We will talk about the concept of K-space here. We will talk about mathematical requirement to generate an image, and then that will give intuition of why frequency encoding and phase encoding are necessary to form an MR image. Then, we change back after explanation of frequency encoding and phase encoding, and then we will change back how they can be used to form an image. The first viewpoint is why do we need frequency encoding and phase encoding in the mathematical viewpoint of MR image. So that is about the concept of K-space, and we'll talk about this mathematically and also with the physical meaning. We talked about the slice selection in the previous video lecture. Now, we have selected a slice position of interest and the thickness to be excited. So what slice selection do is that it's only magnetization in the slice is tipped, excited or resonance by RF pulse and has the transverse magnetization component which is now in phase after applying for the slice refocusing gradient. So as shown here, RF pulse and slice selection gradient, they are combined. Then right after this, we will have a lower signal intensity because all the proton spins excited. Within that excited slice, we will have different precession frequency along the G direction. Then applying for a slice refocusing gradient will make the excited speeds within the slice will have in-phase. Then we will start to see some Free Induction Decay as shown here. And then theoretically, we can acquire data during this period or another analog-to-digital converter or sampling or readout all these terms will be used in interchangeable way. So generally, Free Induction Decay is not directly used for imaging and the reason is FID at this point does not have a spatial information along within the excited play. So it had a spacial information only the G direction or the excited slice direction but it does not have spatial information in the in-plane directions. So we need time for spatial encoding by using gradient course. But there are some exceptions some of the imaging techniques called Spiral imaging or Radial imaging case we directly samples data right after the slice selection. But that is a spatial imaging techniques and other than that the general or the most popular imaging techniques that something we are going to talk now and in this case we generally need time for spatial encoding to get the in-plane information which is converted to image. Okay. So let's consider right after the slice selection period. So the FID signal can be mathematically modulated are represented as shown here. So Free Induction Decay signal S(t) can be mathematically represented as a summation of all the signals and h(x,y ) represent the spatial location information. So that is the image that we want to get so h(x,y) so that is multiplied by e to the minus j two pi f_0 t. So which means this represent the spins precess in the Larmor frequency. So the spins will be precess and then that is induced in the RF coil with the frequency of Larmor frequency two pi f_0 t. And that is going to be applied to all the protons pinned within the space with no differences along x and y if there is no gradient. So all the proton spins along special direction is going to have sinusoidal signal and that is all summed together to form on Free Induction Decay signal. So eventually what we measure from MRI is this signal S(t) but what we need to get is spatial information which is imaging. So we need to get h(x,y) from the measured signal S(t). And then if you see this equation carefully, so this equation can be converted to the Fourier transform domain. So if we can get a two-dimensional Fourier transform of the signal and then we can apply for inverse Fourier transform to acquire the data to get h(x,y). So that is the basic concept of MR imaging. So eventually, MR imaging is not directly on the spatial domain. So MR data is going to be performed on the frequency domain over the spacial image information. Okay. So let's consider the two-dimensional Fourier transform of h(x,y) to be H(u,v). So it's uppercase H and lowercase h. Then that is two-dimensional Fourier transform. I hope you remember the equation, so I'm trying to review here so H(u,v) is going to be the integration of h(x,y) multiply by e to the minus j two pi u x and e to the minus j two pi v y dx dy. So in this case, precession frequency should be a function of x and function of y and that should be multiplied to h(x,y). So in this Free Induction Decay, all the precession frequency is going to be the same with no distinction along x and y, and the goal is to generate this spatially different precession frequency to be applied to the spatial domain. So we need to convert our S(t) into H(u,v). So that means converting this e two minus j two pi f_0 t. So all proton spins precess at the Larmor frequency into we have to convert this into e two minus j two pi u x multiplied by e two minus two pi v y. So that means that can be performed by applying for gradient policy along the other two directions our x and y. And that is going to be about the frequency encoding and phase encoding. Let's talk more carefully about this concept again. So this is the equation for the two-dimensional Fourier transform and we have to generate this signal. To do that, okay so this data acquisition is called K-space so the space of MR data acquisition is called K-space. And the name is, there is some historical reason to put the name as K-space but anyway. So here we replace variables u and v as just k_x and k_y, just replacing the variable. So eventually so MR image is in the spatial domain and its Fourier transform is going to be frequency domain and that is just MR data acquisition domain okay which is called K-space. Let's consider K-space and gradient relationship. So again, so this signal to measure Free Induction Decay right after slice selection is going to be S(t) and that is integration of h(x,y) multiplied by e two minus j two pi f_0 t dx dy and then what gradient does is f(x,y) frequency of precession is going to be modulated in a spatially dependent manner as shown here. So if that is combined with Gamma B_0 so that is with no gradient then it's going to become Gamma hat B_0 and if there is a gradient along x direction and that is going to be changed as a function of x dx plus y dy if there is gradient along y direction and this frequency component f_0 will be replaced by this f(x,y ), if there is a gradient and then let's replace f_0 with f(x,y) and then the signal is going to be h(x,y) multiplied by e two minus j two pi comma B_0 t. So there is Tom, t and that is duration of gradient function. So Gamma B_0 t plus x G_x t_x plus y G_y t_y multiplied by dx dy and then this portion B_0 t is not a function of x and y so it can move out of the integration part and then now the equation have e to the minus j two pi (gamma G_x t_x) x and e two minus j two pi Gamma G_y t_y multiplied by y and here dx dy is duration of gradient and also duration of gradient along x and y. And let's say we acquire the data after applying for this gradient along x and y direction and then the acquired data will have different phase okay that is going to be a function of spatial location x and y, so which is now in the form of a two-dimensional Fourier transform. Okay, so that is required as I mentioned in the the previous slide. So if we compare this equation with a two-dimensional Fourier transform. The term for the Larmor frequency can be demodulated. Then it will be applying from T modulation to the acquired data and this term can be removed. That is very easily done, okay? Then to acquire the signal S(t) is going to be integration of h(x,y) e two minus j two pi (gamma G_x t_x)x and e two minus j two pi (gamma G_y t_y) y so which can be compared to the two-dimensional Fourier transform equations and then we can find the u or k_x part is Gamma G_x t_x and k_y part is going to Gamma G_y t_y, okay? Here G Gamma hat is going to be a constant and what we have to consider the G_x multiplied by t_ x and G_y multiplied by t_y. So this determines the area under the gradient pulsing in the sequence domain, time domain. So G_x t_x, G_y t_y determines the how big area within the gradient pulsing is applied and that determines the location of a frequency domain so or case-based domain same thing. Okay. So here again, so we have K-space domain, frequency domain. If we feel all the data points within the case-based domain and then applying for two-dimensional Fourier transform and then we get MR image, okay? And then again the mathematically k_x and u is going to be Gamma G_x t_x Gamma G_y t_y. So that is going to be determined based on the area under this gradient pulsing, gradient pulsing as shown here, okay? So this area will determine the location on the frequency domain and then if we apply for slice selection and gradient pulsing along x and y and acquire one data points and that these data points will be used to fill this one data point on the frequency domain, okay? And then eventually, we have to fill all the data points within this K-space and then how can we coordinate G_x and G_y an ADC sampling to fill all the K-space points here and that is the concept of frequency encoding and phase encoding that is going to be discussed in the next video lecture.
