Abstract
A theoretical model is used to describe the three-dimensional development of the retinal circulation in the human eye, which occurs after the initial spread of vasculature across the inner surface of the retina. In the model, random sprouting angiogenesis is driven by a growth factor that is produced in tissue at a rate dependent on oxygen level and diffuses to existing vessels. Vessel sprouts connect to form pathways for blood flow and undergo remodeling and pruning. These processes are controlled by known or hypothesized vascular responses to hemodynamic and biochemical stimuli, including conducted responses along vessel walls. The model shows regression of arterio-venous connections on the surface of the retina, allowing perfusion of the underlying tissue. A striking feature of the retinal circulation is the formation of two vascular plexuses located at the inner and outer surfaces of the inner nuclear layer within the retina. The model is used to test hypotheses regarding the formation of these structures. A mechanism based on local production and diffusion of growth factor is shown to be ineffective. However, sprout guidance by localized structures on the boundaries of the inner nuclear layer can account for plexus formation. The resulting networks have vascular density, perfusion and oxygen transport characteristics consistent with observed properties. The model shows how stochastic generation of vascular sprouts combined with a set of biologically based response mechanisms can lead to the generation of a specialized three-dimensional vascular structure with a high degree of organization.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Ahmed J, Pulfer MK, Linsenmeier RA (2001) Measurement of blood flow through the retinal circulation of the cat during normoxia and hypoxemia using fluorescent microspheres. Microvasc Res 62:143–153. https://doi.org/10.1006/mvre.2001.2321
Alberding JP, Secomb TW (2021) Simulation of angiogenesis in three dimensions: application to cerebral cortex. PLoS Comput Biol 17:e1009164. https://doi.org/10.1371/journal.pcbi.1009164
Ambrosi D, Bussolino F, Preziosi L (2005) A review of vasculogenesis models. J Theor Medicine 6:1–19
Anderson AR, Chaplain MA (1998) Continuous and discrete mathematical models of tumor-induced angiogenesis. Bull Math Biol 60:857–899
Arciero J, Harris A, Siesky B, Amireskandari A, Gershuny V, Pickrell A, Guidoboni G (2013) Theoretical analysis of vascular regulatory mechanisms contributing to retinal blood flow autoregulation. Invest Ophthalmol vis Sci 54:5584–5593. https://doi.org/10.1167/iovs.12-11543
Aubert M, Chaplain MA, McDougall SR, Devlin A, Mitchell CA (2011) A continuum mathematical model of the developing murine retinal vasculature. Bull Math Biol 73:2430–2451. https://doi.org/10.1007/s11538-011-9631-y
Balding D, McElwain DL (1985) A mathematical model of tumour-induced capillary growth. J Theor Biol 114:53–73
Bishop JE, Laurent GJ (1995) Collagen turnover and its regulation in the normal and hypertrophying heart. Eur Heart J 16(Suppl C):38–44
Campbell JP, Zhang M, Hwang TS, Bailey ST, Wilson DJ, Jia Y, Huang D (2017) Detailed vascular anatomy of the human retina by projection-resolved optical coherence tomography angiography. Sci Rep 7:42201. https://doi.org/10.1038/srep42201
Cringle SJ, Yu DY, Yu PK, Su EN (2002) Intraretinal oxygen consumption in the rat in vivo. Invest Ophthalmol vis Sci 43:1922–1927
Dorrell MI, Aguilar E, Friedlander M (2002) Retinal vascular development is mediated by endothelial filopodia, a preexisting astrocytic template and specific r-cadherin adhesion. Invest Ophthalmol vis Sci 43:3500–3510
Flegg JA, Menon SN, Maini PK, McElwain DL (2015) On the mathematical modeling of wound healing angiogenesis in skin as a reaction-transport process. Front Physiol 6:262. https://doi.org/10.3389/fphys.2015.00262
Fredrich T, Welter M, Rieger H (2018) Tumorcode : a framework to simulate vascularized tumors. Eur Phys J E Soft Matter 41:55. https://doi.org/10.1140/epje/i2018-11659-x
Fruttiger M (2007) Development of the retinal vasculature. Angiogenesis 10:77–88. https://doi.org/10.1007/s10456-007-9065-1
Gaehtgens P (1971) Radial and longitudinal distensibility of arterial microvessels in the mesentery and their dependence on extravascular structures. Pflugers Arch 330:277–289
Gerhardt H, Golding M, Fruttiger M, Ruhrberg C, Lundkvist A, Abramsson A, Jeltsch M, Mitchell C, Alitalo K, Shima D, Betsholtz C (2003) VEGF guides angiogenic sprouting utilizing endothelial tip cell filopodia. J Cell Biol 161:1163–1177
Gevertz JL, Torquato S (2006) Modeling the effects of vasculature evolution on early brain tumor growth. J Theor Biol 243:517–531. https://doi.org/10.1016/j.jtbi.2006.07.002
Golub AS, Pittman RN (2012) Oxygen dependence of respiration in rat spinotrapezius muscle in situ. Am J Physiol Heart Circ Physiol 303:H47–H56. https://doi.org/10.1152/ajpheart.00131.2012
Hahnfeldt P, Panigrahy D, Folkman J, Hlatky L (1999) Tumor development under angiogenic signaling: a dynamical theory of tumor growth, treatment response, and postvascular dormancy. Cancer Res 59:4770–4775
Hee MR, Izatt JA, Swanson EA, Huang D, Schuman JS, Lin CP, Puliafito CA, Fujimoto JG (1995) Optical coherence tomography of the human retina. Arch Ophthalmol 113:325–332. https://doi.org/10.1001/archopht.1995.01100030081025
Hsu R, Secomb TW (1989) A green’s function method for analysis of oxygen delivery to tissue by microvascular networks. Math Biosci 96:61–78
Iafe NA, Phasukkijwatana N, Chen X, Sarraf D (2016) Retinal capillary density and foveal avascular zone area are age-dependent: quantitative analysis using optical coherence tomography angiography. Invest Ophthalmol vis Sci 57:5780–5787. https://doi.org/10.1167/iovs.16-20045
Jackson ZS, Gotlieb AI, Langille BL (2002) Wall tissue remodeling regulates longitudinal tension in arteries. Circ Res 90:918–925
Ji JW, Mac Gabhann F, Popel AS (2007) Skeletal muscle VEGF gradients in peripheral arterial disease: simulations of rest and exercise. Am J Physiol Heart Circ Physiol 293:H3740–H3749
Leahy C, Radhakrishnan H, Weiner G, Goldberg JL, Srinivasan VJ (2015) Mapping the 3d connectivity of the rat inner retinal vascular network using oct angiography. Invest Ophthalmol vis Sci 56:5785–5793. https://doi.org/10.1167/iovs.15-17210
Levine HA, Pamuk S, Sleeman BD, Nilsen-Hamilton M (2001) Mathematical modeling of capillary formation and development in tumor angiogenesis: penetration into the stroma. Bull Math Biol 63:801–863
Liew G, Wang JJ, Mitchell P, Wong TY (2008) Retinal vascular imaging: a new tool in microvascular disease research. Circ Cardiovasc Imaging 1:156–161. https://doi.org/10.1161/CIRCIMAGING.108.784876
Linsenmeier RA (1986) Effects of light and darkness on oxygen distribution and consumption in the cat retina. J Gen Physiol 88:521–542. https://doi.org/10.1085/jgp.88.4.521
Liotta LA, Saidel GM, Kleinerman J (1977) Diffusion model of tumor vascularization and growth. Bull Math Biol 39:117–128
Lipowsky HH, Zweifach BW (1974) Network analysis of microcirculation of cat mesentery. Microvasc Res 7:73–83
Liu G, Qutub AA, Vempati P, Mac Gabhann F, Popel AS (2011) Module-based multiscale simulation of angiogenesis in skeletal muscle. Theor Biol Med Model 8:6. https://doi.org/10.1186/1742-4682-8-6
Lovasik JV, Kothe AC, Kergoat H (1993) Comparison of noninvasive methods to derive the mean central retinal artery pressure in man. Optom vis Sci 70:1005–1011. https://doi.org/10.1097/00006324-199312000-00003
Lucker A, Secomb TW, Weber B, Jenny P (2017) The relative influence of hematocrit and red blood cell velocity on oxygen transport from capillaries to tissue. Microcirculation. https://doi.org/10.1111/micc.12337
Machado MJ, Watson MG, Devlin AH, Chaplain MA, McDougall SR, Mitchell CA (2011) Dynamics of angiogenesis during wound healing: a coupled in vivo and in silico study. Microcirculation 18:183–197
McDougall SR, Anderson AR, Chaplain MA (2006) Mathematical modelling of dynamic adaptive tumour-induced angiogenesis: clinical implications and therapeutic targeting strategies. J Theor Biol 241:564–589
McDougall SR, Watson MG, Devlin AH, Mitchell CA, Chaplain MA (2012) A hybrid discrete-continuum mathematical model of pattern prediction in the developing retinal vasculature. Bull Math Biol 74:2272–2314. https://doi.org/10.1007/s11538-012-9754-9
Milde F, Bergdorf M, Koumoutsakos P (2008) A hybrid model for three-dimensional simulations of sprouting angiogenesis. Biophys J 95:3146–3160
Milde F, Lauw S, Koumoutsakos P, Iruela-Arispe ML (2013) The mouse retina in 3d: quantification of vascular growth and remodeling. Integr Biol (camb ) 5:1426–1438. https://doi.org/10.1039/c3ib40085a
Murray JD (2003) On the mechanochemical theory of biological pattern formation with application to vasculogenesis. C R Biol 326:239–252. https://doi.org/10.1016/s1631-0691(03)00065-9
Nijboer JM, Van Der Horst IC, Hendriks HG, Ten Duis HJ, Nijsten MW (2007) Myth or reality: hematocrit and hemoglobin differ in trauma. J Trauma 62:1310–1312. https://doi.org/10.1097/TA.0b013e3180341f54
Nunes SS, Greer KA, Stiening CM, Chen HY, Kidd KR, Schwartz MA, Sullivan CJ, Rekapally H, Hoying JB (2010) Implanted microvessels progress through distinct neovascularization phenotypes. Microvasc Res 79:10–20
Owen MR, Alarcon T, Maini PK, Byrne HM (2009) Angiogenesis and vascular remodelling in normal and cancerous tissues. J Math Biol 58:689–721
Paques M, Tadayoni R, Sercombe R, Laurent P, Genevois O, Gaudric A, Vicaut E (2003) Structural and hemodynamic analysis of the mouse retinal microcirculation. Invest Ophthalmol vis Sci 44:4960–4967
Perfahl H, Byrne HM, Chen T, Estrella V, Alarcon T, Lapin A, Gatenby RA, Gillies RJ, Lloyd MC, Maini PK, Reuss M, Owen MR (2011) Multiscale modelling of vascular tumour growth in 3d: the roles of domain size and boundary conditions. PLoS ONE 6:e14790. https://doi.org/10.1371/journal.pone.0014790
Pettet G, Chaplain MA, McElwain DL, Byrne HM (1996) On the rôle of angiogenesis in wound healing. Proc Biol Sci 263:1487–1493. https://doi.org/10.1098/rspb.1996.0217
Popel AS (1989) Theory of oxygen transport to tissue. Crit Rev Biomed Eng 17:257–321
Pries AR, Ley K, Claassen M, Gaehtgens P (1989) Red cell distribution at microvascular bifurcations. Microvasc Res 38:81–101
Pries AR, Reglin B, Secomb TW (2001) Structural adaptation of microvascular networks: functional roles of adaptive responses. Am J Physiol Heart Circ Physiol 281:H1015–H1025
Pries AR, Secomb TW (2008) Blood flow in microvascular networks. In: Tuma RF, Duran WN, Ley K (eds) Handbook of physiology: microcirculation, 2nd edn. Academic Press, San Diego, pp 3–36
Pries AR, Secomb TW (2014) Making microvascular networks work: angiogenesis, remodeling, and pruning. Physiology 29:446–455. https://doi.org/10.1152/physiol.00012.2014
Pries AR, Secomb TW, Gaehtgens P (1995) Structure and hemodynamics of microvascular networks: heterogeneity and correlations. Am J Physiol 269:H1713–H1722
Pries AR, Secomb TW, Gaehtgens P (1998) Structural adaptation and stability of microvascular networks: theory and simulations. Am J Physiol 275:H349–H360
Pries AR, Secomb TW, Gaehtgens P (1999) Structural autoregulation of terminal vascular beds: vascular adaptation and development of hypertension. Hypertension 33:153–161
Pries AR, Secomb TW, Gaehtgens P, Gross JF (1990) Blood flow in microvascular networks. Experiments and simulation. Circ Res 67:826–834
Pries AR, Secomb TW, Gessner T, Sperandio MB, Gross JF, Gaehtgens P (1994) Resistance to blood flow in microvessels in vivo. Circ Res 75:904–915
Risau W (1997) Mechanisms of angiogenesis. Nature 386:671–674
Secomb TW, Alberding JP, Hsu R, Dewhirst MW, Pries AR (2013) Angiogenesis: an adaptive dynamic biological patterning problem. PLoS Comput Biol 9:e1002983. https://doi.org/10.1371/journal.pcbi.1002983
Secomb TW, Hsu R, Dewhirst MW, Klitzman B, Gross JF (1993) Analysis of oxygen transport to tumor tissue by microvascular networks. Int J Radiat Oncol Biol Phys 25:481–489
Secomb TW, Hsu R, Park EY, Dewhirst MW (2004) Green’s function methods for analysis of oxygen delivery to tissue by microvascular networks. Ann Biomed Eng 32:1519–1529
Semenza GL (2001) Regulation of hypoxia-induced angiogenesis: a chaperone escorts VEGF to the dance. J Clin Invest 108:39–40. https://doi.org/10.1172/JCI13374
Stodtmeister R, Ventzke S, Spoerl E, Boehm AG, Terai N, Haustein M, Pillunat LE (2013) Enhanced pressure in the central retinal vein decreases the perfusion pressure in the prelaminar region of the optic nerve head. Invest Ophthalmol vis Sci 54:4698–4704. https://doi.org/10.1167/iovs.12-10607
Tong S, Yuan F (2001) Numerical simulations of angiogenesis in the cornea. Microvasc Res 61:14–27
Travasso RD, Corvera PE, Castro M, Rodriguez-Manzaneque JC, Hernandez-Machado A (2011) Tumor angiogenesis and vascular patterning: a mathematical model. PLoS ONE 6:e19989. https://doi.org/10.1371/journal.pone.0019989
Ucuzian AA, Gassman AA, East AT, Greisler HP (2010) Molecular mediators of angiogenesis. J Burn Care Res 31:158–175. https://doi.org/10.1097/BCR.0b013e3181c7ed82
Vilanova G, Colominas I, Gomez H (2017) A mathematical model of tumour angiogenesis: growth, regression and regrowth. J R Soc Interface 14:20160918. https://doi.org/10.1098/rsif.2016.0918
Wälchli T, Mateos JM, Weinman O, Babic D, Regli L, Hoerstrup SP, Gerhardt H, Schwab ME, Vogel J (2015) Quantitative assessment of angiogenesis, perfused blood vessels and endothelial tip cells in the postnatal mouse brain. Nat Protoc 10:53–74. https://doi.org/10.1038/nprot.2015.002
Watson MG, McDougall SR, Chaplain MA, Devlin AH, Mitchell CA (2012) Dynamics of angiogenesis during murine retinal development: a coupled in vivo and in silico study. J R Soc Interface 9:2351–2364. https://doi.org/10.1098/rsif.2012.0067
Welter M, Rieger H (2010) Physical determinants of vascular network remodeling during tumor growth. Eur Phys J E 33:149–163
Yu DY, Cringle SJ (2001) Oxygen distribution and consumption within the retina in vascularised and avascular retinas and in animal models of retinal disease. Prog Retin Eye Res 20:175–208. https://doi.org/10.1016/S1350-9462(00)00027-6
Yu DY, Cringle SJ (2006) Oxygen distribution in the mouse retina. Invest Ophthalmol vis Sci 47:1109–1112. https://doi.org/10.1167/iovs.05-1118
Zakrzewicz A, Secomb TW, Pries AR (2002) Angioadaptation: Keeping the vascular system in shape. News Physiol Sci 17:197–201
Acknowledgements
This work was supported by National Institutes of Health grants HL034555 and HL133362.
Author information
Authors and Affiliations
Corresponding author
Ethics declarations
Conflict of interest
The authors declare no competing interests.
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.
About this article
Cite this article
Alberding, J.P., Secomb, T.W. Simulation of Angiogenesis in Three Dimensions: Development of the Retinal Circulation. Bull Math Biol 85, 27 (2023). https://doi.org/10.1007/s11538-023-01126-7
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/s11538-023-01126-7