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Index/Radiotherapy/RTH-01

RTH-01

Linear–quadratic survival

S = e^{−αD−βD²}. Cell-survival LQ model.

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RadiobiologyLQFowler

Governing equation

S=e−αD−βD2S=e^{-\alpha D-\beta D^{2}}

D∈[0.2,16]D \in \left[0.2, 16\right]

where

\alpha
Linear α (1/Gy)
\beta
Quadratic β (1/Gy²)
D
Dose (Gy)
S
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.

RTH-01 · dose
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①

Governing law — red

S=e−αD−βD2S=e^{-\alpha D-\beta D^{2}}

D∈[0.2,16]D \in \left[0.2, 16\right]

D=0.2D = 0.2S=0.941S = 0.941

②

Domain of the variable — blue

Dmin⁡=0.2,Dmax⁡=16D_{\min} = 0.2,\quad D_{\max} = 16

D∈[0.2,16]D \in \left[0.2, 16\right]

D=0.2D = 0.2D=0.2D = 0.2

③

Response curve — green

S=e−αD−βD2S=e^{-\alpha D-\beta D^{2}}

D∈[0.2,16]D \in \left[0.2, 16\right]

D=0.2D = 0.2S=0.941S = 0.941

④

Rate of change — amber

dSdD\frac{\mathrm{d}S}{\mathrm{d}D}

D∈[0.2,16]D \in \left[0.2, 16\right]

D=0.299D = 0.299S′=−0.29S^{\prime} = -0.29

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-01 — Linear–quadratic survival) is the form associated with LQ · Fowler. Working symbols: α\alpha, β\beta, DD →\rightarrow SS. α tracks single-hit lethal events; β tracks two-hit repairable damage. α/β is high for early tissue.

Purpose

Purpose: compute SS from α\alpha, β\beta, DD in Radiotherapy via S=e−αD−βD2S=e^{-\alpha D-\beta D^{2}} S = e^{−αD−βD²}. Cell-survival LQ model. Use it when a real radiotherapy question must be answered in SI before a code check.

Live realistic example

In symbols

Live case. Given α=0.300 1/Gy\alpha = 0.300\,\mathrm{1/Gy}, β=0.030 1/Gy2\beta = 0.030\,\mathrm{1/Gy^{2}}, D=2.000 GyD = 2.000\,\mathrm{Gy}, the governing relation S=e−αD−βD2S=e^{-\alpha D-\beta D^{2}} yields S=0.4868 —S = 0.4868\,\mathrm{—}. A survival curve with a shoulder then a steep tail. Move a slider: the numbers are this situation, not a canned story.

Calculator

Inputs

Outputs

  • Surviving fraction S0.4868 —
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RTH-01 · dose
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Narration of this film

A survival curve with a shoulder then a steep tail.

α tracks single-hit lethal events; β tracks two-hit repairable damage. α/β is high for early tissue.

Reading speed

Watch on YouTube