Classical Mathematical Models for Description and Prediction of Experimental Tumor Growth

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.