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Tumor induced angiogenesis is the bridge between benign and malignant tumor growth stages. In this process, growth and migration of endothelial cells build capillaries to supply the tumor with blood for its further growth. Regarding the importance of capillary formation and blood flow in angiogenesis, simulation of this phenomenon plays important role in tumor growth and cancer development studies. In this work, considering intracellular, cellular, and extracellular scales a mathematical model of tumor-induced angiogenesis is used to consider mechanical effects of extracellular matrix on growth and migration of endothelial cells. These effects are matrix density and its fiber length. In this study, to model cellular dynamics, a discrete lattice based Monte Carlo method is used. Results show that migration of endothelial cells and development of capillaries are possible in a specified range of matrix density and matrix fiber length. Based on the results, medium matrix densities and low fiber length provide a suitable environment for capillaries growth and development. The model is a promising tool for modeling tumor induced angiogenesis and is a base for development of models for loop formation and blood flow in capillaries around tumor.

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The use of porous scaffolds for repairing the damaged bone tissues has been increased in recent years. As exploration of the mechanical properties of the scaffolds on the basis of experiments is time consuming and uneconomic, mathematical models are increasingly being introduced into the field, but most of them rely on finite element method and theoretical studies are rarely found in the literature. In this paper, different micromechanical models are presented for obtaining the effective elastic properties of bone scaffolds. Using these models, the mechanical properties of different scaffolds, including ceramic and composite bone scaffolds, are investigated. Single scale and multi-scale modeling approaches are used to simulate the ceramic and composite scaffolds, respectively. Furthermore, because of the wide application of hydroxyapatite in fabrication of bone scaffolds, the mechanical properties of hydroxyapatite scaffolds in different porosities are obtained in the current study by means of the presented methods. Results show that Dewey, self-consistent and differential schemes are the best methods in calculation of the value of Young’s modulus of these scaffolds in porosity ranges of less than 30 %, 30 to 60 % and more than 60 %, respectively. Moreover, self-consistent scheme gives good estimation of the value of Poisson’s ratio of hydroxyapatite scaffolds in different porosities. By obtaining the values of the mechanical properties of the scaffolds in different porosities by these models and using the statistical analysis, the mathematical relationship between the porosity and the mechanical properties of this kind of scaffolds (Young’s modulus and Poisson’s ratio) is obtained.