mri. vector review x y z vector review (2) the dot product the cross product (a scalar) (a vector)...
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Vector Review
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Vector Review (2)
The Dot Product
The Cross Product
)θcos(|| ||zy bababababa zyxx
(a scalar)
)(k
)(j
)(i
yxyx
zxxz
yzzy
abba
baba
bababa
zyx
zyx
bbb
aaa
kji
det (a vector)
)θsin(|| || baba
(a scalar)
MR: Classical Description: Magnetic Moments
NMR is exhibited in atoms with odd # of protons or neutrons.
Spin angular momentum creates a dipole magnetic moment
Spin angular momentum = IIntuitively current, but nuclear spin
operator in quantum mechanics
Planck’s constant / 2
μ
Model proton as a ring of current.
Which atoms have this phenomenon?1H - abundant, largest signal31P23Na
I γ μ
γ= gyromagnetic ratio : the ratio of the dipole moment to angular momentum
MR: Classical Description: Magnetic Fields
Magnetic Fields used in MR:
1) Static main field Bo
2) Radio frequency (RF) field B1
3) Gradient fields Gx, Gy, Gz
μHow do we create and detect these moments?
MR: Classical Description: Magnetic Fields: Bo
1) Static main field Bo
without Bo, spins are randomly oriented.
macroscopically, M 0μ
net magnetizationwith Bo,
a) spins align w/ Bo (polarization)
b) spins exhibit precessional behavior
- a resonance phenomena
MR: Energy of Magnetic Moment
Alignment Convention:
x
y
z
z: longitudinalx,y: transverse
Bo
At equilibrium, μM kMM o
Energy of Magnetic Moment in is equal to the dot product B
ozoz BIγBμ- B-μE
quantum mechanics - quantized states
MR: Energy states of 1H
Hydrogen has two quantized currents,
Bo field creates 2 energy states for Hydrogen where
ozoz BIγBμ- Bμ- Energy of Magnetic Moment in B
21
zI
energy separation o2
γ
21
o
B
Bγ
h
E
resonance frequency fo
MR: Nuclei spin states
There are two populations of nuclei:n+ - called paralleln- - called anti parallel
n+
n-
o2γ
BE hlower energy
higher energy
Which state will nuclei tend to go to? For B= 1.0T
Boltzman distribution: 999993.E/kT
enn
Slightly more will end up in the lower energy state. We call the net difference “aligned spins”. Only a net of 7 in 2*106 protons are aligned for H+ at 1.0 Tesla.(consider 1 million +3 in parallel and 1 million -3 anti-parallel. But...
There is a lot of a water!!!
• 18 g of water is approximately 18 ml and has approximately 2 moles of hydrogen protons
• Consider the protons in 1mm x 1 mm x 1 mm cube.• 2*6.62*1023*1/1000*1/18 = 7.73 x1019 protons/mm3
• If we have 7 excesses protons per 2 million protons, we get .25 million billion protons per cubic millimeter!!!!
Magnetic Resonance: Spins
We refer to these nuclei as spins.
At equilibrium,
- more interesting - What if was not parallel to Bo?
We return to classical physics...
- view each spin as a magnetic dipole (a tiny bar magnet)
oB || M
M
MR: Intro: Classical Physics: Top analogy
Spins in a magnetic field are analogous to a spinning top in a gravitational field.
(gravity - similar to Bo)
r
F
momentumangular dtd
Fr Torque
Top precesses about F
MR:Classical PhysicsView each spin as a magnetic dipole (a tiny bar magnet). Assume we can get dipoles away from B 0 .Classical physics describes the torque of a dipole in a B field as
Torque B
Torque is defined as
)I(
) (
dtd
dtd momentumangular
Bμ)I(
dtd
Multiply both sides by γ Bγμμ
dtd
Now sum over all
Mμ BγMM
dtd
MR: Intro: Classical Physics: Precession
rotates (precesses) about
BγMM
dtd
M
B
Solution to differential equation:
Precessional frequency: γB
is known as the Larmor frequency.Bπ2
γf
TeslaMHz 42.57
π2
γ for 1H
1 Tesla = 104 GaussUsually, Bo = .1 to 3 TeslaSo, at 1 Tesla, fo = 42.57 MHz for 1H
γB or
Other gyromagnetic ratios w/ sensitivity relative to hydrogen
• 13C 10.7MHz/ T, relative sensitivity 0.016
• 31P 17.23 MHz/ T, relative sensitivity 0.066
• 23Na 11.26 MHz/ T, relative sensitivity 0.093
MR: RF Magnetic field
Images & caption: Nishimura, Fig. 3.3
B1 induces rotation of magnetization towards the transverse plane. Strength and duration of B1 can be set for a 90 degree rotation, leaving M entirely in the xy plane.
a) Laboratory frame behavior of M b) Rotating frame behavior of M
MR: RF excitation
By design ,In the rotating frame, the frame rotates about z axis at o radians/sec
o1 B B
x
y
z
1B
M
1) B1 applies torque on M2) M rotates away from z.
(screwdriver analogy)3) Strength and duration of B1 determines torque
This process is referred to as RF excitation.Strength: B1 ~ .1 G
What happens as we leave B1 on?
Bloch Equations – Homogenous Material
It’s important to visualize the components of the vector M
at different times in the sequence.
a) Let us solve the Bloch equation for some interesting cases. In the first case, let’s use an arbitrary M vector, a homogenous material, and consider only the static magnetic field.
b) Ignoring T1 and T2 relaxation, consider the following case.
kBB ˆ0
Bγ
dt
d
MM
Solve
000
det/
B
MMM
kji
M
M
M
dtd zyx
z
y
x
The Solved Bloch Equations
kBB ˆ0
Bγdt
d
MMSolve
0
00
det/
0
0
0
z
xy
yx
zyx
z
y
x
Mdt
d
MBMdt
d
MBMdt
d
B
MMM
kji
M
M
M
dtd
The Solved Bloch Equations
A solution to the series of differential equations is:
Next we allow relaxation.
0
0
0
00
00
100
0)cos()sin(
0)sin()cos(
)(
)(
)(
)(
z
y
x
z
y
x
M
M
M
twtw
twtw
tM
tM
tM
tM
where M0 refers to the initial conditions. M0 refers to the equilibrium magnetization. This solutionshows that the vector M will precess about the B0 field.
MR: DetectionSwitch RF coil to receive mode.
x y
z
Precession of induces EMF in the RF coil. (Faraday’s Law)M
EMF time signal - Lab frame
dt
tMddt
d
))cos((-
- EMF
00
t
Voltage
(free induction decay)
M
for 90 degree excitation
Complex mm is complex.
m =Mx+iMy
Re{m} =Mx Im{m}=My
This notation is convenient:It allows us to represent a two
element vector as a scalar.
Re
Im
m
Mx
My
Transverse Magnetization ComponentThe transverse magnetization relaxes in the Bloch equation
according to
This is a decaying sinusoid. t
Transverse magnetization gives rise to the signal we “readout”.
2T
)ji(
dt
)(d yx MMtM
Solution to this equation is : tiTt eeMtM 02/0)(
MR: Detected signal and Relaxation.Rotating frame
t
Swill precess, but decays.returns to equilibriumM
M
Transverse Component
0Mxy with time constant T2
After 90º, 2T/oxy M)(M tet
MR: Intro: Relaxation: Transverse time constant T2
- spin-spin relaxation
2
xyxy T
MM
dt
d2T/
oxy M)(M tet
T2 values: < 1 ms to 250 ms
What is T2 relaxation?
- z component of field from neighboring dipoles affects the resonant frequencies.- spread in resonant frequency (dephasing) happens on the microscopic level.- low frequency fluctuations create frequency broadening.
Image Contrast:Longer T2’s are brighter in T2-weighted imaging
MR: Relaxation: Some sample tissue time constants: T2
Table: Nishimura, Table 4.2
T2 of some normal tissue types
Tissue T2 (ms)
gray matter 100
white matter 92
muscle 47
fat 85
kidney 58
liver 43
MR: RF Magnetic field
The RF Magnetic Field, also known as the B1 field
To excite equilibrium nuclei ,apply rotating field at o in x-y plane. (transverse plane)
kMM o
Image & caption: Nishimura, Fig. 3.2
B1 radiofrequency field tuned to Larmor frequency and applied in transverse (xy) plane induces nutation (at Larmor frequency) of magnetization vector as it tips away from the z-axis. - lab frame of reference
Bloch Equation Solution: Longitudinal Magnetization Component
1T
k)(
dt
d ozz MMM
The greater the difference from equilibrium, the faster the change
Solution:
)1()( 11 /0
/0 TtTtzz eMeMtM
Initial Mz Return to Equilibrium
pulse RFdegree 90a for 00 zM
Solution: Longitudinal Magnetization Component
initial conditions
equilibrium
Example: What happens with a 180° RF flip?
t
Mo
-Mo
Effect of T1 on relaxation- 180° flip angle
)1()( 11 /0
/0 TtTtzz eMeMtM
)21()(
)1()0()(
)0(
1
11
/0
/0
/
00
Ttz
TtTtzz
zz
eMtM
eMetMtM
MtMM
MR Relaxation: Longitudinal time constant T1
Relaxation is complicated.T1 is known as the spin-lattice, or longitudinal time constant.
1
ozz T
)MM(M
dt
d)1(M)(M 1T/
oztet
T1 values: 100 to 2000 ms
Mechanism:- fluctuating fields with neighbors (dipole interaction)- stimulates energy exchange
n- n+
- energy exchange at resonant frequency.Image Contrast:
- Long T1’s are dark in T1-weighted images- Shorter T1’s are brighter
Is |M| constant?
MR Relaxation: More about T2 and T1
T2 is largely independent of Bo
Solids- immobile spins- low frequency interactions- rapid T2 decay: T2 < 1 ms
Distilled water- mobile spins- slow T2 decay: ~3 s- ice : T2~10 s
T1 processes contribute to T2, but not vice versa.
T1 processes need to be on the order of a period of the resonant frequency.
12 TT
MR: Relaxation: Some sample tissue time constants - T1
Image, caption: Nishimura, Fig. 4.2
fat
liver
kidney
Approximate T1 values as a function of Bo
white matter
gray matter muscle
MR: Detected signal and relaxation after 90 degree RF puls.Rotating frame
t
Swill precess, but decays.returns to equilibriumM
M
Transverse Component
0Mxy with time constant T2
After 90º, 2T/oxy M)(M tet
Longitudinal Component
Mz returns to Mo with time constant T1
After 90º, )1(M)(M 1T/oz
tet
Putting it all together: The Bloch equation
Sums of the phenomena
12 T
k)(
T
)ji( Bγ
dt
d ozyx MMMMM
M
transverse magnetization
precession,RF excitation
longitudinal magnetization
Changes the direction of , but not the length.M
These change the length of only, not the direction.M
B
includes Bo, B1, and G
Now we will talk about affect of G
MR: Intro: So far...What we can do so far:
1) Excite spins using RF field at o
2) Record FID time signal3) Mxy decays, Mz grows4) Repeat.
More about relaxation...
Proton vs. Electron Resonance
Here g is same as
http://hyperphysics.phy-astr.gsu.edu/hbase/nuclear/nmr.html#c1
B = Bohr Magneton N = Nuclear Magneton
Particle SpinLarmor/B
s-1T-1/B
Electron 1/2 1.7608 x 1011 28.025 GHz/T
Proton 1/2 2.6753 x 108 42.5781 MHz/T
Deuteron 1 0.4107 x 108 6.5357 MHz/T
Neutron 1/2 1.8326 x 108 29.1667 MHz/T23Na 3/2 0.7076 x 108 11.2618 MHz/T
31P 1/2 1.0829 x 108 17.2349 MHz/T14N 1 0.1935 x 108 3.08 MHz/T13C 1/2 0.6729 x 108 10.71 MHz/T19F 1/2 2.518 x 108 40.08 MHz/T
http://hyperphysics.phy-astr.gsu.edu/hbase/nuclear/nmr.html#c1