How to schedule VEGF and PD-1 inhibitors in combination cancer therapy?

Abstract

Background: One of the questions in the design of cancer clinical trials with combination of two drugs is in which order to administer the drugs. This is an important question, especially in the case where one agent may interfere with the effectiveness of the other agent.

Results: In the present paper we develop a mathematical model to address this scheduling question in a specific case where one of the drugs is anti-VEGF, which is known to affect the perfusion of other drugs. As a second drug we take anti-PD-1. Both drugs are known to increase the activation of anticancer T cells. Our simulations show that in the case where anti-VEGF reduces the perfusion, a non-overlapping schedule is significantly more effective than a simultaneous injection of the two drugs, and it is somewhat more beneficial to inject anti-PD-1 first.

Conclusion: The method and results of the paper can be extended to other combinations, and they could play an important role in the design of clinical trials with combination therapy, where scheduling strategies may significantly affect the outcome.

Keywords: Anti-PD-1; Anti-VEGF; Combination therapy; PDE model; Scheduling.

Fig. 1

Interaction of immune cells with cancer cells. Sharp arrows indicate proliferation/activation, blocked arrows indicate killing/blocking, and the inverted arrow indicates recruitment/chemoattraction. C: cancer cells, D: dentritic cells, T₁: CD 4⁺ Th1 cells, T₈: CD 8⁺ T cells, Treg: T regulatory cells, Endo: endothelial cells, Ox: Oxygen from the blood. T₁ and T₈ cells and Tregs express PD-1 and PD-L1; tumor expresses PD-L1

Fig. 2

Distribution of cells in space

Fig. 3

Average densities/concentrations, in g/cm³, of all the variables in the model with control case (no drugs). All parameter values are the same as in Tables 3 and 4, for a mouse model

Fig. 4

Growth of tumor volume under treatment with γ_B or γ_A, or combination (γ_B,γ_A). The anti-VEGF or/and the anti-PD-1 treatment started at day 0 and continued for 10 days. a γ_B=3×10⁻⁸g/cm³·day,γ_A=3×10⁻⁸g/cm³·day; b γ_B=3.5×10⁻⁸g/cm³·day,γ_A=0.5×10⁻⁸g/cm³·day. All other parameters are same as in Fig. 3

Fig. 5

Anti-VEGF decreases PD-1 expression on CD 8⁺ T cells. The treatment with anti-PD-1 drug started at day 0 and continued for 30 days with γ_B=2×10⁻⁸g/cm³·day (a) Growth of tumor volume. (b) Expression level of PD-1 on CD 8⁺ T cells

Fig. 6

Tumor volume under schedules S1, S2 and S3 for 4 pairs (γ_B,γ_A). Here γ_B1=9.5×10⁻⁹g/cm³·day,γ_A1=1.2×10⁻¹⁰g/cm³·day,γ_B2=10×10⁻⁹g/cm³·day,γ_A2=1.5×10⁻¹⁰g/cm³·day.a Tumor volume under schedule S1; b Tumor volume under schedule S2; c Tumor volume under schedule S3

Fig. 7

Spatial profiles of the densities of dendritic cells, T cells, endothelial cells and cancer cells with γ_A=1.2×10⁻¹⁰g/cm³·day,γ_B=9.5×10⁻⁹g/cm³·day. a Profiles under schedule S2 at day t=0,8,15 and 16 weeks. b Profiles under schedule S3 at days t=0,8,18 and 19 weeks

Fig. 8

Efficacy maps: The time in weeks (T_crit) at which the tumor volume decreases by 95% from its initial size under treatment with (γ_B,γ_A). a Efficacy map under schedule S1; b Efficacy map under schedule S2; c Efficacy map under schedule S3. The color columns show the time at which the tumor volume was reduced by 95%

Fig. 9

Tumor volume under the schedules S1, S2 and S3 for 4 pairs (γ_B,γ_A). Here γ_B1=9.5×10⁻⁹g/cm³·day,γ_A1=1.2×10⁻¹⁰g/cm³·day,γ_B2=9.5×10⁻⁹g/cm³·day,γ_A2=1.5×10⁻¹⁰g/cm³·day. a Tumor volume under schedule S1; b Tumor volume under schedule S2; c Tumor volume under schedule S3

Fig. 10

Statistically significant PRCC values (p-value <0.01) for tumor volume at day 30