Modeling the effects of a simple immune system and immunodeficiency on the dynamics of conjointly growing tumor and normal cells

Abstract

In this paper, we develop a theoretical contribution towards the understanding of the complex behavior of conjoint tumor-normal cell growth under the influence of immuno-chemotherapeutic agents under simple immune system response. In particular, we consider a core model for the interaction of tumor cells with the surrounding normal cells. We then add the effects of a simple immune system, and both immune-suppression factors and immuno-chemotherapeutic agents as well. Through a series of numerical simulations, we illustrate that the interdependency of tumor-normal cells, together with choice of drug and the nature of the immunodeficiency, leads to a variety of interesting patterns in the evolution of both the tumor and the normal cell populations.

Keywords: Aging; Immune system; Immunodeficiency; Immunotherapy; chemotherapy; normal cell dynamics; tumor cell dynamics; tumor-normal cell interactions.

Figure 1

Green curve: Evolution of normal cell population. Red curve: Evolution of tumor cells. Simulation parameters: r _N=0.4, r _T =0.3, K _T=1.2. 10⁶, K _N=10⁶. A: There is no interaction between normal cells and tumor cells (both populations undergo logistic growth), κ=0, β=0. B: Normal and tumor cells are allowed to interact with each other, κ =0.028, β =1, ρ₀=1, ρ₁=1000, T*=3.10⁵, N₀=1, T₀=1. 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

A: Evolution of Tumor cells. B: Evolution of normal cells. C: Evolution of effector cells. D: Evolution of IL-2 Concentration. The common parameters are the same amount as Figure 1. The new parameters are: a=1, g₂=10⁵, c=0.005, µ₂=0.03, p₁=0.1245, g₁=2*10⁷, p₂=5, g₃=30, µ₃=10. In this figure, Red curve represents the interaction with the immune system. The Black curve represents the interaction of the system with just chemotherapeutic agents when a_T[1-exp^(-MC)]=0.05, and a_N[1-exp^(-MC))=a_E[1-exp^(-MC)]=0.01. The Orange curve represents the interaction with the chemotherapeutic agent when the killing effect of the chemotherapeutic agents on effector and normal cells is minimum. In this case, a_T[1-exp^(-MC)]=0.05 and a_N[1-exp^(-MC)]=a_E[1-exp^(-MC)]=0.001. The Blue curve represents the interaction of the system with the chemotherapeutic agents with the same parameters as the orange graph and the immune boosting agents where a_EE[1-exp^(-MI)]=0.002. The Green curve represents the same case as the blue case with higher dosage of the immune boosting drugs, a_EE[1-exp^(-MI)]=0.004. As explained in the text, the most effective therapy is the case associated to the implementation of the chemotherapeutic agents that majorly kill tumor cells together with effector cells boosting drugs. Some oscillatory behavior can be seen though around the equilibrium when both agents are implemented before reaching the final equilibrium.

Figure 3

In this figure the system behavior under the influence of immunodeficiency viruses is investigated. A: Evolution of Tumor cells. B: Evolution of normal cells. C: Evolution of effector cells. D: Evolution of IL-2 Concentration. E: Evolution of Virus. Same common parameters are implemented as before. The specific parameters are η=3*10⁴, b=5, γ=0.005, α=2.5*10^-4, µ₁ =0.03. The Red curve represents the behavior of the components in the presence of the virus. Black: General chemotherapy is then introduced and then improved to the chemotherapeutic agents with major killing effects on tumor cells. Blue: the effector boosting drugs is added. Green: the dosage of the immune boosting drugs is significantly increases a_EE[1-exp^(-MI)]=0.02 to almost reach to the equilibrium of the system in the absence of any viruses. In dashed-Green line, instead of significantly increasing the dosage of immune boosting drugs to reach to the equilibrium discussed in Figure 2, both chemotherapeutic and immunotherapeutic drugs was increased, a_EE[1-exp^(-MI)]=0.01, a_T[1-exp^(-MC)]=0.05.