Mathematical Modelling of Breast Tumour Development, Treatment and Recurrence

Ph.D. Thesis 2006, University of Dundee

Abstract:

Breast cancer is very common in women throughout the world, being the second leading cause of cancer deaths. About 12% of all women will be affected and about 30%-40% of patients will die, mostly from distant metastatic neoplasms. In recent years, the treatment of breast cancer has shifted from very radical to a more conservative treatment. The increased awareness of this disease, public education and protective screening have enabled early detection of new breast tumours, which enables a more localised treatment and reduces the mortality rate by up to 30%

Total local tumour control can not be assured. About 30% of patients treated with breast conserving surgery alone will develop a local recurrence. Adjuvant conventional external beam radiotherapy reduces the risk of local recurrence to about 10%. However, more than 90% of these local recurrences occur in the tumour bed, which promotes the idea that treatment of the whole breast might not be necessary, and a more localised treatment might be sufficient.

Targeted intra-operative radiotherapy (Targit) is a new concept of partial breast irradiation where single fraction radiotherapy is delivered directly to the tumour bed while the patient is still anaesthetised after surgery. Apart from logistic advantages, this strategy minimises the risk of missing the tumour bed and avoids delay between surgery and radiotherapy. It is presently being compared with the standard fractionated external beam radiotherapy in randomised trials.

The aim of this thesis is to combine different research disciplines to investigate the sources of breast cancer, its development, growth and invasion, and the effectiveness of local treatment with surgery and adjuvant radiotherapy in preventing local recurrence. We will develop different mathematical models of tumour and breast tissue dynamics and combine them with spatial models of tumour growth and invasion. We implement a numerical scheme which solves these equations and develop a visualisation interface to present the results in one, two and three spatial dimensions. We will then apply models of radiotherapy to simulate the response of healthy tissue and tumour cells to irradiation to identify why conventional standard treatment fails to provide local tumour control in 10% of cases. Furthermore, we compare the simulation of standard treatment with simulations of treatment with targeted intra-operative radiotherapy to see if the single high dose of irradiation directed at the tumour bed can prevent local recurrence. The analysis of local recurrence after the different treatments may provide biologically and clinically important information.

Our work has shown that conventional treatment of early breast cancer is likely to eradicate all tumour cells, and hence other sources for local recurrence must exist. Our simulations show that cells in the tumour bed which harbour initial tumour mutations, i.e. loss of heterozygosity on tumour suppressor genes, will give rise to a new tumour within a clinically observable time, and additional genetic instability can be a driving force in the development. The developed model has also been able to predict that targeted intra-operative radiotherapy may be as successful in eradicating stray tumour cells as the conventional treatment, but additionally also eradicate mutated cells in the tumour bed which may ensure a longer relapse-free survival for patients.

The mathematical models presented here are an initial attempt to model a biologically complex phenomenon that has until now received little attention in the literature. Nevertheless, it provides a `proof of principle' that it is possible to produce clinically testable hypotheses on the effects of different approaches of radiotherapy for breast cancer.

Another important aim of this interdisciplinary effort is to develop a solution and visualisation platform for mathematical models which is non-commercial, easy to use, and easy to adapt and to develop further. The visualisation of the simulation results in three spatial dimensions and the interaction of different model variables is often considered too much of a challenge, which forces many researchers to present only one dimensional results instead. An interactive three dimensional visualisation tool, as presented in this thesis, will create a communication platform on which biologists and theoretical oncologists can discuss their models and their results. With an improved dialogue between mathematicians and experimentalists the next generation of models will be more accurate and hopefully lead to ideas for the design of future experiments and new treatments.