Department of Mathematics, Chalmers University of Technology, and Göteborg University

I supervise in ECMI Modelling week 2001, Klagenfurt, Aug. 31-Sept. 9 I suggest the following topics:

A pencil beam model for charged particles

This is an interdisciplinary problem in particle/medical physics concerning electron and photon dose calculations of radiative transfer/radiation treatment.

The project is to model a forward-peaked scattering process in the linear transport equation and study both asymptotic behavior and numerical approximations for the such obtained beam particles.
A front line application is, e.g. in radiation therapy where the objective is twofold: (i) To maximize the deposit of electron dose (energy) inside seek cells (ii) To minimize the deposit of energy in surrounding healthy tissue. In this way one has an optimization (max-min) problem with many challenging and interesting aspects. A more detailed abstract can be found in the following file:

abstract (ps-file) , abstract (pdf-file) .

Implementations are easily done writing MatLab codes. However, knowledge in the following programing/software may be very much useful:
Mathematica, Fortran, PDE tool-box, and if possible DSMC (Direct Simulated Monte-Carlo).

Here are some relevant literature:

M. Asadzadeh,
Streamline diffusion methods for the Fermi and Fokker-Planck equations,
Transport Theory and Statistical Physics, (1997), 319--340.

M. Asadzadeh,
Characteristic Methods for Fokker-Planck and Fermi Pencil Beam Equations,
Proceedings of 21th International Symposium on Rarefied Gas Dynamics,
ed. by R. Brun et al , Vol II, 205--212, Marseille 1998.

M. Asadzadeh,
A posteriori error estimates for the Fokker-Planck and Fermi pencil beam equations,
Math. Models and Methods in Appl. Sci., 10(2000), 737--769.

C. Borgers and E. W. Larsen,
The trasversely integrated scalar flux for a norrowly focused particle beam,
SIAM J. Appl. Math. 55(1955), 1--22.

C. Borgers and E. W. Larsen,
On the accuracy of the Fokker-Planck and Fermi pencil beam equations for charged particle transport,
Med. Phys. 23 (1996), nr. 10, 1749--1759.

C. Borgers and E. W. Larsen,
Asymptotic derivation of the Fermi pencil beam approximation,
Nucl. Sci. Eng. 123(1996), 343--357

D. Jette
Electron Beam Dose Calculations, Radiation therapy physics,
ed by A. Smith, Springer (1995), 95--121

D. Jette,
Electron dose calculation using multiple-scattering theory. A. Guassian multiple-scattering theory.
Med. Phys. 15 (1988), 123--137.

J. Lang,
Adaptive Multilevel Solution of Nonlinear Parabolic PDE System,
Lecture Notes in Computational Science and Engineering, Springer , 16 (2000).

E. W. Larsen,
The amplitude and radius of a radiation beam,
Proceedings of the International Conference on Latest Developments and Fundamental Advances in Radiative Transfer (Los Angeles, CA, 1996). Transport Theory Statist. Phys. 26 (1997), no. 4-5, 533--554.

G. C. Pomraning,
The Fokker-Planck operator as an asymptotic limit,
Math. Models and Methods in Appl. Sci., 2(1992), 21--36.

G. C. Pomraning and A. Prinja,
High-order multiple scattering theories for charged particle transport,
Med. Phys. 23 (1996), nr. 10, 1761--1744.