Modeling transit time distributions in microvascular networks.

The time a red blood cell (RBC) spends in the microvasculature is of prime importance for a number of physiological processes. In this work, we present a methodology for computing an approximation of the so-called transit time distribution (TTD), i.e., the probabilistic description of how long a RBC will reside within the network. As a proof of concept, we apply this methodology to three flavors of the mesh networks. We show that each network type supports multiple distinct steady-state configurations and we present tools for analyzing the associated collection of TTDs, ranging from standard measures like mean capillary transit time (MCTT) and capillary transit time heterogeneity (CTTH) to novel metrics.

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.

Spiral and Rotor Patterns Produced by Fairy Ring Fungi.

A broad class of soil fungi form the annular patterns known as 'fairy rings' and provide one of the only means to observe spatio-temporal dynamics of otherwise cryptic fungal growth processes in natural environments. We present observations of novel spiral and rotor patterns produced by fairy ring fungi and explain these behaviors mathematically by first showing that a well known model of fairy ring fungal growth and the Gray-Scott reaction-diffusion model are mathematically equivalent. We then use bifurcation analysis and numerical simulations to identify the conditions under which spiral waves and rotors can arise. We demonstrate that the region of dimensionless parameter space supporting these more complex dynamics is adjacent to that which produces the more familiar fairy rings, and identify experimental manipulations to test the transitions between these spatial modes. These same manipulations could also feasibly induce fungal colonies to transition from rotor/spiral formation to a set of richer, as yet unobserved, spatial patterns.

Oscillations and Multiple Equilibria in Microvascular Blood Flow.

We investigate the existence of oscillatory dynamics and multiple steady-state flow rates in a network with a simple topology and in vivo microvascular blood flow constitutive laws. Unlike many previous analytic studies, we employ the most biologically relevant models of the physical properties of whole blood. Through a combination of analytic and numeric techniques, we predict in a series of two-parameter bifurcation diagrams a range of dynamical behaviors, including multiple equilibria flow configurations, simple oscillations in volumetric flow rate, and multiple coexistent limit cycles at physically realizable parameters. We show that complexity in network topology is not necessary for complex behaviors to arise and that nonlinear rheology, in particular the plasma skimming effect, is sufficient to support oscillatory dynamics similar to those observed in vivo.

Observations of spontaneous oscillations in simple two-fluid networks.

We investigate the laminar flow of two-fluid mixtures inside a simple network of interconnected tubes. The fluid system is composed of two miscible Newtonian fluids of different viscosity which do not mix and remain as nearly distinct phases. Downstream of a diverging network junction the two fluids do not necessarily split in equal fraction and thus heterogeneity is introduced into network. We find that in the simplest network, a single loop with one inlet and one outlet, under steady inlet conditions, the flow rates and distribution of the two fluids within the network loop can undergo persistent spontaneous oscillations. We develop a simple model which highlights the basic mechanism of the instability and we demonstrate that the model can predict the region of parameter space where oscillations exist. The model predictions are in good agreement with experimental observations.

Local behavioral rules sustain the cell allocation pattern in the combs of honey bee colonies (Apis mellifera).

In the beeswax combs of honey bees, the cells of brood, pollen, and honey have a consistent spatial pattern that is sustained throughout the life of a colony. This spatial pattern is believed to emerge from simple behavioral rules that specify how the queen moves, where foragers deposit honey/pollen and how honey/pollen is consumed from cells. Prior work has shown that a set of such rules can explain the formation of the allocation pattern starting from an empty comb. We show that these rules cannot maintain the pattern once the brood start to vacate their cells, and we propose new, biologically realistic rules that better sustain the observed allocation pattern. We analyze the three resulting models by performing hundreds of simulation runs over many gestational periods and a wide range of parameter values. We develop new metrics for pattern assessment and employ them in analyzing pattern retention over each simulation run. Applied to our simulation results, these metrics show alteration of an accepted model for honey/pollen consumption based on local information can stabilize the cell allocation pattern over time. We also show that adding global information, by biasing the queen's movements towards the center of the comb, expands the parameter regime over which pattern retention occurs.