Nonlinear Analysis 63 (2005) e2237 – e2246 www.elsevier.com/locate/na

Nonlinear behaviors of capillary formation in a deterministic angiogenesis model夡 Shuyu Suna,∗ , Mary F. Wheelera , Mandri Obeyesekereb, c , Charles Patrick Jr.b, c a The Institute for Computational Engineering and Sciences, The University of Texas at Austin,

201 E. 24th Street, Austin, TX 78712, USA b The University of Texas M. D. Anderson Cancer Center, Houston, TX 77030, USA c The University of Texas Center for Biomedical Engineering, Houston, TX 77030, USA

Abstract In this paper, we consider a deterministic approach for modeling angiogenesis. The model equations form a nonlinear coupled system of partial and ordinary differential equations. We propose an efficient, accurate and locally conservative numerical method to solve the nonlinear system. Computational results indicate that the model generates the overall dendritic structure and pattern of the capillary network morphologically similar to those observed in vivo. The influence of the capillary network and the growth factor distribution on each other and their interaction are investigated using numerical simulations. 䉷 2005 Elsevier Ltd. All rights reserved. Keywords: Angiogenesis; Capillary network; Growth factors

1. Introduction Angiogenesis, the outgrowth of new vessels from a pre-existing vasculature, plays an important rule in many mammalian growth processes such as early embryogenesis during the formation of the placenta [6], controlled blood-vessel formation during tissue repair [4,8] and excessive blood-vessel formation during tumor growth [5]. A deep understanding 夡

Supported in part by a grant from the University of Texas Center for Biomedical Engineering.

∗ Corresponding author. Tel.: +1 512 232 7764; fax: +1 512 232 2445.

E-mail address: [email protected] (S. Sun). 0362-546X/$ - see front matter 䉷 2005 Elsevier Ltd. All rights reserved. doi:10.1016/j.na.2005.01.066

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S. Sun et al. / Nonlinear Analysis 63 (2005) e2237 – e2246

of angiogenesis at the capillary level is critical for reparative strategies since the capillary network dictates tissue survival, hemodynamics and mass transport. The angiogenic system is strongly nonlinear and extremely complex, possessing multiple, integrated modulators and feedback loops. The nonlinearity and complexity of the system limits the in vitro and in vivo experiments that may be designed and the amount of nonconfounding information that can be gleaned. A mathematical formulation describing the intercellular growth patterns of capillaries within a tissue is essential for understanding and analyzing these complex phenomena. In this paper, we consider a deterministic approach for modeling angiogenesis proposed in [9]. In this model, the anisotropy of extracellular matrix is reflected by the conductivity of the extracellular matrix for the extension of capillary sprouts. Furthermore, the capillary network is sharply captured by a capillary indicator function. The model equations form a nonlinear coupled system of partial and ordinary differential equations. We propose an efficient, accurate and locally conservative numerical method to solve the equations. The influence of the capillary network and the growth factor distribution on each other and their interaction are investigated using numerical simulations. The remaining parts of this paper are organized as follows. In Section 2, we state the governing equations of angiogenesis. An efficient and locally conservative numerical algorithm is established in Section 3. Angiogeneses from a single parent vessel and from two parent vessels are simulated in Section 4, where dendritic and realistic structures of capillary networks are observed. Finally, a brief summary is given in Section 5 to conclude this paper. 2. The mathematical model We restrict our attention to two spatial dimensions here. The capillary presence is represented by a indicator function n, which is a function of space (x, y) and time t, and has only the value of either 0 or 1 depending on the presence of capillary. The concentration of chemotactic growth factors (CGFs) is denoted by c. We track the individual behavior of each sprout tip in our model. Each individual tip at time t is denoted by (xi (t), yi (t)), where xi (t) and yi (t) are the x and y components of the position of the tip at time t. The collection of all sprout tips at time t is denoted by a set S(t), i.e. S(t) = {(xi (t), yi (t)) for all tips}. We note that the set S might change with time due to sprout branching, extension, and anastomosis discussed below. The capillary indicator function n is closely related to the history of the positions of sprout tips. That is, the value of n is 1 on the trajectories of sprout tips (the trails passed by all sprout tips) and is 0 elsewhere. Mathematically, it can be written as n = F (S), and




1 F (S)(x, y, t) = 0

if (x, y) ∈ otherwise.



0