A mathematical model of the enhancement of tumor vaccine efficacy by immunotherapy

Abstract

TGF-β is an immunoregulatory protein that contributes to inadequate antitumor immune responses in cancer patients. Recent experimental data suggests that TGF-β inhibition alone, provides few clinical benefits, yet it can significantly amplify the anti-tumor immune response when combined with a tumor vaccine. We develop a mathematical model in order to gain insight into the cooperative interaction between anti-TGF-β and vaccine treatments. The mathematical model follows the dynamics of the tumor size, TGF-β concentration, activated cytotoxic effector cells, and regulatory T cells. Using numerical simulations and stability analysis, we study the following scenarios: a control case of no treatment, anti-TGF-β treatment, vaccine treatment, and combined anti-TGF-β vaccine treatments. We show that our model is capable of capturing the observed experimental results, and hence can be potentially used in designing future experiments involving this approach to immunotherapy.

Fig. 1

A diagram of the interactions between the different populations in the mathematical model

Fig. 2

The dynamics of the tumor size in four treatment regimes. Shown are the results of the numerical simulations on top of the experimental data from Terabe et al. (2009)

Fig. 3

A comparison of the dynamics of the tumor size for all treatment regimes

Fig. 4

Simulated population dynamics of the individual populations in the control case: (a) Tumor size (mm²), (b) TGF-β concentration (ng/ml), (c) CTL population (number of), (d) Treg population (number of)

Fig. 5

Simulated population dynamics of the individual populations with combined treatment: (a) Tumor size (mm²), (b) TGF-β concentration (ng/ml), (c) CTL population (number of), (d) Treg population (number of)

Fig. 6

Effector T cell versus tumor size phase portrait when combined treatment is simulated with different levels of tumor antigenicity. Depending on the antigenicity, the tumor load is reduced to near zero for a period of time before the immune response is no longer able to control the tumor (Color figure online)

Fig. 7

Model sensitivity analysis. Done by varying each parameter over a range of values centered around a baseline value and observing the size of the tumor at the end of 30 simulated days. (a) No treatment case: variations of parameters leads to very little changes in the final tumor size. (b) Combined treatment: the system was found to be sensitive to a₁, the parameter quantifying the maximal production rate of TGF-β, c₂, the quantity describing the size at which a tumor begins to produce TGF-β, and f, the quantification of a tumor’s antigenicity

Fig. 8

A simulated tumor growth for a mildly antigenic tumor (f = 0.56)