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 25003/13/2007

Motivation

Principles of NMR

Interactions of spins in B0 field

Principles of 1D MRI

Principles of 2D MRI

2D MRI using the atomic magnetometer.

Applications in progress

o Earth-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

B0

Principles of NMR

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, = B0

M = Mi

RFPulse

B0

Principles of NMR


y

x

zB0

Application of a rf pulse 0=2f0 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 = Mi

M

RFPulse

B0

Principles of NMR


y

x

zB0

y

x

zB0

M = Mi

M precesses in the transverse plane. In the absence of any disturbances, M continues to rotate indefinitely in xy plane.

MM

Time


exp[-it]

RFPulse

B0

Principles of NMR


y

x

zB0

y

x

zB0

M = Mi


Detector

MM

Time


exp[-it]

RFPulse

B0

Principles of NMR


y

x

zB0

y

x

zB0

M = Mi


Detector

MM

Time

Assume: 1) All spins feel same B0.2) No other forces on Mi

(including detection).


exp[-it]

RFPulse

B0

Principles of NMR


y

x

zB0

y

x

zB0

M = Mi

Detector

MM

Time

time, t

sign

al,

s r(t

)Application of a rf pulse 0=2f0 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[-it]

RFPulse

B0

Principles of NMR


y

x

zB0

y

x

zB0

M = Mi

Detector

FT

MM

Time

s r(t

)s r

()

t

Application of a rf pulse 0=2f0 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 = 2f

exp[-it]

(0/2)-1

RFPulse

B0

Principles of NMR


y

x

zB0

y

x

zB0

M = Mi

Detector

(0/2)-1

FT

MM

Time

Boring Spectrum!

0 = 2fs r

(t)

s r(

)

t


exp[-it]

Principles of NMR

y

x

zB0

In Reality:

1) Relaxation (Inherent even if B0 is homogeneous)1) T1: Spins move away from xy plane towards z.2) T2: Spins dephase from each other.3) Chemical Shift.

2) Experimental Design Effects.1) T2*: Field inhomogeneity in B0(x, y, z, t)

1) Could be intentional (e.g., gradient) or not.

Complexity Makes Things Interesting

Principles of NMR

y

x

zB0

T1 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 B0 → lower T1.

2) Math: dM/dt = -(Mz-M0)/T1

1) Solution: Mz = M0 + (Mz(0)-M0)exp(-t/T1)2) 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

T1 Spin Relaxation

Principles of NMR

y

x

zB0

T2 Spin Relaxation: Decay of transverse magnetization, Mxy.

1) T1 plays a role, since as Mxy → Mz, Mxy → 01) But dephasing also decreases Mxy: T2 < T1.

2) T2: Spin-Spin Time Constant1) Variations in Bz with time and position.2) Pertinent fluctuations in Bz are those near dc

frequencies (independent of B0) so that 0 is changed.

3) Molecular motion around the spin of interest.1) Liquids: High Temp more motion, less B, high

T2

2) Solids: slow fluctuations in Bz, extreme T2.3) Bio Tissues: spins bound to large molecules vs.

those free in solution.

3) Math: dM/dt = -Mxy/T2

1) After 90 pulse: Mxy = M0 exp(-t/T2)]

y

x

z

B0+B(r,t)

T2 Spin Relaxation

Mxy

Principles of NMR

y

x

zB0

Comparison of T1 and T2 Spin Relaxation:

y

x

z

B0+B(r,t)

Tissue T1 (ms) T2 (ms)

Gray Matter 950 100

White Matter 600 80

Muscle 900 50

Fat 250 60

Blood 1200 100-200*

*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.

T1/T2 Spin Relaxation

Principles of NMR

y

x

zB0


y

x

z

B0+B(r,t)


Gray Matter 950 100

White Matter 600 80

Muscle 900 50

Fat 250 60

Blood 1200 100-200*

Detector


*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.

Principles of NMR

y

x

zB0


y

x

z

B0+B(r,t)


Gray Matter 950 100

White Matter 600 80

Muscle 900 50

Fat 250 60

Blood 1200 100-200*

*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

Spectrum

T2 << T1

Mxy decays~exp(-t/T2)

Detector

FT 2/T2

Because T2 is independent of B0, higher B0 gives better resolution


s r(t

)

t

Principles of NMR

y

x

zB0

y

x

z

B0+B(r,t)

Inclusion of T1 and T2 Spin Relaxation:

1) Inclusion of mathematical expression:1) Bloch Equation

1

0

2

)(

T

MM

T

MMBM

dt

dM zyx kji

= gyromagnetic ratioT1 = Spin-Lattice (longitudinal-z) relaxation time constantT2 = Spin-Lattice (longitudinal-z) relaxation time constantM0 = Equilibrium Magnetization due to B0 field.i, j, k = Unit vectors in x, y, z directions respectively.


Principles of NMR

y

x

zB0

y

x

z

B0+B(r,t)

Inclusion of T1 and T2 Spin Relaxation:

1) Inclusion of mathematical expression:1) Bloch Equation

1

0

2

)(

T

MM

T

MMBM

dt

dM zyx kji

= gyromagnetic ratioT1 = Spin-Lattice (longitudinal-z) relaxation time constantT2 = Spin-Lattice (longitudinal-z) relaxation time constantM0 = Equilibrium Magnetization due to B0 field.i, j, k = Unit vectors in x, y, z directions respectively.

Precession TransverseDecay

LongitudinalGrowth

Net magnetization is not necessarily constant: e.g., very short T2, long T1.


Principles of NMR

y

x

zB0

Chemical Shift: Nuclei are shielded (slightly) from B0 by the presence of their electron clouds.

1) Effective field felt by a nuclear spin is B0(1-).1) Larmor precession freq, = B0(1-).

1) Shift is often in the ppm range.1) ~500,000 precessions before Mxy = 0

2) Chemical environment determines amount of .1) H2O vs. Fat (fat about 3.5 ppm lower 0)

y

x

z

B0(1-)

Discrete Shift

H HO

+ +

-

H HC

Less Shielding More Shielding

Chemical Shift

Principles of NMR

y

x

zB0


1) Effective field felt by a nuclear spin is B0(1-).1) Larmor precession freq, = B0(1-).

1) Shift is often in the ppm range.1) ~500,000 precessions before Mxy = 0

2) Chemical environment determines amount of .1) H2O vs. Fat (fat about 3.5 ppm lower 0)

y

x

z

B0(1-)

Discrete Shift

H HO

+ +

-

H HC

Detector Less Shielding More Shielding

Chemical Shift

Principles of NMR

y

x

zB0


y

x

z

B0(1-)

Discrete Shift

0

2/T2

Because T2 is independent of B0, higher B0 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

zB0

T2*: B0 Inhomogeneity: Additional decay of Mxy.

1) In addition to T2, which leads to Mxy decay even in a constant B0, application of B0(x, y, z, t) will cause increased dephasing: 1/T2* = 1/T2 + 1/T’, where T’ is the dephasing due only to B0(x, y, z, t).1) T2

* < T2, and depends on B0(x, y, z, t).2) Additional loss of resolution between peaks.

y

x

z

B0+B(r,t)

time,

Field Inhomogeneity

Principles of NMR

y

x

zB0




2) If B0(x, y, z) is not time dependent, then it can be corrected by an echo pulse.

y

x

z

B0+B(r,t)

time,

Field Inhomogeneity

Principles of NMR

y

x

zB0




2) If B0(x, y, z) is not time dependent, then it can be corrected by an echo pulse.

y

x

z

B0+B(r,t)y

x

z

B0+B(r,t)y

x

z

B0+B(r,t)

180x pulse

(x → x, y → –y)

time,

time,

Field Inhomogeneity

Echo!

Principles of NMR

y

x

zB0

T2

T2*

Field Inhomogeneity


3) If echo pulse applied at time, , then echo appears at 2. 1) 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.

180 pulse

applied

Echo

t 90

pulse

’180 pulse

applied

’Echo

s r(t

)

t

Principles of 1DMRI Single B0 – No Spatial Information

Measured response is from all spins in the sample volume. Detector coil probes all space with equal intensity

90 pulse

B0 B0

FID 0

Spectrum

FT 2/T2

If only B0 is present (and homogeneous) all spins remain in phase during precession (as drawn).

- B(x, y, z, t) = B0; thus, (x, y, z) = 0 = B0

time

B0

No Spatial Information(Volume integral)

Detectorcoil

s r(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) = B0 + Gzz

Field strength indicated by line thickness

Gz

Gz = dBz/dzintegrateBz=Gzz

It follows that:B(z=0)=B0



Selective90 pulse

rf=0+Gzz

B(z) = B0 + Gzz


Gz





B(z) = B0 + Gzz


1) Larmor Precession frequency is z-dependent: (z) = B(z) (z) =(B0 + Gzz)(z) = 0 + Gzz

Gz Selective90 pulse

rf=0+Gzz

2) Excite only one plane of z ± z 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

integrateBz=Gzz


FT



B(z) = B0 + Gzz


Gz

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 ± Gzz. Thus, BW = 2Gzz.)

4) To apply a “boxcar” of frequencies ± Gzz, we need the 90 deg excitation profile to be a sinc function in time.1) FT(sinc) = rect

Selective90 pulse

rf=0+Gzz

t 90° z ± z


It follows that:B(z=0)=B0 sinc = (sinx)/x


Pulse Sequence. Shows the relative timing of the RF and gradient pulses.

Selective90

pulse

B(z) = B0 + Gzz

Gz

Pulse Sequence

RF

Gz

0 3 time

0-Gzz


Gradient Echo Pulse. Gradient Echo pulse restores all spins to have the same phase within the slice z.

Selective90

pulse

B(z) = B0 + Gzz

Gz

Pulse Sequence

RF

Gz

Gradient Echo

0

0+Gzz

Before Gradient Echot =

0 3 time

Spins out of phase on xy plane

z

0-Gzz



Selective90

pulse

B(z) = B0 + Gzz

Gz

Pulse Sequence

RF

Gz

Gradient Echo

z

0

0+Gzz


0 time


Top View of xy plane

t=

0 0+Gzz0-Gzz

3



Selective90

pulse

B(z) = B0 + Gzz

Gz

Pulse Sequence

RF

Gz

z

z


After Gradient Echot = 3/2

0 time


Spins all IN phase

Gradient Echot=

t=3

0 0+Gzz0-GzzTop View of xy plane

0-Gzz0

0+Gzz

3



Selective90

pulse

B(z) = B0 + Gzz

Gz



Selective90

pulse

B(z) = B0 + Gzz

Gz

Detectorcoil

time



Selective90

pulse

B(z) = B0 + Gzz

Gz

Detectorcoil

time

FID 0

Spectrum

FT 2/T2No x, y Information, but only spins from the z ± z slice contribute to the signal.

exp(-t/T2)

s r(t

)

t



Selective90

pulse

B(z) = B0 + Gzz

Gz

Detectorcoil

time

FID 0

Spectrum

FT 2/T2No x, y Information, but only spins from the z ± z slice contribute to the signal.

exp(-t/T2)

If we can encode along x and y dimensions, we can iterate for each z slice.

s r(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.

Selective90

pulse in z ± z

z y

xz

z y

x



Selective90

pulse in z ± z

time

z y

xz

Apply x-GradientGx = dBz/dx

Precession Frequency varies with x

z y

x

z

xBz(x) - B0



Selective90

pulse in z ± z

time

z y

xz



z y

x

z

xBz(x) - B0

0 0 + Gxx0 - Gxx

(x)



Selective90

pulse in z ± z

time

z y

xz



z y

x

z

x

0 0 + Gxx0 - Gxx

(x)

Frequency Encoding along x

Bz(x) - B0



z

xBz(x) - B0

0 0 + Gxx0 - Gxx

(x)

Pulse Sequence

RF

Gz

0 time

Gx



z

xDetector

coil

Pulse Sequence

RF

Gz

0 time

Gx

Detect Signal “readout”

Gx on while detecting

Bz(x) - B0

0 0 + Gxx0 - Gxx

(x)



z

xDetector

coil

Apply x-Gradient DURING acquisition.Precession Frequency varies with x.

Bz(x) - B0

0 0 + Gxx0 - Gxx

(x)




z

xDetector

coil

FID

exp(-t/T2*)T2* is based on the intentionally applied gradient.s r

(t)

t

Bz(x) - B0

0 0 + Gxx0 - Gxx

(x)




z

xDetector

coil

FID FT

exp(-t/T2*)T2* is based on the intentionally applied gradient.

0

2/T2*

0 - Gxx 0 + Gxx

s r(t

)

t

Bz(x) - B0

0 0 + Gxx0 - Gxx

(x)

0 + Gxx




Spins at various x positions in space are encoded to a different precession frequency

z

xDetector

coil

FID FT

exp(-t/T2*)T2* is based on the intentionally applied gradient.

0

2/T2*

0 - Gxx 0 + Gxx

s r(t

)

t

Bz(x) - B0

00 - Gxx

(x)

Principles of 1DMRI

90 pulse

Imaging Example

Two Microfluidic Channels. Water only exists in two microfluic channels as shown.

z y

xz

z y

x

Principles of 1DMRI

90 pulse

time

z

xBz(x)

Imaging Example


z y

xz

z y

x

Application of Gx

Principles of 1DMRI

90 pulse

time

z

xBz(x)

1) No spins exist at x=0 where Gx=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


z y

xz

00 - Gxx 0 + Gxx

Image

z y

x

Application of Gx

Principles of 1DMRI

90 pulse

time

z

xBz(x)

1) No spins exist at x=0 where Gx=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


z y

xz

00 - Gxx 0 + Gxx

Image

z y

x

x y

xtGitir dxdyeeyxmts x0),()(

m(x,y) = spin density(x,y)

Application of Gx

Principles of 1DMRI 1DFT Math

Signal is the 1DFT of the line integral along y.

x y




x y


x y

xtGi dxdyeyxmts x),()(

Homodyne the signal (from 0 to 0).



x y


x y



x

xtGi dxexgts x)()( y

dyyxmxg ),()(

Let g(x) = Line integral along y for a given x position.



x y


x y



x

xtGi dxexgts x)()( y

dyyxmxg ),()(

Let g(x) = Line integral along y for a given x position.

2

22

FT)()( tGkx

xtG

i

xx

x

g(x)dxexgts

The homodyned signal is thus the Fourier Transform (along x) of the line integral along y.

Spatial frequencyGxt ~ kx

0 + Gxx

Principles of 1DMRI k-vector perspective

Time Evolution of Spins in an x-Gradient. Spatial frequency, k-vector, changes.

Pulse Sequence

RF

Gz

0 time

Gx

t1 t2

x

Mi(x)

t1

Dephasing across x in time. Rotating frame 0 or relative to x=0

time

00 - Gxx

0 + Gxx


Pulse Sequence

RF

Gz

0 time

Gx

t1 t2

x

Mi(x)

t1


time

00 - Gxx



Pulse Sequence

RF

Gz

0 time

Gx

t1 t2

Mi(x)

t1


time


0 + Gxx00 - Gxxx


Pulse Sequence

RF

Gz

0 time

Gx

t1 t2

Mi(x)

Homo-dynes(t)

t1


time

FID


0 + Gxx00 - Gxxx

0 + Gxx00 - Gxx


Pulse Sequence

RF

Gz

0 time

Gx

tGk xx

2

t1 t2

Mi(x)

Homo-dynes(t)

t1


time

k FID

Spatial frequency encoded by phase

k=0

k: one spatial period


x


Pulse Sequence

RF

Gz

0 time

Gx

t1 t2

Mi(x)

Homo-dynes(t)

t1


time

k FID

k=0


Each Point on FID is a different value of kx


tGk xx

2

0 + Gxx00 - Gxx


x


Pulse Sequence

RF

Gz

0 time

Gx

t1 t2

Mi(x)

Homo-dynes(t)

t1


time

k FID

k=0


Each Point on FID is a different value of kx


tGk xx

2

0 + Gxx00 - Gxxk-vector ~ amount of

spin warping over distance


x

Principles of 1DMRI 2 Approaches to Understand FTThe 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.

tGk xx

2

2/T2*

00 - Gxx 0 + Gxx

Frequency EncodingPrecession: (x)

Phase EncodingPhase(t) ~ Spatial Freq.

FT of FID (time) gives frequency, . depends on position

FT of spatial frequency data, kx, data gives position data, x.Different values of kx 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

x90 pulsed

Plane


Again perform z-slice. Only look at 2D plane. Want to now distinguish spins along y-direction also.

z

y

x

z

y

x90 pulsed

Plane Apply y-Gradient for time ty

Gy = dBz/dy

Gy



z

y

x

z

y

x90 pulsed


Gy = dBz/dyThen Gy turned off

Gy


Phase Encoding. Gy is turned on for a certain time, ty, then off. This generates a difference in phase over y.

All precess

at 0

z

y

x

z

y

x90 pulsed



Gy z

y

x

But spin warped along y by an amount determined by Gyty ~ single ky value

yyy tGk

2

Phase encoding along y

z

y

x


Detect Using Gx. As usual detection occurs with Gx.

All precess

at 0

z

y

x

Bz(x)

z

y

x90 pulsed



Gy z

y

x

Detectorcoil

Detect with Gx

Usual frequency encoding along x

z

y

x



All precess

at 0

z

y

x

Bz(x)

z

y

x90 pulsed



Gy z

y

x

Detectorcoil

Detect with Gx

This time, the magnitude of the signal at each (x-position), corresponds to the intensity of the spatial frequency, ky, encoded by Gy phase encoding step.

z

y

x



All precess

at 0

z

y

x

Bz(x)

Frequency Encoding along x (Gxt)Phase Encoding along y (Gyty)

z

y

x90 pulsed



Gy z

y

x

Detectorcoil

Detect with Gx

This time, the magnitude of the signal at each (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)

Bz(x) - B0

z

x



Phase

Enc

oded

Pulse Sequence

RF

Gz

0 time

Gx

ty

Gy

z

x



Bz(x)Detector

coil

Phase

Enc

oded

Pulse Sequence

RF

Gz

0 time

Gx

Detect Signal

ty

PhaseEncode

Gy

z

x



Bz(x)Detector

coil

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

Enc

oded

Pulse Sequence

RF

Gz

0 time

Gx

Detect Signal

ty

PhaseEncode

Gy


Signal is the 2DFT of the image.

x y

xtGiytGi dxdyeeyxmts xyy ),()( Baseband (Homodyned) signal.



x y


Phase EncodingStep

Gx during recording

of FID



x y


For any given FID, ty is fixed and t is running variable.

Phase EncodingStep

Gx during recording

of FID



x y


2

,2

,FTD2)( yyy

xx

tGk

tGk

y)m(xts


Phase EncodingStep

Gx during recording

of FID

00 - Gxx 0 + Gxx



x y


2

,2

,FTD2)( yyy

xx

tGk

tGk

y)m(xts


Phase EncodingStep

Gx during recording

of FID

00 - Gxx 0 + Gxx

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


k-space perspective. Want to map k-space then take 2DFT. (Each FID samples line in k-space along kx)

kx

ky

Set of data points along the kx axis corresponds to the sampled FID taken with no Gy phase encoding gradient.

Set of data points sampled from the FID with a phase encoding of a given ky (Gyty).

yyy tGk

2 tGk xx

2

Measure FIDkx measured in time

Cha

nge

Gyt

y



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

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.

Rotate forviewing

“FID” along y.



kx

ky What does this data look like?

Image

Rect function

Rec

t fu

nctio

n

Sinc function

Sin

c fu

nctio

n

x

y



kx

ky What does this data look like?

x

Image

Circle function(radially symmetric rect)

Jinc functionRadially symmetric sinc)

y


Updated Pulse Sequence. Want to map k-space then take 2DFT. (Each FID samples line in k-space along kx)

kx

ky Pulse Sequence

RF

Gz

0 time

Gx

tyGy

ty

2ty



kx

ky Pulse Sequence

RF

Gz

0 time

Gx

tyGy

ty

ty



kx

ky Pulse Sequence

RF

Gz

0 time

Gx

Gy

ty

ty

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 kx dimension of (kx). Interval on ky is determined by the change in gradient area (Gyty) between different runs

kx

ky

Sampling rate of k-space

We know that we need enough data to adequately sample the FID in time (kx) dimension

Same principle applies for ky (Gyty) dimension

t, kx

Principles of 2DMRI Field of View

Field of View: Sampling rate of k-space determines the field of view in the object-oriented domain.

kx

ky


x

y

ky

kx

FOVy

=1/ky)

FOVx = 1/kx)

FOV > Image size!Prevent Aliasing

Principles of 2DMRI

kx

ky


ky

kx

Aliasing Issues

Aliasing: If sampling rate is not sufficient, the Field of view will overlap.

Principles of 2DMRI

kx

ky


x

y

ky

kx

FOVy

=1/ky)

FOVx = 1/kx)

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.

kx

ky


x

y

kyNpe

kxNread

y = FOVy/Npe

=(kyNpe)-1

x = FOVx/Nread

=(kxNread)-1

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 FIDNpe: # 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 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.

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, Zcoil= iL, ~ frequency. Thus

significant voltage signals are observed only at high frequencies. (Mxy → icoil. icoil = vsignal/Zcoil.)

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, Gx, Gy, Gz)

2) Storage of one component (Mx, My) along z.

FlowOut

Detector(Analyze stored

component)

Region of Interest(Flow profile, etc.)

Spatial information carried by flow of liquid (H2O) to the detection region.

MRI using the atomic magnetometer

Nitrogen

Magnetometer

Pre-polarization Field (~ 3 kG)

Encoding Field (B0, Bx, By, Bz)

Water Out

H2O

RFPulse

TravelTime

StoragePulse

π/2 π/2 Mt

B0

Remote Detection of NMR/MRIRemote 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

FlowOut

No z-slice performed – use flow and detection time to obtain z-slice.

Detector

Remote Detection

Flow In

FlowOut


Pulse

t

Det

ecto

rs(

t)

Detector

Remote Detection

Flow In

FlowOut


Flow

t

Det

ecto

rs(

t)

Detector

Remote Detection

Flow In

FlowOut

No z-slice performed – use flow (and detection time) to obtain z-slice.

Flow

t

Det

ecto

rs(

t)

Detector

Remote Detection

Flow In

FlowOut


Flow

ttflow

Det

ecto

rs(

t)

Detector

Remote Detection

Flow In

FlowOut


Flow

ttflow

Sig

na

l (n

G)

Time (s)

In absence of diffusion, s(t) is the average of the z-slices in the detection region

Det

ecto

rs(

t)

Detector

In reality, diffusion-weighted z-slice

Encoding2

2

Detection

GPE(x, y)

=(x,-x,y,-y)

a. Pulse sequence: phase encoding

MRI results

x = xt = Gxxt

y = yt = Gyyt

Granwehr, J., et al., PRL 95, 075503 (2005).

kx

kytx,y

tx,y

Sig

na

l (n

G)

Time (s)

Obtain a flow profile for each (kx, ky) 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


x

y

z For a given k-space point, the net magnetization, Mxy, is rotated to a given point on the x,y plane. Need to convert this to a Mz for measurement.


MRI results


x

y

z

x

y

z

2 y Mx → Mz

For a given k-space point, the net magnetization, Mxy, is rotated to a given point on the x,y plane. Need to convert this to a Mz for measurement.


MRI results


x

y

z

x

y

z

2 y Mx → Mz

For a given k-space point, the net magnetization, Mxy, is rotated to a given point on the x,y plane. Need to convert this to a Mz for measurement.

But this only tells you the x-component of Mxy. Need to repeat for y to get the vector Mxy.


MRI results


x

y

z

x

y

z

x

y

z

2 y

2 x

Mx → Mz

My → -Mz

My component of Mxy stored along z for detection


MRI results


x

y

z

x

y

z

x

y

z

2 y

2 x

Mz component detected. Vector addition of signal from /2(x) and /2(y) describes vector Mxy for each point in k-space. This must be repeated for each Gx, Gy (point in k-space) desired. (So far, 2 points per k-space, but why 4?...)

Mx → Mz

My → -Mz

My component of Mxy stored along z for detection

MRI results

x

y

z

x

y

z

2 y Mx → Mz

a. Phase Cycling (Can do storage pulse along y or -y)

MRI results


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)

Mx → Mz

a. Phase Cycling (Cand do storage pulse along y or -y)

x

y

z2 -y

Mx → -Mz

MRI results


x

y

z

x

y

z


allows for data averaging. (Similar to gradiometer; common mode noise rejected) - Set Mz(z=0)=0, then add Mz(y pulse) – Mz(-y pulse)

Mx → Mz


x

y

z2 -y

Mx → -Mz Sig

na

l (n

G)

Time (s)

Mz=-M0

Mz=M0

MRI results


x

y

z

x

y

z


allows for data averaging. (Similar to gradiometer; common mode noise rejected) - Set Mz(z=0)=0, then add Mz(y pulse) – Mz(-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.

Mx → Mz


x

y

z2 -y

Mx → -Mz Sig

na

l (n

G)

Time (s)

Mz=-M0

Mz=M0

a

x

yz

1 mm

z

yx •

b

b. Images of the encoding volume

H2O

MRI results

Images along z are obtained by using the magnetization magnitude from the flow profile after a given flow time.

0.5 s 0.7 s 0.9 s 1.1 s 1.3 s

1.5 s 1.7 s 1.9 s 2.1 s 2.3 s

c. Time-resolved flow images

Sig

na

l (n

G)

Time (s)

H2O

Temporal resolution: 100 msSpatial 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 s 0.6 s 0.8 s 1.0 s 1.2 s

1.4 s 1.6 s 1.8 s 2.0 s 2.2 s


MRI results


MRI results

Resolution: z, 5mm; y, 2.5mm

z

y

0.4 s 0.6 s 0.8 s 1.0 s 1.2 s

1.4 s 1.6 s 1.8 s 2.0 s 2.2 s


MRI results

z

y

0.4 s 0.5 s 0.6 s 0.7 s 0.8 s

Flow Mixing Region


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 2.5mmObject 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 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.

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.

AcknowledgementsShoujun Xu

Louis Bouchard

Pines Group

Alex PinesChristian HiltyJosef GranwehrSabieh AnwarElad HarelAlyse JacobsonAll current ‘nuts

Budker Group

Dmitry BudkerSimon RochesterValeriy YashchukJames HigbieDerek KimballJason StalnakerMisha Babalas

Physics Department Machine ShopChemistry Department Electronic ShopChemistry Department Glass Shop

Good Books: 1) Principles of Magnetic Resonance Imaging, Dwight G. Nishimura, Stanford University2) 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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