Viscoelastic fluid description of bacterial biofilm material properties

A mathematical model describing the constitutive properties of biofilms is required for predicting biofilm deformation, failure, and detachment in response to mechanical forces. Laboratory observations indicate that biofilms are viscoelastic materials. Likewise, current knowledge of biofilm internal structure suggests modeling biofilms as associated polymer viscoelastic systems. Supporting experimental results and a system of viscoelastic fluid equations with a linear Jeffreys viscoelastic stress-strain law are presented here. This system of equations is based on elements of associated polymer physics and is also consistent with presented and previous experimental results. A number of predictions can be made. One particularly interesting result is the prediction of an elastic relaxation time on the order of a few minutes-biofilm disturbances on shorter time scales produce an elastic response, biofilm disturbances on longer time scales result in viscous flow, i.e., nonreversible biofilm deformation. Although not previously recognized, evidence of this phenomenon is in fact present in recent experimental results.     

Invited review liquid crystal models of biological materials and silk spinning

A review of thermodynamic, materials science, and rheological liquid crystal models is presented and applied to a wide range of biological liquid crystals, including helicoidal plywoods, biopolymer solutions, and in vivo liquid crystals. The distinguishing characteristics of liquid crystals (self-assembly, packing, defects, functionalities, processability) are discussed in relation to biological materials and the strong correspondence between different synthetic and biological materials is established. Biological polymer processing based on liquid crystalline precursors includes viscoelastic flow to form and shape fibers. Viscoelastic models for nematic and chiral nematics are reviewed and discussed in terms of key parameters that facilitate understanding and quantitative information from optical textures and rheometers. It is shown that viscoelastic modeling the silk spinning process using liquid crystal theories sheds light on textural transitions in the duct of spiders and silk worms as well as on tactoidal drops and interfacial structures. The range and consistency of the predictions demonstrates that the use of mesoscopic liquid crystal models is another tool to develop the science and biomimetic applications of mesogenic biological soft matter.     

Cell poking: quantitative analysis of indentation of thick viscoelastic layers.

A recently introduced device, the cell poker, measures the force required to indent the exposed surface of a cell adherent to a rigid substratum. The cell poker has provided phenomenological information about the viscoelastic properties of several different types of cells, about mechanical changes triggered by external stimuli, and about the role of the cytoskeleton in these mechanical functions. Except in special cases, however, it has not been possible to extract quantitative estimates of viscosity and elasticity moduli from cell poker measurements. This paper presents cell poker measurements of well characterized viscoelastic polymeric materials, polydimethylsiloxanes of different degrees of polymerization, in a simple shape, a flat, thick layer, which for our purposes can be treated as a half space. Analysis of the measurements in terms of a linear viscoelasticity theory yields viscosity values for three polymer samples in agreement with those determined by measurements on a macroscopic scale. Theoretical analysis further indicates that the measured limiting static elasticity of the layers may result from the tension generated at the interface between the polymer and water. This work demonstrates the possibility of obtaining quantitative viscoelastic material properties from cell poker measurements and represents the first step in extending these quantitative studies to more complicated structures including cells.     

A three-dimensional viscoelastic model for cell deformation with experimental verification

A three-dimensional viscoelastic finite element model is developed for cell micromanipulation by magnetocytometry. The model provides a robust tool for analysis of detailed strain/stress fields induced in the cell monolayer produced by forcing one microbead attached atop a single cell or cell monolayer on a basal substrate. Both the membrane/cortex and the cytoskeleton are modeled as Maxwell viscoelastic materials, but the structural effect of the membrane/cortex was found to be negligible on the timescales corresponding to magnetocytometry. Numerical predictions are validated against experiments performed on NIH 3T3 fibroblasts and previous experimental work. The system proved to be linear with respect to cytoskeleton mechanical properties and bead forcing. Stress and strain patterns were highly localized, suggesting that the effects of magnetocytometry are confined to a region extending <10 microm from the bead. Modulation of cell height has little effect on the results, provided the monolayer is >5 micro m thick. NIH 3T3 fibroblasts exhibited a viscoelastic timescale of approximately 1 s and a shear modulus of approximately 1000 Pa.     

Viscoelastic transition and yield strain of the folded protein

For proteins, the mechanical properties of the folded state are directly related to function, which generally entails conformational motion. Through sub-Angstrom resolution measurements of the AC mechanical susceptibility of a globular protein we describe a new fundamental materials property of the folded state. For increasing amplitude of the forcing, there is a reversible transition from elastic to viscoelastic response. At fixed frequency, the amplitude of the deformation is piecewise linear in the force, with different slopes in the elastic and viscoelastic regimes. Effectively, the protein softens beyond a yield point defined by this transition. We propose that ligand induced conformational changes generally operate in this viscoelastic regime, and that this is a universal property of the folded state.     

On algorithms of evaluation of Fung's relaxation function parameters

The mathematical aspects of the problem of the interpretation of the experimental data on the viscoelastic behaviour of materials by making use of the linear and quasi-linear relaxation model proposed by Fung, are considered. Three idealized cases of the growth of the strain from the zero value of its value a = const., that it will keep at t greater than or equal to tO, are analysed: Case 1--a jump at t = t0, Case 2--a linear law of growth in the interval 0 less than or equal to t less than or equal to t0, Case 3--a parabolic law of growth in the same interval. The exact formulae for calculation of the stress are presented. From them the simple 'small time' and 'great time' asymptotic expressions are derived. These expressions are used for comparison of the Cases 1, 2, 3. An algorithm is suggested for the iterative numerical evaluation of the parameters of the linear model on the basis of the experimental data, corresponding to Case 1, 2 or 3.     

A model for the creep behaviour of tendon

A theoretical model is proposed to explain the viscoelastic behaviour of tendon. The model is based on the hypothesis that the mechanism of tendon deformation is one of shear of the mucopolysaccharide gel between the collagen ribbons in the ‘toe’ region¹ of the stress strain curve followed by fibrillar extension in the ‘linear’ region^2.3. Conventional linear viscoelastic parameters of the constituents of the tendon were used to describe the behaviour of the composite. Certain structural constants of the tendon and an appropriate form for the retardation spectrum of the composite also appear in the model. It was found that the following parameters play an important part in the viscoelastic deformation process; the mean value of the crimp angle of the strained fibres, 0_O; the broadness of the distribution of crimp angles $\text{α}$; the magnitude of the compliance difference for the gel, ΔJ_G and for the fibres, ΔD_F. Excellent agreement between experiment and theory has been observed for a variety of experimental circumstances. Values of 0_O, $\text{α}$ ΔJ_G and ΔD_F were determined for a number of specimens by fitting the model equations to the experimental data. The theory illustrates the expected influence on the viscoelastic properties of tendon which would result from changes in these parameters, which may arise from disease or ageing, for instance. The model also provides a challenge for future experimental work in that an independent determination of the parameters, 0_O, $\text{α}$ ΔJ_G and ΔD_F would confirm or refute the quantitative predictions of the theory presented here.     

Can a rheological muscle model predict force depression/enhancement?

A new phenomenological model of activated muscle is presented. The model is based on a combination of a contractile element, an elastic element that engages upon activation, a linear dashpot and a linear spring. Analytical solutions for a few selected experiments are provided. This model is able to reproduce the response of cat soleus muscle to ramp shortening and stretching and, unlike standard Hill-type models, computations are stable on the descending limb of the force–length relation and force enhancement (depression) following stretching (shortening) is predicted correctly. In its linear version, the model is consistent with a linear force–velocity law, which in this model is a consequence rather than a fundamental characteristic of the material. Results show that the mechanical response of activated muscle can be mimicked by a viscoelastic system. Conceptual differences between this model and standard Hill-type models are analyzed and the advantages of the present model are discussed.     

Non-linear viscoelastic behavior of abdominal aortic aneurysm thrombus

The objective of this work was to determine the linear and non-linear viscoelastic behavior of abdominal aortic aneurysm thrombus and to study the changes in mechanical properties throughout the thickness of the thrombus. Samples are gathered from thrombi of seven patients. Linear viscoelastic data from oscillatory shear experiments show that the change of properties throughout the thrombus is different for each thrombus. Furthermore the variations found within one thrombus are of the same order of magnitude as the variation between patients. To study the non-linear regime, stress relaxation experiments are performed. To describe the phenomena observed experimentally, a non-linear multimode model is presented. The parameters for this model are obtained by fitting this model successfully to the experiments. The model cannot only describe the average stress response for all thrombus samples but also the highest and lowest stress responses. To determine the influence on the wall stress of the behavior observed the model proposed needs to implemented in the finite element wall stress analysis.     

Viscoelastic Properties of a Mixed Culture Biofilm from Rheometer Creep Analysis

The mechanical properties of mixed culture biofilms were determined by creep analysis using an AR1000 rotating disk rheometer. The biofilms were grown directly on the rheometer disks which were rotated in a chemostat for 12 d. The resulting biofilms were heterogeneous and ranged from 35?μm to 50?μm in thickness. The creep curves were all viscoelastic in nature. The close agreement between stress and strain ratio of a sample tested at 0.1 and 0.5 Pa suggested that the biofilms were tested in the linear viscoelastic range and supported the use of linear viscoelastic theory in the development of a constitutive law. The experimental data was fit to a 4-element Burger spring and dashpot model. The shear modulus (G) ranged from 0.2 to 24 Pa and the viscous coefficient (η) from 10 to 3000 Pa. These values were in the same range as those previously estimated from fluid shear deformation of biofilms in flow cells. A viscoelastic biofilm model will help to predict shear related biofilm phenomena such as elevated pressure drop, detachment, and the flow of biofilms over solid surfaces.     

A model of fracture testing of soft viscoelastic tissues

Fracture, or tear, toughness of soft tissues can be computed from the work of fracture divided by the area of new crack surface. For soft tissues without significant plastic deformation, total work, which can be measured experimentally, is composed of the sum of fracture and viscoelastic work. In order to deduce fracture work, a method is needed to estimate viscoelastic work.Two different methods (Ph.D. Dissertation, University of Minnesota, 2000; J. Mater. Sci.: Mater. Med. 12 (2001) 327) have been proposed to estimate viscoelastic work in a fracture test of a soft tissue. The relative merits of these methods are unknown because the true viscoelastic work in an experiment is unknown. In order to characterize the accuracy of these methods, a theoretical model of crack propagation of viscoelastic soft tissue in a tensile test is presented, from which the exact viscoelastic work is calculated. The material is assumed to obey the standard linear solid model.The "exact" solution for the viscoelastic work during the fracture is computed from the model and compared with the work estimated by the two methods. It was found that both methods tend to underestimate the viscoelastic work done, and thus overestimate the fracture work and fracture toughness, although the errors were greater with the Fedewa method. It was further found that low displacement rates can give rise to a "snap" effect, where rapid crack growth can cause a disproportionate amount of viscoelastic energy to be dissipated during unloading. This modeling approach may be useful in evaluating other experimental methods of soft tissue fracture.     

Simulation of quadrupedal locomotion using a rigid body model

Locomotion of the horse is simulated using a mathematical model based on rigid body dynamics. A general method to generate the equations of motion for a two-dimensional rigid body model with an arbitrary number of hinge joints is presented and a numerical solution method, restricted to tree-structured models, is described. Joint movements originating from muscular forces or moments are simulated, but the method also allows that parts of the model follow strictly the pattern of kinematic data. Moment-generators with first-order linear feedback were used as a rotational muscle-equivalent. Ground-hoof interaction forces are approximated by a viscoelastic model and pseudo-Coulomb friction in vertical and horizontal directions respectively. Results of model simulations are compared to experimentally recorded data. Subsequently, adjustments are made to improve the agreement between simulation and experimental results.     

Cross-bridge elasticity in single smooth muscle cells

In smooth muscle, a cross-bridge mechanism is believed to be responsible for active force generation and fiber shortening. In the present studies, the viscoelastic and kinetic properties of the cross-bridge were probed by eliciting tension transients in response to small, rapid, step length changes (delta L = 0.3-1.0% Lcell in 2 ms). Tension transients were obtained in a single smooth muscle cell isolated from the toad (Bufo marinus) stomach muscularis, which was tied between a force transducer and a displacement device. To record the transients, which were of extremely small magnitude (0.1 microN), a high-frequency (400 Hz), ultrasensitive force transducer (18 mV/microN) was designed and built. The transients obtained during maximal force generation (Fmax = 2.26 microN) were characterized by a linear elastic response (Emax = 1.26 X 10(4) mN/mm2) coincident with the length step, which was followed by a biphasic tension recovery made up of two exponentials (tau fast = 5-20 ms, tau slow = 50-300 ms). During the development of force upon activation, transients were elicited. The relationship between stiffness and force was linear, which suggests that the transients originate within the cross-bridge and reflect the cross-bridge's viscoelastic and kinetic properties. The observed fiber elasticity suggests that the smooth muscle cross-bridge is considerably more compliant than in fast striated muscle. A thermodynamic model is presented that allows for an analysis of the factors contributing to the increased compliance of the smooth muscle cross-bridge.     

Viscoelastic behavior of organic materials: consequences of a logarithmic dependence of force on strain rate

The viscoelastic and -plastic behavior of organic materials like bone, tendon or wood, as well as technical polymers, is amply documented. It is usually modeled using linear "Newtonian" friction, i.e., a viscous force proportional to the deformation rate. If the experimental results cannot be fitted with the resulting exponential "Debye" curves, a multitude of relaxation mechanisms or a spectrum of relaxation times is invoked. In this contribution experimental evidence is compiled which indicates that for polymers and organic materials a logarithmic dependence of the deformation force on the deformation rate is more appropriate. The corresponding equation of motion is solved in the quasi-static approximation and the solutions display just the typical deviations from the Debye behavior found experimentally, without any complications from multi-mechanism relaxation.     

Viscoelastic properties of a mixed culture biofilm from rheometer creep analysis

The mechanical properties of mixed culture biofilms were determined by creep analysis using an AR1000 rotating disk rheometer. The biofilms were grown directly on the rheometer disks which were rotated in a chemostat for 12 d. The resulting biofilms were heterogeneous and ranged from 35 microns to 50 microns in thickness. The creep curves were all viscoelastic in nature. The close agreement between stress and strain ratio of a sample tested at 0.1 and 0.5 Pa suggested that the biofilms were tested in the linear viscoelastic range and supported the use of linear viscoelastic theory in the development of a constitutive law. The experimental data was fit to a 4-element Burger spring and dashpot model. The shear modulus (G) ranged from 0.2 to 24 Pa and the viscous coefficient (eta) from 10 to 3000 Pa. These values were in the same range as those previously estimated from fluid shear deformation of biofilms in flow cells. A viscoelastic biofilm model will help to predict shear related biofilm phenomena such as elevated pressure drop, detachment, and the flow of biofilms over solid surfaces.     

Transformation of the viscoelastic functions of calcified tissues and interfacial biomaterials into a common representation

Since both connective and calcified tissues are markedly viscoelastic in nature, an understanding of the behavior of these tissues intrinsically as materials on their own, as well as in composite formation with synthetic implants, is of prime importance in order to predict and anticipate materials' design and function. Thus considerable interest has developed in recent years with respect to measurements of the viscoelastic properties of biological materials. However, attempts to characterize the viscoelasticity of calcified tissues have involved many different experimental procedures; hence results appear in terms of different functions, e.g. relaxation modulus, creep compliance. Since this diversity precludes a simple useful comparison of the results, the present study was initiated so that measured functions could be cast into a common representation, and thus compared. Linear viscoelasticity theory implies definite exact relationships between the functions. Using these relations, experimental results on bone, dentin and implant materials presently used to interface to the natural tissues, e.g. polymethyl methacrylate and high density linear polyethylene, were transformed into the complex dynamic modulus representation. Analysis shows that the results of experiments on bone are not in agreement as to dispersion (i.e. change of modulus with frequency) and its variation with strain. Further, analysis of the internal consistency of some experiments demonstrates a violation of the Boltzmann integral which indicates that linear viscoelasticity (almost invariably assumed by workers in the field) fails for bone in compression. It is concluded that the dynamic behavior of bone is not as well understood as has been thought heretofore; direction is given for future experiments. Contribution No. 59 from the Laboratory for Crystallographic Biophysics; supported by USPHS through NIDR Grant Number 5T1-DE-117-10.     

A new device for measuring the viscoelastic properties of hydrated matrix gels

Determinations of the viscoelastic properties of extracellular matrices (ECMs) are becoming increasingly important for accurate predictive modeling of biological systems. Since the interactions of the cells with the ECM and surrounding fluid (e.g., blood, media) each affect cell behavior; it is advantageous to evaluate the ECM's material properties in the presence of the hydrating fluid. Conventional rheometry methods evaluate the bulk material properties of gel materials while displacing the hydrating liquid film. Such systems are therefore nonideal for testing materials such as ECMs, whose properties change with dehydration. The new patent pending, piezoelectrically actuated linear rheometer is designed to eliminate this problem. It uses a single cantilever to apply an oscillating load to the gel and to sense the gel's deflection. Composed of two thin film piezopolymer layers, the cantilever uses one layer as the actuator, and the second piezopolymer layer to measure the lateral movement of its attached probe. The viscoelastic nature of the ECM adds stiffness and damping to the system, resulting in the attenuation and phase shift of the sensor's output voltage. From these parameters, the ECM's shear storage and loss moduli are then determined. Initial tests on the BioMatrix I and type I collagen ECMs reveal that the first prototype of the piezoelectrically actuated linear rheometer is capable of accurately determining the trend and order of magnitude of an ECM's viscoelastic properties. In this paper, details of the rheometer's design and operating principles are described.     

Interrelation of creep and relaxation: a modeling approach for ligaments

Experimental data (Thornton et al., 1997) show that relaxation proceeds more rapidly (a greater slope on a log-log scale) than creep in ligament, a fact not explained by linear viscoelasticity. An interrelation between creep and relaxation is therefore developed for ligaments based on a single-integral nonlinear superposition model. This interrelation differs from the convolution relation obtained by Laplace transforms for linear materials. We demonstrate via continuum concepts of nonlinear viscoelasticity that such a difference in rate between creep and relaxation phenomenologically occurs when the nonlinearity is of a strain-stiffening type, i.e., the stress-strain curve is concave up as observed in ligament. We also show that it is inconsistent to assume a Fung-type constitutive law (Fung, 1972) for both creep and relaxation. Using the published data of Thornton et al. (1997), the nonlinear interrelation developed herein predicts creep behavior from relaxation data well (R > or = 0.998). Although data are limited and the causal mechanisms associated with viscoelastic tissue behavior are complex, continuum concepts demonstrated here appear capable of interrelating creep and relaxation with fidelity.     

A new method for measuring the yield stress in thin layers of sedimenting blood.

A new method is presented to describe the low shear rate behavior of blood. We observed the response of a thin layer of sedimenting blood to a graded shear stress in a wedge-shaped chamber. The method allows quantitation of the degree of phase separation between red cells and plasma, and extracts the yield stress of the cell phase as a function of hematocrit. Our studies showed that the behavior of normal human blood underwent a transition from a solid-like gel to a Casson fluid. This transition began at the Casson predicted yield stress. The viscoelastic properties of blood were examined at shear stresses below the yield stress. The measured Young's elastic moduli were in good agreement with published data. The yield stress of blood showed a linear dependence on hematocrit up to 60%, and increased more rapidly at higher hematocrit.