RTH-29

TCP snapshot (Poisson)

TCP = exp(−N0 S), S = e^{−αD−βD²}. Zero clonogens left.

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RadiobiologyPoisson TCP

Governing equation

\mathrm{TCP}=\exp\!\left(-N_0 e^{-\alpha D-\beta D^{2}}\right)

$D \in \left[10, 90\right]$

where

N_0: Clonogens N0 (—)
\alpha: Linear α (1/Gy)
\beta: Quadratic β (1/Gy²)
D: Dose (Gy)
TCP: Control probability (—)

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{TCP}=\exp\!\left(-N_0 e^{-\alpha D-\beta D^{2}}\right)

$D \in \left[10, 90\right]$

$D = 10$ $TCP = 0$

Along the curve

②

Domain of the variable — blue

D_{\min} = 10,\quad D_{\max} = 90

$D \in \left[10, 90\right]$

$D = 10$ $D = 10$

Along the curve

③

Response curve — green

\mathrm{TCP}=\exp\!\left(-N_0 e^{-\alpha D-\beta D^{2}}\right)

$D \in \left[10, 90\right]$

$D = 10$ $TCP = 0$

Along the curve

④

Rate of change — amber

\frac{\mathrm{d}TCP}{\mathrm{d}D}

$D \in \left[10, 90\right]$

$D = 10.5$ $TCP^{\prime} = 0$

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-29 — TCP snapshot (Poisson)) is the form associated with Poisson TCP. Working symbols: $N_0$ , $\alpha$ , $\beta$ , $D$ $\rightarrow$ $TCP$ . N0 is the initial clonogen number. A log-kill of 8–10 is a typical curative ambition.

Purpose

Purpose: compute

TCP

from

N_0

\alpha

\beta

D

in Radiotherapy via

\mathrm{TCP}=\exp\!\left(-N_0 e^{-\alpha D-\beta D^{2}}\right)

TCP = exp(−N0 S), S = e^{−αD−βD²}. Zero clonogens left. 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_0 = 1.000e+7\,\mathrm{—}

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

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

D = 60.000\,\mathrm{Gy}

, the governing relation

\mathrm{TCP}=\exp\!\left(-N_0 e^{-\alpha D-\beta D^{2}}\right)

yields

TCP = 1.0000\,\mathrm{—}

. A falling SF and a rising TCP sigmoid. Move a slider: the numbers are this situation, not a canned story.

Calculator

Inputs

Clonogens N0 N_010000000 —Linear α \alpha0.3 1/GyQuadratic β \beta0.03 1/Gy²Dose D60 Gy

Outputs

Control probability TCP1.0000 —

Reading speed

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Narration of this film

A falling SF and a rising TCP sigmoid.

N0 is the initial clonogen number. A log-kill of 8–10 is a typical curative ambition.

Reading speed

Watch on YouTube