Principles of Magnetic Resonance Imaging

Principles of Magnetic Resonance Imaging
J. Peter Mustonen (from David J. Michalak) Presentation for Physics 250 05/01/2008

Interactions of spins in B0 field Principles of 1D-MRI
Outline Motivation Principles of NMR Interactions of spins in B0 field Principles of 1D-MRI Principles of 2D-MRI Summary

Motivation Magnetic Resonance Imaging provides a non-invasive imaging technique. Pros: -No injection of potentially dangerous elements (radioactive dyes) -Only magnetic fields are used for imaging – no x-rays Cons: -Current geometries are expensive, and large/heavy

Principles of NMR B0 Application of prepolarizing magnetic field, B0, aligns the spins in a sample to give a net magnetization, M. M rotates about B0 at a Larmor precession frequency, w0 = gB0 M = SMi

Principles of NMR z B0 B0 M y x
RF Pulse B0 B0 M y x Application of prepolarizing magnetic field, B0, aligns the spins in a sample to give a net magnetization, M. M rotates about B0 at a Larmor precession frequency, w0 = gB0 Application of a rf pulse w0=2pf0 along the x-axis will provide a torque that displaces M from the z axis towards y axis. A certain pulse length will put M right on xy plane M = SMi

Principles of NMR z z B0 B0 exp[-iw0t] B0 M M y y x x
RF Pulse B0 B0 Time exp[-iw0t] B0 M M y y x x Application of prepolarizing magnetic field, B0, aligns the spins in a sample to give a net magnetization, M. M rotates about B0 at a Larmor precession frequency, w0 = gB0 Application of a rf pulse w0=2pf0 along the x-axis will provide a torque that displaces M from the z axis towards y axis. A certain pulse length will put M right on xy plane M precesses in the transverse plane. In the absence of any disturbances, M continues to rotate indefinitely in xy plane. M = SMi

Principles of NMR z z B0 B0 exp[-iw0t] B0 M M y y Detector x x
RF Pulse B0 B0 Time exp[-iw0t] B0 M M y y Detector x x Application of prepolarizing magnetic field, B0, aligns the spins in a sample to give a net magnetization, M. M rotates about B0 at a Larmor precession frequency, w0 = gB0 Application of a rf pulse w0=2pf0 along the x-axis will provide a torque that displaces M from the z axis towards y axis. A certain pulse length will put M right on xy plane M precesses in the transverse plane. In the absence of any disturbances, M continues to rotate indefinitely in xy plane. M = SMi

No other forces on Mi (including detection).
Principles of NMR z z RF Pulse B0 B0 Time exp[-iw0t] B0 M M y y Detector x x Application of prepolarizing magnetic field, B0, aligns the spins in a sample to give a net magnetization, M. M rotates about B0 at a Larmor precession frequency, w0 = gB0 Application of a rf pulse w0=2pf0 along the x-axis will provide a torque that displaces M from the z axis towards y axis. A certain pulse length will put M right on xy plane M precesses in the transverse plane. In the absence of any disturbances, M continues to rotate indefinitely in xy plane. Assume: All spins feel same B0. No other forces on Mi (including detection). M = SMi

RF Pulse B0 B0 Time exp[-iw0t] B0 M M y y Detector x x Application of prepolarizing magnetic field, B0, aligns the spins in a sample to give a net magnetization, M. M rotates about B0 at a Larmor precession frequency, w0 = gB0 Application of a rf pulse w0=2pf0 along the x-axis will provide a torque that displaces M from the z axis towards y axis. A certain pulse length will put M right on xy plane (w0/2p)-1 signal, sr(t) time, t M = SMi

RF Pulse B0 B0 Time exp[-iw0t] B0 M M y y Detector x x Application of prepolarizing magnetic field, B0, aligns the spins in a sample to give a net magnetization, M. M rotates about B0 at a Larmor precession frequency, w0 = gB0 Application of a rf pulse w0=2pf0 along the x-axis will provide a torque that displaces M from the z axis towards y axis. A certain pulse length will put M right on xy plane (w0/2p)-1 sr(t) t FT sr(w) w w0 = 2pf0 M = SMi

RF Pulse B0 B0 Time exp[-iw0t] B0 M M y y Detector x x Application of prepolarizing magnetic field, B0, aligns the spins in a sample to give a net magnetization, M. M rotates about B0 at a Larmor precession frequency, w0 = gB0 Application of a rf pulse w0=2pf0 along the x-axis will provide a torque that displaces M from the z axis towards y axis. A certain pulse length will put M right on xy plane (w0/2p)-1 sr(t) t FT sr(w) w w0 = 2pf0 Boring Spectrum! M = SMi

Complexity Makes Things Interesting
Principles of NMR y x z B0 In Reality: Relaxation (Inherent even if B0 is homogeneous) T1: Spins move away from xy plane towards z. T2: Spins dephase from each other. B0 inhomogeneity. Chemical Shift.

T1 Spin Relaxation Principles of NMR T1 Spin Relaxation: return of the magnetization vector back to z-axis. Spin-Lattice Time Constant: Energy exchange between spins and surrounding lattice. Fluctuations of B field (surrounding dipoles ≈ receivers) at w0 are important. Larger E exchange necessary for larger B0 → longerT1. Math: dM/dt = -(Mz-M0)/T1 Solution: Mz = M0 + (Mz(0)-M0)exp(-t/T1) After 90 pulse: Mz = M0 [1-exp(-t/T1)] M0 = net magnetization based on B0. Mz = component of M0 along the z-axis. t = time y x z B0

T2 Spin Relaxation Principles of NMR T2 Spin Relaxation: Decay of transverse magnetization, Mxy. T1 plays a role, since as Mxy → Mz, Mxy → 0 But dephasing also decreases Mxy: T2 < T1. T2: Spin-Spin Time Constant Variations in Bz with time and position. Pertinent fluctuations in Bz are those near dc frequencies (independent of B0) so that w0 is changed. Molecular motion around the spin of interest. Liquids: High Temp more motion, less DB, high T2 Solids: slow fluctuations in Bz, extreme T2. Bio Tissues: spins bound to large molecules vs. those free in solution. y x z B0 Mxy z B0+DB(r,t) y x

T1/T2 Spin Relaxation Comparison of T1 and T2 Spin Relaxation:
Principles of NMR Comparison of T1 and T2 Spin Relaxation: y x z B0 Tissue T1 (ms) T2 (ms) Gray Matter 950 100 White Matter 600 80 Muscle 900 50 Fat 250 60 Blood 1200 * *200 for arterial blood, 100 for venous blood. B0 = 1.5 T, 37 degC (Body Temp) Magnetic Resonance Imaging: Physical Principles and Sequence Design, Haacke E.M. et al., Wiley: New York, 1999. z B0+DB(r,t) y Math: dM/dt = -Mxy/T2 After 90 pulse: Mxy = M0 exp(-t/T2)] x

T1/T2 Spin Relaxation Comparison of T1 and T2 Spin Relaxation:
Principles of NMR Comparison of T1 and T2 Spin Relaxation: y x z B0 Tissue T1 (ms) T2 (ms) Gray Matter 950 100 White Matter 600 80 Muscle 900 50 Fat 250 60 Blood 1200 * *200 for arterial blood, 100 for venous blood. Magnetic Resonance Imaging: Physical Principles and Sequence Design, Haacke E.M. et al., Wiley: New York, 1999. z T2 << T1 Mxy decays ~exp(-t/T2) B0+DB(r,t) Because T2 is independent of B0, higher B0 gives better resolution y FT 2/T2 sr(t) x t Detector w0 FID Spectrum

T1/T2 Spin Relaxation Inclusion of T1 and T2 Spin Relaxation:
Principles of NMR Inclusion of T1 and T2 Spin Relaxation: Inclusion of mathematical expression: Bloch Equation y x z B0 z g = gyromagnetic ratio T1 = Spin-Lattice (longitudinal-z) relaxation time constant T2 = Spin-Spin (transverse-x/y) relaxation time constant M0 = Equilibrium Magnetization due to B0 field. i, j, k = Unit vectors in x, y, z directions respectively. B0+DB(r,t) y x

T1/T2 Spin Relaxation Inclusion of T1 and T2 Spin Relaxation:
Principles of NMR Inclusion of T1 and T2 Spin Relaxation: Inclusion of mathematical expression: Bloch Equation y x z B0 Precession Transverse Decay Longitudinal Growth z g = gyromagnetic ratio T1 = Spin-Lattice (longitudinal-z) relaxation time constant T2 = Spin-Spin (transverse-x/y) relaxation time constant M0 = Equilibrium Magnetization due to B0 field. i, j, k = Unit vectors in x, y, z directions respectively. B0+DB(r,t) y x Net magnetization is not necessarily constant: e.g., very short T2, long T1.

Chemical Shift Principles of NMR Chemical Shift: Nuclei are shielded (slightly) from B0 by the presence of their electron clouds. Effective field felt by a nuclear spin is B0(1-s). Larmor precession freq, w = gB0(1-s). Shift is often in the ppm range. ~500,000 precessions before Mxy = 0 Chemical environment determines amount of s. H2O vs. Fat (fat about 3.5 ppm lower w0) y x z B0 z 2d- B0(1-s) O C y d+ H d+ H H H x Discrete Shift Detector Less Shielding More Shielding

Chemical Shift Principles of NMR Chemical Shift: Nuclei are shielded (slightly) from B0 by the presence of their electron clouds. y x z B0 Ability to resolve nuclei in different chemical environments is key to NMR 2/T2 z w0(1-s) w0 B0(1-s) Because T2 is independent of B0, higher B0 gives better resolution y x Discrete Shift Detector

Field Inhomogeneity T2*: B0 Inhomogeneity: Additional decay of Mxy.
Principles of NMR T2*: B0 Inhomogeneity: Additional decay of Mxy. In addition to T2, which leads to Mxy decay even in a constant B0, application of dB0(x, y, z, t) will cause increased dephasing: 1/T2* = 1/T2 + 1/T’, where T’ is the dephasing due only to dB0(x, y, z, t). T2* < T2, and depends on dB0(x, y, z, t). Additional loss of resolution between peaks. y x z B0 time, t z B0+dB(r,t) y x

Principles of NMR T2*: B0 Inhomogeneity: Additional decay of Mxy. In addition to T2, which leads to Mxy decay even in a constant B0, application of dB0(x, y, z, t) will cause increased dephasing: 1/T2* = 1/T2 + 1/T’, where T’ is the dephasing due only to dB0(x, y, z, t). T2* < T2, and depends on dB0(x, y, z, t). Additional loss of resolution between peaks. If dB0(x, y, z) is not time dependent, then it can be corrected by an echo pulse. y x z B0 time, t z B0+dB(r,t) y x

Principles of NMR T2*: B0 Inhomogeneity: Additional decay of Mxy. In addition to T2, which leads to Mxy decay even in a constant B0, application of dB0(x, y, z, t) will cause increased dephasing: 1/T2* = 1/T2 + 1/T’, where T’ is the dephasing due only to dB0(x, y, z, t). T2* < T2, and depends on dB0(x, y, z, t). Additional loss of resolution between peaks. If dB0(x, y, z) is not time dependent, then it can be corrected by an echo pulse. y x z B0 time, t z z z B0+dB(r,t) B0+dB(r,t) B0+dB(r,t) y y y 180x pulse (x → x, y → –y) time, t x x x Echo!

Principles of NMR T2*: B0 Inhomogeneity: Additional decay of Mxy. If echo pulse applied at time, t, then echo appears at 2t. Only T’ can be reversed by echo pulsing, T2 cannot be echoed as the field inhomogeneities that lead to T2 are not constant in time or space. 4) Signal after various echo pulsed displayed below. y x z B0 T2 T2* sr(t) t t = 0 90 pulse t 180 pulse applied 2t Echo t’ 180 pulse applied 2(t’-t) Echo

Single B0 – No Spatial Information
Principles of 1DMRI Measured response is from all spins in the sample volume. Detector coil probes all space with equal intensity B0 B0 B0 90 pulse time Detector coil If only B0 is present (and homogeneous) all spins remain in phase during precession (as drawn). - B(x, y, z, t) = B0; thus, w(x, y, z) = w0 = gB0 FT 2/T2 No Spatial Information (Volume integral) sr(t) t FID w0 Spectrum

Slice Selection: z-Gradient
Principles of 1DMRI Slice selection along z-axis. Gradient in z and selective excitation allows detection of a single slice. B(z) = B0 + Gzz Gz Field strength indicated by line thickness Gz = dBz/dz integrate Bz=Gzz It follows that: B(z=0)=B0

Principles of 1DMRI Slice selection along z-axis. Gradient in z and selective excitation allows detection of a single slice. B(z) = B0 + Gzz Gz Selective 90 pulse wrf=w0+gGzz Field strength indicated by line thickness Gz = dBz/dz integrate Bz=Gzz It follows that: B(z=0)=B0

Principles of 1DMRI Slice selection along z-axis. Gradient in z and selective excitation allows detection of a single slice. B(z) = B0 + Gzz Larmor Precession frequency is z-dependent: w(z) = gB(z) w(z) =g(B0 + Gzz) w(z) = w0 + gGzz Gz Selective 90 pulse wrf=w0+gGzz Field strength indicated by line thickness Excite only one plane of z ± Dz by using only one excitation frequency for the 90 pulse. For example, using B0 for excitation: only spins at z=0 get excited. All other spins are off resonance and are not tipped into the transverse plane. Gz = dBz/dz integrate Bz=Gzz It follows that: B(z=0)=B0

Principles of 1DMRI Slice selection along z-axis. Gradient in z and selective excitation allows detection of a single slice. B(z) = B0 + Gzz Gz Selective 90 pulse wrf=w0+gGzz In practice, you must bandwidth match the frequency of the 90 pulse with the desired thickness (Dz) of the z-slice. (i.e., with a linear gradient, the Larmor precession of spins within z = 0 ± Dz oscillate with frequency w0 ± gGzDz. Thus, BW = 2gGzDz.) 4) To apply a “boxcar” of frequencies w ± gGzDz, we need the 90 deg excitation profile to be a sinc function in time. FT(sinc) = rect Field strength indicated by line thickness Gz = dBz/dz integrate Bz=Gzz It follows that: B(z=0)=B0 FT 90° z ± Dz t w sinc = (sinx)/x

Principles of 1DMRI Gradient Echo Pulse. Gradient Echo pulse restores all spins to have the same phase within the slice Dz. B(z) = B0 + Gzz Before Gradient Echo t = t z Gz Selective 90 pulse w0+gGzDz w0 w0-gGzDz Spins out of phase on xy plane Pulse Sequence RF Gradient Echo Gz t 3t/2 time

Principles of 1DMRI Gradient Echo Pulse. Gradient Echo pulse restores all spins to have the same phase within the slice Dz. B(z) = B0 + Gzz Before Gradient Echo t = t z Gz Selective 90 pulse w0+gGzDz w0 w0-gGzDz Spins out of phase on xy plane Pulse Sequence Top View of xy plane w0 w0-gGzDz w0+gGzDz RF t=t Gradient Echo Gz t 3t/2 time

Principles of 1DMRI Gradient Echo Pulse. Gradient Echo pulse restores all spins to have the same phase within the slice Dz. B(z) = B0 + Gzz Before Gradient Echo t = t z Gz Selective 90 pulse w0+gGzDz w0 w0-gGzDz Spins out of phase on xy plane Pulse Sequence Top View of xy plane w0 w0-gGzDz w0+gGzDz RF After Gradient Echo t = 3t/2 t=t Gradient Echo z Gz t=3t/2 t 3t/2 time Spins all IN phase

Principles of 1DMRI Slice selection along z-axis. Gradient in z and selective excitation allows detection of a single slice. B(z) = B0 + Gzz Gz Selective 90 pulse

Principles of 1DMRI Slice selection along z-axis. Gradient in z and selective excitation allows detection of a single slice. B(z) = B0 + Gzz Gz Selective 90 pulse time Detector coil

Principles of 1DMRI Slice selection along z-axis. Gradient in z and selective excitation allows detection of a single slice. B(z) = B0 + Gzz Gz Selective 90 pulse time Detector coil exp(-t/T2) No x, y Information, but only spins from the z ± Dz slice contribute to the signal. FT 2/T2 sr(t) t FID w0 Spectrum If we can encode along x and y dimensions, we can iterate for each z slice.

Precession Frequency varies with x
Frequency Encoding Principles of 1DMRI Perform z-slice. Now only look at 2D plane from now on. Use Gradient along x to generate different Larmor frequencies vs. x-position. y y z z z Selective 90 pulse in z ± Dz time x x x 2Dz Bz(x) - B0 Apply x-Gradient Gx = dBz/dx Precession Frequency varies with x

Precession Frequency varies with x
Frequency Encoding Principles of 1DMRI Perform z-slice. Now only look at 2D plane from now on. Use Gradient along x to generate different Larmor frequencies vs. x-position. y y z z z Selective 90 pulse in z ± Dz time x x x 2Dz Bz(x) - B0 w(x) w0 - gGxx w0 w0 + gGxx Frequency Encoding along x Apply x-Gradient Gx = dBz/dx Precession Frequency varies with x

Frequency Encoding Principles of 1DMRI Perform z-slice. Now only look at 2D plane from now on. Use Gradient along x to generate different Larmor frequencies vs. x-position. z Pulse Sequence RF x Gz Bz(x) - B0 Detector coil Gx w(x) time w0 - gGxx w0 w0 + gGxx Detect Signal “readout” Gx on while detecting

Frequency Encoding Principles of 1DMRI Perform z-slice. Now only look at 2D plane from now on. Use Gradient along x to generate different Larmor frequencies vs. x-position. z T2* is based on the intentionally applied gradient. exp(-t/T2*) sr(t) t x FT Bz(x) - B0 FID Detector coil w(x) w0 - gGxx w0 w0 + gGxx 2/T2* Apply x-Gradient DURING acquisition. Precession Frequency varies with x. w0 - gGxx w0 w0 + gGxx

Frequency Encoding Principles of 1DMRI Perform z-slice. Now only look at 2D plane from now on. Use Gradient along x to generate different Larmor frequencies vs. x-position. z T2* is based on the intentionally applied gradient. exp(-t/T2*) sr(t) t x FT Bz(x) - B0 FID Detector coil w(x) w0 - gGxx w0 w0 + gGxx 2/T2* Apply x-Gradient DURING acquisition. Precession Frequency varies with x. Spins at various x positions in space are encoded to a different precession frequency w0 - gGxx w0 w0 + gGxx

Imaging Example Principles of 1DMRI Two Microfluidic Channels. Water only exists in two microfluic channels as shown. y y Application of Gx z z z 90 pulse time x x Dz x Bz(x)

Imaging Example Principles of 1DMRI Two Microfluidic Channels. Water only exists in two microfluic channels as shown. y y Application of Gx z z z 90 pulse time x x Dz x Bz(x) No spins exist at x=0 where Gx=0 (w0): FT of signal has no intensity at w0. Signal is the line integral along y. (Still no info about y distribution of spins.) Image m(x,y) = spin density(x,y) w0 - gGxx w0 w0 + gGxx

1DFT Math Signal is the 1DFT of the line integral along y.
Principles of 1DMRI Signal is the 1DFT of the line integral along y. Homodyne the signal (from w0 to 0).

Principles of 1DMRI Signal is the 1DFT of the line integral along y. Homodyne the signal (from w0 to 0). Let g(x) = Line integral along y for a given x position.

Principles of 1DMRI Signal is the 1DFT of the line integral along y. Homodyne the signal (from w0 to 0). Let g(x) = Line integral along y for a given x position. Spatial frequency Gxt ~ kx The homodyned signal is thus the Fourier Transform (along x) of the line integral along y.

k-vector perspective Principles of 1DMRI Time Evolution of Spins in an x-Gradient. Spatial frequency, k-vector, changes. SMi(x) w0 - gGxx w0 w0 + gGxx x t1 Pulse Sequence RF time Gz Dephasing across x in time. Rotating frame w0 or relative to x=0 Gx t1 t2 time

k-vector ~ amount of spin warping over distance
k-vector perspective Principles of 1DMRI Time Evolution of Spins in an x-Gradient. Spatial frequency, k-vector, changes. Spatial frequency encoded by phase Homo- dyne s(t) k-vector ~ amount of spin warping over distance SMi(x) w0 - gGxx w0 w0 + gGxx x k=0 t1 k: one spatial period Pulse Sequence RF time k Gz FID Gx Dephasing across x in time. Rotating frame w0 or relative to x=0 t1 t2 time Each Point on FID is a different value of kx

2 Approaches to Understand FT
Principles of 1DMRI The imaging in 1D can be understood in 2 ways: 1) From the received signal perspective: The spins, spatially separated along the x-dimension, are distinguished by the application of a gradient field that makes their Larmor precession vary along x. The FT resolves the difference in frequency and hence position. 2) Homodyned (baseband) signal perspective: As time passes during the application of the gradient, the spins dephase from each other. The amount of dephasing can be represented as a spatial frequency, kx, that increases with measurement time. Frequency Encoding Precession: w(x) 2/T2* FT of FID (time) gives frequency, w. w depends on position w0 - gGxx w0 w0 + gGxx Phase Encoding Phase(t) ~ Spatial Freq. FT of spatial frequency data, kx, data gives position data, x. Different values of kx are probed over time, t.

Apply y-Gradient for time ty
2DFT Principles Principles of 2DMRI Again perform z-slice. Only look at 2D plane. Want to now distinguish spins along y-direction also. y y Gy z z x x 90 pulsed Plane Apply y-Gradient for time ty Gy = dBz/dy Then Gy turned off

2DFT Principles Principles of 2DMRI Phase Encoding. Gy is turned on for a certain time, ty, then off. This generates a difference in phase over y. y y y Gy z z z All precess at w0 x x x 90 pulsed Plane Apply y-Gradient for time ty Gy = dBz/dy Then Gy turned off But spin warped along y by an amount determined by Gyty ~ single ky value Phase encoding along y

2DFT Principles Principles of 2DMRI Detect Using Gx. As usual detection occurs with Gx. y y y Gy z z z All precess at w0 x x x 90 pulsed Plane Apply y-Gradient for time ty Gy = dBz/dy Then Gy turned off Detect with Gx y z x Bz(x) Detector coil Usual frequency encoding along x

2DFT Principles Frequency Encoding along x (Gxt)
Principles of 2DMRI Detect Using Gx. As usual detection occurs with Gx. y y y Gy z z z All precess at w0 x x x 90 pulsed Plane Apply y-Gradient for time ty Gy = dBz/dy Then Gy turned off Detect with Gx y z This time, the magnitude of the signal at each w (x-position), corresponds to the intensity of the spatial frequency, ky, encoded by Gy phase encoding step. (for ky=0 it’s the line integral) x Bz(x) Detector coil Frequency Encoding along x (Gxt) Phase Encoding along y (Gyty)

2DFT Principles Principles of 2DMRI Again perform z-slice. Only look at 2D plane. Want to now distinguish spins along y-direction also. Pulse Sequence z RF Phase Encoded Gz Gy ty x Gx Bz(x) Detector coil time Phase Encode Detect Signal

2DFT Principles Principles of 2DMRI Again perform z-slice. Only look at 2D plane. Want to now distinguish spins along y-direction also. Pulse Sequence z RF Phase Encoded Gz Gy ty x Gx Bz(x) Detector coil time Repeat experiment multiple times varying the Gy gradient strength (or time ty) so that ky receives the same sampling as kx (FID sampling rate). Phase Encode Detect Signal

Gx during recording of FID
2DFT Math Principles of 1DMRI Signal is the 2DFT of the image. Baseband (Homodyned) signal. Phase Encoding Step Gx during recording of FID For any given FID, ty is fixed and t is running variable.

Gx during recording of FID
2DFT Math Principles of 1DMRI Signal is the 2DFT of the image. Baseband (Homodyned) signal. Phase Encoding Step Gx during recording of FID For any given FID, ty is fixed and t is running variable. Intensities at each x correspond to intensity of the ky spatial frequency (applied during phase encoding) at that x position. In other words, the intensity corresponds to 1 pt on the FID taken in the y direction w0 - gGxx w0 w0 + gGxx

2DFT Principles Principles of 2DMRI k-space perspective. Want to map k-space then take 2DFT. (Each FID samples line in k-space along kx) ky Set of data points sampled from the FID with a phase encoding of a given ky (Gyty). Change Gyty kx Set of data points along the kx axis corresponds to the sampled FID taken with no Gy phase encoding gradient. Measure FID kx measured in time

2DFT Principles Principles of 2DMRI k-space perspective. Want to map k-space then take 2DFT. (Each FID samples line in k-space along kx) ky Thus, it is evident that a column of data (at a given x position) on the collection of points in k-space represents the FT of the various Gy values. The data along a line is the FT of the signal in the y direction. kx

2DFT Principles Principles of 2DMRI k-space perspective. Want to map k-space then take 2DFT. (Each FID samples line in k-space along kx) ky Thus, it is evident that a column of data (at a given x position) on the collection of points in k-space represents the FT of the various Gy values. The data along a line is the FT of the signal in the y direction. kx Rotate for viewing “FID” along y.

2DFT Principles Principles of 2DMRI k-space perspective. Want to map k-space then take 2DFT. (Each FID samples line in k-space along kx) Image ky What does this data look like? y Sinc function kx Rect function x Sinc function Rect function

2DFT Principles Principles of 2DMRI k-space perspective. Want to map k-space then take 2DFT. (Each FID samples line in k-space along kx) Image ky What does this data look like? y kx x Jinc function Radially symmetric sinc) Circle function (radially symmetric rect)

2DFT Principles Principles of 2DMRI Updated Pulse Sequence. Want to map k-space then take 2DFT. (Each FID samples line in k-space along kx) ky Pulse Sequence ty RF Gz ty Gy kx Gx 2ty time

2DFT Principles Principles of 2DMRI Updated Pulse Sequence. Want to map k-space then take 2DFT. (Each FID samples line in k-space along kx) ky Pulse Sequence ty RF ty Gz ty Gy kx Gx time

2DFT Principles Principles of 2DMRI Updated Pulse Sequence. Want to map k-space then take 2DFT. (Each FID samples line in k-space along kx) ky Pulse Sequence ty RF ty Gz Gy kx Gx time Representation

Discrete FT Imaging Issues
Principles of 2DMRI Sampling Rate Issues: Real time FID is sampled at various times of interval, Dt, which leads to a sampling rate in the kx dimension of (Dkx). Interval on Dky is determined by the change in gradient area (DGyty) between different runs ky t, kx We know that we need enough data to adequately sample the FID in time (kx) dimension Same principle applies for ky (Gyty) dimension kx Sampling rate of k-space

Field of View Principles of 2DMRI Field of View: Sampling rate of k-space determines the field of view in the object-oriented domain. y FOVx = 1/(Dkx) Dkx ky FOVy =1/(Dky) Dky kx x Sampling rate of k-space FOV > Image size! Prevent Aliasing

Aliasing Issues Principles of 2DMRI Aliasing: If sampling rate is not sufficient, the Field of view will overlap. y FOVx = 1/(Dkx) Dkx ky Dky FOVy =1/(Dky) kx x Sampling rate of k-space FOV > Image size! Prevent Aliasing (Image Overlap)

Resolution Principles of 2DMRI Resolution: Resolution in the object-oriented domain is determined by the extent of k-space measured. dx = FOVx/Nread =(DkxNread)-1 y ky DkxNread dy = FOVy/Npe =(DkyNpe)-1 DkyNpe kx x Sampling rate of k-space Field of View/Resolution ~ # points need to sample (e.g., 25.6 cm image, 1mm resolution: 256 points/dimension, 65.5k points) Nread: # of readout points during FID Npe: # of phase encoding steps

Summary MRI is based on the spatial encoding of spins either through a difference in phase (y) or a difference in Larmor frequency (x): FID in the presence Gx, after a given phase encoding in y, gives a line of points in k-space. FIDs are repeated for a variety of ky values to fill up k-space. 2DFT of k-space gives the image of spin density m(x,y) Limitations. Detection is based on the signal received in a coil. Coil inductor has an impedance, Zcoil= iwL, ~ frequency. Thus significant voltage signals are observed only at high frequencies. (Mxy → icoil. icoil = vsignal/Zcoil.) Requires Large Magnetic fields – cryogenics, homogeneity. Large Fields can lead to signal distortion. Samples containing metals cannot be imaged.

Acknowledgements David Michalak Pines Group Budker Group Alex Pines
Chip Crawford Hattie Ziegler Marcus Donaldson Thomas Theis All current ‘nuts Budker Group Dmitry Budker Micah Ledbetter Good Books: 1) Principles of Magnetic Resonance Imaging, Dwight G. Nishimura, Stanford University 2) Magnetic Resonance Imaging: Physical Principles and Sequence Design, Haacke E.M. et al., Wiley: New York, 1999.

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