1. G Practical MRI 1
Magnetization, Relaxation, Bloch Equation, Basic Radiofrequency (RF) Pulse Shapes

2. 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 th 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 CERN

3. 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)

4. Larmor Frequency Sir Joseph Larmor
The frequency of the precession is known as the Larmor frequency:
where γ is the gyromagnetic ratio, a known constant, unique for each type of atom and B0 is the external magnetic field.
Nuclear Isotope
γ/2π
1H (spin = 1/2)
42.58 [MHz/T]
23Na (spin = 3/2)
11.26 [MHz/T]
13C (spin = 1/2)
10.71 [MHz/T]
Sir Joseph Larmor
11th July th May 1942

5. 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.”

6. 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 B0 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 phase

7. 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:

8. 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 coefficient
Note that precession does not change the magnitude of M, whereas both T1 and T2 relaxation effects do
In species with T2 much shorter than T1, magnetization can temporarily decay to zero prior to its regrowth along the longitudinal direction

9. Longitudinal Relaxation
The longitudinal component of the magnetization behaves according to:

10. Question: What is Mz after a 90° pulse?

11. Longitudinal Relaxation
The longitudinal component of the magnetization behaves according to:
For a 90° pulse Mz(0) = 0 so
T1 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 fluctuations
QUESTION: does T1 depends on field strength? If so, how?

12. Longitudinal Relaxation
The longitudinal component of the magnetization behaves according to:
For a 90° pulse Mz(0) = 0 so
T1 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 fluctuations
T1 lengthen with increasing B0 as greater energy exchange is required at higher frequencies to return to thermal equilibrium

13. Transverse Relaxation
The transverse component of the magnetization behaves according to:

14. Question: What is Mxy after a 90° pulse?

15. Transverse Relaxation
The transverse component of the magnetization behaves according to:
For a 90° pulse Mxy(0) = M0 so
T2 is called the spin-spin time constant and characterizes the decay of the transverse magnetization
T2 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?
T2 <= T1

16. Transverse Relaxation
The transverse component of the magnetization behaves according to:
For a 90° pulse Mxy(0) = M0 so
T2 is called the spin-spin time constant and characterizes the decay of the transverse magnetization
T2 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 component

17. Any questions?

18. 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 B1
B1 is stationary in the rotating frame
B0 can be ignored if B1 on resonance
In the rotating frame the magnetization rotates around B1, as in the laboratory frame it rotates about B0

19. Laboratory Frame vs. Rotating Frame

20. 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:

21. Circularly-Polarized B1 Field
Suppose the carrier frequency of the applied RF field is ωrf, 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:

22. General Case Bloch Equation
Let’s assume a left-circularly polarized B1 field along the x’ axis of the rotating frame:
The Bloch equations become:

23. 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:
= Why ?
No Off-Resonance Effects  ωrf = ω0
We also turn off the RF field before observing the evolution of the magnetization

24. 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:

25. Free Induction Decay (FID)
An initial radiofrequency pulse is applied to spinning protons and tips their vector toward the x-y plane
Amplitude 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.

26. Any questions?

27. 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:
And it has a video introduction on YouTube:

28. Basic Radiofrequency (RF) Pulse Shapes

29. Useful Quantities to Describe RF Pulses
Pulse width (T)
Indicates the duration of the RF pulse
Typically measured in seconds or milliseconds
RF bandwidth (∆f)
A measure of the frequency content of the pulse
FWHM of the frequency profile
Specified in hertz or kilohertz
Flip angle (θ)
Describes the nutation angle produced by the pulse
Measured in radians or degrees
Calculated by finding the area underneath the envelope of the RF pulse
Full Width Half Maximum (FWHM)

30. RF Envelope Denoted with B1(t) and measured in microteslas
Relatively slowly varying function of time, with at most a few zero-crossings per millisecond
The RF pulse played at the transmit coil is a sinusoidal carrier waveform that is modulated (i.e. multiplied) by the RF envelope
The frequency of the RF carrier is typically set equal to the Larmor frequency ± the frequency offset required for the desired slice location

31. RF Envelope vs. RF Carrier
RF envelope - B1(t)
RF carrier
The RF envelope describes the pulse shape, i.e. the magnetic field in the rotating frame

32. 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

33. 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 coil

34. 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 gradient
The bandwidth is broad enough to affect spins with a wide range of resonant frequencies
The pulse length can be very short B1 T

35. 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:
FWHM = 1.22/T
θ = γB1T

36. Problem B1 T
A commercial MR scanner generates a maximum B1 field of 30 μT. What is the width of a hard pulse that gives a 90° flip angle?

37. 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 lobes
An apodization window is usually applied to smooth the slice profile
On commercial scanners, tailored pulses have replaced many sinc pulses, but they remain popular as they are easy to implement
B1(t) t t0
A = Peak RF amplitude

38. Mathematical Description
NL and NR are the number of zero crossing to the left and right of the central peak, respectively
If NL = NR then the sinc pulse is symmetric
With good approximation, the bandwidth is given by:
The dimensionless time-bandwidth product is:

39. Effect of Truncating the Sinc
Because NL and NR are both finite, the sinc pulse has discontinuity in the first derivative at NLt0 and NRt0 :
This results in ringing at the edge of the slice profile
Can be reduced using an apodization window
For NL = NR = N t

40. See you on Thursday!