What is the Best Mathematical Model to Describe Tumor Growth Curves?

Mathematical Modeling of Tumor Growth Curves and Its Application in Biomarker Discovery for Anti-Tumor Drugs

Breaking down the data from our recent research

In last month’s issue of Cancer Research Communications, scientist Drs. Huajun Zhou, Binchen Mao, and Sheng Guo from our Data Science & Bioinformatics Division published a research paper titled “Mathematical Modeling of Tumor Growth in Preclinical Mouse Models with Applications in Biomarker Discovery and Drug Mechanism Studies”.¹

The paper explores the vast amount of tumor growth data in Crown Bioscience's database and applies various mathematical models to fit the data. It was found that the exponential quadratic model provides the best fit for tumor growth curves.

However, when applied to the screening of drug efficacy biomarkers, there was no significant difference between the results of the exponential quadratic model and the commonly used exponential model, indicating that the exponential model is adequate for identifying drug efficacy biomarkers.

By comparing biomarkers identified using the exponential model and non-parametric models with data from mouse studies of some standard-of-care drugs, the study revealed that most of the biomarkers identified were consistent and aligned with the drug mechanisms of action, though there were cases where one model performed better.

This research provides theoretical support and guidance for the analytical processes involved in discovering biomarkers for anti-tumor drug efficacy, thus accelerating the development of personalized anti-tumor drugs in the field.

Tumor inhibition rates: Tumor volume versus growth curves

Before a new anti-tumor drug enters clinical trials, its efficacy is typically validated through tests in tumor cell lines, mouse tumor models, and tumor organoids.

In mouse tumor models, tumor growth inhibition is usually assessed by measuring the tumor volume over a period of 2-4 weeks. Researchers often focus on the tumor volume ratio at the end of the observation period between the treated and control groups (tumor growth inhibition), neglecting the earlier growth curve. This tumor growth curve may contain significant yet overlooked information about tumor growth dynamics.

In our 2019 paper², we revealed that the tumor inhibition rate is not a reliable indicator for screening biomarkers, whereas linear mixed model (LMM) based on exponential growth assumptions and exponential growth rate (eGR) can more accurately identify biomarkers.

However, under the exponential growth assumption, the instantaneous growth rate of the tumor increases indefinitely, which may not hold true in the later stages of tumor growth. Whether a better mathematical model exists to describe tumor growth curves and more efficiently identify biomarkers has not been studied.

Comparing historical mathematical models

Our research utilized tumor growth data from 30,000 mice across various models, including CDX (human-derived tumor cell line mouse xenografts), PDX (human-derived primary tumor mouse xenografts), and syngeneic (mouse-derived tumor cell line mouse xenografts), covering over ten organ sources.

We compared several historical mathematical models:

• the Gompertz model
• logistic model
• monomolecular model
• von Bertalanffy model
• exponential model
• exponential quadratic model (with a time quadratic term added)

We found that the exponential quadratic model provided the best description of tumor growth curves among the six models and could fit up to 87% of the experimental data.

Through tests involving five conventional anti-tumor drugs, we compared biomarkers identified using the exponential model and the exponential quadratic model within a mixed model framework.

Although the exponential quadratic model provided a better fit for the curves, there was no significant difference in the biomarkers identified by the two models.

We also compared methods using mixed models versus those based on the area under the curve-derived exponential growth rate (eGR) for biomarker discovery.

In most cases, the biomarkers identified were consistent across the methods, though there were instances where one method performed better for a specific drug. This suggests that employing multiple methods for biomarker discovery can be advantageous.

Conclusion

Finally, we developed an R package named "TuGroMix," which provides convenience for mouse tumor growth data processing, statistical analysis, and biomarker discovery for researchers in the field.

This research offers methodological support for discovering drug efficacy biomarkers for anti-tumor drugs, potentially aiding pharmaceutical companies in developing more personalized anti-tumor drugs in the future.

Reference

1. Zhou, Huajun, Binchen Mao, and Sheng Guo. "Mathematical Modeling of Tumor Growth in Preclinical Mouse Models with Applications in Biomarker Discovery and Drug Mechanism Studies." Cancer Research Communications 4.8 (2024): 2267-2281.
2. Guo, Sheng, et al. "The design, analysis and application of mouse clinical trials in oncology drug development." BMC cancer 19 (2019): 1-14.