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Classical Mathematical Models for Description and Prediction of Experimental Tumor Growth

Author

Listed:
  • Sébastien Benzekry
  • Clare Lamont
  • Afshin Beheshti
  • Amanda Tracz
  • John M L Ebos
  • Lynn Hlatky
  • Philip Hahnfeldt

Abstract

Despite internal complexity, tumor growth kinetics follow relatively simple laws that can be expressed as mathematical models. To explore this further, quantitative analysis of the most classical of these were performed. The models were assessed against data from two in vivo experimental systems: an ectopic syngeneic tumor (Lewis lung carcinoma) and an orthotopically xenografted human breast carcinoma. The goals were threefold: 1) to determine a statistical model for description of the measurement error, 2) to establish the descriptive power of each model, using several goodness-of-fit metrics and a study of parametric identifiability, and 3) to assess the models' ability to forecast future tumor growth. The models included in the study comprised the exponential, exponential-linear, power law, Gompertz, logistic, generalized logistic, von Bertalanffy and a model with dynamic carrying capacity. For the breast data, the dynamics were best captured by the Gompertz and exponential-linear models. The latter also exhibited the highest predictive power, with excellent prediction scores (≥80%) extending out as far as 12 days in the future. For the lung data, the Gompertz and power law models provided the most parsimonious and parametrically identifiable description. However, not one of the models was able to achieve a substantial prediction rate (≥70%) beyond the next day data point. In this context, adjunction of a priori information on the parameter distribution led to considerable improvement. For instance, forecast success rates went from 14.9% to 62.7% when using the power law model to predict the full future tumor growth curves, using just three data points. These results not only have important implications for biological theories of tumor growth and the use of mathematical modeling in preclinical anti-cancer drug investigations, but also may assist in defining how mathematical models could serve as potential prognostic tools in the clinic.Author Summary: Tumor growth curves display relatively simple time curves that can be quantified using mathematical models. Herein we exploited two experimental animal systems to assess the descriptive and predictive power of nine classical tumor growth models. Several goodness-of-fit metrics and a dedicated error model were employed to rank the models for their relative descriptive power. We found that the model with the highest descriptive power was not necessarily the most predictive one. The breast growth curves had a linear profile that allowed good predictability. Conversely, not one of the models was able to accurately predict the lung growth curves when using only a few data points. To overcome this issue, we considered a method that uses the parameter population distribution, informed from a priori knowledge, to estimate the individual parameter vector of an independent growth curve. This method was found to considerably improve the prediction success rates. These findings may benefit preclinical cancer research by identifying models most descriptive of fundamental growth characteristics. Clinical perspective is also offered on what can be expected from mathematical modeling in terms of future growth prediction.

Suggested Citation

  • Sébastien Benzekry & Clare Lamont & Afshin Beheshti & Amanda Tracz & John M L Ebos & Lynn Hlatky & Philip Hahnfeldt, 2014. "Classical Mathematical Models for Description and Prediction of Experimental Tumor Growth," PLOS Computational Biology, Public Library of Science, vol. 10(8), pages 1-19, August.
  • Handle: RePEc:plo:pcbi00:1003800
    DOI: 10.1371/journal.pcbi.1003800
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    2. Alex Root, 2019. "Mathematical Modeling of The Challenge to Detect Pancreatic Adenocarcinoma Early with Biomarkers," Challenges, MDPI, vol. 10(1), pages 1-15, April.
    3. Ahmed, Najma & Shah, Nehad Ali & Taherifar, Somaye & Zaman, F.D., 2021. "Memory effects and of the killing rate on the tumor cells concentration for a one-dimensional cancer model," Chaos, Solitons & Fractals, Elsevier, vol. 144(C).
    4. Hamzeh Zureigat & Mohammed Al-Smadi & Areen Al-Khateeb & Shrideh Al-Omari & Sharifah Alhazmi, 2023. "Numerical Solution for Fuzzy Time-Fractional Cancer Tumor Model with a Time-Dependent Net Killing Rate of Cancer Cells," IJERPH, MDPI, vol. 20(4), pages 1-13, February.
    5. Gregory Baramidze & Victoria Baramidze & Ying Xu, 2021. "Mathematical model and computational scheme for multi-phase modeling of cellular population and microenvironmental dynamics in soft tissue," PLOS ONE, Public Library of Science, vol. 16(11), pages 1-31, November.
    6. Bulai, I.M. & De Bonis, M.C. & Laurita, C. & Sagaria, V., 2023. "Modeling metastatic tumor evolution, numerical resolution and growth prediction," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 203(C), pages 721-740.
    7. Cabrales, Luis Enrique Bergues & Montijano, Juan I. & Schonbek, Maria & Castañeda, Antonio Rafael Selva, 2018. "A viscous modified Gompertz model for the analysis of the kinetics of tumors under electrochemical therapy," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 151(C), pages 96-110.
    8. Rodrigues, D.S. & Mancera, P.F.A. & Carvalho, T. & Gonçalves, L.F., 2019. "A mathematical model for chemoimmunotherapy of chronic lymphocytic leukemia," Applied Mathematics and Computation, Elsevier, vol. 349(C), pages 118-133.
    9. Bulai, I.M. & De Bonis, M.C. & Laurita, C., 2025. "Numerical solution of metastatic tumor growth models with treatment," Applied Mathematics and Computation, Elsevier, vol. 484(C).
    10. Ella Ya Tyuryumina & Alexey A Neznanov, 2018. "Consolidated mathematical growth model of the primary tumor and secondary distant metastases of breast cancer (CoMPaS)," PLOS ONE, Public Library of Science, vol. 13(7), pages 1-16, July.
    11. Pelayo Martínez-Fernández & Zulima Fernández-Muñiz & Ana Cernea & Juan Luis Fernández-Martínez & Andrzej Kloczkowski, 2023. "Three Mathematical Models for COVID-19 Prediction," Mathematics, MDPI, vol. 11(3), pages 1-16, January.
    12. Charalambos Loizides & Demetris Iacovides & Marios M Hadjiandreou & Gizem Rizki & Achilleas Achilleos & Katerina Strati & Georgios D Mitsis, 2015. "Model-Based Tumor Growth Dynamics and Therapy Response in a Mouse Model of De Novo Carcinogenesis," PLOS ONE, Public Library of Science, vol. 10(12), pages 1-18, December.

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