Mathematical models of tumor volume dynamics in response to radiotherapy

Abstract

From the beginning of the usage of radiotherapy (RT) for cancer treatment, mathematical modeling has been integral to understanding radiobiology and for designing treatment approaches and schedules. There has been extensive modeling of response to RT with the inclusion of various degrees of biological complexity. Here we focus on models of tumor volume dynamics. There has been much discussion on the implications of different models of tumor growth, and it is just important to consider the implications of selecting different models for response to RT. In this study, we compare three models of tumor volume dynamics: (1) exponential growth with RT directly reducing tumor volume, (2) logistic growth with direct tumor volume reduction, and (3) logistic growth with RT reducing the tumor carrying capacity. For all three models, we: performed parameter sensitivity and identifiability analyses; investigated the impact of the parameter sensitivity on the tumor volume trajectories; and examined the rates of change in tumor volume ({Delta}V/{Delta}t) during and RT treatment course. The parameter identifiability and sensitivity analyses revealed the interdependence of the different model parameters and may inform parameter calibration in any further usage of these models. In examining the {Delta}V/{Delta}t trends, we coined a new metric - the point of maximum reduction of tumor volume (MRV) - to quantify the magnitude and timing of the expected largest impact of RT during a treatment course. Ultimately, the results of these analyses help us to better understand the implications of model selection while simultaneously generating many hypotheses about the underlying radiobiology that need to be tested on time-resolved measurements of tumor volume from appropriate pre-clinical or clinical data. The answers to these questions and more detailed study of these and similar models of tumor volume dynamics may enable more appropriate model selection on a disease-site or patient-by-patient basis.