A New Chemotactic Mechanism Governs Long-Range Angiogenesis Induced by Patching an Arterial Graft into a Vein.

Chemotaxis, the migration of cells in response to chemical stimulus, is an important concept in the angiogenesis model. In most angiogenesis models, chemotaxis is defined as the migration of a sprout tip in response to the upgradient of the VEGF (vascular endothelial growth factor). However, we found that angiogenesis induced by performing arterial patch grafting on rabbits occurred under the decreasing VEGFA gradient. Data show that the VEGFA concentration peaked at approximately 0.3 to 0.5 cm away from the arterial patch and decreased as the measurement approaches the patch. We also observed that the new blood vessels formed are twisted and congested in some areas, in a distinguishable manner from non-pathological blood vessels. To explain these observations, we developed a mathematical model and compared the results from numerical simulations with the experimental data. We introduced a new chemotactic velocity using the temporal change in the chemoattractant gradient to govern the sprout tip migration. We performed a hybrid simulation to illustrate the growth of new vessels. Results indicated the speed of growth of new vessels oscillated before reaching the periphery of the arterial patch. Crowded and congested blood vessel formation was observed during numerical simulations. Thus, our numerical simulation results agreed with the experimental data.

Study on the tumor-induced angiogenesis using mathematical models.

We studied angiogenesis using mathematical models describing the dynamics of tip cells. We reviewed the basic ideas of angiogenesis models and its numerical simulation technique to produce realistic computer graphics images of sprouting angiogenesis. We examined the classical model of Anderson-Chaplain using fundamental concepts of mass transport and chemical reaction with ECM degradation included. We then constructed two types of numerical schemes, model-faithful and model-driven ones, where new techniques of numerical simulation are introduced, such as transient probability, particle velocity, and Boolean variables.

Finding Solvable Units of Variables in Nonlinear ODEs of ECM Degradation Pathway Network.

We consider ordinary differential equation (ODE) model for a pathway network that arises in extracellular matrix (ECM) degradation. For solving the ODEs, we propose applying the mass conservation law (MCL), together with a stoichiometry called doubling rule, to them. Then it leads to extracting new units of variables in the ODEs that can be solved explicitly, at least in principle. The simulation results for the ODE solutions show that the numerical solutions are indeed in good accord with theoretical solutions and satisfy the MALs.