Communications in Mathematical Sciences

Volume 15 (2017)

Number 1

Comparison of several reaction and diffusion models of growth factors in angiogenesis

Pages: 1 – 26



Fang Li (Center for Partial Differential Equations, East China Normal University, Shanghai, China)

Xiaoming Zheng (Department of Mathematics, Central Michigan University, Mount Pleasant, Mich., U.S.A.)


We compare three types of mathematical models of growth factor reaction and diffusion in angiogenesis: one describes the reaction on the blood capillary surface, one on the capillary volume, and one on the capillary centerline. Firstly, we explore the analytical properties of these models, including solution regularity and positivity. We prove that the surface-reaction models have smooth and positive solutions and that the volume-reaction models have continuous and positive solutions. The line-reaction models utilize distributions on the capillary centerline to represent the reaction line source. The line-reaction model-I employs the Dirac delta function and the mean value of the growth factor around the centerline, which gives a valid model. The line-reaction model-II and -III use the local value of the growth factor, which either creates a singularity or decouples the reaction from diffusion, thus being invalid. Secondly, we compare the programming complexity and computational cost of these models in numerical implementations. The surface-reaction model is the most complicated and suitable for small domains, while the volume-reaction and line-reaction models are simpler and suitable for large domains with a large number of blood capillaries. Finally, we quantitatively compare these models in the prediction of the growth factor dynamics. It turns out that the volume-reaction and line-reaction model-I agree well with the surface-reaction model for most parameters used in the literature but may differ significantly when the diffusion constant is small.


reaction, diffusion, angiogenesis, growth factors, equations with distribution coefficients, singular solutions

2010 Mathematics Subject Classification

35K57, 35K67, 92B99

Published 10 January 2017