Continuous and discrete mathematical models of tumor-induced angiogenesis.

*(English)*Zbl 0923.92011Angiogenesis is the formation of blood vessels whereby capillary sprouts are formed in response to chemical stimuli. Subsequently they grow, develop and organize themselves into a dendritic structure. A continuous and a discrete probabilistic mathematical model is presented for tumor-induced angiogenesis focussing on three important variables, the endothelial-cell density \(n\), the tumor angiogenic factor (TAF) concentration \(c\), and the fibronectin \(f\). The continuous model is presented as a system of three coupled nonlinear partial differential equations with a no-flux boundary condition where the density \(n\) is given as
\[
\partial n/\partial t=\text{ random motility -- chemotaxis -- hepatotaxis}
\]

\[ \partial f/\partial t=\beta n-\gamma nf,\quad\partial c/\partial t=\eta n c. \] The aims of the model are to examine the relative importance of chemotaxis and haptotaxis in governing the migration of the endothelial cells, to examine capillary network formation with and without proliferation of endothelial cells, and to produce theoretical capillary network structures, from a model based on sound physiological principles with realistic parameter values, which are morphologically similar to those observed in vivo. It is possible to analyze these computed structures for quantitative information such as network expansion rates, loop formation and overall network architecture.

Numerical solutions are obtained from a finite approximation of the system with biologically based boundary and initial conditions. The spatiotemporal evolution of the endothelial cell density and fibronectin uptake is determined from numerical simulation under a variety of chemotaxis/hepatotaxis scenarios producing a wide range of spatiotemporal behavior. On the basis of this continuous model and using the Euler finite difference approximation the authors formulate a discrete model and derive movement probabilities for an individual endothelial cell located at a sprout tip. New capillary sprouts are generated by the branching and the fusion of sprouts by anastomosis. The such fully discretized model is then again simulated to describe the spatiotemporal evolution of the capillary network permitting now the tracking of individual endothelial cells.

\[ \partial f/\partial t=\beta n-\gamma nf,\quad\partial c/\partial t=\eta n c. \] The aims of the model are to examine the relative importance of chemotaxis and haptotaxis in governing the migration of the endothelial cells, to examine capillary network formation with and without proliferation of endothelial cells, and to produce theoretical capillary network structures, from a model based on sound physiological principles with realistic parameter values, which are morphologically similar to those observed in vivo. It is possible to analyze these computed structures for quantitative information such as network expansion rates, loop formation and overall network architecture.

Numerical solutions are obtained from a finite approximation of the system with biologically based boundary and initial conditions. The spatiotemporal evolution of the endothelial cell density and fibronectin uptake is determined from numerical simulation under a variety of chemotaxis/hepatotaxis scenarios producing a wide range of spatiotemporal behavior. On the basis of this continuous model and using the Euler finite difference approximation the authors formulate a discrete model and derive movement probabilities for an individual endothelial cell located at a sprout tip. New capillary sprouts are generated by the branching and the fusion of sprouts by anastomosis. The such fully discretized model is then again simulated to describe the spatiotemporal evolution of the capillary network permitting now the tracking of individual endothelial cells.

Reviewer: Lutz Edler (Heidelberg)

##### MSC:

92C50 | Medical applications (general) |

92C15 | Developmental biology, pattern formation |

92B05 | General biology and biomathematics |