A Computational Pipeline for Patient-Specific Prediction of the Postoperative Mitral Valve Functional State

Abstract

While mitral valve (MV) repair remains the preferred clinical option for mitral regurgitation (MR) treatment, long-term outcomes remain suboptimal and difficult to predict. Furthermore, pre-operative optimization is complicated by the heterogeneity of MR presentations and the multiplicity of potential repair configurations. In the present work, we established a patient-specific MV computational pipeline based strictly on standard-of-care pre-operative imaging data to quantitatively predict the post-repair MV functional state. First, we established human mitral valve chordae tendinae (MVCT) geometric characteristics obtained from five CT-imaged excised human hearts. From these data, we developed a finite-element model of the full patient-specific MV apparatus that included MVCT papillary muscle origins obtained from both the in vitro study and the pre-operative three-dimensional echocardiography images. To functionally tune the patient-specific MV mechanical behavior, we simulated pre-operative MV closure and iteratively updated the leaflet and MVCT prestrains to minimize the mismatch between the simulated and target end-systolic geometries. Using the resultant fully calibrated MV model, we simulated undersized ring annuloplasty (URA) by defining the annular geometry directly from the ring geometry. In three human cases, the postoperative geometries were predicted to 1 mm of the target, and the MV leaflet strain fields demonstrated close agreement with noninvasive strain estimation technique targets. Interestingly, our model predicted increased posterior leaflet tethering after URA in two recurrent patients, which is the likely driver of long-term MV repair failure. In summary, the present pipeline was able to predict postoperative outcomes from pre-operative clinical data alone. This approach can thus lay the foundation for optimal tailored surgical planning for more durable repair, as well as development of mitral valve digital twins.

1 Introduction

Mitral valve regurgitation (MR) is the most common heart valve disease in the United States. It is estimated to affect over 2% of the entire population, with a prevalence of 9.3% in individuals 75 or older, and is expected to double by 2030 due to population aging [1,2]. Furthermore, MR is a major prognostic factor of mortality; the risk of mortality with severe MR at 5 years is 36%, and even moderate MR nearly doubles the risk of mortality in patients with multiple cardiac comorbidities [3–5]. Ischemic MR (IMR) is a subtype of MR occurring secondary to myocardial infarction (MI) as the left ventricle (LV) dilates in response to impaired contractility in the infarcted region [6,7]. LV dilatation displaces the papillary muscles apicolaterally; as a result, the leaflets are tethered and no longer able to completely coapt, which permits the backflow of blood into the left atrium. Additionally, the impaired contractile ability of the LV prevents the mitral annulus from contracting enough to help the mitral leaflets cover the orifice: rather than a healthy 20–25% decrease in annular length, there is only about a 5% decrease. IMR is particularly deadly: even mild degrees of IMR after MI are associated with a significant increase in the risk of mortality and 5-year mortality after MI with IMR is 62% [8–11].

The two main strategies for treating MR of any type are total valve replacement and repair. Mitral valve (MV) repair is presently the preferred option as it preserves the native MV structure and function and has better survival rates compared to replacement [12–16]. In particular, undersized ring annuloplasty (URA) aims to restore sufficient coaptation by reducing the annular orifice area and thereby improving leaflet approximation [17]. However, 6 months after URA repair, up to 30% of patients experience recurrent MR [13], and long-term outcomes remain unpredictable, suboptimal, and poorly understood [18–24]. More recent developments in MV repair include transcatheter approaches such as the MitraClip, which is both a safer option than surgery and has been shown to improve patient outcomes [25,26]. However, its durability is highly dependent on MR etiology, MR severity, and existing comorbidities, and its long-term outcomes remain largely unknown [27–32]. Thus, there is an urgent need for a deeper, quantitative understanding of the functional consequences of MV repair to develop more durable IMR treatments.

The earliest finite element (FE) models of the MV date back to the 1990s [33]. Kunzelman et al. simulated closure of the excised porcine valve and included various tissue properties in their analysis, such as fiber orientation, anisotropic mechanical properties, and variable tissue thickness. Annular and papillary muscle displacements were implemented as Dirichlet boundary conditions, and the simulation was validated using measurements of leaflet coaptation and stress distributions. Subsequent studies by Salgo et al. investigated the influence of annular shape on MV kinematics by modeling the true saddle shape of the MV annulus. They concluded that a saddle shape better-reduced peak stress compared to a flat configuration [34].

With the development of advanced imaging techniques such as computed tomography (CT) and magnetic resonance imaging (MRI), in vivo structural models were developed with boundary conditions extracted directly from patient imaging data [35]. Recently, more complex MV models were created using CT imaging where the mitral valve chordae tendinae (MVCT) insertions, branching patterns, and origins on the papillary muscles were segmented and translated to finite element models [36–38]. The clinical gold standard for MV imaging is real-time three-dimensional (3D) echocardiography (rt-3DE), preferred for its high temporal resolution and contrast. In 2006, a pilot study was conducted to demonstrate the feasibility of reconstructing patient-specific FE meshes of the MV leaflet geometry from clinical echocardiographic datasets [39]. More recently, patient-specific rt-3DE imaging has been used in computational modeling of the MV to assess MV geometric parameters under various MR repair conditions [40–43].

However, despite advancements in rt-3DE imaging technology, this modality cannot adequately resolve the MVCT structures or the full systolic leaflet coaptation zone [44]. Moreover, it is impossible to directly extract patient-specific material properties from rt-3DE imaging. Therefore, the focus of our group has been on developing rt-3DE image-based patient-specific MV models which overcome these limitations and extend MV analysis beyond geometry alone to the valve's full functional behavior. Initially, we developed high-fidelity MV models on both the micro- and macrolevel using ovine in vitro imaging data [45–47]. The end-diastolic (ED) and end-systolic (ES) leaflet geometries and the chordal structure were segmented from micro-CT images, and this highly detailed anatomical information was utilized for model development. We then incorporated local tissue thickness variation, a well-established structural constitutive model, and different MVCT branching patterns into the model [45,48]. Parametric studies demonstrated that our model could accurately capture MV behavior using a functionally equivalent MVCT structure and a constant leaflet thickness. Based on this rigorous foundation, we developed an rt-3DE-based, noninvasive in vivo MV leaflet strain estimation technique, which has since been used to assess the detailed coaptation behavior of MVs after URA repair, predict the recurrence of MR 6 months after URA, and demonstrate the capacity of the MV leaflets to plastically remodel within four weeks after myocardial infarction [44,49,50]. However, the principal limitation of this noninvasive technique is that it cannot be used to predict the MV postoperative state.

Given the critical clinical need for quantitative, patient-specific optimization of MV repair techniques, in this work, we aimed to develop a fully predictive MV repair computational pipeline using standard-of-care pre-operative imaging alone. We will lay out a comprehensive pipeline to first build patient-specific FE models of the full MV apparatus including subannular structures based on pre-operative clinical imaging, appropriate population-averaged human anatomical data, and informed calibration of MV tissue material properties. With this model, we will then simulate URA repair and predict the effects of the procedure on each patient's MV geometry and function. The ability to pre-operatively predict MV repair outcomes on a patient-specific basis will enable the quantitative assessment of repair strategies, tailored treatment plans, and ultimately improved patient outcomes.

2 Methods

2.1 Summary of Approach.

The complete pipeline from model generation to mechanical property calibration to final prediction is summarized in Figs. 1 and 2. Major steps in our approach include the following.

Fig. 1.

A schematic of the data analysis pipeline required to generate the full patient-specific MV apparatus including the subannular structure. The ED and ES-state MV leaflets and all identifiable MVCT origins are segmented from pre-operative rt-3DE imaging. Our previously described shape-morphing method is used to recover the full ES geometry and establish material point correspondence between the ED and ES leaflet meshes. From CT-imaged 3D reconstructions of 5 excised human hearts, we quantified the MVCT origins and insertions, as well as the MV annulus and free edge, which are used to align the points to the same global coordinate system and normalize their locations to enable intersample comparisons. Using the insertion points from all 5 MVs, we defined an MVCT insertion density map on the MV leaflets. By averaging all the MVCT origins points based on their corresponding insertion locations, we defined four representative MVCT origins. A Gaussian weighting scheme is used to map the MVCT displacements quantified from rt-3DE imaging onto the four representative MVCT origins. Finally, the patient-specific full MV apparatus was assembled by combining the pre-operative ED leaflet geometry, the MVCT insertion points as defined by the density map, the ED to ES annular displacement, and the MVCT origin displacements (either the 4 mapped representative displacements or rt-3DE displacements directly.)

Fig. 2.

A flowchart of the simulation pipeline. First, we use the patient-specific full MV apparatus with pre-operative annular displacements to calibrate the MV leaflet and MVCT material properties. We apply a transmural pressure of 100 mmHg on the ventricular surface of the MV leaflets to simulate MV closure (Initial Model). Then, we calculate the MVCT length mismatch between the simulated and ES geometries, and assign this normalized value as the updated MVCT prestrain. This step is looped until the simulation converges (Model 1). Next, we separated the MV leaflets into three zones (anterior clear zone, posterior clear zone, and coaptation region). We calculated the intersurface distance between the simulated and target geometries, and used fminsearch in matlab to minimize this error by optimizing the prestrains in these zones. Again, this step was looped until convergence (Model 2). Finally, we use these full calibrated material properties along with the patient-specific full MV apparatus with updated postoperative annular displacement (ED to annuloplasty ring) to predict the patient-specific postoperative MV state. To validate, we compared the predicted geometry and strain fields with the target ES geometry and deformation.

1. A principal challenge of quantifying the patient-specific MVCT geometry is that current rt-3DE imaging cannot yet adequately resolve the full subannular structure. Therefore, we hypothesized that population-averaged human data acquired from pressurized, CT-imaged ex vivo human hearts could be used in conjunction with patient-specific imaging to develop a practical, high-fidelity of the MV apparatus. This requires population-averaged stochastic information of the human MVCT geometry capable of being used in computational model generation. To accomplish this, we analyzed five CT-imaged excised human hearts and quantified the distribution of MVCT insertions on the MV leaflets and the MVCT origins on the papillary muscles to develop a human MVCT ‘database'. This procedure is detailed in Appendix A. Though previous studies have quantified the morphology of the mitral subannular structure, this is the first to practically detail the distribution of the human MVCT from imaging [51].

2. Next, we segmented pre-operative rt-3DE scans of three patients' MVs immediately before URA and constructed a full finite element model of the ED MV apparatus by integrating the population-averaged MVCT structure with the ED leaflet geometry. We modeled the leaflet tissue response with an initially isotropic structural constitutive material model and the MVCT tissue with an Ogden model fit to human biomechanical data [52].

3. Previous work has demonstrated that the MV leaflets are largely isotropic, likely due to the dense MVCT insertions over almost the complete leaflet area [53]. Thus an initially isotropic material model was used. It should be noted that an anisotropic strain field, such as that observed for the MV, will induce a anisotropic material response.

4. In order to functionally match the patient mechanical behavior response, we calibrated the MV leaflet and MVCT prestrains such that the output of the MV closure simulation matched the shape of the target segmented ES geometry to within 1 mm.

5. We utilized the patient-specific MV apparatus, calibrated material properties, and annuloplasty ring dimensions to simulate the URA repair and quantify the MV leaflet geometry and stress and strain distributions, as well as the MVCT axial stress after repair.

Details are provided in the following.

2.2 Patient-Specific Full Mitral Valve Apparatus

2.2.1 Patient-Specific Mitral Valve Leaflets.

Intra-operative rt-3DE scans were acquired as standard of care during URA MV repair in three patients, providing us with both pre-operative and immediate postoperative imaging data. Full volume images were acquired using Philips ie33 (Philips Medical, Andover, MA) ultrasound system equipped with a 2–7 MHz X7-2t TEE matrix transducer. The scans were imported into Philips qlab with an approximate isotropic resolution of 0.6–0.8 mm and exported as Cartesian DICOM files, then converted to the NIfTI format in ITK-Snap. The pipeline for segmenting and meshing the MV geometry from clinical TEE images has been extensively detailed in Ref. [54]. Briefly, representative frames at ED (open) and ES (closed) were selected from the pre-operative dataset, and a representative ES state frame only was selected from the postoperative dataset. These frames were segmented using an interactive matlab tracing program. The annulus was first rotated to the short-axis view, and the geometric center of the MV orifice was translated to the intersection of the intercommissural and septolateral planes. Several septolateral cross section spanning the full intercommissural diameter of the MV orifice were made at 1 mm intervals, and at each cross section, the anterior and posterior leaflets were traced as separate midsurfaces from the annulus to the free edge. Each image was segmented to include as much of the commissures as possible while using only identifiable leaflet tissue. The points in each traced curve were connected using shape-preserving piecewise cubic spline interpolation. Each curve was then rediscretized into segments of equal arc length and developed into a mesh using two-dimensional (2D) Delaunay triangulation, Poisson-disk sampling, ball-pivoting reconstruction, and Taubin smoothing algorithms. Small regional gaps at the commissures were interpolated when necessary. This procedure resulted in meshes with uniform, unstructured nodal distributions, nodes spaced approximately 1 mm apart, and approximately 2000 elements per MV mesh, which were defined as 3-node triangular shell elements (S3) in Abaqus/Explicit (Dassault Systèmes). The element-wise material directions of the leaflets were defined as previously and summarized here [54]: (1) the transmural direction $\widehat{\mathbf{t}}$ was defined as the face normal vector, oriented outwardly from the leaflet's ventricular surface; (2) the radial direction $\widehat{\mathbf{r}}$ was computed through a 90 deg rotation of $\widehat{\mathbf{t}}$ toward ${\widehat{\mathbf{e}}}_{\mathrm{z}}$ using

$$\widehat{\mathbf{r}}=-(\widehat{\mathbf{t}}·{\widehat{\mathbf{e}}}_{\mathbf{x}}\text{co}t\gamma )·{\widehat{\mathbf{e}}}_{\mathbf{x}}-(\widehat{\mathbf{t}}·{\widehat{\mathbf{e}}}_{\mathbf{y}}\text{co}t\gamma )·{\widehat{\mathbf{e}}}_{\mathbf{y}}+\text{sin}\hspace{0.17em}\gamma ·{\widehat{\mathbf{e}}}_{\mathbf{z}}$$ (1)

where $\gamma ={\text{cos}}^{-1}(\widehat{\mathbf{t}}·{\widehat{\mathbf{e}}}_{\mathbf{z}})$, such that $\widehat{\mathbf{r}}$ lies tangent to the element face and approximately orthogonal to the orthogonal boundary; (3) the circumferential direction was defined as $\widehat{\mathbf{c}}=\widehat{\mathbf{t}}\times \widehat{\mathbf{r}}$ (Fig. 3).

Fig. 3.

The MV local tissue coordinates mapped on a representative patient MV: (a)circumferential direction, $\widehat{\mathbf{c}}$ and (b) radial direction, $\widehat{\mathbf{r}}$

A critical limitation of rt-3DE is that the full coaptation zone cannot be directly visualized, and the two leaflets cannot be distinguished in this region, which precludes the accurate computation of leaflet strain and geometry mismatch [44]. A previously described and extensively validated shape-morphing technique was used to morph the ED leaflet mesh to match the ES leaflet shape, which permits recovery of the full coaptation zone and ensures material point correspondence between the ED and ES state meshes [44,49,50,54]. This fully reconstructed ES geometry was used as the target geometry in the calibration simulations (pre-operative) and for validation (postoperative).

2.2.2 Patient-Specific Identifiable Mitral Valve Chordae Tendinae Origins.

To segment all possible patient-specific MVCT origins, we measured the intercommissural diameter (IC), anteroposterior diameter (AP), and the height of the MVCT origins relative to the annular plane (z_MVCT) in ITK-SNAP in both the ED and ES frames. Then, we multiplied these three parameters with the normalized x, y, and z coordinates of the four representative MVCT origins described in Sec. A.1.2 to align the origins relative to the specific ED MV leaflet geometry (Fig. 14(b)). Next, we used the MVCT insertion density plot defined from the ex vivo human cardiac data to build MVCT insertion points on the patient-specific ED leaflet geometry. In the density plot, the MV leaflets are subidvided into 35 regions, and each region contains a set number of insertions based on the ex vivo data; for each region, we used a random generator in matlab to distribute the allotted number of MVCT insertions such that the minimum distance between two given points was no less than 0.01 mm. These insertion points were then projected onto the patient-specific ED MV leaflet. The insertion points were connected to one of the four MVCT origins depending on their location on the MV leaflet (anterolateral, anteromedial, posterolateral, posteromedial). After defining its connectivity, each individual MVCT was discretized using 1.5 mm long two-node truss elements (T2D3) in Abaqus (Dassault Systèmes).

2.2.3 Patient-Specific Boundary Conditions.

The pre-operative annular boundary condition was prescribed as previously described, namely, assuming a uniform displacement from the ED state to the ES state [54,55]. We also assigned the MVCT origin displacement from ED to ES as a boundary condition by mapping the displacement of the patient-specific rt-3DE segmented origins to the final four representative origins using Gaussian weights. Using the Gaussian function

$$w_{ij}=\frac{1}{\sigma \sqrt{2\pi }}\mathrm{e}\mathrm{xp}(-\frac{1}{2}\frac{{(x-{\mu }_j)}^2}{{\sigma }^2})$$ (2)

we set the variance σ = 10 deg and μ_j to the respective ϕ coordinate in degrees of each representative MVCT origin j = [1, 4]. Then, for each segmented rt-3DE MVCT origin i = [1, n], we calculated its weight w_ij. Using these weights, we computed the displacements x_j, y_j, z_j for each representative MVCT origin by using the segmented MVCT origin displacements x_i, y_i, z_i

$$x_j=\sum _{i=1}^n\frac{w_{ij}{x}_i}{\sum _{i=1}^n{w}_{ij}}$$ (3)

$$y_j=\sum _{i=1}^n\frac{w_{ij}{y}_i}{\sum _{i=1}^n{w}_{ij}}$$ (4)

$$z_j=\sum _{i=1}^n\frac{w_{ij}{z}_i}{\sum _{i=1}^n{w}_{ij}}$$ (5)

The integrated patient-specific ED MV leaflet geometry, MVCT structure, MVCT origin displacements, and annular displacement constitutes the final full patient-specific MV apparatus (Fig. 4).

Fig. 4.

The final patient-specific full MV apparatus, shown in the lateral and atrial views. The ED MV leaflet geometry is segmented from rt-3DE imaging; the MVCT insertions are distributed according to the insertion density map (shown in gray); and the four representative MVCT origins and their ED to ES displacements (guided by the rt-3DE traced MVCT origin displacements) are shown in blue (lateral) and red (medial). The red boundary denotes the pre-operative ES state annulus, and the arrows indicate the annular displacement boundary condition. (Color version online.)

2.3 Pre-Operative Calibration of Mitral Valve Leaflet and Mitral Valve Chordae Tendinae Prestrain

2.3.1 Adjusting Mitral Valve Leaflet and Mitral Valve Chordae Tendinae Prestrain to Tune Local Mechanical Properties.

Previous MV finite element models were developed using high-resolution micro-CT imaging of ovine MVs in the loaded and unloaded states as well as small angle light scattering imaging of the MV leaflets, which provided a high level of detail regarding both the anatomy and tissue properties of the MV structure. However, substantial differences in the material properties of human and ovine MVs have been established [52,56–58], and patient-specific material properties are impossible to obtain using standard-of-care rt-3DE imaging alone. Therefore, rather than matching the MV leaflet and MVCT tissue properties exactly, we aimed to instead tune their prestrains by iteratively simulating MV closure and minimizing the error relative to the target, rt-3DE segmented ES geometry, which would allow us to arrive at functionally equivalent, calibrated material properties for each patient.

The mechanical behavior of the MV leaflet tissues was modeled using a well-established isotropic incompressible hyperelastic structural constitutive model [44,45,49,50,54,59,60]

$$\begin{array}{c}\mathbf{S}=\frac{1}{\pi }{\int }_{-\pi /2}^{\pi /2}{S}_{\text{ens}}\left[E_{\text{ens}}(\theta )\right]\widehat{\mathbf{n}}(\theta )\otimes \hspace{0.17em}\widehat{\mathbf{n}}(\theta )d\theta \hspace{0.17em}\\ +{\mu }_{\mathrm{m}}(\mathbf{I}-C_{33}{\mathbf{C}}^{-1})\end{array}$$ (6)

where S is the second Piola–Kirchhoff (PK2) stress tensor and S_ens is the PK2 stress in a unidirectional ensemble of collagen fibers oriented along $\widehat{\mathbf{n}}=\text{cos}\hspace{0.17em}\theta \hspace{0.17em}\hspace{0.17em}\widehat{\mathbf{c}}+\text{sin}\hspace{0.17em}\theta \hspace{0.17em}\widehat{\mathbf{r}}$. μ_m is the modulus of the ground matrix. The fiber ensemble response was modeled as

$$S_{\text{ens}}(E_{\text{ens}})=\{\begin{array}{cc}{c}_0(e^{c_1\hspace{0.17em}{E}_{\text{ens}}}-1)\hfill & \forall E_{\text{ens}}\le E_{\text{ub}}\\ c_0(e^{c_1\hspace{0.17em}{E}_{\text{ub}}}-1)+c_0{c}_1{e}^{c_1\hspace{0.17em}{E}_{\text{ub}}}(E_{\text{ens}}-E_{\text{ub}})& \forall E_{\text{ens}}>E_{\text{ub}}\end{array}$$ (7)

where c₀ and c₁ are two constants that control the exponential shape of stress–strain curve. E^ub is the upper bound strain and stress–strain curve becomes linear above this limit as the fiber ensembles are recruited. We note that this form of Eq. (6) was chosen so that while it is isotropic in the unstrained state, it does allow for strain-induced anisotropy via fiber rotations and stretch. This is especially the case in the MV where the radial strains are much larger than the circumferential strains.

The prestrain of the MV leaflets was defined in the circumferential and radial material directions as λ_circ and λ_rad, respectively. These two prestrains were utilized to update the deformation matrix Fⁱ⁺¹ = FⁱF^pre after each iteration of the closure simulation, which adjusts the stiffness of the material by shifting the stress–strain curve either right or left. The entire constitutive model as well as the prestrain update was implemented in Abaqus/Explicit using a VUMAT subroutine.

The mechanical response of the MVCT was modeled using an Ogden model, with parameters fitted from data presented in Ref. [52] on the biomechanical properties of aged human MVCT

$$W=\frac{2{\mu }_1}{{\alpha }_1^2}\left({\overline{\lambda }}_1^{{\alpha }_1}+{\overline{\lambda }}_2^{{\alpha }_1}+{\overline{\lambda }}_3^{{\alpha }_1}-3\right)+\frac{1}{D_1}{(J-1)}^2$$ (8)

The model parameters for Eqs. (6), (7), and (8) are summarized below in Table 1.

Table 1.

The model parameters for equations governing the mechanical behavior of the MV leaflets (Eqs. (6), (7)) and the MVCT (Eq. (8)). MVCT: mitral valve chordae tendinae

MV leaflets (Eqs. (6) and (7))		MVCT (Eq. (8))
μm, kPa	10.11	μ 1	20.881
c0, kPa	0.0485	α, MPa	8
c 1	24.26	D1	0.0001
Eub	0.371

The MVCT prestrain was defined in the simulation as thermal expansion and the coefficient of thermal expansion was described as the fractional increase in the length per unit increase of the temperature $\alpha =\frac{1}{L}\hspace{0.17em}\hspace{0.17em}\frac{dL}{dT},$ where L is the current length and T represents temperature. The prestrain of each chordae was then computed as

$${\lambda }_i^{ps}=\text{exp}\hspace{0.17em}[\alpha (T-T_0)]$$ (9)

where T₀ is the reference temperature and T represents current temperature. If λ^ps < 1, the MVCT was shortened. Inversely, the MVCT was lengthened in the axial direction if λ^ps > 1. The MVCT prestrain was implemented in the beginning of the simulation before leaflet pressurization and boundary displacement.

2.3.2 Calibration Simulations.

The full calibration process consists of three steps: (1) Model initiation; (2) Model 1: adjustment of MVCT prestrain; and (3) Model 2: adjustment of MV leaflet prestrain.

1. The patient-specific full MV apparatus as described above defines both the initial geometry (ED leaflet geometry and MVCT structure) and boundary conditions (annular and MVCT origin displacements) for the simulation. A transmural pressure of 100 mmHg was applied on the ventricular surface of the MV leaflets as a loading condition, and the intial material properties were defined as described above (Fig. 5(a)). MV closure was simulated under these conditions and the final ES geometry was used in Model 1 to calibrate the MVCT prestrains.

2. In the second step, we compared the output of the initial model with the target ES geometry and adjusted the MVCT prestrain to minimize the difference between the two geometries (Fig. 5(b)). First, we calculated the length of each chordae on (1) the simulated geometry, L, (2) the initial ED geometry L₀, and (3) the target ES geometry L^∗ by measuring the distance between the same node to the origin position (due to the shape-morphing pipeline used to recover the full target ES geometry, the ED and ES geometries have identical connectivities). The prestrain for each chordae i = 1…m is defined as

   $${\lambda }_i^{ps}=\frac{L*-L+L_0}{L_0}$$ (10)

   As described previously, the MVCT prestrain was handled in Abaqus using thermal expansion, such that each chordae was assigned a temperature T to achieve the desired prestrain using the relation in Eq. (9). Then, another iteration of the simulation was performed using the updated MVCT prestrain, and the l² norm between the simulated and target ES geometry was computed at each node. If the change in mean l² norm was greater than 5%, then the procedure was repeated using new values for L. Otherwise, the prestrains from that iteration were taken as the final MVCT prestrain.

3. Though adjusting the chordal prestrain reduced the mean l² norm considerably, there were still regions with fairly high errors, mainly in the A2 segment. This result is largely due to the fact that there are no MVCT origins near the septum and therefore minimal MVCT in this region of the leaflets that can be adjusted to correct the shape mismatch. Consequently, it is also necessary to correct the MV leaflet prestrain directly. The material properties of the MV leaflets have been shown to be highly heterogeneous, with the anterior belly region found as anisotropic [61,62], and the coaptation region as nearly isotropic, likely due to the high density of MVCT insertions [45,63–65]. Therefore, we separated the MV leaflets into three zones: the anterior and posterior clear zones, and the coaptation region by defining a coaptation zone of 8 mm and separating the anterior and posterior leaflets depending on the direction of the elements' outward normal vectors relative to the y axis (Fig. 5(c)). In each of the three regions, we prescribed circumferential and radial prestrains, and the simulation was performed again. The outputs of the simulation were imported to matlab, where fminsearch was used to find the values of the six prestrains that minimized the l² norm. This procedure was repeated until the convergence criteria were met.

Fig. 5.

The initial model of MV closure. (a) The ramping of the simulation boundary and loading conditions, which include the annular and MVCT origin displacements and transmural pressurization on the ventricular surface of the leaflets, respectively. (b)The MVCT length mismatch between the simulated and target geometries, which forms the basis of the updated MVCT prestrain. (c) The three subregions of the MV leaflets: the anterior clear zone (blue); the posterior clear zone (red); and the coaptation region (gray). (Color version online.)

2.4 Postoperative Prediction.

In a final step, using the fully calibrated material properties from the pre-operative calibration, we simulated URA repair by modifying the annular geometry boundary condition such that the ES state annular dimensions corresponded with the ring type and size each patient received (Fig. 6). We assumed that the transmural pressure was unchanged between the pre-operative and postoperative states; that the MVCT origin displacements were the same after repair; and that the MV leaflet and MVCT prestrains do not change from the pre-operative state. Since the simulation predicts the immediate postoperative state, there has been no time for any significant growth and remodeling of the MV leaflet or myocardial tissue in the left ventricle, so we expect the tissue properties to remain unchanged. Furthermore, though the ring displaces the MV annulus relative to the MVCT origins, the papillary muscles themselves remain unchanged in the URA procedure (unless adjunctive procedures are performed, which was not the case with these three patients). The simulation result was validated by comparing both the geometry and MV leaflet strain fields with the target postoperative ES geometry, which was obtained in a similar manner to the target pre-operative ES geometry via rt-3DE segmentation and then the shape-morphing pipeline to recover the full leaflet coaptation zone.

Fig. 6.

The postoperative annuloplasty ring boundary condition. (a) The Medtronic Profile 3D ring, one of the devices that the 3 patients received. (b) The postoperative annular boundary, from the target post-ES leaflet geometry. (c) The annular displacement from the pre-operative ED state to the ring dimensions.

2.5 Parametric Study of Mitral Valve Chordae Tendinae Origin and Insertion Distributions.

We performed a parametric study on one of the MVs by varying the MVCT insertion and origin points to assess the impact of variable subannular structures on the results. We analyzed the following five configurations:

• C1 (a) 120 insertions distributed using the population-averaged density map.

• (b) Four representative MVCT origins.

• (c) MV leaflets separated into three zones (anterior belly, posterior belly, coaptation region)

• C2 (a) 200 insertions distributed uniformly, with a density of 17.9/cm², following optimal results from a previous publication [66].

• (b) Four representative MVCT origins.

• (c) Three MV leaflet zones.

• C3 (a) 120 insertions distributed using the population-averaged density map.

• (b) Four representative MVCT origins.

• (c) MV leaflets separated into four zones (anterior belly, posterior belly, anterior coaptation, posterior coaptation).

• C4 (a) 120 insertions distributed using the population-averaged density map.

• (b) rt-3DE traced identifiable MVCT origins, where each insertion point connects to its closest origin.

• (c) Three MV leaflet zones.

• C5 (a) 160 insertions distributed using the population-averaged density map.

• (b) rt-3DE traced identifiable MVCT origins.

• (c) Three MV leaflet zones

We initially developed the pipeline as described above with anatomically distributed MVCT insertions and representative MVCT origins with one MV, then conducted the parametric studies to assess how this control configuration performed against the others. Based on the results of this parametric study, we performed the full analysis (from pre-operative calibration to postoperative prediction) of all three MVs using the most optimal configuration (C5).

3 Results

3.1 Pre-Operative Calibration Results.

In the initial model performed without any prestrain, we observed that in the MVCT insertion zone, which is near the free edge of the leaflets, the simulated geometry tends to undershoot the target, suggesting that the material properties overestimate both MVCT and MV leaflet stiffness in this region. Conversely, in the anterior belly region where there are almost no chords, the simulated geometry substantially overshoots the target and does not capture its curvature, implying that in this region the MVCT and MV leaflet stiffness may be underestimated. The mean l² norm between the target and simulated geometries was 1.421 mm, which was substantially greater than our accepted minimum value of 1 mm (roughly the human MV leaflet thickness) (Fig. 7(a)).

Fig. 7.

(a) The l² norm between the simulated geometry and the target ES geometry. The mean intersurface distance is 1.421 mm, and there is a notably large region of mismatch in the anterior belly, likely because there are comparably few insertion points in this region. (b) The l² norm after the updated MVCT prestrain is incorporated and the simulation is looped to convergence. The mean intersurface distance has decreased to 0.832 mm, and while the anterior belly region still demonstrates notable mismatch, both the spread and intensity of the error have decreased. (c) The l² norm after the updated MV leaflet prestrain is incorporated and the simulation is looped to convergence. The mean intersurface distance has dropped to 0.577 mm, which is below our acceptable maximum of 1 mm. Moreover, the error in the anterior belly has significantly decreased.

In the following step, MVCT prestrain was introduced into the simulation following the pipeline described in Sec. 2.3.1. Since the MV apparatus is a highly complex, nonlinear, and coupled structure, at least four running loops were needed to update the MVCT prestrains until convergence was reached. Adjusting the MVCT prestrain permitted a much closer overall match and even better captured the curvature of the anterior leaflet. However, the mean l² norm was 0.832 mm, so we continued to incorporate MV leaflet prestrain (Fig. 7(b)).

Finally, we incorporated the MV leaflet circumferential and radial prestrain in the three regions: anterior belly, posterior belly, and coaptation region. We found that 3 iterations were necessary for convergence. The mean l² norm was 0.577 mm, which is a substantial improvement over model 1 (Fig. 7(c)). Moreover, the anisotropy in the belly zones was validated in that the prestrain in the circumferential direction was much larger than the radial direction. Both prestrains in the coaptation region were around 1, which corresponds with prior findings that this zone is nearly isotropic. The MV leaflet prestrains are listed in Table 2.

Table 2.

The final circumferential and radial prestrains for each of the three subregions of the MV leaflets, and the mean and standard deviation of the MVCT prestrains

Anterior clear zone		Posterior clear zone		Coaptation region		MVCT
λ circ	λ rad	λcirc	λ rad	λ circ	λ rad	λps
1.265	0.963	1.335	0.905	0.967	1.017	−3.909 ± 17.597

The circumferential prestrains are greater than the radial prestrains in both the anterior and posterior clear zones, indicating high anisotropy in these regions. Conversely, both the circumferential and radial prestrains in the coaptation region are near 1, corresponding with previous work demonstrating that this region is nearly isotropic due to its numerous MVCT insertions. MV: mitral valve, MVCT: mitral valve chordae tendinae.

3.2 Parametric Study Results.

The results of the parametric study are summarized in Table 3. We observed that increasing chordal density either in the anatomical or uniform distributions had a substantial improvement in reducing the mean l² norm in the pre-operative calibration (configs. 2 and 5). However, the rt-3DE traced versus representative MVCT origin configurations did not have a major effect (config. 4). Separating the coaptation region by leaflet and assigning independent prestrains also had a limited impact on the results (config. 3). These findings demonstrated that increasing MVCT density in the pre-operative calibration stage may improve errors, and further studies could instead utilize a uniform MVCT insertion density of 17.9/cm² without loss of accuracy.

Table 3.

The final mean pre- and postoperative l² norms of the parametric study on one patient MV for each of the five configurations

	C1	C2	C3	C4	C5
Insertions	Anatomical (120)	Uniform (200)	Anatomical (120)	Anatomical (120)	Anatomical (160)
Origins	Representative	Representative	Representative	rt-3DE	rt-3DE
Coaptation	Combined	Combined	Separated	Combined	Combined
mean l2 norm (pre-op. calibration)	0.577 mm	0.383 mm	0.487 mm	0.456 mm	0.412 mm
mean l2 norm (postop. prediction)	0.892 mm	0.910 mm	0.875 mm	0.885 mm	0.934 mm

With regards to the effect of the different configurations on the postoperative predictions, we noted the least differences between the control configuration and the one with a subdivided coaptation region (config. 3). Similar to the pre-operative results, this outcome indicates that the separated coaptation region will not provide a substantial improvement to the simulation. Moreover, though increasing MVCT density and rt-3DE traced MVCT origins both helped reduce the maximum l² norm, the mean l² norm stayed about the same. Therefore, while the rt-3DE traced MVCT origins and the anatomically distributed MVCT insertions can provide an improved estimation, the representative MVCT origins and uniformly distribution insertion points can be substituted if detailed anatomical and clinical information is not available. Based on these results, we proceeded with config. 5 for our postoperative predictions, and the updated pre-operative calibration results for all 3 MVs are presented in Fig. 8.

Fig. 8.

The final results from the pre-operative calibration using 160 anatomically distributed MVCT and rt-3DE traced MVCT origins for all 3 patient MVs (MV1: recurrent; MV2: recurrent; MV3: nonrecurrent). (a) The simulated geometry overlaid with the target pre-operative ES geometry. (b) The l² norm between the simulated and target geometries, demonstrates close agreement after calibration of the MV leaflet and MVCT prestrains.

3.3 Postoperative Prediction Results.

After incorporating the updated MV leaflet and MVCT prestrains and updating the annular boundary condition to match the annuloplasty ring dimensions, we observed that our predictive simulation was able to match the target post-ES geometry to within 1 mm mean l² norm in all three patients (Fig. 9). We also analyzed the axial S11 stress in the MVCT and noted substantial differences in the chordal stress distributions pre- and postoperatively between a patient who had recurrent MR at 6 months and a patient who did not (Fig. 10(a)). Namely, though both patients had significantly elevated chordal stress on the anterior leaflet prior to repair, in the nonrecurrent patient, this tension was alleviated after the ring was implanted, while in the recurrent patients, the posterior chordal axial stress increased nearly three-fold (Fig. 10(b)). This result confirms findings from other groups [40], and likely results from exaggerated posterior leaflet tethering due to the ring excessively displacing the posterior annulus toward the LV septum.

Fig. 9.

The predicted postoperative geometries using the pre-operative ED MV leaflet geometry, calibrated prestrains, annuloplasty ring boundary condition, anatomically distributed MVCT and rt-3DE traced MVCT origins for all 3 patient MVs (MV1: recurrent; MV2: recurrent; MV3: nonrecurrent). (a) The predicted ES geometry overlaid with the target postoperative ES geometry segmented from rt-3DE. (b) The l² norm between the predicted and target geometries, demonstrating submillimeter agreement.

Fig. 10.

MVCT axial S11 stress in the three patient MVs before and after repair. The MVs are shown in the ventricular view, with MVCT axial stress above 3 MPa highlighted in red and MVCT with stress less than 0.5 MPa not plotted. All three MVs demonstrate high MVCT stress in the anterior leaflet pre-repair. After repair, this stress is offloaded in the nonrecurrent MV, but in both recurrent MVs, this stress is transferred to the posterior leaflet, likely due to exacerbated posterior leaflet tethering imposed by the ring. AL: anterior leaflet; PL: posterior leaflet.

3.4 Mitral Valve Leaflet Stress.

The predictive postoperative simulation is also able to capture a substantial reduction in von Mises leaflet stress after URA repair in all 3 patients (Fig. 11(b)), a finding which has been echoed in previous literature [67,68]. The stress is reduced across all six Carpentier segments in the three MVs, except the A1 segment of the MV2 (Fig. 11(b)), and the reduction is similar between recurrent and nonrecurrent cases. In conjunction with the results of the MVCT stress, these findings suggest that normalization of MV leaflet stress alone may not be an adequate target for repair optimization, and that an integrated approach to the functional behavior of the full MV apparatus is crucial to developing a rigorous understanding of the mechanical consequences of repair techniques and predicting outcomes.

Fig. 11.

The MV leaflet von Mises stress for the three patient MVs (MV1: recurrent; MV2: recurrent; MV3: nonrecurrent). (a) The presurgical von Mises stress distributions. (b) The postsurgical von Mises stress distributions. (c) The relative change in von Mises stress between the pre- and postsurgical states by Carpentier segment for the three patient MVs.

3.5 Validation.

The submillimeter l² norms of the postoperative predicted ES geometries and the target post-ES geometries confirm the technique's capability to reproduce the immediate postoperative geometry with high fidelity. We also compared both the pre-operative and postoperative circumferential and radial strain fields with the strain fields computed using our shape-morphing technique [49,54,60] and found minimal difference in mean strains in both states across both directions, as well as close correspondence of local tissue behavior in all four strain fields (Fig. 12).

Fig. 12.

The circumferential and radial MV leaflet strain fields pre- and post-repair computed using our extensively validated noninvasive strain estimation technique and the full finite-element pipeline. The means of the strain fields correspond closely in both directions, both for the pre-operative simulations and the postoperative predictions.

4 Discussion

4.1 General Overview.

The main objective of the present study was to demonstrate that the patient-specific post-repair MV state can be accurately predicted from strictly pre-operative clinical data. This approach was predicated on a rigorous understanding of MV mechanics previous decade of prior work [45,47–49,54,55, 64,66,69–77]. In the present work, we were motivated by an urgent clinical need for quantitative treatment planning strategies, particularly in the face of rapidly proliferating MV repair approaches and devices, the highly combinatorial nature of MV repair, and suboptimal outcomes reported in major randomized controlled trials. Though this need has spurred prolific inquiry into prognostic factors of repair success, these metrics largely reduce the complexities of MV function and behavior to rt-3DE-based geometric measurements and as such are at once overly restrictive and inadequate [78]. Previous work has demonstrated that even the uniform stress applied by an annuloplasty ring contributes to biosynthetic changes in the MV leaflets, and that altered homeostasis and mechanical loading of the MV can induce plastic deformation of the leaflets, which are likely drivers of repair failure [69,79–81]. Consequently, the ability to predict the functional state of the complete MV apparatus enables a multidimensional and direct assessment of the proposed repair, as well as the possibility for pre-operative optimization of the treatment strategy for each patient.

The novel computational pipeline presented in this study constitutes several significant advancements in the effort toward patient-specific, predictive MV repair simulations, namely:

1. This pipeline is based strictly on human data, and incorporates both patient-specific information from clinical imaging and population-averaged data from an anatomical analysis of imaged ex vivo human hearts.

2. We established detailed distributions of the human MVCT structure, which to the best of our knowledge, was not previously known in humans.

3. We developed a novel calibration method to practically determine the functional patient-specific properties by adjusting MV leaflet and MVCT prestrains, rather than running a full inverse problem.

4. We simulated and predicted the complete post-surgical state of the patient MV, including geometry, deformation, and leaflet and MVCT stress, using standard-of-care pre-operative imaging alone.

4.2 Population-Averaged Distribution of Human Mitral Valve Chordae Tendinae Structure.

As mentioned earlier, a principal challenge of quantifying the patient-specific MVCT geometry is that current rt-3DE imaging cannot yet adequately resolve the full subannular structure. We have previously demonstrated that in order to build a high-fidelity model of the MV, the sub-annular structure is critical for accurate predictions of MV function [45]. The subannular structure is especially crucial for modeling pathological behavior such as leaflet tethering or functional MR which are directly related to the coupling of the LV and MV via the papillary muscles and MVCT [82]. Our previous work has focused on rigorously detailing the ovine MVCT distributions from micro-CT imaging [45,48]. However, there are substantial differences between ovine and human MV anatomy, the most salient being the positions of the papillary muscles and the number of MVCT origins [83]. Furthermore, though morphological studies on human MV anatomy have been conducted on cadaver hearts [51,84–86], the relative positions of various elements of the subannular structure can be distorted in ex vivo measurements, due to tissue prestrain and LV preload in the in vivo heart [87,88].

Therefore, we hypothesized that population-averaged human data acquired from pressurized, CT-imaged ex vivo human hearts could be used in conjunction with patient-specific imaging to develop a practical, high-fidelity model of the MV apparatus. Specifically, we quantified the anatomical MVCT insertion density and representative MVCT origin distributions and integrated these findings with image-derived MV leaflet geometries and MVCT displacements to build the patient-specific MV apparatus model. We noted that the human MVCT insertion distribution differs substantially from the ovine distribution, specifically in that the ovine insertions are more uniformly distributed, while human MVCT insertions are denser in the coaptation zone and relatively sparse in the clear zone [48]. Our parametric study on various configurations for the subannular structure demonstrated that though anatomically distributed MVCT insertions and rt-3DE segmented MVCT origin positions provided the closest overall match with the target geometries, a uniform MVCT distribution, and representative MVCT origin positions can also be used without significant loss of accuracy in situations where the origins are difficult to visualize on clinical imaging (such as due to artifact or non-full volume images).

4.3 Simplified Inverse Modeling Approach.

Additionally, rt-3DE imaging precludes the determination of patient-specific tissue mechanical properties, so we circumvented this limitation by employing simplified inverse modeling approaches to calibrate the functional mechanical behavior of each patient's MV. Initially, we assigned an established structural constitutive model for the MV leaflets and an Ogden model with parameters fitted to aged human MVCT (corresponding to the elevated average age of the MR patient population). Then, we iteratively tuned the prestrains of each MVCT and of three subregions of the MV leaflets to minimize the mismatch of the final simulated geometry and the target pre-operative ES state segmented from clinical imaging. This approach allowed us to model the highly complex behavior of the MV apparatus in a tractable, and most importantly, customizable manner such that we could noninvasively determine effective mechanical properties for each patient. The final prestrains confirmed previously published results on the regional variation of the MV leaflets, notably the high degree of anisotropy in the anterior and posterior belly regions with a nearly isotropic coaptation region (due to the high density of MVCT insertions) [64,65].

4.4 Key Findings.

With this fully calibrated model, we modified the annular boundary displacements to match the annuloplasty ring size and simulated the immediate effect of URA on three patients' MVs. In all patient cases, the simulated pre-operative and predicted postoperative ES geometries were within 1 mm of the target ES geometry, and the predicted MV leaflet strain fields corresponded closely to the target in both the circumferential and radial directions. Additionally, we were able to predict reduced MV leaflet von Mises stress after the URA procedure, which aligns with previous findings in the literature [67,68]. Moreover, we predicted clinically significant differences in MVCT axial stress between nonrecurrent and recurrent patients, namely, in the postoperative shift of MVCT loading due to the ring. In the nonrecurrent patient, the high anterior MVCT stress was almost completely offloaded after repair, while in the recurrent patients, this stress shifted to the posterior leaflet. This shift indicates deleterious posterior leaflet tethering, which has been postulated as a major driver of long-term URA repair failure [40,89]. Our validation with results from a previously established computational pipeline that requires postoperative imaging underscored the capability of the technique presented here to achieve highly similar geometric and deformation results utilizing pre-operative clinical data alone.

4.5 Clinical Implications.

The capacity to predict patient-specific MV functional states has enormous implications in clinical practice, particularly in the field of MV repair in MR, where clinical decision making is complicated by vastly heterogeneous pre-operative presentations and generally unpredictable long-term outcomes. This study represents a new stage in three decades of MV modeling, where patient-specific simulations can inform clinical decision-making, rather than act purely as a post-hoc analysis tool. In particular, though this work focused on URA, which remains a popular repair approach, the technique is entirely repair-agnostic and can be used to optimize any given repair approach, such as transcatheter edge-to-edge repair or MVCT-focused adjunctive procedures [90]. In certain patients, the predictive pipeline may also be used to contraindicate repair entirely and build a quantitative case for total valve replacement.

Though our study focused on the immediate postoperative state, our observations regarding exacerbated posterior chordal stress and posterior leaflet tethering correlate strongly with previous findings explicating a possible mechanism underlying long-term post-repair MR recurrence [40]. Therefore, minimizing posterior chordal stress could potentially be used as a target in patient selection or to optimize the repair configurations (i.e., ring size, shape, etc.) and indeed already motivates adjunctive procedures such as papillary muscle relocation whose aim is to relieve the excess stress imposed on the MVCT or MV leaflets by URA [91]. Recent investigations into leaflet tethering as a potential prognostic factor of recurrence have not been successful in differentiating nonrecurrent and recurrent groups, likely because unindexed, echo-based geometric metrics of MV function are inherently suboptimal due to pre-existing differences in patient size, rt-3DE spatial resolution, and variations in operator error [41]. Moreover, the behavior of the MV, particularly the diseased MV, is highly complex and is unlikely to be meaningfully captured with geometric measurements alone. A functional approach, like the one presented in this study, which relies strictly on the same pre-operative data, may be able to overcome these critical limitations and provide a more nuanced and insightful understanding into MV behavior in disease and after repair.

To our knowledge, this study is also the first to quantify the MVCT insertion and origin distributions using human cardiac imaging data. The results of our analysis enable the practical integration of anatomical MVCT structures with patient-specific geometries, which is particularly useful in clinical applications, where standard-of-care imaging precludes the direct visualization of the patient-specific subannular apparatus. Though our parametric study demonstrated that identifiable MVCT origins segmented from rt-3DE imaging improve the accuracy of the simulation, the difference between the representative and imaged origins is slim, meaning that in situations with time constraints or where the MVCT origins are especially difficult to visualize on rt-3DE imaging, the representative origins could be used instead without significantly impacting the results. These findings bridge the gap between standard-of-care imaging and the need for high-fidelity, integrated simulations of the full MV behavior, which are critical for quantitative patient-specific optimization of long-term repair outcomes.

4.6 Limitations.

We were able to demonstrate excellent agreement with the predicted output and target, both in terms of geometry and deformation for three patients. The human sample size was limited in this study, and larger populations will certainly be needed to evaluate the greater prognostic abilities of this approach. In order to practically analyze a larger database of patient images, semi- or fully-automated segmentation techniques will be necessary [92,93]. Next, though the native human MVCT has a complex branching pattern, we implemented branchless MVCT with constant cross section into this model. However, the objective was not to directly recapitulate the exact patient-specific chordal structure (which cannot be visualized or validated from rt-3DE) but rather to develop a functionally-equivalent structure that can accurately capture the MV closing behavior. Additionally, we created the full number of MVCT as dictated by the population-averaged distribution, but some MVCT carried negligible loads and could be omitted from the model creation. Several iterations of geometric optimization, likely on a case-by-case basis, would be necessary to achieve a more economical distribution. Lastly, because the MVCT origins could not be visualized in the postoperative rt-3DE imaging due to artifact from the implanted ring, we assumed that the MVCT origin displacements do not change immediately after repair, based on an understanding of the mechanics of the repair and existing literature on resultant posterior leaflet tethering. However, with a fully integrated time-evolving LV–MV model of myocardial infarction, we would be able to validate this assumption, and future work is progressing in this direction.

4.7 Conclusions.

In the current study, we developed a full patient-specific MV model including the subannular structures and a predictive MV repair simulation pipeline based strictly on population-averaged data and standard-of-care, pre-operative imaging. We were able to demonstrate a close match between our postoperative predictions and the target segmentations in both geometry and deformation. Additionally, we predicted the consequences of this repair method on the functional behavior of the full MV apparatus and using these results were able to distinguish between patients whose repair did or did not fail 6 months later. Though this pipeline was initially tested on surgical URA repair, we are currently applying it to patients who have undergone repair using contemporary transcatheter approaches. Overall, this pipeline addresses an urgent clinical need for quantitative treatment planning strategies and thus lays the foundation for tailored patient selection and ultimately, a more durable repair.

Appendix A

Appendix A1 Patient-Specific Full Mitral Valve Apparatus

Appendix A 1.1. Ex Vivo Human Hearts: Data Processing.

To quantitatively inform the construction of the human MV subannular apparatus, we studied 3D reconstructions of five excised human hearts from the Visible Heart Laboratories at the University of Minnesota. These excised hearts were fixed in 10% buffered formalin for a period of 24–48 h under 40–50 mmHg hydrostatic pressure to maintain their ED shape. Then, these specimens were CT scanned and 3D models of the complete cardiac anatomy were generated using mimics Software (Materialise, NV) (Fig. 13(a)). In order to analyze the MV subannular structure, we imported the STL files of the cardiac geometries into Paraview (Kitware Inc., Clifton Park, NY), and isolated the full MV apparatus including the annulus, leaflets, MVCT, and papillary muscles from the surrounding myocardium. Next, we manually traced the annulus and free edge, and selected the MVCT insertion points on the MV leaflets, and the MVCT origins on the papillary muscles (Fig. 13(c)). We found that there were approximately 20 origins per heart (Table 4). Due to limitations of the CT imaging, it was impossible to distinguish each individual MVCT insertion, so we supplemented our analysis with data from previous literature and noted that there are approximately 120 insertions in the human heart [51].

Fig. 13.

The MV apparatus anatomy from the rt-3DE patient scans and ex vivo CT-imaged human hearts. (a) The subannular structure of a representative CT-imaged ex vivo cardiac reconstruction. MVCT are colored in cyan, and papillary muscles are highlighted in red. Black arrows indicate a subset of MVCT insertions and origins. (b)Arepresentative patient-specific ED leaflet geometry segmented and meshed from rt-3DE imaging, shown in the atrial view. The Carpentier segments are labeled, and the angular coordinate ϕ is shown. Note that ϕ = 0 approximately bisects the A2 segment. (c) A 2D representation of an ex vivo MV with the Carpentier segments and angular coordinate marked, and its MVCT insertions and corresponding medial and lateral MVCT origins are shown in gray, red, and blue, respectively. (Color version online.)

Table 4.

Number of papillary muscle origins in each human heart

	Posterior	Commissural	Anterior	Total
Heart 1	9	0	9	18
Heart 2	10	1	9	20
Heart 3	10	0	8	18
Heart 4	13	0	9	22
Heart 5	12	0	9	21

To co-register all five hearts, we mapped all the data points to the same global coordinate system as follows. First, we centered the annuli at the origin. Next, we aligned the intercommissural axis to the x axis, such that the positive x direction corresponded to the medial side of the MV. Finally, a plane was fitted through the annular points, and its normal direction was defined as n_ann; this normal vector was then rotated to align with the positive z direction. In order to facilitate analysis of the insertion and origin points relative to the MV geometry, we reparametrized the 3D coordinate system to a 2D mapping system in polar coordinates (ϕ z), where ϕ = 0 was defined as the positive y axis and z was the z position of each point in the 3D global coordinate system (Fig. 13(b)). We fitted two cubic splines through the annular and free edge points respectively, and discretized them into 100 intervals with uniform arc lengths starting at ϕ = 0. The chordal insertion points were projected onto the surface formed between the annular and free edge splines by computing the distance between the annulus and free edge at that insertion's ϕ value and the distance from the insertion to the free edge, and remapping the insertion onto the 2D surface such that the ratio between the two lengths was preserved. Finally, as all five hearts vary in dimension, it was necessary to normalize the MV geometry in order to make valid comparisons across the data. We calculated the intercomissural and anteroposterior diameters, as well as the average z values of all the MVCT origins, which allowed us to normalize all coordinates in the x, y, and z directions, respectively.

Appendix A.1.2. Population-Averaged Mitral Valve Chordae Tendinae Origins and Insertions.

As the MVCT are of low mass and narrow profile, it is not presently possible to reliably segment all MVCT origins in a patient's heart on rt-3DE imaging. Therefore, we generated representative anatomical MVCT origins using the human data from the ex-vivo cardiac geometries. We grouped the MVCT origin positions from Sec. A.1.1 by their respective insertion locations i.e., anterolateral, anteromedial, posterolateral, and posteromedial. By averaging the points within each group, we thus defined four representative MVCT origins (Fig. 14). Similarly, MVCT insertion points cannot be defined using rt-3DE imaging (in whole or in part), so we defined an MVCT insertion density map in order to guide the distribution of the MVCT insertions on given patient-specific MV leaflet geometries (Fig. 15). To account for variation in leaflet size, we calculated a length ratio $\overline{L}$ for each insertion point at its radian position ϕ relative to the length of the leaflet

$$\overline{L}(\varphi )=\frac{z_{\mathrm{insertion}}\left(\varphi \right)-z_{\mathrm{freeedge}}\left(\varphi \right)}{z_{\mathrm{annulus}}\left(\varphi \right)-z_{\mathrm{freeedge}}\left(\varphi \right)}$$

Using this ratio, we then plotted the MVCT insertions based on their radian position relative to the leaflet length and subdivided this map into five groups for each standard Carpentier segment (note that A2 is bisected by ϕ = 0). By quantifying the number of insertion points within each boundary, we were able to generate an insertion density map which dictates the number of insertions in 35 zones across the MV leaflet geometry. As expected, the highest density of insertion points is near the free edge (the maginal chordae) compared to the annulus (basal chordae), and the A2 and P2 segments have more insertions than the A1, A3, P1, and P3 segments.

Fig. 14.

The MVCT origins from both the ex vivo human hearts and a representative patient-specific MV. (a)All the MVCT origins from all five hearts are shown in gray and are grouped based on the location of their respective insertions (anterolateral, anteromedial, posterolateral, posteromedial). The averages of these groups define the four representative origins: the lateral origins are shown in blue; the medial in red; the anterior with filled circle symbols; and the posterior with empty square symbols. (b) A representative patient-specific ED MV leaflet geometry is shown in the atrial view with six identifiable MVCT origins segmented from echo. These origins are tracked from the ED to the ES frames, and their displacements are weighted using a Gaussian function to guide the displacement of the four representative origins. Lateral origins are shown in blue, and medial in red. (Color version online.)

Fig. 15.

The MVCT insertions from all five ex vivo human hearts, and the resulting MVCT insertion density map. (a) All insertions from the five ex vivo human hearts plotted on the averaged 2D MV geometry. Note the higher density of insertions near the free edge, and the relatively sparse clear zone closer to the annulus. (b) The same data plotted on a normalized domain, where $\overline{L}$ represents the distance of a given insertion point from the free edge normalized by the distance of the free edge to the annulus at that ϕ value. The domain is discretized into 35 zones by Carpentier segment and $\overline{L}$ intervals of 0.2. (c) The MVCT insertion map, which dictates how many MVCT are found in each of the 35 regions. This normalized mapping can then be used to generate an anatomical distribution of MVCT insertions on a given patient-specific ED leaflet geometry.

Funding Data

• National Institute of Health/National Heart, Lung, and Blood Institute (Grant Nos. HL129077 and HL119297; Funder ID: 10.13039/100000050).

Nomenclature

λ₁, λ₂, λ₃ =

principal stretches

C =

right Cauchy-Green stress tensor

S =

2nd Piola-Kirchoff stress tensor (PK2)

D₁ =

incompressibilty factor

J =

volume ratio

S_ens =

PK2 stress in the direction of a fiber ensemble