IMG-10

MRI T2 decay

Mxy = M0 e^{−t/T2}. Transverse decay of a spin echo.

Reading speed

MRIBloch T2

Governing equation

M_{xy}=M_0 e^{-t/T_2}

$t \in \left[1, 400\right]$

where

M_0: Initial Mxy (a.u.)
t: Echo time (ms)
T_2: T2 (ms)
M_{xy}: Transverse magnetisation (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.

00:0 / 00:08

①

Governing law — red

M_{xy}=M_0 e^{-t/T_2}

$t \in \left[1, 400\right]$

$t = 1$ $M_{xy} = 0.988$

Along the curve

②

Domain of the variable — blue

t_{\min} = 1,\quad t_{\max} = 400

$t \in \left[1, 400\right]$

$t = 1$ $t = 1$

Along the curve

③

Response curve — green

M_{xy}=M_0 e^{-t/T_2}

$t \in \left[1, 400\right]$

$t = 1$ $M_{xy} = 0.988$

Along the curve

④

Rate of change — amber

\frac{\mathrm{d}M_{xy}}{\mathrm{d}t}

$t \in \left[1, 400\right]$

$t = 3.509$ $M_{xy}^{\prime} = -0.012$

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-10 — MRI T2 decay) is the form associated with Bloch T2. Working symbols: $M_0$ , $t$ , $T_2$ $\rightarrow$ $M_{xy}$ . T2* < T2 because of field inhomogeneity. A 180° pulse refocuses the reversible part.

Purpose

Purpose: compute

M_{xy}

from

M_0

t

T_2

in Medical imaging via

M_{xy}=M_0 e^{-t/T_2}

Mxy = M0 e^{−t/T2}. Transverse decay of a spin echo. 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.}

t = 80.000\,\mathrm{ms}

T_2 = 80.000\,\mathrm{ms}

, the governing relation

M_{xy}=M_0 e^{-t/T_2}

yields

M_{xy} = 0.368\,\mathrm{a.u.}

. A decaying FID / echo envelope. Move a slider: the numbers are this situation, not a canned story.

Calculator

Inputs

Initial Mxy M_01 a.u.Echo time t80 msT2 T_280 ms

Outputs

Transverse magnetisation M_{xy}0.368 a.u.

Reading speed

Watch on YouTube

Free library

Full library

Free PDF / open book

YouTube channels

00:0 / 00:08

Narration of this film

A decaying FID / echo envelope.

T2* < T2 because of field inhomogeneity. A 180° pulse refocuses the reversible part.

Reading speed

Watch on YouTube