Emergence of biological transportation networks as a self-regulated process

Haškovec, Jan; Markowich, Peter A.; Portaro, Simone

doi:10.3934/dcds.2022159

Cited by 6 publications

(5 citation statements)

References 35 publications

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“…Table 1: In this table we see the difference between a traditional ADI scheme and the symmetric version, for different values of the background permeability r. The choice of the parameters, the source function S and the initial datum are defined in Eqs. (12)(13)(14)(15)(16). r = 10 −2 r = 10 −3 r = 10 −4 ADI sym-ADI ADI sym-ADI ADI sym-ADI asymm(C) 6.107e-04 4.390e-08 2.591e-02 1.673e-03 1.536e-01 1.868e-01 asymm(p) 3.8862e-05 4.289e-09 6.137e-02 2.688e-03 2.683e-02 2.808e-02 Moreover, investigating the reasons why we lose the symmetry of the solution, we notice that it is related to the computation of the solution of the Poisson equation ( 1) and in the choice of the background permeability r. When this parameters tends to zero, the condition number of the iteration matrix L increases, up to the order 10 8 .…”

Section: Comparisons Between Adi and Symmetric-adi Methodsmentioning

confidence: 99%

“…The rest of the data are defined in Eqs. (12)(13)(14)(15)(16). In Table 2 we present in the first column, the Wasserstein distance, error W , between one solution with N and N/2 number of grid points.…”

Section: Accuracy Tests: Wasserstein Distancementioning

confidence: 99%

“…In [11] a related PDE-based continuum model was proposed, which consists of a parabolic reaction-diffusion equation for the vector-valued network conductivity, coupled to a Poisson equation for the pressure. The model was subsequently studied in the series of papers [12,13,14,15,16]. A more general model with tensor-valued conductivity was derived in [17].…”

Section: Introductionmentioning

confidence: 99%

“…(13): the module of the variables at final time (a), the flux at final time (b) and the eigenvectors associated to the greatest eigenvalue in absolute value (c). The first row is for r = 10 −2 , the second row for r = 10 −3 and the last one for r = 10 −4 .The rest of the data are defined in Eqs (12)(13)(14)(15)(16)…”

mentioning

confidence: 99%

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Asymmetry and condition number of an elliptic-parabolic system for biological network formation

Astuto¹,

Boffi²,

Haškovec³

et al. 2023

Preprint

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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 non-linear finite difference scheme on a uniform Cartesian grid in a 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 (nonsymmetric) ADI method. Moreover, we study the effect of regularization by isotropic background permeability r > 0, showing that increased condition number of the elliptic problem due to decreasing value of r leads to loss of symmetry. Finally, we perform numerical error analysis of our method in Wasserstein distance.

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Section: Comparisons Between Adi and Symmetric-adi Methodsmentioning

confidence: 99%

Section: Accuracy Tests: Wasserstein Distancementioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

mentioning

confidence: 99%

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Asymmetry and condition number of an elliptic-parabolic system for biological network formation

Astuto¹,

Boffi²,

Haškovec³

et al. 2023

Preprint

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“…In 2013, Hu and Cai [1] proposed a global energy functional approach that considers both material and metabolism costs but driven by local information, such as wall shear stress on the tube walls. This model, referred to as the Cai-Hu model [2], has been also studied in several subsequent papers [3,4,5].…”

Section: Introductionmentioning

confidence: 99%

Finite element discretization of a biological network formation system: a preliminary study

Astuto¹,

Boffi²,

Credali³

2023

Preprint

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A finite element discretization is developed for the Cai-Hu model, describing the formation of biological networks. The model consists of a non linear elliptic equation for the pressure p and a non linear reaction-diffusion equation for the conductivity tensor C. The problem requires high resolution due to the presence of multiple scales, the stiffness in all its components and the non linearities. We propose a low order finite element discretization in space coupled with a semi-implicit time advancing scheme. The code is validated with several numerical tests performed with various choices for the parameters involved in the system. In absence of the exact solution, we apply Richardson extrapolation technique to estimate the order of the method.

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Finite Element Discretization of a Biological Network Formation System: A Preliminary Study

Astuto,

Boffi,

Credali

2024

SEMA SIMAI Springer Series

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Emergence of biological transportation networks as a self-regulated process

Cited by 6 publications

References 35 publications

Asymmetry and condition number of an elliptic-parabolic system for biological network formation

Asymmetry and condition number of an elliptic-parabolic system for biological network formation

Finite element discretization of a biological network formation system: a preliminary study

Finite Element Discretization of a Biological Network Formation System: A Preliminary Study

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