Annals of Mathematical Sciences and Applications

Volume 5 (2020)

Number 1

Mathematical sciences related to theoretical physics, engineering, biology and economics

Guest Editor: Tony Wen-Hann Sheu, National Taiwan University

Simulation of tumor-induced angiogenesis by an HOC approach

Pages: 7 – 39



Jiten C. Kalita (Department of Mathematics, Indian Institute of Technology, Guwahati, Assam, India)

Shubham Goyal (Department of Mathematics, Indian Institute of Technology, Guwahati, Assam, India)

Utkarsh Dixit (Department of Mathematics, Indian Institute of Technology, Guwahati, Assam, India)


Angiogenesis is one of the main processes of vascularization resulting in the formation of capillary sprouts in response to externally supplied chemical stimuli. To date, most of the numerical works simulating the process of angiogenesis has been carried out by lower order accurate schemes like Euler explicit, which may not be good enough to predict the physiological process correctly because of their over diffusive nature. In the present work, we propose a fourth order spatially accurate and second order temporally accurate finite difference scheme for the equations governing the process of angiogenesis from an existing continuous model. For the discrete counterpart of the same model, the coefficients representing the probability density is computed by our high order compact (HOC) data. The proposed scheme is employed to carry out computation of the evolution of endothelial cell migration for three cases: the first two corresponds to the advancement of the density of endothelial cell in time and space mirroring the migration of the endothelial-cells from the parent vessel towards a tumor cell source in the shape of a line, one in the presence of haptotaxis and the other without it. The last deals with the same with a circular tumor cell source with haptotaxis. We also demonstrate that simulation resulting from a lower order accurate scheme might lead to misrepresentation of the physiological process, particularly in the later stages. Contrary to this, in each of the cases, our HOC simulations match excellently with earlier benchmark numerical and some experimental results for all stages of the process, thus confirming the robustness and efficiency of the proposed numerical scheme.


angiogenesis, tumor, sprouting, high-order compact, cell migration

2010 Mathematics Subject Classification

Primary 92Bxx, 92C17. Secondary 76M20.

Received 31 July 2019

Accepted 24 September 2019

Published 27 February 2020