5D – Magnetic Resonance Imaging
In magnetic resonance imaging, we need to measure the NMR signal at specific locations (voxels) in the object.
In order to encode spatial position using frequency, the Larmor fre-quency should vary as a function of spatial position.
- The main magnet in an MRI machine produces a constant mag-netic field of strength B0 in the z direction.
- By imposing a spatial variation in the magnetic field, i.e., a
mag-netic field gradient, different points in 3D space will be subject to
different magnetic field strengths.
- Since ω = γB, we will be able to obtain a range of proton reso-nant frequencies, each dependent on the position of the particular proton within the body.
Slice selection
Magnetic field gradients are generated by three separate gradient coils. - each coil produces a spatially dependent field that varies linearly
and adds to (or subtract from) the main Bz = B0 field - the magnetic field gradients are
G = (Gx, Gy, Gz) = ∂Bz ∂x , ∂Bz ∂y , ∂Bz ∂z (1) (unit gauss/cm or T/m; 1 gauss = 10−4 T; 1 G/cm = 10 mT/m) - the direction of the magnetic field (Bz) is not changed
- there is no additional contribution to the magnetic field at the isocenter (x = 0, y = 0, z = 0) of the magnet
- the total magnetic field is
B = (Bx, By, Bz) = (0, 0, Bz) (2)
i.e., the x and y components are zero, and the z component is given by
The Larmor frequency at a point (x, y, z) is given by
ω(x, y, z) = γBz = γ(B0 +Gxx + Gyy + Gzz) (4)
- slice selection requires: RF pulse + magnetic field gradient - to select a slice parallel to the xy plane:
– set Gz = 0, Gx = Gy = 0
– giving G = (0, 0, Gz); Bz = B0 +Gzz : ω(z) = γ(B0 +Gzz)
– to excite a slice at z = z1, apply an RF pulse of frequency
ω1 = γ(B0 + Gzz1) for a time duration τp
- the RF pulse will cause M0 to tilt away from the longitudinal axis
- to select a slice parallel to the xz plane, set Gy = 0, Gx = Gz = 0
- to select a slice parallel to the yz plane, set Gx = 0, Gy = Gz = 0
- slices at any orienation can be obtained by selecting appropriate combinations of Gx, Gy and Gz.
Example Consider B0 = 1 T, Gz = 1 G/cm , Gx = Gy = 0. Gz = 0.01 T/m Bz = B0 +Gxx + Gyy + Gzz = 1 + 0.01z where z is in m. ω(x, y, z) = γ(B0 +Gxx + Gyy + Gzz) ω(z) = γ(1 + 0.01z) ν(z) = γ–(1 + 0.01z) z (m) −0.3 −0.2 −0.1 0 0.1 0.2 0.3 Bz (T) 0.997 0.998 0.999 1 1.001 1.002 1.003 ν (MHz)) 42.45 42.49 42.54 42.58 42.62 42.66 42.71 6
Consider the situation where Gx = Gy = 0, Gz = 0.
- The Larmor frequencies of the protons as a function of their po-sition in z are
ω(z) = γ(B0 + Gzz)
= ω0 +γGzz (5)
- For a pure sinusoidal excitation with frequency ω = ωc, only
pro-tons on the plane z = zc will respond:
zc = 1
γGz(ωc − ω0) (6)
- To get a reasonable signal level, a slice of finite (non-zero) thick-ness Δz is excited by a range of frequencies Δω around the centre frequency ωc.
- For a rectangular slice profile of thickness Δz = z2−z1 and centre
zc, we require an RF pulse:
Centre frequency: ωc = ωo +γGzzc (7)
Bandwidth: Δω = γGzΔz (8) i.e., the RF pulse contains all the frequencies in the range
ωc − Δω
2 → ωc + Δω
From (6) and (8), the three parameters that are used to control slice position zc and thickness Δz are:
- gradient strength Gz
- RF centre frequency ωc
- RF bandwidth or frequency range Δω
For example, if we increase Gz (Gz,B > Gz,A) with ω0 and Δω un-changed:
- ΔzB < ΔzA
- zc,B < zc,A
i.e., slice thickness is reduced, slice position has moved towards z = 0
Example
A sample is put in a magnetic field with B0 = 1.5 T, and a z gradient
Gz = 3 G/cm is applied. (a) What is the Larmor frequency for the
protons on thez = 0 plane? (b) If we are to image a slab with thickness 0.5 m centred at z = 0, what s the range of Larmor frequencies of the protons in the slab? (c) Suppose that we desire a slice thickness of 2.5mm, what RF frequency range should be excited?
(a) We have
G = (0, 0, Gz)
ν(z) = γ–Bz = γ–(B0 + Gzz)
On the z = 0 plane, ν0 = γ–B0 = 63.87 MHz (b) The magnetic field is
B(z) = B0 +Gzz
Bmin = B0 − Gz × 0.25 = 1.5 T − 7.5 mT = 1.4925 T
Bmax = B0 +Gz × 0.25 = 1.5 T + 7.5 mT = 1.5075 T
So the range of Larmor frequencies within the slab is 63.55 MHz ≤ ν ≤ 64.19 MHz (c) The RF frequency range (bandwidth) is
Δν = γ–GzΔz = γ– × 3 G cm × 2.5 mm = 4.258 kHz G × 3 G cm × 0.25 cm = 3.19 kHz
Practical RF waveforms
After the slice selection gradient has been chosen, the next step is to determine the RF pulse to be applied.
- since the RF pulse is bounded in time, we need to select an ap-propriate envelope function B1e(t)
- the RF pulse may be written as
B1(t) = B1e(t)e−jωrft (9) - e.g., with a pure sine wave and rectangular envelope:
How do we select B1e(t) and ωrf?
- we desire a signal whose centre frequency νc is given by (7) and
contains frequences in the range Δν given by (8):
S(ν) = A rect ν − νc Δν , Δν = γ–GzΔz (10)
- from Fourier transform theory the signal is
s(t) = AΔν sinc(Δνt) ej2πνct (11) - comparing (9) and (11): ωrf = 2πνc = ω0 +γGzzc (12) Be 1(t) = AΔν sinc(Δνt) (13) B1(t) = AΔν sinc(Δνt)e−jωrft (14)
⇒ a rectangular slice profile requires the RF pulse to be modulated
by a sinc function.
- impossible to realise the sinc envelope function because of its in-finite extent
- hence, use a truncated sinc function ⇒ distorted rectangle func-tion in the Fourier domain
- other pulse shapes, e.g., Gaussian pulse, may be used in practice
Thin slices
High spatial resolution images require thin slices, but very thin slices cannot be selected in practice.
Δz = Δν
γ–Gz
- there is an upper limit to the gradient strength (∼ 10 G/cm) - difficult to generate an RF pulse with a very narrow bandwidth
with sufficient power
- a very thin slice implies selection of few protons and hence a weak NMR signal
- in practice, the minimum slice thickness is about 2 mm for a 1.5 T system, and 1 mm on a 3 T system
Summary
To select a slice - thickness Δz - position zc
we apply a gradient Gz together with an RF pulse B1(t) of duration
τp: B1(t) = AΔν sinc(Δνt)e−j2πνct (15) B1e(t) = AΔν sinc(Δνt) (16) Centre frequency: νc = νo +γ–Gzzc (17) Bandwidth: Δν = γ–GzΔz (18) 12
Example
Suppose a 2.5 mm thick slice is excited using a gradient strength of
Gz = 3 G/cm and an RF pulse bandwidth Δν = 3.19 kHz. Sketch
the envelope function of the pulse if τp = 2 ms and a rectangular slice profile is required.
For a rectangular slice profile, the envelope function is
B1e(t) = AΔν sinc(Δνt) where sinc(Δν t) = sinc(3190t) The first zero is at t1 = 1/Δν = 313 μs In terms of t1, τp 2 = 2000/2 313 t1 = 3.19t1
Note that during RF excitation, the protons within the excited slab undergoes forced precession:
- spins at different z experience different Bz
- all the protons within the slab z1 ≤ z ≤ z2 contribute to the measured signal
- Larmor frequencies are dependent on z since ω(z) = ω0 +γGzz - during relaxation, the signal comes from all photons in the slice
z1 ≤ z ≤ z2, which leads to dephasing; e.g.,
at z = z1: ω(z1) = ω0 +γGzz1 at z = z2: ω(z2) = ω0 +γGzz2 - hence the need for refocusing gradients
Example
Given Gz = 1 G/cm, what is the range of frequencies in one slice of
thickness 1.5 mm? We have
Gz = 1 G/cm, γ– = 4.258 kHz/G, Δz = 1.5 mm,
The range of frequencies is Δν = γ–GzΔz = 4.258 kHz G × 1 G cm × 0.15 cm = 639 Hz 14
Refocusing gradients
- During RF excitation, the spins at the lower edge z = z1 of the slice precess more slowly than those at the higher edge z = z2 because of their different Larmor frequencies.
- For the 2.5 mm slice in an earlier example (pg 9),
ν(z1) = 63.55 MHz, ν(zc) = 63.87 MHz, ν(z2) = 64.19 MHz - The spins become out of phase with each other; thus causing Mxy
to decrease.
- The phase that is introduced relative to z = zc can be shown (by
solving the Bloch equations) to be:
φs = ω τp/2 (19)
= γGz(z − zc)τp/2 (20)
where τp is the duration of the RF pulse.
- Since the resulting FID or echo relies on in-phase precession, it is necessary to rephase the spins within the slice (slice rephasing or slice refocusing).
Slice rephasing is achieved by applying a refocusing lobe z gradient waveform, e.g., a constant gradient Gr,z for a duration τr immediately after the RF excitation.
- This gradient (without any accompanying RF excitation) causes a change in the Larmor frequency as a function of z.
- The accumulated phase φr due to the refocusing gradient is
φr = γGr,z(z − zc)τr (21)
- For rephasing, φr+φs = 0, i.e.
Gzτp/2 + Gr,zτr = 0 (22)
which can be achieved by having
Gr,z = −Gz, τr = τp/2 (23)
- The refocusing pulse does not have to be a rectangle; all that is required is that its integral is equal to half the integral of the slice selection gradient, but with opposite sign:
A2 = −1 2A1
Slices at different orientations
Having selected a slice in the excitation phase, the other two dimen-sions must be encoded to produce a 2D image:
- in one of the directions, a frequency-encoding scheme is employed - in the other, a phase-encoding scheme
- this is achieved by appropriately turning on/off the other gradient coils during the data acquisition phase
A simple pulse sequence
Magnetic resonance imaging requires the application of voltage and magnetic pulses of appropriate strengths and at appropriate times. Basic elements of slice selection:
1. A constant z gradient is applied at the same time as an RF wave-form for a time duration τp.
2. At t = τp/2, a negative gradient −Gz is applied to refocus the
spins within the slice.
3. Because of dephasing, the appearance of the FID is delayed until near the conclusion of the refocusing lobe.
4. The ADC (A-to-D converter) is turned on immediately after slice refocusing to measure the FID (signal from all protons in the slice).
Note:
- In reality, the gradient pulses typically look like trapezoids, with finite rise and fall times.
- If no dephasing were present across the selected slice, we would expect the FID to begin at the centre of the RF pulse (t = 0).
