NMD-03

Effective decay constant

λe = λp + λb. Physical plus biological clearance.

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DecayEffective decay

Governing equation

\lambda_e=\lambda_p+\lambda_b

$\lambda_b \in \left[0, 1\right]$

where

\lambda_p: Physical λp (1/h)
\lambda_b: Biological λb (1/h)
\lambda_e: Effective λe (1/h)
T_e: Effective half-life (h)

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

\lambda_e=\lambda_p+\lambda_b

$\lambda_b \in \left[0, 1\right]$

$\lambda_b = 0$ $\lambda_e = 6.001$

Along the curve

②

Domain of the variable — blue

\lambda_b_{\min} = 0,\quad \lambda_b_{\max} = 1

$\lambda_b \in \left[0, 1\right]$

$\lambda_b = 0$ $\lambda_b = 0$

Along the curve

③

Response curve — green

\lambda_e=\lambda_p+\lambda_b

$\lambda_b \in \left[0, 1\right]$

$\lambda_b = 0$ $\lambda_e = 6.001$

Along the curve

④

Rate of change — amber

\frac{\mathrm{d}\lambda_e}{\mathrm{d}\lambda_b}

$\lambda_b \in \left[0, 1\right]$

$\lambda_b = 0.006$ $\lambda_e^{\prime} = -46.86$

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-03 — Effective decay constant) is the form associated with Effective decay. Working symbols: $\lambda_p$ , $\lambda_b$ $\rightarrow$ $\lambda_e$ , $T_e$ . The body never sees a pure physical half-life unless the tracer is trapped (λb → 0).

Purpose

Purpose: compute

\lambda_e

T_e

from

\lambda_p

\lambda_b

in Nuclear medicine via

\lambda_e=\lambda_p+\lambda_b

λe = λp + λb. Physical plus biological clearance. 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

\lambda_p = 0.116\,\mathrm{1/h}

\lambda_b = 0.050\,\mathrm{1/h}

, the governing relation

\lambda_e=\lambda_p+\lambda_b

yields

\lambda_e = 0.1655\,\mathrm{1/h}

T_e = 4.188\,\mathrm{h}

. Two parallel leak rates adding to λe. Move a slider: the numbers are this situation, not a canned story.

Calculator

Inputs

Physical λp \lambda_p0.1155 1/hBiological λb \lambda_b0.05 1/h

Outputs

Effective λe \lambda_e0.1655 1/h
Effective half-life T_e4.188 h

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Narration of this film

Two parallel leak rates adding to λe.

The body never sees a pure physical half-life unless the tracer is trapped (λb → 0).

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Watch on YouTube