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Summary This paper introduces an assumption-lean method that constructs valid and efficient lower predictive bounds for survival times with censored data. We build on recent work by Candès et al. (2023), whose approach first subsets the data to discard any data points with early censoring times and then uses a reweighting technique, namely, weighted conformal inference (Tibshirani et al., 2019), to correct for the distribution shift introduced by this subsetting procedure. For our new method, instead of constraining to a fixed threshold for the censoring time when subsetting the data, we allow for a covariate-dependent and data-adaptive subsetting step, which is better able to capture the heterogeneity of the censoring mechanism. As a result, our method can lead to lower predictive bounds that are less conservative and give more accurate information. We show that in the Type-I right-censoring setting, if either the censoring mechanism or the conditional quantile of the survival time is well estimated, our proposed procedure achieves nearly exact marginal coverage, where in the latter case we additionally have approximate conditional coverage. We evaluate the validity and efficiency of our proposed algorithm in numerical experiments, illustrating its advantage when compared with other competing methods. Finally, our method is applied to a real dataset to generate lower predictive bounds for users’ active times on a mobile app. 

Abstract PurposeThe constrained one‐step spectral CT image reconstruction (cOSSCIR) algorithm with a nonconvex alternating direction method of multipliers optimizer is proposed for addressing computed tomography (CT) metal artifacts caused by beam hardening, noise, and photon starvation. The quantitative performance of cOSSCIR is investigated through a series of photon‐counting CT simulations. MethodscOSSCIR directly estimates basis material maps from photon‐counting data using a physics‐based forward model that accounts for beam hardening. The cOSSCIR optimization framework places constraints on the basis maps, which we hypothesize will stabilize the decomposition and reduce streaks caused by noise and photon starvation. Another advantage of cOSSCIR is that the spectral data need not be registered, so that a ray can be used even if some energy window measurements are unavailable. Photon‐counting CT acquisitions of a virtual pelvic phantom with low‐contrast soft tissue texture and bilateral hip prostheses were simulated. Bone and water basis maps were estimated using the cOSSCIR algorithm and combined to form a virtual monoenergetic image for the evaluation of metal artifacts. The cOSSCIR images were compared to a “two‐step” decomposition approach that first estimated basis sinograms using a maximum likelihood algorithm and then reconstructed basis maps using an iterative total variation constrained least‐squares optimization (MLE+TV). Images were also compared to a nonspectral TV reconstruction of the total number of counts detected for each ray with and without normalized metal artifact reduction (NMAR) applied. The simulated metal density was increased to investigate the effects of increasing photon starvation. The quantitative error and standard deviation in regions of the phantom were compared across the investigated algorithms. The ability of cOSSCIR to reproduce the soft‐tissue texture, while reducing metal artifacts, was quantitatively evaluated. ResultsNoiseless simulations demonstrated the convergence of the cOSSCIR and MLE+TV algorithms to the correct basis maps in the presence of beam‐hardening effects. When noise was simulated, cOSSCIR demonstrated a quantitative error of −1 HU, compared to 2 HU error for the MLE+TV algorithm and −154 HU error for the nonspectral TV+NMAR algorithm. For the cOSSCIR algorithm, the standard deviation in the central iodine region of interest was 20 HU, compared to 299 HU for the MLE+TV algorithm, 41 HU for the MLE+TV+Mask algorithm that excluded rays through metal, and 55 HU for the nonspectral TV+NMAR algorithm. Increasing levels of photon starvation did not impact the bias or standard deviation of the cOSSCIR images. cOSSCIR was able to reproduce the soft‐tissue texture when an appropriate regularization constraint value was selected. ConclusionsBy directly inverting photon‐counting CT data into basis maps using an accurate physics‐based forward model and a constrained optimization algorithm, cOSSCIR avoids metal artifacts due to beam hardening, noise, and photon starvation. The cOSSCIR algorithm demonstrated improved stability and accuracy compared to a two‐step method of decomposition followed by reconstruction.