Table 1.
Models ranked in ascending order of AIC (Akaike information criterion).
Other statistical indices are the log-likelihood estimate (-2LL) and the Bayesian information criterion (BIC). *The reduced Gompertz model is introduced below.
Fig 1.
Population analysis of experimental tumor growth kinetics.
(A) Visual predictive checks assess goodness-of-fit for both structural dynamics and inter-animal variability by reporting model-predicted percentiles (together with confidence prediction intervals (P.I) in comparison to empirical ones. They were obtained by multiple simulations of each model. The time axis was then split into bins and in each interval the empirical percentiles of the observed data were compared with the respective predicted medians and intervals of the simulated data [47]. (B) Prediction distributions. They were obtained by multiple simulations of all individuals in the dataset, excluding the residual error [47]. (C) Individual weighted residuals (IWRES) with respect to time. (D) Observations vs predictions Left: exponential, Center: logistic, Right: Gompertz models.
Table 2.
Fixed effects (typical values) of the parameters of the different models.
Par. = parameter. ω = standard deviation of the random effects. R.S.E. = relative standard errors of the estimates. σ = residual error model parameters. *The reduced Gompertz model is introduced below.
Fig 2.
Individual fits from population analysis.
Three representative examples of individual fits (animal (A), animal (B) and animal (C)) computed with the population approach relative to the exponential (left), the logistic (center) and the Gompertz (right) models.
Fig 3.
Correlation of the Gompertz parameters and diagnostic plots of the reduced Gompertz model from population analysis.
Correlation between the individual parameters of the Gompertz model (A) and results of the population analysis of the reduced Gompertz model: visual predictive check (B), scatter plots of the residuals (C), prediction distribution (D) and examples of individual fits (E).
Fig 4.
Backward predictions computed with likelihood maximization and with Bayesian inference.
Examples of backward predictions of three individuals (A), (B) and (C) computed with likelihood maximization (LM) and Bayesian inference: Gompertz model with likelihood maximization (first row); reduced Gompertz with likelihood maximization (second row); Gompertz with Bayesian inference (third row) and reduced Gompertz with Bayesian inference (fourth row). Only the last three points are considered to estimate the parameters. The grey area is the 95% prediction interval (P.I) and the dotted blue line is the median of the posterior predictive distribution. The red line is the predicted initiation time and the black vertical line the actual initiation time.
Table 3.
Accuracy and precision of methods for prediction of the age of experimental tumors of the three cell lines.
Accuracy was defined as the absolute value of the relative error (in percent). Precision was defined as the width of the 95% prediction interval (PI column, in days). Reported are the means and standard errors (in parenthesis). LM = likelihood maximization.
Fig 5.
Accuracy of the prediction models.
Swarmplots of relative errors obtained under likelihood maximization (A) or Bayesian inference (B) (* p-value < 0.05, ** p-value < 0.01, Levene’s test). (C) Absolute errors: comparison between the different distributions (* p-value < 0.05, ** p-value < 0.01, Wilcoxon test). In (A) three extreme outliers were omitted (values of the relative error were greater than 20) for both the Gompertz and the reduced Gompertz in order to ensure readability. LM = Likelihood Maximization.