# Inverse Square Law in Radiography (SID impact to mAs)

The radiation inverse square law specifies that: the intensity of the radiation goes down by the square of the distance from the source. For instance if you move twice as far from the source the intensity of the radiation will decrease by a factor of 4.

The inverse square law in Radiography specifically describes the fact that x-rays are divergent and spread out as they leave the focal spot like 1/R^2. To maintain constant exposure at the image receptor the inverse square law formula is: mAs_new = mAs * (SID_new/SID)^2, where SID is the source to image distance.

## Are x-ray beams isotropic?

### X-rays are Divergent.

The Source to Image Distance (SID) is an important parameter in x-ray imaging as the x-rays beam is divergent (i.e. spreading out as it leaves the focal spot). The beam divergence leads to the inverse square law in x-ray imaging. The formula for the inverse square law in x-ray imaging is: mAs_new = mAs * (SID_new/SID)^2.

X-rays in medical imaging are generated within an X-ray tube at the focal spot, which is where the electrons are incident onto the anode. The x-rays are generated within the anode material and come back out of the anode and travel towards the patient. Each x-ray travels in a straight line and the x-rays collectively form a cone shape where the x-rays are spreading out as they leave the anode.

As x-rays are traveling out from the anode they are referred to as isotropic since they are spreading out in all directions. This is why it is important that the x-ray tube includes proper collimation so that only the desired region irradiates the patient.

This spreading out of the x-rays as they leave the anode is referred to as a divergent beam since the individual x-rays are spreading out as they travel further from the x-ray anode. If the x-rays all traveled in straight parallel lines the distance of the image receptor from the x-ray source would not impact the intensity of the x-ray beam that is incident on the image receptor.

Since the beam is spreading out the number of x-rays passing through a given area is reduced when it is further from the x-ray source. For instance if you have a koosh ball you can see that towards the center of the kiosk ball there are lots of outshoots in a given area and they decrease as you move further away.

## Why do X-rays spread out like 1/R^2 ?

In this figure we show that as you move further away from the x-ray source within the cone that the x-rays fluency, or the number of x-rays passing through a given area is reducing as 1/R^2 where R is the distance from the x-ray source to the image receptor. This is why this effect is termed the inverse square law and is also written as 1/(SID)^2.

We have a second figure of the same scenario from another angle where we have drawn a few sample x-rays. In reality there are typically at least tens of thousands of x-rays in the beam but we have drawn nine x-rays here as a representative sample.

The intensity of x-rays at distance R is define as I. Then at a distance 2R from the x-ray source the intensity with be I/4, and at a distance of 3R from the x-ray source the intensity of x-rays with be I/9. Thus, you can appreciate that the inverse square law can actually have a significant effect on the intensity of x-rays incident on the image receptor.

Often in Radiography there is a desire to maintain the same exposure on the image receptor. Therefore if we start with a given Source to Image Distance (SID) and given exposure parameters mAs , and then we need to move the image receptor to a new location SID_new we need to compensate by changing the exposure parameters to mAs_new.

## What is the Inverse Square Law

The beam intensity at the image receptor is directly proportional to the exposure parameters mAs. Therefore if we specify that the intensity at the image plane should not be changed we get the equation:

mAs * (1/SID)^2 = mAs_new * (1/SID_new)^2

If we solve this for the ratio of the mAs change we get: mAs_new/mAs=(SID_new/SID)^2.

If we move the original tube current over to the right hand side of the equation we get this relationship:

So, we can see that the conversion factor for the exposure or tube current in radiography to compensate for a change in SID is based on 1/SID^2. To keep the exposure constant when the SID is changed we need to use this conversion factor.

## How do we compensate for SID changes in radiography?

### Conversion factors for SID in Radiography

Taking ratios and squaring them in your head may not be your favorite activity. We have some handy-dandy tables that you can reference when changing the SID. These are just approximates so feel free to do in in a calculator too if you want to have the exactly the conversion value.

In the United States inches are still the most common unit for measuring the distances so the values in this table are provided in inches and represent the original SID on the top and the new SID along the side. The values down the diagonal are all one since the SID didn’t change from the original to the new. This table covers some common changes from 36″ to 60″.

### Wide range of conversion factors for SID from 40″ reference.

In this table a wider range of SID values is considered from 30″ to 96″ where 40″ is the reference value as this is a very common SID. This shows how strong the effect of SID is where the mAs needs to be reduced by roughly a factor of 2 when the SID is reduced to 30″, and the mAs needs to be increased by a factor of 6 when the SID is changed to 96″.

If the SID is changing in the opposite direction the inverse of these factors should be used. For instance if the SID is changed from 30″ to 40″ the factor will be roughly 1/0.5 ~2.