IMG-09

MRI T1 recovery

Mz = M0 (1 − e^{−t/T1}) after saturation. Longitudinal recovery.

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MRIBloch T1

Governing equation

M_z=M_0\left(1-e^{-t/T_1}\right)

$t \in \left[10, 5000\right]$

where

M_0: Equilibrium M0 (a.u.)
t: Time (ms)
T_1: T1 (ms)
M_z: Longitudinal 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.

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①

Governing law — red

M_z=M_0\left(1-e^{-t/T_1}\right)

$t \in \left[10, 5000\right]$

$t = 10$ $M_z = 0.01$

Along the curve

②

Domain of the variable — blue

t_{\min} = 10,\quad t_{\max} = 5000

$t \in \left[10, 5000\right]$

$t = 10$ $t = 10$

Along the curve

③

Response curve — green

M_z=M_0\left(1-e^{-t/T_1}\right)

$t \in \left[10, 5000\right]$

$t = 10$ $M_z = 0.01$

Along the curve

④

Rate of change — amber

\frac{\mathrm{d}M_z}{\mathrm{d}t}

$t \in \left[10, 5000\right]$

$t = 41.38$ $M_z^{\prime} = 9.6\times 10^{-4}$

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-09 — MRI T1 recovery) is the form associated with Bloch T1. Working symbols: $M_0$ , $t$ , $T_1$ $\rightarrow$ $M_z$ . Inversion recovery uses 1 − 2e^{−t/T1}. T1 lengthens at higher B0 and in fluid.

Purpose

Purpose: compute

M_z

from

M_0

t

T_1

in Medical imaging via

M_z=M_0\left(1-e^{-t/T_1}\right)

Mz = M0 (1 − e^{−t/T1}) after saturation. Longitudinal recovery. 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 = 800.000\,\mathrm{ms}

T_1 = 1000.000\,\mathrm{ms}

, the governing relation

M_z=M_0\left(1-e^{-t/T_1}\right)

yields

M_z = 0.551\,\mathrm{a.u.}

. A recovering z-magnetisation after a 90° pulse. Move a slider: the numbers are this situation, not a canned story.

Calculator

Inputs

Equilibrium M0 M_01 a.u.Time t800 msT1 T_11000 ms

Outputs

Longitudinal magnetisation M_z0.551 a.u.

Reading speed

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Narration of this film

A recovering z-magnetisation after a 90° pulse.

Inversion recovery uses 1 − 2e^{−t/T1}. T1 lengthens at higher B0 and in fluid.

Reading speed

Watch on YouTube