RTH-13

Off-axis dose

D(x) = D0 exp(−½ (x/σ)²). A Gaussian look at an off-axis point.

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ProfilesOff-axis dose

Governing equation

D(x)=D_0\exp\!\left(-\tfrac12(x/\sigma)^2\right)

$x \in \left[0, 20\right]$

where

D_0: Central-axis dose (Gy)
x: Off-axis distance (cm)
\sigma: Profile width σ (cm)
D(x): Off-axis 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(x)=D_0\exp\!\left(-\tfrac12(x/\sigma)^2\right)

$x \in \left[0, 20\right]$

$x = 0$ $D(x) = 2$

Along the curve

②

Domain of the variable — blue

x_{\min} = 0,\quad x_{\max} = 20

$x \in \left[0, 20\right]$

$x = 0$ $x = 0$

Along the curve

③

Response curve — green

D(x)=D_0\exp\!\left(-\tfrac12(x/\sigma)^2\right)

$x \in \left[0, 20\right]$

$x = 0$ $D(x) = 2$

Along the curve

④

Rate of change — amber

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

$x \in \left[0, 20\right]$

$x = 0.126$ $D(x)^{\prime} = -0.004$

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-13 — Off-axis dose) is the form associated with Off-axis dose. Working symbols: $D_0$ , $x$ , $\sigma$ $\rightarrow$ $D(x)$ . Inside a flattened field the profile is almost flat; near the penumbra it falls as a Gaussian-like edge.

Purpose

Purpose: compute

D(x)

from

D_0

x

\sigma

in Radiotherapy via

D(x)=D_0\exp\!\left(-\tfrac12(x/\sigma)^2\right)

D(x) = D0 exp(−½ (x/σ)²). A Gaussian look at an off-axis point. 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_0 = 2.000\,\mathrm{Gy}

x = 4.000\,\mathrm{cm}

\sigma = 8.000\,\mathrm{cm}

, the governing relation

D(x)=D_0\exp\!\left(-\tfrac12(x/\sigma)^2\right)

yields

D(x) = 1.765\,\mathrm{Gy}

. A profile, a CAX peak, an off-axis sample. Move a slider: the numbers are this situation, not a canned story.

Calculator

Inputs

Central-axis dose D_02 GyOff-axis distance x4 cmProfile width σ \sigma8 cm

Outputs

Off-axis dose D(x)1.765 Gy

Reading speed

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Narration of this film

A profile, a CAX peak, an off-axis sample.

Inside a flattened field the profile is almost flat; near the penumbra it falls as a Gaussian-like edge.

Reading speed

Watch on YouTube