A Neural Network Finite Element Trileaflet Heart Valve Model Incorporating Multi-body Contact

Abstract

The use of patient-specific computational modeling of cardiovascular diseases has become increasingly popular to improve patient standard of care. Most simulation approaches currently utilize the Finite Element Method (FEM), which is very well established and succeeds in producing high-fidelity results. However, it remains too slow for use in clinical settings, especially when many-query solutions are required to determine optimal therapeutic approaches. As a step towards addressing these demands, we have developed a Neural Network Finite Element (NNFE) approach that greatly accelerates simulations of soft tissue organ function. While the NNFE method utilizes conventional FEM meshes to define the problem geometry, it leverages advancements in neural network architecture design in new GPU-based software tools to solve the governing hyperelastic material PDEs. The NNFE method has recently captured physical contact between a deformable body and a frictionless symmetry plane. In the present work, we extended the NNFE approach to simulate trileaflet heart valve closure as a critical step in moving towards patient-specific applications. Our approach addressed two critical aspects of heart valve simulations: use of 3D solid leaflet models as opposed to shell-based leaflet models, and multi-body contact between the leaflets. We verified the approach by comparing displacements of NNFE simulated closure of a single heart valve leaflet against a frictionless symmetry plane with an identical simulation in tIGAr, the open-source isogeometric analysis extension of FEniCS. The average nodal displacement error was 0.020 mm (0.47% of the maximum displacement). We further evaluated our implementation by varying leaflet collagen fiber directions to mimic physiologically accurate deformation modes. Results of the approach indicated that the observed leaflet deformation patterns agreed well with previous trileaflet simulations. Significant variations in stress were observed transmurally, underscoring the need for solid elements to model leaflet geometry. Computational speed improvements produced an approximately 100 fold speedup, with the NNFE simulations of single leaflet closure taking 0.28 s while its FE counterpart took 61s. Full trileaflet valve models with multi-body contact simulations took approximately 5 seconds, whereas equivalent FEM simulations take several hours. Training the full trileaflet model took approximately 16 hours and was trained over the full functional range of pressure, so that training was only required once for all subsequent simulations. We conclude that the NNFE method can be successfully used to perform rapid simulations of complex 3D soft organ systems, such as the trileaflet heart valve, that involve large deformations, 3D geometries, and multi-body contact. Moreover, the ability to perform post-trained simulations in dramatically shorter time periods underscores the promise of machine learning-based computational mechanics approaches in patient-specific predictive computational models.

1. INTRODUCTION

More than 300,000 heart valve replacement surgeries are performed annually world-wide [1]. Bioprosthetic heart valves (BHV) have important advantages over mechanical valves in that they do not require permanent anticoagulation therapy, operate noiselessly, and have blood flow characteristics similar to the native valve. However, standard surgical BHV durability continues to remain limited to the range of 10–15 years, often achieved only in older patients [2]. Failure results from leaflet structural deterioration mediated by fatigue, tissue mineralization, or both [3]. Despite continued developments in BHV design, in particular transcatheter aortic valve replacement (TAVR) approaches [4], durability remains limited and very difficult to predict. Overall, an ideal heart valve substitute does not yet exist, and robust methods to extend its durability remain largely unexplored. Hence, there is a strong need to develop a greater understanding of the underlying mechanisms involved in BHV durability to improve its present clinical performance.

Given advances in both clinical imaging and computational technologies, simulations are increasingly used to improve patient outcomes in the general cardiovascular field [5–7], as well as during heart valve replacements [8, 9]. All use the Finite Element Method (FEM), as it is a well established, robust approach that provides high-fidelity results [10–12]. However, FEM simulations remain very time consuming to perform. This is especially a concern in a multi-query setting of determining the optimal valve design for a specific patient’s condition, therefore faster simulation approaches are needed.

Recent rapid advances in Neural Network (NN) simulation technologies has sparked an area of research termed Scientific Machine Learning (Sci-ML). The popularity of Sci-ML is due in part to the recent improvements in GPU-based computation [13, 14]. Data-driven approaches that use solutions from forward finite element simulations, which solve for deformation given a prescribed geometry, material parameters, and boundary conditions for a specific boundary value problem, have been developed [15, 16]. An alternative approach attempts to learn the parameter to state map in an discretization-free setting using Neural Operators [17]. Another popular neural-network based approach utilizes direct physics-based training where the neural network is trained using a cost function based on equations governing the mechanics problem [18, 19]. The Deep Ritz method is worth highlighting in this context due to its use of variational approaches, which introduce a new algorithm for solving variational problems using neural network representations of functions in the context of the Ritz method [19]. In general, the advantage of physics-based training approaches is that there is no reliance on acquiring experimental data or generating simulation data for training.

Given the clear advantages of using FE-based physics descriptions, we sought to take advantage of FE discretization for domain mapping, performing numerical integration, and defining boundary conditions. As a promising approach, we have developed a neural network finite element (NNFE) method for soft tissue and organ simulations that can produce highly accurate results within clinically relevant time frames [20–23]. The NNFE approach is a physics-based approach that uses a neural network (NN) to represent the FE nodal displacements. This approach does not rely on data generated from physical experiments or simulations for training; rather, the NNFE model is trained by minimizing the potential energy or a residual derived from the governing equations. The NNFE method has been applied to myocardial tissue mechanics; networks were trained to simulate pressure-volume loops in the left ventricle and triaxial loading of cuboidal myocardium specimens [20–23]. Additionally, a NURBS-based geometric description has recently been adopted [24]. The NNFE approach has demonstrated initial success by executing simulations in short time frames with high accuracy when trained over ranges of boundary conditions. Use of the NNFE approach has a clear advantage for problems such as determining optimal TAVR shape and placement as it offers a more numerically efficient approach for executing the forward simulations required for such a study.

To enhance heart valve modeling fidelity, we sought to move from shell-based approaches to 3D solid representations. The use of solid elements in heart valve simulations is still relatively rare, as most investigators utilize shell based formulations because they are computationally efficient [25–28]. However, shell formulations overlook critical transmural behaviors, especially for layered structures such as natural and replacement heart valve leaflets. Heart valve leaflets are composed of fibrosa, spongiosa, and ventricularis layers with heterogeneous material properties [29, 30]. Capturing the different mechanical responses of these tissues requires solid elements because of their varying material properties. Moreover, some heart valves violate the thin-walled assumption made by shells. Human mitral valves have a maximum annular diameter of 3 cm and a thickness of 3–5 mm [31], which exceeds the maximum thickness to length ratio of 10% made by the thin-walled assumption. NURBS representations of solids are desirable because of their ability to represent geometries with few parameters and the ability to change their polynomial order without changing their geometry. We have conducted initial studies demonstrating basic feasibility [21].

In addition to accurate 3D representations, the integration of multi-body contact is another challenge in heart valve simulations. Heart valve leaflets come into contact during closure because of the surrounding blood’s inertial forces [32], and this contact prevents regurgitation. The interface of the contacting leaflets is assumed to be frictionless, and the force exerted on the leaflet surface is treated as a static load. A potential-based contact penalty term is used as this force is conservative. The force exerted on the leaflets is perpendicular to the leaflet surfaces in their open configuration. Full multi-body contact requires the integration of a search algorithm that detects where contact occurs into the present NNFE framework. In this work, we extended the NNFE approach to capture full multi-body contact in a 3D trileaflet heart valve model. Hertzian-type contact was implemented to provide a potential-based method of enforcing contact between a deformable and a rigid body [24]. Because our loss function is defined using the potential energy of the system, the implementation of contact with a frictionless symmetry plane in the NNFE approach is relatively seamless. This study provided an important step in developing a high-speed NNFE-based method of simulating TAVR function.

2. METHODS

2.1. Overview

In the present work we utilized the NNFE approach [21] to solve for the deformations of a 3D trileaflet heart valve with full multi-body contact between the leaflets. The key advantages of the NNFE method are that it: 1) trains over a range of boundary conditions, 2) utilizes a finite element mesh framework to define the problem geometry, boundary conditions, and perform the necessary numerical integrations, and 3) executes high-fidelity forward simulations rapidly and is thus ideal for multi-query applications [21]. To implement the NNFE method for systems where a potential energy functional exists, training is accomplished by averaging a sample of potential energies over the desired range of boundary conditions (e.g. pressure). For systems where a potential energy does not exist, an approach that minimizes the residual force vector derived from the principle of virtual work is used [20]. The NNFE method is thus ideal for applications such as medical device design and surgical planning where solutions to many different instances of a boundary value problem (BVP) are required in a short time frame.

2.2. Leaflet Geometry

The tri-leaflet heart valve leaflet was generated using a total of seven geometric parameters based on a modification of a parameterized pulmonary valve leaflet [33]. This process will be replicated for an Aortic Valve (AV), the approach developed in this paper should transfer easily between different tri-leaflet heart valves. These parameters include the radius of the base $R_b$, the radius of the commissures $R_c$, leaflet height $H$, and the angle between neighboring coaptation surfaces $\gamma$ (Figure 1). The commissure height $H_s$ is the vertical distance between the midpoint of the free edge the truncated corner of the commissure attachment on the free edge. The valve was further modified to obtain more control over the curvature of the leaflet belly by introducing parameters ${\zeta }_b$, a downward vector from the midpoint of the leaflet free edge that intersects the tangent from the center of the leaflet basal attachment, and ${\theta }_b$, the angle the leaflet belly makes with the horizontal (Figure 1) [34].

Figure 1:

Heart Valve Leaflet Construction. $R_c$: radius of the commissures, $H$: leaflet height, $H_s$: commissure height, and $\gamma$: angle between neighboring coaptation surfaces in (a). ${\zeta }_b$: tangent of the leaflet underside at the midpoint of the free edge, ${\theta }_b$: the angle the leaflet belly makes with the vertical, and $R_b$: radius of the base in (b).

The leaflets were generated with a NURBS-based representation to remain consistent with the geometries previously used in the NNFE approach. A set of geometric parameters (Table 1) are first used to generate a point cloud - a set of points in ${\mathbb{R}}^3$ - representing key points on the leaflet that help define the shape of the free edge, basal attachment and commissure attachments. These points are then fit with a set of curves using the NURBS curve-fitting functionality provided by the open-source Python package NURBS-Python [35]. These curves determine the shape of the free edge, basal attachment, and commissure attachments on the ventricular side of the leaflets. Fitting NURBS curves involves determining the set of control points that either approximate or interpolate a finite set of points. In this case, the point cloud representing the leaflet was interpolated with four connecting curves. A NURBS curve is defined by a linear combination of rational basis functions and control points,

$$C\left(\xi \right)=\sum _{i=1}^n{R}_i^p\left(\xi \right)B_i$$ (1)

where NURBS rational basis functions are defined as $R_i^p\left(\xi \right)=\frac{N_{i,p}\left(\xi \right)w_i}{\sum _{i=1}^n{N}_{i,p}\left(\xi w_i\right.}$. Weights are represented by $w_i$, and $N_{i,p}\left(\xi \right)=\frac{\xi -{\xi }_i}{{\xi }_{i+p}-{\xi }_i}{N}_{i,p-1}\left(\xi \right)+\frac{{\xi }_{i+p+1}-\xi }{{\xi }_{i+p+1}-{\xi }_{i+1}}{N}_{i+1,p-1}\left(\xi \right)$ are B-spline basis functions defined on a non-periodic knot vector. Extending this to surfaces and solids is straightfoward as they are defined by a tensor product of rational basis functions:

$$S\left(\mathit{\xi }\right)=\sum _{i=1}^n{R}_i^p\left(\mathit{\xi }\right)Q_i^p\left(\mathit{\xi }\right)T_i^p\left(\mathit{\xi }\right)B_i$$ (2)

Table 1:

Parameters used to generate heart valve leaflets (Figure 1). Leaflets were identical in the trileaflet heart valve. Note that ${\zeta }_b$ is not varied for the specific leaflet used in this study, but is included as this leaflet generation framework will be used in future work.

$H\left(\mathbf{m}\mathbf{m}\right)$	$H_s\left(\mathbf{m}\mathbf{m}\right)$	$\gamma \left(°\right)$	$R_b\left(\mathbf{m}\mathbf{m}\right)$	$R_c\left(\mathbf{m}\mathbf{m}\right)$	${\zeta }_b$ (unit vec)	${\theta }_b\left(°\right)$	$t\left(\mathbf{m}\mathbf{m}\right)$
10.7	4.23	120	12.0	5.3	[0,0,0]	43.7	0.5

These curves were then connected to create a NURBS surface defining the leaflet with CAD-like functionality provided by the open-source Python package igakit [36]. A ruled surface connecting the basal attachment and leaflet free edge was then constructed. A ruled surface is created by moving a straight line, whose endpoints each lie on separate curves, along these curves. Another ruled surface between the truncated commissure attachments was also constructed. These surfaces were merged to form a NURBS surface representing the leaflet (Figure 2). The leaflet surface was then extruded in the direction of the surface normals $N$ defined by

$${\mathbf{A}}_{\alpha }={\mathbf{X}}_{,\alpha }=\frac{\partial \mathbf{X}}{\partial {\xi }_{\alpha }},N=\frac{{\mathbf{A}}_0\times {\mathbf{A}}_1}{{‖{\mathbf{A}}_0\times {\mathbf{A}}_1‖}_2}$$ (3)

where ${\mathbf{A}}_0$ and ${\mathbf{A}}_1$ are covariant basis vectors, $\mathbf{X}$ is a point in the reference configuration, and $\mathit{\xi }=\left({\xi }_0,{\xi }_1,{\xi }_2\right)$ is a point in the parametric domain. The distance between the original and offset surface generated by the extrusion was determined by the thickness of the valve $t$, which was chosen to be 5.0mm in this study. These solid NURBS igakit geometries were directly read into our FEM solver using the open-source Python package tIGAr [37], whose functionality is further explained in section 3.2.

Figure 2:

Pipeline for Heart Valve Generation with NURBS. A point cloud (the set of blue points) was generated to fit NURBS curves (green arcs) to define the free edge $\left(S_1\right)$, basal attachment ($S_3$), and truncated commissures of the leaflet ($S_2$ and $S_4$) (a). A NURBS surface was then constructed to connect these curves by generating ruled surfaces connecting opposite curves (b). This surface was then extruded to represent the leaflet as a solid by offsetting the surface in the direction of its outward normal (c).

The covariant basis vectors (Equation 3) defining coordinate directions in the computational domain can be used to prescribe the direction of leaflet material axes. We refer to specific material axes prescriptions as fiber trajectories as follows. Preferred (PD) and cross-preferred (XD) fiber directions, which defined a given fiber trajectory, are defined as rotations of the coordinate frame along isoparametric surfaces of the leaflet (Figure 3).

$$\begin{gathered} G_{PD}=\text{cos}\left(\theta \right){\mathbf{A}}_0+\text{sin}\left(\theta \right){\mathbf{A}}_1 \\ {G}_{XD}=-\text{sin}\left(\theta \right){\mathbf{A}}_0+\text{cos}\left(\theta \right){\mathbf{A}}_1 \end{gathered}$$ (4)

Changing the fiber trajectory varies the local direction of anisotropy of the leaflet tissue, strongly affecting the leaflet mechanics, and allows us to study its effect on leaflet stress and strain [38]. Three fiber trajectories were used in this study. They were defined by rotating the leaflet coordinate axes by 0°, $-{15}^{\circ }+{30}^{\circ }{\xi }_0$, and 30° degrees, corresponding to baseline, angled, and chiral fiber trajectories, respectively. Note that ${\xi }_0\in \left[\mathrm{0,1}\right]$ as the parametric domain is defined as a unit cube. For the baseline case, setting $\theta =0^{\circ }$ aligns the preferred and cross-preferred of the fiber directions with the circumferential and radial axes, ${\mathbf{A}}_0$ and ${\mathbf{A}}_1$ respectively, described by the NURBS coordinate system in physical space. The chiral fiber trajectory and circumferential and radial leaflet axes are shown (Figure 5) along with a side-by-side view of baseline and angled fiber trajectories (Figure 4). Again, note that ${\mathbf{A}}_0={\mathbf{A}}_c,{\mathbf{A}}_1={\mathbf{A}}_r$, and ${\mathbf{A}}_2={\mathbf{A}}_t$, where ${\mathbf{A}}_t$ is the transmural, or through-thickness, coordinate direction.

Figure 3:

Coordinate and fiber axes in the reference and current configuration, ${\Omega }_0$ and ${\Omega }_t$ mapped from the parametric domain $\hat{\Omega }$. Maps ${\pi }_0$ and ${\pi }_t$ are defined in Equation 6, coordinate axes (in the reference domain) are defined in Equation 3, and circumferential and radial fiber directions $G_c$ and $G_r$ are defined in Equation 4. Note that an asterisk denotes that a parameter is fixed.

Figure 5:

Examination of leaflet coordinate axes and fiber trajectory (material axis) at a node on the surface of a leaflet mesh (for the chiral case). Fiber trajectory is determined by rotations of the coordinate axes.

Figure 4:

Fiber Trajectory for Baseline, Angled, and Chiral Cases used in leaflets for multi-body contact. Fiber trajectories are shown in red.

2.2.1. Kinematics

Simulating the closure of a tri-leaflet heart valve involves capturing the deformation of an elastic body under a pressure load. In this work, a quasi-static transvalvular pressure load is used to simulate leaflet closure. A brief overview of continuum mechanics is required to understand how the deformation of a heart valve leaflet, a nonlinear elastic material, is modeled. The deformation of an elastic body from its reference (un-deformed) configuration ${\Omega }_0$ to its current configuration ${\Omega }_t$ is described by a deformation map $\phi :{\Omega }_0\to {\Omega }_t,\phi \left(\mathbf{X}\right)=\mathbf{u}\left(\mathbf{X}\right)+\mathbf{X}$ where $\mathbf{X}$ is a material point in the initial (reference) configuration and $u\left(\mathbf{X}\right)$ is its displacement. The deformation gradient $\mathbf{F}=\nabla \phi \left(\mathbf{X}\right)$ is a natural way to describe the strain experienced by a body as it describes how a body deforms locally. The right Cauchy-Green strain tensor $\mathbf{C}={\mathbf{F}}^T\mathbf{F}$ captures only stretching and shearing as ${\mathbf{F}}^T\mathbf{F}={\mathbf{U}}^2$ given the stretch-rotation decomposition of the deformation gradient $\mathbf{F}=\mathbf{R}\mathbf{U}$ [39].

The computation of stretch and stress along the circumferential coordinate of the leaflet requires gradients $\nabla {\pi }_0$ and $\nabla {\pi }_t$. These are gradients of the map from the parametric to reference configuration $\nabla {\pi }_0$ and of the map from the parametric to current configuration $\nabla {\pi }_t$. In a continuum mechanics setting, the deformation gradient $\mathbf{F}$ is the gradient of the map $\phi$ from reference to the deformed configuration of a body in physical space. Below are equations defining these maps, along with their gradients. Note that $\mathbf{X}\in {\Omega }_0,\mathbf{u}+\mathbf{X}=\mathbf{x}\in {\Omega }_t$, and $\mathit{\xi }\in \hat{\Omega }$ (Figure 3).

$${\pi }_0\left(\mathit{\xi }\right)=\mathbf{X}\text{and}{\pi }_t\left(\mathit{\xi }\right)=\mathbf{x}$$ (5)

$$\nabla {\pi }_0=\frac{\partial \mathbf{X}}{\partial \mathit{\xi }}\text{and}\nabla {\pi }_t=\frac{\partial \mathbf{x}}{\partial \mathit{\xi }}$$ (6)

$$\phi \left[{\pi }_0\left(\mathit{\xi }\right)\right]={\pi }_t\left(\mathit{\xi }\right)$$ (7)

The parametric domain $\hat{\Omega }$ is introduced by the usage of NURBS. The maps in Equation 6 are necessary for various computations during a FE simulation using NURBS as a basis for geometry and analysis because we often integrate in the parametric domain. This method of analysis is known as Isogeometric Analysis (IGA) [40]. Note that IGA and FE are used interchangeably in this work as the NNFE approach only requires that the output layer of the Neural Network is defined as the degrees of freedom of the forward problem.

2.2.2. Leaflet Hyperelastic Material Model

We used a hyperelastic strain energy density function to govern the deformation of our model trileaflet heart valves. We used a Fung-like model that was used to model the mechanical behavior of the murine pulmonary valve (mPV) [38]. The model consists of an isotropic term ${\Psi }_{\text{iso}}$ accounting for the extracellular matrix, an anisotropic term ${\Psi }_{\text{aniso}}$ for collagen fibers, and a volumetric penalty term ${\Psi }_{\text{vol}}$ to control the compressibility of the leaflets.

$$\Psi ={\Psi }_{iso}+{\Psi }_{\mathit{\text{aniso}}}+{\Psi }_{\mathit{\text{vol}}}$$ (8)

in where

$$\begin{gathered} {\Psi }_{\mathit{\text{iso}}}=\frac{b_1}{2}\left[e^{b_2{\left(I_1-3\right)}^2}-1\right] \\ {\Psi }_{\mathit{\text{aniso}}}=\frac{c_0}{2}\left(e^Q-1\right) \\ {\Psi }_{\mathit{\text{vol}}}=\kappa \left[J\left(J-2+D_1\right)-D_1\text{log}\left(J\right)-D_1+1\right] \end{gathered}$$ (9)

where $Q=c_1{E}_{PD}^2+c_2{E}_{XD}^2+c_3{E}_{\varphi }^2+c_4{E}_{PD}{E}_{XD}$ and $J=\text{det}F$, and log refers to the natural log. Additionally, $I_1=\text{tr}\left(\mathbf{C}\right)$ is the first invariant. This model introduces a moderate amount of coupling by introducing products of the Green-Lagrange strain tensor $\mathbf{E}$ along user-defined preferred and cross-preferred fiber directions $G_{PD}$ and $G_{XD}$, respectively, where

$$E_c=G_{PD}^T\mathbf{E}{G}_{PD}\text{and}{E}_r=G_{XD}^T\mathbf{E}{G}_{XD}\text{and}{E}_{\varphi }=G_{XD}^T\mathbf{E}{G}_{PD}$$ (10)

and $\mathbf{E}=\frac{1}{2}(\mathbf{C}-\mathbf{I})$. Model parameters $b_1=33.73\text{kPa},b_2=0.97,c_0=11.94\text{kPa},c_1=43.14,c_2=1.19,c_3=3.82$, and $c_4=3.55$ were used in this study [38]. Additionally, $D_1=1{\text{e}}^5$ and $\kappa =10.0\text{kPa}$ were also used. Stretch and stress along fiber directions are defined similarly to strain. Stretch and stress are defined using the right Cauchy-Green stress tensor $\mathbf{C}$ and the cauchy stress tensor $\mathbf{T}=\mathbf{t}\cdot \mathbf{n}$, where $\mathbf{t}$ is the traction at a point with unit vector $\mathbf{n}$ perpendicular a surface through this point. Stretch in the circumferential direction was defined as ${\lambda }_c:=\sqrt{A_c\mathbf{C}{A}_c}$, where $A_c=A_0$, while stress in the circumferential direction is defined as ${\mathbf{T}}_{cc}:=a_c^T\mathbf{T}{a}_c$, where $a_c=a_0$ represents the circumferential coordinate axes in the deformed configuration, ${\mathbf{a}}_i=\frac{\partial \mathbf{x}}{\partial {\xi }_i}$. Coordinate axes in different configurations (reference and current) are used because stretch derived from $\mathbf{C}$ is defined relative to the reference configuration and $\mathbf{T}$ is expressed relative to the current configuration.

2.3. Neural Network Finite Element Method

Here we briefly summarize the neural network finite element method (NNFE), which is fully detailed in [41], and detail extensions required for this problem. The NNFE method is a machine learning approach for rapidly solving parameterized partial differential equations in the many query setting. In general, the idea behind this approach is to design and train a neural network whose inputs are the free parameters of a parameterized partial differential equation, and whose output layer is the degrees of freedom of a finite element solution to the partial differential equation corresponding to the input parameters. The loss function used to train the network is chosen so that it is minimized by networks whose outputs obey the PDE in question for the given input parameters. The PDE residual can be used as this loss function, since it is positive definite and vanishes on solutions, but for many classes of problems, including hyperelasticity, a potential energy functional $\Pi$ exists that is minimized at solutions. This potential energy is typically favorable to use because computing it requires fewer derivatives, and experimental results have shown that it trains networks more rapidly and with better generalization error [42].

To solve the governing PDE, in the classical approach, the minimization of potential energy approach is used [43]. In the absence of contact the potential energy $\Pi$ of the system is described as

$$\Pi ={\int }_{\Omega }\Psi \left(\mathbf{u}\right)dV-p{\int }_{\partial \Omega }\mathbf{u}\cdot \mathbf{n}dA$$ (11)

where $p$ is the transvalvular pressure load, $\Psi$ is the strain energy density function described in Equation 8, $\mathbf{u}$ is the displacement of a domain $\Omega$, and $\mathbf{n}$ are outward normals on the boundary of the domain $\partial \Omega$. In general, $\Psi$ can be treated as an arbitrary strain energy density functional. Minimizing the total potential energy using the directional derivative of $\Pi$, as $\mathbf{u}$ is a function, will solve our system.

$$\underset{\mathbf{u}}{\text{min}}\delta \Pi =D_{\delta \mathbf{u}}\Pi \left(\mathbf{u}\right)={\left.\frac{d}{dϵ}\Pi (\mathbf{u}+ϵ\delta \mathbf{u})\right|}_{ϵ=0}$$ (12)

In this case, we are interested in simulating the deformation of a trileaflet heart valve under a range of pressure loads. We therefore design networks with a single input variable $p$, corresponding to the pressure, and an output layer returning degrees of freedom (DoF) for our previously defined finite element discretizations. We are using an IGA-inspired approach for this problem, so these DoF are the displacements of the control points of the mesh, which we denote as ${\mathbf{u}}_i$.

The size of the output layers of our networks varies from example to example because the output layer is dependent on the discretization of our domain. Though the output layers of our different networks vary based on the specific details of our B-spline discretization, the training details remain largely unchanged. We choose a network architecture that depends on a set of neural network parameters $\left\{{\theta }^i\right\}$. These parameters are the weights and biases, $W$ and $b$, that determine the contribution each neuron has to the network, and thus the degrees of freedom of the system. The network is initialized in a manner such that for all inputs it returns the degrees of freedom corresponding to the identity map. At this point, we note that the predicted control point displacements take the functional form

$${\stackrel{\mathbf{ˆ}}{\mathbf{u}}}^i=f_{NN}\left(p,{\theta }^i\right).$$ (13)

where $f_{NN}$ represents the evaluation of the neural network (Equation 14) and $\stackrel{\mathbf{ˆ}}{\mathbf{u}}$ are control point displacements defined relative to the displacement of the middle layer of control points. We sample pressures using a Halton sequence in our pressure range of interest (0 – 90 mmHg) to obtain a set of training pressures $\left\{p^j\right\}$. These are then passed as input to the NN, yielding a collection of sets of control point displacements $\left\{{\mathbf{u}}_j^i\right\}$, with each set corresponding to an input pressure, ${\stackrel{\mathbf{ˆ}}{\mathbf{u}}}_j^i=f_{NN}\left(p^j,{\theta }^i\right)$. It is important to note that each neural network layer $x_n\in {\mathbb{R}}^{k_n}$, where $k_n$ is the layer width, is defined by a linear combination of highly nonlinear activation functions $\sigma$ and weights $W_n$ and biases $b_n$ for the $n^{\text{th}}$ layer.

$$\begin{gathered} f_{NN}\left(p^j,{\theta }^i\right)=W_n{x}_n+b_n \\ {x}_n=\sigma \left(W_{n-1}{x}_{n-1}+b_{n-1}\right) \end{gathered}$$ (14)

We then use these control point displacements to compute displacements on the body of the leaflet and compute the potential energy associated with each as a function of the network parameters, i.e. we obtain energies $\left\{{\Pi }^j\left({\theta }^i\right)\right\}$. The mean of this set is used as the loss function to train our network $ℒ\left({\theta }^i\right)=\frac{1}{N}\sum _{j=1}^N{\Pi }^j\left({\theta }^i\right)$ over $N$ pressure samples. We automatically differentiate this loss function with respect to the parameters of the NN, and use the Adam optimization algorithm to update the network parameters to minimize the potential energy of the predicted deformations (Figure 6). Adam is a combination of gradient descent with momentum and root-mean squared propagation (RMSP) [44]. For these models, the potential energy consists of three terms, all depending implicitly on the parameters ${\theta }^k$. The potential energy of the strain energy density function ${\Pi }_{\mathit{\text{strain}}}$, computed by integrating Equation 8 over each body ${\Omega }^{\left(i\right)}$, is defined as

$${\Pi }_{\mathit{\text{strain}}}^{\left(i\right)}\left[{\theta }^k\right]={\int }_{{\Omega }^{\left(i\right)}}\Psi dV.$$ (15)

The second term ${\Pi }_{\mathit{\text{pressure}}}$ also depends explicitly on the network’s input $p$, and incorporates the energy associated with the pressure boundary condition on the top leaflet surface

$${\Pi }_{\mathit{\text{pressure}}}^{\left(i\right)}\left[p,{\theta }^k\right]=p{\int }_{{\Gamma }_t^{\left(i\right)}}\mathbf{u}\cdot \mathbf{n}dA,$$ (16)

where ${\Gamma }_t^{\left(i\right)}$ is the traction portion of the $i^{\text{th}}$ body’s boundary, $p$ is the pressure, $\mathbf{u}$ is the predicted displacement, and $\mathbf{n}$ is the outward normal of ${\Gamma }_t^{\left(i\right)}$. The third term ${\Pi }_{\mathit{\text{contact}}}$, the contact potential, is defined later in the text (Equation 19).

Figure 6:

NNFE Training Framework for trileaflet heart valve simulations with multi-body contact. The output layer of the NN defines control point displacements relative to the displacement of the ‘middle layer’ of control points, ${\stackrel{\mathbf{ˆ}}{\mathbf{u}}}_n^{\left(i\right)}$, where ‘middle layer’ refers to the value of control points at ${\xi }_2=0.5$. After mapping these points to actual control point displacements, these then define the displacement on the entire domain $\mathbf{u}\left(\mathit{\xi }\right)$. Other important aspects of training include where the contact search occurs; the contact search algorithm runs after displacements $\mathbf{u}\left(\mathit{\xi }\right)$ are computed to compute the potential energy of a given sample $\Pi \left[p,{\theta }^k\right]$. This step occurs for each sample at each epoch, and is costly.

2.4. NURBS Discretization in the Neural-Network Finite Element Method

It is also important to discuss NURBS in more detail from the perspective of needing to determine the deformation of the domain from the deformation of a finite set of control points. NURBS curves are defined as parametric curves. The same goes for surfaces and volumes. Defining the domain in this parametric coordinate system, with differentiable rational basis functions, makes NURBS geometries analysis-ready. Sets of coordinates in a given direction are referred to as knot vectors $𝒰=\left\{{\xi }_1,{\xi }_2,\dots ,{\xi }_{n+p+1}\right\}$, where $p$ is the polynomial order, and $n$ is the number of basis functions used to define the curve [40]. Given an initial NURBS object with a set of control points and associated knot vector, knots can be inserted in a manner that does not change the geometry of the body that is analogous to mesh refinement.

The displacements of the control points ${\mathbf{B}}_i$ correspond to the coefficients ${\mathbf{u}}_i$ of the linear combination of NURBS basis functions $R_i^p$ that define the displacement of the body. To further elaborate, given control points in the reference and current configuration ${\mathbf{B}}_{i,0}$ and ${\mathbf{B}}_{i,t}$, the displacement ${\mathbf{u}}_t$ of a NURBS curve $C$ at a given point in time is defined as

$${\mathbf{u}}_t\left(\mathit{\xi }\right)=\sum _{i=1}^n{\mathbf{u}}_{i,t}{R}_i^p\left(\mathit{\xi }\right)$$ (17)

Displacement can also be defined as ${\mathbf{u}}_t\left(\mathit{\xi }\right)=C_t\left(\mathit{\xi }\right)-C_0\left(\mathit{\xi }\right)$. This allows us to describe the control points of the deformed configuration of leaflet. If weights ${\mathbf{w}}_i$ of NURBS rational basis functions $R_i^p\left(\mathit{\xi }\right)$ are fixed, then

$$\begin{gathered} {\sum }_{i=1}^n{R}_i^p\left(\mathit{\xi }\right){\mathbf{u}}_{i,t}={\sum }_{i=1}^n{R}_i^p\left(\mathit{\xi }\right){\mathbf{B}}_{i,t}-{\sum }_{i=1}^n{R}_i^p\left(\mathit{\xi }\right){\mathbf{B}}_{i,0} \\ {\sum }_{i=1}^n{R}_i^p\left(\mathit{\xi }\right)\left({\mathbf{u}}_{i,t}-{\mathbf{B}}_{i,t}+{\mathbf{B}}_{i,0}\right)=0 \\ {\mathbf{u}}_{i,t}={\mathbf{B}}_{i,t}-{\mathbf{B}}_{i,0} \end{gathered}$$ (18)

This formulation shows the displacement of the control points is equivalent to the degrees of freedom of our problem $\left({\mathbf{u}}_{i,t}\right)$, and describes how to obtain displacements on our ‘mesh’ from control point displacements. Note that this example was shown in 1-D for brevity and clarity; this holds for NURBS surfaces and solids. Instead of using one basis function $R_i^p\left(\mathit{\xi }\right)$, a product of basis functions in each coordinate direction $R_i^p\left(\mathit{\xi }\right)Q_i^p\left(\mathit{\xi }\right)T_i^p\left(\mathit{\xi }\right)$ is used (Equation 2).

2.5. Contact Formulation in the Neural-Network Finite Element Method

In the following, we incorporate the contact mechanics implementation developed in [24] to model multi-body contact, both between individual leaflets and symmetry planes for the reduced models, and between leaflets for the full trileaflet model. This is done by the addition of a contact penalty energy for each pair of bodies of the form

$${\Pi }_{\mathit{\text{contact}}}^{(i,j)}\left[{\theta }^k\right]=\frac{k}{2}{\int }_{\partial {\Omega }_i}{⟨\frac{\beta }{k}-g^{\left(j\right)}⟩}^{2}dA+\frac{k}{2}{\int }_{\partial {\Omega }_j}{⟨\frac{\beta }{k}-g^{\left(i\right)}⟩}^{2}dA$$ (19)

where $g^{\left(i\right)}$ is the signed distance from $\partial {\Omega }_i$, taking positive values on the exterior of ${\Omega }_i,k$ is a parameter determining the overall strength of the contact force, $\beta$ is a relaxation parameter that can be used to reduce numerical stiffness, and $⟨\cdot ⟩$ is the Macaulay bracket, defined as $⟨x⟩=\text{max}(x,0)$. Note specifically that this energy is positive definite, symmetric in $i$ and $j$, and vanishes when the signed distance between any two points on distinct bodies exceeds the distance $\frac{\beta }{k}$. Additionally, we used $k=1{\text{e}}^7\text{kPa}$ and $\beta =0$ in this study.

The gap functions for our B-Spline heart valve model are approximated in the manner described in [24] using 4 successive bifurcations of each knot interval to generate the array of boundary points. For the single leaflet symmetric simulations, the gap function of the symmetry half-planes, each oriented at an angle of $\pm \frac{\pi }{3}$ from the half-plane defined by $x_1=0$ and $x_2\ge 0$ is given explicitly as

$$g\left(\mathbf{x}={\left[x_1,x_2,x_3\right]}^{\top }\right)=\left\{\begin{array}{ll}‖x_1‖\text{cos}\left(\frac{2\pi }{3}\right)+x_2\text{sin}\left(\frac{2\pi }{3}\right)& ‖x_1‖\text{cos}\left(\frac{\pi }{3}\right)+x_2\text{sin}\left(\frac{\pi }{3}\right)>0\\ -\sqrt{x_1^2+x_2^2}& \text{else}\end{array}\right.$$ (20)

Therefore, the total energy for each training configuration is

$$\Pi \left[p,{\theta }^k\right]=\sum _{i=1}^m\left({\Pi }_{\mathit{\text{strain}}}^{\left(i\right)}\left[{\theta }^k\right]+{\Pi }_{\mathit{\text{pressure}}}^{\left(i\right)}\left[p,{\theta }^k\right]\right)+\sum _{i<j}^m{\Pi }_{\mathit{\text{contact}}}^{(i,j)}\left[{\theta }^k\right],$$ (21)

where $m$ is the number of bodies in each simulation. The loss function used to train the network is therefore

$$ℒ\left[{\theta }^k\right]=\frac{1}{N}\sum _{i=1}^N\Pi \left[p_i,{\theta }^k\right].$$ (22)

2.6. Trileaflet Heart Valve Closure

As previously mentioned, the closure of a trileaflet heart valve provides a challenging and applicable problem to test the incorporation of multi-body contact into the NNFE framework. We seek to capture symmetry-breaking deformation modes with multi-body contact to attempt to capture physiologically accurate heart valve deformations. Capturing multi-body contact with a potential-based formulation is a notoriously difficult problem that often requires the implementation of custom nonlinear solvers to avoid leaving the radius of convergence of Newton’s method, for example [45]. While a Newton step is not used during training, similar difficulties occur during training due to contact and the large deformations the leaflets undergo during closure. To overcome these difficulties, the output layer of the network used defines displacements of control points relative to the displacement of the middle layer of control points. We found that this couples the displacements of the control points in the through-thickness direction of the leaflet, which encourages the leaflet to perform the large translations necessary to close without causing it to invert its domain. This can occur because the pressure only acts on the atrial surface of control points. Additionally, without this modification in the representation of displacements in the output layer, we found that individual layers of elements were prone to large amounts of compression or extension, which created a physically unrealistic deformation.

We ran simulations with the baseline, angled, and chiral fiber trajectories discussed previously to induce physiologically-realistic symmetry-breaking deformation modes. Stretch and stress fields along the material axes were computed to examine how the fiber trajectory influences the mechanical response of the material. Stress in the through-thickness direction of the valve was also computed to examine the affects of using solid elements.

An angled frictionless symmetry plane was used to mimic contact with other leaflets to help pre-load them before multi-body contact occurs. The plane has an angle of 120°, and is defined by Equation 20. In its undeformed state, only the back edges of the leaflet commissure attachments were in contact with this plane. Each leaflet mesh used elements that were linear in the through-thickness direction and quadratic in the radial and circumferential directions. Three elements in the through-thickness direction were used, while seven elements in both the radial and circumferential direction were used. To create the trileaflet heart valve, an instance of a leaflet and symmetry plane were copied and rotated 120° and 240° about the origin. The NURBS solid representation of the leaflet was constructed so that these rotations about the origin cause the truncated commissure attachments on the ventricular side of neighboring leaflets to coincide (Figure 7).

Figure 7:

Trileaflet heart valve geometry used to test multi-body contact in the NNFE framework. Three identical leaflets are joined at their truncated commissure attachments.

It is important to mention that the surface we applied the pressure load to changes during training. The first stage applied the pressure load to the ventricular (top) surface of the leaflet until contact with the frictionless symmetry plane occurs. We applied the pressure load to the atrial (bottom) surface to allow the network to capture large deformations without having to take small steps, as large steps with a pressure applied to the atrial side of the leaflet can cause the top layer of control points to penetrate other elements in the leaflet. Once contact with the frictionless symmetry plane occurred, we applied the pressure to the atrial surface of the leaflet. The commissure attachment edges and basal attachment edges were fixed to have zero displacement. The pressure load applied was normal to the surface of the leaflet in its undeformed configuration, and forward simulations run by the trained network load the atrial surface of the leaflet.

2.6.1. NNFE Neural Network Architecture

A DenseNet architecture with a hyperbolic tangent activation function was used to train a NN to approximate the parameter to state map that governs the deformation of a trileaflet heart valve during closure. 16 training points were defined in a pressure range of $p\in \left[\mathrm{0,90}\right]\text{mmHg}$ during each iteration. The neural network consisted of six layers of 128 neurons, and the Adam optimizer and exponential decay were used for the learning rate with functions from optax for a total of 30,000 epochs. Training was split into three steps of 10,000 epochs each. In the first step, the initial learning rate was $1{\text{e}}^{-8}$. This remained constant for the first half of this step (for the first 5,000 epochs), and then exponentially decayed from $1{\text{e}}^{-8}$ to $1{\text{e}}^{-8}\frac{1}{1{\text{e}}^{1/3}}$ from epochs 5,000 to 10,000. The decay rate is set to $\frac{1}{1{\text{e}}^{1/3}}$ to reduce the learning rate from $1{\text{e}}^{-8}$ to a final value of $1{\text{e}}^{-9}$ in three steps. The pressure load is applied to the ventricular surface of the leaflet during this first step, and symmetry planes are defined for each leaflet to avoid non-physiological deformations in the early stages of training. During the second and third stages of training, the pressure load was applied to the atrial surface of the leaflet and the frictionless symmetry planes were removed. Each of these stages were comprised of 10,000 epochs with a constant learning rate for the first 5,000 epochs, and an exponentially decaying learning rate for the next 5,000 epochs. The decay rate for the second and third stages was the same as the decay rate for the first stage $\left(\frac{1}{1{\text{e}}^{1/3}}\right)$. Additional details regarding the parameters passed to the Adam optimizer are included in the appendix.

2.7. Verification

To verify the NNFE implementation of contact, an IGA simulation of contact between a single leaflet and a frictionless symmetry plane with identical geometry and boundary conditions was run in parallel to its corresponding NNFE simulation. Different quantities of interest such as displacement, stress, and stretch fields were examined. We also computed average and maximum absolute errors between the results obtained from the two methods. Our goal was to achieve minimal loss of accuracy in the NNFE approach when compared to traditional IGA simulations. Multi-body contact simulations were not used for verification because multi-body contact is not a feature of the open-source package FEniCS [46]. We used the functionality provided by tIGAr [37], a FEniCS wrapper, to run simulations with NURBS geometries in an IGA framework, and reached the decision that developing our own multi-body contact code separate from the FEniCS-tIGAr environment was the best path forward. We will briefly explain tIGAr and IGA in this section. A key advancement in the IGA field was the development of the Bezier extraction operator, which maps piecewise Bernstein Polynomials onto a B-Spline basis [47]. tIGAr uses an adaptation of the Bezier Extraction Operator that stores $\left\{{\mathbf{M}}^i\right\}$, the coefficients that help define homogenous scalar IGA basis functions $N_A^{IGA,hom,i}$ as a linear combination of scalar FE basis functions $N_B^{FE,i}$ as sparse matrices.

$$N_A^{IGA,hom,i}=\sum _{B=1}^{\begin{array}{c}{n}_{\mathit{\text{node}}}^{FE,i}\end{array}}{M}_{BA}^i{N}_B^{FE,i}$$ (23)

They are then applied using linear algebra operations in place of writing shape function routines to take extraction operators as input [37]. This is convenient as it leverages existing FE codes and thus requires a minimal amount of refactoring; the FE basis functions are defined in FEniCS.

The deformation of the single leaflet under a quasi-static transvalvular pressure load of 80.0 mmHg was simulated in both the IGA and NNFE frameworks to provide a 1:1 comparison between the two methods. The material model and parameters introduced in section 2.2.2 were used in this study. The same volumetric and contact penalty constants were also used. The definition of the contact plane was also the same; Google JAX datatypes and UFL conditionals were used to replicate the symmetry half-planes defined in the gap function (Equation 20). To verify the simulations, an ${\ell }^2$ norm was used to quantify the displacement error between the IGA and NNFE simulations.

$${‖{\mathbf{u}}_i^{\mathit{\text{NNFE}}}-{\mathbf{u}}_i^{FE}‖}_{{\ell }^2\left(\Omega \right)}≔\sqrt{\sum _{j=0}^2{\left({\left({\mathbf{u}}_i^{\mathit{\text{NNFE}}}\right)}_j-{\left({\mathbf{u}}_i^{FE}\right)}_j\right)}^2}$$ (24)

Where ${\left({\mathbf{u}}_i^{\mathit{\text{NNFE}}}\right)}_j$ corresponds to the displacement at the $i^{\text{th}}$ node in the $j^{\text{th}}$ Cartesian coordinate direction. It is important to note that the NNFE code evaluates integrals in the coordinate domain while tIGAr defines integrals in the reference configuration and evaluates them in the parametric domain through a series of push-forward and pull-back operations under the hood.

As in section 2.6, an angled frictionless symmetry plane was used to mimic contact with other leaflets (Figure 8). The geometry of the plane, the leaflet mesh used, and the pressure load applied to the leaflet were also the same. The number of Finite Elements used to define the geometry in FEniCS was larger than the number of elements used in the NNFE approach, as Finite Element basis functions are described as linear combinations of NURBS basis functions. For scalar fields $x$, the ‘average absolute error’ refers to the average of the quantity $\left|x^{\mathit{\text{NNFE}}}-x^{FE}\right|$ and the ‘maximum absolute error’ refers to its maximum. For displacements, i.e. a vector field, the average absolute error is defined as $\frac{1}{n}\sum _i^n{‖{\mathbf{u}}_i^{\mathit{\text{NNFE}}}-{\mathbf{u}}_i^{FE}‖}_{{\ell }^2\left(\Omega \right)}$, and the maximum absolute error is defined as ${\text{max}}_i{‖{\mathbf{u}}_i^{\mathit{\text{NNFE}}}-{\mathbf{u}}_i^{FE}‖}_{{\ell }^2\left(\Omega \right)}$, where $n$ is the number of nodes in the mesh.

Figure 8:

Single Leaflet with symmetry plane used to mimic contact with other leaflets for verification, isometric and top-down views shown. Commissure attachments are fixed and lie on the frictionless symmetry planes in the reference configuration of the leaflet. Displacements, stretches, and stresses of IGA and NNFE forward simulations for the same BVP are compared.

2.7.1. NNFE Neural Network Architecture

Similar to the trileaflet heart valve simulations (sec 2.6.1), a DenseNet architecture with a hyperbolic tangent activation function was used to train a NN to approximate the parameter to state map that governs the deformation of the pressurized single leaflet against a frictionless symmetry plane. The only differences in architecture were an increased number of training points from 16 to 32, and the use of two training steps of 10,000 epochs each instead of three steps of 10,000 epochs each, where the pressure load was applied to the ventricular surface of the leaflet in the first step, and the atrial surface of the leaflet in the second step. The initial learning rate was still $1{\text{e}}^{-8}$. This remained constant for the first half of this step (for the first 5,000 epochs), and then exponentially decayed from $1{\text{e}}^{-8}$ to $\frac{1}{\sqrt{10}}$ of the initial learning rate during the remaining 5,000 epochs. The same learning rate scheduling was used for the second step, so the final learning rate was also $1{\text{e}}^{-9}$.

3. RESULTS

For the following presentation of the results, the circumferential direction is referred to $A_0$ in the undeformed and $a_0$ in the deformed configurations, respectively. Similarly, the radial and transmural directions refer to the $A_1$ and $A_2$, respectively.

3.1. Full Trileaflet valve Simulations

As mentioned in the Methods we varied the internal structure of the leaflets to see if we could replicate deformation patterns observed in vivo. We examined the difference in the stretch and stress fields with the various prescribed fiber trajectories (Figure 4). The stretches observed in the baseline and angled fiber trajectories were similar, while the chiral fiber trajectory introduced higher stretches along the leaflet free edge. This is because of a slight folding of the leaflet free edge that occurs near the leaflet orifice for the chiral fiber trajectory. Additionally, the change in angle $\Delta \alpha$ of connected edges on the ${\xi }_2$ face closest to the ventricular side of the leaflet was measured. There was a region of large $\Delta \alpha$ (up to 34°) near the commissure attachment observed in the baseline case (Figure 9). This confirms that the deformation of the leaflets is dependent on their internal fiber structure, which indicates that the NNFE approach captures changes in mechanical behavior due to changes in fiber direction.

Figure 9:

Side view of stretches at 80.0 mmHg for Various Fiber Orientations

As with the stretches observed, variations in fiber trajectories produce large variations in the leaflet stresses. Again, this is most obvious with the chiral fiber trajectory. The maximum stress in the radial direction ${\mathbf{T}}_{rr}$ is about 1 MPa, which is 3x larger than the maximum radial stresses for the baseline and angled cases (280 and 300 kPa, respectively). For stress in the shear direction ${\mathbf{T}}_{rc}$, the maximum is about 740 kPa, again $3x$ larger than the maximum shear stress in the baseline (250 kPa) and angled (230 kPa) fiber trajectories (Figure 10). Large stresses occur near the commissure attachments across all fiber trajectories. Additionally, all of these stresses qualitatively reflect those observed in the single leaflet verification case. For the baseline fiber trajectory (which was used in single leaflet verification), maximum stresses in the multibody and single leaflet case are similar, and are compared (Table 2). The nature of the contact in the single leaflet and multi-body cases is different, as contact against a frictionless symmetry plane is more uniform as it is not a deformable body, and the symmetry plane is symmetric about the origin. We do however observe that the largest stresses appear in the circumferential direction for both cases, and the values of these maximum stresses are similar.

Figure 10:

Side view of Cauchy Stresses at 80.0 mmHg for Various Fiber Orientiations

Table 2:

Comparison of maximum stresses in circumferential, radial, and shear directions in single leaflet verification and trileaflet simulations with the baseline fiber trajectory.

Simulation/Max Stress	${\mathbf{T}}_{cc}$	${\mathbf{T}}_{rr}$	${\mathbf{T}}_{rc}$
Single Leaflet	1291 kPa	344 kPa	216 kPa
Trileaflet Valve	3160 kPa	280 kPa	250 kPa

3.1.1. Use of solid elements

To examine the effect of the use of solid elements to model the heart valve leaflets, circumferential stress in the through-thickness direction was examined.

As expected, the circumferential stress varies in the through-thickness direction. This justifies the use of solid elements to model the deformation of a heart valve leaflet (Figure 11). Importantly, forward simulations of multi-body contact run on the order of magnitude of seconds. Shells are known to be more computationally efficient than solids as the dimension reduction decreases the number of degrees of freedom in the discretization of the problem. Given that the execution time of a forward simulation is primarily governed by the architecture of the Neural Network used to approximate the pressure to displacement map, there should be no noticeable difference in runtime when using shells or solids to model the geometry of the leaflet.

Figure 11:

Stress varies nonlinearly in the through-thickness direction, justifying the usage of solid elements.

3.2. NNFE Verification

To verify NNFE results with a FE baseline solution, differences in displacement, stress, and stretch fields of a single leaflet in contact against a frictionless symmetry plane simulated using the NNFE approach and an FE approach were examined. We examined differences in displacement, stretch, and stress fields between the two methods in addition to computing average and maximum absolute errors between the results.

The displacements agreed well, with a maximum ${\ell }^2$ error of 0.052 mm, which was approximately 1% of the maximum displacement experienced by the leaflet (4.25mm) (Figure 12). It should be noted that the radial coordinates of the leaflet are slightly asymmetric about the mid-plane (the plane where ${\xi }_0=0.5$), so the stretches experienced by the leaflet are not perfectly symmetric. Both the NNFE and FE solutions capture this behavior (Figures 13, 14, 15, 16). Additionally, the maximum absolute error for each case is less than 2%, which indicates that the response of the material is captured well by the NNFE approach. Qualitatively, the stress fields appear to be very similar. Large stresses are observed in the leaflet, and the NNFE and FEM approaches capture the same regions of high stress, which occurs at the commissures (Figures 17, 18, 19, 20). These stresses are slightly higher than anticipated because of the relatively coarse discretization of the leaflet used for these simulations. It could also be an artifact of the choice of leaflet geometry and material parameters used as the leaflet parameterization was not designed to minimize stresses.

Figure 12:

NNFE and FE displacements with 80.0 mmHg load applied normal to leaflet reference configuration. Average ${\ell }^2$ error: 0.020 mm, maximum ${\ell }^2$ error: 0.052 mm.

Figure 13:

Circumferential stretch. Average absolute error: 0.178%, max absolute error: 1.73%.

Figure 14:

Radial stretch. Average absolute error: 0.285%, max absolute error: 1.15%.

Figure 15:

Through-thickness stretch. Average absolute error: 0.259%, max absolute error: 1.42%.

Figure 16:

Change in angle $\Delta \alpha$. Average absolute error: 0.23 degrees, max absolute error: 1.67 degrees

Figure 17:

Circumferential stress. Average absolute error: 11.1 kPa, max absolute error: 148 kPa.

Figure 18:

Radial stress. Average absolute error: 3.65 kPa, max absolute error: 26.2 kPa.

Figure 19:

Through-thickness stress. Average absolute error: 5.15 kPa, max absolute error: 80.4 kPa.

Figure 20:

Shear stress. Average absolute error: 2.99 kPa, max absolute error: 30.5 kPa.

3.2.1. NNFE vs. FE (IGA) runtimes

In the present study an NNFE forward simulation for single leaflet with contact took approximately 0.28s, while its FE counterpart took approximately 61s (Table 3). The scaling of runtimes for NNFE and FE solutions with mesh refinement has been demonstrated in our previous work, which showed that NNFE runtime barely changes with an increasing number of degrees of freedom of the problem, while FE simulation runtime directly scales with problem size [21]. This can be explained by the lack of direct scaling of FLOPs with mesh discretization, as only the output layer of the Neural Network used in the NNFE framework is dependent on the number of degrees of freedom of the problem. This is contrary to forward simulations using the Finite Element Method, where the size of the mass matrix scales directly with the number of degrees of freedom $\left(N\right)$. For the multi-body trileaflet scenario, forward simulations took about 1 second to run once they are just in time (JIT)-compiled (which takes approximately 5 seconds). JIT refers to the process of converting Python code to machine code when the program is executed. In the present study an NNFE forward simulation for single leaflet with contact took approximately 0.28s, while its FE counterpart took approximately 61s. The scaling of runtimes for NNFE and FE solutions with mesh refinement has been demonstrated in our previous work, which showed that NNFE runtime barely changes with an increasing number of degrees of freedom of the problem, while FE simulation runtime directly scales with problem size [21]. This can be explained by the lack of direct scaling of FLOPs with mesh discretization, as only the output layer of the Neural Network used in the NNFE framework is dependent on the number of degrees of freedom of the problem. This is contrary to forward simulations using the Finite Element Method, where the size of the mass matrix scales directly with the number of degrees of freedom ($N$). For the multi-body trileaflet scenario, forward simulations took about 1 second to run once they are just in time (JIT)-compiled (which takes approximately 5 seconds). JIT refers to the process of converting Python code to machine code when the program is executed.

Table 3:

Runtime comparison of NNFE vs. FE (IGA) simulations of single leaflet and full trileaflet valve simulations

Simulation/Method	FE (IGA)	NNFE
Single Leaflet	60 s	0.2s
Trileaflet Valve	N/A	2s

4. DISCUSSION

4.1. Study Summary and Key Findings

In the present study we sought to extend our NNFE approach to the simulation of a trileaflet heart valve. This specific application was chosen as it involves several important features of soft organ function, including full 3D geometries, nonlinear hyperelastic material behaviors, and multi-body contact. The use of solid elements allows differences in circumferential stress in the transmural direction to be observed (Figure 11). This is novel as shell-based models of heart valves that fail to capture these transmural effects are traditionally used in these structural mechanics simulations. Moreover, use of NURBS representations of solids are desirable because of their ability to represent geometries with few parameters and the ability to change their polynomial order without changing the geometry. Full multi-body contact was facilitated by the NNFE approach as the loss function of the NN is defined using the potential energy of the system. Deformation patterns of heart valve leaflet with different fiber trajectories matched those observed in previous works [48]. To verify the approach, NNFE simulations of a single leaflet pressurized against a frictionless symmetry plane were compared to IGA simulations using the same model geometry, boundary conditions, and material parameters.

In terms of accuracy, we demonstrated that average difference between displacements computed using a traditional FE/IGA based approach and the NNFE approach was 0.020 mm (0.47%) with a maximum error of 0.052 mm (1.22%) (section 3.2). We also demonstrated that stress and stretch fields agreed well. In terms of performance the NNFE forward simulation of single leaflet contact against a frictionless symmetry plane took only 0.28s, whereas its FE counterpart took 61s. The acceleration of forward simulations relative to traditional FE simulation techniques provides the possibility of using the NNFE approach in a multiple query scenarios. It should be noted that based on earlier studies, the relative performance gains using the NNFE method should improve dramatically as the problem size increases [21].

4.2. Future Use in Biomedical Applications

This work represents a step towards developing high-speed optimization simulations for replacement heart valves such as TAVRs. Such simulation will need to train on several parameters, such as varying leaflet dimensions, thickness, properties, and implantation locations. A NNFE approach is ideal for multi-query scenarios as they are trained over ranges of the design parameter space. This framework can be applied to the optimization of any medical device procedure where similar patient-specific optimization is required. Optimizing surgical procedures can increase the population eligible to receive treatments as current guidelines are often restrictive in an attempt to protect high-risk patients. Guidelines can be loosened if patient-specific responses to proposed procedures can be accurately predicted.

4.3. Broader Implications in Computational Mechanics

This study includes one of the first implementations of a neural network-based multi-body contact framework where the training is not data-driven. Leveraging a physics-based approach is preferred as it does not require conducting costly and time-consuming a-priori numerical experiments for model training. Importantly, this approach is derived from the Deep Ritz method [49, 50]. Working directly with the potential energy of the system over ranges of pressure inputs results in a robust method for solving the governing PDEs. Moreover, contact mechanics problems are notoriously hard to simulate and are computationally expensive. Simulating NNFE multi-body contact in seconds broadens the potential applications of contact mechanics simulation tools by replicating results previously attained on high performance computers on laptops via trained neural networks.

4.4. Future Work

Further development of the NNFE approach is required to capture complex multi-scale interactions to conduct full organ-level simulations. Understanding the organ-level impact of medical device implantation is critical to predict patient-specific response to a proposed procedure. In terms of geometric descriptors, use of T-splines to describe geometry is another future step of our work [51]. T-splines allow for local mesh refinement with far fewer control points, resulting in a higher-fidelity discretization without massively increasing the number of degrees of freedom [52]. Beyond training on boundary conditions, NN can be trained over ranges of material and geometric parameters [53], so that only a single training set is needed. The present study thus provided an important step in developing a high-speed NNFE-based method of simulating soft organ function.