Mechanistic mathematical models (MMs) are crucial for understanding and predicting tumour growth. This study investigates how parameter estimation from experimental data affects the accuracy and interpretation of MM predictions. The research focuses on ordinary differential equation-based MMs, acknowledging the challenges of the inverse problem in parameter estimation, which is highly sensitive to noise. MMs can predict average, specific, or population-level responses depending on the parameter estimation approach. Model identifiability, specifically practical identifiability (unique parameter determination even with noisy data), is a critical concern. A model may lack practical identifiability due to insufficient data, model misspecification, or parameter dependencies. The paper emphasizes that widely distributed parameters might reflect real clinical diversity rather than noise. This research investigates the impact of using different types of MMs of increasing complexity on the parameter identifiability and model prediction accuracy. The influence of censored data (measurements below or above the detection limits) on the parameter estimation is carefully investigated. The use of Bayesian inference, a common approach in parameter estimation for noisy data, is studied. A key focus is on the effect of prior distribution choice on the results. The paper aims to promote a careful consideration of these factors in parameter estimation and to suggest improved methodologies.

The paper reviews existing literature on mechanistic mathematical models (MMs) for tumour growth, highlighting the importance of model identifiability and the challenges of parameter estimation from noisy data. It discusses different approaches to MM prediction, including average, specific, and population-level responses. The literature on Bayesian inference in MM parameter estimation and the handling of censored data are also reviewed. The authors mention existing work demonstrating the log-normal distribution of tumour volume measurements and its implications for parameter estimation. Previous studies on the use of MMs in cancer modeling, including specific model types like the logistic, Gompertz, and generalized logistic models, are referenced.

The study uses five MMs of increasing complexity: Exponential (Exp), Exponential capped (ExpCap), Logistic (Logis), Gompertz (Gomp), and a modified generalized logistic (Rich) model. These models are fit to experimental tumour volume data from a study on Lewis Lung Carcinoma in mice. The data includes censored values representing measurements below the lower limit of detection (LLD) and above the upper limit of detection (ULD). A Markov Chain Monte Carlo (MCMC) method within the phymeme package, using emcee's affine-invariant sampler, is used for Bayesian parameter estimation. The likelihood function is based on the sum of squared residuals of the log10-transformed tumour volumes, reflecting the log-normal distribution of the data. Two different likelihood functions are compared: one that neglects censored data and one that incorporates it using a method adapted from existing literature. The paper examines the effect of two different prior distributions (linear uniform and log-uniform) on the parameter estimations. Profile log-likelihood curves are generated to assess the practical identifiability of each parameter. Parameter sensitivity analysis is performed using central difference approximations of log10 C with respect to log10 parameters. The analysis compares the resulting parameter estimates, considering the maximum likelihood estimate (MLE), maximum a posteriori (MAP) estimate, and 95% credible intervals. The statistical significance of differences is evaluated based on the overlap of 95% credible intervals.

The study found that neglecting censored data leads to overestimation of initial tumour volume and underestimation of carrying capacity. Incorporating censored data produced more realistic estimates, particularly for initial tumour volume, bringing it closer to the expected value based on the number of implanted cells. The choice of prior distribution significantly affected parameter estimation, especially in more complex models with insufficient data. Log-uniform priors appeared more appropriate based on the data’s log-normal nature. The Rich model, being the most complex, exhibited the widest parameter posterior distributions, indicating lower identifiability. The parameter sensitivity analysis showed that different parameters are most influenced by different time points in the data (early vs. late). The analysis demonstrated that reporting only the mode (MAP) and 95% credible intervals of the posterior distributions can be misleading, especially when the distributions are multimodal. The study showed how the different choices of the coefficient used in the generalized logistic model could alter the practical interpretation of the model parameters. Specifically, it was found that the choice of coefficient μ/min(α,1) used in the Rich MM provided the most consistent physical interpretation of the parameters μ and α.

The findings highlight the importance of including censored data in parameter estimation for MMs of tumour growth. Ignoring censored data introduces bias and affects the reliability of model predictions, especially beyond the range of measured data. The study emphasizes the potential for misleading interpretations when only point estimates (mode and credible intervals) are reported instead of full posterior distributions. The impact of prior selection underscores the need for careful consideration of prior knowledge and the data's properties when applying Bayesian methods. The study's results suggest that more complex models might not always be better; the added parameters can decrease identifiability if the data is not sufficiently rich. The choice of MMs could introduce biases on the estimation of the parameters and the model prediction of the tumor growth. The study suggests focusing on model selection approaches such as Akaike or Bayesian information criteria in future work, but notes that the way the parameters are defined and bounded in this study compromises this approach. The selection of priors impacts the final result depending on the amount of data available. This study shows that log-uniform priors provide a suitable choice given the experimental data.

This study demonstrates the critical impact of including censored data and carefully selecting priors in Bayesian parameter estimation for MMs of tumour growth. The methodology proposed provides a more robust framework for parameter estimation and model interpretation. Future work should focus on exploring more sophisticated methods for handling censored data and validating the appropriateness of log-uniform priors in various datasets. A more rigorous model selection approach should also be considered in future work. A clearer visualization of complete posterior distributions is recommended to avoid misleading interpretations.

The study used a single dataset, limiting the generalizability of the findings. The choice of MMs and their parameterizations might not encompass all potential models of tumour growth. The artificial bounds imposed on some parameters could have affected the results and model selection approaches. The use of a log-normal distribution assumption for tumour volume might not be universally applicable across different tumour types or experimental setups.