Frequency-modulated atomic force microscopy localises viscoelastic remodelling in the ageing sheep aorta.

Age-related aortic stiffening is associated with cardiovascular diseases such as heart failure. The mechanical functions of the main structural components of the aorta, such as collagen and elastin, are determined in part by their organisation at the micrometer length scale. With age and disease both components undergo aberrant remodelling, hence, there is a need for accurate characterisation of the biomechanical properties at this length scale. In this study we used a frequency-modulated atomic force microscopy (FM-AFM) technique on a model of ageing in female sheep aorta (young: ~18 months, old: >8 years) to measure the micromechanical properties of the medial layer of the ascending aorta. The novelty of our FM-AFM method, operated at 30kHz, is that it is non-contact and can be performed on a conventional AFM using the ×³cantilever tune' mode, with a spatial (areal) resolution of around 1.6µm(2). We found significant changes in the elastic and viscoelastic properties within the medial lamellar unit (elastic lamellae and adjacent inter-lamellar space) with age. In particular, there was an increase in elastic modulus (Young; geometric mean (geometric SD)=42.9 (2.26)kPa, Old=113.9 (2.57)kPa, P<0.0001), G' and G″ (storage and loss modulus respectively) (Young; G'=14.3 (2.26)kPa, Old G'=38.0 (2.57)kPa, P<0.0001; Young; G″=14.5 (2.56)kPa, Old G″=32.8 (2.52)kPa, P<0.0001). The trends observed in the elastic properties with FM-AFM matched those we have previously found using scanning acoustic microscopy (SAM). The utility of the FM-AFM method is that it does not require custom AFM hardware and can be used to simultaneously determine the elastic and viscoelastic behaviour of a biological sample.

Cochlea's graded curvature effect on low frequency waves.

In the ear, sound waves are processed by a membrane of graded mechanical properties that resides in the fluid-filled spiral cochlea. The role of stiffness grading as a Fourier analyzer is well known, but the role of the curvature has remained elusive. Here, we report that increasing curvature redistributes wave energy density towards the cochlea's outer wall, affecting the shape of waves propagating on the membrane, particularly in the region where low frequency sounds are processed.

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.

Solution of the inverse problem for a linear cochlear model: a tonotopic cochlear amplifier.

The extraordinary fine-tuning characteristic of normal mammalian hearing is attributed to physiological mechanisms collectively known as the cochlear amplifier (CA), which amplifies and sharpens the basilar membrane (BM) vibration response to incoming acoustic pressure oscillations. Electromechanical properties of outer hair cells (OHCs) are believed to be the critical component of the CA, but its "circuitry" as yet remains unknown. Here, the required frequency-space response characteristics of the CA are computationally determined when typical in vivo tuning data are introduced as input to a linear hydroelastic cochlear model whose cross-sectional dynamics are represented by two coupled vibrational degrees of freedom. It is assumed that the CA senses motion at the tectorial membrane (TM) reticular lamina (RL) and applies proportional, equal, and opposite forces to the BM and the RL. The results show the CA to be tonotopically tuned, meaning it conforms to a space-frequency similarity principle like other cochlear dynamical responses. This requires that the active mechanism use information distributed along the cochlear partition. The physiological mechanism responsible for such behavior remains unknown, but here the computed CA characteristics can be qualitatively reproduced by a circuit spanning the length of the cochlea. This does not preclude other mechanisms, but is intended to motivate closer experimental investigation of extracellular and intercellular ionic flow pathways.

Compression, gain, and nonlinear distortion in an active cochlear model with subpartitions.

The propagation of inhomogeneous, weakly nonlinear waves is considered in a cochlear model having two degrees of freedom that represent the transverse motions of the tectorial and basilar membranes within the organ of Corti. It is assumed that nonlinearity arises from the saturation of outer hair cell active force generation. I use multiple scale asymptotics and treat nonlinearity as a correction to a linear hydroelastic wave. The resulting theory is used to explain experimentally observed features of the response of the cochlear partition to a pure tone, including: the amplification of the response in a healthy cochlea vs a dead one; the less than linear growth rate of the response to increasing sound pressure level; and the amount of distortion to be expected at high and low frequencies at basal and apical locations, respectively. I also show that the outer hair cell nonlinearity generates retrograde waves.

Active control of waves in a cochlear model with subpartitions.

Multiscale asymptotic methods developed previously to study macromechanical wave propagation in cochlear models are generalized here to include active control of a cochlear partition having three subpartitions, the basilar membrane, the reticular lamina, and the tectorial membrane. Activation of outer hair cells by stereocilia displacement and/or by lateral wall stretching result in a frequency-dependent force acting between the reticular lamina and basilar membrane. Wavelength-dependent fluid loads are estimated by using the unsteady Stokes' equations, except in the narrow gap between the tectorial membrane and reticular lamina, where lubrication theory is appropriate. The local wavenumber and subpartition amplitude ratios are determined from the zeroth order equations of motion. A solvability relation for the first order equations of motion determines the subpartition amplitudes. The main findings are as follows: The reticular lamina and tectorial membrane move in unison with essentially no squeezing of the gap; an active force level consistent with measurements on isolated outer hair cells can provide a 35-dB amplification and sharpening of subpartition waveforms by delaying dissipation and allowing a greater structural resonance to occur before the wave is cut off; however, previously postulated activity mechanisms for single partition models cannot achieve sharp enough tuning in subpartitioned models.

Elasticity and active force generation of cochlear outer hair cells.

The cochlear outer hair cell is described by a cylindrical membrane model, characterized by area and shear moduli for a passive elastic element and an active tension element dependent on the membrane potential. In passive experiments, these moduli are determined from the pressure-strain relations. The area modulus obtained is 0.07 N m-1, similar to a lipid bilayer and the shear modulus is 0.007 N m-1. These moduli combined with previous active experiments show that the active tension is nearly isotropic and is about 1.6 x 10(-2) N m-1 V-1, resulting in a 0.5 nN/mV force per cell. This implies that the receptor potential for acoustical stimulation produces an active force comparable to the acoustic force applied to the basilar membrane per outer hair cell. This finding supports the hypothesis that the outer hair cell acts as feedback motor in the fine tuning mechanism of the mammalian ear.

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.

Prediction of the cardiac muscle force-velocity relation from its force-time and force-length relations.

The force-velocity relation for cardiac muscle fibers can be calculated from a proposed constitutive law based on force-time and force-length data. The calculated force-velocity relation agrees quite well with the measured force-velocity relation obtained from a quick release of sarcomere controlled rat cardiac trabeculae. The theory confirms the measured linear relationship between maximal velocity of sarcomere shortening and sarcomere length. The implication is that the force-velocity relation is not an independent property, and therefore need not be explicitly included as a rheological element in the constitutive law.

Phasic regional myocardial inflow and outflow: comparison of theory and experiments.

We developed a theory for regional blood flow in the beating heart and validated it with measurements of coronary arterial inflow and venous outflow in the open-chest anesthetized dog. The model used measured aortic, left ventricular, and coronary sinus pressures as input data under control conditions and during long diastoles induced by vagal stimulation. A nonlinear two-compartment lumped model for each transmural layer was obtained by spatial averaging a continuum description of the myocardial microcirculation based on morphometric measurements and appropriate fluid and vascular mechanics principles. The chief results and conclusions of the study are 1) an intramyocardial time constant on the order of 1 s is required to explain the phase opposition between inflow and outflow; 2) capillary and venous perfusion are in phase with arterial pressure, and arterial flow is out of phase with arterial pressure except in superficial intramural layers; 3) subendocardial retrograde systolic flow increases with increased contractility and time constants and decreased arterial pressure; and 4) endocardial capillary and venule volume change by 5.5 and 10%, respectively, during the control cardiac cycle.

On the uniqueness of quasi-static solutions of some linear models of left ventricular mechanics.

We review two models describing the material properties of heart muscle: the fluid-fiber model and the fluid-fiber-collagen model. We show that the fluid-fiber description gives rise to non-uniqueness when used in ventricular modeling while the fluid-fiber-collagen description does not. We derive a general cavity pressure-volume relation for an extended class family of linear models of the heart's left ventricle.

Contraction and relaxation-induced oscillations of the left ventricle of the heart during the isovolumic phases.

A theoretical analysis is presented for the transient dynamical response of the left ventricle of the heart during the isovolumic contraction and relaxation phases of the cardiac cycle. Small oscillations of the left ventricular cavity pressure and wall motion are excited by the initial rates of filling and emptying of the ventricle as well as the rate of change in muscle fiber activation. The analysis applies to the genesis of the first and second heart sounds. The ventricle is modeled as a finite, thick-walled incompressible cylinder having a continuum of imbedded axial and circumferential active muscle fibers, which interacts with a fixed volume of an incompressible, ideal fluid. The solution is obtained using a two-timing asymptotic expansion procedure. The theoretical calculations of left ventricular pressure waveforms compare favorably with published recorded pressure waveforms. The amplitude spectra of computed waveforms contain information concerning the active elastic modulus of the fibers, which is a measure of cardiac contractility.

Wall-thickness and midwall-radius variations in ventricular mechanics.

A fluid-fiber-collagen stress tensor is used to describe the rheology of the left ventricle of the heart. Linear theory is used to find the equilibrium solutions for the end-diastolic and end-systolic states of general axisymmetric shapes that are small perturbations of a thick-walled finite cylinder. The general problem can be studied by superposing the effects of variable midwall radius but constant wall thickness with those of variable wall thickness but constant midwall radius. A Fourier series representation is used to describe the midwall radius and thickness functions. Numerical calculations are performed to determine the deformed geometry and spatial distributions of tissue pressure, stresses, and fiber strains. The calculations proved to be highly accurate when compared to an analytical solution obtained for the special case of no fibers. The results show significant longitudinal differences when compared to results for the cylindrical geometry, with more sensitivity to variation in wall thickness than to variation in midwall radius.

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.

Theoretical analysis of the effects of a radial activation wave and twisting motion on the mechanics of the left ventricle.

The mechanical effects resulting from the normal transmural delay of electrical depolarization of the myocardium are investigated. An activation sequence having a finite radial propagation velocity is introduced into the equations of ventricular mechanics. The resulting system of coupled integral equations is solved using a perturbation method based on the small ratio of transmural propagation time to cardiac period. Numerical calculations are performed using cavity pressure and volume waveforms characteristic of the canine left ventricle (LV), for both simultaneous and delayed activation of fiber layers. The results show that a finite transmural electrical propagation velocity tends to: (i) equalize the transmural distribution of sarcomere length during systole; (ii) equalize the transmural distribution of fiber external work/vol; and (iii) insignificantly affect myocardial tissue pressure. Calculations are also performed to investigate the mechanical effects resulting from the application of an externally applied moment that prevents LV torsion. Those results are highly dependent on the transmural distribution of sarcomere length in the stress-free reference state (unloaded diastole). When we assume a uniform distribution, then normal torsion acting with normal activation delay tends to: (i) increase the magnitude of fiber strain in the subendocardium and decrease it in the subepicardium; (ii) equalize the transmural distribution of fiber external work/vol; and (iii) lower myocardial tissue pressure. The normally occurring transmural delay of activation tends to lessen endocardial O2 demand, while the normally occurring torsion further lessens that demand and improves O2 supply.

Pulse-wave model of brachial arterial pressure modulation in aging and hypertension.

We apply a pulse-wave theory to a model of the human arm arterial system that predicts the changes in the arterial pressure waveform as it traverses the vasculature (increased pulse pressure, sharper main wave, disappearance of the aortic incisura, and appearance of a diastolic dicrotic wave) and also predicts the observed modulation of the waveform during phenylephrine-induced vasoconstriction and nitroglycerin-induced vasodilation. The model considers the arm arterial system as a tapered, distensible tube that bifurcates and then ends in a loop, with sidebranch networks represented by distributed Windkessels. The model uses verifiable values for realistic parameters. We found that the vertical modulation of the dicrotic wave, in humans, decreased with advancing age and with high blood pressure. The model explained these findings in terms of increasing vascular rigidity and decreasing small vessel vasodilator responsiveness. We noted a significant negative correlation between the arterial level of plasma norepinephrine and the amount of modulation of the dicrotic wave after nitroglycerin among subjects 40 yr old or younger, suggesting a sympathetic neurogenic contribution to the vascular abnormalities observed in relatively young patients with essential hypertension.

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.

Pulse-wave propagation in an artery with leakage into small side branches.

Linear pulse propagation theory is applied to a network of elastic tubes. The network is specialized to one having a main branch with many low-admittance side branches. Continuum approximations and an asymptotic analysis of side-branch network input admittance reduce the original network to a tapered leaky tube. The outflow law is equivalent to distributed Windkessels. The conductance and time constant of the Windkessels are related to the geometry of the side-branch networks. Viscous effects are included in the theory. Asymptotic methods are used to determine the dispersion relation and waveform amplitude in the frequency domain. Periodic pressure and flow waveforms are computed using Fourier synthesis.