NMD-27

Bateman parent–daughter

A2 = A10 λ1/(λ2−λ1) (e^{−λ1 t} − e^{−λ2 t}). Growing-in of a daughter.

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DecayBateman

Governing equation

A_2=A_{10}\dfrac{\lambda_1}{\lambda_2-\lambda_1}\left(e^{-\lambda_1 t}-e^{-\lambda_2 t}\right)

$t \in \left[0.1, 200\right]$

where

A_{10}: Parent A0 (MBq)
\lambda_1: Parent λ1 (1/h)
\lambda_2: Daughter λ2 (1/h)
t: Time (h)
A_2: Daughter activity (MBq)

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

A_2=A_{10}\dfrac{\lambda_1}{\lambda_2-\lambda_1}\left(e^{-\lambda_1 t}-e^{-\lambda_2 t}\right)

$t \in \left[0.1, 200\right]$

$t = 0.1$ $A_2 = 10.43$

Along the curve

②

Domain of the variable — blue

t_{\min} = 0.1,\quad t_{\max} = 200

$t \in \left[0.1, 200\right]$

$t = 0.1$ $t = 0.1$

Along the curve

③

Response curve — green

A_2=A_{10}\dfrac{\lambda_1}{\lambda_2-\lambda_1}\left(e^{-\lambda_1 t}-e^{-\lambda_2 t}\right)

$t \in \left[0.1, 200\right]$

$t = 0.1$ $A_2 = 10.43$

Along the curve

④

Rate of change — amber

\frac{\mathrm{d}A_2}{\mathrm{d}t}

$t \in \left[0.1, 200\right]$

$t = 1.357$ $A_2^{\prime} = 88.74$

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-27 — Bateman parent–daughter) is the form associated with Bateman. Working symbols: $A_{10}$ , $\lambda_1$ , $\lambda_2$ , $t$ $\rightarrow$ $A_2$ . The 99Mo/99mTc generator is a Bateman pair with a 66 h parent and a 6 h daughter. Transient equilibrium when λ2 > λ1.

Purpose

Purpose: compute

A_2

from

A_{10}

\lambda_1

\lambda_2

t

in Nuclear medicine via

A_2=A_{10}\dfrac{\lambda_1}{\lambda_2-\lambda_1}\left(e^{-\lambda_1 t}-e^{-\lambda_2 t}\right)

A2 = A10 λ1/(λ2−λ1) (e^{−λ1 t} − e^{−λ2 t}). Growing-in of a daughter. 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

A_{10} = 10000.000\,\mathrm{MBq}

\lambda_1 = 0.011\,\mathrm{1/h}

\lambda_2 = 0.116\,\mathrm{1/h}

t = 24.000\,\mathrm{h}

, the governing relation

A_2=A_{10}\dfrac{\lambda_1}{\lambda_2-\lambda_1}\left(e^{-\lambda_1 t}-e^{-\lambda_2 t}\right)

yields

A_2 = 714.7\,\mathrm{MBq}

. A parent decaying, a daughter rising then falling. Move a slider: the numbers are this situation, not a canned story.

Calculator

Inputs

Parent A0 A_{10}10000 MBqParent λ1 \lambda_10.0105 1/hDaughter λ2 \lambda_20.1155 1/hTime t24 h

Outputs

Daughter activity A_2714.7 MBq

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Narration of this film

A parent decaying, a daughter rising then falling.

The 99Mo/99mTc generator is a Bateman pair with a 66 h parent and a 6 h daughter. Transient equilibrium when λ2 > λ1.

Reading speed

Watch on YouTube