Abstract
Quantitative analysis of tumor growth kinetics has been widely carried out using mathematical models. In the majority of cases, individual or average data were fitted.
Here, we analyzed three classical models (exponential, logistic and Gompertz within the statistical framework of nonlinear mixed-effects modelling, which allowed us to account for inter-animal variability within a population group. We used in vivo data of subcutaneously implanted Lewis Lung carcinoma cells. While the exponential and logistic models failed to accurately fit the data, the Gompertz model provided a superior descriptive power. Moreover, we observed a strong correlation between the Gompertz parameters. Combining this observation with rigorous population parameter estimation motivated a simplification of the standard Gompertz model in a reduced Gompertz model, with only one individual parameter. Using Bayesian inference, we further applied the population methodology to predict the individual initiation times of the tumors from only three measurements. Thanks to its simplicity, the reduced Gompertz model exhibited superior predictive power.
The method that we propose here remains to be extended to clinical data, but these results are promising for the personalized estimation of the tumor age given limited data at diagnosis.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
References
Monolix version 2018R2. Lixoft SAS (2018)
Benzekry, S., et al.: Classical mathematical models for description and prediction of experimental tumor growth. PLoS Comput. Biol. 10(8), e1003800 (2014). https://doi.org/10.1371/journal.pcbi.1003800
Benzekry, S., Tracz, A., Mastri, M., Corbelli, R., Barbolosi, D., Ebos, J.M.L.: Modeling spontaneous metastasis following surgery: an in vivo-in silico approach. Cancer Res. 76(3), 535–547 (2016)
Bertram, J.S., Janik, P.: Establishment of a cloned line of Lewis Lung Carcinoma cells adapted to cell culture. Cancer Lett. 11(1), 63–73 (1980)
Bilous, M., et al.: Quantitative mathematical modeling of clinical brain metastasis dynamics in non-small cell lung cancer. Sci. Rep. 9(1) (2019). https://doi.org/10.1038/s41598-019-49407-3
Brunton, G.F., Wheldon, T.E.: Characteristic species dependent growth patterns of mammalian neoplasms. Cell Tissue Kinet 11(2), 161–175 (1978)
Carpenter, B., et al.: Stan: a probabilistic programming language. J. Stat. Softw. 76(1) (2017). https://doi.org/10.18637/jss.v076.i01
Collins, V.P., Loeffler, R.K., Tivey, H.: Observations on growth rates of human tumors. Am. J. Roentgenol. Radium Ther. Nucl. Med. 76(5), 988–1000 (1956)
Demicheli, R.: Growth of testicular neoplasm lung metastases: tumor-specific relation between two Gompertzian parameters. Eur. J. Cancer 16(12), 1603–1608 (1980). https://doi.org/10.1016/0014-2964(80)90034-1
Frenzen, C.L., Murray, J.D.: A cell kinetics justification for Gompertz’ equation. SIAM J. Appl. Math. 46(4), 614–629 (1986)
Gelman, A.: Bayesian Data Analysis. Chapman & Hall/CRC Texts in Statistical Science, 3rd edn. CRC Press, Boca Raton (2014)
Kramer, A., Calderhead, B., Radde, N.: Hamiltonian Monte Carlo methods for efficient parameter estimation in steady state dynamical systems. BMC Bioinform. 15(1), 253 (2014). https://doi.org/10.1186/1471-2105-15-253
Laird, A.K.: Dynamics of tumor growth. Br. J. Cancer 13, 490–502 (1964)
Lavielle, M.: Mixed Effects Models for the Population Approach: Models, Tasks, Methods and Tools. Chapman & Hall/CRC Biostatistics Series. Taylor & Francis, Boca Raton (2014)
Norton, L.: A Gompertzian model of human breast cancer growth. Cancer Res. 48(24), 7067–7071 (1988)
Norton, L., Simon, R., Brereton, H.D., Bogden, A.E.: Predicting the course of Gompertzian growth. Nature 264(5586), 542–545 (1976). https://doi.org/10.1038/264542a0
Parfitt, A.M., Fyhrie, D.P.: Gompertzian growth curves in parathyroid tumours: further evidence for the set-point hypothesis. Cell Prolif. 30(8–9), 341–349 (1997)
Steel, G.G.: Growth Kinetics of Tumours: Cell Population Kinetics in Relation to the Growth and Treatment of Cancer. Clarendon Press, Oxford (1977)
Steel, G.G.: Species-dependent growth patterns for mammalian neoplasms. Cell Tissue Kinet 13(4), 451–453 (1980)
Vaghi, C., et al.: A reduced Gompertz model for predicting tumor age using a population approach. bioRxiv (2019). https://doi.org/10.1101/670869
Winsor, C.P.: The Gompertz curve as a growth curve. Proc. Natl. Acad. Sci. U.S.A. 18(1), 1–8 (1932)
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2019 Springer Nature Switzerland AG
About this paper
Cite this paper
Vaghi, C. et al. (2019). Population Modeling of Tumor Growth Curves, the Reduced Gompertz Model and Prediction of the Age of a Tumor. In: Bebis, G., Benos, T., Chen, K., Jahn, K., Lima, E. (eds) Mathematical and Computational Oncology. ISMCO 2019. Lecture Notes in Computer Science(), vol 11826. Springer, Cham. https://doi.org/10.1007/978-3-030-35210-3_7
Download citation
DOI: https://doi.org/10.1007/978-3-030-35210-3_7
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-030-35209-7
Online ISBN: 978-3-030-35210-3
eBook Packages: Computer ScienceComputer Science (R0)