Abstract
Understanding the transport of fluid in complex organisms has proven to be a key challenge not only in the medical and biological sciences, but in physics and network theory as well. This is even more so as biologically-inspired design principles have been increasing in popularity, reliably generating solutions to common theoretical and technical problems. On that note, vascular networks in mammalian organs display a magnificent level of self-organization, allowing them to develop and mature, yet miraculously orchestrate the correct transport of oxygen, water, blood etc. This is achieved by a dedicated biochemical feedback system, which is coupled to macroscopic stimuli, such as mechanical stresses. Another important constraint for the morphogenesis of flow networks is their environment, as these networks are spatially embedded. They are therefore exposed to significant constraints with regards to their scalability and dynamical behavior, which are not yet well understood. This thesis addresses the current challenges of network characterization and morphogenesis modeling for three-dimensional embedded networks. In order to derive proper maturation mechanisms, we propose a set of toy models for the creation of non-planar, entangled and reticulated networks. The key mechanisms we focus on in this thesis are flow fluctuation, coupling of pairing structures and metabolite uptake. We show that in accordance with previous theoretical approaches, fluctuation induced nullity can be formulated as a single parameter problem. We demonstrate that the reticulation transition follows a logarithmic law and find plexi with certain topologies to have limited nullity transitions, rendering such plexi intrinsically wasteful in terms of fluctuation generated reticulation. Moreover, we formulate a new coupling model for entangled adapting networks as an approach for vasculature found in the liver lobules, pancreas, kidneys etc. We discuss a model based on local, distance-dependent interactions between pairs of three-dimensional network skeletons. In doing so we find unprecedented delay and breakdown of the fluctuation induced nullity transition for repulsive interactions. In addition we find a new nullity transition emerging for attractive coupling. Next, we study how flow fluctuations and complex metabolic costs can be incorporated into Murray’s Law. Utilizing this law for interpolation, we are able to derive order of magnitude estimation for the parameters in liver networks, suggesting fluctuation driven adaptation to be the dominant factor. We also conclude that attractive coupling is a reasonable mechanism to account for the maintenance of entangled structures. We test optimal metabolite uptake in Kirchhoff networks by evaluating the impact of solute uptake driven dynamics relative to wall-shear stress driven adaptation. Here, we find that a nullity transition emerges in case of a dominant metabolite uptake machinery. In addition to that, we find re-entrant behavior in case of high absorption rates and discover a complex interaction between shear-stress generation and feedback. Nevertheless, we conclude that metabolite uptake optimization is not likely to occur due to radial adaptation alone. We suggest areas for further studies, which should consider absorption rate variation in order to account for realistic uptake profiles. In our outlook, we suggest a complex morphogenesis model for intertwined networks based on the results of this thesis.
