In Silico Synergism and Antagonism of an Anti-tumour System Intervened by Coupling Immunotherapy and Chemotherapy: A Mathematical Modelling Approach

Based on the logistic growth law for a tumour derived from enzymatic dynamics, we address from a physical point of view the phenomena of synergism, additivity and antagonism in an avascular anti-tumour system regulated externally by dual coupling periodic interventions, and propose a theoretical model to simulate the combinational administration of chemotherapy and immunotherapy. The in silico results of our modelling approach reveal that the tumour population density of an anti-tumour system, which is subject to the combinational attack of chemotherapeutical as well as immune intervention, depends on four parameters as below: the therapy intensities D, the coupling intensity I, the coupling coherence R and the phase-shifts Φ between two combinational interventions. In relation to the intensity and nature (synergism, additivity and antagonism) of coupling as well as the phase-shift between two therapeutic interventions, the administration sequence of two periodic interventions makes a difference to the curative efficacy of an anti-tumour system. The isobologram established from our model maintains a considerable consistency with that of the well-established Loewe Additivity model (Tallarida, Pharmacology 319(1):1–7, 2006). Our study discloses the general dynamic feature of an anti-tumour system regulated by two periodic coupling interventions, and the results may serve as a supplement to previous models of drug administration in combination and provide a type of heuristic approach for preclinical pharmacokinetic investigation.