Article Contents
Article Contents

# A new model for the emergence of blood capillary networks

• * Corresponding author

The first two authors contributed equally to this research

• We propose a new model for the emergence of blood capillary networks. We assimilate the tissue and extra cellular matrix as a porous medium, using Darcy's law for describing both blood and interstitial fluid flows. Oxygen obeys a convection-diffusion-reaction equation describing advection by the blood, diffusion and consumption by the tissue. Discrete agents named capillary elements and modelling groups of endothelial cells are created or deleted according to different rules involving the oxygen concentration gradient, the blood velocity, the sheer stress or the capillary element density. Once created, a capillary element locally enhances the hydraulic conductivity matrix, contributing to a local increase of the blood velocity and oxygen flow. No connectivity between the capillary elements is imposed. The coupling between blood, oxygen flow and capillary elements provides a positive feedback mechanism which triggers the emergence of a network of channels of high hydraulic conductivity which we identify as new blood capillaries. We provide two different, biologically relevant geometrical settings and numerically analyze the influence of each of the capillary creation mechanism in detail. All mechanisms seem to concur towards a harmonious network but the most important ones are those involving oxygen gradient and sheer stress. A detailed discussion of this model with respect to the literature and its potential future developments concludes the paper.

Mathematics Subject Classification: Primary: 92C15, 76Z05; Secondary: 92C42, 76S05.

 Citation:

• Figure 1.  A capillary element of length $L_c$ and width $w_c$ with center at $\textbf{X}$ and direction $\boldsymbol{\omega}$

Figure 2.  (A) The function $g \mapsto \psi_1 ( g) = \psi ( ( L_0^c g - 1)/h_c)$ where $\psi$ is defined in (8) models an on/off switch. Its fuzziness region is shadowed in gray. On its left-hand-side the switch is off whereas on its right-hand-side it is on. (B) The function $\rho \mapsto \psi_2 ( \rho) = \psi ( (1 - \rho / \rho_s) / h_s)$ with fuzzy region shadowed in gray. As opposed to (A) the switch is on at the left-hand-side of the shadowed region and it is off on the right-hand-side

Figure 3.  Given a point $\textbf{X}$ in the tissue, the second term of the right-hand-side of the tensors $\textbf{K}$ and $\textbf{D}$ defined in (15) and (16) are computed by summing the tensors $\boldsymbol{\omega}_k \otimes \boldsymbol{\omega}_k$ over all capillary elements $k$ that contain $\textbf{X}$ in their domain $S_k$. For instance, in this sketch, only five (dark-shadowed rods) out of the nine capillary elements are combined to form tensors $\textbf{K}$ and $\textbf{D}$ at $\textbf{X}$

Figure 4.  (A) Geometrical setting for $\Omega_1$, which mimics a cross-section of the tissue in the direction normal to a blood vessel. (B) Geometrical setting for $\Omega_2$ which mimics a cross-section in a plane containing the blood vessel. The dimensions of $\Omega_1$ and $\Omega_2$ are given in Table. 1

Figure 5.  Labeling of boundaries and boundary conditions for the pressure $p$ and oxygen density $\rho$ in $\Omega_1$

Figure 6.  Labeling of boundaries and boundary conditions for the pressure $p$ and oxygen density $\rho$ for $\Omega_2$

Figure 7.  Positions of oxygen particles (red spots) and of capillary elements (blue rods) in the rectangular domain $\Omega_1$ for a realization of the model. As red spots overlay the blue rods, capillary elements lying below the red oxygen particles are present although not seen. All the creation/deletion mechanisms of capillary elements are turned 'On'. All the parameter used are those given in Tables 1 and 2. Pictures (A) to (F) give snapshots at increasing times: $2$ min (A), $4$ min (B), $6$ min (C), $8$ min (D), $10$ min (E), $12$ min (F) after initialization

Figure 8.  Isolines and heatmap of the pressure $p$ in the rectangular domain $\Omega_1$ for the same realization of the model as in Fig. 7. All the creation/deletion mechanisms of capillary elements are turned 'On'. All the parameters used are those given in Tables 1 and 2. Pictures (A) to (F) give snapshots at increasing times: $2$ min (A), $4$ min (B), $6$ min (C), $8$ min (D), $10$ min (E), $12$ min (F) after initialization. The units are given in mmHg

Figure 9.  Heatmap of the Frobenius norm $\gamma$ of the hydraulic conductivity tensor $\boldsymbol{K}$ in the rectangular domain $\Omega_1$ for the same realization of the model as in Fig. 7. All the creation/deletion mechanisms of capillary elements are turned 'On'. All the parameter used are those given in Tables 1 and 2. Pictures (A) to (F) give snapshots at increasing times: $2$ min (A), $4$ min (B), $6$ min (C), $8$ min (D), $10$ min (E), $12$ min (F) after initialization. The units are given in $10^5$ $\mu$m$^2 / (mmHg \, \, min)$

Figure 10.  Two binary decision trees, placed back-to-back. The upper one (I) includes capillary creation by reinforcement while the lower one (II) excludes it. Each tree successively includes or excludes capillary creation by WSS and oxygen gradient (noted $O2\nabla$). At the end of each branch, a typical realization of the model with corresponding inclusion/exclusion of the mechanism is shown. The picture shows the positions of the oxygen particles (red spots) and those of the capillary elements (tiny blue rods). The times for each of the snapshots are the following: 12 min (A), 12 min (B), 19.5 min (C), 30 min (D), 12 min (E), 12 min (F) and 19.5 min (G), after initialization

Figure 11.  Positions of the oxygen particles (red spots) and of the capillary elements (tiny blue rods) in the domain $\Omega_1$ (see caption of Fig. 7 for details) for a realization with mesh-size $\Delta x = \Delta y = 5/8$, the other parameters in Tables 1 and 2 being unchanged. Pictures (A) to (D) give snapshots at increasing times: $2.5$ min (A), $5$ min (B), $7.5$ min (C), $10$ min (D) after initialization

Figure 12.  influence of the pressure gradient. (B) is the same as Fig. 7 (E). (A) is for pressure gradient reduced by $10 \, \%$. (C) is for pressure gradient increased by $10 \, \%$. Positions of oxygen particles (red spots) and of capillary elements (blue rods) in the rectangular domain $\Omega_1$ are plotted at time $10$ min after initialization. All the creation/deletion mechanisms of capillary elements are turned 'On'. All the parameter used are those given in Tables 1 and 2 except for Figs (A) and (C) where the boundary conditions for the pressure are modified as detailed in the text

Figure 13.  influence of the capillary element size. (B) is the same as Fig. 7 (D). (A) is for capillary length $L_c$ divided by $2$. (C) is for capillary length $L_c$ multiplied by $2$. Positions of oxygen particles (red spots) and of capillary elements (blue rods) in the rectangular domain $\Omega_1$ are plotted at time $8$ min after initialization. All the creation/deletion mechanisms of capillary elements are turned 'On'. All the parameter used are those given in Tables 1 and 2 except for Figs (A) and (C) where the capillary length is modified as detailed in the text

Figure 14.  Positions of the oxygen particles (red spots) and of the capillary elements (tiny blue rods) in the domain $\Omega_2$ (see caption of Fig. 7 for details) for a realization of the model. All the creation/deletion mechanisms of capillary elements are turned 'On'. All the parameters used are those given in Tables 1 and 2. Pictures (A) to (F) give snapshots at increasing times: $6.8$ min (A), $13.6$ min (B), $20.4$ min (C), $27.2$ min (D), $34$ min (E), $40.8$ min (F) after initialization

Table 1.  Parameters of the model. In case of estimated parameters ("estim." in the last column), we refer to the corresponding section (indicated in the first column) for the details of this estimation

 Quantity Sym. Value Units Source Geometry 1 (Sect. 2.6) Domain size in $x$-direction $L_x$ $1000$ $\mu \mbox{m}$ estim. Domain size in $y$-direction $L_y$ $2000$ $\mu \mbox{m}$ estim. Oxygen injection region: $L_{\min}$ $950$ $\mu \mbox{m}$ estim. $y$-coordinate of lower end Oxygen injection region: $L_{\max}$ $1050$ $\mu \mbox{m}$ estim. $y$-coordinate of upper end Geometry 2 (Sect. 2.6) Domain size in $x$-direction $L_x$ $2000$ $\mu \mbox{m}$ estim. Domain size in $y$-direction $L_y$ $1000$ $\mu \mbox{m}$ estim. Oxygen injection region: $L_{\min}$ $450$ $\mu \mbox{m}$ estim. $y$-coordinate of lower end Oxygen injection region: $L_{\max}$ $550$ $\mu \mbox{m}$ estim. $y$-coordinate of upper end Blood (Sect. 2.2 & 2.4.4) Pressure at high pressure boundary $p_0$ $37.7$ $\mbox{mmHg}$ [81] Pressure at low pressure boundary $p_1$ $14.6$ $\mbox{mmHg}$ [81] Dynamic viscosity $\mu$ $3.75 \times 10^{-7}$ $\mbox{mmHg min}$ [21] Oxygen and oxygen dynamics (Sect. 2.3) Concentration at injection boundary $\rho_0$ $0.025$ $\mu \mbox{m}^{-2}$ estim. Concentration for linear/nonlinear $\widetilde{\rho}$ $0.1 \times \rho_0$ $\mu \mbox{m}^{-2}$ estim. diffusion shift Maximum consumption rate $\beta_{\rm{sat}}$ $0.01 \times \rho_0$ $\mbox{min}^{-1}\mu \mbox{m}^{-2}$ estim. from [13,75] Michaelis constant $K_m$ $0.5 \times \rho_0$ $\mu \mbox{m}^{-2}$ estim. from [13,75] Capillary elements (Sect. 2.4.1 & 2.5) Length $L_c$ $15$ $\mu \mbox{m}$ [23] Width $w_c$ $4$ $\mu \mbox{m}$ [23] Hydraulic conductivity $\kappa$ $80000$ $\mu \mbox{m}^2 \mbox{min}^{-1}\mbox{mmHg}^{-1}$ estim. Oxygen diffusivity $\Delta$ $200$ $\mu \mbox{m}^2\mbox{min}^{-1}$ estim. Capillary creation: oxygen gradient (Sect. 2.4.2) Maximal creation rate $\nu_c^*$ $0.05$ $\mu \mbox{m}^{-2}\mbox{min}^{-1}$ estim. Oxygen concentration gradient $L_0^c$ $8$ $\mu \mbox{m}$ estim. length threshold Concentration for regularization $\rho^*$ $0.1 \times \rho_0$ $\mu \mbox{m}^{-2}$ estim. of logarithmic sensing Width of sigmoid: oxygen gradient $h_c$ $0.1$ $-$ estim. Oxygen concentration threshold $\rho_s$ $\rho_0$ $\mu \mbox{m}^{-2}$ estim. Width of sigmoid: oxygen concentration $h_s$ $0.1$ $-$ estim. Capillary creation: reinforcement (Sect. 2.4.3) Maximal creation rate $\nu_f^*$ $0.01$ $\mu \mbox{m}^{-2}\mbox{min}^{-1}$ estim. Blood velocity threshold $\bar{u}$ $20$ $\mu \mbox{m } \mbox{min}^{-1}$ estim. Lower oxygen concentration threshold $\underline{\rho}$ $0.1 \times \rho_0$ $\mu \mbox{m}^{-2}$ estim. Upper oxygen concentration threshold $\bar{\rho}$ $0.5 \times \rho_0$ $\mu \mbox{m}^{-2}$ estim. Width of sigmoids $h_f$ $0.1$ $-$ estim. Capillary creation: WSS (Sect. 2.4.4) Maximal creation rate $\nu_w^*$ $0.3$ $\mu \mbox{m}^{-2}\mbox{min}^{-1}$ estim. Width of sigmoid $h_w$ $0.1$ $-$ estim. WSS threshold $\lambda^*$ $3.75 \times 10^{-8}$ mmHg estim. from [56] Capillary removal (Sect. 2.4.5) Removal rate at twice threshold $\nu_r^*$ $30.0$ $\mbox{min}^{-1}$ estim. Hydraulic conductivity threshold $\gamma^*$ $400000$ $\mu \mbox{m}^2 \, \mbox{min}^{-1} \mbox{mmHg}^{-1}$ estim. Tissue (Sect. 2.5) Hydraulic conductivity $k_h$ $400$ $\mu \mbox{m}^2 \mbox{min}^{-1}\mbox{mmHg}^{-1}$ [74] Oxygen diffusivity $\Delta_h$ $10$ $\mu \mbox{m}^2\mbox{min}^{-1}$ [76]

Table 2.  Numerical parameters

 Quantity Symbol Value Unit Finite-element-method for blood flow Mesh size in $x$-direction $\Delta x$ $1.25$ $\mu$m Mesh size in $y$-direction $\Delta y$ $1.25$ $\mu$m SPH particle method for oxygen concentration Particle "mass" $m$ $1.0$ $-$ Smoothing parameter $\eta$ $5.0$ $\mu$m CFL parameter $C$ $0.45$ $-$ Point Poisson process for capillary creation Number of sample points per time step $N_c$ $10^5$ $-$
•  [1] G. Albi, M. Burger, J. Haskovec, P. Markowich and M. Schlottbom, Continuum modeling of biological network formation, in Active Particles, Springer, 1 (2017), 1-48. doi: 10.1007/978-3-319-49996-3_1. [2] C. Amitrano, A. Coniglio and F. Di Liberto, Growth probability distribution in kinetic aggregation processes, Phys. Rev. Lett., 57 (1986), 1016. doi: 10.1103/PhysRevLett.57.1016. [3] D. Balding and D. L. S. McElwain, A mathematical model of tumour-induced capillary growth, J. Theoret. Biol., 114 (1985), 53-73.  doi: 10.1016/S0022-5193(85)80255-1. [4] C. Bardos and E. Tadmor, Stability and spectral convergence of fourier method for nonlinear problems: On the shortcomings of the $2/3$ de-aliasing method, Numer. Math., 129 (2015), 749-782.  doi: 10.1007/s00211-014-0652-y. [5] A. L. Bauer, T. L. Jackson and Y. Jiang, Topography of extracellular matrix mediates vascular morphogenesis and migration speeds in angiogenesis, PLoS Computational Biology, 5 (2009), e1000445, 18pp. doi: 10.1371/journal.pcbi.1000445. [6] E. Boissard, P. Degond and S. Motsch, Trail formation based on directed pheromone deposition, J. Math. Biol., 66 (2013), 1267-1301.  doi: 10.1007/s00285-012-0529-6. [7] S. C. Brenner and R. L. Scott, The Mathematical Theory of Finite Element Methods, vol. 15, Springer Science & Business Media, 2008. doi: 10.1007/978-0-387-75934-0. [8] T. Büscher, A. L. Diez, G. Gompper and J. Elgeti, Instability and fingering of interfaces in growing tissue, New J. Phys., 22 (2020), 083005, 11pp. doi: 10.1088/1367-2630/ab9e88. [9] H. Byrne and M. Chaplain, Mathematical models for tumour angiogenesis: Numerical simulations and nonlinear wave solutions, Bull. Math. Biol., 57 (1995), 461-486.  doi: 10.1007/BF02460635. [10] P. Carmeliet and R. K. Jain, Angiogenesis in cancer and other diseases, Nature, 407 (2000), 249-257.  doi: 10.1038/35025220. [11] A. Chen, J. Darbon, G. Buttazzo, F. Santambrogio and J.-M. Morel, On the equations of landscape formation, Interfaces Free Bound., 16 (2014), 105-136.  doi: 10.4171/IFB/315. [12] A. Chen, J. Darbon and J.-M. Morel, Landscape evolution models: A review of their fundamental equations, Geomorphology, 219 (2014), 68-86.  doi: 10.1016/j.geomorph.2014.04.037. [13] E. Curcio, A. Piscioneri, S. Morelli, S. Salerno, P. Macchiarini and L. De Bartolo, Kinetics of oxygen uptake by cells potentially used in a tissue engineered trachea, Biomaterials, 35 (2014), 6829-6837.  doi: 10.1016/j.biomaterials.2014.04.100. [14] G. Dahlquist and Å. Björck, Numerical Methods in Scientific Computing, Volume i, Society for Industrial and Applied Mathematics, 2008. doi: 10.1137/1.9780898717785. [15] J. T. Daub and R. M. H. Merks, A cell-based model of extracellular-matrix-guided endothelial cell migration during angiogenesis, Bull. Math. Biol., 75 (2013), 1377-1399.  doi: 10.1007/s11538-013-9826-5. [16] P. Degond and S. Mas-Gallic, The weighted particle method for convection-diffusion equations, i, the case of an isotropic viscosity, Mathematics of computation, 53 (1989), 485-507.  doi: 10.2307/2008716. [17] P. Degond and F.-J. Mustieles, A deterministic approximation of diffusion equations using particles, SIAM J. Sci. Stat. Comput., 11 (1990), 293-310.  doi: 10.1137/0911018. [18] Y. Efendiev and T. Y. Hou, Multiscale finite element methods: Theory and applications, vol. 4, Springer Science & Business Media, 2009. doi: 10.1007/978-0-387-09496-0. [19] I. Fischer, J.-P. Gagner, M. Law, E. W. Newcomb and D. Zagzag, Angiogenesis in gliomas: Biology and molecular pathophysiology, Brain pathology, 15 (2005), 297-310.  doi: 10.1111/j.1750-3639.2005.tb00115.x. [20] J. Folkman, Angiogenesis in cancer, vascular, rheumatoid and other disease, Nature Medicine, 1 (1995), 27-30.  doi: 10.1038/nm0195-27. [21] R. L. Fournier,  Basic Transport Phenomena in Biomedical Engineering, CRC press, 2017.  doi: 10.1201/9781315120478. [22] P. A. Galie, D.-H. T. Nguyen, C. K. Choi, D. M. Cohen, P. A. Janmey and C. S. Chen, Fluid shear stress threshold regulates angiogenic sprouting, Proc. Natl. Acad. Sci. USA, 111 (2014), 7968-7973.  doi: 10.1073/pnas.1310842111. [23] B. Garipcan, S. Maenz, T. Pham, U. Settmacher, K. D. Jandt, J. Zanow and J. Bossert, Image analysis of endothelial microstructure and endothelial cell dimensions of human arteries-a preliminary study, Advanced Engineering Materials, 13 (2011), B54-B57. doi: 10.1002/adem.201080076. [24] M. A. Gimbrone Jr, R. S. Cotran, S. B. Leapman and J. Folkman, Tumor growth and neovascularization: An experimental model using the rabbit cornea, Journal of the National Cancer Institute, 52 (1974), 413-427.  doi: 10.1093/jnci/52.2.413. [25] M. S. Gockenbach, Understanding and Implementing The Finite Element Method, Vol. 97, SIAM, 2006. doi: 10.1137/1.9780898717846. [26] D. Goldman and A. S. Popel, A computational study of the effect of capillary network anastomoses and tortuosity on oxygen transport, J. Theoret. Biol., 206 (2000), 181-194.  doi: 10.1006/jtbi.2000.2113. [27] J. A. González, F. J. Rodríguez-Cortés, O. Cronie and J. Mateu, Spatio-temporal point process statistics: A review, Spat. Stat., 18 (2016), 505-544.  doi: 10.1016/j.spasta.2016.10.002. [28] J. A. Grogan, A. J. Connor, J. M. Pitt-Francis, P. K. Maini and H. M. Byrne, The importance of geometry in the corneal micropocket angiogenesis assay, PLoS Computational Biology, 14 (2018), e1006049. doi: 10.1371/journal.pcbi.1006049. [29] J. Haskovec, L. M. Kreusser and P. Markowich, Rigorous continuum limit for the discrete network formation problem, Comm. Partial Differential Equations, 44 (2019), 1159-1185. doi: 10.1080/03605302.2019.1612909. [30] J. Haskovec, P. Markowich and B. Perthame, Mathematical analysis of a pde system for biological network formation, Comm. Partial Differential Equations, 40 (2015), 918-956.  doi: 10.1080/03605302.2014.968792. [31] J. Haskovec, P. Markowich, B. Perthame and M. Schlottbom, Notes on a pde system for biological network formation, Nonlinear Anal., 138 (2016), 127-155.  doi: 10.1016/j.na.2015.12.018. [32] M. B. Hastings and L. S. Levitov, Laplacian growth as one-dimensional turbulence, Phys. D, 116 (1998), 244-252.  doi: 10.1016/S0167-2789(97)00244-3. [33] H. J. Herrmann, Geometrical cluster growth models and kinetic gelation, Physics Reports, 136 (1986), 153-224.  doi: 10.1016/0370-1573(86)90047-5. [34] T. Hillen, M 5 mesoscopic and macroscopic models for mesenchymal motion, J. Math. Biol., 53 (2006), 585-616.  doi: 10.1007/s00285-006-0017-y. [35] D. Hu and D. Cai, Adaptation and optimization of biological transport networks, Phys. Rev. Lett., 111 (2013), 138701. doi: 10.1103/PhysRevLett.111.138701. [36] S. Ichioka, M. Shibata, K. Kosaki, Y. Sato, K. Harii and A. Kamiya, Effects of shear stress on wound-healing angiogenesis in the rabbit ear chamber, Journal of Surgical Research, 72 (1997), 29-35.  doi: 10.1006/jsre.1997.5170. [37] H. Kang, K. J. Bayless and R. Kaunas, Fluid shear stress modulates endothelial cell invasion into three-dimensional collagen matrices, American Journal of Physiology-Heart and Circulatory Physiology, 295 (2008), H2087-H2097. doi: 10.1152/ajpheart.00281.2008. [38] R. Kaunas, H. Kang and K. J. Bayless, Synergistic regulation of angiogenic sprouting by biochemical factors and wall shear stress, Cellular And Molecular Bioengineering, 4 (2011), 547-559.  doi: 10.1007/s12195-011-0208-5. [39] B. Kaur, F. W. Khwaja, E. A. Severson, S. L. Matheny, D. J. Brat and E. G. Van Meir, Hypoxia and the hypoxia-inducible-factor pathway in glioma growth and angiogenesis, Neuro-Oncology, 7 (2005), 134-153.  doi: 10.1215/S1152851704001115. [40] A. B. Langdon, B. I. Cohen and A. Friedman, Direct implicit large time-step particle simulation of plasmas, J. Comput. Phys., 51 (1983), 107-138.  doi: 10.1016/0021-9991(83)90083-9. [41] P. Macklin, S. McDougall, A. R. Anderson, M. A. Chaplain, V. Cristini and J. Lowengrub, Multiscale modelling and nonlinear simulation of vascular tumour growth, Journal of Mathematical Biology, 58 (2009), 765-798.  doi: 10.1007/s00285-008-0216-9. [42] S. Mas-Gallic, Particle approximation of a linear convection-diffusion problem with neumann boundary conditions, SIAM Journal on Numerical Analysis, 32 (1995), 1098-1125.  doi: 10.1137/0732050. [43] M. Matsumoto and T. Nishimura, Mersenne twister: A 623-dimensionally equidistributed uniform pseudo-random number generator, ACM Transactions on Modeling and Computer Simulation (TOMACS), 8 (1998), 3-30.  doi: 10.1145/272991.272995. [44] S. R. McDougall, A. R. Anderson and M. A. Chaplain, Mathematical modelling of dynamic adaptive tumour-induced angiogenesis: Clinical implications and therapeutic targeting strategies, J. Theoret. Biol., 241 (2006), 564-589.  doi: 10.1016/j.jtbi.2005.12.022. [45] G. Mitchison, A model for vein formation in higher plants, Proc. R. Soc. Lond. B, 207 (1980), 79-109.  doi: 10.1098/rspb.1980.0015. [46] G. J. Mitchison, D. E. Hanke and A. R. Sheldrake, The polar transport of auxin and vein patterns in plants, Phil. Trans. R. Soc. Lond. B, 295 (1981), 461-471.  doi: 10.1098/rstb.1981.0154. [47] J. J. Monaghan, Smoothed particle hydrodynamics, Annual Review of Astronomy and Astrophysics, 30 (1992), 543-574.  doi: 10.1007/978-94-011-4780-4_110. [48] M. Müller, D. Charypar and M. Gross, Particle-based fluid simulation for interactive applications, in Proceedings of The 2003 ACM SIGGRAPH/Eurographics Symposium on Computer Animation, Eurographics Association, (2003), 154-159. [49] W. L. Murfee, Implications of fluid shear stress in capillary sprouting during adult microvascular network remodeling, Mechanobiology of the Endothelium, (2015), 166. [50] C. D. Murray, The physiological principle of minimum work: I. the vascular system and the cost of blood volume, Proc. Natl. Acad. Sci. USA, 12 (1926), 207-214.  doi: 10.1073/pnas.12.3.207. [51] V. Muthukkaruppan, L. Kubai and R. Auerbach, Tumor-induced neovascularization in the mouse eye, Journal of the National Cancer Institute, 69 (1982), 699-708. [52] F. Otto, Viscous fingering: An optimal bound on the growth rate of the mixing zone, SIAM Journal on Applied Mathematics, 57 (1997), 982-990.  doi: 10.1137/S003613999529438X. [53] M. R. Owen, T. Alarcón, P. K. Maini and H. M. Byrne, Angiogenesis and vascular remodelling in normal and cancerous tissues, J. Math. Biol., 58 (2009), 689-721.  doi: 10.1007/s00285-008-0213-z. [54] K. J. Painter, Modelling cell migration strategies in the extracellular matrix, J. Math. Biol., 58 (2009), 511-543.  doi: 10.1007/s00285-008-0217-8. [55] S. Paku and N. Paweletz, First steps of tumor-related angiogenesis, laboratory investigation, A Journal of Technical Methods and Pathology, 65 (1991), 334-346. [56] J. Y. Park, J. B. White, N. Walker, C.-H. Kuo, W. Cha, M. E. Meyerhoff and S. Takayama, Responses of endothelial cells to extremely slow flows, Biomicrofluidics, 5 (2011), 022211. doi: 10.1063/1.3576932. [57] N. Paweletz and M. Knierim, Tumor-related angiogenesis, Critical Reviews in Oncology Hematology, 9 (1989), 197-242.  doi: 10.1016/S1040-8428(89)80002-2. [58] R. Penta, D. Ambrosi and A. Quarteroni, Multiscale homogenization for fluid and drug transport in vascularized malignant tissues, Math. Models Methods Appl. Sci., 25 (2015), 79-108.  doi: 10.1142/S0218202515500037. [59] H. Perfahl, H. M. Byrne, T. Chen, V. Estrella, T. Alarcón, A. Lapin, R. A. Gatenby, R. J. Gillies, M. C. Lloyd, P. K. Maini, et al., Multiscale modelling of vascular tumour growth in 3d: The roles of domain size and boundary conditions, PloS One, 6 (2011), e14790. doi: 10.1371/journal.pone.0014790. [60] D. Peurichard, F. Delebecque, A. Lorsignol, C. Barreau, J. Rouquette, X. Descombes, L. Casteilla and P. Degond, Simple mechanical cues could explain adipose tissue morphology, J. Theoret. Biol., 429 (2017), 61-81.  doi: 10.1016/j.jtbi.2017.06.030. [61] L.-K. Phng and H. Gerhardt, Angiogenesis: A team effort coordinated by notch, Developmental cell, 16 (2009), 196-208.  doi: 10.1016/j.devcel.2009.01.015. [62] L. Pietronero and H. Wiesmann, Stochastic model for dielectric breakdown, J. Stat. Phys., 36 (1984), 909-916.  doi: 10.1007/BF01012949. [63] S. Pillay, H. M. Byrne and P. K. Maini, Modeling angiogenesis: A discrete to continuum description, Phys. Rev. E, 95 (2017), 012410, 12pp. doi: 10.1103/physreve.95.012410. [64] A. Pries, T. Secomb and P. Gaehtgens, Structural adaptation and stability of microvascular networks: Theory and simulations, American Journal of Physiology-Heart and Circulatory Physiology, 275 (1998), H349-H360. doi: 10.1152/ajpheart.1998.275.2.H349. [65] A. Pries, T. W. Secomb, T. Gessner, M. Sperandio, J. Gross and P. Gaehtgens, Resistance to blood flow in microvessels in vivo, Circulation Research, 75 (1994), 904-915.  doi: 10.1161/01.RES.75.5.904. [66] P.-A. Raviart, An analysis of particle methods, in Numerical Methods in Fluid Dynamics, Springer, 1127 (1985), 243-324. doi: 10.1007/BFb0074532. [67] W. Risau, Mechanisms of angiogenesis, Nature, 386 (1997), 671-674.  doi: 10.1038/386671a0. [68] A.-G. Rolland-Lagan and P. Prusinkiewicz, Reviewing models of auxin canalization in the context of leaf vein pattern formation in arabidopsis, The Plant Journal, 44 (2005), 854-865. [69] P. G. Saffman and G. I. Taylor, The penetration of a fluid into a porous medium or hele-shaw cell containing a more viscous liquid, Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences, 245 (1958), 312-329.  doi: 10.1098/rspa.1958.0085. [70] M. Schneider, J. Reichold, B. Weber, G. Székely and S. Hirsch, Tissue metabolism driven arterial tree generation, Medical Image Analysis, 16 (2012), 1397-1414.  doi: 10.1016/j.media.2012.04.009. [71] M. Scianna, C. G. Bell and L. Preziosi, A review of mathematical models for the formation of vascular networks, J. Theoret. Biol., 333 (2013), 174-209.  doi: 10.1016/j.jtbi.2013.04.037. [72] T. W. Secomb, J. P. Alberding, R. Hsu, M. W. Dewhirst and A. R. Pries, Angiogenesis: An adaptive dynamic biological patterning problem, PLoS Computational Biology, 9 (2013), e1002983, 12pp. doi: 10.1371/journal.pcbi.1002983. [73] T. C. Skalak and R. J. Price, The role of mechanical stresses in microvascular remodeling, Microcirculation, 3 (1996), 143-165.  doi: 10.3109/10739689609148284. [74] M. A. Swartz and M. E. Fleury, Interstitial flow and its effects in soft tissues, Annu. Rev. Biomed. Eng., 9 (2007), 229-256.  doi: 10.1146/annurev.bioeng.9.060906.151850. [75] G. Takahashi, I. Fatt and T. Goldstick, Oxygen consumption rate of tissue measured by a micropolarographic method, The Journal of general physiology, 50 (1966), 317-335.  doi: 10.1085/jgp.50.2.317. [76] L. Tang, A. L. van de Ven, D. Guo, V. Andasari, V. Cristini, K. C. Li and X. Zhou, Computational modeling of 3d tumor growth and angiogenesis for chemotherapy evaluation, PloS One, 9 (2014), e83962. doi: 10.1371/journal.pone.0083962. [77] R. D. Travasso, E. C. Poiré, M. Castro, J. C. Rodrguez-Manzaneque, and A. Hernández-Machado, Tumor angiogenesis and vascular patterning: A mathematical model, PloS One, 6 (2011), e19989. doi: 10.1371/journal.pone.0019989. [78] J. P. Vila, On particle weighted methods and smooth particle hydrodynamics, Mathematical Models and Methods in Applied Sciences, 9 (1999), 161-209.  doi: 10.1142/S0218202599000117. [79] M. Welter, K. Bartha and H. Rieger, Emergent vascular network inhomogeneities and resulting blood flow patterns in a growing tumor, Journal of Theoretical Biology, 250 (2008), 257-280.  doi: 10.1016/j.jtbi.2007.09.031. [80] M. Welter, K. Bartha and H. Rieger, Vascular remodelling of an arterio-venous blood vessel network during solid tumour growth, Journal of Theoretical Biology, 259 (2009), 405-422.  doi: 10.1016/j.jtbi.2009.04.005. [81] S. A. Williams, S. Wasserman, D. W. Rawlinson, R. I. Kitney, L. H. Smaje and J. E. Tooke, Dynamic measurement of human capillary blood pressure, Clinical Science, 74 (1988), 507-512.  doi: 10.1042/cs0740507. [82] J. Wu, S. Xu, Q. Long, M. W. Collins, C. S. König, G. Zhao, Y. Jiang and A. R. Padhani, Coupled modeling of blood perfusion in intravascular, interstitial spaces in tumor microvasculature, Journal of Biomechanics, 41 (2008), 996-1004.  doi: 10.1016/j.jbiomech.2007.12.008. [83] Y. Xiong, P. Yang, R. L. Proia and T. Hla, Erythrocyte-derived sphingosine 1-phosphate is essential for vascular development, The Journal of Clinical Investigation, 124 (2014), 4823-4828.  doi: 10.1172/JCI77685.

Figures(14)

Tables(2)