3.5 Depth Dose Characteristics: PDD, TMR, TPR & TAR Relationships

Central Axis Depth Dose Physics

As a high-energy photon beam traverses biological tissue, energy is transferred to electrons via photoelectric absorption, Compton scattering, and pair production. These energetic recoil electrons travel through tissue, depositing absorbed dose along their paths. The dose profile along the central axis of the beam exhibits a characteristic curve comprising a build-up region, a point of maximum dose ($d_{\max}$), and an exponential attenuation tail at depth.

100% |       + dmax
     |      / \
80%  |     /   \   Exponential Attenuation Tail
     |    /     \... 
60%  |   /          \...
     |  /  Build-up    \...
40%  | /   Region         \...
     +----------------------------------------> Depth (cm)
     0   dmax (1.5 cm)     10 cm       20 cm

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Depth of Maximum Dose ($d_{\max}$) and Electronic Equilibrium

$d_{\max}$ represents the depth at which kerma (Kinetic Energy Released per Unit Mass) equals absorbed dose, establishing longitudinal charged particle equilibrium (CPE). In the build-up region between the skin surface and $d_{\max}$, kerma is maximum at the surface and decreases exponentially, while absorbed dose rises rapidly as secondary electrons are generated and travel forward into deeper tissue layers. Beyond $d_{\max}$, absorbed dose decreases exponentially due to photon attenuation.

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Percentage Depth Dose (PDD) Characteristics

Mathematical Definition

Percentage Depth Dose (PDD) is defined as the ratio of absorbed dose at depth ($d$) to absorbed dose at reference depth ($d_{\max}$) along the central axis of the beam, expressed as a percentage. PDD is measured at a fixed Source-to-Skin Distance (SSD) (typically 100 cm SSD) in a water phantom.

$$\text{PDD}(d, r, f) = \left( \frac{D_d}{D_{d_{\max}}} \right) \times 100\%$$

Where $d$ is depth, $r$ is field size at the skin surface, and $f$ is SSD.

Physical Parameters Influencing PDD

1. Beam Energy (Directly Proportional): Higher energy photons possess greater penetrating power (lower linear attenuation coefficient $\mu$), shifting $d_{\max}$ deeper into tissue and increasing PDD at depth.
2. Depth (Inversely Proportional): Beyond $d_{\max}$, photon attenuation and beam divergence cause PDD to decay exponentially with increasing depth.
3. Field Size (Directly Proportional): As field size increases, a larger volume of surrounding phantom tissue is irradiated, generating increased phantom scatter radiation reaching the central axis, which increases PDD.
4. Source-Skin Distance (Directly Proportional): Increasing SSD reduces beam divergence according to the Inverse Square Law over the depth interval between $d_{\max}$ and depth $d$, which increases PDD.

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Mayneord F-Factor for Extended Distance Calculations

When a patient is treated at an extended SSD (e.g., treating a total body irradiation or whole-leg sarcoma field at 120 cm SSD instead of 100 cm SSD), PDD increases. The Mayneord F-Factor is derived from the Inverse Square Law to calculate the new PDD ($PDD_2$) at extended distance $f_2$ from the known PDD ($PDD_1$) at distance $f_1$:

$$F = \left( \frac{f_2 + d_{\max}}{f_1 + d_{\max}} \right)^2 \times \left( \frac{f_1 + d}{f_2 + d} \right)^2$$ $$PDD_2 = PDD_1 \times F$$

Clinical Worked Example: A 6 MV photon beam ($d_{\max} = 1.5\text{ cm}$) delivers a PDD of 67.0% at depth $d = 10\text{ cm}$ for $f_1 = 100\text{ cm}$ SSD. Calculate the new PDD at an extended distance $f_2 = 120\text{ cm}$ SSD:

$$F = \left( \frac{120 + 1.5}{100 + 1.5} \right)^2 \times \left( \frac{100 + 10}{120 + 10} \right)^2 = \left( \frac{121.5}{101.5} \right)^2 \times \left( \frac{110}{130} \right)^2 = (1.197)^2 \times (0.846)^2 = 1.433 \times 0.716 = 1.026$$ $$PDD_2 = 67.0\% \times 1.026 = 68.7\%$$

Note on Mayneord F-Factor Limitations: The Mayneord F-Factor works well for small field sizes at moderate depths, but overestimates the PDD increase for large field sizes and deep depths because it ignores the increased phantom scatter generated at extended distances.

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Iso-Distance Dosimetric Parameters: TMR, TPR, and TAR

Tissue Maximum Ratio (TMR)

In Source-to-Axis Distance (SAD) setups (isocentric technique), the machine isocenter remains at a fixed distance (100 cm) while patient depth varies. Using PDD is cumbersome because SSD changes with gantry rotation. Tissue Maximum Ratio (TMR) is defined as the ratio of absorbed dose at depth $d$ to absorbed dose at $d_{\max}$, measured at a constant distance from the radiation source.

$$\text{TMR}(d, r_d) = \frac{D_d}{D_{d_{\max}}}$$

Fundamental Physics Principle: TMR is completely independent of Source-to-Skin Distance (SSD). TMR depends exclusively on beam energy, depth $d$, and field size projected at depth $r_d$.

Tissue Phantom Ratio (TPR) and Beam Quality Specifier ($TPR_{20,10}$)

Tissue Phantom Ratio (TPR) is the ratio of dose at depth $d$ to dose at a fixed reference depth $d_{\text{ref}}$ (typically 5 cm or 10 cm) at a constant source-to-detector distance.

$$\text{TPR}(d, d_{\text{ref}}, r_d) = \frac{D_d}{D_{d_{\text{ref}}}}$$

The ratio of TPR at 20 cm depth to TPR at 10 cm depth in water ($TPR_{20,10}$) serves as the international standard beam quality index under IAEA TRS-398 and AAPM TG-51 protocols to specify photon beam energy independently of electron contamination.

Tissue Air Ratio (TAR)

Tissue Air Ratio (TAR) is the ratio of absorbed dose at depth $d$ in phantom to absorbed dose at the same point in free space (air). TAR is independent of SSD and is used primarily for low-energy photons, Cobalt-60, and radionuclide calculations. At high photon energies (> 10 MV), measuring dose in free space becomes physically unreliable due to large electron contamination cap requirements.