingly, transcutaneous aortic valve replacement (TAVR) devices are utilized for patients
contraindicated for open heart surgery, with growing interest in more general use. While current
TAVR assessments are positive, results to date are generally for <2 years. Since TAVRs are
constructed using thinner XTB leaflets, their mechanical demands are substantially greater than
standard BHV and have associated aggravated durability problems. Paradoxically, further evolution
of TAVR designs require thinner leaflets, which are required to maintain lasting durability even
after being tightly folded for delivery. These requirements increase the already high functional
demands that exist for traditional surgically placed valves. To date, TAVR devices have focused
largely on improved delivery strategies, while the leaflet biomaterials used have remained relatively
unchanged and have the same limited intrinsic durability. Yet, most BHV leaflet research has
focused almost exclusively on mitigation of calcification, This is puzzling as calcification a↵ects less
than half of failed BHV, while tears due to degradation are the predominant mode of failure. Thus,
xenograft design optimization to enhance durability represents a unique and major step forward in
the development of the next generation of BHVs.
3
1.2. On the mechanisms of BHV structural failure
In general, the mechanisms of structural deterioration and eventual failure of BHVs can be gener-
ally divided into two broad categories: i) biologically-driven tissue degeneration and mineralization,
and ii) mechanical fatigue [5]. Mineralization is the accumulation of mineral deposits (mainly calcium
phosphate) within the BHV leaflet tissues, commonly occurring in the commissure region and basal
areas of the cusp [6, 7, 8]. This results in reduced flexibility and weakened tissue, which a↵ects the
normal mechanical functioning of the valve and eventually leads to leaflet tearing and valve failure.
The causes of calcification and techniques to mitigate its e↵ects have been extensively studied in
literature [6, 9, 10, 11, 12]. Calcium phosphate deposits are a result of the reaction between phos-
pholipids in the cell membrane remnants and the calcium in the extracellular matrix (ECM). Thus,
common mitigation strategies involve hindering this reaction, by a) reducing phosphate availability
using reagents such as surfactants and ethanol which extract the phospholipids, or pretreating the
tissue with trivalent metal ions (e.g. FeCl3 and AlCl3) which react with phosphate thereby making
it unavailable for calcification [6, 13, 14] and b) preventing calcium influx using chemical agents that
covalently bind to the bioprosthetic tissue, such as amino-oleic acid (AOA) [6, 15, 16]. Pre-treatment
of the BHV tissue with glutaraldehyde is also known to increase calcification, because of the presence
of residual aldehydes which act as sites for Ca accumulation, and many mitigation strategies attempt
to block these aldehydes [17, 18, 19, 20].
In contrast, while mineralization-reducing methods have been investigated for some time [6, 9, 10,
11], the mechanisms that underlie XTB mechanical fatigue remain poorly understood. Mechanical
fatigue occurs due to cyclic loading of the valve over time and includes damage to the collagen fibers
as well as breakdown of the non-fibrous part of the extracellular matrix (ECM). Fourier transform
infrared spectroscopy results have shown structural damage to collagen fibers at the molecular
level in as little as 50 million cycles [21], which was not detectable at the tissue level at this stage.
Importantly, it has been noted that both calcific and structural damage processes can occur in parallel
or independently [3, 4]. Structural damage has been found to occur in areas of the valve that were
not a↵ected by calcification and are subject to high mechanical forces, suggesting that mechanical
fatigue can independently contribute to structural deterioration of the valve [3, 4, 22, 23, 24, 25].
Other studies have strongly suggested that cyclic stresses may disrupt the collagen architecture and
may induce further calcification [26, 27, 28, 29, 30]. Thus, understanding how mechanical fatigue
4
contributes to BHV structural deterioration may guide the design development of BHVs so that the
degradation due to cyclic stresses can be minimized.
1.3. BHV leaflet plastic behaviors
To clarify the responses of BHV to in vivo loading, we have previously demonstrated that BHV
leaflets can undergo significant changes in geometry and collagen fiber architecture within 50 million
cycles (equivalent to 2 years in vivo operation) (Fig. 1) [31, 32]. These studies indicated that cyclic
loading can lead to permanent deformation in the BHV geometry, especially in the central belly
region which are also the locations where tissue failure has been observed. These and other studies
[33, 34, 35] have demonstrated that BHV XTB can undergo plastic deformations not associated
with damage.
Figure 1: A) The 3D unloaded geometry of a BHV leaflet before and after cyclic loading, with the color indicating thelocal root mean squared curvature. The most significant change in geometry is in the belly region. B) BHV leafletcollagen fiber architecture, showing that the collagen fiber architecture is convected by the dimensional changes. Thegrayscale scale bar shows the orientation index (OI), which is the proportional to the angle containing 50 % of fibers.The lack of changes in the OI suggests that minimal damage to the collagen fiber architecture has occurred.
As stated above, pericardial XTB are the dominant biomaterial used for BHV leaflets. They are
typically fabricated from bovine pericardium and stabilized using an aqueous solution of glutaralde-
hyde, which suppresses immunogenic reactions by crosslinking proteins. The main mechanical e↵ect
of glutaraldehyde treatment is to produce exogenous crosslinks that induce four-fold increase in
5
XTB bending sti↵ness [36], but paradoxically it does not increase the sti↵ness of the collagen fibers
themselves [37]. In addition, it has been shown that exogenous cross-links increase collagen fiber-fiber
mechanical interactions, which can account for up to 30% of the stress in the fully loaded state. The
glutaraldehyde cross-linking process involves a Schi↵-base aldehyde reaction, which is unstable at
room and body temperatures. These cross-links thus constantly undergo a scission-healing process.
The continuous scission-healing process allows an irreversible plastic-like deformation to occur as the
BHV undergoes dynamic loading during operation. This irreversible deformation is not associated
with damage, as it is only a result of change in the unloaded reference configuration. We have
classified the BHV fatigue process into three stages: early (2-5 years), intermediate (2-10 years) and
late (up to failure) (Fig. 2). This is consistent with the significant changes in geometry that occur
within the first 50 million cycles (Fig. 1), where tissue and fiber level damage has not yet reached a
level where it is detectable. The resulting changes in stress patterns begin to induce damage in the
intermediate term, and eventually lead to failure in the late stage. We have noted previously [38]
that plasticity is the dominant e↵ect during the first stage of BHV cyclic loading. Although this
type of plastic behavior is not associated with structural damage, it results in significant changes
in BHV geometry, microstructure, and mechanical properties [31, 32]. This can lead to changes in
stress patterns that can have a significant impact on subsequent structural damage.
1.4. The need for BHV simulations of the fatigue process
Despite their importance, simulation of fatigue e↵ects remain in their infancy, with only a handful
of studies performed. Martin et al. have conducted a series of rigorously done initial studies of the
BHV fatigue process, based on a phenomenological model for long-term fatigue damage with stress
softening and permanent set [39, 40, 41]. However, the constitutive models utilized in these studies
were based on a limited phenomenological modeling approach, with simulations taken from fatigue
information scaled from only 10 million cycles. We have previously shown that plastic deformations
can continue to at least 35 million cycles, and likely require at least 50 million cycles to reach
the full limit [37, 38]. Moreover, there is an upper bound to the plastic deformations, limited by
the deforming local collagen structure. Such mechanistic insights will help optimally utilize XTB
biomaterials and aid in the development of future replacement valve designs. Serrani et al. have
developed a computational model of a polymeric heart valve (PHV) leaflet while simulating the
6
leaflet behavior under a quasi-static pressure load, with the aim of optimizing the PHV structure
[42]. However, they have only considered the e↵ect of a single loading cycle. Furthermore, this
work is focused on polymeric materials which, while behaving similarly at the bulk level, greatly
di↵er in the underlying mechanisms of their respective mechanical behaviors. To our knowledge,
realistic microstructural models of the XTB fatigue process have yet to be incorporated in to current
simulation methodologies.
1.5. Goals of the present study
As a first step towards addressing these goals, we developed a novel numerical simulation
framework for the time evolving properties of BHVs in response to cyclic loading during using a
structurally-based XTB plasticity constitutive model [37, 38]. We utilized this framework to predict
the geometric and structural changes that occurs within the BHV to 50 million cycles, wherein all
plastic deformations will have occurred. To facilitate the time evolving simulation, we utilized an
e↵ective model [43] to represent the response of the plasticity constitutive model [38] at each time
step to considerably improve computational e�ciency. The entire pipeline was implemented in an
isogeometric finite element analysis framework specialized for quasi-static loading conditions, based
on our previous work[44, 45]. Simulation results included time-evolving changes in leaflet shape
(curvature), collagen fiber architecture (orientation and recruitment), as well as rates of change.
Conclusions and future recommendations are then discussed.
7
Figure 2: We speculate that the e↵ects of cyclic loading on BHVs can be divided into three stages: early, intermediateand late. Figure reproduced from [38]
2. METHODS
2.1. Overview
In simulating BHV cyclic loading, we utilized a structurally based plasticity model for time-
dependent behavior. This approach was based on an elastic model of crosslinked tissues that
explicitly incorporated key features of the collagen fiber architecture, namely, recruitment and
rotation, as well as their mechanical interactions augmented by the presence of exogenous cross-links
[37]. The plasticity model then extended this approach by accounting for time-dependent changes
in the exogenously cross-linked (EXL) ’matrix’ (i.e. all non-fibrous tissue components) during
early-to-mid stages of cycling [38]. While shown to be quite accurate and predictive, this approach
is computationally ine�cient due to the presence of quadruple integral terms. We thus utilized
an e↵ective constitutive model in place of the structural constitutive model at each time point
in the actual simulations, with the e↵ective model parameters fit to the structural model output
at each time step (Fig. 3). The material model subroutines were then custom integrated into an
isogeometric finite element code [44, 46, 45], specialized for quasi-static simulations. In the following,
we describing the simulation framework and specific simulations performed.
8
Figure 3: The framework for using the e↵ective constitutive model to improve the e�ciency of using complex meso-or multi-scale models (micro-models) in numerical simulations. Here, A) e↵ective constitutive models act as anintermediate step between micro-models and numerical simulations, where micro-models inform the changes to thee↵ective constitutive model, and the e↵ective constitutive model is then used for the FE simulation. B) An exampleof how this may be implemented for a time-evolving simulation is shown.
2.2. Soft tissue plasticity model
The details of the structural plasticity model have been presented in [38]. To clarify the
presentation, the elastic and plastic modeling sections are presented separately in the following.
2.2.1. Elastic model
In brief, we utilize the fact that pericardial XTBs are a form of exogenously cross-linked soft
collagenous tissues. We thus assumed that there are three contributors to the bulk XTB mechanical
response: 1) the collagen fibers, 2) exogenously cross-linked matrix (i.e. all non-fibrous tissue
components, including cellular remnants), and 3) the mechanical interactions between the collagen
fibers and the cross-linked matrix. We further assumed that, at the bulk level, the XTBs can be
modeled as an incompressible, pseudo-hyperelastic material [47], thus
= �col[ col + int] + (1� �col) m (1)
S = 2@
@C� pC�1 (2)
where is the total strain energy of the tissue, p is the Lagrange multiplier for enforcing incom-
pressibility, C = FT ·F is the right Cauchy-Green deformation tensor, F is the deformation gradient
tensor, S is the second Piola-Kirchho↵ stress tensor, and �col is the mass fraction of collagen fibers,
9
and col, m, and int are the strain energy contributions of the collagen, the matrix and the
interactions respectively. This formulation assumes all the incompressibility e↵ects to be strictly in
the matrix phase only.
The model formulation is based on the classic Lanir-type three-scale homogenization scheme
[48, 49] for general soft tissue structures. In this approach, scale one is the individual fiber, scale
two is an ensemble of fibers that all have a common orientation n, and scale three is the tissue level
wherein the contribution from all fiber ensembles are summed. For scale one, the collagen fibers
were described by a linear relation between the first Piola-Kirchho↵ Stress P and the collagen fiber
stretch ratio �f . As collagen fibers are typically undulated in the unloaded state, they require use
of an e↵ective slack stretch �s, which defines the stretch necessary to straighten the fiber but does
not require detailed knowledge of the unloaded fiber geometry. �s is used in scale two where the
fiber ensemble response formulation is obtained using a probability distribution function of the fiber
slack length, �s(�s). �s(�s) has been defined in detail in [37]. In the final third scale, response from
all fiber ensembles are integrated over all orientations, weighed by the collagen fiber orientation
distribution function (ODF), �✓. In this approach col is given by
col = ⌘col
Z
✓
�✓(✓)
Z �n
1
�s(�s)
✓�n
�s� 1
◆2
d�sd✓. (3)
where ⌘col is the collagen fiber modulus of the fiber. For the cross-linked matrix mechanical response,
we utilized the following modified Yeoh model for the matrix contribution mat
mat =⌘mat
2
✓1
a(I1 � 3)a +
r
b(I1 � 3)b
◆, with 1 < a < b, ab < 2, r � 0. (4)
where ⌘mat is the EXL matrix modulus, a and b are the exponents of the Yeoh model, r is the
relative weight between the two terms and I1 is the first invariant of C. Next, we assumed that all
fiber-fiber and fiber-matrix interactions can be represented at the fiber-ensemble scale level. To
model the interaction term, we use the pseudo invariant I8 = n(↵) ·Cn(�) to capture the mechanical
interaction between two fiber ensembles oriented along n(↵) and n(�) in the reference configuration
(Fig. 4). We note that I8 can be further decomposed into extensional and rotational components
I8 = n(↵) ·Cn(�) = �↵�� cos(↵� �) = Iext8 Irot8 (5)
10
Figure 4: The inter–ensemble interactions can be separated into rotational and extensional e↵ects. Figure reproducedfrom [38].
Iext8 = �↵��, Irot8 = cos(↵� �),
The interaction term, int, is an ensemble level term integrated over all possible pairs of ensembles,
and includes only the Iext8 terms (see [38] for full details), and has the following form
int =⌘int2
Z
↵
Z
�
�✓(↵)�✓(�)
2
4Z �↵
1
Z ��
1
�s(x↵)�s(x�)
�↵��
x↵x�� 1
!2
dx↵dx�
3
5 d↵ d�. (6)
Here, the highlighted blue integrals correspond to the inner stretch terms in blue, while the red
integrals correspond to the outer angular terms in red. We found that the interactions played a
significant role – the interaction term accounted for about 30% of the total tissue stress in the fully
loaded state [37].
2.2.2. Plasticity model
The unstable GLUT cross-linking reactions, which lead to continuous scission-healing in the
EXL matrix, will induce a change in the stress-free reference configuration of the matrix over time.
The embedded collagen fibers will thus convect with the matrix, undergoing rotation and extension.
We refer to this as structural convection of the collagen fiber architecture. It is important to note
that the plasticity was observed only in the matrix, and while the collagen fibers only convect
11
with the matrix deformations; there is no change in the material properties of the collagen fibers
themselves. As the plastic deformation increases, some of the fibers will get recruited. Since the
sti↵ness of the collagen fibers is three orders of magnitude higher than the matrix, they can resist
any further changes in geometry due to continuous scission/healing reactions in the matrix. Thus, a
key takeaway of this model is that the collagen fibers will impose an upper limit on the dimensional
changes of the tissue. This is a direct result of the collagen fiber recruitment resulting from the
increasing plastic strains.
To model this form of plasticity, we utilized a constrained mixture model based on [50]. It is
important to note here that the timescale for the plasticity events is over thousands and millions
of loading cycles. Transient and inertia e↵ects, which occurs in the time frame of a single cycle,
averages out over the course of tens of thousand of cycles and does not play a significant role. It is
likely that the peak stresses and folding that occurs with the dynamics of the leaflet motion plays a
significant role in how damage (deterioration of collagen sti↵ness or tearing of the leaftlet) is accrued,
but this is a whole separate e↵ect to the plasticity modeled and simulated in this paper. As noted
in our previous work [38], damage phenomena occur at an even later stage in comparison to the
plasticity simulated here, and does not play a significant role in this work. In fact, it is especially
important to be able to distinguish and separate the plasticity e↵ect from structural damage so
that form of the structural damage model can be accurately determined. The work done in this
study as a first step to simulating BHV failure cannot be understated. Thus, for plastic e↵ects,
the dominant driver is the average or root mean squared (RMS) loaded state over thousands of
cycles. In brief, after each increment in time, ‘new’ matrix material (i.e. healed in a deformed state)
is created referenced to the current loaded RMS state, while the mass fraction of the ‘old’ matrix
material is reduced, both occurring while conserving the total mass. The response of the matrix at
the bulk level is thus
Sm(t) = b(t) Som +
tZ
0
a(t, ⌧) Snmd⌧, (7)
where b(t) is the proportion of the original amount of the EXL matrix material remaining at time t
with stress contribution Som, and a(t, ⌧) is the proportion of the material newly formed at time ⌧
with stress contribution Snm. The rate of mass transfer between these states was described by first
order kinetics [38]. Next, we account for the convection of collagen fiber architecture with plastic
12
strain by defining the convected fiber recruitment distribution function, t0�s, and the convected
fiber orientation distribution function, t0�✓ in terms of the original distributions and the plastic
deformation t0Fp.
t0�s(
t0Fp, ✓t) = �s[✓0(
t0Fp, ✓t)]
t0�(✓0)
2
t0Jp
, (8)
t0�(✓0) =
qn(✓0) · t0Cp n(✓0),
t0Jp = det(t0Fp)
t0�s(
t0F,�s) =
8>><
>>:
B[�0,�1](y)
t�lb�1�ub, t�ub < y < t�lb
0, otherwise(9)
t�s =�s
t0�(✓)
, y = t�s � t�lb
t�ub � t�lb
where the subscript ‘0’ denotes the 0-cycle referential configuration, the subscript/superscript ‘t’
denotes the structurally convected (plastically deformed) unloaded configuration, and �lb and �ub
are the lower and upper bounds of the slack stretch. Further details about these forms are presented
in [37, 38, 51].
Figure 5: Illustration of the permanent set e↵ect under cyclic uniaxial loading showing A) the relation betweenthe reference configurations during cyclic loading. Figure reproduced from [38], B) the transfer of mass fraction ofthe EXL matrix to the loaded configuration ⌦(s) from the original state ⌦0 and C) the resulting in changes in theunloaded geometry of the tissue.
The final model form is a function of the plasticity rate constant k, the plastic deformation Fp,
the strain history A(t), and the material parameters of the constitutive model in the uncycled state.
13
The input of the model is the applied deformation C referenced to the current unloaded state ⌦P,
given by the plastic deformation Fp from ⌦0 (Fig. 5).
The full model form is
S = S�k,FP,A(⌧),C
�= �col [Scol + Sint] + �matSmat, (10)
where the collagen contribution is
Scol
�k,Fp,A(⌧),C
�= ⌘col
Z
✓
t0�✓[Fp, ✓]
8><
>:
�nZ
1
t0�s
⇥Fp,�s
⇤
�s
✓1
�s� 1
�n
◆d�s
9>=
>;n⌦ n d✓, (11)
the contribution of the interactions is
Sint
�k,Fp,A(⌧),C
�= ⌘int
Z
↵
Z
�
t0�✓
⇥Fp,↵
⇤ t0�✓
⇥Fp, �
⇤(A + B + C + D) d↵ d� (12)
where,
A =
2
64�↵Z
1
��Z
1
2��t0�s[Fp,�s,↵]
t0�s[Fp,�s,�]
�s,↵�s,�
�↵
�s,↵
��
�s,�� 1
!d�s,↵ d�s,�
3
75n(↵)⌦ n(↵)
�↵, (13)
B =
2
64
��Z
1
t0�s[FP,�s,�]
��
�s,�� 1
!2
d�s,�
3
75n(↵)⌦ n(↵)
�↵(14)
C =
2
64�↵Z
1
��Z
1
2��t0�s[FP,�s,↵]
t0�s[FP,�s,�]
�s,↵�s,�
�↵
�s,↵
��
�s,�� 1
!d�s,↵ d�s,�
3
75n(�)⌦ n(�)
��(15)
D =
2
64�↵Z
1
t0�s[FP,�s,↵]
�↵
�s,↵� 1
!2
d�s,↵
3
75n(�)⌦ n(�)
��, (16)
14
and the contribution of the EXL matrix is
Smat
�k,FP,A(⌧),C
�
= ⌘mat
2
64Exp [�k · t]⇣�
I1(FP,A(0))� 3�↵�1
+ r�I1(FP,A(0))� 3
���1⌘
⇥⇣B(FP,A(0))�1 � B�1
33 (FP,A(0))C33C�1⌘
+
tZ
0
k · Exp⇥�k(t� ⌧)
⇤ ⇣�I1(FP,A(⌧))� 3
�↵�1+ r
�I1(FP,A(⌧))� 3
���1⌘
⇥⇣B(FP,A(⌧))�1 � B�1
33 (FP,A(⌧))C33C�1⌘d⌧
3
75 .
(17)
where �✓ =p
n(✓) ·Cn(✓) is the stretch of the fiber ensemble oriented along ✓, and �s,✓ is the
slack stretch of the fiber ensemble oriented along ✓.
2.3. E↵ective material modeling approach
An obvious challenge in using Eqn. (10)-(17) is the substantial computational cost of evaluating
the quadruple integrals. To circumvent this issue, we utilized a recently developed e↵ective soft
tissue constitutive model to perform the actual finite element computations [52]. This basic form
has been shown to able to fully reproduce the response of a wide range of planar soft tissues, along
with a method for robust and fast-convergent parameter estimation. The form of this e↵ective
constitutive model is given by
eff =c0⇣eQ � 1
⌘= c00e
�Qmax
⇣eQ � 1
⌘
Q =b1E2m + b2E
2n + b3E
2� + b4EmEn + b5E
4m + b6E
4n + b7E
3mEn + b8E
2mE
2n
+ b9EmE3n + b10E
4� + b11E
2mE
2� + b12E
2nE
2� + b13EmEnE
2�
(18)
where Qmax = Q(Emax) is a scaling factor that helps reduce parameter covariance by normalizing
the exponential part of the equation (see [52] for details).
The method to determine the e↵ective model parameters are presented in Zhang et al. [52]. The
first step is to generate a set of synthetic data, which should not be done arbitrarily. This is done
15
along a set of optimal loading paths which minimize the covariance between model parameters. We
have shown that optimal loading paths with the use of the scaling factors Qmax is su�cient for
determining a unique set of parameters using gradient algorithms, i.e. this approach improves the
ellipticity about the minimum. The maximum strain energy value, which could be determine from
the plasticity constitutive model, matches closely with the parameter c0 in Eqn. 18 making it an
excellent value for the initial guess, while the exponent parameters bi are generally very consistent in
value over time. As such, parameter estimation for the e↵ective model is generally very quick. The
parameter constraints are likewise presented in Zhang et al. [52]. The e↵ective model su�ciently
preserves the mechanical response of glutaraldehyde pericardium used in this study such that we
were able to obtain similar structural model parameters by fitting to synthetic data generate from
the e↵ective model in comparison to fitting the structural model directly, i.e. the parameter are
within 10�3 in most cases. Some di↵erence are inevitable due to the the lack of complexity in
comparison to the plasticity constitutive model but the overall response is qualitatively preserved.
2.4. Simulation framework
In this next section, we introduce a time dependent simulation framework, which integrates these
models into a holistic setup to simulate in-vivo cyclic loading for BHVs to 50 million cycles. We
divide the simulation framework into the following major stages (Fig. 6):
1. Model formulation and quasi-static simulations of the initial state.
2. Updating the material model parameters for the next time step.
3. Updating the current geometry for the next time step.
Steps 2 and 3 are repeated for each subsequent time step until the maximum time limit is reached.
Details of each step are provided in the following sections.
2.4.1. Initial state model and quasi-static simulation
Next, we first established the following initial (referential) model primary components: 1) The
BHV geometry, 2) the mechanical properties, and 3) mapped collagen fiber architecture. We utilized
an established BHV geometry from [44], bovine pericardium properties from [37], and homogeneous
circumferential aligned collagen fiber orientation distributions. A custom in-house isogeometric finite
16
Figure 6: Flowchart explaining the execution of the di↵erent components of the time dependent simulation framework.
17
element software was used, based on the framework developed by Hsu and co-workers [44, 46, 45].
Briefly, the finite element code was purposed for dynamics and fluid–structure interaction simulations
of heart valves [53, 54, 55], focusing mostly on the tri-leaflet semilunar valves. The tri-leaflet
geometry is based on the commonly used Edwards Pericardial Heart Valve with Kirchho↵–Love shell
elements for the leaflets [46] and finite element solver developed by Hsu et al. [44].
We utilized this code to simulate leaflet deformation under physiological quasi-static transvalvular
pressure of 80 mmHg. A total of 484 Bezier elements were used for each leaflet, with a leaflet
density of 1.0 g/cm3 and a uniform leaflet thickness of 0.386 mm [44]. Contact between leaflets was
handled by a penalty-based approach and imposed at quadrature points of the shell structure [56],
and a clamped boundary condition is applied to the leaflet attachment edge. For simplicity and
consistency, the collagen fiber direction was assumed to be aligned to the circumferential direction
of each leaflet. The bioprosthetic heart valve stent was fixed and undeformable, thus serving as a
stationary reference for the leaflets. A similar validation process as the one presented by Wu et al.
[45] was used to verify the implementation.
2.4.2. Evolution of leaflet material behaviors
The loading history data for the current timestep, as obtained from the FE code, is used to predict
the change in material properties using the plasticity constitutive model. Next, the mechanical
response from the plasticity model is sampled along optimal loading paths and then fitted to equation
18. The parameters of equation 18 are changed in the finite element model and this e↵ective model
is then used to perform the simulation. This e↵ective model (Eqn. 18) acts as an intermediate
between the constitutive model and the finite element model for quasistatic simulation of BHVs (Fig.
3A) and drastically simplifies the implementation of the numerical model by keeping the constitutive
model form the same. The full implementation of the FE model and the boundary conditions are
presented in [53, 54, 55]. Briefly, the stent of the valve are assumed to be rigid and prescribed with
a zero displacement boundary condition and a transvalvular pressure di↵erence, oscillating from 0
to 80 mmHg at 10 Hz, was applied across the closed valve.
2.4.3. Geometry update due to plastic strains
To obtain the plastic strain, we need to use optimization, since the plasticity constitutive model
has no analytical inverse form. The plasticity model can find local change in reference geometry
18
using equation 17 and
FP = argminF
����S⇣k, I,A(⌧),C = FTF
⌘� 0
���� , (19)
Since this is being solved for each element locally, it is not su�cient for generating a geometrically
compatible mesh. To generate a mesh for the updated reference configuration in the following time
step, we will need to perform another FE simulation with load free boundary condition and an
internal residual stress. First, we find the local change in geometry from Eqn. 19. Next, we will
compute an equivalent stress using Eqn. 10, by
SP = S(FP). (20)
This stress, SP, is equivalent of a residual stress corresponding to the local change in geometry,
which is then added to the following weak form
Z
�0
w · ⇢hth@2y
@t2
�����X
d�+
Z
�0
Z hth/2
�hth/2
�E : (S� SP) d⇠3 d�
�Z
�0
w · ⇢hthf d��Z
�t
w · h d� = 0.
(21)
where, y is the midsurface displacement, the derivative @(·)/@t|X holds material coordinates X fixed,
⇢ is the density, S is the 2nd Piola–Kirchho↵ stress, �E is the variation of the Green–Lagrange strain
corresponding to displacement variation w, f is a prescribed body force, hnet is the total traction
from the two sides of the shell, and �0 and �t are the shell midsurfaces in the reference and deformed
configurations respectively [45, 54]. The resulting control point displacements u(x, t+ 1) are then
added to the finite element mesh from the previous time step to generate the new mesh in the new
reference configuration.
2.5. Complete implementation
We utilized custom Python scripts to combine the individual source codes for each component of
the simulation (Fig. 6). The quasi-static simulations were done using the isogeometric finite element
code from Hsu et al. [44, 46, 45], only altering the input parameters and boundary conditions. The
full structural (Eqn. 2) and e↵ective models (Eqn. 18) were implemented in C++ compiled to a
19
Figure 7: Flowchart explaining how the geometry is updated
python module using Cython [57]. The individual components were called in sequence to perform
the plastic strain simulations as follows:
1. Setup a quasistatic FE simulation for the first loading cycle using the initial state information
(Fig. 6, Step 1)
2. Perform FE simulation of the loading cycle and obtain loading history data for the control
points for the current time step (Fig. 6, Step 2)
3. Update structural model locally, compute plastic strain and residual stress from the plastic
strain constitutive model in section 2.2.2 and optimal loading path data for the mechanical
response (Fig. 6, Step 3)
4. Compute new e↵ective model parameters from structural parameters (Fig. 6, Step 3)
5. Setup FE simulation with zero pressure and residual stress from the plastic strain constitutive
model (Fig. 6, Step 4)
6. Perform FE simulation for the next stress free geometry (Fig. 6, Step 4)
7. Setup FE simulation for loading cycle for the next time step using the new e↵ective model
parameters, reference geometry and fiber structure
8. Return to step 2 in Fig. 6 and continue this loop unless final time step is reached
2.6. Evaluating time-evolving changes in BHV leaflet geometry and fiber structure
In the present study we simultaneously predicted changes in the BHV leaflet geometry and
evolution of the collagen fiber structure, for a single cycle and for long-term cyclic loading up to 50
million cycles. To aid in visualizing the e↵ects of these changes we plotted the stretch ratio �, in
20
circumferential and radial directions. To express the resulting shear, we determined the shear angle
↵ from the Green’s strain tensor E using
↵ = sin�1
✓2E12p
1 + 2E11
p1 + 2E22
◆(22)
We also computed the curvature tensor at each node from the NURBS position vector and its
spatial derivatives. From the curvature tensor, we determined the local principal directions and
magnitudes, and subsequently, calculated the Gaussian and mean curvature for the leaflet surface.
To visualize the changes in the collagen fiber architecture with deformation, we utilized a
normalized orientation index ⌫ using
⌫ =�iso � �
�iso, (23)
where, � is the standard deviation of the collagen fiber orientation distribution and �iso is the
standard deviation of a uniform distribution, i.e., the orientation distribution function when the
fibers are distributed uniformly and thus representing an isotropic material. Here, ⌫ = 1 indicates
a perfectly aligned distribution, whereas ⌫ = 0 indicates that the fiber distribution is a uniform
(random) distribution.
Due to the lack of experimental data, we were unable to perform detailed validation. However,
as a basic validation method, we compared the geometric changes predicted by our simulations with
the results in [31, 32], by comparing the ratios hth0
and ht�h0w , where h0 is the initial di↵erence in
height of the highest point of commmissure region, and the triple point, namely the location where
the three leaflets meet, ht is the same height in the unloaded final state, and w is the width of the
BHV leaflet in the initial state (Fig. 1).
3. RESULTS
3.1. Single loading response in the initial and final states.
To compare the BHV responses in the initial (0 cycles) and final (after 50 million cycles) states, we
plotted the circumferential and radial stretch due to one loading cycle in the initial state, referenced
to the unloaded initial state (Fig. 8c, 8d), and the stretch due to one loading cycle in the final state,
referenced to the unloaded final state (Fig. 8e, 8f). In the uncycled state, the highest circumferential
stretch was 1.1344, in the central belly region. The free edge also had high circumferential stretch
21
compared to the surrounding regions, with the maximum value being 1.1315 near this region. The
highest radial stretch was 1.1244, in the lateral belly region of the leaflet. In comparison, after 50
million cycles, the maximum stretch due to one loading cycle was 1.0307 in the circumferential
direction and 1.029 in the radial direction. The BHV exhibited a significantly sti↵er response in the
final loaded state relative to the initial state and a more uniform deformation pattern was observed
in the final state.
3.2. Evolution of BHV leaflet geometry and collagen fiber architecture
To analyze the changes to the BHV leaflet geometry with time, we plotted the circumferential
and radial stretch, the shear angle ↵ and the mean curvature in the unloaded configuration at six
di↵erent time points over the duration of the simulation, referenced to the unloaded uncycled state
(Fig. 9 – Fig. 12). Our results indicated that these dimensional changes slowed down after about 20
million cycles and nearly completely seize after 30 million cycles. The regions that exhibited the
highest plasticity were the belly region and the free edge. After 50 million cycles the maximum
circumferential plastic stretch in the belly region was 1.1209 and in the free edge was 1.1453, while
the maximum radial stretch was 1.1104 in the lateral belly region. By 20 million cycles, the maximum
plastic stretch in circumferential and radial directions had already reached 98% of the maximum
value of plastic stretch at 50 million cycles. The maximum shear angle in the unloaded state at 20
million cycles was 17.64�and at 50 million cycles, it was 17.79�, near the region connected to the
stent. The maximum value of mean curvature was 2.22 at 20 million cycles, and it had increased to
2.31 at 50 million cycles near the center of the free edge. The location and the value of maximum
curvature was very similar to the previous results in [31, 32] (Fig 1), where the center of the free
edge had a maximum mean curvature value of 2.1 at 50 million cycles. Overall, the central region of
the valve had a higher curvature as predicted by our simulations, which agrees with the previous
results.
In addition, we compared the height and width ratios in the initial and final state for the BHV
geometry predicted by our simulations with the BHV geometry for a stented and non-stented design
previously presented in [31, 32] (Table 1). The ratios for the stented BHV design agree with our
simulation results, especially when both height and width are taken into account. While there
is some di↵erence in values for the non-stented design, this is expected due to the di↵erence in
22
Figure 8: (a) Arrows representing circumferential direction in the leaflet, (b) Arrows representing radial directionin the leaflet, (c) Circumferential stretch in uncycled, fully loaded state, referenced to the uncycled, unloadedconfiguration, (d) Radial stretch in uncycled, fully loaded state, referenced to the uncycled, unloaded configuration,(e) Circumferential stretch in cycled, fully loaded state, referenced to cycled, unloaded state, (f) Radial stretch incycled, fully loaded state, referenced to cycled, unloaded state. Note that by 50 million cycles, the circumferentialand radial stretches have been substantially reduced due to the e↵ects of plastic deformation.
23
boundary condition.
hth0
ht�h0w
Experimental result for non-stented BHV design
2.05 0.18
Experimental result forstented BHV design
1.95 0.06
Simulation 1.58 0.08
Table 1: Comparison of height and width ratios for the BHV geometry predicted by our simulations with theexperimental results for BHV geometry for a stented and non-stented design previously presented in [31, 32]
To visualize the corresponding changes in the collagen fiber architecture with time, we plotted
the normalized orientation index, ⌫, at 6 distinct points in time (Fig. 13). The belly region and the
free edge displayed the greatest degree of realignment. In the initial unloaded state, ⌫ was 37.1%,
whereas in the final unloaded state, the fibers had reoriented such that the maximum value of ⌫
was 43.4% at the free edge, and the minimum value of ⌫ was 34.4% in the lateral belly region. This
indicates that the fibers had oriented towards the circumferential region and had become more
aligned in the free edge and central belly region, whereas in the lateral belly region, the fibers had
oriented towards the radial direction.
Additionally, we plotted the circumferential and radial components of S in the loaded state (Fig.
14, Fig. 15). The maximum circumferential stress was 1067 MPa near the commissure region, where
we see high stress concentrations, although the range of stress in the rest of the leaflet was within
0-400 MPa. The maximum radial stress was 358 MPa in the lateral belly region.
24
Figure 9: Time evolution of the circumferential plastic stretch in unloaded state. Here, the plasticity e↵ects werehighest in the belly region and the free edge.
25
Figure 10: Time evolution of the radial plastic stretch in unloaded state. Plasticity was highest in the belly region.
26
Figure 11: Time evolution of the plasticity-induced shear angle ↵ in unloaded state. Shear angle magnitudes werehighest near the leaflet attachments.
27
Figure 12: Time evolution of the Mean curvature in unloaded state. Curvature was highest in the belly region andthe center of the free egde.
28
Figure 13: Time evolution of the collagen fiber Normalized Orientation Index in unloaded state. Significant plasticity-induced collagen fiber reorientation occurred near the belly region and the free edge. Note that these changes werenot associated with any damage mechanisms
29
Figure 14: Time evolution e↵ects on the Scc in the fully loaded state. The high circumferential stress regions were thesame as the regions with highest plasticity in the circumferential direction: the belly region and the free edge. Thesechanges are a coupled results of both the changes in leaflet geometry and tissue properties with cyclic loading.
30
Figure 15: Time evolution e↵ects on the Scc in the fully loaded state. The high circumferential stress regions were thesame as the regions with highest plasticity in the radial direction: the lateral portions of the belly region. Thesechanges are a coupled results of both the changes in leaflet geometry and tissue properties with cyclic loading.
31
4. DISCUSSION
4.1. Major findings
The present study represents the first micro-structural mechanism–driven simulation of XTB
materials in a full BHV simulation framework. Key results indicated that the BHV responds to
cyclic loading in complex ways, including permanent plastic changes to the leaflet geometry which
resulted in substantial changes in leaflet shape (curvature). In addition, at the micro–structural level
change to the underlying collagen fiber architecture (especially alignment) were also substantial,
and also regionally variant.
These ‘plastic’ dimensional changes slowed down asymptotically, with most changes occurring in
the first 20-30 million cycles (Fig. 16B, C). This is roughly equivalent to the first 6-10 months of
in-vivo function. This places enormous importance on knowing this final stable unloaded geometry,
as this is the geometry that the BHV will operate in for the remainder of its life, not the initial BHV
design. We have shown here that we are able to predict this geometry through the plasticity model
as well as the distribution of plastic deformation in the BHV leaflets, which is most significant in the
belly region and the free edge. We also found that the changes in shape induced increased stresses
in both the circumferential and radial directions. Moreover, we observed significant sti↵ening of the
BHV leaflet, resulting in reduction in peak stretches due to a loading cycle (Fig. 16D). In addition,
by utilizing a full structural model formulation we were also able to predict the changes to the
fiber architecture with time under cyclic loading. This capability paves the path for optimizing the
collagen fiber architecture in BHV along with optimizing the overall geometry, thus giving insights
into the microstructural changes and their e↵ects on the function of the BHV over time.
4.2. Computational considerations
The first concern with the use of any structural constitutive model in organ-level computational
simulations is the computational e�ciency. The accurate computation of the integral over the
collagen fiber orientation distribution and the collagen fiber recruitment distribution can result
in 400 to 900 times as many floating point operations as in phenomenological models, making
it intractable in computational simulations with a large number of degrees of freedom. However,
an approach using an e↵ective model [43] can mitigate this problem. The computation of the
structural constitutive model required a minimum 21 gauss quadrature points for �(✓) and 30 gauss
32
Figure 16: A) BHV leaflet with a few key locations highlighted B) Triple point height in the unloaded configurationshowing decay with time C) Plastic stretch in circumferential and radial directions with time at the marked locationsin the BHV leaflet. Most of the dimensional changes occur in the first 20-30 million cycles. D) Peak stretch in thecurrent loaded configuration with reference to the current unloaded configuration with time at a few key locations inthe BHV leaflet. The reduction in peak stretches demonstrates significant sti↵ening in the BHV leaflet.
quadrature points for D(�s). Per evaluation, the structural constitutive model time cost was 2 to 3
orders of magnitude longer to evaluate in comparison to the e↵ective model. Because we updated
the e↵ective model per element, we shifted the cost from the structural constitutive model to the
e↵ective model by virtue of there being 16 times less elements than Gauss points. We also compared
the computation time for both eff (Eqn. 18) and Holzapfel-Gasser-Ogden model [58] for biaxial
simulation of bioprosthetic heart valve tissues and, as expected, found no significant increase in
computational cost. The total elapsed time for eff is 7.58 seconds in comparison to 6.40 seconds
for the Holzapfel-Gasser-Ogden model, which is evidently much faster than any micro-models can
achieve.
Our rationale for why we are not running full FSI is that we are only considering modeling
valve closure. This is because we were interested in simulating the fatigue process, which is driven
primarily by high tensile stresses that occur in the closed state. The e↵ects of hydrostatic forces on
a closed valve can be modeled quite successfully using a uniform pressure load. We have found no
significant di↵erences between structural-only and FSI simulations of valve closure [59]. FSI clearly
33
will take substantially more computing resources to simulate, and since it will not any simulation
accuracy improvements in the present study, we did not consider it.
4.3. The e↵ect of plasticity on bioprosthetic heart valves geometry
The mechanism underlying the observed plasticity in the XTB materials involves the scission-
healing reactions in the matrix due to the Schi↵-base reaction of the GLUT. GLUT treatment has
the important role of suppressing the immune reaction to the BHV, but causes side e↵ects such as
sti↵ening the matrix, accelerating mineralization, and, as we suggested here, inducing plasticity in
the BHV. These continuous scission healing reactions cause the reference configuration to evolve.
As a result, the new reference configuration of the tissue is based on the proportion of the tissue
whose reference configuration is in the original reference configuration and the proportion whose
reference configuration is in the current configuration. The rate at which the proportion of the tissue
undergoes a change in reference configuration is dependent on the proportion of the material still
remaining in the original reference configuration. In theory, this process can go on indefinitely, but
will be limited by the collagen fiber phase, as discussed in the next sections.
4.4. The impact of collagen fiber microstructure on BHV plasticity
The collagen fiber architecture is the major load bearing component in BHVs, and it determines
the current loaded configuration geometry. To underscore its e↵ects, we conducted the following
additional simulations of BHVs with di↵erent collagen fiber architectures: A) uniformly distributed
collagen fibers, B) typical bovine pericardium collagen fiber architecture, and C) native porcine aortic
valve collagen fiber architecture with highly aligned collagen fibers. All simulations significantly
di↵erent maximum in-plane Green-Lagrange strain (MIPE) in the fully pressure-loaded state (Fig.
17). The porcine aortic valve properties result in significant heterogeneities in the deformation
of the leaflets (Fig. 17C). The pericardial valve (Fig. 17B) and the isotropic valve (Fig. 17A)
on the other hand have significantly more homogeneous leaflet deformations, especially from the
top-down view. Both of these undergo approximately the same deformation of 0.2 in MIPE. However,
there is significant di↵erence between the two near the commissure regions of the valve, where
the isotropic case is under significantly higher strain. This zone is susceptible to failure due to
tearing and delamination. It’s clear from this that the collagen fiber architecture has a significant
impact on the fully loaded geometry, which determines the reference configuration that the matrix
34
A) Isotropic B) EXL pericardium C) Porcine aortic valveSi
de v
iew
Top
view
Figure 17: The fully loaded state of the single cycle simulations of intact tri-leaflet valves with the collagen fiberarchitecture of A) a uniform collagen fiber orientation distribution, B) exogenously cross-linked bovine pericardiumvalve, and C) the native porcine aortic valve. A) and B) result in mostly homogeneous stress distributions with A)showing stress concentrations are the commissure regions, while C) results in a very heterogeneous stress distributionand the belly region caving in. The top row shows the side view of the valves at 80 mmHg and the bottom row showsthe top-down view.
is evolving towards. This emphasizes one important reason for the need to understand the collagen
fiber architecture for modeling BHVs under cyclic loading.
The collagen fiber architecture will also limit the amount of plasticity. It is important to
reiterate here that the collagen fiber architecture remains fully intact and undamaged by this plastic
deformation in this phase [38]. Plastic deformation does not continue forever, nor does the shape
of the valve leaflets ever match its fully loaded state. The collagen fiber architecture evolves with
the changes in the configuration of the matrix, shifting in orientation and changing in crimp. The
collagen fibers are sti↵er than the matrix by approximately three orders of magnitudes, and typically
only extend by 3-4%. Thus, when the matrix deforms plastically, it is di�cult for the matrix to exert
significant deformations on the underlying collagen fiber network, it can only reorient the fibers.
The stretching of collagen fibers will pull against the matrix and prevent further plasticity. This
35
is an important mechanism, allowing the final stable geometry to be predicted from the collagen
fiber architecture using our simulations. Thus, it is easy to see the importance of the collagen fiber
architecture in the plasticity of BHVs, and the need for BHV designs to take into account of collagen
fiber architecture to optimize durability.
For clinical applications, the measurement of this collagen fiber orientation distribution of BHV
can be done non-destructively prior to being implanted into a patient. One such technique is
polarized spatial frequency domain imaging [60, 61, 62], which combines polarized light imaging
and spatial frequency domain imaging techniques to perform fiber orientation mapping. Polarized
light imaging is a common technique taking advantage of how the orientation and alignment alters
the scattering distribution of photons to measure the collagen fiber architecture. Changing the
spatial frequency of the polarized light can control the imaging depth by rejecting di↵use photons.
In combination with 3D projection, this o↵ers a path to the full 3 dimensional fiber structural
characterization of BHVs for simulations.
4.5. Structural damage and predicting failure
The mechanism described here for the plasticity of the BHVs is focused on the evolution of
the reference configuration of the matrix and the collagen fiber architecture. A key advantage of
our model is that it utilizes the structural modeling approach and can thus predict the e↵ects of
cyclic stresses at a microstructural level. This allows the computation of the recruitment of collagen
fiber and thus the distribution of crimp and stretch of the collagen fibers in both the unloaded and
loading configuration. This metric for the strain load on collagen fibers over time is an excellent
metric for formulating structural damage at the fiber level. Experimental evidence corroborates with
this theory by showing that the areas that underwent significant changes due to plasticity in our
simulations also undergo significant structural damage [31]. We also found that stress concentrations
were highest in the belly and the commissure region, which are known locations of calcification and
tissue failure. The model predicts that the collagen fibers in areas with significant plastic changes
are also under increased strain in both the loaded and unloaded configuration, and may become
permanently held in a stretched state. The collagen fibers being held in a extended state more often
are more likely to become damaged, where rates of failure after being extended by 7-8% increases
significantly [49, 63]. Coupling this with the plasticity model presented here allows for the prediction
36
Figure 18: BHV fatigue as a multiscale process
of microstructural changes and mechanical response due to fatigue and other damage phenomena, a
step towards a full fatigue damage model capable of predicting device level failure (Fig. 18).
4.6. Potential to improve bioprosthetic heart valve designs
In our simulations, we found that the belly region and the free edge exhibit the highest plasticity.
These regions are also the most common regions of failure due to structural degeneration in BHVs.
Moreover, we also found that the BHV leaflet geometry stabilizes after about 30 million cycles. The
initial geometry can likely have a significant impact on loading of the BHVs. For example, higher
curvatures can increase deformation in the belly region, resulting in increased plastic strains. A
shorter free edge length requires larger circumferential stretch to fully close the BHV. We can use
this knowledge to optimize the initial BHV geometry to reduce the peak stress and make the stress
distribution more uniform in the loaded configuration after plasticity has ceased. This approach
will reduce the stretch that the collagen fibers experience throughout its lifespan, which can reduce
the likelihood of failure in these fibers and in turn improve the durability of the BHV leaflets. For
clinical applications, a long-term clinical study similar to AWT validation studies that explore tissue
strain and leaflet shape would need to be undertaken eventually, especially of explanted BHV to
explore non-failure mechanisms for the changes in BHV shape and material properties, such as done
by Sacks and Schoen [4]. For clinical simulations, the form of the material model is known, as well
as the initial design and geometry. Inverse modeling techniques have already been established to
estimate the initial behaviors [5–8]. These basic information and the approach laid out in this work
should be su�cient to determine in vivo performance.
37
4.7. Future directions in material modeling
The plasticity model and simulation framework developed herein is a simplification of the growth
and remodeling framework by removing the growth component. One potential implication is the
design of devices such as tissue engineered valves, which have the possibility of growing and adaption
to the surrounding environment if seeded with interstitial cells. In addition to the exogenously
crosslinked tissue applications addressed herein, we have observed plasticity like phenomenon in
mitral valve tissue during pregnancy [64]. In that study, our results suggested that much of the
growth and remodeling in the MV leaflet does not begin immediately, but rather undergoes mostly
passive leaflet enlargement until these parameters reach a critically low level, at which point growth
and remodeling are triggered. This initial tissue distension process is very similar in behavior to the
plasticity mechanism outlined in the present work. Thus, the current approach could be applied to
the early phases of soft tissue remodeling, where non-failure mechanisms occur before the onset of
growth of tissue growth and remodeling. This is a key advantage of structural approaches, which
allow us to describe the mechanical response based on real, physically measurable quantities.
5. Limitations
The goal of this work is to establish a proof of concept for the simulation of the e↵ect of plasticity
on BHVs, and produce good agreement with the extant experimental data. We have shown that
our plasticity model can predict the change in geometry of BHVs with cyclic loading as well as the
evolution of the underlying collagen fiber architecture. This can be extended to additional simulations
to explore di↵erent initial geometric and structural conditions in the future. Moreover, a fiber-level
fatigue model can be developed as a starting point beyond the plasticity e↵ects described herein
(e.g. the ’intermediate’ stage shown in Figure 2). However, this first requires su�cient experimental
data to derive model parameters and to validate the model predicted outcomes. Nonetheless, we
underscore that the model developed in the present study agreed well with actual accelerated wear
test studies on BHVs [31, 32]. In particular, we noted similarities in the mean curvature distribution
at the 50 million cycles in the discussion (Fig. 1 vs. Fig. 12).
38
6. Conclusions
We have developed a complete time-dependent framework for the simulation of BHVs under
long-term cyclic loading. This simulation utilizes the predictive mechanism based constitutive model
for the plasticity e↵ect in exogenously crosslinked soft tissues that we previously developed. We have
shown that we can use this simulation to predict the evolving geometry, microstructural changes and
material property changes. These results can then be used to predict regions of increasing likelihood
of structural damage and can be used to optimize the initial design of BHVs based on these factors.
Most important of these e↵ects is that the collagen fiber architecture can play a role in limiting
plasticity, where the straightening of collagen fibers prevents further changes in geometry. Thus,
accounting for the plasticity e↵ect is especially important in the design of BHVs to better improve
their performance and durability.
Funding Source: This study was supported by NIH/NHLBI grant no. R01 HL129077 and R01
HL142504.
39
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