The asymptotics of group Russian roulette

We study the group Russian roulette problem, also known as the shooting problem, defined as follows. We have $n$ armed people in a room. At each chime of a clock, everyone shoots a random other person. The persons shot fall dead and the survivors shoot again at the next chime. Eventually, either everyone is dead or there is a single survivor. We prove that the probability $p_n$ of having no survivors does not converge as $n\to\infty$, and becomes asymptotically periodic and continuous on the $\log n$ scale, with period 1.Comment: 26 pages, 1 figure; Mathematica notebook and output file (calculated exact bounds) are included with the source file

Random walk loop soups and conformal loop ensembles

The random walk loop soup is a Poissonian ensemble of lattice loops; it has been extensively studied because of its connections to the discrete Gaussian free field, but was originally introduced by Lawler and Trujillo Ferreras as a discrete version of the Brownian loop soup of Lawler and Werner, a conformally invariant Poissonian ensemble of planar loops with deep connections to conformal loop ensembles (CLEs) and the Schramm-Loewner evolution (SLE). Lawler and Trujillo Ferreras showed that, roughly speaking, in the continuum scaling limit, ``large'' lattice loops from the random walk loop soup converge to ``large'' loops from the Brownian loop soup. Their results, however, do not extend to clusters of loops, which are interesting because the connection between Brownian loop soup and CLE goes via cluster boundaries. In this paper, we study the scaling limit of clusters of ``large'' lattice loops, showing that they converge to Brownian loop soup clusters. In particular, our results imply that the collection of outer boundaries of outermost clusters composed of ``large'' lattice loops converges to CLE.Comment: 30 pages, 7 figures, to appear in Probab. Theory Related Field

Stochastic domination and weak convergence of conditioned Bernoulli random vectors

For n>=1 let X_n be a vector of n independent Bernoulli random variables. We assume that X_n consists of M "blocks" such that the Bernoulli random variables in block i have success probability p_i. Here M does not depend on n and the size of each block is essentially linear in n. Let X'_n be a random vector having the conditional distribution of X_n, conditioned on the total number of successes being at least k_n, where k_n is also essentially linear in n. Define Y'_n similarly, but with success probabilities q_i>=p_i. We prove that the law of X'_n converges weakly to a distribution that we can describe precisely. We then prove that sup Pr(X'_n <= Y'_n) converges to a constant, where the supremum is taken over all possible couplings of X'_n and Y'_n. This constant is expressed explicitly in terms of the parameters of the system.Comment: 39 pages, 2 figure

Fat fractal percolation and k-fractal percolation

We consider two variations on the Mandelbrot fractal percolation model. In the k-fractal percolation model, the d-dimensional unit cube is divided in N^d equal subcubes, k of which are retained while the others are discarded. The procedure is then iterated inside the retained cubes at all smaller scales. We show that the (properly rescaled) percolation critical value of this model converges to the critical value of ordinary site percolation on a particular d-dimensional lattice as N tends to infinity. This is analogous to the result of Falconer and Grimmett that the critical value for Mandelbrot fractal percolation converges to the critical value of site percolation on the same d-dimensional lattice. In the fat fractal percolation model, subcubes are retained with probability p_n at step n of the construction, where (p_n) is a non-decreasing sequence with \prod p_n > 0. The Lebesgue measure of the limit set is positive a.s. given non-extinction. We prove that either the set of connected components larger than one point has Lebesgue measure zero a.s. or its complement in the limit set has Lebesgue measure zero a.s.Comment: 27 pages, 3 figure

Fast marginal likelihood estimation of penalties for group-adaptive elastic net

Nowadays, clinical research routinely uses omics data, such as gene expression, for predicting clinical outcomes or selecting markers. Additionally, so-called co-data are often available, providing complementary information on the covariates, like p-values from previously published studies or groups of genes corresponding to pathways. Elastic net penalisation is widely used for prediction and covariate selection. Group-adaptive elastic net penalisation learns from co-data to improve the prediction and covariate selection, by penalising important groups of covariates less than other groups. Existing methods are, however, computationally expensive. Here we present a fast method for marginal likelihood estimation of group-adaptive elastic net penalties for generalised linear models. We first derive a low-dimensional representation of the Taylor approximation of the marginal likelihood and its first derivative for group-adaptive ridge penalties, to efficiently estimate these penalties. Then we show by using asymptotic normality of the linear predictors that the marginal likelihood for elastic net models may be approximated well by the marginal likelihood for ridge models. The ridge group penalties are then transformed to elastic net group penalties by using the variance function. The method allows for overlapping groups and unpenalised variables. We demonstrate the method in a model-based simulation study and an application to cancer genomics. The method substantially decreases computation time and outperforms or matches other methods by learning from co-data.Comment: 16 pages, 6 figures, 1 tabl

Machine learning-based analysis of [¹⁸F]DCFPyL PET radiomics for risk stratification in primary prostate cancer

PURPOSE: Quantitative prostate-specific membrane antigen (PSMA) PET analysis may provide for non-invasive and objective risk stratification of primary prostate cancer (PCa) patients. We determined the ability of machine learning-based analysis of quantitative [18F]DCFPyL PET metrics to predict metastatic disease or high-risk pathological tumor features. METHODS: In a prospective cohort study, 76 patients with intermediate- to high-risk PCa scheduled for robot-assisted radical prostatectomy with extended pelvic lymph node dissection underwent pre-operative [18F]DCFPyL PET-CT. Primary tumors were delineated using 50-70% peak isocontour thresholds on images with and without partial-volume correction (PVC). Four hundred and eighty standardized radiomic features were extracted per tumor. Random forest models were trained to predict lymph node involvement (LNI), presence of any metastasis, Gleason score ≥ 8, and presence of extracapsular extension (ECE). For comparison, models were also trained using standard PET features (SUVs, volume, total PSMA uptake). Model performance was validated using 50 times repeated 5-fold cross-validation yielding the mean receiver-operator characteristic curve AUC. RESULTS: The radiomics-based machine learning models predicted LNI (AUC 0.86 ± 0.15, p < 0.01), nodal or distant metastasis (AUC 0.86 ± 0.14, p < 0.01), Gleason score (0.81 ± 0.16, p < 0.01), and ECE (0.76 ± 0.12, p < 0.01). The highest AUCs reached using standard PET metrics were lower than those of radiomics-based models. For LNI and metastasis prediction, PVC and a higher delineation threshold improved model stability. Machine learning pre-processing methods had a minor impact on model performance. CONCLUSION: Machine learning-based analysis of quantitative [18F]DCFPyL PET metrics can predict LNI and high-risk pathological tumor features in primary PCa patients. These findings indicate that PSMA expression detected on PET is related to both primary tumor histopathology and metastatic tendency. Multicenter external validation is needed to determine the benefits of using radiomics versus standard PET metrics in clinical practice

¹⁸F-FDG PET baseline radiomics features improve the prediction of treatment outcome in diffuse large B-cell lymphoma

PURPOSE: Accurate prognostic markers are urgently needed to identify diffuse large B-Cell lymphoma (DLBCL) patients at high risk of progression or relapse. Our purpose was to investigate the potential added value of baseline radiomics features to the international prognostic index (IPI) in predicting outcome after first-line treatment. METHODS: Three hundred seventeen newly diagnosed DLBCL patients were included. Lesions were delineated using a semi-automated segmentation method (standardized uptake value ≥ 4.0), and 490 radiomics features were extracted. We used logistic regression with backward feature selection to predict 2-year time to progression (TTP). The area under the curve (AUC) of the receiver operator characteristic curve was calculated to assess model performance. High-risk groups were defined based on prevalence of events; diagnostic performance was assessed using positive and negative predictive values. RESULTS: The IPI model yielded an AUC of 0.68. The optimal radiomics model comprised the natural logarithms of metabolic tumor volume (MTV) and of SUV(peak) and the maximal distance between the largest lesion and any other lesion (Dmax(bulk), AUC 0.76). Combining radiomics and clinical features showed that a combination of tumor- (MTV, SUV(peak) and Dmax(bulk)) and patient-related parameters (WHO performance status and age > 60 years) performed best (AUC 0.79). Adding radiomics features to clinical predictors increased PPV with 15%, with more accurate selection of high-risk patients compared to the IPI model (progression at 2-year TTP, 44% vs 28%, respectively). CONCLUSION: Prediction models using baseline radiomics combined with currently used clinical predictors identify patients at risk of relapse at baseline and significantly improved model performance. TRIAL REGISTRATION NUMBER AND DATE: EudraCT: 2006–005,174-42, 01–08-2008. SUPPLEMENTARY INFORMATION: The online version contains supplementary material available at 10.1007/s00259-021-05480-3

Whole-body MRI versus an FDG-PET/CT-based reference standard for staging of paediatric Hodgkin lymphoma:a prospective multicentre study

Objectives To assess the concordance of whole-body MRI (WB-MRI) and an FDG-PET/CT-based reference standard for the initial staging in children with Hodgkin lymphoma (HL) Methods Children with newly diagnosed HL were included in this prospective, multicentre, international study and underwent WB-MRI and FDG-PET/CT at staging. Two radiologists and a nuclear medicine physician independently evaluated all images. Discrepancies between WB-MRI and FDG-PET/CT were assessed by an expert panel. All FDG-PET/CT errors were corrected to derive the FDG-PET/CT-based reference standard. The expert panel corrected all reader errors in the WB-MRI DWI dataset to form the intrinsic MRI data. Inter-observer agreement for WB-MRI DWI was calculated using overall agreement, specific agreements and kappa statistics. Concordance for correct classification of all disease sites and disease stage between WB-MRI (without DWI, with DWI and intrinsic WB-MRI DWI) and the reference standard was calculated as primary outcome. Secondary outcomes included positive predictive value, negative predictive value and kappa statistics. Clustering within patients was accounted for using a mixed-effect logistic regression model with random intercepts and a multilevel kappa analysis. Results Sixty-eight children were included. Inter-observer agreement between WB-MRI DWI readers was good for disease stage (kappa= 0.74). WB-MRI DWI agreed with the FDG-PET/CT-based reference standard for determining disease stage in 96% of the patients versus 88% for WB-MRI without DWI. Agreement between WB-MRI DWI and the reference standard was excellent for both nodal (98%) and extra-nodal (100%) staging. Conclusions WB-MRI DWI showed excellent agreement with the FDG-PET/CT-based reference standard. The addition of DWI to the WB-MRI protocol improved the staging agreement

Somatostatin analogues for the prevention of pancreatic fistula after open pancreatoduodenectomy:A nationwide analysis

BACKGROUND: Somatostatin analogues (SA) are currently used to prevent postoperative pancreatic fistula (POPF) development. However, its use is controversial. This study investigated the effect of different SA protocols on the incidence of POPF after pancreatoduodenectomy in a nationwide population. METHODS: All patients undergoing elective open pancreatoduodenectomy were included from the Dutch Pancreatic Cancer Audit (2014-2017). Patients were divided into six groups: no SA, octreotide, lanreotide, pasireotide, octreotide only in high-risk (HR) patients and lanreotide only in HR patients. Primary endpoint was POPF grade B/C. The updated alternative Fistula Risk Score was used to compare POPF rates across various risk scenarios. RESULTS: 1992 patients were included. Overall POPF rate was 13.1%. Lanreotide (10.0%), octreotide-HR (9.4%) and no protocol (12.7%) POPF rates were lower compared to the other protocols (varying from 15.1 to 19.1%, p = 0.001) in crude analysis. Sub-analysis in patients with HR of POPF showed a significantly lower rate of POPF when treated with lanreotide (10.0%) compared to no protocol, octreotide and pasireotide protocol (21.6-26.9%, p = 0.006). Octreotide-HR and lanreotide-HR protocol POPF rates were comparable to lanreotide protocol, however not significantly different from the other protocols. Multivariable regression analysis demonstrated lanreotide protocol to be positively associated with a low odds-ratio (OR) for POPF (OR 0.387, 95% CI 0.180-0.834, p = 0.015). In-hospital mortality rates were not affected. CONCLUSION: Use of lanreotide in all patients undergoing pancreatoduodenectomy has a potential protective effect on POPF development. Protocols for HR patients only might be favorable too. However, future studies are warranted to confirm these findings