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.