[MUSIC] Okay we will talk about the contrast mechanism in gradient echo imaging in this video. Okay, so equation for the gradient echo imaging is a little bit more complicated than the spin echo imaging that we just talked in the previous lectures. So the gradient echo imaging is often performed with a flip angle smaller than 90 degree. Because of that, they usually are even more complicated. But let's try to derive the equations for gradient echo imaging. So after the first alpha pulse, the longitudinal magnetization Mz(t) can be described as M0 cosine alpha, the e to- t/T1 plus M0( 1- e to -t /T1), okay? So if you combine this exponential term, these two terms together, and then that portion is the portion to be recovered, okay. So exponential term, okay, that signal difference is going to be exponentially decayed to zero, and that becomes M0, okay? So you can easily figure out if you combine this term and this term and then see, and then this is M0. So how much signal is lost from the M0? And then that portion recover back the original with time constant T1, okay. You can understand like that. And then let's assume TR is much longer than T2, so we can ignore the transverse magnetization. So we can assume that transverse magnetization is zero at the moment of a new alpha pulse. So even if TR is very short comparable to T2, still we can eliminate the transverse magnetization based on the mechanism called spoiling. But we will not talk about that, but we can assume there is no transverse magnetization for the gradient echo imaging. So at the moment of the second alpha pulse, and then Mz becomes M0 cosine alpha e to -TR/T1 + M0(1- e to -TR/T1). So at the moment of the second alpha pulse, okay, so it's almost the same as this equation, but now this t is replaced by TR. And magnetization just prior to the n-th alpha pulse, will have very similar shape to this equation, but it can be presented as Mz(nTR) and Mz([n-1]TR cosine alpha, e to -TR /T1, + M0(1- e to -TR /T1). Okay this is going to n-th alpha pulse relative to n minus 1's magnetization, okay? So eventually, this imaging is going to be in the steady state which means there will be no signal difference between Mz[n- 1] and Mz[nn], okay. So these two will have the same equation or the same signal intensity which is the deficient of a steady state. So steady state in the steady state, so Mg[n.TR] and Mg([n-1].TR) intensity will be the same. And then we can solve this equation based on this relationship and then that gives signal intensity of Mg, okay, at the steady state to M0, 1- e to -TR/T1 divided by 1- cosine alpha e to -TR/T1. Okay, this is going to be longitudinal magnetization, okay, right before applying for excitation alpha pulse, okay, in the steady state. And then the single intensity or gradient echo is going to be alpha pulse flipping, the signal intensity is going to be the same as transverse longitudinal magnetization multiplied by sine alpha. So that is going to be the signal intensity flipped toward the transverse plane, okay. So that is going to be initial magnetization, right after applying for this alpha pulse, okay? And then that is going to decay during the time called echo time, so we have to multiply e to the -TE/T2*, okay? This is of a similar intensity of gradient echo imaging. Okay, so I'm writing this equation again, okay as shown here. You may not need to understand every detail of this procedure. But you may just remember how this signal, gradient echo imaging signal intensity can be affected by the time constant T1 as shown here, and also T2*, and proton density, okay. So again, this gradient echo imaging is also a function of proton density, and T1 and T2*, okay. And then how can we selectively emphasize one of the above three? And what is the optimal flip angle? So we didn't consider optimal flip angle for the spin echo case, but in case of gradient echo, it's a little bit different. So we have to consider optimal flip angle alpha that maximizes signal intensity. And what is the optimal TR when flip angle alpha is 90 degrees, okay? Okay we may answer for this questions during the next slide, okay. So what is optimal TR when flip angle alpha is 90 degree, and then a long TR, okay, that makes e to -TR/T1 to be close to 0, okay? So that is going to be, that imaging is going to maximize the single intensity. So typically TR of twice or three times longer than T1 will maximize the signal intensity, okay, and minimize these component to be 0, okay. Almost decays to 0, okay. And this long TR increases scan timing also, increased scan time, but that also maximizes signal intensity in the viewpoint of T1 recovery describing this equation. So what is the optimal flip angle alpha that maximizes signal intensity, S(x,y), okay? Okay, that can be calculated based on this equation as shown here. Because depending on the flip angle, so this flip angle of x pulse, the recovery of the signal and also the component flipped on the transverse plane. So which is the determined by sine alpha. And also this cosine alpha term determines how much signal is going to recover. So this is very complicated function of flip angle now, okay? So what is the optimal flip angle alpha that maximizes signal intensity, S(x,y)? So in the viewpoint of TR, we can just increase TR as much as possible, or just twice or three times longer than T1 of tissue or interest and then maximize signal interest. But if the TR is not that long, in between certain values, then that optimal flip angle alpha that maximize the signal, okay? It can be lower than 90 degree, okay? So that's flip angle is called Ernst angle, so Ernst angle flip angle is arc cosine e to -TR/T1, okay? How can you derive that, okay? You can easily derive that, you can apply for differentiation in terms of flip angle alpha to this term, okay, to this equation, to this term. And then you will be able to find that flip angle alpha that maximize this term is arc cosine e to -TR/T1. You can easily derive that by applying differentiation to this equation. So again, for a given shorter TR, okay, shorter than twice or three times the T1 of tissue or interest, okay. TR is shorter than that, then there exists an optimal flip angle which is lower than 90 degree, so that maximizes the MR signals. So because of that gradient echo imaging can be performed faster with the shorter TR. Okay, this is overall gradient echo signal intensity. And the proton density weighted imaging can be performed by making this T, as short as possible, okay. Okay this term, we can make it as short as possible. And then TE as short as possible, then this term will be almost zero, close to this portion becomes zero. So the overall term becomes close to one. So that means almost no T2* contrast. And make TR quite long, so two times or three times of T1 of tissue, or a small flip angle, which is much, it's smaller than the Ernst angle. Then there will be almost no T1 contrast, okay? So now gradient echo imaging is not just a function of a TR, it's also affected by flip angle relative to TR. So the long TR short TR is a relative concept compared to the flip angle in case of gradient echo. If we have flipped very small, flip angle is very small, then you don't have to wait that long for the magnetization to recover back to original. So the TR, long TR, short TR concept is now a relative concept to the flip angle used for imaging. Okay, so for T1-weighted imaging, the concept is similar to spin echo. So short TE, as short as possible, then T2* contrast is going to be gone. And then we may use short TR, okay, in the range of TR or T1 of tissues, or interest, or flip angle is around Ernst angle, okay? And then this T1 contrast is going to be maximized. Well, T2*-weighted imaging can be acquired if we insert echo time to be long enough, okay? In this case, T2, TE is T2* of tissue. So in case of gradient echo, the signal is affected by multi-field imaging, so it's affected by T2*, not T2, okay? Which T2* is a little bit shorter than T2, and it is a function of spatial resolution edges. It's not fixed, it depends on the span of scan resolution, so T2* is mixture of multi-field imaging and T2 effect. So I'm trying to remind the term again. So long TE means signal to signal to spin echo TE similar to T2 of tissue. So in case of gradient echo, long TE is TE is along the T2* of tissues, which maximize T2* contrast. To minimize T1 contrast, we have to use long TR, either twice or three times longer than T1 of tissues or interest when flip angle is around 90, or we can use small flip angle, okay? Which is much smaller than Ernst angle, okay. If TR is a little bit shorter than that. Then this scan condition will be minus T1 contrast, okay. So overall, it's going to be T2*-weighted imaging. So again, it's similar for TR or 1/flip angle and TE. So this is the functions as shown here. So T1-weighted imaging and T2*-weighted imaging and proton density weighted imaging, very similar. So high signal intensity of white matter, which means signal recover faster than grey matter, so it's the brighter. And T2 *-weighted imaging, so the white matter is low intensity, that means signal decay faster than the grey matter. And also the proton density weighted imaging, is the same thing as the spin echo case. So T2* weighed imaging is really useful for detecting the veins or some pathologic conditions such as hemorrhage in the brain. Because they cause faster signal decay, or lower T2, and can also cause multi-field imaging because of the actual hemoglobin, okay? So that causes signal more defaced, okay. So we have good contrasts on the T2*-weighted images. And T1-weighted images can be used for high resolution anatomic imagining,, okay, similar way to the spin echo case.