Effect of Parametric Variation of Chordae Tendineae Structure on Simulated Atrioventricular Valve Closure

Abstract

Purpose:

Many approaches have been used to model chordae tendineae geometries in finite element simulations of atrioventricular heart valves. Unfortunately, current “functional” chordae tendineae geometries lack anatomical fidelity (e.g., branching) that would be helpful when informing clinical decisions. The objectives of this work are (i) to improve synthetic chordae tendineae geometric fidelity to consider branching and (ii) to evaluate how the chordae tendineae geometry affects finite element simulations of valve closure.

Methods:

In this work, we develop an open-source method to construct synthetic chordae tendineae geometries in the SlicerHeart Extension of 3D Slicer. The generated geometries are then used in FEBio finite element simulations of atrioventricular valve function to evaluate how variations in chordae tendineae geometry influence valve behavior. Effects are evaluated using functional and mechanical metrics. Three synthetic geometries were used (healthy mitral valve, mitral valve with annular dilation, and a tricuspid valve) and a patient specific healthy mitral valve.

Results:

Our findings demonstrated that altering the chordae tendineae geometry of a stereotypical mitral valve led to changes in clinically relevant valve metrics (regurgitant orifice area, contact area, and billowing volume) and valve mechanics (first principal strains). Specifically, cross sectional area had influence over 9 out of 10 valve closure metrics, followed by chordae tendineae density, length, radius and branches. The effects of these parameters can be drastic. For example, when primary chord density is decreased ROA increased by 75% relative to baseline.

Conclusion:

This study presents a flexible, open-source method for generating synthetic chordae tendineae with realistic branching structures. Further, we establish relationships between the chordae tendineae geometry and valve functional/mechanical metrics. This research contribution helps enrich our open-source workflow and brings the finite element simulations closer to use in a patient-specific clinical setting.

1. Introduction

Atrioventricular valve (AVV) disease is associated with significant morbidity and mortality in both the developing and developed world. Mitral valve (MVR) and tricuspid valve regurgitation (TVR) are thought to affect 5 million [1] and 1.6 million [2] people in the U.S., respectively. A healthy MV and TV will adequately control the flow of blood between the atria and ventricles on the left and right sides of the heart. Consequently, failure of these AVVs can result in inefficient blood circulation leading to end-organ dysfunction (e.g., liver, lung, kidney), or death. MVR is associated with an increased risk of congestive heart failure and stroke and is the most prevalent form of AVV failure [3, 4]. The TV, while previously described as the “forgotten valve”, has increasingly been the subject of important investigations [5] as part of an effort to better treat TV dysfunction. AVV failure is similarly devastating to both children and adults with congenital heart disease, and in particular to single ventricle patients [6, 7].

The core anatomy of an AVV consists of leaflets, a ring-shaped annulus attaching the leaflets to the heart wall, and chordae tendineae (CDT) that anchor the leaflets to the papillary muscles (PM) in the ventricle wall (Fig. 1 panel (i)). Morphological changes or failure of any of these components can lead to an incompetent valve that allows blood to flow backward from the ventricle to the atrium during ventricular contraction, leading to atrioventricular valve regurgitation (AVVR) [8, 9]. Based on etiology, the factors of AVVR have been categorized as either primary (e.g., direct) or secondary (e.g., indirect) in terms of its effects on the AVV [3, 4, 10]. Myxomatous valve degeneration is an example of primary AVVR that is characterized by elongated CDT and prolapsing leaflets stemming from decreased stiffness of both CDT and leaflet tissue [11, 12]. In contrast, ischemic cardiomyopathy is an example of secondary AVVR related to left ventricular dilation and remodeling [3, 13, 14].

Fig. 1:

Panel (i) shows an unloaded mitral valve with key anatomical features. Panel (ii) shows the four valves used in this study; stereotypical mitral valve (MV), stereotypical annular dilated MV, stereotypical tricuspid valve (TV) and a patient specific MV. Panel (iii) shows the four valve metrics that were identified to quantify valve closure. A MV in its closed state is used to visualize strain, contact area (CA), and billowing volume, while a dilated MV geometry is used to show the regurgitant orifice area (ROA). Strain measurements are represented with a color map, CA, and ROA are shown in red, and the billowing volume is shown as the leaflet above the red annulus plane. On panel (iv) is the method for varying the parameters of the chordae tendineae. Baseline topology can be modified by CDT density, length, number of branches, and branching radius.

In both primary and secondary AVVR cases, impaired CDT can contribute to valve dysfunction and failure given the essential role they play in normal AVV function [15, 16]. In primary AVVR, CDT rupture can result in leaflet prolapse. While in secondary cases, elevated chordal tension in the setting of ventricular dilation, or ischemia-related alterations to the ventricle, can lead to CDT restriction of leaflet motion and reduction in leaflet contact (coaptation). Congenitally abnormal CDT structures in children, such as those seen in parachute or arcade AVV [17, 18], play a deleterious role in valve function. Consequently, chordal tension and direction plays a significant part in the successful repair of complex congenital heart disease with valvular abnormalities such as complete atrioventricular canal and hypoplastic left heart syndrome [6, 15, 19–22].

Given the essential role of CDT in valve function, significant effort has been made to develop therapies directed at replacing or altering chordal geometry to achieve improved AVV function. These interventions are dependent on CDT mechanics, especially as related to their physical classification. Specifically, CDT classified as “primary” are inserted within leaflet free edge and prevent leaflet prolapse during ventricular contraction. Secondary CDT, embedded in leaflet body, support the mechanical interaction between AVV and ventricle, with additional stability within the leaflet basal area provided by tertiary CDT [23–26]. Thus, precise assessment of insertion location, and optimal chordal length, is necessary to understand valve function and to achieve desired therapeutic outcomes. This includes surgical interventions such as: neochordae reconstruction where artificial primary CDT are created to restore function to a prolapsing valve [16]; and PM relocation where CDT-to-leaflet insertion vectors are modified to improve valve function [21, 22, 27–29]).

In this context, realistic representation and fidelity of CDT topology is also critical to emerging computational simulations of AVV function [30]. Notably, finite element (FE) and other multi-physics simulations of AVV function are uniquely positioned to investigate mechanisms of valve dysfunction. In addition, when combined with patient-specific three-dimensional (3D) images (e.g., 3D echocardiography, computed tomography, cardiac magnetic resonance imaging), these computational techniques have the potential to inform iterative preoperative optimization of valve repair “in silico” [30–35]. However, while advances in 3D imaging provide a reliable and noninvasive means of determining valve leaflet structure [15, 36–40], modern clinical imaging cannot yet reliably resolve CDT geometry at their individual chordal insertions with the accuracy and fidelity of “ground truth” geometries achieved from ex vivo and pathological specimens [30, 41, 42].

As a result, significant effort has gone into creating image-informed [27, 43–45] or functionally equivalent [42] CDT topology for the computational simulation of valve function. Most relevant to the present work, Khalighi et al. [42] demonstrated that CDT geometries could be simplified to random insertions of CDT-like structures without branching. These functionally equivalent CDT topologies provided similar predictions as CDT geometries created from high-fidelity ex vivo images of ovine hearts. We developed our open-source finite element simulation framework [8, 46, 47] for atrioventricular valves using these simplified geometries. Although the simplified CDT geometries could predict valve behavior, they do not have realistic features (i.e., branching) necessary to model CDT-related pathology or interventions (e.g., chordal rupture or neochordae). As such, others have created FE models with various implementations of branching CDT geometries [33, 34, 48]. However, the sensitivity of FE simulations of valve function to variations in chordal branching patterns and distributions remains unknown. Furthermore, there is not an existing procedural method for the precise parametric and reproducible creation of specific chordal branching patterns, geometries, and density.

The main objective of this work is to add additional CDT modeling capabilities to our open-source atrioventricular valve simulation framework that combines SlicerHeart and FEBio. Additionally, we aim to provide a baseline understanding of this tool and how CDT influence simulation predictions by: (i) comparing FE simulation results of the more realistic branched CDT geometries to simpler non-branching CDT geometries previously used by ourselves [32, 46] and others [42], (iii) defining how variations in CDT geometry affect FE predictions of parametric AVV closure, (iv) comparing FE simulation results of a patient specific valve to its in-vivo imaging. In order to achieve this analysis we created a novel open-source module in SlicerHeart to procedurally generate chordal branching geometries [40]. We then employed this framework to quantify how variation of five CDT parameters influenced clinically relevant metrics of valve closure metrics in four AVV models.

2. Methods

2.1. Geometry Creation

Three synthetic AVV geometries were created to understand how CDT geometry influences valve closure: a stereotypical healthy MV, the stereotypical MV with a dilated annulus, and a stereotypical TV. Each stereotypical case began with a “textbook” patient image dataset to obtain the annulus and leaflet dimensions. Thereafter, we worked with clinicians to adjust the simulations and ensure the valve closure matched clinical expectations. The complete workflow is outlined in Appendix A. Additionally a healthy MV was constructed from transesophageal 3D echocardiography images of a 13-year old patient using procedures outlined in Laurence et al. [47]. These four geometries can be found in panel (ii) of Figure 1.

We used our new parametric CDT creation module ValveFEMExport to procedurally generate CDT geometries with branching structures. This tool was implemented as a scripted Python module of 3D Slicer within the SlicerHeart extension. The module offers a comprehensive user interface and the necessary logic functions for CDT generation based on user selection. Step-wise schematics are provided in Appendix A. The number of CDT is determined by the product of leaflet region or edge length and user defined density. Branch length is measured by the distance from the insertion point on the NURBS surface to the branching point. Branches are formed by making connections between a branching point and a grid point on the NURBS surface. The radius is the target distance from the insertion point before snapping to the nearest grid point. The AVV and CDT were created in the SlicerHeart extension (www.slicerheart.org) [40] for 3D Slicer [49]. All code is available open source at www.github.com/SlicerHeart.

2.2. Finite Element Simulation

FE analysis of the valve closure was performed using our established workflow in FEBio [32, 46, 47, 50]. The MV and TV CDT were modeled as newly implemented linear truss elements with a prescribed modulus of 150 MPa, cross sectional area (CSA) of 1.0, and 0.44 mm² respectively [24], and discretized into 1 mm segments to allow buckling. The leaflets were discretized using 4-node bilinear quadrilateral shell elements (quad4) with full integration and assumed thickness of 0.396 mm [32]. The nonlinear material behavior of the leaflets was modeled via the nearly incompressible isotropic Lee-Sacks constitutive model [51] with strain energy density ($\psi$) defined as:

$$\psi =\frac{c_0}{2}\left(I_1-3\right)+\frac{c_1}{2}\left(e^{c_2{\left(I_1-3\right)}^2}-1\right)$$ (1)

Here, $c_0=200\mathrm{k}\mathrm{P}\mathrm{a}$, $c_1=2968.4\mathrm{k}\mathrm{P}\mathrm{a}$, $c_2=0.2661$ are material coefficients for the MV and $c_0=10\mathrm{k}\mathrm{P}\mathrm{a}$, $c_1=0.209\mathrm{k}\mathrm{P}\mathrm{a}$, $c_2=9.046$ are material coefficients for the TV [46, 52, 53]. Near incompressibility was enforced with a bulk modulus of 5000 kPa and 10000 kPa for the MV and TV, respectively. $I_1$ is the first invariant of the right Cauchy deformation tensor $\mathbf{C}={\mathbf{F}}^T\mathbf{F}$, and $\mathbf{F}$ is the deformation gradient. The LeeSacks model is uncoupled and the shell formulation uses the three-field formulation to avoid volumetric locking.

Dynamic analysis was performed using the generalized-alpha integration rule, mass damping for both the leaflets and CDT $(C=5000\mathrm{N}\mathrm{s}/\mathrm{m})$, and the FEBio automatic time stepping algorithm $\mathrm{\Delta }{t}_{\mathrm{m}\mathrm{i}\mathrm{n}}={10}^{-8}\mathrm{s}$ and $\mathrm{\Delta }{t}_{\mathrm{m}\mathrm{a}\mathrm{x}}={10}^{-4}\mathrm{s}$). A linearly increasing systolic pressure (100 and 23.7 mmHg for the MV and TV respectively) was applied to the ventricular leaflet surface over 0.005 s [46, 48, 51]. Leaflet contact was modeled using a potential-based formulation [53]. In the parametric valve cases, zero displacement boundary conditions were applied to the annulus and PM tip nodes. The patient specific MV used prescribed displacements for both the annulus and PM tip nodes based on their movement from mid-diastole to mid-systole in 3D echocardiography imaging [46].

2.3. Study Scenarios

We performed several investigations to understand how CDT geometry influences FE predictions of AVV function. First, we used our existing synthetic MV geometry [47] to compare the refined CDT geometry generated using our new approach to the simplified CDT geometry used in our prior work and that of others. The “baseline” MV CDT geometry was then parametrically varied to understand how changes to the CDT structure influence AVV closure. The understanding gained via the stereotypical MV geometry was extended to generate CDT geometries forthe stereotypical MV with annular dilation and TV. Finally, we parametrically varied CDT geometries for a patient specific MV and compared valve closure metrics to those observed in the stereotypical MV, in vivo and FINESSE [47].

2.3.1. Establishing Baseline CDT Geometry for the Synthetic MV

A stereotypical, healthy MV geometry was created to evaluate the effect of CDT geometry on valve closure (see Section 2.1). Our prior CDT generation workflow [46] (based on the work of Khalighi et al. [42]) generated a simplified CDT geometry with 115 insertions. We successfully used our new parametric workflow (see Section 2.1) to obtain 114 insertion points with the following parameters: primary CDT density of 2 chords/cm, secondary CDT density of 3 chords/cm², branch length of 3.5 mm, number of branches as 3 and 4 for primary and secondary CDT respectively, and insertion radius of 1 mm. We compared the FE simulation results using these two CDT geometries (fully simplified and branched) as a first step towards understanding how increased CDT fidelity may influence the FE predictions. It should also be noted that CDT density is quantified in our study in terms of leaflet free edge insertion distance (chords/cm) for primary chords, and leaflet surface area of insertion (chords/cm²) for secondary chords.

2.3.2. Mesh Convergence Study

We performed an initial study to understand how mesh density may influence our findings. In brief, the FE mesh for the synthetic MV was refined using HyperMesh (Altair, Troy, MI), resulting in a wide range of FE mesh densities (590–19424 elements). The MV function was simulated using our FE simulation workflow in FEBio (Section 2.2). The functional and mechanical metrics (see Section 2.4.1) were compared across mesh densities.

2.3.3. Parametric Variations of Primary CDT Geometry

The baseline MV was used to evaluate primary CDT influence on valve closure. The primary CDT density (1–3 chords/cm), branch length (1–10 mm), branch quantity (2–5), and branching radius (0.4–1.8 mm) values were varied using the ValveFEMExport module (Section 2.1). These ranges were chosen based on the minimum available values in the SlicerHeart module and the point when chordae insertion points began overlapping (except for branch length). Additionally, we varied the primary CDT CSA (0.2–2.0 mm²) in FEBio based on values reported by Ross et al. [24].

2.3.4. Parametric Variations of Secondary CDT Geometry

After establishing the role of primary CDT, we used the baseline MV to evaluate how secondary CDT influence valve closure. Both primary and secondary CDT were included based on preliminary simulations that showed primary CDT were needed to prevent localized leaflet free edge prolapse. We varied the secondary CDT density (1–10 chords/cm²), branch length (1–10 mm), branch quantity (1–5), and branch radius (0.4–3 mm). Similar to the primary CDT study, the minimum and maximum values were selected based on minimum values available in the SlicerHeart module and when CDT insertions began overlapping, respectively. Finally, the secondary CDT CSA was varied in FEBio from 0.2–2.0 mm², while keeping the primary CDT constant at 1.0 mm² based on values reported by Ross et al. [24].

2.3.5. Extension to Additional AVV Geometries

Two additional valves were created using the procedure outlined in Section 2.1: a MV with annular dilation (ADMV) and a TV. The ADMV was created by uniformly increasing the annulus of the healthy, synthetic MV by 10% while not altering the leaflet and commissure heights. The TV was adapted from Wu et al. [32] and a generalized annulus was created from Besler et al. [54]. Neither valve had a reference for CDT topology so the baseline CDT values from the normal MV were used and valve closure metrics were evaluated ( 2.3.1). Using the trends observed in the normal MV CDT studies, the CDT parameters were then updated to create more realistic valve closure. This qualified as localized free edge leaflet prolapse for the ADMV and improved regurgitant orifice area (ROA) for the TV.

The patient specific MV was evaluated by starting with the normal MV baseline CDT topology and varying primary and secondary CDT parameters ( 2.3.1). We varied the primary CDT density (1–2.8 chords/cm), branch length (1–10 mm), branch quantity (2–5), branch radius (0.4–2.0 mm), and CSA (0.2–2.0 mm²). The primary CDT were held at the baseline while the secondary CDT were varied by density (1–20 chords/cm²), branch length (1–10 mm), branch quantity (1–10), branch radius (0.4–3.0 mm), and CSA (0.2–2.0 mm²). Additionally, we compared the valve closure metrics between those observed in vivo and our approach to estimate in vivo leaflet strains (FINESSE).

2.4. Analysis

2.4.1. Mechanical and Functional Metrics

We evaluated the effect of CDT geometry using both functional and mechanical metrics. The ROA, contact area (CA), and billowing volume are clinically relevant functional metrics related to valve function that can inform valve repair. Increases in ROA and billowing volume and a decrease in CA are associated with AVVR and valve disease [55, 56]. We used a fully-automated method to accurately quantify the ROA in the Slicer module Orifice Area. The module requires users to draw a closed curve around the closed valve opening, after which the module refines the potential orifice surface using a user-specified number of shrink-wrap iterations. Next, rays are projected over the refined surface to identify streamlines, The ROA is determined by summing the areas associated with rays that do no intersect the valve [32]. The CA was calculated by the surface area of the elements with non-zero traction applied by the contact algorithm [53]. Billowing volume was calculated as the leaflet volume above the annulus plane during ventricular contraction. Slicer’s Parametric Atrioventricular Valve module calculated the three parametric valves’ billow volume. The patient-specific MV needed a separate FINESSE module to calculate its billow volume due to its dynamic annulus (Fig. 1 panel (iii)). Finally, we considered the first principal strain of the leaflets as the mechanical metric owing to its relationship with valve remodeling [3]. Leaflet strain visualizations include the median, 25% and 75% percentiles, and minimum/maximum values, whereas functional metric visualizations report the singular quantified value.

The ordinary least squares functionality of the Python statsmodels module was used to determine the presence of linear, quadratic, or exponential relationships between the valve metrics and CDT parameters. We considered three criteria to evaluate regression quality: (i) the coefficient of determination (R²), (ii) the p-value of each regression coefficient, and (iii) visual inspection, particularly when considering the quadratic relationship (i.e., is the inflection point used to match the data). These findings were used as a quantitative foundation to interpret our results. Note that the regression trends may be similar between the valve geometries; however, the magnitude of the regression coefficients are likely different and should only be used within the context of the stereotypical MV geometry.

2.4.2. CDT Geometry Comparison

We compared FE predictions with varying CDT geometries in three ways to understand: (i) how parameters in the SlicerHeart module altered valve closure, (ii) whether CDT insertion density was the central driver of CDT function, and (iii) whether total CDT CSA was the central driver of CDT function. Normalization was performed separately for each valve geometry. For a given geometry (e.g., stereotypical MV, annular dilation MV, TV, or patient specific MV), we first define a baseline chordal configuration using the the same set of input parameters (2.3.1). We then compute the baseline values of each metric (strain, CA, ROA, billowing volume) for that specific geometry and report all results as percent change relative the that geometry’s own baseline value. The normalization is intended to compare trends of the parameters across different valve models while avoiding differences in magnitudes that appraise form geometry specific anatomy. We compared functional metrics when varying CDT density and branch number to understand if trends were consistent across different CDT geometries with similar insertion point densities. This analysis aimed to test the hypothesis that CDT insertion density will influence valve behavior regardless of which CDT parameter altered the CDT density. This hypothesis is based on prior work by Khalighi et al. [42] that showed drastically simplified CDT geometries provided functionally equivalent behavior as ground truth CDT geometries from ex vivo imaging so long as CDT insertion point density was above a minimum threshold. Finally, we also compared the functional metrics across different CDT geometries with similar total CDT CSA. This analysis sought to test the hypothesis that total CDT CSA will influence valve functional metrics similarly regardless of how the total CDT CSA is varied (i.e., density, branch number, or CSA). The presentation of our results follows this structure.

3. Results

3.1. Primary CDT Geometry

We first altered the primary CDT parameters to examine their effect on valve closure (Fig. 2).

Fig. 2:

Influence of primary CDT parameters (density, length, branches, radius, and CSA) on valve closure metrics. The shaded regions in the strain plots represent the maximum (darker) and inter quartile range (lighter). CA, ROA, and billowing volume were normalized to the MV with baseline CDT topology indicated with a vertical dashed line.

3.1.1. Density

The major trends are that as the primary CDT density increases, the maximum strain decreases from 0.71 – 0.47. Also, the billowing volume generally decreased monotonically from 1 chords/cm (1214.2 mm³ – 75% above baseline) to 3 chords/cm (706.5 mm³ – 2.3% below baseline). There were no observable trends for CA or ROA, but they did have large osculatory changes. The ROA from 1 to 1.2 chords/cm increases from 19.96 mm² (20.0% below baseline) to 29.14 mm² (16.8% above baseline) (Fig. D4).

3.1.2. Length

A well-defined trend for CA and ROA vs. primary CDT length was not observed. However, maximum strain decreased from from 0.59 – 0.52. Billowing volume also decreased from 739.1 mm³ – 7.1% above baseline to 643.3 mm³ – 6.8% below baseline.

3.1.3. Number of Branches

The valve closure metrics generally did not exhibit a trend vs. number of primary CDT branches except in relation to CA. As the number of branches increased CA decreased 78.89 mm² – 3.6% above baseline to 73.98 mm² – 2.84% below baseline.

3.1.4. Branch Radius

Strain and CA exhibited mild sensitivity to radius variation, and this parameter was able to influence ROA and billowing volume, but in general, no trends were observed. ROA reached a maximum at 0.8 mm (23.67 mm² – 4.6% above baseline) and a minimum at 1.6 mm (19.25 mm² – 14.93% below baseline). The billowing volume slightly decreased from 718 mm³ – 4.09% above baseline to 689 mm³ – 0.23% below baseline.

3.1.5. Cross-Sectional Area

Increasing CSA from 0.02 to 0.20 mm² demonstrated influence over all valve closure metrics. Maximum strain increased from 0.53 to 0.56. The CA increased from 75.78 mm² – 3.21% below baseline to 78.99 mm² – 0.9% above baseline. The ROA reached a minimum at 0.2 mm² (23.57 mm² – 15% below baseline) and a maximum at 1.0 mm² (27.79 mm² – at baseline), but with no discernible trend. Billowing volume decreased from 622.3 mm³ – 7.91% above baseline to 689 mm³ – 0.23% below baseline. An internal study (not shown here) demonstrated similar trends when varying CDT elastic modulus, except increased elastic modulus decreased the CA and did not have an effect on primary ROA.

3.2. Secondary CDT Geometry

We also investigated how secondary CDT parameters influence valve closure metrics (Figure 3). In many ways the secondary CDT metrics reflected the findings of the primary CDT (Section 3.1).

Fig. 3:

Influence of secondary CDT parameters (density, length, branches, radius, and CSA) on valve closure metrics. The shaded regions in the strain plots represent the max- and inter-quartile range. CA, ROA, and billowing volume were normalized to the MV with baseline CDT topology indicated with a vertical dashed line.

3.2.1. Density

Increasing the secondary CDT density from 1 chords/cm² to 10 chords/cm² resulted in the maximum first principal strain decreasing from 0.55 to 0.45. However, in contrast to the primary CDT study, increasing the secondary CDT density resulted in a stronger influence over CA, ROA, and billowing volume. The CA decreased from 72.78 mm² – 4.81% above baseline to 50.46 mm² – 27.34% below baseline. ROA moderately varied in relation to secondary CDT density with the minimum density model exhibiting a 16.85% increase above baseline (29.06 mm²), and decreasing to the baseline value (24.87 mm²) at 3 chords/cm² (similar to the primary CDT value). In contrast to the primary CDT variation behavior, the ROA increased from baseline by 31.81% (32.768 mm²) to 1 chords/cm².

3.2.2. Length

Variation of the secondary CDT length produced similar results to the primary study as they both influence strain and billowing volume, and do not influence CA. ROA however, increased from 1 mm (21.51 mm² – 10.19% below baseline) to 10 mm (25.69 mm² – 7.27% above baseline) with non-monotonic variation between extrema.

3.2.3. Number of Branches

Results due to increasing the number of secondary CDT branches did not demonstrate any similarities with the primary CDT study. Here, the maximum strain decreased from 1 branches (0.52) to 5 branches (0.47). Additionally, billowing decreased from 1 branches (533 mm³ – 8.86% above baseline) to 5 branches (482.5 mm³ – 4.09% below baseline).

3.2.4. Branch Radius

Secondary CDT radius demonstrated more influence over all valve closure metrics than in primary CDT radius variation. Maximum strain decreased from 0.4 mm (0.50) to 3 mm (0.48). Contact area peaked at 1.4 mm (73.36 mm² – 5.63% above baseline) and had a minimum at 2.4 mm (58.53 mm² – 15.71% below baseline), but with no apparent trend. ROA generally increased with secondary radius, with a minimum of 22.71 mm² – 8.69% below baseline and a maximum of 27.97 mm² – 12.46% above baseline. The billowing volume decreased from 512.1 mm³ – 4.57% above baseline to 451.5 mm³ – 7.8% below baseline.

3.2.5. Cross-Sectional Area

The secondary CDT CSA variation produced similar results to those observed in the primary CDT CSA study as it demonstrated influence over strain, ROA, and billowing volume, with the exception of it exhibiting only trivial sensitivity on CA. The maximum strain increased from 0.50 to 0.53. The ROA increased from 22.58 mm² – 9.90% below baseline to 25.46 mm² – 1.6% above baseline. The billowing volume decreased from 447.3 mm³ – 14.25% above baseline to 384.1 mm³ – 1.89% below baseline.

3.3. Total CSA

The CDT density, branching, and CSA metrics were re-quantified as a function of total CDT CSA and are presented in Figure 4. There were no consistent trends when varying the density, branches, or CSA of the primary CDT. However, there were consistent decreases in median strain and billowing volume as the total secondary CDT CSA was increased, suggesting total secondary CDT CSA is a crucial consideration to simulations of valve function.

Fig. 4:

The effect of density, branches, and CSA on valve closure metrics based on the total CSA of the CDT. The shaded regions in the strain plots represent the maximum (darker) and inter quartile range (lighter).

3.4. Additional Synthetic Valves

To demonstrate the utility of our new parametric CDT generation tool, the CDT geometries of the ADMV and TV were successfully adjusted from the baseline CDT to improve simulated valve function (Fig. 5). The primary CDT density of the ADMV model was increased to reduce local free edge prolapse, which led to a 13.5% decrease in CA and 17.4% increase in ROA. Additionally, the secondary CDT density of the baseline TV model was decreased to reduce the ROA by 13% and better resemble a functioning, stereotypical TV. Decreasing the secondary CDT also resulted in a minimal change in maximum strain (0.629 – 0.628), decreased CA (4.84%), and increased billowing volume (5.56%). These extensions of our pipeline illustrate how the CDT generation tool can be successfully used for a range of AVV geometries.

Fig. 5:

Lagrange strain color maps comparing The synthetic ADMV and TV at baseline and with adjusted CDT geometry.

3.5. Patient Specific MV

The patient specific MV results are shown in Figures 6 and 7. Primary and secondary density significantly influenced CA, ROA, and billowing; CA and billow volume decreased while conversely ROA increased. Additionally, increasing secondary density decreased strain (0.70 – 0.53). While increasing the length of the primary CDT ROA increased (18%), it had minimal impact on other aspects. Similarly, increasing the number of branches did not substantially affect valve closure with the exception of secondary billow volume which decreased by 20% as the number of branches decreased. Increasing chord branching radius had several substantial impacts on valve closure, particularly secondary ROA, which increased by 229% as the CDT radius increased. CSA had a minor effect on valve closure, but both primary and secondary studies demonstrated a reduction in billowing that resembles the previous chord study.

Fig. 6:

Influence of primary CDT parameters (density, length, branches, radius, and CSA) on patient-specific MV closure metrics. The shaded regions in the strain plots represent the max- and inter-quartile range. CA, ROA, and billowing volume were normalized to the MV with baseline CDT topology indicated with a vertical dashed line.

Fig. 7:

Influence of secondary CDT parameters (density, length, branches, radius, and CSA) on patient-specific MV closure metrics. The shaded regions in the strain plots represent the max- and inter-quartile range. CA, ROA, and billowing volume were normalized to the MV with baseline CDT topology indicated with a vertical dashed line.

We compared the patient specific MV segmentation in Slicer to the FINESSE method published by Laurence et al. [47] and our simulations(Fig. 8). The patient valve is healthy and has no ROA reported. The FINESSE simulation has an ROA of 2.12 mm² and the largest reported ROA from our chord study is 19.99 mm². Despite the non zero values both ROA are under 20 mm², and according to guidelines reported in Otto et al. [56] these MV would not be considered regurgitant. An indicator of valve disease is a billow height larger than 1 mm with Barlow’s disease having a billow volume larger than 1150 mm³ and fibroelastic deficiency (FED) less than 1150 mm³ [55]. The billow hight and volume calculated from the segmentation are 1.2 mm and 161 mm³ respectively. The FINESSE simulation has smaller billowing height and volume at 0.9 mm and 8 mm³. Our chord simulations struggled to get the billowing under control and at a minimum had a billow height and billow volume of 1.4 mm and 177.1 mm³ respectively. Contact area is not quantified in slicer, but the FINESSE simulation has a contact area of 31.14 mm². The primary chord parameters branches, length, radius, and CSA all have a contact area of 31 mm², but the secondary parameters get a smaller area of contact, the largest being 27.6 mm² at 0.2 mm² CSA. The maximum and mean strain values calculated from FINESSE are 0.34 and 0.196, respectively. In contrast, the CDT simulations exhibited a higher maximum strain, ranging from 0.589 to 0.783 across all parameters. The median strain for the primary CDT was higher than that of the secondary CDT studies, ranging from 0.197 to 0.208 and 0.167 to 0.187, respectively. However, both ranges were outside the FINESSE median strain.

Fig. 8:

A segmentation of the patient-specific MV is shown followed by the a strain map of the FINESSE simulation. Strain maps of the primary density study are shown at 1, 2(baseline), and 2.8 chords/cm. Strain maps of the secondary density study are shown at 1, 3(baseline) and 20 chords/cm². Additionally, the cut view is shown comparing FINESSE, primary chord study at 1 chords/cm, and secondary chord study at 20 chords/cm².

4. Discussion

4.1. Overall Findings

In this study, we added new CDT modeling capabilities to our open-source FE simulation framework for atrioventricular heart valves and used this tool to investigate how variations in CDT structure influence AVV closure (e.g., [32, 46, 47]). Our findings demonstrated that altering the CDT geometry of a stereotypical MV led to changes in clinically relevant valve metrics (ROA, CA, and billowing volume) and valve mechanics (first principal strains). Specifically, CSA had the most influence over valve closure metrics, followed by CDT density, length, radius and branches. We then used this information to showcase the flexibility of our new workflow by altering the CDT geometry of two additional valve topologies (MV with annular dilation and TV) to improve FE predictions. The patient specific MV was able to achieve a wide range of values based on the changing parameters, but did not achieve a reliable replication of the mid-systolic segmentation. The present investigation is a substantial improvement of our open-source, image-derived modeling capabilities and, to our knowledge, this is the first study to parametrically investigate how CDT branching influences FE predictions of AVV function.

4.2. Existing Literature

There has been great progress in computational predictions of AVV closure over the past two decades [27, 33, 33, 42, 43, 45, 48, 52, 57–60], yet accurate reconstruction of in vivo CDT geometries remains an unresolved challenge. This has led to research groups adopting various methods (see Table 1) in which the CDT geometries are: (i) accurately reconstructed from ex vivo samples, (ii) informed from clinical images and numerical optimization, (iii) artificially generated based on assumed CDT properties, or (iv) completely disregarded or simplified to force or displacement boundary conditions acting on the leaflets. Here, we review each classification of CDT geometry fidelity to contextualize our newly developed CDT generation tool.

Table 1:

Previous literature on computational modeling of AVVs grouped by CDT implementation in FE modeling. Abbreviations: CDT - Chordae Tendineae, Img. Der. - Image Derived, Img. Inf. - Image Informed, Synth. - Synthetic, Y - Yes, N - No, AL - Anterior Leaflet, PL - Posterior Leaflet, SL - Septal Leaflet, BSM - Basal, Strut, Marginal, Het. - Heterogeneous, CSA - Cross-Sectional Area, MV - Mitral Valve, TV - Tricuspid Valve.

Publication	CDT Fidelity	CDT Insertion Classification	Branching Points	CSA (mm2)	CDT Origins	CDT Arc	Valve
Drach et al. 2018 [61]	Img. Der.	Y	Het.	Het.	12†	Y	Ex Vivo MV
Mathur et al. 2022 [34]	Img. Der.	AL, PL, SL	1	0.33, 0.32, 0.40	4	Y	Ex Vivo TV
Laurence et al. 2020 [33]	Img. Der.	Y	3	0.17	3	Y	Ex Vivo TV
Wang et al. 2013 [57]	Img. Inf.	BSM	1	0.71, 2.05, 0.38	21	N	In Vivo MV
Pham et al. 2017 [44]	Img. Inf.	BSM	1	0.71, 2.05, 0.38	18	Y	In Vivo MV
Kong et al. 2018 [43]	Img. Inf.	Y	1	0.95	†	Y	In Vivo TV
Khalighi et al. 2019 [42]	Synth.	N	0	Constant†	2	N	Ex Vivo MV
Johnson et al. 2021 [48]	Synth.	Y	0	0.166	4	Y	Parametric TV
Stevanella et al. 2010 [52]	Synth.	Y	0	0.171	3	Y	In Vivo TV
Wenk et al. 2010 [62]	Synth.	Y	0	Constant†	4†	N	In Vivo MV
Wu et al. 2023 [32]	Synth.	Y	0	N/A	3	N	In Vivo TV

^†Not reported. Note: Values were interpreted from figures where possible.

4.2.1. High-Fidelity Chordae Reconstruction

Ex vivo methods are currently the only option to fully capture complex CDT geometry. For example, Drach et al. [61] generated micro-CT scans of an explanted ovine MV to fully reconstruct the valve, annulus, and CDT in both unloaded/loaded configurations. The full reconstruction of the CDT origins, branching, and CSA allowed accurate predictions of the MV closure and mechanics in healthy, diseased, and repaired configurations. Explanted valves also enable researchers to measure, count, and virtually reconstruct CDT geometry in the absence of high-fidelity imaging [33, 34], although this method may be more prone to user error and bias. In general, the main challenge precluding these methods for clinical application is the need of an explanted valve; however, photon-counting CT is an emerging clinical imaging modality that may allow for accurate virtual reconstruction of in vivo CDT structure [63].

4.2.2. Image-Informed Chordae Reconstruction

To partially overcome the challenges of obtaining complete in vivo CDT geometries, some groups have used the partial visualization of CDT geometries in clinical images to inform CDT geometry and function. Most notably, this has been achieved with multi-slice CT scans that can resolve the PM, CDT origins, and some degree of CDT topology and leaflet insertion points [43, 44, 57]. This information is used to infer the CDT geometry with critical parameters (e.g., length) optimized by comparing FE predictions of valve closure to the image-derived leaflet surface. Although this approach can successfully match in vivo valve behavior, it is possible that patient-specific leaflet behaviors or other simulation discrepancies are implicitly embedded in the optimized CDT parameter(s).

4.2.3. Purely Artificial Chordae Construction

In contrast to image-informed CDT reconstruction, purely artificial methods disregard any information possibly inferred from clinical images and synthetically construct CDT geometries. These efforts were largely motivated by the work of Khalighi et al. [42] which iteratively simplified a high-resolution micro-CT scan of an ovine heart (see also [61]) to understand the minimum required CDT fidelity to emulate CDT function. Their foundational work demonstrated that the CDT structure can be significantly simplified to have (i) no branches, (ii) constant CSA, (iii) combined CDT origins (i.e., PMs), and (iv) random leaflet insertions so long as the insertion density was above a critical value. Later works by our group [32, 46] used this when developing our open-source FEBio workflow for heart valves. Other groups established their own methods for parametrically defining chordae structure [48] or creating artificial CDT geometries [52].

4.2.4. Emulating Chordae Behavior

A sub-class of FE simulations focused on matching predicted leaflet closure to clinical images have entirely removed the CDT geometry and emulated CDT behavior via force or displacement boundary conditions. The FE shape enforcement work of Rego et al. [64] compared forced-based methods for emulating chordae behavior and found similar performance whether applied to the entire leaflet, the leaflet rough zone, or the leaflet free edge. We used these findings to remove CDT from our recent development of an open-source FE shape enforcement method in FEBio [47]. Alternatively, Ross et al. [65] demonstrated a method to directly prescribe leaflet free edge motion from clinical images which substantially improved optimization search results when estimating in vivo leaflet properties. This method performs best when the FE prediction is being matched to an image-derived leaflet surface via shape enforcement or material optimization, but a chordae emulating force needs careful tuning to ensure no unnecessary tensions are applied to the leaflets. Furthermore, this method cannot be used when altering the valve geometry (i.e., simulating diseased-driven changes or surgical repair), limiting clinical application only to estimation of in vivo leaflet strains or material properties.

4.2.5. The Present Study

The new workflow presented herein has expanded upon purely artificial methods for reconstructing CDT geometry by including CDT branching with precise parametric control over the branching properties (i.e., branch length and branch number). We have used our open-source CDT generation tool to provide the first parametric assessment of how CDT geometry influences AVV closure, which will support future FE simulations of AVV function. The current framework also serves as a valuable foundation for generating CDT geometries with more morphological considerations (e.g., varying CSA) or CDT geometries informed by clinical images. Therefore, we have intentionally developed our CDT generation tools open-source in the SlicerHeart extension for 3D Slicer to (i) provide an open-source foundation for other research groups to expand upon our parametric generation of CDT geometries and (ii) allow smooth integration with clinical imaging data and the open-source FEBio software.

Certain CDT parameters had more influence over valve closure metrics than others. CDT density consistently had some of the largest variations from baseline values. This may be due to the fact that the density parameter causes a restructuring of the CDT topology, whereas parameters such as number of branches and branching radius only have a localized effect restricted to the insertion region. The large variations caused by the density parameter may also be due to the fact that material property changes such as total CSA can have large changes with the changing number of CDT. The restructuring of the CDT topology by changing the density can specifically have a large impact on the commissure folding which can influence strain and CA.

The parameters that showed the most consistent trends between the stereotypical MV and the patient-specific MV were chordal density and CSA. Specifically, decreaseing primary and secondary density reduced maximum leaflet strain in both models. We also observed that secondary density reduced contact area in both the stereotypical and patient specific MV. In contrast, for parameters such as primary radius, we did not observe a consistent trend across strain, CA, ROA, or billowing in either model.

Additionally, we varied the elastic modulus to evaluate its influence on valve closure. We found that the CSA and elastic modulus had similar trends for max strain, primary CA, and billowing, with no trend related to secondary ROA. However, increased elastic modulus decreased CA (CSA had no trend) and increased CSA increased primary ROA (elastic modulus had no trend).

No one parameter of CDT geometries generated for the patient specific valve were were able to match all aspects of valve closure metrics with the in vivo imaging or FINESSE valve. All parameters were able to create a competent valve that would not be classified as regurgitant based on its ROA. None of the parameters were able to reduce the billow volume to reflect what was seen in the imaging of the valve. Conversely the CA remained too low compared the the FINESSE model. Finally, the median strain only reflected the strain seen in FINESSE in certain parameters, but the maximum strain remained too high for all. This study finds that there would require manual tuning of individual parameters to faithfully represent an in vivo AVV.

4.3. Toward Clinical Application

Realistic modeling of chordal structures is important to future simulations of valve function, valve dysfunction, and valve repair. For example, alteration of chordal structure, such as the introduction of “neo-chords” is an important part of many clinically applied valve repairs [66]. A significant question in such repairs is how many and how long should the neo-chords be, and modeling of such interventions requiring modeling of CDT topologies beyond the simple, non-branching, functionally-equivalent frameworks demonstrated in Khalighi et al. [42]. In contrast, our framework allows the reproducible creation and refinement of more realistic chordal branching which allows simulation of these more complex chordal interventions. Currently, accurate “subject-specific” chordal models can only reliably be obtained from direct inspection or micro-ct imaging of static ex vivo models [30]. However, emerging technology such as photon-counting CT may provide the fidelity to accurately visualize the majority of the chordal structure [63]. As such technology evolves, greater fidelity of chordal modeling can be incorporated into patient-specific studies.

4.4. Study Limitations

The study completed above sought to incorporate all information currently available in the literature; however, simplifying assumptions were still required. The topological models of the AVVs were approximated by merging in vivo imaging with ex vivo data reported in the literature.

In addition to the geometry, the leaflets were simplified with a homogeneous thickness taken from ex vivo data. Similarly, the leaflet constitutive model was taken from ex vivo mechanical testing as there is no current method to extract patient-specific material properties from in vivo 3D imaging. Our group and others are developing in-silico tools for these purposes [65, 67]. We used a linear material model as a simplifying assumption for our CDT, but studies have shown that the Ogden material model would better represent CDT tissue [68].

Finally, commissure folding is challenging to model and slight alterations to the CDT geometry may cause relatively large variations in the localized commissural buckling. We believe this may contribute to the fluctuations through out our study, specifically in the mesh density study and stereotypical MV (Fig. C3, D4). The mesh density CA and billowing volume had fluctuations that did not provide clear mesh convergence, but these effects should be minimized in our study results since the same mesh density was used throughout. Additionally, the unpredictable folding of the commissure do not represent meaningful differences in the closed valve configuration. Visual inspection reveals remarkably similar strain contours (besides localized hot spots near CDT insertions that commonly occur in valve simulations), closed valve geometries and leaflet billowing profiles across all mesh densities. In the stereotypical MV primary density CA and ROA had large oscillatory effects, however a visual evaluation of the valve showed minor observable changes. In the future, we can extend our methods to image-informed biomechanics (e.g., [47, 67]) and optimization methodology to improve simulation performance in these valve regions.

Availability of data and material.

The simulation models used in this study are available in the FEBio Model Repository.