# Magnification and Blurring Effects for Radiographers and Radiologic Technologists (with Focal Spot Blur Formula)

Magnification occurs in x-ray imaging because the x-rays are divergent or spread out from the x-ray source. Therefore, the object will appear larger on the detector than the true object size. Magnification in radiography is defined as (Image Size/Object Size) and is equal to the (SID/SOD) which is the source to image distance divided by the source to object distance.

# Magnification

## What is the magnification in Radiography?

We will start with an analogy that almost everyone will be familiar with from their childhood or parenthood. If you have a flashlight pointed at the wall you can make a shadow puppet with your hand such as a bird or a bunny.

Remember that as you move closer to the flashlight, the projection of the bunny is going to be larger on the wall. Then as your fingers move further away from the flashlight, the projection of the bunny is going to end up smaller. This process is called magnification as objects closer to the source are going to be larger (if the image position is kept fixed). Since visible light and x-rays travel in straight lines the magnification phenomena is the same.

The light comes out of a light bulb or flashlight in all directions is called a divergent. In the same way the x-rays coming out of the x-ray tube are divergent as well (i.e. going in all directions within the collimated region).

Instead of a shadow puppet we are concerned with imaging something that is residing inside of the patient. We can think about an object that is in a plane within the patient parallel to the imaging receptor (e.g. the detector). In reality we are concerned with making images of the heart, lungs or bones but lets start by thinking about a simple object with is just a straight line as this is simplest to think about.

The definition of the magnification is the relationship between the object plane and the image plane. The magnification is defines as the (Image Size)/(Object Size). Since the x-rays are spreading out (i.e. diverging) the magnification will always be a number that is greater than 1 (i.e. the image size will always be larger than the object size).

## How is magnification in Radiography dependent on the object position?

Since we often have control of the magnification in radiography exams it is important to understand the distances that control the magnification. In the figure below we define the Source to Object Distance (SOD) and the Source to Image Distance (SID) (note sometimes you may see others call this the Source to Detector Distance (SDD), this is referring to the same distance).

In the figure below you can see that we can make one triangle that has the object size and the SOD, and another triangle with the image size and the SID. As shown these triangles are similar triangles. The SID is proportional to this image size, in the same way that this SOD is proportional to this object size. Therefore, the ratio of the sides are equal, i.e Image Size/Object Size = SID/SOD. You can see that this is our definition of magnification, so the magnification=SID/SOD.

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We can also solve this relationship for other variables. Most typically we would want to solve for the image size or the object size.

For instance, if you are taking a chest radiograph and the true object is 10mm long what will be its length in the image plane if the SOD is 170 cm and the SID is 180 cm?

The image size=(SID/SOD)*Object Size, which will be (180/170)*10 = 10.6mm. From this is example you can see that typically chest radiography will have a relatively small magnification and the size of the object in the image plane will only be slightly larger than the true size.

Another example would be an interventional angiography scenario where the SOD is 50cm and the SOD is 100cm, with the same object length of 10mm.

For this case the image size=(SID/SOD)*Object Size, which will be (100/50)*10 = 20mm. From this is example you can see that typically chest radiography will have a relatively small magnification and the size of the object in the image plane will be significantly larger than the true object size.

At a high level these clinical radiography scenarios have a low magnification as it is typically desirable to have the detector fairly close to the patient:

• Mammography

On the other hand these clinical scenarios typically have a high magnification:

• Angiography / Cardiac Cath labs
• Magnification Mammography

Obviously, it is also possible to solve for the object size as well if you know the image size. You will just be multiplying by the inverse of the magnification so that you can find the true object length if measure it on the image plane. As we mentioned above these assume that the object is lying in a plane parallel to the detector.

Thus far we have been assuming that the object in the center of the x-ray field (i.e. it is on the iso ray)now is something that’s sitting right at the ISO ray. If instead of the object being right on the iso ray it is moved a little bit higher, would that change the magnification or will the magnification stay the same?

Then the answer is that as long as the orientation of this object stays the same and the distance, namely the source to object distance (SOD) stays the same, the magnification will stay the same as well. As seen in the figure above the size of two images will be the same even if the object is moved off of the iso ray.

The size of the object projected onto the image plane will not be changed if the object is moved but still is in the same plane. So if something is moved up or down it is not as important as if it is moved toward/away from the source, or if it is rotated. We will next discuss the effect of rotation on the x-ray radiograph (i.e. the x-ray projection).

## Is the object position the only important factor in projection imaging?

Finally, we wanted to point out that in addition to the position within the scan field of view that there are other parameters which strongly affect the appearance on an x-ray radiograph. Namely, if the object is rotated there will be significantly different projections.

There are several factors which make reading radiographs difficult including overlapping anatomy and difficulty distinguishing low contrast structures. Additionally, the dependence of the radiograph on the rotation of the objects makes image interpretation more challenging. As you can see from the figure if two objects are rotated, they could project either the same or very differently depending on the direction of the rotation.

If a tomographic modality like CT is used instead of x-ray radiography these rotation artifacts can be eliminated because there are many measurements from different orientations. There are many use cases of x-ray radiography so it is important to keep these geometric effects in mind as you are setting up for each x-ray exposure.

To conclude we have discussed geometric effects in x-ray radiography including magnification and rotation effects. These effects are similar to the example that we mentioned at the onset of making shadow puppets on the wall with your fingers. So, feel free to â€˜studyâ€™ at home by making your own shadows on the wall and change the position and orientation of your fingers to see how it affects the projection on the wall.

• The magnification is defined as the ratio of the (image size) / (object size).
• The magnification can be calculated as (SID)/(SOD)
• The projections are also strongly dependent on the rotation of the objects, unlike CT.

# Focal Spot Blurring

## X-Ray Radiography Geometrical Unsharpness (Blurring)

In Radiography an important consideration is the image sharpness. Here we will discuss the blurring in an x-ray radiograph that is due to the focal spot. This is sometimes referred to as geometrical unsharpness. We have a separate post which discusses the resolution limits due to a digital detector, and the quantification or measurement of image resolution (https://www.howradiologyworks.com/digital-sampling-concepts/). So here we will focus on the effect of the x-ray focal spot.

## Why are there multiple focal spots on radiography systems?

It would be nice if we could use one focal spot for all anatomies but unfortunately, we need to select the best focal spot to use for each x-ray exposure.

The reason is that there are always trade-offs in physical systems and while we’d like to have the focal spot be as small as possible for sharp (crisp) imaging there is also the need to get enough x-rays through the patient to get a proper exposure at the detector.

But if the focal spot is very small, it’s possible that we won’t have enough x-rays to make it through the patient, and thus we won’t get a good image. So, for each anatomy and clinical indication (i.e. what are we trying to see in the image) there will be an optimal focal spot. That is why it is important to be aware of the trade-off that you are making when you select the focal spot for a given exposure.

In both cases (small or large spot), we are using what is called the â€˜Line Focus Principleâ€™, which we discuss in more detail later. But briefly here: the electrons are emitted from the cathode and incident on the anode over a given region and then the outcoming x-rays have a narrower width based on the angle of the anode (target).

The patient is going to see a smaller cross-section here and that helps because we want to spread out the heat on the target so that we don’t melt the target.

We would like the effective spot to be small, but if we can’t get enough x-ray flux to get through the patient, then we’re going to need to use a larger spot.

## A perfect world (i.e. an ideal point x-ray source)

First imagine an ideal point source, even though we canâ€™t really make one in reality. In this case our x-rays are coming from one point. Then imagine there is an object in the body that we would like to image as shown here. Since our x-rays travel in straight lines if the edge of the object is well defined the edge in the image will also be well defined as all the x-rays are coming from the same location.

If we have a nice perfect point for our focal spot, it’s going to look nice and sharp on our detector because there will be a crisp line delineating both sides of this structure as shown in the figure.

An ideal focal spot would be a single point and would lead to crisp edges in the x-ray images.

## The real world (i.e. an actual source)

In reality there is a finite size of each focal spot. Because our focal spot has a finite size the x-rays are actually coming out over a 2D region and not just a single point. In our illustrations we will just draw one dimension (from top to bottom) but into the screen there is also another dimension to the focal spot (which has the same effect on blurring the image in that direction).

To visualize the extremes of the focal spot blurring we can think about x-rays originating from both ends (the top and the bottom). This shows you that if we want to image an edge, that the edge isn’t going to be perfectly crisp because the image will have a gradual transition rather than the sharp transition that occurred in the case of the ideal focal spot. This effect occurs on the top and the bottom as shown in this figure (and likewise occurs on the left and the right as well in the image although not pictured here).

Since the x-rays passing through the patient are coming from the whole region of the focal spot this blurring effect depends upon the size of the focal spot. The larger the focal spot size is, the more blurring that will occur on the detector.

In this figure you can see that instead of it just being a very sharp crisp edge, it’s going to be a gradient edge where there is a gradual transition at the edge rather than being nice and sharp edge. That blurring region is typically called the penumbra of the x-ray beam, and the region that is fully blocked behind the object is termed the umbra.

In an actual system there is blurring due to the focal spot (i.e. the penumbra).

## Image Unsharpness from Focal Spot Quantified

Next let’s quantify this blurring as we have mentioned that it is dependent upon the size of the focal spot, but also clearly depends on the system geometry (i.e. how close the object and image are to the x-ray source.

We will define the unsharpness in the image due to the penumbra as â€˜Uâ€™, and since it is dependent on the system geometry we will also define some important distances.

As we mentioned above the focal spot has a finite size in both dimensions, but we show just one here in the figure for simplicity. We define the effective focal spot size as â€˜fâ€™ in this figure. Then SID is the source to the image distance and SOD is the source to the object distance (i.e. the portion of the patient being imaged).

The object to image distance OID is then the SID minus the source to object distance.

If we draw lines from the edge of the focal spot we can make similar triangles. From these triangles we can see that f/SID = U/OID. If we solve for U we get U=f*OID/SID. Thus the unsharpness in the image directly relates to f (focal spot size) and then we multiply by the ratio of the object to image distance divided by the source to image distance.

So, that gives us our geometric blurring or the level of the unsharpness due to the focal spot. You can do simple calculations if for instance, imagine the focal spot had a size of 3 millimeters. Imagine that the object that were interested in, was halfway in between our x-ray tube and our imaging plane where our detector sits. So, if our object is halfway in between, then these two are going to be the same and if the focal spot that we said was three millimeters, then the unsharpeness of the detector is three millimeters.

You can do different calculations depending on if you change the magnification to see how that affects the penumbra. As the object is moved closer to the source (leaving the source to image plane fixed) the penumbra becomes larger. Luckily, the rest of the image becomes larger as well.

The most important thing that controls the unsharpeness there, in addition to the magnification term is the size of the focal spot itself. That’s why we discussed above that there is a desire to have a small focal spot.

But we know that in physical systems we want to have the acquisition done relatively quickly, so the patient doesn’t have a chance to move and we need to get enough x-rays going through the patient so that we can make a good image. That is the tradeoff between spatial resolution (small spot) and available x-rays (large spot), and this is why x-ray systems typically have multiple focal spot sizes.

# Line Focus Principle in X-ray Imaging

## What is the line focus principle and why is it used on x-ray tubes?

In this section we will be describing the line focus principle for x-ray tubes, which enables more heat deposition and a small effective focal spot to be used in imaging. The small focal spot provides high resolution and depositing heat over a larger area enables higher mA settings.

If you need a review or are not familiar with how x-rays are generated in a medical x-ray tube, we have a separate post where we describe the physical mechanisms responsible for x-ray generation and the mechanism for x-ray generation (i.e. electrons accelerated and run into a heavy metal).

The electrons are boiled off from a cathode and pulled from the cathode to the anode by the tube potential (i.e. the kVp). As discussed above there are typically multiple physical focal spots on an x-ray or CT system where the size of the electron beam is determined largely by the size of the filament (where the electrons are boiling off).

In most radiography and CT x-ray tubes the anode will be rotating so that the electrons will be incident on different areas of the anode (so as not to melt the anode). That is another useful trick like the line focus principle to enable more electrons to be incident on the anode without the target material melting.

The purpose of the line focus principle is to enable a higher exposure (i.e. more x-rays) while still having a small focal. In the figure below you can see that the size of the actual focal spot is controlled by the size of the electron beam coming from the cathode and the target angle. In the figure below you can see that the width of the electrons beam can be larger than the width of the x-ray beam.

Remember that the electrons are coming from a two-dimensional surface and in this figure, we just show a cut through the tube where the electrons are coming from a line. The other dimension we can think of as into the screen. In that dimension the line focus principle does not have an impact and the size of the electron beam is the same as the size of the x-ray beam. So if this figure is showing the length of the focal spot, then the other dimension into the screen is the width of the focal spot.

The major point here is that the length of the actual focal spot and the effective size of the spot as it travels through the patient and is incident on the detector are different, and thus a larger actual length can be used to spread the heat of the electron beam over a bigger surface.

## What is the relationship between the actual and effective focal spots size?

If we consider this geometry you see that the actual focal spot and the effective focal spot are both sides of a right triangle. The figure below shows the same triangle, and now the tube target angle has been added to the triangle.

If we remember from middle/high school geometry that the definition of sin is the opposite side over the hypotenuse. So the sin of the target angle is equal to the effective spot divided by the actual spot. If we want to solve for the effective spot we see that it is the actual spot multiplied by the sin of the target angle.

In practice the tube angles are typically between 6 and 20 degrees for an x-ray tube. So, the sine of theta is between 0.1 and 0.34.

This means that the maximum difference between the effective focal spot length and the actual focal spot length would be a factor of 0.1 (i.e. the actual focal spot is 10 times larger than the effective focal spot). Then for the tubes with the largest angles the difference is roughly 1/3 (i.e. the actual focal spot is 3 times the effective focal spot size).

## What is the tube loading gain and how is it related to the target angle?

Another term that is used in referring to this same effect is the loading gain. The loading gain quantifies how much more energy that can be deposited in the tube due to the line focus principle.

In the figure below you can see that we start with the same exact equation. The only difference is that now we solve for the Actual Spot Size divided by the Effective Spot size as this quantifies the factor in heat spreading or tube loading which is enabled by the line focus principle.

If we solve for the loading gain we find that it is equal to 1 over the sin of the target angle. This is just another way of looking at the same effect and from the range of target angles that we calculated above we remember that sin of the target angle varied from 0.1 to 0.34. Therefore, then 1 over sine of the target angle will vary from 3 to 10. Thus, the line focus principle actually enables additional tube loading (ie. more heat via higher mA or kVp) by a factor of 3 to 10 depending on the target angle.

That ability to deposit heat means we can get more flux without melting the tube. One downside or one negative effect of having very shallow angles is something called the heel effect or the anode heal effect. It gets worse at smaller angles because the smaller the target angle the longer the path of target material that needs to be traversed on the anode side. This is discussed in more detail in a separate post we have on the heel effect (ADD LINK when post done).

As we mentioned above there are multiple focal spots on the most systems enabling you to select the best focal spot for a given clinical task. Each of the focal spots on the system will take advantage of the line focus principle in the same way as the target angle is the same for multiple focal spots.

In your clinical exposures when you are imaging an extremity (e.g. a hand or foot) it will be desirable to use a small focal spot as this can achieve the highest resolution possible and does not typically require a very high exposure.

On the other hand, if you need the x-rays to get through a more attenuating anatomy such as a large abdomen you will need to use the larger focal spot. So, depending on the clinical task you will use the fact that there are different cathodes on the system in order to generate the best images for your patients.

• The line focus principle enables higher tube loading (i.e. higher mA/kVp) while maintaining a small focal spot size.
• The loading factor for x-ray tubes describes how much more heat can be deposited due to the line focus principle and ranges from 3-10 for x-ray tubes.

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