During development, a nearly homogeneous capillary mesh remodels into a hierarchical, interdigitating pair of arterial and venous trees. In 1999, Fleury and Schwartz proposed that this remodeling is Laplacian growth: the capillary plexus acts as a porous medium with blood pressure satisfying ΔP=0\Delta P = 0, arteries and veins are perfectly conducting equipotentials, and vessel growth follows wall shear stress, v=−k∇Pv = -k\nabla P — making vasculogenesis formally identical to diffusion-limited aggregation (DLA) and the dielectric breakdown model. They stated what would be needed before the model could be considered a complete description of vasculogenesis: the mechano-transduction mechanisms converting shear stress into vessel enlargement; capillary regression during pruning and its effect on the apparent fractal dimension; anisotropy of the vascular wall; the realistic case of large vessels with finite hydraulic resistance (known to produce more space-filling patterns); multi-layer three-dimensional geometry requiring source terms in the Laplace equation; and the precise fractal dimension of vascular trees versus the canonical 1.7 of DLA. The authors' own follow-ups incorporated pruning/disconnection and tissue growth (2000–2006) and modeled three-dimensional interdigitating arborescences (2000); Lorthois and Cassot (2010) substantially resolved the fractal-dimension question for healthy networks (superposition of a space-filling capillary mesh and quasi-fractal trees); and Richard et al. (2018) demonstrated experimentally and computationally that finite resistance from arterial flattening drives arteriovenous interlacing. The surviving open core is the completion and quantitative validation of the model; the full scope of what remains open is defined by the six deliverables of the Problem Statement below, which the follow-ups noted above address only in part.

Background

In vertebrate embryos the first vessels arise by vasculogenesis: endothelial cells (the cells lining blood vessels) assemble into a nearly homogeneous lattice of fine capillaries — a capillary plexus — which over days is remodeled into a hierarchical tree of thick arteries and veins that interdigitate without forming direct shunts. Genetic and chemical cues (for example gradients of the growth factor VEGF, vascular endothelial growth factor, a diffusible signal that guides vessel sprouting) guide sprouting, but by themselves they do not explain the hierarchy of vessel sizes or the paired, mutually avoiding arrangement of arteries and veins. A second class of explanations is mechanical: endothelial cells are known to be exquisitely sensitive to fluid wall shear stress, the frictional force per unit area that flowing blood exerts on the vessel lining. Fleury and Schwartz (1999) proposed that this sensitivity alone can self-organize the vascular tree. Their idealization: if the capillary plexus is treated as a uniform porous medium, in which flow velocity is proportional to the local pressure gradient (Darcy's law), the blood pressure P satisfies Laplace's equation ΔP=0\Delta P = 0 — the same equation as the electrostatic potential around a conductor. Flow in each capillary is Poiseuille flow (the parabolic profile of viscous flow in a tube), so the wall shear stress in a capillary of radius R is τ=−(R/2)∇P\tau = -(R/2)\nabla P, proportional to the local pressure gradient. If the probability (or speed) for a capillary segment to be replaced by a larger vessel is proportional to wall shear stress — a local growth law v=−k∇Pv = -k\nabla P, with k a mechano-transduction time scale — then the growing arterial and venous trees are Laplacian growth fronts, mathematically identical to the dielectric breakdown model (DBM) introduced by Niemeyer, Pietronero and Wiesmann (1984) for electrical discharge patterns, and to diffusion-limited aggregation (DLA), the canonical process introduced by Witten and Sander (1981) in which random walkers stick irreversibly to a growing aggregate and generate sparse branched clusters that are fractal — their mass within a radius rr grows as rDr^D rather than as r2r^2, the exponent DD being the fractal dimension — with D≈1.7D \approx 1.7 in two dimensions. Treating arteries and veins as perfectly conducting (zero pressure drop) 'electrodes', the model reproduced the gross organization of the vasculature of the chick yolk sac (the membrane surrounding the embryo in which the first functional vessels form) and predicted a spontaneous arteriovenous tropism (vessels of one kind grow toward the other) and hierarchy. The physics literature had also shown that the perfectly conducting limit is a singular idealization: Grier and Mueth (1993) demonstrated for DLA-type growth that when the growing aggregate itself carries finite dissipation (finite resistance), the growth stabilizes a dense radial morphology — far more space-filling than DLA — a qualitative change of pattern class. The 1999 paper presented the model as a limiting case and enumerated what would be required to make it a complete description of vasculogenesis: the detailed mechano-transduction by which endothelial cells convert shear stress into vessel enlargement and remodeling; capillary regression (the disappearance of capillary segments during the developmental pruning of redundant vessels) and how it modifies the apparent fractal dimension and pattern; anisotropy of the vascular wall (non-uniform mechanical properties); the realistic case of large vessels with finite resistance instead of being perfectly conducting; the case of multiple planar vascular layers connected by vertical capillaries (as in the retina), which requires a source term in the Laplace equation; and the precise fractal dimension of vascular trees, which may differ from the canonical 1.7 of DLA because of finite-medium effects, source terms, or the other modifications — together with the quantitative interplay of these factors with chemotactic gradients (spatial gradients of diffusing chemical signals that direct vessel growth) and mechanical strain of the tissue

Problem Statement

The problem is to extend the shear-driven Laplacian-growth model of vasculogenesis beyond its 1999 limiting case into a complete, quantitatively validated physical description of vascular-network self-organization. Concretely, determine: (1) the mechano-transduction law by which wall shear stress is converted into vessel enlargement and remodeling — whether the assumed local law v=−k∇Pv = -k\nabla P holds quantitatively in the developing vasculature, with which functional form and effective mechano-transduction coupling; (2) how capillary regression (the disappearance of capillary segments during pruning) modifies the predicted vascular pattern and its apparent fractal dimension; (3) how anisotropy of the vascular wall (non-uniform mechanical properties of the medium in which the pressure field is solved) alters the predicted trees; (4) the pattern class and its fractal dimension when large vessels are treated with finite hydraulic resistance instead of as perfectly conducting equipotentials — expected from dissipative Laplacian-growth physics to be more space-filling; (5) the model's behavior in realistic multi-planar/three-dimensional geometries (capillary layers connected by vertical capillaries, as in the retina), where the pressure field acquires source terms in the Laplace equation; (6) the fractal dimension(s) of the resulting vascular trees as modified by finite-medium effects, source terms and the extensions above, in quantitative comparison with measured vascular networks. These deliverables are coupled components of one objective: to establish whether hemodynamic self-organization, expressed by an extended Laplacian-growth model, quantitatively suffices to generate the observed architecture of arterial and venous trees growing through a capillary mesh. The deliverable list (1)–(6) defines the full scope of the sought complete description. Components (2) and (5) have published partial models by the same group (pruning and three-dimensional interlaced aggregation, 2000-2006), and component (6) has the conceptual fractal analysis of healthy vascular networks by Lorthois and Cassot (2010), which address these questions only in part. Each deliverable concerns the following for the extended model, where 'pattern statistics' means the fractal dimension and the space-filling fraction of the grown arterial and venous trees: (1) a quantitatively specified functional form of the shear-to-enlargement growth law and its effective coupling, validating or replacing v=−k∇Pv = -k\nabla P; (2) how capillary regression changes the pattern statistics of the predicted trees; (3) how wall anisotropy changes the pattern statistics of the predicted trees; (4) the pattern class under finite hydraulic resistance — space-filling versus DLA-like — together with its pattern statistics; (5) the pattern statistics of the trees grown in the multi-layer three-dimensional geometry; and (6) the pattern statistics of the resulting trees under the combined (2)-(5) extensions, compared with at least one measured vascular network from the cited literature. The pattern statistics of interest are those of each mechanism acting alone and of the model with all (2)-(5) extensions applied.

References

  1. Vincent Fleury and Laurent Schwartz, "Diffusion limited aggregation from shear stress as a simple model of vasculogenesis", Fractals 7(1):33-39, 1999. DOI: 10.1142/S0218348X99000050, https://doi.org/10.1142/S0218348X99000050
  2. Vincent Fleury and Laurent Schwartz, "Modelisation of 3-D microvasculature by interlaced diffusion limited aggregation", Fractals 8(3):255-259, 2000. DOI: 10.1142/S0218348X00000317, https://doi.org/10.1142/S0218348X00000317
  3. Vincent Fleury, "Branching morphogenesis in a reaction-diffusion model", Physical Review E 61(4):4156-4160, 2000. DOI: 10.1103/PhysRevE.61.4156, https://doi.org/10.1103/PhysRevE.61.4156
  4. V. Fleury and L. Schwartz, "A Link Between Dendritic Growth and Remodeling of Blood Vessels", in Branching in Nature, Springer, 2001, pp. 293-303. DOI: 10.1007/978-3-662-06162-6_17, https://doi.org/10.1007/978-3-662-06162-6_17
  5. Ferdinand le Noble, V. Fleury, A. Pries, P. Corvol, A. Eichmann and R. S. Reneman, "Control of arterial branching morphogenesis in embryogenesis: go with the flow", Cardiovascular Research 65(3):619-628, 2005. DOI: 10.1016/j.cardiores.2004.09.018, https://doi.org/10.1016/j.cardiores.2004.09.018
  6. T.-H. Nguyen, A. Eichmann, F. le Noble and V. Fleury, "Dynamics of vascular branching morphogenesis: the effect of blood and tissue flow", Physical Review E 73:061907, 2006. DOI: 10.1103/PhysRevE.73.061907, https://doi.org/10.1103/PhysRevE.73.061907
  7. S. Lorthois and F. Cassot, "Fractal analysis of vascular networks: insights from morphogenesis", Journal of Theoretical Biology 262(4):614-633, 2010. DOI: 10.1016/j.jtbi.2009.10.037, https://doi.org/10.1016/j.jtbi.2009.10.037
  8. A. Al-Kilani, S. Lorthois, T.-H. Nguyen, F. le Noble, A. Cornelissen, M. Unbekandt, O. Boryskina, L. Leroy and V. Fleury, "During vertebrate development, arteries exert a morphological control over the venous pattern through physical factors", Physical Review E 77:051912, 2008. DOI: 10.1103/PhysRevE.77.051912, https://doi.org/10.1103/PhysRevE.77.051912
  9. Vincent Fleury, Mathieu Unbekandt, Alia Al-Kilani and Thi-Hanh Nguyen, "The Textural Aspects of Vessel Formation during Embryo Development and Their Relation to Gastrulation Movements", Organogenesis 3(1):49-56, 2007. DOI: 10.4161/org.3.1.3238, https://doi.org/10.4161/org.3.1.3238
  10. Sophie Richard, Amanda Brun, Antonio Tedesco, Benjamin Gallois, Naoual Taghi, Philippe Dantan, Johanne Seguin and Vincent Fleury, "Direct imaging of capillaries reveals the mechanism of arteriovenous interlacing in the chick chorioallantoic membrane", Communications Biology 1:235, 2018. DOI: 10.1038/s42003-018-0229-x, https://doi.org/10.1038/s42003-018-0229-x
  11. David G. Grier and Daniel Mueth, "Dissipation, geometry, and the stability of the dense radial morphology", Physical Review E 48:3841, 1993. DOI: 10.1103/PhysRevE.48.3841, https://doi.org/10.1103/PhysRevE.48.3841
  12. T. A. Witten and L. M. Sander, "Diffusion-Limited Aggregation, a Kinetic Critical Phenomenon", Physical Review Letters 47:1400, 1981. DOI: 10.1103/PhysRevLett.47.1400, https://doi.org/10.1103/PhysRevLett.47.1400
  13. L. Niemeyer, L. Pietronero and H. J. Wiesmann, "Fractal Dimension of Dielectric Breakdown", Physical Review Letters 52:1033, 1984. DOI: 10.1103/PhysRevLett.52.1033, https://doi.org/10.1103/PhysRevLett.52.1033
  14. Laurent Risser, Franck Plouraboué, Alexandre Steyer, Peter Cloetens, Géraldine Le Duc and Caroline Fonta, "From Homogeneous to Fractal Normal and Tumorous Microvascular Networks in the Brain", Journal of Cerebral Blood Flow & Metabolism 27:293-303, 2007. DOI: 10.1038/sj.jcbfm.9600332, https://doi.org/10.1038/sj.jcbfm.9600332
  15. A. R. Pries, T. W. Secomb and P. Gaehtgens, "Structural adaptation and stability of microvascular networks: theory and simulations", American Journal of Physiology-Heart and Circulatory Physiology 275:H349-H360, 1998. DOI: 10.1152/ajpheart.1998.275.2.h349, https://doi.org/10.1152/ajpheart.1998.275.2.h349
  16. L. Schwartz, J. da Veiga Moreira and M. Jolicoeur, "Physical forces modulate cell differentiation and proliferation processes", Journal of Cellular and Molecular Medicine 22(2):738-745, 2018. DOI: 10.1111/jcmm.13417, https://doi.org/10.1111/jcmm.13417
  17. Barry R. Masters, "Fractal Analysis of the Vascular Tree in the Human Retina", Annual Review of Biomedical Engineering 6:427-452, 2004. DOI: 10.1146/annurev.bioeng.6.040803.140100, https://doi.org/10.1146/annurev.bioeng.6.040803.140100