Exercise 2-1 · T2 Mapping

Quantitative MRI mOOC

T₂ Mapping · Exercise 2-1

RF phase and spin echoes

Pulse phase: which way does each RF phase (±x, ±y) turn the magnetization?
Refocusing: after the 90°, which pulse brings the spins back together?
Spin echo: when and where does the echo form?
One echo: does a single echo measure T₂ or T₂^*?

The controls: B₁ along ±x or ±y

One spin, on resonance, with no relaxation. Each arrow key turns on B₁ along one axis:

Key	B₁ along
→	+x
←	−x
↑	+y
↓	−y

Hold a key: the magnetization rotates around B₁. Let go: B₁ turns off and the magnetization stays where it is.

Try: hold → for half a second (45° about +x), then hold ↑. The tip traces the rim of a cone around +y: M isn’t perpendicular to the new axis.

The blue trail is the tip’s path. On this slide the arrow keys drive B₁: use Space or Page Down for the next slide.

Click and drag to reorient

Game: hit the targets

(a) Steer M into each target, using as little RF as you can.

Reaching a target ends the pulse, and the next target appears. There is no clock. Two scores, both lower is better:

RF pulses: each key press is one pulse. It lasts until you let go, or until M reaches the target.
Total angle: every degree of every pulse. The RF energy deposited (SAR) grows with it.

On this slide the arrow keys drive B₁: use Space or Page Down for the next slide.

Click and drag to reorient

A game on the same controls: steer M into each target in turn (level 1: +y, −y, +y, +x, then +z; level 2: +y, (1,1,1)/√3, (1,1,−1)/√3, +x, (1,−1,1)/√3, then (−1,−1,1)/√3). A target counts as soon as the tip of M enters it; that ends the pulse (B1 turns off, even if the key is still held), and the next target appears. Two scores, both lower is better: the number of RF pulses (key presses), and the total angle turned, which stands for the RF energy deposited (SAR), since B1 is always the same strength here. When the level is done, the slide shows the expert score (6 pulses, 604°), and View solution replays it. 6 pulses: one per target plus one, because +x is on no circle through +y (pulses about x keep M in the yz plane, and a pulse about y leaves M near +y). 604°: ← to +y, → back past +z to −y, ← back to +y, → 73° to near +z, ↑ to +x, ↓ back to +z. Reversing rather than going on, and turning near +z, saves degrees, because the targets only need to be touched. Level 2 is about cones: when M is not perpendicular to B1, its tip sweeps a cone. (1,1,1)/√3 to (1,1,−1)/√3 is a single 90° pulse about +y (y stays at 0.58), and so is the last move, (1,−1,1)/√3 to (−1,−1,1)/√3. Its targets are red and half the size. Expert score: 9 pulses, 530°. Space or Page Down moves on; the arrow keys are captured on this slide.

A fan of spins, TE/2 after the 90°

(b) Which pulse brings the spins back together at TE?

The 90° about +x tipped M onto −y. A time TE/2 has passed since: the spins have fanned out by about 60°, the red ones precessing faster and the blue ones slower. The black arrow, M, is their sum.

Flip angle 180°

An arrow key turns every spin by the flip angle about its axis. The pulse is instantaneous: the spins don’t precess while it turns them.

No relaxation on this slide. The arrow keys apply pulses here: use Space or Page Down for the next slide.

Click and drag to reorient

A fan of 25 spins, TE/2 after a 90° about +x: their phases are Gaussian with σ = 15° (about 60° from edge to edge), red for the spins that precess faster than the centre frequency, blue for the slower ones. The flip angle is set on the slider (180° to start), and each arrow key applies it about ±x or ±y, animated, with the scan clock stopped (the pulse is instantaneous: no precession during it); Wait TE/2 lets the spins precess for TE/2 (each gains the same phase as in the first TE/2), and Wait TE for twice that, and |Mxy| is the size of their sum in the transverse plane. With 180° about ±x, the fan is mirrored onto +y and comes back together there at TE (|Mxy| back to 100%); with 180° about ±y, it is mirrored about the y axis and refocuses on −y. Below 180°, part of each spin is left behind or tipped out of the plane, and the echo at TE is smaller. Waiting longer after the echo, the fan spreads out again. No relaxation, so the only loss is from the pulse.

What we saw · 1

A pulse’s phase is the axis M turns around

Perpendicular to the axis, M sweeps a circle; otherwise, a cone.
Along the axis, M doesn’t move.

Lesson of the controls slide: the pulse phase (+x, −x, +y, −y) is the rotation axis. The cone is the key picture for what follows: a pulse acts differently on magnetization depending on where M sits relative to the pulse axis, which is why the axis of the refocusing pulses matters once they are not exactly 180° (CP versus CPMG in Exercise 2-3). In full: The phase sets where B1 points in the transverse plane: +x, −x, +y or −y. M turns around that axis (right-hand rule); −x turns it the opposite way to +x. Same angle, different axis, different result: from +z, 90° about +x puts M on −y; 90° about +y puts it on +x. M sweeps a great circle only when it is perpendicular to the axis. Otherwise it sweeps a cone around the axis: after 45° about +x, M is halfway between +z and −y, and a pulse about +y swings its tip around a cone. M along the axis doesn’t move at all: after 90° about +x, M is on −y, and a pulse about ±y leaves it there. (Exercise 2-3 comes back to this.)

A spin echo

Press Play: the 90° fires on its own at t = 0 and the spins fan out. Then press an arrow whenever you like: a 180° fires at once, and an SE marker shows where the echo will form.

Pulse sequence

Signal

Spins Fan

Click and drag to reorient · V switches the view.

What we saw · 2

A 180° refocuses the fan

The echo forms at TE = 2 × t₁₈₀.
Only T₂ decay is lost: e^−TE/T₂.

Measuring T₂ (or T₂*) from one echo

(d) Measure once with the 180° and once without: which gives T₂, and which T₂*?

Press ▶ Play. The 90° fires by itself at t = 0.
When the ? reaches the black line (at TE/2), press any arrow key: that fires the 180°.
The signal is measured at ACQ (at TE), and that one value gives T₂ = −TE / ln S.

If you see a ?, this is when you want to try and press. Don’t press at all, and you measure T₂* instead.

Pulse sequence

Signal

Spins

Click and drag to reorient · V switches the view.

A single-echo measurement, tutorial style: the symbols are those of the echo-train game in Exercise 2-3. The 90° fires on its own at t = 0; the red dashed square marks TE/2 = 10 ms, where a press of any arrow key fires the 180° (a press inside the square fires exactly on the beat); the spin echo is measured at ACQ, TE = 20 ms. With S0 = 1 (the signal just after the 90°), one value is enough: S = e^(−20/80) = 0.78 and T2 = −TE / ln S = 80 ms. Without a press, nothing refocuses the spins, and the signal at TE has decayed with T2* instead: the fan is Lorentzian, the same as in Exercise 2-3’s game (T2′ = 19.1 ms), so T2* = 1/(1/T2 + 1/T2′) = 15.4 ms: S(TE) = 0.27, and the slide reports about 15 ms. A press outside the square fires the 180° then and there: the echo then misses TE and the estimate is off. T2 = 80 ms, T1 = ∞, 3 ms of scan per second. Space moves on; the arrow keys and Enter are captured on this slide.

Lessons

Spin echoes

Phase = rotation axis: a circle if M is perpendicular to it, a cone otherwise.
Spin echo at 2 × t₁₈₀: only T₂ decay is lost.
One echo, one value: T₂ = −TE / ln S; without the 180°, the same gives T₂*.

Exercise 2-2: stimulated echoes →

RF phase and spin echoes

The controls: B1 along ±x or ±y

Game: hit the targets

A fan of spins, TE/2 after the 90°

A pulse’s phase is the axis M turns around

A spin echo

A 180° refocuses the fan

Measuring T2 (or T2*) from one echo

Spin echoes

The controls: B₁ along ±x or ±y

Measuring T₂ (or T₂*) from one echo