In this paper, we carry out a travelling-wave analysis of a model of tumour invasion with degenerate, cross-dependent diffusion. We consider two types of invasive fronts of tumour tissue into extracellular matrix (ECM), which represents healthy tissue. These types differ according to whether the density of ECM far ahead of the wave front is maximal or not. In the former case, we use a shooting argument to prove that there exists a unique travelling wave solution for any positive propagation speed. In the latter case, we further develop this argument to prove that there exists a unique travelling wave solution for any propagation speed greater than or equal to a strictly positive minimal wave speed. Using a combination of analytical and numerical results, we conjecture that the minimal wave speed depends monotonically on the degradation rate of ECM by tumour cells and the ECM density far ahead of the front.
The increasing cancer burden on society has motivated the development of new therapies to complement existing ones. Hyperthermia has garnered significant interest, showing noteworthy potential to act synergistically with radiotherapy. In this paper, we extend a spatially-resolved mathematical model for tumour growth proposed by Greenspan (Stud. Appl. Math., 1972), first incorporating treatments of hyperthermia alone and then combined with radiation. We use this first model to compare the Jung (Jung, Radiat. Res., 1986) and AlphaR (Brüningk et al. Int. J. Hyperth., 2018) survival functions, which we use to represent hyperthermia treatment effects. We also compare this model’s predictions to experimental data and data generated by a cellular automaton model developed by Brüningk et al. (J. Royal Soc. Interface, 2018). Model simulations do not align with the data, underlining modifications required to enhance the accuracy of our model’s predictions. We lastly investigate the benefits of a combination therapy, specifically showing that treatment efficacy differences between a uni-modal and a multi-modal therapy increase with tumour hypoxia. Even though our model requires improvements, our work suggests that tumour composition may be important for clinical applications in terms of predicting tumour response to combined hyperthermia and radiotherapy.