Chemotherapy in conjoint aging-tumor systems: some simple models for addressing coupled aging-cancer dynamics

Abstract

Background: In this paper we consider two approaches to examining the complex dynamics of conjoint aging-cancer cellular systems undergoing chemotherapeutic intervention. In particular, we focus on the effect of cells growing conjointly in a culture plate as a precursor to considering the larger multi-dimensional models of such systems. Tumor cell growth is considered from both the logistic and the Gompertzian case, while normal cell growth of fibroblasts (WI-38 human diploid fibroblasts) is considered as logistic only.

Results: We demonstrate, in a simple approach, how the interdependency of different cell types in a tumor, together with specifications of for treatment, can lead to different evolutionary patterns for normal and tumor cells during a course of therapy.

Conclusions: These results have significance for understanding appropriate pharmacotherapy for elderly patients who are also undergoing chemotherapy.

Figure 1

Blue curve: Evolution of normal cells. Purple curve: Evolution of tumor cells. Common parameters: r_N = 0.4, r_T = 0.3, K_T = 1.2.10⁶, K_N = 10⁶. Left: There is no interaction between normal cells and tumor cells (both populations undergo logistic growth), k = 0, β = 0. Right: Normal and tumor cells are allowed to interact with each other, k = 1, β = 2, ρ₀ = 1, ρ₁ = 1000, T* = 3.10⁵, N₀ = 1, T₀ = 1. The mini-window magnifies the behavior of normal and tumor cells close to the critical size of the tumor. As the size of the tumor cells T exceed the critical size, T* (dashed line), the size of normal cells N starts decreasing.

Figure 2

The evolution of normal cells and tumor cells during the phase of therapy. The drug is considered to be static. First row: the drug does not have any effects on normal cells, a_N(1 - e^mu) = 0, and a_T(1 - e^mu) = 0.01 (red), 0.05 (green), 0.1 (black). Second row: the drug kills both normal and tumor cells with more killing strength on the tumor cells. Blue represents the untreated system when a_N(1 - e^mu) = 0 = a_T(1 - e^mu) = 0. From there, the response of the tumor cells is considered to be constant, a_T(1 - e^mu) = 0.1, while a variation is considered for the response of normal cells as: a_N(1 - e^mu) = 0.01 (red), 0.05, (green), 0.1 (black). Third row: the drug kills both normal and tumor cells with more killing strength on normal cells. blue is untreated system when a_N(1 - e^mu) = 0 = a_T(1 - e^mu) = 0, From there, the response of the normal cells is considered to be constant, a_N(1 - e^mu) = 0.1, while a variation is considered for the response of the normal cells as: a_T(1 - e^mu) = 0.01 (red), 0.05, (green), 0.1 (black). The rest of the parameters are similar to the common parameter introduced in Figure [1].

Figure 3

The evolution of normal cells and tumor cells during the phases of therapy. The drug is considered to be dynamic and its concentration diffuses exponentially over time. u₀ is the initial value of the drug and d is the decaying rate, and m is linked to pharmacokinetics and considered to be 1 in this study. The drug does not have any effect on normal cells, a_N(1 - e^mu) = 0, a_T = 0.1. First row: The evolution of normal and tumor cells is simulated for different drug decaying rates. u₀ = 1, and untreated (blue), d = 0.1 (red), 0.5 (green), 1 (black), and 2 (brown). As can be seen, the system tends to behave as untreated as the decaying rate increases. Second row: Same parameters in the first row except the initial value of the drug is increased, u₀ = 3, which maintains the diffusion behavior of the drug leading to slower growth for the tumor cells and a delay in entering the inhibition phase for the normal cells. The rest of the parameters are similar to the common parameter introduced in Figure [1].

Figure 4

Model 2. Top-Blue: Evolution of normal cells, Purple: Evolution of tumor cells. Common parameters: r_N = 0.4, h = 10⁵, γ = 0.083, K_N = 10⁶. Top left: there is no interaction between normal cells and tumor cells k = 0, β = 0. Tumor cells show a higher increasing rate at the beginning compared to the normal cells. Top right: normal and tumor cells are allowed to interact, k = 1, β = 2, ρ₀ = 1, ρ₁ = 1000, T* = 3.10⁵, N₀ = 1, T₀ = 1. Fast growing tumor cells force the normal passes the critical size of tumor cells quickly and force normal cells enter to the inhibition phase in a short time. Down: The evolution of normal cells and tumor cells during the phase of therapy. The drug is considered to be static. Down Left: the drug does not have any effects on normal cells, a_N(1 - e^mu) = 0, and a_T(1 - e^mu) = 0.0 (Blue-untreated system), 0.1 (red), and 0.17 (green). Effect of the drug in slower growth of tumor cells and a delay in entering the inhibition phase for the normal cells can be detected.