Basic Principles of magnetism

The nuclei of many kinds of atoms, commonly hydrogen, are tiny magnets. In the earth’s magnetic field they line up to some extent just as you walk around. When you walk past a piece of iron they’ll flop around in different directions. Think of us as having microscopic compass needles precessing (spinning on their axes like gyroscopes) in an orderly direction. To make an MR image, this tendency of the nuclei to line up in the direction of a magnetic field can be manipulated and measured. Since the nuclei from different regions of the body can be made to precess at different frequencies (their magneto-resonance frequencies), the electromagnetic energy at these frequencies yields signals that are location dependent. Computer images can be calculated, enhanced, and displayed. MRI is safe because only a very tiny amount of energy is absorbed or emitted, corresponding to the amount of energy in radio waves, to which we are constantly exposed. MRI does not affect any chemical processes. It doesn’t change molecules at all. The atomic nuclei within the molecules just report what is happening.

In the presence of a static magnetic field (B₀), the atomic nuclei possess energy which differs depending on their orientation( ΔE). ΔE determines the strength of the signal and is related to the resonance frequencies (ν), by Planck’s constant (h). (Planck's constant h = 6.626x10^-34 J.s ).

Δ_E  = hv     (1)

The size of ΔE and v depend on the size of the static magnetic field, (B₀), and the magnetogyric ratio, (ϒ), a characteristic of each kind of atomic nucleus, as shown in equation 2. This is the Larmor (Joseph Larmor, 1857 - 1942) relationship. The Larmor equation (2) is fundamental to all of nuclear magnetic resonance (NMR) and its subfield, MRI.

v = (ϒ/2π )B₀    (2) ( Larmor equation: Frequency rate of precession is proportional to the strength of magnetic field -  ಎ=  ϒ* B )

Together these two equations determine that

 ΔE =  hv = h (ϒ/2π )B₀    (3) ( This equation means that, the more the strength of the magnet, the higher the RF and the better the contrast and the resolution of the MRI data).

Basic Principles of MRI

The hydrogen (1^H) atom inside body possess “spin”
• In the absence of external magnetic field, the spin directions of all atoms are random and cancel each other.
• When placed in an external magnetic field, the spins align with the external field.
• By applying an rotating magnetic field in the direction orthogonal to the static field, the spins can be pulled away from the z-axis with an angle \alpha
• The bulk magnetization vector rotates around z at the Larmor frequency (precess)
• The precession relaxes gradually, with the xy-component reduces in time, z-component increases
• The xy component of the magnetization vector produces a voltage signal, which is the NMR signal we measure

What is spin?

• Spin is a fundamental property of nature like electrical charge or mass. Spin comes in multiples of 1/2 and can be + or -. Protons, electrons, and neutrons possess spin. Individual unpaired electrons, protons, and neutrons each possesses a spin of ½ or - ½.
• Two or more particles with spins having opposite signs can pair up to eliminate the observable manifestations of spin.
• In nuclear magnetic resonance, it is unpaired nuclear spins that are of importance.

Nuclear Spin

• A nucleus consists of protons and neutrons • When the total number of protons and neutrons (=mass number A) is odd or the total number of protons is odd, a nucleus has an angular momentum (\phi) and hence spin – Ex. Hydrogen (1^H) (1 proton), 13^C • The spin of a nucleus generates a magnetic filed, which has a magnetic moment (\mu) • The spin causes the nucleus behave like a tiny magnet with a north and south pole Angular momentum vs magnetic moment

Nuclear spin system

• Collection of identical nuclei in a given sample of material (also known as spin packet, a voxel in the imaged volume)
• In the absence of external magnetic field, the spin orientations of the nuclei are random and cancel each other
• When placed in a magnetic field, the microscopic spins tend to align with the external field, producing a net bulk magnetization aligned with the external field.

Nuclear Magnetization

• Put sample in external magnetic field: B₀=B₀ž
• Spins align in one of two directions:

-54⁰ of ž "up"  (Low energy state)
-180-54⁰ off ž "down" (high energy state)
• Slight preference for "up" direction: N-/N+ = e-E/kT
• Sample becomes magnetized.
• Magnetization vector:

Precession

 

Spins PRECESS at a single frequency (w₀), but incoherently − they are not in phase, so that the sum of x-y components is 0, with net magnetization vector in z direction W0=\gamma B_0: Larmor freq. Equilibrium value: M₀ : same direction as B₀ and depends on x=(x,y,z) only. Magnitude: M₀

How to make spins in phase?

Irradiating with a rotating magnetic field B_1 of frequency w₀, causes spins to precess coherently, or in phase, generating a xy-component

Process involved in MRI

• Put patient in a static field B_0 (much stronger than the earth’s field)
• (step 1) Wait until the nuclear magnetization reaches an equilibrium (align with B_0)
• Applying a rotating magnetic field B_1 (much weaker than B_0) to bring M to an initial angle \alpha with B_0 (rotating freq=Larmor freq.)
• M(t) precess around B_0 at Larmor frequency around B_0 axis (z direction) with angle \alpha
• The component in z increases in time (longitudinal relaxation) with time constant T1
• The component in x-y plane reduces in time (transverse relaxation) with time constant T2
• Measure the transverse component at a certain time after the excitation (NMR signal)
• Go back to step 1
• By using different excitation pulse sequences, the signal amplitude can reflect mainly the proton density, T1 or T2 at a given voxel

 

Evolution of magnetization when a time varying magnetic field is applied

• M=M(x,t)
• Relation to bulk angular momentum J : M=  J
• Focus on small sample → voxel
  - M=M(t)
  - Equations of motion = Bloch equations
• M(t) experiences a torque when an external magnetic field B(t) is applied
  torque is   = MxB
• Torque is related to angular momentum
  =dJ/dt
• Eliminate J to yield
  dM(t)/dt =  M(t) x B(t)
• Valid for "short" times
  Using the right hand rule, M will rotate around z if M is not aligned with z
   

  	i     j    k
  MxB =	Mx  My  Mz
  	Bx  By  Bz
  	= (MyBz-MzBy ) i + (MzBx-MxBz  ) j+ ( MxBy-MyBx )k

Direction of MxB follows “right hand” rule

Solution under a Static Field with an Initial Angle

• B(t)=[0,0,B_0]
• MxB = M_y B_0 i - M_x B_0 j + 0 k
• dM_x/dt = M_y B_0
• dM_y/dt = - M_x B_0

Precession Due to a Static Field with an Initial Angle

Let B(t) = B0; M(0) angel α  with ź Then Mx(t) = M0 sin α cos (- B0t + Ø ) My(t) = M0 sin α sin (- B0t + Ø ) Mz(t) = M0 cos α where M0 = |M(0)| Ø  arbitrary Precession with Larmor frequency  ಎ0 =  B0  or  v0 =  B0 This is the frequency of the photon which would cause a transition between the two energy levels of the spin. B0=1.5T, \gamma=42.58 MHz/T, v0=63.9 MHz

Longitudinal and Transverse Components

M(t) = (M_x(t), M_y(t), M_z(t))
Think of M(t) with two components:
Longitudinal magnetization - M_z(t) - No change
Transverse magnetization - M_xy(t) = M_x(t) + _jM_y(t) - Rapidly rotating

NMR Signal

 

• The rapidly rotating transverse magnetization (M_xy) creates a radio frequency excitation within the sample. • If we put a coil of wire outside the sample, the RF excitation will induce a voltage signal. • In MRI, we measure this voltage signal. • Voltage produced is (Faraday’s Law of Induction)
	Br(r) is field produced at r by unit direct current in coil around sample.

 

Source of MR Contrast

• Different tissues vary in T1, T2 and PD (proton density)
• The pulse sequence parameters can be designed so that the captured signal magnitude is mainly influenced by one of these parameters
• Pulse sequence parameters
– Tip angle \alpha
– Echo time TE
– Pulse repetition time TR

 

Simplification

• B^r(r)=B^r
• Longitudinal magnetization changes too slow
• Transverse magnetization dominates
• Final expression:
  |V| = ಎ₀V_sM₀ sin ∝B^r  
  Recall  ಎ₀ = ϒB₀, M_o = ( B₀ ϒ²h² /4kT ). P_D
  Therefore |V| ∝ B₀², P_D
   

 

How to tilt M to an initial angle?

• Applying a circularly polarized (rotating) magnetic field B_1(t) in the x-y plane with the same Larmor frequency forces the magnetization vector to tilt down to the x-y plane – B_1(t) has two orthogonal components, in x and y directions respectively, and is produced by using quadrature RF coil – Simplest envelop B_1,e is a rectangular pulse • Motion of M(t) is spiral

 

Circulatory Polarized Magnetic Field

 

Tip Angle

• If M is parallel to z-axis before the RF excitation pulse, the tip angle after the excitation (with duration \tau_p) is: 
• If B_1^e(t) is rectangular  ∝ = ϒB₁^p
• Pulse that leads to \alpha=\pi/2 is called “\pi over 2 pulse”, which
elicits the largest transverse component M_xy, and hence largest
NMR signal
• Pulse that leads to \alpha=\pi is called “\pi pulse” or inverse pulse,
which is used to induce spin echo (later)
• The excitation pulse (envelop of B_1(t)) is also called “an alpha
pulse”

 

Relaxation

• Magnetization cannot precess forever
• Two independent relaxation processes
• Transverse relaxation -  ≡spin-spin relaxation
• Longitudinal relaxation - ≡spin-lattice relaxation
• Detailed prosperities differ in tissues - Gives rise to tissue contrast.

 

Longitudinal Relaxation

 

• The magnetization vectors tend to return to equilibrium state (parallel to B_0)		= M_0 cos\alpha = 0 for \pi/2 pulse	In the laboratory frame, M takes a spiraling path back to its equilibrium orientation. But here in the rotating frame, it simply rotates in the y ׳-z ׳ plane. The z component of M, Mz, grows back into its equilibrium value, exponentially: Mz = |M|(1 - e-t/T1)

 

Transverse Relaxation

• The strength of the magnetic field in the immediate environment of a 1H nucleus is not homogeneous due to presence of other nucleus (and their interactions) • Hence the Larmor frequencies of nearby nuclides are slightly different (some spins faster, some slower) – Spin-spin interactions • This causes dephasing of the xy components of the magnetization vectors, leading to exponential decay of M_ • T_2 is called transverse relaxation time, which is the time for M_xy to decrease by 1/e. • Also called spin-spin relaxation time • T2 is much smaller than T1 – For tissue in body, T2: 25-250ms, T1: 250-2500 ms

Free Induction Decay

 

• The voltage signal (NMR signal) produced by decaying M_xy also decays • This is called free induction decay (FID), and is the signal we measure in MRI

 

T2 Star Decay

• Received signal actually decays faster than T_2 (having a shorter relaxation time T_2^*) • Caused by fixed spatial variation of the static field B_0 due to imperfection of the magnet – Accelerates the dephasing of magnetization vectors – Note that T2 is caused by spatial variation of the static field due to interactions of nearby spins • The initial decay rate is governed by T_2^* , but the later decay by T_2.

Formation of spin echo

• By applying a 180 degree pulse, the dephased spins can recover their coherence, and form an echo signal

 

RF Pulse Sequence and Corresponding NMR Signal

 

Spin echo sequence

• Multiple π pulses create “Carr-Purcell-Meiboom-Gill (CPMG)” sequence
• Echo Magnitude Decays with time constant T2

Bloch Equations

Typical Brain Tissue Parameters

	Proton Density	T2 (ms)	T1 (ms)
White matter	0.61	67	510
Gray matter	0.69	77	760
CSF	1.00	280	2650
Weighting TR TE T1 Short Short T2 Long Long PD Long  Short
	• Signal at equilibrium proportional to PD   • Long TR: – Minimizes effects of different degrees of saturation (T1 contrast) – Maximizes signal (all return to equilibrium) • Short TE: – Minimizes T2 contrast – Maximizes signal	• Long TR: – Minimizes influence of different T1 • Long TE: – Maximizes T2 contrast – Relatively poor SNR	• Short TR: – Maximizes T1 contrast due to different degrees of saturation – If TR too long, tissues with different T1 all return equilibrium already • Short TE: – Minimizes T2 influence, maximizes signal

 

Summary: Process Involved in MRI

• Put patient in a static field B_0 in z-direction
• (step 1) Wait until the bulk magnetization reaches an equilibrium (align with B_0)
• Apply a rotating field (alpha pulse) in the xy plane to bring M to an initial angle \alpha with B_0. Typically \alpha=\pi/2
• M(t) precesses around B_0 (z direction) at Larmor freq. with angle \alpha
• The component in z increases in time (longitudinal relaxation) with time constant T1
• The component in x-y plane reduces in time (transverse relaxation) with time constant T2
• Apply \pi pulse to induce echo to bring transverse components in phase to increase signal strength
• Measure the transverse component at different times (NMR signal), to deduce T1 or T2
• Go back to step 1
• By using different excitation pulse sequences (differing in TE, TR, \alpha), the signal amplitude can reflect mainly the proton density, T1 or T2 at a given voxel

 

Transforming the acquired MRI data to images

 

MRI task is to acquire k-space image then transform to a spatial-domain image. kx is sampled (read out) in real time to give N samples. ky is adjusted before each readout. MRI image is the magnitude of the Fourier transform of the k-space image.

Gradient Echo Imaging

•Signal is generated by magnetic field refocusing mechanism only (the use of negative and positive gradient) •Signal intensity is governed by                       S = So e-TE/T2* •Can be used to measure T2* value of the tissue •R2* = R2 + R2ih +R2ph  (R2=1/T2) •Used in 3D and BOLD fMRI

 

Annotations

1. The Larmor frequency f0 ( Center frequency is the frequency rate at which protons spin (precess) with just static magnetic field) for a 1.5 T magnet is 1.5(T)*42.56(MHz T-1) = 63.8 MHz, for 3T= 128 MHz, for 7T = 300 MHz and for 11.7T = 500 MHz. The gyromagnetic ratio here is for ¹H
2. One gauss is defined as one maxwell per square centimeter; it equals 1×10−4 tesla.
3. Typical values
   10⁻⁹–10⁻⁸ gauss: the human brain magnetic field
   0.31–0.58 gauss: the Earth's magnetic field on its surface
   25 gauss: the Earth's magnetic field in its core
   50 gauss: a typical refrigerator magnet
   100 gauss: a small iron magnet
   2000 gauss: a small neodymium-iron-boron (NIB) magnet
   15,000-30,000 gauss: a medical magnetic resonance imaging electromagnet
   10¹²–10¹³ gauss: the surface of a neutron star
   4×10¹³ gauss: the quantum electrodynamic threshold
   10¹⁵ gauss: the magnetic field of some newly created magnetars
   10¹⁷ gauss: the upper limit to neutron star magnetism, no known object in the universe can generate a stronger magnetic field
4. The relation between FOV and gradient direction: FOV is defined as bandwidth divided by the gradient times and gyromagnetic ration
5. Bandwidth is defined as follows: BW = 1/ ΔT,
6.  Δk = 1/FOV where  k is the spatial frequency of k-space and Δk are cycles per distance
    

The gyromagnetic ratio is dependent upon the atoms:

Nucleus	γ / 106 rad s−1 T−1	γ/2π / MHz T−1
1H	267.513	42.576
2H	41.065	6.536
3He	-203.789	-32.434
7Li	103.962	16.546
13C	67.262	10.705
14N	19.331	3.077
15N	-27.116	-4.316
17O	-36.264	-5.772
19F	251.662	40.053
23Na	70.761	11.262
31P	108.291	17.235
129Xe	-73.997	-11.777