BB24: Continuum and multiscale modelling of vascular tumour growth

Background

Establishing mathematical models for the growth of vascular tumours is important to gain a better understanding of tumours and to study the effect of treatments like radio- or chemotherapy. Ideally, a model could help to identify the right treatment depending on the kind of tumour a patient has and the state it is in.

By the time tumours are detected they are usually vascularised. This is because without vasculature, tumours cannot obtain enough nutrients to maintain growth. Hence, angiogenesis, i.e. the formation of new blood vessels from existing ones, is considered a hallmark of all solid cancers [1].

Due to the importance of angiogenesis and the vasculature for tumour growth, targeting the vasculature itself has become a treatment strategy. However, this has not led to the success hoped for. One reason is that an efficient vasculature is important for the delivery of drugs, so if the vasculature is destroyed further treatment can become less efficient [2-4].

A mathematical model of vascular tumour growth should help to decide in which situations one should target the vasculature, and when this can be counterproductive. Furthermore, the combined effect of drugs or radiotherapy targeting the vasculature and the tumour should also be understood from the model.

Techniques and Challenges

Tumour growth involves a huge number of scales, ranging from the molecular basis of DNA mutations and epigenetic changes, cell signalling and changes in the cell cycle via growth processes and interactions of individual cells, to the behaviour of the macroscopic tumour and its interaction with the host body. Each level can be modelled in a variety of different ways, and linking the different scales together is a formidable challenge. For example, we use continuum models for macroscopic modelling of different cell types, cellular automaton models for individual cells and Ordinary Differential Equations (ODEs) for incorporating variables affecting the cell cycle. Putting them all together leads to multiscale models operating at different temporal and spatial scales (reviews of tumour growth models can be found in [5, 6]).

Future Plans

We are planning to extend existing multiscale models like [7] to, for example, include interstitial flow and study drug delivery and the effect of chemotherapy. We also plan to study the effect of radiotherapy on the tumour and the vasculature. Furthermore, we want to study continuum limits of discrete models. This can be useful as it is impossible computationally to model all individual cells and their internal mechanisms in a macroscopic tumour. Hence, we might use discrete models in interacting and proliferating regions and continuum models for the rest of the tissue, such that a good compromise between the accuracy and speed of computation is reached.

The models studied will be validated with data obtained from experimental collaborators at Roche and Moffitt.

References

[1] Hanahan D., Weinberg R.A.: The hallmarks of cancer, Cell 100.1: 57-70, 2000

[2] Carmeliet P., Jain R.K.: Angiogenesis in cancer and other diseases, Nature: 249-257, 2000

[3] Potente M., Holger G., Carmeliet P.: Basic and therapeutic aspects of angiogenesis, Cell 146.6: 873-887, 2011

[4] Jain, R.K.: Normalization of tumor vasculature: an emerging concept in antiangiogenic therapy, Science 307.5706: 58-62, 2005

[5] Byrne, H.M.: Dissecting cancer through mathematics: from the cell to the animal model, Nature Reviews Cancer 10.3: 221-230, 2010

[6] Byrne, H.M. et al.: Modelling aspects of cancer dynamics: a review, Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 364.1843: 1563-1578, 2006

[7] Perfahl, H. et al.: Multiscale modelling of vascular tumour growth in 3D: the roles of domain size and boundary conditions, PloS one 6.4: e14790, 2011