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Self-regulated biological transportation structures with general entropy dissipation: 2D case and leaf-shaped domain

Clarissa Astuto, Peter Markowich, Simone Portaro, Giovanni Russo

TL;DR

The paper develops a 2D gradient-flow model for self-regulating biological transport networks, coupling nonlinear elliptic equations for pressure $p$ and an auxiliary variable $\\sigma$ to a parabolic equation for the conductivity tensor $\\mathbb{C}$, derived from the entropy-dissipation energy $E[\\mathbb{C}]$ with a general entropy generator $\\Phi$. It proves local well-posedness in Hölder spaces using Schauder theory and semigroup arguments, and implements a ghost nodal finite element method with level-set unfitted discretization to compute steady states after parameter scaling that reduces the focus to the ratio of diffusion to metabolism ${\\widetilde{D}}, {\\widetilde{\\nu}}$. Numerical experiments on circular and leaf-shaped domains reveal rich branching patterns whose complexity grows as the background permeability parameter $r$ decreases and as ${\\widetilde{D}}$ and ${\\widetilde{\\nu}}$ are varied, including symmetry breaking and potential fractal-like branching. The work provides a rigorous and computationally tractable framework for exploring entropy-driven pattern formation in leaf venation-like networks and demonstrates the method’s capability to capture intricate 2D morphologies in biologically inspired domains.

Abstract

In recent years, the study of biological transportation networks has attracted significant interest, focusing on their self-regulating, demand-driven nature. This paper examines a mathematical model for these networks, featuring nonlinear elliptic equations for pressure and an auxiliary variable, and a reaction-diffusion parabolic equation for the conductivity tensor, introduced in \cite{portaro2022emergence}. The model, based on an energy functional with diffusive and metabolic terms, allows for various entropy generating functions, facilitating its application to different biological scenarios. We proved a local well-posedness result for the problem in Hölder spaces employing Schauder and semigroup theory. Then, after a suitable parameter reduction through scaling, we computed the numerical solution for the proposed system using a recently developed ghost nodal finite element method \cite{astuto2024nodal}. An interesting aspect emerges when the solution is very articulated and the branches occupy a wide region of the domain.

Self-regulated biological transportation structures with general entropy dissipation: 2D case and leaf-shaped domain

TL;DR

The paper develops a 2D gradient-flow model for self-regulating biological transport networks, coupling nonlinear elliptic equations for pressure and an auxiliary variable to a parabolic equation for the conductivity tensor , derived from the entropy-dissipation energy with a general entropy generator . It proves local well-posedness in Hölder spaces using Schauder theory and semigroup arguments, and implements a ghost nodal finite element method with level-set unfitted discretization to compute steady states after parameter scaling that reduces the focus to the ratio of diffusion to metabolism . Numerical experiments on circular and leaf-shaped domains reveal rich branching patterns whose complexity grows as the background permeability parameter decreases and as and are varied, including symmetry breaking and potential fractal-like branching. The work provides a rigorous and computationally tractable framework for exploring entropy-driven pattern formation in leaf venation-like networks and demonstrates the method’s capability to capture intricate 2D morphologies in biologically inspired domains.

Abstract

In recent years, the study of biological transportation networks has attracted significant interest, focusing on their self-regulating, demand-driven nature. This paper examines a mathematical model for these networks, featuring nonlinear elliptic equations for pressure and an auxiliary variable, and a reaction-diffusion parabolic equation for the conductivity tensor, introduced in \cite{portaro2022emergence}. The model, based on an energy functional with diffusive and metabolic terms, allows for various entropy generating functions, facilitating its application to different biological scenarios. We proved a local well-posedness result for the problem in Hölder spaces employing Schauder and semigroup theory. Then, after a suitable parameter reduction through scaling, we computed the numerical solution for the proposed system using a recently developed ghost nodal finite element method \cite{astuto2024nodal}. An interesting aspect emerges when the solution is very articulated and the branches occupy a wide region of the domain.
Paper Structure (8 sections, 1 theorem, 61 equations, 14 figures, 2 tables, 2 algorithms)

This paper contains 8 sections, 1 theorem, 61 equations, 14 figures, 2 tables, 2 algorithms.

Table of Contents

  1. Introduction
  2. Model
  3. Existence of solutions of the multi-dimensional problem in Hölder space
  4. Variational formulation
  5. Space discretization
  6. Time discretization
  7. Numerical results
  8. Conclusions

Key Result

Theorem 1

Let $S \in C^{0,\alpha}(\Bar{\Omega})$ for some $\alpha \in (0, 1)$, $D=0$ and $\Phi \in C^4(\mathbb{R})$. Then, the problem (eq:poisson--eq:sigma_law) subject to initial condition eq:ic - with $\mathbb{C}_0(x) \ge \beta$ - admits a local in time solution $\mathbb{C} \in X_T$ for some $T>0$, where t

Figures (14)

  • Figure 1: Discretization of the computational domain. $\Omega$ is the green region inside the unit square $R$. (a): classification of the grid points: the blue points are the internal ones while the red circles denote the ghost points. (b): points of intersection between the grid and the boundary $\Gamma$ (see the definition of $A$ and $B$ in Algorithm \ref{['alg_ab']}).
  • Figure 2: Grid before and after snapping technique. (a): representation of the cell related to the internal point $P$ (blue points), whose distance from $\Gamma$ is less than $h^2$; (b): zoom-in of the shape of the domain, after the grid point $P$ has changed its classification, from internal to ghost point (red circles).
  • Figure 3: (a): representation of the indices $\{k_0, \ldots, k_{m-1}\}$, the edges $\{l_0,\ldots,l_{m-1} \}$ of a generic element $K$ and the intersection points $A$ and $B$ from Algorithm\ref{['alg_ab']}. (b): scheme of the three quadrature points (circles) for each edge $l_i, \, i = 0,\cdots,4$. The squared points represent the vertices $P_i,\, i = 0,\cdots,4,$ of the polygon $\mathcal{P}$.
  • Figure 4: Configuration of the solution in a circular domain, changing the number $N$ of cells. All the parameters are in Table \ref{['tab:parameters']}, and for the Fisher case we choose $\widetilde{D} = 2.5\cdot 10^{-5}, \widetilde{\nu} = 0.1$. In all cases, the stable radially symmetric solution has eight main branches, i.e. it is symmetric with respect to rotations by $k\pi/4$, $k\in\mathbb{Z}$.
  • Figure 5: Snapshots of the solution at various times, illustrating the progressive loss of symmetry. In these plots $N = 800$ and the other parameters are in Table \ref{['tab:parameters']}, with $\widetilde{D} = 2.5\cdot 10^{-5}$ and $\widetilde{\nu} = 0.1$.
  • ...and 9 more figures

Theorems & Definitions (4)

  • Remark 1
  • Theorem 1
  • proof
  • Remark 2