Communications on Applied Mathematics and Computation >

2025 , Vol. 7 >Issue 1: 78 - 94

DOI: https://doi.org/10.1007/s42967-023-00297-3

Asymmetry and Condition Number of an Elliptic-Parabolic System for Biological Network Formation

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  • 1 Applied Mathematics and Computational Sciences Department, King Abdullah University of Science and Technology, 4700 Thuwal, Saudi Arabia;
    2 Department of Mathematics "F. Casorati", University of Pavia, 27100 Pavia, Italy;
    3 Department of Mathematics, University of Vienna, 1090 Vienna, Austria;
    4 Department of Mathematics and Computer Science, University of Catania, 95125 Catania, Italy

Received date: 2023-01-31

  Revised date: 2023-06-07

  Accepted date: 2023-07-05

  Online published: 2025-04-21

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Abstract

We present results of numerical simulations of the tensor-valued elliptic-parabolic PDE model for biological network formation. The numerical method is based on a nonlinear finite difference scheme on a uniform Cartesian grid in a two-dimensional (2D) domain. The focus is on the impact of different discretization methods and choices of regularization parameters on the symmetry of the numerical solution. In particular, we show that using the symmetric alternating direction implicit (ADI) method for time discretization helps preserve the symmetry of the solution, compared to the (non-symmetric) ADI method. Moreover, we study the effect of the regularization by the isotropic background permeability r > 0, showing that the increased condition number of the elliptic problem due to decreasing value of r leads to loss of symmetry. We show that in this case, neither the use of the symmetric ADI method preserves the symmetry of the solution. Finally, we perform the numerical error analysis of our method making use of the Wasserstein distance.

Key words： Bionetwork formation; Cai-Hu model; Leaf venation; Asymmetry; Conditioning number; Finite-difference scheme; Semi-implicit; Symmetric alternating direction implicit (ADI); Wasserstein distance

Cite this article

Clarissa Astuto, Daniele Boffi, Jan Haskovec, Peter Markowich, Giovanni Russo . Asymmetry and Condition Number of an Elliptic-Parabolic System for Biological Network Formation[J]. Communications on Applied Mathematics and Computation, 2025 , 7(1) : 78 -94 . DOI: 10.1007/s42967-023-00297-3

References

1. Albi, G., Artina, M., Foransier, M., Markowich, P.: Biological transportation networks: modeling and simulation. Anal. Appl. 14, 185–206 (2016). https:// doi. org/ 10. 1142/ S0219 53051 54000 59
2. Albi, G., Burger, M., Haskovec, J., Markowich, P., Schlottbom, M.: Continuum Modelling of Biological Network Formation. Active Particles, vol. I: Theory, Models, Applications. Birkhäuser-Springer, Boston (2017)
3. Alcubierre, M., Schutz, B.F.: Time symmetric ADI and casual reconnection. In: International Workshop on Numerical Relativity. Cambridge University Press, Cambridge (1992)
4. Ambrosio, L., Gigli, N., Savaré, G.: Gradient Flows: in Metric Spaces and in the Space of Probability Measures. Springer Sci. Bus. Media, Basel (2005). https:// link. sprin ger. com/ book/ 10. 1007/ b1370 80
5. Astuto, C., Boffi, D., Credali, F.: Finite element discretization of a biological network formation system: a preliminary study. arXiv: 2303. 10625 (2023)
6. Astuto, C., Boffi, D., Haskovec, J., Markowich, P., Russo, G.: Comparison of two aspects of a PDE model for biological network formation. Math. Comput. Appl. 27, 87 (2022). https:// doi. org/ 10. 3390/ mca27 050087
7. Carrillo, J.A., Toscani, G.: Wasserstein metric and large-time asymptotics of nonlinear diffusion equations. In: New Trends in Mathematical Physics: In Honour of the Salvatore Rionero 70th Birthday— Proceedings of the International Meeting (2004). https:// doi. org/ 10. 1142/ 97898 12702 319_ 0022
8. Chen, Q., Jiang, L., Li, C., Hu, D., Bu, J., Cai, D., Du, J.: Haemodynamics-Driven Developmental Pruning of Brain Vasculature in Zebrafish. Public Library of Science, San Francisco (2012). https:// doi. org/ 10. 1371/ journ al. pone. 00476 17
9. Eichmann, A., Le Noble, F., Autiero, M., Carmeliet, P.: Guidance of vascular and neural network formation. Curr. Opin. Neurobiol. 15, 108–115 (2005). https:// doi. org/ 10. 1016/j. conb. 2005. 08. 012
10. Fang, D., Jin, S., Markowich, P., Perthame, B.: Implicit and semi-implicit numerical schemes for the gradient flow of the formation of biological transport networks. SMAI J. Comput. Math. 5, 229–249 (2019). https:// doi. org/ 10. 5802/ smai- jcm. 59 11. Hacking, W.J., Van Bavel, E., Spaan, J.A.: Shear stress is not sufficient to control growth of vascular networks: a model study. Am. J. Physiol. Heart Circ. Physiol. 270, H364–H375 (1996). https:// doi. org/
10. 1152/ ajphe art. 1996. 270.1. H364
12. Haskovec, J., Markowich, P., Pilli, G.: Tensor PDE model of biological network formation. Commun. Math. Sci. 20, 1173–1191 (2022)
13. Haskovec, J., Markowich, P., Perthame, B.: Mathematical analysis of a PDE system for biological network formation. Comm. Partial Differential Equations 40, 918–956 (2015). https:// doi. org/ 10. 1080/ 03605 302. 2014. 968792
14. Haskovec, J., Markowich, P., Perthame, B., Schlottbom, M.: Notes on a PDE system for biological network formation. Nonlinear Anal. 138, 127–155 (2016). https:// doi. org/ 10. 1016/j. na. 2015. 12. 010
15. Haskovec, J., Markowich, P., Portaro, S.: Emergence of biological transportation networks as a selfregulated process. Discrete Contin. Dyn. Syst. 43, 1499–1515 (2022). https:// doi. org/ 10. 3934/ dcds. 20221 59
16. Hu, D.: Optimization, Adaptation, and Initialization of Biological Transport Networks. Notes from Lecture 1, 3–1 (2013)
17. Hu, D., Cai, D.: Adaptation and optimization of biological transport networks. Phys. Rev. Lett. 111, 138701 (2013). https:// doi. org/ 10. 1103/ PhysR evLett. 110. 138101
18. Hu, D., Cai, D.: An optimization principle for initiation and adaptation of biological transport networks. Commun. Math. Sci. 17, 1427–1436 (2019)
19. Hu, D., Cai, D., Rangan, A.V.: Blood vessel adaptation with fluctuations in capillary flow distribution. PLoS One 7, 45444 (2012). https:// doi. org/ 10. 1371/ journ al. pone. 00447 97
20. Malinowski, R.: Understanding of leaf development—the science of complexity. Plants 2, 396–415 (2013). https:// doi. org/ 10. 3390/ plant s2040 541
21. Michel, O., Biondi, J.: Morphogenesis of neural networks. Neural Process. Lett. 2, 9–12 (1995). https:// doi. org/ 10. 1007/ BF023 09873
22. Otto, F.: Double Degenerate Diffusion Equations as Steepest Descent. Bonn University (1996). https:// books. google. com. sa/ books? id= oxLdG wAACA AJ
23. Otto, F.: The geometry of dissipative evolution equations: the porous medium equation. Comm. Partial Differential Equations 26, 101–174 (2001). https:// doi. org/ 10. 1081/ PDE- 10000 2243
24. Peaceman, D.W., Rachford, H.H., Jr.: The numerical solution of parabolic and elliptic equations. J. Soc. Ind. Appl. Math. 3, 28–41 (1955)
25. Pohl, U., Holtz, J., Busse, R., Bassenge, E.: Crucial role of endothelium in the vasodilator response to increased flow in vivo. Hypertension 8, 37–44 (1986). https:// doi. org/ 10. 1161/ 01. hyp.8. 1. 37
26. Pries, A.R., Secomb, T.W., Gaehtgens, P.: Structural adaptation and stability of microvascular networks: theory and simulations. Am. J. Physiol. Heart Circ. Physiol. 275, H349–H360 (1998). https:// doi. org/ 10. 1152/ ajphe art. 1998. 275.2. H349
27. Raudino, A., Grassi, A., Lombardo, G., Russo, G., Astuto, C., Corti, M.: Anomalous sorption kinetics of self-interacting particles by a spherical trap. Commun. Comput. Phys. 31, 707–738 (2022). https:// doi. org/ 10. 4208/ cicp. OA- 2021- 0101
28. Santambrogio, F.: Euclidean, metric, and Wasserstein gradient flows: an overview. Bull. Math. Sci. 7, 87–154 (2017). https:// doi. org/ 10. 1007/ s13373- 017- 0104-8
29. Sedmera, D.: Function and Form in the Developing Cardiovascular System. Cardiovascular Research. Oxford University Press, Oxford (2011). https:// doi. org/ 10. 1093/ cvr/ cvr228
30. Villani, C.: Topics in Optimal Transportation. American Mathematical Society, Providence (2003)
31. Villani, C.: Optimal Transport: Old and New. Springer, Berlin (2009)
32. Villani, C.: Couplings and changes of variables. In: Optimal Transport. Grundlehren der mathematischen Wissenschaften, vol 338. Springer, Berlin, Heidelberg (2009). https:// doi. org/ 10. 1007/ 978-3- 540- 71050-9_1
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