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In JoVE (1)
Other Publications (31)
- American Journal of Physiology. Heart and Circulatory Physiology
- Developmental Dynamics : an Official Publication of the American Association of Anatomists
- Journal of Biomechanics
- Annals of Biomedical Engineering
- Journal of Biomechanical Engineering
- Developmental Biology
- Journal of Biomechanical Engineering
- Developmental Dynamics : an Official Publication of the American Association of Anatomists
- The International Journal of Developmental Biology
- Developmental Dynamics : an Official Publication of the American Association of Anatomists
- Anatomy and Embryology
- Annals of Biomedical Engineering
- Journal of Biomechanical Engineering
- Anatomical Record (Hoboken, N.J. : 2007)
- Biomechanics and Modeling in Mechanobiology
- Biomechanics and Modeling in Mechanobiology
- Biomechanics and Modeling in Mechanobiology
- Computer Methods in Biomechanics and Biomedical Engineering
- Journal of Biomechanical Engineering
- Journal of Biomechanical Engineering
- Philosophical Transactions. Series A, Mathematical, Physical, and Engineering Sciences
- Biomechanics and Modeling in Mechanobiology
- Journal of Biomechanical Engineering
- Journal of Biomechanical Engineering
- Development (Cambridge, England)
- Annals of the New York Academy of Sciences
- Optics Letters
- Journal of Biomechanical Engineering
- Journal of Biomechanical Engineering
- Annals of Biomedical Engineering
- Journal of the Royal Society, Interface / the Royal Society
Articles by Larry A. Taber in JoVE
JoVE General
Tracking Morphogenetic Tissue Deformations in the Early Chick Embryo
1Department of Biomedical Engineering, Washington University, 2Institute for Information Transmission Problems, Russian Academy of Sciences, 3Department of Mechanical Engineering and Materials Science, Washington University
This article describes surface labeling and ex ovo tissue culture in the early chick embryo. Techniques amenable to time-lapse bright field, fluorescence, and optical coherence tomography imaging are presented. Tracking surface labels with high spatiotemporal resolution enables kinematic quantities such as morphogenetic strains (deformations) to be calculated in both two and three dimensions.
Other articles by Larry A. Taber on PubMed
Intramyocardial Pressure Measurements in the Stage 18 Embryonic Chick Heart
Intramyocardial pressure (IMP) and ventricular pressure (VP) were measured in the trabeculating heart of the stage 18 chick embryo (3 days of incubation). Pressure was measured at several locations across the ventricle using a fluid-filled servo-null system. Maximum systolic and minimum diastolic IMP tended to be greater in the dorsal wall than in the ventral wall, but transmural distributions of peak active (maximum minus minimum) IMP were similar in both walls. Peak active IMP near midwall was similar to peak active VP, but peak active IMP in the subepicardial and subendocardial layers was four to five times larger. These results suggest that the passive stiffness of the dorsal wall is greater than that of the ventral wall and that during contraction the inner and outer layers of both walls generate more contractile force and/or become less permeable to flow than the middle part of the wall. Measured pressures likely correspond to regional variations in wall stress that may influence morphogenesis and function in the embryonic heart.
Cardiac Looping in Experimental Conditions: Effects of Extraembryonic Forces
The chick embryo is a popular experimental model used to study the mechanisms of cardiac looping. To facilitate oxygen transport, researchers typically culture the embryo on the surface of the medium. Such preparations, however, expose the embryo and the heart to surface tension that is not present in ovo. This study investigates the influence that surface and extraembryonic membrane tensions have on looping morphology. To eliminate surface tension, we developed a technique in which the embryo is cultured under a thin layer of fluid. To eliminate membrane tension, the membrane was removed. Our results show that both tensions can affect looping, with surface tension potentially having a much greater effect. Moreover, we show that surface tension can alter results in one classic looping experiment.
Regional Epicardial Strain in the Embryonic Chick Heart During the Early Looping Stages
Epicardial strains were measured in Hamburger-Hamilton stage 11 and 12 embryonic chick hearts (1.6-2.0 days of incubation). These stages include part of the early phase of cardiac looping, as the initially straight heart tube bends and twists to form a curved c-shaped tube. By analyzing the motion of microbeads placed on the myocardial surface, we measured strains near the outer curvature, in the central region, and near the inner curvature of the primitive ventricle. No significant differences in strain were found between stages. Relative to end diastole, all three regions shortened by about 10% during systole in the circumferential direction, and the outer curvature shortened longitudinally by about 5%. In contrast, and unlike strains in older hearts, the inner curvature and central regions elongated by approximately 5-10% in the longitudinal direction during systole. These results are consistent with microstructural data and suggest that the material properties of the outer curvature are relatively isotropic, whereas the properties of the central and inner curvature regions are orthotropic, with contractile stress exerted primarily in the circumferential direction.
Mechanical Asymmetry in the Embryonic Chick Heart During Looping
Cardiac looping, which begins with ventral bending and rightward rotation of the primitive heart tube, is an essential morphogenetic event that occurs early in vertebrate development. The biophysical mechanism that drives this process is unknown. It has been speculated that increased stiffness along the dorsal side of the ventricle combined with an intrinsic cardiac force causes the heart to bend. There is no experimental support for this hypothesis, however, since little is known about regional mechanical properties of the heart during looping. We directly measured diastolic stiffness of the inner curvature (IC), outer curvature (OC), and dorsal-ventral sides of the stage 12 chick heart by microindentation. The IC of intact hearts was found to be significantly stiffer than either the OC or the sides. which were of similar stiffness. Isolated cardiac jelly, which is a thick, extracellular matrix compartment underlying the myocardium, was approximately an order of magnitude softer than intact hearts. The results of a computational model simulating the indentation experiments, combined with the stiffness measurements, suggests the regional variation in stiffness is due to the material properties of the myocardium. A second model shows that a relatively stiff IC may facilitate bending of the heart tube during looping.
On the Effects of Residual Stress in Microindentation Tests of Soft Tissue Structures
Microindentation methods are commonly used to determine material properties of soft tissues at the cell or even sub-cellular level. In determining properties from force-displacement (FD) data, it is often assumed that the tissue is initially a stress-free, homogeneous, linear elastic half-space. Residual stress, however, can strongly influence such results. In this paper, we present a new microindentation method for determining both elastic properties and residual stress in soft tissues that, to a first approximation, can be regarded as a pre-stressed layer embedded in or adhered to an underlying relatively soft, elastic foundation. The effects of residual stress are shown using two linear elastic models that approximate specific biological structures. The first model is an axially loaded beam on a relatively soft, elastic foundation (i.e., stress-fiber embedded in cytoplasm), while the second is a radially loaded plate on a foundation (e.g., cell membrane or epithelium). To illustrate our method, we use a nonlinear finite element (FE) model and experimental FD and surface contour data to find elastic properties and residual stress in the early embryonic chick heart, which, in the region near the indenter tip, is approximated as an isotropic circular plate under tension on a foundation. It is shown that the deformation of the surface in a microindentation test can be used along with FD data to estimate material properties, as well as residual stress, in soft tissue structures that can be regarded as a plate under tension on an elastic foundation. This method may not be as useful, however, for structures that behave as a beam on a foundation.
The Role of Mechanical Forces in Dextral Rotation During Cardiac Looping in the Chick Embryo
Cardiac looping is a vital morphogenetic process that transforms the initially straight heart tube into a curved tube normally directed toward the right side of the embryo. While recent work has brought major advances in our understanding of the genetic and molecular pathways involved in looping, the biophysical mechanisms that drive this process have remained poorly understood. This paper examines the role of biomechanical forces in cardiac rotation during the initial stages of looping, when the heart bends and rotates into a c-shaped tube (c-looping). Embryonic chick hearts were subjected to mechanical and chemical perturbations, and tissue stress and strain were studied using dissection and fluorescent labeling, respectively. The results suggest that (1) the heart contains little or no intrinsic ability to rotate, as external forces exerted by the splanchnopleure (SPL) and the omphalomesenteric veins (OVs) drive rotation; (2) unbalanced forces in the omphalomesenteric veins play a role in left-right looping directionality; and (3) in addition to ventral bending and rightward rotation, the heart tube also bends slightly toward the right. The results of this study may help investigators searching for the link between gene expression and the mechanical processes that drive looping.
Material Properties and Residual Stress in the Stage 12 Chick Heart During Cardiac Looping
During the morphogenetic process of cardiac looping, the initially straight cardiac tube bends and twists into a curved tube. The biophysical mechanisms that drive looping remain unknown, but the process clearly involves mechanical forces. Hence, it is important to determine mechanical properties of the early heart, which is a muscle-wrapped tube consisting primarily of a thin outer layer of myocardium surrounding a thick extracellular matrix compartment known as cardiac jelly. In this work, we used microindentation experiments and finite element modeling, combined with an inverse computational method, to determine constitutive relations for the myocardium and cardiac jelly at the outer curvature of stage 12 chick hearts. Material coefficients for exponential strain-energy density functions were found by fitting force-displacement and surface displacement data near the indenter Residual stress in the myocardium also was estimated. These results should be useful for computational models of the looping heart.
Role of Actin Polymerization in Bending of the Early Heart Tube
During cardiac c-looping, the heart transforms from a straight tube into a c-shaped tube, presenting the first evidence of left-right asymmetry in the embryo. C-looping consists of two primary deformation components: ventral bending and dextral rotation. This study examines the role of actin polymerization in bending of the heart tube. Exposure of stage 9-11 chick embryos to low concentrations of the actin polymerization inhibitors cytochalasin D (5 nM-2.0 microM) and latrunculin A (LA; 25 nM-2.0 microM) suppressed looping in a stage- and concentration-dependent manner in both whole embryos and isolated hearts. Local exposure of either the dorsal or ventral sides of isolated hearts to LA also inhibited looping, but less than global exposure, indicating that both sides contribute to the bending mechanism. Taken together, these data suggest that ongoing actin polymerization is required for the bending component of cardiac c-looping, and we speculate that polymerization-driven myocardial cell shape changes cause this deformation.
Biophysical Mechanisms of Cardiac Looping
During early embryogenesis, the heart is a single, relatively straight tube which bends and twists (loops) rightward to create the basic plan of the mature heart. Despite intensive study for many decades, the biophysical mechanisms which drive and regulate cardiac looping have remained poorly understood. This review discusses, from a historical perspective, studies of looping mechanics and various theories which have been proposed for this complex process. Then, based on recently acquired data, a new biomechanical hypothesis is proposed for the rst phase of looping (c-looping). Understanding morphogenetic mechanisms would facilitate research devoted to preventing and treating congenital heart malformations caused by looping abnormalities.
Morphogenetic Adaptation of the Looping Embryonic Heart to Altered Mechanical Loads
The biophysical mechanisms that drive and regulate cardiac looping are not well understood, but mechanical forces likely play a central role. Previous studies have shown that cardiac torsion, which defines left-right directionality, is caused largely by forces exerted on the heart tube by a membrane called the splanchnopleure (SPL). Here we show that, when the SPL is removed from the embryonic chick heart, torsion is initially suppressed. Several hours later, however, normal torsion is restored. This delayed torsion coincides with increased myocardial stiffness, especially on the right side of the heart. Exposure to the myosin inhibitor Y-27632 suppressed both responses, suggesting that the delayed torsion is caused by an abnormal cytoskeletal contraction. This hypothesis is supported further by computational modeling. These results suggest that the looping embryonic heart has the ability to adapt to changes in the mechanical environment, which may play a regulatory role during morphogenesis.
Myosin-based Contraction is Not Necessary for Cardiac C-looping in the Chick Embryo
During the initial phase of cardiac looping, known as c-looping, the heart bends and twists into a c-shaped tube with the convex outer curvature normally directed toward the right side of the embryo. Despite intensive study for more than 80 years, the biophysical mechanisms that drive and regulate looping remain poorly understood, although some investigators have speculated that differential cytoskeletal contraction supplies the driving force for c-looping. The purpose of this investigation was to test this hypothesis. To inhibit contraction, embryonic chick hearts at stages 10-12 (10-16 somites, 33-48 h) were exposed to the myosin inhibitors 2,3-butanedione monoxime (BDM), ML-7, Y-27632, and blebbistatin. Experiments were conducted in both whole embryo culture and, to focus on bending alone, isolated heart culture. Measurements of heart stiffness and phosphorylation of the myosin regulatory light chains showed that BDM, Y-27632, and blebbistatin significantly reduced myocardial contractility, while ML-7 had a lesser effect. None of these drugs significantly affected looping during the studied stages. These results suggest that active contraction is not required for normal c-looping of the embryonic chick heart between stages 10 and 12.
Computational Model for Early Cardiac Looping
Looping is a vital event during early cardiac morphogenesis, as the initially straight heart tube bends and twists into a curved tube, laying out the basic pattern of the future four-chambered heart. Despite intensive study for almost a century, the biophysical mechanisms that drive this process are not well understood. To explore a recently proposed hypothesis for looping, we constructed a finite element model for the embryonic chick heart during the first phase of looping, called c-looping. The model includes the main structures of the early heart (heart tube, omphalomesenteric veins, and dorsal mesocardium), and the analysis features realistic three-dimensional geometry, nonlinear passive and active material properties, and anisotropic growth. As per our earlier hypothesis for c-looping, actin-based morpho-genetic processes (active cell shape change, cytoskeletal contraction, and cell migration) are simulated in specific regions of the model. The model correctly predicts the initial gross morphological shape changes of the heart, as well as distributions of morphogenetic stresses and strains measured in embryonic chick hearts. The model was tested further in studies that perturbed normal cardiac morphogenesis. The model, taken together with the new experimental data, supports our hypothesis for the mechanisms that drive early looping.
Computational Model for the Transition from Peristaltic to Pulsatile Flow in the Embryonic Heart Tube
Early in development, the heart is a single muscle-wrapped tube without formed valves. Yet survival of the embryo depends on the ability of this tube to pump blood at steadily increasing rates and pressures. Developmental biologists historically have speculated that the heart tube pumps via a peristaltic mechanism, with a wave of contraction propagating from the inflow to the outflow end. Physiological measurements, however, have shown that the flow becomes pulsatile in character quite early in development, before the valves form. Here, we use a computational model for flow though the embryonic heart to explore the pumping mechanism. Results from the model show that endocardial cushions, which are valve primordia arising near the ends of the tube, induce a transition from peristaltic to pulsatile flow. Comparison of numerical results with published experimental data shows reasonably good agreement for various pressure and flow parameters. This study illustrates the interrelationship between form and function in the early embryonic heart.
Optical Coherence Tomography As a Tool for Measuring Morphogenetic Deformation of the Looping Heart
Optical coherence tomography (OCT) was used to investigate morphogenesis of the embryonic chick heart during the first phase of looping (c-looping), as the heart bends and twists into a c-shaped tube. The present study focuses on the morphomechanical effects of the splanchnopleure (SPL), a membrane that has been shown to play a major role in cardiac torsion by pressing against the ventral surface of the heart. Without the SPL, rightward torsion (rotation) is delayed. The images show that compressive forces exerted by the SPL alter the shapes of the heart tube and primitive atria, as well as their spatial relationships. The SPL normally holds the heart in the plane of the embryo and forces cardiac jelly (CJ) out of adjacent regions in the atria. When the SPL is removed, cross-sections become more circular, CJ is more uniformly distributed, and the heart displaces ventrally. In addition, OCT-based morphogenetic strain maps were measured during looping by tracking the three-dimensional motions of microspheres placed on the myocardium. The spatial-temporal patterns of the strains correlated well with the observed behavior of the heart, including delayed torsion that occurs in SPL-lacking embryos. These results illustrate the potential of OCT as a tool in studies of morphogenesis, as well as provide a better understanding of the mechanical forces that drive cardiac looping.
Growth and Remodeling in a Thick-walled Artery Model: Effects of Spatial Variations in Wall Constituents
A mathematical model is presented for growth and remodeling of arteries. The model is a thick-walled tube composed of a constrained mixture of smooth muscle cells, elastin and collagen. Material properties and radial and axial distributions of each constituent are prescribed according to previously published data. The analysis includes stress-dependent growth and contractility of the muscle and turnover of collagen fibers. Simulations were conducted for homeostatic conditions and for the temporal response following sudden hypertension. Numerical pressure-radius relations and opening angles (residual stress) show reasonable agreement with published experimental results. In particular, for realistic material and structural properties, the model predicts measured variations in opening angles along the length of the aorta with reasonable accuracy. These results provide a better understanding of the determinants of residual stress in arteries and could lend insight into the importance of constituent distributions in both natural and tissue-engineered blood vessels.
Theoretical Study of Beloussov's Hyper-restoration Hypothesis for Mechanical Regulation of Morphogenesis
Computational models were used to explore the idea that morphogenesis is regulated, in part, by feedback from mechanical stress according to Beloussov's hyper-restoration (HR) hypothesis. According to this hypothesis, active tissue responses to stress perturbations tend to restore, but overshoot, the original (target) stress. To capture this behavior, the rate of growth or contraction is assumed to depend on the difference between the current and target stresses. Stress overshoot is obtained by letting the target stress change at a rate proportional to the same stress difference. The feasibility of the HR hypothesis is illustrated by models for stretching of epithelia, cylindrical bending of plates, invagination of cylindrical and spherical shells, and early amphibian development. In each case, an initial perturbation leads to an active mechanical response that changes the form of the tissue. The results show that some morphogenetic processes can be entirely self-driven by HR responses once they are initiated (possibly by genetic activity). Other processes, however, may require secondary mechanisms or perturbations to proceed to completion.
Computational Modeling of Morphogenesis Regulated by Mechanical Feedback
Mechanical forces cause changes in form during embryogenesis and likely play a role in regulating these changes. This paper explores the idea that changes in homeostatic tissue stress (target stress), possibly modulated by genes, drive some morphogenetic processes. Computational models are presented to illustrate how regional variations in target stress can cause a range of complex behaviors involving the bending of epithelia. These models include growth and cytoskeletal contraction regulated by stress-based mechanical feedback. All simulations were carried out using the commercial finite element code ABAQUS, with growth and contraction included by modifying the zero-stress state in the material constitutive relations. Results presented for bending of bilayered beams and invagination of cylindrical and spherical shells provide insight into some of the mechanical aspects that must be considered in studying morphogenetic mechanisms.
Computational Study of Growth and Remodelling in the Aortic Arch
Opening angles (OAs) are associated with growth and remodelling in arteries. One curiosity has been the relatively large OAs found in the aortic arch of some animals. Here, we use computational models to explore the reasons behind this phenomenon. The artery is assumed to contain a smooth muscle/collagen phase and an elastin phase. In the models, growth and remodelling of smooth muscle/collagen depends on wall stress and fluid shear stress. Remodelling of elastin, which normally turns over very slowly, is neglected. The results indicate that OAs generally increase with longitudinal curvature (torus model), earlier elastin production during development, and decreased wall stiffness. Correlating these results with available experimental data suggests that all of these effects may contribute to the large OAs in the aortic arch. The models also suggest that the slow turnover rate of elastin limits longitudinal growth. These results should promote increased understanding of the causes of residual stress in arteries.
A New Method for Measuring Deformation of Folding Surfaces During Morphogenesis
During morphogenesis, epithelia (cell sheets) undergo complex deformations as they stretch, bend, and twist to form the embryo. Often these changes in shape create multivalued surfaces that can be problematic for strain measurements. This paper presents a method for quantifying deformation of such surfaces. The method requires four-dimensional spatiotemporal coordinates of a finite number of surface markers, acquired using standard imaging techniques. From the coordinates of the markers, various deformation measures are computed as functions of time and space using straightforward matrix algebra. This method accommodates sparse randomly scattered marker arrays, with reasonable errors in marker locations. The accuracy of the method is examined for some sample problems with exact solutions. Then, the utility of the method is illustrated by using it to measure surface stretch ratios and shear in the looping heart and developing brain of the early chick embryo. In these examples, microspheres are tracked using optical coherence tomography. This technique provides a new tool that can be used in studies of the mechanics of morphogenesis.
On Modeling Morphogenesis of the Looping Heart Following Mechanical Perturbations
Looping is a crucial early phase during heart development, as the initially straight heart tube (HT) deforms into a curved tube to lay out the basic plan of the mature heart. This paper focuses on the first phase of looping, called c-looping, when the HT bends ventrally and twists dextrally (rightward) to create a c-shaped tube. Previous research has shown that bending is an intrinsic process, while dextral torsion is likely caused by external forces acting on the heart. However, the specific mechanisms that drive and regulate looping are not yet completely understood. Here, we present new experimental data and finite element models to help define these mechanisms for the torsional component of c-looping. First, with regions of growth and contraction specified according to experiments on chick embryos, a three-dimensional model exhibits morphogenetic deformation consistent with observations for normal looping. Next, the model is tested further using experiments in which looping is perturbed by removing structures that exert forces on the heart--a membrane (splanchnopleure (SPL)) that presses against the ventral surface of the heart and the left and right primitive atria. In all cases, the model predicts the correct qualitative behavior. Finally, a two-dimensional model of the HT cross section is used to study a feedback mechanism for stress-based regulation of looping. The model is tested using experiments in which the SPL is removed before, during, and after c-looping. In each simulation, the model predicts the correct response. Hence, these models provide new insight into the mechanical mechanisms that drive and regulate cardiac looping.
Towards a Unified Theory for Morphomechanics
Mechanical forces are closely involved in the construction of an embryo. Experiments have suggested that mechanical feedback plays a role in regulating these forces, but the nature of this feedback is poorly understood. Here, we propose a general principle for the mechanics of morphogenesis, as governed by a pair of evolution equations based on feedback from tissue stress. In one equation, the rate of growth (or contraction) depends on the difference between the current tissue stress and a target (homeostatic) stress. In the other equation, the target stress changes at a rate that depends on the same stress difference. The parameters in these morphomechanical laws are assumed to depend on stress rate. Computational models are used to illustrate how these equations can capture a relatively wide range of behaviours observed in developing embryos, as well as show the limitations of this theory. Specific applications include growth of pressure vessels (e.g. the heart, arteries and brain), wound healing and sea urchin gastrulation. Understanding the fundamental principles of tissue construction can help engineers design new strategies for creating replacement tissues and organs in vitro.
Residual Stress in the Adult Mouse Brain
This work provides direct evidence that sustained tensile stress exists in white matter of the mature mouse brain. This finding has important implications for the mechanisms of brain development, as tension in neural axons has been hypothesized to drive cortical folding in the human brain. In addition, knowledge of residual stress is required to fully understand the mechanisms behind traumatic brain injury and changes in mechanical properties due to aging and disease. To estimate residual stress in the brain, we performed serial dissection experiments on 500-mum thick coronal slices from fresh adult mouse brains and developed finite element models for these experiments. Radial cuts were made either into cortical gray matter, or through the cortex and the underlying white matter tract composed of parallel neural axons. Cuts into cortical gray matter did not open, but cuts through both layers consistently opened at the point where the cut crossed the white matter. We infer that the cerebral white matter is under considerable tension in the circumferential direction in the coronal cerebral plane, parallel to most of the neural fibers, while the cerebral cortical gray matter is in compression. The models show that the observed deformation after cutting can be caused by more growth in the gray matter than in the white matter, with the estimated tensile stress in the white matter being on the order of 100-1,000 Pa.
A New Method to Measure Cortical Growth in the Developing Brain
Folding of the cerebral cortex is a critical phase of brain development in higher mammals but the biomechanics of folding remain incompletely understood. During folding, the growth of the cortical surface is heterogeneous and anisotropic. We developed and applied a new technique to measure spatial and directional variations in surface growth from longitudinal magnetic resonance imaging (MRI) studies of a single animal or human subject. MRI provides high resolution 3D image volumes of the brain at different stages of development. Surface representations of the cerebral cortex are obtained by segmentation of these volumes. Estimation of local surface growth between two times requires establishment of a point-to-point correspondence ("registration") between surfaces measured at those times. Here we present a novel approach for the registration of two surfaces in which an energy function is minimized by solving a partial differential equation on a spherical surface. The energy function includes a strain-energy term due to distortion and an "error energy" term due to mismatch between surface features. This algorithm, implemented with the finite element method, brings surface features into approximate alignment while minimizing deformation in regions without explicit matching criteria. The method was validated by application to three simulated test cases and applied to characterize growth of the ferret cortex during folding. Cortical surfaces were created from MRI data acquired in vivo at 14 days, 21 days, and 28 days of life. Deformation gradient and Lagrangian strain tensors describe the kinematics of growth over this interval. These quantitative results illuminate the spatial, temporal, and directional patterns of growth during cortical folding.
Automatic Generation of User Material Subroutines for Biomechanical Growth Analysis
The analysis of the biomechanics of growth and remodeling in soft tissues requires the formulation of specialized pseudoelastic constitutive relations. The nonlinear finite element analysis package ABAQUS allows the user to implement such specialized material responses through the coding of a user material subroutine called UMAT. However, hand coding UMAT subroutines is a challenge even for simple pseudoelastic materials and requires substantial time to debug and test the code. To resolve this issue, we develop an automatic UMAT code generation procedure for pseudoelastic materials using the symbolic mathematics package MATHEMATICA and extend the UMAT generator to include continuum growth. The performance of the automatically coded UMAT is tested by simulating the stress-stretch response of a material defined by a Fung-orthotropic strain energy function, subject to uniaxial stretching, equibiaxial stretching, and simple shear in ABAQUS. The MATHEMATICA UMAT generator is then extended to include continuum growth by adding a growth subroutine to the automatically generated UMAT. The MATHEMATICA UMAT generator correctly derives the variables required in the UMAT code, quickly providing a ready-to-use UMAT. In turn, the UMAT accurately simulates the pseudoelastic response. In order to test the growth UMAT, we simulate the growth-based bending of a bilayered bar with differing fiber directions in a nongrowing passive layer. The anisotropic passive layer, being topologically tied to the growing isotropic layer, causes the bending bar to twist laterally. The results of simulations demonstrate the validity of the automatically coded UMAT, used in both standardized tests of hyperelastic materials and for a biomechanical growth analysis.
Mechanics of Head Fold Formation: Investigating Tissue-level Forces During Early Development
During its earliest stages, the avian embryo is approximately planar. Through a complex series of folds, this flat geometry is transformed into the intricate three-dimensional structure of the developing organism. Formation of the head fold (HF) is the first step in this cascading sequence of out-of-plane tissue folds. The HF establishes the anterior extent of the embryo and initiates heart, foregut and brain development. Here, we use a combination of computational modeling and experiments to determine the physical forces that drive HF formation. Using chick embryos cultured ex ovo, we measured: (1) changes in tissue morphology in living embryos using optical coherence tomography (OCT); (2) morphogenetic strains (deformations) through the tracking of tissue labels; and (3) regional tissue stresses using changes in the geometry of circular wounds punched through the blastoderm. To determine the physical mechanisms that generate the HF, we created a three-dimensional computational model of the early embryo, consisting of pseudoelastic plates representing the blastoderm and vitelline membrane. Based on previous experimental findings, we simulated the following morphogenetic mechanisms: (1) convergent extension in the neural plate (NP); (2) cell wedging along the anterior NP border; and (3) autonomous in-plane deformations outside the NP. Our numerical predictions agree relatively well with the observed morphology, as well as with our measured stress and strain distributions. The model also predicts the abnormal tissue geometries produced when development is mechanically perturbed. Taken together, the results suggest that the proposed morphogenetic mechanisms provide the main tissue-level forces that drive HF formation.
The Role of Mechanical Forces in the Torsional Component of Cardiac Looping
During early development, the initially straight heart tube (HT) bends and twists (loops) into a curved tube to lay out the basic plan of the mature heart. The physical mechanisms that drive and regulate looping are not yet completely understood. This paper reviews our recent studies of the mechanics of cardiac torsion during the first phase of looping (c-looping). Experiments and computational modeling show that torsion is primarily caused by forces exerted on the HT by the primitive atria and the splanchnopleure, a membrane that presses against the ventral surface of the heart. Experimental and numerical results are described and integrated to propose a hypothesis for cardiac torsion, and key aspects of our hypothesis are tested using experiments that perturb normal looping. For each perturbation, the models predict the correct qualitative response. These studies provide new insight into the mechanisms that drive and regulate cardiac looping.
In Vivo Photoacoustic Imaging of Transverse Blood Flow by Using Doppler Broadening of Bandwidth
A method is proposed to measure transverse blood flow by using photoacoustic Doppler broadening of bandwidth. By measuring bovine blood flowing through a plastic tube, the linear dependence of the broadening on the flow speed was validated. The blood flow of the microvasculature in a mouse ear and a chicken embryo (stage 16) was also studied.
Opening Angles and Material Properties of the Early Embryonic Chick Brain
Mechanical forces play an important role during brain development. In the early embryo, the anterior end of the neural tube enlarges and differentiates into the major brain subdivisions, including three expanding vesicles (forebrain, midbrain, and hindbrain) separated by two constrictions. Once the anterior neuropore and the spinal neurocoel occlude, the brain tube undergoes further regional growth and expansion in response to increasing cerebrospinal fluid pressure. Although this is known to be a response to mechanical loads, the mechanical properties of the developing brain remain largely unknown. In this work, we measured regional opening angles (due to residual stress) and stiffness of the embryonic chick brain during Hamburger-Hamilton stages 11-13 (approximately 42-51 h incubation). Opening angles resulting from a radial cut on transverse brain slices were about 40-110 deg (depending on region and stage) and served as an indicator of circumferential residual stress. In addition, using a custom-made microindentation device and finite-element models, we determined regional indentation stiffness and material properties. The results indicate that the modulus is relatively independent of position and stage of development with the average shear modulus being about 220 Pa for stages 11-13 chick brains. Information on the regional material properties of the early embryonic brain will help illuminate the process of early brain morphogenesis.
Axons Pull on the Brain, but Tension Does Not Drive Cortical Folding
During human brain development, the cerebral cortex undergoes substantial folding, leading to its characteristic highly convoluted form. Folding is necessary to accommodate the expansion of the cerebral cortex; abnormal cortical folding is linked to various neurological disorders, including schizophrenia, epilepsy, autism, and mental retardation. Although this process requires mechanical forces, the specific force-generating mechanisms that drive folding remain unclear. The two most widely accepted hypotheses are as follows: (1) Folding is caused by differential growth of the cortex and (2) folding is caused by mechanical tension generated in axons. Direct evidence supporting either theory, however, is lacking. Here we show that axons are indeed under considerable tension in the developing ferret brain, but the patterns of tissue stress are not consistent with a causal role for axonal tension. In particular, microdissection assays reveal that significant tension exists along axons aligned circumferentially in subcortical white matter tracts, as well as those aligned radially inside developing gyri (outward folds). Contrary to previous speculation, however, axonal tension is not directed across developing gyri, suggesting that axon tension does not drive folding. On the other hand, using computational (finite element) models, we show that differential cortical growth accompanied by remodeling of the subplate leads to outward folds and stress fields that are consistent with our microdissection experiments, supporting a mechanism involving differential growth. Local perturbations, such as temporal differences in the initiation of cortical growth, can ensure consistent folding patterns. This study shows that a combination of experimental and computational mechanics can be used to evaluate competing hypotheses of morphogenesis, and illuminate the biomechanics of cortical folding.
Mechanical Stress As a Regulator of Cytoskeletal Contractility and Nuclear Shape in Embryonic Epithelia
The mechano-sensitive responses of the heart and brain were examined in the chick embryo during Hamburger and Hamilton stages 10-12. During these early stages of development, cells in these structures are organized into epithelia. Isolated hearts and brains were compressed by controlled amounts of surface tension (ST) at the surface of the sample, and microindentation was used to measure tissue stiffness following several hours of culture. The response of both organs was qualitatively similar, as they stiffened under reduced loading. With increased loading, however, the brain softened while heart stiffness was similar to controls. In the brain, changes in nuclear shape and morphology correlated with these responses, as nuclei became more elliptical with decreased loading and rounder with increased loading. Exposure to the myosin inhibitor blebbistatin indicated that these changes in stiffness and nuclear shape are likely caused by altered cytoskeletal contraction. Computational modeling suggests that this behavior tends to return peak tissue stress back toward the levels it has in the intact heart and brain. These results suggest that developing cardiac and neural epithelia respond similarly to changes in applied loads by altering contractility in ways that tend to restore the original mechanical stress state. Hence, this study supports the view that stress-based mechanical feedback plays a role in regulating epithelial development.
Damped and Persistent Oscillations in a Simple Model of Cell Crawling
A very simple, one-dimensional, discrete, autonomous model of cell crawling is proposed; the model involves only three or four coupled first-order differential equations. This form is sufficient to describe many general features of cell migration, including both steady forward motion and oscillatory progress. Closed-form expressions for crawling speeds and internal forces are obtained in terms of dimensionless parameters that characterize active intracellular processes and the passive mechanical properties of the cell. Two versions of the model are described: a basic cell model with simple elastic coupling between front and rear, which exhibits stable, steady forward crawling after initial transient oscillations have decayed, and a poroelastic model, which can exhibit oscillatory crawling in the steady state.
