Notes on Meeting in Baltimore. Nov 12-13, 2001

Our discussions focused mostly on optimization issues in IMRT: Intensity-modulated radiation treatment planning. We found many areas that require further investigation, and in which improved optimization techniques could have a large impact on practice.
The main treatment device we discussed was a linear accelerator equipped with a multileaf collimator (MLC). Terminology for various treament schemes associated with this device is roughly as follows.

Terminology

1) Step-and-shoot.

The beam is aimed at the tumor from a number of different angles (say, 6). At each angle the MLC is set to a number of different shapes (say, 3) and a beam of a certain weight is delivered. The allowable shapes are dictated by the physical characteristics of the MLC (see below). The shapes used at each angle may or may not conform to the shape of the actual tumor.

2) "Conventional".

Here, the beam is aimed from a number of different angles, but the shape of the MLC aperture is chosen to be conformal to the "beam's-eye view" of the tumor. The beam may be filtered through a wedge that attenuates the radiation across the beam, in a fashion that can be modeled by a fixed angle (orientation of the wedge) and an affine function. Usually, one beam is delivered from each angle. Sometimes, two beams are delivered (one with and one without the wedge, or with the wedge oriented in two different ways). A scheme of this type is the most popular one in current use.

3) IMAT.

In this scheme, the gantry is moved during delivery of the beam. The aperture shape may be varied continuously during the move. It has been found that IMAT can be approximated by a step-and-shoot procedure in which beams are delivered from angles along the path separated by about 10 degrees. Hence IMAT can be modeled and optimized by similar techniques to IMRT, except that there are additional constraints on the aperture shapes, due to the fact that the speed at which collimator leaves can move between successive approximating positions is limited.

More details

An additional degree of freedom is that there may be two to four different beam types that can be used (electron and photon, different beam energies.)
Data for the model consist of "dose matrices" associated with each pencil beam. Each such matrix indicates the total radiation delivered to each voxel in the target area by a "pencil beam" of unit weight with a given orientation. The amount of data for a real target is potentially huge: There are approximately 100 pencil beams that can be delivered from each direction, around 36 candidate directions, and up to four beam types. For the 20-cm diameter water cylinder, used for algorithm prototyping, the dose matrices for a single direction have been precalculated. A rotational transformation can be applied to obtain the dose matrices for other angles. For a "real" target, the Monte Carlo technique must be used to calculated complete dose matrices from every candidate direction. For some parts of the modeling and optimization, it may be possible to use approximate dose matrices obtained via a much cheaper calculation (TERMA ??).
The objective is essentially to deliver a prescribed dose to the target area, while avoiding overdosing of nearby "sensitive structures" and of surrounding normal tissue. Treatment planning aims to achieve this goal by choosing the variables in the model appropriately. These variables vary according to the type of treatment (see above) but may include the following:
- angles at which the beam is to be delivered;
- weight of each beam;
- aperture shapes at each angle;
- use of wedge and its orientation;
- energy of beam to be used.
In current practice, the objective function often takes the form of a weighted least-squares deviation from target dose values. Constraints are sometimes imposed on dosage in a given region. In addition, there are constraints of the form "no more than 30% of the target can receive a dose of more than 20 Gy" that are easy to specify via the dose-volume histogram (DVH). These are however difficult to model from an optimization viewpoint.

Total Body Irradiation

Another somewhat different problem we discussed was total-body irradiation (TBI) used in treatment of leukemia patients prior to bone marrow transplants. In the perferred mode of delivery, the patient is laid flat and a wide beam is scanned over him from toe to head. Since the distance of aperture to the point of incidence of the beam on the patient's body changes with angle of the beam, and since the volums of body exposed to the beam varies with body width/thickness at the point of incidence, the amount of radiation delivered to the body varies with angle. The objective is to vary the rate of scan so that a uniform dose is delivered to all parts of the body.

State of Art

The recent paper [1] optimizes a step-and-shoot plan by fixing 6 beam angles, then choosing 3 aperture shapes and beam weights to be delivered from each angle, via a simulated annealing approach. Candidate changes in each variable were rejected immediately if they resulted in infeasible MLC apertures. The technique for handling DVH constraints was to change the objective function between iterations, applying higher weights to those voxels in which dose "just exceeds" the preferred limit in an attempt to force the right proportion of them to drop below the preferred limit. The algorithm resulted in a very effective plan, involving many fewer MLC aperture shapes than a plan developed by a commercial vendor. Since 7 seconds are required on the equipment in use at UMB to change MLC leaf configurations, there is a large reduction in treatment time.
Michael reported on a leaf sequencing algorithm developed by Boland et al. It overcomes the difficulties associated with specifying allowable aperture shapes by defining a network that essentially hard-wires the allowbility constraints. Flows through the network define valid configurations of the MLC leaves. A prescribed dose can be solved as a network flow problem. Michael has a simple implementation in GAMS. The objective in this problem is the total dose delivered, NOT the number of different shapes used. A depth-first search can be used to "back out" the different shapes from the solution of the network flow problem.