07 MRI

FLASH / GRE spoiled signal

S = M₀ sinθ (1−E1) / (1−cosθ E1) · e^{−TE/T2*}, E1 = e^{−TR/T1}.

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Simulation

FLASH / GRE spoiled 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=M_0\sin\theta\frac{1-E_1}{1-\cos\theta\,E_1}e^{-TE/T_2^*}

Variables

T1 · T1msT2* · T2*msTR · TRmsTE · TEmsθ · Flip angle°

Results

S/M_0
Relative signal
0.1678
θ_E
Ernst angle
30.6038°

Curve

Explanation

S=M_0\sin\theta\frac{1-E_1}{1-\cos\theta\,E_1}e^{-TE/T_2^*}

What it means

Spoiled gradient echo (FLASH, T1-FFE, SPGR) destroys leftover xy each TR, so the steady state is a T1-weighted longitudinal recovery sampled at flip angle θ. The Ernst angle maximises S for a given TR/T1 (see the Ernst calculator). Long TR or tiny θ → proton-density; short TR and large θ → T1 weighting. T2* decay over TE is the remaining exponential. 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

Enter T1, T2*, TR, TE (ms) and flip angle (degrees). Compare two tissues by running twice (fat vs muscle, Gd-enhanced vs native). Default is a 1.5 T T1-weighted abdomen: T1=800, T2*=40, TR=120, TE=4, θ=70°. Change one input and watch the curve and the simulation follow.

Symbols

T1T1800 ms
T2*T2*40 ms
TRTR120 ms
TETE4 ms
θFlip angle70 °

Worked example

A typical case from the default values: T1 = 800 ms (T1); T2* = 40 ms (T2*); TR = 120 ms (TR); TE = 4 ms (TE); θ = 70 ° (Flip angle). Substituting into the relation gives S/M_0 = 0.1678; θ_E = 30.6038 °. These are teaching numbers — align them with your machine.

Typical values give

S/M_0 = 0.1678
θ_E = 30.6038°

Where it comes from

The displayed formula is the working relation. S = M₀ sinθ (1−E1) / (1−cosθ E1) · e^{−TE/T2*}, E1 = e^{−TR/T1}. Usual reference: Ernst / Haase (FLASH) / McRobbie. Derive it in the specialty lesson, then return here to pin the numbers.

Reference: Ernst / Haase (FLASH) / McRobbie

Assumptions & limits

Perfect spoiling (RF + gradient). Coherent GRE (FISP, FFE) keeps xy and needs a different formula (T2/T1). Flip-angle errors (B1) change both contrast and the Ernst peak.

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. Perfect spoiling (RF + gradient). Coherent GRE (FISP, FFE) keeps xy and needs a different formula (T2/T1). Flip-angle errors (B1) change both contrast and the Ernst peak.

Keep this

Name the nucleus and the field before you quote a Larmor frequency. Perfect spoiling (RF + gradient). Coherent GRE (FISP, FFE) keeps xy and needs a different formula (T2/T1). Flip-angle errors (B1) change both contrast and the Ernst peak.

In this specialty

MRI physics