Open validation and generalization program for the interlaced diffusion-limited-aggregation model of 3-D microvasculature formation (Fleury-Schwartz, 2000)

The interlaced diffusion-limited-aggregation (DLA) model of Fleury and Schwartz (Fractals 8:255-259, 2000) proposes that a random 3-D capillary plexus matures into two interdigitated, non-shunting arborescent (arterial and venous) vasculatures under two local, flow-driven rules: irreversible enlargement of capillaries selected by a Laplacian pressure field (growth velocity $v=-k\,\mathrm{grad}\,P$, mathematically equivalent to DLA / dielectric-breakdown growth), and irreversible regression of capillary bonds experiencing persistently low shear, with the arterial tree grown first and the venous tree grown on the remaining plexus. The model leaves open: the molecular mechanism and quantitative parametrization of shear-stress transduction into vessel enlargement and of the regression threshold; the incorporation and quantified consequences of realistic extensions (cross-section-dependent flow resistance, multiple arterial/venous rudiments, finite delays, reversible kinetics); sensitivity of morphology to lattice discretization, boundary conditions $P_{0}$/$P_{1}$, and $k$; applicability to post-infarct and tumor angiogenesis; and scalability to larger, anatomically realistic 3-D domains. Later work has advanced the underlying physics and biology — notably a 2018 experimental mechanism of arteriovenous interlacing based on arterial-tip flattening and tissue swelling, and the identification of endothelial mechanosensory machinery — but no published study has implemented the listed extensions, characterized the model's parameter sensitivity, scaled it to realistic 3-D domains, or tested its predictive power in pathological angiogenesis. The validation-and-generalization program therefore remains open.

Background

Formation of the microvasculature poses a patterning problem: an initially disorganized, roughly uniform lattice of tiny vessels — the capillary plexus, assembled by vasculogenesis (the in-situ assembly of vessels) — must be remodeled into a hierarchical network in which branched arterial trees deliver blood to, and branched venous trees drain it from, a distal capillary bed where gas and nutrient exchange occurs. In organs such as the chick chorioallantoic membrane (CAM, the extra-embryonic respiratory membrane of the chick embryo) the mature pattern shows a striking interdigitation of arteries and veins: the two trees approach and interlace over the same territory without forming direct arteriovenous shunts, that is, connections that short-circuit the capillary bed. Shunts are detrimental because they divert perfusion away from exchange vessels.

A long experimental tradition going back to Thoma's observations on embryos holds that hemodynamic forces drive this remodeling: vessels carrying high flow enlarge, and vessels starved of flow regress. The key local signal is wall shear stress, the frictional force per unit area that flowing blood exerts on the endothelial cells lining a vessel; in a narrow tube under Poiseuille (parabolic) flow, shear stress is proportional to the local pressure gradient. Quantitative "structural adaptation" models in this spirit, notably that of Pries, Reglin and Secomb, update vessel diameters according to hemodynamic (shear, pressure) and metabolic stimuli and reproduce many properties of real microvascular networks. However, a model study by Hacking, VanBavel and Spaan concluded that a purely shear-based reversible adaptation rule is not sufficient to control the growth of vascular networks: their simulated networks failed to develop realistic branching hierarchies.

Against this background Fleury and Schwartz proposed a different idealization. In a 1999 paper they treated the capillary plexus as a network of high-resistance bonds in which the blood pressure $P$ satisfies the Laplace equation $\Delta P = 0$ (steady flow in a resistance network), with Dirichlet boundary conditions (values imposed directly on the solution) $P=P_{0}$ at an arterial rudiment and $P=P_{1}$ at a venous rudiment, and postulated a growth velocity field $v = -k\,\mathrm{grad}\,P$ with $k$ a positive constant. This is mathematically the structure of Laplacian growth processes from statistical physics: diffusion-limited aggregation (DLA), a stochastic process in which random walkers attach to a growing cluster and generate fractal trees, and the dielectric-breakdown model of spark propagation between electrodes. Since shear stress is proportional to the pressure gradient, growth along $\mathrm{grad}\,P$ implements shear-driven enlargement: a small subset of capillaries is selected and irreversibly enlarged from radius $r$ to radius $R$ with $R\gg r$. Two-dimensional DLA simulations in a chick yolk-sac geometry reproduced observed tree patterns, including a mutual "tropism" of arterial and venous rudiments analogous to spark propagation between two electrodes.

The 2000 paper that defines the present problem extended this to three dimensions and to two interlaced trees by adding: (i) irreversible capillary regression — removal of capillary bonds that experience persistently low shear, which prunes the side branches attached to the enlarging tree; and (ii) sequential growth — the arterial tree is grown first from an arterial rudiment, then the venous tree is grown on the remaining plexus. A Monte-Carlo simulation (stochastic, random-walk-based growth) on a $40\times40\times7$ lattice of capillary segments, in which 500 segments were converted to arterioles and 50 terminal tips retained, produced two branched, interdigitated trees with few loops and few direct arteriovenous shunts, the venules showing a statistical tendency to grow toward arterial tips. The authors presented this as the first computational construction of a 3-D organ-scale vasculature from an initially random plexus using only local shear sensing and flow-computed pressure gradients; the model as published still lacks: the molecular and quantitative basis of shear-stress transduction; several physical extensions; any characterization of parameter sensitivity; any application to post-infarct or tumor angiogenesis; and any demonstration of scalability beyond the small demonstration lattice. Experimental work shortly afterwards (le Noble et al. 2004) confirmed that blood flow regulates arterial-venous differentiation in the chick yolk sac, supporting the model's biological premise, while the same group's later theoretical and imaging studies (2006-2018) developed additional mechanisms — tissue deformation and prepatterns, and arterial-tip flattening with tissue swelling — that bear on the same phenomenology.

Problem Statement

The source paper — Fleury and Schwartz, "Modelisation of 3-D microvasculature by interlaced diffusion limited aggregation" (Fractals 8(3):255-259, 2000) — proposes that a random 3-D capillary plexus matures into two interdigitated, non-shunting arborescent (arterial and venous) vasculatures under two local, flow-driven rules: irreversible enlargement of capillaries selected by a Laplacian pressure field ($\Delta P = 0$ with Dirichlet conditions $P=P_{0}$ at the arterial rudiment and $P=P_{1}$ at the venous rudiment, growth velocity $v=-k\,\mathrm{grad}\,P$), and irreversible regression of capillary bonds experiencing persistently low shear, with the arterial tree grown first and the venous tree grown on the remaining plexus. The open problem — the work needed to validate, generalize, and apply this mechanism — is:

(1) Parametrize the biology of the growth rule: the molecular and cellular mechanisms by which endothelial cells transduce shear stress into irreversible capillary enlargement, and the quantitative form of the low-shear regression threshold.

(2) Incorporate and quantify the consequences of realistic physical extensions of the model — flow resistance explicitly dependent on vessel cross-section, multiple incoming arterial or venous rudiments, finite time delays before capillary regression, and reversible versus irreversible transformation kinetics — in particular their effects on branching statistics and on the incidence of arteriovenous shunts.

(3) Characterize the sensitivity of the resulting vascular morphology to the choices of lattice discretization, boundary-condition values ($P_{0}$ and $P_{1}$), and the proportionality constant $k$ in $v=-k\,\mathrm{grad}\,P$.

(4) Test the model's applicability and predictive power for post-infarct angiogenesis and for abnormal tumor-induced angiogenesis, where the endothelial response to shear may be altered, against available experimental data.

(5) Establish the scalability and robustness of the discrete Monte-Carlo implementation to larger, anatomically realistic three-dimensional domains and to biologically measured parameter ranges.

These five components jointly serve one objective: to determine whether the shear-driven selection-plus-regression mechanism is a valid, robust, and quantitatively predictive explanation of 3-D microvascular maturation, or whether it fails, and if so where and why. The determination is made across the admissible parameter domain — the growth and regression kinetics, lattice discretizations, boundary values $P_{0}$/$P_{1}$, and constant $k$ that lie within the biologically measured ranges fixed by components (1) and (5) — and on 3-D domains at least as large as the source's demonstration lattice: it asks whether the simulated plexus matures into two interdigitated arborescent trees that are non-shunting and whose quantified morphological signatures (branching statistics, arteriovenous shunt incidence, degree of interdigitation, preserved capillary exchange region) reproduce the corresponding measured signatures of the biological targets — the chick chorioallantoic membrane and embryonic yolk-sac vasculatures of the source and its experimental follow-ups and, for component (4), the post-infarct and tumor vasculatures — or whether the mechanism fails to do so, and where. Reproduction of the measured signatures on a proper sub-domain of the admissible parameter domain while failing elsewhere would leave the mechanism valid only on that sub-domain of validity, with a complementary sub-domain of failure and an identifiable reason for the failure. The morphological metrics together with their measured reference values and tolerances, and the admissible parameter ranges, enter through operational definitions, each anchored to the cited experimental literature with stated provenance, that are fixed by components (1) and (5). Because the admissible domain is continuous and the growth process is stochastic, the outcome depends on the sampling resolution over the domain, the number of independent Monte-Carlo realizations, and the tolerance and confidence adopted, which the study must state and justify rather than rely on an unstated notion of covering the domain throughout, complemented by analytical demonstration where possible. Components (1) and (5) are therefore prerequisites, and the remaining components are independent work items. The mechanism is refuted by demonstrating that it cannot produce two interdigitated non-shunting trees within the admissible parameter domain.

References

1. V. Fleury and L. 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 (primary source of this problem)
2. V. Fleury and L. 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
3. W. J. Hacking, E. VanBavel and J. A. Spaan, "Shear stress is not sufficient to control growth of vascular networks: a model study", American Journal of Physiology 270(1 Pt 2):H364-H375 (1996). DOI: 10.1152/ajpheart.1996.270.1.H364, https://doi.org/10.1152/ajpheart.1996.270.1.H364
4. A. R. Pries, B. Reglin and T. W. Secomb, "Structural adaptation of microvascular networks: functional roles of adaptive responses", American Journal of Physiology - Heart and Circulatory Physiology 281(3):H1015-H1025 (2001). DOI: 10.1152/ajpheart.2001.281.3.H1015, https://doi.org/10.1152/ajpheart.2001.281.3.H1015
5. F. le Noble, D. Moyon, L. Pardanaud, L. Yuan, V. Djonov, R. Matthijsen, C. Breant, V. Fleury and A. Eichmann, "Flow regulates arterial-venous differentiation in the chick embryo yolk sac", Development 131(2):361-375 (2004). DOI: 10.1242/dev.00929, https://doi.org/10.1242/dev.00929
6. E. Tzima, M. Irani-Tehrani, W. B. Kiosses, E. Dejana, D. A. Schultz, B. Engelhardt, G. Cao, H. DeLisser and M. A. Schwartz, "A mechanosensory complex that mediates the endothelial cell response to fluid shear stress", Nature 437(7057):426-431 (2005). DOI: 10.1038/nature03952, https://doi.org/10.1038/nature03952
7. 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(6):061907 (2006). DOI: 10.1103/PhysRevE.73.061907, https://doi.org/10.1103/PhysRevE.73.061907
8. V. Fleury, M. Unbekandt, A. Al-Kilani and T.-H. 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
9. 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(5):051912 (2008). DOI: 10.1103/PhysRevE.77.051912, https://doi.org/10.1103/PhysRevE.77.051912
10. 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(3):405-422 (2009). DOI: 10.1016/j.jtbi.2009.04.005, https://doi.org/10.1016/j.jtbi.2009.04.005
11. 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
12. S. Richard, A. Brun, A. Tedesco, B. Gallois, N. Taghi, P. Dantan, J. Seguin and V. 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
13. S. M. A. Aghamirmohammadali, R. Bozorgmehry Boozarjomehry and M. Abdekhodaie, "Modelling of retinal vasculature based on genetically tuned parametric L-system", Royal Society Open Science 5(5):171639 (2018). DOI: 10.1098/rsos.171639, https://doi.org/10.1098/rsos.171639
14. R. Amare, A. A. Bahadori and S. Eckels, "Representing unsegmented vessels using available vascular data for bioheat transfer simulation", Frontiers in Thermal Engineering 5:1536410 (2025). DOI: 10.3389/fther.2025.1536410, https://doi.org/10.3389/fther.2025.1536410
15. A. Hahn, J. Bode, T. Kruwel, G. Solecki, S. Heiland, M. Bendszus, B. Tews, F. Winkler, M. O. Breckwoldt and F. T. Kurz, "Glioblastoma multiforme restructures the topological connectivity of cerebrovascular networks", Scientific Reports 9:11757 (2019). DOI: 10.1038/s41598-019-47567-w, https://doi.org/10.1038/s41598-019-47567-w
16. O. Iizuka, S. Kawamura, A. Tero, A. Uemura and T. Miura, "Remodeling mechanisms determine size distributions in developing retinal vasculature", PLoS ONE 15(10):e0235373 (2020). DOI: 10.1371/journal.pone.0235373, https://doi.org/10.1371/journal.pone.0235373
17. S. Jafari Nivlouei, A. Guerra, J. Belinha, N. Mangir, S. MacNeil, C. Salgado, F. J. Monteiro and R. Natal Jorge, "Angiogenesis Dynamics: A Computational Model of Intravascular Flow Within a Structural Adaptive Vascular Network", Biomedicines 12(12):2845 (2024). DOI: 10.3390/biomedicines12122845, https://doi.org/10.3390/biomedicines12122845