Transcription
Hello and welcome back. In the next few talks, we're going to be covering Doppler ultrasound physics, and we'll start off today by looking at some of the basics: the Doppler effect, Doppler shift, the Doppler equation, and then rounding things off by looking at angle correction in our Doppler ultrasound machine and some of the pitfalls that we can encounter when trying to angle correct.
Now, when we create a wave from a stationary source, we get successive peaks in that wave heading out into the tissues. Now, the wavelength of those waves, the difference in distance between successive peaks, remains the same if our source is stationary. The number of successive wavelengths that pass a particular point per second also remains the same. The frequency of our wave remains the same as long as our source of that wave is stationary, and the time taken between successive peaks, the period of the wave, remains the same.
Now, if that source of sound was to move, we will get a change in the frequency of that wave. As that sound source moves, we can see that in the direction in which it's moving, we get an increase in wave frequency. The number of waves passing a particular point per second increases. The pitch of our wave will increase. As that frequency increases, our wavelength, as well as our period, decrease.
Now, the opposite is also true. Where the sound source is moving away, the wavelength gets longer, our period gets longer, and our frequency decreases. Our pitch of this wave decreases. So, it's the movement of that wave that changes the frequency of the wave.
Now, we can see that the difference between this original frequency and the frequency once that sound source moves is what's known as the Doppler shift. Now, we have a source of the sound wave as well as receivers of the sound wave. Now, it's this observed change in frequency that is known as our Doppler shift. And to calculate the Doppler shift, which is a change in frequency, and we measure that in Hertz, is the frequency that we receive, this change frequency, minus the transmitted frequency from our source. So, what was the original frequency? What is the new frequency? The difference between those two is what's known as our Doppler shift.
Now, our Doppler shift here will be a positive value. Our frequency received is higher than our frequency transmitted. The Doppler shift on the other side here will be a negative value. We will have a negative Doppler shift. Our original frequency, our transmitter frequency, is higher than the frequency that is being received on this side. So, the Doppler shift is a result of the Doppler effect.
In order for Doppler shift to occur, either the source of our sound is moving, the receiver or the receptor for that sound is moving, or there is a reflector of that sound that is moving within the ultrasound field. Now, we can use this phenomenon in ultrasound when trying to measure the speed of something within our ultrasound field. So, if we take this example here, we have our ultrasound machine and we have blood within a vessel that is moving at a specific velocity.
Now, if that blood is moving towards the ultrasound machine, ultrasound pulses that are incident on that blood will return back to the ultrasound machine at a higher frequency. So, let's have a look at what that looks like here. Blood's moving towards the machine, the reflected ultrasound pulse, the echo coming back, has a higher frequency. We have a positive Doppler shift.
Now, this change in frequency, the Doppler shift, is proportional to the velocity of that blood. And the value that we get back here is a scalar value. It has magnitude but no direction, and that value we get back here is a speed value that we can calculate. Now, as that blood moves faster and faster, the change in frequency will become greater. We get a higher and higher frequency returning echo.
Now, the opposite is also true. If that blood was moving away from our ultrasound transducer, we would get a negative Doppler shift. The frequency of that ultrasound wave returning to our ultrasound machine will be less than the original frequency here. Now, I've mentioned the Doppler shift is proportional to the velocity of that blood, but we have no directional component here. We only have magnitude that we are measuring. The magnitude of Doppler shift, and we need to break these down into separate components because our ultrasound frequency coming in towards the moving object is at an angle of incidence to the movement of that object.
Now, if we were to get rid of that angle, have our ultrasound machine directly in line with the movement of blood here, we can then calculate the speed of the blood moving here because it's directly proportional to that Doppler shift. We can use an equation. The Doppler shift is two times the frequency of our transmitted ultrasound pulse times by the speed of the blood moving towards the ultrasound machine over the sum of both the speed of the blood and the speed of sound in that tissue.
Now, when we look at this fraction here, the speed of blood moving in tissues never really gets higher than 200 centimeters per second, and the speed of sound in centimeters per second is 154,000 centimeters per second. We see we are dealing with very different magnitudes here. And as a result, this speed of the red blood cells in our denominator here is often ignored because it's such a small fraction of the speed of sound of ultrasound within our tissues. So, often you'll see this equation written like this.
Now, I'm not going to get into why, but this two here is a coefficient within our Doppler shift equation, and that's needed when we have a moving reflector. It's got to do with the speed of sound of both our incident ultrasound wave as well as the speed of the reflector shifting that ultrasound wave in the returning echo. Now, here we are using the speed of the blood moving within the vessel and seeing that that speed is related to our Doppler shift.
In the previous examples, we had an angle to this ultrasound machine. Now, because we can break the system down into multiple different components, the velocity of our blood as well as the Doppler shift at a specific angle, we can use trigonometry to calculate this velocity value if we know this angle of incidence or our Doppler angle here. Now, what we are trying to calculate is the velocity here, and what we actually have the known value is our Doppler shift heading towards the ultrasound machine. We can create a right-angled triangle here with our Doppler angle. And if we take the adjacent side of this right-angled triangle and we're trying to calculate the hypotenuse here, we can use the cosine of this angle to have our adjacent, or our Doppler shift, over the velocity of the blood traveling through the blood vessel. And if we rearrange that equation, the velocity of blood traveling through the blood vessel times by the cosine of this angle will give us our Doppler shift, or the speed that we had calculated from our ultrasound machine.
Now, when we looked at that initial equation, we had the speed of blood moving through tissue over the speed of sound. Now, we can plug in this new formula that we've calculated here, which compensates for an angle of incidence, to get what's known as our Doppler equation. Now, we can see here that the Doppler equation and the Doppler shift equation that we calculated in our previous slide are essentially the same. Our Doppler shift is two times the frequency of our transmitted ultrasound pulse times by the velocity of the moving component within our ultrasound image times by essentially the hypotenuse, which is our velocity of the reflector within our ultrasound image, and the cosine of the angle theta here.
Now, velocity is no longer a scalar. It has magnitude and direction. What we have here is a vector, and this cosine of theta allows us to add that directional component. We move from a scalar magnitude value to a vector, a magnitude with direction. This is replacing the speed of the reflector that we had in our previous equation and compensating for this angle of incidence. And then we divide by the speed of sound, just as we saw in our previous calculation.
Now, what we are trying to calculate when looking at Doppler ultrasound is the velocity. This is our unknown. The machine will calculate the Doppler shift. We know the speed of sound within tissue. We know the angle of incidence that we program into our ultrasound image, and we know the frequency of our transducer. The unknown here is our velocity. So, we can rearrange this equation to have velocity as the variable that we are calculating here.
Now, I've not seen in exams anyone being asked to actually calculate the velocity of blood flow within a tissue, but what we do need to understand is the relationships between these variables and the Doppler shift as well as velocity. We can see here that as the velocity of our blood changes, as the velocity increases, our Doppler shift will increase. The faster that blood is moving towards our ultrasound machine, the higher the Doppler shift value. The speed of sound is constant within tissues, and the frequency of our ultrasound machine is also constant.
Now, as the angle of incidence increases, as this theta value increases, the cosine theta value will decrease. And I'm going to show you later a table showing you those values. So, as our angle of incidence increases, or the Doppler angle increases, our Doppler shift will decrease. The magnitude of this scalar value will decrease.
So, let's have a look at that cosine value as this angle changes. So, I've given three examples here. We've got a shallow angle, a small Doppler angle, a slightly steeper angle, and then a 90-degree angle here. We can see that when our ultrasound probe is directly in line with the blood flow, our angle of incidence is zero degrees, our cosine theta will be one. The Doppler shift that is observed is directly proportional to the velocity of that blood.
Now, as we increase that angle of incidence, our cosine theta value will decrease. So, we can see that an increasing Doppler angle will result in a decrease in Doppler shift. Now, when our transducer is at a 30-degree angle to the flow of blood, the calculated Doppler shift here, or the calculated velocity from our ultrasound machine, will be 0.87 of the actual value of the velocity of the blood traveling in our blood vessel. And when that angle gets to 60 degrees, the Doppler shift here will be half the value of the actual velocity of the blood traveling through tissues.
Now, because we are dealing with cosine here, where we have an adjacent side over our hypotenuse side, we cannot have the ultrasound machine being 90 degrees or perpendicular to the flow of blood. We won't be able to measure any Doppler shift here. Our cosine theta value will be zero, and subsequently, our Doppler shift will be zero.
Now, at very steep angles, we run into refraction problems here, and the ultrasound wave is having to travel through the blood vessel wall. And generally, we like an angle between 30 and 60 degrees in order to get an accurate calculation that doesn't have the problem of refraction but also doesn't have this issue of angle correction here. Now, because we've calculated a cosine theta value here, the velocity that we are calculating should be accurate. This is what's known as angle correction. The Doppler shift that is calculated by the ultrasound machine at a certain angle is then accounted for, and that velocity can be calculated. This is what's known as angle correction.
Now, not all angle correction is created equal. Let's take an example here where we have an ultrasound pulse incident on a blood vessel with a changing Doppler angle. Now, using the Doppler shift that is calculated by this ultrasound machine here, we get a speed of 50 centimeters per second calculated by this ultrasound machine. Now, depending on what this angle is, the velocity of blood that we calculate will be different. If we have this ultrasound machine here at zero degrees to the flow of blood, the calculated velocity of that flow of blood will be 50 centimeters per second. If the ultrasound machine is at 60 degrees to this blood vessel and we measure a frequency shift proportional to 50 centimeters per second here, our actual velocity of blood will be 100 centimeters per second. Our cosine of theta at 60 degrees is 0.5, the value of the actual velocity here. So, we take 50 centimeters per second divided by 0.5, get our actual velocity of blood, which will be 100 centimeters per second. At 80 degrees, if we are measuring a Doppler frequency shift that's proportional to 50 centimeters per second, our actual velocity of blood within the blood vessel will be just shy of 300 centimeters per second. So, we can see that using this cosine theta value will give us accurate representations of what the actual velocity of blood here is.
Now, the issue comes is when we are setting our ultrasound machine and telling it what angle of incidence we are using. We're never going to have that perfectly accurate. So, if we're using a 30-degree angle and we're off by 5 degrees, say we say to the ultrasound machine, I'm measuring at a 30-degree angle, but in fact, we're measuring at a 35-degree angle, the calculated value at 30 degrees and the actual value at 35 degrees is very similar. We've maybe got 55 centimeters per second and 60 centimeters per second.
Now, as this angle increases and we're off by an angle of 5 degrees here, we can see that our actual value and our calculated values start becoming very different. And above 60, there's a real exponential curve to this graph. If we were measuring at an angle of incidence of 80 degrees and our estimation was only 5 degrees off, the calculated velocity of that blood will be over 100 centimeters per second faster, just based on that five-degree angle difference. And that's why we want to use angles of incidence that are 30 to 60 degrees.
Another way of looking at this is by taking a constant velocity in a blood vessel. Say the blood is traveling through that blood vessel at 100 centimeters per second, and we are changing the angle of incidence here. The angle of incidence is represented on the left-hand side of our graph. The velocity of the blood within this blood vessel, the actual velocity, is 100 centimeters per second. Now, if we were off by just three degrees on this angle of incidence, we think we're at 45 degrees, but in fact, we're at 48 degrees, our calculated, our estimated velocity will be 105.7. We've got an error of 5.7 percent. At 60 degrees, if we were three degrees off, our error would be just over 10 percent.
Now, we can get away with these values. These are close enough to the values we are trying to measure within our blood vessel. When we're at 80 degrees, a small error, a three-degree error, because of the steepness of this graph, will result in a 45 difference between our estimated velocity and the actual velocity of the blood within this blood vessel. So, as that angle increases, our estimated velocity, if we make any error on that angle, it's going to be well off our actual velocity within the blood vessel. And that's why we want to use angles between 30 and 60 degrees.
The Doppler effect, Doppler shift, and the Doppler equation are common exam questions, and I've included them in the question bank that I've linked in the first line in the description below. We've covered the basic underlying concepts now of Doppler ultrasound. Next, we're going to be looking at the specific types of Doppler ultrasound that we can use in clinical imaging, and we're going to see which types of ultrasound is best to use in different clinical scenarios. So, until then, I'll see you all in the next talk. Goodbye, everybody.