| Sitting at the interface of biology and engineering, biomechanics and mechanobiology are important areas where computational modeling is applied to study how mechanics and biological processes influence and regulate each other. In the continuum regime of biomechanics and mechanobiology, inverse problems often arise in applications such as biomechanical imaging (BMI) and cell traction force microscopy (TFM). In BMI, typically the tissue properties, for instance, the shear modulus or the nonlinear elastic parameter, are reconstructed, given experimentally measured full interior displacement field. BMI is extensively studied in tissue mechanical property quantification. On the other hand, in cell TFM, the traction on a cell's surface is recovered, given experimentally measured displacement field in the extracellular matrix, where dense tracking beads are embedded. Cell traction is crucial in understanding stem cell differentiation, cancer cell metastasis, embryonic morphogenesis, etc. For both BMI and TFM, the displacement field is obtained from various imaging modalities, such as magnetic resonance (MR), ultrasound, optical coherence tomography (OCT), confocal laser scanning microscopy (CLSM) and stimulated emission depletion (STED) microscopy, where the resolution ranges from millimeters to nanometers.;Both problems described above are governed by the same system of elliptic PDEs that enforce mechanical equilibrium for an elastic material. In this thesis, we apply an optimization framework and treat the above two types of inverse problems as PDE-constrained optimization problems and solve them with one general algorithmic framework. In the language of PDE-constrained optimization problems BMI leads to a parameter identification problem, and TFM leads to a source identification problem. For the parameter identification problem, we demonstrate the iterative reconstruction of shear modulus with displacement data from OCT. Moreover, we apply and demonstrate the utility of adaptive mesh refinement and domain decomposition in efficiently solving this problem. We validate these methods with tissue-mimicking phantoms and ex-vivo and in-vivo biological tissues in 2D and 3D. For the source identification problem, we pose and implement a novel formulation that accounts for finite deformation and material nonlinearity in 3D. The algorithm is applied to in-silico problems and the error incurred in making the linear elastic assumption is quantified. It is also applied to determine the tractions exerted by live cells on their surroundings. All computations were performed using our in-house FORTRAN code, nonlinear adjoint coefficients estimator (NLACE) which is parallelized on shared-memory machines. |
