# Mathematical modeling of the synergetic effect between radiotherapy and immunotherapy

## TL;DR

This paper introduces a mathematical model to study how timing of radiation treatments can enhance the synergy between radiotherapy and immunotherapy for better tumor control.

## Contribution

A novel discrete-time model is developed to quantify the synergy between PULSAR therapy and immunotherapy based on T cell dynamics and treatment timing.

## Key findings

- The model accurately replicates the PULSAR effect observed in animal studies with a root mean square error of 287 mm3.
- Extended radiation intervals (ten days) combined with immunotherapy significantly improved tumor responses in mice.
- The model provides a tool for designing personalized ultra-fractionated stereotactic adaptive radiation trials.

## Abstract

The synergy between radiotherapy and immunotherapy plays a pivotal role in enhancing tumor control and treatment outcomes. To explore the underlying mechanisms of synergy, we investigated a novel treatment approach known as personalized ultra-fractionated stereotactic adaptive radiation (PULSAR) therapy, which emphasizes the impact of radiation timing. Unlike conventional daily treatments, PULSAR delivers high-dose radiation in spaced intervals over weeks or months, enabling tumors to adapt and potentially enhancing synergy with immunotherapy. Drawing on insights from small-animal radiation studies, we developed a discrete-time model based on multiple difference equations to elucidate the temporal dynamics of tumor control driven by both radiation and the adaptive immune response. By accounting for the migration and infiltration of T cells within the tumor microenvironment, we established a quantitative link between radiation therapy and immunotherapy. Model parameters were estimated using a simulated annealing algorithm applied to training data, and our model achieved high accuracy for the test data with a root mean square error of 287 mm3. Notably, our framework replicated the PULSAR effect observed in animal studies, revealing that longer intervals between radiation treatments in the context of immunotherapy yielded enhanced tumor control. Specifically, mice receiving immunotherapy alongside radiation pulses delivered at extended intervals, ten days, showed markedly improved tumor responses, whereas those treated with shorter intervals did not achieve comparable benefits. Moreover, our model offers an in-silico tool for designing future personalized ultra-fractionated stereotactic adaptive radiation trials. Overall, these findings underscore the critical importance of treatment timing in harnessing the synergy between radiotherapy and immunotherapy and highlight the potential of our model to optimize and individualize treatment protocols.