Optimisation in Radiotherapy

I first discovered this surprising application of linear programming while looking through annual conference papers of the NZ Operations Research Society (ORSNZ) and have since had the opportunity of talking about the problem with Matthias Ehrgott from Auckland University's Dept of Engineering Science. It is a good example of the way new problems are being presented to mathematicians to work on in this increasingly technological world.

In early external beam radiotherapy treatments, a whole area of the body would be irratiated equally by the beam. Obviously, the maximum radiation dose that could be given to a tumor site has been restricted by the tolerance and sensitivity of the surrounding nearby healthy tissues. An improvement was to create a filter in front of the beam for each patient so as to shape the beam to the tumour. The development of a device called a multileaf collimator (MLC), with moveable 5 mm thick blades (leaves) of lead/tungsten that can block off sections of the beam, together with improvements in imaging techniques, has lead to 3-Dimensional Conformal Radiotherapy, where shaped radiation beams are aimed from several angles of exposure to intersect at the tumour, providing a much larger absorbed dose there than in the surrounding, healthy tissue.

Intensity-Modulated Radiation Therapy (IMRT) is a refinement on conformal radiotherapy, where the intensity beam is controlled, or modulated within the given area, using the MLC. Use of IMRT is growing in more complicated body sites, such as the neck and prostate. Auckland has had this high tech equipment for two or three years.

(Pictured right: multi-leaf collimator)

IMRT in brief:
•Beams (photon/ electron) produced by the linear accelerator are focused on tumour from different directions (3 to 9)
•Intensity across each beam can be modulated by multi-leaf collimator (effectively, beam divided into large no. of beamlets/ bixels)
•Aim:
Focus radiation so that enough dose is delivered to tumour (unlike normal cells, cancerous cells with damaged DNA can’t reproduce) while limiting dose to critical organs and healthy tissue.

The 3 optimisation problems:

1. The geometry problem: What angles (beam directions) should be used?

2. Finding optimal beam intensities for each angle

3. Optimising the delivery schedule

The beam intensity problem can be formulated as a very large linear programme:

•Multi-Objective function:
One for tumour, critical organ(s) and normal tissue (minimise underdosing of the tumour and to minimise overdosing healthy organs and other tissues)
•Variables:
Let xi be the intensity at bixel i
(the MLC can have 40 leaves and 40 stops, so up to 40 × 40 = 1600 variables for each beam direction)
•Constraints:
The region of the body is divided into 3-D volume elements (voxels), and so one constraint for each voxel, given by dose levels of oncologists prescription
(order of 100,000 constraints)
Note: The optimisation computes a set of possible treatment plans for which less overdosing of healthy organs implies more underdosing of the tumour and vice versa. It assists the planner to select one such plan that is best for the patient.

Once the intensities are determined, there is still the problem of how they can be efficiently delivered using the collimator settings, so that the number of shapes and total radiation time is minimised. Effectively, this boils down to the problem of how to decompose a (40 × 40 ) matrix into the sum of matrices whose non-zero elements are identical. No algorithm exists for minimising the number of matrices in the decomposition, so here we have a new problem for computer folks.

The highly complex IMRT treatment has thrown up challenging problems for mathematicians working in OR. Matthias Ehrgott has been working on the multi-objective linear programming problem mentioned above. His time and assistance in giving me an insight into this work has been greatly appreciated.