NMD-20

Biological excretion

f(t) = 1 − e^{−λb t}. Fraction excreted by a single biological compartment.

Reading speed

BiokineticsBiological excretion

Governing equation

f(t)=1-e^{-\lambda_b t}

$t \in \left[0.1, 24\right]$

where

\lambda_b: Biological λb (1/h)
t: Time (h)
f: Excreted fraction (—)

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.

00:0 / 00:08

①

Governing law — red

f(t)=1-e^{-\lambda_b t}

$t \in \left[0.1, 24\right]$

$t = 0.1$ $f = 0.039$

Along the curve

②

Domain of the variable — blue

t_{\min} = 0.1,\quad t_{\max} = 24

$t \in \left[0.1, 24\right]$

$t = 0.1$ $t = 0.1$

Along the curve

③

Response curve — green

f(t)=1-e^{-\lambda_b t}

$t \in \left[0.1, 24\right]$

$t = 0.1$ $f = 0.039$

Along the curve

④

Rate of change — amber

\frac{\mathrm{d}f}{\mathrm{d}t}

$t \in \left[0.1, 24\right]$

$t = 0.25$ $f^{\prime} = 0.362$

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-20 — Biological excretion) is the form associated with Biological excretion. Working symbols: $\lambda_b$ , $t$ $\rightarrow$ $f$ . Urinary excretion of MAG3 or DTPA is almost a single exponential after the plasma peak.

Purpose

Purpose: compute

f

from

\lambda_b

t

in Nuclear medicine via

f(t)=1-e^{-\lambda_b t}

f(t) = 1 − e^{−λb t}. Fraction excreted by a single biological compartment. 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_b = 0.400\,\mathrm{1/h}

t = 2.000\,\mathrm{h}

, the governing relation

f(t)=1-e^{-\lambda_b t}

yields

f = 0.551\,\mathrm{—}

. A rising cumulative urine curve. Move a slider: the numbers are this situation, not a canned story.

Calculator

Inputs

Biological λb \lambda_b0.4 1/hTime t2 h

Outputs

Excreted fraction f0.551 —

Reading speed

Watch on YouTube

Free library

Full library

Free PDF / open book

YouTube channels

00:0 / 00:08

Narration of this film

A rising cumulative urine curve.

Urinary excretion of MAG3 or DTPA is almost a single exponential after the plasma peak.

Reading speed

Watch on YouTube