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We propose in this article a model describing the dynamic of a system of adipocytes, structured by their sizes. This model takes into account the differentiation of a population of mesenchymal cells into preadipocytes and of preadipocytes into adipocytes; the differentiation rates depend on the mean adipocyte radius. The considered equations are therefore ordinary differential equations, coupled with an advection equation, the growth rate of which depends on food availability and on the total surface of adipocytes. Since this velocity is discontinuous, we need to introduce a convenient notion of solutions coming from Filippov theory. We are consequently able to determine the stationary solutions of the system, to prove the existence and uniqueness of solutions and to describe the asymptotic behavior of solutions in some simple cases. Finally, the parameters of the model are fitted thanks to some experimental data and numerical simulations are displayed; a spatial extension of the model is studied numerically.

Purpose: Although recent regulations improved conditions of laboratory animals, their use remains essential in cancer research to determine treatment efficacy. In most cases, such experiments are performed on xenografted animals for which tumor volume is mostly estimated from caliper measurements. However, many formulas have been employed for this estimation and no standardization is available yet. Methods: Using previous animal studies, we compared all formulas used by the scientific community in 2019. Data were collected from 93 mice orthotopically xenografted with human breast cancer cells. All formulas were evaluated and ranked based on correlation and lower mean relative error. They were then used in a Gompertz quantitative model of tumor growth. Results: Seven formulas for tumor volume estimation were identified and a statistically significant difference was observed among them (ANOVA test, p < 2.10−16), with the ellipsoid formula (1/6 π × L × W × (L + W)/2) being the most accurate (mean relative error = 0.272 ± 0.201). This was confirmed by the mathematical modeling analysis where this formula resulted in the smallest estimated residual variability. Interestingly, such result was no longer valid for tumors over 1968 ± 425 mg, for which a cubic formula (L x W x H) should be preferred. Main findings: When considering that tumor volume remains under 1500mm3, to limit animal stress, improve tumor growth monitoring and go toward mathematic models, the following formula 1/6 π × L × W x (L + W)/2 should be preferred. [ABSTRACT FROM AUTHOR]

Tumor growth curves are classically modeled by means of ordinary differential equations. In analyzing the Gompertz model several studies have reported a striking correlation between the two parameters of the model, which could be used to reduce the dimensionality and improve predictive power. We analyzed tumor growth kinetics within the statistical framework of nonlinear mixed-effects (population approach). This allowed the simultaneous modeling of tumor dynamics and inter-animal variability. Experimental data comprised three animal models of breast and lung cancers, with 833 measurements in 94 animals. Candidate models of tumor growth included the exponential, logistic and Gompertz models. The exponential and—more notably—logistic models failed to describe the experimental data whereas the Gompertz model generated very good fits. The previously reported population-level correlation between the Gompertz parameters was further confirmed in our analysis (R2> 0.92 in all groups). Combining this structural correlation with rigorous population parameter estimation, we propose a reduced Gompertz function consisting of a single individual parameter (and one population parameter). Leveraging the population approach using Bayesian inference, we estimated times 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 (prediction error) was 12.2% versus 78% and mean precision (width of the 95% prediction interval) was 15.6 days versus 210 days, for the breast cancer cell line. These results demonstrate the superior predictive power of the reduced Gompertz model, especially when combined with Bayesian estimation. They offer possible clinical perspectives for personalized prediction of the age of a tumor from limited data at diagnosis. The code and data used in our analysis are publicly available at https://github.com/cristinavaghi/plumky. Author summary: Mathematical 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. [ABSTRACT FROM AUTHOR]