Mathematical modeling of the effects of potassium ions on the tumor immune system and virotherapy.

To understand how extracellular potassium ions influence both anti-tumor immunity and virotherapy in cancer progression, we construct two mathematical models informed by our experimental results. For the tumor immune system, our analysis shows a high concentration of extracellular potassium ions diminishes the killing rate of tumor cells by immune cells. Our model confirms that the stimulation coefficient of immune cells by tumor cells remains a crucial parameter in the presence of extracellular potassium ions, which largely controls the overall tumor growth. For virotherapy, we obtain a formula for the basic viral reproduction number which combines several parameters including potassium ions. When this number is greater than one, virotherapy achieves some partial success, where the tumor load is an increasing function of the potassium ion concentration. Therefore, the tumor load is reduced if the potassium ion concentration is lowered. The delay parameter of the viral lytic cycle also affects the basic reproduction number, and we find a critical delay time which determines when the reproduction number is greater than one. We performed some numerical analysis. A two-parameter bifurcation analysis reveals a positive correlation between viral burst size and the potassium-ion absorption rate, suggesting that higher absorption rates can enhance the success of virotherapy.