07 MRI

Spin-echo signal

S ∝ ρ (1 − e^{−TR/T1}) e^{−TE/T2} for a 90°–180° SE sequence.

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Simulation

Spin-echo signal — Change the numbers; the scene follows.

Where it works

MRI scanner

RF coil

At the RF coil — flip angle, SAR, SNR, receive bandwidth, and the pulse sequence.

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Formula

S \propto \rho\,(1-e^{-TR/T_1})e^{-TE/T_2}

Variables

ρ · Proton densitya.u.T₁ · T1msT₂ · T2msTR · Repetition timemsTE · Echo timems

Results

S
Relative signal
0.3377a.u.
PD term
PD/T1 term
0.9179
T2 term
T2 term
0.3679

Explanation

S \propto \rho\,(1-e^{-TR/T_1})e^{-TE/T_2}

What it means

The classic spin-echo signal S ∝ ρ (1−e^{−TR/T1}) e^{−TE/T2} is the product of a T1 recovery term and a T2 decay term. It is still the mental model for choosing TR/TE even when the sequence is TSE/FSE. This is a working relation in MRI physics.

Where it is used

Clinically it sits on the MRI scanner — RF coil. At the RF coil — flip angle, SAR, SNR, receive bandwidth, and the pulse sequence. MRI physics lives in the magnet, the gradient, and the voxel: Larmor, Ernst, diffusion, and SAR. These relations decide whether a sequence is possible, safe, and worth the time.

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How to use it

T1-w: short TR (~500 ms), short TE (~10 ms). T2-w: long TR (>2000), long TE (~80–100). PD: long TR, short TE. The three outputs split those terms. Change one input and watch the curve and the simulation follow.

Symbols

ρProton density1 a.u.
T₁T1800 ms
T₂T280 ms
TRRepetition time2,000 ms
TEEcho time80 ms

Worked example

A typical case from the default values: ρ = 1 a.u. (Proton density); T₁ = 800 ms (T1); T₂ = 80 ms (T2); TR = 2,000 ms (Repetition time); TE = 80 ms (Echo time). Substituting into the relation gives S = 0.3377 a.u.; PD term = 0.9179; T2 term = 0.3679. These are teaching numbers — align them with your machine.

Typical values give

S = 0.3377a.u.
PD term = 0.9179
T2 term = 0.3679

Where it comes from

The displayed formula is the working relation. S ∝ ρ (1 − e^{−TR/T1}) e^{−TE/T2} for a 90°–180° SE sequence. Usual reference: McRobbie. Derive it in the specialty lesson, then return here to pin the numbers.

Reference: McRobbie

Assumptions & limits

90°–180°, no stimulated echoes, no magnetisation-transfer in TSE trains. Flip angles other than 90° need the Ernst/GRE formula.

Pitfalls

γ for ¹H is not γ for ¹³C. Ernst angle needs the true T1 at that field, not a 1.5 T table used at 3 T. SAR scales with B₀² and flip² — a 3 T copy of a 1.5 T protocol is not automatically legal. 90°–180°, no stimulated echoes, no magnetisation-transfer in TSE trains. Flip angles other than 90° need the Ernst/GRE formula.

Keep this

Name the nucleus and the field before you quote a Larmor frequency. 90°–180°, no stimulated echoes, no magnetisation-transfer in TSE trains. Flip angles other than 90° need the Ernst/GRE formula.

In this specialty

MRI physics