Solution of the Fokker–Planck Pencil Beam Equation for Electrons by the Laplace Transform Technique

While many medical physicists understand the basic principles underlying Monte Carlo codes such as EGS [Ka00], Geant [Wr01], and MCNP [Br93], there is less appreciation of the capabilities of deterministic methods which in principle can provide comparable accuracies to Monte Carlo. Only within the last years have serious studies been made on the appliance of deterministic calculations to medical physics applications. The most versatile and widely used deterministic methods are the P N approximation [Da57]; [SeViPa00], the S N method (discrete ordinates method) [ViBa95]; [ViSeBa95], and their variants [SeVi94]; [RoViVo06]. The method of discrete ordinates has been used successfully in neutral particle applications [D096]; [Da92] and gamma ray transport calculations for many years. The calculations for these two types of radiation are done very similarly, since they are both neutral particles. On the other hand, to our knowledge, the P N approximation has not yet been applied in the solution of the charged particle pencil beam transport equation. Pencil beam equations are used to model, e.g., problems of collimated electron and photon particles penetrating piecewise homogeneous regions. The collisions between the beam particles and particles from beams with different directions cause deposit of some part of the energy carried by the beams at the collision sites. To obtain a desired “amounts of energy deposited at certain parts of the target region” (dose) is of crucial interest in radiative cancer therapy.