Model Comparison and Uncertainty Quantification in Tumor Growth

Gustavo Taiji Naozuka, Emanuelle Arantes Paixão, João Vitor Oliveira Silva, Maurício Pessoa da Cunha Menezes, Regina Cerqueira Almeida

Abstract


Mathematical and computational modeling have been increasingly applied in many areas of cancer research, aiming to improve the understanding of tumorigenic mechanisms and to suggest more effective therapy protocols. The mathematical description of the tumor growth dynamics is often made using the exponential, logistic, and Gompertz models. However, recent literature has suggested that the Allee effect may play an important role in the early stages of tumor dynamics, including cancer relapse and metastasis. For a model to provide reliable predictions, it is necessary to have a rigorous evaluation of the uncertainty inherent in the modeling process. In this work, our main objective is to show how a model framework that integrates sensitivity analysis, model calibration, and model selection techniques can improve and systematically characterize model and data uncertainties. We investigate five distinct models with different complexities, which encompass the exponential, logistic, Gompertz, and weak and strong Allee effects dynamics. Using tumor growth data published in the literature, we perform a global sensitivity analysis, apply a Bayesian framework for parameter inference, evaluate the associated sensitivity matrices, and use different information criteria for model selection (First- and Second-Order Akaike Information Criteria and Bayesian Information Criterion). We show that such a wider methodology allows having a more detailed picture of each model assumption and uncertainty, calibration reliability, ultimately improving tumor mathematical description. The used in vivo data suggested the existence of both a competitive effect among tumor cells and a weak Allee effect in the growth dynamics. The proposed model framework highlights the need for more detailed experimental studies on the influence of the Allee effect on the analyzed cancer scenario.

Keywords


Predictive oncology; Inverse problem; Allee effect; Logistic model; Gompertz model; Exponential model

Full Text:

PDF

References


WHO, “World Health Organization. Cancer.” www.who.int/cancer/en/, Feb 2019.

INCA, “Instituto Nacional de Câncer. Estatísticas de câncer.” www.inca.gov.br/numeros-de-cancer, Nov 2018.

R. Brady and H. Enderling, “Mathematical models of cancer: when to predict novel therapies, and when not to,”Bulletin of Mathematical Biology, vol. 81, pp. 3722 – 3731, 2019.

P. Altrock, F. Michor, and L. Liu, “The mathematics of cancer: integrating quantitative models,”Nature Reviews, vol. 15, pp. 730–745, 2015.

A. Saltelli, M. Ratto, T. Andres, F. Campolongo, J. Cariboni, D. Gatelli, M. Saisana, and S. Tarantola,Global sensitivity analysis: the primer. John Wiley & Sons, 2008.

A. Saltelli, K. Aleksankina, W. Becker, P. Fennell, F. Ferretti, N. Holst, S. Li, and Q. Wu, “Why so many published sensitivity analyses are false: A systematic review of sensitivity analysis practices,”Environmental Modelling & Software, vol. 114, pp. 29 – 39, 2019.

A. C. M. Resende, “Sensitivity analysis as a tool for tumor growth modeling,” Master’s thesis, Laboratório Nacional de de Computação Científica (LNCC/MCTIC), Petrópolis/RJ, Brazil, 2016.

H. Murphy, H. Jaafari, and H. M. Dobrovolny, “Differences in predictions of ODE models of tumor growth: a cautionary example,”BMC Cancer, vol. 16, no. 1, p. 163, 2016.

D. J. Warne, R. E. Baker, and M. J. Simpson, “Using experimental data and information criteria to guide model selection for reaction–diffusion problems in mathematical biology,”Bulletin of Mathematical Biology, vol. 81, no. 6, pp. 1760–1804, 2019.

J. T. Oden, E. E. Prudencio, and A. Hawkins-Daarud, “Selection and assessment of phenomenological models of tumor growth,”Mathematical Models and Methods in Applied Sciences, vol. 23, no. 7, pp. 1309–1338, 2013.

E. A. B. F. Lima, J. T. Oden, D. A. Hormuth, T. E. Yankeelov, and R. C. Almeida, “Selection, calibration, and validation of models of tumor growth,”Mathematical Models and Methods in Applied Sciences, vol. 26, no. 12, pp. 2341–2368, 2016.

F. Pianosi, K. Beven, J. Freer, J. W. Hall, J. Rougier, D. B. Stephenson, and T. Wagener, “Sensitivity analysis of environmental models: A systematic review with practical workflow,”Environmental Modelling & Software, vol. 79, pp. 214 – 232, 2016.

T. Toni, D. Welch, N. Strelkowa, A. Ipsen, and M. P. Stumpf, “Approximate bayesian computation scheme for parameter inference and model selection in dynamical systems,”Journal of The Royal Society Interface, vol. 6, no. 31, pp. 187–202, 2009.

J. da Costa, H. Orlande, and W. da Silva, “Model selection and parameter estimation in tumor growth models using approximate bayesian computation – ABC,”Computational and Applied Mathematics, vol. 37, pp. 2795 – 2815, 2017.

P. Kirk, T. Thorne, and M. P. Stumpf, “Model selection in systems and synthetic biology,”Current Opinion in Biotechnology, vol. 24, no. 4, pp. 767 – 774, 2013.

J. Liepe, P. Kirk, S. Filippi, T. Toni, C. P. Barnes, and M. P. H. Stumpf, “A framework for parameter estimation and model selection from experimental data in systems biology using approximate bayesian computation,”Nature Protocols, vol. 9, pp. 439 – 456, 2014.

K. P. Burnham and D. R. Anderson,Model Selection and Multimodel Inference – A Practical Information-Theoretic Approach. New York: Springer-Verlag, 2nd ed., 2002.

K. P. Burnham and D. R. Anderson, “Multimodel inference: Understanding AIC and BIC in model selection,”Sociological Methods & Research, vol. 33, no. 2, pp. 261 – 304, 2019.

K. S. Korolev, J. B. Xavier, and J. G. Gore, “Turning ecology and evolution against cancer,”Nature Reviews Cancer, vol. 14, pp. 371 – 380, 2014.

D. Basanta and A. R. A. Anderson, “Exploiting ecological principles to better understand cancer progression and treatment,”Interface Focus, vol. 3, no. 4, pp. 1–11, 2013.

F. Courchamp, L. Berec, and J. Gascoigne,Allee effects in ecology and conservation. Oxford University Press, 2008.

F. Courchamp, T. Clutton-Brock, and B. Grenfell, “Inverse density dependence and the allee effect,”Trends in Ecology & Evolution, vol. 14, no. 10, pp. 405–410, 1999.

K. E. Johnson, G. Howard, W. Mo, M. K. Strasser, E. A. B. F. Lima, S. Huang, and A. Brock, “Cancer cell population growth kinetics at low densities deviate from the exponential growth model and suggest an Allee effect,”PLOS Biology, vol. 17(8), p. e3000399, 2019.

Z. Neufeld, W. v. Witt, D. Lakatos, J. Wang, B. Hegedus, and A. Czirok, “The role of Allee effect in modelling post resection recurrence of glioblastoma,” PLOS Computational Biology, vol. 17, pp. 1–14, 2017.

K. Böttger, H. Hatzikirou, A. Voss-Böhme, E. A. Cavalcanti-Adam, M. A. Herrero, and A. Deutsch, “An emerging Allee effect is critical for tumor initiation and persistence,”PLOS Computational Biology, vol. 3, pp. 1–14, 2015.

P. Feng, Z. Dai, and D. Wallace, “On a 2D model of avascular tumor with weak Allee effect,”Journal of Applied Mathematics, vol. 2019, pp. 1–13, 2019.

A. Worschech, N. Chen, Y. A. Yu, and et al., “Systemic treatment of xenografts with vaccinia virus GLV-1h68 reveals the immunologic facet of oncolytic therapy,”BMC Genomics, vol. 10, pp. 301–301, 2009.

A. Rohatgi, “WebPlotDigitizer version: 4.1.” https://automeris.io/WebPlotDigitizer/, Jan 2018.

R. J. LeVeque,Finite Difference Methods for Ordinary and Partial Differential Equations. Phyladelphia, PA: Society for Industrial and Applied Mathematics, 2007.

F. Campolongo, S. Tarantola, and A. Saltelli, “Tackling quantitatively large dimensionality problems,”Computer physics communications, vol. 117, no. 1-2, pp. 75–85, 1999.

G. Box and G. Tiao,Bayesian inference in statistical analysis. Addison-Wesley series in behavioral science: Quantitative methods, Addison-Wesley Pub. Co., 1973.

E. E. Prudêncio and K. W. Schulz, “The parallel C++ statistical library ‘QUESO’: Quantification of Uncertainty for Estimation, Simulation and Optimization,” in Euro-Par 2011: Parallel Processing Workshops, pp. 398–407, Springer, 2012.

P. Gregory,Bayesian Logical Data Analysis for the Physical Sciences: A Comparative Approach with MathematicaR©Support. Cambridge University Press, 2005.

A. Cintrón-Arias, H. T. Banks, A. Capaldi, and A. L. Lloyd, “A sensitivity matrix based methodology for inverse problem formulation,”Journal of Inverse and Ill-Posed Problems, pp. 545–564, 2009.

M. Delitala and M. Ferraro, “Is the Allee effect relevant in cancer evolution and therapy?,”AIMS Mathematics, vol. 5, no. 6, pp. 7649–7660, 2020.


Article Metrics

Metrics Loading ...

Metrics powered by PLOS ALM

Refbacks

  • There are currently no refbacks.



Trends in Computational and Applied Mathematics

A publication of the Brazilian Society of Applied and Computational Mathematics (SBMAC)

 

Indexed in:

                       

 

Desenvolvido por:

Logomarca da Lepidus Tecnologia