Doppler Effect, Equation, and Angle Correction in Ultrasound Physics

[00:00:00]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 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 from 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 now 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 insination 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 adapter 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 y but this 2 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 insulation 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 angle triangle and we're trying to calculate the hypotenuse here we can use the cosine of this angle to have our adjacent or our adopter 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 intonation to get what's known as our Doppler equation now we can see here that the doctor equation and the Doppler shift equation that we calculated in our previous side 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 insulation 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 insulation 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 incination 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 insulation 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 insulation 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 intonation 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 now 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 now 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 a hundred 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 insulation 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 of 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 insulation 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 intonation that are 30 to 60 degrees now 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 insulation here the angle of intonation is represented on the left-hand side of our graph the velocity of the blood within this blood vessel the actual velocity is a hundred centimeters per second now if we were off by just three degrees on this angle of intonation we think we're at 45 degrees but in fact we're at 48 degrees our calculator 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 now 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