R3.5: Linear Quadratic Equation

The linear quadratic equation is the most widely accepted method of fitting the survival of cells following radiation to an equation. It is given by:

\begin{align} S(D)=e^{-(\alpha D + \beta D^2)} \end{align}

Where S is the number of surviving cells following a dose of D, and α and β describe the linear and quadratic parts of the survival curve. The α and β constants vary between different tissues and tumours.
The α term describes the linear component to the curve. Therefore, the cell death which results from the α component increases linearly with dose.
The β term describes the quadratic part of the curve. As the dose increases, the cell death resulting from the β constant increases in proportion to the square of the dose.

A useful term is the α/β ratio. This is the dose, in Gray, when the number of cells killed by the linear component α is equal to the cell kill from the quadratic β constant. Tissues which have a higher α/β ratio (that is, more linear killing than quadratic killing) tend to have a more linear slope when plotted on the logarithmic scale. Tissues with a low α/β ratio generally have a parabolic shape.
The reason for the close association between cell kill and the linear quadratic equation and α/β ratio is not fully understood. It was initially thought that the linear component related to double strand breaks caused by a single track of radiation, and that the quadratic component related two separate radiation tracks causing separate single strand breaks in such close proximity that a double strand break occurred. However it is argued that the likelihood of two separate radiation tracks interacting within the proximity of the same section
of DNA is highly improbable at normal dose rates.