NMD-26

Effective half-life

Te = Tp Tb / (Tp + Tb). Harmonic combination of physical and biological half-lives.

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DecayEffective half-life

Governing equation

T_e=T_p T_b/(T_p+T_b)

$T_b \in \left[0.2, 500\right]$

where

T_p: Physical half-life (h)
T_b: Biological half-life (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

T_e=T_p T_b/(T_p+T_b)

$T_b \in \left[0.2, 500\right]$

$T_b = 0.2$ $T_e = 0.194$

Along the curve

②

Domain of the variable — blue

T_b_{\min} = 0.2,\quad T_b_{\max} = 500

$T_b \in \left[0.2, 500\right]$

$T_b = 0.2$ $T_b = 0.2$

Along the curve

③

Response curve — green

T_e=T_p T_b/(T_p+T_b)

$T_b \in \left[0.2, 500\right]$

$T_b = 0.2$ $T_e = 0.194$

Along the curve

④

Rate of change — amber

\frac{\mathrm{d}T_e}{\mathrm{d}T_b}

$T_b \in \left[0.2, 500\right]$

$T_b = 3.343$ $T_e^{\prime} = 0.465$

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-26 — Effective half-life) is the form associated with Effective half-life. Working symbols: $T_p$ , $T_b$ $\rightarrow$ $T_e$ . Always Te < min(Tp, Tb). A trapped tracer (Tb → ∞) has Te = Tp.

Purpose

Purpose: compute

T_e

from

T_p

T_b

in Nuclear medicine via

T_e=T_p T_b/(T_p+T_b)

Te = Tp Tb / (Tp + Tb). Harmonic combination of physical and biological half-lives. 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

T_p = 6.010\,\mathrm{h}

T_b = 10.000\,\mathrm{h}

, the governing relation

T_e=T_p T_b/(T_p+T_b)

yields

T_e = 3.754\,\mathrm{h}

. Two half-lives squeezing into a shorter Te. Move a slider: the numbers are this situation, not a canned story.

Calculator

Inputs

Physical half-life T_p6.01 hBiological half-life T_b10 h

Outputs

Effective half-life T_e3.754 h

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Narration of this film

Two half-lives squeezing into a shorter Te.

Always Te < min(Tp, Tb). A trapped tracer (Tb → ∞) has Te = Tp.

Reading speed

Watch on YouTube