RTH-30

α/β from two isoeffects

α/β = (n2 d2² − n1 d1²)/(n1 d1 − n2 d2). Two equal-BED schedules.

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RadiobiologyIsoeffect α/β

Governing equation

\alpha/\beta=\dfrac{n_2 d_2^2-n_1 d_1^2}{n_1 d_1-n_2 d_2}

$d_2 \in \left[0.5, 20\right]$

where

n_1: Fractions 1 (—)
d_1: Dose/fx 1 (Gy)
n_2: Fractions 2 (—)
d_2: Dose/fx 2 (Gy)
\alpha/\beta: Recovered α/β (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

\alpha/\beta=\dfrac{n_2 d_2^2-n_1 d_1^2}{n_1 d_1-n_2 d_2}

$d_2 \in \left[0.5, 20\right]$

$d_2 = 0.5$ $\alpha/\beta = -2.065$

Along the curve

②

Domain of the variable — blue

d_2_{\min} = 0.5,\quad d_2_{\max} = 20

$d_2 \in \left[0.5, 20\right]$

$d_2 = 0.5$ $d_2 = 0.5$

Along the curve

③

Response curve — green

\alpha/\beta=\dfrac{n_2 d_2^2-n_1 d_1^2}{n_1 d_1-n_2 d_2}

$d_2 \in \left[0.5, 20\right]$

$d_2 = 0.5$ $\alpha/\beta = -2.065$

Along the curve

④

Rate of change — amber

\frac{\mathrm{d}\alpha/\beta}{\mathrm{d}d_2}

$d_2 \in \left[0.5, 20\right]$

$d_2 = 0.623$ $\alpha/\beta^{\prime} = -0.073$

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-30 — α/β from two isoeffects) is the form associated with Isoeffect α/β. Working symbols: $n_1$ , $d_1$ , $n_2$ , $d_2$ $\rightarrow$ $\alpha/\beta$ . Withers' isoeffect: n1 d1 (1+d1/(α/β)) = n2 d2 (1+d2/(α/β)). Solve for α/β.

Purpose

Purpose: compute

\alpha/\beta

from

n_1

d_1

n_2

d_2

in Radiotherapy via

\alpha/\beta=\dfrac{n_2 d_2^2-n_1 d_1^2}{n_1 d_1-n_2 d_2}

α/β = (n2 d2² − n1 d1²)/(n1 d1 − n2 d2). Two equal-BED schedules. 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_1 = 30.000\,\mathrm{—}

d_1 = 2.000\,\mathrm{Gy}

n_2 = 5.000\,\mathrm{—}

d_2 = 6.000\,\mathrm{Gy}

, the governing relation

\alpha/\beta=\dfrac{n_2 d_2^2-n_1 d_1^2}{n_1 d_1-n_2 d_2}

yields

\alpha/\beta = 2.00\,\mathrm{Gy}

. Two fractionation calendars meeting at one BED. Move a slider: the numbers are this situation, not a canned story.

Calculator

Inputs

Fractions 1 n_130 —Dose/fx 1 d_12 GyFractions 2 n_25 —Dose/fx 2 d_26 Gy

Outputs

Recovered α/β \alpha/\beta2.00 Gy

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Narration of this film

Two fractionation calendars meeting at one BED.

Withers' isoeffect: n1 d1 (1+d1/(α/β)) = n2 d2 (1+d2/(α/β)). Solve for α/β.

Reading speed

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