Richard S. Chadwick

Determination of Elastic Moduli of Thin Layers of Soft Material Using the Atomic Force Microscope

We address three problems that limit the use of the atomic force microscope when measuring elastic moduli of soft materials at microscopic scales. The first concerns the use of sharp cantilever tips, which typically induce local strains that far exceed the linear material regime. We show that this problem can be alleviated by using microspheres as probes, and we establish the criteria for their use. The second relates to the common use of the Hertz contact mechanics model, which leads to significant errors when applied to thin samples. We develop novel, simple to use corrections to apply for such cases. Samples that are either bonded or not bonded to a rigid substrate are considered. The third problem concerns the difficulty in establishing when contact occurs on a soft material. We obtain error estimates for the elastic modulus resulting from such uncertainty and discuss the sensitivity of the estimation methods to error in contact point. The theoretical and experimental results are compared to macroscopic measurements on poly(vinyl-alcohol) gels.

Nonpolarized signaling reveals two distinct modes of 3D cell migration

The elastic behavior of the 3D extracellular matrix determines the relative polarization of intracellular signaling and whether cells migrate using lamellipodia or lobopodia.

Focal delivery during direct infusion to brain: role of flow rate, catheter diameter, and tissue mechanics

Direct interstitial infusion is a technique capable of delivering agents over both small and large dimensions of brain tissue. However, at a sufficiently high volumetric inflow rate, backflow along the catheter shaft may occur and compromise delivery. A scaling relationship for the finite backflow distance along this catheter in pure gray matter (x(m)) has been determined from a mathematical model based on Stokes flow, Darcy flow in porous media, and elastic deformation of the brain tissue: x(m) = constant Q(o)(3)R(4)r(c)(4)G(-3)mu(-1) 1/5 [corrected] = volumetric inflow rate, R = tissue hydraulic resistance, r(c) = catheter radius, G = shear modulus, and mu = viscosity). This implies that backflow is minimized by the use of small diameter catheters and that a fixed (minimal) backflow distance may be maintained by offsetting an increase in flow rate with a similar decrease in catheter radius. Generally, backflow is avoided in rat gray matter with a 32-gauge catheter operating below 0.5 microliter/min. An extension of the scaling relationship to include brain size in the resistance term leads to the finding that absolute backflow distance obtained with a given catheter and inflow rate is weakly affected by the depth of catheter tip placement and, thus, brain size. Finally, an extension of the model to describe catheter passage through a white matter layer before terminating in the gray has been shown to account for observed percentages of albumin in the corpus callosum after a 4-microliter infusion of the compound to rat striatum over a range of volumetric inflow rates.

A Biophysical Model of Cochlear Processing: Intensity Dependence of Pure Tone Responses.

A mathematical model of cochlear processing is developed to account for the nonlinear dependence of frequency selectivity on intensity in inner hair cell and auditory nerve fiber responses. The model describes the transformation from acoustic stimulus to intracellular hair cell potentials in the cochlea. It incorporates a linear formulation of basilar membrane mechanics and subtectorial fluid-cilia displacement coupling, and a simplified description of the inner hair cell nonlinear transduction process. The analysis at this stage is restricted to low-frequency single tones. The computed responses to single tone inputs exhibit the experimentally observed nonlinear effects of increasing intensity such as the increase in the bandwidth of frequency selectivity and the downward shift of the best frequency. In the model, the first effect is primarily due to the saturating effect of the hair cell nonlinearity. The second results from the combined effects of both the nonlinearity and of the inner hair cell low-pass transfer function. In contrast to these shifts along the frequency axis, the model does not exhibit intensity dependent shifts of the spatial location along the cochlea of the peak response for a given single tone. The observed shifts therefore do not contradict an intensity invariant tonotopic code.

The influence of cochlear shape on low-frequency hearing

The conventional theory about the snail shell shape of the mammalian cochlea is that it evolved essentially and perhaps solely to conserve space inside the skull. Recently, a theory proposed that the spirals graded curvature enhances the cochleas mechanical response to low frequencies. This article provides a multispecies analysis of cochlear shape to test this theory and demonstrates that the ratio of the radii of curvature from the outermost and innermost turns of the cochlear spiral is a significant cochlear feature that correlates strongly with low-frequency hearing limits. The ratio, which is a measure of curvature gradient, is a reflection of the ability of cochlear curvature to focus acoustic energy at the outer wall of the cochlear canal as the wave propagates toward the apex of the cochlea.

Effects of collagen microstructure on the mechanics of the left ventricle

The microstructure of the collagen sheath or weave surrounding a myocyte and the collagen struts interconnecting neighboring myocytes is incorporated into a fluid-fiber-collagen continuum description of the myocardium. The sheaths contribute to anisotropic elasticity, whereas the struts contribute to an isotropic component. Elastic moduli of the composite myocyte-sheath complex and the strut matrix are estimated from existing passive biaxial loading data from sheets of canine myocardium. The contribution of the sheath to the elasticity of the myocyte-sheath complex is critically dependent on the helical pitch angle. Calculations for a cylindrical model of the left ventricle using both a fluid-fiber and fluid-fiber-collagen stress tensor show that the collagen strut matrix tends to limit muscle fiber lengthening; increase myocardial tissue pressure during systole, with endocardial tissue pressure exceeding left ventricular pressure; decrease tissue pressure during diastole, and thus facilitate myocardial blood flow; and aid filling during ventricular relaxation by providing a suction effect that relies on a release of stored elastic energy from the previous contraction. Calculations show that this energy is stored mostly in the collagen struts.

AXISYMMETRIC INDENTATION OF A THIN INCOMPRESSIBLE ELASTIC LAYER

Axisymmetric indentation of an incompressible elastic layer by a frictionless rigid sphere is considered. Estimates of the contact radius and the force required to produce a given indentation are given when the thickness of the layer is small compared to the contact radius. The elastic layer is either bonded to or slips along a rigid substrate. The estimates are obtained by asymptotically matching a lubrication-type expansion valid in the contact region to an edge layer expansion studied using the Wiener--Hopf technique. A substantially larger force is required to equally indent a bonded layer compared to a slipping layer. In the former case the force is inversely proportional to the cube of the layer thickness, rather than to the layer thickness itself, if the layer is free to slip.

Effects of coiling on the micromechanics of the mammalian cochlea

The cochlea transduces sound-induced vibrations in the inner ear into electrical signals in the auditory nerve via complex fluid–structure interactions. The mammalian cochlea is a spiral-shaped organ, which is often uncoiled for cochlear modelling. In those few studies where coiling has been considered, the cochlear partition was often reduced to the basilar membrane only. Here, we extend our recently developed hybrid analytical/numerical micromechanics model to include curvature effects, which were previously ignored. We also use a realistic cross-section geometry, including the tectorial membrane and cellular structures of the organ of Corti, to model the apical and basal regions of a guinea-pig cochlea. We formulate the governing equations of the fluid and solid domains in a curvilinear coordinate system. The WKB perturbation method is used to treat the propagation of travelling waves along the coiled cochlear duct, and the O(1) system of the governing equations is solved in the transverse plane using finite-element analysis. We find that the curvature of the cochlear geometry has an important functional significance; at the apex, it greatly increases the shear gain of the cochlear partition, which is a measure of the bending efficiency of the outer hair cell stereocilia.

Noncontact microrheology at acoustic frequencies using frequency-modulated atomic force microscopy.

We report an atomic force microscopy (AFM) method for assessing elastic and viscous properties of soft samples at acoustic frequencies under non-contact conditions. The method can be used to measure material properties via frequency modulation and is based on hydrodynamics theory of thin gaps we developed here. A cantilever with an attached microsphere is forced to oscillate tens of nanometers above a sample. The elastic modulus and viscosity of the sample are estimated by measuring the frequency-dependence of the phase lag between the oscillating microsphere and the driving piezo at various heights above the sample. This method features an effective area of pyramidal tips used in contact AFM but with only piconewton applied forces. Using this method, we analyzed polyacrylamide gels of different stiffness and assessed graded mechanical properties of guinea pig tectorial membrane. The technique enables the study of microrheology of biological tissues that produce or detect sound.

Influence of sickle hemoglobin polymerization and membrane properties on deformability of sickle erythrocytes in the microcirculation.

The rheological properties of normal erythrocytes appear to be largely determined by those of the red cell membrane. In sickle cell disease, the intracellular polymerization of sickle hemoglobin upon deoxygenation leads to a marked increase in intracellular viscosity and elastic stiffness as well as having indirect effects on the cell membrane. To estimate the components of abnormal cell rheology due to the polymerization process and that due to the membrane abnormalities, we have developed a simple mathematical model of whole cell deformability in narrow vessels. This model uses hydrodynamic lubrication theory to describe the pulsatile flow in the gap between a cell and the vessel wall. The interior of the cell is modeled as a Voigt viscoelastic solid with parameters for the viscous and elastic moduli, while the membrane is assigned an elastic shear modulus. In response to an oscillatory fluid shear stress, the cell--modeled as a cylinder of constant volume and surface area--undergoes a conical deformation which may be calculated. We use published values of normal and sickle cell membrane elastic modulus and of sickle hemoglobin viscous and elastic moduli as a function of oxygen saturation, to estimate normalized tip displacement, d/ho, and relative hydrodynamic resistance, Rr, as a function of polymer fraction of hemoglobin for sickle erythrocytes. These results show the transition from membrane to internal polymer dominance of deformability as oxygen saturation is lowered. More detailed experimental data, including those at other oscillatory frequencies and for cells with higher concentrations of hemoglobin S, are needed to apply fully this approach to understanding the deformability of sickle erythrocytes in the microcirculation. The model should be useful for reconciling the vast and disparate sets of data available on the abnormal properties of sickle cell hemoglobin and sickle erythrocyte membranes, the two main factors that lead to pathology in patients with this disease.