A novel hybrid model of tumor control, combining pulse surveillance with tumor size-guided therapies

Comprehensive cancer therapy protocols are generally administered periodically (at discrete time points) and often determined by tumor size. In the literature, a class of mathematical models are proposed, including therapeutic protocols either with pulse therapy at fixed time points or with state-dependent therapy. However, it is clear in practice (in both clinical and experimental contexts) that the treatment process is much more complex combining pulse surveillance (pulse therapy) with a threshold policy (state-dependent implementation of therapy). Here, we propose a novel mathematical model describing tumor size-guided pulse therapies at fixed time points, which is more in line with the design of clinical protocols for cancer therapies. In studying the dynamics of the proposed model, we find a new kind of periodic solution for dynamical systems with pulse intervention, which we denote as a (l+m)T periodic solution. In more detail, the study of the existence and stability of semi-trivial periodic solutions uncovers very rich dynamics, including the existence and stability of high order (order-k (k>1)) effector cell-free periodic solutions. We obtain analytical expressions for these periodic soluations, and also establish the existence and stability of order-1 tumor-free periodic solution. Furthermore, we discuss the existence and stability of positive periodic solutions. We initially establish the existence of stable order-1 periodic solutions through carefully considering the bifurcation near the effector cell-free periodic solution. Our numerical simulations confirm three possibilities: (1) that two different positive periodic solutions can be bi-stable, (2) that the positive equilibrium can be bi-stable with a positive periodic solution, or (3) that the positive equilibrium is tri-stable with two different positive periodic solutions. Our main results show that no tumor-free periodic solution exists when the threshold policy is feasible, and this may be a reason why frequently a tumor can be only reduced to a certain low level and not eliminated completely. One of the key considerations in the threshold schedule is to determine an optimal threshold of tumor size to balance the therapy frequency (or the times of therapy) and the amplitude of the tumor size, furthermore the optimal therapy should also be individual-based (or personalized). We believe that our modelling approach can be widely applied into many other fields of life science, and provide critical guidance for the implementation of control strategies.