Mathematical Model of X-ray Attenuation
This is a computer simulation of what happens when an X-ray is passing through something. Here that something is the so-called Shepp-Logan phantom, a rough model of a cross-section of a head. The purple graph shows the intensity of the X_ray at each time. The two things available to us in a practical measurement are
Let us study X-ray attenuation using a very simple example. Here we have a homogeneous block of material, carefully constructed so that its width is the half-thickness for this particular radiation we use. Then half of the photons travelling inside the block are absorbed or scattered and do not arrive at the detector. Therefore, the final intensity I1 is half of the initial intensity I0.
We continue studing X-ray attenuation using another very simple example. Here we have two homogeneous blocks of material, both half-thickness wide. Then half of the photons travelling inside each block are absorbed or scattered. Therefore, the final intensity I1 is a quarter of the initial intensity I0.
We combine the previous observations in a simplified study of X-ray attenuation based on photon counts. Here we send one thousand X-ray photons to the detector through free space. Ignoring quantum mechanical stuff leading to Poisson-distributed photon counting noise, the ideal detector counts precisely 1000 arriving photons.
Here we send three distinct X-rays to the detector, one of them through free space and two of them encountering one or two blocks of material. All of the rays start off with 1000 photons. In the first case, all 1000 make it to the detector. In the second case, exactly half of the photons are lost and the detector counts 500 photons. In the third case, there are 500 photons left after the first block, and half of those are further absorbed or scattered inside the second block. The final count is then 250 photons.
We counteract the exponential attenuation law by taking logarithm of the photon counts. Our aim is to arrive at line integrals of an attenuation function, so we expect to see zero for the ray passing through free space.
We subtract the logarithms of photon counts from the number 6.9 corresponding to the free space ray. The result is zero for the free space, something (0.7) for the case of one block, and double that something (1.4) for the case of two blocks. We have successfully transformed our photon count data into line integrals appearing in the Radon transform.
Now we can show how the line integral develops as the X-ray is travelling inside the block of material. Of course, in practice we would only know the final line integral.
Finally, here is an illustration of the line integral evolving as the X-ray passes through the Shepp-Logan phantom.