An integrated inverse model-experimental approach to determine soft tissue three-dimensional constitutive parameters: application to post-infarcted myocardium

Abstract

Knowledge of the complete three-dimensional (3D) mechanical behavior of soft tissues is essential in understanding their pathophysiology and in developing novel therapies. Despite significant progress made in experimentation and modeling, a complete approach for the full characterization of soft tissue 3D behavior remains elusive. A major challenge is the complex architecture of soft tissues, such as myocardium, which endows them with strongly anisotropic and heterogeneous mechanical properties. Available experimental approaches for quantifying the 3D mechanical behavior of myocardium are limited to preselected planar biaxial and 3D cuboidal shear tests. These approaches fall short in pursuing a model-driven approach that operates over the full kinematic space. To address these limitations, we took the following approach. First, based on a kinematical analysis and using a given strain energy density function (SEDF), we obtained an optimal set of displacement paths based on the full 3D deformation gradient tensor. We then applied this optimal set to obtain novel experimental data from a 1-cm cube of post-infarcted left ventricular myocardium. Next, we developed an inverse finite element (FE) simulation of the experimental configuration embedded in a parameter optimization scheme for estimation of the SEDF parameters. Notable features of this approach include: (i) enhanced determinability and predictive capability of the estimated parameters following an optimal design of experiments, (ii) accurate simulation of the experimental setup and transmural variation of local fiber directions in the FE environment, and (iii) application of all displacement paths to a single specimen to minimize testing time so that tissue viability could be maintained. Our results indicated that, in contrast to the common approach of conducting preselected tests and choosing an SEDF a posteriori, the optimal design of experiments, integrated with a chosen SEDF and full 3D kinematics, leads to a more robust characterization of the mechanical behavior of myocardium and higher predictive capabilities of the SEDF. The methodology proposed and demonstrated herein will ultimately provide a means to reliably predict tissue-level behaviors, thus facilitating organ-level simulations for efficient diagnosis and evaluation of potential treatments. While applied to myocardium, such developments are also applicable to characterization of other types of soft tissues.

Appendices

Appendix A: Validation of the analysis pipeline

In this appendix, we provide the results for the validation of the simulation-experimental pipeline using isotropic and anisotropic synthetic specimens as described in Sect. 2.6.2.

1.1 Isotropic gel

Single tension and simple shear tests with flat plate attachment on the same specimen (see Fig. 5b) led to matched properties. The FE model (see Fig. 5c) equipped with (single-parameter) neo-Hookean material description provided reliable fit to the stress–stretch data of single tension test (see Fig. 12a) with $r^{2}=0.99$. The estimated value of unknown shear modulus was 35.77 kPa. This value was used in FE simulations to compare against the simple shear test date, and an overall good agreement was found with $r^{2}=0.96$ (Fig. 12b). The agreement was particularly good for smaller deformations (up to 0.04 shear).

1.2 Transversely isotropic rubber

The FE model (see Fig. 5f) equipped with transversely isotropic Fung material model (Eq. 26) was used to fit the stress–stretch data from the first set of tests comprising of two single tension tests and two simple shear tests in the transverse and longitudinal directions. The overall fit was very good (Fig. 13a, b) with $r^{2}=0.97$. The FE model with the estimated values (Table 5) were used to compare against the data from the second set of tests comprising of equibiaxial test in longitudinal and transverse directions, the perpendicular simple shear, and the single compression in the transverse direction (Fig. 13c, d, e). The overall fit was satisfactory to validate the robustness of the device for mechanical testing of anisotropic materials. This result also underscores how optimal design of experiments can be used to derive model parameters with good predictive capabilities.

Table 5 Estimated values of the Fung parameters for synthetic fiber-reinforced rubber

Fig. 14

3D surfaces of the objective function $\varPhi$ [defined in (12)] for the energy function (28). Results for a Set 1, b Set 2, and c Set 3, as defined in “Appendix B”

Appendix B

In this appendix, we discuss a simple example of optimal design of mechanical testing to characterize the mechanical behavior of a transversely isotropic solid based on a three-parameter constitutive model. Our attention is restricted to D-optimality criterion and we show how this criterion can guide the experiments to optimally estimate the material constants and avoid identifiability issues. We assume that the solid material with preferred direction along $b_3$ is characterized by following energy function

$$\begin{aligned} W({\mathbf{E}})=\frac{1}{2}c_0 \left\{ {\hbox {exp}\left[ {c_1 \left( E_{11}^2 +E_{22}^2 \right) +c_3 E_{33}^2 } \right] -1} \right\} , \end{aligned}$$

(28)

where $c_1$ and $c_3$ characterize the in-plane and out-of-plane stiffness of the material, respectively. In a similar manner as discussed in Sect. 2.4, we use the objective function $\varPhi$ to estimate the parameters. To be able to visualize the objective function in three-dimensional space, we assign a value to $c_0$ and use mechanical tests to estimate $c_1$ and $c_3$. We compare the values of the D-optimality criterion $\varDelta _\mathrm{D}$, defined in (20), for three sets each of which is composed of two deformation protocols as follows:

• Set 1: Simple shear XY and simple shear ZX (see Fig. 2)

• Set 2: Single tension X and simple shear ZX

• Set 3: Single tension X and single tension Z

We calculated the objective function for numerical values $c_0 =3 \hbox { kPa}$, $c_1 =2$, and $c_3 =5$ (Fig. 14). A zero value for $\varDelta _\mathrm{D}$ indicates that $c_1$ and $c_3$ are not uniquely identifiable (Fig. 14a) through Set 1, and the increasing value of $\varDelta _\mathrm{D}$ for Sets 2 and 3 reflects the modifying stability of the optimization process (Fig. 14b, c).