I have a medical physics problem - I want to maximise the dose absorbed by a brain tumour whilst minimising the dose in the rest of the brain, especially certain organs, such as the pituitary gland, optic nerves and cochlea (organs at risk, OAR). The dose is delivered by syringes which have 3 different entry points, placing spheres of radiation within the tumour along their paths (imagine the syringes create a cylinder in which the spheres are placed, so I will have 3 cylinders containing spheres of radiation).So, my goal function :a target absorbed dose and homogeneity for the tumourminimise absorbed dose for OAR (with an absolute upper limit)minimise absorbed dose for normal brain tissue (with upper limit that is lower than for the OAR)And the variables I can change include:Entry points of syringes Directions of syringesWhere to place spheres along the syringe's path (spacing along the cylinder)At some later point, I would like to implement variable sphere size between a minimum and maximum volume, but for now I'll keep it simple...I am completely new to inverse problem solving algorithms and am at a complete loss as to where to start looking for answers. Any help or pointers towards good material to read would be greatly appreciated.EDIT: I should have mentioned that the spheres emit almost purely beta radiation.
Algorithms for radiation treatment planning
pde;algorithms;reference request;inverse problem;optimal control
This problem is actually more of an optimal control problem for a partial differential equation. As a starting point, I would recommend the following books:F. Trltzsch, Optimal control of partial differential equations, AMS, 2010M. Hinze, R. Pinnau, M. Ulbrich, S. Ulbrich, Optimization with PDE constraints, SpringerThe first one in particular is an excellent textbook.Your specific problem, however, is far from the standard examples treated in textbooks: First, your equation is the radiative transport equation, which behaves very different from standard second-order equations. There are some groups working on this topic, though: You should look at the papers of Rick Barnard, Martin Frank and Michael Herty, e.g., http://arxiv.org/abs/1105.5261. There's also a paper by J. Tervo, M. Vauhkonen, E. Boman.Second, your variables involve the geometry of the problem, which leads to a so-called shape optimization problem. Here the standard reference would beM.C. Delfour, J.-P. Zolsio, Shapes and Geometries, SIAM, 2011J. Haslinger, R. A. E. Mkinen, Introduction to Shape Optimization: Theory, Approximation, and Computation, SIAM, 2003There's also recent work by Volker Schulz. (I'm not an expert in this area, so these are very selective references to give you some keywords to search for.)