# A Derivation of the PDD-TPR Conversion Equation

### Intro & Purpose

Both the percent depth dose (PDD) and the tissue phantom ratio (TPR) describe how the delivered dose changes along the beam's central axis as a function of depth. While either equation can be used to calculate monitor units (MU) or dose using the corrections-based formulae, the PDD is ideal for surface-based treatments while the TPR is ideal for isocentric-based calculations. This is primarily because of the way they are measured. For PDD measurements, the SSD is kept constant and the detector moves to measure dose vs. depth. On the other hand, for TPR measurements the SAD is kept constant, meaning that the water tank must be moved (or filled/drained) to measure dose vs. depth. One could measure TPR values, but typically in practice only PDD values are measured, and then converted to TPR values. In this lab we will derive the relationship between the two depth-dose ratios.

### Setup Conditions & Question

Consider the two setups shown in the figure below. Setup 1 by itself is adequate for PDD measurement: keep the SDD (f) constant, move the detector to different depths d, and measure the dose relative to that at a reference depth d_0. To measure the TPR we need to compare the dose at depth d to a reference depth d_0, but with the SAD kept constant. So we also need setup 2, which displays the measurement conditions for TPR reference depth dose. Given this setup and measurements, derive the relationship between the TPR and PDD:

TPR(d,r_d) = \frac{PDD(d,r,f)}{100} \frac{S_p(r_{d_0})}{S_p(r_d)} \frac{(f+d)^2}{(f+d_0)^2}

### Solution

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