Principles of Magnetic Resonance Imaging David J. Michalak Presentation for Physics 250 03/13/2007.

Principles of Magnetic Resonance Imaging David J. Michalak Presentation for Physics 250 03/13/2007

 Motivation  Principles of NMR  Interactions of spins in B 0 field  Principles of 1D MRI  Principles of 2D MRI  2D MRI using the atomic magnetometer.  Applications in progress oEarth-field MRI for microfluidics  Summary Outline

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 Motivation www.nlm.nih.gov

B0B0 Principles of NMR Application of prepolarizing magnetic field, B 0, aligns the spins in a sample to give a net magnetization, M. M rotates about B 0 at a Larmor precession frequency,  =  B 0 M =  M i

RF Pulse B0B0 Principles of NMR Application of prepolarizing magnetic field, B 0, aligns the spins in a sample to give a net magnetization, M. M rotates about B 0 at a Larmor precession frequency,  =  B 0 y x z B0B0 Application of a rf pulse  0 =2  f 0 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 =  M i M

RF Pulse B0B0 Principles of NMR Application of prepolarizing magnetic field, B 0, aligns the spins in a sample to give a net magnetization, M. M rotates about B 0 at a Larmor precession frequency,  =  B 0 y x z B0B0 y x z B0B0 M =  M i M precesses in the transverse plane. In the absence of any disturbances, M continues to rotate indefinitely in xy plane. MM Time Application of a rf pulse  0 =2  f 0 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 exp[- i   t]

RF Pulse B0B0 Principles of NMR Application of prepolarizing magnetic field, B 0, aligns the spins in a sample to give a net magnetization, M. M rotates about B 0 at a Larmor precession frequency,  =  B 0 y x z B0B0 y x z B0B0 M =  M i M precesses in the transverse plane. In the absence of any disturbances, M continues to rotate indefinitely in xy plane. Detector MM Time Application of a rf pulse  0 =2  f 0 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 exp[- i   t]

RF Pulse B0B0 Principles of NMR Application of prepolarizing magnetic field, B 0, aligns the spins in a sample to give a net magnetization, M. M rotates about B 0 at a Larmor precession frequency,  =  B 0 y x z B0B0 y x z B0B0 M =  M i M precesses in the transverse plane. In the absence of any disturbances, M continues to rotate indefinitely in xy plane. Detector MM Time Assume: 1)All spins feel same B 0. 2)No other forces on M i (including detection). Application of a rf pulse  0 =2  f 0 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 exp[- i   t]

RF Pulse B0B0 Principles of NMR Application of prepolarizing magnetic field, B 0, aligns the spins in a sample to give a net magnetization, M. M rotates about B 0 at a Larmor precession frequency,  =  B 0 y x z B0B0 y x z B0B0 M =  M i Detector MM Time time, t signal, s r (t) Application of a rf pulse  0 =2  f 0 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 (  0 /2  ) -1 exp[- i   t]

RF Pulse B0B0 Principles of NMR Application of prepolarizing magnetic field, B 0, aligns the spins in a sample to give a net magnetization, M. M rotates about B 0 at a Larmor precession frequency,  =  B 0 y x z B0B0 y x z B0B0 M =  M i Detector FT MM Time sr(t)sr(t) sr()sr() t Application of a rf pulse  0 =2  f 0 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  0 = 2  f   exp[- i   t] (  0 /2  ) -1

RF Pulse B0B0 Principles of NMR Application of prepolarizing magnetic field, B 0, aligns the spins in a sample to give a net magnetization, M. M rotates about B 0 at a Larmor precession frequency,  =  B 0 y x z B0B0 y x z B0B0 M =  M i Detector (  0 /2  ) -1 FT MM Time Boring Spectrum!  0 = 2  f  sr(t)sr(t) sr()sr() t  Application of a rf pulse  0 =2  f 0 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 exp[- i   t]

Principles of NMR y x z B0B0 In Reality: 1)Relaxation (Inherent even if B 0 is homogeneous) 1) T 1 : Spins move away from xy plane towards z. 2) T 2 : Spins dephase from each other. 3) Chemical Shift. 2)Experimental Design Effects. 1) T 2 *: Field inhomogeneity in B 0 ( x, y, z, t ) 1)Could be intentional (e.g., gradient) or not. Complexity Makes Things Interesting

Principles of NMR y x z B0B0 T 1 Spin Relaxation: return of the magnetization vector back to z -axis. 1)Spin-Lattice Time Constant: 1) Energy exchange between spins and surrounding lattice. 2) Fluctuations of B field (surrounding dipoles ≈ receivers) at  0 are important. Larger E exchange necessary for larger B 0 → lower T 1. 2)Math: dM/dt = -(M z -M 0 )/T 1 1)Solution: M z = M 0 + (M z (0)-M 0 )exp(-t/T 1 ) 2)After 90 pulse: M z = M 0 [1-exp(-t/T 1 )] M 0 = net magnetization based on B 0. M z = component of M 0 along the z-axis. t = time T 1 Spin Relaxation

Principles of NMR y x z B0B0 T 2 Spin Relaxation: Decay of transverse magnetization, M xy. 1)T 1 plays a role, since as M xy → M z, M xy → 0 1)But dephasing also decreases M xy : T 2 < T 1. 2)T 2 : Spin-Spin Time Constant 1) Variations in B z with time and position. 2) Pertinent fluctuations in B z are those near dc frequencies (independent of B 0 ) so that  0 is changed. 3) Molecular motion around the spin of interest. 1)Liquids: High Temp more motion, less  B, high T 2 2)Solids: slow fluctuations in B z, extreme T 2. 3)Bio Tissues: spins bound to large molecules vs. those free in solution. 3)Math: dM/dt = -M xy /T 2 1)After 90 pulse: M xy = M 0 exp(-t/T 2 )] y x z B 0 +  B(r,t) T 2 Spin Relaxation M xy

Principles of NMR y x z B0B0 Comparison of T 1 and T 2 Spin Relaxation: y x z B 0 +  B(r,t) Tissue T 1 (ms)T 2 (ms) Gray Matter950100 White Matter60080 Muscle90050 Fat25060 Blood1200100-200* *200 for arterial blood, 100 for venous blood. B 0 = 1.5 T, 37 degC (Body Temp) Magnetic Resonance Imaging: Physical Principles and Sequence Design, Haacke E.M. et al., Wiley: New York, 1999. T 1 /T 2 Spin Relaxation

Principles of NMR y x z B0B0 Comparison of T 1 and T 2 Spin Relaxation: y x z B 0 +  B(r,t) Tissue T 1 (ms)T 2 (ms) Gray Matter950100 White Matter60080 Muscle90050 Fat25060 Blood1200100-200* Detector T 1 /T 2 Spin Relaxation *200 for arterial blood, 100 for venous blood. B 0 = 1.5 T, 37 degC (Body Temp) Magnetic Resonance Imaging: Physical Principles and Sequence Design, Haacke E.M. et al., Wiley: New York, 1999.

Principles of NMR y x z B0B0 Comparison of T 1 and T 2 Spin Relaxation: y x z B 0 +  B(r,t) Tissue T 1 (ms)T 2 (ms) Gray Matter950100 White Matter60080 Muscle90050 Fat25060 Blood1200100-200* *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. FID 00 Spectrum T 2 << T 1 M xy decays ~exp(-t/T 2 ) Detector FT 2/T 2 Because T 2 is independent of B 0, higher B 0 gives better resolution T 1 /T 2 Spin Relaxation sr(t)sr(t) t

Principles of NMR y x z B0B0 y x z B 0 +  B(r,t) Inclusion of T 1 and T 2 Spin Relaxation: 1)Inclusion of mathematical expression: 1)Bloch Equation  = gyromagnetic ratio T 1 = Spin-Lattice (longitudinal- z ) relaxation time constant T 2 = Spin-Lattice (longitudinal- z ) relaxation time constant M 0 = Equilibrium Magnetization due to B 0 field. i, j, k = Unit vectors in x, y, z directions respectively. T 1 /T 2 Spin Relaxation

Principles of NMR y x z B0B0 y x z B 0 +  B(r,t) Inclusion of T 1 and T 2 Spin Relaxation: 1)Inclusion of mathematical expression: 1)Bloch Equation  = gyromagnetic ratio T 1 = Spin-Lattice (longitudinal- z ) relaxation time constant T 2 = Spin-Lattice (longitudinal- z ) relaxation time constant M 0 = Equilibrium Magnetization due to B 0 field. i, j, k = Unit vectors in x, y, z directions respectively. PrecessionTransverse Decay Longitudinal Growth Net magnetization is not necessarily constant: e.g., very short T 2, long T 1. T 1 /T 2 Spin Relaxation

Principles of NMR y x z B0B0 Chemical Shift: Nuclei are shielded (slightly) from B 0 by the presence of their electron clouds. 1)Effective field felt by a nuclear spin is B 0 (1-  ). 1)Larmor precession freq,  =  B 0 (1-  ). 1)Shift is often in the ppm range. 1)~500,000 precessions before M xy = 0 2)Chemical environment determines amount of . 1)H 2 O vs. Fat (fat about 3.5 ppm lower  0 ) y x z B 0 (1-  ) Discrete Shift H H O ++ ++  - H H C Less Shielding More Shielding Chemical Shift

Principles of NMR y x z B0B0 Chemical Shift: Nuclei are shielded (slightly) from B 0 by the presence of their electron clouds. 1)Effective field felt by a nuclear spin is B 0 (1-  ). 1)Larmor precession freq,  =  B 0 (1-  ). 1)Shift is often in the ppm range. 1)~500,000 precessions before M xy = 0 2)Chemical environment determines amount of . 1)H 2 O vs. Fat (fat about 3.5 ppm lower  0 ) y x z B 0 (1-  ) Discrete Shift H H O ++ ++  - H H C Detector Less Shielding More Shielding Chemical Shift

Principles of NMR y x z B0B0 Chemical Shift: Nuclei are shielded (slightly) from B 0 by the presence of their electron clouds. y x z B 0 (1-  ) Discrete Shift 00 2/T 2 Because T 2 is independent of B 0, higher B 0 gives better resolution Detector  0 (1-  ) Ability to resolve nuclei in different chemical environments is key to NMR Chemical Shift

Principles of NMR y x z B0B0 T 2 *: B 0 Inhomogeneity: Additional decay of M xy. 1)In addition to T 2, which leads to M xy decay even in a constant B 0, application of  B 0 ( x, y, z, t ) will cause increased dephasing: 1/T 2 * = 1/T 2 + 1/T’, where T’ is the dephasing due only to  B 0 ( x, y, z, t ). 1) T 2 * < T 2, and depends on  B 0 ( x, y, z, t ). 2) Additional loss of resolution between peaks. y x z B 0 +  B(r,t) time,  Field Inhomogeneity

Principles of NMR y x z B0B0 T 2 *: B 0 Inhomogeneity: Additional decay of M xy. 1)In addition to T 2, which leads to M xy decay even in a constant B 0, application of  B 0 ( x, y, z, t ) will cause increased dephasing: 1/T 2 * = 1/T 2 + 1/T’, where T’ is the dephasing due only to  B 0 ( x, y, z, t ). 1) T 2 * < T 2, and depends on  B 0 ( x, y, z, t ). 2) Additional loss of resolution between peaks. 2)If  B 0 ( x, y, z ) is not time dependent, then it can be corrected by an echo pulse. y x z B 0 +  B(r,t) time,  Field Inhomogeneity

Principles of NMR y x z B0B0 T 2 *: B 0 Inhomogeneity: Additional decay of M xy. 1)In addition to T 2, which leads to M xy decay even in a constant B 0, application of  B 0 ( x, y, z, t ) will cause increased dephasing: 1/T 2 * = 1/T 2 + 1/T’, where T’ is the dephasing due only to  B 0 ( x, y, z, t ). 1) T 2 * < T 2, and depends on  B 0 ( x, y, z, t ). 2) Additional loss of resolution between peaks. 2)If  B 0 ( x, y, z ) is not time dependent, then it can be corrected by an echo pulse. y x z B 0 +  B(r,t) y x z y x z 180 x pulse ( x → x, y → – y ) time,  Field Inhomogeneity Echo!

Principles of NMR y x z B0B0 T2T2 T2*T2* Field Inhomogeneity T 2 *: B 0 Inhomogeneity: Additional decay of M xy. 3)If echo pulse applied at time, , then echo appears at 2 . 1)Only T’ can be reversed by echo pulsing, T 2 cannot be echoed as the field inhomogeneities that lead to T 2 are not constant in time or space. 4)Signal after various echo pulsed displayed below.  180 pulse applied  Echo t  90 pulse  ’ 180 pulse applied   ’  Echo sr(t)sr(t) t

Principles of 1DMRI Single B 0 – No Spatial Information Measured response is from all spins in the sample volume. Detector coil probes all space with equal intensity 90 pulse B0B0 B0B0 FID 00 Spectrum FT 2/T 2 If only B 0 is present (and homogeneous) all spins remain in phase during precession (as drawn). - B( x, y, z, t ) = B 0 ; thus,  ( x, y, z ) =  0 =  B 0 time B0B0 No Spatial Information (Volume integral) Detector coil sr(t)sr(t) t

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

Principles of 1DMRI Slice Selection: z-Gradient Slice selection along z -axis. Gradient in z and selective excitation allows detection of a single slice. Selective 90 pulse  rf =  0 +  G z z B(z) = B 0 + G z z Field strength indicated by line thickness GzGz G z = dB z /dz integrate B z =G z z It follows that: B(z=0)=B 0

Principles of 1DMRI Slice Selection: z-Gradient Slice selection along z -axis. Gradient in z and selective excitation allows detection of a single slice. B(z) = B 0 + G z z Field strength indicated by line thickness 1)Larmor Precession frequency is z- dependent:  (z) =  B(z)  (z) =  (B 0 + G z z)  (z) =  0 +  G z z GzGz Selective 90 pulse  rf =  0 +  G z z 2)Excite only one plane of z ±  z by using only one excitation frequency for the 90 pulse. For example, using B 0 for excitation: only spins at z=0 get excited. All other spins are off resonance and are not tipped into the transverse plane. G z = dB z /dz integrate B z =G z z It follows that: B(z=0)=B 0

FT Principles of 1DMRI Slice Selection: z-Gradient Slice selection along z -axis. Gradient in z and selective excitation allows detection of a single slice. B(z) = B 0 + G z z Field strength indicated by line thickness GzGz 3)In practice, you must bandwidth match the frequency of the 90 pulse with the desired thickness (  z) of the z-slice. (i.e., with a linear gradient, the Larmor precession of spins within z = 0 ±  z oscillate with frequency  0 ±  G z  z. Thus, BW = 2  G z  z.) 4)To apply a “boxcar” of frequencies  ±  G z  z, we need the 90 deg excitation profile to be a sinc function in time. 1)FT(sinc) = rect Selective 90 pulse  rf =  0 +  G z z t  90° z ±  z G z = dB z /dz integrate B z =G z z It follows that: B(z=0)=B 0 sinc = (sinx)/x

Principles of 1DMRI Slice Selection: z-Gradient Pulse Sequence. Shows the relative timing of the RF and gradient pulses. Selective 90 pulse B(z) = B 0 + G z z GzGz Pulse Sequence RF GzGz 0  3   time

0-Gzz0-Gzz Principles of 1DMRI Slice Selection: z-Gradient Gradient Echo Pulse. Gradient Echo pulse restores all spins to have the same phase within the slice  z. Selective 90 pulse B(z) = B 0 + G z z GzGz Pulse Sequence RF GzGz Gradient Echo 00 0+Gzz0+Gzz Before Gradient Echo t =  0  3   time Spins out of phase on xy plane z

0-Gzz0-Gzz Principles of 1DMRI Slice Selection: z-Gradient Gradient Echo Pulse. Gradient Echo pulse restores all spins to have the same phase within the slice  z. Selective 90 pulse B(z) = B 0 + G z z GzGz Pulse Sequence RF GzGz Gradient Echo z 00 0+Gzz0+Gzz Before Gradient Echo t =  0  time Spins out of phase on xy plane Top View of xy plane t=t= 00 0+Gzz0+Gzz 0-Gzz0-Gzz 3  

Principles of 1DMRI Slice Selection: z-Gradient Gradient Echo Pulse. Gradient Echo pulse restores all spins to have the same phase within the slice  z. Selective 90 pulse B(z) = B 0 + G z z GzGz Pulse Sequence RF GzGz z z Before Gradient Echo t =  After Gradient Echo t = 3  /2 0  time Spins out of phase on xy plane Spins all IN phase Gradient Echo t=t= t=3t=3 00 0+Gzz0+Gzz 0-Gzz0-Gzz Top View of xy plane 0-Gzz0-Gzz 00 0+Gzz0+Gzz 3  

Principles of 1DMRI Slice Selection: z-Gradient Slice selection along z -axis. Gradient in z and selective excitation allows detection of a single slice. Selective 90 pulse B(z) = B 0 + G z z GzGz

Principles of 1DMRI Slice Selection: z-Gradient Slice selection along z -axis. Gradient in z and selective excitation allows detection of a single slice. Selective 90 pulse B(z) = B 0 + G z z GzGz Detector coil time

Principles of 1DMRI Slice Selection: z-Gradient Slice selection along z -axis. Gradient in z and selective excitation allows detection of a single slice. Selective 90 pulse B(z) = B 0 + G z z GzGz Detector coil time FID 00 Spectrum FT 2/T 2 No x, y Information, but only spins from the z ±  z slice contribute to the signal. exp(-t/T 2 ) sr(t)sr(t) t

Principles of 1DMRI Slice Selection: z-Gradient Slice selection along z -axis. Gradient in z and selective excitation allows detection of a single slice. Selective 90 pulse B(z) = B 0 + G z z GzGz Detector coil time FID 00 Spectrum FT 2/T 2 No x, y Information, but only spins from the z ±  z slice contribute to the signal. exp(-t/T 2 ) If we can encode along x and y dimensions, we can iterate for each z slice. sr(t)sr(t) t

Principles of 1DMRI Frequency Encoding Perform z-slice. Now only look at 2D plane from now on. Use Gradient along x to generate different Larmor frequencies vs. x-position. Selective 90 pulse in z ±  z z y x  z z y x

Principles of 1DMRI Frequency Encoding Perform z-slice. Now only look at 2D plane from now on. Use Gradient along x to generate different Larmor frequencies vs. x-position. Selective 90 pulse in z ±  z time z y x  z Apply x-Gradient G x = dB z /dx Precession Frequency varies with x z y x z x B z (x) - B 0

Principles of 1DMRI Frequency Encoding Perform z-slice. Now only look at 2D plane from now on. Use Gradient along x to generate different Larmor frequencies vs. x-position. Selective 90 pulse in z ±  z time z y x  z Apply x-Gradient G x = dB z /dx Precession Frequency varies with x z y x z x B z (x) - B 0 00  0 +  G x x  0 -  G x x (x)(x)

Principles of 1DMRI Frequency Encoding Perform z-slice. Now only look at 2D plane from now on. Use Gradient along x to generate different Larmor frequencies vs. x-position. Selective 90 pulse in z ±  z time z y x  z Apply x-Gradient G x = dB z /dx Precession Frequency varies with x z y x z x 00  0 +  G x x  0 -  G x x (x)(x) Frequency Encoding along x B z (x) - B 0

Principles of 1DMRI Frequency Encoding 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 x B z (x) - B 0 00  0 +  G x x  0 -  G x x (x)(x) Pulse Sequence RF GzGz 0 time GxGx

Principles of 1DMRI Frequency Encoding 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 x Detector coil Pulse Sequence RF GzGz 0 time GxGx Detect Signal “readout” G x on while detecting B z (x) - B 0 00  0 +  G x x  0 -  G x x (x)(x)

Principles of 1DMRI Frequency Encoding 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 x Detector coil Apply x-Gradient DURING acquisition. Precession Frequency varies with x. B z (x) - B 0 00  0 +  G x x  0 -  G x x (x)(x)

Principles of 1DMRI Frequency Encoding Perform z-slice. Now only look at 2D plane from now on. Use Gradient along x to generate different Larmor frequencies vs. x-position. Apply x-Gradient DURING acquisition. Precession Frequency varies with x. z x Detector coil FID exp(-t/T 2 *) T 2 * is based on the intentionally applied gradient. sr(t)sr(t) t B z (x) - B 0 00  0 +  G x x  0 -  G x x (x)(x)

Principles of 1DMRI Frequency Encoding Perform z-slice. Now only look at 2D plane from now on. Use Gradient along x to generate different Larmor frequencies vs. x-position. Apply x-Gradient DURING acquisition. Precession Frequency varies with x. z x Detector coil FID FT exp(-t/T 2 *) T 2 * is based on the intentionally applied gradient. 00 2/T 2 *  0 -  G x x  0 +  G x x sr(t)sr(t) t B z (x) - B 0 00  0 +  G x x  0 -  G x x (x)(x)

 0 +  G x x Principles of 1DMRI Frequency Encoding Perform z-slice. Now only look at 2D plane from now on. Use Gradient along x to generate different Larmor frequencies vs. x-position. Apply x-Gradient DURING acquisition. Precession Frequency varies with x. Spins at various x positions in space are encoded to a different precession frequency z x Detector coil FID FT exp(-t/T 2 *) T 2 * is based on the intentionally applied gradient. 00 2/T 2 *  0 -  G x x  0 +  G x x sr(t)sr(t) t B z (x) - B 0 00  0 -  G x x (x)(x)

Principles of 1DMRI 90 pulse Imaging Example Two Microfluidic Channels. Water only exists in two microfluic channels as shown. z y x zz z y x

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

Principles of 1DMRI 90 pulse time z x Bz(x)Bz(x) 1)No spins exist at x=0 where G x =0 (  0 ): FT of signal has no intensity at  0. 2)Signal is the line integral along y. (Still no info about y distribution of spins.) Imaging Example Two Microfluidic Channels. Water only exists in two microfluic channels as shown. z y x zz 00  0 -  G x x  0 +  G x x Image z y x Application of G x

Principles of 1DMRI 90 pulse time z x Bz(x)Bz(x) 1)No spins exist at x=0 where G x =0 (  0 ): FT of signal has no intensity at  0. 2)Signal is the line integral along y. (Still no info about y distribution of spins.) Imaging Example Two Microfluidic Channels. Water only exists in two microfluic channels as shown. z y x zz 00  0 -  G x x  0 +  G x x Image z y x m( x,y ) = spin density( x,y ) Application of G x

Principles of 1DMRI 1DFT Math Signal is the 1DFT of the line integral along y.

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

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

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

 0 +  G x x Principles of 1DMRI k-vector perspective Time Evolution of Spins in an x-Gradient. Spatial frequency, k-vector, changes. Pulse Sequence RF GzGz 0 time GxGx t1t1 t2t2 x Mi(x)Mi(x) t1t1 Dephasing across x in time. Rotating frame  0 or relative to x =0 time 00  0 -  G x x

 0 +  G x x Principles of 1DMRI k-vector perspective Pulse Sequence RF GzGz 0 time GxGx t1t1 t2t2 x Mi(x)Mi(x) t1t1 Dephasing across x in time. Rotating frame  0 or relative to x =0 time 00  0 -  G x x Time Evolution of Spins in an x-Gradient. Spatial frequency, k-vector, changes.

Principles of 1DMRI k-vector perspective Pulse Sequence RF GzGz 0 time GxGx t1t1 t2t2 Mi(x)Mi(x) t1t1 Dephasing across x in time. Rotating frame  0 or relative to x =0 time Time Evolution of Spins in an x-Gradient. Spatial frequency, k-vector, changes.  0 +  G x x 00  0 -  G x x x

Principles of 1DMRI k-vector perspective Pulse Sequence RF GzGz 0 time GxGx t1t1 t2t2 Mi(x)Mi(x) Homo- dyne s(t) t1t1 Dephasing across x in time. Rotating frame  0 or relative to x =0 time FID Time Evolution of Spins in an x-Gradient. Spatial frequency, k-vector, changes.  0 +  G x x 00  0 -  G x x x

 0 +  G x x 00  0 -  G x x Principles of 1DMRI k-vector perspective Pulse Sequence RF GzGz 0 time GxGx t1t1 t2t2 Mi(x)Mi(x) Homo- dyne s(t) t1t1 Dephasing across x in time. Rotating frame  0 or relative to x =0 time k FID Spatial frequency encoded by phase k=0 k: one spatial period Time Evolution of Spins in an x-Gradient. Spatial frequency, k-vector, changes. x

Principles of 1DMRI k-vector perspective Pulse Sequence RF GzGz 0 time GxGx t1t1 t2t2 Mi(x)Mi(x) Homo- dyne s(t) t1t1 Dephasing across x in time. Rotating frame  0 or relative to x =0 time k FID k=0 k: one spatial period Each Point on FID is a different value of k x Time Evolution of Spins in an x-Gradient. Spatial frequency, k-vector, changes.  0 +  G x x 00  0 -  G x x Spatial frequency encoded by phase x

Principles of 1DMRI k-vector perspective Pulse Sequence RF GzGz 0 time GxGx t1t1 t2t2 Mi(x)Mi(x) Homo- dyne s(t) t1t1 Dephasing across x in time. Rotating frame  0 or relative to x =0 time k FID k=0 k: one spatial period Each Point on FID is a different value of k x Time Evolution of Spins in an x-Gradient. Spatial frequency, k-vector, changes.  0 +  G x x 00  0 -  G x x k-vector ~ amount of spin warping over distance Spatial frequency encoded by phase x

Principles of 1DMRI 2 Approaches to Understand FT 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, k x, that increases with measurement time. 2/T 2 * 00  0 -  G x x  0 +  G x x Frequency Encoding Precession:  (x) Phase Encoding Phase(t) ~ Spatial Freq. FT of FID (time) gives frequency, .  depends on position FT of spatial frequency data, k x, data gives position data, x. Different values of k x are probed over time, t.

Principles of 2DMRI 2DFT Principles Again perform z-slice. Only look at 2D plane. Want to now distinguish spins along x- and y-directions. z y x 90 pulsed Plane

Principles of 2DMRI 2DFT Principles Again perform z-slice. Only look at 2D plane. Want to now distinguish spins along y-direction also. z y x z y x 90 pulsed Plane Apply y-Gradient for time t y G y = dB z /dy GyGy

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

Principles of 2DMRI 2DFT Principles Phase Encoding. G y is turned on for a certain time, t y, then off. This generates a difference in phase over y. All precess at  0 z y x z y x 90 pulsed Plane Apply y-Gradient for time t y G y = dB z /dy Then G y turned off GyGy z y x But spin warped along y by an amount determined by G y t y ~ single k y value Phase encoding along y

z y x Principles of 2DMRI 2DFT Principles Detect Using G x. As usual detection occurs with G x. All precess at  0 z y x Bz(x)Bz(x) z y x 90 pulsed Plane Apply y-Gradient for time t y G y = dB z /dy Then G y turned off GyGy z y x Detector coil Detect with G x Usual frequency encoding along x

z y x Principles of 2DMRI 2DFT Principles Detect Using G x. As usual detection occurs with G x. All precess at  0 z y x Bz(x)Bz(x) z y x 90 pulsed Plane Apply y-Gradient for time t y G y = dB z /dy Then G y turned off GyGy z y x Detector coil Detect with G x This time, the magnitude of the signal at each  (x-position), corresponds to the intensity of the spatial frequency, k y, encoded by G y phase encoding step.

z y x Principles of 2DMRI 2DFT Principles Detect Using G x. As usual detection occurs with G x. All precess at  0 z y x Bz(x)Bz(x) Frequency Encoding along x (G x t) Phase Encoding along y (G y t y ) z y x 90 pulsed Plane Apply y-Gradient for time t y G y = dB z /dy Then G y turned off GyGy z y x Detector coil Detect with G x This time, the magnitude of the signal at each  (x-position), corresponds to the intensity of the spatial frequency, k y, encoded by G y phase encoding step. (for k y =0 it’s the line integral)

B z (x) - B 0 z x Principles of 2DMRI 2DFT Principles Again perform z-slice. Only look at 2D plane. Want to now distinguish spins along y-direction also. Phase Encoded Pulse Sequence RF GzGz 0 time GxGx tyty GyGy

z x Principles of 2DMRI 2DFT Principles Again perform z-slice. Only look at 2D plane. Want to now distinguish spins along y-direction also. Bz(x)Bz(x) Detector coil Phase Encoded Pulse Sequence RF GzGz 0 time GxGx Detect Signal tyty Phase Encode GyGy

z x Principles of 2DMRI 2DFT Principles Again perform z-slice. Only look at 2D plane. Want to now distinguish spins along y-direction also. Bz(x)Bz(x) Detector coil Repeat experiment multiple times varying the G y gradient strength (or time t y ) so that k y receives the same sampling as k x (FID sampling rate). Phase Encoded Pulse Sequence RF GzGz 0 time GxGx Detect Signal tyty Phase Encode GyGy

Principles of 1DMRI 2DFT Math Signal is the 2DFT of the image. Baseband (Homodyned) signal.

Principles of 1DMRI 2DFT Math Signal is the 2DFT of the image. Baseband (Homodyned) signal. Phase Encoding Step G x during recording of FID

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

Principles of 1DMRI 2DFT Math Signal is the 2DFT of the image. Baseband (Homodyned) signal. For any given FID, t y is fixed and t is running variable. Phase Encoding Step G x during recording of FID 00  0 -  G x x  0 +  G x x

Principles of 1DMRI 2DFT Math Signal is the 2DFT of the image. Baseband (Homodyned) signal. For any given FID, t y is fixed and t is running variable. Phase Encoding Step G x during recording of FID 00  0 -  G x x  0 +  G x x Intensities at each x correspond to intensity of the k y 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

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

Principles of 2DMRI 2DFT Principles k-space perspective. Want to map k-space then take 2DFT. (Each FID samples line in k-space along k x ) kxkx kyky 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 G y values. The data along a line is the FT of the signal in the y direction.

Principles of 2DMRI 2DFT Principles k-space perspective. Want to map k-space then take 2DFT. (Each FID samples line in k-space along k x ) kxkx kyky 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 G y values. The data along a line is the FT of the signal in the y direction. Rotate for viewing “FID” along y.

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

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

Principles of 2DMRI 2DFT Principles Updated Pulse Sequence. Want to map k-space then take 2DFT. (Each FID samples line in k-space along k x ) kxkx kyky Pulse Sequence RF GzGz 0 time GxGx tyty GyGy tyty 2t y

Principles of 2DMRI 2DFT Principles Updated Pulse Sequence. Want to map k-space then take 2DFT. (Each FID samples line in k-space along k x ) kxkx kyky Pulse Sequence RF GzGz 0 time GxGx tyty GyGy tyty tyty

Principles of 2DMRI 2DFT Principles Updated Pulse Sequence. Want to map k-space then take 2DFT. (Each FID samples line in k-space along k x ) kxkx kyky Pulse Sequence RF GzGz 0 time GxGx GyGy tyty tyty Representation

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

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

Principles of 2DMRI kxkx kyky Sampling rate of k-space kyky kxkx Aliasing Issues Aliasing: If sampling rate is not sufficient, the Field of view will overlap.

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

Principles of 2DMRI Resolution Resolution: Resolution in the object-oriented domain is determined by the extent of k-space measured. kxkx kyky Sampling rate of k-space x y  k y N pe  k x N read  y = FOV y /N pe =(  k y N pe ) -1  x = FOV x /N read =(  k x N read ) -1 Field of View/Resolution ~ # points need to sample (e.g., 25.6 cm image, 1mm resolution: 256 points/dimension, 65.5k points) N read : # of readout points during FID N pe : # of phase encoding steps

Summary 1.MRI is based on the spatial encoding of spins either through a difference in phase (y) or a difference in Larmor frequency (x): 1.FID in the presence G x, after a given phase encoding in y, gives a line of points in k-space. FIDs are repeated for a variety of k y values to fill up k- space. 1.2DFT of k-space gives the image of spin density m(x,y) 2.Limitations. 1.Detection is based on the signal received in a coil. 1.Coil inductor has an impedance, Z coil = i  L, ~ frequency. Thus significant voltage signals are observed only at high frequencies. (M xy → i coil. i coil = v signal /Z coil.) 2.Requires Large Magnetic fields – cryogenics, homogeneity. 1.Large Fields can lead to signal distortion. Samples containing metals cannot be imaged 3.Atomic magnetometry – large fields not necessary. Remote detection can be used so that imaging can be performed in the presence of metals.

Remote Detection Flow In 1)Spatial Encoding (90 pulse, G x, G y, G z ) 2)Storage of one component (M x, M y ) along z. Flow Out Detector (Analyze stored component) Region of Interest (Flow profile, etc.) Spatial information carried by flow of liquid (H 2 O) to the detection region.

MRI using the atomic magnetometer Nitrogen Magnetometer Pre-polarization Field (~ 3 kG) Encoding Field ( B 0, B x, B y, B z ) Water Out H2OH2O

RF Pulse Travel Time Storage Pulse π/2 M t B0B0  Remote Detection of NMR/MRI 1)dc magnetometer means NO FID recorded. 2) k-space in must be measured point-by-point. -Very slow! Imaging takes much more time! -But z-slice obtained all at once by flow profile. Encoding Region

Remote Detection Flow In Flow Out No z-slice performed – use flow and detection time to obtain z-slice. Detector

Remote Detection Flow In Flow Out No z-slice performed – use flow and detection time to obtain z-slice. Pulse t Detector s(t) Detector

Remote Detection Flow In Flow Out No z-slice performed – use flow and detection time to obtain z-slice. Flow t Detector s(t) Detector

Remote Detection Flow In Flow Out No z-slice performed – use flow (and detection time) to obtain z-slice. Flow t Detector s(t) Detector

Remote Detection Flow In Flow Out No z-slice performed – use flow and detection time to obtain z-slice. Flow tt flow Detector s(t) Detector

Remote Detection Flow In Flow Out No z-slice performed – use flow and detection time to obtain z-slice. Flow tt flow Signal (nG) Time (s) In absence of diffusion, s(t) is the average of the z-slices in the detection region Detector s(t) Detector In reality, diffusion- weighted z-slice

Encoding  2  2 Detection G PE (x, y)  =(x,-x,y,-y) a. Pulse sequence: phase encoding MRI results  x =  x t =  G x xt  y =  y t =  G y yt Granwehr, J., et al., PRL 95, 075503 (2005). kxkx kyky t x,y Signal (nG) Time (s) Obtain a flow profile for each (k x, k y ) point and repeat for all points in k-space - Actually, 4 flow profiles are obtained for each point in k-space… For x,y dimensions:

a. Phase Cycling (90 storage pulse along x or y) MRI results Granwehr, J., et al., PRL 95, 075503 (2005). x y z For a given k-space point, the net magnetization, M xy, is rotated to a given point on the x,y plane. Need to convert this to a M z for measurement.

a. Phase Cycling (90 storage pulse along x or y) MRI results Granwehr, J., et al., PRL 95, 075503 (2005). x y z x y z  2 y M x → M z For a given k-space point, the net magnetization, M xy, is rotated to a given point on the x,y plane. Need to convert this to a M z for measurement.

a. Phase Cycling (90 storage pulse along x or y) MRI results Granwehr, J., et al., PRL 95, 075503 (2005). x y z x y z  2 y M x → M z For a given k-space point, the net magnetization, M xy, is rotated to a given point on the x,y plane. Need to convert this to a M z for measurement. But this only tells you the x-component of M xy. Need to repeat for y to get the vector M xy.

a. Phase Cycling (90 storage pulse along x or y) MRI results Granwehr, J., et al., PRL 95, 075503 (2005). x y z x y z x y z  2 y  2 x M x → M z M y → -M z M y component of M xy stored along z for detection

a. Phase Cycling (90 storage pulse along x or y) MRI results Granwehr, J., et al., PRL 95, 075503 (2005). x y z x y z x y z  2 y  2 x M z component detected. Vector addition of signal from  /2(x) and  /2(y) describes vector M xy for each point in k-space. This must be repeated for each G x, G y (point in k-space) desired. (So far, 2 points per k-space, but why 4?...) M x → M z M y → -M z M y component of M xy stored along z for detection

MRI results x y z x y z  2 y M x → M z a. Phase Cycling (Can do storage pulse along y or -y)

MRI results Granwehr, J., et al., PRL 95, 075503 (2005). x y z x y z  2 y 1) Repeating the 90 storage pulse along y and –y allows for data averaging. (Similar to gradiometer; common mode noise rejected) M x → M z a. Phase Cycling (Cand do storage pulse along y or -y) x y z  2 -y M x → -M z

MRI results Granwehr, J., et al., PRL 95, 075503 (2005). x y z x y z  2 y 1) Repeating the 90 storage pulse along y and –y allows for data averaging. (Similar to gradiometer; common mode noise rejected) - Set M z (z=0)=0, then add M z (y pulse) – M z (-y pulse) M x → M z a. Phase Cycling (Cand do storage pulse along y or -y) x y z  2 -y M x → -M z Signal (nG) Time (s) M z =-M 0 M z =M 0

MRI results Granwehr, J., et al., PRL 95, 075503 (2005). x y z x y z  2 y 1) Repeating the 90 storage pulse along y and –y allows for data averaging. (Similar to gradiometer; common mode noise rejected) - Set M z (z=0)=0, then add M z (y pulse) – M z (-y pulse) 2) Repeat for x and –x storage pulses to get 4 flow profiles (x, y, -x, -y) for each point in k-space. M x → M z a. Phase Cycling (Cand do storage pulse along y or -y) x y z  2 -y M x → -M z Signal (nG) Time (s) M z =-M 0 M z =M 0

a x y z 1 mm z y x b b. Images of the encoding volume H2OH2O MRI results Images along z are obtained by using the magnetization magnitude from the flow profile after a given flow time.

0.5 s0.7 s0.9 s1.1 s1.3 s 1.5 s1.7 s1.9 s2.1 s2.3 s c. Time-resolved flow images Signal (nG) Time (s) H2OH2O Temporal resolution: 100 ms Spatial resolution: 1.6 mm x 1.6 mm x 4.7 mm MRI results Z-sampling

Resolution: z, 5mm; y, 2.5mm z y 0.4 s0.6 s0.8 s1.0 s1.2 s 1.4 s1.6 s1.8 s2.0 s2.2 s c. Time-resolved flow images MRI results

c. Time-resolved flow images MRI results Resolution: z, 5mm; y, 2.5mm z y 0.4 s0.6 s0.8 s1.0 s1.2 s 1.4 s1.6 s1.8 s2.0 s2.2 s

c. Time-resolved flow images MRI results z y 0.4 s0.5 s0.6 s0.7 s0.8 s Flow Mixing Region

c. Time-resolved flow images MRI results High-field MRI (300 MHz) of flow in a porous metallic sample. The images show only the inlet and outlet, while imaging of the sample region (marked by the red box) is not possible. LMRI of flow in a porous metallic sample in the Earth’s field 1.4 s1.2 s1.0 s0.8 s0.6 s0.4 s In-plane resolution: 2.5mm x 2.5mm Object size: 12 mm diameter, 12 mm length Stainless Steel Porous Sample

Summary 1.MRI is based on the spatial encoding of spins either through a difference in phase (y) or a difference in Larmor frequency (x): 1.FID in the presence G x, after a given phase encoding in y, gives a line of points in k-space. FIDs are repeated for a variety of k y values to fill up k-space. 1.2DFT of k-space gives the image of spin density m(x,y) 2.MRI can be performed using the atomic magnetometer. 1.Image takes much longer to achieve! Measurement of FID in Gx not yet possible. 2.Hopefully rf magnetometry can solve this problem.

Acknowledgements Shoujun Xu Louis Bouchard Pines Group Alex Pines Christian Hilty Josef Granwehr Sabieh Anwar Elad Harel Alyse Jacobson All current ‘nuts Budker Group Dmitry Budker Simon Rochester Valeriy Yashchuk James Higbie Derek Kimball Jason Stalnaker Misha Babalas Physics Department Machine Shop Chemistry Department Electronic Shop Chemistry Department Glass Shop 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.

Principles of Magnetic Resonance Imaging David J. Michalak Presentation for Physics 250 03/13/2007.

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Principles of Magnetic Resonance Imaging David J. Michalak Presentation for Physics 250 03/13/2007.

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Presentation on theme: "Principles of Magnetic Resonance Imaging David J. Michalak Presentation for Physics 250 03/13/2007."— Presentation transcript:

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