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 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.

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 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.

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

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

.

ℏ with m=-j,-j+1,...j.

The energy difference between UP and DOWN states depends both on magnetic field strength B and gyromagnetic ratio

TheZeemaneffectforparticleswithspinjInthepresenceofatime-

independentexternalmagneticfieldBofmagnitudeBtheparticlecanoccupytwodifferentenergystates“spinup”and“spindown”

TheenergydifferencebetweenbothstatesisproportionaltoB

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 laterFor proton, for

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/oKB0(Tesla) Excess spin

0.15 495

0.35 11551.0 32951.5 4945

4.0 13,200

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.

where is the number of protons in the voxel.

Operator 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.

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.

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

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 Eup

. For

B= 1 T,

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 is time independent.

Denoting 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 the 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: •The 90

o 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.

•The 180o or inverse pulse. is

rotated to negative z-axis:

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, CH

3 , ...). As a result

spins rotate at slightly different angular velocity.

DephasingofmagnetizationwithtimefollowingaoRFexcitation

aAttallspinsareinphasephasecoherence

bAfteratimeT2dephasingresultsinadecreaseofthetransverse

componenttoofitsinitialvaluecUltimatelythespinsareisotropicallydistributedandthereisnonetmagnetizationleft

t=0 t=T2

t=∞

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

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 whichis 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

aThespin-

spinrelaxationprocessforwaterandfatAftertT2thetransversema

gnetizationhasdecreasedtoofitsvalueattAftertT2o

nlyoftheinitialvalueremainsbThespin-

latticerelaxationprocessforwaterandfatAftertT1thelongitudinal

magnetizationhasreachedofitsequilibriumvalueAftertT1

ithasreached

SchematicoverviewofaNMRexperimentTheRFpulsecreatesanettransversemagnetizationduetoenergyabsorptionandphasecoherenceAftertheRFpulsetwodistinctrelaxationphenomenaensurethatthedynamicthermalequilibriumisreachedagain

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.

aThecoilalongthehorizontalaxismeasuresacosine

bthecoilalongtheverticalaxismeasuresasine

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-, z- directions) magnetic fields onto z-direction main field. Nobel prize 2003. Allows a slice or volume selection.Slicesinanydirectioncanbeselectedby

applyinganappropriatelinearmagneticfieldgradientThisdynamicsequenceshowsthefourcardiacchamberstogetherwiththeheartvalvesinaplaneparalleltothecardiacaxis

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 thicknesscontains a well defined range of processing

frequencies

Principleofslice-selectionAnarrow-bandedRFpulsewithbandwidthBW∆ωisappliedinthepresenceofaslice-selectiongradientThesameprincipleappliestoslab-selectionbutthebandwidthoftheRFpulseisthenmuchlargerSlabsareusedinDimaging

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 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:

leading to t-depend M:

Signal in receiver:

where magnetization density

Define

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.

Whenapositivegradientinthex-

directionisappliedathespatialfrequencykxincrease

sb

Relaxation effects can be included in the expression of S(k). Using inverse FT one can restore the density.

Illustrationofthek-theoremaModulusoftherawdata measuredbytheMRimagingsystemfordisplaypurposesthelogarithmofthemodulusisshownbModulusoftheimageobtainedfromaDInverse Fourier Transform oftherawdataina

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 kmin

X<1, or kmin

<

1/X. Also we need to resolve all slices

kmax

>N/X. So the condition is

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

The Spin Echo Pulse sequence

180o

90o

RF

Gy G

xSignal(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.

is the constant time when Gy is on.

In practical imaging one changes Gy in integer steps:

leading to the ladder in the plot.

aTheDFLASHpulsesequenceisaGE (gradient - echo)sequenceinwhichaspoilergradientisappliedimmediatelyafterthedatacollectioninordertodephasetheremainingtransversemagnetizationbThetrajectoryofthek-vectorobtainedwiththepulseschemeina

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.

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

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

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

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.

GhostingisacharacteristicartifactcausedbyperiodicmotionInthisT-weightedSEimageoftheheartbreathingheartbeatsandpulsatingbloodvesselsyieldghostingandblurringNotethatbloodflowyieldsatotaldephasingandthusdarkbloodvessels

Magnetic Resonance Angiography.

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

to zero:

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

SchematicillustrationofaDFLASHbasedTime Of Flight sequenceFirstorderflowrephasinggradientsareappliedinthevolume-selectionandfrequency-encodingdirectionstopreventthedephasingthatotherwisewouldbecausedbythecorrespondingoriginalgradients

An example of practical sequence

Bonus of the procedure - all stationary components loose coherence and so the signal

from them becomes wicker.

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!!!

aIllustrationoftheMIPalgorithmAprojectionviewofaDdatasetisobtained bytakingthemaximumsignalintensityalongeachrayperpendiculartotheimagebMIPofaDMRAdatasetofpartofthebrain

ba

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

MIPprojectionsofahighresolutionDtime-of-flightacquisitionoftheintracranialarterieswithoutcontrastADimpressionisobtainedbycalculatingMIPsfromsubsequentdirectionsaroundthevasculartreeandquicklydisplayingthemoneaftertheother.

mra_cor

mra_ax

To study a complex flow of blood a contrast with enriched concentration of protons is injected. In the videos contrastenhancedDMRangiographyofthethoracicvesselsaAxialbsagittalccoronalviewanddmaximumintensityprojection.

Image of tissue surrounding vessel can be manually striped off

Normal runoff MRA in a 30 year old male