Noll (2006) MRI Notes 3: page 1 Notes on MRI, Part III The 3rd Dimension - Z The 3D signal equation can be written as follows: ),(),(),(),,())()()((2exp(),,()(tkwtkvtkuzyxzyxwvuMdxdydzztkytkxtkizyxmts====++−=∫∫∫π where M(u,v,w) is the 3D FT of m(x,y,z). In the spin-warp method for 2D acquisition, one line at a time is acquired in the 2D Fourier domain (or k-space). This method is easily extended to 3D by using phase encoding in two dimensions (rather than 1) and frequency encoding in the remaining dimension: This results in the acquisition of a cubic data set one line at a time:Noll (2006) MRI Notes 3: page 2 The sampling requirements and spatial resolution requirements are the same as they would be for the 2D spin-warp method (FOVz = 1/Δkz; Δz = 1/Wkz). If there are Ny and Nz samples in the y and z directions, respectively, then the total time to acquire the 3D volume is Ny*Nz*TR. For example, for Ny = Nz =128 and TR = 33 ms, the overall image acquisition time is 9 min – rather long! Slice Selective Excitation The most common approach for dealing with the 3rd (z) dimension is to use slice selective excitation. This is done by applying a z-gradient so that the resonance frequency varies in the z-direction and applying a bandpass RF pulse to excite only the those spins whose resonant frequency lies within the band: We will again solve the Bloch equations for this specific case. We will let B1 be a time-varying magnetic field rotating at ω0. For this analysis, we’ll let the rotating frame be at ωframe = ω0. ')()()sin)(cos()(1,001iBjiB11tBttttBteff=+=ωω A z-gradient is applied, so the component in the z-direction is: kBkBzz)()()()(.,0zGzzGBzzeffz⋅=⋅+= and the net effective applied field is: kiB )(')(1zGtBzeff⋅+=Noll (2006) MRI Notes 3: page 3 The Bloch equation for this case reduces to the following: roteffrotrotMBMM⎥⎥⎥⎦⎤⎢⎢⎢⎣⎡−−=×=0)(0)(00011tBtBzGzGdtdzzγγγγγ What we would like to know is how the magnetization, Mrot, varies as a function of z position following the application of the specified B1 field. This is, in general, a very difficult equation to solve because it is non-linear. Small Tip Angle Approximation One particularly useful approach to the solution to the above Bloch equation is to use the “small tip angle approximation.” Here, we assume the B1 produces a small net rotation angle, say, ∫°< )30( 6)(1πγdttB In this case, we can assume the z component of the magnetization, mz, is approximately equal to m0 during the RF pulse. Essentially, we are saying that: 1)(cos01≈⎟⎟⎠⎞⎜⎜⎝⎛∫tdBττγ Under this assumption, 0=dtdmz, mz(t) = m0, and thus mz(t) = m0. The above Bloch equation can then be rewritten as: ⎥⎥⎥⎦⎤⎢⎢⎢⎣⎡⎥⎥⎥⎦⎤⎢⎢⎢⎣⎡−=0,,1000)(000mmmtBzGzGdtdrotyrotxzzγγγrotM We now would like to solve for mxy,rot(z,t) = mx,rot(z,t) + i my,rot(z,t). We can then write a differential equation using for the transverse component: 01,01,,,,,)()(mtBizmGimtBizmGizmGdtdmidtdmdtdmrotxyzrotxzrotyzrotyrotxrotxyγγγγγ+−=+−=+=Noll (2006) MRI Notes 3: page 4 Observe that iγGzz is a constant with respect to time and thus we have a first order differential equation with a driving function iγB1(t)m0. For initial condition, mxy,rot(z,t) = 0, the solution to this differential equation can be shown be: ∫−=tzztGirotxydBzGieimtzmz010,)()exp(),(ξξγξγγ Again, we want the solution to this differential equation at the time of the end of the RF pulse, which we define as τ, mxy,rot(z,τ): We now make a variable substitution, 2/τξ−=s . We also assume that the RF pulse that is symmetrical (even) around τ/2 and that it is zero outside of the interval [0,τ]. The magnetization can now be described as: {}zGxzGizzGizzGizzGirotxyzzzzzsBFemidsszGisBemidssBzsGiemidssBszGieimzmπγτγτγτττγτττγτγπγπτγτγγτγτγτ2112/012/02/2/12/02/2/10,)2/(22exp)2/()2/()exp()2/())2/(exp(),(=−−∞∞−−−−−−+=⎟⎟⎠⎞⎜⎜⎝⎛⎟⎠⎞⎜⎝⎛+=+=++=∫∫∫ For symmetrical RF pulses, the forward and inverse FT are the same. Thus, under the small tip angle approximation, the slice profile (variation of mxy,rot with z) is related to the spectrum of the RF pulse: {}zGfrotxyztBFzmπγτ21,)(),(=∝Noll (2006) MRI Notes 3: page 5 and zGfzπγ2= is the conversion between spectrum and the z location: We’re almost there, but we still have some undesired phase variation in the z-direction, )2/exp(τγzGiz−, the can lead to undesired phase destruction when integrated by the RF coil. How is this fixed? We simply apply a negative Gz for a period τ/2. This is often called a slice rephasing pulse. This will result in phase accumulation of: ()2/exp),(exp2/3τγγττzGidttzBiz=⎟⎟⎠⎞⎜⎜⎝⎛Δ−∫ and thus:Noll (2006) MRI Notes 3: page 6 {}zGxrotxyzsBFmizmπγτγτ210,)2/()2/3,(=+= There is a k-space picture to this. For this, we assume that the RF pulse occurs instantaneously a the center of the pulse (at τ/2) and we begin accumulation area in k-z after that point. By applying a negative gradient for the same duration as the last half of the pulse, the areas cancel and the k-space location in the z direction is returned to the origin. Notice for an RF pulse applied along the real (i’) axis, the magnetization will end up along the imaginary (j’)axis. Also observe the flip angle at the center of the slice is: {}012/2/1)2/()2/(=−+=+=∫zsBFdssBτγτγαττ Previously, we discussed the that transverse magnetization after an α pulse was m0sinα, but for small α, sinα ~ α. So here, the transverse magnetization is also m0α. Example – the sinc RF pulse Consider an RF pulse roughly in the form: ⎟⎠⎞⎜⎝⎛−=TtAtB2/sinc)(1τ which has a spectrum: {}TBWBWfATsBF /1 where,rect)2/(1=⎟⎠⎞⎜⎝⎛=+τ The magnetization will be: ⎟⎠⎞⎜⎝⎛Δ=⎟⎠⎞⎜⎝⎛=zzmiBWzGATmizmzrotxyrect2rect)2/3,(00,απγγτNoll (2006) MRI Notes 3: page 7 where the slice thickness is zGBWzγπ2=Δ and flip angle is ATγα=. Putting Slice Selection with the Signal Equation Let’s define our slice profile function: {}zGfztBFzpπγτγ21)2/()(=+= then )()2/3,(0,zpimzmrotxy=τ Now we go back to the case where we have a 3D distribution of magnetization by substituting im0 = m(x,y,z) and putting it into the signal equation (again the RF coil integrates across the object):