Mathematical modelling of tumour growth and treatment effect of hyperthermia and radiotherapy.

我的研究旨在研究不同的连续统数学模型，这些模型描述了实体肿瘤对高温和放疗治疗的生长和反应。我扩展了由格林斯潘提出的肿瘤生长的空间解析数学模型。达成。数学。（1972），包括单独热疗和联合放疗的效果。该模型是一个移动边值问题，其中根据肿瘤边界上固定氧源的浓度分布发展出径向对称的肿瘤。肿瘤由增生边缘、缺氧环和坏死核心组成，这些层的边界由局部氧水平决定。扩展的格林斯潘模型帮助我解决了我的项目的两个主要目标。我首先将模型的预测与实验数据和由Bruningk等人开发的元胞自动机（CA）模型生成的数据进行了比较。接口,2018)。使用计算方法，我进行了与数据不一致的模型模拟，这表明该模型不能准确预测高温对肿瘤生长的影响。我相信改变简化的生物学假设可以改进建模方法。其次，我探讨了热疗和放疗联合治疗的好处。我们的模型预测，单模式和多模式治疗之间的治疗效果差异随着治疗时缺氧肿瘤体积的增加而线性增加。当我收到证明模型不足的数据时，我已经完成了这个分析。我仍然讨论了获得的结果，以突出一个有趣的问题：一个合适的数学模型能否证明肿瘤组成是治疗效果的关键？我的博士项目将建立在上述工作的基础上，通过发展更详细的肿瘤生长连续模型，解释高温和放疗引起的细胞死亡调节过程的差异。辐射诱导的细胞死亡不是一个瞬时的生物过程，因为大多数受辐射的细胞在尝试和失败有丝分裂后死亡。因此，我们的模型中要包括的一个关键特征是辐射和细胞死亡之间的时间延迟。细胞也可以衰老，即不增殖但有活力，这也应该纳入模型。前面提到的CA模型解释了这些细节，我有兴趣将它与我们的模型进行比较。受已发表的实验结果的启发，我先前假设热量会降低缺氧细胞对辐射的抵抗力。然而，有实验证据支持热疗和放疗的共生作用的其他假设。例如，由于pH值较低，热优先针对缺氧环境中的细胞，而不是缺氧，或者热辐射通过再氧化使肿瘤环境变得敏感。我认为，确保新模型包括对高温诱导的细胞死亡的更详细的数学描述，以解释不同的机制，将是有益的。另一个需要考虑的因素是细胞死亡后的命运：它们是否被驱逐到瘤外环境中？它们会留在肿瘤内吗？如果是的话，他们是否会加入坏死的核心？所做的假设可以显著地改变模型，这就是为什么尝试不同的选择很重要。新的模型将评估和验证使用体外数据的实验从三维无血管肿瘤球体。然后可以比较批准的模型并用于预测体内血管肿瘤的治疗反应。利用这些预测，我可以研究各种结合热疗和放疗的治疗方案，并确定是否存在最佳方法。该项目属于EPSRC数学生物学研究领域。

My research aims to investigate different continuum mathematical models that describe the growth and response of solid tumours to treatment with hyperthermia and radiotherapy.I extended a spatially-resolved mathematical model for tumour growth proposed by Greenspan (Stud. Appl. Math., 1972), incorporating the effect of hyperthermia treatment alone and combined with radiation. The model is a moving boundary-value problem where a radially-symmetric tumour is developed according to the concentration profile of a fixed oxygen source on the tumour's boundary. Tumour composition consists of a proliferating rim, a hypoxic annulus and a necrotic core, and these layers' boundaries are determined by local oxygen levels. The extended Greenspan model helped me address the two main goals of my project. I first compared the model's predictions to experimental data and data generated by a cellular automaton (CA) model developed by Bruningk et al. (J. Royal Soc. Interface, 2018). Using a computational approach, I ran model simulations that did not align with the data, suggesting that this model cannot accurately predict the effect of hyperthermia on tumour growth. I believe altering the simplifying biological assumptions made could improve the modelling approach. Secondly, I explored the benefits of a combination therapy of hyperthermia and radiation. Our model predicted that treatment efficacy differences between uni- and multi-modal therapies increase linearly with hypoxic tumoural volume at the time of treatment. I had finalised this analysis when I received the data that established the inadequacy of the model. I still discussed the results obtained to highlight an interesting question : can an appropriate mathematical model demonstrate that tumour composition is key for treatment efficacy ?My DPhil project will build upon the work described above through the development of more detailed continuum models of tumour growth that account for differences in the processes regulating cell death due to hyperthermia and radiotherapy. Radiation-induced cell death is not an instantaneous biological process as most irradiated cells die after attempting and failing mitosis. Therefore, a key feature to include in our models is a time delay between radiation and cell death. Cells can alternatively become senescent, i.e. non-proliferative but viable, which should also be incorporated in the models. The aforementioned CA model accounts for these details and I would be interested in comparing it with our models.Motivated by published experimental results, I previously assumed that heat reduces the resistance of hypoxic cells to radiation. However, there is experimental evidence supporting other hypotheses about the symbiotic action of hyperthermia and radiotherapy. For example, heat preferentially targets cells in hypoxic environments due to the lower pH, rather than a lack of oxygen, or heat radiosensitises the tumoural environment by reoxygenating it. I think it would be infomative to ensure the new models include a more detailed mathematical description of hyperthermia-induced cell death that accounts for the different mechanisms at play. Another factor to consider is the fate of cells upon their death : are they expelled into the extratumoural environment ? Do they remain within the tumour ? If so, do they join the necrotic core or not ? The assumptions made can significantly alter the model, which is why it is important to trial the different options. New models will be assessed and validated using in vitro data from experiments on 3D avascular tumour spheroids. Approved models can then be compared and used to predict the treatment response of in vivo vascular tumours. Using these predictions, I could investigate various therapeutic protocols combining hyperthermia and radiotherapy and establish whether there is an optimal method.This project falls within the EPSRC Mathematical Biology research area.