Basic Principles of Magnetic Resonance

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Jorge Jovicich [email protected]

Basic Principles of Magnetic Resonance

I) Historical Background

II) An MR experiment

- Overview

- Can we scan the subject?

- The subject goes into the magnet

- Brief RF pulses are applied

- The MR signal

- Summary

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Overview of a MRI procedure

Magnetic field

Tissue protons align with magnetic field

(equilibrium state) RF pulses Protons absorb RF energy (excited state) Relaxation processes

Protons emit RF energy (return to equilibrium state)

NMR signal detection Relaxation processes Repeat Spatial encoding using magnetic field gradients Subject Safety screening

RAW DATA MATRIX Fourier transform IMAGE

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Can we scan the subject?

Safety Issues

• Not everybody can undergo a MR examination

J. Jovicich, MIT

• Screening is mandatory to ensure safety / quality

during the MR examination

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A typical MRI scanner

The main magnetic field:

• is VERY strong ( 3T ~ 30,000 times the earth’s magnetic field) • is A L W A Y S O N !! (superconducting currents) • it’s strength decays ~ 1 / r3_{(r is the distance from the center)}

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Can we scan the subject?

Safety Issues

• Risks:

- Ferromagnetic materials (unsafe)

- will be subject to force / torque in the main magnetic field - risks associated with motion / displacements

- active biomedical implants may not function properly

From: www.symplyphysics.com/flying_objects.html J. Jovicich, MIT

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Can we scan the subject?

Safety Issues

- Non ferromagnetic materials (safe but give image distortions) • Risks:

Sagitall MRI of a normal female subject

With hair rubber band* Without hair rubber band

* The two ends of the rubber band are joined by a ~ 2 mm cooper clamp. J. Jovicich, MIT

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• Risks:

- Non ferromagnetic materials (safe but give image distortions) - Claustrophobia (subject unhappy → image distortions)

Can we scan the subject?

Safety Issues

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• Risks:

- Non ferromagnetic materials (safe but give image distortions)

- Acoustic protection (mandatory)

- Claustrophobia (subject unhappy → image distortions)

- Movement during examination (potential problem for dynamic studies)

Can we scan the subject?

Safety Issues

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Overview of a MRI procedure Subject

Safety screening

Magnetic field

We will introduce the following concepts:

• equilibrium magnetization

• dynamics of the magnetization

• rotating coordinate system

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The subject goes into the magnet...

O H H Water molecules Hydrogen nucleus: magnetic moment r

: uniform static magnetic field

: static macroscopic magnetization

B

r

0

M

r

0

Two energy states:

Low energy state High energy state Anti-parallel Parallel Precession frequency: o = Bo B_o (E₁) (E₀)

∆

E

=

E

₁

−

E

₀

∆

E

=

h

0 Y B_o M_o Single nucleus J. Jovicich, MIT

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The subject goes into the magnet… (continued)

A group of protons

M_o

Net magnetization Mo

Equilibrium magnetization Mo: - M_z aligned with Bo - M_xy = 0 Y B_o M_o

N

₁

N

₀

=

exp

−

h B

0

kT



 



 

N₁ = # of protons in state E₁ N₀ = # of protons in state E₀ k : Boltzmann’s constant T : temperature

Spin states distribution

E₁ E_o Energy states B_o ωo = γ Bo

∆

E

=

h

0 J. Jovicich, MIT

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Reminder of three main steps in MRI

0) Equilibrium (M_o along B_o)

1) RF excitation

(tip M_o away from equil.)

2) Precession of M_xy induces signal (dephasing for a time TE)

3) Return to equilibrium (recovery time TR)

Dynamics of

magnetization

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Dynamics of the magnetization

• Equation of motion of in external

• Classical mechanics formalism

• First, model a single nucleus

• Then, add up for sample

• Finally, consider relaxation (Bloch equations)

M

r

_B

r

_ext

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• Mechanical moment

⇔

angular momentum (spin)

• Magnetic moment

⇔

spin

• Magnetic moment and magnetic field interaction

r

T

=

d

r

L

dt

r

=

L

r

T

r

=

r

×

B

r

ext

Equation of motion for a single nucleus:

r

d r

_{( t )}

dt

=

r

( t )

×

r

B

_{ext ( t )}

B

_ext

Precession of

about

with frequency

r

B

r

ext

=

B

_ext J. Jovicich, MIT

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Equation of motion for the magnetization vector:

• Assuming no interaction between nuclei

M

r

=

r

0

+

r

1

+

r

2

+

...

=

r

i

∑

M

r

d

M

r

_{( t )}

dt

=

r

M

_{( t )}

×

B

r

_{ext ( t)}

B

_ext

Precession of

about

with frequency

r

B

_ext

=

B

_ext

• And since for each nuclei

d r

_{i( t )}

dt

=

r

i (t )

×

r

B

_{ext ( t )} (spin excess)

M

r

J. Jovicich, MIT

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To describe the magnetization we need

to choose a coordinate systems

z x y z = z’ x’ y’

Laboratory coordinate system Rotating coordinate system

ωo = γ Bo

M(t) M’(t)

with: B_ext(t) = B₀+ B₁(t) _{with: B}_eff_{(t) = B}₁_’(t)

B_o B_o

d

M

r

_{( t )}

dt

=

r

M

_{( t )}

×

B

r

_{ext ( t)}

d

M

r

'( t )

dt

=

r

M

'( t )

×

r

B

_{eff ( t )}

On-resonance spins: static Off-resonance spins:

ωo

ωo + ∆ω ∆ω

Example:

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The NMR signal

• We need net transverse magnetization: - precession of M_xy about B₀

- rotating magnetic field

- induces current in a coil: MR signal

M_xy

B_o

• At equilibrium no signal:

- static longitudinal magnetization M_o

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• With an RF pulse on resonance we can rotate M₀

The NMR signal (continuation)

B_o

B₁’

• Dynamics:

• B₁’(t): B’₁ constant and ⊥ to B₀

•M’ precesses about B₁ with ω₁= γ B’₁ • Flip angle θ: Rotating frame: ' 1 B t = γ ∆ θ ∆ dM r ' ( t ) dt = r M '( t ) × r B _{eff ( t )} B_eff(t) = B₁’(t)

• Signal relaxation, system goes back to equilibrium M_z → M_o (T₁ relaxation)

M_xy → 0 (signal loss, T₂ & T₂* relaxation) • Relaxation mechanisms give tissue contrast

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The NMR signal (continuation)

Schematic representation: Radio-frequency pulse

(oscillating B₁(t) rotating at ω₀)

NMR signal

(transverse magnetization decay)

‘Free Induction Decay’

T₂* decay

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T₁ or spin-lattice relaxation (longitudinal magnetization)

• Mz defined by spin excess population between two energy states • Mz recovery → transitions between spin states

• transitions → fluctuating transverse field on resonance → molecular motion • exponential recovery (T₁≈ 100 - 3000 ms, longer for higher B_o)

Relaxation mechanisms

dM

_z

dt

=

M

_o

−

M

_z

T

₁ time M_z M_o Mz=0 after 90o RF pulse Mz=Mo at equilibrium TR

Repetition time (TR): time allowed for recovery, defines T₁ contrast

B_o

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T₂ or spin-spin relaxation (transverse magnetization)

• incoherent exchange of energy between spins

• molecular motion → fluctuations in local B_z → resonance frequency variations • dephasing of transverse magnetization → signal decay

• exponential decay (T₂ ≈ 70 - 1000 ms)

Relaxation mechanisms (continued)

2 T M dt dM_xy _xy − = time M_xy M_osinθ spins dephase M_xy NMR signal TE

Echo time (TE): time allowed for dephasing, defines T₂ contrast

M_xy max after 90o RF pulse M_xy =0 at equilibrium J. Jovicich, MIT

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Relaxation mechanisms (continued)

T₂* relaxation

• dephasing of transverse magnetization due to both: - microscopic molecular interactions (T₂)

- spatial variations of the external main field ∆B (tissue/air, tissue/bone interfaces)

• exponential decay (T₂* ≈ 30 - 100 ms, shorter for higher B_o)

time M_xy M_osin T₂ T₂*

1 T

₂

*

=

1 T

₂

+ ∆Β

2

tissue air B = µ µ₀ H Boundary conditions Field distortions (∆B) TE J. Jovicich, MIT

Basic Principles of Magnetic Resonance