M. Lustig, EECS UC Berkeley

Principles of MRIEE225E / BIO265

Lecture 05

Instructor: Miki LustigUC Berkeley, EECS

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M. Lustig, EECS UC Berkeley

What is this?

• `

The first NMR spectrum of ethanol 1951.

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M. Lustig, EECS UC Berkeley

Today

• Last time:–Linear systems, Fourier Transforms, Sampling

• Today:–Ch 3. Overview

• Classical description of MRI• Basic Imaging

• Homework Due tonight!

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M. Lustig, EECS UC Berkeley

Spatial Frequency

• Vinyl Record–Transforms a temporal signal to a spatial signal

BioE 265 HW #1

Prof. Steve ConollyDue: Friday 8 Feb 2008

1. Consider an old-fashioned LP gramophone phonograph. These were the preferred media for recording music from about 1910 to the late 1980's. Wikipedia has an interesting entry on this media: http://en.wikipedia.org/wiki/Vinyl_record. Below you can see a close in shot of ~8 grooves on a 78 RPM recording, obtained from an LBNL Report 51983, 26-March-2003 Vitaliy Fadeyev and Carl Haber, publication on bSpace. The transverse undulations are transduced into music. 1a) What is the spatial frequency of a 2 kHz tone at the outside radius (4.75 inches)?1b) What is the spatial frequency of a 2 kHz tone at the inside radius (1.875 inches)?

2. Proofs:2a) Prove the stretch or scaling theorem for 2D Fourier Transforms. That is, show that

F{f(x/a, y/b)} = |ab|F (akx, bky).

2b) Derive the shift theorem for 2D FTs.2c) Derive the convolution theorem for 2D FTs. 2d) Derive the derivative theorem for 1D FTs. That is, find a simple expression for the FT of df(x)/dx in terms of F(k).

3.) Linear Differential Equation with constant coefficients: in class we showed that if h(x) is the solution to a LDE with an impulse input then the solution to a general input, u(x), is given by u(x)*h(x). Re-derive this result but use FT's throughout. You will need to use the derivative theorem from 2d above.

http://offtosognefjord.tumblr.com

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M. Lustig, EECS UC Berkeley

What is the frequency?

(-1,-1)

(1,1)

(1,-1)

(1,1)

(-1,1)

a) fx=1, fy=2b) fx=2, fy=1

c) fx = 4, fy=2d) fx=2, fy=4

cos(2⇡(f

x

x+ f

y

y))

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M. Lustig, EECS UC Berkeley

What is the Temporal Frequency?

(-1,-1)

(1,1)

(1,-1)

(1,1)

(-1,1)

a) cos(2π8t)b) cos(2π8t2)

c) cos(2π4t)d) cos(2π4t2)

Vinyl rotates at 1 Hz

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M. Lustig, EECS UC Berkeley

What is the Temporal Frequency?

(-1,-1)

(1,1)

(1,-1)

(1,1)

(-1,1)

a) cos(2π100t)b) cos(2π100t2)

c) cos(2π40t)d) none of the answers

Vinyl rotates at 1 Hz

Aliasing!

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M. Lustig, EECS UC Berkeley

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M. Lustig, EECS UC Berkeley

Classical Description of MR

• Atoms with odd # of protons/neutrons have nuclear spin angular momentum–Intrinsic QM property (ch. 4)–Also intrinsic magnetic moment

• Like Spinning magnetic dipoles• In biological tissue:

– Mostly 1H in H2O– Sometimes 31P, 13C, 23Na - Exotic

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M. Lustig, EECS UC Berkeley

Classical Description of MR

• Interaction of magnetization with 3 fields– B0 - Main field ⇒ Polarization and resonance

– B1 - RF field ⇒ Signal production + reception

– G - Gradient fields ⇒ Spatial encoding

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M. Lustig, EECS UC Berkeley

B0 - Main Field

• Produces polarization of sample M0

• Resonance at Larmor frequency

• For Protons:

• Others:• 23Na• 13C• 15N

z~M0

B0

! = ��B0

Gyromagnetic Ratio�

2⇡= 4.257 KHz/G

� = 2⇡4.257 Krad/G

�

2⇡= 1.127 KHz/G

�

2⇡= 1.071 KHz/G

�

2⇡= �0.43 KHz/G

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M. Lustig, EECS UC Berkeley

Typical B0

0.1T 4.2MHz Very Low!

0.5T 21MHz Low (permanent/resistive)

1.5T 63MHz “High” Diagnostic (superconducting)

3T 127MHz “High” Diagnostic (superconducting)

4T 170MHz Rare

7/9.4T 300/400 MHz Very High - research only

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M. Lustig, EECS UC Berkeley

B0 Field

• For Spatial/Spectral Localization we require homogeneity

• This is: 64Hz @ 1.5T– Pretty remarkable!

�B0 ⇠ 1ppmover 40cm3 FOV

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M. Lustig, EECS UC Berkeley

Why Resonance

• For a bar magnet–Torque, but no resonance–Missing angular momentum

• Resonance is like a spinning-top

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M. Lustig, EECS UC Berkeley

B1 - RF Field

• Can’t Directly Detect M0

– • Resonance is the key!

– B0 is DC while spins resonate ⇒ Detection!

– Sample resonates at ω0=-γB0

• Excite magnetization off the z direction– Apply RF field at ω0=-γB0 in the x-verse plane– Has to be resonant to do something

Q: Why?A: Huge field!Minduced = µ�1

0 V �B � ⇡ 4 · 10�9

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M. Lustig, EECS UC Berkeley

RF Excitation

• In the lab frame B1(t) = Ay

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M. Lustig, EECS UC Berkeley

RF excitation

• In the lab frame B1(t) = Ae�i!0t

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M. Lustig, EECS UC Berkeley

RF excitation

• In the rotating frame @ω0 B1

(t) = Ayrot

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M. Lustig, EECS UC Berkeley

RF Excitation

• In the rotating frame: Precession about B1

• Typical B1’s: 0.14 - 0.35G ± (10%/20%) accuracy @(1.5T/3T)• Duration 1-3ms which is a long time at 64MHz!• Peak power 20KW!

!0 = �B1 = 4.257 · 0.16 = 0.68 KHz

0.367ms 90� ) 23000 rotations

lab rotating

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M. Lustig, EECS UC Berkeley

Relaxation

• T1 : Longitudal relaxation ~ 10-2000ms• T2 : Transverse relaxation ~ 10-300ms• Main source of contrast (Later!)• T2 < T1 Always!

Time

Tra

nsve

rse

Mxy

e-t/T2

Time

1-e-t/T1

Long

itud

inal

Mz

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M. Lustig, EECS UC Berkeley

RF Reception

• Precession enduces EMF in coil: Faraday’s law ⇒ Free induction decay

V (t)

t

FID

t

After demodulation

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M. Lustig, EECS UC Berkeley

Gradient Fields

• B1 has poor localization λ@64MHz ~ 0.5m in tissue• Instead encode position in frequency

• Small concomitant fields Bx, By are also created. These do not contribute much to precession - fields are NOT oscillating at Larmor freq.

!0 !(x) = !0 + �G

x

x

5-5

~

G =

@Bz

@x

,

@Bz

@y

,

@Bz

@z

�

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M. Lustig, EECS UC Berkeley

Gradient Fields

• Typical #– G = 1-10 G/cm = 10-100 mT/m = 4.2-42 KHz/cm– Waveforms in audio frequency– Slew-Rate = 15-20 G/cm/ms

• Safety concern is in dB/dt• Peripheral nerve stimulation can happen

–Big Amplifiers: 1200 Volts, 200 Amps

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