# Program Information

## Correction of the Translational and Rotational Errors in the Zenith- and Azimuth-Angle Directions

### J Chang^{1}*, (1) Northwell Health, Lake Success, NY

## Presentations

**MO-L-GePD-J(B)-1 (Monday, July 31, 2017) 1:15 PM - 1:45 PM Room: Joint Imaging-Therapy ePoster Lounge - B**

**Purpose:**PTV (planning target volume) margin is traditionally added around the CTV (clinical target volume) in the x-, y- and z-directions that are inconvenient for analyzing and compensating for the rotational errors. The purpose of this study is to develop a new correction scheme that corrects the translational and rotational errors directly in the directions of rotation, i.e., the zenith- and azimuth-angle (aθ and aφ) directions.

**Methods:**The translational (S) and rotational (R) errors algong the aθ and aφ directions were assumed independent random motion vectors following the two-dimensional (2D) normal distribution with zero mean. The combined error S+R also followed the 2D normal distribution with zero mean and standard deviations (SDs) equal to the square root of the sum of the variance of the S and R along the aθ and aφ directions. The probability density function (p.d.f.) of S+R was derived, followed by a change of variables to the polar coordinates and integrating the joint p.d.f. with respect to the angular coordinate to obtain the marginal p.d.f. for random motion P in the radial direction.

**Results:**P follows the Chi distribution with two degrees of freedom (DOF), or P2 follows the Chi-square distribution with two DOF. To make sure the CTV coverage probability is larger than 1-α for significance level α (e.g., 5%), PTV margins were needed in aθ and aφ directions equal to the product of SD of S+R in aθ and aφ directions and the corresponding χ2 value with two DOF at significance level α. Since the SDs of S+R increase with the distance between the CTV and isocenter, the PTV margins also increase with this distance in proportion.

**Conclusion:**A mathematical framework has been developed for correcting setup uncertainties in the direction of rotation. This framework is mathematically simple and conceptually straightforward to implement.

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