Frequency Encoding and Phase Encoding

I’m always searching for ways to illustrate concepts using “simple” analytical examples (I’ll let you decide whether or not this example is simple). Today, I present analytical examples of frequency and phase encoding during magnetic resonance imaging. Russ Hobbie and I discuss MRI in Chapter 18 of Intermediate Physics for Medicine and Biology.

1. Introduction

Our goal is to understand how the measured MRI signal changes when magnetic field gradients are present. These gradients are essential for “encoding” information about the spatial distribution of spins in the frequency and phase of the signal. To simplify our discussion, we make several assumptions:

• The radio-frequency π/2 and π pulses, used to rotate the spins into the x-y plane and then create an echo, are so brief that the spins rotate instantaneously compared to all other time scales. Similarly, any slice selection gradient G_z = dB_z/dz exists only during the radio-frequency pulses. We won’t include G_z in our drawings of pulse sequences. 
• We ignore relaxation, so the longitudinal and transverse time constants T₁ and T₂ are infinite.
• Despite ignoring relaxation, the spins do dephase leading to a free induction decay with time constant T₂*. Dephasing is caused by a distribution of spin frequencies, corresponding to small-scale static heterogeneities of the magnetic field. We assume that the spin frequencies ω have the distribution
  
  The peak frequency ω_o is the Larmor frequency equal to γB_o, where γ is the gyromagnetic ratio and B_o is the main magnetic field. The time constant τ indicates the width of the frequency distribution.
  
  
  
• The spins are distributed uniformly along the x axis from -Δx to +Δx.
  

2. Spin-Echo

The spin-echo pulse sequence, with no gradients and no frequency or phase encoding, is similar to Fig. 18.24 in IPMB. Our pulse sequences consist of three functions of time. The radio-frequency (RF) pulses are shown on the first line; the time between the π/2 and π pulses is T_E/2. The magnetic field gradient in the x direction, G_x = dB_z/dx, is indicated in the second line; for this first example G_x is zero. The recorded signal, M_x, is in the third line.

Our goal is to calculate M_x(t). During the time between the two radio frequency pulses, we calculate the signal by integrating the precessing spins over x and ω

In this case the x integral is trivial: the integrand does not depend on x. We can solve the ω integral analytically using the u-substitution u=τ(ω-ω_o), the cosine addition formula cos(A+B) = cosA cosB – sinA sinB, and the definite integral

The resulting free induction decay (FID) is

where τ corresponds to T₂*. The exponential shape of the free induction decay arises from the particular form of our spin distribution. The wider the distribution of frequencies, the faster the decay.

The spins accumulate phase relative to those precessing at the Larmor frequency. Just before the π pulse the extra phase is (ω-ω_o)T_E/2. The π pulse changes the sign of this phase, or in other words adds an additional phase -(ω-ω_o)T_E. After the π pulse the signal is

The x integral is again trivial and the ω integral produces an echo

which peaks at t = T_E and decays with time constant τ.

3. Phase Encoding

Phase encoding adds a gradient field G_x of duration T between the radio-frequency π/2 and π pulses. It shifts the phase of the spins by different amounts at different x locations (thus, position information is encoded in the phase of the signal). This phase shift is then reversed by the π pulse.

The trickiest part of calculating M_x(t) is keeping track of the phase shifts: (ω-ω_o)t is the phase shift up to time t because of the distribution of frequencies, -(ω-ω_o)T_E arises because the spins are flipped by the π pulse, γG_xxT is caused by the phase-encoding gradient, and -2γG_xxT is again from flipping by the π pulse. During the echo the signal simplifies to

We can solve both the x and ω integrals by repeatedly using the cosine addition formula (it is tedious but not difficult; I leave the details to you), and find

The amplitude of the echo depends on the factor sin(γG_xΔxT)/ (γG_xΔxT). For a G_x of zero this factor is one and the result is the same as for the spin-echo. If we repeat this pulse sequence with different values of G_x and measure the amplitude of each echo, we can trace out the function sin(γG_xΔxT)/ (γG_xΔxT), which is the Fourier transform of the spin distribution as a function of position.

4. Frequency Encoding

To do frequency encoding, we add a readout gradient G_x that is on during the echo and lasts a time T, like in Fig. 18.26 of IPMB. In addition, we include a prepulse of opposite polarity and half duration just before the readout, to cancel any extra phase shift accumulated during the echo. (Russ and I discuss this extra lobe of the G_x pulse when analyzing Fig. 18.29c, but we get its sign wrong).

The free induction decay and the phase reversal caused by the π-pulse are the same as in the spin-echo example. Once G_x begins the result differs. The frequency again depends on x. The phase shifts are: (ω-ω_o)t because of the distribution of frequencies, -(ω-ω_o)T_E from the π pulse, -γG_xxT/2 caused by the prepulse, and γG_xx(t-(T_E -T/2)) during readout. The recorded signal simplifies to

The echo during the readout gradient is (you really must fill in the missing steps yourself to benefit from this post)

The envelope of the echo is the product of two terms, which are both functions of time: An exponential e^-|t-T_E|/τ that has the shape of the echo with no gradient, and a factor sin(γG_xΔx(t-T_E))/ (γG_xΔx(t-T_E)). The amplitude of the echo at t=T_E is the same as if G_x were zero, but the shape of the echo has changed because of the time-dependent factor containing the gradient. The function containing the sine is the Fourier transform of the spin distribution. Therefore, the extra time-dependent modulation of the echo by G_x contains information about the spatial distribution of spins.

5. Conclusion

What do we learn from this example? A phase-encoding gradient changes the amplitude of the echo but not its shape. A frequency-encoding gradient, on the other hand, changes the shape but not the amplitude. Both can be written as a modulated Larmor-frequency signal. In the pulse sequences shown above, the Larmor frequency is drawn too low in order to make the figure clearer. In fact, the Larmor frequencies in MRI are many megahertz, and thousands of oscillations occur during the free induction decay and echo.

I analyzed both phase encoding and frequency encoding in the x direction and considered each individually, because I wanted to compare and contrast their behavior. In practice, frequency encoding is performed using a G_x gradient in the x direction and phase encoding with a G_y gradient in the y direction, mapping out the two-dimensional Fourier transform of the spin distribution (see IPMB for more). 

Until I did this calculation I never completely understood what the shape of the echo looks like during readout. I hope it helps you as much as it helped me. Enjoy!