A mesoscopic model of biological transportation networks

Abstract

We introduce a mesoscopic model for natural network formation processes, acting as a bridge between the discrete and continuous network approach proposed in [17]. The models are based on a common approach where the dynamics of the conductance network is subject to pressure force effects. We first study topological properties of the discrete model and we prove that if the metabolic energy consumption term is concave with respect to the conductivities, the optimal network structure is a tree (i.e., no loops are present). We then analyze various aspects of the mesoscopic modeling approach, in particular its relation to the discrete model and its stationary solutions, including discrete network solutions. Moreover, we present an alternative formulation of the mesoscopic model that avoids the explicit presence of the pressure in the energy functional.AMSC: 35B36; 92C42; 35K55; 49J20

Emergence of biological transportation networks as a self-regulated process

“…A significant problem for proving well-posedness of the system (3.3), (3.5), (3.6) with general convex entropy loss densities Φ = Φ(w) is the fact that we are not able to establish preservation of nonnegativity of the tensor D = D(t). However, modeling considerations [1,2,11,20,21] motivate us to make the ansatz (1.16) for D, namely…”

“…where D = D(t, x) and t is the time-like variable induced by the gradient flow. A particular case of this type of process in the context of biological applications (e.g., leaf venation in plants) is the network formation problem introduced in [26] and further analyzed in the series of papers [1,2,11,20,21,22,31,39,40]. Here the quantity D = D(t, x) represent the tensor-valued local conductivity of the network, which is understood as a continuous porous medium.…”

“…A generic example is represented by neural (brain) tissue in animals and humans. However, a quick inspection of equation (1.14), or its straightforward modification accounting for the presence of the metabolic term in (1.15), reveals that, due to lack of a minimum principle, it does not guarantee preservation of positive (semi)definitness of the tensor D. In a future work, we shall examine the well-posedness of the system in the case of small sources S = S(x) and/or small time t. Another option, inspired by [1,2,11,20,21], is to make the ansatz…”

Tensor PDE model of biological network formation

“…Detailed mathematical analysis of the system (1.9)-(1.10) with M (s) := s γ was carried out in the series of papers [1,2,8,9] and in [16,20,21], while its various other aspects were studied in [4,[10][11][12][13].…”

Implicit and Semi-implicit Numerical Schemes for the Gradient Flow of the Formation of Biological Transport Networks

“…DMS-1522184, DMS-1107291: RNMS KI-Net, and NSFC grant No. 31571071. studied in [13,14,1,11,4,12]. To aim at the understanding of the growth and formation of these biological networks, we consider the PDE version in this work, which is in fact a lot more challenging in terms of numerical simulations.…”