Phase Encoding Gradient MRI | MRI Signal Localisation | MRI Physics Course #9

[00:00:00]hello everybody and welcome back so this is the  third talk in a three-part Series where we're looking at how exactly we localize signal within  an MRI image now what have we done so far well first we've seen how we can select a specific  slice along the z-axis using what's known as a slice selection gradient and this spins within  that slice will be resonating or processing in Phase with one another we've then looked at how  we can create a frequency differential along the x-axis of that slice and use that frequency  encoding gradient to delineate where signal is coming from along that x-axis based on the  frequencies of those spins now when we apply this frequency encoding gradient we apply it  over the time that we are sampling that signal now as you can see this frequency encoding  gradient will cause the spins to process faster at this end of the x-axis than they  will at this end of the x-axis and there will be a gradient of frequencies as we move along  that x-axis now when we apply that frequency encoding gradient see how this spins on this  edge of the slice or processing faster than the spins on the other end of the slice now we  can measure the net magnetization Vector of the entire slice over a period of time and that's  what's known as the data acquisition time now that data acquisition time happens during this  frequency encoding gradient now we can sample that multiple times converting this analog signal into  a digitized signal a digital signal and we get discrete digital values for each point along that  analog signal and the number of times we sample that analog signal will determine the number of  frequencies we can delineate along that x-axis Now using this signal that we've recorded during  data acquisition we can perform what is known as a one-dimensional inverse Fourier transformation  now what that Fourier transformation does is it calculates the frequencies that are  responsible for that net magnetization vector the change of that net magnetization Vector  over time is unique for a specific subset of frequencies with varying different amplitudes  now based on the frequencies that we've teased out of this net magnetization Vector signal  we can place those signals along the x-axis because the different frequencies correspond to  a different x-axis location now in the previous talk in order to avoid some confusion I used this  example here which is technically incorrect you can see that when we are acquiring the signal  we get this rephasing and then defacing at a free induction Decay rate of the signal now what  causes that rephasing and defacing at te what is this 180 degree radio frequency pulse that 180  degree radio frequency pulse as we looked at in the slide selection talk causes the spins to  D phase and then re-phase exactly at te and we get that rephasing causing an increase in signal  and then we get that free induction Decay now the increase in Signal allows us to account for the  local magnetic field in homogeneities and get a c signal at a te that is much more similar  to a true T2 Decay now that is confusing to you go back to the slide selection and the  frequency encoding gradient talks now because we get that increase and decrease in Signal the  net magnetization Vector that we read out from this slide is actually going to look a lot more  like this where we get an increase in signal up to t e and then a decrease in Signal that's  occurring because of those re-phasing within the slice now we saw that we can sample that  analog signal multiple different times and I used a very small sample number when we looked at  frequency encoding in fact we can sample that many more times we often use 128 or 256 different  samples during that data acquisition period now the frequencies that are contributing to  this signal remain the same they're based on that x-axis frequency encoding gradient now we  can take this signal the combination of all these different frequencies and we can delineate those  frequencies and organize them in a way that go from low frequency to high frequency the higher  frequency net magnetization vectors are going to correspond to this region on the slice the lower  frequencies will be at the other end of the slice what we've done here is we've converted  a time-based domain where we're sampling that analog signal over time during that frequency  encoding gradient and we've used a one-dimensional Fourier transformation to encode these specific  frequencies that are contributing to that image now the frequencies we order from low to high  which will give us the x-axis signals of the slice that we've selected it's going to give us  the entire column signal along that particular slice now as I've mentioned that transformation is  an inverse one-dimensional Fourier transformation Now using this single data acquisition during this  frequency encoding gradient what we're able to do is create an image based on the signal coming  from the entire column at the different x-axis locations now in order to create this image here  we've only passed through the sequence once we've used this data that we've acquired over time and  used it to delineate the different frequencies the unique combination of frequencies that will  give us this analog signal here now we've got no way of knowing where that signal is coming from  along the y-axis and this talk we're going to see how we can delineate those signals based on y-axis  location now let's go back to the slice we look at the slice at a period of time where we haven't yet  applied our frequency encoding gradients and we've Switched Off The Slice selection gradient as well  as our radio frequency pulse what's happening in this slice at this given period of time well the  spins in this slice are resonating all in Phase with one another at the same frequency the only  magnetization that this slice is experiencing at the moment is our main magnetic field and  we've seen that processional frequency is proportional to the magnetic field and if it's  only experiencing the main magnetic field all of these spins will be processing in Phase with  one another now spins that are spinning in Phase with one another will provide a sinusoidal signal  that can be measured out remember we have yet to apply that frequency encoding gradient now these  spins are accumulating because they're in Phase with one another giving us a net magnetization  Vector that looks like this now if we were trying to calculate the y-axis contributions to this net  magnetization Vector we would see that the signal coming from each location along those y-axis would  be in Phase with one another and they would be processing at the same frequency you can see that  it's the accumulation of these different y-axis components that's giving us this net magnetization  Vector that we're measuring remember we're only measuring that one cumulative signal that is  represented by this red line here now what we need to do is introduce some differentiation  along the y-axis now in order to do that we can look at our slice within the Cartesian plane  here we have applied a frequency encoding gradient along the x-axis now we need to apply some sort  of gradient along the y-axis in order to introduce some differences based on y-axis location now the  way we do this is by applying what's known as a phase encoding gradient now the phase encoding  gradient can't induce frequency changes along the y-axis our x-axis localization using this  frequency encoding gradient only works if the frequencies at a specific x-axis location are the  same in that entire y-axis column so we now can't introduce frequency changes along this y-axis  so how do we get about doing this well what we do is apply a gradient between the 90 and the  180 degree radio frequency pulses that magnetic field gradient is happening here in the y-axis  using these gradient coils now there are multiple different ways that we can represent this gradient  we can represent this gradient using this blue line here where we see that the magnetic field  at the top end of our slice is increased by a certain amount and at the bottom end of the slice  is decreased by a certain amount we've applied a gradient along this y-axis we can also represent  this gradient using this color change here you will see there's a specific point along this  gradient where the net magnetization will be our main magnetic field we've neither added nor  subtracted any net magnetization to this central part of our image now most commonly you'll often  see it represented using this symbol here where we are adding magnetization to the upper half of  our slice and subtracting magnetization from the main magnetic field to the lower half of our slice  again there is a point where there is no change in magnetization and that's what's known as the null  point which we'll see is really important when we go about localizing that signal later now what  we're going to do is apply this phasing coding gradient for a specific period of time and you can  see that the application of this phase encoding gradient for this short period of time causes the  spins to D phase based on their y-axis location because the magnetic field strength is stronger  as we head out to the peripheries here we will get more dephasing of these spins than  we do as we get closer to the null point you'll see at the no point there is no dephasing  of those spins they are still processing at the Llama frequency that's based on the main magnetic  field and you'll see that relative to this spins at the null point we will get the phasing of  these spins in the anti-clockwise direction now we've defased these spins  based on their y-axis location now once we've turned off that phase encoding  gradient what's going to happen to these spins well they will be experiencing the main magnetic  field they will continue to process based on the main magnetic field strength so let's turn off  this phase encoding gradient and see what happens to these spins they are now processing at a rate  that's proportional to the main magnetic field that's the only magnetic field that is influencing  the slice at this given period of time you can see now that the frequencies are all the same  the frequencies are the Llama frequency that's proportional to the main magnetic field what's  different is we've introduced some phase change based on the y-axis location of these spins now  let's see what happens to the net magnetization Vector that we measure from the entire slice when  we apply a phase encoding gradient remember when these spins were processing in phase and we  hadn't applied a phasing coding gradient the y-axis contributions were all in Phase with one  another now look what happens to these y-axis contributions as we apply that phase encoding  gradient you see now that these are out of phase with one another the peaks of these signals no  longer line up now look what happened to the net magnetization Vector that we measured as we  apply that phase encoding gradient you see we get loss of net magnetization Vector signal because of  that dephasing and transverse magnetization is a function of how in Phase those spins are with one  another now once that phase encoding gradient is Switched Off all of those spins all of those net  magnetization vectors are going to process at the Llama frequency all we've done now is introduce  phase difference based on y-axis location now what we're going to do is we're going to apply  a frequency encoding gradient to that slice now what is that frequency encoding gradient going  to do to these spins well the spins or the net magnetization vectors at the far end of the slice  are going to process at a faster rate than those at the near end of the slice we are going to now  introduce a frequency encoding gradient along that slice now as we introduce that frequency encoding  gradient you can see how this spins on the right hand side of our image are processing faster than  those at the left hand side and then we can go and measure the net magnetization Vector of that  entire slice now this net magnetization Vector can still be sampled during our data acquisition  period and we can get discrete values over time if you look at each column of spins here although  it looks like a disorganized chaos and mess you can see that each column still has the same  frequency as all of those spins along that x-axis column the difference here is that the  phase encoding gradient has applied some memory here we've got the phasing of these spins based on  their y-axis location although their frequencies are the same depending on where they're located  along the x-axis now we can take this data acquisition that we've got here and correlate it  to the specific phase encoding gradient that we used now if we compare that to the signal that we  generated without phase encoding we can see that we've acquired two different data Acquisitions  the first was the net magnetization Vector over time without any phase change in the y-axis  Direction the second data set that we've acquired has now taken the net magnetization Vector over  time but it's required that when there's been a certain amount of phase encoding applied in the  y-axis Direction now both of these are encoding for the same image the anatomy that we're Imaging  in that specific slice hasn't changed at all the only thing that's changed between these two data  acquisition periods is the amount of phase that we are introducing into the y-axis we can use either  one of these data sets to do a one-dimensional inverse Fourier transformation and calculate the  x-axis locations and the amplitude of the signal at each x-axis location now we can repeat this by  using a larger phase encoding gradient and what we can do is increase the magnetic field strength  along that y-axis location creating a larger phase encoding gradient in the y-axis Direction  now look what happens to these net magnetization vectors when we apply an increased phase encoding  gradient we're applying a stronger phase encoding gradient along the y-axis we can see that these  spins now or these net magnetization vectors have de-phased even further and we're getting  a further reduction in signal remember when we acquired this one we saw there was a reduction in  signal and you can kind of see that reduction in Signal when you compare these two data acquisition  points the signal here is lower than the signal when there wasn't any phasing again now what's  happened is based on the location in the y-axis here we have gotten even more de-phasing than we  had in our first example now hopefully you can see here that The Closer the spins are to the null  point where there's no change in magnetization Vector the less phase change there will be and  as we change the strength of the phasing coding gradient through multiple different iterations  those that are near the periphery of the slice are going to experience more phase change than  those closer to the null points on the slice and it's that degree or phase change as we change the  phase encoding gradient that is going to help us to localize that y-axis signal at least in part  and you can see that those that experience more phase change that signal is likely coming from  a y-axis location that is further away from the null Point again we can apply a frequency encoding  gradient here and as we are applying that gradient measure out another signal now you can see that  that signal is even less at this specific phase strength because these spins are even more out of  phase with one another however we can still use the signal that we've acquired over a period of  time as we're converting that analog signal into a digital signal we can still do a one-dimensional  inverse for your transformation and get x-axis locations with specific signals for the entire  Columns of that x-axis now not only can we apply a phase encoding gradient along the y-axis in  One Direction we can actually apply it in the opposite direction where the lower part of our  slice is now gaining magnetic field strength and the upper part of our slice is losing magnetic  field strength relative to that main magnetic field and as we apply that gradient here we can  see we get dephasing in the opposite direction we are still losing signal here again we allow  time to pass and we apply a frequency encoding gradient along the x-axis of our slice as we  apply that frequency encoding gradient we can measure the signal the net magnetization Vector  signal of the entire slice and get another data acquisition now the data that we've acquired here  corresponds to this phase change now it turns out that we can repeat the step multiple times to  acquire multiple different phase change datas and the number of phases that we use the number  of phase encoding steps that we use determines the number of pixels that we can delineate in the  y-axis of the picture that we're trying to create now I said when we use the frequency encoding  gradient we only required one cycle here from the 90 degree RF pulse to the time of repetition  in order to apply another phase encoding step we need to repeat the sequence over again at first we  applied NO phase encoding gradient we got our data here from the frequency encoding step and then  we waited to tr we allowed those spins to gain longitudinal magnetization before then flipping  them to 90 degrees and repeating the process once we flip those to 90 degrees we then applied  at different phase encode ingredients acquired a different data set at our frequency encoding  gradient we then again waited till our time of repetition before repeating the sequence again  with a different phase encoding gradient you can see that for each additional phase encoding  gradient we need to repeat the sequence therefore in order to add resolution in the y-axis Direction  in the phase encoding Direction it takes much longer because we need to repeat the cycle over  and over again for each phase encoding gradient that we are applying to our sample now if we  look at the data that we've acquired so far we've used four different phase encoding steps  we can then use different magnitudes of phase encoding and apply that signal that we've  acquired to a matrix here and each time we use a different phase strength we can generate a  different signal here and the way that we organize the signal that we're generating is based on the  amount of phase that we use to acquire that signal by convention we acquire the unfazed sample and  then each time we repeat the cycle we introduce a small amount of phase in both the positive and  negative directions so we first then apply a small amount of positive phase and acquire this line of  data in our next cycle we apply a small amount of negative phase and we acquire this line of data  we then repeat this process over and over again until we've done the number of phase encoding  steps that allows us to get the resolution that we want on the y-axis of our image now what we are  creating here is not an image we're not creating pixels the grayscale values here represents a data  point a numerical value and as those data points that we're going to plug into formulas later  that we can use then to generate our image now once we've acquired enough phase encoding steps to  give us the y-axis resolution that we want we will ultimately create what is known as k space now the  number of rows that we've included in k space here will equal the number of phase encoding steps  that we have done and it's those phase encoding steps that will determine our y-axis resolution  in the image now each line of this case space represents the net magnetization Vector change  over a given period of time and we can use that one-dimensional Fourier transformation to take  that data from the individual row and transform it into frequency based or x-axis location based  data and we can do that for each and every phase encoding step that we've used in our sequence  here so we can convert k space data which is time-based data each point along the x-axis in k  space represents the net magnetization Vector of the entire slice at a given period in time and we  can convert that into a frequency based or x-axis location based data set here this process is a  one-dimensional inverse Fourier transformation in both of these the only thing that differs between  the various different rows here is the amount of phase that we have applied in the sequence here  now you can see that signal gets stronger and it gets weaker as we head along in time that's  representing the signal phasing and dephasing you can also see that signal gets weaker as we  head out to the peripheries that's representing the amount of de-phasing that we have applied  as we have applied stronger and stronger phase encoding gradients now if we look at both k space  and this one-dimensional Fourier transform we can see that the values along the x-axis here are  coming from the data acquisition time that is happening during the frequency encoding gradient  we can also see that the rows that are generating k space are occurring at varying different phase  encoding steps that is how we create this k-space data now we can use this k space data as well  as this one-dimensional Fourier transformation data combine those data sets to give us ultimately  the image that we're trying to create now I want to take you through a very basic process that is  going to show you how we can combine these data sets in order to create this image now the way  that we're going to describe this is a much more simplified version of the actual process that is  happening in the background we are going to take only two data points along a specific x-axis  location when in fact often we're using 256 different data points and comparing them to one  another now the process that we're going to use to delineate those two separate points is applicable  to the process that's used to delineate 256 different points except in our example we'll only  have two variables in actual practical sense when we're generating MRI images in real life there are  256 different variables now much of what I'm going to explain to you here is adapted from Dr Alan  elster at mriquestions.com and I'm going to link those articles below go and check them out he has  a very good way of explaining these and I've just adapted these to show you a slightly different way  of how we can go about calculating where signal is coming from at each location on our image so what  we want to do is take two separate pixels here that have the same x-axis location and try and  delineate the actual pixel values for these two locations now where exactly in this data set is  the signal coming from in these two pixels well we have organized the signal in this data set based  on x-axis location so the signals here will impart come from these two pixels that we are trying to  calculate now I'm going to take you through a set of examples to show you at least in theory how  we can separate these two signals now in order to do this we're going to make two assumptions the  first assumption that we're going to make is all of the signal coming from this specific x-axis  location is only coming from these two pixels now we know that isn't true we know the signal on  that x-axis location is coming from 256 different pixels but for our example we're going to assume  that it's only coming from those two pixels the second assumption that we're going to make  is that we can calculate the degree of phasing that is required for the spins in either these  two pixels to be 180 degrees out of phase with one another now this isn't a false assumption  we know the phase encoding gradients that we're applying to the slides that we've selected and  we know the location of the pixels that we are trying to calculate the signal value for and  there is a mathematical formula that will allow us to calculate which degree of phasing based  on the location of these two pixels will cause the spins to be 180 degrees out of phase or one  another we're not going to actually calculate that but it can be done so let's have a look  at these two in closer detail now as we looked at before depending on the y-axis location of  these particular pixels they will experience different degrees of dephasing as we expose them  to different levels of phase encode ingredients now in order to calculate that degree of phasing  we first need to figure out what is the pixel value when these spins are perfectly in Phase with  one another we first need that in our data set now when will these pixels be perfectly in Phase with  one another there will be no phase change along the y-axis when we don't apply a phase encode  ingredient so when we don't apply a phase encoding gradient we know that we generate the frequency  data at the center of this frequency data set here this data set at this given point here represents  all of the net magnetization vectors delineates it into frequencies when we haven't applied any  phase encoding along the y-axis this particular data point here represents all of the signal  coming from the x-axis column along our image now that signal coming from the x-axis column in  our image we're making the assumption that that signal is only coming from these two pixels of  Interest it also represents the signal that is coming over the entire period of data acquisition  time remember we've used all of that data that we generated in k space to Fourier transform  and create this frequency based location here now there'll only be one period of time along our  pulse sequence when the spins will be perfectly in Phase with one another and that's te remember  we get this increase in Signal as those spins are re-facing with one another and they're perfectly  in phase at te not only are they perfectly in Phase a te based on our slide selection gradient  they are also perfectly in Phase that t based on the frequency encoding gradient remember we apply  a short negative or D phasing frequency encoding gradient before we apply the frequency encoding  gradient allowing those spins to re-phase as we're changing their frequencies and it turns out  at exactly the middle of this frequency encoding gradient is when all those spins will be perfectly  in Phase with one another they will have differing frequencies but for that brief moment in time the  net magnetization vectors will all be pointing in the same way now the second assumption we  made is that we know a specific amount of phase application that will cause the spins in  these two pixels to be perfectly out of phase with one another and again that can be calculated  mathematically now we've got two separate signal values one when we had no phase encoding gradients  and one when we've got a specific phase encoding gradient that means the spins in these two pixels  are 180 degrees out of phase with one another now remember the data that we are acquiring here  at this given point represents all of the signal that is measured throughout the entire data  acquisition period remember k space is taking that analog signal over a period of time and we're  using all of that case-based data to frequency encode and make the data that is acquired along  this line of the frequency encoding data space so this data point here represents all of the  signal coming from the x-axis at this location over the entire data acquisition sample so we  can't use this data point alone because this data point is the accumulation of signal over time what  we need to then do is look at the k space data as well as the frequency encoding data and we can  see that k space data will have a specific net magnetization Vector at t e that represents the  entire net magnetization Vector of the slice at te now we don't want the data from the entire slice  and we don't want the data from the entire data acquisition period and we can use both this  data point and this data point to delineate the signal contribution from that x-axis  location here at that specific given period of time and this in part is the process involved  in two-dimensional Fourier transformation where we take these two data sets and create the image  that we're eventually looking at on our computer screen now we don't need to know this process  in detail but we need to know that it's the combination of these two data points that gives  us these specific signal from that x-axis column at a given period of time so let's go about  using these in an actual practical example we've taken that point in time where we've used  our k space and this frequency encoding data to get a set measurable signal that we can calculate  that's coming from this entire x-axis column here now again we're making the assumption that this  signal is only coming from these two pixels in fact they're coming from 256 different pixels  along the y-axis here now we don't know the individual signal contributions from these two  separate pixels but we do know what the signal is based on the calculation that we've made from  the entire x-axis column here and that signal can be given a specific value a specific numerical  value for this example it's arbitrary we're going to use 14. now we know that the signal is  a combination of both of these signals here we're assuming that these pixels are the only thing  that's contributing to Signal along the x-axis now the amplitude from our first signal it  added to the amplitude of our second signal will give us the amplitude of the total signal  that we're measuring from this x-axis point now these net magnetization vectors have the  same frequency because they're at the same x-axis location they're at the same frequency encoding  gradient location not only do they have the same frequency but they're in Phase with one another  because we haven't supplied a phase encoding gradient so we've generated a formula here where  we can add the signal amplitude from signal one to the signal amplitude from signal 2 to get a  numerical data point that we've calculated from our frequency encoding and our k space data we can  then manipulate this formula to isolate signal 2 here we've moved signal one across and we can  see that signal 2 is equal to 14 the amplitude of the signal that we've calculated minus signal  one minus the contribution from our first signal now remember that signal 2 is equal to 14 minus  signal one let's look at the second example where we've applied a specific amount of phasing or D  phasing gradient to the y-axis here which means that the pixels here are completely 180 degrees  out of phase with one another we can then use this data point in combination with our k space  to figure out what signal is being generated at this x-axis location and here we get a different  numerical value because there's been de-phasing that signal coming from this x-axis location  is going to be less than our original signal now this signal is a combination of both the  signal from signal 1 and the signal from signal 2. they are still at the same frequency because  they're along the same x-axis location but they're at different phases now 180 degree outer  phase because of this phase encoding gradient we can see that the frequencies remain the  same but they're now 180 degrees out of phase this signal that we've now calculated  is a combination of both signal one minus the amplitude of signal 2. so if we take  signal one and take away the amplitude of signal 2 we will get this value that we've calculated here  now remember in the previous example we calculated what signal 2 was signal 2 was equal to 14 minus  signal one so we can substitute that value in here signal one minus signal 2 which is 14 minus signal  one will give us the amplitude of this measured signal that we have at this x-axis location we  can then solve for this formula we take minus 14 across so it'll be 6 plus 14 will give us 20 and  Signal 1 minus minus signal 1 will be Sigma 1 plus signal one we get an equation here that we can  calculate signal one plus signal one equals 20. so from this formula we can see that signal one the  signal coming from one of our pixels must be 10.

[00:34:18]because 10 plus 10 will equal 20. substituting  that signal one value in here 10 minus what will give us 6 10 minus 4 will give us 6. what we've  done now is we've calculated the signal 2 value we've got a pixel value here an arbitrary number  of 10 and a pixel value here of four we have now managed to differentiate the signal along the  y-axis based on this dephasing comparing this D phasing to when our spins had no phase difference  and hopefully you can see from this two pixel example how we can extrapolate this out to 256  different variables in our equation and use both the frequency encoding data and the k space data  in combination to get signal values based on the x-axis location and based on a specific period of  time when we are acquiring that data acquisition now the maths behind this is not important the  core concept here is that we can delineate why access values based on their degree of D phasing  that we apply along the y-axis of the slice now as we change the amount of D phasing along the  y-axis of our slice those signals coming from the peripheries of our slice will experience a greater  degree of dephasing than those near the middle of our slice and it's that degree of dephasing that  we measure in this output signal that allows us to eventually calculate where the signal is coming  from along the y-axis so to summarize now what we've done is we've used a simple pulse sequence  using multiple different gradients to allow us to select a specific slice allow us to delineate the  signal based on frequencies along the x-axis and then use multiple phase encoding steps along the  y-axis combining all of those data sets to allow us to create our final image here now in order to  create this frequency encoding data and ultimately create this image here we get all of the data  from this data set here which is known as k space so you can see that just from storing the data  in k space we can ultimately create individual images for each slice that we are generating in  our MRI image and each slice has a different case based data set now in the next talk we're  going to have a closer look at k space and how different regions of K space contribute to  different features in this image here we know that any individual point in k space represents  the net magnetization Vector of the entire slice at a given period of time but one data point  is not enough to create this image here we need all of these data points in combination to  create the MRI image now I know that this phase encoding step is very difficult conceptually to  understand if you are interested in this stuff go through this lecture multiple times and make  sure you have that understanding in your mind if these concepts are too complicated and you  don't have the time to fully understand this understand that it's the degree of D phasing  and the signal loss from that defacing that determines where signal is located along the  y-axis of our image now these concepts are commonly asked in exams and if you are studying  for a specific exam I've LinkedIn question Bank below that I've curated you can go and test  yourself using that question bank if you're preparing for a specific exam otherwise I'll see  you in the next talk where we're going to dive deeper into this data acquisition set known as  k space I'll see you all there goodbye everybody