Spatial Confinement Causes Lifetime Enhancement and Expansion of Vortex Rings with Positive Filament Tension

We study the impact of spatial confinement on the dynamics of three-dimensional excitation vortices with circular filaments. In a chemically active medium we observe a decreased contraction of such scroll rings and even expanding ones, despite of their positive filament tension. We propose a kinematical model which takes into account the interaction of the scroll ring with a confining Neumann boundary. The model reproduces all experimentally observed regimes of ring evolution, and correctly predicts the results obtained by numerical simulations of the underlying reaction-diffusion equations

Three-Dimensional Autonomous Pacemaker in the Photosensitive Belousov-Zhabotinsky medium

In experiments with the photosensitive Belousov-Zhabotinsky reaction (PBZR) we found a stable three-dimensional organizing center that periodically emits trigger waves of chemical concentration. The experiments are performed in a parameter regime with negative line tension using an open gel reactor to maintain stationary non-equilibrium conditions. The observed periodic wave source is formed by a scroll ring stabilized due to its interaction with a no-flux boundary. Sufficiently far from the boundary, the scroll ring expands and undergoes the negative line tension instability before it finally develops into scroll wave turbulence. Our experimental results are reproduced by numerical integration of the modified Oregonator model for the PBZR. Stationary and breathing self-organized pacemakers have been found in these numerical simulations. In the latter case, both the radius of the scroll ring and the distance of its filament plane to the no-flux boundary after some transient undergo undamped stable limit cycle oscillations. So far, in contrary to their stationary counterpart, the numerically predicted breathing autonomous pacemaker has not been observed in the chemical experiment

Global Flows with Invariant Measures for the Inviscid Modified SQG Equations

We consider the family known as modified or generalized surface quasi-geostrophic equations (mSQG) consisting of the classical inviscid surface quasi-geostrophic (SQG) equation together with a family of regularized active scalars given by introducing a smoothing operator of nonzero but possibly arbitrarily small degree. This family naturally interpolates between the 2D Euler equation and the SQG equation. For this family of equations we construct an invariant measure on a rough $L^2$-based Sobolev space and establish the existence of solutions of arbitrarily large lifespan for initial data in a set of full measure in the rough Sobolev space.Comment: 18 page

Appearance Modelling and Reconstruction for Navigation in Minimally Invasive Surgery

Minimally invasive surgery is playing an increasingly important role for patient care. Whilst its direct patient benefit in terms of reduced trauma, improved recovery and shortened hospitalisation has been well established, there is a sustained need for improved training of the existing procedures and the development of new smart instruments to tackle the issue of visualisation, ergonomic control, haptic and tactile feedback. For endoscopic intervention, the small field of view in the presence of a complex anatomy can easily introduce disorientation to the operator as the tortuous access pathway is not always easy to predict and control with standard endoscopes. Effective training through simulation devices, based on either virtual reality or mixed-reality simulators, can help to improve the spatial awareness, consistency and safety of these procedures. This thesis examines the use of endoscopic videos for both simulation and navigation purposes. More specifically, it addresses the challenging problem of how to build high-fidelity subject-specific simulation environments for improved training and skills assessment. Issues related to mesh parameterisation and texture blending are investigated. With the maturity of computer vision in terms of both 3D shape reconstruction and localisation and mapping, vision-based techniques have enjoyed significant interest in recent years for surgical navigation. The thesis also tackles the problem of how to use vision-based techniques for providing a detailed 3D map and dynamically expanded field of view to improve spatial awareness and avoid operator disorientation. The key advantage of this approach is that it does not require additional hardware, and thus introduces minimal interference to the existing surgical workflow. The derived 3D map can be effectively integrated with pre-operative data, allowing both global and local 3D navigation by taking into account tissue structural and appearance changes. Both simulation and laboratory-based experiments are conducted throughout this research to assess the practical value of the method proposed