RTH-22

Wedge angle

θ = arctan((z_toe − z_heel)/x). Tilt of an isodose between toe and heel.

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WedgesICRU wedge angle

Governing equation

\theta=\arctan\!\left((z_{\mathrm{toe}}-z_{\mathrm{heel}})/x\right)

$z_{toe} \in \left[2, 25\right]$

where

z_{toe}: Toe isodose depth (cm)
z_{heel}: Heel isodose depth (cm)
x: Separation (cm)
\theta: Wedge angle (°)

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

\theta=\arctan\!\left((z_{\mathrm{toe}}-z_{\mathrm{heel}})/x\right)

$z_{toe} \in \left[2, 25\right]$

$z_{toe} = 2$ $\theta = -30.96$

Along the curve

②

Domain of the variable — blue

z_{toe}_{\min} = 2,\quad z_{toe}_{\max} = 25

$z_{toe} \in \left[2, 25\right]$

$z_{toe} = 2$ $z_{toe} = 2$

Along the curve

③

Response curve — green

\theta=\arctan\!\left((z_{\mathrm{toe}}-z_{\mathrm{heel}})/x\right)

$z_{toe} \in \left[2, 25\right]$

$z_{toe} = 2$ $\theta = -30.96$

Along the curve

④

Rate of change — amber

\frac{\mathrm{d}\theta}{\mathrm{d}z_{toe}}

$z_{toe} \in \left[2, 25\right]$

$z_{toe} = 2.145$ $\theta^{\prime} = 4.267$

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-22 — Wedge angle) is the form associated with ICRU wedge angle. Working symbols: $z_{toe}$ , $z_{heel}$ , $x$ $\rightarrow$ $\theta$ . ICRU quotes the angle between the 50 % isodose (often at 10 cm) and the normal to CAX.

Purpose

Purpose: compute

\theta

from

z_{toe}

z_{heel}

x

in Radiotherapy via

\theta=\arctan\!\left((z_{\mathrm{toe}}-z_{\mathrm{heel}})/x\right)

θ = arctan((z_toe − z_heel)/x). Tilt of an isodose between toe and heel. Use it when a real radiotherapy question must be answered in SI before a code check.

Live realistic example

In symbols

Live case. Given

z_{toe} = 12.000\,\mathrm{cm}

z_{heel} = 8.000\,\mathrm{cm}

x = 10.000\,\mathrm{cm}

, the governing relation

\theta=\arctan\!\left((z_{\mathrm{toe}}-z_{\mathrm{heel}})/x\right)

yields

\theta = 21.8\,\mathrm{^{\circ}}

. A tilted isodose across a wedged field. Move a slider: the numbers are this situation, not a canned story.

Calculator

Inputs

Toe isodose depth z_{toe}12 cmHeel isodose depth z_{heel}8 cmSeparation x10 cm

Outputs

Wedge angle \theta21.8 °

Reading speed

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Narration of this film

A tilted isodose across a wedged field.

ICRU quotes the angle between the 50 % isodose (often at 10 cm) and the normal to CAX.

Reading speed

Watch on YouTube