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Avascular Tumor Growth Modelling: Physical Insights to Skin Cancer

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Mathematical Oncology 2013

Abstract

In this chapter I present the state-of-the-art theoretical models for avascular tumor growth which are well established nowadays. I focus on models able to treat morphologic instabilities and phase segregation, two typical features of skin cancer for example melanoma. Contrary to experiments made in vitro on growing colonies, I show that the geometry of melanoma confined in the epidermis in the early stages of tumor growth suppresses the necrotic core and is responsible of inhomogeneities due to aggregation of cancerous cells. A relatively simple model consisting in the adaptation of the two-phase mixture model is enough to explain the morphologies of the tumor not only qualitatively but also quantitatively. Despite the complexity of the nonlinear partial differential equations that results from this model, I also present analytical treatments based on the techniques of nonlinear physics and W.K.B approximation to explain the observed structures in dermatology.

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Notes

  1. 1.

    http://www.ncbi.nlm.nih.gov/ Pubmed. The number of items in oncology has been determined using the keywords “cancer” and “tumor.” The number of articles referring to a mathematical modelling was determined by adding the keywords “modelling” and “mathematical model.” Search on 29th June 2012.

  2. 2.

    In the case of a diffusion-limited growth one often chooses \(\varGamma = an-\delta\) [97]. This non-uniform growth including a uniform apoptosis term δ in the tumor was introduced by McElwain and Morris in 1978 [101] according to experiments of Sutherland and Durand [134] showing that a dormant state was reached without the appearance of necrotic cell loss in the tumor center.

  3. 3.

    For avascular tumors, one often chooses \(S = -\delta _{n}n\), that is, an oxygen consumption proportional to the oxygen concentration as introduced by Deakin [49], according to the experiments of Sutherland and Durand [134] that show an inconsistency with the model of Greenspan. Such a relationship is well justified when the oxygen concentration is low.

  4. 4.

    By definition \(F_{ij} = \partial x_{i}/\partial X_{j}\) with \(\boldsymbol{X}\) identifying points of the solid in the initial state Ω 0 and \(\boldsymbol{x}\) in the current state Ω t [51, 53].

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Acknowledgements

This chapter is a summary of results about tumor modelling obtained in Ecole Normale Supérieure during these last years. I strongly acknowledge three Ph.D students, Julien Dervaux, Clément Chatelain and Thibaut Balois, for their intensive collaboration on this very rich topic between mathematics, nonlinear and soft matter physics and biology. This work is supported in part by AAP INSERM 2012.

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  1. Laboratoire de Physique Statistique, Ecole Normale Supérieure, UPMC Univ Paris 06, Université Paris Diderot, CNRS, 24 rue Lhomond, 75005, Paris, France

    Martina Ben Amar

  2. Institut Universitaire de Cancérologie, Faculté de médecine, Université Pierre et Marie Curie-Paris 6, 91 Bd de l’Hôpital, 75013, Paris, France

    Martina Ben Amar

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  1. International Prevention Research Institute, Lyon, France

    Alberto d'Onofrio

  2. Istituto di Analisi dei Sistemi ed Informatica "Antonio Ruberti"—CNR, Rome, Italy

    Alberto Gandolfi

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Amar, M.B. (2014). Avascular Tumor Growth Modelling: Physical Insights to Skin Cancer. In: d'Onofrio, A., Gandolfi, A. (eds) Mathematical Oncology 2013. Modeling and Simulation in Science, Engineering and Technology. Birkhäuser, New York, NY. https://doi.org/10.1007/978-1-4939-0458-7_3

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