MRI

1
MRI
2
Vector Review
z
x
y
3
Vector Review (2)

• The Dot Product

(a scalar)
The Cross Product
(a vector)
(a scalar)
4
MR Classical Description Magnetic Moments

• NMR is exhibited in atoms with odd of protons
  or neutrons.

Intuitively current, but nuclear spin operator
in quantum mechanics
Spin angular momentum
Plancks constant / 2?
Spin angular momentum creates a dipole magnetic
moment
gyromagnetic ratio the ratio of the dipole
moment to angular momentum
Which atoms have this phenomenon? 1H -
abundant, largest signal 31P 23Na
Model proton as a ring of current.
5
MR Classical Description Magnetic Fields
How do we create and detect these moments?

• Magnetic Fields used in MR
• 1) Static main field Bo
• 2) Radio frequency (RF) field B1
• 3) Gradient fields Gx, Gy, Gz

6
MR Classical Description Magnetic Fields Bo

• 1) Static main field Bo
• without Bo, spins are randomly oriented.
• macroscopically,

net magnetization
with Bo, a) spins align w/ Bo (polarization)
b) spins exhibit precessional behavior - a
resonance phenomena
7
Reference Frame
z
y
x
8
MR Energy of Magnetic Moment

• Alignment Convention

Bo
x
z longitudinal x,y transverse
z
y
At equilibrium,
Energy of Magnetic Moment in is equal to the
dot product
quantum mechanics - quantized states
9
MR Energy states of 1H
Energy of Magnetic Moment in

• Hydrogen has two quantized currents,
• Bo field creates 2 energy states for Hydrogen
  where

energy separation
resonance frequency fo
10
MR Nuclei spin states

• There are two populations of nuclei
• n - called parallel
• n- - called anti parallel

n-
higher energy
n
lower energy
Which state will nuclei tend to go to? For B 1.0T
Boltzman distribution
Slightly more will end up in the lower energy
state. We call the net difference aligned
spins. Only a net of 7 in 2106 protons are
aligned for H at 1.0 Tesla. (consider 1 million
3 in parallel and 1 million -3 anti-parallel.
But...
11
There is a lot of a water!!!

• 18 g of water is approximately 18 ml and has
  approximately 2 moles of hydrogen protons
• Consider the protons in 1mm x 1 mm x 1 mm cube.
• 26.6210231/10001/18 7.73 x1019 protons/mm3
• If we have 7 excesses protons per 2 million
  protons, we get .25 million billion protons per
  cubic millimeter!!!!

12
Magnetic Resonance Spins

• We refer to these nuclei as spins.
• At equilibrium,
• - more interesting -
• What if was not parallel to Bo?
• We return to classical physics...
• - view each spin as a magnetic dipole (a tiny
  bar magnet)

13
MR Intro Classical Physics Top analogy

• Spins in a magnetic field are analogous to a
  spinning top in a gravitational field.

(gravity - similar to Bo)
Top precesses about
14
MRClassical Physics

• View each spin as a magnetic dipole (a tiny bar
  magnet). Assume we can get dipoles away from B0
  .Classical physics describes the
• torque of a dipole in a B field as

Torque
Torque is defined as
Multiply both sides by
Now sum over all
15
Partial Bloch Equation Describes interaction of
M and B

• Above Portion of the Bloch Equation
• Explains how to change the direction of the
  magnetization vector M with applied magnetic
  fields, B.

16
Bloch Equations Homogenous Material

• Its important to visualize the components of the
  vector M
• at different times in the sequence.

1. Let us solve the Bloch equation for some
   interesting cases. In the first case, lets use
   an arbitrary M vector, a homogenous material,
   and consider only the static magnetic field.
2. Ignoring T1 and T2 relaxation, consider the
   following case.

Solve
17
The Solved Bloch Equations
Solve
18
The Solved Bloch Equations

• A solution to the series of differential
  equations is

where M0 refers to the initial conditions. (M0
refers to the equilibrium magnetization when no
RF has been applied for some time. Some time
would be several T1 relaxation intervals) Here,
19
Try two initial conditions
1.First mimics spins in equilibrium position
along z
2.First mimics spins right after a 90 degree RF
excitation
20
Solution to 1)
1.First mimics spins in equilibrium position
along z
Solution
Conclusion Since M and B start in the same
direction, there cross product is zero. Nothing
will change
21
Solution to 2
1.First mimics spins right after a 90 degree RF
excitation
Conclusion This solution describes a circular
path for the transverse magnetization. M and B
are constantly perpendicular. This drives a
circular motion.
22
Sample Torso Coil
y
z
x
23
MR Classical Physics Precession
Solution to differential equation

• rotates (precesses) about

Precessional frequency
is known as the Larmor frequency.
or
for 1H
Usually, Bo .1 to 3 Tesla So, at 1 Tesla, fo
42.57 MHz for 1H
1 Tesla 104 Gauss
24
MR RF Magnetic field
The RF Magnetic Field, also known as the B1
field To excite nuclei
, apply rotating field at ?o in x-y plane.
(transverse plane)
B1 radiofrequency field tuned to Larmor frequency
and applied in transverse (xy) plane induces
nutation (at Larmor frequency) of magnetization
vector as it tips away from the z-axis. - lab
frame of reference
Image caption Nishimura, Fig. 3.2
25
MR RF Magnetic field (Laboratory and Rotating
Frame)
a) Laboratory frame behavior of M
b) Rotating frame behavior of M

• B1 induces rotation of magnetization towards the
  transverse plane. Strength and duration of B1
  can be set for a 90 degree rotation, leaving M
  entirely in the xy plane.
• See Proton Procession under RF excitation on
  webpage animation

Images caption Nishimura, Fig. 3.3
26
MR RF General Excitation (Rotating Frame)

• By design ,
• In the rotating frame, the frame rotates about z
  axis at ?o radians/sec

z
1) B1 applies torque on M 2) M rotates away from
z. (screwdriver analogy) 3) Strength and
duration of B1 determines flip angle
y
x
This process is referred to as RF
excitation. What determines flip angle?
Typical B1 Strength B1 .1 G. A field of .1 G
for 1 ms (T above) will produce approximately a
90 degree pulse
27
Viewing RF Pulse as a Rotation Matrix about X axis
What happens for
What happens for
28
After a flip of angle a

• Initial conditions
• After flip of angle a

z
a
x
29
MR Detection

• Switch RF coil to receive mode.

z
y
x
M
EMF time signal - Lab frame
Voltage
t
for 90 degree excitation
(free induction decay)
30
Complex m

• m is complex.
• m MxiMy
• Rem Mx ImmMy
• This notation is convenient
• It allows us to represent a two element vector as
  a scalar.

Im
m
My
Re
Mx
31
Transverse Magnetization Component

• The transverse magnetization relaxes in the
  Bloch equation according to

Solution to this equation is
This is a decaying sinusoid.
t
Transverse magnetization gives rise to the
signal we readout.
32
MR Detected signal and Relaxation.

• Rotating frame

will precess, but decays.
S
t
Transverse Component
with time constant T2
After 90º,
33
MR Relaxation Transverse time constant T2

• - spin-spin relaxation

T2 values lt 1 ms to 250 ms What is T2
relaxation? - z component of field from
neighboring dipoles affects the resonant
frequencies. - spread in resonant frequency
(dephasing) happens on the microscopic
level. - low frequency fluctuations create
frequency broadening. Image Contrast Longer
T2s are brighter in T2-weighted imaging, darker
in T1- weighted imaging
34
MR Relaxation Some sample tissue time
constants T2
T2 of some normal tissue types
Tissue T2 (ms)
gray matter 100
white matter 92
muscle 47
fat 85
kidney 58
liver 43

• Table Nishimura, Table 4.2

35
Bloch Equation Solution Longitudinal
Magnetization Relaxation Component
The greater the difference from equilibrium, the
faster the change
Solution
Return to Equilibrium
Initial Mz Doesnt have to be 0!
36
Solution Longitudinal Magnetization Component
equilibrium
initial conditions
Example What happens with a 180 RF flip?
Effect of T1 on relaxation - 180 flip angle
Mo
t
-Mo
37
T1 Relaxation
38
MR Relaxation Longitudinal time constant T1

• Relaxation is complicated.
• T1 is known as the spin-lattice, or longitudinal
  time constant.

T1 values 100 to 2000 ms Mechanism -
fluctuating fields with neighbors (dipole
interaction) - stimulates energy exchange n-
n - energy exchange at resonant
frequency. Image Contrast - Long T1s are dark
in T1-weighted images - Shorter T1s are
brighter Is M constant?
39
MR Relaxation More about T2 and T1

• T2 is largely independent of Bo
• Solids
• - immobile spins
• - low frequency interactions
• - rapid T2 decay T2 lt 1 ms
• Distilled water
• - mobile spins
• - slow T2 decay 3 s
• - ice T210 ?s
• T1 processes contribute to T2, but not vice
  versa.
• T1 processes need to be on the order of a period
  of the resonant frequency.

40
MR Relaxation Some sample tissue time constants
- T1
Approximate T1 values as a function of Bo
gray matter
muscle
white matter
kidney
liver
fat

• Image, caption Nishimura, Fig. 4.2

41
Components of M after Excitation
Laboratory Frame
42
MR Detected signal and relaxation after 90
degree RF puls.

• Rotating frame

will precess, but decays. returns to equilibrium
S
t
Transverse Component
with time constant T2
After 90º,
Longitudinal Component
Mz returns to Mo with time constant T1
After 90º,
43
MR Contrast Mechanisms
T2-Weighted Coronal Brain
T1-Weighted Coronal Brain
44
Putting it all together The Bloch equation

• Sums of the phenomena

precession, RF excitation
transverse magnetization
longitudinal magnetization
Changes the direction of , but not the
length.
These change the length of only, not the
direction.
includes Bo, B1, and
Now we will talk about affect of
45
MR Polarization and Excitation

• What we can do so far
• 1) Excite spins using RF field at ?o
• 2) Record FID time signal
• 3) Mxy decays, Mz grows
• 4) Repeat.
• Now, we will work to understand spatial encoding
  of the signal