Constitutive material modeling of cell: a micromechanics approach

Strain-dependent elastography of cancer cells reveals heterogeneity and stiffening due to attachment

Because cells vary in thickness and in biomechanical properties, the use of a constant force trigger during atomic force microscopy (AFM) stiffness mapping produces a varied nominal strain that can obfuscate the comparison of local material properties. In this study, we measured the biomechanical spatial heterogeneity of ovarian and breast cancer cells by using an indentation-dependent pointwise Hertzian method. Force curves and surface topography were used together to determine cell stiffness as a function of nominal strain. By recording stiffness at a particular strain, it may be possible to improve comparison of the material properties of cells and produce higher contrast representations of cell mechanical properties. Defining a linear region of elasticity that corresponds to a modest nominal strain, we were able to clearly distinguish the mechanics of the perinuclear region of cells. We observed that, relative to the lamelopodial stiffness, the perinuclear region was softer for metastatic cancer cells than their nonmetastatic counterparts. Moreover, contrast in the strain-dependent elastography in comparison to conventional force mapping with Hertzian model analysis revealed a significant stiffening phenomenon in the thin lamellipodial region in which the modulus scales inversely and exponentially with cell thickness. The observed exponential stiffening is not affected by relaxation of cytoskeletal tension, but finite element modeling indicates it is affected by substrate adhesion. The novel cell mapping technique explores cancer cell mechanical nonlinearity that results from regional heterogeneity, which could help explain how metastatic cancer cells can show soft phenotypes while simultaneously increasing force generation and invasiveness.

Impact of physiological loads of arterial wall on nucleus deformation in endothelial cells: A computational study

INTRODUCTION

Computational modeling can enhance the understanding of cell mechanics. To achieve this, finite element models of endothelial cells were proposed with shapes mimicking their natural state inside the endothelium within the cardiovascular system. Implementing the recently proposed bendo-tensegrity concept, these models consider flexural (buckling) as well as tensional/compressional behavior of microtubules and also incorporate the waviness of intermediate filaments.

MATERIALS AND METHODS

Four different models were created (flat and domed hexagons, both regular and elongated in the direction of blood flow) and loaded by biaxial deformation, blood pressure, and shear load from blood flow - natural physiological conditions of the arterial endothelium - aiming to investigate the "in situ" mechanical response of the cell.

RESULTS

The impact of individual components of loads on the nucleus deformation (more specifically on the first principal strain) potentially influencing mechanotransduction was investigated and the role of the cytoskeleton and its constituents in the mechanical response of the endothelial cell was assessed. The results show (i) the impact of pulsating blood pressure on cyclic deformations of the nucleus, which increase substantially with decreasing axial pre-stretch of the cell, (ii) the importance of relatively low shear stresses in the cell response and nucleus deformation.

CONCLUSION

Not only the pulsatile blood pressure but also the wall shear stress may induce significant deformation of the nucleus and thus trigger remodelation processes in endothelial cells.

Finite Element Simulations of Mechanical Behaviour of Endothelial Cells

Biomechanical models based on the finite element method have already shown their potential in the simulation of the mechanical behaviour of cells. For instance, development of atherosclerosis is accelerated by damage of the endothelium, a monolayer of endothelial cells on the inner surface of arteries. Finite element models enable us to investigate mechanical factors not only at the level of the arterial wall but also at the level of individual cells. To achieve this, several finite element models of endothelial cells with different shapes are presented in this paper. Implementing the recently proposed bendotensegrity concept, these models consider the flexural behaviour of microtubules and incorporate also waviness of intermediate filaments. The suspended and adherent cell models are validated by comparison of their simulated force-deformation curves with experiments from the literature. The flat and dome cell models, mimicking natural cell shapes inside the endothelial layer, are then used to simulate their response in compression and shear which represent typical loads in a vascular wall. The models enable us to analyse the role of individual cytoskeletal components in the mechanical responses, as well as to quantify the nucleus deformation which is hypothesized to be the quantity decisive for mechanotransduction.

Biomechanical Heterogeneity of Living Cells: Comparison between Atomic Force Microscopy and Finite Element Simulation

Atomic force microscopy (AFM) indentation is a popular method for characterizing the micromechanical properties of soft materials such as living cells. However, the mechanical data obtained from deep indentation measurements can be difficult and problematic to interpret as a result of the complex geometry of a cell, the nonlinearity of indentation contact, and constitutive relations of heterogeneous hyperelastic soft components. Living MDA-MB-231 cells were indented by spherical probes to obtain morphological and mechanical data that were adopted to build an accurate finite element model (FEM) for a parametric study. Initially, a 2D-axisymmetric numerical model was constructed with the main purpose of understanding the effect of geometrical and mechanical properties of constitutive parts such as the cell body, nucleus, and lamellipodium. A series of FEM deformation fields were directly compared with atomic force spectroscopy in order to resolve the mechanical convolution of heterogeneous parts and quantify Young's modulus and the geometry of nuclei. Furthermore, a 3D finite element model was constructed to investigate indentation events located far from the axisymmetric geometry. In this framework, the joint FEM/AFM approach has provided a useful methodology and a comprehensive characterization of the heterogeneous structure of living cells, emphasizing the deconvolution of geometrical structure and the true elastic modulus of the cell nucleus.

A shear assay study of single normal/breast cancer cell deformation and detachment from poly-di-methyl-siloxane (PDMS) surfaces

This paper presents the results of a combined experimental and analytical/computational study of viscoelastic cell deformation and detachment from poly-di-methyl-siloxane (PDMS) surfaces. Fluid mechanics and fracture mechanics concepts are used to model the detachment of biological cells observed under shear assay conditions. The analytical and computational models are used to compute crack driving forces, which are then related to crack extension during the detachment of normal breast cells and breast cancer cells from PDMS surfaces that are relevant to biomedical implants. The interactions between cells and the extracellular matrix, or the extracellular matrix and the PDMS substrate, are then characterized using force microscopy measurements of the pull-off forces that are used to determine the adhesion energies. Finally, fluorescence microscopy staining of the cytosketelal structures (actin, micro-tubulin and cyto-keratin), transmembrane proteins (vimentin) and the ECM structures (Arginin Glycine Aspartate - RGD) is used to show that the detachment of cells during the shear assay experiments occurs via interfacial cracking between (between the ECM and the cell membranes) with a high incidence of crack bridging by transmembrane vinculin structures that undergo pull-out until they detach from the actin cytoskeletal structure. The implications of the results are discussed for the design of interfaces that are relevant to implantable biomedical devices and normal/cancer tissue.

Multiscale FEM simulations of cross-linked actin network embedded in cytosol with the focus on the filament orientation

The present contribution focuses on the application of the multiscale finite element method to the modeling of actin networks that are embedded in the cytosol. These cell components are of particular importance with regard to the cell response to external stimuli. The homogenization strategy chosen uses the Hill-Mandel macrohomogeneity condition for bridging 2 scales: the macroscopic scale that is related to the cell level and the microscopic scale related to the representative volume element. For the modeling of filaments, the Holzapfel-Ogden β-model is applied. It provides a relationship between the tensile force and the caused stretches, serves as the basis for the derivation of the stress and elasticity tensors, and enables a novel finite element implementation. The elements with the neo-Hookean constitutive law are applied for the simulation of the cytosol. The results presented corroborate the main advantage of the concept, namely, its flexibility with regard to the choice of the representative volume element as well as of macroscopic tests. The focus is particularly placed on the study of the filament orientation and of its influence on the effective behavior.

Nuclear lamin A/C harnesses the perinuclear apical actin cables to protect nuclear morphology

The distinct spatial architecture of the apical actin cables (or actin cap) facilitates rapid biophysical signaling between extracellular mechanical stimuli and intracellular responses, including nuclear shaping, cytoskeletal remodeling, and the mechanotransduction of external forces into biochemical signals. These functions are abrogated in lamin A/C-deficient mouse embryonic fibroblasts that recapitulate the defective nuclear organization of laminopathies, featuring disruption of the actin cap. However, how nuclear lamin A/C mediates the ability of the actin cap to regulate nuclear morphology remains unclear. Here, we show that lamin A/C expressing cells can form an actin cap to resist nuclear deformation in response to physiological mechanical stresses. This study reveals how the nuclear lamin A/C-mediated formation of the perinuclear apical actin cables protects the nuclear structural integrity from extracellular physical disturbances. Our findings highlight the role of the physical interactions between the cytoskeletal network and the nucleus in cellular mechanical homeostasis.

Vaginal Changes Due to Varying Degrees of Rectocele Prolapse: A Computational Study

Pelvic organ prolapse (POP), downward descent of the pelvic organs resulting in a protrusion of the vagina, is a highly prevalent condition, responsible for 300,000 surgeries in the U.S. annually. Rectocele, a posterior vaginal wall (PVW) prolapse of the rectum, is the second most common type of POP after cystocele. A rectocele usually manifests itself along with other types of prolapse with multicompartment pelvic floor defects. To date, the specific mechanics of rectocele formation are poorly understood, which does not allow its early stage detection and progression prediction over time. Recently, with the advancement of imaging and computational modeling techniques, a plethora of finite element (FE) models have been developed to study vaginal prolapse from different perspectives and allow a better understanding of dynamic interactions of pelvic organs and their supporting structures. So far, most studies have focused on anterior vaginal prolapse (AVP) (or cystocele) and limited data exist on the role of pelvic muscles and ligaments on the development and progression of rectocele. In this work, a full-scale magnetic resonance imaging (MRI) based three-dimensional (3D) computational model of the female pelvic anatomy, comprising the vaginal canal, uterus, and rectum, was developed to study the effect of varying degrees (or sizes) of rectocele prolapse on the vaginal canal for the first time. Vaginal wall displacements and stresses generated due to the varying rectocele size and average abdominal pressures were estimated. Considering the direction pointing from anterior to posterior side of the pelvic system as the positive Y-direction, it was found that rectocele leads to negative Y-direction displacements, causing the vaginal cross section to shrink significantly at the lower half of the vaginal canal. Besides the negative Y displacements, the rectocele bulging was observed to push the PVW downward toward the vaginal hiatus, exhibiting the well-known "kneeling effect." Also, the stress field on the PVW was found to localize at the upper half of the vaginal canal and shift eventually to the lower half with increase in rectocele size. Additionally, clinical relevance and implications of the results were discussed.

An integrated enhancement and reconstruction strategy for the quantitative extraction of actin stress fibers from fluorescence micrographs

BACKGROUND

The stress fibers are prominent organization of actin filaments that perform important functions in cellular processes such as migration, polarization, and traction force generation, and whose collective organization reflects the physiological and mechanical activities of the cells. Easily visualized by fluorescence microscopy, the stress fibers are widely used as qualitative descriptors of cell phenotypes. However, due to the complexity of the stress fibers and the presence of other actin-containing cellular features, images of stress fibers are relatively challenging to quantitatively analyze using previously developed approaches, requiring significant user intervention. This poses a challenge for the automation of their detection, segmentation, and quantitative analysis.

RESULT

Here we describe an open-source software package, SFEX (Stress Fiber Extractor), which is geared for efficient enhancement, segmentation, and analysis of actin stress fibers in adherent tissue culture cells. Our method made use of a carefully chosen image filtering technique to enhance filamentous structures, effectively facilitating the detection and segmentation of stress fibers by binary thresholding. We subdivided the skeletons of stress fiber traces into piecewise-linear fragments, and used a set of geometric criteria to reconstruct the stress fiber networks by pairing appropriate fiber fragments. Our strategy enables the trajectory of a majority of stress fibers within the cells to be comprehensively extracted. We also present a method for quantifying the dimensions of the stress fibers using an image gradient-based approach. We determine the optimal parameter space using sensitivity analysis, and demonstrate the utility of our approach by analyzing actin stress fibers in cells cultured on various micropattern substrates.

CONCLUSION

We present an open-source graphically-interfaced computational tool for the extraction and quantification of stress fibers in adherent cells with minimal user input. This facilitates the automated extraction of actin stress fibers from fluorescence images. We highlight their potential uses by analyzing images of cells with shapes constrained by fibronectin micropatterns. The method we reported here could serve as the first step in the detection and characterization of the spatial properties of actin stress fibers to enable further detailed morphological analysis.

Computational analysis of morphologies and phase transitions of cross-linked, semi-flexible polymer networks

The eukaryotic cytoskeleton is a protein fibre network mainly consisting of the semi-flexible biopolymer F-actin, microtubules and intermediate filaments. It is well known to exhibit a pronounced structural polymorphism, which enables intracellular processes such as cell adhesion, cell motility and cell division. We present a computational study on cross-linked networks of semi-flexible polymers, which offers a detailed analysis of the network structure and phase transitions from one morphology to another. We elaborate the morphological differences, their mechanical implications and the order of the observed phase transitions. Finally, we present a perspective on how the information gained in our simulations can be exploited in order to build both flexible and accurate, microstructurally informed, homogenized constitutive models of the cytoskeleton.

Coupled Simulation of Heart Valves: Applications to Clinical Practice

The last few decades have seen great advances in the understanding of heart valves, and consequently, in the development of novel treatment modalities and surgical procedures for valves afflicted by disease. This is due in part to the profound advancements in computing technology and noninvasive medical imaging techniques that have made it possible to numerically model the complex heart valve systems characterized by distinct features at different length scales and various interacting processes. In this article, we highlight the importance of explicitly coupling these multiple scales and diverse processes to accurately simulate the true behavior of the heart valves, in health and disease. We examine some of the computational modeling studies that have a direct consequence on clinical practice.

Structural finite element analysis to explain cell mechanics variability

The ability to model the mechanical responses of different cell types presents many opportunities to tissue engineering research to further identify changes from physiological conditions to disease. Using a previously validated finite element cell model we aim to show how variation of the material properties of the intracellular components affects cell response after compression and shearing. A parametric study was performed to understand the key mechanical features from different cell types, focussing on specific cytoskeleton components and prestress. Results show that actin cortex does not have a mechanical role in resisting shearing loading conditions. The sensitivity analysis predicted that cell force to compression and shearing is highly affected by changes in cortex thickness, cortex Young's modulus and rigidity of the remaining cytoplasm. Variation of prestress affects mainly the response of cells under shear loads and the model defines a relationship between cell force and prestress depending on the specific loading conditions, which is in good agreement with in vitro experiments. The results are used to make predictions that can relate mechanical properties with cell phenotype to be used as guidelines for individual cytoskeletal structures for future modelling efforts of the structure-function relationships of living cells.

A linear programming approach to reconstructing subcellular structures from confocal images for automated generation of representative 3D cellular models

This paper presents a novel computer vision algorithm to analyze 3D stacks of confocal images of fluorescently stained single cells. The goal of the algorithm is to create representative in silico model structures that can be imported into finite element analysis software for mechanical characterization. Segmentation of cell and nucleus boundaries is accomplished via standard thresholding methods. Using novel linear programming methods, a representative actin stress fiber network is generated by computing a linear superposition of fibers having minimum discrepancy compared with an experimental 3D confocal image. Qualitative validation is performed through analysis of seven 3D confocal image stacks of adherent vascular smooth muscle cells (VSMCs) grown in 2D culture. The presented method is able to automatically generate 3D geometries of the cell's boundary, nucleus, and representative F-actin network based on standard cell microscopy data. These geometries can be used for direct importation and implementation in structural finite element models for analysis of the mechanics of a single cell to potentially speed discoveries in the fields of regenerative medicine, mechanobiology, and drug discovery.

A computational approach to understand phenotypic structure and constitutive mechanics relationships of single cells

The goal of this study is to construct a representative 3D finite element model (FEM) of individual cells based on their sub-cellular structures that predicts cell mechanical behavior. The FEM simulations replicate atomic force microscopy (AFM) nanoindentation experiments on live vascular smooth muscle cells. Individual cells are characterized mechanically with AFM and then imaged in 3D using a spinning disc confocal microscope. Using these images, geometries for the FEM are automatically generated via image segmentation and linear programming algorithms. The geometries consist of independent structures representing the nucleus, actin stress fiber network, and cytoplasm. These are imported into commercial software for mesh refinement and analysis. The FEM presented here is capable of predicting AFM results well for 500 nm indentations. The FEM results are relatively insensitive to both the exact number and diameter of fibers used. Despite the localized nature of AFM nanoindentation, the model predicts that stresses are distributed in an anisotropic manner throughout the cell body via the actin stress fibers. This pattern of stress distribution is likely a result of the geometric arrangement of the actin network.

Advances in cationic photopolymerization

This review discusses cationic UV-curing processes of vinyl ethers, propenyl ethers, and epoxy monomers. Cationic photopolymerization based on photogeneration of acid from onium salts induced by UV light and consecutive polymerization initiated by photogenerated acid was first proposed at the end of the 1970s. The process engendered high interest both in academia and in industry. Cationic photoinduction presents some advantages over comparable radical-mediated processes, particularly the absence of inhibition by oxygen, low shrinkage, and good adhesion, and mechanical properties of the UV-cured materials. Moreover, the monomers employed are generally less toxic and irritant than acrylates and methacrylates, which are widely used in radical photopolymerization. In this overview, particular emphasis is given to our recent contributions to the field of cationic photopolymerization for different classes of monomers.

A finite element study of micropipette aspiration of single cells: effect of compressibility

Micropipette aspiration (MA) technique has been widely used to measure the viscoelastic properties of different cell types. Cells experience nonlinear large deformations during the aspiration procedure. Neo-Hookean viscohyperelastic (NHVH) incompressible and compressible models were used to simulate the creep behavior of cells in MA, particularly accounting for the effect of compressibility, bulk relaxation, and hardening phenomena under large strain. In order to find optimal material parameters, the models were fitted to the experimental data available for mesenchymal stem cells. Finally, through Neo-Hookean porohyperelastic (NHPH) material model for the cell, the influence of fluid flow on the aspiration length of the cell was studied. Based on the results, we suggest that the compressibility and bulk relaxation/fluid flow play a significant role in the deformation behavior of single cells and should be taken into account in the analysis of the mechanics of cells.

An Eulerian/XFEM formulation for the large deformation of cortical cell membrane

Most animal cells are surrounded by a thin layer of actin meshwork below their membrane, commonly known as the actin cortex (or cortical membrane). An increasing number of studies have highlighted the role of this structure in many cell functions including contraction and locomotion, but modelling has been limited by the fact that the membrane thickness (about 1 μm) is usually much smaller than the typical size of a cell (10-100 μm). To overcome theoretical and numerical issues resulting from this observation, we introduce in this paper a continuum formulation, based on surface elasticity, that views the cortex as an infinitely thin membrane that can resists tangential deformation. To accurately model the large deformations of cells, we introduced equilibrium equations and constitutive relations within the Eulerian viewpoint such that all quantities (stress, rate of deformation) lie in the current configuration. A solution procedure is then introduced based on a coupled extended finite element approach that enables a continuum solution to the boundary value problem in which discontinuities in both strain and displacement (due to cortical elasticity) are easily handled. We validate the approach by studying the effect of cortical elasticity on the deformation of a cell adhering on a stiff substrate and undergoing internal contraction. Results show very good prediction of the proposed method when compared with experimental observations and analytical solutions for simple cases. In particular, the model can be used to study how cell properties such as stiffness and contraction of both cytoskeleton and cortical membrane lead to variations in cell's surface curvature. These numerical results show that the proposed method can be used to gain critical insights into how the cortical membrane affects cell deformation and how it may be used as a means to determine a cell's mechanical properties by measuring curvatures of its membrane.

Biomechanics of single cortical neurons

This study presents experimental results and computational analysis of the large strain dynamic behavior of single neurons in vitro with the objective of formulating a novel quantitative framework for the biomechanics of cortical neurons. Relying on the atomic force microscopy (AFM) technique, novel testing protocols are developed to enable the characterization of neural soma deformability over a range of indentation rates spanning three orders of magnitude, 10, 1, and 0.1 μm s(-1). Modified spherical AFM probes were utilized to compress the cell bodies of neonatal rat cortical neurons in load, unload, reload and relaxation conditions. The cell response showed marked hysteretic features, strong non-linearities, and substantial time/rate dependencies. The rheological data were complemented with geometrical measurements of cell body morphology, i.e. cross-diameter and height estimates. A constitutive model, validated by the present experiments, is proposed to quantify the mechanical behavior of cortical neurons. The model aimed to correlate empirical findings with measurable degrees of (hyper)elastic resilience and viscosity at the cell level. The proposed formulation, predicated upon previous constitutive model developments undertaken at the cortical tissue level, was implemented in a three-dimensional finite element framework. The simulated cell response was calibrated to the experimental measurements under the selected test conditions, providing a novel single cell model that could form the basis for further refinements.

Validation of computational models in biomechanics

The topics of verification and validation have increasingly been discussed in the field of computational biomechanics, and many recent articles have applied these concepts in an attempt to build credibility for models of complex biological systems. Verification and validation are evolving techniques that, if used improperly, can lead to false conclusions about a system under study. In basic science, these erroneous conclusions may lead to failure of a subsequent hypothesis, but they can have more profound effects if the model is designed to predict patient outcomes. While several authors have reviewed verification and validation as they pertain to traditional solid and fluid mechanics, it is the intent of this paper to present them in the context of computational biomechanics. Specifically, the task of model validation will be discussed, with a focus on current techniques. It is hoped that this review will encourage investigators to engage and adopt the verification and validation process in an effort to increase peer acceptance of computational biomechanics models.

Nanomechanics of polymer gels and biological tissues: A critical review of analytical approaches in the Hertzian regime and beyond

We survey recent progress in the application of nanoindentation to characterize the local mechanical properties of polymer gels and biological tissues. We review the theories, analytical models based thereon, and data processing techniques commonly used to determine elastic properties of these classes of materials by instrumented nanoindentation. Examples from the testing of synthetic and biological gels are used to illustrate the limitations of existing theories and approaches. Emphasis is placed on the need for contact mechanics models that more accurately represent the large-strain behaviour of soft matter.