Gravitational allocation on the sphere.

Given a collection L of n points on a sphere [Formula: see text] of surface area n, a fair allocation is a partition of the sphere into n parts each of area 1, and each is associated with a distinct point of L. We show that, if the n points are chosen uniformly at random and if the partition is defined by a certain "gravitational" potential, then the expected distance between a point on the sphere and the associated point of L is [Formula: see text] We use our result to define a matching between two collections of n independent and uniform points on the sphere and prove that the expected distance between a pair of matched points is [Formula: see text], which is optimal by a result of Ajtai, Komlós, and Tusnády.

Efficacy and safety of regenerative cell therapy for pulmonary arterial hypertension in animal models: a preclinical systematic review protocol.

BACKGROUND: Pulmonary arterial hypertension (PAH) is a rare disease (15 cases per million) that is characterized by widespread loss of the pulmonary microcirculation and elevated pulmonary vascular resistance leading to pathological right ventricular remodeling and ultimately right heart failure. Regenerative cell therapies (i.e., therapies involving cells with stem or progenitor-like properties) could potentially restore the effective lung microcirculation and provide a curative therapy for PAH. Preclinical evidence suggests that regenerative cell therapy using endothelial progenitor cells or mesenchymal stem cells may be beneficial in the treatment of PAH. These findings have led to the completion of a small number of human clinical trials, albeit with modest effect compared to animal studies. The objective of this systematic review is to compare the efficacy and safety of regenerative cell therapies in preclinical models of PAH as well as assess study quality to inform future clinical studies. METHODS: We will include preclinical studies of PAH in which a regenerative cell type was administered and outcomes compared to a disease control. The primary outcome will be pulmonary hemodynamics as assessed by measurement of right ventricular systolic pressure and/or mean pulmonary arterial pressure. Secondary outcomes will include mortality, survival, right ventricular remodeling, pulmonary vascular resistance, cardiac output, cardiac index, pulmonary acceleration time, tricuspid annular systolic excursion, and right ventricular wall thickness. Electronic searches of MEDLINE and EMBASE databases will be constructed and reviewed by the Peer Review of Electronic Search Strategies (PRESS) process. Search results will be screened independently in duplicate. Data from eligible studies will be extracted, pooled, and analyzed using random effects models. Risk of bias will be assessed using the SYstematic Review Centre for Laboratory animal Experimentation (SYRCLE) risk of bias tool, and individual study reporting will be assessed according to an itemized checklist based on the Animal Research: Reporting of In vivo Experiments (ARRIVE) guidelines. DISCUSSION: This systematic review will examine the efficacy and safety of regenerative cell therapy in preclinical models of PAH. As well, the literature will be assessed for study quality and risk of bias. The results will guide the design of future clinical trials and preclinical animal studies. SYSTEMATIC REVIEW REGISTRATION: CAMARADES ( http://www.dcn.ed.ac.uk/camarades/SyRF/Protocols.htm ).