Numerical solution of the Dirichlet initial boundary value problem for the heat equation in exterior 3-dimensional domains using integral equations

A numerical method for the Dirichlet initial boundary value problem for the heat equation in the exterior and unbounded region of a smooth closed simply connected 3-dimensional domain is proposed and investigated. This method is based on a combination of a Laguerre transformation with respect to the time variable and an integral equation approach in the spatial variables. Using the Laguerre transformation in time reduces the parabolic problem to a sequence of stationary elliptic problems which are solved by a boundary layer approach giving a sequence of boundary integral equations of the first kind to solve. Under the assumption that the boundary surface of the solution domain has a one-to-one mapping onto the unit sphere, these integral equations are transformed and rewritten over this sphere. The numerical discretisation and solution are obtained by a discrete projection method involving spherical harmonic functions. Numerical results are included.

Apoptotic cell‐associated ICAM‐3 in the phagocytic removal of apoptotic cells

Apoptosis is a highly controlled cell death programme that culminates in the exposure of molecular ‘flags’ at the dying cell surface that permit recognition and removal by viable phagocytes. Failure to efficiently remove dying cells can lead to devastating inflammatory and autoimmune disorders. The molecular mechanisms underlying apoptotic cell surface changes are poorly understood. Our previous work has shown an apoptosis-associated functional change in ICAM-3 (a heavily glycosylated, leukocyte-restricted Immunoglobulin Super-Family member) resulting in a molecular ‘flag’ to mediate corpse removal. Here we detail apoptosis-associated changes in ICAM-3 and define their role in ICAM-3’s novel function in apoptotic cell clearance. We show ICAM-3 functions to tether apoptotic leukocytes to macrophages via an undefined receptor. Though CD14 has been suggested as a possible receptor for apoptotic cell-associated ICAM-3, we demonstrate ICAM-3 functions for apoptotic cell clearance in the absence of CD14. Furthermore, we demonstrate leukocytes display early changes in cell surface glycosylation and a marked reduction in ICAM-3, a change that correlates reduced cell volume throughout apoptosis. This loss of ICAM-3 occurs via shedding of ICAM-3 in microparticles (‘apoptotic bodies’). Such microparticles are potent chemoattractants for macrophages. Notably, microparticles from ICAM-3-deficient leukocytes are significantly less chemoattractive than microparticles from their ICAM-3-replete counterparts. These data support the hypothesis that ICAM-3 acts as an apoptotic cell-associated ligand to tether dying cells to phagocytes in a CD14-independent manner. Furthermore our data suggest that released ICAM-3 may promote the recruitment of phagocytes to sites of apoptosis.

Expression of VEGF-C and activation of its receptors VEGFR-2 and VEGFR-3 in trophoblast

Placental villous development requires the co-ordinated action of angiogenic factors on both endothelial and trophoblast cells. Like vascular endothelial growth factor (VEGF), VEGF-C increases vascular permeability, stimulates endothelial cell proliferation and migration. In the present study, we investigated the expression of VEGF-C and its receptors VEGFR-3 and VEGFR-2 in normal and intrauterine growth-restricted (IUGR) placenta. Immunolocalisation studies showed that like VEGF and VEGFR-1, VEGF-C, VEGFR-3 and VEGFR-2 co-localised to the syncytiotrophoblast, to cells in the maternal decidua, as well as to the endothelium of the large placental blood vessels. Western blot analysis demonstrated a significant decrease in placental VEGF-C and VEGFR-3 protein expression in severe IUGR as compared to gestationally-matched third trimester pregnancies. Conditioned medium from VEGF-C producing pancreatic carcinoma (Suit-2) and endometrial epithelial (Hec-1B) cell lines caused an increased association of the phosphorylated extracellular signal regulated kinase (ERK) in VEGFR-3 immunoprecipitates from spontaneously transformed first trimester trophoblast cells. VEGF121 caused dose-dependant phosphorylation of VEGFR-2 in trophoblast cells as well as stimulating DNA synthesis. In addition, premixing VEGF165 with heparin sulphate proteoglycan potentiated trophoblast proliferation and the association of phospho-ERK with the VEGFR-2 receptor. VEGF165-mediated DNA synthesis was inhibited by anti-VEGFR-2 neutralising antibody. The results demonstrate functional VEGFR-2 and VEGFR-3 receptors on trophoblast and suggest that the decreased expression of VEGF-C and VEGFR-3 may contribute to the abnormal villous development observed in IUGR placenta.