Ruthenium(II)-arene complexes with biotin-containing ligands were prepared so that a novel drug delivery system based on tumor-specific vitamin-receptor mediated endocytosis could be developed. The complexes were characterized by spectroscopic methods and their in vitro anticancer activity in cancer cell lines with various levels of major biotin receptor (COLO205, HCT116 and SW620 cells) was tested in comparison with the ligands. In all cases, coordination of ruthenium resulted in significantly enhanced cytotoxicity. The affinity of Ru(II) -biotin complexes to avidin was investigated and was lower than that of unmodified biotin. Hill coefficients in the range 2.012-2.851 suggest strong positive cooperation between the complexes and avidin. To estimate the likelihood of binding to the biotin receptor/transporter, docking studies with avidin and streptavidin were conducted. These explain, to some extent, the in vitro anticancer activity results and support the conclusion that these novel half-sandwich ruthenium(II)-biotin conjugates may act as biological vectors to cancer cells, although no clear relationship between the cellular Ru content, the cytotoxicity, and the presence of the biotin moiety was observed.

The form of topological derivatives for an integral shape functional is derived for a class of semilinear elliptic equations. The convergence of finite element approximation for the topological derivatives is shown and the error estimates in the L ∞ norm are obtained. The results of numerical experiments which confirm the theoretical convergence rate are presented.

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Abstract. Framework for shape and topology sensitivity analysis in geometrical domains with cracks is established for elastic bodies in two spatial dimensions. Equilibrium problem for elastic body with cracks is considered. Inequality type boundary conditions are prescribed at the crack faces providing a non-penetration between the crack faces. Modelling of such problems in two spatial dimensions is presented with all necessary details for further applications in shape optimization in structural mechanics. In the paper, general results on the shape and topology sensitivity analysis of this problem are provided. The results are interesting on its own. In particular, the existence of the shape and topological derivatives of the energy functional is obtained. It is shown, in fact, that the level set type method [4] can be applied to shape and topology opimization of the related variational inequalities for elasticity problems in domains with cracks, with the nonpenetration condition prescribed on the crack faces. The results presented in the paper can be used for numerical solution of shape optimization and inverse problems in structural mechanics.Key words. Crack with non-penetration, shape sensitivity, derivative of energy functional, topological derivative AMS subject classifications. Primary 35J85, 74K20 Secondary 35J25, 74M151. Introduction. Shape optimization requires few mathematical results, in the framework of modelling and numerical solution, for any specific class of problems governed by partial differential equations of mathematical physics. Usually, we need to show the well posedness of the specific problem, and also we can propose a numerical method for the effective solution procedure. Hence, in order to solve a shape optimization problem we are obliged to have the results on• the existence and continuous dependence with respect to the shape of solutions to the model, which may result in the existence of optimal shapes, • the differentiability of solutions with respect to the boundary variations, which imply the existence of shape gradients and leads to some necessary conditions for optimality, of the first order and possibly of the second order which leads to the Newton method of shape optimization, • and in addition, perform the asymptotic analysis of the related boundary value problem in singularly perturbed geometrical domains and derive the form of the topological derivative for the shape functional of interest, which allows for the topology changes in the process of numerical optimization, if necessary, • and finally, we may device a numerical method and show its efficiency in numerical examples, and its convergence form the mathematical point of view.