Consider X-rays incident on a block of tissue of rectangular section. Also, let us regard the tissue as having six "layers" as shown in the figure below.
Each "layer" stops a certain fraction of all the rays that arrive. For example, if the attenuation coefficient is 0.5/layer, then each layer stops half the rays. Thus if 128 photons are incident, and the thickness of the tissue is 6 layers, then we have:
Therefore, the attenuation of the tissue is (0.5) to the power of the number of layers.
This can be written formally as:
where L is the layer number.
The notion of layers is simply a convenience for illustration. Actually, the photons are stopped with depth. The same expression can be written in its more common form, using the depth of the tissue d as a continuous variable, and using the Greek letter mu to denote the depth dependent attenuation.
When tissues with different attenuation coefficients are in the path of a photon beam, the total attenuation of the beam depends on the distance traversed through each type of tissue and the attenuation coefficient of that type of tissue. For example, if we consider the following situation where we have tissue B of thickness 'b' inside another tissue A of thickness 'a':
Photon beam P is attenuated by an amount:
and photon beam Q is attenuated by an amount:
The following figure shows a point source, a flat object and a screen. From the geometry it is clear that if the object moves closer to the source the image formed on the screen becomes bigger, and if the object moves closer to the screen the image becomes smaller, until it becomes the same size as the object when the object is at the screen. The magnification is thus: (a+b)/a
The following figure shows a diffuse source and a point object. Here the source is magnified or reduced depending on the ratio of the source-to-object distance and the object-to-screen distance. The magnification of the source is b/a
When we have a finite source and a finite object, the image formed is:
the source scaled by b/a spatially convolved with the object scaled by (a+b)/a
When we have a solid object we can consider it to be made up of several parallel planes. Each plane produces an image as described above. The total image is the superposition of all of them.
The following simulation shows a hollow cylinder placed between an X-ray source and a screen. The cyclinder can be rotated, translated and moved using the scroll bars. After each re-positioning of the cylinder redraw the calculated images, and study the effects of depth dependent magnification, orientation and position.
A slice from a plane parallel to the screen is also displayed to see how orientation affects the image formation. The position of the slice can be selected by the user.
Simulation of X-ray image formation
In computerized tomography the object is regarded as comprised of small cells (or volume elements or voxels), which attenuate the X-ray. The attenuation coefficients of these volume elements or voxels is computed to reconstruct the internal structure of the object.
There are several methods used for such reconstruction. We shall consider one method called the Fourier transform method since the concepts have been developed in earlier modules.
Consider the object shown in the picture below. Several X-ray images of it are taken using pairs of X-rays sources and recording screens - A, B, C, D are shown. For convenience of calculation, the X-ray source is taken to be comprised of parallel rays as shown in the figure.
The object has a sinusoidally varying attenuation coefficient in the diagonal direction.
The images obtained in each recording instant will reflect the total attenuation seen by each ray. In all the images except 'C', the recorded signal will be constant - since the total variation of white through gray to black is same for all the rays. In 'C' the recorded signal will be a sinusoid. Therefore, if we take the Fourier transform of the recorded signals, we will get the 2-D Fourier transform of the entire object as indicated in the figure below. The inverse Fourier transform will then give the desired 2-D image.
There are other reconstruction techniques for computerised tomography - like, the back projection method and the arithmetic reconstruction technique. Each of the techniques has some advantages and disadvantages.
In principle, subtraction of one digital image from another is straightforward. Since the intensity of all the pixels is stored numerically we only have to subtract corresponding pixels in the two images. This method of digital image subtraction is useful in removing the background from a picture.
In digital subtraction angiography, two images are taken - one is a normal digitized X-ray and the second is a digitized X-ray with a contrast agent injected into the blood vessels.
The attenuation coefficient of the images are of the form:
and
where the subscript '1' denotes the biological tissue and subscript '2' denoted the contrast agent. We can now take the logarithm and then subtract to obtain the image of the contrast agent filled blood vessels.
Suresh Devasahayam
Departments of Physiology, and Physical Medicine & Rehabilitation
Christian Medical College, Vellore
July, 2002