RTH-02

Fractionated BED

BED = n d (1 + d/(α/β)). n fractions of d Gy.

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RadiobiologyBEDFowler

Governing equation

\mathrm{BED}=nd\left(1+\dfrac{d}{\alpha/\beta}\right)

$d \in \left[0.5, 20\right]$

where

n: Fractions (—)
d: Dose per fraction (Gy)
\alpha/\beta: α/β (Gy)
BED: BED (Gy)

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

\mathrm{BED}=nd\left(1+\dfrac{d}{\alpha/\beta}\right)

$d \in \left[0.5, 20\right]$

$d = 0.5$ $BED = 15.75$

Along the curve

②

Domain of the variable — blue

d_{\min} = 0.5,\quad d_{\max} = 20

$d \in \left[0.5, 20\right]$

$d = 0.5$ $d = 0.5$

Along the curve

③

Response curve — green

\mathrm{BED}=nd\left(1+\dfrac{d}{\alpha/\beta}\right)

$d \in \left[0.5, 20\right]$

$d = 0.5$ $BED = 15.75$

Along the curve

④

Rate of change — amber

\frac{\mathrm{d}BED}{\mathrm{d}d}

$d \in \left[0.5, 20\right]$

$d = 0.623$ $BED^{\prime} = 33.74$

Along the curve

Lecture brief

Historical brief

The linear-quadratic model, BED, tissue-maximum ratio and Clarkson scatter turned radiotherapy into fraction arithmetic. MU calculation on these sheets is that clinical closed form. This sheet (RTH-02 — Fractionated BED) is the form associated with BED · Fowler. Working symbols: $n$ , $d$ , $\alpha/\beta$ $\rightarrow$ $BED$ . Iso-survival comparison of schedules. Late tissue uses α/β ≈ 3 Gy, tumour ≈ 10 Gy.

Purpose

Purpose: compute

BED

from

n

d

\alpha/\beta

in Radiotherapy via

\mathrm{BED}=nd\left(1+\dfrac{d}{\alpha/\beta}\right)

BED = n d (1 + d/(α/β)). n fractions of d Gy. Use it when a real radiotherapy question must be answered in SI before a code check.

Live realistic example

In symbols

Live case. Given

n = 30.000\,\mathrm{—}

d = 2.000\,\mathrm{Gy}

\alpha/\beta = 10.000\,\mathrm{Gy}

, the governing relation

\mathrm{BED}=nd\left(1+\dfrac{d}{\alpha/\beta}\right)

yields

BED = 72.00\,\mathrm{Gy}

. A calendar of fractions and a BED bar. Move a slider: the numbers are this situation, not a canned story.

Calculator

Inputs

Fractions n30 —Dose per fraction d2 Gyα/β \alpha/\beta10 Gy

Outputs

BED BED72.00 Gy

Reading speed

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Narration of this film

A calendar of fractions and a BED bar.

Iso-survival comparison of schedules. Late tissue uses α/β ≈ 3 Gy, tumour ≈ 10 Gy.

Reading speed

Watch on YouTube