Cancer is one of the leading causes of death in the world. In the UK alone, around 1000 cases are diagnosed every day and, according to the NHS, almost half of those treated for cancer have radiotherapy as part of their treatment plan. An integral part of any effective treatment planning system is rapid, accurate, and robust prediction of the distribution of radiation dose delivered to the tumour, the surrounding tissue, and vital organs, for a given beam configuration. These predictions are derived by simulating radiation transport through the patient's body, as represented by CT image data, governed by an appropriate model of the physical interaction between the radiation and the tissue.
The current gold standard is the Monte Carlo (MC) approach, which considers many particle histories travelling through the tissue. However, the `error' in the MC simulations decreases very slowly: adding an extra decimal place to the accuracy takes 100 times as long, which means that the generation of sufficiently accurate predictions is simply too slow. This issue is commonly addressed by exploiting simpler, less realistic models, which can be very quickly simulated; however, validation of such reduced-physics models still typically involves benchmarking with MC. Clearly, this is not a very satisfactory situation; indeed, there is an urgent need to develop the next generation of computational tools, which are not only more efficient than MC, but provide reliable information regarding the size of the error in the numerical solution, thereby providing rigorous solution verification for these safety-critical applications.
An alternative to stochastic MC algorithms, is the linear Boltzmann transport equation (LBTE); here, radiation is treated as a continuous quantity with mathematical expressions describing its generation and its `flow' through and absorption by the patient. This absorption is important: sufficient absorption occurring in the correct location leads to the desired tumour control, however, too much in other places can lead to side-effects/complications. The LBTE can be directly discretized to yield a convenient computer representation which, in contrast to MC, leads to an alternative approximation which may directly exploit the smoothness of the underlying differential operator. In this way, the numerical error can be controlled in a much more efficient manner, relative to the computational effort required by MC algorithms, which does not exploit the smoothness of the underlying analytical solution. We wish to address the key challenges in this approach: to deal efficiently with the high-dimensional nature of the LBTE and the geometrical complexity of the human body. Methods of this type have been employed in the medical physics community, showing great promise, but they are still relatively new, have yet to take advantage of the full range of techniques which can be used to improve accuracy and efficiency, and are embedded in proprietary software.
We propose to develop precise, efficient, and robust algorithms for solving the LBTE, using methods ideally suited to problems of this type and which are amenable to adaptive computation. We will therefore develop an algorithm optimally configured to exploit smooth regions and yet able to adapt to handle regions when tissue properties change rapidly, targeted at efficient approximation of prescribed quantities of (clinical) interest, e.g., dose-to-organ. The underlying numerical method we will employ will naturally provide a framework within which errors compared to an exact representation of the specified physics can be quantified, so that rigorous solution verification can be achieved. Furthermore, we will work with Altnagelvin Hospital, to ensure that our software is tested fairly and comprehensively against existing simulation software in collaboration with end-users, and that its development is guided by the clinical protocols it would ultimately have to follow.
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