Modelling tumor growth and cancer dynamics

Christophe LETELLIERLouise VIGER
Fabrice DENISClément DRAGHI
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Cancer is known to be impredictible for a given patient. A natural question is therefore whether it is due to a very large sensitivity to initial conditions or to a lack of knowledge of the underlying processes. The relationship between cancer and chaos is thus a question often addressed. Nevetheless, one should distinguish the spatial growth of tumor - sometimes investigated using the paradigms of fractals - and the evolution of a population of tumor cells (possibly chaotic).

We choose to start our investigations of tumor growth by considering the model describing interactions between host, immune and tumor cells as proposed by de Pillis & Radunskaya [1]. Initially chosen because it was the single one we found to produce chaotic attractors [2], this model is one of the very rare models taking into account the host (healthy) cells. This is a very relevant characteristics due to the growing interest devoted to the role of the micro-environment [3].

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Fig. 1 : Chaotic attractor produced by the de Pillis-Radunskaya model. Differential embeddings induced by each of the three variables.

Our first contribution was to investigate the dynamics of de Pillis & Radunskaya’s model, showing that it qualitatively corresponds to clinical features [4] ; we also showed that the best variable to observe cancer dynamics is associated with the population of host cells and not, as expected, the population of tumor cells (the symbolic observability coefficients are reported in Fig. 1). Using this result, we designed a follow-up for patients treated for a lung cancer based on a self-evaluation of their global status by weakly filling a form [5] [6]. In Fig. 2, the weekly filled form by a patient who was treated by chemioradiotherapy for a lung cancer : the relapse was detected by our clinical follow-up two months before the routine imaging. Treating earlier relapse allows more efficient therapy and increases the survival.

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Fig. 2 : Weekly filled form of a patient (65 years, 86 kg), smoker, no sport, treated by chemiotherapy, Probability relapse = 75%.

During the development of tumor, there are roughy two phases. When the tumor is small, there is no neovessels to supply oxygen to tumor cells ; the tumor is said to be avascular and is growth is rather slow. With a larger size, the oxygen is no longer in sufficient quantity to allow its growth, and tumor cells stimulate endothelial cells responsible to the production of new vessels toward the tumor, thus carrying oxygen to tumor cells : this is the so-called angiogenic switch (Fig. 3). The tumor is then said to be a vascular tumor. Its growth is then faster than before the angiogenic switch. Moreover, the connection of the tumor to blood vessels allows a spread of malignant cells through the whole body ; when one finds an appropriate nest, it starts to create a colony, leading to a new tumor, called metastatis.

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Fig. 3 : To satisfy their demand in oxygen, tumor cells stimulate endothelial cells for creating new blood vessels, carrying oxygen up to the tumor.

We therefore developed an extended version of de Pillis-Radunskaya’s model including the endothelial cells [7]. This model allows to reproduce the angiogenic switch, that is, the transport of oxygen up to the tumor, leading to a faster tumor growth, and inducing a possible metastatic spread. Depending on the growth rate of endothelial cells and for the same initial conditions, we thus showed that for a short duration, the population evolves in a non distinguishable manner and then the population of endothelial cells starts to significantly increase, leading to a boosted proliferation of tumor cells (Fig. 4). This apparent sensitivity to initial conditions is in fact a sensitivity to parameter values of the underlying dynamics. There is a threshold value for the endothelial cell growth rate beyond which, the angiogenic switch is possible.

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Fig. 4 : Evolution of the population of host, immune, tumor and endothelial cells for two different values of the endothelial cell growth rate. In (a), the tumor remains avascular. Contrary to this, in (b), the tumor becomes vascularized and the metastatic spread is possible.

[1] L. G. de Pillis & A. Radunskaya, A mathematical tumor model with immune resistance and drug therapy : an optimal control approach, .lournal of Theoretical Medicine. 3, 79-100, 2001.

[2] M. Itik & S. P. Banks, Chaos in a three-dimensional cancer model, International Journal of Bifurcation & Chaos, 20 (1), 71-79, 2010.

[3] M. J. Bissell & W. C. Hines, Why don’t we get more cancer ? A proposed role of the microenvironment in restraining cancer progression, Nature Medicine, 17 (3), 320-329, 2011.

[4] C. Letellier, F. Denis & L. A. Aguirre, What can be learned from a chaotic cancer model ?, Journal of Theoretical Biology, 322, 7-16, 2013.

[5] F. Denis, L. Viger, A. Charron, E. Voog & C. Letellier, Detecting lung cancer relapse using self-evaluation forms weekly filled at home : the sentinel follow-up, Supportive Care in Cancer, 22 (1), 79-85, 2013.

[6] F. Denis, L. Viger, A. Charron, E. Voog, O. Dupuis, Y. Pointreau & C. Letellier, Detection of lung cancer relapse using self-reported symptoms transmitted via an Internet Web-application : pilot study of the sentinel follow-up, Support Care Cancer, 22 (6), 1467-1473, 2014.

[7] L. Viger, F. Denis, M. Rosalie & C. Letellier, A cancer model for the angiogenic switch, Journal of Theoretical Biology, 360, 21-33, 2014.

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