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.

A POROELASTIC MODEL FOR CELL CRAWLING INCLUDING MECHANICAL COUPLING BETWEEN CYTOSKELETAL CONTRACTION AND ACTIN POLYMERIZATION.

Much is known about the biophysical mechanisms involved in cell crawling, but how these processes are coordinated to produce directed motion is not well understood. Here, we propose a new hypothesis whereby local cytoskeletal contraction generates fluid flow through the lamellipodium, with the pressure at the front of the cell facilitating actin polymerization which pushes the leading edge forward. The contraction, in turn, is regulated by stress in the cytoskeleton. To test this hypothesis, finite element models for a crawling cell are presented. These models are based on nonlinear poroelasticity theory, modified to include the effects of active contraction and growth, which are regulated by mechanical feedback laws. Results from the models agree reasonably well with published experimental data for cell speed, actin flow, and cytoskeletal deformation in migrating fish epidermal keratocytes. The models also suggest that oscillations can occur for certain ranges of parameter values.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.