The effect of fibre cell remodelling on the power and optical quality of the lens

Vertebrate eye lenses are uniquely adapted to form a refractive index gradient (GRIN) for improved acuity, and to grow slowly in size despite constant cell proliferation. The mechanisms behind these adaptations remain poorly understood. We hypothesize that cell compaction contributes to both. To test this notion, we examined the relationship between lens size and shape, refractive characteristics and the cross-sectional areas of constituent fibre cells in mice of different ages. We developed a block-face imaging method to visualize cellular cross sections and found that the cross-sectional areas of fibre cells rose and then decreased over time, with the most significant reduction occurring in denucleating cells in the adult lens cortex, followed by cells in the embryonic nucleus. These findings help reconcile differences between the predictions of lens growth models and empirical data. Biomechanical simulations suggested that compressive forces generated from continuous deposition of fibre cells could contribute to cellular compaction. However, optical measurements revealed that the GRIN did not mirror the pattern of cellular compaction, implying that compaction alone cannot account for GRIN formation and that additional mechanisms are likely to be involved.

Mechanical forces in cerebral cortical folding: a review of measurements and models

Folding of the cerebral cortical surface is a critical process in human brain development, yet despite decades of indirect study and speculation the mechanics of the process remain incompletely understood. Leading hypotheses have focused on the roles of circumferential expansion of the cortex, radial growth, and internal tension in neuronal fibers (axons). In this article, we review advances in the mathematical modeling of growth and morphogenesis and new experimental data, which together promise to clarify the mechanical basis of cortical folding. Recent experimental studies have illuminated not only the fundamental cellular and molecular processes underlying cortical development, but also the stress state and mechanical behavior of the developing brain. The combination of mathematical modeling and biomechanical data provides a means to evaluate hypothesized mechanisms objectively and quantitatively, and to ensure that they are consistent with physical law, given plausible assumptions and reasonable parameter values.

A cortical folding model incorporating stress-dependent growth explains gyral wavelengths and stress patterns in the developing brain

In humans and many other mammals, the cortex (the outer layer of the brain) folds during development. The mechanics of folding are not well understood; leading explanations are either incomplete or at odds with physical measurements. We propose a mathematical model in which (i) folding is driven by tangential expansion of the cortex and (ii) deeper layers grow in response to the resulting stress. In this model the wavelength of cortical folds depends predictably on the rate of cortical growth relative to the rate of stress-induced growth. We show analytically and in simulations that faster cortical expansion leads to shorter gyral wavelengths; slower cortical expansion leads to long wavelengths or even smooth (lissencephalic) surfaces. No inner or outer (skull) constraint is needed to produce folding, but initial shape and mechanical heterogeneity influence the final shape. The proposed model predicts patterns of stress in the tissue that are consistent with experimental observations.

Perspectives on biological growth and remodeling

The continuum mechanical treatment of biological growth and remodeling has attracted considerable attention over the past fifteen years. Many aspects of these problems are now well-understood, yet there remain areas in need of significant development from the standpoint of experiments, theory, and computation. In this perspective paper we review the state of the field and highlight open questions, challenges, and avenues for further development.

A 2-D model of flow-induced alterations in the geometry, structure, and properties of carotid arteries

Evidence from diverse investigations suggests that arterial growth and remodeling correlates well with changes in mechanical stresses from their homeostatic values. Ultimately, therefore, there is a need for a comprehensive theory that accounts for changes in the 3-D distribution of stress within the arterial wall, including residual stress, and its relation to the mechanisms of mechanotransduction. Here, however, we consider a simpler theory that allows competing hypotheses to be tested easily, that can provide guidance in the development of a 3-D theory, and that may be useful in modeling solid-fluid interactions and interpreting clinical data. Specifically, we present a 2-D constrained mixture model for the adaptation of a cylindrical artery in response to a sustained alteration in flow. Using a rule-of-mixtures model for the stress response and first order kinetics for the production and removal of the three primary load-bearing constituents within the wall, we illustrate capabilities of the model by comparing responses given complete versus negligible turnover of elastin. Findings suggest that biological constraints may result in suboptimal adaptations, consistent with reported observations. To build upon this finding, however, there is a need for significantly more data to guide the hypothesis testing as well as the formulation of specific constitutive relations within the model.

Theoretical and experimental study of growth and remodeling in the developing heart

A theoretical model is presented for growth and remodeling in the developing embryonic heart. The model is a thick-walled tube composed of two layers of orthotropic pseudoelastic material. The analysis includes stress and strain dependent volumetric growth, with changes in material properties specified to reflect the evolving structure of the heart wall. For use in model validation, experimental measurements of ventricular opening angles are reported for 3-4-day old chick embryos under control and pressure overload conditions. Owing to changes in residual stress in the overloaded heart, the opening angle decreased from 31 +/- 10 degrees to -8 +/- 12 degrees (mean +/- SD) within 12 h and then increased slightly. The opening angle at 12 h was significantly less than the control value. With an appropriate choice of parameters, the model yields reasonable agreement with these and other published opening angle data, as well as with temporal changes in lumen radius, wall thickness, epicardial strains, and pressure-volume curves during development before and after birth.

Approach to quantify the mechanical behavior of the intact embryonic chick heart

The heart undergoes remarkable changes during embryonic development due to genetic programing and epigenetic influences such as mechanical loads. An important goal is to develop mathematical models that describe and predict the influence of mechanics on cardiac development, yet the data needed to develop such models remain scant. In particular, prior data from embryonic hearts have come from one-dimensional tests, which reveal well the general characteristic behaviors but are not sufficient for quantifying the complex material behavior exhibited by most soft tissues. We developed a computer-controlled system for quantifying in vitro the multiaxial, regionally dependent mechanical response of the intact embryonic chick heart to controlled distension pressures; such tests have not been accomplished heretofore but promise to contribute to our understanding for they better mimic the native loading conditions and geometry. Calibration of the device indicated that distending the hearts using low flow rates avoids significant viscous losses and thereby allows pressure to be measured directly over small ranges (0-2 mmHg) with good resolution (0.01 mm Hg). Likewise, an on-line video system allows two-dimensional strains to be measured regionally by tracking the motions of triplets of closely spaced (100 microm) microspheres to reasonable resolution (2.5 microm/pixel). Illustrative data from 18 hearts show the utility of the new system, confirm previous findings with regard to the strong viscoelastic response of the stage 18 embryonic chick heart, and provide guidance for the design of future experiments.

Biomechanics of cardiovascular development

It long has been known that mechanical forces play a role in the development of the cardiovascular system, but only recently have biomechanical engineers begun to explore this field. This paper reviews some of this work. First, an overview of the relevant biology is discussed. Next, a mechanical theory is presented that can be used to model developmental processes. The theory includes the effects of finite volumetric growth and active contractile forces. Finally, applications of this and other theories to problems of cardiovascular development are discussed, and some future directions are suggested. The intent is to stimulate further interest among engineers in this important area of research.

Stress-modulated growth, residual stress, and vascular heterogeneity

A simple phenomenological model is used to study interrelations between material properties, growth-induced residual stresses, and opening angles in arteries. The artery is assumed to be a thick-walled tube composed of an orthotropic pseudoelastic material. In addition, the normal mature vessel is assumed to have uniform circumferential wall stress, which is achieved here via a mechanical growth law. Residual stresses are computed for three configurations: the unloaded intact artery, the artery after a single transmural cut, and the inner and outer rings of the artery created by combined radial and circumferential cuts. The results show that the magnitudes of the opening angles depend strongly on the heterogeneity of the material properties of the vessel wall and that multiple radial and circumferential cuts may be needed to relieve all residual stress. In addition, comparing computed opening angles with published experimental data for the bovine carotid artery suggests that the material properties change continuously across the vessel wall and that stress, not strain, correlates well with growth in arteries.

Theoretical model for myocardial trabeculation

During the morphogenetic process of myocardial trabeculation, most of the cardiac jelly of the initially smooth-walled heart is replaced by sponge-like muscle. The mechanisms that drive and regulate this important process are poorly understood. Using a theoretical model, we examined the possible role that cytoskeletal contraction plays during the initial stages of trabeculation. The myocardium is modeled as a thin viscoelastic membrane consisting of contractile (stress) fibers embedded in an isotropic incompressible matrix, with the interaction of myocardial cells and cardiac jelly fibers providing long-range mechanical effects. The stress fibers are assumed to behave like smooth muscle and to normally operate on the descending limb of their stress-stretch curve. Mechanical instability due to the effectively negative stiffness then leads to the creation of pattern. As a first approximation, computations were carried out for a flat rectangular membrane with stress fibers aligned along a single direction. The computed deformation patterns depend strongly on the magnitude and anisotropy of the long-range effects. Given plausible assumptions about the mechanical properties of the embryonic heart, the model predicts trabecular patterns similar to those observed in the embryo, including the development of circumferential ridges and relatively thin regions ("holes") in the trabecular sheets.

Investigating Murray's law in the chick embryo

According to the optimization principle known as Murray's law, the blood vessel geometry at a bifurcation satisfies the relation alpha = (D3(1) + D3(2))/D3(0) = 1, where D0, D1, and D2 are the diameters of the parent and two daughter vessels, respectively. Previous investigations have shown that mature blood vessels adhere to this law fairly closely. The purpose of this study was to test Murray's law in the developing extraembryonic blood vessels of 2-4 day-old chick embryos. Vessel diameters were measured manually using image analysis software. The measurements for the group of all vessels at all studied stages (n = 449) gave alpha = 1.01+/-0.34 (mean +/- SD), and the value of alpha is similar at all stages. These results indicate that Murray's law holds in the chick embryo, even before medial smooth muscle becomes functional, suggesting that blood vessels follow the same basic morphogenetic rules throughout life.

Mechanical aspects of cardiac development

Heart development depends on a dynamic interaction between genetic and epigenetic factors. This paper discusses some of the biomechanical processes that help shape the heart in the embryo. First, an overview is given of some of the critical events that occur during cardiac development. Next, mechanics and modeling strategies are discussed for the morphogenetic processes of cardiac tube formation, cardiac looping, myocardial trabeculation, septation, valve formation, and muscle-fiber alignment. Finally, some considerations for future work in this area are listed.

Biomechanical growth laws for muscle tissue

It is generally accepted that growth of muscle tissue depends in part on biomechanical factors. However, the precise relationships that govern mechanically induced growth are not known. This paper uses available data to propose a set of biomechanical growth laws for striated and smooth muscle. For striated muscle fibers, transverse and longitudinal growth are hypothesized to depend on the active and passive fiber stress, respectively. For smooth muscle fibers in arteries, transverse growth is assumed to depend on the fiber stress (active behavior is ignored), with longitudinal growth depending on both fiber stress and the shear stress on the endothelium due to blood flow. In both types of muscle, the rate of growth is assumed to depend linearly on the stresses. Relatively simple models for skeletal muscle, the heart, and arteries are used to show that the proposed growth laws can predict many of the known characteristics of muscle growth during development and following load perturbations in the mature animal.

A model for aortic growth based on fluid shear and fiber stresses

Stress-modulated growth in the aorta is studied using a theoretical model. The model is a thick-walled tube composed of two pseudoelastic, orthotropic layers representing the intima/media and the adventitia. Both layers are assumed to follow a growth law in which the time rates of change of the growth stretch ratios depend linearly on the local smooth muscle fiber stress and on the shear stress due to blood flow on the endothelium. Using finite elasticity theory modified to include volumetric growth, we computed temporal changes in stress, geometry, and opening angle (residual strain) during development and following the onset of sudden hypertension. For appropriate values of the coefficients in the growth law, the model yields results in reasonable agreement with published data for global and local growth of the rat aorta.

An optimization principle for vascular radius including the effects of smooth muscle tone

An optimization principle is proposed for the regulation of vascular morphology. This principle, which extends Murray's law, is based on the hypothesis that blood vessel diameter is controlled by a mechanism that minimizes the total energy required to drive the blood flow, to maintain the blood supply, and to support smooth muscle tone. A theoretical analysis reveals that the proposed principle predicts that the optimum shear stress on the vessel wall due to blood flow increases with blood pressure. This result agrees qualitatively with published findings that the fluid shear stress in veins is significantly smaller than it is in arteries.

Passive stress-strain measurements in the stage-16 and stage-18 embryonic chick heart

The first stress-strain measurements on embryonic cardiovascular tissue are described here, obtained from cyclic uniaxial loading of the primitive ventricle. An excised ventricular segment from Hamburger/Hamilton stage-16 or stage-18 chicks (2-1/2 and 3 days of a 21-day incubation period) was mounted longitudinally between two small wires in oxygenated Krebs-Henseleit cardioplegia solution. One wire was attached to an ultrasensitive force transducer and the other to a Huxley micromanipulator controlled by remote motor drive. A real-time video tracking system calculated three myocardial surface strains based on the positions of three surface markers while the heart was deformed in a triangular wave pattern. Force transducer output was filtered, digitally sampled, and stored with strains and time. Results were plotted as strain (longitudinal, circumferential, shear, and principal) versus time, stress versus time, and stress versus longitudinal strain. The stress-strain curves were nonlinear, even at low strain levels. The hysteresis loops were large; mean hysteresis energy as a proportion of total cycle stored strain energy was 36 percent (stage 16) and 41 percent (stage 18). We created a finite element model of the ventricle and fit the model behavior to the experimental behavior to determine parameters for a stage-18 pseudoelastic strain-energy function of exponential form. The calculated exponential parameter is significantly lower than that found in corresponding uniaxial studies of mature myocardium, possibly indicating the lower fiber content of the immature tissue. The results of this study are the first step in characterizing material properties for comparisons with later developmental stages and with impaired and altered myocardium. The long-term goal is to aid in identifying the biomechanical factors regulating growth and morphogenesis.

Theoretical study of stress-modulated growth in the aorta

A theoretical model is presented for stress-modulated growth in the aorta. The model consists of a pseudoelastic tube composed of two layers representing the intima/media and the adventitia. Finite volumetric growth is included by letting the time-rate of change of the zero-stress dimensions of each volume element depend linearly on the local stresses. After analysing the model, we examine its fundamental growth response under changes in loads, material properties, and growth parameters. The behavior of the model is quite sensitive to changes in material nonlinearity and in the coefficients of the growth law. Next, growth of the aorta is simulated during development and maturity. For an appropriate choice of the parameters, the model exhibits patterns of growth that agree qualitatively with known characteristics of aortic growth. Comparison of model results with published experimental data during hypertension in the rat shows good agreement in the time course of the vessel radii and residual strain. Finally, the implications of the results are discussed in the context of deducing a general mechanical growth law for soft tissues. The proposed model should be useful in studies to determine the biomechanical factor that regulates growth.

Mechanics of ventricular torsion

Recent research suggests that left ventricular torsion is an important indicator of cardiac function. We used two theoretical models to study the mechanics of this phenomenon: a compressible cylinder and an incompressible ellipsoid of revolution. The analyses of both models account for large- strain passive and active material behavior, with a muscle fiber angle that varies linearly from endocardium to epicardium. Relative to the end- diastolic configuration, the predicted torsion exhibits several experimentally observed features, including a peak near end systole, rapid untwisting during isovolumic relaxation, and increased twist near the apex. The magnitude of the twist is sensitive to the fiber architecture, the ventricular geometry, and the compressibility and contractility of the myocardium. In particular, the model predicts that the systolic twist increases with increasing compressibility, contractility, and wall thickness, while it decreases with increasing cavity volume. The peak twist approximately doubles (from about 0.02 to 0.04 rad cm(-1)) with a doubling of myocardial compressibility or with a change in the endocardial/epicardial muscle fiber angles from 90/ -90 degrees to 60/ -60 degrees. The twist is less sensitive to changes in contractility and ventricular geometry. These findings provide a basis for interpreting measurements of ventricular torsion in the clinical setting.

A model for stress-induced growth in the developing heart

Mechanical loads affect growth and morphogenesis in the developing heart. Using a theoretical model, we studied stress-modulated growth in the embryonic chick ventricle during stages 21-29 (4-6 days of a 21-day incubation period). The model is a thick-walled, compressible, pseudoelastic cylinder, with finite volumetric growth included by letting the rate of change of the local zero-stress configuration depend linearly on the Cauchy stresses. After investigating the fundamental behavior of the model, we used it to study global and local growth in the primitive ventricle due to normal and abnormal cavity pressures. With end-diastolic pressure taken as the growth-modulating stimulus, correlating theoretical and available experimental results yielded the coefficients of the growth law, which was assumed to be independent of time and loading conditions. For both normal and elevated pressures, the predicted changes in radius and wall volume during development were similar to experimental measurements. In addition, the residual stress generated by differential growth agreed with experimental data. These results suggest that wall stress may be a biomechanical factor that regulates growth in the embryonic heart.

Mechanics of cardiac looping

During the early stages of embryonic development, the heart is a smooth-walled, muscle-wrapped tube that bends and rotates in a vital, but poorly understood, morphogenetic process called looping. Since looping involves biomechanical forces, this paper examines two mechanically based hypotheses for the bending component of cardiac looping. The first hypothesis is that an initial tension in or near the dorsal mesocardium (DM), a longitudinal structure along the outside of the ventricle, drives the deformation. To relieve the bending stresses in the tube, the myocytes change shape passively, and then they deform actively to continue the process to completion of a full loop. In the second hypothesis, contraction of circumferentially arranged actin macrofilaments produces circumferential compression and longitudinal expansion (due to incompressibility) of the myocytes. The DM locally constrains the longitudinal deformation, forcing the tube to bend. The feasibility of these hypotheses was evaluated using theoretical models and published experimental results. The models, which consist of beams composed of two layers representing the DM and the ventricular myocardium, show that the hypotheses are consistent with most of the known data, but further studies are necessary. In this regard, the models provide a conceptual framework for designing experiments to investigate the mechanics of looping.

Epicardial strains in embryonic chick ventricle at stages 16 through 24

Embryonic cardiac development depends, in part, on the local biomechanical environment. Tracking the motions of microspheres attached to the embryonic chick ventricle, we computed two-dimensional epicardial strains at Hamburger-Hamilton stages 16, 18, 21, and 24 (2.5, 3.5, 4.0, and 4.5 days, respectively, of a 21-day incubation period). First, in a cross-sectional study, strains were measured in separate embryos at each stage (n > or = 19 per stage). Then, in a longitudinal study, strains were measured serially on the same heart, with the eggs resealed and reincubated between successive stages (n > or = 4 per stage). Although the heart undergoes major changes in mass, morphology, and loading during the studied stages, both studies showed that peak circumferential and longitudinal strains relative to end diastole were similar in magnitude (0.13 to 0.16) and did not change significantly across the stage range. The peak principal strains also showed no significant changes, with magnitudes of approximately 0.11 and 0.18. The shear strains were small, and their signs varied from one heart to another. These results suggest that wall strain is maintained within a relatively narrow range during primary cardiac morphogenesis.

A nonliner poroelastic model for the trabecular embryonic heart

A theoretical model is presented for the primitive right ventricle of the stage 21 chick embryo. At this stage of development, the wall of the heart is trabecular with direct intramyocardial blood flow. The model is a pressurized fluid-filled cylinder composed of a porous inner layer of isotropic myocardium and a relatively thin compact outer layer of transversely isotropic myocardium. The analysis is based on nonlinear poroelasticity theory, modified to include residual strain and muscle activation. Correlating theoretical and experimental pressure-volume loops and epicardial strains gives first-approximation constitutive relations for stage 21 embryonic myocardium. The results from the model suggest three primary conclusions: (1) Some muscle fibers likely are aligned in the compact layer, with a fiber angle approximately + 10 deg from the circumferential direction. (2) Blood is drawn into the wall of the ventricle during diastolic filling and isovolumic contraction and is squeezed out of the wall during systolic ejection, giving a primitive intramyocardial circulation before the coronary arteries form. As the heart rate increases, the transmural blood-flow velocity increases, but the volume of blood exchanged with the lumen per beat decreases. (3) Residual strain affects transmural stress distributions, producing nearly uniform stresses in the porous layer, where the peak end-systolic stress occurs. These results improve our understanding of the relation between form and function in the developing heart and provide directions for biological experiments to study cardiac morphogenesis.

Mechanical effects of looping in the embryonic chick heart

During early embryonic development, the heart bends into a curved tube in a vital morphogenetic process called looping. Since looping involves poorly understood biomechanical forces that are difficult to measure, this paper presents a theoretical model for the tubular chick heart, whose development is similar to that of the human heart. Representing the basic morphology of the looped ventricle, the model is a thick-walled, isotropic, pressurized curved tube composed of three layers representing the myocardium, cardiac jelly, and endocardium. The model is analyzed with nonlinear elasticity theory, modified to include residual strain and muscle activation, and material properties are determined by correlating theoretical and experimental pressure-volume relations. The results show that longitudinal curvature significantly influences the biomechanical behavior of the embryonic heart. As the curvature increases, the compliance of the tube increases, especially at end systole. Stress concentrations, which develop in the endocardium during diastole and in the myocardium during systole, also increase with the curvature. The largest wall stress during the cardiac cycle occurs near the beginning of systolic ejection in the myocardial layer at the inner curvature of the tube. Relative to end diastole, the model predicts epicardial strains that are nearly equal in the circumferential and meridional directions, in agreement with experimental measurements. These results provide insight into the interrelation between biomechanical forces and morphogenesis during cardiac looping.

Residual strain in the ventricle of the stage 16-24 chick embryo

Residual stress and strain, i.e., the stress and strain remaining in a solid when all external loads are removed, may be produced in biological tissues by differential growth. During cardiac development, residual stress and strain may play a role in cardiac morphogenesis by affecting ventricular wall stress. After a transmural radial cut, a passive ventricular cross section opens into a sector, and the size of the opening angle provides a measure of the circumferential residual strain. Residual strains were characterized in this manner for the apical region of the diastolic embryonic chick heart for Hamburger-Hamilton stages 16, 18, 21, and 24 (approximately 2.5, 3.5, 4.0, and 4.5 days, respectively, of a 21-day incubation period). The average opening angle at these stages was 107 +/- 10 degrees, 79 +/- 10 degrees, 73 +/- 11 degrees, and 74 +/- 7 degrees, respectively (n > or = 5 for each stage). These measured angles were correlated with changes in ventricular morphology. Scanning electron micrographs of the apex revealed that the wall of the ventricle is smooth at stage 16. Then at stage 18, myocardial trabeculae develop, forming ridges with primarily a circumferential orientation. By stage 21, the trabeculae develop into a mesh, giving the ventricular wall a spongelike appearance, and the preferred orientation is lost by stage 24. The large decrease in opening angle between stages 16 and 18 corresponded to the onset of trabeculation, which is the greatest change in form during the studied stages. We speculate that residual strain is an important biomechanical factor during cardiac morphogenesis.

Cardiac mechanics in the stage-16 chick embryo

A theoretical model is presented for the tubular heart of the stage-16 chick embryo (2.3 days of a 21-day incubation period). The model is a thick-walled, pseudoelastic cylindrical shell composed of three isotropic layers: the endocardium, the cardiac jelly, and the myocardium. The analysis is based on a shell theory that accounts for large deformation, material nonlinearity, residual strain, and muscle activation, with material properties inferred from available experimental data. We also measured epicardial strains from recorded motions of microspheres on the primitive right ventricles of stage-16 white Leghorn chick embryos. Relative to end diastole, peak axial and circumferential Lagrange strains occurred near end systole and had similar values. The magnitudes of these strains varied along the longitudinal axis of the heart (-0.16 +/- 0.08), being larger near the ends of the primitive right ventricle and smaller near midventricle. The in-plane shear strain was less than 0.05. Comparison of theoretical and experimental strains during the cardiac cycle shows generally good agreement. In addition, the model gives strong stress concentrations in the myocardial layer at end systole.

On a nonlinear theory for muscle shells: Part II--Application to the beating left ventricle

This paper specializes the nonlinear laminated-muscle-shell theory developed in Part I to cylindrical geometry and computes stresses in arteries and the beating left ventricle. The theory accounts for large strain, material nonlinearity, thick-shell effects, torsion, muscle activation, and residual strain. First, comparison with elasticity solutions for pressurized arteries shows that the accuracy of the shell theory increases as transmural stress gradients and the shell thickness decrease. Residual strain reduces the stress gradients, lowering the error in the predicted peak stress in thick-walled arteries (R/t = 2.8) from about 30 to 10 percent. Second, the canine left ventricle is modeled as a thick-walled laminated cylinder with an internal pressure. Each layer is composed of transversely isotropic muscle with a fiber orientation based on anatomical data. Using a single pseudostrain-energy density function (with time-varying coefficients) for passive and active myocardium, the model predicts strain distributions that agree fairly well with published experimental measurements. The results also show that the peak fiber stress occurs subendocardially near the beginning of ejection and that residual strains significantly alter stress gradients within each lamina, but the magnitude of the peak fiber stress changes by less than 20 percent.

On a nonlinear theory for muscle shells: Part I--Theoretical development

This paper presents a theory for studies of the large-strain behavior of biological shells composed of layers of incompressible, orthotropic tissue, possibly muscle, of arbitrary orientation. The intrinsic equations of the laminated-shell theory, expressed in lines-of-curvature coordinates, account for large membrane [O(1)] and moderately large bending and transverse shear strains [O(0.3)], nonlinear material properties, and transverse normal stress and strain. An expansion is derived for a general two-dimensional strain-energy density function, which includes residual stress and muscle activation through a shifting zero-stress configuration. Strain-displacement relations are given for the special case of axisymmetric deformation of shells of revolution with torsion.

The possible role of poroelasticity in the apparent viscoelastic behavior of passive cardiac muscle

This paper investigates the contribution of extracellular fluid flow to the apparent viscoelastic behavior of passive cardiac muscle. The muscle is modeled as an incompressible, isotropic, poroelastic solid saturated by an incompressible viscous fluid. Based on Biot's linear and nonlinear consolidation theories, solutions are presented for general time-dependent uniaxial loading of unconfined cylindrical muscle specimens. The nonlinear analysis includes the effects of large strain, material nonlinearity, and strain-dependent permeability. The computed results show that, for axial stretch ratios greater than 1.1, the changing permeability and the loading rate strongly affect the total stress relaxation and the short-time relaxation rate. Comparisons of theoretical and published experimental results show that extracellular fluid flow can account for several observed biomechanical features of passive myocardium, including the insensitivity of stress-strain curves to loading rate and of stress-relaxation curves to the amount of stretch. Theoretical hysteresis loops, however, are too small. Thus, both poroelastic and tissue viscoelastic effects must be considered in studies of passive cardiac muscle.

Large deformation mechanics of the enucleated eyeball

Large deformation of enucleated pig eyeballs under rigid cylindrical indenters was studied analytically and experimentally. The analytic model for the eyeball consists of a fluid-filled spherical membrane composed of an incompressible, elastic material with an exponential strain energy function. The Rayleigh-Ritz technique provided an approximate solution via a potential energy formulation. Comparison with results from tests on eyeballs and a water-filled rubber (Mooney-Rivlin) shell shows good agreement at large deflection, where membrane action dominates. Due to the highly nonlinear stress-strain relations for the sclera, the load remains relatively small until the indenter displacement approaches 40-60 percent of the eyeball radius, and then the load increases rapidly. Depending on the indenter size, either a perforation or a rupture type of failure occurs.

Cochlear model including three-dimensional fluid and four modes of partition flexibility

The WKB solution is developed for the analysis of a straight box cochlear model which includes four modes of partition displacement, simulating the motion of the bony shelf and arches of Corti, as well as the pectinate zone of the basilar membrane. The theory is similar to that previously used for the 1-mode model with scalar quantities now replaced by 4-vectors. Calculations are carried out for the guinea pig cochlea with stiffness computed mainly from the anatomy and assumed physiological values for the materials. Results show that the stiffness is such that the amplitude and phase of the basilar membrane response are not significantly altered from those given by the 1-mode model. For primates and some other mammals, the bony shelf is substantially weaker than in the guinea pig and causes a much more rapid accumulation of phase along the basilar membrane. Thus, with anatomically and physiologically consistent parameters, the model yields good correlation in phase and amplitude with the in vivo measurements which have been made in the squirrel monkey by Rhode [J. Acoust. Soc. Am. 64, 158-176 (1978)] as well as in the guinea pig by Wilson and Johnstone [J. Acoust. Soc. Am. 57, 705-723 (1975)] and Rhode [Basic Mechanisms in Hearing (Academic, New York, 1973), pp. 49-63].

Three-dimensional model calculations for guinea pig cochlea

The WKB approximation was used for calculations of the pure tone response of a straight box model of the guinea pig cochlea with square scale cross sections and the fluid density and viscosity of water. Only one mode of elastic deformation of the partition was considered, corresponding to a flexible pectinate zone of the basilar membrane (BM) with rigid bony shelf, arches, and spiral ligament. Four distributions of pectinate zone transverse bending stiffness were considered, corresponding to volume compliances: (1) CB, measured by Békésy in the guinea pig post mortem, (2) CB/4, (3) CPL, deduced from Békésy's point load measurements in a human, with BM thickness inversely proportional to the width and rescaled for the guinea pig, and (4) 10CPL. We also considered various values of the relative longitudinal stiffness of the basilar membrane and the condition of drained or filled scala tympani. When compared to in vivo and post-mortem measurements of the guinea pig, the model results lead to the conjecture that the transverse fibers of the basilar membrane decrease in stiffness with time post mortem, while the ground substance increases in stiffness. Calculations using the compliance CB/4, with the ST drained with zero longitudinal BM stiffness give a response similar in location, peak shape, and phase to the in vivo capacitance probe measurements of Wilson and Johnstone [J., Acoust. Soc. Am 57, 705--23 (1975)]. Calculations for the ST filled and closed show a BM amplitude similar in location and shape to the spiral ganglion cell threshold curves obtained by Robertson and Johnstone [J. Acoust. Soc. Am. 57, 466--469 (1979)] from abnormal cochleas without outer hair cells. This indicates that the normal peak neural stimulation occurs about 1 mm apical of the BM peak amplitude. Naturally, the discrepancies between the postulated physical model and the cochlea prevent firm conclusions about cochlear function.

Comparison of WKB calculations and experimental results for three-dimensional cochlear models

The WKB asymptotic method is applied to the calculation of cochlear models with square scala cross section, for which the fluid motion is fully three dimensional. The analysis begins with the exact solution for wave propagation in a duct with constant properties. This solution is somewhat tedious but straightforward, since it requires a Fourier series expansion across the duct. Then with the formulation of Whitham [Linear and Nonlinear Waves (Wiley, New York, 1974)], the approximate solution is readily generated for the duct with properties which vary slowly along the length. Numerical calculations are carried out for the experimental models of Cannel [Ph.D. thesis, Univ. of Warwick (1969)] and Helle [Dr.-Ing. disser., Technische Univ., Müchen (1974)] who furnish quantitative details of both "basilar membrane" response and model parameters. Without any free parameters for adjusting, the present WKB solution shows quite satisfactory agreement with the experimental model results. Computer time is reasonable; the calculation of displacement envelope and phase at a number of stations along the cochlea for a given frequency requires only one second of CPU time. Thus the credibility and practically of the approach is established for the investigation of yet more realistic and more elaborate cochlear models.

Comparison of WKB and finite difference calculations for a two-dimensional cochlear model

There are many points of uncertainty in the subject of cochlear models. In this paper only the question of efficient computing methods is addressed. For the cochlear model with a one-dimensional approximation for the fluid motion, Zweig, Lipes, and Pierce [J. Acoust. Soc. Am. 59, 975-982 (1976)] have shown that the WKB method agrees well with a direct numerical integration. For the two-dimensional fluid model, Neely [E.D. thesis, California Institute of Technology, Pasadena, CA (1977)] has shown that a direct finite difference solution is an order of magnitude faster than the integral equation approach used by Allen [J. Acoust. Soc. Am 61, 110-119 (1977)]. In the present work, a formal WKB solution is derived following Whitham [Linear and Nonlinear Waves (Wiley, New York, 1974)]. The advantage of this formulation is simplicity, but the disadvantage is that no error estimate is available. We find that the numerical results from the WKB solution agree well with those of Neely (1977), while the computer time is reduced by another order of magnitude. Thus, the WKB method seems to offer the satisfactory accuracy, efficiency, and flexibility for treating the more realistic cochlear models.