An Alternative to the Linear-Quadratic Model

In Section 16.9 of Intermediate Physics for Medicine and Biology, Russ Hobbie and I discuss the linear-quadratic model.

The linear-quadratic model is often used to describe cell survival curves… We use it as a simplified model for DNA damage from ionizing radiation.

Suppose you plate cells in a culture and then expose them to x-rays. In the linear-quadratic model, the probability of cell survival, P, is

P = e^{–αD–βD2}

where D is the dose (in grays) and α and β are constants. At large doses, the quadratic term dominates and P falls as P = e^–βD2. In some experiments, however, at large doses P falls exponentially. It turns out that there is another simple model—called the multi-target single-hit (MTSH) model—describing how P depends on D in survival curves,

P = 1 – (1 –e^–α'D)^N

Let’s compare and contrast these curves. They both have two parameters: α and β for the linear-quadratic model, and α' and N in the MTSH model. Both give P = 1 if D is zero (as they must). They both fall off more slowly at small doses and then faster at large doses. However, while the linear-quadratic model falls off at large dose as e^–βD2, the MTSH model falls off exponentially (linearly in a semilog plot).

If α'D is large, then the exponential is small. We can expand the polynomial using (1 – x)^N = 1 – N x + …, keep only the first two terms, and then use some algebra to shown that at large doses P = N e^–α'D. If you extrapolate this large-dose behavior back to zero dose, you get P = N, which provides a simple way to determine N.

Below is a plot of both curves. The blue curve is the linear-quadratic model with α = 0.1 Gy^-1 and β = 0.1 Gy^-2. The gold curve is the MTSH model with α’=1.2 Gy^-1 and N = 10. The dashed gold line is the extrapolation of the large dose behavior back to zero dose to get N. 

If the survival curve falls off exponentially at large doses use the MTSH model. If it falls off quadratically at large doses use the linear-quadratic model. Sometimes the data doesn’t fit either of these simple toy models. Moreover, often P is difficult to measure when it’s very small, so the large dose behavior is unclear. The two models are based on different assumptions, none of which may apply to your data. Choosing which model to use is not always easy. That’s what makes it so fun.