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.     

Power-law rheology analysis of cells undergoing micropipette aspiration

Accurate quantification of the mechanical properties of living cells requires the combined use of experimental techniques and theoretical models. In this paper, we investigate the viscoelastic response of suspended NIH 3T3 fibroblasts undergoing micropipette aspiration using power-law rheology model. As an important first step, we examine the pipette size effect on cell deformation and find that pipettes larger than ~7 μm are more suitable for bulk rheological measurements than smaller ones and the cell can be treated as effectively continuum. When the large pipettes are used to apply a constant pressure to a cell, the creep deformation is better fitted with the power-law rheology model than with the liquid drop or spring-dashpot models; magnetic twisting cytometry measurement on the rounded cell confirms the power-law behavior. This finding is further extended to suspended cells treated with drugs targeting their cytoskeleton. As such, our results suggest that the application of relatively large pipettes can provide more effective assessment of the bulk material properties as well as support application of power-law rheology to cells in suspension.     

Mechanical behavior of human embryonic stem cell pellet under unconfined compression

As a prelude to the understanding of mechanotransduction in human embryonic stem cell (hESC) differentiation, the mechanical behavior of hESCs in the form of cell pellet is studied. The pellets were tested after 3 or 5 weeks of cell culture in order to demonstrate the effect of the duration of cell culture on the mechanical properties of the pellets. A micromechanical tester was used to conduct unconfined compression on hESC pellet, and experimental, numerical, and analytical methods were combined to determine the mechanical properties of hESC pellet. It is assumed that the mechanical behavior of hESC pellets can be described by an isotropic, linear viscoelastic model consisting of a spring and two Maxwell units in parallel, and the Poisson’s ratio of the hESC pellet is constant based on pellet deformation in the direction perpendicular to the compression direction. Finite element method (FEM) simulation was adopted to determine the values of Poisson’s ratio and the five parameters contained in the viscoelastic model. The variations of Poisson’s ratio and the initial elastic modulus are found to be larger compared with those of the four other parameters. Results show that longer duration of cell culture leads to higher modulus of hESC pellet. The effect of pellet size error on the values of mechanical parameters determined is studied using FEM simulation, and it is found that the effect of size error on Poisson’s ratio and initial elastic modulus is much larger than that on the other parameters.     

Passive deformation analysis of human leukocytes

The following analysis presents an experimental and theoretical study of the passive viscoelastic behavior of human leukocytes. Individual neutrophils in EDTA were observed both during their partial aspiration into a small micropipette and after expulsion from a large micropipette where the cell had been totally aspirated and deformed into a sausage shape. To analyze the data, a passive model of leukocyte rheology has been developed consisting of a cortical shell containing a Maxwell fluid which describes the average properties of the cell cytoplasm. The cortical shell represents a crosslinked actin layer near the surface of the cell and is assumed to be under pre-stressed tension. This model can reproduce the results of experiments using micropipette for both short-time small deformation and slow recovery data after large deformation. In addition, a finite element scheme has been established for the same model which shows close agreement with the analytical solution.     

A computational model for the transit of a cancer cell through a constricted microchannel

We propose a three-dimensional computational model to simulate the transient deformation of suspended cancer cells flowing through a constricted microchannel. We model the cell as a liquid droplet enclosed by a viscoelastic membrane, and its nucleus as a smaller stiffer capsule. The cell deformation and its interaction with the suspending fluid are solved through a well-tested immersed boundary lattice Boltzmann method. To identify a minimal mechanical model that can quantitatively predict the transient cell deformation in a constricted channel, we conduct extensive parametric studies of the effects of the rheology of the cell membrane, cytoplasm and nucleus and compare the results with a recent experiment conducted on human leukaemia cells. We find that excellent agreement with the experiment can be achieved by employing a viscoelastic cell membrane model with the membrane viscosity depending on its mode of deformation (shear versus elongation). The cell nucleus limits the overall deformation of the whole cell, and its effect increases with the nucleus size. The present computational model may be used to guide the design of microfluidic devices to sort cancer cells, or to inversely infer cell mechanical properties from their flow-induced deformation.

     

Leukocyte deformability: finite element modeling of large viscoelastic deformation.

An axisymmetric deformation of a viscoelastic sphere bounded by a prestressed elastic thin shell in response to external pressure is studied by a finite element method. The research is motivated by the need for understanding the passive behavior of human leukocytes (white blood cells) and interpreting extensive experimental data in terms of the mechanical properties. The cell at rest is modeled as a sphere consisting of a cortical prestressed shell with incompressible Maxwell fluid interior. A large-strain deformation theory is developed based on the proposed model. General non-linear, large strain constitutive relations for the cortical shell are derived by neglecting the bending stiffness. A representation of the constitutive equations in the form of an integral of strain history for the incompressible Maxwell interior is used in the formulation of numerical scheme. A finite element program is developed, in which a sliding boundary condition is imposed on all contact surfaces. The mathematical model developed is applied to evaluate experimental data of pipette tests and observations of blood flow.     

Mechanics and deformation of the nucleus in micropipette aspiration experiment

Robust biomechanical models are essential for the study of nuclear mechanics and deformation and can help shed light on the underlying mechanisms of stress transition in nuclear elements. Here, we develop a computational model for an isolated nucleus undergoing micropipette aspiration. Our model includes distinct components representing the nucleoplasm and nuclear envelope. The nuclear envelope itself comprises three layers: inner and outer nuclear membranes and one thicker layer representing the nuclear lamina. The nucleoplasm is modeled as a viscoelastic Maxwell material with a single time constant, while a modified Maxwell model, equivalent to a spring and a dashpot in series and both in parallel with a spring, is adopted for the inner and outer nuclear membranes. The nuclear envelope layer is taken as a linear elastic material. The proposed computational model, validated using experimental observations of Guilak et al. [2000. Viscoelastic properties of the cell nucleus. Biochemical and Biophysical Research Communications 269, 781-786] and Deguchi et al. [2005, Flow-induced hardening of endothelial nucleus as an intracellular stress-bearing organelle. Journal of Biomechanics 38, 1751-1759], is employed to study nuclear mechanics and deformation in micropipette aspiration and to shed light on the contribution of individual nuclear components on the response. The results indicate that the overall response of an isolated nucleus in micropipette aspiration is highly sensitive to the apparent stiffness of the nuclear lamina. This observation suggests that micropipette aspiration is an effective technique for examining the influence of various kinds of alteration in the nuclear lamina, such as mutations in the gene encoding lamin A, and also structural remodeling due to mechanical perturbation.     

Glioma-derived mutations in isocitrate dehydrogenase 2 beneficial to traditional chemotherapy

Sonoporation is a promising drug delivery technique with great potential in medicine. However, its applications have been limited mostly by the lack of understanding its underlying biophysical mechanism, partly due to the inadequacy of the existing models for coupling with highly sensitive imaging techniques to directly observe the actual precursor events of cell–microbubble interaction under low intensity ultrasound. Here, we introduce a new in vitro method utilizing capillary-microgripping system and micro-transducer to achieve maximum level of experimental flexibility for capturing real time highly magnified images of cell–microbubble interaction, hitherto unseen in this context. Insonation of isolated single cells and microbubbles parallel with high speed microphotography and fluorescence microscopy allowed us to identify dynamic responses of cell-membrane/microbubble in correlation with sonoporation. Our results showed that bubble motion and linear oscillation in close contact with the cell membrane can cause local deformation and transient porosity in the cell membrane without rupturing it. This method can also be used as an in situ gene/drug delivery system of targeted cells for non-invasive clinical applications.     

Modeling of microstructural kinematics during simple elongation of central nervous system tissue

Damage to axons and glial cells in the central nervous system (CNS) white matter is a nearly universal feature of traumatic brain injury, yet it is not clear how the tissue mechanical deformations are transferred to the cellular components of the CNS. Defining how cellular deformations relate to the applied tissue deformation field can both highlight cellular populations at risk for mechanical injury, and define the fraction of cells in a specific population that will exhibit damage. In this investigation, microstructurally based models of CNS white matter were developed and tested against measured transformations of the CNS tissue microstructure under simple elongation. Results show that axons in the unstretched optic nerves were significantly wavy or undulated, where the measured axonal path length was greater than the end-to-end distance of the axon. The average undulation parameter--defined as the true axonal length divided by the end-to-end length--was 1.13. In stretched nerves, mean axonal undulations decreased with increasing applied stretch ratio (lambda)--the mean undulation values decreased to 1.06 at lambda = 1.06, 1.04 at lambda = 1.12, and 1.02 at lambda = 1.25. A model describing the gradual coupling, or tethering, of the axons to the surrounding glial cells best fit the experimental data. These modeling efforts indicate the fraction of the axonal and glial populations experiencing deformation increases with applied elongation, consistent with the observation that both axonal and glial cell injury increases at higher levels of white matter injury. Ultimately, these results can be used in conjunction with computational simulations of traumatic brain injury to aid in establishing the relative risk of cellular structures in the CNS white matter to mechanical injury.     

Three-dimensional finite element modeling of pericellular matrix and cell mechanics in the nucleus pulposus of the intervertebral disk based on in situ morphology

Nucleus pulposus (NP) cells of the intervertebral disk (IVD) have unique morphological characteristics and biologic responses to mechanical stimuli that may regulate maintenance and health of the IVD. NP cells reside as single cell, paired or multiple cells in a contiguous pericellular matrix (PCM), whose structure and properties may significantly influence cell and extracellular matrix mechanics. In this study, a computational model was developed to predict the stress–strain, fluid pressure and flow fields for cells and their surrounding PCM in the NP using three-dimensional (3D) finite element models based on the in situ morphology of cell–PCM regions of the mature rat NP, measured using confocal microscopy. Three-dimensional geometries of the extracellular matrix and representative cell–matrix units were used to construct 3D finite element models of the structures as isotropic and biphasic materials. In response to compressive strain of the extracellular matrix, NP cells and PCM regions were predicted to experience volumetric strains that were 1.9–3.7 and 1.4–2.1 times greater than the extracellular matrix, respectively. Volumetric and deviatoric strain concentrations were generally found at the cell/PCM interface, while von Mises stress concentrations were associated with the PCM/extracellular matrix interface. Cell–matrix units containing greater cell numbers were associated with higher peak cell strains and lower rates of fluid pressurization upon loading. These studies provide new model predictions for micromechanics of NP cells that can contribute to an understanding of mechanotransduction in the IVD and its changes with aging and degeneration.     

Disparity estimation through Green’s functions of matching equations

Binocular disparities arise from positional differences of scene features projected in the two retinae, and constitute the primary sensory cue for stereo vision. Here we introduce a new computational model for disparity estimation, based on the Green’s function of an image matching equation. When filtering a Gabor-function-modulated signal, the considered Green’s function yields a similarly modulated but shifted version of the original signal. Since a Gabor function models the receptive field of a cortical simple cell, the Green’s kernel thus allows the simulation of relative shifts between the cell’s left and right binocular inputs. A measure of the local degree of matching of such shifted inputs can then be introduced which affords disparity estimation in a similar manner to the energy model of the complex cortical cells. We have therefore effectively reformulated, in physiologically plausible terms, an image matching approach to disparity estimation. Our experiments show that the Green’s function method allows the detection of disparities both from random-dot and real-world stereograms. Partially supported by CNPq-Brazil.     

A study and model of the role of the Renshaw cell in regulating the transient firing rate of the motoneuron

 This study sought to investigate the role of the Renshaw cell with respect to transient motoneuron firing. By studying the cat motoneuron and Renshaw cell, several low-order lumped parameter models were developed that simulate the known characteristics of the injected input current vs. firing rate. The neuron models in the Renshaw cell inhibition configuration were tuned to fit experimental data from cat motoneurons. Models included both linear versions and those with sigmoidal nonlinearities. Results of the simulation indicate that the motoneuron itself provides the adaptation seen in its firing rate and that the Renshaw cell’s role is primarily to fine-tune the motoneuron’s adaptation process. Received: 23 July 1993/Accepted in revised form: 9 February 1994     

Effect of colchicine on viscoelastic properties of neutrophils

The effect of colchicine (15-60 micrograms/ml) on the viscoelastic properties of human neutrophils was studied by the micropipette technique. The small deformation of the neutrophil in response to a step aspiration pressure was analyzed by using a three-element model in which an elastic element, K1, is in parallel with a Maxwell element composed of another elastic element, K2, in series with a viscous element, mu. Colchicine treatment of neutrophils caused decreases in K2 and mu without affecting K1. The results indicate that the integrity of the microtubules plays a significant role in providing the viscoelastic resistance (as represented by the Maxwell element in the model) of neutrophils to deforming stress.     

Waddington redux: models and explanation in stem cell and systems biology

Stem cell biology and systems biology are two prominent new approaches to studying cell development. In stem cell biology, the predominant method is experimental manipulation of concrete cells and tissues. Systems biology, in contrast, emphasizes mathematical modeling of cellular systems. For scientists and philosophers interested in development, an important question arises: how should the two approaches relate? This essay proposes an answer, using the model of Waddington’s landscape to triangulate between stem cell and systems approaches. This simple abstract model represents development as an undulating surface of hills and valleys. Originally constructed by C. H. Waddington to visually explicate an integrated theory of genetics, development and evolution, the landscape model can play an updated unificatory role. I examine this model’s structure, representational assumptions, and uses in all three contexts, and argue that explanations of cell development require both mathematical models and concrete experiments. On this view, the two approaches are interdependent, with mathematical models playing a crucial but circumscribed role in explanations of cell development.     

Shear viscoelasticity of suspensions of biological cells with viscoelastic membrane II

To discuss the relaxation phenomena of biological cell suspensions, we calculate the complex intrinsic viscosity of dispersions of spherical cells with viscoelastic membrane as a function of the frequency taking account of interfacial tension at both the interfaces of the membrane. The Maxwell model and two kinds of the three-parameter models are used to describe the viscoelasticity of the cell membrane. The results are computed mainly for the Maxwell model similarly in case of the Voigt Model (Abe, K., Takano, Y. and Sakanishi, A. Biorheology 21 405-414, 1984). The computed results of the four models, the Voigt, the Maxwell and the two kinds of the three-parameter viscoelastic models, are compared with the experimental data.     

A computationally efficient strategy to estimate muscle forces in a finite element musculoskeletal model of the lower limb

Concurrent multiscale simulation strategies are required in computational biomechanics to study the interdependence between body scales. However, detailed finite element models rarely include muscle recruitment due to the computational burden of both the finite element method and the optimization strategies widely used to estimate muscle forces. The aim of this study was twofold: first, to develop a computationally efficient muscle force prediction strategy based on proportional-integral-derivative (PID) controllers to track gait and chair rise experimental joint motion with a finite element musculoskeletal model of the lower limb, including a deformable knee representation with 12 degrees of freedom; and, second, to demonstrate that the inclusion of joint-level deformability affects muscle force estimation by using two different knee models and comparing muscle forces between the two solutions. The PID control strategy tracked experimental hip, knee, and ankle flexion/extension with root mean square errors below 1°, and estimated muscle, contact and ligament forces in good agreement with previous results and electromyography signals. Differences up to 11% and 20% in the vasti and biceps femoris forces, respectively, were observed between the two knee models, which might be attributed to a combination of differing joint contact geometry, ligament behavior, joint kinematics, and muscle moment arms. The tracking strategy developed in this study addressed the inevitable tradeoff between computational cost and model detail in musculoskeletal simulations and can be used with finite element musculoskeletal models to efficiently estimate the interdependence between muscle forces and tissue deformation.     

An orthotropic viscoelastic model for the passive myocardium: continuum basis and numerical treatment

This study deals with the viscoelastic constitutive modeling and the respective computational analysis of the human passive myocardium. We start by recapitulating the locally orthotropic inner structure of the human myocardial tissue and model the mechanical response through invariants and structure tensors associated with three orthonormal basis vectors. In accordance with recent experimental findings the ventricular myocardial tissue is assumed to be incompressible, thick-walled, orthotropic and viscoelastic. In particular, one spring element coupled with Maxwell elements in parallel endows the model with viscoelastic features such that four dashpots describe the viscous response due to matrix, fiber, sheet and fiber-sheet fragments. In order to alleviate the numerical obstacles, the strictly incompressible model is altered by decomposing the free-energy function into volumetric-isochoric elastic and isochoric-viscoelastic parts along with the multiplicative split of the deformation gradient which enables the three-field mixed finite element method. The crucial aspect of the viscoelastic formulation is linked to the rate equations of the viscous overstresses resulting from a 3-D analogy of a generalized 1-D Maxwell model. We provide algorithmic updates for second Piola–Kirchhoff stress and elasticity tensors. In the sequel, we address some numerical aspects of the constitutive model by applying it to elastic, cyclic and relaxation test data obtained from biaxial extension and triaxial shear tests whereby we assess the fitting capacity of the model. With the tissue parameters identified, we conduct (elastic and viscoelastic) finite element simulations for an ellipsoidal geometry retrieved from a human specimen.     

Human stem cells for CNS repair

Although most peripheral tissues have at least a limited ability for self-repair, the central nervous system (CNS) has long been known to be relatively resistant to regeneration. Small numbers of stem cells have been found in the adult brain but do not appear to be able to affect any significant recovery following disease or insult. In the last few decades, the idea of being able to repair the brain by introducing new cells to repair damaged areas has become an accepted potential treatment for neurodegenerative diseases. This review focuses on the suitability of various human stem cell sources for such treatments of both slowly progressing conditions, such as Parkinson’s disease, Huntington’s disease and multiple sclerosis, and acute insult, such as stroke and spinal cord injury. Despite stem cell transplantation having now moved a step closer to the clinic with the first trials of autologous mesenchymal stem cells, the effects shown are moderate and are not yet at the stage of development that can fulfil the hopes that have been placed on stem cells as a means to replace degenerating cells in the CNS. Success will depend on careful investigation in experimental models to enable us to understand not just the practicalities of stem cell use, but also the underlying biological principles.     

Genetic control theory of developmental events

An earlier theory of cell differentiation and morphogenesis (Wassermann, 1972, 1973, 1978) is combined with the genetic control model of Davidson and Britten (e.g. 1979). The resulting new theory suggests how, bysystematic process algorithms, specifically enumerated combinations of batteries of structural genes can become switched on in particularly enumerated cells, via battery-specific enumerable regulator genes. The systematization is idealized. Up to a certain stage of development in each mitotically arising cell a unique cell-specific combination of structural genes called ‘marker genes’ is active. Marker genes are assumed to code for cell-specifying marker proteins (CSMPs) which permit cells carrying related markers to recognize each other, thus permitting specific cell sorting.Batteries of marker genes could ensure great developmental precision and can safeguard—via redundancies of CSMP types—against accidental loss or detrimental mutational modification of CSMPs or marker genes, respectively. This paper is much concerned with cell lineage in relation to ‘microdifferentiation’, where ‘microdifferentiation’ of a cell refers to a cell's active marker genes and its syntheses of CSMPs. A drastic distinction is made between ‘microdifferentiation’ and ‘gross’ differentiation of a cell, where the same ‘gross’ differentiation may be shared by a large number of cells that could each be uniquely ‘microdifferentiated’. Typical ‘gross’ differentiation could manifest itself in tissue specificity, whereas, up to certain stages of development, all cells of the same gross differentiation type (say tissue specificity) could each be uniquely ‘microdifferentiated’. The theory also assumes that at certain stages of the developmental process some (or in some organisms all) of the previously uniquely specified cells could give rise to small (or occasionally large) clones of equispecified cells, some of which might form clusters that represent complete ‘morphogenetic fields’ Tentative implementation mechanisms are proposed which suggest how the theory could operate in molecular biological terms. In particular, CSMPs could endow cell surface membranes with a highly specific protein network, and an associated equally specific cell surface coat. It is suggested how these highly specified cell surface coats and other systems could provide an ‘extracellular guidance network’ which could help to direct cells to attain energetically optimal locations relative to each other based on the matching of their surface specificities. In numerous experimental situations, where normally present optimal matching of cells is excluded, ‘alternative matching’ based on experiment-specific suboptimal matching could explain many data, notably in experimental development neurobiology (Wassermann, 1978).     

Cell assembly dynamics in detailed and abstract attractor models of cortical associative memory

Summary During the last few decades we have seen a convergence among ideas and hypotheses regarding functional principles underlying human memory. Hebb’s now more than fifty years old conjecture concerning synaptic plasticity and cell assemblies, formalized mathematically as attractor neural networks, has remained among the most viable and productive theoretical frameworks. It suggests plausible explanations for Gestalt aspects of active memory like perceptual completion, reconstruction and rivalry. We review the biological plausibility of these theories and discuss some critical issues concerning their associative memory functionality in the light of simulation studies of models with palimpsest memory properties. The focus is on memory properties and dynamics of networks modularized in terms of cortical minicolumns and hypercolumns. Biophysical compartmental models demonstrate attractor dynamics that support cell assembly operations with fast convergence and low firing rates. Using a scaling model we obtain reasonable relative connection densities and amplitudes. An abstract attractor network model reproduces systems level psychological phenomena seen in human memory experiments as the Sternberg and von Restorff effects. We conclude that there is today considerable substance in Hebb’s theory of cell assemblies and its attractor network formulations, and that they have contributed to increasing our understanding of cortical associative memory function. The criticism raised with regard to biological and psychological plausibility as well as low storage capacity, slow retrieval etc has largely been disproved. Rather, this paradigm has gained further support from new experimental data as well as computational modeling.