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Incompressible flows are characterised by turbulent, chaotic behaviour. The current mathematical understanding of the driving equations, the Euler and Navier- Stokes equations, is plagued by the absence of definitive existence, regularity, and uniqueness results. Numerical approximations are often diverging and lack the complete mathematical foundations necessary to provide stability and convergence estimates. A possible remedy for these problems may be provided by ensembles of solu- tions of the Euler and Navier-Stokes equations. Ensembles of solutions provide a framework for the existence of generalized solutions which carry the notion of ran- domness, whilst providing, at the same time, foundations in which we can carry out limits of approximations, thus proving convergence of our numerical approxim- ations. In this work, we explore the efficient numerical approximation of ensembles of incompressible flows within two main frameworks: the first consists of so-called admissible measure valued solutions for inviscid flows, the second will be built on statistical solutions for viscous fluids. We explore the concept of multilevel Monte Carlo simulations applied to the approximation of ensembles of incompressible flows, and we formulate a new effi- cient and converging scheme for the computation of statistics of the ensemble of solutions. We are particularly interested in multi level approaches: algorithms for which the varying granularity of the approximation of an individual solution can be ex- ploited to yield more efficient computations. We will also exploit our numerical algorithms to investigate the uniqueness of a particular type of solution for the 2 dimensional Euler equations, the Delort class of solutions.

Azathioprine has been extensively used in the management of inflammatory bowel diseases. It might cause pancreatic damage in the form of either asymptomatic elevation in serum amylase/lipase or overt acute pancreatitis. Here we report the case of a 61-year-old patient with ulcerative colitis who had been treated with azathioprine for three years, achieving clinical remission. During treatment he presented an asymptomatic elevation of serum pancreatic enzymes, without any signs of pancreatitis at imaging. This evidence brought us to reassess the drug dosage, without achieving a normalization of biochemical analysis. Autoimmune pancreatitis was excluded. One year after the suspension of azathioprine, we still face persistent high levels of amylase/lipase. Normalization of enzymatic values in patients who develop intolerance to azathioprine, in the form of either asymptomatic elevation in serum amylase/lipase or overt acute pancreatitis, is usually achieved in about two months after stopping drug intake. Asymptomatic elevation in serum pancreatic enzymes in the absence of pancreatic disease is reported in the literature and defined as “Gullo’s syndrome,” but nobody of the subjects studied had been treated in the past with pancreatotoxic drugs. Might this case be defined as “benign pancreatic hyperenzymemia”?

BackgroundThough the precise criteria for accessing LT are consistently being applied, HCC recurrence (HCC-R_LT) still affects more than 15% of the patients. We analyzed the clinical, histopathological, and biological features of patients with HCC to identify the predictive factors associated with cancer recurrence and survival after LT.MethodsWe retrospectively analyzed 441 patients with HCC who underwent LT in our center. Overall, 70 (15.8%) of them developed HCC-R_LT. We matched them by age at transplant and etiology with 70 non-recurrent patients. A comparable cohort from the Liver Transplant Centre of Bologna served as validation. The clinical and biochemical characteristics and pre-LT criteria (Milan, Metroticket, Metroticket_AFP, and AFP model) were evaluated. Histological analysis and immunohistochemistry for angiopoietin-2 in the tumor and non-tumor tissue of explanted livers were performed. Patients’ follow-up was until death, last clinical evaluation, or 31 December 2021. In patients with HCC-R_LT, the date of diagnosis of recurrence and anatomical site has been reported; if a biopsy of recurrence was available, histologic and immunohistochemical analyses were also performed.ResultsPatients were followed up for a mean period of 62.7 ± 54.7 months (median, 39 months). A higher risk of HCC-R_LT was evident for factors related indirectly (AFP) or directly (endothelial angiopoietin-2, microvascular invasion) to biological HCC aggressiveness. In multivariate analysis, only angiopoietin-2 expression was independently associated with recurrence. Extremely high levels of endothelial angiopoietin-2 expression were also found in hepatic recurrence and all different metastatic locations. In univariate analysis, MELD, Metroticket_AFP Score, Edmondson–Steiner grade, microvascular invasion, and endothelial angiopoietin-2 were significantly related to survival. In multivariate analysis, angiopoietin-2 expression, Metroticket_AFP score, and MELD (in both training and validation cohorts) independently predicted mortality. In time-dependent area under receiver operating characteristic curve analysis, the endothelial angiopoietin-2 expression had the highest specificity and sensitivity for recurrence (AUC 0.922, 95% CI 0.876–0.962, p < 0.0001).ConclusionsEndothelial angiopoietin-2 expression is a powerful independent predictor of post-LT tumor recurrence and mortality, highlighting the fundamental role of tumor biology in defining the patients’ prognosis after liver transplantation. The great advantage of endothelial angiopoietin-2 is that it is evaluable in HCC biopsy before LT and could drive a patient’s priority on the waiting list.