1. Lecture 1: Magnetic Resonance
Review from last lecture:
Static field is Bo
RF field is B1
Magnetization is
Larmor frequency z
 = MHz/T for 1H y x

2. RF Excitation & Detection
- excite spin system out of equilibrium
- B1 cos(ot) field
- Is it not doing anything when cos(ot) = 0?
- See
- Tips M away from Bo
- creates Mxy component
- Mxy precesses in transverse plane.
- RF coil detects passing , i.e. time-varying creates EMF
useful
waste
Sr(t)
Free Induction Decay (FID) t

3. RF Excitation & Detection: Diagramsz y x y . . z x
Excitation: rotating frame y x
To Receiver
Detection: . RF
detection coils z .
To Receiver
Problem:
RF coils are sensitive to transverse component of How do we localize to a spatial location when RF field is sensitive to entire volume?

4. Field diagram Simplified Drawing of Basic Instrumentation.
Body lies on table encompassed by
coils for static field Bo,
gradient fields (two of three shown),
and radiofrequency field B1.
Image, caption: copyright Nishimura, Fig. 3.15

5. Gradient Fields
The gradient field is the last magnetic field we have to discuss.
- key for imaging
- Paul Lauterbur
Gradient coils are designed to create an additional B field that varies linearly across the scanner as shown below when current is driven into the coil. The slope of linear change is known as the gradient field and is directly proportional to the current driven into the coil.
The value of Bz varies in x linearly. x Bz Bo
slope = Gx
Whole Body Scanners:
Gx = 1-4 G/cm
(10-40 mT/m)
Gx can be considered as the magnitude of the gradient field, or, if you
like, as the current level being driven into the coil.

6. Gradient Fields Apply Gx during a FID Frequency is proportional to Bz
slope = Gx Bo x

7. Gradient Fields: Encoding
Signal(t) Sr f t fo
The Fourier transform of the signal gives us the projection of the object. The signal detected by the coil, a function of time (t),has spatial information encoded into it. Bz
so,
slope = Gx Bo x

8. Gradient Fields: Example
Assume a 40 cm wide torso.
Let Gx = 0.5 G/cm
What is the frequency range across the body?

9. Gradient Fields: Example
Assume a 40 cm wide object.
Gx = 0.5 G/cm
At 1 T,
Bandwidth of interest is small compared to MHz.

10. Gradient Fields: Matlab example
sr(t) x
m(x) t
sr(t) t
s(t)
m(x) x

11. Gradient Fields
Gy changes field strength of B field in z direction as a function of y
Gz changes field strength of B field in z direction as a function of z

12. Gradient Fields
We can write this all together by looking at G as a vector. Each element refers to the gradient field present at any time ( current in the coil at that time).

13. 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

14. Review: Cosine Theorem (What is a mixer? )
- Consider a pulse A(t) that is multiplied by cos(ot). This is called modulation .
A(t) is called the envelope function.
o is the carrier frequency.
Mixer
A(t) cos(ot)
A(t)
cos(ot) Sr
Frequency response of RF pulse
o = 2fo fo f

15. Imaging methods Lets consider 2D imaging
Slice excitation or selective excitation
We first spoke of B1(t) as a rectangle, an on/off pulse,
a) Now modulate the carrier cos(ot) by B1(t) where B1(t) is a sinc .
F.T.
B1(t)
Mixer
B1(t) cos(ot)
F.T.
B1(t)
cos(ot) o

16. Lets consider 2D imaging Slice excitation or selective excitation
Imaging methods
Lets consider 2D imaging
Slice excitation or selective excitation
b) Simultaneously, we apply a gradient Gz. z  z 
slope = Gz
F.T.
B1(t) cos(ot) o
Set bandwidth of
is slice thickness.

18. Gradient illustration
The effects of the main magnetic field and the applied slice gradient. In this example, the local magnetic field changes in one-Gauss increments accompanied by a change in the precessional frequency.
L31, slides repeated here: first imaging method, basic procedure, projection reconstruction.
Image, caption: copyright Proruk & Sawyer, GE Medical Systems Applications Guide, Fig. 11

19. Basic Procedure 1) Selectively excite a slice (z)
- time? .4 ms to 4 ms
- thickness? 2 mm to 1 cm
2) Record FID, control Gx and Gy
- time? 1 ms to 50 ms
3) Wait for recovery
- time? 5 ms to 3s
4) Repeat for next measurement.
- measurements? 128 to 512
- in just 1 flip

20. Projection Reconstruction
1) Record FID with constant gradient on.
Signal(t) or sr(t)
Object Sr y t x f fo
2) Above, frequency maps to spatial pos. x with f0 corresponds to x = 0
3) Shift above projection to shift f0 to 0.
4) For projection angle , turn on Gx and Gy at same time. y  x
5) Sequence through all  from 0 to 180
6) Use convolution back projection to create image.

21. Imaging Method #2: Fourier Method
Consider a rectangular, homogenous object. Think of arrows to left as the direction of transverse magnetization in the x,y plane immediately after a 90 degree RF pulse is applied.
We can refer to the magnitude and direction of the transverse magnetization as a complex number.
Our goal is to determine these complex values throughout the object. The magnitude of these values represent the proton density of the object. These magnitude values will form the desired image. We use the variable m to represent this complex value. y x

22. Imaging Method #2: Fourier Methody
Spins will start out in phase. Next,
record FID with constant x gradient on.
Spins will stay in phase in y, but not in x. x Sr
Signal(t) or sr(t) t f fo ky kx
F(u,v) space is known as k-space in MR. Axes are kx, ky. If we consider a z dimension,we have a kz dimension also.

23. Imaging Method #2 2D Fourier Transform (2D FT) - most popular by far
- can’t do this with x-ray
Methodology:
For every measurement, record FID w/ only Gx on. Demodulate sr(t) down to baseband to get s(t) ( Shift from lab to rotating frame)
But prior to FID (the readout)
- also apply Gy for time y
- relative frequency w/ Gy on is (y)= Gyy
After y, relative phase in y direction has changed.
Relative phase=
Phase varies linearly with position in y.
Let’s break this down and look at the signal and magnetization after this y gradient without the Gx gradient.

24. Imaging Method #2: Encoding in y direction.
Break this down: Think of phase as the direction of in the x,y plane. Look at magnetization immediately after gradient pulse on Gy but before any Gx readout
proton
density y y
spins after y seconds of Gy gradient. x x
Gy= 0 Spins in phase after excitation. y
Signal detected by coil will be a constant level, described as. x

25. Imaging Method #2: Projections as Fourier Transforms
Look at the expression for the signal the coil will see at this point

26. Imaging Method #2: Excitation and gradient application.
Think of phase as the direction of in the x,y plane. y
proton
density
spins after y
of Gy gradient.
Z grad x RF
X Grad Gy y
Y Grad ty x

27. Imaging Method #2: Projections as Fourier Transforms (2)
Putting it all together...
After applying the Gy gradient, apply a Gx readout gradient and record the varying sr(t) signal. Demodulate to baseband. This is one experiment.
Perform several experiments by varying Gyy to induce different spatial frequency weightings in y. Each experiment gives the values of the spatial frequency domain for a red line in k-space below. This is the 2D Fourier Transform method of imaging.
Consider the 64 x 8 box to the right. A series of
MR experiments as described above were performed. To simplify
visualization, a 1D FFT in the horizontal direction was done on each experiment before display. The results
are shown on the bottom where each row is a separate experiment with a different Y direction phase weighting. ky
Note on transparency: Ch. 3 G.E. handout kx

28. Imaging Method #2: Projections as Fourier Transforms (2)
Putting it all together...
F.T-1
in x
F.T.-1 in y,
m(x,y)
Note on transparency: Ch. 3 G.E. handout ky
Each red line is an experiment with a different Gy ( or ty). This is the 2D Fourier Transform method of imaging. kx