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 tumour
  • minimise 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 syringes
  • Where 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.

  • 3
    $\begingroup$ Hi Richard Brown, and welcome to scicomp! Your question seems to be akin to "how do I solve/learn to solve inverse problems", and that is a very big question with massive scope. It would be helpful if you could narrow the focus of your question a bit. It would also help if you write out your equations (objective, constraints, etc...) in your question. $\endgroup$
    – Paul
    Jan 30, 2015 at 17:43

3 Answers 3


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:

The 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 be

There'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.)

  • $\begingroup$ Thanks, these are some good references that I'll continue to work through. However, I should have mentioned in the question that beta radiation is used, so to a first order approximation I can assume that all the energy is absorbed in a few mm radius of the spheres containing the radiation. In that case, does my problem become simply a shape optimisation problem? $\endgroup$ Feb 2, 2015 at 10:13
  • $\begingroup$ If I'm understanding you correctly, you are saying that (to a good approximation) there is no real transport of radiation. In this case, you have a pure geometric optimization problem, which is a different kettle of fish (though not necessarily any easier). $\endgroup$ Feb 7, 2015 at 21:40

In addition to Brian's reference, there was also a review article in SIAM Review a few years ago that summarized the state of research at the time: http://epubs.siam.org/doi/abs/10.1137/S0036144598342032


There has been a lot of research in this area over the last 20 years. It's appropriate to start by using search engines such as Google Scholar and Web of Science to look for survey and review articles. For example, you might look at



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