Linear and nonlinear shear wave propagation in viscoelastic media

This dissertation covers material pertaining to direct and inverse problems for viscoealstic materials with applications to pulmonary fluids (e.g. mucus) and other biological materials. This research extends a classic viscoelastic characterization technique, originally developed by John D. Ferry, to small volume (micro-liter) samples of fluid through mathematical modeling and computation in the development of the micro-parallel plate rheometer (MPPR). Ferry’s inverse characterization protocol measures the attenuation length and wave length of shear waves in birefringent synthetic polymers, and uses the exact solution for a linear viscoelastic fluid undergoing periodic deformation in a semi-infinite domain to obtain formulas relating these values to the linear viscoelastic (material) parameters. In this dissertation, Ferry’s exact solution is extended to include both finite depth effects that resolve counter-propagating waves, and nonlinear effects that arise from Giesekus and upper-convected Maxwell constitutive equations. These generalizations of Ferry’s solution are used in the development of the MPPR, which generates a shear wave in a finite depth domain and uses micro-bead tracking and nonlinear regression on fluid displacement time series to extend Ferry’s protocol of viscoelastic characterization. The final topic of this dissertation is a predictive model for stress communication and filtering across viscoelastic layers. By focusing on stress signals arriving at either boundary in a finite depth domain, this work identifies a remarkable structure in boundary extreme stress signals which could play a key role in regulating certain aspects of lung function. This structure is derived from harmonic resonance in the elastic solid material limit, and demonstrated to persist and diverge from this limit as a function of a dimensionless parameter (the ratio of the attenuation length to the wave length). Analytical results demonstrate that this structure persists with respect to all material and driving parameters and in more general boundary value problems

Statistical multi-moment bifurcations in random delay coupled swarms

We study the effects of discrete, randomly distributed time delays on the dynamics of a coupled system of self-propelling particles. Bifurcation analysis on a mean field approximation of the system reveals that the system possesses patterns with certain universal characteristics that depend on distinguished moments of the time delay distribution. Specifically, we show both theoretically and numerically that although bifurcations of simple patterns, such as translations, change stability only as a function of the first moment of the time delay distribution, more complex patterns arising from Hopf bifurcations depend on all of the moments

Rare Event Extinction on Stochastic Networks

We consider the problem of extinction processes on random networks with a given structure. For sufficiently large well-mixed populations, the process of extinction of one or more state variable components occurs in the tail of the quasi-stationary probability distribution, thereby making it a rare event. Here we show how to extend the theory of large deviations to random networks to predict extinction times. In particular, we use the theory to find the most probable path leading to extinction. We apply the methodology to epidemic models and discover how mean extinction times scale with epidemiological and network parameters in Erdos-Renyi networks. The results are shown to compare quite well with Monte Carlo simulations of the network in predicting both the most probable paths to extinction and mean extinction times.Comment: 5 pages 6 figure

Intervention-Based Stochastic Disease Eradication

Disease control is of paramount importance in public health with infectious disease extinction as the ultimate goal. Although diseases may go extinct due to random loss of effective contacts where the infection is transmitted to new susceptible individuals, the time to extinction in the absence of control may be prohibitively long. Thus intervention controls, such as vaccination of susceptible individuals and/or treatment of infectives, are typically based on a deterministic schedule, such as periodically vaccinating susceptible children based on school calendars. In reality, however, such policies are administered as a random process, while still possessing a mean period. Here, we consider the effect of randomly distributed intervention as disease control on large finite populations. We show explicitly how intervention control, based on mean period and treatment fraction, modulates the average extinction times as a function of population size and rate of infection spread. In particular, our results show an exponential improvement in extinction times even though the controls are implemented using a random Poisson distribution. Finally, we discover those parameter regimes where random treatment yields an exponential improvement in extinction times over the application of strictly periodic intervention. The implication of our results is discussed in light of the availability of limited resources for control.Comment: 18 pages, 10 Figure

Extensions of the Ferry shear wave model for active linear and nonlinear microrheology

The classical oscillatory shear wave model of Ferry et al. [J. Polym. Sci. 2:593-611, (1947)] is extended for active linear and nonlinear microrheology. In the Ferry protocol, oscillation and attenuation lengths of the shear wave measured from strobe photographs determine storage and loss moduli at each frequency of plate oscillation. The microliter volumes typical in biology require modifications of experimental method and theory. Microbead tracking replaces strobe photographs. Reflection from the top boundary yields counterpropagating modes which are modeled here for linear and nonlinear viscoelastic constitutive laws. Furthermore, bulk imposed strain is easily controlled, and we explore the onset of normal stress generation and shear thinning using nonlinear viscoelastic models. For this paper, we present the theory, exact linear and nonlinear solutions where possible, and simulation tools more generally. We then illustrate errors in inverse characterization by application of the Ferry formulas, due to both suppression of wave reflection and nonlinearity, even if there were no experimental error. This shear wave method presents an active and nonlinear analog of the two-point microrheology of Crocker et al. [Phys. Rev. Lett. 85: 888 - 891 (2000)]. Nonlocal (spatially extended) deformations and stresses are propagated through a small volume sample, on wavelengths long relative to bead size. The setup is ideal for exploration of nonlinear threshold behavior

Spatial stress and strain distributions of viscoelastic layers in oscillatory shear

One of the standard experimental probes of a viscoelastic material is to measure the response of a layer trapped between parallel surfaces, imposing either periodic stress or strain at one boundary and measuring the other. The relative phase between stress and strain yields solid-like and liquid-like properties, called the storage and loss moduli, respectively, which are then captured over a range of imposed frequencies. Rarely are the full spatial distributions of shear and normal stresses considered, primarily because they cannot be measured except at boundaries and the information was not deemed of particular interest in theoretical studies. Likewise, strain distributions throughout the layer were traditionally ignored except in a classical protocol of Ferry, Adler and Sawyer, based on snapshots of standing shear waves. Recent investigations of thin lung mucus layers exposed to oscillatory stress (breathing) and strain (coordinated cilia), however, suggest that the wide range of healthy conditions and environmental or disease assaults lead to conditions that are quite disparate from the “surface loading” and “gap loading” conditions that characterize classical rheometers. In this article, we extend our previous linear and nonlinear models of boundary stresses in controlled oscillatory strain to the entire layer. To illustrate non-intuitive heterogeneous responses, we characterize experimental conditions and material parameter ranges where the maximum stresses migrate into the channel interior

A mechanochemical model for auto-regulation of lung airway surface layer volume

We develop a proof-of-principle model for auto-regulation of water volume in the lung airway surface layer (ASL) by coupling biochemical kinetics, transient ASL volume, and homeostatic mechanical stresses. The model is based on the hypothesis that ASL volume is sensed through soluble mediators and phasic stresses generated by beating cilia and air drag forces. Model parameters are fit based on available data on human bronchial epithelial cell cultures. Simulations then demonstrate that homeostatic volume regulation is a natural consequence of the processes described. The model maintains ASL volume within a physiological range that modulates with phasic stress frequency and amplitude. Next, we show that the model successfully reproduces the responses of cell cultures to significant isotonic and hypotonic challenges, and to hypertonic saline, an effective therapy for mucus hydration in cystic fibrosis patients. These results compel an advanced airway hydration model with therapeutic value that will necessitate detailed kinetics of multiple molecular pathways, feedback to ASL viscoelasticity properties, and stress signaling from the ASL to the cilia and epithelial cells