# A Computational Framework for Atrioventricular Valve Modeling using Open-
Source Software

Wensi Wu
Department of Anesthesiology and
Critical Care Medicine,
Children’s Hospital of Philadelphia,
Philadelphia, PA 19104
<EMAIL_ADDRESS>
&Stephen Ching
Department of Anesthesiology and
Critical Care Medicine,
Children’s Hospital of Philadelphia,
Philadelphia, PA 19104
<EMAIL_ADDRESS>
&Steve A. Maas
Department of Biomedical Engineering, and
Scientific Computing and Imaging Institute,
University of Utah,
Salt Lake City, UT 84112
<EMAIL_ADDRESS>
&Andras Lasso
Laboratory for Percutaneous Surgery,
Queen’s University,
Kingston, ON
<EMAIL_ADDRESS>
&Patricia Sabin
Department of Anesthesiology and
Critical Care Medicine,
Children’s Hospital of Philadelphia,
Philadelphia, PA 19104
<EMAIL_ADDRESS>
&Jeffrey A. Weiss
Department of Biomedical Engineering, and
Scientific Computing and Imaging Institute,
University of Utah,
Salt Lake City, UT 84112
<EMAIL_ADDRESS>
&Matthew A. Jolley
Department of Anesthesiology and
Critical Care Medicine,
Division of Pediatric Cardiology,
Children’s Hospital of Philadelphia,
Philadelphia, PA 19104
<EMAIL_ADDRESS>

###### Abstract

Atrioventricular valve regurgitation is a significant cause of morbidity and
mortality in patients with acquired and congenital cardiac valve disease.
Image-derived computational modeling of atrioventricular valves has advanced
substantially over the last decade and holds particular promise to inform
valve repair in small and heterogeneous populations which are less likely to
be optimized through empiric clinical application. While an abundance of
computational biomechanics studies have investigated mitral and tricuspid
valve disease in adults, few studies have investigated application to
vulnerable pediatric and congenital heart populations. Further, to date,
investigators have primarily relied upon a series of commercial applications
that are neither designed for image-derived modeling of cardiac valves, nor
freely available to facilitate transparent and reproducible valve science. To
address this deficiency, we aimed to build an open-source computational
framework for the image-derived biomechanical analysis of atrioventricular
valves. In the present work, we integrated an open-source valve modeling
platform, SlicerHeart, and an open-source biomechanics finite element modeling
software, FEBio, to facilitate image-derived atrioventricular valve model
creation and finite element analysis. We present a detailed verification and
sensitivity analysis to demonstrate the fidelity of this modeling in
application to 3D echocardiography-derived pediatric mitral and tricuspid
valve models. Our analyses achieved excellent agreement with those reported in
the literature. As such, this evolving computational framework offers a
promising initial foundation for future development and investigation of valve
mechanics, in particular collaborative efforts targeting the development of
improved repairs for children with congenital heart disease.

_K_ eywords atrioventricular valves $\cdot$ uncertainty analysis $\cdot$ valve
mechanics $\cdot$ contact potential $\cdot$ finite element modeling $\cdot$
open-source

##  1 Introduction

Multi-modality imaging, including 3D echocardiography(3DE), has transformed
adult mitral valve surgery by capturing the full, complex geometry of the
valve in real time, providing an intuitive view of the functioning valve
directly to the surgeon. While informative, 3D visualization alone is
insufficient for quantitative assessment and analysis of the valve. The
development of image-derived mitral valve computer modeling tools has
partially unlocked this potential, allowing precise, quantitative comparison
of normal valves to dysfunctional valves, greatly improving the understanding
of the 3D structural correlates of adult mitral valve dysfunction [1, 2, 3,
4]. However, correlation cannot infer causality: it does not elucidate the
physical basis for why valves fail over time or allow the testing and
comparison of novel repair strategies. The application of image-derived finite
element modeling has begun to provide this capability, bringing forth the
potential to determine patient-specific structural contributors to valve
stress, strain, and failure, as well as the development of the optimal repair
for an individual patient [5, 6, 7, 8, 9, 10]. Specifically, computational
modeling allows for investigations of valve function and assessment of leaflet
stress and strain that would otherwise be unobtainable using conventional in
vitro methods or clinical trials [9].

Significant advances in FEM modeling of the mitral and tricuspid valve have
occurred over the last decade [11, 12, 13, 5, 9, 14]. Notably, several
barriers to the application of FEM to the modeling of atrioventricular valves
from human images have been significantly addressed through these studies.
Initial FEM models were constructed from micro-CT images of ex-vivo animal and
cadaveric human hearts. These high-resolution images clearly demonstrated the
papillary and chordal support structure for the valve and provided a detailed
roadmap for incorporation of those structures into the valve model. However,
while valve leaflets and papillary muscle heads are clearly visible in living
human 3DE and tomographic imaging, the individual chords are not reliably
visualized. Thankfully, Khalighi et. al. compared simplified distributed
chordal models to “ground truth” models based on micro-CT and showed that for
chord density greater than 15 chords/cm2, the results were functionally
equivalent to the ground truth models [8]. This study opened the potential to
utilize living human clinical images as the basis for FEM to investigate valve
leaflet stress and strain. A second advance has been the development of
realistic constitutive models and specialized FEM techniques to support the
requirements necessary for the modeling of valve biomechanics [15, 13, 9].
Finally, the complexity associated with modeling thin pliable structures and
obtaining stable solutions (convergence) when modeling complex valve leaflet
motion has been challenging [11, 16]. Kamensky et al. recently described a
potential-based contact algorithm that is particularly well suited for valve
leaflet modeling and has been subsequently validated for modeling the
tricuspid valve [13, 17, 18]. The combination of these advances brings the
field to a state where FEM techniques may be meaningfully applied to derived
from 3D images of atrioventricular valves in living humans.

Building on this capability, previous investigations have demonstrated that
perturbations of valve structure result in differences in leaflet stress and
strain in image-derived models of the adult mitral valve, and more recently,
the tricuspid valve [11, 12, 13, 9, 14]. Notably, elevated leaflet stress and
strain are associated with valve failure, pathologic changes in valve leaflets
(leaflet prolapse, chordal rupture), and changes in valve leaflet gene
expression [4, 19, 20, 21]. Further, finite element method (FEM)
investigations in the mitral and tricuspid valve have demonstrated how both
image derived and parametric valve models can be “surgically” altered to
precisely investigate the effect of anatomic variation and surgical
interventions upon leaflet stress, strain, and coaptation [16, 22, 15, 9, 5].
These modeling capabilities may be particularly well suited to the
optimization of valve repair techniques in small and heterogeneous
populations, such as children with congenital heart disease, who do not
benefit from empiric validation of valve repair techniques through high-volume
application.

However, the majority of methods described to date leverage serial application
of multiple different commercial software platforms that are not fully
modifiable or configurable by the end user. A different commercial tool is
used to import 3D data, create models, incorporate models in FEM software, and
run simulations. Further, each commercial tool is not open or transparent as
to its methodology, limiting reproducibility and extension of functionality.
Similarly, suitable contact and constitutive models for valve modeling are not
typically provided in such packages. As such, there is the need for the
development of an open-source pipeline for computational valve modeling to
catalyze open and transparent valve science. In a major step toward an
integrated image-derived valve modeling pipeline, we implemented a robust
constitutive model and a novel contact potential formulation in the FEBio
finite element software 111www.febio.org, [23, 24, 25] and applied it to 3DE
image-derived models of pediatric mitral and tricuspid valves. We then
performed a detailed sensitivity analysis of the effect of varying modeling
parameters upon leaflet stress and strain to verify the newly implemented
framework.

##  2 Methods

In the present work, we integrated and employed open-source platforms to
facilitate the computational modeling process: SlicerHeart
222github.com/SlicerHeart [26, 27], 3D Slicer 333www.slicer.org [28], and
FEBio 3.5.1 [23, 24, 25]. The current pipeline consisted of two major
procedures. First, we constructed 3DE-derived FEM models of atrioventricular
valves using SlicerHeart and 3D Slicer. Second, we imported those valve models
and performed FEM analyses to assess the mechanical responses of the valves
using FEBio. Notably, while we utilized 3DE in this study, this workflow is
fundamentally applicable to images created using computed tomography (CT) or
cardiac magnetic resonance imaging (MRI).

###  2.1 Data Import and Valve Model Creation

This study was approved by the Institutional review board at the Children’s
Hospital of Philadelphia. 3DE of mitral and tricuspid valves were identified
from an existing database at the Children’s Hospital of Philadelphia. The
mitral valve was based on a 15 year-old male with normal heart anatomy and no
mitral regurgitation. The tricuspid valve was based on a 3 year-old male with
hypoplastic left heart syndrome with no significant valve regurgitation.
Images had been acquired on a Philips Epiq system (Philips Medical, Andover,
MA). The 3DE volume data (DICOM) were imported into 3D Slicer using the
Philips 4D US DICOM patcher module in SlicerHeart, as previously described
[27].

The valve model creation pipeline is described in Fig. 1. First, the Valve
Annular Modeling module in SlicerHeart was used to define the annulus and free
edge of each valve. Then, the mitral and tricuspid valves were segmented to
create valve models using the Valve Segmentation Module in SlicerHeart [27].
The finite element valve models were created by first defining the annular
contour curve using a 24-control point periodic spline. A second periodic
control point spline was constructed along the free edges of the leaflets to
model the open area of the valve. The resulting models were imported into
Autodesk Fusion 360 (AutoDesk, San Rafael, CA). A Non-Uniform rational
B-Spline (NURBS) surface was then lofted between the two curves to create the
leaflets and valve surface. Control points on the NURBS surface were further
edited to match the native geometry of the segmentation of the valve. This was
the only part of the workflow that is not currently open-source, and the
development of leaflet medial surface extraction and NURBS editing is underway
in SlicerHeart to eliminate this need.

A custom Python scripted workflow was preliminarily created in SlicerHeart to
distribute the chordae evenly along the leaflets (Fig. 2). The chordae
tendineae are not reliably identifiable from 3DE images, but the papillary
muscle origins can readily be seen. The locations of the papillary muscle tips
were identified in the 3DE image within 3D Slicer, and the registered
coordinates were used to model the chordal origins as a single point per
papillary muscle, as demonstrated by Khaligi et al [12].

Figure 1: Finite element modeling (FEM) pipeline from 3DE image: valve
construction process from 3DE image to FEM model. (A) Define annulus and free
edge control points, (B) create leaflet segmentation, (C) create splines, (D)
initialize Non-Uniform Rational B-Spline (NURBS) surface, (E) adjust NURBS
surface to segmentation, and (F) project chordae onto leaflets. Figure 2:
Finite element modeling pipeline from 3D echocardiographic image: chordae
tendineae modeling. (A) Identify papillary muscles, (B) define chordal
insertion area, and (C) project chordae.

###  2.2 Constitutive Model of Atrioventricular Valve Leaflets

Atrioventricular valve leaflets consist of three layers of tissue: fibrosa,
spongiosa, and atrialis. The fibrosa is the thickest layer of the three, and
is directed towards the ventricular chamber, while the atrialis layer, as the
name suggests, is directed towards the atria. The fibrosa layer is composed
primarily of type I collagen fibers and provides the most mechanical support
when subjected to flexural tension; the spongiosa layer is made of highly
hydrated glycosaminoglycans (GAGs) and serves as a lubrication layer for the
fibrosa and atrialis layers; the atrialis layer contains collagen and radially
aligned elastin fibers and sets out to provide support for reducing radial
strains when the valve undergoes physiological loading [29, 30, 9].

In the present work we adopted an incompressible, isotropic, and hyperelastic
constitutive model for the atrioventricular valve leaflet tissue, namely the
Lee-Sacks constitutive model [11]. While numerous studies have shown that the
heart valve tissues exhibit anisotropic behaviors (i.e., the heart valve
leaflets experience higher principal strain in the radial direction than in
the circumferential direction)[31, 32, 11, 16, 9], obtaining collagen fiber
orientation data to accurately reflect the anisotropic characteristic feature
of the leaflet tissue is a challenging task. Additionally, Wu et al.
demonstrated that the effects of anisotropy on heart valve leaflet coaptation
and orifice are small compared to the sizes of the leaflets [33, 18]. The Lee-
Sacks isotropic material model [11] provided a simple and computationally
efficient formulation [33] and was used for approximating the biomechanic
response of the leaflet tissue in the present work. The second Piola-Kirchhoff
stress, $\mathbf{S}$, of the Lee-Sacks soft tissue model was formulated using
an isotropic Fung-type material, wherein the contributions of the
extracellular matrix and collagen fiber network were represented additively by
neo-Hookean and exponential terms, respectively, to capture the nonlinear
stress-strain response:

$\mathbf{S}=2\frac{\delta\psi_{el}}{\delta\mathbf{C}}-\lambda_{p}\mathbf{C}^{-1}.\\\
$ (1)

Here, $\mathbf{C}$ is the right Cauchy-Green deformation tensor, $\lambda_{p}$
is a Lagrange multiplier for ensuring material incompressibility
($\lambda_{p}$= 5000 is applied in the present work), and $\psi_{el}$ is an
elastic strain energy function. The hyperelastic strain energy function was
expressed as:

$\psi_{el}=\frac{c_{0}}{2}(I_{1}-3)+\frac{c_{1}}{2}(\exp{c_{2}(I_{1}-3)^{2}}),$
(2)

where $c_{0}$, $c_{1}$, and $c_{2}$ are material coefficients, and
$I_{1}=tr\mathbf{C}$. Therefore, the term
$\frac{\delta\psi_{el}}{\delta\mathbf{C}}$ in Eq. 1 becomes:

$\frac{\delta\psi_{el}}{\delta\mathbf{C}}=\frac{1}{2}(c_{0}+2c_{1}c_{2}(I_{1}-3)\exp{c_{2}(I_{1}-3)^{2}})\mathbf{I}.$
(3)

###  2.3 Chordae Tendineae Modeling

The chordae tendineae play a critical role in ensuring proper heart valve
closure; these branch-like collagenous tissues connect the leaflets to the
papillary muscle heads to prevent leaflet prolapse into the left atrium during
ventricular contraction. However, while individual chordae can be visualized
via ex-vivo CT scan, individual chordae cannot be be reliably visualized using
clinically derived 3DE in humans[9]. Therefore, the geometric modeling of
chordae tendineae from 3DE derived models relies on strategic simplifications
to circumvent modeling limitations while accurately preserving their
biological function. In the present work, we opted for a simplified, but
robust, chordal topology approach introduced by Khalighi et al [12]. Khalighi
et al demonstrated that a uniformly distributed, branchless chordal model with
$15\pm 2$ chords/cm2 was able to reproduce the ground truth results in
predicting mitral valve closure to a high degree of accuracy. We uniformly
distributed at least 17 chords/cm2 over the free edge of our valve models. The
chords were modeled as tension-only 2-node linear springs that connect the
leaflet insertion points to the papillary muscle tips. The mechanical behavior
of the springs was considered mathematically within FEBio with a nonlinear
force-displacement response where the tension was equal to zero up to a
defined stretch value and the springs behaved linearly once the stretch
threshold was reached.

###  2.4 Contact Potential

Atrioventricular valve leaflet contact can be particularly difficult to
realistically model. Kamensky et al. recently described a potential-based
contact formulation [13]. This contact formulation was implemented in FEBio
after preliminary investigation found it to be particularly well suited to
allowing the valve leaflets to slide past one another, avoiding unrealistic
penetration of opposed leaflets. The contact potential between the two objects
of interest was approximated by the contact potential energy expressed as:

$E_{c}=\int_{\Omega_{0}^{2}}\int_{\Omega_{0}^{1}}\phi(r_{12})\,d\mathbf{X}_{1}d\mathbf{X}_{2},$
(4)

where $\mathbf{X}_{1}$ and $\mathbf{X}_{2}$ are in the reference
configurations of the two contact bodies denoted by $\Omega_{0}^{1}$ and
$\Omega_{0}^{2}$ respectively, $\phi(r_{12})$ is a contact potential kernel,
and $r_{12}$ denotes the Euclidean distance between two contact points.

The contact potential was described by the following force-separation law:

$\begin{gathered}-\phi^{{}^{\prime}}(r_{12})=\begin{cases}\frac{k_{c}}{(r_{12})^{p}}-c_{2},&r_{2}<r_{in},\\\
c_{1}(r_{12}-r_{out})^{2},&r_{in}\leq r_{12}<r_{out},\\\
0,&\text{otherwise},\end{cases}\end{gathered}$ (5)

where $k_{c}$ is a dimensionless scaling factor for the contact force,
$r_{in}$ is the inner distance that governs the transition between parabolic
and higher order regions of the contact potential, $r_{out}$ is the outer
distance that defines the boundary of the contact surface, and $p$ is the
power of the contact potential. We refer interested readers to [13, 18] for
detailed information regarding the numerical implementation of the contact
potential.

###  2.5 Solution Procedure

The valve medial surface was discretized into 4-node linear quadrilateral
(Quad4) shell elements [34]. We applied fixed displacement boundary conditions
to the outer circumference of the annulus edge and the papillary muscle tips.
We deformed the model from diastolic to systolic configuration by prescribing
a physiologically realistic systolic ventricular pressure orthogonal to the
ventricular surface of the leaflets. In contact models, it is not unusual for
the simulation to fail to converge because of unrealistic high frequency modes
in the numerical solution. We employed a mass-damping scheme by applying a
damping matrix $C=2000$ Ns/m to the model to suppress any such spurious
oscillations. The implicit Newmark time integration scheme, along with an
automatic time stepping algorithm, was used to facilitate time advancement and
maintain nonlinear solution convergence in the transient dynamic simulation.
The displacement and energy residual tolerances in FEBio were set to 0.001 and
0.1, respectively.

Fig. 3A and 3B provide a geometric visualization of the mitral valve model.
The mitral model was discretized into 735 elements, 2965 elements, and 11689
elements for mesh convergence analysis. The material coefficients used for the
mitral valve were: $c_{0}=200$ kPa; $c_{1}=2968.4$ kPa; and $c_{2}=0.2661$
[11]. We assumed 0.396 mm uniform thickness over the leaflets.

Figure 3: Geometric representation of image-derived mitral and tricuspid valve
finite element models. (A) Model of open mitral valve with annulus, leaflets,
and chordae tendineae defined, (B) model of closed mitral valve with leaflet
regions defined, (C) model of an open tricuspid valve with annulus, leaflets,
and chordae tendineae defined, and (D) model of a closed tricuspid valve with
leaflets defined. Mitral valve annular circumference was 12.3 cm and annular
area projected onto the least squares annular plane was 11.0 cm2. Tricuspid
valve annular circumference was 13.1 cm and annular area projected onto the
least squares annular plane was 12.0 cm2.

We applied 100 mm Hg ventricular pressure to the ventricular surface of the
mitral valve. The pressure was first increased linearly over 0.005 s
(consistent with [11]) to initiate the dynamic simulations. The pressure was
then kept constant until the valve reached steady-state (full closure). The
pressure evolution as a function of time was:

$\begin{gathered}p(t)=100~{}\text{mm Hg}\begin{cases}t/T,&t<T~{}\text{s},\\\
1,&\text{otherwise},\end{cases}\end{gathered}$ (6)

where $T$ is the ramp-up time scale and was set to 0.005 s. Once $T$ is
reached, the pressure load is maintained until steady-state. The initial time
step increment was set to $\Delta t=0.0001$ s. We terminated the simulation at
0.02 s. A total of 57 chords were attached to the mitral valve. The chordal
force as a function of leaflet displacement was described by a step function
in which zero tethering force was applied for the initial 5 mm stretch
threshold to accurately reflect the chordal slack length. The chords were each
prescribed a 30 mN tension force after the stretch threshold. The contact
potential parameters for the fine mesh were: $k_{c}=1$, $p=4$, $R_{in}=0.2$
mm, and $R_{out}=0.5$ mm.

A visualization of the tricuspid valve model is shown in Fig. 3C and 3D. The
mesh densities for the coarse, medium, and fine meshes consisted of 761, 3044,
and 12176 elements, respectively. The material coefficients used for the
tricuspid valve were: $c_{0}=10$ kPa; $c_{1}=0.209$ kPa; and $c_{2}=9.046$
[31, 13]. We assumed 0.396 mm uniform thickness over the leaflets. The peak
systolic pressure value, 23.7 mm Hg [31, 5], was applied to the tricupsid
model:

$\begin{gathered}p(t)=23.7~{}\text{mm
Hg}\begin{cases}t/0.01,&t<0.01~{}\text{s},\\\
1,&\text{otherwise}.\end{cases}\end{gathered}$ (7)

We attached a total of 41 chords to the tricuspid valve; 20 mN chordal tension
force was applied to each chord. The contact potential parameters for the fine
mesh were: $k_{c}=1$, $p=4$, $R_{in}=0.25$ mm, and $R_{out}=0.4$ mm.

All numerical simulations were performed on a high performance computing
system located at the Children’s Hospital of Philadelphia, consisting of Intel
Xeon CPU E5-2680 v3 computing nodes. Each node contains 24 cores, with 2.5 GHz
core speed and 128 GB RAM per node. The CPU time of the mitral and tricuspid
valve fine mesh models were 14 and 34 minutes, respectively.

###  2.6 Convergence Analysis

We examined the mean, 75${}^{\textrm{th}}$ percentile, and
95${}^{\textrm{th}}$ percentile of the $1^{\textrm{st}}$ principal stress and
strain, together with the valve closure profiles of the three mesh models to
assess the convergence. The mean values represent the average stress/strain
over all elements. The 75${}^{\textrm{th}}$ percentile and
95${}^{\textrm{th}}$ percentile refer to the values that cover 75$\%$ and
95$\%$ of the stresses/strains, respectively. Mesh convergence was determined
when the percentage of $L^{2}$ relative error norms reduced to less than
$5\%$. The $L^{2}$ relative error norms were defined as:

$L^{2}\text{ relative error
norm}=\sqrt{\frac{\sum_{i=0}^{N}(S_{ref,i}-S_{test,i})^{2}}{\sum_{i=0}^{N}S_{ref,i}^{2}}}\times
100,$ (8)

where $S_{ref,i}$ represents the $i^{\textrm{th}}$ index of the stress or
strain time history with the fine mesh, $S_{test,i}$ represents the
$i^{\textrm{th}}$ index of the stress or strain time history with either the
coarse or medium mesh, and $N$ represents the total number of time steps.

###  2.7 Uncertainty Analysis

Uncertainty analysis is an indispensable part of computational modeling. With
numerous parameters involved in valve modeling, there is a need to better
understand the role each parameter plays in affecting the final model
responses. As such, rigorous sensitivity analyses were performed to identify
the effect of the parameters and inform the robustness and predictive
capability of the numerical models and approaches. Ultimately, such analyses
correlate in vivo and in silico data such that reasonable tolerances and
margins of error can be determined should such models be used in a
translational setting.

We carried out two uncertainty analysis approaches (traditional and
statistical) with the fine mesh mitral and tricuspid models. In the
traditional approach, we explored the influence of the modeling parameters
individually, while in the statistical approach, we utilized a polynomial
chaos expansion (PCE) function to quantify the uncertainty in the FEM model
[35]. The parameters under consideration were the material coefficients in the
Lee-Sacks constitutive models (coefficient $c_{0}$ in the neo-Hookean term,
and coefficients $c_{1}$ and $c_{2}$ in the isotropic exponential term) [11].
Additionally, we explored the influence of the chordal stretch threshold and
tension, as previous works suggested chordal rest length had significant
impacts on valve closures [36, 37, 38]. We sampled five data points uniformly
distributed within the range of $\pm 50\%$ of the baseline values used in the
mesh convergence analyses.

We leveraged UncertainSCI for the statistical uncertainty analysis procedure,
a Python-based toolkit that harnesses modern techniques to estimate model and
parametric uncertainty, with a particular emphasis on needs for biomedical
simulations and applications [35]. This was achieved by interfacing
UncertainSCI with the FEBio solver using via a Python subroutine within FEBio
and FEBioUncertainSCI 444github.com/febiosoftware/FEBioUncertainSCI.
UncertainSCI used a polynomical chaos expansion function with a weighted
approximate Fekete points (WAFP) method to randomly generate collocation
points within the user-specified n-dimensional space (where the number of
dimensions is determined by the number of input parameters) for sensitivity
quantification. Unlike the traditional approach, where the parameters of
interest were varied one at a time, UncertainSCI allowed for multiple
variables such that the influence of input interactions may be examined.
Furthermore, UncertainSCI provided quantitative measures of the uncertainties
from the input parameters by computing the relative variance that each
parameter contributed to the total variance, namely the first-order Sobel
index. This enables better evaluation and comparison of the model output
uncertainty from the material constants.

##  3 Results

We present verification and sensitivity analysis results of two image-derived
valve models (mitral and tricuspid) to demonstrate the feasibility and
robustness of the open-source software in the present work. For each valve
model, we considered three mesh densities (coarse, medium, and fine). The
length and width of each element were divided exactly in half within each
level of refinement – resulting in a fourfold increase in mesh density as we
refined the models. After we established the convergence of the FEM models, we
used the fine mesh models to perform uncertainty quantification to study the
effect of modeling input parameters on the biomechanical responses of
atrioventricular valves.

###  3.1 Mitral Valve Verification

The means, standard deviations, 75${}^{\textrm{th}}$ percentile, and
95${}^{\textrm{th}}$ percentile of the stresses and strains of the whole
mitral valve are shown in Fig. 4C and 4D. We did not see substantial
differences in stresses and strains related to mesh density. The sum of
chordal tethering force on both papillary muscles was 6.84 N.

Each mitral valve leaflet is anatomically divided into three scallop regions.
Fig. 4E and 4F present the 95${}^{\textrm{th}}$ percentile regional stresses
and strains with the fine mesh. The results suggested that the anterior
leaflet experiences a higher stress concentration (about two times higher than
the posterior leaflet). The stress levels among the three scallop regions
within each leaflet were similar. In our mitral model the region A2
experienced lower strains compared to regions A1 and A3. However, region P2
presented relatively higher strains in comparison to regions P1 and P3.

Figure 4: Mitral valve stress and strain responses. (A) Stress profile on the
mitral valve at steady state, (B) strain profile on the mitral valve at steady
state, (C) stress responses with coarse, medium, and fine meshes (shaded areas
indicate standard deviations), (D) strain responses with coarse, medium, and
fine meshes (shaded areas indicate standard deviations), (E)
95${}^{\textrm{th}}$ percentile 1${}^{\textrm{st}}$ principal stress responses
on various mitral valve regions, and (F) 95${}^{\textrm{th}}$ percentile
1${}^{\textrm{st}}$ strain responses on various mitral valve regions. Results
suggested that the anterior leaflet experiences higher stress and strain
concentrations than the posterior leaflet.

We evaluated a cross section of the mitral valve to assess valve closure of
the three meshes (Fig. 5). The closing profiles were nearly identical among
the three meshes, which suggested that the mesh densities used in the present
work were sufficient to capture the valve closing behavior.

Figure 5: Mitral valve closing profiles. (A) Location at which the slices were
made (red line), and (B) valve closure configurations for coarse, medium, and
fine meshes at the anterior-posterior coaptation. Nearly identical closing
profiles suggested that all of the tested mesh densities were sufficient to
capture valve closing behavior.

The convergence behavior was confirmed by the $L^{2}$ relative error norms,
where we observed substantial error reduction with higher mesh density (Table
1). Region P1 reported the highest percentage error in mesh refinement level
from medium to fine, with $4.89\%$ relative error in mean principal stress.
Nonetheless, this relative error was below the $5\%$ threshold; indicating
reasonable agreement between the medium and fine meshes. The mean,
$75^{\textrm{th}}$ percentile, and $95^{\textrm{th}}$ percentile stresses of
the whole mitral with the fine mesh at steady state were 252.29 kPa, 336.45
kPa, and 546.22 kPa, respectively. The corresponding mean, $75^{\textrm{th}}$
percentile, and $95^{\textrm{th}}$ percentile strains at steady state were
0.21, 0.25, and 0.34, respectively. Regional stresses and strains are reported
in Appendix A Table 3.

|  |  $1^{\textrm{st}}$ principal stress |  $1^{\textrm{st}}$ principal strain
---|---|---|---
| Mesh refinement | mean |  $75^{\textrm{th}}$ percentile |  $95^{\textrm{th}}$ percentile | mean |  $75^{\textrm{th}}$ percentile |  $95^{\textrm{th}}$ percentile
Mitral | coarse to medium | 3.27 | 1.93 | 1.72 | 1.46 | 2.09 | 1.22
medium to fine | 0.92 | 0.74 | 0.53 | 0.39 | 0.77 | 0.81
| coarse to medium | 3.76 | 2.19 | 2.61 | 5.36 | 1.79 | 1.83
A1 | medium to fine | 0.99 | 0.90 | 0.73 | 1.66 | 0.61 | 0.50
A2 | coarse to medium | 1.98 | 1.97 | 1.18 | 0.96 | 1.19 | 0.90
medium to fine | 0.65 | 0.52 | 0.73 | 0.45 | 0.42 | 0.88
| coarse to medium | 2.07 | 1.16 | 2.10 | 1.02 | 1.05 | 1.04
A3 | medium to fine | 1.16 | 1.09 | 0.70 | 1.17 | 0.73 | 0.63
P1 | coarse to medium | 12.12 | 9.21 | 10.22 | 9.80 | 10.60 | 6.69
medium to fine | 4.89 | 3.37 | 3.79 | 2.88 | 3.00 | 3.10
| coarse to medium | 10.17 | 8.14 | 7.16 | 5.44 | 5.88 | 7.15
P2 | medium to fine | 2.10 | 2.05 | 2.41 | 0.92 | 1.36 | 1.70
P3 | coarse to medium | 4.03 | 5.84 | 11.22 | 5.03 | 6.74 | 1.22
medium to fine | 1.66 | 2.11 | 2.69 | 1.29 | 1.67 | 0.81

Table 1: $L^{2}$ relative error norms (%) in various sections of the mitral
valve. All $L^{2}$ relative error norms were under $5\%$ in the medium to fine
mesh refinement level, with the highest error norm ($4.89\%$) in the mean
$1^{\textrm{st}}$ principal stress in region P1.

###  3.2 Tricuspid Valve Verification

The stress and strain responses of the tricuspid valve are shown in Fig. 6.
Fig. 6E suggests that the anterior leaflet experienced the highest stress
concentration, followed by the posterior leaflet, and lastly the septal
leaflet. On the other hand, the $95^{\textrm{th}}$ percentile of strains were
nearly identical among the leaflets (Fig. 6F). Similarly to the mitral valve,
we created three cross sections through the tricuspid valve to assess the
closure configurations between leaflets (Fig. 7). Differences in valve closure
profiles were observed with the models. The septal leaflet deformation with
the coarse mesh showed significantly different characteristics in comparison
to the two finer meshes. This suggested that the mesh density played an
important role in capturing complex curvatures, as demonstrated in our
specific image-derived tricuspid model. The sum of chordal tethering force on
all papillary muscles was 2.38 N.

Figure 6: Tricuspid valve stress and strain responses. (A) Stress profile on
the tricuspid valve at steady state, (B) strain profile on the tricuspid valve
at steady state, (C) stress responses with coarse, medium, and fine meshes
(shaded areas indicate standard deviations), (D) strain responses with coarse,
medium, and fine meshes (shaded areas indicate standard deviations), (E)
95${}^{\textrm{th}}$ percentile 1${}^{\textrm{st}}$ principal stress responses
on various tricuspid valve leaflets, (F) 95${}^{\textrm{th}}$ percentile
1${}^{\textrm{st}}$ principal strain responses on various tricuspid valve
regions. While results suggested that the anterior leaflet experienced higher
stress concentrations than the posterior and septal leaflets,
95${}^{\textrm{th}}$ percentile 1${}^{\textrm{st}}$ principal strains were
nearly identical among the leaflets. Figure 7: Tricuspid valve closing
profiles. (A) Locations at which the slices were made (red lines); valve
closure configurations for coarse, medium, and fine meshes at the (B)
anterior-posterior coaptation, (C) anterior-septal coaptation, and (D)
posterior-septal coaptation. Septal leaflet deformation with the coarse mesh
showed significantly different characteristics in comparison to the two finer
meshes, suggesting that the mesh density played an important role in capturing
complex curvatures, as demonstrated in the tricuspid model.

The $L^{2}$ relative error norms of the tricuspid model are presented in Table
2. The highest percentage error (in mesh refinement level from medium to fine)
was found in the posterior leaflet, with $4.8\%$ in the $95^{\textrm{th}}$
percentile principal stress. All relative errors were below $5\%$. Therefore,
the fine mesh of tricuspid model was a converged mesh. The mean,
$75^{\textrm{th}}$ percentile, and $95^{\textrm{th}}$ percentile stresses of
the whole tricuspid valve at steady state were 64.83 kPa, 82.62 kPa, and
127.27 kPa, respectively. The mean, $75^{\textrm{th}}$ percentile, and
$95^{\textrm{th}}$ percentile strains at steady state were 0.51, 0.59, and
0.63, respectively; stresses and strains on individual leaflet are reported in
Appendix A Table 4.

|  |  $1^{\textrm{st}}$ principal stress |  $1^{\textrm{st}}$ principal strain
---|---|---|---
| Mesh refinement | mean |  $75^{\textrm{th}}$ percentile |  $95^{\textrm{th}}$ percentile | mean |  $75^{\textrm{th}}$ percentile |  $95^{\textrm{th}}$ percentile
Tricuspid | coarse to medium | 6.94 | 3.80 | 5.18 | 1.38 | 2.22 | 1.65
medium to fine | 3.33 | 2.63 | 2.05 | 0.59 | 0.86 | 1.62
| coarse to medium | 5.25 | 1.17 | 2.46 | 1.39 | 0.78 | 2.04
Anterior | medium to fine | 4.28 | 1.49 | 4.02 | 0.80 | 0.88 | 1.85
Posterior | coarse to medium | 5.58 | 4.80 | 9.56 | 1.10 | 2.59 | 3.77
medium to fine | 3.27 | 3.12 | 4.80 | 0.56 | 1.09 | 1.52
| coarse to medium | 10.78 | 5.73 | 3.57 | 3.33 | 3.59 | 2.92
Septal | medium to fine | 2.25 | 2.41 | 1.78 | 0.64 | 1.04 | 0.76

Table 2: $L^{2}$ relative error norms (%) in various sections of the tricuspid
valve. All $L^{2}$ relative error norms were under $5\%$ in the medium to fine
mesh refinement level, with the highest error norm ($4.8\%$) in the
$95^{\textrm{th}}$ percentile of the $1^{\textrm{st}}$ principal stress in the
posterior leaflet.

###  3.3 Traditional Sensitivity Analysis

The $95^{\textrm{th}}$ percentile of the 1${}^{\textrm{st}}$ principal
stresses and strains at steady state are presented in Fig. 8. The mitral and
tricuspid valve modeling parameters demonstrated different influences on the
stress and strain responses in the valve leaflets. Among the material
coefficients for the mitral valve, the material coefficients in the isotropic
exponential term ($c_{1}$ and $c_{2}$) had the most significant effect on both
stresses and strains – with approximately the same minimum, maximum, and
interquartile range (Fig. 8A and 8B); material coefficient $c_{0}$ had an
inconsequential influence on the stresses, but had a strong influence on the
strain response. Additionally, Fig. 8A and 8B show that the chordal stretch
threshold and tension force had opposite effects in stresses and strains. We
observed that a higher chordal stretch threshold lead to higher stresses and
strains; contrarily, increasing chordal tension forces yielded reduced mitral
valve mechanical responses.

An opposite behavior regarding the effect of the chordal stretch threshold and
tension force was seen in the tricuspid valve stress responses (Fig. 8C).
Specifically, great chordal tension and lower chordal stretch threshold
correspond to higher leaflet stress. Furthermore, Fig. 8D indicates the
chordal stretch threshold and tension force had insignificant effects on
tricuspid strains. In terms of material coefficients, Fig. 8C and 8D suggest
that the tricuspid model was most sensitive to the coefficient $c_{2}$ (in
agreement with the mitral model). In contrast, coefficients $c_{0}$ and
$c_{1}$ had negligible effects.

Figure 8: The resulting sensitivity in the 95${}^{\textrm{th}}$ percentile
1${}^{\textrm{st}}$ principal stresses and strains from various modeling
parameters. (A) Sensitivity of stress in the mitral valve, (B) sensitivity of
strain in the mitral valve, (C) sensitivity of stress in the tricuspid valve,
and (D) sensitivity of strain in the tricuspid valve. The main box-plots
demonstrate the spread of skewness of the stresses and strains for each
individual modeling parameter. The line plots display the relations between
the modeling parameters and the stresses/strains. Results suggested that the
mitral valve stresses and strains were most sensitive to material coefficients
$c_{1}$ and $c_{2}$, with $c_{0}$ displaying additional influence on strains;
tricuspid valve stresses and strains were most sensitive to material
coefficient $c_{2}$. For the mitral valve, increased chordal stretch threshold
led to higher stress/strain and increased chordal tension led to lower
stress/strain, whereas the tricuspid valve displayed the opposite responses
for stress and negligible responses for strain.

The systolic configurations of the mitral and tricuspid valves at steady state
are presented in Fig. 9 and 10 to assess the influence of chordae modeling
parameters on valve closure. Models with higher stretch threshold or lower
chordal tension force displayed noticeable billowing, or leaflet prolapse into
the atrium. Alternatively, models with early or excessive tensioning of the
chordae demonstrated poor coaptation, indicating inadequate valve closure, and
potential regurgitation.

Figure 9: Mitral valve systolic configurations resulting from variations in
chordae modeling parameters. (A) Systolic configurations subject to various
stretch threshold with a fixed chordal tension at 30 mN; and (B) systolic
configurations subject to various chordal tension with a fixed stretch
threshold at 5 mm. Higher stretch threshold or lower chordal tension force led
to models with noticeable billowing. Figure 10: Tricuspid valve systolic
configurations resulting from variations in chordae modeling parameters. (A)
Systolic configurations subject to various stretch threshold with a fixed
chordal tension at 20 mN; and (B) systolic configurations subject to various
chordal tension with a fixed stretch threshold at 5 mm. Higher stretch
threshold or lower chordal tension force led to models with noticeable
billowing.

###  3.4 Statistical Sensitivity Analysis

To facilitate the statistical sensitivity analysis, we applied a fourth
polynomial order PCE function to identify the sampling points in the material
parameter space for FEM analyses. We first assessed the sensitivity indices
from interactions between parameters (i.e., the output uncertainty due to
variations from two or more parameters). The relative variances obtained were
small – less than 0.005 – indicating that the material parameters contribute
independently to the uncertainty of the model output.

The total sensitivities of the material constants were reported in Fig. 11.
The statistical results suggested material coefficient $c_{1}$ was the most
dominant, with Sobel index 0.71 for stresses and 0.47 for strains for the
mitral valve. Similar to the findings in the traditional approach, material
constant $c_{0}$ had negligible effects on mitral valve stresses (Fig. 11A),
but it was the second highest contributor to the total output variance on
mitral valve strains (Fig. 11B). In the tricuspid model, material coefficient
$c_{2}$ contributed the most to the model uncertainty – $c_{2}$ had a Sobel
index of 0.91 and 0.95 in stresses and strains, respectively. In contrast, the
Sobel indices for material coefficients $c_{0}$ and $c_{1}$ were well below
0.1 (Fig. 11C and 11D).

Figure 11: The main sensitivity of the stresses and strains as a function of
material constants using UncertainSCI. (A) Uncertainty in mitral valve stress,
(B) uncertainty in mitral valve strain, (C) uncertainty in tricuspid valve
stress, and (D) uncertainty in tricuspid valve strain. Bar plots denote the
first-order Sobel indices and box plots depict the spread of stresses and
strains of 45 UncertainSCI samples with fixed polynomial chaos expansion order
4 and fixed random number generator seed 0. Results were consistent with the
traditional approach, with $c_{1}$ and $c_{2}$ influencing mitral valve
stresses and strains (with additional influence of $c_{0}$ on strains), and
$c_{2}$ influencing tricuspid valve stresses and strains.

##  4 Discussion

Our long term goal is to develop a robust, open-source computational framework
for the modeling of atrioventricular valves from 3D images to inform valve
repair in children with congenital heart disease. We have taken the first step
to achieve this by integrating and extending established open-source tools for
cardiac image processing and finite element analysis [28, 26, 27, 23, 24, 25,
35]. This work forms an initial foundation for the collaborative development
and application of biomechanical modeling to investigate the effect of image
derived, patient-specific valve structure on leaflet stress and strain. This,
in turn, may inform the design and application of more durable valve repairs
[39]. While our driving application is congenital heart disease, the tools
described are fundamentally applicable to any population.

In seminal work on the mitral valve, Votta et al. observed that the stresses
on the leaflet belly were in the range of 130 to 540 kPa on the anterior
leaflet and 60 to 279 kPa on the posterior leaflet[40] at the systolic
pressure of 120 mm Hg; Wang et al. reported a maximum principal stress of 160
kPa on the anterior leaflet midsection at 110 mm Hg peak systolic pressure
[32]; Lee et al. found the maximum radial and circumferential stresses on
anterior belly to be 509.5$\pm$38.4 kPa and 301.4$\pm$12.2 kPa at 90 mm Hg
peak transvalvular pressure[11]. In our study, the 95${}^{\textrm{th}}$
percentile principal stresses on the anterior and posterior belly at 100 mm Hg
peak pressure were 536.16 kPa and 300.47 kPa, respectively. Our results agreed
well with those reported in the literature; the differences in the stress
values can be explained by differences in image-derived model structure and
modeling approaches, including different material constitutive models,
transvalvular pressure, and chordal tension force. As demonstrated in the
sensitivity analysis, variations in chordal tension force and stretch
threshold can lead to more than 50 kPa difference in principal stresses. In
addition, our analyses suggested that overall, the anterior leaflet had higher
principal stresses than the posterior leaflet, consistent with prior studies.

Of the previous studies on the FEM modeling of tricuspid valves, Stevanella et
al. reported higher maximum principal anterior leaflet stress (430 kPa) than
posterior leaflet (120 kPa) at 23.7 mm Hg [31]; Kong et al. reported average
principal stress of 37-80 kPa, 25-91 kPa, 24-63 kPa on the anterior,
posterior, and septal leaflets, respectively, at mid-systole under the same
peak transvalvular pressure [5]; Laurence et al. reported the von Mises stress
of 24.7$\pm$7.9 kPa (anterior), 30.6$\pm$10.9 kPa (posterior), and
41.9$\pm$8.6 kPa (septal) on the tricuspid leaflet belly at 25 mm Hg peak
pressure [41]. Our tricuspid stress values agreed well with Kong et al.’s
approximations. The discrepancies between our results and Stevanella et al.
may be due to differences both the geometry of the model and the material
properties of their model. (Stevanella et al. applied mitral valve material
properties for the tricuspid FEM models.) Meanwhile, Laurence et al.’s
tricuspid model was approximately a third smaller than ours, which contributed
to the slightly lower stresses reported in their work. In addition to the
stresses, Stevanella et al. reported circumferential and radial strains of
0.13-0.16 and 0.25-0.30 on the anterior leaflet belly [31]; Kong et al.
reported average principal strains of 0.19-0.26, 0.07-0.17, and 0.11-0.21 on
the anterior, posterior, and septal leaflets, respectively [5]; and Laurence
et al. reported maximum principal strains of 0.33$\pm$0.07 (anterior),
0.41$\pm$0.06 (posterior), and 0.44$\pm$0.03 (septal) on the tricuspid leaflet
belly. Our strain predictions were slightly higher than the reported values.
Given that our sensitivity analysis suggested the chordal stretch threshold
and tension force were inconsequential to tricuspid valve strains, the
differences observed could be due a combination of differences in constitutive
models and valve geometry.

The valve geometries in the present work were derived from images but
simplified to create representative models for this initial investigation.
However, anatomically accurate valve geometries are critical in determining
the stress profiles on the mitral leaflets [12, 9, 41]. For example, Jimenez
et al. suggested that the annular geometry had direct consequence to the
chordal force distribution [42]. Sacks et al. [9] observed that the segment A2
had the highest stresses on some models. While we also observed high stress
concentration on segment A2, in our specific mitral model segments A1 and A3
had slightly higher stresses – possibly due to variations in the annular and
leaflet geometry in our specific model. Notably, the stress distribution of
our mitral valve FEM model agreed well with the model in a previous study with
similar mitral annular shape [43]. This highlights the importance of image-
derived patient-specific FEM models in order to generate insights relevant to
that specific valve geometry.

Computational modeling of atrioventricular valve dynamics has historically
been a challenge due to numerical instability that arises from leaflet
contact. Frequently, researchers had to sacrifice solution accuracy (as
obtaining a converged solution was difficult) by terminating simulations based
on a fixed number of iterations within each time step rather than by a set
residual [44, 45]. Although highly refined commercial packages such as LS-DYNA
555http://www.lstc.com/products/ls-dyna and ABAQUS
666https://www.3ds.com/products-services/simulia/products/abaqus made
obtaining converged solutions possible, any such simulations required a time
step of $10^{-6}$ s or less and took more than 20 days to run with explicit
time integration scheme [46]. Kamensky’s volume potential approach [13]
introduced a novel modeling method in contact mechanics, providing a robust
and computationally efficient strategy to overcome the challenges encountered
in atrioventricular valve FEM modeling. We have now implemented this powerful
methodology in FEBio. Together with the implicit time integration scheme
available in FEBio, our models were able to achieve satisfactory systolic
configurations in well under an hour using a single CPU with 24 cores.

Computational models of atrioventricular valves have become increasingly
accurate at representing physical reality. However, these simulations often do
not capture the impact of parameter uncertainty in their predictions [47]. As
such, we performed uncertainty analysis using both traditional and statistical
approach to determine the most significant tissue material constants and
chordal configuration on the biomechanical response of our mitral and
tricuspid valve FEM models. Our results indicated that the material constants
in the Lee-Sacks model, as well as the chordal stretch threshold and tension,
may have varying degrees of influence in image-derived models of the tricuspid
and mitral valve. This highlights 1) the need for population specific, and
possibly patient-specific tissue constitutive models for atrioventricular
valves, and 2) the importance of anatomically accurate chordal length and
properties for the most accurate assessment of the leaflet stress and strain.

Consistent with our analysis, prior work has demonstrated that valve dynamics
are highly influenced by chordal length and stretch threshold [37, 38]. As
such, substantial effort has been dedicated to accurately reconstructing the
geometry and topology of chordae tendineae using high-resolution micro-CT
imaging of static excised animal hearts [32, 16, 9]. Notably, micro-CT cannot
currently be applied to living humans or beating hearts. Further, it is not
currently feasible to visualize individual chordae in living humans using
readily available 4D imaging techniques such as 3DE. As such, several
approaches have used to approximate chordal length and geometry in the absence
of a detailed knowledge of chordal structure. Mansi et al. approximated the
chordal length using the distance between papillary muscle tips and leaflet
free edge in end diastole [37]. Kong et al. first assumed the chordae were
straight and branch-less, then iteratively adjusted the chordal length until
the FEM leaflet model matched CT images of the leaflets in systole. [5].
Khalighi et al. developed a simplified, but functionally equivalent, framework
for modeling the chordae tendineae when they cannot be visualized in the 3D
image [8]. Specifically, Khalighi et al. proposed to approximate the mitral
chordae tendineae topology and geometry as branch-less chords uniformly
distributed on the leaflets. The chordal lengths were estimated as the
distance from the papillary muscle tips to the leaflet insertion points at
systolic configuration. While this method provided accurate leaflet stress and
strain response for mitral valve, validation in application to the tricuspid
valve has yet be demonstrated. In our work, we adopted a similar approach as
Khalighi et al. to model the topology of the chordae tendineae. The applied
chordal tension force was obtained through an iterative process until the
total tethering force on the papillary muscle tips agreed with those reported
in the literature and created realistic valve closure in comparison to the
images from which they were derived. The total tethering forces in our mitral
valve and tricuspid valve FEM models were 6.84 N and 2.38 N, respectively.
These values fall within the range of total tethering forces in the
literature, which were 4 to 13.5 N for the mitral valve [9, 48, 40, 31], and
2.02 to 4.95 N for the tricuspid valve [5].

Finally, we demonstrated the integration of a recently developed statistical
uncertainty analysis toolbox, UncertainSCI, into FEBio. Our UnscertainSCI-
based statistical analysis agreed well with the findings from traditional
analysis, which demonstrated the fidelity of the WAFP-based PCE method for
uncertainty quantification. This WAFP-based PCE function was able to
efficiently quantify the sensitivity of model input parameters to model
response with significantly fewer sample points than standard Monte Carlo
methods [35]. In addition, this approach offered further insights into the
exact uncertainty measures from each input parameters, which were not
quantifiable using the traditional approach. This provided additional
information regarding the sensitivity that each parameter induced in the
model. We computed the main sensitivity analysis with various PCE orders and
random number generators (Appendix B). We did not observe significant
differences in the sensitivity indices among the PCE orders and random number
generators. This indicates the reliability of the WAFP-based PCE method for
predicting sensitivities in our present work and in future applications.

##  5 Limitations and Future Work

Valve geometry contributes greatly to the biomechanical function of the valve.
In the future, we hope to perform more clinically relevant studies, including
comparison of dysfunctional valves to normal valves. In addition, the
alteration of valve structure can be used to understand the effect of valve
geometry on valve stress and strain. Finally, “surgical" alteration of such
valves, emulating existing and novel repair techniques could be used to
optimize and inform surgical repairs before they are attempted in humans.

To the authors’ knowledge, there is currently no open-source, ready-to-use
mesh generation package for lofting valve surfaces from contour curves. While
the main components of model creation are open-source, we utilized a
commercial CAD program to mesh the leaflets. In the future, we hope to expand
the capability to create and alter image-derived valve shell models directly
within SlicerHeart [49, 18].

Finally, we utilized constitutive models derived from adult mitral and
tricuspid valves. While this allows realistic comparison to existing work, and
a means for comparison of the valve biomechanics with variation of a baseline
structure, it is likely that the adult-derived constitutive models do not
accurately describe the mechanical properties of the wide range of ages and
valve types present in congenital heart disease. Further work is needed to
develop a framework for the development of age and pathology-specific
constitutive models for this diverse population, as well as refinement of
applicable leaflet shape fitting approaches to mitigate this impediment to
translational application when precise constitutive models are not available
[39].

##  6 Conclusion

We describe the preliminary implementation of an integrated image-to-FEM-
modeling workflow for the biomechanical modeling of atrioventricular valves.
While the driving application underlying the development of this evolving
open-source framework was to inform a more disciplined and rigorous approach
to the assessment of valve failure in children congenital heart disease, it is
fundamentally applicable to a wide range of valve science. Our initial stress
and strain results yielded excellent agreement compared with the literature
and we provided a detailed sensitivity analysis of the FEM modeling parameters
using both traditional and statistical methods. Future work will focus on
optimization, validation, and application to investigate the biomechanics of
dysfunctional atrioventricular valves.

##  7 Acknowledgment

This work was supported by NIH R01HL153166, R01GM083925, U24EB029007, Big
Hearts to Little Hearts, a Children’s Hospital of Philadelphia (CHOP) Frontier
Fund (Pediatric Valve Center), The Cora Topolewski Fund at CHOP Pediatric
Valve Center, and the Canarie Research Software foundation.

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##  Appendix A Appendix A: Additional Verification Results

Here, we report on additional stress and strain data for the mitral and
tricuspid FEM models.

|  |  $1^{\textrm{st}}$ principal stress |  $1^{\textrm{st}}$ principal strain
---|---|---|---
| mean |  $75^{\textrm{th}}$ percentile |  $95^{\textrm{th}}$ percentile | mean |  $75^{\textrm{th}}$ percentile |  $95^{\textrm{th}}$ percentile
Mitral | 252.29 | 336.45 | 546.22 | 0.21 | 0.25 | 0.34
A1 | 489.89 | 572.02 | 635.20 | 0.31 | 0.35 | 0.38
A2 | 474.85 | 504.18 | 536.16 | 0.27 | 0.29 | 0.32
A3 | 522.06 | 566.63 | 618.20 | 0.31 | 0.34 | 0.37
P1 | 244.96 | 260.34 | 276.73 | 0.22 | 0.23 | 0.25
P2 | 267.06 | 282.82 | 300.47 | 0.23 | 0.24 | 0.24
P3 | 248.22 | 264.90 | 276.13 | 0.21 | 0.23 | 0.25

Table 3: $1^{\textrm{st}}$ principal stresses and strains on the mitral valve
with the fine mesh. We observed the highest $95^{\textrm{th}}$ percentile 1st
principal stress and strain in region A1.

|  |  $1^{\textrm{st}}$ principal stress |  $1^{\textrm{st}}$ principal strain
---|---|---|---
| mean |  $75^{\textrm{th}}$ percentile |  $95^{\textrm{th}}$ percentile | mean |  $75^{\textrm{th}}$ percentile |  $95^{\textrm{th}}$ percentile
Tricuspid | 64.83 | 82.62 | 127.27 | 0.51 | 0.59 | 0.63
Anterior | 75.53 | 101.44 | 142.88 | 0.52 | 0.61 | 0.64
Posterior | 68.76 | 86.71 | 112.94 | 0.51 | 0.58 | 0.62
Septal | 51.48 | 65.08 | 90.20 | 0.51 | 0.58 | 0.61

Table 4: $1^{\textrm{st}}$ principal stresses and strains on the tricuspid
valve with the fine mesh. We observed the highest $95^{\textrm{th}}$
percentile 1st principal stress and strain in the anterior leaflet.

##  Appendix B Appendix B: FEBioUncertainSCI Sensitivity

Here, we report on the sensitivity of FEBioUncertainSCI to the polynomial
order and random number generator.

Figure 12: The main sensitivity of the mitral valve stresses and strains as a
function of material constants with varying polynomial chaos expansion (PCE)
orders using UncertainSCI. (A) Uncertainty in valve stress with 30 samples and
PCE order 3, (B) uncertainty in valve stress with 45 samples and PCE order 4,
(C) uncertainty in valve stress with 66 samples and PCE order 5, (D)
uncertainty in valve strain with 30 samples and PCE order 3, (E) uncertainty
in valve strain with 45 samples and PCE order 4, and (F) uncertainty in valve
strain with 66 samples and PCE order 5. Bar plots denote the first-order Sobel
indices and box plots depict the spread of stresses and strains of
UncertainSCI samples with a fixed random seed of 0. We did not observe
significant differences in the first-order Sobel indices. Figure 13: The main
sensitivity of the mitral valve stresses and strains as a function of material
constants with varying random number generator seeds using UncertainSCI. (A)
Uncertainty in valve stress with random seed 10, (B) uncertainty in valve
stress with random seed 20, (C) uncertainty in valve stress with random seed
40, (D) uncertainty in valve strain with random seed 10, (E) uncertainty in
valve strain with random seed 20, and (F) uncertainty in valve strain with
random seed 40. Bar plots denote the first-order Sobel indices and box plots
depict the spread of stresses and strains of 45 UncertainSCI samples with
fixed polynomial chaos expansion order 4. We did not observe significant
differences in the first-order Sobel indices. Figure 14: The main sensitivity
of the tricuspid valve stresses and strains as a function of material
constants with varying polynomial chaos expansion (PCE) orders using
UncertainSCI. (A) Uncertainty in valve stress with 30 samples and PCE order 3,
(B) uncertainty in valve stress with 45 samples and PCE order 4, (C)
uncertainty in valve stress with 66 samples and PCE order 5, (D) uncertainty
in valve strain with 30 samples and PCE order 3, (E) uncertainty in valve
strain with 45 samples and PCE order 4, and (F) uncertainty in valve strain
with 66 samples and PCE order 5. Bar plots denote the first-order Sobel
indices and box plots depict the spread of stresses and strains of
UncertainSCI samples with a fixed random seed of 0. We did not observe
significant differences in the first-order Sobel indices. Figure 15: The main
sensitivity of the tricuspid valve stresses and strains as a function of
material constants with varying random number generator seeds using
UncertainSCI. (A) Uncertainty in valve stress with random seed 10, (B)
uncertainty in valve stress with random seed 20, (C) uncertainty in valve
stress with random seed 40, (D) uncertainty in valve strain with random seed
10, (E) uncertainty in valve strain with random seed 20, and (F) uncertainty
in valve strain with random seed 40. Bar plots denote the first-order Sobel
indices and box plots depict the spread of stresses and strains of 45
UncertainSCI samples with fixed polynomial chaos expansion order 4. We did not
observe significant differences in the first-order Sobel indices.