Numerical methods for radiation transport and radiotherapy treatment planning

该项目的动机来自于癌症放疗治疗背景下的临床挑战。当辐射穿过活组织时，它会积聚能量，导致DNA损伤，阻止肿瘤细胞复制。由于辐射也会破坏非肿瘤细胞的DNA，所以尽可能不伤害肿瘤周围的健康组织是很重要的。如果肿瘤位于重要器官附近，这一点就更加重要了。为了确保计划的放射治疗过程向肿瘤提供足够高的辐射剂量，同时保留重要的周围器官，对可能的治疗配置进行优化程序。该项目的主要目标包括研究目前在临床环境中使用的辐射传输模型背后的物理假设，并在准确性、鲁棒性和计算效率方面开发改进的放射治疗计划数值方法。起点将是一个简化的一维辐射输运模型，该模型可以等效地表述为单个粒子水平上的随机模型，也可以表示为描述许多粒子的平均宏观行为的偏微分方程（PDE）模型。我们的目标是将这个模型扩展到三个空间维度，并在相关的地方包括进一步的物理行为。通过利用可应用于随机或PDE模型的不同数值方法的范围，我们的目标是开发新的更有效的数值方法来优化放疗治疗计划问题。这可能包括使用串行化或并行化来加速特定计算机体系结构上的数值模拟，例如GPU（图形处理单元）。我们还旨在研究使用强稳定性保持（SSP）数值方法，它在时间步长之间的空间中保持解的单调性。这些方法在模拟守恒定律或一般物理系统时很有用，因为它们产生的数值解仍然遵循物理定律。由于本项目的重点是研究放射治疗计划的数值和数学方法，因此不需要收集数据。该方法包括编写代码和在编程语言Python中进行数值模拟，以及与数值模拟相关的收敛结果的理论数学分析。提高用于治疗计划的数值方法的速度和准确性将对如何进行癌症治疗产生积极影响，更准确地计划治疗有助于确保将肿瘤周围健康组织的辐射暴露降至最低。更快的治疗计划软件将使临床医生能够根据患者生理变化更频繁地重新校准治疗计划，而不是目前在实践中所做的，这有助于最大限度地减少治疗中的不确定性。

The motivation for this project comes from clinical challenges in the context of radiotherapy for cancer treatment. When radiation moves through living tissue it deposits energy, which causes DNA damage, stopping the tumour cells from replicating. Since radiation will also damage the DNA in non-tumour cells, it is important to as far as possible spare the healthy tissue surrounding the tumour. This becomes even more important if the tumour is located near a vital organ. To ensure that a planned course of radiotherapy treatment delivers a high enough radiation dose to the tumour, whilst sparing important surrounding organs, an optimisation procedure is carried out over possible treatment configurations. The main aims of this project include investigating the physical assumptions behind models for radiation transport that are currently used in a clinical setting, and developing improved numerical methods for radiotherapy treatment planning in terms of accuracy, robustness, and computational efficiency. The starting point will be a simplified one-dimensional model for radiation transport, which can be equivalently formulated either as a stochastic model on the individual particle level, or as a partial differential equation (PDE) model, which describes the average macroscopic behaviour of many particles. We aim to extend this model to three spatial dimensions, and to include further physical behaviour where relevant. By leveraging the range of different numerical methods which can be applied to either the stochastic or PDE model, we aim to develop new more efficient numerical approaches to the optimisation problem of radiotherapy treatment planning. This may include using serialisation or parallelisation to accelerate our numerical simulations on specific computer architechtures, such as on a GPU (graphics processing unit). We also aim to investigate the use of strong stability preserving (SSP) numerical methods, which preserve the monotonicity of a solution in space between timesteps. These methods are useful when simulating conservation laws or physical systems in general, as they yield numerical solutions that still obey the laws of physics.As the focus of this project is on investigating numerical and mathematical methods for radiotherapy treatment planning, no data collection will be necessary. The methodology includes writing code and conducting numerical simulations in the programming language Python, and theoretical mathematical analysis of convergence results related to the numerical simulations. Improving the speed and accuracy of the numerical methods used for treatment planning will have a positive impact on how cancer treatment is carried out, with a more accurately planned treatment helping to guarantee a minimised radiation exposure of healthy tissue surrounding the tumour. Faster treatment planning software would enable clinicians to recalibrate the treatment plan to changes in patient physiology more often than what is currenlty done in practice, helping to minimise uncertainties in the delivered treatment.