A Generalized Linear-Quadratic Model for Radiosurgery, Stereotactic Body Radiation Therapy, and High–Dose Rate Brachytherapy

A generalized mathematical model for the relation between radiation dose and tumor cell death enables better treatment planning and dose schedule designs for current targeted high-dose radiation therapies in cancer. Better Aim Through Better Math More like a laser-guided missile than a conventional bomb, current radiation therapy for cancer hits small targets and tries to minimize collateral damage. Better imaging and delivery, and new ways to keep patients immobilized, have improved our ability to irradiate small, defined areas. Because the doses can be higher when the radiation beams are highly focused (and normal tissue is less likely to suffer), new radiation therapies are given in fewer, larger doses than the long series of low-dose treatments used in the past. But the use of large doses has been problematic, because the calculations used historically to devise the dose and schedule for therapy do not work well with large shots of radiation. Wang et al. have now derived a general equation that applies to both low- and high-dose treatment situations, and so will improve further the effectiveness of current radiation treatments for cancer. When radiation oncologists develop a treatment plan for a cancer patient, they make assumptions about how much radiation is needed to kill the tumor cells. Guided by past experiments, the linear-quadratic equation has defined this dose-response relationship for decades. But this equation does not account for the fact that at high radiation doses there is less sublethal damage to DNA (and more lethal damage). The resulting error in the calculation grows as the doses get higher, causing the effectiveness of a given amount of radiation to be overestimated, potentially leading to inadequate treatment of patients. The general form of the linear-quadratic equation (gLQ) that is derived here by Wang et al. is valid at both low and high doses and accurately calculates the amount of sublethal damage to DNA. They show that the traditional LQ model is an instance of the general model, as is another model used for high radiation doses, called the target model. Experiments taken from the literature, which measure the effects of a wide dose range of radiation on cells grown in vitro and in animals, confirm that the gLQ model accurately predicts the killing effects of radiation through the whole dose range (up to 13 Gy, the highest amount given to animals). When used to predict high-dose responses, the standard LQ model does not conform to the actual data. The gLQ will be a boon to physicians for designing ever more sophisticated, specialized high-dose cancer therapies. With this approach, unusual treatment regimes that use one or a few large doses can be more accurately administered, and the abundant clinical experience in the low-dose range can be extrapolated to high doses. The gLQ, combined with newer image-guided radiation therapy, should markedly enable improvements in radiation ablation of solid cancers. Conventional radiation therapy for cancer usually consists of multiple treatments (called fractions) with low doses of radiation. These dose schemes are planned with the guidance of the linear-quadratic (LQ) model, which has been the most prevalent model for designing dose schemes in radiation therapy. The high-dose fractions used in newer advanced radiosurgery, stereotactic radiation therapy, and high–dose rate brachytherapy techniques, however, cannot be accurately calculated with the traditional LQ model. To address this problem, we developed a generalized LQ (gLQ) model that encompasses the entire range of possible dose delivery patterns and derived formulas for special radiotherapy schemes. We show that the gLQ model can naturally derive the traditional LQ model for low-dose and low–dose rate irradiation and the target model for high-dose irradiation as two special cases of gLQ. LQ and gLQ models were compared with published data obtained in vitro from Chinese hamster ovary cells across a wide dose range [0 to ~11.5 gray (Gy)] and from animals with dose fractions up to 13.5 Gy. The gLQ model provided consistent interpretation across the full dose range, whereas the LQ model generated parameters that depended on dose range, fitted only data with doses of 3.25 Gy or less, and failed to predict high-dose responses. Therefore, the gLQ model is useful for analyzing experimental radiation response data across wide dose ranges and translating common low-dose clinical experience into high-dose radiotherapy schemes for advanced radiation treatments.