Pulse repetition frequency ultrasound confusion

[Taylor_1989]

Homework Statement

Q:[/B]An ultrasound department seeks your advice regarding scans they perform to visualise and confirm foetal heart beating. They wish to know what imaging frame rate should be set in order to visualise the heart movement reliably but achieve the maximum possible reduction in ultrasound exposure of the foetus. They explain that the typical scan conditions are a foetal heartrate of 150 beats per minute, imaged using a maximum scan display depth of 10 cm.

You research the scanner and discover that:

(a) the line density for imaging is fixed at 100 lines per frame

(b) the pulse repetition frequency is affected by a minimum delay of 200 microseconds at the end of each pulse-echo cycle for the scanner to process the returning signals before the next pulse-echo cycle is commenced and

(c) the scanner is designed to run initially at the maximum frame rate achievable for the depth of scan that is set, but that the frame rate may then be adjusted down by the operator.

Note: The frame rate is expressed in frames per second or fps.

Which of the following is the best advice to give? Please select one answer only

a.
Adjust the frame rate down to 10 fps and achieve an exposure reduction of approximately 87 %

b.
Adjust the frame rate down to 5 fps and achieve an exposure reduction of approximately 94 %

c.
Adjust the frame rate down to 5 fps and achieve an exposure reduction of approximately 83 %

d.
Adjust the frame rate down to 10 fps and achieve an exposure reduction of approximately 67 %

e.
Adjust the frame rate down to 2.5 fps and achieve an exposure reduction of approximately 97 %

f.
Adjust the frame rate down to 2.5 fps and achieve an exposure reduction of approximately 92 %

Homework Equations

[/B]
Image depth

$$PRF=1540/(2*D_{max}) [1]$$

PRF = frame rate x line density [2]

The Attempt at a Solution

Here what I know the inverse of the PRF will give the me pulse echo period, which is the time between each pulse, so as I understand the transducer send out a plus and then switched off until the next plus which is the PEP (pulse echo period), now this has to be greater than the return time of the pulse which in this case it is $1/7700=129.8701299\mu s$ value was obtained from using the [1] so as the time it would take the sound wave to travel is $t=0.2/1540=130\mu s$ so it only by a fraction which is why I think the $200\mu s$ is included. so given this I did the following:

I then converted this into the pulse echo which is 1/7700Hz
after this I add the 200 microsecond to it as this is the delay in the transducer I believe between pluses so I did the following: 1/7700Hz +1/5000hz=127/385000 taking the inverse gave me

3031.496063hz
I then took 3031.496063hz and dived it by 100 as that the line density so I could get a frame rate which was 30.31496063fps
I then took the percentage decrease of each one with this value i.e
((30.31496063-10)/30.31496063)*100=67.0129870%
((30.31496063-5)/30.31496063)*100=83.50649351%
((30.31496063-2.5)/30.31496063)*100=91.7532675%
which give me 3 percentages that are in the ans s my issue is it say in the ans that it approximate which leaves me in a problem with either rounding 91.7 to 92 or choose the 67.012 as this seems to be the most correct with the ans to pick as there is no rounding involved. Can anyone see a problem with the way I have approached this problem?

sorry for the mix of latex my I have been having issue with my keyboard layout, so I just had to finish the rest of like so.

 

[tnich]

The speed of sound in human tissue is 1540 in what units? I suggest that you write out the units in your PRF equation and make sure you are getting an answer that makes sense.
I suggest that you make an effort to state your answers to an appropriate number of significant digits.
What is the time between the initiation of one pulse and the initiation of the next pulse? The inverse of that value should be your pulse repetition frequency.
How often will you need to observe the foetal heart beat to visualize it reliably?

 

[tnich]

. . . which give me 3 percentages that are in the ans s my issue is it say in the ans that it approximate which leaves me in a problem with either rounding 91.7 to 92 or choose the 67.012 as this seems to be the most correct with the ans to pick as there is no rounding involved.

In spite of my questions before, I think you have correctly determined the maximum PRF.
If you observe the heart at 2.5fps, how much does the image change from frame to frame?
How about if you observe it at 5fps?

 

[Taylor_1989]

The speed of sound in human tissue is 1540 in what units? I suggest that you write out the units in your PRF equation and make sure you are getting an answer that makes sense.
I suggest that you make an effort to state your answers to an appropriate number of significant digits.
What is the time between the initiation of one pulse and the initiation of the next pulse? The inverse of that value should be your pulse repetition frequency.
How often will you need to observe the foetal heart beat to visualize it reliably?

So I will now correct my working and make it more clear my apologise, it was very late and I was sloppy with my workings.

$$PRF=\frac{1540\:ms^{-1}}{2\cdot \left(0.1m\right)}=7700Hz [1]$$

$$PEP=$$\frac{1540ms^{-1}}{2\cdot (0.1m}=129.8\mu s [2]$$

Now the time take for the ultrasound to travel over a complete distance of $0.2m$ then the total time take will be:

$t=\frac{0.2m}{1540ms^{-1}}=129.8\mu s [3]$

As this is approximately the same as the FPS the this is where I believe the $200\mu s$ comes in. Adding this [3] I can the calulte the correct maximum frame rate, which is:

$$(PEP)=129.8\mu s+200\mu s =329.8\mu s [4]$$

$$PRF=1/PEP=3031.5Hz [5]$$

So now to to calculate the maximum frame rate I dived [5]/100 to get the following: $30.3Fps$

using what you gave as a hint:

In spite of my questions before, I think you have correctly determined the maximum PRF.
If you observe the heart at 2.5fps, how much does the image change from frame to frame?
How about if you observe it at 5fps?

I then started to think about it now my thoughts are this, that the as the heart is beating at 150bmp this give a 2.5fps so the image will not change between pluse, if the ultrasound is set to a 2.5fps but at 5fps there will be addtional 2.5fps which mean the image will appear more clear? I think. But with the maximum FPS that I have calculated and the question want the maximum possible reduction then

f.
Adjust the frame rate down to 2.5 fps and achieve an exposure reduction of approximately 92 %

the f would be the correct ans, have I thought of this correctly?

Edit I don't know why I am getting a math processing error, I have check my latex but don't see nothing wrong?

 

[tnich]

I then started to think about it now my thoughts are this, that the as the heart is beating at 150bmp this give a 2.5fps so the image will not change between pluse, if the ultrasound is set to a 2.5fps but at 5fps there will be addtional 2.5fps which mean the image will appear more clear? I think. But with the maximum FPS that I have calculated and the question want the maximum possible reduction then the f would be the correct ans, have I thought of this correctly?

You problem statement says, "They wish to know what imaging frame rate should be set in order to visualise the heart movement reliably but achieve the maximum possible reduction in ultrasound exposure of the foetus." Will they be able to visualise the heart movement at 2.5fps? This issue is called "aliasing". I suggest that you do some research on aliasing to determine what is an appropriate PRF.

 

[Taylor_1989]

I suggest that you do some research on aliasing to determine what is an appropriate PRF

I was just reading when revving my problem I was reading an article and I have understood correctly the, there is some called the Nquist condition which is used to calculate the PRF needed to avoid the aliasing condition, am I correct in saying the PRF need to be 2 times the frequency of the heartbeat for this condition to be avoided, which in this case I am wrong and it should be turned down to 5fps. I also think now that now they would not be able to see the hear beat a 2.5fps as i think the would just so a standing image?

 

[tnich]

I was just reading when revving my problem I was reading an article and I have understood correctly the, there is some called the Nquist condition which is used to calculate the PRF needed to avoid the aliasing condition, am I correct in saying the PRF need to be 2 times the frequency of the heartbeat for this condition to be avoided, which in this case I am wrong and it should be turned down to 5fps. I also think now that now they would not be able to see the hear beat a 2.5fps as i think the would just so a standing image?

Yes, the frame rate needs to be at least twice the heartbeat frequency.
Even at twice the frequency it is possible to see aliasing. Imagine you are observing a sine wave $f(t) = sin(2πft+φ)$ at regular intervals. If the PRF is $2f$ and the pulses are at times $t_n=\frac n {2f} - \frac φ {2πf}$, then at each pulse, the amplitude of the wave is 0 and there is no apparent difference from one image to the next.
However, in this case with a frame rate of $2f$ the frame covers half of the period of the heartbeat. The phase angle $φ$ varies over the whole image (each line of the image represents a different phase angle), so in comparing successive frames only a small part of the image might look similar. A PRF of twice the heartbeat frequency should be adequate to see motion.

 

[Taylor_1989]

First of all I would like to say thank you! I can start to see what is going on now, with respect to how things are working, iv been trying to tackle this problem for nearly a day in half. By just viewing in terms of the function you have given has cleared a lot up once again much appreciated.