RTH-04

Tissue–maximum ratio

TMR(d) = D(d)/D(dmax) at fixed SAD. The SAD cousin of PDD.

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Beam dataKhan TMR

Governing equation

\mathrm{TMR}(d)=e^{-\mu(d-d_{\max})}

$d \in \left[0.5, 40\right]$

where

\mu: Attenuation μ (1/cm)
d: Depth (cm)
d_{max}: Depth of maximum (cm)
TMR: Tissue–maximum ratio (—)

The figure, part by part

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①

Governing law — red

\mathrm{TMR}(d)=e^{-\mu(d-d_{\max})}

$d \in \left[0.5, 40\right]$

$d = 0.5$ $TMR = 1.051$

Along the curve

②

Domain of the variable — blue

d_{\min} = 0.5,\quad d_{\max} = 40

$d \in \left[0.5, 40\right]$

$d = 0.5$ $d = 0.5$

Along the curve

③

Response curve — green

\mathrm{TMR}(d)=e^{-\mu(d-d_{\max})}

$d \in \left[0.5, 40\right]$

$d = 0.5$ $TMR = 1.051$

Along the curve

④

Rate of change — amber

\frac{\mathrm{d}TMR}{\mathrm{d}d}

$d \in \left[0.5, 40\right]$

$d = 0.748$ $TMR^{\prime} = -0.052$

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-04 — Tissue–maximum ratio) is the form associated with Khan TMR. Working symbols: $\mu$ , $d$ , $d_{max}$ $\rightarrow$ $TMR$ . Scatter is referred to the field size at depth. TMR ≈ exp(−μ(d−dmax)) for a first look.

Purpose

Purpose: compute

TMR

from

\mu

d

d_{max}

in Radiotherapy via

\mathrm{TMR}(d)=e^{-\mu(d-d_{\max})}

TMR(d) = D(d)/D(dmax) at fixed SAD. The SAD cousin of PDD. Use it when a real radiotherapy question must be answered in SI before a code check.

Live realistic example

In symbols

Live case. Given

\mu = 0.050\,\mathrm{1/cm}

d = 10.000\,\mathrm{cm}

d_{max} = 1.500\,\mathrm{cm}

, the governing relation

\mathrm{TMR}(d)=e^{-\mu(d-d_{\max})}

yields

TMR = 0.6538\,\mathrm{—}

. An isocentric phantom, two depths, a ratio. Move a slider: the numbers are this situation, not a canned story.

Calculator

Inputs

Attenuation μ \mu0.05 1/cmDepth d10 cmDepth of maximum d_{max}1.5 cm

Outputs

Tissue–maximum ratio TMR0.6538 —

Reading speed

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Narration of this film

An isocentric phantom, two depths, a ratio.

Scatter is referred to the field size at depth. TMR ≈ exp(−μ(d−dmax)) for a first look.

Reading speed

Watch on YouTube