Modelling the influence of treatment time on the biological effectiveness of single radiosurgery treatments: derivation of protective dose modification factors

Objective: To provide simpler models for adjusting total dose to compensate for significant variations in central nervous system radiosurgical treatment times, which vary and will influence treatment bioeffectiveness. At present, no allowance is made for time variations. A framework of simpler equations would allow radiosurgical outcomes to be analysed with respect to treatment time, and a system for dose adjustments between radioisotope and linac-based techniques with different treatment durations. Methods: The standard biological effective dose (BED) equations for fractionated and protracted radiations have been combined, using biexponential DNA repair kinetics, to provide the following equation: BED = x.nd (1 + (nd/k - dk) f (mu T-1) + d/k f (mu 1t)) + (1 - x). nd (1 + (nd/k - dk) f (mu T-2) + dk/f (mu(2)t)) for n isocentres (or subfractions), each treated to a variable dose d in time t, the overall time-being, T, mu 1, mu 2, are fast and slow repair rate coefficients, with partition factors of x and (1-x), respectively and k is the alpha/beta ratio, with f(mu T) being the function that summates sublethal damage repair. Thus, repair during the period of irradiation and in the time interval between each isocentre can be taken into account. Simpler monoexponential and linear models are also used. Results: The results obtained using simpler models are compared with those obtained using more complex retrospective Gamma Knife BED treatment planning by Millar et al. (2015) in a group of 23 patients on a 13 Gy physical isodose surface. The above equation provides a BED value around 3% above their minimum values, 4% below their average value and 10% below their maximum BED values. Changes in isocentre numbers used, due to treatment plan complexity, can influence total treatment time, producing variations in the BED-time data: instead of a unique curve for each n value, in aggregate form the data (ranging from around 20 to 140 min treatment times) can be fitted by monoexponential time functions and further approximated to a linear function for more rapid estimations. Worked examples show how dose can then be tailored to the expected treatment times in order to obtain isoeffective treatments for central nervous system tissues. Conclusion: The models allow better analysis of radio-surgical treatment time data and guidance to the choice of dose to match the overall time. Although this study is based on Gamma Knife treatments, in principle the methods will also apply to any radiosurgical technique, so that dose-time compensations can be made between differing techniques. Advances in knowledge: The new BED equation-based framework is relevant to analyse and optimise radiosurgical treatments.