Estimating stresses driving tissue flows using a Stokes inverse problem

We propose an inverse problem to derive the stress distributions that drive the tissue flows during gastrulation in the epiblast of the chick embryo, from measurements of the tissue velocity fields at different stages of development. We assume that the embryonic tissue can be described as a highly viscous fluid, characterized by the Stokes equations. Using the theory of the optimal control, the stress distributions are determined by minimizing an objective functional, which is constructed such that it can match the numerical velocity of the flow with the experimental velocity data by choosing the stress as the control variable. The Lagrange multiplier method is utilized to derive the optimality system. The finite element method is used to approximate these partial differential equations numerically and we use the conjugate gradient algorithm to solve the optimal control problem.