Real-Life Math
For the student, a mistake in math can cause embarrassment at the
blackboard. But for Mike Patterson and other medical physicists, a mistake
in math has much more serious consequences.
As a medical physicist,
Patterson calculates the radiation doses for cancer patients undergoing radiation
treatment.
"There have been incidents where patients have been killed
by radiation therapy accidents," he says.
Such incidents are rare.
But the delivery of excess radiation can damage normal tissues such as the
skin or spinal cord. On the other hand, if too little radiation reaches the
tumor, the chances of controlling it are reduced.
You are a medical
physicist. You show up for work early in the morning and begin the usual tests
to make sure everything is in running order and the day's treatments
are going to be safe.
You perform a test measurement to ensure the
treatment machine output is correct. If it is, the machine should deliver
a dose of 2.00 gray (Gy). A gray is the unit of radiation dose and is equal
to 1 joule (J) per kilogram (kg).
To measure the output of the radiation
treatment unit, you use an ionization chamber, a small capsule filled with
air. When radiation passes through the chamber, it ionizes some of the air
molecules. These positively and negatively charged molecules can be collected
and measured as an electrical charge.
The charge is proportional to
the radiation dose. But it also depends on how many air molecules are in the
chamber. And the number of air molecules in the chamber depends on environmental
conditions, specifically the temperature (t) and pressure (p) because the
chamber is not sealed.
You need to correct for this effect. The chamber
is calibrated for a specific set of conditions:
t = 22 C
p = 760
millimeters (mm) Hg
If you use the chamber under these conditions,
the radiation dose is calculated this way:
Radiation dose = chamber
reading x calibration factor
The chamber reading is 1.83. The
calibration factor for this chamber is 1.112. That means:
Radiation
dose = 1.83 x 1.112
Radiation dose = 2.03 Gy
However,
you notice that the room's temperature (t) is 24 degrees today and the
pressure (p) in the room is 740 mm Hg. These conditions are different than
the calibration conditions. You're going to need to multiply your chamber
reading by a correction factor to get the right number.
Here's
the formula:
Correction factor = (calibrated pressure / new pressure)
x [(273 + new temperature) / 295]
Now we have:
Radiation
dose = chamber reading x calibration factor x correction factor
What
is the actual output of the treatment machine? Do you think it would be safe
to use the machine?