Mathematics Awareness Week 1994

Mathematics and Medicine

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The Mathematics of Medical Imaging
(One in a series of six articles on Mathematics and Medicine being distributed by the Joint Policy Board for Mathematics in celebration of Mathematics Awareness Week 1994.)

If a gospel of medical imaging exists, it might read, "In the beginning, there was the X-ray." Now, nearly a century after German physicist Konrad Roentgen burned the first X-ray image into a photographic plate, the technology available for detecting disease and deformity in the body and tissues has advanced far beyond vacuum tubes and chemical emulsions.

The frontier of medical imaging now lies in tomography -- the art of figuring out what's inside the body by probing it from the outside. For this, physicians and medical researchers rely on an alphabet soup of technologies with names like CAT, MRI, and PET.

Computers are vital in tomography. With their ability to make many calculations quickly, computers transform vast streams of raw data gathered by scanning machines into vivid images of tumors, infections, internal bleeding, and other life-threatening conditions.

But if computers are the eyes of medical imaging, then mathematics is certainly its brain. The Nobel Committee recognized this by awarding half of the 1979 prize for physiology or medicine to U.S. physicist Allan Cormack. He worked out the mathematics of what we now call computer-assisted tomography, or CAT scanning. Cormack and British researcher Godfrey Hounsfield, who shared the prize, each independently figured out a way to reconstruct the internal structures of the body from essentially flat, two-dimensional X-ray images.

"When you look at the role of mathematics in medical imaging, the greatest influence by far has been in the development of X-ray computed tomography," notes Jerry L. Prince, an electrical engineer with the Center for Biomedical Visualization at Johns Hopkins University in Baltimore.

In CAT scanning, an X-ray scanner probes the patient from multiple angles, gathering information on the density of internal organs and tissues. In contrast, magnetic resonance imaging, or MRI, uses magnetic fields and radio beams to scan the body for differences in the chemical makeup of its tissues and structures.

In both CAT and MRI, the actual image is pieced together by mathematical recipes called algorithms. These algorithms contain the step-by-step instructions that tell a computer how to reconstruct a person's insides from an incoherent mush of scanner data. In a sense, a reconstruction algorithm works like a police sketch artist, combining information gathered from many points of view into a single portrait.

Moreover, in addition to its use in image reconstruction, mathematics is important in other aspects of medical tomography, says neuroscientist Barry Horwitz of the National Institutes of Health in Bethesda, Maryland. Researchers use mathematics to find out if a proposed new imaging technique is likely to work, before it's tested on actual patients.

For example, mathematical modeling played an important role in the development of a method for using positron emission tomography, or PET scanning, to map the brain's metabolism of glucose. Brain cells digest glucose as an energy source, so by determining where glucose concentrates, scientists can figure out which parts of the brain are involved in basic mental functions like learning and memory.

To develop glucose-based PET scanning, researchers created a mathematical model to simulate the technique's basic biochemical principles. The model showed that a radioactively labelled tracer chemical, designed to be very similar in structure to glucose, would be able to wend its way from the bloodstream into nerve cells just like the real thing.

In an actual PET scan, this tracer is injected into the patient. As the tracer is taken up by brain cells, the PET scanner measures its radioactivity. A reconstruction algorithm converts this information into an image of glucose metabolism in the brain.

In a medical situation, the influence of mathematics in tomography usually ends when an individual scan shows the location of a tumor or a leaking artery. In medical research, however, there's an additional role for mathematics: image analysis. Researchers routinely use mathematical techniques to compare tomographic scans of different people with similar diseases, identifying common features that may prove useful both in the laboratory and the clinic.

Horwitz and colleagues have undertaken such comparative studies in an effort to develop an MRI technique for detecting subtle differences in the brains of people with Alzheimer's disease. The researchers hope to use these differences as the basis for early diagnosis of this disease.

Back to Math Awareness Week 1994

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