To model the relation between myocardial deformation and mechanical responses, an appropriate constitutive model is required to describe its passive mechanical behavior. Such a constitutive model expresses the Cauchy stress tensor σ as a general function of the deformation gradient tensor F parameterized by a few material parameters ω, (1)σ=H(F;ω).

In this study, we consider the ventricular myocardium as a passive, heterogeneous, nonlinear, hyperelastic, and nearly incompressible material. Using the deformation gradient F, the right Cauchy-Green tensor can be calculated as, (2)C=FTF.

Strain invariants can be calculated from C. For instance, the first invariant I1 is (3)I1=tr(C).

A unit vector a0 is typically used to describe a preferred material direction in the reference configuration, which introduces material anisotropy. Thus, the pseudo-invariant I4, which characterizes the deformation along the fiber directions is defined by (4)I4=a0⋅(Ca0).

In the case of myocardium tissues, the myofiber, sheet, and sheet-normal directions are needed to fully describe the anisotropic constitutive relations. Hence, the unit vectors in the fiber, sheet, and sheet-normal directions are f0, s0, and n0, respectively. Accordingly, the pseudo-invariants I4f, I4s, I4n for in the fiber, sheet, and sheet-normal directions can be defined, respectively by (5)I4f=f0⋅(Cf0),I4s=s0⋅(Cs0),I4n=n0⋅(Cn0).

Hyperelastic constitutive relations typically focus on prescribing a specific form of the strain energy density function. For passive myocardium tissues, the Holzapfel-Ogden model (HO) [30] is a popular choice. In this study, we consider a simplified strain energy density function Ψ in which fiber-sheet and fiber-sheet-normal coupling terms are not included. With local preferred fiber direction f0, the deviatoric part Ψ¯ of strain energy density function Ψ is given by (6)Ψ¯=a2bexp[b(I¯1−3)]+af2bf{exp[bf(I¯4f−1)2]−1}where material parameters a (kPa) and b characterize the stiffness of the underlying non-collagenous and non-muscular matrix. Material parameters af (kPa) and bf reflect the stiffening behavior in the myofiber direction. The above 4 parameters are positive material constants. The volumetric strain energy density is (7)U(J)=k2(J2−12−ln⁡J)where k is the bulk modulus and is fixed to be 105 in this study. J=det(F) stands for the Jacobian of deformation gradient. The above derivations lead to the total strain energy function given by (8)Ψ=Ψ¯+U

The goal of the inverse problem is to identify the four unknown material constitutive parameters ω=[a,b,af,bf] of LV from dynamic clinical image data during end-diastole. Given the HO constitutive model, the Cauchy stress tensor σ can be computed as the derivative of the strain energy density function with respect to the deformation gradient tensor F using the following expression: (9)σ=J−1∂(Ψ)∂FFT

Conventionally, the Cauchy stress tensor σ in Eq. (9) is derived manually for a specific strain energy function. In contrast, the PyTorch environment enables efficient computation of σ through automatic differentiation (namely autograd) without requiring model-specific derivations. In the present study, autograd was employed to compute the Cauchy stress tensor σ following Eq. (9), which was employed in the FEA-only and DNN-FEA implementation introduced in the subsequent sections.

In this work, nonlinear FEA was employed to solve forward and inverse hyperelasticity problems. Here, we briefly summarize the standard FEA method used to discretize continuous kinematics and static equilibrium equations. Such discretization is established through interpolation using shape functions, for example, the displacement u of a specified location in an element can be obtained via (10)u=∑i=1nNi(ξ1,ξ2,ξ3)uiwhere Ni is the shape function corresponding to node i as a function of element natural coordinates (ξ1,ξ2,ξ3). Ni interpolates nodal displacement ui inside the element. n is the number of nodes in the element. The deformation gradient tensor can be obtained as (11)F=∑i=1nxi⨂∂Ni∂X

Hence, the rate of deformation d can also be discretized by using (12)d=12∑i=1n(vi⨂∂Ni∂x+∂Ni∂x⨂vi)where vi is the velocity at node i. In FEA, equilibrium is often expressed in the weak form of principal of virtual work. Without body force, the total virtual work done by the residual force within a deformable body defined by volume V and boundary area ∂V can be expressed as [42]: (13)δW=∫Vσ:δddV−∫∂VtδvdAwhere δd is the virtual rate of deformation, δv is the virtual velocity, and t is the traction forces per unit area acting on ∂V. Discretizing Eq. (13) for a virtual nodal velocity δvi of element m, and recognizing that δd=1/2∑i=1n(δvi⨂∂Ni∂x+∂Ni∂x⨂δvi), and δv=∑i=1nNiδvi, resulted in [42]: (14)δWim=δvi⋅(Tim−Gim)where Tim=∫Vmσ∂Ni∂xdV is the internal equivalent nodal force, and Gim=∫∂VmNitdA is the external equivalent nodal force. The virtual work of the contributions from all elements containing node i (m = 1 to Mi) can be obtained by (15)δWi=∑m=1MiδWim=δvi(Ti−Gi)where Ti=∑m=1MiTim and Gi=∑m=1MiGim are the assembled internal and external equivalent nodal force, respectively. The residual force at each node is (16)Ri=Ti−Gi

The virtual work equation Eq. (15) must be satisfied for any arbitrary virtual nodal velocities, consequently, the equilibrium condition is satisfied when the residual force at each node vanishes, i.e., Ri=0.

Traditionally, the FEA solution of Eq. (16) is obtained using the Newton-Raphson iterative procedure with a prescribed tolerance on the maximum residual nodal forces. In a forward nonlinear FEA problem, given the undeformed geometry, material parameters, and boundary conditions, the nodal displacements ui are incrementally solved with traction forces gradually applied in each load step. The tangent stiffness matrix K=∂R∂u must be explicitly computed at each iteration for the Newton-Raphson method. In contrast, an inverse nonlinear FEA problem solves for the material parameters ωm at each element, given the undeformed and deformed geometries along with boundary conditions. However, computing the tangent stiffness matrix with respect to material parameters, i.e., K=∂R∂ω, is challenging. Using finite differences for each parameter can make the process extremely computationally expensive. Traditionally, accelerated inverse solutions require manual derivation of an adjoint solution [43–45] to obtain K, which can be tedious and often limited to a specific constitutive relation. Fortunately, our recently developed PyTorch-FEA framework [39] addresses these challenges by enabling efficient inverse material parameter identification without the need to manually derive the tangent stiffness matrix. Specifically, the goal is to determine an optimal set of parameters ωm at each element by minimizing the loss function defined in terms of residual forces: (17)L(ωm)=1N∑i=1N‖Ri‖22where N is the total number of nodes. ‖◼‖2 denotes the L2 norm. Minimizing this loss yields the optimized material parameters. As shown in Fig. 1, by taking advantage of PyTorch’s automatic differentiation (autograd), gradients of the loss with respect to ωm are computed directly, which avoids manual adjoint method derivations. Changing the constitutive relation can be achieved simply by substituting a different strain energy density function. Moreover, PyTorch-FEA enables optimization with state-of-the-art algorithms such as limited-memory Broyden-Fletcher-Goldfarb-Shanno (L-BFGS) and supports efficient GPU acceleration. In this study, the FEA-only inverse material parameter identification was completed using PyTorch-FEA on a NVIDIA A800 GPU using the L-BFGS optimizer.

The optimization problems associated with identification of nonlinear hyperelastic material parameters often encounter difficulties known as local optima, i.e., multiple sets of material parameters can yield similar near-minimal loss values [46]. This challenge is amplified in heterogeneous material parameter identification due to the large number of unknowns (number of parameters in the constitutive model × number of elements). The problem is further complicated by input noise, such as noises resulting from medical image modalities [47], which propagates to the deformed and undeformed geometries.

To address these issues and mitigate the impact of heterogeneous unknown fields, traditional approaches often incorporate regularization terms (e.g., L1 and L2 regularization or smoothness-based priors) into the loss function to penalize large spatial parameter differences and enforce smoothness [48–50]. However, these explicit penalty terms regularize the solution toward prior assumptions such as smoothness of the identified material parameter fields.

In this study, we employed a DNN-FEA integration strategy to regularize the material parameter field without relying on explicit priors. Specifically, a multilayer perceptron (MLP) was used as a universal function approximator to represent the spatial distribution of constitutive parameters as a function of the undeformed reference coordinates X, i.e., (18)ω(X)=DNNaNN(X)where aNN represents the MLP learnable parameters. Therefore, the material parameters for each element, ωm, can be computed at the center of element m, X¯m. Instead of solving for the parameters independently at each element, the MLP weights are updated to represent the spatial distribution of parameters. According to the universal function approximation theorem, a properly constructed MLP can approximate any spatial distribution function, which provides a flexible regularization effect without prescribing explicit priors.

In the DNN-FEA inverse workflow, the same loss function based on FEA nodal residual force (Eq. (17)) was employed without the need for additional regularization terms, (19)L(aNN)=1N∑i=1N‖Ri‖22

The ability of DNN to regularize parameter distributions arises from its continuous differentiability, which inherently links the material parameters of adjacent elements within the MLP. Since ω is expressed as a function of aNN, the loss function is consequently also a function of these weights. As shown in Fig. 2, the DNN was used to compute material parameters at each element ωm. A sinusoidal activation function was used in the MLP. The network architecture (number of layers and units per layer, and activation function) was selected via grid search. The learnable parameters aNN are optimized by minimizing the loss function (Eq. (19)) using the L-BFGS algorithm. Training proceeds until the maximum parameter difference between successive epochs falls below a termination tolerance (e.g., 10−10). In this study, DNN training was performed on the Nvidia A800 GPU.

In a previous study, we retrospectively collected pre-surgical ECG-gated cardiac CT images from a patient with aortic stenosis who subsequently underwent transcatheter aortic valve replacement (TAVR). We developed a deformation-based deep learning framework, namely DeepCarve (Deep Cardiac Volumetric Mesh) [52], to automatically generate patient-specific volumetric meshes for FEA-based TAVR deployment simulations. In the present study, the DeepCarve framework was used to generate a surface mesh of the patient’s left ventricle (LV) at the early-diastolic phase from the patient-specific CT scans. Since this is the phase when diastolic filling has just begun and the intraventricular pressure is typically near zero, the geometry was considered to represent the undeformed, zero-pressure reference state for subsequent analyses.

The LV surface geometry was cut at both the mitral and aortic annuli and represented by triangular elements. To convert the surface mesh into a volume mesh, tetrahedral elements were employed to fill the LV myocardium. However, 4-node linear tetrahedral elements are unsuitable for nonlinear large-deformation FEA because they lead to a constant deformation gradient within each element [53,54]. Therefore, following surface mesh generation, the LV volume mesh was constructed using 10-node quadratic tetrahedral elements generated by using the ‘tetra mesh’ tool in Altair HyperMesh, which are well suited for large deformation analysis. Finally, fiber orientations representing a simplified myocardial architecture were defined on the undeformed LV volume mesh. Myofibers were assumed to be aligned circumferentially, which represents the orientation of mid-wall fibers. The sheet-normal direction was defined as the surface normal of the outer LV wall, and the sheet direction was computed as the cross product of the myofiber and sheet-normal directions.

Synthetic heterogeneous material property distributions (Distributions 1 and 2) were assigned to the undeformed LV geometry to generate the deformed end-diastolic configuration. A cylindrical coordinate system was employed to define the spatial variation of material parameters, which were assumed to vary along the circumferential (θ) and axial (z) directions but remain constant along the radial, thickness (r) direction. To represent typical spatial nonuniformity [55], periodic patterns were introduced into the material parameter distributions using a summation of sinusoidal functions. The following distribution was used for Distribution 1: (20a)ωp(z,θ)=ω¯p+λ(sin⁡10θ)(0.2sin⁡(z)+0.3sin⁡(0.1z)+0.5sin⁡(0.01z)where ωp (a=1,2,3,4) is one of the material parameters (a,b,af,bf). ω¯p denotes the baseline material parameters, which were derived from MRI-based measurement of healthy human subjects in a previous study [56] and are shown in Table 1. λ is a tunable constant controlling the amplitude of the periodic fluctuations, which was adjusted to ensure that ω is always positive. For Distribution 1, λ was set to be 0.2, −0.5, 0.5, −0.5, respectively, for a,b,af,bf. The resulting range of ωp is shown in Table 1. Similarly, Distribution 2 was generated to represent slightly larger spatial patterns: (20b)ωp(z,θ)=ω¯p+λ(sin⁡5θ)(0.2sin⁡(0.5z)+0.3sin⁡(0.1z)+0.5sin⁡(0.01z)where ω¯p was derived from cyclic simple shear experiments of porcine heart tissues [57]. λ was set to be 0.5, −0.5, 0.5, −0.5, respectively, for a,b,af,bf in Distribution 2 to enforce that ω is always positive. The resulting range of ωp in Distribution 2 is also shown in Table 1.

Starting from the undeformed LV geometry derived from patient CT scans and the assigned heterogeneous material property distribution, a forward finite element simulation was performed to generate the deformed end-diastolic LV geometry under a uniform intraventricular pressure. The pressure was set to P=30 mmHg consistent with typical values of healthy and diseased individuals [22]. The forward FEA simulation was implemented using PyTorch (Fig. 3). Briefly, forward FEA computed the nodal displacements ui(t) by minimizing the residual force (Eq. (16)) as loss in each pseudo time step t. The intraventricular pressure was gradually ramped to 30 mmHg, such that at step t, P(t)=t⋅P. Convergence at each time step was considered achieved when the ratio between maximum residual nodal force and averaged internal equivalent nodal forces fell below the prescribed tolerance of 0.005. If the solution failed to converge at a given time step, the pseudo-time increment was adaptively reduced and the displacement solution ui(t) was reattempted. For displacement boundary conditions, all nodes located on the mitral and aortic annuli were constrained in all degrees of freedom throughout the simulation.

For validation, both FEA and DNN-FEA were employed to identify heterogeneous passive material parameters of the left ventricle (LV) from the undeformed and deformed geometries. The identified material parameter distributions obtained from FEA and DNN-FEA were compared against the ground-truth distributions used to generate the geometries. To quantitatively evaluate the accuracy of the inverse identification, the average and maximum relative errors of each HO material parameter, denoted as ωp, were computed across all elements: (21)εpm=|ω^pm−ωpm|ω¯p (22)εpavg=1M∑m=1M|ω^pm−ωpm|ω¯p (23)εpmax=max|ω^pm−ωpm|ω¯pwhere ω^pm and ωpm represent the identified and ground-truth parameters of element m, respectively, and ω¯p=1M∑m=1Mωpm denotes the mean ground-truth parameter across all elements.

To further compare the identified and ground-truth mechanical responses, stress-strain curves were generated under simulated biaxial loading conditions using both the identified and ground-truth material parameters. Three stretch-controlled loading protocols were considered, varying the ratio between myofiber strain (Eff) and sheet (Ess) strain as Eff:Ess=0.5,1,2. The corresponding Cauchy stress and Green strain responses were visualized in the myofiber and sheet directions. Performance was quantified by first generating stress-strain curves based on the average identified parameters (ω¯^) and the average ground-truth parameters (ω¯) material parameters. Additionally, stress-strain curves were simulated for representative spatial locations within the LV geometry to illustrate local variations in model performance.

Both the FEA-only and DNN-FEA inverse solvers were optimized using the L-BFGS algorithm implemented in PyTorch [58,59]. The following hyperparameters were used in the numerical validation: history size = 20, line-search type = strong Wolfe, maximum iterations per step = 1, gradient tolerance = 1×10−10, parameter-change tolerance = 1×10−10, and learning rate = 1.0. No preconditioning was used. The L-BFGS optimization procedure is fully deterministic.

To determine the optimal MLP architecture for the DNN-FEA inverse approach, a grid search was performed by varying both the number of hidden layers and the number of units per layer using validation data with Distribution 1. The results of this search are summarized in Table 2. The best-performing DNN architecture was selected based on the following four criteria: (1) the minimized loss function value (total residual forces), (2) computation time per epoch, (3) the average error ε¯avg, and (4) maximum error ε¯max across all material parameters. To balance these trade-offs, an overall score was calculated by normalizing each criterion between its minimum and maximum values and then summing the normalized criteria. Based on this evaluation, the final architecture selected consisted of 12 hidden layers with 256 units per layer. This DNN architecture was applied to identify heterogenous material parameters in Distribution 2.

Several activation functions (sinusoidal, softplus with sigmoid in the output layer, ReLU, and ELU) were compared using the best-performing DNN architecture. As shown in Table 3, the sinusoidal activation function achieved the best overall score and was retained for subsequent analysis.

Both the FEA and DNN-FEA approaches were applied to the numerical validation dataset with Distribution 1. The identified spatial distributions of the four material parameters are shown in Fig. 4, alongside the ground-truth distributions. While FEA produced parameter fields with similar spatial patterns resembling the ground truth, noticeable deviations in parameter values were observed. These discrepancies are likely attributable to local optima in the optimization of the loss function. In contrast, DNN-FEA yielded spatial fields that were visually much closer to the ground truth for all parameters. Quantitative evaluation is summarized in Table 4, which reports the average and maximum errors for each parameter. The largest errors were observed for parameter a, where FEA produced errors several times higher than those of the other parameters. By comparison, DNN-FEA consistently maintained average errors below 2.2%, which demonstrated substantially greater accuracy. Fig. 5 further illustrates the spatial error distributions for each parameter. Overall, the closer agreement with ground truth distribution and the lower errors indicate that the DNN-FEA framework consistently outperformed the FEA method.

To further investigate the mechanism behind the superior performance of DNN-FEA in the validation case with material parameter Distribution 1, we visualized the residual force-based loss function values and the average errors εpavg of each material parameter during the solution process over 20,000 epochs for both the FEA and DNN-FEA methods. The results are shown in Fig. 6. For all four parameters, DNN-FEA exhibited a steady reduction in loss values, accompanied by a progressive and consistent decrease in average parameter error following the same trend. In contrast, the FEA method showed an inconsistent reduction in average error, even though the loss function decreased quickly to a small value. For parameters b and bf, the average error initially increased slightly before decreasing. This discrepancy between residual force loss and parameter error suggests that the FEA-only solution was likely trapped in local optima. These findings demonstrate that DNN-FEA provides an effective regularization effect in inverse material parameter identification, which helped to avoid suboptimal solutions.

To evaluate the identified myocardial mechanical response against the ground-truth response in the validation case with material parameter Distribution 1, we simulated stress-strain behaviors in both the myofiber and sheet directions. Fig. 7 compares the stress-strain curves obtained using the spatially averaged material parameters ω¯^ identified by the FEA and DNN-FEA methods, identified by the FEA and DNN-FEA methods, as well as those obtained from the spatially averaged ground-truth parameters ω¯, under three biaxial strain ratios between the myofiber and sheet directions (0.5, 1.0, and 2.0). The DNN-FEA model showed a closer match to the ground-truth response, while the FEA average response also agreed reasonably well.

To compare heterogeneous mechanical responses at different spatial locations in the validation case with material parameter Distribution 1, Fig. 8 shows equibiaxial stress-strain curves at four representative elements on the LV geometry. At elements 1, 2, and 4, DNN-FEA more closely matched the ground-truth curves in both the myofiber and sheet directions compared to FEA. At element 3, FEA slightly outperformed DNN-FEA in the sheet direction, while DNN-FEA achieved a better match in the fiber direction. Across all sampled locations, DNN-FEA closely approximated the ground-truth responses, with R2 > 0.942, whereas FEA showed weaker agreement, particularly in the sheet direction (R2 = −0.617 and 0.100 at element 1 and 2). The corresponding material parameters at these locations are listed in Table 5. These results confirm that DNN-FEA yields identified estimates closer to ground-truth values and more effectively captures spatially varying material properties.

Using the DNN architecture identified through grid search with material parameter Distribution 1, the DNN-FEA framework was applied to the numerical validation dataset with material parameter Distribution 2 to identify heterogeneous material properties. For comparison, the FEA method in PyTorch was also included. The identified spatial distributions of the four material parameters obtained by FEA, DNN-FEA, and the ground truth are shown in Fig. 9. As illustrated in Fig. 10, DNN-FEA achieved close agreement with the ground truth, whereas the FEA method produced noticeable errors. Table 6 further demonstrated that DNN-FEA resulted in substantially lower average and maximum errors for each parameter compared to FEA. These results demonstrate that the DNN architecture obtained through grid search may be applied to identify different spatial material property distributions.

To assess the performance of the proposed inverse DNN-FEA framework under noisy conditions, we performed an additional experiment using deformed geometry with Gaussian noise. Noise was introduced during the forward FEA simulation by applying non-uniform pressure loading, in which spatially varying noises were added to the uniform intraventricular pressure (P = 30 mmHg) with 1% standard deviation. The undeformed and noisy deformed geometries were then used as inputs for inverse identification using the same DNN-FEA configuration and hyperparameters as in case 1. Fig. 11 summarizes the identified material parameters under noisy inputs. Table 7 demonstrates that the DNN-FEA framework achieves relatively low εpavg across all 4 material parameters.

To quantify the sensitivity of the DNN-FEA workflow to pressure mismatch, we repeated the inverse identification process using uniform pressures of 28, 29, 31, and 32 mmHg as inputs, while keeping the same input geometries generated under a pressure of 30 mmHg. Quantitative comparisons are summarized in Table 8. These results demonstrate that small pressure mismatches (±1–2 mmHg) do not significantly degrade the performance of the proposed framework.

An additional sensitivity analysis was performed using bulk modulus values of k=5×104 and k=2×105 in forward FEA simulations. The resulting differences in displacement were quantified relative to the case with k=1×105. The results, summarized in the Table 9, indicate that the FEA outcomes are nearly independent of the specific choice of k.