So far, we talked about the cases when our body, if it is outside MRI machine and inside MRI machine. So, when our bodies are located outside MRI machine, the proton spins, which are source of MR signal. Those proton spins have a spin angular momentum and magnitude moment. And they are randomly oriented, so there will be no signal sources. There are signal sources, but it will not generate any component. But when our body is placed inside the MRI machine, then those spins will be aligned along the direction or parallel to the field, and the anti-parallel to the field. And also, those proton spins will precess or wobble around the axis of main magnetic field direction. Each protons that magnetization component will not be zero because there will be small number of spin excess that contributes to non-zero net magnetization. Those two components are the case of when our body is placed inside MRI machine. And then now, we will talk about what happens when we apply for RF energy to our body. That is about the magnetic resonance. So, this is microscopic viewpoint of magnetic resonance. We have two states inside the MRI machine. So, the spins with parallel to the field and anti-parallel to the field, and there are small [inaudible] with spin excess. And those states with parallel to the field had low energy state, and anti-parallel to the state are high energy state. When RF pulse is applied with the frequency of Larmor frequency, which is precession frequency over the protons spins. And then RF energy is applied to that same frequency over precession frequency, then those spins with the low energy states move to high energy state; meaning, the spins with parallel to the field move to the state of anti-parallel to the field. And this property is a quantum mechanical properties which cannot be explained with a conventional physics. So, when RF energy with the same frequency as the precession frequency, Larmor frequency, is applied through the transmission RF coil, and proton spins pick up energy and go from the low energy state to high energy state. So, it should be noted that proton spins can exist only in two states, so parallel to the field or anti-parallel to the field. And then, this quantum mechanical explanation of MR phenomenon is a little bit difficult to understand, but the easier way to understand MR physics and MR signal is to explain the concept with net magnetization. And then, what will happen to the net magnetization? So, the net magnetization is about the summation of these proton that spins. And then, we will not care about individual magnetization. So, this net magnetization gives much better understanding of MR machine. So, before talking about the net magnetization, we have to introduce a concept of rotating reference frame. And the reason is, the net magnetization itself rotates at a frequency over Larmor frequency. If it is along the Z direction, it looks like not rotate, but it's a little bit off-center. Then, it also wobbles in the same way as the individual proton spins. This net magnetization itself rotates at Larmor frequency, so we have to introduce a new coordinate. It is sometimes convenient, not sometimes but in many cases, it's convenient to express and visualize the evolution of magnetic vector in a frame of reference called rotating reference frame, and that is rotating at the Larmor frequency. So, we are introducing a new axis called X prime and Y prime. And Z direction is the same as the physical Z direction, the direction of main magnetic field. But X prime and Y prime, in plane direction itself is rotating at the Larmor frequency which can be represented in this equation. So, the transverse magnetization is rotating at Larmor frequency omega zero. So, it's much simpler to understand that magnetization dynamics. So, in this rotating reference frame, transverse component itself is rotating. So, the net magnetization looks like a stationary in this rotating reference frame. [inaudible] then rotating at the Larmor frequency. So, it's much easier to understand and much easier to visualize the dynamics of net magnetization. The dynamics of magnetizations will be represented in the rotating reference frame throughout this course, unless otherwise specified. Let's go back to the magnetic resonance phenomena again. So, I'm going to explain the concept of a quantum mechanical viewpoint, and also classical physical viewpoint of the net magnetization simultaneously in this field. There are spin excess. And then, when we apply for RF energy at the same frequency of this precession frequency, then RF energy will decrease the net longitudinal magnetization overall. The reason is, those spins parallel to the field will moved to the anti-parallel to the field. So, the longitudinal component will get reduced. And then the thing happening is, this net transverse magnetization start to appear because those spins become a little bit in-phase. And then, we start to observe a transverse component. And this is just a general phenomenon. This is just physical phenomenon that people observed. So, this energy is converted to the longitudinal direction, and then these energy conversion generate transverse component and longitudinal component gradually reduced. And then, if we're applying for this RF energy continuously and then, until that longitudinal component are completely disappear, then this transverse component will get maximized. And then at this point, we can detect a signal in the receiver RF coil which is called a free induction decay. So, this FID signal, can be maximized by stopping application of the RF energy when the longitudinal magnetization becomes zero. And this RF pulse is called 90 degree RF pulse. And since the original longitudinal magnetization is flipped towards transverse plane completely and that maximizes a signal. So when we stop applying for the RF energy, and this magnetization will recover back to original. So in the viewpoint of transverse component, and it will get the decay to zero and the longitudinal component will get recovered back to original, and this procedure is called relaxation. So in the viewpoint MR imaging, only this transverse component can be detected and used for MR imaging. And the reason is the majority of the photon spins are aligned along parallel or anti-parallel to direction. We only represent the spin excess. But majority of them are aligned along longitudinal directions so they cancel each other. So, this number of spin excess is much smaller than the majority of the spins cancel each other. So we cannot detect any signal component along z direction. Well we can detect signal only on the transverse plane which has orthogonal component to the B_0. So this can be only detected and the useful MR imaging. So the net magnetization eventually returns back to original state which is called relaxation. So from the moment I'll stop applying for the RF energy to the relaxation back to the original, so during these states we can detect MR signal, which is called free induction decay. So, here is the representation of this magnetization phenomenon in the viewpoint of rotating reference frame and laboratory frame in comparison. So here, we use z prime and x prime, y prime, and z, x and y. So the RF energy should to be applied along the transverse plane. So x prime or y prime direction, and the RF energy is often denoted as a B_1 in comparison to B_0 which is a main magnetic field direction, z direction. And this B_1 component, RF energy component should be applied along transverse direction, either x prime direction or y prime direction with some angle in between. And sequence programmers can manipulate these phases of the RF pass it on the rotating reference frame. So, this is the frame, the therapy's rotating at the Larmor frequency, and we can manipulate the transmission phase of the RF energy. And in the laboratory frame, the trajectory is in a spiral form. So if z, x and y are in the stationary frame and this net magnetization looks like moving in a spiral form as in this figure, which is difficult to understand. And in fact, as I mentioned here, the sequence programmer can manipulate transmission phase and also reception phase of the signal. Both of them can be manipulated and can be controlled. So in majority of the cases we use these rotating reference frame to represent MR signal dynamics. So for the z longitudinal component, if we apply for RF energy P_1 field along x prime direction, then based on the right hand rule the magnetization will move along the y prime direction as shown here, based on that right hand rule. So what happens if we apply the RF energy along the z direction? Well then, while transverse direction x prime while y prime directions and then, it will simply change the precision speed, and will not cause magnetization. And this change will be very minor compared to the very strong extra magnetic fields. So, there will be almost no change. And also there will be almost no magnetic resonance. So it is important to place RF coils, both transmission and reception, such that RF fields generated by the RF coils are perpendicular to B_0. So most of the RF coils are already designed in that way. But if we use a very special surface coil which can be placed for the imaging, and then those for instance ring type surface coil can be provided by the RF vendor, MR vendor. And then this type of RF coil should be placed in a way perpendicular to the main line filed direction. So that RF fields generated by the RF coils will be perpendicular to the main magnetic field direction to have maximum signal. If they are slightly tilted, then the signal will be reduced. And only we get that signal a component projection along the orthogonal direction of the B_0. If it gets to parallel, the ring type coil, the symmetric axis is parallel to the B_0, and then we cannot detect any signal. I'm going to talk about the concept of flip angle here. So the final flip angle and the phase of net magnetization depends on both amplitude and the duration of the RF pulse B_1. So here B_1 can be represented as an envelope of P_1 multiplied by precision frequency. In case of laboratory frame, stationary laboratory frame. But if we see this concept into the rotating reference frame, and then the B_1 can be represented, we can just switch to only the envelope of the RF path. And that pi over two or 90 degrees RF pulse, can be generated if we turn off the RF after the net magnetization has moved down into the transverse plane, and then that maximize the signal. And then pi or 180 degree RF pulse can be generated if we apply twice as long as or twice as high as the 90 degree RF pulse. And then magnetization is flipped along the minus z direction. And then this RF pulse is called pi or 180 degree RF pulse. So both of them are used for MR imaging in many cases. We'll talk about that much later. And then the concept of flip angle is alpha. It can be presented as a summation of all the B_1 field as a function of time. So that is multiplied by total magnetization gamma, and then this represent flip angle of net magnetization. So, this is related to this equation and as shown here. So the original net magnetization and z component will be flipped following toward the transverse component or if we apply for more energy, then that may flip even more. So the angle between the original net magnetization z direction and the eventual net magnetization location and this angle is called flip angle. And so this concept can be considered if we apply for RF energy along x prime direction. And this net magnetization is going to flip toward y prime direction. Well, if we keep applying for energy, and this net magnetization will keep moving. And then what happens if we apply for the energy more than 180 degree flipping? And then this magnetization is going to keep rotate following with along the axis of x prime. So it will keep rotating, it will keep applying for the energy. And this rotation speed is determined following the Larmor equation. In this case, omega one equals rotation frequency or speed. Omega one equals gamma B_1. So in case B_1 is the RF energy applied along x prime direction. So this Larmor equation is applicable along the B_1 field too and which is very similar to the period field which is omega zero equals gamma B_0. So this flip angle can be calculated based on this omega one equals gamma B_1 equation. So omega one multiplied by time tau corresponds to the flip angle of this magnetization. So this equation is derived based on this Larmor equation in the direction of B_1. So the flip angle alpha, can be manipulated by this operator, and we can determine how high should the flip angle for best MR image.