  Cardiac muscle tissue during relaxation is commonly modelled 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.
