Interactive X-Ray Transmission Calculator for Radiologic Technologists (Beer’s Law Equation)

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This is a calculator the transmission, (fraction of the x-ray beam), that will penetrate different thickness of materials such as water, bone, air and lead. This calculator is useful to get a sense of what fraction of the x-ray beam will pass through different objects. The calculator has the option for a few energies of x-rays and we describe in the article the relationship between the x-ray attenuation and the x-ray energy.

X-ray Transmission

In x-ray imaging the contrast comes from the difference between the x-ray transmission through different types of materials. If you haven’t already checked it out please see our post on x-ray and CT contrast generation.

In this calculator we want to give you a feel for how much of a given x-ray beam will be transmitted through objects of different materials. The idea is that as an x-ray technologist or radiographer than you can be familiar with relative x-ray transmission.

For instance, if you have 10 cm of water and an x-ray beam of 50 keV, would 50% of the photons make it through or 10% or somewhere in between. After going through this material and using this rad calc yourself you should be able to confidently answer these types of questions.

Beer’s Law

Two terms “X-Ray Transmission” and “X-Ray Attenuation” are different ways to describe the same thing – how x-rays pass through the body of patient. While X-Ray Transmission quantifies the x-rays that pass through the body, X-Ray Attenuation describes x-rays stopped in the body. Transmission and Attenuation can be expressed as a number from 0 to 1. (Transmission=1-Attenuation, or if percentage values are being used [% Transmission=100% – % Attenuation]).

Let’s imagine a simple model: x-rays of a single energy (i.e. monoenergetic) passing through one single material and the x-rays are parallel as they go through the material. One important parameter in this model is how thick is the material.

These x-rays have an intensity which is the number of x-rays incident on the material (over a given area and a given time).

In real x-ray systems it is slightly more complex as there are many energies in an x-ray spectrum, but this exercise will help provide us with the basic intuition on x-ray attenuation.

If intensity of the incoming x-rays is Iand output intensity is I. The transmission will be: I/ I0.

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The question that we ask here is how does the transmission change depending on the type of material. Some materials of interest in the human body are: water, air, fat, and bone.

Next we’re going to talk about Beer’s Law. Do you think it has anything to do with this nice sampling arrangement of brews here? Unfortunately, it’s not related to this kind of beer. Beer’s Law has several names including Beer-Lambert and Beer–Lambert–Bouguer law. The additional names come from different individuals who made contributions to the discovery of this basic physic behavior. It was discovered  first in chemistry for optical photons (i.e. regular light), but the same behavior applies for x-rays as well. We’ll just call it Beer’s Law for simplification in this post.

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Beer’s Law relates to two variables – input intensity, I0, and output intensity – I. The multiplying term here will always be a number between 0 and 1. This is just saying that the output intensity will be the same or lower compared with the input intensity.This coefficient in the exponential function µ (pronounced mu) is dependent on the material and the energy of the x-rays.

The intensity of output x-rays depends on the thickness of material as well. The µ and thickness are both in the exponent. Higher µ means that more x-rays are absorbed by the material and larger thickness, x, also means more x-rays absorbed.

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The linear attenuation coefficient, µ, can be separated in two pieces in order to see how x-ray absorption depends on density of material. The linear attenuation coefficientis separated into the mass attenuation coefficient µ/ρ and the density ρ.

The mass attenuation coefficient depends on material and the x-ray energy. The density depends on material and is simply mass/volume. More dense materials stop x-rays at a higher rate.

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When there is sufficient flux of x-rays through the patient, there is more contrast at the lower energies and that’s especially true for materials like bone and calcium. This is because the x-ray attenuation coefficient is higher at lower energies ( Link to description of photoelectric effect).

Another fact that we wanted to note here is that fat is less dense than water (e.g. oil floats in water), and for that reason it has a lower x-ray attenuation. Other important materials of interest in the body are water and muscle. Muscle has a little bit higher x-ray attenuation than water.

Rad X-ray Transmission Calculator Examples

Here are some demonstrations of how to use the rad x-ray transmission calculator (above).

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You can select the inputs for Beer’s law from some materials of interest. We also provide the x-ray energy as an option so that you can compare the transmission for different x-ray energies.

The equation that we implement in this calculator is here, and the parameters come from the NIST website.

Let’s look at some examples here.

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Most of the human body is made up of water so we start with water. Let’s look at the thickness of 10 cm (100 mm) in terms of what percentage of x-rays 50 keV photons entering will be coming out when they’re incident on 10 cm or 100 mm of water.

About 10% of those x-rays are going to make it through. This represents the size of a small child? 10 cm is very small in terms of the size of a water equivalent individual. Most individuals are going to be significantly larger than this.

Let’s start thinking about how much attenuation is there going to be as the x-rays are passing through water. So, 90% of those x-rays are going to be stopped and 10% are going to make it through.

We don’t think you need to remember specific mass attenuation values but rather just get a feeling of “if 50keV x-rays pass through 10 cm of water, and on the other end only 10% of the x-rays make it through.” So, for a regular patient, we’re going to be dealing with significantly more attenuation than that, and hence even lower transmission.

In the first example we looked at water and we’re dealing with 50 keV and 10% made it through. So what’s going to happen if we switch these from 50 keV x-rays to 80 keV x-rays? Are more or less x-rays are going to make it through?

If you’re a practicing technologist, you’ll know that when we need to have more penetration through the patient, we go to higher keVs (namely higher kVps on the system, that lead to more high keV photons).

So if we go to higher keV, we now have 50% more transmission (15% compared with 10%).

We went from 10% to more than 15%, so an increase of 50% in terms of the amount of x-rays that are going to be making it through that amount of water. That’s an important point that we want to emphasize, higher energies leads to higher transmission.  

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If the photon energy is increased this leads to, higher transmission, i.e. higher penetration through the object.

We can see that that’s happening because our mass attenuation coefficient is decreasing when we change energy. This is why more x-rays pass through at higher energies.

One very interesting material is lead. Lead is used as a material to block the x-rays coming through (i.e. shielding). If we look at 1 mm of lead, and we’re talking about 50 keV, then we’re going to have 0.01% transmission. This is true because lead is very dense and has a very high mass attenuation coefficient. That is why lead is used for shielding because in comparison with most materials much less lead is needed to stop the x-rays. That is why it’s typically used in shielding applications.

Summary

Please play around with this tool for yourself and do some thought experiments as to where the attenuation is coming from – is it coming from the density of the material or is it coming from the mass attenuation.

Then on your own you can see how different is blood from water, how different is blood from air, and just kind of play around and get a sense for yourself of the relationship between these different materials and also how the transmission (attenuation) varies as a function of energy.

That’ll tell you a lot about why we use certain types of energies for different applications. For instance, bone is going to be significantly more attenuating than water. The bone is more attenuating not because of its density but because of its x-ray attenuation properties (i.e. high Z material). Also, important to note is that the x-ray transmission is lower especially at low energies. This can be seen from the plot above as well where bone attenuation increases significantly for lower energies.

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Rad Take-home Points: Try the calculator yourself to get a feeling for the relationships and the examples provided here:

  • Even water is relatively highly attenuating (Typically 10% or less of the primary beam will make it through the patient).
  • To increase the penetration the x-ray energy should be increased.
  • High Z materials like bone are more attenuating than water and have very significant increase in attenuation for lower x-ray energies.
  • Very dense high Z materials like lead stop x-rays much better than common materials in the body.
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