

MCM: The Mathematical Contest in Modeling
ICM: The Interdisciplinary Contest in Modeling
2003
MCM Problems
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:
 determine what size boxes to use
 determine how many boxes to use
 determine how the boxes will be stacked
 determine if any modifications to the boxes would help
 generalize to different combined weights (stunt person &
motorcycle) and different jump heights
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 welldefined, 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 highenergy photon
beams from linear accelerators.
The gamma knife unit delivers a single high dose of ionizing radiation
emanating from 201 cobalt60 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, threedimensional 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.
 Minimize the dose gradient across the target volume.
 Match specified isodose contours to the target volumes.
 Match specified dosevolume constraints of the target and
critical organ.
 Minimize the integral dose to the entire volume of normal
tissues or organs.
 Constrain dose to specified normal tissue points below tolerance
doses.
 Minimize the maximum dose to critical volumes.
In gamma unit treatment planning, we have the following constraints:
 Prohibit shots from protruding outside the target.
 Prohibit shots from overlapping (to avoid hot spots).
 Cover the target volume with effective dosage as much as possible.
But at least 90% of the target volume must be covered by shots.
 Use as few shots as possible.
Your tasks are to formulate the optimal treatment planning for a
gamma knife unit as a spherepacking 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.
2003
ICM Problem
PROBLEM C:
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