ABSTRACT
As applied to radiation transport applications in radiotherapy and dosimetry, the Monte
Carlo method provides a numerical solution to the Boltzmann transport equation (e.g. Kase
and Nelson 1978; Duderstadt and Martin 1979) that directly employs the fundamental micro-
scopic physical laws of electron-atom and photon-atom interactions. Monte Carlo simulation
faithfully reproduces the individual particle tracks, in a statistical sense, within current knowl-
edge of the physical laws: the scattering and absorption cross-sections. The radiation fields’
macroscopic features (e.g. the average track-length per incident photon in a given volume of
space) are computed as an average over many individual particle simulations or histories. If the
true average
x exists and the distribution in x has a true finite variance, s
, the Central
Limit Theorem (Lindeberg 1922; Feller 1967) for energies 0.001 MeV up to 50 MeV guar-
antees that the Monte Carlo estimator for
x, that is referred to here as h xi, can be made
arbitrarily close to
x by increasing the number, N, of particle histories simulated. Moreover,
the Central Limit Theorem predicts that the distribution of h xi is Gaussian, characterised by a
variance s
that may be simply estimated in the simulation. The Central Limit Theorem also
predicts that in the limitN/N, s
/0. This limiting result is also proven by the Strong Law of
Large Numbers (Feller 1967).
