Computerized Tomography: The New Medical X-Ray Technology

Abstract

Computerized X-ray tomography is a completely new way of using X-rays for medical diagnosis. It gives physicians a more accurate way of seeing inside the human body and permits safe, convenient, and quantitative location of tumors, blood clots and other conditions which would be painful, dangerous, or even impossible to locate by other methods. Although each tomography machine costs hundreds of thousands of dollars, hundreds of tomography machines are already in use. A mathematical algorithm to convert X-ray attenuation measurements into a cross-sectional image plays a central role in tomography. Sophisticated mathematical analysis using Fourier transforms has led to algorithms which are much more accurate and efficient than the algorithm used in the first commercial tomography machines. We show how some of the algorithms in actual use have been developed. We also discuss some related mathematical theorems and open questions. 1. Introduction. In computerized tomography, X-ray transmission measurements are recorded on a computer memory device rather than on film, and a sophisticated mathematical algorithm is applied. This produces a numerical description of tissue density as a function of position within a thin slice through the body. The physician examines this function by use of visual displays. In the ordinary medical use of X-rays, the picture is something like a shadow; any feature in line with denser bone tissue tends to be blocked out. In other words, if we could make a great many pictures, each of a thin slice perpendicular to the beam of X-rays, the actual X-ray picture is formed by superposition of all these hypothetical pictures, i.e., it is a kind of multiple exposure. Com- puterized X-ray tomography provides a picture of a single thin slice through the body, without superposition. The word tomography is related to the Greek word tomos meaning cut or slice. Imagine a thin slice, say through the head, perpendicular to the main body axis. Several hundred parallel X-ray pencil beams are projected through the head in the plane of this slice, and the attenuation of X-rays in each beam is measured separately and recorded. (In earlier machines a single beam has been used by translating it parallel to itself within the plane; some of the later machines discussed in Section 3 use fan rather than parallel arrays of beams.) Another set of parallel beams is used within the same plane but at an angle of perhaps 10 or so with the first set, and measurements are taken again. The process is repeated until measurements have been taken for a grid covering all directions in the plane. An elaborate calculation then permits approximate reconstruction of the X-ray attenuation density as a function of position within the slice. In appropriate units, tissue density in the head varies roughly between 1.0 and 1.05 with the exception of bone which has a density of about 2. Some features of medical interest are indicated by variations of density as small as .005. Reconstructing tissue density with adequate accuracy at a sufficiently fine grid of points is thus a challenging project. Mathematically we may describe the problem as follows. Consider a fixed plane through the body. Let f(x,y) denote the density at the point (x,y), and let L be any line in the plane. Suppose we direct a thin beam of X-rays into the body along L, and measure how much the intensity is attenuated by going through the body. It is easy to see that the logarithm of the attenuation factor is given