Mathematical model approach to describe tumour response in mice after vaccine administration and its applicability to immune-stimulatory cytokine-based strategies

Abstract

Immunotherapy is a growing therapeutic strategy in oncology based on the stimulation of innate and adaptive immune systems to induce the death of tumour cells. In this paper, we have developed a population semi-mechanistic model able to characterize the mechanisms implied in tumour growth dynamic after the administration of CyaA-E7, a vaccine able to target antigen to dendritic cells, thus triggering a potent immune response. The mathematical model developed presented the following main components: (1) tumour progression in the animals without treatment was described with a linear model, (2) vaccine effects were modelled assuming that vaccine triggers a non-instantaneous immune response inducing cell death. Delayed response was described with a series of two transit compartments, (3) a resistance effect decreasing vaccine efficiency was also incorporated through a regulator compartment dependent upon tumour size, and (4) a mixture model at the level of the elimination of the induced signal vaccine (k 2) to model tumour relapse after treatment, observed in a small percentage of animals (15.6%). The proposed model structure was successfully applied to describe antitumor effect of IL-12, suggesting its applicability to different immune-stimulatory therapies. In addition, a simulation exercise to evaluate in silico the impact on tumour size of possible combination therapies has been shown. This type of mathematical approaches may be helpful to maximize the information obtained from experiments in mice, reducing the number of animals and the cost of developing new antitumor immunotherapies.

Fig. 1

Individual raw data profiles. a–g C57BL/6 mice were injected with 5 × 10⁵ TC1 cells (n = 13–19 mice per group) on day 0. A unique dose of PBS or of 50 μg of CyaA-E7 was intravenously administered on day 4, 7, 11, 18, 25 or 30 (yellow line) (8). h–i C57BL/6 mice were injected with 5 × 10⁵ MC38 cells (n = 12–21 mice per group) on day 0. A unique dose of PBS or a hydrodynamic injection of 10 μg of a plasmid codifying for murine IL-12 was administered on day 23 (yellow line) (25). Tumour size, computed as the mean of two perpendicular diameters, over time is represented for each mouse and each dosing group included in the study; 2 mm was considered as the limit of quantification (dashed line)

Fig. 2

Schematic representation and population performance of the mathematical model selected. After VAC administration and through a transit compartment (TRAN), the vaccine triggers a signal (SVAC) able to decrease tumour size (Ts). Two different populations (responders and non-responders) at the level of the SVAC elimination were identified. An inhibition of vaccine efficacy induced by a regulator compartment (REG) controlled by tumour size was detected. The behaviour of the different model compartment over time for both populations, along with the percentage of inhibition (%INH) induced by REG over k₃ versus the amount in the regulator compartment under no vaccine administration, and highlighting the REG amount at relevant time points, are presented. k ₁ first-order rate constant controlling vaccine elimination and transit between compartments; k ₂ first-order rate constant accounting for SVAC degradation; λ zero-order rate constant of tumour growth; k ₃ the vaccine efficacy second-order rate constant; k ₄ the first-order rate constant controlling the regulator compartment dynamics; REG ₅₀ amount in the regulator compartment needed to inhibit vaccine activity by a half; γ the shape of that inhibitory process. TAD time after dose (vaccine) administration. Tdose day of vaccine administration

Fig. 3

Individual model predictions. Tumour size observations (points) and individual model predictions of a mouse that respond (light grey) or do not respond (dark grey) to the vaccine are presented for the different CyaA-E7 dosing groups. 2 mm was considered as the limit of quantification (dashed line)

Fig. 4

Visual and numerical predictive check to evaluate final model performance. Simulated tumour size measurements above the limit of quantification (upper panels) and percentage of data below the limit of quantification (lower panel) versus raw data (points) are plotted over time for CyaA-E7 (orange) or IL-12 (red) dosing groups. Grey areas in the upper panels represent the 90% prediction interval of the simulated median. Grey areas in the lower panels represent the 90% prediction interval of the simulated percentage of data below the limit of quantification. Solid and dashed black lines are the simulated and raw median respectively. 2 mm was considered as the limit of quantification (red dashed line)

Fig. 5

Vaccine efficacy evaluation. Probability of cure at the end of the experiment was estimated for 1000 simulated studies for both tested therapies CyaA-E7 (left panel) and IL-12 (right panel). Simulated median was plotted against raw probability of cure for each dosing group in the training dataset (points). Grey shadow represents 90% prediction interval of the simulated data. Blue triangle represents probability of cure of an independent study not included in the analysis (external evaluation with the validation dataset)

Fig. 6

Generalization of the model. a Sensitivity analysis of the model representing the percentage of change in tumour size when one model parameter at a time is modified. b–d Predicted tumour response over time for different theoretical drugs when administered in combination with CyaA-E7 vaccine on day 25 after tumour cell inoculation. A range of efficacy parameters is explored for each simulated scenario (see Supplementary material)