01 Therapy

Radiological path length

Effective water depth d_eff = Σ ρ_i Δx_i (physical-density or relative-stopping-power scaling).

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Simulation

Radiological path length — Change the numbers; the scene follows.

Where it works

Proton gantry

Nozzle / snout

At the snout: residual proton range after the range shifter and aperture.

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Formula

d_{\mathrm{eff}}=\sum_i \rho_i^{\mathrm{rel}}\Delta x_i

Variables

Δx₁ · Thickness 1cmρ₁ · Relative density 1Δx₂ · Thickness 2cmρ₂ · Relative density 2Δx₃ · Thickness 3cmρ₃ · Relative density 3

Results

d_eff
Water-equivalent depth
8cm
d_phys
Physical depth
14cm
d_eff/d_phys
Ratio
0.5714

Explanation

d_{\mathrm{eff}}=\sum_i \rho_i^{\mathrm{rel}}\Delta x_i

What it means

Heterogeneity correction in a 1-D ray: each slab of thickness Δx and relative density (or relative stopping power, RSP, for protons) contributes ρ Δx to the water-equivalent path. Lung (ρ≈0.25) shortens the effective depth — dose goes up, range goes longer. Bone (ρ≈1.4–1.8) does the opposite. MV photon Batho/equivalent-TAR and proton RSP all start from this sum. This is a working relation in Radiotherapy.

Where it is used

Clinically it sits on the Proton gantry — Nozzle / snout. At the snout: residual proton range after the range shifter and aperture. 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.

Proton gantry · Open this machine

How to use it

Enter up to three slabs (thickness + relative density). Unused slabs: set Δx=0. Example: 4 cm tissue (1.0) + 8 cm lung (0.25) + 2 cm tissue = 4+2+2 = 8 cm water — a 14 cm physical depth. Change one input and watch the curve and the simulation follow.

Symbols

Δx₁Thickness 14 cm
ρ₁Relative density 11
Δx₂Thickness 28 cm
ρ₂Relative density 20.25
Δx₃Thickness 32 cm
ρ₃Relative density 31

Worked example

A typical case from the default values: Δx₁ = 4 cm (Thickness 1); ρ₁ = 1 (Relative density 1); Δx₂ = 8 cm (Thickness 2); ρ₂ = 0.25 (Relative density 2); Δx₃ = 2 cm (Thickness 3); ρ₃ = 1 (Relative density 3). Substituting into the relation gives d_eff = 8 cm; d_phys = 14 cm; d_eff/d_phys = 0.5714. These are teaching numbers — align them with your machine.

Typical values give

d_eff = 8cm
d_phys = 14cm
d_eff/d_phys = 0.5714

Where it comes from

The displayed formula is the working relation. Effective water depth d_eff = Σ ρ_i Δx_i (physical-density or relative-stopping-power scaling). Usual reference: Khan / ICRU 78 (RSP for protons). Derive it in the specialty lesson, then return here to pin the numbers.

Reference: Khan / ICRU 78 (RSP for protons)

Assumptions & limits

1-D equivalent path. Does not transport scatter (photons) or nuclear secondaries (protons). Relative electron density for photons ≠ RSP for protons ≠ relative stopping power for carbon. Use the quantity matching the beam.

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. 1-D equivalent path. Does not transport scatter (photons) or nuclear secondaries (protons). Relative electron density for photons ≠ RSP for protons ≠ relative stopping power for carbon. Use the quantity matching the beam.

Keep this

Name the SSD, energy, and field size with every PDD or TMR you quote. 1-D equivalent path. Does not transport scatter (photons) or nuclear secondaries (protons). Relative electron density for photons ≠ RSP for protons ≠ relative stopping power for carbon. Use the quantity matching the beam.

In this specialty

Radiotherapy