Mathematical Modelling of Radiotherapy Strategies for Early Breast Cancer

H Enderling^1*, ARA Anderson¹, MAJ Chaplain¹, AJ Munro², JS Vaidya^2*

¹Division of Mathematics, ²Department of Surgery and Molecular Oncology
University of Dundee, Dundee, UK

^*Correspondence: Heiko Enderling and Jayant S Vaidya

J. Theor. Biol. 241(1), 158-171, 2006

Abstract:
Targeted Intraoperative radiotherapy (Targit) is a new concept of partial breast irradiation where single fraction radiotherapy is delivered directly to the tumour bed. 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. In this paper we present a mathematical model for the growth and invasion of a solid tumour into a domain of tissue (in this case breast tissue), and then a model for surgery and radiation treatment of this tumour. We use the established linear quadratic (LQ) model to compute the survival probabilities for both tumour cells and irradiated breast tissue and then simulate the effects of conventional external beam radiotherapy and Targit. True local recurrence of the tumour could arise either from stray tumour cells, or the tumour bed that harbours morphologically normal cells having a predisposition to genetic changes, such as a loss of heterozygosity (LOH) in genes that are crucial for tumourigenesis, e.g. tumour suppressor genes (TSGs). Our mathematical model predicts that the single high dose of radiotherapy delivered by Targit would result in eliminating all these sources of recurrence, whereas the fractionated external beam radiotherapy would eliminate stray tumour cells, but allow (by virtue of its very schedule) the cells with LOH in TSGs or cell-cycle checkpoint genes to pass on low-dose radiation-induced DNA damage and consequently mutations that may favour the development of a new tumour. The mathematical model presented here is an initial attempt to model a biologically complex phenomenon that has until now received little attention in the literature and 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.

Keywords:
Mathematical Modelling, Tumour Invasion, Breast Cancer, Radiotherapy, Targit

[Simulation videos are at: http://www.maths.dundee.ac.uk/~heikman/targit ]

Solid tumour growth and invasion

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Figure 5c: Spatio-temporal evolution of solid tumour growth and invasion in Phase I of our model framework from a computational simulation of the model. The tumour cells (red) proliferate and invade the surrounding tissue (green). The MDEs (solid blue line) degrade the tissue to make space for the tumour to grow into.

Breast conserving surgery

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Figure 6: Animation showing the simulation of surgery for early breast cancer. Tumour cell density is red, healthy tissue density is green and MDE concentration is the blue line. When the dense tumour diameter (dominance of tumour cells over tissue cells) reaches 2 cm, surgery is simulated and all cells and enzymes in the area of high tumour tissue density are removed. The presence of some tumour cells in the tissue after surgery (the small brown triangle at the bottom left of the green area near the x-axis) emphasizes the need of radiotherapy following breast conserving surgery.

Regrow due to tumour cells have been left behind after surgery

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Figure 7: Video showing the spatio-temporal evolution of the recurrence of a breast tumour (red) when there are a few tumour cells left behind in the tissue (green) after the surgery as shown in Figure 6. The matrix degrading enzyme concentration is plotted as the blue line.

Simulation results of conventional EBRT treatment

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Figure 8: Video showing the simulation of conventional external beam radiotherapy treatment following breast conserving surgery. Fractions with doses of 2 Gy each (solid black line in later frames) are delivered to the whole domain eradicating all tumour cells (brown) but also harming the healthy (green) breast tissue.

Simulation results of Targit

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Figure 9: Video showing the results of simulating radiotherapy treatment post-surgery using the new Targit method. The solid black line shows the initial distribution throughout the tissue of the radiation delivered by Targit. The rapid attenuation of the radiation dose protects tissue at larger distances from the applicator. Later frames show the domain after irradiation. We see that all the tumour cells are killed. However, note that the damage to the breast tissue due to irradiation is very localised.

Irradiation of cells harbouring LOH with conventional EBRT

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Figure 11: Video showing the simulation of standard fractionated radiotherapy following breast conserving surgery and its effect on cells with LOH. As can be seen from the plots, this standard treatment does indeed eradicate all tumour cells, but fails to eliminate cells with LOH on TSGs in the immediate vicinity of the tumour (yellow area), since these mutated cell-cycle check-point genes lack the detection of radiation induced genetic damage.

Irradiation of cells harbouring LOH with Targit

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Figure 12: Video showing the simulation of a single high dose of Targit radiation (solid black line) delivered to areas adjacent to the primary tumour and its effect on cells with LOH. As can be seen, Targit eliminates tissue cells that previously surrounded the (red) tumour and may have shared its clonal origin and may have harboured cells with LOH (yellow area). The (green) tissue is only affected near the tumour bed. Tissue at distant sites is spared damage.

Last modified: Wed Jul 7 14:06:21 GMT Daylight Time 2006