Mathematical Contest in Modeling, 2003

The event

The Mathematical Contest in Modeling is an annual event, sponsored by the Consortium for Mathematics and Its Applications, in which students at colleges and universities all over the world are asked to develop and analyze mathematical models of open-ended, practical problems for which no direct solutions are known. Participants work in teams of three and are permitted to use libraries, computers, and other inanimate sources of knowledge and inspiration. The organizers of the contest propose three problems; over the four days of the contest, each team selects one of these problems, designs, implements, and analyzes a model, and writes a substantial report presenting its results.

The nineteenth Mathematical Contest in Modeling was held on February 6-10, 2003. Grinnell fielded four teams this year:

The problems

Here are the problems that COMAP posed this year. The team comprising Zelealem Yilma and Ananta Tiwari chose to work on problem B, the other three teams on problem A.

Problem A: The stunt person

An exciting action scene in a movie is going to be filmed, and you are the stunt coordinator! A stunt person on a motorcycle will jump over an elephant and land in a pile of cardboard boxes to cushion their fall. You need to protect the stunt person, and also use relatively few cardboard boxes (lower cost, not seen by camera, etc.)

Your job is to:

Note that, in “Tomorrow Never Dies”, the James Bond character on a motorcycle jumps over a helicopter.

Problem B: Gamma knife treatment planning

Stereotactic radiosurgery delivers a single high dose of ionizing radiation to a radiographically well-defined, small intracranial 3D brain tumor without delivering any significant fraction of the prescribed dose to the surrounding brain tissue. Three modalities are commonly used in this area; they are the gamma knife unit, heavy charged particle beams, and external high-energy photon beams from linear accelerators.

The gamma knife unit delivers a single high dose of ionizing radiation emanating from 201 cobalt-60 unit sources through a heavy helmet. All 201 beams simultaneously intersect at the isocenter, resulting in a spherical (approximately) dose distribution at the effective dose levels. Irradiating the isocenter to deliver dose is termed a “shot.” Shots can be represented as different spheres. Four interchangeable outer collimator helmets with beam channel diameters of 4, 8, 14, and 18 mm are available for irradiating different size volumes. For a target volume larger than one shot, multiple shots can be used to cover the entire target. In practice, most target volumes are treated with 1 to 15 shots. The target volume is a bounded, three-dimensional digital image that usually consists of millions of points.

The goal of radiosurgery is to deplete tumor cells while preserving normal structures. Since there are physical limitations and biological uncertainties involved in this therapy process, a treatment plan needs to account for all those limitations and uncertainties. In general, an optimal treatment plan is designed to meet the following requirements.

  1. Minimize the dose gradient across the target volume.
  2. Match specified isodose contours to the target volumes.
  3. Match specified dose-volume constraints of the target and critical organ.
  4. Minimize the integral dose to the entire volume of normal tissues or organs.
  5. Constrain dose to specified normal tissue points below tolerance doses.
  6. Minimize the maximum dose to critical volumes.

In gamma unit treatment planning, we have the following constraints:

  1. Prohibit shots from protruding outside the target.
  2. Prohibit shots from overlapping (to avoid hot spots).
  3. Cover the target volume with effective dosage as much as possible. But at least 90% of the target volume must be covered by shots.
  4. Use as few shots as possible.

Your tasks are to formulate the optimal treatment planning for a gamma knife unit as a sphere-packing problem, and propose an algorithm to find a solution. While designing your algorithm, you must keep in mind that your algorithm must be reasonably efficient.

Results

This year, 492 teams submitted complete entries. Eleven of these entries were judged Outstanding; seventy others, Meritorious. One hundred fifty-three received Honorable Mentions in COMAP's report. The remaining 258 teams were classified as Successful Participants.

The team of Atanas Djumaliev, Rajendra Magar, and Dessislava Dimova earned the rating of Honorable Mention from COMAP's judges. The other three teams were ranked as Successful Participants.

This document is available on the World Wide Web as

http://www.math.grinnell.edu/mcm-2003.xhtml