Exercise 2-3 · T2 Mapping

Quantitative MRI mOOC

T₂ Mapping · Exercise 2-3

Measuring T₂ with an echo train

How an echo train measures T₂ in ideal conditions, what imperfect pulses do to it, and how to overcome them with the CPMG condition.

Perfect pulses: which phase patterns give the best (and worst) T₂?
Slight flip-angle errors (B₁ = 0.89): which pulse patterns are best and worst now?
Large flip-angle errors (B₁ = 0.67): even with the best pattern, which echoes would you fit?
Sandbox: what do you measure without any refocusing pulses?

Refresher: the controls and the symbols

Each arrow key fires a pulse about its axis.

starts the sequence.

?	Press now: fire a pulse when it reaches the black line.
	The black line: now.
↑	A pulse that fired on the beat (any press inside the square does).
missed	No press in time.
excitation	Fires by itself at t = 0.
refocusing	One of your presses.
ACQ	The signal is measured here.

“Phase”Hero game: Echo trains

Here B₁ = 1: the excitation is exactly 90°, and every refocusing pulse exactly 180°.

(a) Select the phase directions that will provide the best T₂ measurement. Time your pulses to hit the boxes.

Pulse sequence

Signal

Spins Fan

Click and drag to reorient · V switches the view.

Settings · T₂ = 80 ms, T₁ = ∞, echoes every 20 ms. The spins’ frequency offsets (the fan):

Equally spaced: evenly over ±16.7 Hz; the signal decays as a sinc.

Gaussian: σ = 11.8 Hz; the signal decays as e^{−(2πσt)²/2}.

Lorentzian: half-width γ = 8.3 Hz; the signal decays as e^−t/T₂′, with T₂′ = 1/(2πγ) = 19.1 ms.

The game, with B1 = 1: the excitation is exactly 90° and every refocusing pulse exactly 180° (the B1 games later in the deck change that). The question at the top: which phase directions give the best T2 measurement? (With perfect pulses, any: the next slide’s hint leads there.) Press Play (or Enter), top right. The settings box under the spins defines the three fans: equally spaced over ±16.7 Hz, Gaussian with σ = 11.8 Hz, Lorentzian with half-width γ = 8.3 Hz (T2′ = 1/(2πγ) = 19.1 ms); with T2 = 80 ms, a Lorentzian fan without refocusing decays with T2* = 1/(1/T2 + 1/T2′) = 15.4 ms. The 90° fires on its own; then hit an arrow key each time a red box reaches the line: ten 180° pulses, ten echoes (τ = 10 ms, TE = 20, 40, … 200 ms). A press inside a box fires exactly on the beat. Fixed settings: perfect 180° pulses, T2 = 80 ms, T1 infinite, 6 ms of scan per second, a Lorentzian fan of spins to start (the Fan menu switches it), as wide as an equally spaced fan drifting ±60° by each 180°. The signal at each echo time is saved for the next slide, which fits T2 to it. Space moves on; the arrow keys are captured on this slide.

Measurements and the T₂ fit

The ten echo amplitudes from the game, shown first on their own; Fit then adds a mono-exponential fit S0·e^(−TE/T2) (least squares on the amplitudes, so missed echoes count as low points rather than breaking the fit), the fitted T2, and how far it is from the true 80 ms. Once the fit is shown, Hint suggests trying other combinations of pulse directions (arbitrary, always the same, alternating): with perfect pulses the results do not change, only the timing matters. Its second suggestion, the Lorentzian fan with no 180° pulses at all: the fit then gives T2*, not T2. A Lorentzian spread of frequencies makes the signal decay as e^(−t/T2′), here with T2′ ≈ 19 ms, so 1/T2* = 1/T2 + 1/T2′ gives T2* ≈ 15.4 ms (the fit gives about 15.6 ms). The other fans do not give a clean T2*: the equally spaced fan decays as a sinc, with ripples, and the Gaussian fan as a Gaussian, so an exponential fit to them means little. Try again goes back to the game and resets it.

What we saw · 1

Perfect pulses give T₂ exactly

Pure spin echoes: the fit gives 80.0 ms.
Only the timing matters, not the pulse axis.
Without any refocusing pulses, the fit with the Lorentzian fan gives T₂* (about 15 ms), not T₂.

Intermezzo: B₁ in a real brain

B₁ is the transmit field relative to its nominal value, and it is rarely 1 everywhere. Where B₁ = 0.89, a 90° pulse is only 80° and a 180° is 160°.

This is a measured B₁ map of a brain at 3 T, from the book’s B₁ mapping chapter (Figure 4.13, filtered). Grey has no value.

Contour lines every 0.1 (the highest value here is 1.18, so there is no 1.2 line):

B₁	a 90° is	a 180° is
0.8	72°	144°
0.9	81°	162°
1	90°	180°
1.1	99°	198°

An intermezzo before the B1 games: what B1 looks like in practice. The map is the filtered actual-flip-angle (AFI) B1 map from the book’s B1 mapping chapter (Figure 4.13, right; a 7×7×7 median filter attenuates Gibbs ringing). Drawn with the book’s colours (0.7 red to 1.3 blue, white at 1); the few pixels inside the head that have no value (stored as 0) are grey. Hovering over the map shows B1 at that pixel. Contour lines every 0.1 from 0.8 to 1.1 (B1 = 1 thicker, white on the map), drawn across the grey pixels from their neighbours; there is no 1.2 line because the map peaks at 1.18. In this slice B1 runs from 0.73 to 1.18: 4% of the brain is below 0.8, 11% between 0.8 and 0.9, 30% between 0.9 and 1, 33% between 1 and 1.1, and 23% above 1.1. The next game’s B1 = 0.89 sits just inside the 0.9 line. B1 = 0.67, the 60°/120° of the game after next, does not occur in this brain at 3 T; it does at 7 T (where B1 falls off strongly away from the centre of the head) and in the body at 3 T.

Echo trains with B₁ = 0.89

Here B₁ = 0.89 of its nominal value, so every flip angle comes out short: the 90° is now 80°, and each 180° is now 160°. Crushers flank each 160°, as in a real sequence.

(b) Which phase directions give the best T₂ now? Time your pulses to hit the boxes.

Pulse sequence

Signal

Spins Fan

Click and drag to reorient · V switches the view.

The same game as before, with B1 inhomogeneity: B1 = 0.89 (the transmit field is 0.89 of its nominal value), so the excitation is 80° instead of 90° and every refocusing pulse is 160° instead of 180°. Same timing (τ = 10 ms, ten echoes), T2 = 80 ms, T1 infinite, ±60° fan. Crushers around each refocusing pulse: the leftover longitudinal magnetization from the 80° excitation (cos 80° ≈ 17%) is tipped partly into the transverse plane by every 160°; that part only sees the second crusher lobe and is spoiled, while the rest stays along z. Echo magnetization that a 160° parks along z keeps its crusher phase and still comes back as stimulated echoes. With 160° pulses the choice of axis now matters: CPMG (all ↑ or all ↓, along the magnetization after a 90° about +x) keeps the echoes up; CP (all → or all ←) lets the errors pile up. Space moves on; the arrow keys are captured on this slide.

Measurements and the T₂ fit, with B₁ inhomogeneity

The ten measurements from the B1-inhomogeneity game (80° excitation, 160° refocusing), shown first on their own; Fit then adds the mono-exponential fit S0·e^(−TE/T2), the fitted T2 and its difference from the true 80 ms. With CPMG phases the train stays clean but the fit comes out a little long (about 81 ms when every pulse is on the beat): part of the signal spends time along z, where nothing decays here (T1 infinite). With CP phases the train breaks up and the fit means little. After the first fit, "Fit without echo 1" refits echoes 2–10 (about 81.4 → 80.1 ms with CPMG on the beat), with echo 1 greyed out; "Fit all echoes" goes back. The hint asks two things. CP versus CPMG: CPMG’s train stays clean, CP’s falls apart (the CPMG condition, next slide). And single-echo spin echoes: each echo then gets one 160°, which scales it by the same sin²(80°) = 0.97 whatever the TE, with no stimulated echoes, so the fitted T2 is unbiased: the bias comes from the echo train, not from the imperfect pulse alone. Try again goes back to the B1 game and resets it. Export saves the run as JSON: the settings, each pulse as pressed (key, axis, timing, in the box or not), the ten measurements at full precision, and both fits.

What we saw · 2

The CPMG condition

Refocus about the axis M lies on after the 90° (+y after a 90° about +x): each pulse’s error is undone by the next, and the train stays clean.
CPMG (↑): about 81 ms, slightly long from stimulated echoes. CP (→): the train falls apart.

A larger B₁ error

Now B₁ = 0.67 of its nominal value: the 90° is now 60°, and each 180° is now 120°. Crushers flank each 120°.

Pulse sequence

Signal

Spins Fan

Click and drag to reorient · V switches the view.

Measurements and the T₂ fit, larger B₁ error

The ten measurements from the 60°/120° game with crushers, shown first on their own; Fit adds S0·e^(−TE/T2), the fitted T2 and its difference from the true 80 ms. Then the fit can be redone without echo 1, on the even echoes only, or on the odd echoes only, one at a time, with the echoes left out greyed; "Fit all echoes" goes back. Without echo 1: echo 1 is the spin echo alone (sin²(60°) = 75% of what a 180° would give), while from echo 2 on, stimulated echoes add in, so dropping echo 1 removes most of the bias (about 93 → 84 ms with CPMG on the beat). Even or odd echoes only: the CPMG train zig-zags between odd and even echoes, and each subset is biased its own way. None of these simple fixes removes the bias at 120°; Milford et al. 2015 (PLOS ONE) and McPhee & Wilman 2018 (JMRI) report the same, and recommend fitting a model of the sequence (EPG, Bloch simulation). Hint: in the z–x side view (the pulses are about +y, so they tip the fanned-out spins between x and z), at echo 1 about a third of the magnetization is still tipped out along z (mean |Mz| ≈ 0.3), so echo 1 is low (about 84% of ideal); the second pulse tips it back and echo 2 is on the curve (about 101%). The odd echoes include echo 1, the lowest point, which pulls the odd fit to about 95 ms; without echo 1 the odd echoes give about 87 ms, close to the even ones (about 84 ms). In this game the later echoes do not alternate much: echo 1 makes most of the odd–even difference. Try again goes back to that game and resets it. Export saves the run as JSON.

What we saw · 3

Big B₁ errors: no simple fit fixes it

All echoes about 93 ms; without echo 1 about 84; even echoes about 84.
The fix: fit a model that includes stimulated echoes, such as an extended phase graph (EPG) simulation of the echo train.

Lessons

Measuring T₂ with an echo train

Perfect pulses: T₂ exactly, whatever the phase directions.
The CPMG condition: refocus about the axis M lies on after the 90°; CP falls apart with imperfect pulses.
Large B₁ errors: dropping echo 1 or fitting even echoes helps, but not enough.
The fix: model the stimulated echoes in the fit (extended phase graph, EPG).
No refocusing: the signal decays with T₂*, not T₂.

Next: a sandbox with every setting open.

Sandbox: every setting

(d) Play the Lorentzian fan with no refocusing pulses at all: what does the fit give?

Excitation ° Refocusing ° Crushers T₂ ms T₁ ms T₁ = ∞ τ ms Echoes Dephasing ± ° Fan Speed ms/s

Pulse sequence

Signal, with the T₂ envelope e^−t/T₂ (dashed)

Spins

Click and drag to reorient · V switches the view.

The sandbox: the same game with every setting open. Excitation and refocusing angles (the excitation is about +x), crushers around each refocusing pulse (they spoil what a refocusing pulse tips into the plane from magnetization that was never transverse), T2, T1 (or infinite), τ, the number of echoes, the dephasing of the fan by each refocusing pulse, an even or Gaussian fan, and the speed. The signal panel shows the true T2 envelope, e^(−t/T2), dashed. Changes apply when Play is pressed (and show at once while no run is going). Before a run, the line under Play suggests experiments (CP against CPMG at several angles, crushers on and off, the Lorentzian fan without refocusing to check 1/T2* = 1/T2 + 1/T2′). When a run ends, it gives the T2 fitted to all echoes, without echo 1, to the even and to the odd echoes; Export saves the run as JSON. Space moves on; the arrow keys and Enter are captured on this slide.

Measuring T2 with an echo train

Refresher: the controls and the symbols

“Phase”Hero game: Echo trains

Measurements and the T2 fit

Perfect pulses give T2 exactly

Intermezzo: B1 in a real brain

Echo trains with B1 = 0.89

Measurements and the T2 fit, with B1 inhomogeneity

The CPMG condition

A larger B1 error

Measurements and the T2 fit, larger B1 error

Big B1 errors: no simple fit fixes it

Measuring T2 with an echo train

Sandbox: every setting

Measuring T₂ with an echo train

Measurements and the T₂ fit

Perfect pulses give T₂ exactly

Intermezzo: B₁ in a real brain

Echo trains with B₁ = 0.89

Measurements and the T₂ fit, with B₁ inhomogeneity

A larger B₁ error

Measurements and the T₂ fit, larger B₁ error

Big B₁ errors: no simple fit fixes it

Measuring T₂ with an echo train