A Hybrid Mathematical Model of Tumor-Induced Angiogenesis with Blood Perfusion

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1 TSINGHUA SCIENCE AND TECHNOLOGY ISSNll ll09/10llpp Volume 19, Number 6, December 2014 A Hybrid Mathematical Model of Tumor-Induced Angiogenesis with Blood Perfusion Junping Meng, Shoubin Dong, Liqun Tang, and Yi Jiang Abstract: Angiogenesis, the growth of new blood vessel from existing ones, is a pivotal stage in cancer development, and is an important target for cancer therapy. We develop a hybrid mathematical model to understand the mechanisms behind tumor-induced angiogenesis. This model describes uptake of Tumor Angiogenic Factor (TAF) at extracellular level, uses partial differential equation to describe the evolution of endothelial cell density including TAF induced proliferation, chemotaxis to TAF, and haptotaxis to extracellular matrix. In addition we also consider the phenomenon of blood perfusion in the micro-vessels. The model produces sprout formation with realistic morphological and dynamical features, including the so-called brush border effect, the dendritic branching and fusing of the capillary sprouts forming a vessel network. The model also demonstrates the effects of individual mechanisms in tumor angiogenesis: Chemotaxis to TAF is the key driving mechanisms for the extension of sprout cell; endothelial proliferation is not absolutely necessary for sprout extension; haptotaxis to ExtraCellular Matrix (ECM) gradient provides additional guidance to sprout extension, suggesting potential targets for anti-angiogenic therapies. Key words: tumor angiogenesis; ExtraCellular Matrix (ECM); capillary network; partial differential equation 1 Introduction Despite tremendous progress in the fighting against cancer in recent decades, the disease remains a huge health challenge in both the developed and developing countries. Understanding how cancer develop has a Junping Meng is with the School of Electronic & Information Engineering, South China University of Technology, Guangzhou , China. junpingmeng@163.com. Shoubin Dong is with the School of Computer Science & Engineering, South China University of Technology, Guangzhou , China. sbdong@scut.edu.cn. Liqun Tang is with the School of Civil Engineering & Transportation, South China University of Technology, Guangzhou , China. lqtang@scut.edu.cn. Yi Jiang is with the Department of Mathematics & Statistics, Georgia State University, Atlanta, GA 30303, USA. yjiang12@gsu.edu. To whom correspondence should be addressed. Manuscript received: ; accepted: profound and far reaching significance for it can not only help to decrease the incidence and mortality rate, but also guide clinical diagnosis and treatment. Tumorinduced angiogenesis, the growth of new blood vessel from existing ones due to tumor, is one key step in cancer development, marking the transition from benign to potentially malignant tumor. We propose to a new mathematical model that aims to help understand this important stage of cancer development. Since the initial proposal by Folkman in the early 1970s [1], much research has focused on understanding the mechanisms of angiogenesis and on targeting angiogenesis as a means of cancer therapy. The healthy body controls angiogenesis by balancing pro- and anti-angiogenic factors [2]. This balance is disrupted in many pathologies including cancer [1]. The growth of tumor is limited by nutrient supply and waste accumulation [3]. Hypoxic or oxygen deprived cells inside tumors respond to the environmental changes by up-regulating their production of Tumor

2 Junping Meng et al.: A Hybrid Mathematical Model of Tumor-Induced Angiogenesis with Blood Perfusion 649 Angiogenic Factors (TAFs), the most important ones being the Vascular Endothelial Growth Factor family (VEGFs). The TAFs diffuse into surrounding tissue, establish a concentration gradient, and create a proangiogenic environment. Endothelial cells that line the nearby blood vessel, when activated by TAFs, release proteases to degrade the blood vessel wall, leave the vessel wall, and migrate into the tissue. The active endothelial cells proliferate and migrate upgradient toward the high concentration of TAFs because they are chemotactic to TAF, and form a capillary sprout. It has been proposed that only the leading endothelial cells migrate, termed tip cell, while the cells behind the sprout tip follow the path and proliferate [3]. Additionally, the migration of endothelial cells requires biomechanical interactions between cells and the extracellular matrix (ECM), collagen and fibronectin are the major proteins of ECM. Gradient of cell-ecm binding also guides endothelial cells migration toward higher cell-ecm binding, as contact guidance and haptotaxis. The endothelial cells could also secrete Matrix Metallo Proteinases (MMPs), which degrade the ECM to pave a path when the ECM is too dense. The sprout formation requires complicated multifaceted and multiscale coordination of processes from molecular, cellular, to tissue levels. The past couple of decades have seen many mathematical models of angiogenesis. Anderson and Chaplain [4] modeled cornea angiogenesis using a discrete-continuous model where the tip cells were discretization of a set of continuous equations. The model assumed that the migration of endothelial cells is influenced by three factors: random motility, chemotaxis in response to TAF released by the tumor, and haptotaxis in response to fibronectin gradients in the extracellular matrix. Similar to Ref. [4], Harrington et al. [5] modeled tumor-induced angiogenesis, using reaction-diffusion equations for dynamics of TAF and inhibitor concentrations and endothelial cells were discrete object following growth rules depending on TAF and inhibitor concentrations. Sun et al. [6] used a two-dimensional (2-D) discrete model of capillary formation, where each endothelial cell moves following equations of motion subject to both chemotaxis and haptotaxis. Das et al. [7] used a three-dimensional (3-D) lattice model, where endothelial cells are point particles on the lattice, and reaction-diffusion equations govern the dynamics of VEGF, ECM, and MMP. Bauer et al. [8] used the Cellular Potts Model (CPM) for a more detailed cell-based model of tumorinduced angiogenesis. The CPM describes spatially extended endothelial cells and their behaviors and interactions with their environment, including explicit, spatially inhomogeneous ECM, while a continuous equation describes the TAF dynamics. An extended model [9] highlights the effect of ECM topography on the spatial coordination of endothelial cells into sprouts. Shirinifard et al. [10] also used the CPM framework to simulate the long-term process of tumor angiogenesis and vascular tumor growth. Schugart et al. [11] developed a most sophisticated model of woundhealing angiogenesis. This model includes reactiondiffusion equations to describe the spatiotemporal dynamics of capillary tips, capillary sprouts, fibroblasts, inflammatory cells, inflammatory cells, and oxygen and ECM, respectively. The micro-vessels formed in tumor angiogenesis are not as well-regulated as normal vessels, lacking the support and maintenance of other cells such as smooth muscle cells and pericytes. These vessels are immature and irregular, have shunts blockages, and often are leaky. Stéphanou et al. [12] considered the influence of blood flow in vascular tumor growth. Cai et al. [13] modeled tumor avascular growth, angiogenesis, and blood perfusion. These models considered vessel remodeling instead of beginning stage of angiogenesis. As tumor angiogenesis is a continuous and long process, lasting a few weeks to a few years, the immature characteristics of the initial tumor vessels have a profound impact on further tumor angiogenesis. In particular the leaky nature of tumor vessels modifies the environment for both the endothelial cells and the tumor cells. This impact of leaky vessel on the later stage of tumor angiogenesis has not been studied, and is the focus of this paper. We aim to model the tumor-induced angiogenesis with blood perfusion during tumor angiogenesis. 2 Mathematical Model We developed a hybrid discrete-continuous model for tumor angiogenesis, with a discrete cell model on lattice for active endothelial sprout tip cells, and continuous equations describing the endothelial cell density field, as well as TAF and fibronectin fields. This treatment is based on the assumption that the motion of an individual endothelial cell located at the tip of a capillary sprout governs the motion of the whole sprout, similar to that in Ref. [4]. In vivo experiments using implant human

3 650 Tsinghua Science and Technology, December 2014, 19(6): tumor in mouse [14, 15] clearly demonstrated that the tip cells of the vessel sprout show little random motility, because the movement of individual cells is constrained by the surrounding tissue [16]. This observation allows us to omit the influence of random motility on endothelial cell migration. Contrary to some previous models (e.g., Ref. [4]), where only the tip cell is allowed to proliferate, and some other models (e.g., Ref. [8]), where only the cells behind the tip are allowed to proliferate, we simply assume that all endothelial cells of the new micro-vessel can proliferate. We denote the endothelial cell density by n and the equation for endothelial cell density D kn 1 C c nrc r.nrf / (1) It shows that the variation of the endothelial cell density with time t depends on the proliferation of endothelial cells with a proliferation rate k, the chemotaxis to the TAF (c), and the haptotaxis to fibronectin (f ). In Eq. (1), is the chemotaxis coefficient, and is the haptotaxis coefficient, both are positive; reflects the chemotaxis sensitivity to TAF and is also a positive constant. This model of endothelial cell density is different from all previous models where proliferation and migration were combined as a diffusion term of the cell density. We included the proliferation term explicitly, disentangling the proliferation and migration effects, which allowed us to separately examine the different mechanisms on angiogenesis. We can rewrite Eq. (1) in terms " rc D k C 1 C c 2 rc 1 C c C rf # r 2 c r 2 f n 1 C c rn (2) The TAF diffuses in the tissue and is absorbed by the endothelial cells. We describe the dynamics of TAF with this D D cr 2 c nc (3) where c is the TAF concentration, D c is the diffusion coefficient of TAF, is the rate of the TAF uptake by the endothelial cells. Because diffusion timescale is much faster than those for cell migration and molecular update by the cells, we can safely assume that the distribution of TAF is at steady state except at the location and time when cell uptake occurs. Hence we can simplify the equation for the TAF concentration D nc Assuming the source of TAF is a tumor of 1-2 mm in size, and the hypoxic cells locate around 0.3 mm from the center, outside the necrotic core [17], we chose the initial distribution( of TAF concentration as 1; 0 r 0:3I c.x; y; 0/ D.1:3 r/ 2 (5) ; r > 0:3 where r is the distance to tumor center. The initial distribution of TAF is shown in Fig. 1a. Fibronectin, representing the ECM, can be produced and degraded by endothelial cell, remodeled because of migration of cells. But for simplicity, as we focus on the TAF and blood perfusion on angiogenesis, we assumed that the distribution of fibronectin is at a steady D 0 We borrowed the initial concentration profile of fibronectin from Ref. [4], f.x; y; 0/ D k 1 e x2 (7) where k 1 and are positive constants. This initial distribution of fibronectin is shown in Fig. 1b, where the fibronectin concentration is higher at the parent vessel than near the tumor. We also examined another scenario (Fig. 1c) where the fibronectin concentration is higher near the tumor than near the parent vessel. Our simulation domain is a lattice with grid size equals to 10 m, which makes the domain size 22 mm 2. The grid size corresponds to approximately the size of one endothelial cell. A solid tumor spheroid is located at the right side of the domain (not shown), serving as a source of the TAF, and a parent vessel is located at the left side of the domain (also not shown). Four sprout tips were present at the parent vessel as the initial endothelial cell density (Fig. 1d). We also assume that the capillary sprout cannot grow out the square domain, and use a constant value boundary condition, or Dirichlet boundary condition for cell density at all boundaries. Similar to the Anderson & Chaplain model [4], we used this set of rules for the sprout: A sprout tip can branch when (1) its age exceeds the branching age age th, (2) the endothelial cell density is greater than a threshold level n b, and (3) sufficient space is available around the sprout tip. After branching the old sprout stops growing, while the two new sprout tips begin to grow with their ages set to zero.

4 Junping Meng et al.: A Hybrid Mathematical Model of Tumor-Induced Angiogenesis with Blood Perfusion 651 Fig. 1 Initial distributions of TAF concentration (a), fibronectin concentration (b, c), and endothelial cell density (d), in a simulation domain. The initial 4 sprout tip cells are at 0.3 mm, 0.8 mm, 1.3 mm, and 1.6 mm along the parent vessel. We also modeled anastomosis, the formation of loops by capillary sprouts, which is another important feature of angiogenesis. We allowed two types of anastomosis: tip-to-tip anastomosis and tip-to-sprout anastomosis. In tip-to-tip anastomosis, when a sprout tip meets another sprout tip, both the tips stop growing and form a loop. In tip-to-sprout anastomosis, when a sprout tip meets another existing sprout, the tip fuses with the sprout and stops growing. The model simulation follows the pseudo code in Algorithm 1. At every time step, each active sprout tip checks its neighboring empty sites. On a square lattice a tip has 4 nearest neighbors, but only 3 sites ahead of the tip are available, because a tip is always automatically followed by a sprout: All the sites where the tip has been is where the sprout is. The endothelial cell density n is set to be 1, the maximum value, for the sprout body. This treatment reflects the immature and leaky nature of the tumor vessel sprouts. The tip will only extend into a site where the endothelial cell density exceeds a threshold density n th, reflecting the requirement that the formation of micro-vessel requires a certain number of endothelial cells. If there are more than one site that have high enough cell density, then Algorithm 1 Algorithm description of the model for For every time step do Solve equations (2) and (4), update endothelial cell density and TAF concentration for For every sprout tip do if the state of the sprout tip is activated then if the branching condition is satisfied then The original sprout tip stops grow and two new sprout tips appear with age=0 else if there is room for a sprout tip to move into then Move into the position where the endothelial cell density exceeds the threshold value and is the highest, and increase the age of this sprout tip by 1 if the neighbor position has been occupied by a sprout tip then Tip-to-tip anastomosis happens and this tip becomes inactive end if else Tip-to-sprout anastomosis happens and this tip becomes inactive end if end if end if end for end for

5 652 Tsinghua Science and Technology, December 2014, 19(6): the sprout tip will choose the position with the highest density. In addition, the tip will check the branching and anastomosis rules to decide if it will branch or fuse with a neighboring sprout or tip. If a branching event should happen, the old tip extends into two new positions as two new tips with age=0. If a tip-to-tip or a tip-to-sprout anastomosis should happen, the tips involved become inactive. The cell density for the site of the old, inactive tip is set of maximum n=1. We then solve for Eqs. (2) and (4), update the TAF concentration field, c, as well and the endothelial cell density field, n. The new tips are then ready to explore its neighborhood again for the next time step. 3 Simulation Results Table 1 lists the parameters used in our simulations. The values of threshold endothelial cell densities, n b and n th, for sprout tip branching and extension, respectively, were chosen through trial and error to ensure the sprouts are continuous and without fragmentation that is nonbiological and is a possible simulation artifact. We simulated the evolution of micro-vessel morphology for 15 days, which is the typical timescale for tumor vasculature to develop. Figure 2 shows the snapshots of a simulation from the initial four sprout tips. In the beginning days, the sprout tips extend in parallel, before the sprouts start to spread to cover the space between initial sprouts. We can understand Table 1 Parameter values used in simulations. Parameter Value Source 0.6 Ref. [4] 0.38 Ref. [4] 0.34 Ref. [4] 0.1 Ref. [4] k 2 Ref. [18] k Ref. [4] 0.45 Ref. [4] age th 0.38 Ref. [4] n b 0.3 Estimated n th 0.3 Estimated this initial parallel growth as follows: As can be seen in Fig. 1a, the TAF concentration gradient along the y-axis, near the parent vessel, is shallow, as if the TAF is from a line source (or very large tumor). The endothelial cell tips can only sense the TAF gradient along the x-axis. Hence the tips extend up-gradient toward tumor along the x-axis until the gradient along the y-axis is large enough, when the tips start to extend along the y-axis and the sprouts bend toward the center. As the sprouts extend toward the tumor, more tips become old enough to branch. When sprout density increases and they extend close to each other, anastomosis also happens. The simulation reproduces the so-called brush border effect : the closer to the tumor, the denser the sprouts. The fibronectin concentration field does not have any gradient along the Fig. 2 Snapshots of a typical angiogenesis simulation showing the evolution of capillary network from initial 4 sprout tips, in 15 days. Simulation domain is or 22 mm 2.

6 Junping Meng et al.: A Hybrid Mathematical Model of Tumor-Induced Angiogenesis with Blood Perfusion 653 y-axis, hence has no influence on sprout morphological dynamics. As its gradient is in the opposite direction of the TAF, fibronectin acts to hold back the sprout tip cells and slows down sprout extension. Now with such a tumor angiogenesis model in place, we could examine these mechanisms individually using the model. We call the simulations that generated Fig. 2 as baseline, and the parameters used as baseline parameters. First we turned off chemotaxis by setting the chemotaxis coefficient to zero and keeping the rest the same as in baseline. The sprouts did not grow at all (results not shown), suggesting that chemotaxis to TAF is an essential mechanism for angiogenesis. Next we turned off cell proliferation, while keeping the rest the same as those in baseline parameters. Figure 3 shows the snapshots of capillary network in the absence of cell proliferation. After the initial few days (about 6 days), the micro-vessel barely grows: the extension of sprouts stagnates. Two possible explanations are: Angiogenesis cannot occur in the absence of cell proliferation, as suggested by Sholley et al. [19] in their experimental model, or without proliferation, the chemotaxis and haptotaxis effects, working in opposite directions, cancel each other in this model. We address this question below. To determine the effect of haptotaxis on the sprout formation, we turned off endothelial cells haptotaxis to fibronectin by setting =0. Figure 4 shows the evolution of capillary network formation without haptotaxis. The striking difference is that the sprout cells reach the tumor in about 4 days, which is much faster compared to the baseline simulation (Fig. 2). This result indicates Fig. 3 Snapshots of capillary network formation in the absence of cell proliferation, with k=0. Parameters are otherwise the same as in Fig. 2. Fig. 4 Snapshots of capillary network formation in the absence of haptotaxis (=0), all other parameters are the same as in baseline simulation, Fig. 2.

7 654 Tsinghua Science and Technology, December 2014, 19(6): that haptotaxis to a fibronectin gradient could slow down the speed of sprout extension, which is quite easy to understand because the gradient for fibronectin is higher near the parent vessel and lower at the tumor (Fig. 1b). Haptotaxis to such a gradient would guide the sprout extension toward the parent vessel, away from the tumor. This original setting from Ref. [4] was designed to match the timing of the vessels reaching tumor in about 2 weeks. In the other scenario where fibronectin gradient is in the opposite direction, i.e., fibronectin is denser near the tumor (Fig. 1c), the simulations showed different results. When haptotaxis and chemotaxis work together, the sprouts extend a lot faster and can reach the tumor in 4.5 days (Fig. 5), resulting in less tortuous and less branchy vessels. If we turned off endothelial cell proliferation (Fig. 6), we see that different from Fig. 3, the sprouts are able to keep growing and reach the tumor. With haptotaxis and chemotaxis working together to grow the sprouts, the extension speeds with and without endothelial cell proliferation turn out to be very similar (compare Figs. 5 and 6). This result clearly indicates that cell proliferation is not absolutely essential in sprout extension. Cell proliferation would rescue the sprouts if chemotaxis and haptotaxis work in opposite directions causing the sprout growth stagnation; otherwise, the cell density changes due to chemotaxis and haptotaxis, by cell migration and cell stretching, is sufficient to generate the elongated and continuous sprouts, without cell proliferation. Figure 7 shows the snapshots of the capillary network Fig. 5 Snapshots of capillary network formation when haptotaxis and chemotaxis work together to extend sprout. Fig. 6 Snapshots of capillary network formation when haptotaxis and chemotaxis work together in the absence of cell proliferation with k=0. Fig. 7 Snapshots of capillary network in the absence of both cell proliferation and haptotaxis.

8 Junping Meng et al.: A Hybrid Mathematical Model of Tumor-Induced Angiogenesis with Blood Perfusion 655 in the absence of endothelial cell proliferation and haptotaxis. The capillary network did not stall, but the sprout tips extend slower than in Figs. 5 and 6. This result further proves that endothelial cell proliferation is not necessary to start the capillary formation. Chemotaxis to TAF gradient alone can result in sprout extension. Haptotaxis to fibronectin and endothelial cell proliferation both will facilitate the process. 4 Disscusion Tumor-induced angiogenesis is a highly important step in cancer development. We developed a mathematical model to help shed light into the mechanisms of tumor angiogenesis. The model is a hybrid discrete and continuous model, where the tip cells are discretized cell density modeled as individual cells, and the cell density, TAF, and fibronectin are three continuous fields following partial differential equations. The new features of this model include (1) a complete and explicit separation of individual mechanisms, including proliferation, TAF chemotaxis, and fibronectin haptotaxis, and (2) considering the phenomenon of blood perfusion after the initial formation of micro-vessels. The simulation results reproduced brush border effects when all mechanisms are considered. When we examined the effects of each mechanism by shutting off individual mechanisms in the simulations, we found that chemotaxis to TAF is the key driving mechanisms for the extension of sprout cell, and without chemotaxis (or TAF) angiogenesis cannot happen; TAF gradient is essential in activating endothelial cells. This result agrees with the current understanding that angiogenesis starts with the overexpression of TAFs, and that targeting TAF (e.g., using bevacizumab, the antibody for vascular endothelial growth factor) is a major component of cancer chemotherapy. We showed that endothelial proliferation is not absolutely necessary for sprout extension, provided that chemotaxis and haptotaxis work together to cause cell migration and stretching. This conclusion was exaggerated in this model because we do not consider cell size in the differential equations nor as points on the grid. If we consider the realistic cell size and shape, a cell can only be stretched to a certain length (or aspect ratio). Endothelial cell proliferation would be necessary to ensure successful tumor angiogenesis. We also have a clear understanding of the effect of haptotaxis: Its gradient provides additional guidance to sprout extension. When the ECM gradient is opposite of TAF gradient, turning off haptotaxis enhances angiogenesis; when the ECM gradient is the same as TAF gradient, turning off haptotaxis slows down angiogenesis. The role of ECM in cell migration is multi-fold. On the one hand, the ECM provides contact guidance for migrating cells, helping the migration; on the other hand, the cells will need to bind to the ECM when they migrate, and the binding and unbinding of cell surface receptor (integrins) and ECM proteins take time and energy. More recent research also pointed out that ECM mediates molecular signals between cells, inducing cell differentiation and physiological changes [20]. These different actions have opposite effects on cell migration speed. Hence there is an optimal ECM density where the cell migrates most efficiently. In addition, ECM is heterogeneous, and its topography alone can influence cell migration and sprout formation [9]. Our treatment of haptotaxis to ECM in this model is a simple consideration of gradient following. More sophisticated model of cell-ecm interactions, especially in biomechanics and mechanosensing will allow for more realistic understanding of ECM in tumor angiogenesis. The same understanding also applies to tumor invasion, where the basic biology of cell-ecm interactions between tumor cell and ECM should be the same as between endothelial cells and ECM. In summary, we have developed a mathematical model of tumor angiogenesis that offers some new insight into tumor angiogenesis. Further development in terms of cell-ecm interaction will be necessary to make the model more realistic. Also the model is completely deterministic; it would be interesting to investigate the role of stochasticity, in both cell decision-making and cell migration, in tumor angiogenesis. Acknowledgements This work was supported by the National Natural Science Foundation of China (No ). References [1] J. Folkman, Tumor angiogenesis: A possible control point in tumor growth, Annals of Internal Medicine, vol. 82, no. 1, pp , 1995.

9 656 Tsinghua Science and Technology, December 2014, 19(6): [2] P. Carmeliet and R. K. Jain, Angiogenesis in cancer and otherdiseases, Nature, vol. 407, no. 6801, pp , [3] H. Gerhardt, M. Golding, M. Fruttiger, C. Ruhrberg, A. Lundkvist, A. Abramsson, M. Jeltsch, C. Mitchell, K. Alitalo, D. Shima, et al., VEGF guides angiogenic sprouting utilizing endothelial tip cell filopodia, The Journal of Cell Biology, vol. 161, no. 6, pp , [4] A. R. A. Anderson and M. A. J. Chaplain, Continuous and discrete mathematical models of tumor-induced angiogenesis, Bulletin of Mathematical Biology, vol. 60, no. 5, pp , [5] H. A. Harrington, M. Maier, L. Naidoo, N. Whitaker, and P. G. Kevrekidis, A hybrid model for tumor-induced angiogenesis in the cornea in the presence of inhibitors, Mathematical and Computer Modelling, vol. 46, no. 3, pp , [6] S. Y. Sun, M. F. Wheeler, M. Obeyesekere, and C. Patrick Jr, Nonlinear behaviors of capillary formation in a deterministic angiogenesis model, Nonlinear Analysis: Theory, Methods & Applications, vol. 63, no. 5, pp. e2237- e2246, [7] A. Das, D. Lauffenburger, H. Asada, and R. D. Kamm, A hybrid continuum discrete modelling approach to predict and control angiogenesis: Analysis of combinatorial growth factor and matrix effects on vessel-sprouting morphology, Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences, vol. 368, no. 1921, pp , [8] A. L. Bauer, T. L. Jackson, and Y. Jiang, A cell-based model exhibiting branching and anastomosis during tumorinduced angiogenesis, Biophysical Journal, vol. 92, no. 9, pp , [9] A. L. Bauer, T. L. Jackson, and Y. Jiang, Topography of extracellular matrix mediates vascular morphogenesis and migration speeds in angiogenesis, PLoS Computational Biology, vol. 5, no. 7, p. e , [10] A. Shirinifard, J. S. Gens, B. L. Zaitlen, N. J. Popławski, M. Swat, and J. A. Glazier, 3D multi-cell simulation of tumor growth and angiogenesis, PLoS One, vol. 4, no. 10, p. e7190, [11] R. C. Schugart, A. Friedman, R. Zhao, and C. K. Sen, Wound angiogenesis as a function of tissue oxygen Junping Meng received her master s degree in computer science and engineering in 2014 from South China University of Technology. Her bachelor degree was received from Hebei University in tension: A mathematical model, Proceedings of the National Academy of Sciences, vol. 105, no. 7, pp , [12] A. Stéphanou, S. R. McDougall, A. R. A. Anderson, and M. A. J. Chaplain, Mathematical modelling of the influence of blood rheological properties upon adaptative tumour-induced angiogenesis, Mathematical and Computer Modelling, vol. 44, no. 1, pp , [13] Y. Cai, S. X. Xu, J. Wu, and Q. Long, Coupled modelling of tumour angiogenesis, tumour growth and blood perfusion, Journal of Theoretical Biology, vol. 279, no. 1, pp , [14] N. Paweletz and M. Knierim, Tumor-related angiogenesis, Critical Reviews in Oncology/Hematology, vol. 9, no. 3, pp , [15] S. Paku and N. Paweletz, First steps of tumor-related angiogenesis, Laboratory Investigation; A Journal of Technical Methods and Pathology, vol. 65, no. 3, pp , [16] M. A. Rupnick, C. L. Stokes, S. K. Williams, and D. A. Lauffenburger, Quantitative analysis of random motility of human microvessel endothelial cells using a linear underagarose assay, Laboratory Investigation; A Journal of Technical Methods and Pathology, vol. 59, no. 3, pp , [17] B. S. Kuszyk, F. M. Corl, F. N. Franano, D. A. Bluemke, L. V. Hofmann, B. J. Fortman, and E. K. Fishman, Tumor transport physiology: Implications for imaging and imaging-guided therapy, American Journal of Roentgenology, vol. 177, no. 4, pp , [18] A. Lesart, B. Van Der Sanden, L. Hamard, F. Estève, and A. Stéphanou, On the importance of the submicrovascular network in a computational model of tumour growth, Microvascular Research, vol. 84, no. 2, pp , [19] N. M. Sholley, G. P. Ferguson, H. R. Seibel, J. L. Montour, and J. D. Wilson, Mechanisms of neovascularization, Vascular sprouting can occur without proliferation of endothelial cells, Laboratory Investigation; A Journal of Technical Methods and Pathology, vol. 51, no. 6, pp , [20] R. O. Hynes, The extracellular matrix: Not just pretty fibrils, Science, vol. 326, no. 5957, pp , Shoubin Dong is a professor of School of Computer Science and Engineering of South China University of Technology (SCUT). Her main research areas include high performance computing, big data processing, and next generation Internet. Dr. Dong received her PhD degree in electronic engineering in 1994 from University of Science and Technology of China (USTC). She was a visiting scholar of School of Computer Science of Carnegie Mellon University (CMU) during She is now the deputy director of Communication and Computer Network Laboratory (CCNL) of Guangdong Province, China.

10 Junping Meng et al.: A Hybrid Mathematical Model of Tumor-Induced Angiogenesis with Blood Perfusion 657 Liqun Tang is a professor in the Department of Engineering Mechanics at South China University of Technology, China. He received his PhD degree from University of Science and Technology of China in He was ever a research associate in the Hong Kong Polytechnic University and visiting scholar in the Texas A&M University. His research interests are mechanics of materials, damage mechanics, and meso mechanics. He focuses on: (1) meso-structural features of heterogeneous materials and numerical modelling; (2) long-term health monitoring of structures and structural damage identification; (3) constitutive relations of soft matters; and (4) new kinds of engineering materials and their constitutive relations. Yi Jiang received her PhD degree in physics from University of Notre Dame in 1998, and was a research scientist at Los Alamos National Laboratory until she joined Georgia State University in 2011, where she is an associate professor in the Department of Mathematics and Statistics. Her research lies in the general direction of mathematical biology and biophysics. Her current focus is in developing multi-scale models of cancer development, including tumor growth, angiogenesis, invasion, and therapy.