Penalties for Deviations from Prescribed Dose

The histogram-based approach may easily be used to implement an objective function that penalizes deviations from prescribed doses on a voxel-by-voxel basis. A weighted-sum-of-squared-deviations objective function could be implemented simply by specifying:

where is the prescripted is a weight indicating the relative importance of getting as close to the prescribed dose as possible for the particular tissue type r. (Note that we don't actually need to use a finite number M of bins for this objective function; we could simply iterate over all the voxels in instead.)

Such an objective function should cause the optimizer to:

• Keep the doses close to the prescribed doses, particularly in critical tissue or other tissue with high weights .
• This should naturally lead to high dose gradients at tissue boundaries, particularly boundaries between tumor tissue and normal critical tissue.

Alternatively, we could formulate even simpler objective functions with penalties proportional to the amount of normal tissue dosed over a certain threshold, plus additional penalties proportional to the amount of tumor tissue receiving doses under another threshold. Or, we could allow the practitioner to graphically specify more complicated penalty functions for specific deviations from the prescribed dose via the use of splines.

Any of the above possible objective functions could be supplemented with an additional term that explicitly penalizes nonuniform dose distributions throughout the tumor. For example, some multiple of the standard deviation of the dose values found in the tumor tissue's histogram could be subtracted from the rest of the objective function.