Physica A 462 (2016) 560–568 Contents lists available at ScienceDirect Physica A journal homepage: www.elsevier.com/locate/physa Phase transitions in tumor growth: III vascular and metastasis behavior J.A. Llanos-Pérez a , J.A. Betancourt-Mar a,* , G. Cocho b , R. Mansilla c , José Manuel Nieto-Villar d,e,a,** a Mexican Institute of Complex Systems, Tamaulipas, Mexico b Departamento de Sistemas Complejos del Instituto de Física de la UNAM, Mexico c Centro de Investigaciones Interdisciplinarias en Ciencias y Humanidades, UNAM, Mexico d Department of Chemical-Physics, M.V. Lomonosov Chemistry Division, Faculty of Chemistry, University of Havana, Havana 10400, Cuba e H. Poincaré Group of Complex Systems, Physics Faculty, University of Havana, Havana 10400, Cuba highlights • Cancer as an open, complex, self-organizing nonlinear dynamic system. • Metastasis dynamics may exhibit a Shilnikov’s chaos. • The entropy production rate as a Lyapunov function for cancer growth. • The epithelial-to-mesenchymal transition appears as phase transition. article info Article history: Received 30 November 2015 Received in revised form 16 March 2016 Available online 21 June 2016 Keywords: Phase transition Vascular and metastasis tumor growth Entropy production Chaos and complexity abstract We propose a mechanism for avascular, vascular and metastasis tumor growth based on a chemical network model. Vascular growth and metastasis, appear as a hard phase transition type, as ‘‘first order’’, through a supercritical Andronov–Hopf bifurcation, emergence of limit cycle and then through a cascade of bifurcations type saddle-foci Shilnikov’s bifurcation. Finally, the thermodynamics framework developed shows that the entropy production rate, as a Lyapunov function, indicates the directional character and stability of the dynamical behavior of tumor growth according to this model. © 2016 Elsevier B.V. All rights reserved. 1. Introduction Cancer is a generic name given to a complex network of interactions of malignant cells which have lost their specialization and control over normal growth. This network of malignant cells could be considered as a nonlinear dynamical system, self-organized in time and space, far from thermodynamic equilibrium, exhibiting high complexity [1], robustness [2] and adaptability [3]. * Corresponding author. ** Corresponding author at: Department of Chemical-Physics, M.V. Lomonosov Chemistry Division, Faculty of Chemistry, University of Havana, Havana 10400, Cuba. E-mail addresses: betancourt@mics.edu.mx (J.A. Betancourt-Mar), nieto@fq.uh.cu (J.M. Nieto-Villar). http://dx.doi.org/10.1016/j.physa.2016.06.086 0378-4371/© 2016 Elsevier B.V. All rights reserved.