Modelling pulsed immunotherapy of tumour–immune interaction

• We develop a novel mathematical model that describes the tumour–immune interaction with pulsed immunotherapy.
• The existence and stability of the tumour free periodic solution are addressed.
• The effects of ACI associated or not with IL-2 on immunotherapy are investigated numerically in detail.
• The results showed that the tumour can be eradicated or controlled with combined therapies.

We develop a mathematical model that describes the tumour–immune interaction and the effect on it of pulsed immunotherapy, based on the administration of adoptive cellular immunotherapy (ACI) combined with interleukin-2 (IL-2). The stability conditions for the tumour-free periodic solution are provided with different frequencies of ACI applications and IL-2 infusions. Furthermore, the effects of period, dosage and times of drug deliveries on the amplitudes of the tumour-free periodic solution were investigated. The most feasible immunotherapy strategy was determined by comparing immunotherapy with ACI treatment with or without IL-2. However, to investigate how to enhance the efficacy of chemotherapy (radiotherapy) and reduce its side-effects, we developed a model involving periodic applications of immunotherapy with chemotherapy (radiotherapy) applied only when the density of the tumour reached a given threshold. The results revealed that the initial densities, the effector cell: tumour cell ratios, the periods TT and a given critical number of tumour cells CTCT are crucial for cancer treatment, which confirms that it is important to customize treatment strategies for individual patients.