Download Biomedical Imaging II

Biomedical Imaging I
Class 9 – Magnetic Resonance Imaging (MRI)
Physical Theory
11/09/05
BMI I FS05 – Class 9 “MRI Physic” Slide 1
Magnetic Resonance in a Nutshell
Hydrogen Nuclei (Protons)
Axis of Angular Momentum
(Spin), Magnetic Moment
BMI I FS05 – Class 9 “MRI Physics” Slide 2
Magnetic Resonance in a Nutshell
Spins PRECESS at
a single frequency
(f0), but incoherently
− they are not in
phase
External Magnetic Field
BMI I FS05 – Class 9 “MRI Physics” Slide 3
Magnetic Resonance in a Nutshell
Irradiating with a
(radio frequency)
field of frequency f0,
causes spins to
precess coherently,
or in phase
BMI I FS05 – Class 9 “MRI Physics” Slide 4
Magnetic Field I
magnetic field lines
S
By staying in the interior
region of the field, we can
ignore edge effects.
But how do we describe
magnetic fields and field
strengths quantitatively?
N
BMI I FS05 – Class 9 “MRI Physics” Slide 5
Magnetic Field II
S
B
If the charge is crossing
magnetic field lines, it
experiences a force F.
v
F
An electric charge q
moves between the N
and S poles with velocity
v.
q
(Perhaps better to put it
the other way: if the
charge experiences a
force, then a magnetic
field B is present!)
N
BMI I FS05 – Class 9 “MRI Physics” Slide 6
Magnetic Field III
Operationally, magnetic field is
defined in terms of q, v and F,
according to the formula
S
B
F = qvB.
q
Notice, the force is the crossproduct, or vector product of qv
and B. Thus F is  both v and
B.
Recall that ab = |a||b|sin n, where
 is the angle between a and b and
the direction of n is determined by the
right-hand rule.
i
j
k
Alternatively, a  b  a x
ay
az
bx
by
bz
F
v
N
  a y b z  a z b y  i   a z b x  a x b z  j  a x b y  a y b x  k
BMI I FS05 – Class 9 “MRI Physics” Slide 7
Magnetic Field IV
F[N] = q[A-s]v[m-s-1]B
For consistency, units of B must be N-(A-m)-1
1 N-(A-m)-1  1 T (tesla)
If a current of 1 A flows in a direction perpendicular to the
field lines of a 1 T magnetic field, each one-meter length
of moving charges will experience a magnetic force of 1
N
B goes by several different names in physics literature:
Magnetic field
Magnetic induction
Magnetic induction vector
Magnetic flux density
BMI I FS05 – Class 9 “MRI Physics” Slide 8
Magnetic Pole Strength, Magnetic Moment I
qm  |F|/|B| = magnetic pole
strength.
S
Units are N/N-(A-m)-1 = A-m
B
F1

θ

F1 and F2 are a force couple,
and as such exert a net
torque on the bar magnet
 = ×F = ×qmB = qm ×B
m  qm = magnetic
moment, or magnetic dipole
moment [A-m2].
F2
N
So, 
m×B
=
BMI I FS05 – Class 9 “MRI Physics” Slide 9
Magnetic Pole Strength, Magnetic Moment II
Recall also that, in general, 
dL/dt
S
F1
B

θ

F2
N
[N-m]
=
L = angular momentum [kg-m2-s-1]
(Analogy to Newton’s second law: F [N] =
dp/dt, where p [kg-m-s-1] = linear
momentum)
Definition of magnetic moment as
product of distance and pole strength is
analogous to electric dipole moment
definition (i.e., product of separated
charge and distance). But it is somewhat
fictitious, given that there are no
magnetic monopoles.
Note that we could define m without invoking the intermediate concept of
magnetic pole strength: m  (|F|/|B|).
BMI I FS05 – Class 9 “MRI Physics” Slide 10
Magnetic Moment III
S
A loop carrying current I is placed in
a uniform magnetic field.
F2
B

b
I
a
F1
N
|F1| = |F2| = Ia|B|,
a force couple that generates
a net torque:
There is no magnetic force on the
loop segments in which the current
flows || the field lines. There is a
magnetic force on the segments in
which the current is  the field lines.
The magnitude of the force can be
computed from F = qv×B, when we
recall that charge [A-s] times velocity
[m-s-1] equals current [A] times
length [m].
|| = (Ia|B|)(bsin)
distance)
(force
times
= IA|B|sin (A = ab = loop area)
BMI I FS05 – Class 9 “MRI Physics” Slide 11
Magnetic Moment III
A loop carrying current I is placed in
a uniform magnetic field.
S
B
F2
n 


b
I
a
F1
N
|F1| = |F2| = Ia|B|,
a force couple that generates
a net torque:
m = IAn, magnetic moment
For a N-turn coil, m = NIAn
There is no magnetic force on the
loop segments in which the current
flows || the field lines. There is a
magnetic force on the segments in
which the current is  the field lines.
The magnitude of the force can be
computed from F = qv×B, when we
recall that charge [A-s] times velocity
[m-s-1] equals current [A] times
length [m].
|| = (Ia|B|)(bsin)
distance)
(force
times
= IA|B|sin (A = ab = loop area)
 = IAn×B  m×B
BMI I FS05 – Class 9 “MRI Physics” Slide 12
Magnetization: Definition, Relation to Magnetic Moment
A material of volume V [m3] has magnetic moment m [A-m2]
Its magnetization M is its magnetic moment per unit volume:
M  m/V [A-m-1]
BMI I FS05 – Class 9 “MRI Physics” Slide 13
Angular Momentum  Magnetic Moment
Particle of charge q and mass m (do not
confuse with m!), moving at speed |v| in a
circular orbit of radius |r|.
Orbital period: T = 2|r|/|v|
Current: I = q/T = q|v|/(2|r|)
r
Magnetic moment:
|m| = IA = [q|v|/(2|r|)](|r|2)
= ½q|v||r|
Angular momentum:
|L| = m|v||r| = m(½q|v||r|)/(½q)
= 2m|m|/q
(Recall that L = mr×v)
BMI I FS05 – Class 9 “MRI Physics” Slide 14
Magnetogyric Ratio
If a charged particle has non-zero angular momentum, then it also has a
magnetic moment (and vice versa), and m || L.
  |m|/|L| = (½q|v||r|)/(m|v||r|) = q/(2m)
 is called the magnetogyric ratio
 [A-s-kg-1] is inversely proportional to particle’s
mass-to-charge ratio
r
Notice that units of  can be rearranged to:
A-s-kg-1 = A-m-s2-kg-1-m-1-s-1 = (A-m)-N-1-s-1
= s-1-[N-(A-m)-1]-1 = Hz-T-1
Now rotate plane of
current loop, and
place it in a uniform
magnetic field:
m
B0
m precesses about an
axis parallel to field lines,
but with what frequency?
BMI I FS05 – Class 9 “MRI Physics” Slide 15
Magnetogyric Ratio and Precession Frequency
m
B0
 = 2f |B0|
Proportionality constant is
the magnetogyric ratio!
f = |B0|
(Some authors define  so
that  = |B0|; be aware!)
Thus for macroscopic, or classical, cycling currents, precession frequency
is inversely proportional to mass-to-charge ratio.
For quantum mechanical cycling currents (e.g., electrons, protons,
neutrons, many types of atomic nuclei), relationship is more complicated,
but same qualitative trend is seen. Among atomic nuclei, precession
frequency trends downward as atomic number Z increases.
BMI I FS05 – Class 9 “MRI Physics” Slide 16
Angular Momentum (Spin) of Atomic Nuclei I
Every atomic nucleus has a spin quantum number, s
Permissible values of s depend on mass number A.
Odd A: s may be 1/2, 3/2, 5/2, …
Even A: s may be 0, 1, 2, …
The magnitude of the intrinsic angular momentum, or
spin, S that corresponds to a given value of s is |S| =
s  s  1
h
2 , where h is Planck’s constant
The direction of S can not be precisely defined. The
most we can say is that the component of S in a given
direction is equal to ms , where permissible values of ms
are -s, -s+1,…,s-1,s.
BMI I FS05 – Class 9 “MRI Physics” Slide 17
Angular Momentum (Spin) of Atomic Nuclei II
S  s  s  1 , Sz  ms
B0, z
So, if s 
S 
1
2
1
+0.5
3
(e.g., H, H):
3
2
, Sz  
1
2
or +
1
2
1
3
 54.74
  cos1

0
-0.5
B0, z
And if s  1 (e.g., H):
2
S  2 , Sz   ,0, or +
+1
1
0
1  cos1
1
2
 45
 2  90
-1
BMI I FS05 – Class 9 “MRI Physics” Slide 18
Alignment of 1H Nuclei in a Magnetic Field
B0
m
mz
Protons must orient themselves
such that the z-components of their
magnetic moments lie in one of the
two permissible directions
m
mz
What about direction of m?
Correct quantum mechanical description
is that m does not have an orientation,
but is delocalized over all directions that
are consistent with fixed value of mz.
mz
For the purpose of predicting/interpreting
the interaction of m with radiation, we
can think of m as a well-defined vector
rapidly precessing about z-direction.
mz
What is the
precession
frequency?
BMI I FS05 – Class 9 “MRI Physics” Slide 19
Orientational Distribution of 1H Nuclei
B0
m
mz
Protons must orient themselves
such that the z-components of their
magnetic moments lie in one of the
two permissible directions
m
mz
What fraction of nuclei are
in the “up” state and what
fraction are “down”?
The orientation with mz aligned with B0 has lower potential energy, and
is favored (North pole of nuclear magnet facing South pole of external
field).
The fractional population of the favored state increases with increasing
|B0|, and increases with decreasing (absolute) temperature T.

N
Boltzmann distribution: down  e
N up
 h B0
kT
,
k  1.381  10 23 J - K -1
h  6.626  1034 J - s
BMI I FS05 – Class 9 “MRI Physics” Slide 20
Transitions Between Spin States (Orientations) I
QM result: energy difference between the “up” and “down” states of
mz is ΔE0 = h|B0|
As always, frequency of radiation whose quanta (photons) have
precisely that amount of energy is f0 = ΔE0/h
So, f0 = |B0|
Different nuclei have different values of . (Units of  are MHz/T.)
1H:
 = 42.58; 2H:  = 6.53; 3H:  = 45.41
13C:
 = 10.71
31P:
 = 17.25
23Na:
 = 11.27
39K:
 = 1.99
19F:
 = 40.08
BMI I FS05 – Class 9 “MRI Physics” Slide 21
Transitions Between Spin States II
The frequency f0 that corresponds to the energy difference between
the spin states is called the Larmor frequency.
The Larmor frequency f0 is the (apparent) precession frequency for
m about the magnetic field direction.
(In QM, the azimuthal part of the proton’s wave function precesses
at frequency f0, but this is not experimentally observable.)
Three important processes occur:
hf0
hf0
+
Absorption
+


+
Stimulated
emission
2hf0
+
+

+
Spontaneous
emission
(Relaxation)
BMI I FS05 – Class 9 “MRI Physics” Slide 22
Transitions Between Spin States III
hf0
hf0
+
Absorption
+


+
Stimulated
emission
2hf0
+
+

+
Spontaneous
emission
(Relaxation)
The number of 1H nuclei in the low-energy “up” state is slightly
greater than the number in the high-energy “down” state.
Irradiation at the Larmor frequency promotes the small excess of
low-energy nuclei into the high-energy state.
When the nuclei return to the low-energy state, they emit radiation at
the Larmor frequency.
The radiation emitted by the relaxing nuclei is the NMR signal that is
measured and later used to construct MR images.
BMI I FS05 – Class 9 “MRI Physics” Slide 23
Saturation
Suppose the average time required for an excited nucleus to return
to the ground state is long (low relaxation rate, long excited-state
lifetime)
If the external radiation is intense or is kept on for a long time,
ground-state nuclei may be promoted to the excited state faster than
they can return to the ground state.
Eventually, an exact 50/50 distribution of nuclei in the ground and
excited states is reached
At this point the system is saturated. No NMR signal is produced,
because the rates of “up”→“down” and “down”→“up” transitions are
equal.
BMI I FS05 – Class 9 “MRI Physics” Slide 24
Radiation ↔ Rotating Magnetic Field I
S
Static
magnetic field
B0
Imagine that we
replace the EM
field with…
Sinusoidal EM
field
z
N
y
x
BMI I FS05 – Class 9 “MRI Physics” Slide 25
Radiation ↔ Rotating Magnetic Field II
S
B0
…two more magnets,
whose fields are 
B0, that rotate, in
opposite directions, at
the Larmor frequency
N
BMI I FS05 – Class 9 “MRI Physics” Slide 26
Radiation ↔ Rotating Magnetic Field III
Simplified bird’s-eye view of counter-rotating magnetic field vectors
t=0
1/(8f0)
1/(4f0)
3/(8f0)
1/(2f0)
5/(8f0)
3/(4f0)
7/(8f0)
1/f0
So what does resulting B vs. t look like?
This time-dependent field is called B1
BMI I FS05 – Class 9 “MRI Physics” Slide 27
Rotating Reference Frame I
Coordinate system
rotated about z axis
Original (laboratory)
coordinate system
B0
(1-10 T)
z, z’
z
counter-rotating
magnetic fields
 y
y
y’
x’
x
resultant field,
sinusoidally varying
in x direction
x
Instead of a constant rotation
angle , let  = 2f0t = 0t
x’ = ysin + xcos = -ysin0t + xcos0t
y’ = ycos - xsin = ycos0t + xsin0t
BMI I FS05 – Class 9 “MRI Physics” Slide 28
Rotating Reference Frame II
Rotating coordinate system,
observed from laboratory frame
B0
(1-10 T)
z, z’
Rotating coordinate system,
observed from within itself
But what is the
magnitude of B0 in
this reference frame?
 y
B
0
z’
y’
y’
x’
x
x’
This magnetic field, rotating at
2f0, can
be field,
ignored;
These axes are rotating
This
magnetic
B1, its
is
frequency
is too and
high has
to
in the xy plane, with
fixed
in direction
induce magnitude:
transitions ~0.01
between
frequency f0
constant
T
orientational states of the
protons’ magnetic moments
BMI I FS05 – Class 9 “MRI Physics” Slide 29
Excursion: Bloch Equations I
For an individual atomic nucleus, dL/dt = m×B
L – angular momentum, m – magnetic moment, B – magnetic
field
dL/dt = m×B = dm/dt
Summing over all nuclei gives the corresponding equation for the
bulk (macroscopic) magnetization: dM/dt = M×B
i
j
k
M  B  Mx
My
Mz
Bx
By
Bz
  M y Bz  Mz By  i   Mz Bx  M x Bz  j   M x By  M y Bx  k
The net magnetic field B is the vector sum of the static longitudinal
field and the counter-rotating transverse fields. In the laboratory
frame, these sum to: (B1cosω0t + B1cosω0t + 0)i + (B1sinω0t B1sinω0t + 0)j + (0 + 0 + B0)k.
BMI I FS05 – Class 9 “MRI Physics” Slide 30
Excursion: Bloch Equations II
Combining the preceding equations, we have:
dM/dt = [(MyBz – MzBy)i + (MzBx – MxBz)j + (MxBy – MyBx)k],
and Bx = 2|B1|cosω0t, By = 0, Bz = |B0|
So the three components of dM/dt are:
What do
these mean?
dMx/dt = My|B0|,
dMy/dt = (2Mz|B1|cosω0t - Mx|B0|),
dMz/dt = -2My|B1|cosω0t
Then Bloch assumed that there are two relaxation processes (i.e.,
spin-lattice and spin-spin), and that these are first-order, with time
constants T1 and T2. So the final form of the Bloch equations are:
BMI I FS05 – Class 9 “MRI Physics” Slide 31
Excursion: Bloch Equations III
Bloch equations:
dMx/dt = My|B0| - Mx/T2,
dMy/dt = (2Mz|B1|cosω0t - Mx|B0|) – My/T2,
dMz/dt = -2My|B1|cosω0t - (Mz – M0)/T1
These are three coupled ordinary linear differential equations.
Can be solved exactly, if laboriously
Tell us exactly how the magnetization responds to an EM field, of
any duration, strength, and frequency
• The quantity ω0 in the equations can actually be any
frequency (“off-resonance” rotation), doesn’t have to be the
Larmor frequency.
Now we are able to answer question from Slide 29:
What is |B0| in the reference frame rotating at the Larmor
frequency (“on-resonance” rotation)?
BMI I FS05 – Class 9 “MRI Physics” Slide 32
Effective Field I
M = Mxi + Myj + Mzk
dM/dt = (Mx/t)i + Mx(i/t) + (My/t)j + My(j/t) +
(Mz/t)k
+ Mz(k/t)
= [(Mx/t)i + (My/t)j + (Mz/t)k] + [Mx(i/t) +
My(j/t)
+ Mz(k/t)]
i/t = (ω×i)/(2), j/t = (ω×j)/(2), k/t = (ω×k)/(2)
• ω is the angular frequency vector
dM/dt = (dM/dt)fixed = M/t + ω×(Mxi + Myj + Mzk)/(2)
= (M/t)rot + (ω×M)/(2)
As shown previously, (dM/dt)fixed = M×B
So, (M/t)rot = M×B - (ω×M)/(2) = M×B + (M×ω)/(2)
= M×(B + ω/(2))  M×Beff
The apparent, or effective field in a rotating reference frame is
different from that in the laboratory frame
BMI I FS05 – Class 9 “MRI Physics” Slide 33
Effective Field II
The apparent, or effective field in a rotating reference frame is
different from that in the laboratory frame
Starting with a homogeneous static longitudinal field B0, add a
transverse field B1 that rotates in the x-y plane with frequency
f = /(2). In the frame that rotates at frequency f, the effective field
is Beff = B0 + ω/(2) + B1
If  = 0 (f = f0), then the effective longitudinal field is zero!
Beff = B1, the transverse field is all the field there is
Magnetization M precesses about B1 with frequency f1 = |B1|
If the B1 field is present for time tp, then the resulting tip angle is
 = 2 |B1|tp
BMI I FS05 – Class 9 “MRI Physics” Slide 34
Relaxation I
From Slide 31, what are spin-lattice relaxation and spinspin relaxation?
What do time constants T1 and T2 mean?
“Lattice” means the material (i.e., tissue) the 1H nuclei
are embedded in
1H
nuclei are not the only things around that have
magnetic moments
• Other species of nuclei
• Electrons
A 1H magnetic moment can couple (i.e., exchange
energy) with these other moments
BMI I FS05 – Class 9 “MRI Physics” Slide 35
Spin-Lattice Relaxation I
Spin-lattice interactions occur whenever a physical
process causes the magnetic field at a 1H nucleus to
fluctuate
Spin-lattice interactions cause the perturbed distribution
of magnetic moments (i.e., tipped bulk magnetization) to
return to equilibrium more rapidly
Types of spin-lattice interaction
Magnetic dipole-dipole interactions
Electric quadrupole interactions
Chemical shift anisotropy interactions
Scalar-coupling interactions
Spin-rotation interactions
Look ’em up!
What is the T1 time constant associated with these
processes?
BMI I FS05 – Class 9 “MRI Physics” Slide 36
Spin-Lattice Relaxation II
What is the T1 time constant associated with spin-lattice
interactions?
Recall that static
field
direction
defines z, z‫׳‬
B0
At equilibrium, M
point in z‫ ׳‬direction
z‫׳‬
y‫׳‬
x‫׳‬
BMI I FS05 – Class 9 “MRI Physics” Slide 37
Spin-Lattice Relaxation III
What is the T1 time constant associated with spin-lattice
interactions?
B0
Now impose
a
Then
turn the
transverse field
transverse
magnetic field
off
z‫׳‬
y‫׳‬
x‫׳‬
…and
tip
magnetization
towards the
plane
the
x‫׳‬-y‫׳‬
BMI I FS05 – Class 9 “MRI Physics” Slide 38
Spin-Lattice Relaxation IV
What is the T1 time constant associated with spin-lattice
interactions?
In the laboratory frame, M
takes a spiralling path back to
its equilibrium orientation. But
here in the rotating frame, it
simply rotates in the y‫׳‬-z‫׳‬
plane.
B0
z‫׳‬
Mz
M
y‫׳‬
x‫׳‬
The z component of M, Mz, grows back
into its equlibrium value, exponentially:
Mz = |M|(1 - e-t/T1)
BMI I FS05 – Class 9 “MRI Physics” Slide 39
Relaxation II
From Slide 31, what are spin-lattice relaxation and spinspin relaxation?
What do time constants T1 and T2 mean?
A 1H magnetic moment can couple (i.e., exchange
energy with) the magnetic moments of other 1H nuclei in
its vicinity
These are called “spin-spin coupling”
Spin-spin interactions occur when the magnetic field at a
given 1H nucleus fluctuates
Therefore, should the rates of these interaction
depend on temperature? If so, do they increase or
decrease with increasing temperature?
BMI I FS05 – Class 9 “MRI Physics” Slide 40
Spin-Spin Relaxation I
What is the T2 time constant associated with spin-spin
interactions?
B0
z‫׳‬
Mtr
Mz
If there were no spin-spin
coupling,
the
transverse
component of M, Mtr, would
decay to 0 at the same rate
as Mz returns to its original
orientation
M
y‫׳‬
x‫׳‬
What are the effects of
spin-spin coupling?
BMI I FS05 – Class 9 “MRI Physics” Slide 41
Spin-Spin Relaxation II
W hat are the effects of spin-spin coupling?
z‫׳‬
B0
Mz
y‫׳‬
x‫׳‬
Because the magnetic fields
at individual 1H nuclei are not
exactly B0, their Larmor
frequencies are not exactly
f0 .
But the frequency of the
rotating reference frame is
exactly f0. So in this frame M
appears to separate into
many magnetization vectors
the precess about z‫׳‬.
Some of them (f < f0) precess
counterclockwise
(viewed
from above), others (f > f0)
precess clockwise.
BMI I FS05 – Class 9 “MRI Physics” Slide 42
Spin-Spin Relaxation III
W hat are the effects of spin-spin coupling?
B0
z‫׳‬
When M is completely dephased, Mtr is 0, even though
Mz has not yet grown back
completely: Mtr = 0, Mz < |M|
Mz
y‫׳‬
x‫׳‬
Within a short time, M is
completely de-phased. It is
spread out over the entire
cone defined by cosθ = Mz/|M|
This also shows why T2
can not be >T1. It must be
the case that T2  T1. In
practice, usually T2 << T1.
Mtr decreases exponentially,
with time constant T2:
Mtr = Mtr0 e-t/T2
BMI I FS05 – Class 9 “MRI Physics” Slide 43
Relaxation III
In this example, T1 = 0.5 s
In this example, T2 = 0.2 s
BMI I FS05 – Class 9 “MRI Physics” Slide 44
Effect of B0 Field Heterogeneity
What is the common element in spin-spin and spin-lattice
interactions?
They require fluctuations in the strength of the magnetic
field in the immediate environment of a 1H nucleus
If the static B0 field itself is not perfectly uniform, its
spatial heterogeneity accelerates the de-phasing of the
bulk magnetization vector
The net, or apparent, decay rate of the transverse
magnetization is 1/T2*  1/T2 + |B0|.
T2* (“tee-two-star”) has a spin-spin coupling contribution
and a field inhomogeneity contribution
T2* < T2 always, and typically T2* << T2
BMI I FS05 – Class 9 “MRI Physics” Slide 45