PULSAR Effect: Revealing potential synergies in combined radiation therapy and immunotherapy via differential equations

PULSAR (personalized ultrafractionated stereotactic adaptive radiotherapy) is a form of radiotherapy method where a patient is given a large dose or “pulse” of radiation a couple of weeks apart rather than daily small doses. The tumor response is then monitored to determine when the subsequent pulse should be given. Pre-clinical trials have shown better tumor response in mice that received immunotherapy along with pulses spaced 10 days apart. However, this was not the case when the pulses were 1 or 4 days apart. Therefore, a synergistic effect between immunotherapy and PULSAR is observed when the pulses are spaced out by a certain number of days. In our study, we aimed to develop a mathematical model that can capture the synergistic effect by considering a time-dependent weight function that takes into account the spacing between pulses. We determined feasible parameters by fitting murine tumor volume data of six treatment groups via simulated annealing algorithm. Applying these parameters to the model we simulated 4000 trials with varying sequencing of pulses. These simulations indicated that if pulses were spaced apart by at least 9 days the tumor volume was about 200 $m{m}^3$ to 250 $m{m}^3$ smaller when treated with PULSAR combined with immunotherapy. We successfully demonstrate that our model is simple to implement and can generate tumor volume data that is consistent with the pre-clinical trial data. Our model has the potential to aid in the development of clinical trials of PULSAR therapy.