Model Microvascular Networks Can Have Many Equilibria.

We show that large microvascular networks with realistic topologies, geometries, boundary conditions, and constitutive laws can exhibit many steady-state flow configurations. This is in direct contrast to most previous studies which have assumed, implicitly or explicitly, that a given network can only possess one equilibrium state. While our techniques are general and can be applied to any network, we focus on two distinct network types that model human tissues: perturbed honeycomb networks and random networks generated from Voronoi diagrams. We demonstrate that the disparity between observed and predicted flow directions reported in previous studies might be attributable to the presence of multiple equilibria. We show that the pathway effect, in which hematocrit is steadily increased along a series of diverging junctions, has important implications for equilibrium discovery, and that our estimates of the number of equilibria supported by these networks are conservative. If a more complete description of the plasma skimming effect that captures red blood cell allocation at junctions with high feed hematocrit were to be obtained empirically, then the number of equilibria found by our approach would at worst remain the same and would in all likelihood increase significantly.

A physical sciences network characterization of circulating tumor cell aggregate transport.

Circulating tumor cells (CTC) have been implicated in the hematogenous spread of cancer. To investigate the fluid phase of cancer from a physical sciences perspective, the multi-institutional Physical Sciences-Oncology Center (PS-OC) Network performed multidisciplinary biophysical studies of single CTC and CTC aggregates from a patient with breast cancer. CTCs, ranging from single cells to aggregates comprised of 2-5 cells, were isolated using the high-definition CTC assay and biophysically profiled using quantitative phase microscopy. Single CTCs and aggregates were then modeled in an in vitro system comprised of multiple breast cancer cell lines and microfluidic devices used to model E-selectin mediated rolling in the vasculature. Using a numerical model coupling elastic collisions between red blood cells and CTCs, the dependence of CTC vascular margination on single CTCs and CTC aggregate morphology and stiffness was interrogated. These results provide a multifaceted characterization of single CTC and CTC aggregate dynamics in the vasculature and illustrate a framework to integrate clinical, biophysical, and mathematical approaches to enhance our understanding of the fluid phase of cancer.

Multiple equilibrium states in a micro-vascular network.

We use a simple model of micro-vascular blood flow to explore conditions that give rise to multiple equilibrium states in a three-node micro-vascular network. The model accounts for two primary rheological effects: the Fåhraeus-Lindqvist effect, which describes the apparent viscosity of blood in a vessel, and the plasma skimming effect, which governs the separation of red blood cells at diverging nodes. We show that multiple equilibrium states are possible, and we use our analytical and computational tools to design an experiment for validation.

Bistability in a simple fluid network due to viscosity contrast.

We study the existence of multiple equilibrium states in a simple fluid network using Newtonian fluids and laminar flow. We demonstrate theoretically the presence of hysteresis and bistability, and we confirm these predictions in an experiment using two miscible fluids of different viscosity-sucrose solution and water. Possible applications include blood flow, microfluidics, and other network flows governed by similar principles.

Blood flow in microvascular networks: a study in nonlinear biology.

Plasma skimming and the Fahraeus-Lindqvist effect are well-known phenomena in blood rheology. By combining these peculiarities of blood flow in the microcirculation with simple topological models of microvascular networks, we have uncovered interesting nonlinear behavior regarding blood flow in networks. Nonlinearity manifests itself in the existence of multiple steady states. This is due to the nonlinear dependence of viscosity on blood cell concentration. Nonlinearity also appears in the form of spontaneous oscillations in limit cycles. These limit cycles arise from the fact that the physics of blood flow can be modeled in terms of state dependent delay equations with multiple interacting delay times. In this paper we extend our previous work on blood flow in a simple two node network and begin to explore how topological complexity influences the dynamics of network blood flow. In addition we present initial evidence that the nonlinear phenomena predicted by our model are observed experimentally.

Oscillations in a simple microvascular network.

We have identified the simplest topology that will permit spontaneous oscillations in a model of microvascular blood flow that includes the plasma skimming effect and the Fahraeus-Lindqvist effect and assumes that the flow can be described by a first-order wave equation in blood hematocrit. Our analysis is based on transforming the governing partial differential equations into delay differential equations and analyzing the associated linear stability problem. In doing so we have discovered three dimensionless parameters, which can be used to predict the occurrence of nonlinear oscillations. Two of these parameters are related to the response of the hydraulic resistances in the branches to perturbations. The other parameter is related to the amount of time necessary for the blood to pass through each of the branches. The simple topology used in this study is much simpler than networks found in vivo. However, we believe our analysis will form the basis for understanding more complex networks.