Some implications of linear-quadratic-linear radiation dose-response with regard to hypofractionation

Recent technological advances enable radiation therapy to be delivered in a highly conformal manner to targets located almost anywhere in the body. This capability has renewed the clinical interest in hypofractionation wherein the tumor is delivered a few fractions of very large dose per fraction. Extrapolating clinical experience from conventional regimens to fractions of high dose is important to designing hypofractionated treatments. The concept of biologically effective dose (BED) based on the linear-quadratic (LQ) formulation e(-(alphaD+betaD2) is a useful tool for intercomparing conventional fractionations but may be hampered if the value of alpha/beta is dose range dependent and/or when extrapolating to fractions of high dose because the LQ curve bends continuously on the log-linear plot. This does not coincide with what is observed experimentally in many clonogenic cell survival studies at high dose wherein radiation dose-response relationships more closely approximate a straight line. Intercomparison of conventional fractionations with hypofractionated regimens may benefit from BED calculations which instead use a dose range independent linear-quadratic-linear (LQ-L) formulation which better fits the experimental data across a wider range of dose. The dosimetric implications of LQ-L are explored using a simple model which requires only the specification of a dose D(T) at which the LQ curve transitions to final linearity and the log(e) cell kill per Gy in the final linear portion of the survival curve at high dose. It is shown that the line tangent to the LQ curve at transition dose D(T) can often be used to approximate the final slope of the dose response curve. When D(T) = 2alpha/ beta Gy, the line tangent to the LQ curve at D(T) intersects the e(-alphaD) and e(-betaD2) curves at dose alpha/ beta Gy and also closely fits the linear response in the high dose region of some classic in vitro cell survival curves for which the value of alpha/beta is low. It is hypothesized that D(T) will increase as the magnitude of alpha/beta increases. Examples are presented illustrating how to recognize LQ-L behavior in multifraction isoeffect studies of late responses such as spinal cord and lung. When planning hypofractionated regimens involving reactions with low alpha/beta, recognizing LQ-L behavior could be important because the dose-response is likely to transition to final linearity within the contemplated range of hypofractional doses.