IMG-12

Spin-echo signal

S = M0 (1 − e^{−TR/T1}) e^{−TE/T2}. The classical SE contrast.

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MRISpin echo

Governing equation

S=M_0\left(1-e^{-TR/T_1}\right)e^{-TE/T_2}

$TE \in \left[2, 200\right]$

where

M_0: M0 (a.u.)
TR: TR (ms)
T_1: T1 (ms)
TE: TE (ms)
T_2: T2 (ms)
S: SE signal (a.u.)

The figure, part by part

Each numbered marker wears the color of its card. Hover a card to light the marker. Tap a marker to jump to that card. The slider walks a red point along the curve; Animate drawing traces it like a pen.

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①

Governing law — red

S=M_0\left(1-e^{-TR/T_1}\right)e^{-TE/T_2}

$TE \in \left[2, 200\right]$

$TE = 2$ $S = 0.453$

Along the curve

②

Domain of the variable — blue

TE_{\min} = 2,\quad TE_{\max} = 200

$TE \in \left[2, 200\right]$

$TE = 2$ $TE = 2$

Along the curve

③

Response curve — green

S=M_0\left(1-e^{-TR/T_1}\right)e^{-TE/T_2}

$TE \in \left[2, 200\right]$

$TE = 2$ $S = 0.453$

Along the curve

④

Rate of change — amber

\frac{\mathrm{d}S}{\mathrm{d}TE}

$TE \in \left[2, 200\right]$

$TE = 3.245$ $S^{\prime} = -0.006$

Along the curve

Lecture brief

Historical brief

Larmor precession, CT Beer projections, SNR and Nyquist sampling are why MRI and CT images exist as numbers. The lab computes frequency, dose and resolution limits. This sheet (IMG-12 — Spin-echo signal) is the form associated with Spin echo. Working symbols: $M_0$ , $TR$ , $T_1$ , $TE$ , $T_2$ $\rightarrow$ $S$ . Short TE / short TR is T1-weighted; long TE / long TR is T2-weighted. PD sits at short TE, long TR.

Purpose

Purpose: compute

S

from

M_0

TR

T_1

TE

T_2

in Medical imaging via

S=M_0\left(1-e^{-TR/T_1}\right)e^{-TE/T_2}

S = M0 (1 − e^{−TR/T1}) e^{−TE/T2}. The classical SE contrast. Use it when a real medical imaging question must be answered in SI before a code check.

Live realistic example

In symbols

Live case. Given

M_0 = 1.000\,\mathrm{a.u.}

TR = 500.000\,\mathrm{ms}

T_1 = 800.000\,\mathrm{ms}

TE = 15.000\,\mathrm{ms}

T_2 = 80.000\,\mathrm{ms}

, the governing relation

S=M_0\left(1-e^{-TR/T_1}\right)e^{-TE/T_2}

yields

S = 0.3853\,\mathrm{a.u.}

. A 90–180 pair and an echo at TE. Move a slider: the numbers are this situation, not a canned story.

Calculator

Inputs

M0 M_01 a.u.TR TR500 msT1 T_1800 msTE TE15 msT2 T_280 ms

Outputs

SE signal S0.3853 a.u.

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Narration of this film

A 90–180 pair and an echo at TE.

Short TE / short TR is T1-weighted; long TE / long TR is T2-weighted. PD sits at short TE, long TR.

Reading speed

Watch on YouTube