Note on uncertainty in Monte Carlo dose calculations and its relation to microdosimetry

The Type A standard uncertainty in Monte Carlo (MC) dose calculations is usually determined using the "history by history" method. Its applicability is based on the assumption that the central limit theorem (CLT) can be applied such that the dispersion of repeated calculations can be modeled by a Normal distribution. The justification for this assumption, however, is not obvious. The concept of stochastic quantities used in the field of microdosimetry offers an alternative approach to assess uncertainty. This leads to a new and simple expression.

The value of the MC determined absorbed dose is considered a random variable which is comparable to the stochastic quantity specific energy, z. This quantity plays an important role in microdosimetry and in the definition of the quantity absorbed dose, D. One of the main features of z is that it is itself the product of two other random variables, specifically of the mean dose contribution in a 'single event' and of the mean number of such events. The term 'single event' signifies the sum of energies imparted by all correlated particles to the matter in a given volume. The similarity between the MC calculated absorbed dose and the specific energy is used to establish the 'event by event' method for the determination of the uncertainty. MC dose calculations were performed to test and compare both methods.

It is shown that the dispersion of values obtained by MC dose calculations indeed depend on the product of the mean absorbed dose per event, and the number of events. Applying methods to obtain the variance of a product of two random variables, a simple formula for the assessment of uncertainties is obtained which is slightly different from the 'history by history' method. Interestingly, both formulas yield indistinguishable results. This finding is attributed to the large number of histories used in MC simulations. Due to the fact that the values of a MC calculated absorbed dose are the product of two approximately Normal distributions it can be demonstrated that the resulting product is also approximately normally distributed.

The event by event approach appears to be more suitable than the history by history approach because it takes into account the randomness of the number of events involved in MC dose calculations. Under the condition of large numbers of histories, however, both approaches lead to the same simple expression for the determination of uncertainty in MC dose calculations. It is suggested to replace the formula currently used by the new expression. Finally, it turned out that the concept and ideas that were developed in the field of microdosimetry already 50 years ago can be usefully applied also in MC calculations.