On the monolithic and staggered solution of cell contractility and focal adhesion growth

The mechanical response of cells to stimuli tightly couples biochemical and biomechanical processes, which describe fundamental properties such as cell growth and reorientation. Cells interact continuously with their external surroundings, the extracellular matrix (ECM), by establishing a bond between cell and ECM through the formation of focal adhesions. Focal adhesions are made up of integrins, which are mechanosensitive proteins and responsible for the communication between the cell and the ECM. The governing biochemomechanical processes can be modeled by means of a continuum approach considering mechanical and thermodynamic equilibrium to describe cell contractility and focal adhesion growth. The immanent multiphysical character of cell mechanics involves important aspects such as the coupling of fields of different scales and corresponding interface conditions that are sensitive to the solution of the governing numerical problem. These aspects become even more relevant when considering a feedback loop among the multiphysical solutions fields. In this contribution, we consider solution properties and sensitivity aspects of a nonlinear mechanical continuum model for the prognosis of stress fiber growth and reorientation incorporating a mechanosensitive feedback loop. We provide the governing equations of a Hill model-based stress fiber growth, which is coupled to a thermodynamical approach modeling the focal adhesions. Furthermore, a variational formulation including the algebraic equations is derived for staggered and monolithic solution approaches and the reaction-diffusion equation that models the feedback mechanism. We test both schemes with regard to reliability, accuracy, and numerical efficiency for different model parameters and loading scenarios. We present algorithmic aspects of the considered solution schemes and reveal their robustness with regard to model refinement in space and time and finally provide an assessment of their overall solution performance for multiphysics problems in the context of cell mechanics.