Universal scaling laws rule explosive growth in human cancers


Most physical and other natural systems are complex entities that are composed of a large number of interacting individual elements. It is a surprising fact that they often obey the so-called scaling laws that relate an observable quantity to a measure of the size of the system. Here, we describe the discovery of universal superlinear metabolic scaling laws in human cancers. This dependence underpins increasing tumour aggressiveness, owing to evolutionary dynamics, that leads to an explosive growth as the disease progresses. We validated this dynamic using longitudinal volumetric data of different histologies from large cohorts of patients with cancer. To explain our observations we tested complex, biologically inspired mathematical models that describe the key processes that govern tumour growth. Our models predict that the emergence of superlinear allometric scaling laws is an inherently three-dimensional phenomenon. Moreover, the scaling laws that we identified allowed us to define a set of metabolic metrics with prognostic value, which adds clinical utility to our findings.

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Fig. 1: A superlinear scaling law governs glucose uptake and proliferation in human cancers.
Fig. 2: Explosive longitudinal volumetric dynamics of untreated malignant human tumours.
Fig. 3: Stochastic mesoscale models with evolutionary dynamics lead to superlinear scaling laws in silico.
Fig. 4: Scaling laws allow for classification of patients with cancer into prognostic groups.

Data availability

Source data for Figs. 1,2,4 are available for this paper. All other data that support the plots within this paper and other findings of this study are available from the corresponding author upon reasonable request.

Code availability

The mesoscopic simulator code is available for download from http://matematicas.uclm.es/molab/DiscrSimulator1.zip. Source data are provided with this paper.


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This research has been supported by the James S. McDonnell Foundation 21st Century Science Initiative in Mathematical and Complex Systems Approaches for Brain Cancer (collaborative awards 220020560 and 220020450), Ministerio de Economía y Competitividad/FEDER, Spain (grant no. MTM2015-71200-R), Junta de Comunidades de Castilla-La Mancha (grant no. SBPLY/17/180501/000154). Research in the Brain Metastasis Group is supported by MINECO grant MINECO-Retos SAF2017-89643-R (M.V.), Bristol-Myers Squibb Melanoma Research Alliance Young Investigator Award 2017 (498103) (M.V.), Beug Foundation’s Prize for Metastasis Research 2017 (M.V.), Fundación Ramón Areces (CIVP19S8163) (M.V.), Worldwide Cancer Research (19-0177) (M.V.), H2020-FETOPEN (828972) (M.V.), Fundació La Marató de tv3 (141), Clinic and Laboratory Integration Program CRI Award 2018 (54545) (M.V.), AECC Coordinated Translational Groups 2017 (GCTRA16015SEOA) (M.V.), LAB AECC 2019 (LABAE19002VALI) (M.V.), La Caixa INPhINIT Fellowship (LCF/BQ/IN17/11620028) (P.G.-G.), La Caixa-Severo Ochoa International PhD Program Fellowship (LCF/BQ/SO16/52270014) (L.Z.). M.V. is a member of the European Molecular Biology Organization Young Investigators programme (4053). We would like to acknowledge J. Cervera and J. C. Peñalver from the IVO Foundation (Valencia, Spain).

Author information




V.M.P.-G. designed the research, collected and processed data, performed the fittings, and developed and simulated the mathematical models. J.J.B. collected data, processed data, performed fitting tasks and developed the automatic segmentation algorithm. J.P.-B. collected and processed data. O.L.-T. collected and processed data and fitted the longitudinal volumetric data. M.V., L.Z., P.G.-G., P.S.-G., E.H.-S.M. and R.H. performed the experiments in the animal models. E.A., P.S., M.M., D.A., A.O.d.M., A.A.R., F.V. and A.F.H.M. collected data. G.A.J.L. collected and processed data. M.P.-C. processed data. G.F.C., J.J.B., J.J., J.B.-B., Y.A. and A.M. developed and simulated the mathematical models. D.M.-G. performed the survival studies. A.M.G.V. designed the research, and collected and processed data. V.M.P.-G. and G.F.C. drafted the manuscript. All of the authors edited and approved the final manuscript.

Corresponding author

Correspondence to Víctor M. Pérez-García.

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Extended data

Extended Data Fig. 1 Two human cancer animal models display superlinear growth dynamics.

Two human cancer animal models display superlinear growth dynamics. Group 1 (G1) data correspond to untreated nude mice injected with the human lung adenocarcinoma brain tropic model H2030-BrM (see methods). Group data (G2) correspond to primary glioma cells (L0627) expressing the luciferase reporter gene injected into the brain of nude mice (see methods). Bioluminiscence images for G1 for some mice are shown in panel A. Total tumour mass growth curves for G1 showed superlinear dynamics with best fitting exponent β = 1.25 (for G2 it was β = 1.3). (B, upper panel). Errors relative to best fit were found to be substantially smaller with the optimal superlinear fits than for both the linear and sublinear fits (exponents 1 and 0.75 respectively) (B, lower panel).

Supplementary information

Supplementary Information

Supplementary text, figures, tables and references.

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Source data

Source Data Fig. 1

Data of TLA versus MTV for the different tumour histologies considered.

Source Data Fig. 2

Data of tumour volume versus time for the different patient groups and tumour histologies shown in the figure.

Source Data Fig. 4

Survival data for the different groups and tumour histologies shown in the figure.

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Pérez-García, V.M., Calvo, G.F., Bosque, J.J. et al. Universal scaling laws rule explosive growth in human cancers. Nat. Phys. (2020). https://doi.org/10.1038/s41567-020-0978-6

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