# The effects of leaflet material properties on the simulated function of
regurgitant mitral valves

Wensi Wu
Department of Anesthesiology and
Critical Care Medicine,
Division of Pediatric Cardiology,
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>
&Patricia Sabin
Department of Anesthesiology and
Critical Care Medicine,
Children’s Hospital of Philadelphia,
Philadelphia, PA 19104
<EMAIL_ADDRESS>
&Devin W. Laurence
Department of Anesthesiology and
Critical Care Medicine,
Division of Pediatric Cardiology,
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>
&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

Advances in three-dimensional imaging provide the ability to construct and
analyze finite element (FE) models to evaluate the biomechanical behavior and
function of atrioventricular valves. However, while obtaining patient-specific
valve geometry is now possible, non-invasive measurement of patient-specific
leaflet material properties remains nearly impossible. Both valve geometry and
tissue properties play a significant role in governing valve dynamics, leading
to the central question of whether clinically relevant insights can be
attained from FE analysis of atrioventricular valves without precise knowledge
of tissue properties. As such we investigated 1) the influence of tissue
extensibility and 2) the effects of constitutive model parameters and leaflet
thickness on simulated valve function and mechanics. We compared metrics of
valve function (e.g., leaflet coaptation and regurgitant orifice area) and
mechanics (e.g., stress and strain) across one normal and three regurgitant
mitral valve (MV) models with common mechanisms of regurgitation (annular
dilation, leaflet prolapse, leaflet tethering) of both moderate and severe
degree. We developed a novel fully-automated approach to accurately quantify
regurgitant orifice areas of complex valve geometries. We found that the
relative ordering of the mechanical and functional metrics was maintained
across a group of valves using material properties up to 15% softer than the
representative adult mitral constitutive model. Our findings suggest that FE
simulations can be used to qualitatively compare how differences and
alterations in valve structure affect relative atrioventricular valve function
even in populations where material properties are not precisely known.

_K_ eywords valve function $\cdot$ valve mechanics $\cdot$ tissue properties
$\cdot$ uncertainty analysis $\cdot$ valvular regurgitation

##  1 Introduction

Recent advances in cardiac imaging, such as 3D echocardiography (3DE), offer a
noninvasive means to evaluate the structure of dysfunctional heart valves [1,
2]. 3D images allow intuitive visualization of valve anatomy, identification
of mechanisms of valve dysfunction, and quantification of the size of the
valve leak (e.g., regurgitant orifice area (ROA)). In addition, 3D images
enable the association of quantitative metrics of 3D valve structure to
valvular function (degree of regurgitation) [3, 4, 5, 6, 7, 8], which in turn
may inform valve repair [9, 10, 11, 12, 13]. However, association cannot infer
causation, and it is not feasible to perform a randomized controlled trial of
each proposed valve intervention due to the heterogeneity of clinical
populations and the duration and cost to complete such studies [14].

3D image-derived computational approaches have the potential to meet the need
for valve structural assessment and identification of the optimal repair for
an individual patient. Over the last 20 years, extensive progress has been
made in FE modeling of the mitral valve (MV), and more recently, the tricuspid
valve [15, 16, 17, 18, 14, 19]. These investigations have used 3D image-
derived models to investigate how variations in valve structure influence
valve closure as well as leaflet stress and strain in adult atrioventricular
valves [20, 14, 21, 22]. In addition, a model of a dysfunctional valve can be
realistically altered using computer-aided design (CAD) and FE analysis to
precisely simulate the effect of adult atrioventricular valve repairs such as
annuloplasty, leaflet resection, papillary muscle redirection, addition of
prosthetic chords, and transcatheter edge-to-edge repair [23, 24, 25, 26, 17,
27, 28]. These seminal works provide a roadmap for the controlled and
reproducible virtual manipulation of leaflet structure, annular structure, and
papillary muscle head position beginning from an initial valve model. FE
simulations can then be applied to determine the effect of variations in both
native valve anatomy and virtual interventions on leaflet coaptation, stress
and strain, relative to a baseline valve model.

While simulation of customized valve repairs is promising, patient-specific
leaflet material properties cannot yet be noninvasively obtained, and the
effect of variation in material properties on metrics of repair quality
(leaflet coaptation, stress and strain) remains unknown. Notably, material
properties may vary across individuals, and most modeling is currently
performed using material parameters derived from animal or adult cadaveric
tissue [14, 29, 30, 31]. Such variation is particularly relevant to the
application of valve modeling in children, where valve leaflet material
properties have not been described, and may differ across age and disease.
Given both valve geometry and material properties are critical factors
affecting leaflet mechanics and valve function [32, 33, 16, 14, 21], if the
effect of variation in material properties is markedly greater than that of
geometry, then evolving computationally derived optimizations of valve repair
are less likely to be clinically informative.

As such, we examined the effect of material properties on functional (leaflet
coaptation area and ROA) and biomechanical (leaflet stress and strain) metrics
across one normal (non-regurgitant) MV and six regurgitant MV model variants
with different mechanisms of regurgitation. We explored the influence of
material properties on the relative magnitude of functional and biomechanical
metrics across this range of valve geometry. We hypothesized that the relative
ordering of the magnitude of functional and mechanical metrics of a group of
valve models would be preserved across a range of leaflet material properties.
This in turn would suggest that meaningful relative comparisons of valve
function and mechanics could be obtained across variations of valve geometry
(e.g., different interventions to repair a valve) even in the absence of
precise knowledge of tissue material properties.

##  2 Methods

We aimed to evaluate the effects of varying soft tissue properties on
mechanical and functional metrics of dysfunctional MVs using FE analysis. MV
anatomy is shown in Fig. 1. We first explored the influence of tissue
extensibility on the mechanical and functional metrics of MVs with common
mechanisms of regurgitation (annular dilation, posterior leaflet prolapse in
the P2 region, and posterior leaflet tethering) across varied grades of
regurgitation. Subsequently, we studied the effects of individual material
parameters and leaflet thickness on the mechanical and functional metrics
across severely regurgitant MVs. Fig. 2 provides an overview of the valve
types considered in the present work. The FE model creation and analysis
procedure are summarized in the following subsections. Interested readers may
refer to Wu et al. [22] for more details regarding the computational modeling
framework.

Figure 1: Mitral valve geometry and anatomy. (A) Open mitral valve with
annulus, free edge, leaflets, and chordae tendineae shown; (B) Closed mitral
valve with regions of the anterior leaflet (A1, A2, A3) and posterior
leaflet(P1, P2, P3) labeled. Figure 2: Overview. We created models of one
normal and three regurgitant MV with different mechanisms of regurgitation (
posterior leaflet tethering, posterior leaflet prolapse in the P2 region, and
symmetric annular dilation). For the three dysfunctional morphologies
(Tethered, P2 prolapse, Annular dilation) we created morphologies with two
degrees of regurgitation (moderate and severe). We used these MV morphologies
to examine the effects of tissue extensibility and the individual material
coefficient on the mechanical and functional metrics. Visualization of chordae
omitted for clarity.

###  2.1 Finite element modeling procedure

The computational analyses in the present work leveraged three open-source
platforms: SlicerHeart 111github.com/SlicerHeart [9, 34], 3D Slicer
222www.slicer.org [35], and FEBio 333www.febio.org [36, 37, 38]. Use of 3DE
images for this work was approved by the Institutional Review Board at the
Children’s Hospital of Philadelphia.

To create a baseline FE mesh emulating realistic MV geometry, we used
SlicerHeart to segment the valve leaflets of a normal adolescent male’s MV
from 3DE images. Then, we defined periodic control point splines at the
annulus and free edge to extract the medial surface from the valve
segmentation. Finally, we lofted a NURBS grid surface between the splines and
fitted the grid surface to the valve medial surface. From this baseline
geometry, the NURBS grid surface was edited to create a normal MV with full
coaptation upon closure in FEBio. Three abnormal MV variants were created with
further adjustments to the NURBS grid surface, papillary muscle tip placement,
and chordae tension: (1) for posterior leaflet tethering, chordae length was
increased by lowering papillary muscle tips and chordae tension was increased,
(2) for posterior leaflet prolapse, chordae tension was decreased in the P2
region, and (3) for annular dilation, the diameter of the annulus curve was
increased without lengthening the leaflets. For each of these three
morphologies, models with two degrees of regurgitation severity (moderate,
severe) were created.

As tissue anisotropy has negligible effects on the global leaflet deformation
response [39], we adopted the incompressible, isotropic, hyperelastic Lee-
Sacks constitutive model [15] to model the MV leaflet tissue for its
simplicity and efficient formulation. The Lee-Sacks model is characterized by
an unconstrained constitutive model in which the contributions of the
extracellular matrix and collagen fiber network were approximated by combining
neo-Hookean and exponential terms. Details of the constitutive model
formulation can be found in [15, 17].

The FE meshes were discretized into 4-node linear quadrilateral (Quad4) shell
elements [40]. Pinned boundary conditions were applied to the annulus edge and
the papillary muscle tips to restrict spatial translations in the valve
models. A systolic pressure (100 mmHg) was prescribed to the ventricular
surface of the leaflets to simulate valve closure. The leaflet contact was
modeled using a potential-based formulation [17]. The chords were modeled as
previously described [22]. Dynamic analyses were performed on the valve models
using an implicit Newmark time integration scheme.

###  2.2 Mechanical and functional metrics

The mechanical and functional metrics considered include the average
$1^{\text{st}}$ principal stress and strain, contact area, and ROA. The
contact area for the Kamensky contact [17] was calculated by integrating the
contact surface over the elements with non-zero traction applied to its
integration points based on a force-separation law defined in Kamensky et al.
[17].

We developed a new fully-automated method to accurately quantify the ROA by
coupling a shrink-wrapping method [41, 42, 43] with raycasting. This novel
technique can capture multiple regurgitant orifices, and we have released it
open-source in the Orifice Area module of SlicerHeart extension in 3D Slicer
[34]. The procedures are shown in Fig. 3. In summary, we first imported the
systolic frame of the valve FE model to 3D Slicer and inflated the surface
mesh by the leaflet thickness. This step helps improve the approximation
accuracy of the orifice opening. Second, we placed landmark points near the
orifice opening to create a closed contour (Fig. 3A). Third, we warped a flat
disk template to the closed contour to generate a continuous surface [44]
represented the red and blue patches combined in Fig. 3B-D. The red patches
indicated areas where the continuous surface is in contact with the leaflet
surface, whereas the blue patches represented the potential orifice surface.
Fourth, we iteratively smoothed (via Laplacian method), remeshed (via
Approximated Centroidal Voronoi Diagrams algorithm [45]), and displaced (in
the direction of gradient vectors determined from the leaflet surface) the
continuous surface towards the orifice opening, until the change in orifice
surface area between iterations fell within a user-defined tolerance. The
shrink-wrapping method may overestimate the potential orifice area in cases
where the regurgitant opening is situated in a narrow and deep crease between
the anterior and posterior leaflets. As such, we generated 400 rays in the
surface normal direction of a 50-degree cone over the potential orifice
surfaces to filter out invalid patches (Fig. 3F). In other words, any patches
with rays intersecting the leaflet surface were considered invalid orifice
areas. Finally, we summed up all valid patches as the final, confirmed
effective ROAs (Fig. 3F). From our experience, we found 400 rays from each
patch are sufficient for identifying small openings within reasonable
computational time. Users may use fewer rays if desired.

Figure 3: ROA computation procedure. (A) Create a closed contour near the
valve opening. (B) SlicerHeart automatically generates a continuous surface.
The red region indicates the area where the continuous surface is in contact
with the leaflet surface; blue indicates the potential orifice surface. (C to
D) the potential orifice surface gradually descends toward the orifice opening
in the valve model by a user-defined number of shrink-wrapping iterations. (E)
Generate 400 rays (red lines) over the potential orifice surface to identify
the streamlines that pass through the valve without intersecting the leaflets.
(F) Compute the effective ROA by summing the areas around the rays that do not
intersect the leaflet surface.

###  2.3 Numerical derivation of material variants

While there are representative material models to capture leaflet tissue
properties in adults, characterizing tissue properties in children remains a
challenging task due to the inaccessibility of human tissue. This limitation
hinders meaningful studies of valve functions and closure. As many studies
suggested that heart valve tissue is softer in children [46, 47, 48, 49], we
performed uniaxial FE analysis on a single-element FE model and numerically
derived five sets of material coefficients that yielded various degrees of
increased extensibility from the adult mitral leaflet properties (Fig. 4).
These models were used to represent the range of leaflet properties in
children in the interim of developing more precise tissue models.

Figure 4: Material variants. We derived five material variants, each
representing various degrees of tissue extensibility, by performing a uniaxial
test using FE analysis. The material constants are shown in (A), their
corresponding stress-strain curves are shown in (B), and the percentage
difference of the stretch ratio, $\lambda$, between each material variant and
the mitral model (C).

###  2.4 Uncertainty analysis

In addition to evaluating the influence of tissue extensibility on the
mechanical and functional metrics of a diverse set of regurgitant MVs, we also
examined the sensitivity of individual material coefficients ($c_{0}$,
$c_{1}$, and $c_{2}$ in the Lee-Sack constitutive model) and leaflet thickness
on the mechanical and functional outcomes of the normal and severely
regurgitant MVs. We uniformly sampled five data points in the range of $\pm
50\%$ of the reference value for each model parameter (Fig. 1). Then, we
performed FE analyses by varying one parameter at a time, while keeping the
reference values for the remaining parameters, to identify the sensitivity of
the mechanical and functional metrics concerning individual model parameters.

Table 1: Uncertainty analysis sample points. We sampled a total of five points
uniformly between the minimum and maximum values for each parameter to
identify the sensitivity of the mechanical and functional metrics with
perturbation in the material constants and leaflet thickness.
---

##  3 Results

In Section 3.1, we altered material extensibility to examine the effects of
tissue stretches on the mechanical and functional metrics for pathologic
valves. In Section 3.2, we performed an uncertainty analysis for one normal
and three severely regurgitant valves to discern the effects of individual
material model parameters and the corresponding mechanical and functional
metrics at valve closure.

###  3.1 Mechanical and functional metrics comparative studies

####  3.1.1 Similar regurgitant MV geometries

We applied five sets of material coefficients to regurgitant valves of
moderate and severe regurgitant grades to assess the influence of tissue
extensibility on the mechanical and functional metrics in MVs of similar
geometries. We report our findings for the tethered MV in this section to save
space; readers may refer to Appendix A for the remaining findings.

The top and side profiles of mechanical and functional metrics for the
tethered mitral are shown in Fig. 5. We observed that the softer material
coefficients led to higher 1${}^{\text{st}}$ principal stress and strain, more
contact area, and less ROA. In addition, as the regurgitation severity
increased (i.e., ROA), so did the stress and strain. Again, the color red in
the contact area panels represents the contact area, and the color blue in the
ROA panels represents the ROA.

Figure 5: Mechanical and functional metric profiles for tethered models. We
observed an increase in $1^{\text{st}}$ principal stress, $1^{\text{st}}$
principal strain, and contact area but a decrease in ROA in the softer
material variants. Additionally, valves with severe regurgitation had higher
stress, strain, and regurgitant area, but less contact area.

We ranked the mechanical and functional metrics from lowest to highest by
regurgitation severity (Table 2A) and by material extensibility (Table 2B). In
both cases, rankings across the mechanical and functional metrics were mostly
maintained. A minor inconsistency in the ordering was observed, but the
differences in the numerical values in situations with inconsistent ordering
were negligible, possibly due to numerical errors, as shown in Appendix A.
Altogether, this suggested that the relative ordering of the mechanical and
functional metrics was maintained across diseased valves of similar
geometries, for the range of material properties considered in the present
work. That said, imprecise approximation of in vivo leaflet tissue
extensibility does not affect the comparative mechanical and functional
outcomes across MVs of similar geometries, so long as the same material
parameters were used across the population.

Table 2: Mechanical and functional metric rankings for tethered models. A) We
ranked the mechanical and functional metrics by regurgitation severity. (B) We
ranked the mechanical and functional metrics by material variant. The relative
ordering of the mechanical and functional metrics was mostly preserved in both
scenarios. M-Mitral; 1-Variant 1; 2-Variant 2; 3-Variant 3; 4-Variant 4;
5-Variant 5.
---

####  3.1.2 Disparate regurgitant MV geometries

To understand whether valve geometries or material properties dominate the
mechanical and functional metrics, we applied the five sets of material
parameters to one normal and three severely regurgitated MVs (tethered, P2
prolapse, and annular dilation) representing a diverse range of MV geometries.

The top and side views of average $1^{\text{st}}$ principal stress, average
$1^{\text{st}}$ principal strain, contact area, and ROA for each of the
abnormal valves are shown in Fig. 6. Note, the color red in the contact area
panels represent the contact area, and the color blue in the ROA panels
represents the ROA. Across the four MVs, we observed higher stress, strain,
and contact area, but lower ROA in material models with higher extensibility.
While the stress and strain patterns were consistent across different material
models within each type of regurgitant valve, the stress and strain profiles
were significantly different across the regurgitant valves. The stresses and
strains were highest in the A1 and A3 regions of the P2 prolapse and tethered
valves, and A1 region of the annular dilation valves. This inconsistency
suggested that valve geometries (i.e., the annular geometry, leaflet geometry,
and subvalvular apparatus) had a considerable effect on the stress and strain
profiles of the valves.

Figure 6: Mechanical and functional metric profiles for normal and severely
regurgitant MVs. The average $1^{\text{st}}$ principal stress, average
$1^{\text{st}}$ principal strain, contact area, and ROA of the normal and
three diseased MVs were shown. Variants 1, 3, and 5 represent tissue material
variants shown in Fig. 4, with variant 1 being the stiffest and 5 being the
softest among the numerically derived material variants. Across the four
valves, the $1^{\text{st}}$ principal stress, $1^{\text{st}}$ principal
strain, and contact area increased with softer tissue; the opposite for the
ROA. Regarding stress/strain patterns, the highest stress and strain were
found in the A1 and A3 regions of the P2 prolapse and tethered valves, and A1
region of the annular dilation valves.

In Table 3, we ranked the mechanical and functional metrics from lowest to
highest by mitral pathology (Table 3A) and by material models (Table 3B). When
ranking the metrics, we sorted the values from lowest to highest. In Table 3A,
we could not identify a clear pattern when ranking the contact area,
suggesting valve geometry had the greatest impact on the contact area among
the mechanical and functional metrics. However, we observed excellent ranking
consistency for the average 1${}^{\text{st}}$ principal stress, strain, and
ROA with extensibility from mitral to Model 3. In other words, the relative
ordering of the metrics of valve mechanics and ROA were maintained for
material models up to approximately 15% more stretchable than the mitral
material properties. This finding indicated that the comparative mechanical
and functional outcomes were still consistent in MVs of significantly
dissimilar geometries, so long the true in vivo tissue softness was within 15%
of the representative MVs material parameters.

In Table 3B, the relative ordering of the mechanical and functional metrics
was mostly preserved across the mitral pathology. This finding was consistent
with our intuition that softer tissue yields higher stresses, strains, and
contact area ratio but a lower ROA. The numerical values of the mechanical and
functional metrics were provided in Appendix B.

Table 3: Mechanical and functional metric rankings. (A) We ranked the
mechanical and functional metrics by regurgitant MVs; the rankings were most
consistent in average 1${}^{\text{st}}$ principal stress, strain, and ROA for
material variants stiffer than variant 4. (B) We ranked the mechanical and
functional metrics by material variant; the rankings were consistent across
the valves. N-normal; PP-P2 prolapse; T-tethered; AD-annular dilation.
M-Mitral; 1-Variant 1; 2-Variant 2; 3-Variant 3; 4-Variant 4; 5-Variant 5.
---

###  3.2 Uncertainty analyses for dysfunctional MVs

Fig. 7 shows the effects of material coefficients ($c_{0}$, $c_{1}$, $c_{2}$)
and leaflet thickness on the average $1^{\rm{st}}$ principal stress and strain
at valve closure. Leaflet thickness dominated the mechanical and functional
metrics across the four valves compared to material constants $c_{0}$,
$c_{1}$, and $c_{2}$. In particular, material constants had practically no
effects on leaflet stresses. The line plots that demonstrated changes in
mechanical and functional metrics concerning model parameters can be found in
Appendix C. We observed thicker leaflets had lower leaflet stresses, strains,
and contact area, but higher ROA.

Figure 7: Sensitivity results. We evaluated the sensitivity of the mechanical
and functional metrics to perturbation in model parameters for four valves:
normal, tethered, P2 prolapse, and annular dilation. The box plots represented
the spread of skewness in the mechanical and functional metrics for each
modeling parameter. These results indicated the mechanical and functional
metrics were most sensitive to leaflet thickness, consistent across the four
valves. Additionally, all three material constants had a negligible effect on
leaflet stresses.

Fig. 8 presents the sliced view of the dysfunctional valves to assess the
effects of model parameters on leaflet deformation at systole. Our results
indicated leaflet deformation was relatively insensitive to tissue coefficient
$c_{0}$. However, noticeable differences in leaflet deformation were observed
in situations with varying $c_{1}$, $c_{2}$, and leaflet thickness; leaflet
deformation reduced with higher $c_{1}$, $c_{2}$, and thicker leaflets.

Figure 8: Cross-sectional views of mitral models in systole. Cross-sectional
views of the FE models were shown. The leaflet deformation was relatively
insensitive to material coefficient $c_{0}$, moderately sensitive to material
coefficients $c_{1}$ and $c_{2}$, and most sensitive to leaflet thickness.

##  4 Discussion

###  4.1 Overall findings

In the present work, we sought to understand how uncertainty and variation in
heart valve leaflet mechanical properties influence FE-derived predictions of
valve function and mechanics. We found that the relative ordering of the
magnitude of functional and mechanical metrics across range of MV morphologies
was consistent when a tissue extensibility within 15% of the baseline model
was employed. This suggests that geometry is the dominant determinant of
metrics of valve function and mechanics within this range of variation in
material properties.

We explored how variations in material coefficients influenced FE metrics and
comparisons across functional and dysfunctional MV morphology. As expected we
noted that each individual MV geometry had similar stress and strain contours,
with differing magnitudes, as constitutive model parameters were varied.
Despite the clear influence of varied constitutive model parameters on the FE
predictions, the relative ordering of the functional metrics was maintained
across a range of leaflet material properties within 15% of the baseline
mitral model stretch ratio. Further, the relative ordering of functional
metrics for dysfunctional valves with increasing regurgitation severity was
preserved across the leaflet material properties defined in this work. These
results suggest that meaningful qualitative insights into the immediate (i.e.,
functional metrics) and long-term (i.e., mechanical metrics) valve function
can be obtained in the absence of precise knowledge of leaflet tissue
properties.

We also explored the effects of uncertainty in tissue thickness and
constitutive model parameters on the FE-derived valve metrics. Our results
showed that the mechanical metrics (i.e., tissue stress and strain) and the
functional metrics (i.e., valve contact area and ROA) were most sensitive to
the tissue thickness. Interestingly, we noted that all three constitutive
model parameters had a similar influence on the simulation results. These
trends were consistent for all four MV geometries, but the magnitude of
variation differed for each metric with tissue thickness having the most
profound impact on the predicted leaflet stress.

###  4.2 Comparisons with existing literature

Previous studies have used intricate animal models and experimental systems to
determine heart valve leaflet mechanical properties for developing high-
fidelity simulation tools. We noted that our predicted MV leaflet average
stresses for the normal model ($265.69$ kPa) were within a similar range as
the previous studies of Lee et al. [15] ($240$-$600$ kPa) and Rim et al. [50]
($150-300$ kPa). Furthermore, our predicted average principal strains
($0.220$) were within the range of previous experimental findings of Lee et
al. [51] and Amini et al. [52] for porcine valves ($10$-$35\%$), but were
larger than the strains determined by Rausch et al. [53] for sheep valves
($4$-$7\%$). Slight differences may be attributed to differing transvalvular
pressures, constitutive model forms, assumed boundary conditions (e.g.,
dynamic annulus), and the lack of pre-strains.

There has been an emerging interest in using cardiac imaging to determine the
functional mechanical properties of heart valves and better understand valve
disease. Our predicted strains were larger than a recent clinical study that
used cardiac imaging to determine functional MV leaflet strains of $7$-$9\%$
[54]. However, our numerical results echoed their observation that diseased
valves have larger strains than healthy valves. This suggests that underlying
valve morphology is the most important determinant of valve deformation, and
further illustrates that FE simulations may be used as an exploratory tool
without precise a priori knowledge of constitutive model parameters.
Ultimately, this expands on the versatility of image-derived investigations,
such as the recent work by Narang et al. [55] that found differences in
leaflet shear strains for adult MVs with recurrent ischemic mitral
regurgitation.

###  4.3 Clinical implications

In this work, we sought to explore extending the application of FE methods to
clinical applications (e.g., comparison of different repair strategies) where
exact or patient-specific tissue material properties are unknown. Our results
show that FE modeling predictions can be used to understand relative disease
severity and the effect of alterations to the leaflet mechanical environment
so long as material property parameters are maintained across the comparison
group and fall within the range of $\sim 15\%$ of the baseline model. For
example, comparing different valve repairs for atrioventricular valve disease
could significantly contribute to the optimization of valve repairs in small
and heterogeneous populations less likely to benefit from iterative clinical
refinement, such as children with congenital heart disease. Future inverse
methods leveraging patient-specific images may inform these developments, but
the present work has illustrated that FE analysis can be used to derive
meaningful comparisons in parallel with the maturation of our understanding of
material properties of pediatric valves.

An additional contribution of this work is a new method we have implemented in
SlicerHeart that couples a shrink-wrapping method [41, 42, 43] with raycasting
to automatically and reproducibly determine the ROA of heart valves. Previous
simulation studies relied on simplifying the closed valve geometry to two-
dimensional slices [21] or sampling different perspectives to find the largest
orifice area [28]. Although both methods may provide reasonable approximations
of the ROA, they are orientation dependent, may fail to accurately capture the
complex closed valve geometry (e.g., multiple jets), and consequently often
misrepresent the true ROA. Our methodology overcomes these limitations by
employing an algorithm that is invariant to the view angle and can consider
multiple regurgitant jets across the valve.

###  4.4 Study limitations and future directions

This study includes several convenient simplifications that do not detract
from the quality of the findings. First, we used an isotropic form of the Lee-
Sacks constitutive model with adult MV parameters that cannot capture the
typical anisotropic mechanical behaviors of heart valve leaflets. Fortunately,
Wu et al. [39] demonstrated that this anisotropy plays a minimal role in
predicting large-scale tissue deformations and subsequent geometry-based
functional metrics (e.g., contact area). We can explore the effect of
anisotropy and differences due to age [56] in future work. Second, we
simplified the boundary conditions in the FE simulations so that the annulus
and papillary muscle tips were pinned rather than dynamic, from ventricular
contraction and relaxation, as they would be in vivo. Considering these sub-
valvular components as dynamic rather than static may reduce the predicted
leaflet stresses [50], but this will not affect our comparisons as all FE
simulations shared this assumption. Third, it is known that the heart valve
leaflets have regionally heterogeneous properties (e.g., thickness [51]) that
were not considered in the present study, which may eventually confound
comparisons between healthy and diseased valve function. Finally, this is an
exploratory study seeking to expand the applicability of computational tools
to scenarios with no a priori knowledge of heart valve properties, but future
applications of this open-source computational framework require further
experimental validation.

##  5 Conclusion

This study has provided the first insight into how uncertainty and variation
in constitutive model parameters influence FE simulations of atrioventricular
valve function. FE-derived geometry and mechanical metrics were more sensitive
to uncertainty in the tissue thickness than the Lee-Sacks constitutive model
parameters. In addition, relative ordering of metrics of valve function and
mechanics remained consistent across valve models when varying constitutive
model parameters so long as the stretches remained within $15\%$ of the
reference MV leaflet stretches. As such, FE simulations of atrioventricular
valve function may be used to explore how differences and alterations in valve
structure affect heart valve function even in populations where material
properties are not precisely known.

##  6 Acknowledgment

This work was funded by the Cora Topolewski Cardiac Research Fund At the
Children’s Hospital of Philadelphia (CHOP), The Pediatric Valve Center
Frontier Program at CHOP, the Additional Ventures Single Ventricle Research
Fund, an Additional Ventures Single Ventricle Research Fund Expansion Award,
and the National Institutes of Health (NIH R01 HL153166, R01 GM083925, and U24
EB029007). In addition, both WW and DWL were supported by NHLBI T32 HL007915.

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##  Appendix A Appendix A: Additional results for regurgitant MVs with similar
geometries

Figure 9: Profiles of mechanical and functional metrics the for P2 prolapse
models. We observed an increase in $1^{\text{st}}$ principal stress and
strain, and contact area but a decrease in ROA as material variants became
softer. Additionally, models with higher regurgitation had higher stress,
strain, regurgitant area, and lower contact area. Figure 10: Mechanical and
functional metric profiles for annular dilation models. We observed an
increase in $1^{\text{st}}$ principal stress, $1^{\text{st}}$ principal
strain, and contact area but a decrease in ROA as the material variants became
softer. The models appeared to have higher stresses on the posterior leaflets
but lower on the anterior leaflets. The strains were similar across models
with varying degrees of regurgitation. Lastly, we observed higher regurgitant
and lower contact areas as regurgitation severity increased. Table 4:
Numerical values associated with mechanical and functional metrics. The values
of each regurgitant metric for the P2 prolapse, tethered, and annular dilation
with normal, moderate, and severe regurgitation were presented.
---
Table 5: Mechanical and functional metric rankings by regurgitation severity.
The relative ordering of each regurgitant metric by regurgitation severity was
mostly maintained across the P2 prolapse, tethered, and annular dilation
models. The differences between their values were rather insignificant in a
few instances where the ordering was flipped.
---
Table 6: Mechanical and functional metric rankings by material variants. The
relative ordering of each regurgitant metric by material variants was mostly
maintained across the P2 prolapse, tethered, and annular dilation models. The
differences between their values were rather insignificant in a few instances
where the ordering was flipped. M-Mitral; 1-Variant 1; 2-Variant 2; 3-Variant
3; 4-Variant 4; 5-Variant 5.)
---

##  Appendix B Appendix B: Additional results for regurgitant MVs with
disparate geometries

Table 7: Numerical values associated with mechanical and functional metrics.
The values of each mechanical and functional metric for the diseased MVs were
presented. The relative orderings of the mechanical and functional metrics by
MVs were consistent in average 1${}^{\text{st}}$ principal stress, strain, and
ROA for material variants stiffer than variant 4. The relative ordering of the
mechanical and functional metrics by material variants was consistent across
the valves.
---

##  Appendix C Appendix C: Additional sensitivity analysis results

Figure 11: Sensitivity results. The material constants appeared to have
negligible effects on the average principal stress. Meanwhile, higher leaflet
thickness yielded lower leaflet stress. Further, higher leaflet thickness,
$c_{0}$, $c_{1}$, and $c_{2}$ led to lower leaflet strain and contact area but
higher ROA.