Description of Magnetic Resonance Imaging (continued)

 

Detection (I): RF tips the magnetization in the transverse plane

 

The B1 component of a perpendicular RF pulse at the Larmor frequency applied for a π/2 duration tips the longitudinal magnetization Mz by 90° and converts it to a transverse magnetization Mxy

 

Following the growth of the net magnetization Mo, an electromagnetic frequency, and specifically a radiofrequency pulse, with a B1 oscillating magnetic field component is applied perpendicularly to the main magnetic field Bo. The radiofrequency is chosen to be equal to the precession/Larmor frequency of the spins. Alternatively, the pulsing frequency can be equal to the Larmor frequency. Initially, the net magnetization Mo is aligned with Bo but following the torque exerted by B1, it will be tipped out of alignment. Only when the frequency is identical to that of the precession can Mo and B1 remain locked together in an appropriate relation for tipping and energy exchange.

 

We can determine the effect of the torque of the changing magnetic field that is applied perpendicularly as mentioned in this reference by initiating our analysis with the relationships:

 

B1(x) = B1 cos(ωt) and B1(y) = - B1 sin(ωt)

 

It is proven that the magnetic field will tip the magnetization (flip the spins) for a specific angle which depends on the duration of the pulse.

 

This angle is termed flip/tip angle (φ) and is given by φ = ω*T = γ*B*T where T is the duration of the pulse.

 

As mentioned in the above reference, if we wish to flip the spin by 90°(π/2) with a perpendicular pulse when it is given that B1 = 0.2 G = 2*10-5 T, we have to apply the pulse for a duration of 294 μs.

 

The above pulse is termed a 90° pulse. Due to this the initial magnetisation Mo (Mz) is converted to a transverse magnetization Mxy. The transverse magnetization Mxy will precess at the Larmor frequency. Note that as a significant alignment of the spins had been mediated previously upon the effect of the main magnetic field Bo (z axis), when the spins will be tipped in the xy plane, they will be precessing in a synchronized manner demonstrating phase coherence. In other words, the polarization that had been mediated previously by the Bo is sustained upon the transition to the xy plane.

 

 

 

Detection (II): Signal emission during transverse magnetization decay

 

Following the end of the RF pulse, the transverse magnetization Mxy decays exponentially during T2 relaxation providing the Magnetic Resonance signal

 

If we stop the radiofrequency pulse, in the absence of the B1 torque which was sustaining precession of spins in a coherent way, individual spins will be influenced by:

(a) the magnetic effect of neighbouring spins, as well as potential collisions with them and

(b) the inhomogeneities of the magnetic field.

 

Due to these influences they will start dephasing, that is the phase of their rotational motion will be altered as a deceleration or a phase “lag” will be induced.

 

As a result, a progressively decreasing number of spins will precess with the same phase, in a coherent manner and the Mxy magnetisation will be decreased. This process of decay of the transverse magnetization Mxy is termed T2 relaxation or "spin-spin relaxation" and is accompanied by energy emission. It was modelled by Felix Bloch in 1946 as a simple exponential decay (similarly to a radio-isotope) with time constant T2.

 

Mxy = Mo e-t/T2

 

T2 is the time required for the transverse magnetization to decrease to approximately 37% (1/e) of its initial value.

 

By convention, we use time constant T2 to indicate the effect of spin-spin interactions and T2* to include the magnetic field inhomogeneities.

 

 

Figure 1: Decay of Mxy component of magnetization. Image: Allen D. Elster, MRIquestions.com (ref).

 

 

 

Analysis of the Free Induction Decay (FID) signal

 

If we considered hypothetically that during the emission phase the Mxy magnetization was sustained, then we could imagine that each precession of its vector near the coil would generate a voltage with the form of a sine wave with frequency identical to the Larmor frequency and a stable amplitude. However, due to the decay of the Mxy magnetization which consists of the decrease of the number of spins precessing with the same phase, a dampening will be introduced and we will obtain a damped sine wave during energy emission. As a result, the generated signal (Figure 2) termed "Free Induction Decay" (originally termed "Nuclear Induction Decay" or "Free Induction" signal) will be an exponentially damped sine wave of the form:

 

[sin ωot] e-t/T2*

 

where ωo is the Larmor frequency and T2* is the time constant.   

 

We use time constant T2 to indicate the effect of spin-spin interactions and T2* to include the magnetic field inhomogeneities.

 

Figure 2: Free-Induction Decay (FID) - Wikipedia - By Nmr_fid_good_shim.svg: GyroMagicianderivative work: Imalipusram - This file was derived from: Nmr fid good shim.svg:, CC BY-SA 3.0, https://commons.wikimedia.org/w/index.php?curid=30312230

 

 

Figure 3: (Figure 2.3 from Jan G. Krummenacker 2012 thesis).

Above: 90°(x) pulse, followed by FID. Signal decays exponentially with time T2.

Below: Fourier Transform of FID provides a spectrum which shows a signal at the frequency offset between the Larmor frequency ω0 and the spectrometer reference frequency.

As mentioned at the Jan G. Krummenacker 2012 thesisthe simplest NMR experiment or Fourier Transform-NMR experiment consists of applying a 90°(x) pulse and recording the induced signal on the coil. The 90°(x) will rotate the magnetization in the xy plane, where it will precess with the Larmor frequency and will decay exponentially in the process which is termed free induction decay or FID. The FID oscillates with the frequency offset between the Larmor frequency ω0 and the spectrometer reference frequency ω and decays exponentially with T2* (asterisk due to the inclusion of spin-spin interactions and field inhomogeneity). Fourier transform of the FID provides a spectrum with a resonance line at the frequency offset. It is noted that in an NMR spectrum, there is important information about the sample is contained in the frequency offset ωo − ω, such as the chemical shift δ.

 

 

The signal is acquired in a process termed signal acquisition by the instrumentation and is analyzed by a Fourier transform for frequency identification. The Fourier transform of a signal whose amplitude/intensity changes with time, i.e. a signal which is a function of time and is therefore found in the time domain provides a signal which is a function of frequency or the frequency domain signal which we know as spectrum. The FID is a signal whose intensity varies with time whereas a spectrum displays how intensity varies with frequency.

 

The Fourier transform of a simple sinusoidal function is one frequency and therefore it can be mentioned that it is a line that is infinitely sharp. The Fourier transform of an exponential function is a Lorentzian function (i.e. an exponential function in the time domain is a Lorentzian function in the frequency domain).

 

As an FID is an exponential function and specifically an exponential decay of frequency, we do not have one frequency but many different frequencies. Its Fourier transform is not an infinitely sharp line but it has a shape. Its lines have a distribution (cf. shape) that is termed a Lorentzian distribution or a Lorentzian function (ref.). In spectroscopy we refer to this as spectral line broadening (ref.). It is noted that a similar case is the Gaussian distribution, which is known as the bell curve of the normal distribution.

 

In this case, for the Lorentzian distribution, as for other distributions, we define the following parameters: po is the position of the maximum (peak) corresponding to the transition energy E (p is a position), xo is the location parameter specifying the location of the peak of the distribution, γ is the scale parameter which specifies the half-width at half-maximum (HWHM) and alternatively 2γ is the full width at half maximum (FWHM) (ref.).

 

 

Figure 4: Comparison of Gaussian (red) and Lorentzian (blue) standardized line shapes. The HWHM (w/2) is 1. From WikipediaAs mentioned in Wikipedia, an atomic transition is associated with a specific amount of energy, E. However, when this energy is measured by means of some spectroscopic technique, the line is not infinitely sharp, but has a particular shape. Numerous factors can contribute to the broadening of spectral lines. A principal source of broadening is lifetime broadening. According to the uncertainty principle the uncertainty in energy, ΔE and the lifetime, Δt, of the excited state are related by:

 

ΔE Δt ≦ ћ

 

This determines the minimum possible line width. As the excited state decays exponentially in time this effect produces a line with Lorentzian shape in terms of frequency (or wavenumber).

A Lorentzian line shape function can be represented as

 

L=1/(1+x^2),

 

where L signifies a Lorentzian function standardized, for spectroscopic purposes, to a maximum value of 1; x is a subsidiary variable defined as

 

x=p°-p/(w/2)

 

where p0 is the position of the maximum (corresponding to the transition energy E), p is a position, and w is the full width at half maximum (FWHM), the width of the curve when the intensity is half the maximum intensity (this occurs at the points p = p0±w/2). The unit of p0, p and w is typically wavenumber or frequency. The variable x is dimensionless and is zero at p=p0.

 

Additional reference: https://en.wikipedia.org/wiki/Full_width_at_half_maximum

The relationship of FWHM with relaxation time is discussed at this reference https://www.chem.wisc.edu/areas/reich/nmr/08-tech-01-relax.htm.

ν½ = 1/(π T2)

For protons, T2 is usually between 1 and 10 seconds

T2 = 1 sec, ν½ = 1/π = 0.3 Hz

T2 = 10 sec, ν½ = 1/10 π = 0.03 Hz

 

(for T2=0.1 sec, ν½=3 Hz)

 

 

 

We can imagine two cases following the stop of the RF:

  1. The precessing spins receive strong magnetic influences from the neighbouring spins or even collisions and as a result a constantly increasing number of spins stop precessing coherently and decrease the Mxy magnetization. In this case, due to the abrupt decrease of the number of spins, the decay (FID) is very short or in other words we have a short T2*. We can expect increased variability in the phases of the spins and their frequencies.

  2. The precessing spins receive slight magnetic influences from the neighbouring spins and as a result a very slowly increasing number of spins stop precessing coherently and decrease the Mxy magnetization. In this case the decay (FID) is very long or in other words we have a very long T2*. We can expect moderate variability in the phases of the spins and their frequencies.

 

From the above, we can conclude that the shorter the decay (FID), or the smaller the T2*, the more increased is the broadening of the spectroscopy lines as demonstrated in Figure 5 (ref.).

 

 

 

Figure 5: From ref. Figure 4-6. The shorter the decay (FID), the more increased is the broadening of the spectroscopy lines


 

The linewidth of the spectroscopy/resonance line correlates with T2*: for a Lorentzian line the full-width-half-maximum (“FWHM”) is 1/(πT2*). It is the lifetime of spin coherence and therefore T2 - referring to gain and loss of magnetization in the xy-direction - which governs the actual width.

 

Practically, in order to obtain an NMR spectrum, applying one pulse and recording one FID does not suffice as the NMR signal is very weak and noisy. In order to improve the signal-to-noise ratio (“SNR”) of the resonance signal, the experiment is repeated several times and the result is averaged, which results in a SNR improvement factor of √ N for N averaging scans.

 

It is necessary that each repetition starts with the magnetization being at its thermal equilibrium value and therefore longitudinal relaxation of time T1 must be reestablished. Therefore, time between scans has to be longer than T1 and in principle ≈ 5 · T1.