IMG-33

Doppler shift

Blood-flow US.

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ImagingDoppler

Governing equation

\\Delta f=2 f_0 v\\cos\\theta/c

$v \in \left[0.01, 5\right]$

where

f_0: f0 (MHz)
v: Flow (m/s)
\theta: Angle (°)
\Delta f: df (kHz)

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

\\Delta f=2 f_0 v\\cos\\theta/c

$v \in \left[0.01, 5\right]$

$v = 0.01$ $\Delta f = 0.056$

Along the curve

②

Domain of the variable — blue

v_{\min} = 0.01,\quad v_{\max} = 5

$v \in \left[0.01, 5\right]$

$v = 0.01$ $v = 0.01$

Along the curve

③

Response curve — green

\\Delta f=2 f_0 v\\cos\\theta/c

$v \in \left[0.01, 5\right]$

$v = 0.01$ $\Delta f = 0.056$

Along the curve

④

Rate of change — amber

\frac{\mathrm{d}\Delta f}{\mathrm{d}v}

$v \in \left[0.01, 5\right]$

$v = 0.041$ $\Delta f^{\prime} = 5.624$

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-33 — Doppler shift) is the form associated with Doppler. Working symbols: $f_0$ , $v$ , $\theta$ $\rightarrow$ $\Delta f$ . Blood-flow US.

Purpose

Purpose: compute

\Delta f

from

f_0

v

\theta

in Medical imaging via

\\Delta f=2 f_0 v\\cos\\theta/c

Blood-flow US. 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

f_0 = 5.000\,\mathrm{MHz}

v = 0.500\,\mathrm{m/s}

\theta = 30.000\,\mathrm{^{\circ}}

, the governing relation

\\Delta f=2 f_0 v\\cos\\theta/c

yields

\Delta f = 2.812\,\mathrm{kHz}

. Blood-flow US. Move a slider: the numbers are this situation, not a canned story.

Calculator

Inputs

f0 f_05 MHzFlow v0.5 m/sAngle \theta30 °

Outputs

df \Delta f2.812 kHz

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Narration of this film

Blood-flow US.

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