Periodically Pulsed Immunotherapy in a Mathematical Model of Tumor-Immune Interaction

Abstract

Immunotherapy is one of the most recent approaches in cancer therapy. In this paper, a mathematical model of tumor-immune interaction with periodically pulsed immunotherapy, which is described by impulsive differential equations, is considered. The ODE system is turned into a discrete-time dynamical system for bifurcation analysis. A mathematical analysis is performed to determine the minimum dosage for successful treatment. An adaptive grid method is then developed to identify the fixed points and their bifurcations. The effects of continuous and pulsed treatment strategies are compared. The interindividual variability is studied by one-parameter and two-parameter bifurcation diagrams. Increasing the strength of the immune response improves the outcome of the treatment if the immune response is weak. However, it becomes a drawback when the strength of the immune response increases over a certain threshold. Multiple attractors exist so that a treatment may result in a tumor-free state, a large tumor, or a middle-sized tumor depending on initial conditions. It is believed that the numerical method proposed in this paper can be applied to a class of mathematical models of periodically pulsed drug therapies.

Modelling pulsed immunotherapy of tumour–immune interaction

“…Besides, the tumour can be eradicated in a short treatment period or by a large dosage of s 1 alone (see Fig. 4(a), (c) and (e)), but the IL-2 treatment alone cannot clear the tumour, in agreement with previous research [3,20,37]. ).…”

“…Note that the dynamics are very sensitive to parameter c [20,37]. What we want to address is how it will impact the efficacy of pulsed immunotherapy for Case 1.…”

“…Without treatment, malignant tumour cells can grow uncontrollably and often metastasize from their initial site to other parts of the body with fatal consequences. Common therapies include surgery, radiotherapy, chemotherapy, immunotherapy or combinations thereof [37]. Immunotherapy, used to stimulate a strong immune response to target tumours, has become one of the most common approaches in cancer therapy [4,29,30].…”

“…Recently, the mathematical model of tumour-immune interaction developed by Kirschner and Panetta has been re-considered with pulsed immunotherapy described by impulsive differential equations [37] and a bifurcation analysis related to key parameters and its biological implications were discussed briefly. Note that Adoptive Cellular Immunotherapy (ACI) refers to the injection of cultured effector cells that have anti-tumour activity into the tumour site [20], so that ACI acts directly on the tumour cells.…”

“…Assume τ n+1 − τ n ≡ T p for all n (n ∈ N , if n = 1, then this special case is considered by Wei and Lin [37]), where T p is the period of the applications of ACI. For this case, system (1) is said to be a T p periodic system if there exists a positive integer k N such that…”