Scaling laws and evolutionary dynamics in cancer: Results and open mathematical problems

Many natural systems are complex entities composed of a large number of interacting individual elements. It is remarkable that they often obey the so-called scaling laws. These ‘laws’ relate an observable quantity to a measure of the size of the system, such as its volume or mass [1]. In this talk I will describe universal scaling laws in human cancers [2] and how they imply an increase of tumor aggressiveness with time that leads to an explosive growth as the disease progresses. The observations can be understood using different types of biologically-inspired mathematical models [2,3], some of them displaying finite-time blow-up. Most of the observed phenomena can be described using different types of models based on nonlocal partial differential equations. The mathematical approaches lead to the definition of different biomarkers of the disease aggressiveness that have been validated using cancers imaging data [4].

I will also discuss several open mathematical problems of relevance arising in the context of this research.

 

[1] West G, Scale: The Universal Laws of Life and Death in Organisms, Cities and Companies. Penguin (2018).

[2] V. M. Pérez-García et al, Universal scaling laws rule explosive growth in human cancers, Nature Physics 16, 1232-1237 (2020).

[3] J. Jiménez-Sánchez, A. Martínez-Rubio, A. Popov, J. Pérez-Beteta, Y. Azimzade, D. Molina-García, J. Belmonte-Beitia, G. F. Calvo, V. M. Pérez-García. A mesoscopic simulator to uncover heterogeneity and evolutionary dynamics in tumors. PLOS Computational Biology 17(2) e1008266 (2021).

[4] J. Jiménez-Sánchez, J. J. Bosque, G. A. Jiménez-Londoño, D. Molina-García, A. Martínez-Rubio, J. Pérez-Beteta, C. Ortega-Sabater, A. F. Honguero-Martínez, A. M. García-Vicente, G. F. Calvo, V. M. Pérez-García. Evolutionary dynamics at the tumor edge reveals metabolic imaging biomarkers. Proceedings of the National Academy of Sciences 118(6) e2018110118 (2021).