RTH-17

Photon build-up

D(z) = Dmax (1 − e^{−z/z0}). Charged-particle equilibrium growing in.

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Beam dataBuild-up

Governing equation

D(z)=D_{\max}\left(1-e^{-z/z_0}\right)

$z \in \left[0, 5\right]$

where

D_{max}: Dose at dmax (Gy)
z: Depth (cm)
z_0: Build-up length (cm)
D(z): Build-up dose (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

D(z)=D_{\max}\left(1-e^{-z/z_0}\right)

$z \in \left[0, 5\right]$

$z = 0$ $D(z) = 0$

Along the curve

②

Domain of the variable — blue

z_{\min} = 0,\quad z_{\max} = 5

$z \in \left[0, 5\right]$

$z = 0$ $z = 0$

Along the curve

③

Response curve — green

D(z)=D_{\max}\left(1-e^{-z/z_0}\right)

$z \in \left[0, 5\right]$

$z = 0$ $D(z) = 0$

Along the curve

④

Rate of change — amber

\frac{\mathrm{d}D(z)}{\mathrm{d}z}

$z \in \left[0, 5\right]$

$z = 0.031$ $D(z)^{\prime} = 3.165$

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-17 — Photon build-up) is the form associated with Build-up. Working symbols: $D_{max}$ , $z$ , $z_0$ $\rightarrow$ $D(z)$ . dmax deepens with energy: ~1.5 cm at 6 MV, ~3 cm at 15 MV. Skin sparing is the clinical gift.

Purpose

Purpose: compute

D(z)

from

D_{max}

z

z_0

in Radiotherapy via

D(z)=D_{\max}\left(1-e^{-z/z_0}\right)

D(z) = Dmax (1 − e^{−z/z0}). Charged-particle equilibrium growing in. Use it when a real radiotherapy question must be answered in SI before a code check.

Live realistic example

In symbols

Live case. Given

D_{max} = 2.000\,\mathrm{Gy}

z = 0.500\,\mathrm{cm}

z_0 = 0.600\,\mathrm{cm}

, the governing relation

D(z)=D_{\max}\left(1-e^{-z/z_0}\right)

yields

D(z) = 1.131\,\mathrm{Gy}

. A rising curve from the skin to dmax. Move a slider: the numbers are this situation, not a canned story.

Calculator

Inputs

Dose at dmax D_{max}2 GyDepth z0.5 cmBuild-up length z_00.6 cm

Outputs

Build-up dose D(z)1.131 Gy

Reading speed

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Narration of this film

A rising curve from the skin to dmax.

dmax deepens with energy: ~1.5 cm at 6 MV, ~3 cm at 15 MV. Skin sparing is the clinical gift.

Reading speed

Watch on YouTube