NMD-19

Narrow-beam attenuation

I = I0 e^{−μx}. SPECT/PET attenuation of a photon path.

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AttenuationNarrow-beam μ

Governing equation

I=I_0 e^{-\mu x}

$x \in \left[0.5, 40\right]$

where

I_0: Unattenuated rate (cps)
\mu: Linear attenuation (1/cm)
x: Path length (cm)
I: Transmitted rate (cps)

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

I=I_0 e^{-\mu x}

$x \in \left[0.5, 40\right]$

$x = 0.5$ $I = 927.7$

Along the curve

②

Domain of the variable — blue

x_{\min} = 0.5,\quad x_{\max} = 40

$x \in \left[0.5, 40\right]$

$x = 0.5$ $x = 0.5$

Along the curve

③

Response curve — green

I=I_0 e^{-\mu x}

$x \in \left[0.5, 40\right]$

$x = 0.5$ $I = 927.7$

Along the curve

④

Rate of change — amber

\frac{\mathrm{d}I}{\mathrm{d}x}

$x \in \left[0.5, 40\right]$

$x = 0.748$ $I^{\prime} = -134.1$

Along the curve

Lecture brief

Historical brief

MIRD schema, in-vivo decay, uptake fractions and PET coincidence turned unsealed sources into organ dose. The sheets compute activity, residence and well-counter geometry. This sheet (NMD-19 — Narrow-beam attenuation) is the form associated with Narrow-beam μ. Working symbols: $I_0$ , $\mu$ , $x$ $\rightarrow$ $I$ . μ_water(140 keV) ≈ 0.15 cm⁻¹, μ_water(511 keV) ≈ 0.096 cm⁻¹. Chang's method uses a uniform μ map.

Purpose

Purpose: compute

I

from

I_0

\mu

x

in Nuclear medicine via

I=I_0 e^{-\mu x}

I = I0 e^{−μx}. SPECT/PET attenuation of a photon path. Use it when a real nuclear medicine question must be answered in SI before a code check.

Live realistic example

In symbols

Live case. Given

I_0 = 1000.000\,\mathrm{cps}

\mu = 0.150\,\mathrm{1/cm}

x = 12.000\,\mathrm{cm}

, the governing relation

I=I_0 e^{-\mu x}

yields

I = 165.3\,\mathrm{cps}

. A slab, a fading photon beam. Move a slider: the numbers are this situation, not a canned story.

Calculator

Inputs

Unattenuated rate I_01000 cpsLinear attenuation \mu0.15 1/cmPath length x12 cm

Outputs

Transmitted rate I165.3 cps

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Narration of this film

A slab, a fading photon beam.

μ_water(140 keV) ≈ 0.15 cm⁻¹, μ_water(511 keV) ≈ 0.096 cm⁻¹. Chang's method uses a uniform μ map.

Reading speed

Watch on YouTube