THAT'S MATHS:Sophisticated mathematical processing is used to produce useful images, helping to save lives, writes PETER LYNCH
Many lives are saved each year through a synergistic combination of engineering, computing, physics, medical science and mathematics. This combination is CT imaging, or “computed tomography”, which is now an essential tool for medical diagnosis.
The story began in 1895, when Wilhelm Roentgen made the first radiograph using what he called X-rays. These high-energy electromagnetic beams can penetrate body tissues where light cannot reach. Internal organs can be examined non-invasively and abnormalities located with precision. For his trailblazing work, Roentgen was awarded the first Nobel Prize in Physics in 1901.
The power and utility of X-ray imaging has been greatly expanded by combining X-rays with computer systems to generate three-dimensional images of organs of the body.
The diagnostic equipment used to do this is called a CT scanner (or Cat scanner). The word tomography comes from the Greek “tomos”, or slice, and a CT scan is made by combining X-ray images of cross-sections or slices through the body. From these, a 3-D representation of internal organs can be built up.
Radiologists can use CT scans to examine all the major parts of the body, including the abdomen, chest, heart, and head. In a CT scan, multiple X-ray images are taken from different directions. The X-ray data is then fed into a tomographic reconstruction program to be processed by a computer. The image reconstruction problem is essentially a mathematical procedure.
The deduction of the tissue structure from the X-rays is done using a technique first devised by an Austrian mathematician, Johann Radon. He was motivated by purely theoretical interests when, in 1917, he developed the operation now known as the Radon transform. He could not have anticipated the great utility of his work in the practical context of CT. Reconstruction techniques have grown in sophistication, but are still founded on Radon’s work.
As they pass through the body, X-rays are absorbed to differing degrees by body tissues of varying optical density. The total attenuation, or dampening, is expressed as a “line integral”, the sum of the absorptions along the path of the X-ray beam. The more tissue along the path, and the denser that tissue, the less intense the beam becomes. The challenge is to determine the patterns of normal and abnormal tissue from the outgoing X-rays.
If the X-ray patterns were uncorrupted, the mathematical conversion to 3-D images would be straightforward. In reality, there is always noise present, and this introduces difficulties: Radon’s “inverse transform” is very unstable and error-prone, so a stable modification of the method, known as “filtered back-projection”, is used. More accurate algorithms have been developed in recent years, and research in this area is continuing.
Applications of tomography are not confined to medicine. The technique is also used in non-destructive materials testing, both in large-scale engineering and in the manufacture of microchips. It is also used to compute ozone concentrations in the atmosphere from satellite data.
In addition to CT, there are numerous other volume-imaging techniques. Electron tomography uses a beam of electrons in place of the X-rays, ocean-acoustic tomography uses sound waves, and seismic tomography analyses waves generated by Earth movements to understand geological structures. All involve sophisticated mathematical processing to produce useful images from raw data.
Peter Lynch is professor of meteorology at University College Dublin. He blogs at thatsmaths.com
