Mathematics of Gravitational Lensing: Multiple Imaging and Magnification

The mathematical theory of gravitational lensing has revealed many generic and global properties. Beginning with multiple imaging, we review Morse-theoretic image counting formulas and lower bound results, and complex-algebraic upper bounds in the case of single and multiple lens planes. We discuss recent advances in the mathematics of stochastic lensing, discussing a general formula for the global expected number of minimum lensed images as well as asymptotic formulas for the probability densities of the microlensing random time delay functions, random lensing maps, and random shear, and an asymptotic expression for the global expected number of micro-minima. Multiple imaging in optical geometry and a spacetime setting are treated. We review global magnification relation results for model-dependent scenarios and cover recent developments on universal local magnification relations for higher order caustics.Comment: 25 pages, 4 figures. Invited review submitted for special issue of General Relativity and Gravitatio

Clinico-therapeutical considerations on the arterial embolisms of the lower limbs

[No abstract available

Ultrasound echo-enhancers in the evaluation of endovascular prostheses.

Whereas conventional vascular surgical procedures are routinely monitored by ultrasound imaging, endovascular prostheses are usually monitored by radiographical imaging techniques. The aim of this study was to determine the safety, patient acceptance and role of ultrasound echo-enhancers (SHU 508 A) in the evaluation of endovascular devices. Nine patients were studied: six had stents (four in the internal carotid artery and two in the renal artery) and three had endoluminal prostheses (one in the abdominal aorta, one in the subclavian and one in the iliac artery). Endovascular patency and correct placement were studied with contrast angiography and ultrasound examination with and without contrast enhancement. Patients underwent angiography at the end of the endovascular procedure, and ultrasound examinations on the first postoperative day. Ultrasound contrast media (SHU 508 A) was injected through an antecubital vein at a rate of 1 ml/s. A total of 13 injections were given in nine patients. Ultrasonograms were obtained with 7.5 and 3.5 MHz transducers (Acuson 128 XP) and recorded on videotape for off-line visual inspection. Baseline and echo-enhanced sequences were assessed by two independent observers. None of the patients reported side-effects during or after the injection of the ultrasound echo-enhancer. Postoperative angiography showed endovascular patency and correct placement in all patients. The baseline ultrasound examination confirmed endoprosthesis patency in seven of nine patients: none revealed endoprosthesis malplacement. Contrast-enhanced ultrasound examination confirmed endoprosthesis patency. In two cases, the echo-contrast examination revealed persisting flow within the aneurysmal sac: in another patient it showed the incomplete adhesion of the distal portion of the endoprosthesis to the arterial wall. The preliminary findings suggest that ultrasound echo-enhancers are safe to use and induce no side-effects. Echo-enhanced sequences often provide additional information on the technical success of endovascular procedures. They promise to be useful in follow-up studies for assessing the functioning of endoprostheses

Contrast Carotid Ultrasound for the Detection of Unstable Plaques with Neoangiogenesis: A Pilot Study

Objectives: To evaluate whether contrast ultrasonography can be used to distinguish asymptomatic from symptomatic carotid plaques and provide insight into underlying pathophysiological differences. Design: Contrast Carotid ultrasound was performed in both symptomatic and asymptomatic patients referred for carotid endarterectomy. Materials and methods: Of 77 consecutive patients referred for carotid artery evaluation, 64 underwent carotid endarterectomy for asymptomatic cerebrovascular disease and 9 underwent urgent surgery for acute neurological deficits with hemiparesis. The endarterectomy specimens were assessed immunohistologically. Results: In all 9 patients undergoing urgent surgery, contrast ultrasonography showed the accumulation of diffuse microbubble contrast at the base of the carotid plaque. This pattern was observed only in 1/64 of the patients undergoing surgery for asymptomatic carotid disease. Immunohistologically staining of the endarterectomy specimens showed that the area of microbubble contrast at the base of the symptomatic plaques was associated with an increased number of small diameter (20-30 μm) microvessels staining for vascular endothelial growth factor (VEGF). Conclusions: Contrast carotid ultrasonography may allow the identification of microvessels with neoangiogenesis at the base of carotid plaques, and differentiate symptomatic from asymptomatic plaques. © 2009 European Society for Vascular Surgery