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Background Current standard of care for metastatic castration-sensitive prostate cancer supplements androgen deprivation therapy with either docetaxel, second-generation hormonal therapy, or radiotherapy. We aimed to evaluate the efficacy and safety of abiraterone plus prednisone, with or without radiotherapy, in addition to standard of care. Methods We conducted an open-label, randomised, phase 3 study with a 2 x 2 factorial design (PEACE-1) at 77 hospitals across Belgium, France, Ireland, Italy, Romania, Spain, and Switzerland. Eligible patients were male, aged 18 years or older, with histologically confirmed or cytologically confirmed de novo metastatic prostate adenocarcinoma, and an Eastern Cooperative Oncology Group performance status of 0-1 (or 2 due to bone pain). Participants were randomly assigned (1: 1:1:1) to standard of care (androgen deprivation therapy alone or with intravenous docetaxel 75 mg/m(2) once every 3 weeks), standard of care plus radiotherapy, standard of care plus abiraterone (oral 1000 mg abiraterone once daily plus oral 5 mg prednisone twice daily), or standard of care plus radiotherapy plus abiraterone. Neither the investigators nor the patients were masked to treatment allocation. The coprimary endpoints were radiographic progression-free survival and overall survival. Abiraterone efficacy was first assessed in the overall population and then in the population who received androgen deprivation therapy with docetaxel as standard of care (population of interest). This study is ongoing and is registered with ClinicalTrials.gov, NCT01957436. Findings Between Nov 27, 2013, and Dec 20, 2018, 1173 patients were enrolled (one patient subsequently withdrew consent for analysis of his data) and assigned to receive standard of care (n=296), standard of care plus radiotherapy (n=293), standard of care plus abiraterone (n=292), or standard of care plus radiotherapy plus abiraterone (n=291). Median follow-up was 3.5 years (IQR 2.8-4.6) for radiographic progression-free survival and 4.4 years (3.5-5.4) for overall survival. Adjusted Cox regression modelling revealed no interaction between abiraterone and radiotherapy, enabling the pooled analysis of abiraterone efficacy. In the overall population, patients assigned to receive abiraterone (n=583) had longer radiographic progression-free survival (hazard ratio [HR] 0.54, 99.9% CI 0.41-0.71; p

This paper deals with the numerical solution of the Heston partial differential equation that plays an important role in financial option pricing, Heston (1993, Rev. Finan. Stud. 6). A feature of this time-dependent, two-dimensional convection-diffusion-reaction equation is the presence of a mixed spatial-derivative term, which stems from the correlation between the two underlying stochastic processes for the asset price and its variance. Semi-discretization of the Heston PDE, using finite difference schemes on a non-uniform grid, gives rise to large systems of stiff ordinary differential equations. For the effective numerical solution of these systems, standard implicit time-stepping methods are often not suitable anymore, and tailored time-discretization methods are required. In the present paper, we investigate four splitting schemes of the Alternating Direction Implicit (ADI) type: the Douglas scheme, the Craig & Sneyd scheme, the Modified Craig & Sneyd scheme, and the Hundsdorfer & Verwer scheme - each of which contains a free parameter. ADI schemes were not originally developed to deal with mixed spatial-derivative terms. Accordingly, we first discuss the adaptation of the above four ADI schemes to the Heston equation. Subsequently, we present various numerical examples with realistic data sets from the literature, where we consider European call options as well as down-and-out barrier options. Combined with ample theoretical stability results for ADI schemes that have recently been obtained in In 't Hout & Welfert (2007, Appl. Numer. Math.), we arrive at three ADI schemes that all prove to be very effective in the numerical solution of the Heston PDE with a mixed derivative term., Paper has been published