Page 289 - Biomedical Engineering and Design Handbook Volume 2, Applications

CHAPTER 10

THE PRINCIPLES OF X-RAY

COMPUTED TOMOGRAPHY

Peter Rockett
Oxford University, Oxfordshire, England

Ge Wang
University of Iowa, Iowa City, Iowa

10.1 INTRODUCTION 267 10.4. THE MICROTOMOGRAPHY
10.2. THE INTERACTION OF X-RAYS SYSTEM 291
WITH MATTER 269 REFERENCES 314
10.3. THE MATHEMATICAL MODEL 275

10.1 INTRODUCTION

The object of x-ray computed tomography is to reproduce, in the form of digital data, the internal
structure of three-dimensional bodies with as little distortion as possible. This is achieved by the
mathematical reconstruction of a series of two-dimensional images produced by the projection of
an x-ray beam through the body in question. In this respect the form and functioning of x-ray
computed tomography systems are closely linked to the physics of the interaction of x-ray photons
with the material body and to the principles that govern the mathematical process of image recon-
struction. In depth these are complex matters, but the fundamentals can be simply explained to
provide a sound basis for further study. 1–4
The relative values of absorption and transmission of adjacent elemental ray paths in an x-ray
beam provide the contrast in a projected image. The associated interaction of the x-rays with the
material body can be defined in terms of collision cross sections that describe the probability of
5
particular events occurring along the ray path. Here, the principal effects, for x-ray energies below
1 MeV, are identified in Sec. 10.2 as the photoelectron event and the Compton scattering event. 6,7
These effects are combined to define a linear attenuation coefficient m(x) that describes the variation
of photon intensity Φ along the ray path x according to the exponential relation Φ=
m(x) dl), where Φ is the initial intensity of the ray and l is the path length.
Φ exp(–∫ l
0 0
Taking account of the role of attenuation in the formation of image contrast, the method by which
the material features along the ray path are reconstructed can be mathematically formulated in detail.
In this process it is recognized that the two-dimensional object function f(x, y) is represented by the
attenuation coefficient m(x, y). A key assumption in this process is that x-ray photons travel in straight
lines and that diffraction effects may be ignored. Given the energetic nature of x-ray photons, this is
a reasonable approach and enables considerable simplification of the underlying principles that are

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