Many important biological and physical problems involve either the interaction of a fluid with an elastic body, or the flow of a fluid through a deformable, porous medium. These problems are known as fluid-structure interactions and fluid-solid mixture problems, respectively. In most of these applications, the ultimate goal is the optimization or optimal control of the considered process, as well as related sensitivity analysis (with respect to relevant physical/biological parameters) investigations. Examples include the design of small-scale unmanned aircrafts and morphing aircraft wings, like robo-bees, minimizing turbulence in blood flow within a stenosed or stented artery, or optimizing the pressure of the blood flow and investigating the influence of pertinent biological parameters in the case of the lamina cribrosa in the eye, where these factors are believed to be related to the development of glaucoma. This research program addresses sensitivity analysis, control, and optimization-related problems for moving boundary fluid-elasticity interactions and fluid flows through biological tissues. The project will focus on the particular example of the coupling between biomechanics and hemodynamics in the lamina cribrosa in the eye, where performing sensitivity analysis with respect to important parameters (like blood flow and intraocular and retrolaminar pressures) will further the understanding of the cause and progression of glaucoma, and enable novel means for preventing or treating glaucoma. This research will also be applicable to other biological fluid-solid mixtures such as cartilages, bones, and engineered tissues.
Moving boundary fluid-elasticity interactions and porous media flows are nonlinear dynamical systems with Banach space parameters, and most questions related to their sensitivity analysis and control are open in the field. This project will (i) advance the theory of optimal control problems subject to these nonlinear systems by developing the theoretical framework for sensitivity analysis and optimality conditions; (ii) perform sensitivity analysis on the lamina cribrosa model to identify the important biological parameters, and use the obtained results to develop and address possible control strategies in the development of glaucoma. Well-posedness of optimal solutions, Frechet differentiability of cost functionals and state with respect to both distributed and boundary controls, and derivation and well-posedness analysis of adjoint systems will be addressed. These results will provide the starting point for all sensitivity calculations, parameter estimation, and derivative-based optimization algorithms.
