Optimization

The radiotherapy treatment planning problem can mathematically be reformulated as an optimization problem: deliver a sufficient amount of dose to the tumour to eradicate the malignant cells, then minimize the harmful dose to the healthy organs or tissues.

There are basically 2 types of approaches: unconstrained and constrained optimization. The advantages of unconstrained optimization is that it can be extremely fast, and the implementation is relatively easy. However, real-life problems always have constraints: there are always limitations on resources, achievements, etc. These restrictions are usually reformulated into a composite objective function, where you are not optizing on what you really want. You may need several reoptimizations, comparing each result with the real problem, and heuristically adjusting the parameters of the unconstrained objective function.

This may be a good idea for some type of problems, as it can give you acceptable results within limited time. However, the result may be far away from the optimum of the problem you intend to solve, and heuristics are difficult to develop when the problem becomes more and more complex.

I myself am a profound proponent of constrained optimization. I want to have the maximum salivary gland sparing possible, and nothing less! For that reason, it is also convenient to know whether or not the maximum sparing was achieved (i.e. if the optimality conditions were met). And yes, to solve a constrained optimization problem you need to sequentially solve several unconstrained optimization problems, but then in a sophisticated manner, proven to converge.

The following series demonstrate the practical advantages of constrained optimization, by optimizing a one-dimensional dose-distribution (or folding a spline, if you prefer that visual).

Step 1: minimum tumour dose
Here the minimum dose is optimized up to a certain level. The green dot on the left is a maximum constraint for an organ. The optimization terminates once the goal is achieved, and in this case, the constraint is inactive: the result would be identical without having a constraint on the organ.

Step 2: maximum tumour dose
Here the dose in the middle of the tumour is reduced. At some point, the constraint on the green organ at the left becomes active.

Step 3: reduce organ dose
A final step is to further reduce the dose to the organ.

And that is how intuitive it works!