Contributions to the inverse problem of radiation transport

• Radiation field inside an empty shell source.
• Direct and inverse problem for this idealization. Analytical solution of the integral equation.
• Zero collision flux when the shell source is filled with a material media.
• Applications to medical physics and nondestructive assay.
• Numerical algorithm and analytical results verified with Monte Carlo calculations of fluxes and doses.

We present an integral equation that describes the uncollided particle flux for the case of an inward spherical shell source of radius R. This is a reasonable description, for example, of a point source that moves on a spherical surface located at distance R from the target of a radiation treatment. The additional assumption of conditions for radial symmetry allows the derivation of an integral equation that relates the scalar flux to the description of the beam source as function of the angle between the direction of the source particles and the normal to the sphere.Analytical and numerical solutions for this integral equation are successfully compared with, respectively, known analytical results and with Monte Carlo simulations. The integral equation can then be used for solutions of the inverse problem: given the flux obtain the source, i.e. the shape of the beam. A numerical algorithm was developed for this purpose as well as an analytical solutions based on the solution of the integral equation by the use of the Laplace transform.The optimal shape for the beam is then obtained based on the constraint that the source has to be positive and finite everywhere, allowing the design of appropriate collimators for the beams. Monte Carlo calculations as a function of the number of collisions show that the uncollided flux for the beam so determined behaves as expected and that penumbra effects due to multiple collisions are sufficiently small (∼20%) to consider the beam as a good first guess for an iterative procedure for the design, for example, of 3-D conformal radiotherapy treatment.