01 Therapy

Equivalent square

Map a rectangular field to a square of similar scatter (Sterling).

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Simulation

Equivalent square — Change the numbers; the scene follows.

Where it works

Linear accelerator

Collimator / MLC

At the jaws / MLC: field size on the collimator sets equivalent square, penumbra, and Sc,p.

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Formula

s = \frac{4A}{P} = \frac{2ab}{a+b}

Variables

a · Lengthcmb · Widthcm

Results

s
Equivalent square
9.6cm
A
Area
96cm²
P
Perimeter
40cm

Explanation

s = \frac{4A}{P} = \frac{2ab}{a+b}

What it means

Sterling’s equivalent square s = 4A/P = 2ab/(a+b) maps a rectangle to a square that produces nearly the same scatter (hence similar PDD, TMR, and Sp). A 10×15 field ≈ 12 cm square. This is a working relation in Radiotherapy.

Where it is used

Clinically it sits on the Linear accelerator — Collimator / MLC. At the jaws / MLC: field size on the collimator sets equivalent square, penumbra, and Sc,p. Radiotherapy equations sit at the console and in the bunker: output, depth dose, equivalent square, and the monitor units that treat the patient. Hand-calc them beside the TPS, never instead of a commissioned plan.

Linear accelerator · Open this machine

How to use it

Enter length and width of the collimator opening (or blocked equivalent). Use s to look up output factors and PDD. For circles, s ≈ 0.9 × diameter is sometimes used. Change one input and watch the curve and the simulation follow.

Symbols

aLength12 cm
bWidth8 cm

Worked example

A typical case from the default values: a = 12 cm (Length); b = 8 cm (Width). Substituting into the relation gives s = 9.6 cm; A = 96 cm²; P = 40 cm. These are teaching numbers — align them with your machine.

Typical values give

s = 9.6cm
A = 96cm²
P = 40cm

Where it comes from

The displayed formula is the working relation. Map a rectangular field to a square of similar scatter (Sterling). Usual reference: Sterling et al. / Khan. Derive it in the specialty lesson, then return here to pin the numbers.

Reference: Sterling et al. / Khan

Assumptions & limits

Rectangles only. Irregular and MLC fields need Clarkson sector integration or a TPS equivalent-square. Elongation beyond ~4:1 degrades the approximation.

Pitfalls

Never mix PDD from one SSD with TMR from another without converting. Field size at the surface is not the size at isocentre. A hand MU is a check, not a treatment. Rectangles only. Irregular and MLC fields need Clarkson sector integration or a TPS equivalent-square. Elongation beyond ~4:1 degrades the approximation.

Keep this

For rectangles. Irregular fields need Clarkson or sector integration.

In this specialty

Radiotherapy