Modeling the Effects of Space Structure and Combination Therapies on Phenotypic Heterogeneity and Drug Resistance in Solid Tumors

Abstract

Histopathological evidence supports the idea that the emergence of phenotypic heterogeneity and resistance to cytotoxic drugs can be considered as a process of selection in tumor cell populations. In this framework, can we explain intra-tumor heterogeneity in terms of selection driven by the local cell environment? Can we overcome the emergence of resistance and favor the eradication of cancer cells by using combination therapies? Bearing these questions in mind, we develop a model describing cell dynamics inside a tumor spheroid under the effects of cytotoxic and cytostatic drugs. Cancer cells are assumed to be structured as a population by two real variables standing for space position and the expression level of a phenotype of resistance to cytotoxic drugs. The model takes explicitly into account the dynamics of resources and anticancer drugs as well as their interactions with the cell population under treatment. We analyze the effects of space structure and combination therapies on phenotypic heterogeneity and chemotherapeutic resistance. Furthermore, we study the efficacy of combined therapy protocols based on constant infusion and bang–bang delivery of cytotoxic and cytostatic drugs.

Appendices

Appendix 1: Qualitative Mathematical Justification for Phenotypic Selection

From a mathematical standpoint, using the considerations drawn in Lorz et al. (2011), Mirrahimi and Perthame (2014) and Perthame and Barles (2008), the long-term dynamics of the concentration points $X(t,r)$ can be formally characterized by evaluating

$$\begin{aligned} \lim _{t \rightarrow \infty } R\big (x=X(t,r),\rho (t,r),c_{1,2}(t,r),s(t,r)\big ) = 0. \end{aligned}$$

In order to verify it numerically, we write the following identity

$$\begin{aligned} R\big (X(t,r),\rho (t,r),c_{1,2}(t,r),s(t,r)\big ) = 0, \end{aligned}$$

which implies, due to the definitions of the functions $p$ and $\mu _1$ provided in Sect. 3,

$$\begin{aligned} X(t,r)&= \frac{1+\mu _2 c_2(t,r)}{a_2 s(t,r) - b_2 c_1(t,r) [1+\mu _2 c_2(t,r)]} \bigg [ \frac{s(t,r)}{1+\mu _2 c_2(t,r)}(a_1 + a_2) \\&-\, d \rho (t,r) - (b_1 + b_2) c_1(t,r) \bigg ]. \end{aligned}$$

Figure 12 shows that the curves $X(t,r)$ obtained from the formula above for $t$ equal to the final time of simulations are in good agreement with the positions of the maximum points $x_M(t,r)$ of $n(t,r,x)/\rho (t,r)$ at the same time instant.

Fig. 12

Qualitative mathematical justification for phenotypic selection. Plots of $X(t,r)$ (solid lines) and positions of the maximum points $x_M(t,r)$ of $n(t,r,x)/\rho (t,r)$ ($\cdot$) for $C_{1,2}(\cdot ) = 0$ and $t=50$ (left panel); $C_1(\cdot ) = 0, C_2(\cdot ) = C_2 > 0$ and $t=200$ (center panel); $C_1(\cdot ) = C_1 > 0, C_2(\cdot ) = 0$ and $t=120$ (right panel). The unit of time is days

Appendix 2: Values of the Parameters

Figure 2: $C^0=0.005, \varepsilon =0.005, S_1 = 12, \alpha _s=0.08, \alpha _{c_1}=0.08, \alpha _{c_2}=0.2, \gamma _s=1, \gamma _{c_1}=1, \gamma _{c_2}=1, d=14.48$; (a) , $C_1(\cdot ) = 0, C_2(\cdot ) = 0, \mu _2=800$; (b) $C_1(\cdot ) = 0, C_2(\cdot ) = 3, \mu _2=8$; (c) $C_1(\cdot ) = 3, C_2(\cdot ) = 0, \mu _2=800$.

Figure 3: $C^0=0.005, \varepsilon =0.005, S_1 = 12, \alpha _s=0.08, \alpha _{c_1}=0.08, \alpha _{c_2}=0.2, \gamma _s=1, \gamma _{c_1}=1, \gamma _{c_2}=1, d=14.48$; (left) $C_1(\cdot ) = 3, C_2(\cdot ) = 0, \mu _2=800$; (right) $C_1(\cdot ) := 0, C_2(\cdot ) := 3, \mu _2=8$.

Figure 4: $C^0=0.005, \varepsilon =0.005, S_1 = 12, \alpha _s=0.08, \alpha _{c_1}=0.08, \alpha _{c_2}=0.2, \gamma _s=1, \gamma _{c_1}=1, \gamma _{c_2}=1, d=14.48, \mu _2=8, C_1 = 4, C_2(\cdot ) = 0$.

Figure 5: $C^0=0.05, \varepsilon =0.025, S_1 = 1.6, \alpha _s=0.08, \alpha _{c_1}=0.08, \alpha _{c_2}=0.2, \gamma _s=1, \gamma _{c_1}=1, \gamma _{c_2}=1, d=14.48, \mu _2=800, C_a = 2.4, C_b = 3.6, C_d = 6, C_2(\cdot ) = 0$.

Figure 7: $C^0=0.005, \varepsilon =0.005, S_1 = 12, \alpha _s=0.08, \alpha _{c_1}=0.08, \alpha _{c_2}=0.2, \gamma _s=1, \gamma _{c_1}=1, \gamma _{c_2}=1, d=14.48, \mu _2=800, C_1(\cdot ) = 0, C_2(\cdot ) = 0$.

Figures 8 and 9: $C^0=0.005, \varepsilon =0.005, S_1 = 12, \alpha _s=0.08, \alpha _{c_1}=0.08, \alpha _{c_2}=0.2, =1, \gamma _{c_1}=1, \gamma _{c_2}=1, d=14.48, \mu _2=800, C_a = 4, C_b = 15$.

Figure 10: $C^0=0.005, \varepsilon =0.005, S_1 = 12, \alpha _s=0.08, \alpha _{c_1}=0.08, \alpha _{c_2}=0.2, \gamma _s=1, \gamma _{c_1}=1, \gamma _{c_2}=1, d=14.48, \mu _2=800, C_a = 0.24, C_b = 0.90$.

Figure 11: $C^0=0.005, \varepsilon =0.005, S_1 = 12, \alpha _s=0.08, \alpha _{c_1}=0.08, \alpha _{c_2}=0.2, \gamma _s=1, \gamma _{c_1}=1, \gamma _{c_2}=1, d=14.48, \mu _2=800, C_d = 0.24, C_e = 0.90$.

Figure 12: $C^0=0.005, \varepsilon =0.005, S_1 = 12, \alpha _s=0.08, \alpha _{c_1}=0.08, \alpha _{c_2}=0.2, \gamma _s=1, \gamma _{c_1}=1, \gamma _{c_2}=1, d=14.48$; (left) $C_1(\cdot ) = 0, C_2(\cdot ) = 0, \mu _2=800$; (center) $C_1(\cdot ) = 0, C_2(\cdot ) = 3, \mu _2=8$; (right) $C_1(\cdot ) = 3, C_2(\cdot ) = 0, \mu _2=800$.