Nuclear magnetic resonance

Lecture 21

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 NMRBloch, 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:

4He, 14N, 16O.

hh h

h

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: µ= γSStrictly speaking, in QM this is the operator relation, µand are operators of magnetic moment and spin.S

is the gyromagnetic ratioγ

Nuclear Spin - Energy in magnetic fieldProjection 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.

S2 = S2z +S2x +S2y = S(S+1)h2

sh,(s!1)h, ...(!s+1)h,!sh

s=12h,sz =±1

2h

which is always larger than

< S2z >! S2h2

Magnetic moment /Spin interacts with magnetic field.

Energy of interaction is

Corresponding term in the Hamiltonian is

E =!!µ·!B

H ! = "!µ· !B

leading to correction to the energy of the state

For a particle like a proton with s=1/2 there are two possible energy values

.

ΔE =!µB0 =!mγB0 ℏ with m=-j,-j+1,...j.

Eup =!12hγB0 Edown =

12hγB0

γ

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 time-independent 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.

Eup!Edown = hγB0The frequency of the photon with energy hω0

equal to difference of these energies is ω0 = γB0

This is a resonance condition, and ω0is the Larmor (angular frequency)

ω0 = γB0 - very important formula which significance will become clear later

For proton, for B0 = 1Tesla,ω0 = 42.6MHz

ΔE = 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:

N-/N+ = e-ΔE/kT where:

ΔE is the energy difference between the spin states

k is Boltzmann's constant (1.3805x10-23 J/Kelvin) T is the temperature in Kelvin.

At physiologic temperature approximately only ~3 in 106 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 eV/oK

B0(Tesla) Excess spin

0.15 495

0.35 11551.0 32951.5 4945

4.0 13,200

Bo

Alignment in an Applied Magnetic Field

The stronger the field, the larger the net magnetization and the bigger the MR signal !!!!

This is a dynamical equilibrium where individual protons

have nearly random orientation.

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.

!M0 =ns∑i=1

!µi,

where ns is the number of protons in the voxel.

d !M0dt

= !M0! γ!B

Operator !M0 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.

< !M02! >

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

!J

!µ= γ!J

!B= (0,0,B0)!J and!µ

The potential energy E is

E =!!µ·!B=!µB0 cosθ=!γJB0 cosθ

E is minimal if are parallel.

In difference from QM the z projection of J can take any values between -J and J.

Since the potential energy depends only on z coordinate it is clear that it corresponds to a

torque force acting on the atom.

!B and!µ

mg

M B0

In a Static Field• Gyroscope in a

gravitational field• Proton in a magnetic

field

T=-mg

Analogy - torque for gyroscope and proton.

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 ω0 of this

precession is proportional to B0. For positive γ the precession is

clockwise.

Motion equation: in Classical Mechanics d!Jdt

=!τ

where!τ is the net torque acting on the system;

in our case!τ=!µ!!B. Combining with !µ= γ!Jd!µdt

=!µ! γ!B.

we obtain:

Solution of this eqn is µx(t) = µx(0)cos(ω0t)+µy(0)sin(ω0t)

µy(t) = µy(0)cos(ω0t)!µx(0)sin(ω0t)µz(t) = µz(0)

with ω0 = γB0 which is exactly the same frequency as we

obtained in QM!!!

The constants µx(0), µy(0), µz(0) are values of

components at t=0.

Let µxy(t) = µx(t)+ iµy(t), and µxy(0) = µx(0)+ iµy(0).

Here I use complex variables. i is imaginary unit.

µxy(t) = µxy(0)eiω0t

Hence for positive γ , the transverse component of rotates clockwise about z’=z axis with Larmor frequency. This motion is called precession.

!µ

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.

ω0 !µ

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

Eup!Edown = hγB0If the photon with the resonance energy

ω0 = γB0

is absorbed by the system spin can flip with system being excited to a higher level Eup. For B= 1 T,

ω0 = 43.57 MHz

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)

!B1(t) = B1(cos(ω0t),!sin(ω0t),0)= B1 cos(ω0t)! iB1 sin(ω0t),0) = B1e!iω0t.

where B1 is time independent.

d !Mdt

= !M! γ(!B+ !B1(t).

Denoting as !M 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.

ω0

At the same time !B1 is stationary in the rotating frame.

Hence the motion of relative to !M !B1in the rotating frame is the same as relative to !M !B

in the case we considered before( in the stationary frame). Consequently, processes around with the precession frequency

!M !B1

ω1 = γB1

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 :

!M!M

α

α=Z t

0γB1dτ= γB1τ= ω1t.

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 .

!B1

!B1

Two important flip angles:

• The 90o pulse: brings along y’-axis !M

!M = (0,M0,0)

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.

!M

• The 180o or inverse pulse. is rotated to negative z-axis:

!M

!M = (0,0,!M0)

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 component.

(see also animation in the folder - images_mri/spinmovie.ppt)

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 H2O, -OH, CH3 , ...). As a result spins rotate at slightly

different angular velocity.

Dephasing of magnetization with time following a 90o RF excitation. (a) At t=0, all spins are in phase (phase coherence). (b) After a time T2,

dephasing results in a decrease of the transverse component to 37% of its initial value. (c) Ultimately, the spins are isotropically distributed and there is no net magnetization left.

t=0 t=T2 t=∞

Dephasing process can be described by a first order decay model:

!Mtr(t) = !Mtr(0)e!t/T2

T2 depends on the tissue: T2 =50ms for fat and 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).

Spin-Lattice Relaxation(b) causes the longitudinal component of the net magnetization to change from M0 cosθ which

M0

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).

Ml(t) =M0 cosαe!t/T1 +M0(1! e!t/T1)

T1 = 100 ms for fat;

T1 = 2000 ms for water

(a) The spin-spin relaxation process for water and fat. After t = T2, the

transverse magnetization has decreased to 37% of its value at t = 0. After t = 5T2, only 0.67% of the initial value remains. (b) The spin-lattice relaxation

process for water and fat. After t = T1, the longitudinal magnetization has

reached 63% of its equilibrium value. After t = 5T1, it has reached 99.3%.

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

Consider 900 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 thevertical axis measures a sine.

This is for stationary frame. In moving frame s(t) = Ae!t/T2

s(t) = Ae!t/T2e!iω0tsx(t) = Ae!t/T2 cos(!ω0t)

sy(t) = Ae!t/T2 sin(!ω0t)

Imaging

The detected signal in the case described above does not carry spacial information. New idea: superimpose linear gradient (x-, y-, z- directions) 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

!G= (Gx,Gy,Gz) =!0,0,

∂Bz∂z

"

Gradients on millitesla/meter are used.ω(z) = γ(B0+Gzz)

Hence a slice/ slab of thickness Δz

contains a well defined range of processing frequencies

Δω= γGZΔz

Let us consider an example of slices perpendicular to the z-axis (though any direction can be used).

Linear gradient field is characterized by

!G= (Gx,Gy,Gz) =!0,0,

∂Bz∂z

"

Gradients on millitesla/meter are used.ω(z) = γ(B0+Gzz)

Hence a slice/ slab of thickness Δzcontains a well defined range of processing frequencies

Δω= γGZΔz

Principle of slice-selection. A narrow-banded RF pulse with bandwidth BW = ∆ω 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.

Δz=BWγGZ

Constrains:

A very thin slice - too few protons - too weak signal. Small Signal /Noise ratio.

Gradient cannot be larger than (10-40) mT/m.

Hard to generate narrow RF pulse.

Position encoding: the !k theorem

After RF pulse there is a transverse component of magnetization in every point in (x,y). To encode position on the slice additional gradient is applied in x,y plane. For simplicity consider x direction only.

In rotating frame this generates rotation of magnetization with a frequency which depends on x:

ω(x) = γGxx

leading to t-depend M: Mtr(x,y,z) =Mtr(x,y,0)e!iγGxxt

Signal in receiver: s(t) =Z ∞

!∞

Z ∞

!∞ρ(x,y)e!iγGxxtdxdy

where ρ(x,y)

kx =γ2πGxt

magnetization density

Define

s(t) = S(kx) =Z ∞

!∞

Z ∞

!∞ρ(x,y)e!i2πkxxdxdy

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 kx increases (b).

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).

S(kx,ky)

Basic pulse sequences

What k range is necessary? Let us consider the object of length X and a measurement involving taking N slices.

When doing Fourier transform we need several waves in the object. Hence kminX<1, or kmin< 1/X. Also we need to

resolve all slices kmax>N/X. So the condition is

kmax > 1/ΔX , kmin < 1/X

Many strategies (>100) for scanning k space. I will discuss few of them briefly.

The Spin Echo Pulse sequence

180o90o

RF

GyGx

Signal(a) (b)

Gz

ky

kx

Figure (b) shows trajectory of k. (a) describes G, RF, Signal

Slice selection gradient is applied Gz together with 90 and 180 degree 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 of s(t) which depends on time, t.

φ(y)

φ(y) = γGyTphis the constant time when Gy is on.Tph

In practical imaging one changes Gy in integer steps:

Gy = mgy 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.

Tss

3D GE image of the brain, shown as a coronal, sagittal and transaxial sequence.

mprage_cor.avi, mprage_sag.avi, mprage_trans.avi,

φ(y,z) = γ(mgyyTph+ngzzTss)

Gx

Gy

Alternate strategy – spiral, much more efficient k-space coverage

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 Φm =Z TE

0γ!G(t) ·!r(t)dt

Φm =!r0Z TE

0γ!G(t)dt+!v0

Z TE

0γ!G(t)tdt+!a0

Z TE

0γ!G(t)

t2

2dt+ ...

!r(t) =!r(0)+d!rdt

(0)t+ ... =!r(0)+!v(0)t+!a(0)t2/2+ ...Expanding in Taylor series

or Φm =!r0!m(0)+!v(0)!m(1)+!a(0)!m(2)+ ...

where !ml =Z TE

0γ!G(t)

tl

l!dt

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 T1-weighted 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 depictedvessels in the neck as signal voids

Magnetic Resonance Angiography.

Main idea: choose the pulses which will put the lowest moments

!ml =Z TE

0γ!G(t)

tl

l!dt to zero: !m0 = 0, !m1 = 0

For example: use (b) instead of (a).

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.

An example of practical sequence

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 flipangles allowed

blood which flowedthrough the 2D

image plane to bedepicted as being

brighter thanstationary 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!!!

(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.

(b)(a)

To visualize the results one often uses maximum intensity projection (MIP)

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.

Willis_mip.ave

mra_sag

mra_mipmra_cor

mra_ax

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 are presented.

http://www.uth.tmc.edu/radiology/publish/mra/55_male.html

Image of tissue surrounding vessel can be manually striped off

Normal runoff MRA in a 30 year old male