Finding the number of cells that proliferate and die is dependent on the time difference considered. Assuming the cell cycle length of a cancer cell is approximately 24 hours (1 day), the probability that the cell divides within this time frame is almost 100% [1], [2]. This assumption is based on the fact that most cancer cells exhibit rapid proliferation, a characteristic that simplifies tumor growth modeling while preserving its rigor.

The number of cancer cells in a given time is related to difference between the birth and death rate of cancer cells. From equation (1) below, alpha represents the fraction of proliferation, beta represents the fraction of deaths, 'c' represents the cancer cell population and lambda represents the net population growth rate. (i.e alpha - beta) . A positive lambda value results in a graph that depicts increasing population growth, a negative lambda value results in a graph that depicts decreasing population growth and a lambda value of zero indicates a stable population.

Explanation of Model and Equations

Figure 1: Graph of equation 1 for different values of lambda (from the paper, page 6)

Dynamic Tumor Growth Rates

Tumor cells initially grow fast, but as the tumor gets bigger, its growth slows down. This change in growth behavior is due to resource limitations like space, oxygen, and nutrients. To capture this, dynamic growth rates, were used, where the rate of tumor growth depends on the current size of the tumor.[3]

Equation (2) depicts the logistic growth model. As 'c' is less than 'K', the tumor grows almost exponentially and as 'c' approaches 'K', the growth slows down and eventually stops. (K is the carrying capacity)

Figure 2a and 2b: Graph of Logistic Growth Model (from the paper, page 7)

Equation (3) below shows the Gompertz model equation. It fits experimental tumor growth data better than the logistic model.

Growth slows earlier and more sharply as the tumor approaches the carrying capacity, 'K'

Figure 3a and 3b: Graph of Gompertz model (from the paper, page 8)

Click the tumor treatment tab to access the explanation of the tumor treatment model in the paper

Tumor Treatment

References

[1] Dana-Farber Cancer Institute, "What is the cell cycle and how is it related to cancer?" Dana-Farber Insight Blog, 2023. [Online]. Available: https://blog.dana-farber.org/insight/2023/05/what-is-the-cell-cycle-and-how-is-it-related-to-cancer/. Accessed: Mar. 21, 2025.

[2] BMC Cancer, "Cell cycle duration in tumor cells: Evidence from SL2 tumors," BMC Cancer, vol. 5, no. 122, 2005. [Online]. Available: https://bmccancer.biomedcentral.com/articles/10.1186/1471-2407-5-122. Accessed: Mar. 21, 2025.