A new cell-based FE model for the mechanics of embryonic epithelia

In order to overcome a significant stiffening artefact associated with current finite element (FE) models for the mechanics of embryonic epithelia, two new FE formulations were developed. Cell–cell interfacial tensions γ are represented by constant-force rod elements as in previous models. However, the viscosity of the cytoplasm with its embedded organelles and filament networks is modeled using viscous triangular elements, it is modeled using either radial and circumferential dashpots or an orthogonal dashpot system rather than the viscous triangular elements typical of previous two-dimensional FE models. The models are tested against tissue (epithelium) stretching because it gives rise to significant changes in cell shape and against cell sorting because it involves high rates of cell rearrangement. The orthogonal dashpot system is found to capture cell size and shape effects well, give the model cells characteristics that are consistent with those of real cells, provide high computational efficiency and hold promise for future three-dimensional analyses.     

A three-dimensional finite element model for the mechanics of cell-cell interactions

Technical challenges, including significant ones associated with cell rearrangement, have hampered the development of three-dimensional finite element models for the mechanics of embryonic cells. These challenges have been overcome by a new formulation in which the contents of each cell, assumed to have a viscosity mu, are modeled using a system of orthogonal dashpots. This approach overcomes a stiffening artifact that affects more traditional models, in which space-filling viscous elements are used to model the cytoplasm. Cells are assumed to be polyhedral in geometry, and each n-sided polygonal face is subdivided into n triangles with a common node at the face center so that it needs not remain flat. A constant tension gamma is assumed to act along each cell-cell interface, and cell rearrangements occur through one of two complementary topological transformations. The formulation predicts mechanical interactions between pairs of similar or dissimilar cells that are consistent with experiments, two-dimensional simulations, contact angle theory, and intracellular pressure calculations. Simulations of the partial engulfment of one tissue type by another show that the formulation is able to model aggregates of several hundred cells without difficulty. Simulations carried out using this formulation suggest new experimental approaches for measuring cell surface tensions and interfacial tensions. The formulation holds promise as a tool for gaining insight into the mechanics of isolated or aggregated embryonic cells and for the design and interpretation of experiments that involve them.     

Cell-level finite element studies of viscous cells in planar aggregates

A new cell-level finite element formulation is presented and used to investigate how epithelia and other planar collections of viscous cells might deform during events such as embryo morphogenesis and wound healing. Forces arising from cytoskeletal components, cytoplasm viscosity, and cell-cell adhesions are included. Individual cells are modeled using multiple finite elements, and cell rearrangements can occur. Simulations of cell-sheet stretching indicate that the initial stages of sheet stretching are characterized by changes in cell shape, while subsequent stages are governed by cell rearrangement. Inferences can be made from the simulations about the forces that act in real cell sheets when suitable experimental data are available.     

Asymmetric cell division of a triangular halophilic archaebacterium

Abstract The cell division of the halophilic archaebacterium, Haloarcula japonicus , which has a characteristic triangular shape in high salt concentration media, was analysed by time lapse microscopic cinematography. Cell division on an agar medium occured on average every 3.7 h. Cell plates were laid down asymmetrically, generating triangular or rhomboid shape daughter cells which then separated. Cell plate formation was clearly observed because the cells are flat and thin enough to see through using a conventional light microscope.     

A Mechanistic Collective Cell Model for Epithelial Colony Growth and Contact Inhibition

We present a mechanistic hybrid continuum-discrete model to simulate the dynamics of epithelial cell colonies. Collective cell dynamics are modeled using continuum equations that capture plastic, viscoelastic, and elastic deformations in the clusters while providing single-cell resolution. The continuum equations can be viewed as a coarse-grained version of previously developed discrete models that treat epithelial clusters as a two-dimensional network of vertices or stochastic interacting particles and follow the framework of dynamic density functional theory appropriately modified to account for cell size and shape variability. The discrete component of the model implements cell division and thus influences cell size and shape that couple to the continuum component. The model is validated against recent in vitro studies of epithelial cell colonies using Madin-Darby canine kidney cells. In good agreement with experiments, we find that mechanical interactions and constraints on the local expansion of cell size cause inhibition of cell motion and reductive cell division. This leads to successively smaller cells and a transition from exponential to quadratic growth of the colony that is associated with a constant-thickness rim of growing cells at the cluster edge, as well as the emergence of short-range ordering and solid-like behavior. A detailed analysis of the model reveals a scale invariance of the growth and provides insight into the generation of stresses and their influence on the dynamics of the colonies. Compared to previous models, our approach has several advantages: it is independent of dimension, it can be parameterized using classical elastic properties (Poisson’s ratio and Young’s modulus), and it can easily be extended to incorporate multiple cell types and general substrate geometries.     

Comparison of analytical and inverse finite element approaches to estimate cell viscoelastic properties by micropipette aspiration

The viscoelastic properties of cells are important in predicting cell deformation under mechanical loading and may reflect cell phenotype or pathological transition. Previous studies have demonstrated that viscoelastic parameters estimated by finite element (FE) analyses of micropipette aspiration (MA) data differ from those estimated by the analytical half-space model. However, it is unclear whether these differences are statistically significant, as previous studies have been based on average cell properties or parametric analyses that do not reflect the inherent experimental and biological variability of real experimental data. To determine whether cell material parameters estimated by the half-space model are significantly different from those predicted by the FE method, we implemented an inverse FE method to estimate the viscoelastic parameters of a population of primary porcine aortic valve interstitial cells tested by MA. We found that inherent differences between the analytical and inverse FE estimation methods resulted in statistically significant differences in individual cell properties. However, in cases with small pipette to cell radius ratios and short loading periods, model-dependent differences were masked by experimental and cell-to-cell variability. Analytical models that account for finite cell-size and loading rate further relaxed the experimental conditions for which accurate cell material parameter estimates could be obtained. These data provide practical guidelines for analysis of MA data that account for the wide range of conditions encountered in typical experiments.     

A power-law rheology-based finite element model for single cell deformation

Physical forces can elicit complex time- and space-dependent deformations in living cells. These deformations at the subcellular level are difficult to measure but can be estimated using computational approaches such as finite element (FE) simulation. Existing FE models predominantly treat cells as spring-dashpot viscoelastic materials, while broad experimental data are now lending support to the power-law rheology (PLR) model. Here, we developed a large deformation FE model that incorporated PLR and experimentally verified this model by performing micropipette aspiration on fibroblasts under various mechanical loadings. With a single set of rheological properties, this model recapitulated the diverse micropipette aspiration data obtained using three protocols and with a range of micropipette sizes. More intriguingly, our analysis revealed that decreased pipette size leads to increased pressure gradient, potentially explaining our previous counterintuitive finding that decreased pipette size leads to increased incidence of cell blebbing and injury. Taken together, our work leads to more accurate rheological interpretation of micropipette aspiration experiments than previous models and suggests pressure gradient as a potential determinant of cell injury.     

Stretching and Relaxation of Malaria-Infected Red Blood Cells

The invasion of red blood cells (RBCs) by malaria parasites is a complex dynamic process, in which the infected RBCs gradually lose their deformability and their ability to recover their original shape is greatly reduced with the maturation of the parasites. In this work, we developed two types of cell model, one with an included parasite, and the other without an included parasite. The former is a representation of real malaria-infected RBCs, in which the parasite is treated as a rigid body. In the latter, where the parasite is absent, the membrane modulus and viscosity are elevated so as to produce the same features present in the parasite model. In both cases, the cell membrane is modeled as a viscoelastic triangular network connected by wormlike chains. We studied the transient behaviors of stretching deformation and shape relaxation of malaria-infected RBCs based on these two models and found that both models can generate results in agreement with those of previously published studies. With the parasite maturation, the shape deformation becomes smaller and smaller due to increasing cell rigidity, whereas the shape relaxation time becomes longer and longer due to the cell’s reduced ability to recover its original shape.     

Stretching and Relaxation of Malaria-Infected Red Blood Cells

The invasion of red blood cells (RBCs) by malaria parasites is a complex dynamic process, in which the infected RBCs gradually lose their deformability and their ability to recover their original shape is greatly reduced with the maturation of the parasites. In this work, we developed two types of cell model, one with an included parasite, and the other without an included parasite. The former is a representation of real malaria-infected RBCs, in which the parasite is treated as a rigid body. In the latter, where the parasite is absent, the membrane modulus and viscosity are elevated so as to produce the same features present in the parasite model. In both cases, the cell membrane is modeled as a viscoelastic triangular network connected by wormlike chains. We studied the transient behaviors of stretching deformation and shape relaxation of malaria-infected RBCs based on these two models and found that both models can generate results in agreement with those of previously published studies. With the parasite maturation, the shape deformation becomes smaller and smaller due to increasing cell rigidity, whereas the shape relaxation time becomes longer and longer due to the cell’s reduced ability to recover its original shape.     

Confocal-based cell-specific finite element modeling extended to study variable cell shapes and intracellular structures: The example of the adipocyte

This communication extends the recently reported cell-specific finite element (FE) method in Slomka and Gefen (2010) in which geometrically realistic FE cell models are created from confocal microscopy scans for large deformation analyses. The cell-specific FE method is extended here in the following aspects: (i) we demonstrate that cell-specific FE is versatile enough to deal with cells of substantially different geometrical shapes. The examples of an “elongated” pre-adipocyte and a “round” mature adipocyte are used to demonstrate this feature. (ii) We demonstrate that cell-specific FE can be used to analyze the mechanical behavior of cells that incorporate complex intracellular structures and are subjected to large deformations—again through the example of an adipocyte which contains a multitude of lipid droplets, each having a different size and shape. By demonstrating feasibility of inclusion of such inhomogeneities in the cytoplasm, the present work paves the way for modeling cellular organelles such as Golgi bodies, lysosomes and mitochondria in mechanically loaded cells using cell-specific FE.     

A Gauss-Kronrod-Trapezoidal integration scheme for modeling biological tissues with continuous fiber distributions

Fibrous biological tissues may be modeled using a continuous fiber distribution (CFD) to capture tension–compression nonlinearity, anisotropic fiber distributions, and load-induced anisotropy. The CFD framework requires spherical integration of weighted individual fiber responses, with fibers contributing to the stress response only when they are in tension. The common method for performing this integration employs the discretization of the unit sphere into a polyhedron with nearly uniform triangular faces (finite element integration or FEI scheme). Although FEI has proven to be more accurate and efficient than integration using spherical coordinates, it presents three major drawbacks: First, the number of elements on the unit sphere needed to achieve satisfactory accuracy becomes a significant computational cost in a finite element (FE) analysis. Second, fibers may not be in tension in some regions on the unit sphere, where the integration becomes a waste. Third, if tensed fiber bundles span a small region compared to the area of the elements on the sphere, a significant discretization error arises. This study presents an integration scheme specialized to the CFD framework, which significantly mitigates the first drawback of the FEI scheme, while eliminating the second and third completely. Here, integration is performed only over the regions of the unit sphere where fibers are in tension. Gauss–Kronrod quadrature is used across latitudes and the trapezoidal scheme across longitudes. Over a wide range of strain states, fiber material properties, and fiber angular distributions, results demonstrate that this new scheme always outperforms FEI, sometimes by orders of magnitude in the number of computational steps and relative accuracy of the stress calculation.     

The Y1 receptor for NPY: a key modulator of the adaptive immune system

Growing evidence suggests that the neuropeptide Y (NPY) system plays an important role in the immune system. Yet, little is known about the expression of NPY and receptors in the immune system. Moreover, original contradicting results have confused the picture and hampered a clear understanding of its role in the immune system. The use of Y(1) receptor-deficient mice, combined with advanced methods to investigate immune functions, have provided the solution to the problem raised by previous disparities. From results obtained using Y(1)-deficient mice (Y(1)(-/-)), we uncovered a bimodal role for Y(1) on immune cells. Y(1) expression on antigen-presenting cells (APC) is essential for their function as T cell priming elements. Conversely, Y(1) signaling in T cells plays a regulatory role without which T cells are hyper-responsive. The opposite role of Y(1) on APC and T cells has reconciled previous disparities by showing that signaling via Y(1) protects against inflammation by inhibiting T cell responses, whereas Y(1)(-/-) mice are protected in the same inflammatory models due to defective APCs.     

Motility of the spirochete Leptospira

Spirochetes are a group of bacteria with a unique ultrastructure and a fascinating swimming behavior. This article reviews the hydrodynamics of spirochete motility, and examines the motility of the spirochete Leptospira in detail. Models of Leptospira motility are discussed, and future experiments are proposed. The outermost structure of Leptospira is a membrane sheath, and within this sheath are a helically shaped cell cylinder and two periplasmic flagella. One periplasmic flagellum is attached subterminally at either end of the cell cylinder and extends partway down the length of the cell. In swimming cells, each end of the cell may assume either a spiral or a hook shape. Translational cells have the anterior end spiral shaped, and the posterior end hook shaped. In the model of Berg et al., the periplasmic flagella are believed to rotate between the sheath and the cell cylinder. Rotation of the anterior periplasmic flagellum causes the generation of a gyrating spiral-shaped wave. This wave is believed sufficient to propel the cells forward in a low-viscosity medium. The cell cylinder concomitantly rolls around the periplasmic flagella in the opposite direction--which allows the cell to literally screw through a gel-like viscous medium without slippage. This model is presented, and it is contrasted to previous models of Leptospira motility.     

Prediction of femoral strength using 3D finite element models reconstructed from DXA images: validation against experiments

Computed tomography (CT)-based finite element (FE) models may improve the current osteoporosis diagnostics and prediction of fracture risk by providing an estimate for femoral strength. However, the need for a CT scan, as opposed to the conventional use of dual-energy X-ray absorptiometry (DXA) for osteoporosis diagnostics, is considered a major obstacle. The 3D shape and bone mineral density (BMD) distribution of a femur can be reconstructed using a statistical shape and appearance model (SSAM) and the DXA image of the femur. Then, the reconstructed shape and BMD could be used to build FE models to predict bone strength. Since high accuracy is needed in all steps of the analysis, this study aimed at evaluating the ability of a 3D FE model built from one 2D DXA image to predict the strains and fracture load of human femora. Three cadaver femora were retrieved, for which experimental measurements from ex vivo mechanical tests were available. FE models were built using the SSAM-based reconstructions: using only the SSAM-reconstructed shape, only the SSAM-reconstructed BMD distribution, and the full SSAM-based reconstruction (including both shape and BMD distribution). When compared with experimental data, the SSAM-based models predicted accurately principal strains (coefficient of determination >0.83, normalized root-mean-square error <16%) and femoral strength (standard error of the estimate 1215 N). These results were only slightly inferior to those obtained with CT-based FE models, but with the considerable advantage of the models being built from DXA images. In summary, the results support the feasibility of SSAM-based models as a practical tool to introduce FE-based bone strength estimation in the current fracture risk diagnostics.     

Integrating force-sensing and signaling pathways in a model for the regulation of wing imaginal disc size

The regulation of organ size constitutes a major unsolved question in developmental biology. The wing imaginal disc of Drosophila serves as a widely used model system to study this question. Several mechanisms have been proposed to have an impact on final size, but they are either contradicted by experimental data or they cannot explain a number of key experimental observations and may thus be missing crucial elements. We have modeled a regulatory network that integrates the experimentally confirmed molecular interactions underlying other available models. Furthermore, the network includes hypothetical interactions between mechanical forces and specific growth regulators, leading to a size regulation mechanism that conceptually combines elements of existing models, and can be understood in terms of a compression gradient model. According to this model, compression increases in the center of the disc during growth. Growth stops once compression levels in the disc center reach a certain threshold and the compression gradient drops below a certain level in the rest of the disc. Our model can account for growth termination as well as for the paradoxical observation that growth occurs uniformly in the presence of a growth factor gradient and non-uniformly in the presence of a uniform growth factor distribution. Furthermore, it can account for other experimental observations that argue either in favor or against other models. The model also makes specific predictions about the distribution of cell shape and size in the developing disc, which we were able to confirm experimentally.     

Differential polarization imaging. III. Theory confirmation. Patterns of polymerization of hemoglobin S in red blood sickle cells.

In this paper we test the predictions of the differential polarization imaging theory developed in the previous two papers. A characterization of the patterns of polymerization of hemoglobin in red blood cells from patients with sickle cell anemia is presented. This system was chosen because it is relatively easy to handle and because previous studies have been done on it. A differential polarization microscope designed and built in our laboratory was used to carry out this study. This microscope uses an image dissector camera, a photoelastic modulator, and a phase-lock amplifier. This design represents a substantial modification with respect to the instrumentation used in the previous results communicated on this system. Therefore, the results presented here also permit us to confirm the validity of our conclusions. On the basis of the differential polarization images obtained, models of the patterns of polymerization of the hemoglobin S inside the sickle cells are proposed and their M12 and regular images are calculated by the theory. Good agreement between those models and the experimental systems is found, as well as with the results previously reported.     

Statistical shape and appearance models for fast and automated estimation of proximal femur fracture load using 2D finite element models

Estimating the risk of osteoporotic fractures is an important diagnostic step that needs to be taken before medicinal treatment. Densitometry-based criteria are normally used in clinical practice for this purpose. However, densitometry-based techniques could not explain all low-energy fractures. As patient-specific finite element (FE) models allow for consideration of other parameters (e.g. load conditions) that are known to be associated with fracture, they are considered promising candidates for more accurate fracture risk estimation. Nevertheless, they are often time consuming, expensive, and complex to build and may need the type of expertise that is not normally available in clinical settings. In this study, we report the development of an automated platform for estimating proximal femur fracture loads using patient-specific 2D FE models generated using dual-energy x-ray absorptiometry (DXA) scans. First, a statistical shape and appearance model (SSAM) is built using DXA scans of patients screened for osteoporosis following a low energy fracture. SSAM is then used together with Active Appearance Models (AAM) for automated segmentation of the proximal femur from new unseen DXA scans. The mean point-to-curve error of the automated procedure, i.e. 1.2–1.4 mm, is shown to be only slightly larger than the intra-observer variability of manual segmentation, i.e. 1.0 mm. Moreover, the developed platform automatically meshes the segmented shape, assigns density-based mechanical properties, assigns loads and boundary conditions, submits the 2D FE model for solution, and performs post-processing of the 2D FE simulation data to determine fracture loads. The fracture loads predicted using the manually generated and automatically generated 2D FE models are shown to be very close with a mean difference of around 8.8%. Repeated measures ANOVA showed no significant differences between the fracture loads calculated using FE models manually generated by three independent observers and those calculated using the automatically generated FE models (p>0.05).     

Three-Dimensional Numerical Model of Cell Morphology during Migration in Multi-Signaling Substrates

Cell Migration associated with cell shape changes are of central importance in many biological processes ranging from morphogenesis to metastatic cancer cells. Cell movement is a result of cyclic changes of cell morphology due to effective forces on cell body, leading to periodic fluctuations of the cell length and cell membrane area. It is well-known that the cell can be guided by different effective stimuli such as mechanotaxis, thermotaxis, chemotaxis and/or electrotaxis. Regulation of intracellular mechanics and cell’s physical interaction with its substrate rely on control of cell shape during cell migration. In this notion, it is essential to understand how each natural or external stimulus may affect the cell behavior. Therefore, a three-dimensional (3D) computational model is here developed to analyze a free mode of cell shape changes during migration in a multi-signaling micro-environment. This model is based on previous models that are presented by the same authors to study cell migration with a constant spherical cell shape in a multi-signaling substrates and mechanotaxis effect on cell morphology. Using the finite element discrete methodology, the cell is represented by a group of finite elements. The cell motion is modeled by equilibrium of effective forces on cell body such as traction, protrusion, electrostatic and drag forces, where the cell traction force is a function of the cell internal deformations. To study cell behavior in the presence of different stimuli, the model has been employed in different numerical cases. Our findings, which are qualitatively consistent with well-known related experimental observations, indicate that adding a new stimulus to the cell substrate pushes the cell to migrate more directionally in more elongated form towards the more effective stimuli. For instance, the presence of thermotaxis, chemotaxis and electrotaxis can further move the cell centroid towards the corresponding stimulus, respectively, diminishing the mechanotaxis effect. Besides, the stronger stimulus imposes a greater cell elongation and more cell membrane area. The present model not only provides new insights into cell morphology in a multi-signaling micro-environment but also enables us to investigate in more precise way the cell migration in the presence of different stimuli.     

Modeling cell apoptosis for simulating three-dimensional multicellular morphogenesis based on a reversible network reconnection framework

Morphogenesis in multicellular organisms is accompanied by apoptotic cell behaviors: cell shrinkage and cell disappearance. The mechanical effects of these behaviors are spatiotemporally regulated within multicellular dynamics to achieve proper tissue sizes and shapes in three-dimensional (3D) space. To analyze 3D multicellular dynamics, 3D vertex models have been suggested, in which a reversible network reconnection (RNR) model has successfully expressed 3D cell rearrangements during large deformations. To analyze the effects of apoptotic cell behaviors on 3D multicellular morphogenesis, we modeled cell apoptosis based on the RNR model framework. Cell shrinkage was modeled by the potential energy as a function of individual cell times during the apoptotic phase. Cell disappearance was modeled by merging neighboring polyhedrons at their boundary surface according to the topological rules of the RNR model. To establish that the apoptotic cell behaviors could be expressed as modeled, we simulated morphogenesis driven by cell apoptosis in two types of tissue topology: 3D monolayer cell sheet and 3D compacted cell aggregate. In both types of tissue topology, the numerical simulations successfully illustrated that cell aggregates gradually shrank because of successive cell apoptosis. During tissue shrinkage, the number of cells in aggregates decreased while maintaining individual cell size and shape. Moreover, in case of localizing apoptotic cells within a part of the 3D monolayer cell aggregate, the cell apoptosis caused the global tissue bending by pulling on surrounding cells. In case of localizing apoptotic cells on the surface of the 3D compacted cell aggregate, the cell apoptosis caused successive, directional cell rearrangements from the inside to the surface. Thus, the proposed model successfully provided a basis for expressing apoptotic cell behaviors during 3D multicellular morphogenesis based on an RNR model framework.     

Differential polarization imaging. III. Theory confirmation. Patterns of polymerization of hemoglobin S in red blood sickle cells.

In this paper we test the predictions of the differential polarization imaging theory developed in the previous two papers. A characterization of the patterns of polymerization of hemoglobin in red blood cells from patients with sickle cell anemia is presented. This system was chosen because it is relatively easy to handle and because previous studies have been done on it. A differential polarization microscope designed and built in our laboratory was used to carry out this study. This microscope uses an image dissector camera, a photoelastic modulator, and a phase-lock amplifier. This design represents a substantial modification with respect to the instrumentation used in the previous results communicated on this system. Therefore, the results presented here also permit us to confirm the validity of our conclusions. On the basis of the differential polarization images obtained, models of the patterns of polymerization of the hemoglobin S inside the sickle cells are proposed and their M12 and regular images are calculated by the theory. Good agreement between those models and the experimental systems is found, as well as with the results previously reported.