Gaussian process regression of HDR prostate biopsy dose for improved modelling of localization-induced uncertainty

Biopsy-level dose interpretation after high-dose-rate prostate brachytherapy is limited by localization uncertainty, and separate voxelwise Monte Carlo summaries do not explicitly account for the axial spatial correlation present along a biopsy core.

To develop and internally evaluate a biopsy-level Gaussian process regression framework for assigning along-core high-dose-rate prostate brachytherapy dose under localization uncertainty, and to test whether explicit modelling of axial spatial correlation improves uncertainty characterization relative to voxelwise Monte Carlo summaries treated independently.

We analyzed 27 MR-informed, transrectal-ultrasound (TRUS)-guided transperineal template biopsies from 15 patients treated with high-dose-rate prostate brachytherapy. For each biopsy voxel, Monte Carlo localization perturbations were propagated to generate dose summaries and voxel-specific variances. Biopsy-specific empirical semivariograms were fitted with candidate covariance kernels, and ordinary-mean heteroscedastic Gaussian process posteriors were computed along each core. Internal evaluation included latent-profile residual diagnostics, kernel-sensitivity analysis, and blocked cross-validation using contiguous held-out axial segments scored on held-out points only.

For the selected baseline kernel, Matérn $\nu =1.5$ , the across-biopsy median voxel-averaged standard deviation was 6.91 Gy for the Monte Carlo summaries and 2.60 Gy for the latent GP posterior. The median within-biopsy reduction was 59.1%, with a median shrinkage ratio of 2.45. Fitted length scales were heterogeneous across biopsies, with median ${\hat{\ell }}_b=15.35$ mm, while nugget fractions were typically small. Cohort standardized residuals were near-centered with conservative spread. In blocked cross-validation, predictive performance on held-out points remained favorable, with median root-mean-square error of 1.99 Gy, median mean absolute error of 1.58 Gy, and median negative log predictive density of 3.28. Among the tested kernels, held-out predictive error favored Matérn $\nu =1.5$ .

Along-core dose interpretation after localization uncertainty should use a spatially correlated model rather than separate voxelwise summaries alone. This Gaussian process framework provides a covariance-informed posterior dose representation with held-out internal validation and offers a practical basis for biopsy-level probabilistic quality assurance and future dose-response analyses.