Theoretical model for myocardial trabeculation

Necking and failure of constrained 3D microtissues induced by cellular tension

In this paper we report a fundamental morphological instability of constrained 3D microtissues induced by positive chemomechanical feedback between actomyosin-driven contraction and the mechanical stresses arising from the constraints. Using a 3D model for mechanotransduction we find that perturbations in the shape of contractile tissues grow in an unstable manner leading to formation of "necks" that lead to the failure of the tissue by narrowing and subsequent elongation. The magnitude of the instability is shown to be determined by the level of active contractile strain, the stiffness of the extracellular matrix, and the components of the tissue that act in parallel with the active component and the stiffness of the boundaries that constrain the tissue. A phase diagram that demarcates stable and unstable behavior of 3D tissues as a function of these material parameters is derived. The predictions of our model are verified by analyzing the necking and failure of normal human fibroblast tissue constrained in a loop-ended dog-bone geometry and cardiac microtissues constrained between microcantilevers. By analyzing the time evolution of the morphology of the constrained tissues we have quantitatively determined the chemomechanical coupling parameters that characterize the generation of active stresses in these tissues. More generally, the analytical and numerical methods we have developed provide a quantitative framework to study how contractility can influence tissue morphology in complex 3D environments such as morphogenesis and organogenesis.

Patterns of muscular strain in the embryonic heart wall

The hypothesis that inner layers of contracting muscular tubes undergo greater strain than concentric outer layers was tested by numerical modeling and by confocal microscopy of strain within the wall of the early chick heart. We modeled the looped heart as a thin muscular shell surrounding an inner layer of sponge-like trabeculae by two methods: calculation within a two-dimensional three-variable lumped model and simulated expansion of a three-dimensional, four-layer mesh of finite elements. Analysis of both models, and correlative microscopy of chamber dimensions, sarcomere spacing, and membrane leaks, indicate a gradient of strain decreasing across the wall from highest strain along inner layers. Prediction of wall thickening during expansion was confirmed by ultrasonography of beating hearts. Degree of stretch determined by radial position may thus contribute to observed patterns of regional myocardial conditioning and slowed proliferation, as well as to the morphogenesis of ventricular trabeculae and conduction fascicles. Developmental Dynamics 238:1535-1546, 2009. (c) 2009 Wiley-Liss, Inc.

Computational models of myocardial endomysial collagen arrangement

Collagen extracellular matrix is one of the factors related to high passive stiffness of cardiac muscle. However, the architecture and the mechanical aspects of the cardiac collagen matrix are not completely known. In particular, endomysial collagen contribution to the passive mechanics of cardiac muscle as well as its micro anatomical arrangement is still a matter of debate. In order to investigate mechanical and structural properties of endomysial collagen, we consider two alternative computational models of some specific aspects of the cardiac muscle. These two models represent two different views of endomysial collagen distribution: (1) the traditional view and (2) a new view suggested by the data obtained from scanning electron microscopy (SEM) in NaOH macerated samples (a method for isolating collagen from the other tissue). We model the myocardial tissue as a net of spring elements representing the cardiomyocytes together with the endomysial collagen distribution. Each element is a viscous elastic spring, characterized by an elastic and a viscous constant. We connect these springs to imitate the interconnections between collagen fibers. Then we apply to the net of springs some external forces of suitable magnitude and direction, obtaining an extension of the net itself. In our setting, the ratio forces magnitude /net extension is intended to model the stress /strain ratio of a microscopical portion of the myocardial tissue. To solve the problem of the correct identification of the values of the different parameters involved, we use an artificial neural network approach. In particular, we use this technique to learn, given a distribution of external forces, the elastic constants of the springs needed to obtain a desired extension as an equilibrium position. Our experimental findings show that, in the model of collagen distribution structured according to the new view, a given stress /strain ratio (of the net of springs, in the sense specified above) is obtained with much smaller (w.r.t. the other model, corresponding to the traditional view) elasticity constants of the springs. This seems to indicate that by an appropriate structure, a given stiffness of the myocardial tissue can be obtained with endomysial collagen fibers of much smaller size.

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.

Transitions in ventricular activation revealed by two-dimensional optical mapping

While cardiac function in the mature heart is dependent on a properly functioning His-Purkinje system, the early embryonic tubular heart efficiently pumps blood without a distinct specialized conduction system. Although His-Purkinje system precursors have been identified using immunohistological techniques in the looped heart, little is known whether these precursors function electrically. To address this question, we used high-resolution optical mapping and fluorescent dyes with two CCD cameras to describe the motion-corrected activation patterns of 76 embryonic chick hearts from tubular stages (stage 10) to mature septated hearts (stage 35). Ventricular activation in the tubular looped heart (stages 10-17) using both calcium-sensitive fluo-4 and voltage-sensitive di-4-ANEPPS shows sequentially uniform propagation. In late looped hearts (stages 18-22), domains of the dorsal and lateral ventricle are preferentially activated before spreading to the remaining myocardium and show alternating regions of fast and slow propagation. During stages 22-26, action potentials arise from the dorsal ventricle. By stages 27-29, action potential breakthrough is also observed at the right ventricle apex. By stage 31, activation of the heart proceeds from foci at the apex and dorsal surface of the heart. The breakthrough foci correspond to regions where putative conduction system precursors have been identified immunohistologically. To date, our study represents the most detailed electrophysiological characterization of the embryonic heart between the looped and preseptated stages and suggests that ventricular activation undergoes a gradual transformation from sequential to a mature pattern with right and left epicardial breakthroughs. Our investigation suggests that cardiac conduction system precursors may be electrophysiologically distinct and mature gradually throughout cardiac morphogenesis in the chick.

Mechanisms of pattern formation in development and evolution

We present a classification of developmental mechanisms that have been shown experimentally to generate pattern and form in metazoan organisms. We propose that all such mechanisms can be organized into three basic categories and that two of these may act as composite mechanisms in two different ways. The simple categories are cell autonomous mechanisms in which cells enter into specific arrangements ('patterns') without interacting, inductive mechanisms in which cell communication leads to changes in pattern by reciprocal or hierarchical alteration of cell phenotypes ('states') and morphogenetic mechanisms in which pattern changes by means of cell interactions that do not change cell states. The latter two types of mechanism can be combined either morphostatically, in which case inductive mechanisms act first, followed by the morphogenetic mechanism, or morphodynamically, in which case both types of mechanisms interact continuously to modify each other's dynamics. We propose that this previously unexplored distinction in the operation of composite developmental mechanisms provides insight into the dynamics of many developmental processes. In particular, morphostatic and morphodynamic mechanisms respond to small changes in their genetic and microenvironmental components in dramatically different ways. We suggest that these differences in 'variational properties' lead to morphostatic and morphodynamic mechanisms being represented to different extents in early and late stages of development and to their contributing in distinct ways to morphological transitions in evolution.