The Fermi pencil beam formula and the higher-order multiple scattering
theory due to Jette are shown to result from a perturbative treatment
of the linear Boltzmann equation with Fokker-Planck scattering. Using
asymptotic one-dimensional solutions for the transverse integrated (s
pherical) fluence as well as its variance, approximate higher-order pe
ncil beam theories are constructed. These simple and explicit formulae
are shown, by comparison with benchmark Monte Carlo results, to be si
gnificantly more accurate than the Fermi and Jette equations, particul
arly at large distances from the beam axis. (C) 1996 American Associat
ion of Physicists in Medicine.