Aside from the geometric assumptions of infinite homogeneous slabs, the pencil beam
model does not handle changes in scatter from lateral heterogeneities; deviation of up to 5% can
be expected for low energy photon beams, and 14% for high energy photon beams8,9
1.4 Solving the Boltzmann Equation
At the most fundamental level, radiation transport simulation involves determining the
distribution of the particles (in physical space, direction, time, and energy) after they have
traversed some distance into a medium. Formally, a way to begin this discussion is by writing
the general equation for particle transport from kinetic theory of gases. This theory does not
explicitly treat individual particles, but rather describes the aggregate behavior of a continuous
particle distribution in a statistical manner. Let n be the expected number of particles within a
differential volume element d3r at position r with a velocity (which is directly related to its
kinetic energy) that is within a differential velocity d3v, at time t. Then, if a general solution is
found for the "particle density function" n(r, v, t) in phase space, then the transport problem is
solved.
When transporting the radiation through an arbitrary volume V, bounded with a surface
area S, the rate of change of number of particles is due to flow out through the surface of V,
collision events within V, or sources (scattering and independent sources) of particles located
inside of V. (all the terms of rate of change; commonly the first two terms are losses and the last
term is production)
chan ge in )= to flow + to collisionsI + to sources
nat time t houhSin V in V