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Classical Mathematical Models for Description and Prediction of Experimental Tumor Growth
Sébastien Benzekry, Clare Lamont, Afshin Beheshti, Amanda Tracz, John M. L. Ebos, Lynn Hlatky, Philip Hahnfeldt
TL;DR
The paper addresses limited comparative evidence on how broadly used mathematical models describe and predict in vivo tumor growth. It evaluates nine models across lung and breast tumor systems using measurement-error modeling, fit and identifiability analyses, and forecasting tests. Gompertz and power-law models were preferred for parsimonious lung description, while breast prediction favored the exponential-linear model; a priori parameter information substantially improved lung forecasts.
Problem
Comprehensive comparisons of mathematical tumor-growth models’ descriptive power are limited, and predictive power is rarely evaluated despite its relevance to clinical and preclinical applications.
Method
The study compares nine growth models across syngeneic lung and orthotopic breast tumor datasets, incorporating a measurement-error model, goodness-of-fit and identifiability analyses, and truncated-curve forecasting.
Results
Gompertz and power-law models gave the most parsimonious and identifiable lung descriptions, while breast prediction favored the exponential-linear model and a priori parameter information improved lung forecasts.
Takeaways & Limitations
Model choice should depend on tumor dataset and task, and parameter-distribution information can materially improve future-growth prediction.
Takeaways & Limitations
Prediction improvements from a priori information partly reflected the homogeneity of the growth data, particularly the LLC data.
Abstract
from arXiv · showhide
Despite internal complexity, tumor growth kinetics follow relatively simple macroscopic laws that have been quantified by mathematical models. To resolve this further, quantitative and discriminant analyses were performed for the purpose of comparing alternative models for their abilities to describe and predict tumor growth. For this we used two in vivo experimental systems, an ectopic syngeneic tumor (Lewis lung carcinoma) and an orthotopically xenografted human breast carcinoma. The goals were threefold: to 1) determine a statistical model for description of the volume measurement error, 2) establish the descriptive power of each model, using several goodness-of-fit metrics and a study of parametric identifiability, and 3) assess the models ability to forecast future tumor growth. Nine models were compared that included the exponential, power law, Gompertz and (generalized) logistic formalisms. The Gompertz and power law provided the most parsimonious and parametrically identifiable description of the lung data, whereas the breast data were best captured by the Gompertz and exponential-linear models. The latter also exhibited the highest predictive power for the breast tumor growth curves, with excellent prediction scores (greater than 80$\%$) extending out as far as 12 days. In contrast, for the lung data, none of the models were able to achieve substantial prediction rates (greater than 70$\%$) further than the next day data point. In this context, adjunction of a priori information on the parameter distribution led to considerable improvement of predictions. 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 clinical setting.
Introduction
The study quantitatively compares mathematical tumor-growth models for describing and forecasting two in vivo tumor systems, while accounting for measurement error and parameter identifiability. It finds that model performance depends on the dataset and evaluation goal, with predictive performance generally more limited than descriptive fit.
- Study design: The study compares nine exponential, power-law, Gompertz, and logistic-family models using two in vivo tumor-growth datasets.The datasets comprise syngeneic Lewis lung carcinoma and orthotopic human breast carcinoma xenografts.
- Study design: The analysis evaluates measurement error, goodness of fit, parameter parsimony, practical identifiability, and future-growth prediction.Models were assessed both for descriptive adequacy and for forecasting from truncated growth curves.
- Descriptive power: The generalized logistic model achieved the best pure least-squares fits for both datasets, reflecting its structural flexibility.Its flexibility also introduced more parameters than the preferred parsimonious models.
- Descriptive power: For lung tumors, Gompertz and power-law models provided similarly descriptive but more parsimonious and identifiable fits than more flexible alternatives.The generalized logistic, dynamic carrying-capacity, and von Bertalanffy models showed poorer parameter identifiability.
Text S1: Numerical procedures for parameters estimation
The procedures combine bounded numerical fitting, initialization sensitivity checks, and practical identifiability analysis to evaluate model estimation reliability.
- Numerical procedures: Analytical model formulas were used whenever available, while the dynamic CC model was solved numerically with Matlab's ode45.
- Numerical procedures: lsqcurvefit with a trust-region algorithm was used for most fits, while generalized logistic fitting used fminsearch with the Nelder–Mead algorithm.Convergence was systematically checked and was effective for all parameter estimations performed.
- Identifiability analysis: Different initializations were systematically tested to assess practical identifiability and sensitivity to starting values.The study used the LLC data set for this sensitivity analysis and explored a compact parameter-space region around population-fit means.
- Model comparison: The generalized logistic model fitted both data sets accurately but showed low identifiability, whereas exponential 1 and logistic models fit poorly across most metrics.
B. Examples of sharp saturation of the generalized logistic model
Figure S3 illustrates breast-tumor predictions from the exponential-linear model using five data points, evaluated at the second-next-day point and across the future curve.
- The figure compares prediction success at the second-next-day point (OK2) with success over the global future curve (OKglob).Both criteria use normalized error smaller than 3.
- Predictions were generated from five data points using the exponential-linear model.