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Nuclear magnetic resonance
Lecture 22
A very brief history
Stern and Gerlach – atomic beam experiments
Isidor Rabi – molecular beam exp.; nuclear magnetic
moments (angular momentum)
Felix Bloch & Edward Purcell – NM resonance
Ernst & Anderson – FT NMR (FT=Fourier transform)
Ernst, Aue, Jeener, et al – 2D FT NMR
Bloch, Purcell and Ernst have been awarded the Nobel Prize
for their work
Lauterbur & Mansfield – NMR imaging - the Nobel
prize 2003 (adding space coordinate)
Actually first body images are due to Raymond Damadian
- who discovered different spin relaxation times for tumors.
In quantum mechanics orbital momentum, L, is
quantized in units of
, so it takes discreet
values of L=(0,1,2...)
where
=h/2π and h
is the Plank constant.
In addition an elementary particle can have internal
orbital motion - spin, S, which also takes discreet
values. It is quantized in half units of . Spin quantum
number J - 0,1/2, 1, 3/2.
Proton, neutron have spin 1/2, while nuclei can have
a wide range of spins. We discussed that most of
the stable nuclei are even-even. In such nuclei
spins of protons as well as spins of neutrons are
oriented in opposite directions resulting in the total
spin equal zero:
4
14
16
Spin 1/2 charged point-like particles have magnetic
moment which can be calculated in the Dirac theory
and (more accurately) in quantum electrodynamics.
Protons and neutron have internal quark structure
leading to modification of the magnetic moment and in
particular to non zero magnetic moment for the
neutron.
Spinning charged particle or charged particle having
orbital motion can be considered as a small magnet
generated by a closed current.
Magnetic fields of two nucleons with spins in opposite
directions cancel:
Hence only nuclei with unpaired nucleons have magnetic properties.
Nuclear magnetic moment is proportional to spin:
Strictly speaking, in QM this is the operator relation,
and
are operators of magnetic moment and spin.
is the gyromagnetic ratio
Nuclear Spin - Energy in magnetic field
Projection of spin to a given direction is also
quantized:
corresponding to magnetic quantum number, m,
changing between -j and j.
For the nucleon
Spin cannot be precisely directed in say z direction,
there is always a bit of wobbling.
which is always larger than
Magnetic moment /Spin interacts with magnetic
field.
Energy of interaction is
Corresponding term in the Hamiltonian is
leading to correction to the energy of the state
ℏ with m=-j,-j+1,...j.
For a particle like a proton with s=1/2 there are two
possible energy values
.
The energy difference between UP and DOWN states
depends both on magnetic field strength B and
gyromagnetic ratio
The Zeeman effect for particles with spin j = 1/2 . In the presence of a timeindependent external magnetic field B of magnitude B0, the particle can
occupy two different energy states, “spin up” ( ) and “spin down” ( ). The
energy difference between both states is proportional to B0.
The frequency of the photon with energy
equal to difference of these energies is
This is a resonance condition, and
is the Larmor (angular frequency)
- very important formula which significance
will become clear later
For proton, for
DE = hfL
•Protons moving from low to high energy state require radiofrequency.
•Protons moving from high to low energy release radiofrequency.
State Population Distribution
Boltzmann statistics provides the population distribution for these
two states:
- + -ΔE/kT
N /N = e
ΔE
where:
is the energy difference between the spin states
k is Boltzmann's constant (1.3805x10
-23
J/Kelvin)
T is the temperature in Kelvin.
6
At physiologic temperature approximately only ~3 in 10 excess protons are in the
low energy state for one Tesla field.
Net Magnetization
Nlower/Nhigher=exp(-ΔE/kT)
Example: take 1 billion protons at room
temperature(37oC) = k=8.62 x 10-5
oK
eV/
B (Tesla)
Excess spin
0
0.15
0.35
1.0
1.5
495
1155
3295
4945
4.0
13,200
Alignment in an Applied Magnetic Field
Bo
This is a
dynamical
equilibrium where
individual protons
have nearly
random
The stronger the field, the
larger the net magnetization and
the bigger the MR signal !!!!
Important theorem: for description of dynamical
equilibrium for a larger collection of protons (sufficiently
large voxel) the expected behavior of a large number of
spin is equivalent to the classical behavior of the net
magnetization vector representing the sum of individual
spins.
where
Operator
is the number of protons in the voxel.
satisfies equation:
Due to symmetry around z axis in the dynamical
equilibrium expectation value of the vector M has only z
component parallel to B. At the same time expectation
value of
is not equal to zero. Hence it is
instructive to consider also classical picture of the
interaction of magnetic dipole with magnetic field.
Classical consideration.
Consider motion of an atom with angular momentum
and associated magnetic moment μ in the external
magnetic field
. The vectors
are
parallel and
The potential energy E is
E is minimal if
are parallel.
Since the potential energy depends only on
z coordinate it is clear that it corresponds to
a torque force acting on the atom.
In difference from QM the z projection of J can take any
values between -J and J.
Analogy - torque for gyroscope and proton.
In a Static Field
• Gyroscope in a
gravitational field
• Proton in a magnetic
field
M
T=-mg
mg
B0
If a particle with angular momentum J and magnetic moment μ is
suspended without friction in an external magnetic field B , a
precession about B occurs. The angular frequency ω of this
0
precession is proportional to B . For positive γ the precession
is
0
clockwise.
Motion equation: in Classical Mechanics
is the net torque acting on the system;
Combining with
we obtain:
Solution of this eqn is
which is exactly the same frequency as we
obtained in QM!!!
The constants
are values of
components at t=0.
Here I use complex variables. i is imaginary unit.
Hence for positive γ , the transverse component
of
rotates clockwise about z’=z axis with
Larmor frequency. This motion is called
precession.
Rotating frame.
Further simplification: use of rotating frame with
coordinate axes x’,y’,z’ that rotate clockwise with
frequency
in which
stands still. In this
frame an effective magnetic field is zero.
Disturbing the dynamic equilibrium: The RF field
We discussed above that if the system is placed in the
field of strength B, the energy splitting of the levels is
given by
If the photon with the resonance energy
is absorbed by the system spin can flip with system
being excited to a higher level E . For B= 1 T,
up
RF wave can be generated by sending alternative
currents in two coils positions along the x- and y-axes
of the coordinate system (in electronics - quadrature
transmitter)
where
Denoting
is time independent.
as net magnetization
To solve this equation we switch to the rotating frame
which we discussed before.
It is the frame which rotates with angular velocity
In this frame the field B does not act on M.
At the same time
is stationary in the rotating frame
Hence the motion of relative to
in the rotating frame is the same as
relative to
in the case we considered before( in the stationary
frame). Consequently,
processes around
with th
precession frequency
At t=0 the effective magnetic field lies along the x’ axis,
and it rotates
away from z=z’ axis along the circle in
z’,y’ plane to the y’ axis. The angle between z-axis
and
is called the flip angle
:
Hence it is possible to rotate M by any flip angle. If the
up-time of the RF field is halved,
should be doubled,
which implies a quadrupling of the delivered power. Due
to electric part of EM field substantial part of it is
transformed to heat which limits the increase of
.
•
Two important flip angles:
o
The 90 pulse: brings
along y’-axis
There is no longitudinal magnetization. Both levels are
occupied with the same probability. If pulse is stopped
when this angle is reached,
in the rest frame will
rotate clockwise in x-y plane.
•
o
The 180 or inverse pulse.
rotated to negative z-axis:
is
QM - the majority of spins occupy the highest energy
level.
The magnetic field interacts independently with
different nuclei, hence the rotations of different nuclei
are coherent. This phenomenon is called phase
coherence. It explains why in nonequilibrium
conditions magnetization vector can have transverse
(see also animation in the folder - images_mri/spinmovie.ppt)
component.
After RF is switched off the process of the return to
the dynamical equilibrium: relaxation starts.
(a) Spin-spin relaxation is the phenomenon which causes the
disappearance of the transverse component of the magnetization
vector. On microscopic level it is due slight variations in the
magnetic field near individual nuclei because of different
chemical environment (protons can belong to H O, -OH, CH ,
2
...). As a result spins rotate at slightly different angular
velocity.3
t=∞
ot=T2
Dephasing of magnetization with time following a 90 RF excitation.
(a) At t=0, all spins are in phase (phase coherence).
(b) After a time T , dephasing results in a decrease of the transverse component to
2
37% of its initial value.
(c) Ultimately, the spins are isotropically distributed and there is no net magnetization
left.
t=0
Dephasing process can be described by a first order
decay model:
depends on the tissue: T =50ms for fat and
2
1500 ms for water. See next slide. Spin -spin
interaction is an entropy phenomenon. The disorder
increases, but there is no change in the energy
(occupancy of two levels does not change).
T
2
(b)
Spin-Lattice Relaxation
causes the longitudinal componen
of the net magnetization to change from
which
is the value of the longitudinal (z) component right
after the RF pulse to
This relaxation is the result of the interaction of the spin
with the surrounding macromolecules (lattice). Process
involves de-excitation of nuclei from a higher energy
level - leading to some heat release (much smaller than
in RF). One can again use the first order model with
spin-lattice relaxation time (see figure in next slide).
100 ms for fat;
2000 ms for water
(a) The spin-spin relaxation process for water and fat. After t = T , the
transverse magnetization has decreased to 37% of its value at t = 20. After t =
5T , only 0.67% of the initial value remains. (b) The spin-lattice relaxation
2
process
for water and fat. After t = T , the longitudinal magnetization has
reached 63% of its equilibrium value.1 After t = 5T , it has reached 99.3%.
1
Schematic overview of a NMR experiment. The RF pulse creates a net
transverse magnetization, due to energy absorption and phase coherence.
After the RF pulse, two distinct relaxation phenomena ensure that the
dynamic (thermal) equilibrium is reached again.
Signal detection and detector
0
Consider 90 pulse. Right after RF each voxel has
a net magnetization vector which rotates clockwise (
in the rest frame). This leads to an induced current in
the antenna (coil) placed around the sample. To
increase signal to noise ratio (SNR) two coils in
quadrature are used.
(a) The coil along the horizontal
axis measures a cosine
(b) the coil along the
vertical axis measures a sine.
This is for stationary frame. In moving frame
Imaging
The detected signal in the case described above does
not carry spacial information.
New idea: superimpose linear gradient (x-, y-, zdirections) magnetic fields onto z-direction main
field. Nobel prize 2003. Allows a slice or volume
selection.
Slices in any direction can be selected by
applying an appropriate linear magnetic
field gradient. This dynamic sequence shows
the four cardiac chambers together with the
heart valves in a plane parallel to
the cardiac axis
Let us consider an example of slices perpendicular
to the z-axis (though any direction can be used).
Linear gradient field is characterized by
Gradients on millitesla/meter are used.
Hence a slice/ slab of thickness
contains a well defined range of processing
frequencies
Let us consider an example of slices perpendicular
to the z-axis (though any direction can be used).
Linear gradient field is characterized by
Gradients on millitesla/meter are used.
Hence a slice/ slab of thickness
contains a well defined range of processing
frequencies
Principle of slice-selection. A narrow-banded RF pulse
with bandwidth BW = ∆ is applied in the presence of a
slice-selection gradient. The same principle applies to
slab-selection, but the bandwidth of the RF pulse is then
much larger. Slabs are used in 3D imaging.
Constrains:
Gradient cannot be larger than (10-40) mT/m.
Hard to generate narrow RF pulse.
A very thin slice - too few protons - too weak signal.
Small Signal /Noise ratio.
Position encoding: the
theorem
After RF pulse there is a transverse component of
magnetization in every point in (x,y). To encode positio
on the slice additional gradient is applied in x,y plane. F
simplicity consider x direction only.
In rotating frame this generates rotation of
magnetization with a frequency which depends on
x:
leading to t-depend M:
Signal in receiver:
where
Define
magnetization density
Can generalize to the case of field depending on x,
y,z dependent gradient. Signal in different moments
t measures FT for different k’s.
When a positive gradient in the x-direction is applied (a), the
spatial frequency k increases (b).
x
Relaxation effects can be included in the expression of
S(k). Using inverse FT one can restore the density.
Illustration of the k-theorem: (a) Modulus of the raw data
measured
by the MR imaging system (for display purposes, the logarithm of the
modulus is shown) (b) Modulus of the image obtained from a 2D Inverse
Fourier Transform of the raw data in (a).
Basic pulse sequences
What k range is necessary? Let us consider the object
of length X and a measurement involving taking N
slices. we need several waves
When doing Fourier transform
in the object. Hence k
X<1, or k
< 1/X. Also we
min
min
need to resolve all slices k
>N/X. So the condition is
max
Many strategies (>100) for scanning k
space. I will discuss few of them briefly.
The Spin Echo Pulse sequence
o
90
k
y
o
180
RF
G
G z
yG
x
Signal
k
x
(a)
(b)
Figure (b) shows trajectory of k. (a) describes G, RF, Signal
Slice selection gradient is applied G together with 90 and 180 degree
z
pulses.
It leads to compensation of dephasing at t=2TE. Gz leads to
dephasing which can be compensated by change of sign of Gz - instead
one extend a bit the second Gz pulse. Ladder represents phase-encoding
gradient Gy
It leads to a y-dependent phase shift
on time, t.
of s(t) which depends
is the constant time when Gy is on.
In practical imaging one changes Gy in integer steps:
leading to the ladder in the plot.
(a) The 2D FLASH pulse sequence is a GE (gradient - echo) sequence in
which a spoiler gradient is applied immediately after the data collection in
order to dephase the remaining transverse magnetization.
(b) The trajectory of the k-vector obtained with the pulse scheme in (a)
In 3D imaging one has to use two phase encoding
gradient ladders:
where
is the on - time of the gradient in the slab
selection direction.
mprage_cor.avi,mprage_sag.avi,mprage_trans.avi,
3D GE image of the brain, shown as a coronal,
sagittal and transaxial sequence.
Alternate strategy – spiral,
much more efficient k-space coverage
G
x
G
y
Imaging of moving spins.
Previous discussion assumed that atoms do not move. Problem in
case of blood, breathing,... as during the pulse atoms moves
relative to the magnetic field and hence the resonance frequency
will change.
The total phase shift at time TE is
Expanding in Taylor series
or
where
This motion induced dephasing is another source of dephasing - it leads to
suppression of signal from blood vessels and to artifacts like ghosting.
Ghosting is a characteristic artifact caused by periodic motion. In this T1weighted SE image of the heart, breathing, heartbeats and pulsating blood
vessels yield ghosting and blurring. Note that blood flow yields a total
dephasing and thus dark blood vessels.
Vessel Signal Voids
Early multi-slice spin echo images depicted
vessels in the neck as signal voids
Magnetic Resonance Angiography.
Main idea: choose the pulses which will put the lowest moments
to zero:
For example: use (b) instead of (a).
An example of practical sequence
Schematic illustration of a 3D FLASH based Time Of Flight sequence. First
order flow rephasing gradients are applied in the volume-selection and
frequency-encoding directions to prevent the dephasing that otherwise would be
caused by the corresponding original gradients.
Bonus of the procedure - all stationary components
loose coherence and so the signal from them becomes
wicker.
Bright Blood Images
Using gradient
(field) echo images
with partial flip
angles allowed
blood which flowed
through the 2D
image plane to be
depicted as being
brighter than
stationary tissue.
MRI allows to study various types of motion:
diffusion 0.01-0.1 mm/s,
perfusion 0.1-1 mm/s,
cerebrospinal fluid flow 1mm/s -1cm/s,
venous flow 1-10 cm/s,
arterial flow 10-100 cm/s,
stenotic flow 1-10 m/s. SIX orders of magnitude!!!
To visualize the results one often uses maximum intensi
projection (MIP)
(a)
(b)
(a) Illustration of the MIP algorithm. A projection view of a 3D data set is
obtained by taking the maximum signal intensity along each ray perpendicular
to the image. (b) MIP of a 3D MRA data set of part of the brain.
MIP projections of a high resolution 3D time-of-flight acquisition of the
intracranial arteries (without contrast). A 3D impression is obtained by
calculating MIPs from subsequent directions around the vascular tree and
quickly displaying them one after the other.
mra_ax
mra_cor
To study a complex
flow of blood a
contrast with
enriched
concentration of
protons is injected.
In the videos
contrast enhanced
3D MR
angiography of the
thoracic vessels.
(a) Axial, (b)
sagittal (c) coronal
view and (d)
maximum intensity
projection.
Normal runoff MRA in a 30 year old male
Image of tissue surrounding vessel can be manually striped off