Mathematical methods for computing soft tissue deformations for surgical simulation and medical image registration

Research output: ThesisDoctoral Thesis

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Abstract

[Truncated abstarct] Medical imaging is an important tool in modern image guided surgery systems that help the surgeon localize with precision the target and therefore improve the clinical outcome of surgery. During a surgical intervention, a soft organ changes its shape and position, making the usage of pre-operative trajectory planning impossible. The trajectory must be updated so it corresponds to the new position and shape of the organ, information that can be partially obtained using lower quality intra-operative images. The intra-operative images must be registered to the high-resolution pre-operative images in order to match the features in the two image sets. The registration process must be very fast, as the computations are done intra-operative, as well as very accurate. Registration of soft tissue is much more difficult than rigid registration, as it requires knowledge about local deformations. The computing of the soft tissue deformations is done using a biomechanical model. This model is constructed based on the high resolution pre-operative images and is updated using displacements recovered from the lower quality intra-operative images. The deformed shape of the organ is then computed using the Finite Element Method. The motivation for this thesis is the need to find the best mathematical methods for performing these computations. The following aspects of the Finite Element Method are considered: formulation used, time integration method, elements used in the mesh, steady state solution methods and contacts. In order to ensure the accuracy of the results, the developed algorithms must handle nonlinear material models, nearly incompressible materials, large deformations and geometric nonlinearities. As an application example, the proposed algorithms are combined in order to simulate the brain shift during surgery. The Total Lagrangian formulation of the Finite Element Method, where all variables are referred to the un-deformed state of the system, was found to be the most appropriate for fast simulations as many quantities involved in the computation are constant and therefore can be pre-computed. It also offers benefits in implementing the hyper-elastic constitutive laws usually used for defining soft tissue behaviour. The explicit time integration method offered the best results in terms of speed, as it does not require any solution of large systems of equations and allows straightforward treatment of nonlinearities. However, the method is only conditionally stable, and therefore the time stepping must be controlled in order to ensure convergence. In order to meet the real time requirements low order under-integrated elements, such as the linear tetrahedron and the under-integrated linear hexahedron, are used in the mesh. These elements are very simple but they also pose some numerical problems. The linear tetrahedron becomes very stiff in case of nearly incompressible materials (volumetric locking) and the under-integrated linear hexahedron has zero energy modes (hourglass modes) because of the one-point spatial integration scheme used. Improved algorithms for handling volumetric locking and hourglass control are presented...
Original languageEnglish
QualificationDoctor of Philosophy
Publication statusUnpublished - 2010

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