RTH-32

Hypoxia OER survival

S_h = exp(−α D/OER − β D²/OER²). Oxygen-rescaled LQ.

Reading speed

RadiobiologyOERGray

Governing equation

S_h=\exp\!\left(-\dfrac{\alpha D}{\mathrm{OER}}-\dfrac{\beta D^{2}}{\mathrm{OER}^{2}}\right)

$D \in \left[0.2, 16\right]$

where

\alpha: Linear α (1/Gy)
\beta: Quadratic β (1/Gy²)
D: Dose (Gy)
OER: Oxygen enhancement ratio (—)
S_h: Hypoxic surviving 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

S_h=\exp\!\left(-\dfrac{\alpha D}{\mathrm{OER}}-\dfrac{\beta D^{2}}{\mathrm{OER}^{2}}\right)

$D \in \left[0.2, 16\right]$

$D = 0.2$ $S_h = 0.979$

Along the curve

②

Domain of the variable — blue

D_{\min} = 0.2,\quad D_{\max} = 16

$D \in \left[0.2, 16\right]$

$D = 0.2$ $D = 0.2$

Along the curve

③

Response curve — green

S_h=\exp\!\left(-\dfrac{\alpha D}{\mathrm{OER}}-\dfrac{\beta D^{2}}{\mathrm{OER}^{2}}\right)

$D \in \left[0.2, 16\right]$

$D = 0.2$ $S_h = 0.979$

Along the curve

④

Rate of change — amber

\frac{\mathrm{d}S_h}{\mathrm{d}D}

$D \in \left[0.2, 16\right]$

$D = 0.299$ $S_h^{\prime} = -0.106$

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-32 — Hypoxia OER survival) is the form associated with OER · Gray. Working symbols: $\alpha$ , $\beta$ , $D$ , $OER$ $\rightarrow$ $S_h$ . OER ≈ 2.5–3 for photons; closer to 1 for high-LET. Hypoxic cores resist standard fractions.

Purpose

Purpose: compute

S_h

from

\alpha

\beta

D

OER

in Radiotherapy via

S_h=\exp\!\left(-\dfrac{\alpha D}{\mathrm{OER}}-\dfrac{\beta D^{2}}{\mathrm{OER}^{2}}\right)

S_h = exp(−α D/OER − β D²/OER²). Oxygen-rescaled LQ. Use it when a real radiotherapy question must be answered in SI before a code check.

Live realistic example

In symbols

Live case. Given

\alpha = 0.300\,\mathrm{1/Gy}

\beta = 0.030\,\mathrm{1/Gy^{2}}

D = 2.000\,\mathrm{Gy}

OER = 2.800\,\mathrm{—}

, the governing relation

S_h=\exp\!\left(-\dfrac{\alpha D}{\mathrm{OER}}-\dfrac{\beta D^{2}}{\mathrm{OER}^{2}}\right)

yields

S_h = 0.7949\,\mathrm{—}

. Two survival curves, aerobic and hypoxic. Move a slider: the numbers are this situation, not a canned story.

Calculator

Inputs

Linear α \alpha0.3 1/GyQuadratic β \beta0.03 1/Gy²Dose D2 GyOxygen enhancement ratio OER2.8 —

Outputs

Hypoxic surviving fraction S_h0.7949 —

Reading speed

Watch on YouTube

Free library

Full library

Free PDF / open book

YouTube channels

00:0 / 00:08

Narration of this film

Two survival curves, aerobic and hypoxic.

OER ≈ 2.5–3 for photons; closer to 1 for high-LET. Hypoxic cores resist standard fractions.

Reading speed

Watch on YouTube