Combination therapy of cancer with cancer vaccine and immune checkpoint inhibitors: A mathematical model

Abstract

In this paper we consider a combination therapy of cancer. One drug is a vaccine which activates dendritic cells so that they induce more T cells to infiltrate the tumor. The other drug is a checkpoint inhibitor, which enables the T cells to remain active against the cancer cells. The two drugs are positively correlated in the sense that an increase in the amount of each drug results in a reduction in the tumor volume. We consider the question whether a treatment with combination of the two drugs at certain levels is preferable to a treatment by one of the drugs alone at 'roughly' twice the dosage level; if that is the case, then we say that there is a positive 'synergy' for this combination of dosages. To address this question, we develop a mathematical model using a system of partial differential equations. The variables include dendritic and cancer cells, CD4+ and CD8+ T cells, IL-12 and IL-2, GM-CSF produced by the vaccine, and a T cell checkpoint inhibitor associated with PD-1. We use the model to explore the efficacy of the two drugs, separately and in combination, and compare the simulations with data from mouse experiments. We next introduce the concept of synergy between the drugs and develop a synergy map which suggests in what proportion to administer the drugs in order to achieve the maximum reduction of tumor volume under the constraint of maximum tolerated dose.

Fig 1. Interaction of immune cells with tumor cells.

Sharp arrows indicate proliferation/activation, blocked arrows indicate killing/blocking, and dashed lines indicate proteins on T cells. GM-CSF activates dendritic cells; activated dendritic cells produce IL-12; IL-12 activates naive CD4⁺ and CD8⁺ T cells; activated CD4⁺ T cells (Th1) produce IL-2 which induces proliferation of activated CD4⁺ and CD8⁺ T cells. Activated CD4⁺ and CD8⁺ T cells kill cancer cells. Activated CD4⁺ and CD8⁺ T cells express PD-1 and PD-L1, and cancer cells express PD-L1. The complex PD-1-PD-L1 inhibits the function of active CD4⁺ and CD8⁺ T cells.

Fig 2. Average densities/concentrations of all the variables in the model in the control case (no drugs).

Figures in the first and second panels and the first figure in the third panel show the average density or concentration of each species in the model. The last three figures in the third panel show the total average density of all cells, the growth of tumor radius and the growth of tumor volume, respectively. All parameter values are the same as in Tables 2 and 3.

Fig 3. Spatial distribution of density of cancer cells and T₁ + T₈ cells at time t = 15, 30, 60 days.

All parameter values are the same as in Fig 2.

Fig 4. The growth of tumor radius R(t) during the administration of GM-CSF-secreting vaccine and anti-PD-1 drug.

(a) GM-CSF-secreting vaccine is administered at rate γ_G = 0.87 × 10⁻¹⁰ g/cm³ ⋅ day and anti-PD-1 drug is administered at rate γ_A = 2 × 10⁻¹⁰ g/cm³ ⋅ day. (b) GM-CSF-secreting vaccine is administered at rate γ_G = 0.87 × 10⁻¹⁰ g/cm³ ⋅ day and anti-PD-1 drug is administered at rate γ_A = 3 × 10⁻¹⁰ g/cm³ ⋅ day.

Fig 5. The growth of tumor radius R(t) during the administration of GM-CSF-secreting vaccine and anti-PD-1 drug.

GM-CSF-secreting vaccine is administered at rate γ_G = 3.84 × 10⁻¹⁰ g/cm³ ⋅ day and anti-PD-1 drug is administered at rate γ_A = 1 × 10⁻¹⁰ g/cm³ ⋅ day.

Fig 6. Drug efficacy map and synergy map.

(a) Efficacy map: The color column shows the efficacy E(γ_G, γ_A) when γ_G varies between 0 − 4.8 × 10⁻¹⁰ g/cm³ ⋅ day and γ_A varies between 0 − 4 × 10⁻¹⁰ g/cm³ ⋅ day. (b) Synergy map: The color column shows the synergy σ(γ_G, γ_A) when γ_G varies between 0 − 2.4 × 10⁻¹⁰ g/cm³ ⋅ day and γ_A varies between 0 − 2 × 10⁻¹⁰ g/cm³ ⋅ day. For a given γ_G, the optimal synergy of the combined therapy (γ_G, γ_A) occurs when (γ_G, γ_A) lies on the solid curve.

Fig 7. Average densities of cancer cells and T cells.

(a) The average density of cancer cells (C), and (b) the average density of T cells (T₁ + T₈), under the combination of the drugs with (γ_G, γ_A), where γ_G varies between 0 − 4.8 × 10⁻¹⁰ g/cm³ ⋅ day and γ_A varies between 0 − 4 × 10⁻¹⁰ g/cm³ ⋅ day.

Fig 8. Statistically significant PRCC values (p-value < 0.01) for R(t) at day 60.