4.3 Simulation results and comparison to experiments
4.3.1 T-cell inhibition and threshold for therapy success
4 A stochastic model for melanoma T-cell therapy
small tumours. Both of these scenarios do not fit the experimental findings, where tumours both respond well to treatment and escape the therapy, with a tendency of better treatment success in small tumours.
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Figure 4.9: Growth curves of tumours undergoing ACTMETi therapy according to the old model, generated by simulations for different initial tumour sizes, shown as tumour diameter [mm]. T-cell efficiency either low (A) or high (B). Vertical lines mark beginning of METi injections, injection of Pmel-1 T-cells, and end of METi injections. Dashed lines indicate tumours undergoing eradication.
An explanation for the type of behaviour that we see in the simulations is the fact that, in the old model, T-cells and differentiated WT melanoma cells interact in a predator-prey like manner, where the T-cell proliferation and melanoma cell killing rates are proportional to the number of differentiated WT melanoma cells and T-cells, i.e. of the formbCDNDif fNCD8
andkDif fNDif fNCD8. As a result, T-cell activity increases with the size of the targeted cell population and large tumours trigger a more effective treatment. Independent of the initial conditions, the system converges to the same stable state balancing T-cells and melanoma cells. Depending on the parameters for T-cell efficiency, this stable state is either at a high level of melanoma cells, causing a tumour relapse, or a low level, resulting in containment of the tumour. This can be compared to the discussion of the simpler pure predator-prey system (1.7) in Section 1.2. Figure 1.1B shows the same fluctuating convergence towards an equilibrium state that depends on the rate at which predators kill their prey, which is the equivalent of T-cell efficiency. The relapse of very small tumours under high T-cell efficiency is caused by the extinction of T-cells due to random fluctuations, as shown in Figure 4.10.
To counteract this effect, and since T-cells cannot physically proliferate infinitely fast, we include aspects of negative feedback within the immune system into our new model. In-flammatory cytokines like INF-γ promote the up-regulation of negative immune checkpoint molecules (PD-L1) on melanoma cells, which inhibit the immune reaction. To model this, we introduce the factor (1−hNCyto)+ into the rates for T-cell proliferation and melanoma cell killing, which decreases T-cell activity as the number of cytokines increases until eventually, at a cytokine level ofNCyto = 1/h, the T-cells are shut down. Since the cytokines themselves are secreted at T-cell proliferation in our model, this constitutes a negative feedback loop within the immune system that limits the efficacy of treatment.
4.3 Simulation results and comparison to experiments
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Figure 4.10:Simulation (and log-plot) of the evolution of different cell/molecule types for for tumours undergoing ACTMETitherapy in a very small tumour for high T-cell efficiency. Vertical lines mark beginning of METi injections, injection of Pmel-1 T-cells, and end of METi injections.
Simulations using this adjusted model recapitulated the tumour growth kinetics of HCmel12 WT melanomas treated with ACTMETi from [87] as shown in Figure 4.11.
When comparing the experimantal and simulated data, note that tumours with a diameter below 1 mm will likely not be detected in the experiments. Tumours that are marked as undergoing eradication in Figure 4.11 are more likely to exists at a small population size, contained by the T-cell population and undergoing predator-prey like oscillations. This is predicted by the mathematical model and has also been witnessed in experiments, see [152], where this state was described as a so-called immune equilibrium. While our simulations
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Figure 4.11:(A) Tumour growth curves from experimental data published in Glodde et al. [87]. (B) Tumour growth curves generated by simulations for different initial tumour sizes. Shown as tumour diameter [mm], vertical lines mark beginning of METi injections, injection of Pmel-1 T-cells, and end of METi injections. Dashed lines indicate tumours undergoing eradication.
can recapitulate the size of the tumour at treatment onset and the approximate duration of tumour remission and eradication, it is of note that they also predicted a critical threshold for treatment success. Below a critical tumour size at treatment onset (roughly 6 mm), ACTMETi achieves long-term melanoma control or eradication. Above this threshold, the T-cells cannot contain the tumour and a remission occurs. This is in line with the experimental
4 A stochastic model for melanoma T-cell therapy
data.
In the corresponding deterministic system, this threshold marks a critical point. Slight perturbations at the time of treatment onset amplify and lead to very different paths of tumour evolution. In our simulations, we can witness both courses of treatment for the same initial conditions, see Figure 4.12. This is a case of stochastic behaviour in the interior of the state space, where no subpopulation is small and in danger of dying out due to fluctuations.
However, simulations also show that the fate of the melanoma cell population is decided early on, due to fluctuations in the beginning of tumour growth that determine whether the tumour size at treatment onset is sub- or super-critical.
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Figure 4.12: Simulations of the evolution of different cell/molecule types for tumours undergoing ACTMETitherapy. Both trajectories are generated with the same initial conditions that produce tumours near the critical threshold for size at treatment onset. Vertical lines mark beginning of METi injections, injection of Pmel-1 T-cells, and end of METi injections.
Overall, our mathematical modelling approach for pure WT melanomas was able to identify T-cell inhibition as a key mechanism in the immune response and predict a critical threshold for therapy success.