Basic Concept of MRI

Chun Yuan

Magnetic Moment

• Magnetic dipole and magnetic moment• Nuclei with an odd number of protons or

neutrons have a net magnetic moment (spin)• Most common nuclei

which have magnetic moments are:– 1H, 2H, 7Li, 13C, 19F,

23Na, 31P, and 127I

Electron Proton

Neutron

_

_

++

++

No External Magnetic Field

• In the absence of an external magnetic field• The nuclei align randomly• The nuclei produce no net magnetization

External Magnetic Field (B0)

• The nuclei align in 1 of 2 positions depending on energy state

• Low energy nuclei align with the field in parallel position

• High energy nuclei alignagainst the field in antiparallel position

B0

Increasing B0

• As B0 increases more nuclei align in the parallel low energy position

B0

Net Magnetization Vector

• A net magnetization vector is formed– Pairs of parallel and antiparallel nuclei cancel– The magnetic moments of the unpaired nuclei

create a sum effect called net magnetization vector

– Only the unpaired nuclei participate in the MR signal

B0

Net Magnetization Vector

• The net vector is the sum of all of the parallel, unpaired, low energy protons– The strength is the SUM of the magnetic strengths of

the individual protons– The direction is the SUM of

the polar directions of the individual protons

– In the low energy state the net vector aligns along the longitudinal or Z axis and is called Mz

B0 Mz

Precession in B0

• They wobble like gyroscope– Thermal agitation prevents the nuclei from aligning

perfectly with B0 so the nuclei actually align at an angle– As B0 attempts to pull the nuclei into perfect alignment

the conflicting forces cause the nuclei to precessB0

The Larmor Equation

• The larmor equation calculates the frequency of precession– Precessional frequency depends on

• The type of nucleus• The strength of the external magnetic field

= wgBo

Omega or PrecessionalFrequency

Gamma orGyromagnetic

Ratio

ExternalMagnetic Field

Strength

Gyromagnetic Ratio g• The gyromagnetic ratio yields frequency at 1 Tesla• The GMR is unique for each type of nucleus

GMR in MHz29.1642.5806.5310.7003.0840.0511.2611.0917.24

Nucleusn1H2H13C14N19F23Na27Al31P

Example

Most MR scanners operate at 1.5 T. What is the Larmor Frequency of protons at this field strength?

f0 = 0 / 2

= B0 / 2

= (42.58 MHz/T)(1.5 T)= 63.84 MHz

There’s No Signal Yet

• Mz CANNOT BE MEASURED WHEN ALIGNED WITH B0• Mz must be moved away from B0 in order to generate

a signal• How do we move

Mz away from B0?

B0 Mz

RF Excitation

• The frequency of the RF energy must match the frequency of the precessing nuclei in order to transfer energy

• The magetic field exerted by the RF energy is called B1• B1 must be transmitted perpendicular to B0

B0

=

= w gBo

B1

Resonance

• In the presence of B1, low energy nuclei absorb energy and shift to high energy state

B0

B1

MzB0

B1

Shift of the Net Magnetization

• The direction of net vector shifts as the individual nuclei transition to high energy– The RF pulse is labelled according to shift it creates in the net

magnetization– A 90 degree pulse moves the net magnetization 90 degrees– How far does a 180 degree pulse move the net magnetization?– When the net magnetization is in the transverse plane it is called Mxy

B0

B1 Mxy

Flip Angle

• Magnetization is tipped using a radiofrequency pulse – Frequency of RF pulse is ω0

– Magnitude of RF pulse is B1(t)

– Total tip angle is α=γ∫B1(t) dt– α=90º (π/2) maximizes signal– α=180º (π) called an inversion pulse

• M essentially precesses around B1(t) with an instantaneous frequency of = B(t)

x

y

z

M

B1(t)

M0

Example

• An RF field B1exp{-j 0t} is applied to a sample where B1 = 50 milligauss. How long must it be applied to produce a tip of 90º? B1 t = /2

t = /(2B1)

= /(2 (2 x 42.58 MHz/T)(0.05 Gauss x 10-4 T/Gauss))

= 1.17 milliseconds

Relexation• WHEN B1 IS REMOVED THE NUCLEI EMIT ENERGY AND SHIFT BACK TO

LOW ENERGY STATE• THE TRANSITION BACK TO LOW ENERGY STATE IS CALLED RELAXATION• AFTER EMITTING ENERGY THE NUCLEI RETURN TO PARALLEL ALIGNMENT

B0

Mxy

Faraday’s Law of Induction

• 3 CRITERIA MUST BE MET TO GENERATE A SIGNAL– A conductor – A magnetic field– Motion of the magnetic field

in relation to the conductor • IN MR– The RF coil provides the conductor– And Mxy provides the moving

magnetic field because it precesses

x

y

z

M

Mxy

Mz

Antenna

s(t) Mxy(t)

Free Induction Decay (FID)

)sin( 0/

02 tess Tt

• In the 90-FID pulse sequence, net magnetization is rotated down into the XY plane with a 90o pulse.

• The net magnetization vector begins to precess about the +Z axis.

• The magnitude of the vector also decays with time.

Bloch Equation

• The Block equation relate the time evolution of magnetization to – the external magnetic fields, – relaxation times (T1 and T2), – the molecular self-diffusion coefficient (D).

· g is the gyromagnetic ratio– depends on nucleus– For proton /2g p = 42.58 MHz/Tesla

MDT

zMM

T

yMxMBM

dt

Md zyx

2

1

0

2

ˆ)(ˆˆ

Rotation Reference Frame

x y x' y'

z

y

x

z

y

x

M

M

M

tt

tt

M

M

M

100

0cossin

0sincos

00

00

'

'

'

z

tMdt

tMd

dt

tMd

ˆ

)()()('

0

External Magnetic Fields

• Static Magnetic Field B0

• RF Magnetic Field B1

)(0ˆ)('

00

0

whenzMMdt

Md

dt

Md

BMdt

Md

x y x' y'

00001

01

ˆ)cos(sinˆ)cos(ˆ)(

'

))((

BztytxtBBwhere

BMMdt

Md

dt

Md

BtBMdt

Md

rfrfeff

eff

z

T1 Relaxation• T1 relaxation is also known as thermal or spin-lattice relaxation• T1 relaxation involves an energy exchange--excited nuclei release energy

and return to equilibrium • T1 relaxation causes recovery of the net magnetization to the longitudinal

axis

11 //0

1

0 )0(1)(,)()( Tt

zTt

zzz eMeMtM

T

tMM

dt

tdM

Mz

tShort T1 Long T1

M0

63%

ExampleFor a sample with T1 = 1 second, how long after a 180 degree pulsewill the net magnetization be 0?

z

M

z

M

z

M0

z

M

Mz(t) = M0(1 - e -t/T1) + Mz(0) e -t/T1

0 = M0(1 - e -t/T1) - M0 e -t/T1

0 = 1 - 2e -t/T1

t = T1 ln2 = 0.69 seconds

T2 Relaxation• T2 relaxation is also known as thermal or spin-spin relaxation• T2 relaxation involves the loss of phase coherence and is

caused by the local magnetic field• T2 relaxation causes dephasing of the net magnetization in

the transverse plane

2/

2

)0()(,)()( Tt

xyxyxyxy eMtMT

tM

dt

tdM

T2

37%

T2* (Star) Relaxation

• Two factors contribute to the decay of transverse magnetization.– molecular interactions (said to lead to a pure T2 molecular effect)

– variations in Bo (said to lead to an inhomogeneous T2 effectThe combination of these two factors is what actually results in the decay of transverse magnetization.

• The combined time constant is called T2 star and is given the symbol T2*. The relationship between the T2 from molecular processes and that from inhomogeneities in the magnetic field is as follows.

inhoTTT 22*

2 /1/1/1

Relaxation and contrast

• Relaxation time T1, T2 and T2* vary with– Field strength – Temperature– Tissue types– In vitro vs. in vivo– Age

• Fundamentally important for generating contrastAt 1.5T:

Gray matter White Matter CSFT1 (ms) 520 390 2000

T2 (ms) 100 90 300

proton density (relative) 10.5 11 10.8

Images with Different Contrast

ExampleSuppose an degree RF pulse is applied every TR seconds for a long time. What is the steady-state magnitude of Mxy immediately after excitation assuming TR >> T2

Let M(n-) be the magnetization just before the nth RF pulse and M(n+) be the magnetization just after the pulse. Because TR >> T2, we know

Mxy(n-) = 0. Therefore,

Mxy(n+) = Mz(n-) sin and Mz(n+) = Mz(n-) cos

T1 relaxation gives

Mz([n+1]-) = M0(1 - e -TR/T1) + Mz(n+) e -TR/T1

...

TR

RF

SolutionAt steady state, M(n-) = M([n+1]-)

Mz(n-) = M0(1 - e -TR/T1) + Mz(n+) e -TR/T1

Mz(n-) = M0(1 - e -TR/T1) + Mz(n-) cos e -TR/T1

Thus,Mz(n-) = M0(1 - e -TR/T1) / (1- cos e -TR/T1)

and

Mxy(n+) = Mz(n-) sin = M0 sin (1 - e -TR/T1) / (1- cos e -TR/T1)

(This equation comes in handy for analyzing MR imaging because images require multiple RF excitations and this equation is useful for optimizing )

Spin EchoThe basic MRI sequence is called “spin echo”. The RFexcitation for spin echo is as follows:

Sketch its response, where TE is on the order of several times T2*

We know we get an FID in response to the 90 degree pulse:

But, what does the 180 degree pulse do?

90º 180º

TE/2

RF

T2*

Spin EchoRecall dephasing gives:

After the 180 degree pulse, the faster spins trail the slower ones:

Thus, the spins “rephase”, then dephase again:

(Note: Only dephasing due to T2* can be rephased. T2 relaxation is affected by random processes. Thus, the echo is lower in amplitude than the original FID)

x

y slower

faster

x

y

slower

faster 180º

T2*

90º 180º

T2* T2*

T2

TE

s(t)

RF

Spin Echo

Spin echo signal for =90From previous slide, with =90:

Mxy(0) = M0 (1 - e -TR/T1)

Adding T2 relaxation gives:

Mxy(TE) = M0 (1 - e -TR/T1) e -TE/T2

“protondensity (PD)”

T1

weightingT2

weighting

PD weighting T1 weighting T2 weightingT1 long short longT2 short short long

MR Image Formation

• Magnetic Field Gradient

• Three key concepts in MRI formation:– slice selection– frequency encoding– phase encoding

Slice Selection

• Goal: Excite (Mz -> Mxy) in a well defined slice of tissue

• Application of RF pulse and gradient field– Energy deposition at selective frequencies

Excite this slice only

RF

Gz

Diagram of Slice Selection

Gz

RF bandwidth

slice thickness depends on RF pulse bandwidth

slice thickness depends on Gradient strength

widerBW

widerslice

steepergradient

narrowerslice

B0

BW = (/2)Gz z

Pulse shape

• Slice profile Fourier Transform of the RF pulse shape• Square pulse:

• Better choice: sinc pulse

FT

RF Pulse Slice Profile

FT

RF Pulse Slice Profile

BW ~ 1/DT

Slice Profile and RF Pulse

Example

What duration should an RF pulse be to excite a 1 mm slice of tissue using a gradient strength of 5 Gauss/cm (assume bandwidth (Hz) 1/duration (sec)).

Required bandwidth is

BW = (/2)Gz z

= (42.58 MHz/T)(5 x 10-4 T/cm)(0.1 cm)

= 2.1 kHz

T = 1/BW

= 0.47 msec

Slice Dephasing

Total dephasing roughlyequivalent to half thearea of the gradient

Can be fixed with anegative gradient withhalf the area:

Gz

Phase Encoding• Phase encoding gradient is imposed before acquisition• While the gradient is on the nuclei precess at different frequencies• When the gradient is turned off the nuclei return to precessing at the

same frequency but their phase has been shifted relative to their gradient position

Gp Gp Gp

Phase Encoding Equation

TGkwhere

dyeyxM

dyeyxM

dyeyxMkxS

y

yik

yTGi

yiy

y

2

2

)(

),(

),(

),(),(

Frequency Encoding

Goal: Map Mxy(x,y) within the slice or “image plane”

• Application of gradient field Gx after slice selection– Position along x axis encoded by frequency– applied during data acquisition– Centered at echo

Gf

Frequency Encoding Equation

TGktGkwhere

dxdyeeyxM

dxekxS

dxekxS

dxekxSkkS

fyfx

yikxik

xiky

xtGiy

xiyyx

yx

x

f

,

),(

),(

),(

),(),(

22

2

2

)(

Note: Signal acquired in kx, ky space is a Fourier transform of M(x,y), so image M(x,y) can be reconstructed with inversed Fourier transform.