Adjoint multi-start-based estimation of cardiac hyperelastic material parameters using shear data

Abstract

Cardiac muscle tissue during relaxation is commonly modeled as a hyperelastic material with strongly nonlinear and anisotropic stress response. Adapting the behavior of such a model to experimental or patient data gives rise to a parameter estimation problem which involves a significant number of parameters. Gradient-based optimization algorithms provide a way to solve such nonlinear parameter estimation problems with relatively few iterations, but require the gradient of the objective functional with respect to the model parameters. This gradient has traditionally been obtained using finite differences, the calculation of which scales linearly with the number of model parameters, and introduces a differencing error. By using an automatically derived adjoint equation, we are able to calculate this gradient more efficiently, and with minimal implementation effort. We test this adjoint framework on a least squares fitting problem involving data from simple shear tests on cardiac tissue samples. A second challenge which arises in gradient-based optimization is the dependency of the algorithm on a suitable initial guess. We show how a multi-start procedure can alleviate this dependency. Finally, we provide estimates for the material parameters of the Holzapfel and Ogden strain energy law using finite element models together with experimental shear data.

Keywords: Adjoint equation; Cardiac mechanics; Hyperelasticity; Multi-start optimization; Parameter estimation.

Fig. 1

Stress–strain relations, numbered 1 through 6, obtained from simple shearing experiments performed on $3\text{mm}\times 3\text{mm}\times 3\text{mm}$ cubes of myocardium extracted from 6 porcine hearts. The modes are ordered from highest to lowest stiffness in each experiment. The data originate from the study Dokos et al. (2002), but were not published in the subsequent article. In Experiment 4 the data for one of the NS–NF curves were copied into the other before we received it, so the two curves lie here on top of one another

Fig. 2

Finite element representation of cubes of cardiac tissue undergoing simple shear in the NS mode. The bottom of the cube is fixed, and the top displacement is given. Left homogeneous deformation with a constant shear angle. Right finite element solution on a $6\times 6\times 6$ mesh. The plot shows the value of the NS-component of the right Cauchy–Green strain tensor $\mathbf{C}$

Fig. 3

Comparison of optimized model stress–strain curves with experimental data. The dots are interpolated experimental data at Gauss points, the solid lines show the output of the finite element models with $N=8$ elements per edge of the cube

Fig. 4

Gradient efficiency: ratio of gradient evaluation runtime over single Newton iteration runtime for increasing linear system sizes