- Physical Review Journals
All JournalsPhysics Magazine

PRX Life

Reuse & Permissions

It is not necessary to obtain permission to reuse this article or its components as it is available under the terms of the Creative Commons Attribution 4.0 International license. This license permits unrestricted use, distribution, and reproduction in any medium, provided attribution to the author(s) and the published article's title, journal citation, and DOI are maintained. Please note that some figures may have been included with permission from other third parties. It is your responsibility to obtain the proper permission from the rights holder directly for these figures.

Export citation

Export citation

Choose format for download:

Download Citation
  • Open Access

Hierarchical Loop Stabilization in Periodically Driven Elastic Networks

Purba Chatterjee1 and Eleni Katifori1,2

PDF

PRX Life 4, 013006 – Published 13 January, 2026

DOI: https://doi.org/10.1103/gb8b-7yt3

Abstract

Network remodeling, or adaptation, in the presence of periodically driven forcings has hereto remained largely unexplored, despite the fact that a broad class of biological transport networks, e.g., animal vasculature, depends on periodic driving (pulsatility of the heart) to maintain flow. Here we propose a new adaptation model that incorporates the effect of short-term flow pulsatility into the long-term remodeling signal for periodically driven elastic networks and is consistent with optimality principles of dissipation and metabolic cost minimization. Using this model to adapt hierarchical elastic networks with multiple levels of looping, we show that very different network architectures are possible at steady state depending on the driving frequency of the pulsatile source and the geometric asymmetry of the paths between the externally driven nodes of the network. Specifically resonant frequencies are shown to prioritize the stabilization of fully looped structures or higher level loops proximal to the source, whereas antiresonant frequencies predominantly stabilize loop-less structures or lower-level loops distal to the source. Thus, this model offers a mechanism that can explain the stabilization of phenotypically diverse loopy network architectures in response to source pulsatility under physiologically relevant conditions and in the absence of other known loop stabilization mechanisms, such as random fluctuations in the load or perfusion homogenization.

View figure in article

Physics Subject Headings (PhySH)

Collections

This article appears in the following collection:

PRX Life Collection on adaptation and learning

PRX Life presents a special collection on ‘Biological systems that learn’, curated by the organizers of the National Institute for Theory and Mathematics in Biology (NITMB) workshop together with early-career researchers. The papers in this collection illustrate how living systems acquire functions, showing it is useful to describe them as tunable matter with many individually adjustable degrees of freedom characterizing microscopic properties. Drawing inspiration from AI and non-biological systems that can acquire desired functions, the featured research highlights the timeliness of the topic and its potential to drive unifying frameworks for emergent collective function in living tunable matter.

Every article in this collection is subject to a rigorous peer review process, upholding the high standards applied to all papers in the journal. The PRX Life editorial team manages peer review and makes all editorial decisions.

Article Text

Supplemental Material

References (61)

  1. T. Nelson and N. Dengler, Leaf vascular pattern formation, Plant Cell 9, 1121 (1997).
  2. M. F. Laguna, S. Bohn, and E. A. Jagla, The role of elastic stresses on leaf venation morphogenesis, PLoS Comput. Biol. 4, e1000055 (2008).
  3. H. Ronellenfitsch and E. Katifori, Phenotypes of vascular flow networks, Phys. Rev. Lett. 123, 248101 (2019).
  4. E. Katifori, G. J. Szöllősi, and M. O. Magnasco, Damage and fluctuations induce loops in optimal transport networks, Phys. Rev. Lett. 104, 048704 (2010).
  5. F. Corson, Fluctuations and redundancy in optimal transport networks, Phys. Rev. Lett. 104, 048703 (2010).
  6. F. Kaiser, H. Ronellenfitsch, and D. Witthaut, Discontinuous transition to loop formation in optimal supply networks, Nat. Commun. 11, 5796 (2020).
  7. A. Pries, T. Secomb, and P. Gaehtgens, Structural adaptation and stability of microvascular networks: Theory and simulations, Am. J. Physiol. Heart Circ. Physiol. 275, H349 (1998).
  8. A. R. Pries and T. W. Secomb, Modeling structural adaptation of microcirculation, Microcirculation 15, 753 (2008).
  9. D. Hu and D. Cai, Adaptation and optimization of biological transport networks, Phys. Rev. Lett. 111, 138701 (2013).
  10. H. Ronellenfitsch and E. Katifori, Global optimization, local adaptation, and the role of growth in distribution networks, Phys. Rev. Lett. 117, 138301 (2016).
  11. S.-S. Chang and M. Roper, Microvascular networks with uniform flow, J. Theor. Biol. 462, 48 (2019).
  12. G. Gounaris, M. Jovchevska, M. R. Garcia, and E. Katifori, The central role of metabolism in vascular morphogenesis, arXiv:2111.04657.
  13. F. Kramer and C. D. Modes, Biological flow networks: Antagonism between hydrodynamic and metabolic stimuli as driver of topological transitions, Phys. Rev. Res. 5, 023106 (2023).
  14. S. Marbach, N. Ziethen, and K. Alim, Vascular adaptation model from force balance: Physarum polycephalum as a case study, New J. Phys. 25, 123052 (2023).
  15. R. D. Travasso, C. A. Penick, R. R. Dunn, and E. C. Poiré, Predicting cardiac frequencies in mammals, Sci. Rep. 15, 7017 (2025).
  16. J. M. Harazny, C. Ott, U. Raff, J. Welzenbach, N. Kwella, G. Michelson, and R. E. Schmieder, First experience in analyzing pulsatile retinal capillary flow and arteriolar structural parameters measured noninvasively in hypertensive patients, J. Hypertens. 32, 2246 (2014).
  17. S. Fancher and E. Katifori, Mechanical response in elastic fluid flow networks, Phys. Rev. Fluids 7, 013101 (2022).
  18. S. Fancher and E. Katifori, Tradeoffs between energy efficiency and mechanical response in fluid flow networks, arXiv:2102.13197.
  19. A. Cheng, C. A. Williamitis, and M. S. Slaughter, Comparison of continuous-flow and pulsatile-flow left ventricular assist devices: Is there an advantage to pulsatility? Ann. Cardiothorac. Surg. 3, 573 (2014).
  20. C. R. Bartoli, G. A. Giridharan, K. N. Litwak, M. Sobieski, S. D. Prabhu, M. S. Slaughter, and S. C. Koenig, Hemodynamic responses to continuous versus pulsatile mechanical unloading of the failing left ventricle, ASAIO J. 56, 410 (2010).
  21. G. Muhire, M. F. Iulita, D. Vallerand, J. Youwakim, M. Gratuze, F. R. Petry, E. Planel, G. Ferland, and H. Girouard, Arterial stiffness due to carotid calcification disrupts cerebral blood flow regulation and leads to cognitive deficits, J. Am. Heart Assoc. 8, e011630 (2019).
  22. J. Gräwer, C. D. Modes, M. O. Magnasco, and E. Katifori, Structural self-assembly and avalanchelike dynamics in locally adaptive networks, Phys. Rev. E 92, 012801 (2015).
  23. R. Waszkiewicz, J. B. Shaw, M. Lisicki, and P. Szymczak, Goldilocks fluctuations: Dynamic constraints on loop formation in scale-free transport networks, Phys. Rev. Lett. 132, 137401 (2024).
  24. L. Gyllingberg, Y. Tian, and D. J. Sumpter, A minimal model of cognition based on oscillatory and current-based reinforcement processes, J. R. Soc. Interface. 22, rsif.2024.0402 (2025).
  25. P. Chatterjee, S. Fancher, and E. Katifori, Interplay between short and long time scales in adapting, periodically driven elastic flow networks, Phys. Rev. Res. 6, 043015 (2024).
  26. A. Tero, T. Saigusa, R. Kobayashi, and T. Nakagaki, Rules for biologically inspired adaptive network design, Science 327, 439 (2010).
  27. W. J. Hacking, E. VanBavel, and J. A. E. Spaan, Shear stressis not sufficient to control growth of vascular networks: A model study, Am. J. Physiol. 270, H364 (1996).
  28. A. G. Rolland-Lagan and P. Prusinkiewicz, Reviewing models of auxin canalization in the context of leaf vein pattern formation in arabidopsis, Plant J. 44, 854 (2005).
  29. K. van Berkel, R. J. de Boer, B. Scheres, and K. ten Tusscher, Polar auxin transport: Models and mechanisms, Development 140, 2253 (2013).
  30. C. D. Murray, The physiological principle of minimum work applied to the angle of branching of arteries, J. Gen. Physiol. 9, 835 (1926).
  31. T. F. Sherman, On connecting large vessels to small. The meaning of Murray's law, J. Gen. Physiol. 78, 431 (1981).
  32. G. S. Kassab and Y.-C.B. Fung, The pattern of coronary arteriolar bifurcations and the uniform shear hypothesis, Ann. Biomed. Eng. 23, 13 (1995).
  33. M. Zamir, J. Medeiros, and T. Cunningham, Arterial bifurcations in the human retina, J. Gen. Physiol. 74, 537 (1979).
  34. G. S. Kassab, Scaling laws of vascular trees: Of form and function, Am. J. Physiol. Heart Circ. Physiol. 290, H894 (2006).
  35. Y. Huo and G. S. Kassab, The scaling of blood flow resistance: From a single vessel to the entire distal tree, Biophys. J. 96, 339 (2009).
  36. D. J. Taylor, H. Saxton, I. Halliday, T. Newman, D. Hose, G. S. Kassab, J. P. Gunn, and P. D. Morris, Systematic review and meta-analysis of Murray's law in the coronary arterial circulation, Am. J. Physiol. Heart Circ. Physiol. 327, H182 (2024).
  37. Y. Zhou, G. S. Kassab, and S. Molloi, On the design of the coronary arterial tree: A generalization of Murray's law, Phys. Med. Biol. 44, 2929 (1999).
  38. A. Kamiya and T. Togawa, Optimal branching structure of the vascular tree, Bull. Math. Biophys. 34, 431 (1972).
  39. H. N. Mayrovitz and J. Roy, Microvascular blood flow: Evidence indicating a cubic dependence on arteriolar diameter, Am. J. Physiol. Heart Circ. Physiol. 245, H1031 (1983).
  40. L. A. Taber, S. Ng, A. M. Quesnel, J. Whatman, and C. J. Carmen, Investigating Murray's law in the chick embryo, J. Biomech. 34, 121 (2001).
  41. A. D. Hughes, Optimality, cost minimization and the design of arterial networks, Artery Res. 10, 1 (2015).
  42. C. Silva and A. H. Reis, Scaling relations of branching pulsatile flows, Int. J. Therm. Sci. 88, 77 (2015).
  43. P. R. Painter, P. Edén, and H.-U. Bengtsson, Pulsatile blood flow, shear force, energy dissipation and Murray's law, Theor. Biol. Med. Modell. 3, 31 (2006).
  44. M. G. Taylor, The elastic properties of arteries in relation to the physiological functions of the arterial system, Gastroenterology 52, 358 (1967).
  45. M. Shumal, M. Saghafian, E. Shirani, and M. Nili-AhmadAbadi, Association of Murray's law with atherosclerosis risk: Numerical validation of a general scaling law of arterial tree, Comput. Biol. Med. 186, 109741 (2025).
  46. J.-Y. Hahn, H.-C. Gwon, S. U. Kwon, S.-H. Choi, J.-H. Choi, S. H. Lee, K.-P. Hong, J. E. Park, and D. K. Kim, Comparison of vessel geometry in bifurcation between normal and diseased segments: Intravascular ultrasound analysis, Atherosclerosis 201, 326 (2008).
  47. A. W. Schoenenberger, N. Urbanek, S. Toggweiler, R. Seelos, P. Jamshidi, T. J. Resink, and P. Erne, Deviation from Murray's law is associated with a higher degree of calcification in coronary bifurcations, Atherosclerosis 221, 124 (2012).
  48. A. Kamiya, R. Bukhari, and T. Togawa, Adaptive regulation of wall shear stress optimizing vascular tree function, Bull. Math. Biol. 46, 127 (1984).
  49. Q. Chen, L. Jiang, C. Li, D. Hu, J.-w. Bu, D. Cai, and J.-l. Du, Haemodynamics-driven developmental pruning of brain vasculature in zebrafish, PLoS Biol. 10, e1001374 (2012).
  50. J. R. Banavar, F. Colaiori, A. Flammini, A. Maritan, and A. Rinaldo, Topology of the fittest transportation network, Phys. Rev. Lett. 84, 4745 (2000).
  51. M. Durand, Architecture of optimal transport networks, Phys. Rev. E 73, 016116 (2006).
  52. S. Bohn and M. O. Magnasco, Structure, scaling, and phase transition in the optimal transport network, Phys. Rev. Lett. 98, 088702 (2007).
  53. H. Kurz, Physiology of angiogenesis, J. Neuro-Oncol. 50, 17 (2000).
  54. D. Hu, D. Cai, and A. Rangan, Blood vessel adaptation with fluctuations in capillary flow distribution, PLoS ONE 7, e45444 (2012).
  55. A. Eichmann, L. Yuan, D. Moyon, F. Lenoble, L. Pardanaud, and C. Bréant, Vascular development: From precursor cells to branched arterial and venous networks, Int. J. Dev. Biol. 49, 259 (2005).
  56. M. Scianna, C. G. Bell, and L. Preziosi, A review of mathematical models for the formation of vascular networks, J. Theor. Biol. 333, 174 (2013).
  57. Y. Qi, S.-S. Chang, Y. Wang, C. Chen, K. I. Baek, T. Hsiai, and M. Roper, Hemodynamic regulation allows stable growth of microvascular networks, Proc. Natl. Acad. Sci. USA 121, e2310993121 (2024).
  58. M. Zamir and R. Budwig, Physics of pulsatile flow, Appl. Mech. Rev. 55, B35 (2002).
  59. See Supplemental Material at https://link.aps.org/supplemental/10.1103/gb8b-7yt3 for details of the flow-dynamics model, network construction, and additional results.
  60. W. K. Ang and P. W. Jowitt, Some new insights on informational entropy for water distribution networks, Eng. Optim. 37, 277 (2005).
  61. P. Chatterjee, Remodeling simulations (2025), https://github.com/purbachat/PulsatileAdaptation.

Outline

Information

PRX Life 4, 013006– Published 13 January, 2026
  • Received 25 March 2025
  • Accepted 2 December 2025
Crossmark - Check for updates button
Creative Commons Logo

Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article's title, journal citation, and DOI.

Published by the American Physical Society

Sign In to Your Journals Account

Filter

Filter

Article Lookup

Enter a citation