Tensor PDE model of biological network formation

Abstract

We study an elliptic-parabolic system of partial differential equations describing formation of biological network structures. The model takes into consideration the evolution of the permeability tensor under the influence of a diffusion term, representing randomness in the material structure, a decay term describing metabolic cost and a pressure force. A Darcy's law type equation describes the pressure field. In the spatially two-dimensional setting, we present a constructive, formal derivation of the PDE system from the discrete network formation model in the refinement limit of a sequence of unstructured triangulations. Moreover, we show that the PDE system is a formal L 2gradient flow of an energy functional with biological interpretation, and study its convexity properties. For the case when the energy functional is convex, we construct unique global weak solutions of the PDE system. Finally, we construct steady state solutions in one-and multi-dimensional settings and discuss their stability properties.

Emergence of biological transportation networks as a self-regulated process

“…Moreover, it is easy to prove that E is C 2 on its domain of definition. Obviously, L 2 α (Ω; S d ) is a closed and convex subset of L 2 (Ω; S d ) and E is convex and lower semicontinuous on L 2 (Ω; S d ) (see the proof of Proposition 3.3 in [23] for the lower semicontinuity proof). The statement then follows from standard theory, see e.g.…”

“…This is mitigated by adding a relaxation term for D to the right-hand side of (3.10). Following the suggestion of [20,21,23,26], we choose the power-law |D| γ−2 D with γ > 1. Setting ϕ ≡ 0 for simplicity, we arrive at the following stationary version of (3.10),…”

“…where the parameter α > 0 is the metabolic coefficient. The energy is constrained by (1.1), (1.2), representing the local mass conservation −∇ • q = S. We refer to [18,19,23] for a derivation of the system from the discrete graph-based model of [26]. A crucial observation about (1.1), (1.2), (1.7) made in [23] is that for γ ≥ 1 it is a convex functional in D. Consequently, by standard theory [3] we obtain the existence and uniqueness of the corresponding L 2 -gradient flow,…”

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

“…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]. The two modeling approaches were compared numerically in [18].…”

“…The two modeling approaches were compared numerically in [18]. This paper focuses on numerical treatment of the tensor-valued PDE model of [17], where the permeability tensor appearing in the Poisson equation is regularized by adding a multiple rI of the identity matrix, with r > 0. We discretize the system in space using a finitedifference scheme on a two-dimensional Cartesian grid.…”