G16.4427 Practical MRI 1 - PowerPoint Presentation

Slide1

G16.4427 Practical MRI 1

Magnetization,

Relaxation, Bloch Equation,

Basic Radiofrequency

(RF) Pulse ShapesSlide2

Felix Bloch

“ The magnetic moments of nuclei in normal matter will results in a nuclear paramagnetic polarization upon establishment of equilibrium in a constant magnetic field. It is shown that a radio-frequency (RF) field at right angles to the constant field causes a forced precession of the total polarization around the constant field with decreasing latitude as the

Larmor

frequency approaches adiabatically the frequency of the RF field. Thus there results a component of the nuclear polarization in right angles to both the constant and the RF field and it is shown that under normal laboratory conditions this component can induce observable voltages.”

23rd October 1905 - 10th September 1983

Before emigrating in the US and becoming the first professor of theoretical physics at Stanford, he worked in Europe with Heisenberg, Pauli, Bohr and Fermi.

In 1946 he proposed the Bloch equations

In 1952 he shared the Nobel prize with Purcell

From 1954 to 1955 he served as the first

director-general of CERNSlide3

Nuclear Magnetization

“ The magnetic moments of nuclei in normal matter will results in a nuclear paramagnetic polarization upon establishment of equilibrium in a constant magnetic field...”

Nuclei with an odd number of protons and/or

neutrons have an intrinsic spin

They behave as tiny magnets with the magnetic moment proportional to the spin

The proportionality constant is the gyromagnetic

ratio At equilibrium the nuclei align with the

external magnetic field, but they also precess

around its axis (the torque is perpendicular to

the magnetic moment and so to the spin)Slide4

Larmor Frequency

The frequency of the precession is known as the

Larmor

frequency:

11th July 1857 - 19th May 1942

Sir Joseph Larmor

where

γ

is the

gyromagnetic

ratio, a known constant, unique for each type of atom and

B

0

is the external magnetic field.

Nuclear Isotope

γ/2π

1

H (spin = 1/2)

42.58 [MHz/T]

23

Na (spin = 3/2)

11.26

[MHz/T]

13

C (spin = 1/2)

10.71 [MHz/T]Slide5

RF Excitation

“ ... It is shown that a radio-frequency (RF) field at right angles to the constant field causes a forced precession of the total polarization around the constant field with decreasing latitude as the

Larmor

frequency approaches adiabatically the frequency of the RF field. Thus there results a component of the nuclear polarization in right angles to both the constant and the RF field and it is shown that under normal laboratory conditions this component can induce observable voltages.”Slide6

MR Signal

To obtain an MR signal we need to excite the the spins out of the equilibrium (i.e. we must tip the net magnetization away from the

B

0 direction)We apply an RF pulse B1 in the x-y plane, tuned at the resonance (i.e. Larmor) frequency of the spins

The magnetization will rotate at the Larmor frequency  the nuclei precess in phaseSlide7

Precession

At thermal equilibrium,

M

and B0 will be pointed in the same direction. The torque applied to a dipole momentum μ in the presence of B0 is:

Multiplying both sides by γ and summing over a unit volume:Slide8

The Bloch Equations

The dynamics of nuclear magnetization are described

phenomenologically

by the Bloch equations:The last term accounts for transfer of magnetization by diffusion, with D = molecular self-diffusion coefficientNote that precession does not change the magnitude of M

, whereas both T1 and T2 relaxation effects doIn species with T2 much shorter than T1, magnetization can temporarily decay to zero prior to its regrowth along the longitudinal directionSlide9

Longitudinal Relaxation

The longitudinal component of the magnetization behaves according to:Slide10

Question:

What is

M

z after a 90° pulse?Slide11

Longitudinal Relaxation

The longitudinal component of the magnetization behaves according to:

For a 90° pulse

Mz(0) = 0 soT1 is called the spin-lattice time constant and characterizes the return to equilibrium along the

z-direction (i.e. direction of B0) T1 involves the exchange of energy between the nuclei and the surrounding lattice, due to xy-component field fluctuationsQUESTION: does T1

depends on field strength? If so, how?Slide12

Longitudinal Relaxation

The longitudinal component of the magnetization behaves according to:

For a 90° pulse

Mz(0) = 0 soT1 is called the spin-lattice time constant and characterizes the return to equilibrium along the

z-direction (i.e. direction of B0) T1 involves the exchange of energy between the nuclei and the surrounding lattice, due to xy-component field fluctuationsT1

lengthen with increasing B0 as greater energy exchange is required at higher frequencies to return to thermal equilibriumSlide13

Transverse Relaxation

The transverse component of the magnetization behaves according to:Slide14

Question:

What is

M

xy after a 90° pulse?Slide15

Transverse Relaxation

The transverse component of the magnetization behaves according to:

For a 90° pulse

Mxy(0) = M0 soT2

is called the spin-spin time constant and characterizes the decay of the transverse magnetizationT2 depends on the same xy-component field fluctuation that accounts for T1 relaxation, but also on z-component field fluctuations QUESTION: What’s the result of that?Slide16

Transverse Relaxation

The transverse component of the magnetization behaves according to:

For a 90° pulse

Mxy(0) = M0 soT2

is called the spin-spin time constant and characterizes the decay of the transverse magnetizationT2 depends on the same xy-component field fluctuation that accounts for T1 relaxation, but also on z-component field fluctuations (

T2 ≤ T1)The effect manifests as a loss of phase coherence (

dephasing) of the transverse componentSlide17

Any questions?Slide18

Rotating Frame

A rotating frame is a coordinate system whose transverse plane is rotating clockwise at an angular frequency

ω

In MRI, we commonly visualize the RF excitation using a frame that rotates with B1B1 is stationary in the rotating frameB0 can be ignored if

B1 on resonanceIn the rotating frame the magnetization rotates around B1, as in the laboratory frame it rotates about B0Slide19

Laboratory Frame vs. Rotating FrameSlide20

Magnetization in the Rotating Frame

The rate of change of the magnetization in the rotating reference frame (ignoring relaxation) is:

In the rotating frame, the equation describing the behavior of

M takes the same form as the original Bloch equation but now influenced by an effective field Beff:Slide21

Circularly-Polarized B1 Field

Suppose the carrier frequency of the applied RF field is

ω

rf, then the quadrature field is:

If the excitation is at the Larmor frequency, the rapidly oscillating field is transformed into a much simpler form in the rotating frame:Slide22

General Case Bloch Equation

Let’s assume a left-circularly polarized B

1

field along the x’ axis of the rotating frame:The Bloch equations become:Slide23

Small Tip-Angle Approximation

It is easier to solve the Bloch equation after making the following assumptions:

At equilibrium

Mrot = [0 0 M0] (initial condition)RF pulse is weak leading to a small tip angle θ

< 30°The Bloch equations become:

= 0 Why ?

We also turn off the RF field before observing the evolution of the magnetization

No Off-Resonance Effects



ω

rf

= ω

0

Slide24

Solution of the Bloch Equation

The transverse and longitudinal components are decoupled:

We are usually interested in the transverse component, as it determines the time signal detected:Slide25

Free Induction Decay (FID)

An initial radiofrequency pulse is applied to spinning protons and tips their vector toward the

x-y

planeAmplitude of signal (S0) is proportional to net vector in x-y plane (maximum signal for 90° RF pulse)An exponentially decaying signal is then received in the absence of any gradients. Slide26

Any questions?Slide27

Bloch Simulators

There are many free educational MRI software that allows to simulate the evolution of magnetization for different MRI experiments.

This one can be run directly from the web at:

http://www.drcmr.dk/BlochSimulatorAnd it has a video introduction on YouTube:

http://www.drcmr.dk/blochSlide28

Basic Radiofrequency (RF) Pulse ShapesSlide29

Useful Quantities to Describe RF Pulses

Pulse width (

T

)Indicates the duration of the RF pulseTypically measured in seconds or millisecondsRF bandwidth (∆f)A measure of the frequency content of the pulse

FWHM of the frequency profileSpecified in hertz or kilohertzFlip angle (θ)Describes the nutation angle produced by the pulseMeasured in radians or degreesCalculated by finding the area underneath the envelope of the RF pulseSlide30

RF Envelope

Denoted with

B

1(t) and measured in microteslasRelatively slowly varying function of time, with at most a few zero-crossings per millisecondThe RF pulse played at the transmit coil is a sinusoidal carrier waveform that is modulated (i.e. multiplied) by the RF envelopeThe frequency of the RF carrier is typically set equal to the Larmor frequency ± the frequency offset required for the desired slice locationSlide31

RF Envelope vs. RF Carrier

RF envelope -

B

1(t)

RF carrier

The RF envelope describes the pulse shape, i.e. the magnetic field in the rotating frameSlide32

Specific Absorption Rate (SAR)

RF pulses deposit RF energy that can cause unwanted heating of the patient

This heating is measured by SAR (Watts/Kg)

In the clinical range of field strengths: SAR ∝B02 θ2 ∆fSlide33

Specific Absorption Rate (SAR)

RF pulses deposit RF energy that can cause unwanted heating of the patient

This heating is measured by SAR (Watts/Kg)

In the clinical range of field strengths: SAR ∝B02 θ2 ∆f

The energy involved in tipping the spins is negligible compared to the energy dissipate as heat. As a result:The spatial distribution of SAR is not concentrated near the selected slice, but covers the entire sensitive region of the RF transmit coilSlide34

Rectangular Pulses

A rectangular (or hard) pulse is shaped like a rectangular window function in the time domain

Used when no spatial selection is required

They are played without a concurrent gradientThe bandwidth is broad enough to affect spins with a wide range of resonant frequenciesThe pulse length can be very short

B

1

TSlide35

Properties of Rectangular Pulses

In the small flip angle approximation, the corresponding frequency profile is a SINC

As the first zero-crossing of the SINC is the inverse of the corresponding pulse width, a small

T means that a hard pulse flips spins over a wide bandwidth The flip angle is directly proportional to the amplitude (B1) and the width (

T) of the pulse:θ = γB1T

FWHM = 1.22/TSlide36

Problem

B

1

T

A commercial MR scanner generates a maximum B

1

field of 30 μT. What is the width of a hard pulse that gives a 90° flip angle?Slide37

SINC Pulses

Widely used for selective excitations. Why?

In practice, the

sinc pulse must have finite duration, so it is truncated except for the central lobe and few side lobesAn apodization window is usually applied to smooth the slice profileOn commercial scanners, tailored pulses have replaced many sinc pulses, but they remain popular as they are easy to implement

B

1

(t)

t

t

0

t

0

A

= Peak RF amplitudeSlide38

Mathematical Description

N

L

and NR are the number of zero crossing to the left and right of the central peak, respectivelyIf NL = N

R then the sinc pulse is symmetricWith good approximation, the bandwidth is given by:The dimensionless time-bandwidth product is:Slide39

Effect of Truncating the Sinc

Because

N

L and NR are both finite, the sinc pulse has discontinuity in the first derivative at NL

t0 and NRt0 :This results in ringing at the edge of the slice profile

Can be reduced using an apodization window

For N

L

=

N

R

= N

tSlide40

See you on Thursday!