CTDI Weighting, Pitch and DLP Calculations

Mathematical Formulations: CTDI100, CTDIw, CTDIvol & DLP

1. $CTDI_{100}$ (Single-Axis 100 mm Index)

$CTDI_{100}$ represents the absorbed dose integrated over the $100\text{ mm}$ length of the pencil chamber divided by the total nominal beam collimation width for a single axial gantry rotation:

$$CTDI_{100} = \frac{1}{N \times T} \int_{-50\text{ mm}}^{+50\text{ mm}} D(z) \, dz$$

• $N$: Number of simultaneous tomographic channels / detector rows acquired in a single rotation.
• $T$: Nominal physical thickness of each acquired tomographic slice.
• $N \times T$: Total nominal collimation width of the primary x-ray beam along the z-axis (in $\text{mm}$).
• $D(z)$: Longitudinal dose distribution profile as a function of position $z$.
• Units: Milligray ($\text{mGy}$).
• Measurements: Acquired separately at the central axis ($CTDI_{100,\text{center}}$) and in the four peripheral holes, which are averaged ($CTDI_{100,\text{periphery}}$).

2. Weighted CTDI ($CTDI_w$)

Because absorbed dose varies radially across the phantom, $CTDI_w$ combines the central and peripheral measurements into a single value representing the average absorbed dose within a single cross-sectional transverse slice:

$$CTDI_w = \frac{1}{3} CTDI_{100, \text{center}} + \frac{2}{3} CTDI_{100, \text{periphery}}$$

The one-third central and two-thirds peripheral weights approximate a cross-sectional phantom average. They are conventional weighting factors, not an exact patient-organ dose calculation.

3. Volume CTDI ($CTDI_{\text{vol}}$)

While $CTDI_w$ quantifies average dose within a single contiguous rotation, modern clinical protocols acquire continuous helical volumes or axial scans with variable spacing. Volume CTDI normalizes $CTDI_w$ for table translation speed by dividing by Pitch:

$$CTDI_{\text{vol}} = \frac{CTDI_w}{\text{Pitch}} = CTDI_w \times \frac{N \times T}{I}$$

• Pitch ($P$): Defined as table travel per gantry rotation ($I$) divided by total beam collimation ($N \times T$):

$$\text{Pitch} = \frac{I}{N \times T}$$

• Dose Scaling with Pitch:
  • When $\text{Pitch} = 1.0$: $CTDI_{\text{vol}} = CTDI_w$.
  • When $\text{Pitch} > 1.0$ (pitch extension): The helical x-ray helix stretches, gaps exist between turns, and $CTDI_{\text{vol}}$ decreases inversely with pitch (provided tube current is not automatically compensated).
  • When $\text{Pitch} < 1.0$ (overlapping helix): Overlapping radiation loops increase $CTDI_{\text{vol}}$ above $CTDI_w$.

DLP is a dose-length index in mGy·cm, not energy in joules. For an illustrative constant-output series, multiply CTDIvol by the stated scan length. Scanner-reported DLP and the accounting for overranging follow the implemented dose-reporting method; do not assume every displayed range is identical to the irradiated extent.

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Step-by-Step Worked Clinical Calculations

Worked Example 1: Calculating $CTDI_w$ from Acrylic Phantom Readings

During routine annual physics testing of an adult body protocol, a medical physicist measures the following $CTDI_{100}$ values in a $32\text{ cm}$ PMMA phantom:

• Center hole ($CTDI_{100,\text{center}}$): $11.4\text{ mGy}$
• 12:00 peripheral hole: $24.8\text{ mGy}$
• 3:00 peripheral hole: $25.6\text{ mGy}$
• 6:00 peripheral hole: $24.2\text{ mGy}$
• 9:00 peripheral hole: $25.4\text{ mGy}$

Step 1: Calculate the average peripheral dose ($CTDI_{100,\text{periphery}}$):

$$CTDI_{100,\text{periphery}} = \frac{24.8 + 25.6 + 24.2 + 25.4}{4} = \frac{100.0}{4} = 25.0\text{ mGy}$$

Step 2: Apply the $1/3$ to $2/3$ weighting formula:

$$CTDI_w = \frac{1}{3}(11.4\text{ mGy}) + \frac{2}{3}(25.0\text{ mGy}) = 3.8\text{ mGy} + 16.67\text{ mGy} = 20.47\text{ mGy}$$

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Worked Example 2: Calculating $CTDI_{\text{vol}}$ with Pitch Adjustment

The technologist configures a helical scan using the protocol from Example 1 ($CTDI_w = 20.47\text{ mGy}$). The beam collimation is $64 \times 0.625\text{ mm} = 40\text{ mm}$, and the table feed is set to $50\text{ mm}$ per gantry rotation.

Step 1: Calculate the helical pitch:

$$\text{Pitch} = \frac{I}{N \times T} = \frac{50\text{ mm}}{40\text{ mm}} = 1.25$$

Step 2: Calculate $CTDI_{\text{vol}}$:

$$CTDI_{\text{vol}} = \frac{CTDI_w}{\text{Pitch}} = \frac{20.47\text{ mGy}}{1.25} = 16.38\text{ mGy}$$

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Worked Example 3: Calculating Total DLP for a Multiphase Abdomen/Pelvis Scan

A patient undergoes a multiphase liver CT consisting of an unenhanced liver series followed by a portal venous phase abdomen/pelvis series:

• Series 1 (Liver only): $CTDI_{\text{vol}} = 18.0\text{ mGy}$, scan length $L = 22\text{ cm}$.
• Series 2 (Abdomen/Pelvis): $CTDI_{\text{vol}} = 16.0\text{ mGy}$, scan length $L = 48\text{ cm}$.

Step 1: Calculate DLP for Series 1:

$$DLP_1 = 18.0\text{ mGy} \times 22\text{ cm} = 396\text{ mGy} \cdot \text{cm}$$

Step 2: Calculate DLP for Series 2:

$$DLP_2 = 16.0\text{ mGy} \times 48\text{ cm} = 768\text{ mGy} \cdot \text{cm}$$

Step 3: Calculate cumulative examination DLP:

$$DLP_{\text{total}} = 396 + 768 = 1164\text{ mGy} \cdot \text{cm}$$

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Physics reference: AAPM CT dosimetry guidance.