March  2021, 16(1): 91-138. doi: 10.3934/nhm.2021001

A new model for the emergence of blood capillary networks

1. 

Department of Mathematics, North Carolina State University, Raleigh, NC 27695, USA

2. 

MathNeuro Team, Inria Sophia-Antipolis Mediterrannee, 2004 Routes des Lucioles, BP93, 06902 Valbonne Cedex, France

3. 

INRIA Paris, 2, rue Simone Iff, 75589 Paris Cedex 12, France

4. 

Université de Toulouse-INPT-UPS, Institut de Mécanique des Fluides, 31000 Toulouse, France

5. 

STROMALab, Université de Toulouse, Inserm U1031, EFS, INP-ENVT, UPS, CNRS ERL5311, Toulouse, France

6. 

Department of Mathematics, Imperial College London, London SW7 2AZ, UK

* Corresponding author

The first two authors contributed equally to this research

Received  February 2020 Revised  November 2020 Published  December 2020

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.

Citation: Pedro Aceves-Sanchez, Benjamin Aymard, Diane Peurichard, Pol Kennel, Anne Lorsignol, Franck Plouraboué, Louis Casteilla, Pierre Degond. A new model for the emergence of blood capillary networks. Networks & Heterogeneous Media, 2021, 16 (1) : 91-138. doi: 10.3934/nhm.2021001
References:
[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[6]

E. BoissardP. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[10]

P. Carmeliet and R. K. Jain, Angiogenesis in cancer and other diseases, Nature, 407 (2000), 249-257.  doi: 10.1038/35025220.  Google Scholar

[11]

A. ChenJ. DarbonG. ButtazzoF. Santambrogio and J.-M. Morel, On the equations of landscape formation, Interfaces Free Bound., 16 (2014), 105-136.  doi: 10.4171/IFB/315.  Google Scholar

[12]

A. ChenJ. 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.  Google Scholar

[13]

E. CurcioA. PiscioneriS. MorelliS. SalernoP. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[19]

I. FischerJ.-P. GagnerM. LawE. 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.  Google Scholar

[20]

J. Folkman, Angiogenesis in cancer, vascular, rheumatoid and other disease, Nature Medicine, 1 (1995), 27-30.  doi: 10.1038/nm0195-27.  Google Scholar

[21] R. L. Fournier, Basic Transport Phenomena in Biomedical Engineering, CRC press, 2017.  doi: 10.1201/9781315120478.  Google Scholar
[22]

P. A. GalieD.-H. T. NguyenC. K. ChoiD. M. CohenP. 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.  Google Scholar

[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.  Google Scholar

[24]

M. A. Gimbrone JrR. S. CotranS. 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.  Google Scholar

[25]

M. S. Gockenbach, Understanding and Implementing The Finite Element Method, Vol. 97, SIAM, 2006. doi: 10.1137/1.9780898717846.  Google Scholar

[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.  Google Scholar

[27]

J. A. GonzálezF. J. Rodríguez-CortésO. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[30]

J. HaskovecP. 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.  Google Scholar

[31]

J. HaskovecP. MarkowichB. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[36]

S. IchiokaM. ShibataK. KosakiY. SatoK. 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.  Google Scholar

[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.  Google Scholar

[38]

R. KaunasH. 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.  Google Scholar

[39]

B. KaurF. W. KhwajaE. A. SeversonS. L. MathenyD. 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.  Google Scholar

[40]

A. B. LangdonB. 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.  Google Scholar

[41]

P. MacklinS. McDougallA. R. AndersonM. A. ChaplainV. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[44]

S. R. McDougallA. 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.  Google Scholar

[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.  Google Scholar

[46]

G. J. MitchisonD. 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.  Google Scholar

[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.  Google Scholar

[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. Google Scholar

[49]

W. L. Murfee, Implications of fluid shear stress in capillary sprouting during adult microvascular network remodeling, Mechanobiology of the Endothelium, (2015), 166. Google Scholar

[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.  Google Scholar

[51]

V. MuthukkaruppanL. Kubai and R. Auerbach, Tumor-induced neovascularization in the mouse eye, Journal of the National Cancer Institute, 69 (1982), 699-708.   Google Scholar

[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.  Google Scholar

[53]

M. R. OwenT. AlarcónP. 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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[58]

R. PentaD. 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.  Google Scholar

[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.  Google Scholar

[60]

D. PeurichardF. DelebecqueA. LorsignolC. BarreauJ. RouquetteX. DescombesL. 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.  Google Scholar

[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.  Google Scholar

[62]

L. Pietronero and H. Wiesmann, Stochastic model for dielectric breakdown, J. Stat. Phys., 36 (1984), 909-916.  doi: 10.1007/BF01012949.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[65]

A. PriesT. W. SecombT. GessnerM. SperandioJ. 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.  Google Scholar

[66]

P.-A. Raviart, An analysis of particle methods, in Numerical Methods in Fluid Dynamics, Springer, 1127 (1985), 243-324. doi: 10.1007/BFb0074532.  Google Scholar

[67]

W. Risau, Mechanisms of angiogenesis, Nature, 386 (1997), 671-674.  doi: 10.1038/386671a0.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[70]

M. SchneiderJ. ReicholdB. WeberG. 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.  Google Scholar

[71]

M. SciannaC. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[75]

G. TakahashiI. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[79]

M. WelterK. 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.  Google Scholar

[80]

M. WelterK. 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.  Google Scholar

[81]

S. A. WilliamsS. WassermanD. W. RawlinsonR. I. KitneyL. H. Smaje and J. E. Tooke, Dynamic measurement of human capillary blood pressure, Clinical Science, 74 (1988), 507-512.  doi: 10.1042/cs0740507.  Google Scholar

[82]

J. WuS. XuQ. LongM. W. CollinsC. S. KönigG. ZhaoY. 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.  Google Scholar

[83]

Y. XiongP. YangR. 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.  Google Scholar

show all references

References:
[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[6]

E. BoissardP. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[10]

P. Carmeliet and R. K. Jain, Angiogenesis in cancer and other diseases, Nature, 407 (2000), 249-257.  doi: 10.1038/35025220.  Google Scholar

[11]

A. ChenJ. DarbonG. ButtazzoF. Santambrogio and J.-M. Morel, On the equations of landscape formation, Interfaces Free Bound., 16 (2014), 105-136.  doi: 10.4171/IFB/315.  Google Scholar

[12]

A. ChenJ. 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.  Google Scholar

[13]

E. CurcioA. PiscioneriS. MorelliS. SalernoP. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[19]

I. FischerJ.-P. GagnerM. LawE. 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.  Google Scholar

[20]

J. Folkman, Angiogenesis in cancer, vascular, rheumatoid and other disease, Nature Medicine, 1 (1995), 27-30.  doi: 10.1038/nm0195-27.  Google Scholar

[21] R. L. Fournier, Basic Transport Phenomena in Biomedical Engineering, CRC press, 2017.  doi: 10.1201/9781315120478.  Google Scholar
[22]

P. A. GalieD.-H. T. NguyenC. K. ChoiD. M. CohenP. 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.  Google Scholar

[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.  Google Scholar

[24]

M. A. Gimbrone JrR. S. CotranS. 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.  Google Scholar

[25]

M. S. Gockenbach, Understanding and Implementing The Finite Element Method, Vol. 97, SIAM, 2006. doi: 10.1137/1.9780898717846.  Google Scholar

[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.  Google Scholar

[27]

J. A. GonzálezF. J. Rodríguez-CortésO. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[30]

J. HaskovecP. 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.  Google Scholar

[31]

J. HaskovecP. MarkowichB. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[36]

S. IchiokaM. ShibataK. KosakiY. SatoK. 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.  Google Scholar

[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.  Google Scholar

[38]

R. KaunasH. 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.  Google Scholar

[39]

B. KaurF. W. KhwajaE. A. SeversonS. L. MathenyD. 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.  Google Scholar

[40]

A. B. LangdonB. 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.  Google Scholar

[41]

P. MacklinS. McDougallA. R. AndersonM. A. ChaplainV. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[44]

S. R. McDougallA. 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.  Google Scholar

[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.  Google Scholar

[46]

G. J. MitchisonD. 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.  Google Scholar

[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.  Google Scholar

[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. Google Scholar

[49]

W. L. Murfee, Implications of fluid shear stress in capillary sprouting during adult microvascular network remodeling, Mechanobiology of the Endothelium, (2015), 166. Google Scholar

[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.  Google Scholar

[51]

V. MuthukkaruppanL. Kubai and R. Auerbach, Tumor-induced neovascularization in the mouse eye, Journal of the National Cancer Institute, 69 (1982), 699-708.   Google Scholar

[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.  Google Scholar

[53]

M. R. OwenT. AlarcónP. 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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[58]

R. PentaD. 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.  Google Scholar

[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.  Google Scholar

[60]

D. PeurichardF. DelebecqueA. LorsignolC. BarreauJ. RouquetteX. DescombesL. 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.  Google Scholar

[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.  Google Scholar

[62]

L. Pietronero and H. Wiesmann, Stochastic model for dielectric breakdown, J. Stat. Phys., 36 (1984), 909-916.  doi: 10.1007/BF01012949.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[65]

A. PriesT. W. SecombT. GessnerM. SperandioJ. 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.  Google Scholar

[66]

P.-A. Raviart, An analysis of particle methods, in Numerical Methods in Fluid Dynamics, Springer, 1127 (1985), 243-324. doi: 10.1007/BFb0074532.  Google Scholar

[67]

W. Risau, Mechanisms of angiogenesis, Nature, 386 (1997), 671-674.  doi: 10.1038/386671a0.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[70]

M. SchneiderJ. ReicholdB. WeberG. 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.  Google Scholar

[71]

M. SciannaC. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[75]

G. TakahashiI. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[79]

M. WelterK. 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.  Google Scholar

[80]

M. WelterK. 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.  Google Scholar

[81]

S. A. WilliamsS. WassermanD. W. RawlinsonR. I. KitneyL. H. Smaje and J. E. Tooke, Dynamic measurement of human capillary blood pressure, Clinical Science, 74 (1988), 507-512.  doi: 10.1042/cs0740507.  Google Scholar

[82]

J. WuS. XuQ. LongM. W. CollinsC. S. KönigG. ZhaoY. 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.  Google Scholar

[83]

Y. XiongP. YangR. 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.  Google Scholar

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} $
Table. 1">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 $
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 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
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 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
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 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
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 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
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 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
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 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
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">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]
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 $ $ - $
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 $ $ - $
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