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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 Angular Momentum
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.
Spin Angular Momentum
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
Principles of Medical Imaging – Shung, Smith and Tsui
The spin angular momentum yields a magnetic moment
zm Lμ
Energy Levels of Spins and B0
Principles of Medical Imaging – Shung, Smith and Tsui
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
Principles of Medical Imaging – Shung, Smith and Tsui
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
Principles of Medical Imaging – Shung, Smith and Tsui
Signal Processing of Free Induction Decay
Principles of Medical Imaging – Shung, Smith and Tsui
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)
Principles of Medical Imaging – Shung, Smith and Tsui
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
Principles of Medical Imaging – Shung, Smith and Tsui
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
Magnetic Resonance Image Formation- Slice Selection
MRI: Basic Principles and Applications - Brown, Semelka
How do we excite only a slice of spins?
Fourier Pairs!
time frequency
sinc rect
FT
Magnetic Resonance Image Formation- Slice Selection
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
Magnetic Resonance Image Formation- Phase Encoding
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:
Magnetic Resonance Image Formation- Phase Encoding
x’
y’
z’
Magnetic Resonance Image Formation- Phase Encoding
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
Structure of MRI Data: k-space
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
Structure of MRI Data: k-space
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