On spherical Monte Carlo simulations for multivariate normal probabilities

The calculation of multivariate normal probabilities is of great importance in many statistical and economic applications. In this paper we propose a spherical Monte Carlo method with both theoretical analysis and numerical simulation. We start by writing the multivariate normal probability via an inner radial integral and an outer spherical integral using the spherical transformation. For the outer spherical integral, we apply an integration rule by randomly rotating a predetermined set of well-located points. To find the desired set, we derive an upper bound for the variance of the Monte Carlo estimator and propose a set which is related to the kissing number problem in sphere packings. For the inner radial integral, we employ the idea of antithetic variates and identify certain conditions so that variance reduction is guaranteed. Extensive Monte Carlo simulations on some probabilities confirm these claims.

On spherical Monte Carlo simulations for multivariate normal probabilities

The calculation of multivariate normal probabilities is of great importance in many statistical and economic applications. In this paper we propose a spherical Monte Carlo method with both theoretical analysis and numerical simulation. We start by writing the multivariate normal probability via an inner radial integral and an outer spherical integral using the spherical transformation. For the outer spherical integral, we apply an integration rule by randomly rotating a predetermined set of well-located points. To find the desired set, we derive an upper bound for the variance of the Monte Carlo estimator and propose a set which is related to the kissing number problem in sphere packings. For the inner radial integral, we employ the idea of antithetic variates and identify certain conditions so that variance reduction is guaranteed. Extensive Monte Carlo simulations on some probabilities confirm these claims.

Morphologic Platelet Changes in Vitro and During Thrombus Formation in the Rat

Using high power Nomarski optics the shape change of thrombocytes can be shown. Platelets appear in their native shape as flat discs, less than 25% show pseudopodes. Depending on the time after blood sampling and on incubation temperature platelets in PRP or citrated blood swell and form tentacles. The addition of ADP to PRP induces the formation of aggregates. Single platelets form large vesicles rupturing and releasing granulated material. The remaining platelet material fuses. Bencyclan affects platelet morphology by inducing a spherical transformation, which is paralleled by the inhibition of platelet adhesiveness, spreading and aggregation. Observations in small mesenteric vessels of the rat show platelets in their native shape under stasis conditions. Vascular lesions are produced with a focused laserbeam (Hadron 513 biolaser). Immediately after the lesion platelets stick to the site of the microburn. Within seconds these platelets swell and form protrusions. After 3 - 10 min the vessel is occluded by a thrombus, of platelets, which undergo further swelling. Later the thrombus is partially or completely swept away and the vessel is recanalized. Irreversible fusion of platelets is rarely observed. These morphologic platelet changes differ markedly from those observed during in vitro aggregation. Injection of a new antithrombotic substance (Bay G 6575) diminishes the adhesion of platelets on the vessel lesion. The morphologic changes of single platelets (primary shape change) probably represent basic processes in hemostasis and thrombus formation.

Morphologic Platelet Changes in Citrate Blood and PRP

Using high power Nomarski optics the film presents the shape change of thrombocytes in vitro. Their native shape can be studied by immediate fixation of blood at venepuncture. Platelets appear as flat discs. Less than 25% show pseudopodes. In citrate blood and PRP a pro-qressing shape change occurs depending on time after blood sampling and on incubation temperature. Platelets tend to swell and tentacle formation continues with time. At room temperature 90% of thrombocytes have undergone these changes within 60 min. The addition of ADP (10-6 molar) to PRP induces the formation and rupture of large vesicles in single platelets. Thrombocytes are aggregating, many of the aggregated platelets show the same large vesicles. They are bursting and the release of a granular material can be observed. The remaining platelet material is fusing to an unstructured mass. Bencyclan (2 × 10-4 molar) affects platelet morphology by inducing a spherical transformation, which is paralleled by the inhibition of platelet adhesiveness, spreading and aggregation. Finally a platelet is shown spreading on a glass surface. The mor-nhologic changes of single platelets (primary shape change) demonstrated in the film probably represent basic processes in hemostasis and thrombus formation.