# Universal Coordinates and Parametrization Methods for Mitral Valve Dynamic Simulations

Pierre-FrédéricVillard<fnref n="1" />,<fnref n="2" />*, Peter Hammer<fnref n="3" />, Marie-Odile Berger<fnref n="1" />, and Hao Gao<fnref n="4" />

**Abstract.** The mitral valve plays a critical role in unidirectional blood flow within the heart, but is prone to pathologies requiring surgical intervention. Computational modeling offers promising solutions to improve surgical outcomes. However, challenges remain due to its computing time and difficult patient-specific data integration. This study introduces a parameterized framework for modeling the mitral valve, incorporating a universal coordinate system to standardize geometry across datasets. An initial feasibility study of this framework was demonstrated through dynamic mitral valve simulations using an immersed boundary method. These results highlight the potential of this parameterization approach to be used for deep learning or to enhance segmentation, particularly in cases of incomplete or noisy data.

**Keywords:** Mitral valve · Fluid-structure interaction · anisotropy.

## 1 Introduction

The mitral valve (MV) is one of the four cardiac valves, playing a critical role in maintaining unidirectional blood flow within the heart. It prevents blood regurgitation from the left ventricle to the left atrium during ventricular systole. Structurally, the mitral valve consists of two leaflets (anterior and posterior) that are held in place by chordae when the valve is closed. These chordae are anchored to both the leaflets and the papillary muscles. The leaflets are connected to the left ventricle via the annulus, forming a highly dynamic and interconnected system.

Unfortunately, the mitral valve is susceptible to various pathologies that can compromise its function, often necessitating surgical intervention. Predicting the outcomes of such surgeries remains a significant challenge due to the valve's complex anatomy and biomechanics. Consequently, considerable research efforts have focused on the development of computer-based simulations using biomechanical models to improve predictions of treatment efficacy and optimize surgical planning [9,6].

<footnote n="1">¹ Université de Lorraine, CNRS, Inria, LORIA, Nancy, France</footnote>

<footnote n="2">² Harvard School of Engineering and Applied Sciences, Cambridge, MA, USA</footnote>

<footnote n="3">³ Harvard Medical School, Boston, MA, USA</footnote>

<footnote n="4">⁴ University of Glasgow, Glasgow, UK</footnote>

<footnote>* Corresponding author: pierrefrederic.villard@loria.fr</footnote>
Accurately modeling the behavior of the mitral valve is highly complex due to its anisotropic properties, the influence of blood flow, and intricate boundary conditions such as ventricular contraction, annular dilation, and chordal tension [8]. Current computational models, while advanced, often suffer from excessively long processing times, rendering them impractical for direct use in clinical treatment planning [2,9]. Additionally, there is limited research focusing on geometries derived from patient-specific data, further constraining the applicability of these models in personalized medicine.

Extracting the geometry of the mitral valve from medical imaging for patient-specific simulations presents significant challenges. The valve's structures are difficult to visualize in clinical images. While the leaflets can sometimes be delineated, this requires dynamic imaging to identify coaptation regions accurately [3]. Only a subset of chordae tendineae is typically visible, often limited to the anchoring points at the papillary muscles. Some studies have explored geometry extraction under controlled conditions [12,10], but these approaches are not directly transferable to clinical imaging, further highlighting the gap between current research and clinical applications. A further challenge is to incorporate subject-specific collagen fibers in MV leaflets, which can not be acquired with in vivo imaging techniques, though ex vivo experiments can measure those collagen fibers, but not applicable to patients. Thus, a rule-based approach that takes into account ex vivo measurements is highly desirable.

This study proposes the development of a parameterized framework for modeling the mitral valve and its sub-valvular apparatus, a universal normalized coordinate system is further introduced aiming to standardize representations across different datasets and facilitate the application of learning techniques as demonstrated in prior works [4]. Another key contribution is the integration of this universal modeling approach with a novel parameterization of mitral valves. This parameterization method is designed to strike a balance between simplicity—allowing straightforward adaptation to real-world patient data—and precision, ensuring sufficient anatomical fidelity. The proposed parameterization holds significant potential to facilitate segmentation processes, particularly in scenarios involving incomplete or noisy clinical data. By leveraging the universal model and parameterized representation, this approach aims to provide robust tools for improving the quality of geometry extraction, facilitating patient-specific simulations, and bridging the gap between computational modeling and clinical application.

# 2 Methodology

## 2.1 Total MV parametrization

In this section, we present a method to parameterize the mitral valve. The proposed strategy involves defining a geometric model characterized by a minimal set of parameters that can be readily measured on an actual valve. This model serves two purposes: it facilitates the segmentation of clinical data by fitting an equivalent 3D model and enables the generation of diverse and realistic valve
representations. These synthetic valves can be employed in biomechanical simu-
lations to generate data, which can be utilized for deep learning analyses.

The first step involves modeling the valve in a flattened configuration, resem-
bling a dissected and unfolded valve. This approach allows for the measurement
of the circumference Cvalve and the geometry of the individual lobes. This study
is inspired by the work of [11]. The lobes are represented by four semi-ellipses de-
fined by their minor and major axes, aᵢ and bᵢ, respectively, where i ∈ {1, 2, 3, 4}.
The first three ellipses correspond to the posterior part of the valve, while the
fourth represents the anterior part (see Figure 1.a). This flat representation is
then projected onto a cylinder with an elliptical cross-section. The circumference
Cvalve is given by ∑ᵢ aᵢ. The minor radius rminor of the elliptical cross-section is
defined as Cvalve/(2π), and the major radius rmajor is determined using the ratio
between the two axes κratio as rmajor = κratio · rminor. Additionally, a thickness
for the valve tvalve is defined. The projection onto the elliptical cylinder is illus-
trated in Figure 1.b. To facilitate the attachment of the chords and to achieve
an open configuration closer to physiological reality, the curvature of the valve
is parameterized using a function Δρ(zᵢ). This function describes a change in
radius Δρᵢ as a function of the height zᵢ, as illustrated in Figure 1.c). Here,
ρᵢ represents the radius at each point i in polar coordinates, and α and β are
parameters that could be fitted on the real valve.

<figure><img src="image_1.png" /><figcaption>**Fig. 1.** Mitral valve parametrization: a. flatten shape, b. Projection on an elliptic cylinder and c. Z-dependent curvature.</figcaption></figure>

The chords are parameterized by four points representing the papillary mus-
cles, a list of insertion point positions on the leaflet, and the slack ratio. For
simplification, the initial model assumes an architecture without branching. The
slack ratio introduces slack in the chords at the open state, ensuring proper valve
closure. This is achieved by automatically adding a midpoint, enabling spline-
based modeling. In order to map insertion points that could be measured on a
real leaflet, we used universal coordinates, as explained in the following.

## 2.2 Leaflets parameterization with universal coordinates

Here we introduce a universal coordinate system for MV leaflets, inspired by
the concept of universal ventricular coordinates in cardiac modeling, as first
introduced by Bayer et al. [1]. Their framework provides a method to describe anatomical structures within the heart using a generic coordinate system. By adopting a similar principle, our work extends this idea to the mitral valve, ensuring consistent spatial representation across varying datasets. As shown in Figure 2, a series of Laplace equations ($\Delta u = 0$ with $u$ the universal coordinate) are solved to create a local coordinate system for each point of the computational mesh. Specifically, for the MV leaflets, we design a circumferential-radial-normal system. To define the circumferential coordinate, we separate the anterior and posterior leaflets with two separated boundaries: SepL and SepR as shown in Figure 2.a. Using the Laplace-Dirichlet method by setting $u = 0$ on SepL and $u = 180$ on SepR, we then can numerically compute $u$ for each mesh node. For the node on the posterior leaflet, we redefine $u = 360 - u$, thus the circumferential coordinate lies in $[0, 360]$, see Figure 2.b. The local circumferential direction is further defined as $v_c = \frac{\partial u}{\partial x} / \left| \frac{\partial u}{\partial x} \right|$. In a similar way, we solve $\Delta w = 0$ for the radial coordinate by setting $w = 0$ on the annulus ring and $w = 1$ on the free edge of the leaflets, see Figure 2.c. The local radial direction is then defined as $v_r = \frac{\partial w}{\partial x} / \left| \frac{\partial w}{\partial x} \right|$. To obtain an orthogonal local coordinate system, we first compute the normal direction $v_n = v_c \times v_r$. Figure 2.d shows this local coordinate system, which can be further used to generate rule-based collagen fibers, see an example in Figure 2.e. This universal coordinate system $(u, w)$ can be further used to establish a mapping between different MVs, i.e. locations with the same $(u, w)$.

<figure><img src="image_2.png" /><figcaption>Fig. 2. Computing a universal coordinate system in MV leaflets, from which rule-based fibers can be generated based on that coordinate system.</figcaption></figure>

## 2.3 MV dynamic simulation

To demonstrate the effectiveness of our proposed parameterization, we further simulate MV dynamics from opening to closure using the immersed boundary
with finite element approach (IB/FE) described in [7]. In brief, the MV is attached to a housing structure and then further mounted to a rigid straight tube. A physiological transvalvular pressure is applied to one end of the tube and the other end is held with zero pressure. The leaflets are characterized using an invariant-based strain energy function with two families of fibers using the universal coordinate system as detailed in Section 2.2.

# 3 Results and discussion

## 3.1 Parametrization

Various steps are necessary to fit our parametrization to a real MV. As a proof-of-concept study, we parametrize an existing MV model that was obtained from multi-slice CT images of a normal MV from a 61-year old male [5]. A 3D mesh was extracted from it in order to measure the parameters see Figure 3.a. It further has a CT-derived chordae including the marginal, basal and strut chordae. First, a Principal Component Analysis (PCA) was used to compute the eigenvectors of the point cloud of the valve (see Figure 3.b). The first two principal components define a plane capturing the maximum variance of the data. Projecting the points onto this plane provides a 2D representation of the valve in the axial view. An ellipse is then fitted to the convex hull of the 2D point cloud by optimizing the parameters using a nonlinear least-squares method (see Figure 3.c). This approach enables to measure the dimensions of the ellipse, $r_{minor}$ and $r_{major}$, as well as finding a coordinate system that maps the real valve to a parametric valve centered at the origin and oriented along the ellipse directions. The dimensions of the semi-ellipses for each lobe, $a_i$ and $b_i$, are measured on the geometry of the real valve using geodesic paths. The same method is applied to measure the curvature at the midpoint of each lobe. Figure 3.d illustrates the three lobes of the posterior leaflet, showing the manually clicked points, the corresponding semi-ellipses in red, and the curvature measurement path in blue.

<figure><img src="image_3.png" /><figcaption>**Fig. 3.** Real valve parametrization: a. the original MV model from [5], b. Position and orientation with PCA, c. ellipse fitting and d. semi-automatic lobes measurements</figcaption></figure>

To parameterize the chordae from real data, the attachment points to the papillary muscles and the valve leaflets are manually identified and annotated directly on the geometry of the real valve. An illustration of this selection process is shown in Figure 4.a, where a spline-based representation is utilized. The
insertion points on the leaflets are then converted into universal coordinates, en-
abling their mapping to the leaflets of the parameterized valve, as explained in 2.
The spline serves as a guide for extrusion, starting from a circular cross-section
with a diameter measured directly from the real data. Figure 4.b illustrates the
result for a selected chordae segment attached to the anterior valve leaflet. The
overall result of the parameterization process is shown in Figure 4.c, with both
the parameterized and real valve geometries. The current approach models the
annulus as a plane rather than its saddle shape, placing the parametric valve
annular plane midway through the actual leaflet. Regarding the universal coordi-
nates, multiple mapping strategies can be considered. While the annulus (w = 0)
is widely accepted, the free edges (w = 1) depend on the precision of the seg-
mentation process. It is important to establish clear and robust segmentation
guidelines to ensure consistency and reliability across different datasets.

<figure><img src="image_4.png" /><figcaption>**Fig. 4.** Results on chordae generation. a. Manual annotation and spline representation on the real valve. b. Generation of the chordae geometry, illustration on parts of the chordae and part of the anterior leaflet (black: papillary muscle barycenter, red: midpoint, green: attachment points, cyan anterior leaflet, blue: chordae expressed as splines and gray: contour of a chordae. c. comparison between real (blue wireframe) and parametric (red surface) geometries.</figcaption></figure>

**3.2 The IB/FE MV modeling**

Figure 5 shows the simulated MV dynamics with rule-based generated fibers. It can be seen that the MV can open successfully (Figure 5.a), with maximum displacements on the free edges of the anterior and posterior leaflets, the less motion near the commissure region, which might be due to the presence of chordae. Figure 5.b shows the closure state of the MV in the middle of ventricular systole. Both leaflets are pushed back towards the left atrium with much reduced orifice and straightened chordae.

Using the real MV (Figure 3), we further simulated its dynamics with dif-
ferent rule-based fibers by making use of the universal coordinate system, they
are 0°, 18° and 36°. Fiber angle 0° means all fibers align circumferentially, often
found in the literature [7], the angle 18° is based on the data from [13], and the
<figure><img src="image_5.png" /><figcaption>**Fig. 5.** MV dynamics simulated with the immersed boundary methods: a. the fully-opened state, b. the final closure state. Colored by displacements, blue to red represents low to high.</figcaption></figure>

angle 36° indicates more separated fiber families than 18°. The MV can open successfully with nearly identical opening configurations. Figure 6 shows the first principal stress when the MV is fully closed, there are noticeable differences that can be found with very different peak values. For example, the highest first principal stress is found for the fiber angle 18°, which is then reduced when the fiber angle is 36°. This might be caused by the complex dynamics of the leaflet and the tethering effect of chordae during systole, further studies are needed to elucidate how fiber orientations affect the leaflet dynamics. It can also be found that the free edge on the posterior is compressed the most with fiber angle 36°. The universal coordinate system proposed in this study will facilitate the incorporation of a realistic collagen fiber network towards patient-specific MV modeling.

<figure><img src="image_6.png" /><figcaption>**Fig. 6.** Simulated MV dynamics using the real MV from [5] with different fiber angles: 0°, 18° and 36° at fully-opened and fully-closed states.</figcaption></figure>

# 4 Conclusion

This study presents a novel parameterized framework for modeling the mitral valve. By integrating a universal coordinate system and parameterization, the proposed approach could facilitate the standardization of MV geometries across datasets and improves the segmentation of noisy or incomplete clinical imaging data. The framework demonstrates its potential through dynamic simulations of MV behavior. The parameterization not only enhances the anatomical fidelity of computational models but also incorporates patient-specific fiber orientations.
The use of real clinical data in our study represents an encouraging step to-
ward validation. Unlike generic valve models commonly found in the literature,
our approach is based on actual patient-specific valves, making it more represen-
tative of real-world anatomy and bringing it closer to clinical applicability. We
show that our model replicates key functional behaviors of the valve, including
closure and stress distribution, using a real valve as a reference.

The universal coordinate system for the MV leaflet is part of the param-
eterization procedure, it serves several purposes: (1) to construct a rule-based
collagen architecture as we have demonstrated in this study; (2) to provide a
mapping scheme when comparing different MV geometries. In fact, we have
used this universal coordinates to map the insertion points of the chordae from
the real mitral valve to the parametric MV, see Section 3.1.

Future work will focus on refining the parametrization, including optimizing
of chordae structure, including the saddle shape of the annulus and getting more
accurate curvature of the lobe. We also plan to apply this parameterization
method to our image database, which includes both healthy and pathological
valves, to ensure its robustness and scalability.

**Acknowledgments.** Hao Gao and Pierre-Frédéric Villard are grateful for the funding provided by the British Council and the Going Global Global Partnerships - Springboard Programme (No. 1270441886).

**Disclosure of Interests.** The authors have no competing interests to declare that are relevant to the content of this article.

References

1. Bayer, J., Prassl, A.J., Pashaei, A., Gomez, J.F., Frontera, A., Neic, A., Plank, G., Vigmond, E.J.: Universal ventricular coordinates: A generic framework for describing position within the heart and transferring data. Medical image analysis **45**, 83–93 (2018)

2. Caballero, A., et al.: New insights into mitral heart valve prolapse after chordae rupture through fluid-structure interaction computational modeling. Scientific reports **8**(1), 1–14 (2018)

3. Carnahan, P., Ginty, O., Moore, J., Lasso, A., Jolley, M.A., Herz, C., Eskandari, M., Bainbridge, D., Peters, T.M.: Interactive-automatic segmentation and modelling of the mitral valve. In: Coudière, Y., Ozenne, V., Vigmond, E., Zemzemi, N. (eds.) Functional Imaging and Modeling of the Heart. pp. 397–404. Springer International Publishing, Cham (2019)

4. Dabiri, Y., Mahadevan, V.S., Guccione, J.M., Kassab, G.S.: Machine learning used for simulation of mitraclip intervention: A proof-of-concept study. Frontiers in Genetics **14** (2023), https://api.semanticscholar.org/CorpusID:257441895

5. Feng, L., Qi, N., Gao, H., Sun, W., Vazquez, M., Griffith, B.E., Luo, X.: On the chordae structure and dynamic behaviour of the mitral valve. IMA journal of applied mathematics **83**(6), 1066–1091 (2018)

6. Gao, H., Feng, L., Qi, N., Berry, C., Griffith, B.E., Luo, X.: A coupled mitral valve—left ventricle model with fluid-structure interaction. Medical engineering & physics **47**, 128–136 (2017)
7. Gao, H., Ma, X., Qi, N., Berry, C., Griffith, B.E., Luo, X.: A finite strain nonlinear human mitral valve model with fluid-structure interaction. International journal for numerical methods in biomedical engineering 30(12), 1597–1613 (2014)

8. Gao, H., Qi, N., Feng, L., Ma, X., Danton, M., Berry, C., Luo, X.: Modelling mitral valvular dynamics-current trend and future directions. International Journal for Numerical Methods in Biomedical Engineering 33(10), e2858 (2017)

9. Khaledian, N., Villard, P.F., Hammer, P.E., Perrin, D.P., Berger, M.O.: Influence of Anisotropy on Fluid-Structure Interaction Simulations of Image-Based and Generic Mitral Valves. In: LNCS FIMH 2023: Functional Imaging and Modeling of the Heart. vol. 13958, pp. 455–464. Springer, Lyon, France (Jun 2023). https://doi.org/10.1007/978-3-031-35302-4_47, https://hal.science/hal-04117508

10. Panicheva, D., Villard, P.F., Hammer, P.E., Perrin, D., Berger, M.O.: Automatic extraction of the mitral valve chordae geometry for biomechanical simulation. International Journal of Computer Assisted Radiology and Surgery pp. 709–720 (2021)

11. Shen, X., Wang, T., Cao, X., Cai, L.: The geometric model of the human mitral valve. PLOS ONE 12(8), e0183362 (2017)

12. Toma, M., Bloodworth, C., Einstein, D., Pierce, E., Cochran, R., Yoganathan, A., Kunzelman, K.: High resolution subject-specific mitral valve imaging and modeling: Experimental and computational methods. Biomechanics and Modeling in Mechanobiology 15 (12 2016). https://doi.org/10.1007/s10237-016-0786-1

13. Wang, Q., Sun, W.: Finite element modeling of mitral valve dynamic deformation using patient-specific multi-slices computed tomography scans. Annals of biomedical engineering 41, 142–153 (2013)