Mathematical model predicts response to radiotherapy of grade II gliomas
L. A. Pérez-Romasanta, J. Belmonte-Beitia, A. Martínez-González, G. F. Calvo, V. M. Pérez-García
Reports of Practical Oncology and Radiotherapy 18, S63 (2013)
Abstract
Background. We present a mathematical model for grade II glioma progression and response to radiotherapy (RT) able to predict the long-time response to treatment. Materials and methods. The model describes the evolution of the tumor cell density as a function of time-space incorporating: (i) tumor cell infiltration, as diffusion coefficient D accounting for the cellular motility measured in mm2/day and (ii) proliferation with an average rate r (1/day). The response to radiation is modelled as the evolution of the density of damaged cells that are assumed to complete an average number of k mitosis before dying [Typical parameters: diffusion around 0.0075 mm2/day and proliferation rates in the range r = 0.01–0.001 day?1]. The fraction of tumor cells damaged by a radiation dose is estimated by the L-Q model. Different radiotherapy schemes were simulated including the standard one of 54 Gy in 30 fractions of 1.8 Gy over a time range of 6 weeks. Results. The model output was compared with recently published clinical results (Pallud et al, Neuro-Oncology, 2012) and with those of clinical trials (Van den Bert et al, Lancet, 2005; Shaw et al, J. Clin. Oncol., 2002) with excellent agreement. Proliferation rates determine the response to RT. The smaller the proliferation rate, the longer the progression-free interval and survival rates. Highly proliferative tumors respond earlier to RT but bear an adverse prognosis. Cell motility does not significantly affect early response, but has a relevant impact on survival. Deferring radiotherapy or splitting doses does not affect survival. This concept justifies splitted treatment strategies. The response of the tumor can be fed into the model to provide information regarding proliferation rate and rough estimates for the time of transition to malignancy. The mathematical analysis of the model also gives an equation for the tumor time of birth. Conclusions. The model provides an explanation to published observations and suggests novel radiation therapy strategies potentially useful.