Individualized growth prediction of mice skin tumors with maximum likelihood estimators

A calibration and uncertainty quantification analysis of classical, fractional and multiscale logistic models of tumour growth

BACKGROUND AND OBJECTIVE

The validation of mathematical models of tumour growth is frequently hampered by the lack of sufficient experimental data, resulting in qualitative rather than quantitative studies. Recent approaches to this problem have attempted to extract information about tumour growth by integrating multiscale experimental measurements, such as longitudinal cell counts and gene expression data. In the present study, we investigated the performance of several mathematical models of tumour growth, including classical logistic, fractional and novel multiscale models, in terms of quantifying in-vitro tumour growth in the presence and absence of therapy. We further examined the effect of genes associated with changes in chemosensitivity in cell death rates.

METHODS

The multiscale expansion of logistic growth models was performed by coupling gene expression profiles to the cell death rates. State-of-the-art Bayesian inference, likelihood maximisation and uncertainty quantification techniques allowed a thorough evaluation of model performance.

RESULTS

The results suggest that the classical single-cell population model (SCPM) was the best fit for the untreated and low-dose treatment conditions, while the multiscale model with a cell death rate symmetric with the expression profile of OCT4 (Sym-SCPM) yielded the best fit for the high-dose treatment data. Further identifiability analysis showed that the multiscale model was both structurally and practically identifiable under the condition of known OCT4 expression profiles.

CONCLUSIONS

Overall, the present study demonstrates that model performance can be improved by incorporating multiscale measurements of tumour growth when high-dose treatment is involved.

Gompertz tracking of the growth trajectories of the human-liver-cancer xenograft-tumors in nude mice

BACKGROUND AND OBJECTIVE

The accurate tracking of the growth trajectories of the human-liver-cancer xenograft-tumors in nude mice is an important prerequisite for the effective use of relevant trial results. Our objective is first to find out whether the Gompertz model can accurately track the growth trajectories of the xenograft-tumors in the non-target control group, Gsk3β knockdown, and c-Myc knockdown groups; secondly, to verify the effect of knocking down Gsk3β or c-Myc on the growth of xenograft-tumors and reveal the mechanism; finally, to demonstrate the 100-day Gompertz growth trajectory, which is a complete growth process with two phases and three stages.

METHODS

The 18 male specific-pathogen-free (SPF) BALB/c nude mice were randomly divided into three groups and different interventions were performed to establish the non-target control, Gsk3β knockdown, and c-Myc knockdown groups. The volumes of the xenograft-tumors were measured from day 14 to day 30 after transplantation. The first 30-days and the whole 100-days of Gompertz growth trajectories of the xenograft-tumors were obtained respectively, and the growth assessment indicators of each group were calculated based on the parameters of the Gompertz model.

RESULTS

1) The Gompertz model can accurately track the growth trajectories of xenograft-tumors in the non-target control, Gsk3β knockdown, and c-Myc knockdown groups; 2) knocking down Gsk3β or c-Myc can inhibit the growth of xenograft-tumors. It is the combined effect of growth-promoting factor, growth inhibitory factor, and the delay of angiogenesis, of which the delay of angiogenesis plays a decisive role; 3) the 100-day Gompertz growth trajectory can provide complete information about the two phases and three stages of xenograft-tumor growth. Therefore, it is strongly recommended that nude mouse trials be extended from 30 days (currently widely accepted) to 100 days.

CONCLUSIONS

The Gompertz model can well reveal the growth pattern of the human-liver-cancer xenograft-tumors in nude mice. Combined with the growth assessment indicators obtained from the Gompertz model parameters, one can further clarify the mechanism that affects the growth of xenograft-tumors. The Gompertz tracking of the growth trajectories of the human-liver-cancer xenograft-tumors in nude mice has broad application prospects in the fields of basic research, drug verification, and clinical treatment, etc.

The initial engraftment of tumor cells is critical for the future growth pattern: a mathematical study based on simulations and animal experiments

Background

Xenograft mouse tumor models are used to study mechanisms of tumor growth and metastasis formation and to investigate the efficacy of different therapeutic interventions. After injection the engrafted cells form a local tumor nodule. Following an initial lag period of several days, the size of the tumor is measured periodically throughout the experiment using calipers. This method of determining tumor size is error prone because the measurement is two-dimensional (calipers do not measure tumor depth). Primary tumor growth can be described mathematically by suitable growth functions, the choice of which is not always obvious. Growth parameters provide information on tumor growth and are determined by applying nonlinear curve fitting.

Methods

We used self-generated synthetic data including random measurement errors to research the accuracy of parameter estimation based on caliper measured tumor data. Fit metrics were investigated to identify the most appropriate growth function for a given synthetic dataset. We studied the effects of measuring tumor size at different frequencies on the accuracy and precision of the estimated parameters. For curve fitting with fixed initial tumor volume, we varied this fixed initial volume during the fitting process to investigate the effect on the resulting estimated parameters. We determined the number of surviving engrafted tumor cells after injection using ex vivo bioluminescence imaging, to demonstrate the effect on experiments of incorrect assumptions about the initial tumor volume.

Results

To select a suitable growth function, measurement data from at least 15 animals should be considered. Tumor volume should be measured at least every three days to estimate accurate growth parameters. Daily measurement of the tumor volume is the most accurate way to improve long-term predictability of tumor growth. The initial tumor volume needs to have a fixed value in order to achieve meaningful results. An incorrect value for the initial tumor volume leads to large deviations in the resulting growth parameters.

Conclusions

The actual number of cancer cells engrafting directly after subcutaneous injection is critical for future tumor growth and distinctly influences the parameters for tumor growth determined by curve fitting.