Open-source virtual bronchoscopy for image guided navigation

This thesis describes the development of an open-source system for virtual bronchoscopy used in combination with electromagnetic instrument tracking. The end application is virtual navigation of the lung for biopsy of early stage cancer nodules. The open-source platform 3D Slicer was used for creating freely available algorithms for virtual bronchscopy. Firstly, the development of an open-source semi-automatic algorithm for prediction of solitary pulmonary nodule malignancy is presented. This approach may help the physician decide whether to proceed with biopsy of the nodule. The user-selected nodule is segmented in order to extract radiological characteristics (i.e., size, location, edge smoothness, calcification presence, cavity wall thickness) which are combined with patient information to calculate likelihood of malignancy. The overall accuracy of the algorithm is shown to be high compared to independent experts' assessment of malignancy. The algorithm is also compared with two different predictors, and our approach is shown to provide the best overall prediction accuracy. The development of an airway segmentation algorithm which extracts the airway tree from surrounding structures on chest Computed Tomography (CT) images is then described. This represents the first fundamental step toward the creation of a virtual bronchoscopy system. Clinical and ex-vivo images are used to evaluate performance of the algorithm. Different CT scan parameters are investigated and parameters for successful airway segmentation are optimized. Slice thickness is the most affecting parameter, while variation of reconstruction kernel and radiation dose is shown to be less critical. Airway segmentation is used to create a 3D rendered model of the airway tree for virtual navigation. Finally, the first open-source virtual bronchoscopy system was combined with electromagnetic tracking of the bronchoscope for the development of a GPS-like system for navigating within the lungs. Tools for pre-procedural planning and for helping with navigation are provided. Registration between the lungs of the patient and the virtually reconstructed airway tree is achieved using a landmark-based approach. In an attempt to reduce difficulties with registration errors, we also implemented a landmark-free registration method based on a balanced airway survey. In-vitro and in-vivo testing showed good accuracy for this registration approach. The centreline of the 3D airway model is extracted and used to compensate for possible registration errors. Tools are provided to select a target for biopsy on the patient CT image, and pathways from the trachea towards the selected targets are automatically created. The pathways guide the physician during navigation, while distance to target information is updated in real-time and presented to the user. During navigation, video from the bronchoscope is streamed and presented to the physician next to the 3D rendered image. The electromagnetic tracking is implemented with 5 DOF sensing that does not provide roll rotation information. An intensity-based image registration approach is implemented to rotate the virtual image according to the bronchoscope's rotations. The virtual bronchoscopy system is shown to be easy to use and accurate in replicating the clinical setting, as demonstrated in the pre-clinical environment of a breathing lung method. Animal studies were performed to evaluate the overall system performance.

Uniformly convergent finite element methods for singularly perturbed parabolic partial differential equations

This thesis is concerned with uniformly convergent finite element methods for numerically solving singularly perturbed parabolic partial differential equations in one space variable. First, we use Petrov-Galerkin finite element methods to generate three schemes for such problems, each of these schemes uses exponentially fitted elements in space. Two of them are lumped and the other is non-lumped. On meshes which are either arbitrary or slightly restricted, we derive global energy norm and L2 norm error bounds, uniformly in the diffusion parameter. Under some reasonable global assumptions together with realistic local assumptions on the solution and its derivatives, we prove that these exponentially fitted schemes are locally uniformly convergent, with order one, in a discrete L∞norm both outside and inside the boundary layer. We next analyse a streamline diffusion scheme on a Shishkin mesh for a model singularly perturbed parabolic partial differential equation. The method with piecewise linear space-time elements is shown, under reasonable assumptions on the solution, to be convergent, independently of the diffusion parameter, with a pointwise accuracy of almost order 5/4 outside layers and almost order 3/4 inside the boundary layer. Numerical results for the above schemes are presented. Finally, we examine a cell vertex finite volume method which is applied to a model time-dependent convection-diffusion problem. Local errors away from all layers are obtained in the l2 seminorm by using techniques from finite element analysis.

Uniformly convergent finite element and finite difference methods for singularly perturbed ordinary differential equations

This thesis is concerned with uniformly convergent finite element and finite difference methods for numerically solving singularly perturbed two-point boundary value problems. We examine the following four problems: (i) high order problem of reaction-diffusion type; (ii) high order problem of convection-diffusion type; (iii) second order interior turning point problem; (iv) semilinear reaction-diffusion problem. Firstly, we consider high order problems of reaction-diffusion type and convection-diffusion type. Under suitable hypotheses, the coercivity of the associated bilinear forms is proved and representation results for the solutions of such problems are given. It is shown that, on an equidistant mesh, polynomial schemes cannot achieve a high order of convergence which is uniform in the perturbation parameter. Piecewise polynomial Galerkin finite element methods are then constructed on a Shishkin mesh. High order convergence results, which are uniform in the perturbation parameter, are obtained in various norms. Secondly, we investigate linear second order problems with interior turning points. Piecewise linear Galerkin finite element methods are generated on various piecewise equidistant meshes designed for such problems. These methods are shown to be convergent, uniformly in the singular perturbation parameter, in a weighted energy norm and the usual L2 norm. Finally, we deal with a semilinear reaction-diffusion problem. Asymptotic properties of solutions to this problem are discussed and analysed. Two simple finite difference schemes on Shishkin meshes are applied to the problem. They are proved to be uniformly convergent of second order and fourth order respectively. Existence and uniqueness of a solution to both schemes are investigated. Numerical results for the above methods are presented.

Spinoza on method and freedom

The thesis as a whole argues that Spinoza’s Ethics in both method and content is aimed at the normal, partly rational person. Chapter 1 is on Spinoza’s writing style, finding that rather than being arid and technical, it aims to convince the reader by means of various rhetorical techniques, so does not assume an already rational reader. The following chapters of Part 1 examine whether the Ethics’ use of the synthetic geometric method exposes it to Descartes’ critique of that method in the “Second Replies” to his Meditations, that it is not suitable for pedagogy. This involves a consideration of the role of the TIE, finding in that early text not the analytic wing of a two-part analytic-synthetic method, but rather a defence and necessitation of a stand-alone synthetic method. Part 2 of the thesis develops this study of Spinoza’s writing for the common man to consider whether he is writing about the common man. This is done by examining one of the seemingly most abstract propositions in the Ethics, 4P72, which claims that a free man will not deceive even to save his own life. The study examines who exactly is this “free man” and what is his role in the Ethics. The study looks at the examples of free men in the TTP and at the concept of the model in the Ethics, and finds that rather than the free man being an impossible ideal which we can aim at but never achieve, everyone is free to some extent, and that even normal people are at times “the free man”.