Three-dimensional prostate position estimation with a single x-ray imager utilizing the spatial probability density

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In radiotherapy, target motion during treatment delivery can be managed either by motion inclusive margins or by gating or tracking based on intrafraction target position monitoring. If radio-opaque fiducial markers are used the required three-dimensional (3D) target position signal for gating or tracking can be obtained by simultaneous acquisition of two x-ray images from different angles. However, most treatment machines do not have such stereoscopic imaging capability. Alternatively, the 3D target position may be estimated with a single imager (monoscopic imaging) although it only provides the projected target position in the two dimensions of the imager plane. In this study, we developed a probability-based method to estimate the unresolved motion component parallel to the imager axis from the projected motion. A 3D Gaussian probability density was assumed for the target position. Projection of the target into a certain point on the imager means that it is located on the ray line that connects this point with the focus point of the x-ray source. The 1D probability density along this line was calculated from the 3D probability density and its expectation value was used as the estimate for the unresolved position. The mathematical framework of the method was developed including analytical expressions for the estimated unresolved component as a function of resolved components and for the estimation uncertainty. Use of the method was demonstrated for prostate in a simulation study of monoscopic imaging. First, the required 3D probability density was constructed as a population average from a data set consisting of 536 continuous prostate position tracks from 17 patients recorded at 10 Hz. Next, monoscopic imaging at a fixed imaging angle and imaging frequency was simulated for each prostate track. Estimated 3D prostate tracks were constructed from the simulated projection images by the proposed method and compared with the actual tracks in order to determine the root-mean-square (rms) error. The simulations were performed with imaging angles in the range from 0° to 180° (relative to vertical) and imaging frequencies in the range from 0.1 s (corresponding to continuous imaging) to 600 s (corresponding to no intrafraction imaging). For comparison, simulations were also performed with stereoscopic imaging, where perfect position determination in all three directions was assumed, and with monoscopic imaging without estimation of the unresolved motion, where the motion component along the imager axis was assumed to be zero. For continuous imaging, the accuracy of monoscopic imaging was limited by the uncertainty in the unresolved position estimation. The resulting vector rms error for the population corresponded closely to the theoretically derived estimation uncertainty. The estimation did not improve the accuracy of lateral monoscopic imaging, but it reduced the population rms error from 1.59 mm to 1.11 mm for vertical imaging. This improvement was most prominent for outlying tracks with large unresolved motion. Stereoscopic imaging was clearly superior to monoscopic imaging for high frequency imaging. For less frequent imaging, the accuracy of both monoscopic and stereoscopic imaging decreased due to target motion between images. Since this was most prominent for stereoscopic imaging, the difference in accuracy between monoscopic and stereoscopic imaging decreased with increasing imaging period. In conclusion, a method for estimation of the 3D target position from 2D projections has been developed and its use has been demonstrated in a simulation study of monoscopic prostate tracking.