A simple mathematical model of cyclic hypoxia and its impact on hypofractionated radiotherapy

There is evidence that the population of cells that experience fluctuating oxygen levels ("acute," or, "cyclic" hypoxia) are more radioresistant than chronically hypoxic ones and hence, this population may determine radiotherapy (RT) response, in particular for hypofractionated RT, where reoxygenation may not be as prominent. A considerable effort has been devoted to examining the impact of hypoxia on hypofractionated RT; however, much less attention has been paid to cyclic hypoxia specifically and the role its kinetics may play in determining the efficacy of these treatments. Here, a simple mathematical model of cyclic hypoxia and fractionation effects was worked out to quantify this.

Cancer clonogen survival fraction was estimated using the linear quadratic model, modified to account for oxygen enhancement effects. An analytic approximation for oxygen transport away from a random network of capillaries with fluctuating oxygen levels was used to model inter-fraction tissue oxygen kinetics. The resulting survival fraction formula was used to derive an expression for the iso-survival biologically effective dose (BED), BED_iso-SF . These were computed for some common extra-cranial hypofractionated RT regimens.

Using relevant literature parameter values, inter-fraction fluctuations in oxygenation were found to result in an added 1-2 logs of clonogen survival fraction in going from five fractions to one for the same nominal BED (i.e., excluding the effects of oxygen levels on radiosensitivity). BED_iso-SF 's for most ultra-hypofractionated (five or fewer fractions) regimens in a given tumor site are similar in magnitude, suggesting iso-efficacy for common fractionation schedules.

Although significant, the loss of cell-killing with increasing hypofractionation is not nearly as large as previous estimates based on the assumption of complete reoxygenation between fractions. Most ultra-hypofractionated regimens currently in place offer sufficiently high doses to counter this loss of cell killing, although care should be taken in implementing single-fraction regimens.