Population modeling of tumor growth curves, the reduced Gompertz model and prediction of the age of a tumor
C. Vaghi (1,2), A. Rodallec (3), R. Fanciullino (3), J. Ciccolini (3), J. Mochel (4), M. Mastri (5), J. ML Ebos (5), C. Poignard (1), S. Benzekry (1,2)
(1) MONC team, Inria Bordeaux Sud-Ouest, France (2) Institut de Mathématiques de Bordeaux, France, (3) SMARTc, Center for Research on Cancer of Marseille, France, (4) Iowa State University, Department of Biomedical Sciences, Ames, USA, (5) Roswell Park Comprehensive Cancer Center, Buffalo, NY, USA
Introduction:
Tumor growth curves are classically modeled by ordinary differential equations. In analyzing the Gompertz model several studies have reported a striking correlation between the two parameters of the model [1–6]. Although this observation is still under debate, it might imply a constant maximal tumor size within a given species [3]. We analyzed tumor growth kinetics within the statistical framework of nonlinear mixed-effects (population approach). This allowed for the simultaneous modeling of tumor dynamics and inter- animal variability. Moreover, we computed the population parameters and used these as prior information to predict individual tumor initiation given only three measurements. This question is of fundamental importance in the clinic since the age of a tumor can be used as a proxy for determination of the invisible metastatic burden at diagnosis [14].
Objectives:
- test the descriptive power of different tumor growth models within a population
- study the correlation between the parameters of the Gompertz model within a population and define a novel, simplified model (the reduced Gompertz)
- use the estimated population parameters to perform individual predictions of tumor initiation
Methods:
The experimental data comprised two animal models of breast and lung cancers, with a total of 182 measurements in 86 animals. Nonlinear mixed effects modeling was used to compare different tumor growth equations, namely the Exponential, Logistic and Gompertz models [7]. Moreover, combining the correlation between the two parameters of the Gompertz model with rigorous population parameter estimation, we propose a novel reduced Gompertz model with only one individual parameter. We then considered the problem of predicting the initiation time of a tumor from only three late measurements, comparing the results arising from Bayesian inference and from likelihood maximization. We used the stochastic approximation of the EM algorithm (SAEM) for population analysis [8], implemented in the Monolix software (version 2018R2, Lixoft) and the Hamiltonian Monte Carlo algorithm [9,10] implemented in Stan [11] for Bayesian inference.
Results:
Population analysis: Confirming previous results [12], the Exponential and the Logistic models failed to describe the experimental data whereas the Gompertz model generated very good fits. The correlation between the Gompertz parameters was confirmed in our analysis. At the population level, the SAEM algorithm estimated a correlation of the random effects equal to 0.981. At the individual level, the two parameters were also highly correlated (R2 > 0.96 in all groups). This suggested a reduction of the number of degrees of freedom of the Gompertz model. The proposed reduced Gompertz model had one parameter with random effects and one parameter with fixed effects within the population. The latter suggested a characteristic constant of tumor growth within a given animal model [4]. We assessed the descriptive power of the reduced Gompertz model and found that performances were similar to the two-parameters Gompertz equation.
Prediction of the age of individual tumors: Thanks to its simplicity, the reduced Gompertz model showed superior predictive power. In addition, drastic improvements were observed when leveraging population priors using Bayesian inference compared to likelihood maximization, both in terms of accuracy (mean error 12.7% versus 88.5% for the breast data and 9.4% versus 67.6% for the lung data) and precision (mean value 15.6 days versus 242 days for the breast data, 7.34 days versus 84.8 days for the lung data). The Gompertz model exhibited a lack of parameter identifiability when likelihood maximization was applied. Using Bayesian inference, the accuracy was significantly better, but precision still inadequate. Overall, the combination of the reduced Gompertz model with Bayesian inference clearly outperformed the other methods for prediction of the age of experimental tumors.
Conclusions:
The method that we proposed here remains to be extended to clinical data, although it will not be possible to have a firm confirmation since the natural history of neoplasms since their inception cannot be observed. Nevertheless, the encouraging results obtained here could allow to give approximate estimates. Personalized estimations of the age of a given patient’s tumor would yield important epidemiological insights and could also be informative in routine clinical practice [13].
References:
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