The Origin and Development of the Geometrical Ideas in Arabic Mathematics : The Synopsis of the Geometrical Works of al-Quhi

Arabic Mathematics has been characterized as algebra. Compared with this, Arabic geometry had not influence on the later mathematics, and has not been studied so much. However without this geometry, no solution of cubic equations has not completed in Arabic mathematics. We sketch here the synopsis of the geometrical works of Abu Sahl al-Quhl (second half of the tenth century), 'one of the most eminent mathematicians in Iraq', and investigate the origin and development of his geometrical ideas. Thirty three mathematical works are attributed to him, and almost of them are geometrical. His ideas were from Archimedes, Euclid and Apollonius. The opus magnum of the last one is indispensable for al-Quhl's works, and in the field of conic sections he contributed much. He completed the lacuna of the Greek mathematics, and developed it further. For showing this aspect four treatises are presented with partial translations. 'On Tangent Circles' investigated Apollonian circle problems further, and 'On the Trisection of Angle' solved the famous problem by Apollonian conic sections. 'On the Motion' was a unique treatise in Arabic mathematics, for it dealt with infinity which had been avoided in Greek mathematics. 'On the Perfect Compass (an instrument to draw conies by continuous moving)' gave an idea on the new classification of curves, which anticipates the seventeenth-century European mathematics. The problems and method which he used seems to be analytical and purely Greek, and he might be called as the last Greek-style mathematician. The atmosphere where he studied shows that Arabic science developed under a kind of patronage, and the manuscripts containing his treatises shows that Greek geometry was well established at his times. In conclusion, geometry flourished in Arabic world of the tenth century, and its results were over the Greek ones, and might be compared to the early modern mathematics in Europe

Induction of lymphatic endothelial cell differentiation in embryoid bodies

The molecular mechanisms that regulate the formation of the lymphatic vascular system remain poorly characterized. Whereas studies in embryonic stem (ES) cells have provided major new insights into the mechanisms of blood vessel formation, the development of lymphatic endothelium has not been previously observed. We established embryoid bodies (EBs) from murine ES cells in the presence or absence of lymphangiogenic growth factors. We found that lymphatic endothelial cells develop at day 18 after EB formation. These cells express CD31 and the lymphatic lineage markers Prox-1 and Lyve-1, but not the vascular marker MECA-32, and they frequently sprout from preexisting blood vessels. Lymphatic vessel formation was potently promoted by VEGF-A and VEGF-C but not by bFGF. Our results reveal, for the first time, that ES cells can differentiate into lymphatic endothelial cells, and they identify the EB assay as a powerful new tool to dissect the molecular mechanisms that control lymphatic vessel formation

Targeting Tissue Factor to Tumor Vasculature to Induce Tumor Infarction

Besides its central functional role in coagulation, TF has been described as being operational in the development of malignancies and is currently being studied as a possible therapeutic tool against cancer. One of the avenues being explored is retargeting TF or its truncated extracellular part (tTF) to the tumor vasculature to induce tumor vessel occlusion and tumor infarction. To this end, multiple structures on tumor vascular wall cells have been studied at which tTF has been aimed via antibodies, derivatives, or as bifunctional fusion protein through targeting peptides. Among these targets were vascular adhesion molecules, oncofetal variants of fibronectin, prostate-specific membrane antigens, vascular endothelial growth factor receptors and co-receptors, integrins, fibroblast activation proteins, NG2 proteoglycan, microthrombus-associated fibrin-fibronectin, and aminopeptidase N. Targeting was also attempted toward cellular membranes within an acidic milieu or toward necrotic tumor areas. tTF-NGR, targeting tTF primarily at aminopeptidase N on angiogenic endothelial cells, was the first drug candidate from this emerging class of coaguligands translated to clinical studies in cancer patients. Upon completion of a phase I study, tTF-NGR entered randomized studies in oncology to test the therapeutic impact of this novel therapeutic modality

Anticancer Therapy by Tumor Vessel Infarction with Polyethylene Glycol Conjugated Retargeted Tissue Factor

tTF-NGR consists of the extracellular domain of tissue factor and the peptide GNGRAHA, a ligand of the surface protein aminopeptidase N and of integrin α_vβ₃. Both surface proteins are upregulated on endothelial cells of tumor vessels. tTF-NGR shows antitumor activity in xenografts and inhibition of tumor blood flow in cancer patients. We performed random TMS­(PEG)₁₂ PEGylation of tTF-NGR to improve the antitumor profile of the molecule. PEGylation resulted in an approximately 2-log step decreased procoagulatory activity of the molecule. Pharmacokinetic studies in mice showed a more than 1-log step higher mean area under the curve. Comparison of the LD₁₀ values for both compounds and their lowest effective antitumor dose against human tumor xenografts showed an improved therapeutic range (active/toxic dose in mg/kg body weight) of 1/5 mg/kg for tTF-NGR and 3/>160 mg/kg for TMS­(PEG)₁₂ tTF-NGR. Results demonstrate that PEGylation can significantly improve the therapeutic range of tTF-NGR