# Image-based simulation of mitral valve dynamic closure including anisotropy

Nariman Khaledian, Pierre-Frédéric Villard, Peter Hammer, Douglas Perrin,

Marie-Odile Berger

► To cite this version:

Nariman Khaledian, Pierre-Frédéric Villard, Peter Hammer, Douglas Perrin, Marie-Odile Berger. Image-based simulation of mitral valve dynamic closure including anisotropy. Medical Image Analysis, 2025, 99, pp.103323. 10.1016/j.media.2024.103323. hal-04774575

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Submitted on 13 Nov 2024

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# Image-based Simulation of Mitral Valve Dynamic Closure including Anisotropy

Nariman Khaledian¹, Pierre-Frédéric Villard¹, Peter E. Hammer²,³, Douglas P. Perrin²,³, and Marie-Odile Berger¹

¹Université de Lorraine, CNRS, Inria, LORIA, Nancy, France

²Department of Cardiac Surgery, Boston Children's Hospital, Boston, MA, USA

³Harvard Medical School, Boston, MA, USA

## Abstract

Simulation of the dynamic behavior of mitral valve closure could improve clinical treatment by predicting surgical procedures outcome. We propose here a method to achieve this goal by using the immersed boundary method. In order to go toward patient-based simulation, we tailor our method to be adapted to a valve extracted from medical image data. It includes investigating segmentation process, smoothness of geometry, case setup and the shape of the left ventricle. We also study the influence of leaflet tissue anisotropy on the quality of the valve closure by comparing with an isotropic model. As part of the anisotropy analysis, we study the influence of the principal material direction by comparing methods to obtain them without dissection.

Results show that our method can be scaled to various image-based data. We evaluate the mitral valve closure quality based on measuring bulging area, contact map, and flow rate. The results show also that the anisotropic material model more precisely represents the physiological characteristics of the valve tissue. Furthermore, results indicate that the orientation of the principal material direction plays a role in the effectiveness of the valve seal.

## 1 Introduction

The mitral valve (MV) provides a one-way flow of oxygenated blood from the left atrium to the left ventricle. The mitral valve has two leaflets (anterior and posterior) attached to the annulus. They have a complex microstructure whose collagen fibers are the main load carrier. A network of chordae tendineae is fixed to the leaflets and attaches distally to the papillary muscles. It contributes to close the valve during systole, preventing blood from leaking into the left atrium and ensuring that all blood leaving the left ventricle is ejected via the aortic valve.

Mitral valve insufficiency is a common cause of heart failure, with mitral regurgitation being the most common type of heart valve disease. Structural defects of the MV, such as leaflets that are too long or too stiff, and chordae that are elongated, are implicated in the development of many MV diseases. Surgery may be needed to repair the valve and restore proper closure. However, there is no consensus on the best procedure for a given patient, and the outcome of surgery is highly dependent on the experience of the surgeon [AvKS21]. For these reasons, the development of patient-based computational simulations of MV behaviour has the potential to improve the clinician's understanding of a given MV and to help the surgeon choose the appropriate surgical procedure [Cha10, KGS$^{+}$14, SSD$^{+}$19]. However, to date,
no clinical imaging modality is capable of visualiz-
ing the complete MV structures. Therefore, we rely
in this study on microCT data of explanted porcine
hearts which allow visualisation of both the leaflets
and the chordae.

Though significant progress has been realized in
the field of computational simulations of the heart,
obtaining realistic patient-based simulations of the
MV is limited by the difficulty of (i) obtaining an
accurate geometry of leaflets and chordae, (ii) mod-
eling the material model and leaflet microstructure
realistically, and (iii) obtaining appropriate bound-
ary conditions.

Due to these difficulties, most work to date has not
aimed at processing any subject-specific data. With
regards to the valve geometry, many studies have fo-
cused on generic valves or simplified and smooth ge-
ometries, which greatly simplify the true anatomical
complexity. While these simulations allow us to un-
derstand the function of the valve, they are of little
use as a diagnosis or surgical aid. Some studies used
real valves but made use of devices that facilitated
image acquisition and segmentation. For example, in
[RFS+13], sensors were attached to the valve in or-
der to get an approximation of the geometry, while
[TBE+16] used a set up that facilitated the acquisi-
tion and the segmentation of the valve. Other studies
like [FGG+19, FGQ+21] used a single image-based
patient data in their studies and inferred the geome-
try by careful manual delineation of the smooth im-
age data. Finally, the set of chordae is usually not
derived from real data but is often either set arbi-
trarily or imported from another dataset. Such ap-
proaches are not compatible with the idea of per-
forming patient-based simulation from clinical data,
where it is necessary to work with imperfect seg-
mentation of the valve (non-smooth shape, irregular
mesh) or of the leaflets, potentially leading to diver-
gence of the simulation during high MV deformation.
It must be noted that simplification of the geometry
is a concern not only for the valve but also for the left
ventricle in which the MV is immersed, and which is
used to define the boundary conditions. It is often
represented by a cylinder fitted on the annulus, but
we show in the following that this simplified model is
too restrictive for real data.

In contrast to these methods, we propose a simula-
tion framework capable of producing realistic blood
flow simulations and of dealing with any complex
image-based MV geometry, the complexity arising
both from the intrinsic complexity of the shape of
the valve and from the irregularities in the mesh due
to the segmentation process (either manual or auto-
matic).

Similarly to most work of the state of the art, only
the quality of the valve closure in simplified condi-
tions is analyzed, even if the shape of the left ven-
tricle is deforming during the cardiac cycle and the
papillary muscles are pulling down the chordae.

Besides geometry, the choice of the appropriate
constitutive model and of its material parameters is
of paramount importance for obtaining simulations
that fairly represent the actual mechanical response
of leaflet tissues. Despite the complexity of leaflet mi-
crostructure and although several works have demon-
strated the high anisotropy of the leaflet, the sim-
plified isotropic model is frequently used in the lit-
erature [DPDFS05, MGG<sup>+</sup>13, LAG<sup>+</sup>14, AAAB<sup>+</sup>22,
PG20, ETZA<sup>+</sup>21, KFN17, KVB22]. However, there
is a trend in favour of anisotropic models which bet-
ter suit the complexity of the leaflet structure. As
the anisotropic behavior of the MV is generally at-
tributed to the presence of collagen fibers within the
tissues, there seems to be a consensus towards the
use of dispersion-type anisotropic constitutive mod-
els such as the Holzapfel-Gasser-Ogden (HGO) model
[GOH06] which takes into account not only the mean
orientation of the collagen fibers but also the amount
by which they are dispersed around this mean. The
choice of an anisotropic model along with its param-
eters is generally based on its ability to fit a set of
biaxial tension experiments conducted on specimens
of tissues [DADH23]. However due to regional varia-
tions of the anisotropy rates — the ratio of the stiff-
ness between two different directions - on the leaflets
[LSP<sup>+</sup>23, LRJ<sup>+</sup>19], convergence of these models is
not guaranteed when complex MV are considered.
This motivated us to compare the dynamic closure of
MV with isotropic or anisotropic constitutive models.
In our study, HGO is used as the anisotropic model
where we consider the local mean orientation to be
parallel to the annulus boundary, which is a common
choice for modelling fibers. However various methods
have been developed to determine the fiber orienta-
tion without necessitating dissection of specimen of
the MV. We thus investigate in this article their im-
pact on the MV simulation.

To summarize, we propose a fluid-structure sim-
ulation framework based on the immersed bound-
ary methods. We describe the various adaptations
we had to make to take into account complex valve
shapes. We show that this framework can perform ac-
curate simulations of MV closure for several subject-
specific valves and also produce an accurate map of
contact, which allows the assessment of the quality
of valve closure. To the best of our knowledge, no
previous works have addressed such a variety of real
valves for FSI simulation. Simulations have been re-
alized both for isotropic and anisotropic models, in
order to compare the impact of these models on com-
plex shapes. We also compare different fiber-based
anisotropic models which do not necessitate dissec-
tion.

Results show that the use of anisotropy confers a
greater realism to the simulation than the isotropic
model whatever the fiber direction model used and
the metrics employed. This study shows, however,
that most of the metrics are global in the sense that
they qualify the required shape or behavior of the
valve, but they are not sufficiently specific to vali-
date one fiber model over another. We discuss in
conclusion possible ways to acquire direct measure-
ments such as the height of the coaption surface or
measurements of the pressure at some points. These
quantities, which are estimated by our method via
the contact map, will thus be of invaluable help to
validate complex MV models The paper is organized
as follows: we first explore the state-of-the-art solu-
tions existing for addressing the challenges mentioned
above in section 2. Following that, in section 3 we de-
scribe the material models and their implementation,
then, we provide details concerning the geometry and
case setup. Results are presented and discussed in
section 4.

2 Related works

## 2.1 Material model

In the literature, various constitutive models have
been used to model the anisotropic behavior of the
MV leaflets. Most commonly used anisotropic con-
stitutive models for biological tissues are often de-
scribed by a strain energy function with exponen-
tial form [WMG96, AMDM09, ASMD10, DBB04,
GOH06, MNY98, FFP79]. The hyperelastic behav-
ior of the biological tissue is captured in the strain
energy function by the exponential term. The con-
stitutive models can be categorized based on the vari-
ables used in their strain energy function, or based on
the terms used in the strain energy function (more
known as coupled and decoupled). In the following,
we will first discuss the categorization based on the
variables used, followed by the categorization based
on the coupled and decoupled terms.

The strain energy function in the anisotropic con-
stitutive models can be described based on strain,
anisotropic invariant, and fiber dispersion. The first
constitutive model based on strain is presented by
[FFP79]. In this model, the strain energy function
is an exponential term with a direct relation to the
strain in different directions. In anisotropic constitu-
tive models based on anisotropic invariants, for ex-
ample in [HGO00, MNY98, HSGR05], the strain en-
ergy function is defined based on invariants of the
strain tensor. Invariants are scalar quantities derived
from the strain tensor that remain unchanged un-
der a coordinate transformation. They provide in-
sight into the material’s deformation independent of
the coordinate system being used. In this approach,
the microstructure of the tissue is taken into ac-
count, but the fibers are considered to be parallel
to each other. In the models based on fiber disper-
sion [Lan79, Lan83, GOH06] a dispersion factor is
specified which describes how fibers are scattered in
the tissue compared to the main direction of the fiber
family. These factors also influence the practicality
of the model. An extensive study on the fitting per-
formance of twelve different anisotropic constitutive
models was described in [DADH23]. The study con-
cluded that the anisotropic models based on the dis-
persion of the fibers show better fitting performance compared to the other categories mentioned.

Another approach to categorize the anisotropic constitutive models is as coupled [MNY98, FFP79] and decoupled [GOH06]. [MNY98] proposed an anisotropic constitutive model specific for MV leaflets in which the strain energy function consists of a single term describing the material behavior in all directions. In decoupled models, such as the HGO constitutive model [GOH06], the strain energy function consists of one isotropic and one anisotropic term which in combination describes the desired anisotropic behavior. In [RFS¹¹³] performance of both coupled and decoupled anisotropic models was compared to in vivo measurement on ovine data with shell structure simulations. The study indicated that the average nodal displacement error relative to in vivo measurements is lower when utilizing the decoupled HGO model. Therefore, it concluded that the decoupled HGO model serves as a more effective constitutive model for simulating mitral valve behavior.

As we discussed above, several constitutive models have been used for incorporating anisotropic and isotropic materials. However, the HGO, general Fung, and Ogden constitutive models are frequently utilized in the literature. Therefore, for the isotropic model, we used the third-order Ogden hyperelastic constitutive model. Moreover, to incorporate anisotropy, we first studied the process of obtaining the material parameters of the decoupled HGO model and the coupled generalized Fung [FFP79] anisotropic hyperelastic material model. Then we investigated the influence of anisotropy on MV behavior by comparing the results of the isotropic model with the anisotropic one. In section 3, we explore the details of these models.

In some studies, for example in [LRJ¹¹⁹], experimental measurements show that the anisotropy rate is not uniform, meaning that the anisotropy rates depend on the region where the specimen is extracted for the stress-strain test. In this study, they took samples from six different regions of the MV anterior leaflet tissue and measured their stress-stretch response. The results show a noticeable difference in the anisotropy rate in different regions of the leaflet (see Fig.1). This can be explained by considering

leaflet tissue as a composite of fiber families running through an isotropic base material and assuming that fiber orientation dictates the anisotropy rate. The assumption that the fiber orientation dictates the anisotropy rate is mentioned in [PS09]. This idea is in line with the physiological characteristics of the MV leaflet microstructure. The MV leaflet comprises four primary layers, namely atrialis, spongiosa, fibrosa, and ventricularis [MRR10], with the fibrosa layer comprising crimped collagen fibers that function as a load-bearing mechanism and strengthen the tissue in its direction. Therefore, in MV simulation, it is important to have a fiber map that defines the fiber direction in each region of the leaflet.

<figure><img src="image_2.png" /><figcaption>Figure 1: The anisotropy rate changes depending on the region where the specimen is obtained. Data reported in [LRJ¹¹⁹].</figcaption></figure>

In many of the studies in the literature, fibers are modeled by implementing the fiber map extracted from observing the micro-structure of the leaflets as described in [SSHP22, LAG¹¹⁴, CGD¹¹²]. Usually, the fiber map is extracted by using small-angle light scattering (SALS) to observe the microstructure of the leaflet [SHB⁰⁰²]. The fiber map is unique for each specimen and obtaining it is time-consuming work and needs resources and expertise in conducting experimental studies.

To have a practical model, that can be used for any patient-specific case, simplification in modeling different aspects of the MV is inevitable. In the majority of the studies, they simply directed the fibers to be parallel to the plane surface crossing the annulus ring as in [KMP19, SVR09, VCV⁰⁰⁸].

Numerous studies have sought to develop a fiber model by observing the MV leaflet microstructure. In [CKC⁰⁹¹], they extracted the fiber orientation in
the MV anterior leaflet tissue by using polarized light microscopy (PLD). By observing the fiber map they proposed (see Fig.2), we realize that in the middle of the leaflets, the fibers seem to be parallel to the annulus direction, and close to the commissures, which are the regions between the anterior and posterior leaflets, being perpendicular to the free edge, the edge of the leaflet which is not attached to the heart wall. Another example is the experimental study on the in vivo data presented in [SHB+02], reporting that the fibers in the central region of the leaflet are oriented in the circumferential direction of the leaflet. An interesting observation is that the anisotropy rate presented in [LRJ+19] (see Fig.1) appears to be in agreement with the fiber map proposed in [CKC+91] (see Fig.2). For instance, in Fig.1, sections B and E exhibit a higher anisotropy rate than regions A, D, C, and F. Furthermore, by considering these sections in the fiber map shown in Fig.2, it becomes apparent that in areas of high anisotropy, fibers align more parallel to the circumferential direction, while in other areas, they are more obliquely oriented. This idea is independently implemented in the literature. For example, in [NKI+13], they used a fiber model that in the central region of the leaflets are oriented in the circumferential direction, but they become radial, perpendicular to the annulus ring, near the commissures.

<figure><img src="image_3.png" /><figcaption>Figure 2: Map of fibers for MV anterior leaflet obtained based on PLD data [CKC+91].</figcaption></figure>

A prominent effort in obtaining a generalized fiber map was undertaken [EKR+05]. They concluded that the anisotropy rate used to describe the mechanical response of the MV leaflet influences the MV function. They used the SALS data reported

in [CKC+91] to determine the average directions of collagen fibers. They created a standard template for fiber maps which has been widely used in the literature [RFS+13, EKR+04, PHS09, SPN11, PSSH10, EDPJ+10].

One major limitation of the models we specified so far is their lack of consideration for the mechanical response of the MV leaflet in the peak systole, when the leaflets are under the maximum tension. This oversight becomes clear when considering that the fibers are load-bearing structures. Their primary role is to bear the additional load of the leaflet. Ideally, their direction should be oriented toward where they are mostly needed, to reduce excess strain. At the cellular level, it has been studied that the heart valve cell stiffness increases in the direction of applied stress [MYL+06]. Studies such as [DCBB08, Lau12] have explored the remodelling of heart valve collagen fibers at the macroscopic level. These studies have found that the stress-based approach, which involves re-orienting collagen fibers based on the maximum principal stress at each point on the leaflet, leads to deformed shapes that are more consistent with experimental observations. In addition, an experimental study in [LZL+15] explored the affine kinematic notion, which proposes that crimped collagen fibers in heart valve tissue deform and rotate to align with the macroscopic deformation of the tissue. This study confirmed that, at the homogenized tissue level, fiber orientation does indeed deform in accordance with the affine kinematic model.

In this study, we compared the MV behavior based on these fiber models and took a step further to study other rule-based fiber models that take into account the physiological traits of the MV leaflet. These fiber models are described in the section 3.

## 2.2 Geometry and case setup

One of the main reasons why many studies use generic or simplified models is that these representations facilitate applicability in various scenarios, reduce the complexities of the simulation and facilitates the convergence of the numerical schemes. Generic MV geometry is suitable for scientific understanding, helping with the identification of fundamental
MV behavior traits [AAAB$^{+22}$, AEHE19, ESH14]. Generic or simplified models MV can also be used to discern the impact of various factors such as material properties [KMP19] or numerical method [LDSB10] without the complexity of the geometry interfering in the analysis.

With the progress made in simulation, it has become important to move towards patient-based geometries with the goal to replicate specific valve dynamics. However, since it is impossible to acquire leaflets and chordae with clinical modalities, microCT images are used to image the leaflet and chordae from ovine or porcine ex-vivo hearts. In addition it is difficult to segment these thin organs, either manually or automatically, and the segmented organ is often prone to irregularities.

To alleviate this problem, an ex-vivo setup is used in [TBE$^{+16}$] to obtain clean and high-resolution subject-specific MV imaging which was used subsequently in many studies of his team [TBP$^{+17}$, SSD$^{+19}$, TEBI$^{+20}$]. More commonly, studies are carried out on a single dataset on which a smooth segmentation of the leaflets has been carried out very carefully [GFQ$^{+17}$, CMMS20, MCM$^{+17}$, BGG$^{+19}$, AEHE19, FGG$^{+19}$, FGQ$^{+21}$].

Obtaining the tree shapes of the chordae is even more complex and many works have used generic evenly distributed chordae tendineae [GFQ$^{+17}$] or adapted existing data to a specific patient [BGG$^{+19}$]. An exception is [PVH$^{+21}$] who proposed an automatic method from chordae segmentation. Most of the time, the annulus is approximated with a circular or elliptic shape or cubic Hermitian splines [RFS$^{+13}$] though its real shape is much more complex [PVH$^{+21}$, TEBI$^{+20}$]. Only a small number of papers have addressed MV simulation with leaflet and chordae geometry based on medical imaging as [FQG$^{+18}$] who investigated how chordae structure impacts the MV dynamics and [RT23] in the context of prolapsus study. In our study, the geometry of both leaflet and chordae structures are obtained from microCT datasets.

In the FSI simulation setup, the MV mesh is immersed inside an Eulerian domain representing the LV. For an accurate simulation of the MV dynamic closure, it is essential to model the fluid effects on

the leaflets [LDSB10] and the surrounding anatomic structures: annulus, chordae, papillary muscle, and ventricle. In the literature on MV FSI simulation, the representation of the LV shape can be classified into three distinct categories (see Fig.3). In the first category, the MV is mounted inside the LV geometry extracted from medical imaging (Fig.3.a). Typically, for the closure of the valve during the cardiac cycle, no pressure inlet is defined in this setup, and the pressure is built up by squeezing the ventricle geometry which leads to the valve closure as explained in [CMMS20, GFQ$^{+17}$, BGG$^{+19}$]. In the second approach (Fig.3.b), a U-shaped geometry, which is a simplified version of the LV is utilized, as demonstrated in studies such as [AEHE19, LDZSB11]. During the closure of the valve, the fluid enters from the left side of the domain and passes through the MV. In the final approach, the MV is placed inside a tubular geometry [TEBI$^{+20}$, OÖB$^{+21}$, TBP$^{+17}$]. Depending on the placement of the annulus ring, a tube can be fitted on the annulus ring as in [DYS$^{+19}$] (Fig.3.d), or housing is used to seal the area between the annulus ring and the cylinder walls as in [TEBI$^{+20}$, OÖB$^{+21}$, TBP$^{+17}$] (Fig.3.c).

<figure><img src="image_4.png" /><figcaption>Figure 3: Simplified ventricle models: LV geometry extracted from medical images (a), simplified ventricle shape, or U-shape (b, and two different representations of the tubular volume: large tube with housing (c) and fitted tube (d). During the closure, fluid inlets are in green, and fluid outlets are in red.</figcaption></figure>

MV behavior with image-based LV has already been investigated in studies such as [GFQ$^{+17}$, CMMS20, MCM$^{+17}$, BGG$^{+19}$, AEHE19]. However,
the practicality of implementing such geometries for patient-specific models needs further investigation. In one study in particular, [LDSB10], they conducted a comparison between a U-shape and a tubular ventricle. The results showed that although the stress pattern on the leaflet was similar in both cases, the magnitude and closed state differed. Note however that the FSI approach used in this study was based on the ALE method, which was not able to investigate the quality of the closure and the coaptation area due to the limitation of the method in capturing the contact between the leaflets.

The implementation of image-based LV geometry requires segmentation of the LV motion in temporal data in order to define the LV displacement as a boundary condition. Most of the time, such data is not available, or not easy to obtain accurately. In addition, the added value of results based on such data has not been demonstrated. On the other hand, with a simplified LV geometry, validation experiments can be designed more easily in a controlled lab setting. Therefore, a simpler tubular representation of LV is preferred in the literature for investigating the MV behavior as in [DYS<sup>+</sup>19, TEBI<sup>+</sup>20, OÖB<sup>+</sup>21].

In the case of the tubular volume, a first advantage of using a fitted tube compared to the larger cylinder is the computation time. The size of the fluid domain in FSI simulation affects computation time, and using a fitted tube can have a noticeable impact on this aspect. In addition, using a large cylinder with housing compared to a fitted tube might lead to a more accurate representation of the force distribution on the leaflet, taking into account the impact of other organs and the blood motion pushing the MV inward during the closure. In addition, image-based geometries may be irregular, with an annulus shape neither elliptic nor planar. The use of a fitted tube is often impossible, since the MV in the open state may lie outside the cylinder defined by the annulus. The large cylinder can be seen as an intermediate solution between a fitted tube, which is not realistic, and a complete model of the ventricle which is difficult to obtain and implement, and sometimes not practical. Therefore, in our simulations, we considered both the fitted tube and the large cylinder as a representation of LV to investigate the influence of

the material characteristics on MV behavior.

# 3 Methods

## 3.1 Patient-based framework

In our study, we implemented one generic valve and four image-based geometries, to manifest that our model is not tailored only for one set of data, contrary to the work of several groups [TBP<sup>+</sup>17, GFQ<sup>+</sup>17]. The image-based geometry is extracted from a 3D computed tomography (microCT) scan of an explanted porcine heart. During the imaging, the valve was pressurized by inserting an air pump through the aorta with regulated pressure similar to the physiological condition of the ventricle. All the porcine valves used for the imaging closed naturally ex-vivo without orifice holes. A summary of the data we used for our simulation is shown in table 1. The leaflets of P-1, P-2, P-3 and P-4 were manually segmented on medical images as well as the chordae for P-1, P-2, P-3. The chordae of P-4 were automatically segmented using the method introduced on [PVH<sup>+</sup>21]. In this work, the segmented chordae were compared to handmade architecture delineation and showed an 87.% accuracy rate. In addition, in the context of structural simulation, it was shown that the simulation of the MV with the segmented architecture was similar to results obtained with manually delineated architecture. In the case of anisotropy, we illustrate the results with the P-1 valve. Results on other data-based valves follow the same trends.

### 3.1.1 Case setup

The biomechanical model of the mitral valve consisted of the valve immersed in a blood domain modeled by a tube. The ends of the cords were connected to fixed nodes located on the papillary muscles and leaflets. The blood velocity was defined as zero with respect to the surfaces of the leaflets and the tube wall, which is the common way of modeling the fluid-structure interface. Blood was simulated as an incompressible, isothermal, Newtonian fluid, with dy-
<table><caption>Table 1: Summary of the data we used for our study of MV simulation.</caption><thead><tr><th></th><th>Generic valve</th><th>P-1</th><th>P-2</th><th>P-3</th><th>P-4</th></tr></thead><tbody><tr><td>Origin:</td><td>geometry-based</td><td>image-based</td><td>image-based</td><td>image-based</td><td>image-based</td></tr><tr><td>Acquisition state:</td><td></td><td>open state</td><td>open state</td><td>open state</td><td>close state</td></tr><tr><td>Chordae extraction:</td><td></td><td>manual</td><td>manual</td><td>manual</td><td>automatic</td></tr><tr><td>Mesh size:</td><td>20k</td><td>30k</td><td>30k</td><td>26k</td><td>38k</td></tr><tr><td>Image:</td><td colspan="5">[Image of a fitted tube with a generic valve]</td></tr></tbody></table>

namic viscosity and density values taken from the literature. This assumption has been made following recent studies showing that blood viscosity is constant in heart and large vessels. It should be considered as non-Newtonian only if the cross-section of the flow is very small [MdTQV18, FKF18]. Time-dependent transvalvular pressure based on real measurements extracted from [Lev13] was applied at the bottom of the domain to simulate the pressure difference between the left ventricle and the left atrium. The simulation began with the systolic peak and continued through to the diastolic peak of the cardiac cycle. An adaptive time step dependent on the residuals was used. Chords were modeled as linear elastic beams with a behavior law based on experimental data [MES+11]. It is difficult to resolve the fine structure of chordae near the leaflet attachments. To model each of these regions of chordae attachment, a progressively decreasing distribution of forces was applied to a group of leaflet nodes, thus avoiding the concentration of forces on a single node and potentially preventing distortion of the element. More information about the setup can be found in [KVB22] where an isotropic model was used on the generic valve.

Fig.4 shows the FSI case setup with a fitted tube in the case of both image-based and generic valves. The tube in the generic valve is relatively simple with an elliptic base, but the image-based valve has a more

<figure><img src="image_5.png" /><figcaption>Figure 4: Case set-up showing the image-based (left) and generic (right) MV mounted inside a fitted tube.</figcaption></figure>

complex annulus shape. The tube base is the shape of the annulus ring to prevent blood from passing over the annulus. The tube direction is normal to the annulus plane and is determined thanks to a principal component analysis on the vertices of the annulus ring.

### 3.1.2 Adaptation to patient-based data

As mentioned in [KVB22], the generic MV geometry was discretized and represented by elements using quadratic interpolation. It is better for modeling structures with curved surfaces such as the leaflets and more accurate for integrating the strain energy over the element. However, adapting the method to patient-based data require changing the element types. The quadratic elements introduce some diffi-
culties related to convergence in problems involved with complex contact and deformation. In addition, this type of element needs more nodes to define which requires more computational resources to solve. We then used elements using linear interpolation for data-based valves.

The LV representation with a fitted tube is the simplest representation and works well for a generic valve with planar annulus. However, a difficulty is that image-based acquired leaflets may be partly outside due to the tube axis being defined as orthogonal to the annulus plane. The other issue is that the fitted tube imperfectly reflects the fact that the valve is immersed in a large volume of blood in the ventricle. We thus also considered implementing a large cylinder as the inner boundary of the left ventricle in our case setup. A large cylinder needs an additional structure to obstruct the flow between the annulus ring and the cylinder walls and define it as an elastic material with a high Young's modulus. The expected outcome from using a large cylinder and a seal would be a more accurate representation of the force exerted on the MV leaflets during closure by other organs, as it captures the effect of flow pushing the leaflets from the sides. Fig.5 displays the cross-section view of the implementation of P-1 image-based valve in both a fitted tube and a larger cylinder.

<figure><img src="image_6.png" /><figcaption>Figure 5: The image-based MV geometry inside the large cylinder (a.), and inside the fitted tube (b.). Orange: valve, red: chordae, blue: additional geometry to obstruct the flow, and light green: pinned annulus region.</figcaption></figure>

The image-based geometries are extracted from the segmentation of microCT images, and it leads to various challenges compared to generic geometries. The leaflets have non-regular contours with

some noise. Many folds can occur during the closing process. In this context, we investigated the influence of the smoothness rate of the image-based MV surface extracted from the segmentation process. We used Laplacian filtering to reduce the roughness of the image-based leaflet surface while preserving the whole geometry [Sor05]. Fig.6 shows the image-based MV geometry before and after applying the filter. The average absolute distance of each node in the smoothed valve, to the original geometry is 0.145mm. In both cases, 1mm uniform thickness is specified for the MV leaflet. In section 4, we compared the valve closure in both the original image-based geometry and the smoothed case to investigate the impact of smoothness on MV closure.

There are other case setup factors that could affect the simulation outcome, however, our focus was solely on the aforementioned aspects.

<figure><img src="image_7.png" /><figcaption>Figure 6: Image-based MV before (a.) and after (b.) applying the smoothing filter.</figcaption></figure>

### 3.1.3 Technical framework

There are several open-source platforms and commercial solvers that implement immersed boundary methods for solving fluid problems. We have chosen Abaqus because it contains all the features necessary to model the mitral valve closure such as hyperelastic material, large deformation, complex geometry from medical imaging, interaction between the fluid and structure, and contact between the structure walls. Another advantage is that the solver is implicit, which is more suited for large deformations and time step handling.
One of the key components that help closure in real valves is using an anisotropic model. We will see how it has been implemented within the Abaqus framework in the next section.

## 3.2 Constitutive law

### 3.2.1 Anisotropy model

As previously stated, modeling anisotropy for the leaflet is important. It is possible to measure an anisotropic behavior from biaxial testing on valve samples. This only allows adjusting the 2D parameters, thin tissue does not allow out-of-plane measurement of the membrane deformations, so shear data in the direction perpendicular to the surface of the sample is not available. The HGO model-based approach dispenses with these data. It is not the case from other models like the Fung model [FFP79] for instance. The HGO constitutive model was originally used to describe the mechanical behavior of arterial walls [GOH06]. The HGO strain energy function (ψ) is:

$$ \psi = C_{10}(I_1 - 3) + \frac{K_1}{2K_2} \sum_{\alpha=1}^{N} [\exp(K_2 <E_\alpha>^2) - 1] + \frac{1}{K} \left[ \frac{J^2 - 1}{2} - \ln J \right]. \quad (1) $$

The strain energy function in the HGO model is "decoupled", consisting of two separate terms defining the mechanical response of the tissue, and one term addressing the compressibility. In this model, the anisotropic behavior is introduced by families of fibers running through a uniform medium. The uniform medium is considered isotropic and the first term of the equation 1, is addressing the isotropic part. The second term introduces the anisotropy to the model by defining the specifics of the fibers embedded in the uniform medium. K is the bulk modulus, a measure of a substance's resistance to uniform compression, and is described as:

$$ K = V \frac{dP}{dV}, \quad (2) $$

V is the initial volume of the material, $dP$ indicates the change in pressure, and $dV$ denotes the change in volume. Materials with a high bulk modulus are relatively incompressible, meaning that they don't change volume much under pressure. For example, water has a relatively high bulk modulus, which makes it almost fully incompressible. On the other hand, gases typically have a much lower bulk modulus, making them much more compressible. J is the total volume ratio, calculated based on how much the volume of the material is changing from the initial to the final state. In the case of incompressible material, J is equal to one. It is worth noting that as most of the MV tissue is made of water, the MV leaflet is considered incompressible. Therefore, for MV tissue, the third term in equation 1 will be zero.

The isotropic part is captured using the linear neo-Hookean model. The $C_{10}$ coefficient in the linear part of the equation 1 dictates the characteristics of the uniform medium. N denotes the number of fiber families. The distribution of fibers in the MV leaflet is complex, some studies [WS13] have considered two families of fibers for the MV leaflet. In our case, we used a simpler representation of the constitutive model by incorporating only one family of fibers, N = 1, to dictate the direction of the anisotropy in the leaflet tissue. Both $K_1$ and $K_2$ are specifying the characteristics of the fibers. In the stress-strain curve, $K_1$, also known as fiber modulus, dictates the slope of the curve at the beginning of the plot, and $K_2$ (also known as the fiber exponential coefficient) dictates the nonlinearity of the relationships [HOB20]. The effective stretch of the fibers $<E_\alpha>$ is active only when the fibers are stretched. This behavior is enforced by:

$$ <E_\alpha> = \frac{|E_\alpha| + E_\alpha}{2}. \quad (3) $$

According to this equation, during compression, the term $<E_\alpha>$ will be equal to zero and the fibers will not contribute to the mechanical response of the tissue. In the case of fiber stretching, this term is defined as:

$$ E_\alpha = \kappa(I_1 - 3) + (1 - 3\kappa)(I_{4(\alpha\alpha)} - 1), \quad (4) $$
$I_1$ is the first deviatoric strain invariant, and $I_4(\alpha\alpha)$ is the pseudo-invariant related to the fibers. $\kappa$ is the dispersion factor that determines how dispersed the fiber family is. The value of $\kappa$ can be between zero, for parallel fiber families, to 1/3, for randomly distributed fibers. No shear measurement are directly needed in the HGO model [GOH06], the stress-strain response in the principal directions alone suffices to determine the material parameters.

### 3.2.2 Material parameters

To obtain the HGO parameters, we decided to employ an approach already used in the literature which is based on inverse finite element analysis [ABVA16]. In this approach, instead of minimizing the prediction error from the strain energy function, the error between the results from a finite element analysis and a set of experimental data is conducted. The finite element analysis is usually a simulation of a tension test on a simple rectangular cube. The material model is already implemented in the inverse finite element analysis and a fitness function is described to minimize the prediction error. The optimization process is done with the MCalibration software<fnref n="1" /> which is connected to the Abaqus solver to execute the inverse finite element analysis calculations.

The solver uses a normalized mean absolute difference (NMAD) fitness function on PK1 stress S as:

$$NMAD = 100 \times \frac{|S^{exp} - S^{p}|}{max(S^{exp}, S^{p})} \quad (5)$$

The stress values, derived from both the experimental and simulated data, are represented by the expressions $S^{exp}$ and $S^{p}$, respectively. This fitness function has been optimized with a genetic algorithm for minimization. It considers more than a hundred plausible initial values, and keeps the one that provides the smallest residual. Visual comparison between the stress-strain response from the finite element simulation and the experimental data simplifies the task of setting initial values for the optimization. The parameters computed with this method are reported in table.2. In Fig.7, the dashed lines depict

the mechanical response of the HGO model, as simulated in Abaqus using the specified parameters. The solid lines show the input measurement data reported in [JLK+18]. By observing Fig.7, The computed mechanical response fits for both directions in most of the stress range, except for stress values exceeding 200kPa in the second direction (represented by the red dashed line). We deemed that acceptable for this study based on the observation that the majority of the stress distribution on the leaflet falls below the 200kPa threshold.

<figure><img src="image_8.png" /><figcaption>Figure 7: The result of the inverse finite element analysis procedure: Showing the mechanical response of the HGO model with the optimized parameters (dashed lines) compared to the input data (solid lines) [JLK+18].</figcaption></figure>

<table><caption>Table 2: Material parameters for the HGO model, derived from experimental data reported in [JLK+18].</caption><tr><td>Mat. param.:</td><td>C<sub>10</sub>[Pa]</td><td>K<sub>1</sub>[Pa]</td><td>K<sub>2</sub>[Pa]</td><td>κ</td></tr><tr><td>Opt. value:</td><td>2272</td><td>4000</td><td>150</td><td>0.25</td></tr></table>

### 3.2.3 Equivalent isotropic model

For comparison with an isotropic constitutive model, we modeled the mechanical response of the leaflet as a homogeneous incompressible 3rd-order Ogden hyperelastic material. The Ogden hyperelastic material is a widely used constitutive model in computational mechanics, which provides a mathematical description of the strain-energy density of a material as a function of its deformation state. The strain energy func-

<footnote n="1">¹https://polymerfem.com/mcalibration/</footnote>
tion, W, depends on the strain measures that show the deformation. The most commonly used strain measure in the Ogden model is the principal stretch ratio which is described as follows:

$$ \mathbf{W} = \sum_{i=1}^{N} \frac{\mu_i}{\alpha_i} (\lambda_1^{\alpha_i} + \lambda_2^{\alpha_i} + \lambda_3^{\alpha_i} - 3). \quad (6) $$

$\lambda_1$, $\lambda_2$, and $\lambda_3$, are the principal stretch ratios, $\mu_i$ and $\alpha_i$ are material parameters that are determined by fitting experimental data on the strain energy function. By incorporating $N = 3$, the constitutive model has a higher degree of flexibility in capturing the non-linear stress-strain behavior of materials over a wider range of deformations.

For the acquisition of the material parameters, we used the built-in curve fitting capability of Abaqus for extracting material parameters. To have isotropic material parameters comparable with the anisotropic one, first, we set the dispersion coefficient, $\kappa$, in equation 1 to the maximum, 1/3, which makes the HGO constitutive model act as isotropic in the leaflet surface. The dashed line in Fig.8 shows the mechanical response of the HGO model if $\kappa = 1/3$. The stress-strain response is then extracted and used as input data to obtain the material parameters in the third-order Ogden hyperelastic constitutive model. The material parameters are reported in table 3. We used these parameters to compute the isotropic response of the MV. This allowed for a comparative analysis of the MV dynamic closure when subjected to both isotropic and anisotropic material models.

<figure><img src="image_9.png" /><figcaption>Figure 8: Comparison of the HGO mechanical response in case of $\kappa = 1/3$, which results in planar isotropic characteristics, against the stress-strain data (solid lines) reported in [JLK$^{+}$18].</figcaption></figure>

<table><caption>Table 3: Material parameters for the 3rd-order Ogden hyperelastic material model, derived from experimental data reported in [JLK$^{+}$18].</caption><thead><tr><th></th><th>$i = 1$</th><th>$i = 2$</th><th>$i = 3$</th></tr></thead><tbody><tr><td>$\mu$:</td><td>-481836</td><td>484166</td><td>742</td></tr><tr><td>$\alpha$:</td><td>-19.4</td><td>-19.3</td><td>-25</td></tr></tbody></table>

### 3.2.4 Anisotropy direction

In Section 2, we examined the current methods for determining fiber orientation in leaflets. This involved three main strategies: directly observing the leaflet's microstructure, utilizing simplified models, and fiber models based on physical computation. In the simplified approach, the fiber model often involved adopting a template for fiber orientation derived from a specific MV sample [EKR$^{+}$05] or assuming fibers to be parallel to the annulus plane surface. Furthermore, in studies employing computational physics to infer fiber orientation models, they specified the fiber direction to align with the direction of the maximum principal stress and strain. In our study, we examined rule-based fibers, which are characterized by various strategies for their formation and orientation. These fiber models are parallel, stress-based, affine model and randomly distributed fibers.

In the majority of the studies, they simply directed the fibers to be parallel to the surface plane approximating the annulus as in [KMP19, SVR09, VCV$^{+}$08]. We used the parallel model which we considered the fibers to be aligned with the annulus surface.

To consider the mechanical response of the MV leaflet tissue in specifying the microstructure of the leaflets, we used two fiber models based on the principal stress and principal strain directions. First, we launched the simulation by randomly distributing the fibers within the leaflet tissue. From the simulation results, for each leaflet element, we captured the positive principal stress and strain during peak systole. Using 3D vector manipulation, we projected these principal directions from the MV final state back to its initial state when the valve was open. Finally, we
implemented these directions in a diffusion algorithm based on the Gaussian smoothing approach to have a slightly more uniform and smoother fiber map.

The complex distribution of fibers on the leaflets has led some recent studies to suggest that the MV leaflet tissue is in fact resembling an isotropic behavior [LRJ+19, LSP+23]. A highly dispersed fiber orientation results in a material model with a planar isotropic mechanical response. As the fibers lie randomly on the leaflet surface, it prevents the excessive stretch of the leaflet tissue, while preserving the flexibility of the leaflet surface for bending and movement. We took this possibility into account and specified one of our fiber models to be randomly distributed on the surface of the leaflet.

Both the stress-based and affine models were linked to the stress map. Fig 9 shows the stress map for the stress-based model. A zoom on fiber directions and corresponding stress values in the black rectangle is shown in Fig 10. On this figure, each line shows a comparison of the fiber direction and the stress. Parallel and random fiber models are independent of the stress while stress-based and affine models are not. We can also see the influence of the chordae attachment points on the leaflet.

<figure><img src="image_10.png" /><figcaption>Figure 9: Global view of the stress map and zoom location for Fig 10</figcaption></figure>

# 4 Results and discussion

Our aim was to perform simulations of MV closure for a range of healthy valves. Various setups were tested such as isotropic and anisotropic models, geometric smoothing and ventricle geometry. We also compared

<figure><img src="image_11.png" /><figcaption>Figure 10: Fiber directions with chordae in red vs stress distribution. Parallel to the annulus (a), random distribution (b), stress-based (c) and affine model (d)</figcaption></figure>

different techniques of defining anisotropic directions without the need of dissecting the valve.

## 4.1 Metrics

To evaluate our models, we used the criteria classically used and in the first place the success of the effective closure of the valve, which is often assessed by the flow rate [CWG+19, FQG+18, GMQ+14]. The stress distribution is another common metric to qualitatively assess the coherence of the simulation. Besides these two criteria, we also used the contact map, which we introduced in our previous work [KVB22], which allowed us to quantify the quality of the MV closure. This map allowed us to evaluate the coaptation areas as well as the quality (strength) of the
valve closure.

Furthermore, our evaluation took into account the alignment of our results with physiological behaviors reported in clinical studies. Specifically, we considered factors such as minimal valve bulging[JHG$^{+}$18], a characteristic of healthy valves, and the balanced distribution of forces between primary and secondary chordae throughout the cardiac cycle, as studied in the literature using ex vivo data [PIMW$^{+}$20]. Both bulging and force distribution over time served as essential evaluation criteria in our experiments.

It should be noted that these criteria are global in the sense that they qualify the required shape of the valve, an expected distribution of forces or its proper functioning via the leakage volume. While these metrics allowed us to assess the overall functioning of the valve, and in particular the interest of an anisotropic model compared to an isotropic model, they were not sufficient to assess the contribution of fibre models in detail. Only direct pressure measurements at the contact level, for example, would allow us to go further in the comparison of complex models, as we will discuss in section 4.5

## 4.2 Comparison between isotropy and anisotropy models

We compare in this section the simulations obtained with and without considering anisotropy in the case of a generic and an image-based valve. We used parallel fibers to model the anisotropic directions because it is the more common model used in the literature. We used a contact map, bulging volume, and measurements from the literature (chordae force), to compare the behavior and patterns emerging in the simulation. All the results from this section are based on MV inside a fitted tube as the LV using the data p-1.

The mesh resolutions for the fitted tube LV and the generic MV leaflets were set at 80*<i>k</i>* cubic elements and 20*<i>k</i>* tetrahedron elements, respectively. In addition, the simulation for the image-based MV with the fitted tube consisted of 80*<i>k</i>* cubic elements in the fluid domain and 30*<i>k</i>* tetrahedron elements for the leaflets. The simulation took 14 hours on an Intel Xeon 3.60 GHz workstation with 12 processors.

### 4.2.1 Bulging volume

Fig.11 presents a comparison of the bulging volume for both the anisotropic and isotropic constitutive models, across image-based and generic valves. To calculate this metric, we computed the volume between the surface approximating the annulus and the valve deformed surface at the end of the simulation. In Fig.11, the red surface shows where the bulging occurs, and the green region is the part of the tissue leaflet that drops below the annulus curved surface.

<figure><img src="image_12.png" /><figcaption>Figure 11: Top row: bulging on the generic valve with (a.) isotropic and (b.) anisotropic constitutive model. Bottom row: bulging on the image-based valve P-1 with (a.) isotropic and (b.) anisotropic constitutive model</figcaption></figure>

Both for generic and image-based cases, the use of the isotropic constitutive model induced a higher bulging of the leaflet (see Fig.11.a and Fig.11.c), whereas anisotropy showed a decreased bulging. This fact is in good agreement with clinical knowledge which states that “A normal mitral valve has a large surface of coaptation with the free edge positioned low below the plane of the orifice” [CAF10]. In the image-based MV with the anisotropic constitutive model, the bulging was the least, and the saddle-shape curvature of the leaflet in the final state of the simulation was apparent.
### 4.2.2 Chordae forces

Another observation was the ability of our model to reproduce the imbalanced chordae forces between primary and secondary chordae, a phenomenon previously quantified in an ex vivo study by [PIMW+20] and visually depicted in Fig.12.a. Our simulations during closure have to be compared to this graph in the time interval [0.1 – 0.2] seconds. As indicated by the data in Figure 12.b and Figure 12.c, the order of chordae forces across different categories is expected to align with the ex-vivo measurements (Figure 12.a). During the closure phase, the ex-vivo data showed an increasing trend globally in chordae forces. This trend was replicated in both isotropic (Fig.12.b) and anisotropic (Fig.12.c) simulations, with two exceptions: the posterior secondary chordae in the isotropic case, and the posterior primary chordae in the anisotropic case. In the isotropic model, we observed that the force magnitudes for the posterior secondary chordae fell below those of the primary chordae. On the contrary, the order of chordae forces in different categories was respected in the simulation with anisotropy. Furthermore, the magnitude of chordae forces in this case was more similar to the values reported in the ex-vivo measurement.

<figure><img src="image_13.png" /><figcaption>Figure 12: Chordae force distribution on ex-vivo data from [PIMW+20] (a.), simulation with isotropic (b.), and anisotropic (c.) constitutive model with data P-1.</figcaption></figure>

### 4.2.3 Contact

When analyzing the results from the contact map, there is no ground truth in how exactly the contact between the leaflets should occur. However, in our analysis, we favored a more uniform shape of the coaptation area, a larger coaptation area, and stronger contact pressure. These criteria are open to debate and more studies on this subject are needed. In addition, experts in the field seem to have different opinions on how the contact between the leaflets should occur.

The map of contact obtained with the isotropic and anisotropic constitutive model is presented in Fig.13. The coaptation area in the case of the isotropic constitutive model (Fig.13.a) was very irregular. The coaptation strap near the annulus in this case was caused by high bulging folding the leaflet in this region and creating a crease near the annulus. In the case of the anisotropic constitutive model, the coaptation area was more regular and parallel to the annulus.

<figure><img src="image_14.png" /><figcaption>Figure 13: Contact map with isotropic (a.) and anisotropic constitutive model (b.) with data P-1.</figcaption></figure>

The contact pressure distribution in both cases appears to be evenly spread across their coaptation areas. Nonetheless, the isotropic case exhibited slightly stronger contacts, an observation that aligns with its larger bulging volume. To have a better understanding of the contact pressure distribution, we studied the probability of contact pressure distribution by us-
ing a cumulative map, i.e. $P(\|pressure\| > t)$, which is presented in Fig.14. According to this curve, higher contact pressure amplitudes were more common in the isotropic case, specifically when the amplitude is between 20 to 60 kPa.

<figure><img src="image_15.png" /><figcaption>Figure 14: Cumulative distribution function (CDF) of the contact pressure in both isotropic and anisotropic constitutive models.</figcaption></figure>

Despite the irregular coaptation area in the isotropic case, the flow rate in both instances exhibited a comparable behavior (see Fig.15). As pressure accumulates from the bottom of the tubular LV, the flow rate reaches its maximum at the same time frame, with the anisotropic case showing a slightly lower peak. Following that, the flow rate drops to zero in both cases and starts to oscillate. The coaptation area and flow rate measurements both confirmed that the valve was sealed in both the isotropic and anisotropic cases. However, a high bulging volume and irregular coaptation area in the isotropic case underscored its limited capacity to properly simulate the dynamic closure of the MV.

<figure><img src="image_16.png" /><figcaption>Figure 15: Flow rate in both isotropic and anisotropic constitutive models.</figcaption></figure>

In summary of these simulation results, we con-

clude that an anisotropy model is better suited to model the valve closure, particularly in order to have a method that can be applied to any patient based data.

## 4.3 Influence of geometry and fluid volume

In this part, we used the anisotropic constitutive model with parallel fibers and image-based valves to investigate the influence of smoothing the segmented geometry in the simulation results. By using an In- tel Xeon 3.60 GHz workstation with 4 processors, the simulation was completed in around 50 hours in av- erage with a large cylinder comprising 200k cubic elements in the fluid domains and 30k tetrahedron elements for the leaflets. The resolutions for both the fluid and structural numerical domains were cho- sen to ensure mesh independence in the simulation. The width of the large cylinder was adapted for each data to include the valve with a small margin from the tubular LV walls. The height of the tube is sub- jective. We considered 35mm for the length of the lower part (from the inlet to the annulus surface) to mimic the LV size. This will give enough clearance between the inlet and the free edge of the leaflet. The length of the upper part, which represents the left atrium, was taken as 15mm to have a clearance between the leaflet's top surface and the outlet. The tests have been performed on the four image-based valves of Table 1.

### 4.3.1 Smoothed leaflet

To investigate the influence of smoothing the seg- mented MV, we compared the simulation results of both the original image-based and the smoothed ver- sion. The purpose of investigating the smoothness of the leaflets was to understand how the roughness of the extracted geometry from the segmentation pro- cess influences the simulation outcome. Depending on the quality of the segmentation process and the resolution of the geometry extracted, the roughness of the extracted MV leaflet from the segmentation process can vary [CMB+21]. Therefore, we applied smoothing with an average absolute distance of 0.145
mm on the leaflet of the image-based geometry. This approach aimed to reduce surface roughness on the valve leaflets while maintaining the leaflets essential characteristics, such as the size of the opening orifice and the original curvature. The opened state of the image-based valve in both the original shape and the smoothed version is shown in Fig. 6.

Our experiments showed that smoothing enhances the contact between the leaflets and slightly improved the bulging volume (see Fig. 16). In the contact map depicted in Fig. 17, the coaptation area in the smoothed case was larger compared to the original image-based valve. Furthermore, as illustrated in the cumulative distribution function (CDF) plot in Fig. 18, the contact pressure magnitude was notably higher in the case of the smoothed valve. In general, the roughness of the surface obtained from the segmentation process influenced the coaptation and contact pressure, but the overall shape of the MV in the peak systole was not affected. Better contact and coaptation are expected by smoothing the leaflets, as this process reduces the surface noise on the geometry. We used a weak smoothing since larger values may comprise the valve ability to close properly, resulting in closure with orifice holes.

<figure><img src="image_17.png" /><figcaption>Figure 16: Bulging on the (a.) image-based with fitted tube and anisotropic model and (b.) smoothed valve with the fitted tube and anisotropic model on data P-1.</figcaption></figure>

<figure><img src="image_18.png" /><figcaption>Figure 17: Contact maps for the image-based valve with the fitted tube setup and anisotropic model (a.), the smoothed valve with the fitted tube setup and anisotropic model (b.), and the smoothed valve with the large cylinder setup and anisotropic model (c.) on data P-1.</figcaption></figure>

### 4.3.2 Using the large cylinder as the tubular left ventricle

For the case of the large cylinder, the presence of the additional part blocking the flow passing over the an-nulus, caused some instabilities as vibration on the leaflet during the closure. It may be due to the in-terface between the leaflet and this housing part. We
<figure><img src="image_19.png" /><figcaption>Figure 18: Cumulative distribution function (CDF) of the contact pressure with anisotropic model.</figcaption></figure>

minimized this numerical instability by refining the resolution of the elements in the large cylinder in order to have more element layers in the axial direction of the cylinder. The influence of this vibration is visible in the flow rate shown in Fig.19. The closure was achieved in all cases. The fluctuation shown in the flow rate was due to the vibration of the valve pushing the flow outward at the bottom of the large cylinder. Simulation with a larger cylinder was performed on P-1, P-2, P-3 and P-4. Video of a smoothed valve closure with large cylinders for P-1, P-2 and P-4 are accessible through the links <fnref n="1" />, <fnref n="2" /> and <fnref n="3" />. The latest video on P-4 shows that our method of simulating MV closure is even scalable to image-based data with chordae automatically segmented.

<figure><img src="image_20.png" /><figcaption>Figure 19: Flow rate shown for image-based in the fitted tube, smoothed valve in the fitted tube, smoothed valve in the large cylinder. All cases are with the anisotropic model.</figcaption></figure>

The stress distribution in the closed state for the smoothed valve in the case setup with the fitted tube and the large cylinder is illustrated in Fig.20. It can be seen that neither the fitted tube (Fig.20.a) nor the large cylinder representing the LV (Fig. 20.b) exhibited any visible orifice hole. Notably, the stress patterns and their magnitudes were consistent across all three scenarios. However, a minor distinction was observed: the stress magnitudes in the case of the large cylinder were slightly lower than those found in the fitted tube. Furthermore, areas of high stress near the annulus ring and chordae attachment points appeared which are shown in red. Regarding the quality of closure, the final state of the smoothed valve in both cases of fitted tube and large cylinder appeared to exhibit a relatively flat profile, as indicated by the bulging volume illustrated in Fig.21.

Fig.22 shows the visualization of the P-2 and P-4 valves in the closed state. In this figure, the bulging volume and the saddle shape of the MV in the closed state are reported. The simulation revealed a noticeably large bulging volume in the P-4 valve. This can be attributed to the fact that P-4 was segmented in the closed state, in which the leaflets were already stretched, and consequently, the dynamic closure of the P-4 will further stretch the valve.

<figure><img src="image_21.png" /><figcaption>Figure 20: Top view of the stress distribution in case of image-based valve with fitted-tube and anisotropic model (a.) and the image-based valve with large cylinder and anisotropic model (b.) on data P-1.</figcaption></figure>

According to the contact map shown in Fig.17, the use of a large cylinder increased the size of the coaptation area. In the simulation with the fitted tube,

<footnote n="1">¹https://youtube.com/shorts/6lEgS0Y8_ic</footnote>

<footnote n="2">²https://youtube.com/shorts/tDscocfHlco</footnote>

<footnote n="3">³https://youtube.com/shorts/Nf08_JgvS_4</footnote>
<figure><img src="image_22.png" /><figcaption>Figure 21: Bulging on the (a.) smoothed valve with the fitted tube and anisotropic model, (b.) and smoothed valve with the large cylinder and anisotropic model on data P-1.</figcaption></figure>

<figure><img src="image_23.png" /><figcaption>Figure 22: Bulging on the (a.) P-2 with the large cylinder and anisotropic model, and (b.) P-4 with large cylinder and anisotropic model.</figcaption></figure>

as pressure rises, there is a tendency for the fluid to infiltrate the contact area, leading to a separation of the free edges of the leaflets (as seen in Fig.17.a and Fig.17.b). In contrast, the simulation involving the larger tube provided a more accurate representation, as the large volume of blood in the ventricle exerted pressure on the leaflets, thereby ensuring that the free edges remain closed as shown in Fig.17.c. In addition, the contact pressure in the simulation with a large cylinder was notably stronger compared to the simulation with the fitted tube (see Fig.18).

In summary of these simulation results, we conclude that smoothing the leaflet geometry given by manual segmentation and using a large cylinder with a margin after the annulus helps to perform database mitral valve simulation that can reach normal, competent closure.

## 4.4 Influence of microstructure of the leaflets

In this section, we detail the simulation results for each mentioned fiber model. There is no ground truth data available to know what the pressure is on the leaflets during peak systole. We compared then the fiber models by determining which model's results aligned most closely with the anticipated physiological traits of the MV dynamic closure. From a physiological standpoint, the optimal microstructure should provide maximum flexibility, enabling the leaflet to move seamlessly throughout the cardiac cycle. It should also ensure effective coaptation and contact pressure, while minimizing tissue bulging and stretching [CAF10]. Consequently, in our simulation, we aimed to identify the fiber model that best exhibits these mechanical characteristics.

We employed the HGO constitutive model already described in section 3 for these simulations. All the simulations presented in this section were based on the smoothed version of the image-based valve mounted in the large cylinder as the tubular left ventricle. In the figures presented in this section, a through f respectively depict the results for fiber models based on the following criteria: parallel to the annulus, stress-based, affine model, and random distribution.

### 4.4.1 Closed state

Fig.23 shows the strain distribution on the leaflet tissue at peak systole for all the fiber models. In the case of parallel fiber, a significant stretch was seen. Conversely, other fiber models exhibited more moderate tissue stretching, with the random fiber model exhibiting the least stretch.

This observation correlated with the bulging volume illustrated in Fig.24. Parallel fiber model exhibited extensive bulging, while the random fiber model presented the least bulging volume among all the fiber models. Excessive strain leading to pronounced leaflet bulging can be symptomatic of a compromised valve. We assume that the strain distribution should be small across the leaflet [CAF10], minimizing unnecessary stretching throughout the cardiac cycle.
Therefore, according to these two metrics, parallel fiber model performed poorly compared to the random, stress-based, and affine fiber models.

<figure><img src="image_24.png" /><figcaption>Figure 23: Distribution of the logarithmic strain over the leaflet tissue at peak systole in the case of fiber models based on: Parallel to the annulus (a), random distribution (b), stress-based (c) and affine model (d).</figcaption></figure>

### 4.4.2 Contact

Fig.25 presents the contact map across all the fiber models. Notably, the parallel model exhibited the largest coaptation area, while the random model showed the smallest. Although there were minor variations in the shape and size of the coaptation area among most models, these differences were relatively subtle. The exception to this pattern was the parallel model, which deviated significantly in both the shape and size of the coaptation area compared to the others. The surface size of the coaptation area in all cases was uniformly distributed close to the free edge of the leaflets, with no significant discontinuity, which showed sealed closure. The contact

<figure><img src="image_25.png" /><figcaption>Figure 24: Bulging volume at peak systole in the case of fiber models based on: Parallel to the annulus (a), random distribution (b), stress-based (c) and affine model (d). The red region shows the bulging volume where the valve stretches above the annulus surface.</figcaption></figure>

pressure in the stress-based model was weaker compared to other models. Parallel fiber models manifested notably large contact pressures. This observation is consistent with the fact that both models demonstrated a comparably larger bulging volume. According to the coaptation area reported in Fig.25, the contact pressure and coaptation area in random, and affine models fell within a similar range.

<figure><img src="image_26.png" /><figcaption>Figure 25: Contact map showing the coaptation area and the contact pressure in case of fiber models based on: Parallel to the annulus (a), random distribution (b), stress-based (c) and affine model (d).</figcaption></figure>
### 4.4.3 Chordae forces

Considering the distribution of the chordae forces during the MV dynamic closure, we observed that in parallel, random, stress-based, and affine fiber models, the sequence of each chordae category aligned with the order observed in the ex vivo measurement data reported in [PIMW+20] (see Fig.26).

<figure><img src="image_27.png" /><figcaption>Figure 26: The chordae force during the dynamic closure of the MV in case of fiber models based on: Parallel to the annulus (a), random distribution (b), stress-based (c) and affine model (d). The figure at the bottom shows the ex vivo measurement of the chordae force during the cardiac cycle reported in [PIMW+20].</figcaption></figure>

In summary, we compared fiber distribution methods based on valve bulging in the closed position, contact pressure and force distribution on the chordae. We conclude that the pattern of fiber distribution is important in determining features of the closed valve, although the absence of any real ground truth tem- pers this conclusion.

## 4.5 Conclusion

While most works on dynamic simulation of the mitral valve have been dedicated to a single, often simplified model, we have investigated how to improve simulation frameworks to be able to scale up and handle subject-specific data that are more complex in shape and whose segmentation (manual or algorithmic) may result in irregularities in the leaflets or chordae. Limitations of our study include use of porcine data for our image-based models and use of simplified geometry (i.e., cylindrical) for the flow domain, however these are simply inputs to our workflow that can be easily altered to reflect specific data. The paper describes various contributions to achieving this goal, focusing on the nature of the finite elements to be used, the need to use housing around the annulus, and the benefits of a small amplitude smoothing to achieve convergent simulations, both in the case of isotropic models (which was expected) and anisotropic models (which was less expected). The anisotropic model enabled us to obtain valve closure simulations with normal physiological features (e.g., no bulging) and consistent with clinical knowledge of chordae force distribution.

As the anisotropy of the valve is due to the collagen fibers that make it up, we also studied the influence of more complex fiber models than a uniformly parallel model. With a view to future clinical use, we restricted ourselves to models that did not require dissection.

Our study did not, however, reveal one model that was better than the others, as all these models had a similar level of performance in terms of the valve quality criteria used. Despite the potential clinical interest of the contact map to evaluate the quality of the valve closure, and though the contact map reveals differences in intensity which may be locally strong, we were unable to use these maps to compare different material models, as there is scarce clinical or experimental data on coaptation surface area or forces in normal mitral valves. While clinicians recognize the value of a generous coaptation surface between leaflets in a well-functioning valve, specifics of the size, shape, and magnitude of the contact map for normal valves are poorly understood.
However, recent studies have shown that intraop-
erative direct measurement of the coaptation height
is now possible thanks to a miniaturized fiberscope
technology[RKE+24]. This opens the way in the near
future towards real-time, direct measurements at the
closed state, which could help to differentiate the im-
pact of complex fiber pattern models and, in general,
to allow proper assessment of computational mod-
els. Another interesting avenue of research towards
improved quantitative evaluation of the models we
want to investigate is the acquisition of temporal 3D
sequences of the valve on ex vivo porcine hearts taken
for instance from stereo videography or ultrasound
which may allow temporal comparison of computa-
tional model predictions of valve dynamics across the
full cardiac cycle.

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