MRI Physics: FID, Relaxation, and Signal Processing

Rotating Mxy changes the magnetic flux through the coil, and Faraday's law turns changing flux into a voltage (V = −dΦ/dt).
Like a spinning magnet next to a wire loop (a dynamo). Mz alone is static, so it induces nothing.

The decaying, oscillating voltage recorded right after an excitation pulse: free induction decay.
"Free" = no pulses on, "induction" = Faraday's law, "decay" = coherence is being lost. Its oscillation shows which frequencies are present; its envelope shows how fast Mxy disappears (T2* for an unrefocused FID).

It subtracts f_ref, so the displayed signal only oscillates at the small difference Δf = f_L − f_ref.
It is like viewing in a frame rotating at f_ref. The spins still precess at ~600 MHz; only the display changes. 0 Hz on the spectrum means "at f_ref".
Trap: changing f_ref does not change B0 or the spins.

The voltage is compared with two references 90° apart, which keeps both amplitude and phase (and the sign of Δf).
s(t) = s_R(t) + i·s_I(t). With one channel you could not tell +Δf from −Δf. This is quadrature detection, not to be confused with a quadrature coil.

The Fourier transform turns the time signal into a spectrum: how much signal sits at each frequency offset.
A component oscillating at Δf in the FID becomes a peak at Δf. Same data, two views. Several frequencies (e.g., several metabolites) → several peaks.

For a correctly phased peak, the real part is a symmetric absorption peak; the imaginary part is an antisymmetric dispersion curve crossing zero at the same frequency.
They are two views of one resonance, not two peaks. The absorption peak is used to read position and width.

It adjusts the overall signal phase so the real spectrum shows a clean absorption peak.
With the wrong phase, absorption and dispersion mix into a lopsided peak whose position and width are hard to read.

FWHM = 1/(π·T2*): fast decay → broad peak, slow decay → narrow peak.
FWHM is the width in Hz between the two points where the phased real (absorption) peak crosses half its maximum height. Worked: T2* = 10 ms → 1/(π × 0.010 s) ≈ 32 Hz; doubling T2* halves it.
Trap: not the width at the base, and not read off the dispersion curve.

Height depends on width: for the same area, a broader peak is shorter.
The area (integral) tracks the number of contributing spins. Peak height also changes with gain and flip angle.

Regrowth of Mz toward M0 as spins hand energy to their surroundings and repopulate the lower energy level.
Also called spin–lattice relaxation. Short T1 → magnetization is ready for the next excitation sooner.

Irreversible loss of transverse phase coherence caused by random, fluctuating spin–spin interactions.
Neighbouring spins jiggle each other's local field randomly, so phases drift apart. The arrows fan out and cancel; Mxy shrinks though the spins are still there. An echo cannot undo it.

T2* is the decay you actually see in an FID: intrinsic T2 plus extra dephasing from static field non-uniformity. T2* ≤ T2.
1/T2* = 1/T2 + 1/T2′. The T2′ part is static (each spin keeps its own slightly wrong frequency), so a spin echo can undo it. Better shimming → T2* closer to T2. In tissue usually T2* ≤ T2 < T1.

No: they happen at the same time but independently. Mxy decays (T2) while Mz regrows (T1).
Because T2 is much shorter than T1 in tissue, Mxy is usually gone long before Mz has recovered.
Trap: "Mxy disappears because M has returned to z."

Mz(t) = M0(1 − e^(−t/T1)): 63% back at t = T1, > 99% at 5·T1. Mxy(t) = M0·e^(−t/T2): 37% left at t = T2.
The "wait 5 T1" rule is why fully relaxed acquisitions use TR ≈ 5 × T1.

Pulsing again before Mz has recovered, so less Mz is available to tip and each repetition gives less signal.
After a few repeats a smaller steady signal is reached. Short-T1 tissue recovers more between pulses and stays brighter. Longer TR or shorter T1 reduces saturation.

Prepare Mz, wait a chosen delay, then tip it with a readout pulse: the signal size tells you how much Mz had recovered.
Repeat for several delays and fit the recovery curve.

1) 180° inverts Mz to −M0.
2) Wait TI while Mz recovers.
3) Readout pulse: signal ∝ Mz(TI).
4) Repeat for several TI and fit Mz(TI) = M0(1 − 2e^(−TI/T1)).
Starting from −M0 doubles the range of the recovery curve compared with starting from zero.

Acquire spin echoes at several TE, plot echo-centre amplitude vs TE, fit S = S0·e^(−TE/T2).
The 180° pulse refocuses static dephasing at each echo centre, so only intrinsic T2 remains. Keep TR, gain and flip angles identical between TE values.

T2 = 40 ms / ln 2 ≈ 58 ms.
Ratio: 50/100 = e^(−40/T2). Taking the ratio cancels the unknown starting signal S0.

From the decay of an unrefocused FID, or from gradient echoes at several TE (S = S0·e^(−TE/T2*)).
No 180° pulse, so the static field spread stays in the signal. Easier than T2, but the value depends on how good the shim is.