The delivery of an amount of radiation to a certain tumour volume will not always lead to the same results. This is the random nature of cell kill – and although the specific outcome is unknown the probability of a particular outcome can be determined with some accuracy. At some doses, the risk of a reaction approaches zero, and at higher doses the chance of reaction may approach 100%.

The shape of the curve which describes tumour control or normal tissue complications charted against dose is sigmoid in shape. This was first observed in 1936 by Holthusen. He hypothesised that the difference was due to individual patient sensitivity to radiation.

The dose response curve can be modelled through several techniques.

## Poisson Model

The Poisson model was proposed in 1961 (Munro & Gilbert) and states that “the object of treating a tumour by radiotherapy is to damage every single potentially malignant cell to such an extent that it cannot continue to proliferate.” A formula for calculating the probability of tumour cure after irradiation of N cells was created.

Poisson statistics are used to determine the probability of an event occurring in relation to the known average number of events occurring. The events may be over time, or in the case of radiotherapy, per dose.

The formula for determining the probability of a certain number of events (k), where the average number of events λ is:

For instance, if it is known that 10 cars pass through a toll gate every minute on average, the chance that only 2 cars pass would be given by:

(2)When enough radiation is delivered to the tumour mass that 1 lethal hit would be expected per cell, the likely percentage of cells receiving one (or more) lethal hits is 63%. Therefore, 37% of cells would receive no lethal hits and therefore the tumour would most likely not be cured. The probability of zero surviving clonagens is given by the tumour control probability (TCP) which is equal to where – λ is the average number of surviving clonagens. For the above example, if 100 clonagens were present within the tumour, an average of 37 clonagens would survive. Therefore the TCP would be e^{-37}.

As the lethal hits per clonagen increases, the TCP also increases. The increase occurs rapidly once the lethal hits per clonagen exceeds 3 – 4 (likelihood of surviving clonagens begins to approach zero at this point).