07 MRI

T1 recovery and T2 decay

Mz after saturation and Mxy after a 90° pulse.

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Simulation

T1 recovery and T2 decay — Change the numbers; the scene follows.

Where it works

MRI scanner

Patient in bore

In the tissue inside the bore — relaxation, fat/water, contrast, flow, magic angle.

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Formula

M_z(t)=M_0(1-e^{-t/T_1}),\quad M_{xy}(t)=M_0 e^{-t/T_2}

Variables

M₀ · Equilibrium magnetisationa.u.T₁ · Longitudinal relaxationmsT₂ · Transverse relaxationmst · Timems

Results

M_z
Longitudinal magnetisation
0.2212a.u.
M_xy
Transverse magnetisation
0.0821a.u.

Curve

Explanation

M_z(t)=M_0(1-e^{-t/T_1}),\quad M_{xy}(t)=M_0 e^{-t/T_2}

What it means

After a 90° pulse, longitudinal magnetisation recovers as M_z = M₀(1−e^{−t/T1}) and transverse magnetisation decays as M_xy = M₀ e^{−t/T2}. T1 is seconds-to-hundreds of ms (tissue, field); T2 is tens of ms and always ≤ T1. This is a working relation in MRI physics.

Where it is used

Clinically it sits on the MRI scanner — Patient in bore. In the tissue inside the bore — relaxation, fat/water, contrast, flow, magic angle. 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.

MRI scanner · Open this machine

How to use it

The dual curve is the picture behind SE contrast: short TE / short TR → T1 weighting; long TE / long TR → T2 weighting; short TE / long TR → PD. Change one input and watch the curve and the simulation follow.

Symbols

M₀Equilibrium magnetisation1 a.u.
T₁Longitudinal relaxation800 ms
T₂Transverse relaxation80 ms
tTime200 ms

Worked example

A typical case from the default values: M₀ = 1 a.u. (Equilibrium magnetisation); T₁ = 800 ms (Longitudinal relaxation); T₂ = 80 ms (Transverse relaxation); t = 200 ms (Time). Substituting into the relation gives M_z = 0.2212 a.u.; M_xy = 0.0821 a.u.. These are teaching numbers — align them with your machine.

Typical values give

M_z = 0.2212a.u.
M_xy = 0.0821a.u.

Where it comes from

The displayed formula is the working relation. Mz after saturation and Mxy after a 90° pulse. Usual reference: McRobbie. Derive it in the specialty lesson, then return here to pin the numbers.

Reference: McRobbie

Assumptions & limits

Inversion recovery uses 1−2e^{−TI/T1}, not this saturation-recovery form. T2* (GRE) is shorter than T2 (see T2* calculator).

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. Inversion recovery uses 1−2e^{−TI/T1}, not this saturation-recovery form. T2* (GRE) is shorter than T2 (see T2* calculator).

Keep this

Name the nucleus and the field before you quote a Larmor frequency. Inversion recovery uses 1−2e^{−TI/T1}, not this saturation-recovery form. T2* (GRE) is shorter than T2 (see T2* calculator).

In this specialty

MRI physics