On the stability of homogeneous solutions to some aggregation models
R. Kowalczyk A. Gamba L. Preziosi
Vasculogenesis, i.e. self-assembly of endothelial cells leading to capillary network formation, has been the object of many experimental investigations in recent years, due to its relevance both in physiological and in pathological conditions. We performed a detailed linear stability analysis of two models of in vitro vasculogenesis, with the aim of checking their potential for structure formation starting from initial data representing a continuum cell monolayer. The first model turns out to be unstable at low cell densities, while pressure stabilizes it at high densities. The second model is instead stable at low cell densities. Detailed information about the instability regions and the structure of the critical wave numbers are obtained in several interesting limiting cases. We expect that altogether, this information will be useful for further comparison of the two models with experiments.
keywords: Burgers' equation. Angiogenesis
Modeling cell movement in anisotropic and heterogeneous network tissues
A. Chauviere T. Hillen L. Preziosi
Cell motion and interaction with the extracellular matrix is studied deriving a kinetic model and considering its diffusive limit. The model takes into account the chemotactic and haptotactic effects, and obtains friction as a result of the interactions between cells and between cells and the fibrous environment. The evolution depends on the fibre distribution, as cells preferentially move along the fibre direction and tend to cleave and remodel the extracellular matrix when their direction of motion is not aligned with the fibre direction. Simulations are performed to describe the behavior of an ensemble of cells under the action of a chemotactic field and in the presence of heterogeneous and anisotropic fibre networks.
keywords: Mathematical model Fiber network. Cell motion
Modeling the motion of a cell population in the extracellular matrix
A. Chauviere L. Preziosi T. Hillen
The paper aims at describing the motion of cells in fibrous tissues taking into account the interaction with the network fibers and among cells, chemotaxis, and contact guidance from network fibers. Both a kinetic model and its continuum limit are described.
keywords: contact guidance. continuum model cell motion chemotaxis
Exogenous control of vascular network formation in vitro: a mathematical model
V. Lanza D. Ambrosi L. Preziosi
The reconstitution of a proper and functional vascular network is a major issue in tissue engineering and regeneration. The limited success of current technologies may be related to the difficulties to build a vascular tree with correct geometric ratios for nutrient delivery. The present paper develops a mathematical model suggesting how an anisotropic vascular network can be built in vitro by using exogenous chemoattractant and chemorepellent. The formation of the network is strongly related to the nonlinear characteristics of the model.
keywords: Mathematical Modelling Tissue Engineering. Pattern Formation Vascularization

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