References
- 1. B. Alberts, A. Johnson, J. Lewis, M. Raff, K. Roberts, and P. Walter, Molecular Biology of the Cell, 4th ed. Garland Science, 2002.
- 2. H. Osada and T. Takahashi, Genetic alterations of multiple tumor suppressors and oncogenes in the carcinogenesis and progression of lung cancer, Oncogene, vol. 21, pp. 7421–7434, 2002.
- 3. W. Mueller-Klieser, Tumor biology and experimental therapeutics, Crit. Rev. Oncol. Hematol., vol. 36, pp. 123–139, 2002.
- 4. P. Vaupel and M.Hockel, Blood supply, oxygenation status and metabolic micromilieu of breast cancers: characterization and therapeutic relevance (review), Int. J. Oncol., vol. 17, pp. 869–879, 2000.
- 5. J. M. Brown, Tumor microenvironment and the response to anticancer therapy, Cancer Biol. Ther., vol. 1, pp. 453–458, 2002.
- 6. S. S. Cross, Fractals in pathology, J. Pathol., vol. 182, pp. 1–8, 1997.
- 7. G. Landini, Y. Hirayama, T. J. Li, and M. Kitano, Increased fractal complexity of the epithelial connective tissue interface in the tongue of 4nq0-treated rats, Pathol. Res. Pract., vol. 196, pp. 251– 258, 2000.
- 8. A. Balter, R. M. H. Merks, N. J. Poplawski, M. Swat, and A. J. Glazier, The Glazier-Graner-Hogeweg model: extensions, future directions, and opportunities for further study, in Single-Cell-Based Models in Biology and Medicine (A. R. A. Anderson, M. A. J. Chaplain, and K. A. Rejniak, eds.), Mathematics and Biosciences in Interactions, pp. 151–167, Birkaüser, 2007.
- 9. J. A. Glazier and F. Graner, Simulation of the differential adhesion driven rearrangement of biological cells, Phys. Rev. E, vol. 47, pp. 2128–2158, 1993.
- 10. J. A. Glazier, A. Balter, and N. J. Poplawski, Magnetization to morphogenesis: a brief history of the Glazier-Graner-Hogeweg model, in Single-Cell-Based Models in Biology and Medicine (A. R. A. Anderson, M. A. J. Chaplain, and K. A. Rejniak, eds.), Mathematics and Biosciences in Interactions, pp. 79–106, Birkaüser, 2007.
- 11. F. Graner and J. A. Glazier, Simulation of biological cell sorting using a two dimensional extended Potts model, Phys. Rev. Lett., vol. 69, pp. 2013–2017, 1992.
- 12. M. Scianna and L. Preziosi, Multiscale developments of the cellular Potts model, Multiscale Model. Simul., vol. 10, pp. 342–382, 2012.
- 13. E. Ising, Beitrag zur theorie des ferromagnetismus, Z. Physik., vol. 31, p. 253, 1925.
- 14. R. B. Potts, Some generalized order-disorder transformations, Proc. Camb. Phil. Soc., vol. 48, pp. 106– 109, 1952.
- 15. R. M. H. Merks and P. Koolwijk, Modeling morphogenesis in silico and in vitro: Towards quantitative, predictive, cell-based modeling, Math. Model. Nat. Phenom., vol. 4, pp. 149–171, 2009.
- 16. M. Scianna, L. Munaron, and L. Preziosi, A multiscale hybrid approach for vasculogenesis and related potential blocking therapies, Prog. Biophys. Mol. Biol., vol. 160, pp. 450–462, 2010.
- 17. S. Turner and J. A. Sherratt, Intercellular adhesion and cancer invasion: A discrete simulation using the extended potts model, J. Theor. Biol., vol. 216, pp. 85–100, 2002.
- 18. N. Metropolis, A. Rosenbluth, M. N. Rosenbluth, A. H. Teller, and E. Teller, Equation of state calculations by fast computing machines, J. Chem. Phys., vol. 21, pp. 1087–1092, 1953.
- 19. M. S. Steinberg, Does differential adhesion govern self-assembly processes in histogenesis? equilibrium configurations and the emergence of a hierarchy among populations of embryonic cells, J. Exp. Zool., vol. 173, pp. 395–433, 1970.
- 20. S. Huang and D. E. Ingber, The structural and mechanical complexity of cell-growth control, Nat. Cell Biol., vol. 1, pp. 131–138, 1999.
- 21. N. J. Savill and P. Hogeweg, Modelling morphogenesis: From single cells to crawling slugs, J. Theor. Biol., vol. 184, pp. 118–124, 1997.
- 22. G. Murphy and J. Gavrilovic, Proteolysis and cell migration: Creating a path?, Curr. Opin. Cell Biol., vol. 11, pp. 614–621, 1999.
- 23. A. Colombi, M. Scianna, and A. Tosin, Differentiated cell behavior: a multiscale approach using measure theory, J. Math. Biol., 2015, in press. doi: 10.1007/s00285-014-0846-z.10.1007/s00285-014-0846-z
- 24. A. Colombi, M. Scianna, and L. Preziosi, A measure-theoretic model for collective cell migration and aggregation, Math. Model. Nat. Phenom., vol. 1, no. 10, pp. 32–63, 2015.
- 25. E. Cristiani, B. Piccoli, and A. Tosin, Multiscale modeling of granular flows with application to crowd dynamics, Multiscale Model. Simul., vol. 9, no. 1, pp. 155–182, 2011.
- 26. B. Piccoli and F. Rossi, Transport equation with nonlocal velocity in Wasserstein spaces: convergence of numerical schemes, Acta Appl. Math., vol. 124, no. 1, pp. 73–105, 2013.
- 27. B. Piccoli and A. Tosin, Time-evolving measures and macroscopic modeling of pedestrian flow, Arch. Ration. Mech. Anal., vol. 199, no. 3, pp. 707–738, 2011.
- 28. A. Tosin and P. Frasca, Existence and approximation of probability measure solutions to models of collective behaviors, Netw. Heterog. Media, vol. 6, no. 3, pp. 561–596, 2011.
- 29. R. Gatenby, K. Smallbone, P. Maini, F. Rose, J. Averill, R. Nagle, L. Worrall, and R. Gillies, Cellular adaptations to hypoxia and acidosis during somatic evolution of breast cancer, Br. J. Cancer, vol. 97, pp. 646–653, 2007.
- 30. J. Smolle, Fractal tumor stromal border in a nonequilibrium growth model, Anal. Quant. Cytol. Histol., vol. 20, pp. 7–13, 1998.
- 31. S. M. Wise, J. S. Lowengrub, H. B. Frieboes, and V. Cristini, Three-dimensional multispecies nonlinear tumor growth–i model and numerical method, Int. J. Oncol., vol. 253, pp. 524–543, 2008.
- 32. A. R. A. Anderson, A. M. Weaver, P. T. Cummings, and V. Quaranta, Tumor morphology and phenotypic evolution driven by selective pressure from the microenvironment, Cell, vol. 127, no. 5, pp. 905–915, 2006.