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Algorithms for Nonlinear Finite Element-based Modeling of Soft-tissue Deformation and Cutting

Advances in robotics and information technology are leading to the development of virtual reality-based surgical simulators as an alternative to the conventional means of medical training. Modeling and simulation of medical procedures also have numerous applications in pre-operative and intra-operative surgical planning as well as robotic (semi)-autonomous execution of surgical tasks. Surgical simulation requires modeling of human soft-tissue organs. Soft-tissues exhibit geometrical and material nonlinearities that should be taken into account for realistic modeling of the deformations and interaction forces between the surgical tool and tissues during medical procedures. However, most existing work in the literature, particularly for modeling of cutting, use linear deformation models. In this thesis, modeling of two common surgical tasks, i.e. palpation and cutting, using nonlinear modeling techniques has been studied. The complicated mechanical behavior of soft-tissue deformation is modeled by considering both geometrical and material nonlinearities. Large deformations are modeled by employing a nonlinear strain measure, the Green-Lagrange strain tensor, and a nonlinear stress-strain curve is employed by using an Ogden-based hyperelastic constitutive equation. The incompressible property of soft-tissue material during the deformation is enforced by modifying the strain energy function to include a term iv that penalizes changes in the object’s area/volume. The problem of simulating the tool-tissue interactions using nonlinear dynamic analysis is formulated within a total Lagrangian framework. The finite element method is utilized to discretize the deformable object model in space and an explicit time integration is employed to solve for the resulting deformations. In this thesis, the nonlinear finite element analysis with the Ogden-based constitutive equation has also been applied to the modeling of soft-tissue cutting. Element separation and node snapping are used to create a cut in the mesh that is close to the tool trajectory. The external force applied on the object along the tool direction is used as a physical cutting criterion. The possibility of producing degenerated elements by node snapping that can cause numerical instability in the simulation is eliminated by remeshing the local elements when badly shaped elements are generated. The remeshing process involves retriangulation of the local elements using the Delaunay function and/or moving a node depending on what is needed in order to generate elements with the required quality. Extensive simulations have been carried out in order to evaluate and demonstrate the effectiveness of the proposed modeling techniques and the results are reported in the thesis. A two-dimensional object with a concentrated external force has been considered in the simulations.