A reduced Gompertz model for predicting tumor age using a population approach

Abstract

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.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 interanimal variability. Experimental data comprised three animal models of breast and lung cancers, with 843 measurements in 94 animals. Candidate models of tumor growth included the Exponential, Logistic and Gompertz. The Exponential and -more notably -Logistic models failed to describe the experimental data whereas the Gompertz model generated very good fits. The population-level correlation between the Gompertz parameters was further confirmed in our analysis (R 2 > 0.96 in all groups). Combining this structural correlation with rigorous population parameter estimation, we propose a novel reduced Gompertz function consisting of a single individual parameter. Leveraging the population approach using bayesian inference, we estimated the time of tumor initiation using three late measurement timepoints. The reduced Gompertz model was found to exhibit the best results, with drastic improvements when using bayesian inference as compared to likelihood maximization alone, for both accuracy and precision. Specifically, mean accuracy was 12.1% versus 74.1% and mean precision was 15.2 days versus 186 days, for the breast cancer cell line.These results offer promising clinical perspectives for the personalized prediction of tumor age from limited data at diagnosis. In turn, such predictions could be helpful for assessing the extent of invisible metastasis at the time of diagnosis. Author summaryMathematical models for tumor growth kinetics have been widely used since several decades but mostly fitted to individual or average growth curves. Here we compared three classical models (Exponential, Logistic and Gompertz) using a population approach, which accounts for inter-animal variability. The Exponential and the Logistic models failed to fit the experimental data while the Gompertz model showed excellent descriptive power. Moreover, the strong correlation between the two parameters of the Gompertz equation motivated a simplification of the model, the reduced Gompertz model, with a single individual parameter and equal descriptive power. Combining the mixed-effects approach with Bayesian inference, we predicted the age of individual tumors with only few late measurements. Thanks to its simplicity, the reduced Gompertz model showed superior predictive power. Although our method remains to be extended to clinical data, these results are promising for the personalized estimation of the age of a tumor from limited measurements at diagnosis. Such predictions could contribute to the development of computational models for metastasis.

Descriptive and prognostic value of a computational model of metastasis in high-risk neuroblastoma

“…Our findings suggested growth to be exponential and fast, with primary tumors reaching 10 cm tumors in less than 3 months. This contrasts with previous studies where tumor growth kinetics were shown to slow down and deviate from an exponential pattern, following rather a gompertzian pattern (25,54). This is probably explained by the embryonic nature of this cancer, which makes it particularly aggressive.…”

Population Modeling of Tumor Growth Curves, the Reduced Gompertz Model and Prediction of the Age of a Tumor

“…In [20] other two data sets (two animal models of breast cancer, measured by volume and fluorescence) are considered for the analysis with equivalent results.…”

Uncertainty-based Gompertz growth model for tumor population and its numerical analysis

“…It has already been established that dealing with parameter inaccuracy is not always suitable due to a lack of comprehensive knowledge or estimation failure. A basic technique of coping with Gompertz equation uncertainties ( 6) is utilized to obtain these parameter estimations by utilizing the equation (7) to calculate the average approximations and to assess the complexity [42,43,[45][46][47].…”