Some say that the risk following radiation exposure should follow a quadratic model. But the data often looks linear. I wonder if this could be because of uncertainty in the dose which has a tendency to smooth and flatten the dose response curve, making it linear?

set.seed(0)
data <- data.frame(n=1:100) %>%
  mutate(true_dose = 0:99,
         dose_error = exp(rnorm(100, mean=0, sd=0.3)),  # e ^ (normal error)
         measured_dose = true_dose * dose_error,
         effect = true_dose^2)

ggplot(data) +
  geom_point(aes(true_dose, effect)) + 
  geom_point(aes(measured_dose, effect), color='red') + 
  geom_smooth(aes(measured_dose, effect), color='red') + 
  xlab(label = 'dose')

plot of chunk unnamed-chunk-1

Figure: Dose response with noise

Dose (x) vs effect (y) for a quadratic response (black) and another quadratic response where noise has been added to the dose.

Result

Amazingly, this seems to explain two big mysteries in radiobiology:

  1. Dose response looks linear if there is noise in the dose even though it is actually quadratic. This explains why atomic bomb survivors seem to have a linear dose responses.
  2. Dose response seems to atenuate at high doses if there is noise in the dose. Traditionally this has been attributed to cell killing.

The point here is that dose response follows the linear quadratic model. Deviation from this is an artifact of noisy dose measurements!