Basics of Magnetic Resonance Imaging. Angular Momentum Orbital Angular Momentum Principles of...

Basics of Magnetic Resonance Imaging

Angular Momentum

vrmL

Orbital Angular Momentum

Principles of Medical Imaging – Shung, Smith and Tsui

Angular Momentum

Spin Angular Momentum

Spin is an intrinsic propertyof the nucleons (protons and neutrons) in a nucleus

HOWEVER –

The name doesn’t mean thatspin results from the nucleons rotating about an axis!!!

http://svs.gsfc.nasa.gov/vis/a000000/a001300/a001319/


Spin is quantized – it can only take certain values

Iz mL IIIImI ,...2,1,

Here I is the total spin quantum number of the nucleus. The proton has I = ½. Lz is the angular momentum due to that spin.


To get the total spin of a nucleus we add up (separately) the spins of the protons and neutrons

Only nuclei with an odd number of protons or neutrons will be visible to MRI

We do pairwise addition:

7 protons 8 neutrons

Note: 14N has spin 1.

15N has spin ½ .

Alignment of Spins in a Magnetic Field, B0


The spin angular momentum yields a magnetic moment

zm Lμ

Energy Levels of Spins and B0


E1 = B0

E2 = -B0

Energy Levels and Spin

E = E1 - E2 = 2 B0

= (h/2) B0 for spin-1/2 particles

B0 is the main magnetic field

Energy Level Population and Field Strength

2,000,0001,000,002

999,999 999,995

1,000,005

999,987

1,000,014

B0 = 0n = 0

B0 = 0.5Tn = 3

B0 = 1.5Tn = 10

B0 = 4.0Tn = 27

Spins are distributed according to the Boltzmann distribution

kTElowerupper eNN /)/(

Larmor Frequency

0B


Excitation Energy and Frames of Reference

B0

B1

Beff

x

z

y

x

z

lab frame

y

rotating frame

B0 = main magnetic fieldB1 = applied field (pulse)Beff = vector sum B0+B1

Net Magnetization, M, and the Rotating Frame

B1

MM

y

x

z

y

x

z

B0 0B1 = 0

B0 0B1 0

While B1 0, Mprecesses around B1

Net Magnetization, M, and the Rotating Frame

We turn B1 “on” by a applying radiofrequency (RF) to the sample at the Larmor frequency. This is a resonant absorption of energy.

If we leave B1 on just long enough for M to rotate into the x - y plane, then we have applied a “90 pulse”. In this case, Nupper = Nlower.

If we leave B1 on just long enough for M to rotate along the -z axis, then we have applied a “180 pulse” (inversion). In this case, Nupper = Nlower + M.

Free Induction Decay

What is the effect of applying a 90 pulse?

RF

/2

time

Free Induction Decay

The effect of a 90 pulse is to rotate M into the x - y (transverse) plane. If we place a detection coil (a loop of wire) perpendicular to the transverse place, we will detect an inducedcurrent in the loop as M precesses by (in the lab frame).

Minitial

y

x

z

MfinalB1


Signal Processing of Free Induction Decay


We can characterize the signal by its:AmplitudePhaseFrequency

Fourier Transform

Signal Processing of Free Induction Decay

We see that after a 90 pulse, we get a cosinusoidal signal.To quantitatively describe the signal we

calculate its Fourier transform.

(think: Larmor frequency)


http://www.med.harvard.edu/JPNM/physics/didactics/improc/intro/fourier3.html

Fourier Transform of Time Domain Data

Image Contrast (I)

We can detect the signal from water molecules in the body.

Can we make an image?

Will it be a useful image?

Relaxation Processes

Fortunately for us, the signal we get from water moleculesin the body depends on their local environment.

Spins can interact by exchanging or losing energy (or both).

As in all spectroscopy methods, we put energy into the system and we then detect the emitted energy to learn about the composition of the sample.

We then use some variables to characterize the emission of energy which (indirectly) tell us about the environment ofthe spins.

Image Contrast!

Relaxation Processes (T2)1. Spin-spin relaxation time (T2): when spins interact With each others magnetic field, they can exchange energy (perform a spin flip). They can lose phase coherence, however. Only affects Mxy.

Signal without T2

interaction betweenspins

Signal including T2

interactions betweenspins

http://irm-francophone.com/htm/signal.htm

T2*

T2*???

T2* and T2

T2 is an intrinsic property of the sample. This is what we are interested in to use for contrast generation.

T2* is the time constant of the decay of the free induction decay. It is related to the intrinsic T2 in the following way:

litysusceptibi2

ityinhomogene22

*2

1111

TTTT

depends upon the local magnetic field

sampleonly

T2* and T2

litysusceptibi2

ityinhomogene22

*2

1111

TTTT

Not a randomprocess

randomprocess

Inhomogeneity term - dephasing due to magnet (B0) imperfections depends upon position

Susceptibility term - dephasing due to the interaction of different sample regions with B0

(depends upon position)

/2 pulse

(delay)

Relaxation Processes (T2)

T2*

Effect of Spin Coherence on Signal

x

y

z

x

y

z

http://irm-francophone.com/htm/signal.htm

Reverse

Start

Irreversible versus Reversible http://www.cchem.berkeley.edu/demolab/images/HahnEchoSpinRes.htm

Hahn Spin Echo Pulse Sequence

http://www.chem.queensu.ca/FACILITIES/NMR/nmr/webcourse/t2.htm

http://www.esr.ethz.ch/intro/spinecho.html

Hahn Spin Echo and T2

Echo spacing ,

Signal

We can calculate T2 by changing the echo spacing, , and recording the signal at 2.

2/2)2( TeS

http://spiff.rit.edu/classes/phys273/exponential/exponential.html

Spin-Lattice Relaxation, T1

To look at the behavior of the longitudinal component of M (Mz), we start by putting M along the -z axis and then read it out with a 90 pulse.

)21( 1/0

TTIz eMM

Spin-Lattice Relaxation, T1

Energy levels and Inversion

Equilibrium

Net Magnetization:

After Inversion

= =

Relaxation Processes (T1)

pulse

/2 pulse/2 pulse

long TIshort TI

Image Contrast and T1

In (a) the TI is chosen to null the signal from curve [ii], while the TI in (b) nulls out [i]

http://www.fmrib.ox.ac.uk/~stuart/thesis/chapter_2/section2_4.html

Now Can We Make A Useful Image?

Magnetic Resonance Imaging

Magnetic Resonance Image Formation

What do we need?

1. Ability to image thin slices

2. No projections - image slices with arbitrary orientation

3. Way to control the spatial resolution

4. Way to carry over the spectroscopic contrast mechanismsto imaging.

Magnetic Resonance Image Formation

0B

How can we spatially encode this signal?

)sin( tAsignal


Magnetic Resonance Image Formation- Slice Selection

Magnetic field gradients: B = B(position)

zGBB zeff 0 0BzGz

)( 0 zGB zeff precessional frequency is now a function of position

zGz

)(2

= frequency bandwidthz = slice thickness


MRI: Basic Principles and Applications - Brown, Semelka

How do we excite only a slice of spins?

Fourier Pairs!

time frequency

sinc rect

FT


So far we have a slab of tissue whose spins are excited. The next step is to place a grid over the slab and define pixels.

Magnetic Resonance Image Formation- Phase Encoding

yGBB yeff 0

)(2 0 yGB yeff

Center of magnet

Gyeff < 0

eff > 0


ytGy

Apply gradient for a finite duration = (y)

(the phase of M over each region of the sample dependsupon it’s position)

This is because the gradient makes each spin precess withan angular frequency that depends on it’s position. Forthe duration of the gradient, t, spins move faster or slowerthan 0 depending upon where they are. After the gradientis turned off, all spins again precess at 0.

The phase accumulated during time, t, is:


x’

y’

z’


x’

y’

z’

x’

y’

z’

x’

y’

z’

M after a 90pulse

Gy = 0 Gy < 0 Gy > 0

Turn on the phase encoding gradient

Magnetic Resonance Image FormationFrequency Encoding (Readout)

How do we spatially encode the frequency of the signal?

Can we turn on another gradient?When? And for how long?

Now we need to encode the x-direction...

We apply a gradient while the signal is being acquiredas the spin-echo is being formed.

timeGx

/2

Signal

RF

Therefore, the precessional frequency is a function of position

Magnetic Resonance Image FormationFrequency Encoding (Readout)

Magnetic Resonance Image FormationSpin-echo Pulse Sequence

http://spl.harvard.edu:8000/pages/papers/zientara/fast/fastimaging.html

?(3)

?(2)

? (1)

Phase Encoding:

Each acquisitionis separately encodedwith a different phase.

The sum total of the Nacquisitions is called the ‘k-space’ data.

(1)

MRI: Basic Principles and Applications - Brown, Semelka

The Effect of Frequency Encoding on the Signal (dephasing gradient)(2)

The effect of having the gradient on during the time that the magnetization is moving from the z-axis to the y-axis is to curve the path. In general, one needs to apply a slice refocussing gradient of opposite magnitude after the RF pulse so that the spins are in phase at the end of the pulse. The area of the negative gradient must be one half the area of the slice selection gradient pulse.

x

y

z

Slice-select refocusinggradient

(3)

Structure of MRI Data: k-space

view -128

view -55

view 40

k-space256 x 256 points

row 40

row -55

row -128

A/D, 256 points

kx = frequencyky = phase

http://www.indyrad.iupui.edu/public/lectures/mri/iu_lectures/mri_homepage.htm


What is k-space?

The time-domain signal that we collect from eachspatially encoded spin echo gets put in a matrix.

This data is called k-space data and is a space of spatial frequencies in an image.

To get from spatial frequencies to image spacewe perform a 2-D Fourier Transform of the k-space data.

General intensity level is represented by low spatial frequencies; detail is represented by high spatial freq’s.

low spatial frequencieslow spatial frequencies

high spatial frequencieshigh spatial frequencies

allfrequencies

allfrequencies

http://www.indyrad.iupui.edu/public/lectures/mri/iu_lectures/mri_homepage.htm

FT

http://www.jsdi.or.jp/~fumipon/mri/K-space.htm

k-space data image data


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