Optimum inactivation dose and indices of radiation response based on the linear quadratic survival equation

For typical tumor-cell dose-response curves, the efficiency ratio, i.e., the ratio between the fraction of cells killed and the radiation dose administered, is a continuously decreasing function of dose. However, if the survival curve is sufficiently 'shouldered', this ratio has a maximum value at a dose greater than zero. In radiotherapy, one possible criterion for the ideal dose per session is a high value of the efficiency ratio for the targeted cells, but a low value for surrounding healthy cells. Efficiency ratios can be derived from dose-response relationships. Any linear quadratic dose-survival curve of the form S=exp(-αD+βD2) can be completely described by two parameters, s and φ, where s=α+√β and φ=√β/s. The former parameter is an index of radiosensitivity, and the latter is an index of curve shape. Using these indices, the ratio of fraction of inactivated cells to dose can be calculated and its maximum, as dose varies, determined. For values of φ greater than 0.55, this ratio has a maximum when the dose is approximately 1/s. However, for values of φ less than 0.4, this ratio is greatest when the dose is zero. Since φ varies widely among different cell lines, it may be possible to optimize radiotherapeutic dose-fractionation regimes using these indices. The parameterization of dose-survival relationships in terms of s and φ also simplifies conceptualization of the survival-curve characteristics. Both the mean inactivation dose and the dose required to reduce survival to 1/e are approximately equal to 1/s. The fraction surviving a dose of 1/s falls between exp(-1) and exp(-3/4). This dose is approximately equal to the mean inactivation dose; for human tumor cells, its mean is approximately 2 Gy. The dose corresponding to a fraction surviving equal to 1/e, or ln(S)=-1, falls between 1/s and 5/4 s for all non-negative values of α and β. (orig.)