We can use the Poisson distribution to model radiation hit in target cells. \[P(n)=\frac{\lambda^ne^{-\lambda}}{n!}\] where \(\lambda\) is the average number of events occurred in a unit time and n is the specific number of events occurred in a unit time.

If each “hit” is assumed to result in cell inactivation, then the probability of survival is the probability of not being hit, P(0). \[P(0)=\frac{\lambda^0e^{-\lambda}}{0!}=e^{-\lambda}\] We define a dose \(D_0\) that delivers, on average, one lethal event per target. This dose will result in \(P(0)=e^{-1}=0.37\) survival. \(D_0\) is often called the mean lethal dose.

Linear-Quadratic Model

The linear-quadratic (L-Q) equation is the most wildly accepted method of fitting the survival of cells following radiation. It is given by \[S=e^{-(\alpha D+\beta D^{2})}\] Where S is the number of surviving cells following a dose of D, and \(\alpha\) and \(\beta\) describe the linear and quadratic parts of the survival curve. The \(\alpha\) and \(\beta\) constants vary between different tissues and tumors.

Multi-Target Model

In a multi-target (M-T) model, each cell contains n distinct and identical targets and all n targets must be inactivated to kill the cell. The probability that a target is not hit is \(e^{-\frac{D}{D_0}}\). The probability that a target is hit is \(1-e^{-\frac{D}{D_0}}\). The probability that all n targets are hit is \((1-e^{-\frac{D}{D_0}})^n\). Therefore the probability that all n targets will not be hit, i.e., the probability of survival, is \[S=1-(1-e^{-\frac{D}{D_0}})^n\]

Here this clonogenic assay shows the survival of FaDu cells following radiation and we are going to fit the dose response curve to the L-Q and M-T models.

Load and view data.

Transforming the L-Q equation to \(lnS=-\alpha D-\beta D^2\) reveals that we can use a polynomial regression to get starting estimates.

model_poly <- lm(-log(surviving.fraction)~-1+radiation+I(radiation^2), data=data) #-1 is to set the intercept to 0
summary(model_poly)
model_lq<- nls(surviving.fraction~exp(-a*radiation-b*radiation^2),data=data, start = list(a=coef(model_poly)[1],b=coef(model_poly)[2]))
summary(model_lq)

To fit the M-T model, I don’t have a way of estimating the starting values. Try a few numbers to feed the model.

model_mt <- nls(surviving.fraction ~ 1-(1-exp(-radiation/D0))^n, data=data, start = list(D0=2,n=2))
summary(model_mt)

To create the fitted regression lines

# create more "radiation" points for the fitted line
newdata <- data.frame(radiation = rep(seq(0,8, length = 100),2), model=rep(c("lq","mt"), each=100))

# predict surviving fractions for each radiation dose based on each model
newdata$prediction <- c(predict(model_lq,newdata)[1:100], predict(model_mt,newdata)[1:100])

Final plot with ggplot2

So the FaDu cell line has \(\alpha=\) 0.21, \(\beta=\) 0.06, \(D_0=\) 1.52, and n=2.38.