Mechanical models for living cells--a review

Microfluidic Applications in Prostate Cancer Research

Prostate cancer is a disease in which cells in the prostate, a gland in the male reproductive system below the bladder, grow out of control and, among men, it is the second-most frequently diagnosed cancer (other than skin cancer). In recent years, prostate cancer death rate has stabilized and, currently, it is the second-most frequent cause of cancer death in men (after lung cancer). Most deaths occur due to metastasis, as cancer cells from the original tumor establish secondary tumors in distant organs. For a long time, classical cell cultures and animal models have been utilized in basic and applied scientific research, including clinical applications for many diseases, such as prostate cancer, since no better alternatives were available. Although helpful in dissecting cellular mechanisms, these models are poor predictors of physiological behavior mainly because of the lack of appropriate microenvironments. Microfluidics has emerged in the last two decades as a technology that could lead to a paradigm shift in life sciences and, in particular, controlling cancer. Microfluidic systems, such as organ-on-chips, have been assembled to mimic the critical functions of human organs. These microphysiological systems enable the long-term maintenance of cellular co-cultures in vitro to reconstitute in vivo tissue-level microenvironments, bridging the gap between traditional cell cultures and animal models. Several reviews on microfluidics for prostate cancer studies have been published focusing on technology advancement and disease progression. As metastatic castration-resistant prostate cancer remains a clinically challenging late-stage cancer, with no curative treatments, we expanded this review to cover recent microfluidic applications related to prostate cancer research. The review includes discussions of the roles of microfluidics in modeling the human prostate, prostate cancer initiation and development, as well as prostate cancer detection and therapy, highlighting potentially major contributions of microfluidics in the continuous march toward eradicating prostate cancer.

An in-silico study on the mechanical behavior of colorectal cancer cell lines in the micropipette aspiration process

Cancer alters the structural integrity and morphology of cells. Consequently, the cell function is overshadowed. In this study, the micropipette aspiration process is computationally modeled to predict the mechanical behavior of the colorectal cancer cells. The intended cancer cells are modeled as an incompressible Neo-Hookean visco-hyperelastic material. Also, the micropipette is assumed to be rigid with no deformation. The proposed model is validated with an in-vitro study. To capture the equilibrium and time-dependent behaviors of cells, ramp, and creep tests are respectively performed using the finite element method. Through the simulations, the effects of the micropipette geometry and the aspiration pressure on the colorectal cancer cell lines are investigated. Our findings indicate that, as the inner radius of the micropipette increases, despite the increase in deformation rate and aspirated length, the time to reach the equilibrium state increases. Nevertheless, it is obvious that increasing the tip curvature radius has a small effect on the change of the aspirated length. But, due to the decrease in the stress concentration, it drastically reduces the equilibrium time and increases the deformation rate significantly. Interestingly, our results demonstrate that increasing the aspiration pressure somehow causes the cell stiffening, thereby reducing the upward trend of deformation rate, equilibrium time, and aspirated length. Our findings provide valuable insights for researchers in cell therapy and cancer treatment and can aid in developing more precise microfluidic.

Mechanical characterization of regenerating Hydra tissue spheres

Hydra vulgaris, long known for its remarkable regenerative capabilities, is also a long-standing source of inspiration for models of spontaneous patterning. Recently it became clear that early patterning during Hydra regeneration is an integrated mechanochemical process whereby morphogen dynamics is influenced by tissue mechanics. One roadblock to understanding Hydra self-organization is our lack of knowledge about the mechanical properties of these organisms. In this study, we combined microfluidic developments to perform parallelized microaspiration rheological experiments and numerical simulations to characterize these mechanical properties. We found three different behaviors depending on the applied stresses: an elastic response, a viscoelastic response, and tissue rupture. Using models of deformable shells, we quantify their Young's modulus, shear viscosity, and the critical stresses required to switch between behaviors. Based on these experimental results, we propose a description of the tissue mechanics during normal regeneration. Our results provide a first step toward the development of original mechanochemical models of patterning grounded in quantitative experimental data.

Single-Cell Mechanics: Structural Determinants and Functional Relevance

The mechanical phenotype of a cell determines its ability to deform under force and is therefore relevant to cellular functions that require changes in cell shape, such as migration or circulation through the microvasculature. On the practical level, the mechanical phenotype can be used as a global readout of the cell's functional state, a marker for disease diagnostics, or an input for tissue modeling. We focus our review on the current knowledge of structural components that contribute to the determination of the cellular mechanical properties and highlight the physiological processes in which the mechanical phenotype of the cells is of critical relevance. The ongoing efforts to understand how to efficiently measure and control the mechanical properties of cells will define the progress in the field and drive mechanical phenotyping toward clinical applications.

Size Matters: Rethinking Hertz Model Interpretation for Cell Mechanics Using AFM

Cell mechanics are a biophysical indicator of cell state, such as cancer metastasis, leukocyte activation, and cell cycle progression. Atomic force microscopy (AFM) is a widely used technique to measure cell mechanics, where the Young modulus of a cell is usually derived from the Hertz contact model. However, the Hertz model assumes that the cell is an elastic, isotropic, and homogeneous material and that the indentation is small compared to the cell size. These assumptions neglect the effects of the cytoskeleton, cell size and shape, and cell environment on cell deformation. In this study, we investigated the influence of cell size on the estimated Young's modulus using liposomes as cell models. Liposomes were prepared with different sizes and filled with phosphate buffered saline (PBS) or hyaluronic acid (HA) to mimic the cytoplasm. AFM was used to obtain the force indentation curves and fit them to the Hertz model. We found that the larger the liposome, the lower the estimated Young's modulus for both PBS-filled and HA-filled liposomes. This suggests that the Young modulus obtained from the Hertz model is not only a property of the cell material but also depends on the cell dimensions. Therefore, when comparing or interpreting cell mechanics using the Hertz model, it is essential to account for cell size.

Critical review of single-cell mechanotyping approaches for biomedical applications

Accurate mechanical measurements of cells has the potential to improve diagnostics, therapeutics and advance understanding of disease mechanisms, where high-resolution mechanical information can be measured by deforming individual cells. Here we evaluate recently developed techniques for measuring cell-scale stiffness properties; while many such techniques have been developed, much of the work examining single-cell stiffness is impacted by difficulties in standardization and comparability, giving rise to large variations in reported mechanical moduli. We highlight the role of underlying mechanical theories driving this variability, and note opportunities to develop novel mechanotyping devices and theoretical models that facilitate convenient and accurate mechanical characterisation. Moreover, many high-throughput approaches are confounded by factors including cell size, surface friction, natural population heterogeneity and convolution of elastic and viscous contributions to cell deformability. We nevertheless identify key approaches based on deformability cytometry as a promising direction for further development, where both high-throughput and accurate single-cell resolutions can be realized.

A mechanical transition from tension to buckling underlies the jigsaw puzzle shape morphogenesis of histoblasts in the Drosophila epidermis

The polygonal shape of cells in proliferating epithelia is a result of the tensile forces of the cytoskeletal cortex and packing geometry set by the cell cycle. In the larval Drosophila epidermis, two cell populations, histoblasts and larval epithelial cells, compete for space as they grow on a limited body surface. They do so in the absence of cell divisions. We report a striking morphological transition of histoblasts during larval development, where they change from a tensed network configuration with straight cell outlines at the level of adherens junctions to a highly folded morphology. The apical surface of histoblasts shrinks while their growing adherens junctions fold, forming deep lobules. Volume increase of growing histoblasts is accommodated basally, compensating for the shrinking apical area. The folded geometry of apical junctions resembles elastic buckling, and we show that the imbalance between the shrinkage of the apical domain of histoblasts and the continuous growth of junctions triggers buckling. Our model is supported by laser dissections and optical tweezer experiments together with computer simulations. Our analysis pinpoints the ability of histoblasts to store mechanical energy to a much greater extent than most other epithelial cell types investigated so far, while retaining the ability to dissipate stress on the hours time scale. Finally, we propose a possible mechanism for size regulation of histoblast apical size through the lateral pressure of the epidermis, driven by the growth of cells on a limited surface. Buckling effectively compacts histoblasts at their apical plane and may serve to avoid physical harm to these adult epidermis precursors during larval life. Our work indicates that in growing nondividing cells, compressive forces, instead of tension, may drive cell morphology.

Electromechanical interactions between cell membrane and nuclear envelope: Beyond the standard Schwan's model of biological cells

We investigate little-appreciated features of the hierarchical core-shell (CS) models of the electrical, mechanical, and electromechanical interactions between the cell membrane (CM) and nuclear envelope (NE). We first consider a simple model of an individual cell based on a coupled resistor-capacitor (Schwan model (SM)) network and show that the CM, when exposed to ac electric fields, acts as a low pass filter while the NE acts as a wide and asymmetric bandpass filter. We provide a simplified calculation for characteristic time associated with the capacitive charging of the NE and parameterize its range of behavior. We furthermore observe several new features dealing with mechanical analogs of the SM based on elementary spring-damper combinations. The chief merit of these models is that they can predict creep compliance responses of an individual cell under static stress and their effective retardation time constants. Next, we use an alternative and a more accurate CS physical model solved by finite element simulations for which geometrical cell reshaping under electromechanical stress (electrodeformation (ED)) is included in a continuum approach with spatial resolution. We show that under an electric field excitation, the elongated nucleus scales differently compared to the electrodeformed cell.

Experimental measurement and numerical modeling of deformation behavior of breast cancer cells passing through constricted microfluidic channels

During the multistep process of metastasis, cancer cells encounter various mechanical forces which make them deform drastically. Developing accurate in-silico models, capable of simulating the interactions between the mechanical forces and highly deformable cancer cells, can pave the way for the development of novel diagnostic and predictive methods for metastatic progression. Spring-network models of cancer cell, empowered by our recently proposed identification approach, promises a versatile numerical tool for developing experimentally validated models that can simulate complex interactions at cellular scale. Using this numerical tool, we presented spring-network models of breast cancer cells that can accurately replicate the experimental data of deformation behavior of the cells flowing in a fluidic domain and passing narrow constrictions comparable to microcapillary. First, using high-speed imaging, we experimentally studied the deformability of breast cancer cell lines with varying metastatic potential (MCF-7 (less invasive), SKBR-3 (medium-high invasive), and MDA-MB-231 (highly invasive)) in terms of their entry time to a constricted microfluidic channel. We observed that MDA-MB-231, that has the highest metastatic potential, is the most deformable cell among the three. Then, by focusing on this cell line, experimental measurements were expanded to two more constricted microchannel dimensions. The experimental deformability data in three constricted microchannel sizes for various cell sizes, enabled accurate identification of the unknown parameters of the spring-network model of the breast cancer cell line (MDA-MB-231). Our results show that the identified parameters depend on the cell size, suggesting the need for a systematic procedure for identifying the size-dependent parameters of spring-network models of cells. As the numerical results show, the presented cell models can simulate the entry process of the cell into constricted channels with very good agreements with the measured experimental data.

Subcellular mechano-regulation of cell migration in confined extracellular microenvironment

Cell migration is a highly coordinated cellular event that determines diverse physiological and pathological processes in which the continuous interaction of a migrating cell with neighboring cells or the extracellular matrix is regulated by the physical setting of the extracellular microenvironment. In confined spaces, cell migration occurs differently compared to unconfined open spaces owing to the additional forces that limit cell motility, which create a driving bias for cells to invade the confined space, resulting in a distinct cell motility process compared to what is expected in open spaces. Moreover, cells in confined environments can be subjected to elevated mechanical compression, which causes physical stimuli and activates the damage repair cycle in the cell, including the DNA in the nucleus. Although cells have a self-restoring system to repair damage from the cell membrane to the genetic components of the nucleus, this process may result in genetic and/or epigenetic alterations that can increase the risk of the progression of diverse diseases, such as cancer and immune disorders. Furthermore, there has been a shift in the paradigm of bioengineering from the development of new biomaterials to controlling biophysical cues and fine-tuning cell behaviors to cure damaged/diseased tissues. The external physical cues perceived by cells are transduced along the mechanosensitive machinery, which is further channeled into the nucleus through subcellular molecular linkages of the nucleoskeleton and cytoskeleton or the biochemical translocation of transcription factors. Thus, external cues can directly or indirectly regulate genetic transcriptional processes and nuclear mechanics, ultimately determining cell fate. In this review, we discuss the importance of the biophysical cues, response mechanisms, and mechanical models of cell migration in confined environments. We also discuss the effect of force-dependent deformation of subcellular components, specifically focusing on subnuclear organelles, such as nuclear membranes and chromosomal organization. This review will provide a biophysical perspective on cancer progression and metastasis as well as abnormal cellular proliferation.

Computational simulation of applying mechanical vibration to mesenchymal stem cell for mechanical modulation toward bone tissue engineering

Evaluation of cell response to mechanical stimuli at in vitro conditions is known as one of the important issues for modulating cell behavior. Mechanical stimuli, including mechanical vibration and oscillatory fluid flow, act as important biophysical signals for the mechanical modulation of stem cells. In the present study, mesenchymal stem cell (MSC) consists of cytoplasm, nucleus, actin, and microtubule. Also, integrin and primary cilium were considered as mechanoreceptors. In this study, the combined effect of vibration and oscillatory fluid flow on the cell and its components were investigated using numerical modeling. The results of the FEM and FSI model showed that the cell response (stress and strain values) at the frequency of 30 H z mechanical vibration has the highest value. The achieved results on shear stress caused by the fluid flow on the cell showed that the cell experiences shear stress in the range of 0 . 1 - 10 Pa . Mechanoreceptors that bind separately to the cell surface, can be highly stimulated by hydrodynamic pressure and, therefore, can play a role in the mechanical modulation of MSCs at in vitro conditions. The results of this research can be effective in future studies to optimize the conditions of mechanical stimuli applied to the cell culture medium and to determine the mechanisms involved in mechanotransduction.

Generative models of morphogenesis in developmental biology

Understanding the mechanism by which cells coordinate their differentiation and migration is critical to our understanding of many fundamental processes such as wound healing, disease progression, and developmental biology. Mathematical models have been an essential tool for testing and developing our understanding, such as models of cells as soft spherical particles, reaction-diffusion systems that couple cell movement to environmental factors, and multi-scale multi-physics simulations that combine bottom-up rule-based models with continuum laws. However, mathematical models can often be loosely related to data or have so many parameters that model behaviour is weakly constrained. Recent methods in machine learning introduce new means by which models can be derived and deployed. In this review, we discuss examples of mathematical models of aspects of developmental biology, such as cell migration, and how these models can be combined with these recent machine learning methods.

Probing mechanical interaction of immune receptors and cytoskeleton by membrane nanotube extraction

The role of force application in immune cell recognition is now well established, the force being transmitted between the actin cytoskeleton to the anchoring ligands through receptors such as integrins. In this chain, the mechanics of the cytoskeleton to receptor link, though clearly crucial, remains poorly understood. To probe this link, we combine mechanical extraction of membrane tubes from T cells using optical tweezers, and fitting of the resulting force curves with a viscoelastic model taking into account the cell and relevant molecules. We solicit this link using four different antibodies against various membrane bound receptors: antiCD3 to target the T Cell Receptor (TCR) complex, antiCD45 for the long sugar CD45, and two clones of antiCD11 targeting open or closed conformation of LFA1 integrins. Upon disruption of the cytoskeleton, the stiffness of the link changes for two of the receptors, exposing the existence of a receptor to cytoskeleton link-namely TCR-complex and open LFA1, and does not change for the other two where a weaker link was expected. Our integrated approach allows us to probe, for the first time, the mechanics of the intracellular receptor-cytoskeleton link in immune cells.

A Rate-Dependent Cell Microinjection Model Based on Membrane Theory

As an effective method to deliver external materials into biological cells, microinjection has been widely applied in the biomedical field. However, the knowledge of cell mechanical property is still inadequate, which greatly limits the efficiency and success rate of injection. Thus, a new rate-dependent mechanical model based on membrane theory is proposed for the first time. In this model, an analytical equilibrium equation between the injection force and cell deformation is established by considering the speed effect of microinjection. Different from the traditional membrane-theory-based model, the elastic coefficient of the constitutive material in the proposed model is modified as a function of the injection velocity and acceleration, effectively simulating the influence of speeds on the mechanical responses and providing a more generalized and practical model. Using this model, other mechanical responses at different speeds can be also accurately predicted, including the distribution of membrane tension and stress and the deformed shape. To verify the validity of the model, numerical simulations and experiments were carried out. The results show that the proposed model can match the real mechanical responses well at different injection speeds up to 2 mm/s. The model presented in this paper will be promising in the application of automatic batch cell microinjection with high efficiency.

Microstructural and cellular characterisation of the subchondral trabecular bone in human knee and hip osteoarthritis using synchrotron tomography

OBJECTIVE

It is unclear if different factors influence osteoarthritis (OA) progression and degenerative changes characterising OA disease in hip and knee. We investigated the difference between hip OA and knee OA at the subchondral bone (SCB) tissue and cellular level, relative to the degree of cartilage degeneration.

DESIGN

Bone samples were collected from 11 patients (aged 70.4 ± 10.7years) undergoing knee arthroplasty and 8 patients (aged 62.3 ± 13.4years) undergoing hip arthroplasty surgery. Trabecular bone microstructure, osteocyte-lacunar network, and bone matrix vascularity were evaluated using synchrotron micro-CT imaging. Additionally, osteocyte density, viability, and connectivity were determined histologically.

RESULTS

The associations between severe cartilage degeneration and increase of bone volume fraction (%) [- 8.7, 95% CI (-14.1, -3.4)], trabecular number (#/mm) [- 1.5, 95% CI (-0.8, -2.3)], osteocyte lacunar density (#/mm3) [4714.9; 95% CI (2079.1, 7350.6)] and decrease of trabecular separation (mm) [- 0.07, 95% CI (0.02, 0.1)] were found in both knee and hip OA. When compared to knee OA, hip OA was characterised by larger (µm3) but less spheric osteocyte lacunae [47.3; 95% CI (11.2, 83.4), - 0.04; 95% CI (-0.06, -0.02), respectively], lower vascular canal density (#/mm3) [- 22.8; 95% CI (-35.4, -10.3)], lower osteocyte cell density (#/mm2) [- 84.2; 95% CI (-102.5, -67.4)], and less senescent (#/mm2) but more apoptotic osteocytes (%) [- 2.4; 95% CI (-3.6, -1.2), 24.9; 95% CI (17.7, 32.1)], respectively.

CONCLUSION

SCB from hip OA and knee OA exhibits different characteristics at the tissue and cellular levels, suggesting different mechanisms of OA progression in different joints.

Tumour growth: An approach to calibrate parameters of a multiphase porous media model based on in vitro observations of Neuroblastoma spheroid growth in a hydrogel microenvironment

To unravel processes that lead to the growth of solid tumours, it is necessary to link knowledge of cancer biology with the physical properties of the tumour and its interaction with the surrounding microenvironment. Our understanding of the underlying mechanisms is however still imprecise. We therefore developed computational physics-based models, which incorporate the interaction of the tumour with its surroundings based on the theory of porous media. However, the experimental validation of such models represents a challenge to its clinical use as a prognostic tool. This study combines a physics-based model with in vitro experiments based on microfluidic devices used to mimic a three-dimensional tumour microenvironment. By conducting a global sensitivity analysis, we identify the most influential input parameters and infer their posterior distribution based on Bayesian calibration. The resulting probability density is in agreement with the scattering of the experimental data and thus validates the proposed workflow. This study demonstrates the huge challenges associated with determining precise parameters with usually only limited data for such complex processes and models, but also demonstrates in general how to indirectly characterise the mechanical properties of neuroblastoma spheroids that cannot feasibly be measured experimentally.

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.

Mechanotransduction in tumor dynamics modeling

Mechanotherapy is a groundbreaking approach to impact carcinogenesis. Cells sense and respond to mechanical stimuli, translating them into biochemical signals in a process known as mechanotransduction. The impact of stress on tumor growth has been studied in the last three decades, and many papers highlight the role of mechanics as a critical self-inducer of tumor fate at the in vitro and in vivo biological levels. Meanwhile, mathematical models attempt to determine laws to reproduce tumor dynamics. This review discusses biological mechanotransduction mechanisms and mathematical-biomechanical models together. The aim is to provide a common framework for the different approaches that have emerged in the literature from the perspective of tumor avascularity and to provide insight into emerging mechanotherapies that have attracted interest in recent years.

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.

Valve interstitial cells under impact load, a mechanobiology study

Understanding the relationship between mechanobiology and the biosynthetic activities of the valve interstitial cells (VICs) in health and disease under severe dynamic loading conditions is of particular interest. The purpose of this study is to further understand the mechanobiology of heart valve leaflet tissue and the VICs under impact forces. Two novel computational and experimental platforms were developed to study the effect of impact load on the VICs to monitor for apoptosis. The first objective was to design and develop an apparatus to experimentally study viability (apoptosis) of the porcine heart valve leaflet tissue VICs in the aortic position under controlled impact forces. Apoptosis was assessed based on terminal transferase dUTP nick end-labelling (TUNEL) assay. The second objective was to develop a computational platform to estimate the stress and strain fields in the vicinity of VICs when the tissue experiences impact forces. A nonlinear finite element (FE) model with an anisotropic, hyperelastic and heterogeneous material model for the matrix and cells was developed. Preliminary results confirm that interstitial cells are successfully resistant to impact loads up to 30 times more than normal physiological conditions. Additionally, the structure and composition of heart valve leaflet tissue provides a mechanical shield for VICs protecting them from excessive mechanical forces such as impact loads. Although, the entire tissue may experience excessive stresses, which may lead to structural damage, the stresses around and near VICs remain consistency low. Results of this study may be used for heart valve leaflet tissue-engineering, as well as further understanding the mechanobiology of the VICs in health and disease.

The role of the cortex in indentation experiments of animal cells

We present a model useful for interpretation of indentation experiments on animal cells. We use finite element modeling for a thorough representation of the complex structure of an animal cell. In our model, the crucial constituent is the cell cortex-a rigid layer of cytoplasmic proteins present on the inner side of the cell membrane. It plays a vital role in the mechanical interactions between cells. The cell cortex is modeled by a three-dimensional solid to reflect its bending stiffness. This approach allows us to interpret the results of the indentation measurements and extract the mechanical properties of the individual elements of the cell structure. During the simulations, we scan a broad range of parameters such as cortex thickness and Young's modulus, cytoplasm Young's modulus, and indenter radius, which define cell properties and experimental conditions. Finally, we propose a simple closed-form formula that approximates the simulated results with satisfactory accuracy. Our formula is as easy to use as Hertz's function to extract cell properties from the measurement, yet it considers the cell's inner structure, including cell cortex, cytoplasm, and nucleus.

Measuring biological materials mechanics with atomic force microscopy - Determination of viscoelastic cell properties from stress relaxation experiments

Cells are complex, viscoelastic bodies. Their mechanical properties are defined by the arrangement of semiflexible cytoskeletal fibers, their crosslinking, and the active remodeling of the cytoskeletal network. Atomic force microscopy (AFM) is an often-used technique for the study of cell mechanics, enabling time- and frequency-dependent measurements with nanometer resolution. Cells exhibit time-dependent deformation when stress is applied. In this work, we have investigated the stress relaxation of HeLa cells when subjected to a constant strain. We have varied the applied force (1, 2, 4, and 8 nN) and pause time (1, 10, and 60 s) to check for common assumptions for the use of models of linear viscoelasticity. Then, we have applied three models (standard linear solid, five element Maxwell, power law rheology) to study their suitability to fit the datasets. We show that the five element Maxwell model captures the stress relaxation response the best while still retaining a low number of free variables. This work serves as an introduction and guide when performing stress relaxation experiments on soft matter using AFM. RESEARCH HIGHLIGHTS: Cells exhibit linear viscoelastic properties when subjected to stress relaxation measurements at the studied different forces and times. The stress relaxation is best described by a five element Maxwell model. All three used models capture a softening and fluidization of cells when disrupting actin filaments.

Molecular determinants of intrinsic cellular stiffness in health and disease

In recent years, the role of intrinsic biophysical features, especially cellular stiffness, in diverse cellular and disease processes is being increasingly recognized. New high throughput techniques for the quantification of cellular stiffness facilitate the study of their roles in health and diseases. In this review, we summarized recent discovery about how cellular stiffness is involved in cell stemness, tumorigenesis, and blood diseases. In addition, we review the molecular mechanisms underlying the gene regulation of cellular stiffness in health and disease progression. Finally, we discussed the current understanding on how the cytoskeleton structure and the regulation of these genes contribute to cellular stiffness, highlighting where the field of cellular stiffness is headed.

Anisotropy profoundly alters stress fields within contractile cells and cell aggregates

Many biological phenomena such as cell proliferation and death are correlated with stress fields within cells. Stress fields are quantified using computational methods which rely on fundamental assumptions about local mechanical properties. Most existing methods such as Monolayer Stress Microscopy assume isotropic properties, yet experimental observations strongly suggest anisotropy. We first model anisotropy in circular cells analytically using Eshelby's inclusion method. Our solution reveals that uniform anisotropy cannot exist in cells due to the occurrence of substantial stress concentration in the central region. A more realistic non-uniform anisotropy model is then introduced based on experimental observations and implemented numerically which interestingly clears out stress concentration. Stresses within the entire aggregate also drastically change compared to the isotropic case, resulting in better agreement with observed biomarkers. We provide a physics-based mechanism to explain the low alignment of stress fibers in the center of cells, which might explain certain biological phenomena e.g., existence of disrupted rounded cells, and higher apoptosis rate at the center of circular aggregates.

Impedance-Enabled Camera-Free Intrinsic Mechanical Cytometry

Mechanical properties of single cells are important label-free biomarkers normally measured by expensive and complex imaging systems. To unlock this limit and allow mechanical properties comparable across different measurement platforms, camera-free intrinsic mechanical cytometry (CFIMC) is proposed for on-the-fly measurement of two major intrinsic mechanical parameters, that is, Young's modulus E and fluidity β, of single cells. CFIMC adopts a framework that couples the impedance electrodes with the constriction channel spatially, so that the impedance signals contain the dynamic deformability information of the cell squeezing through the constriction channel. Deformation of the cell is thus extracted from the impedance signals and used to derive the intrinsic mechanical parameters. With reasonably high throughput (>500 cells min^-1 ), CFIMC can successfully reveal the mechanical difference in cancer and normal cells (i.e., human breast cell lines MCF-10A, MCF-7, and MDA-MB-231), living and fixed cells, and pharmacological perturbations of the cytoskeleton. It is further found that 1 µM level concentration of Cytochalasin B may be the threshold for the treated cells to induce a significant cytoskeleton effect reflected by the mechanical parameters. It is envisioned that CFIMC provides an alternative avenue for high-throughput and real-time single-cell intrinsic mechanical analysis.

Constitutive Equations for Analyzing Stress Relaxation and Creep of Viscoelastic Materials Based on Standard Linear Solid Model Derived with Finite Loading Rate

The viscoelastic properties of materials such as polymers can be quantitatively evaluated by measuring and analyzing the viscoelastic behaviors such as stress relaxation and creep. The standard linear solid model is a classical and commonly used mathematical model for analyzing stress relaxation and creep behaviors. Traditionally, the constitutive equations for analyzing stress relaxation and creep behaviors based on the standard linear solid model are derived using the assumption that the loading is a step function, implying that the loading rate used in the loading process of stress relaxation and creep tests is infinite. Using such constitutive equations may cause significant errors in analyses since the loading rate must be finite (no matter how fast it is) in a real stress relaxation or creep experiment. The purpose of this paper is to introduce the constitutive equations for analyzing stress relaxation and creep behaviors based on the standard linear solid model derived with a finite loading rate. The finite element computational simulation results demonstrate that the constitutive equations derived with a finite loading rate can produce accurate results in the evaluation of all viscoelastic parameters regardless of the loading rate in most cases. It is recommended that the constitutive equations derived with a finite loading rate should replace the traditional ones derived with an infinite loading rate to analyze stress relaxation and creep behaviors for quantitatively evaluating the viscoelastic properties of materials.

Non-invasive measurement of nuclear relative stiffness from quantitative analysis of microscopy data

The connection between the properties of a cell tissue and those of the single constituent cells remains to be elucidated. At the purely mechanical level, the degree of rigidity of different cellular components, such as the nucleus and the cytoplasm, modulates the interplay between the cell inner processes and the external environment, while simultaneously mediating the mechanical interactions between neighboring cells. Being able to quantify the correlation between single-cell and tissue properties would improve our mechanobiological understanding of cell tissues. Here we develop a methodology to quantitatively extract a set of structural and motility parameters from the analysis of time-lapse movies of nuclei belonging to jammed and flocking cell monolayers. We then study in detail the correlation between the dynamical state of the tissue and the deformation of the nuclei. We observe that the nuclear deformation rate linearly correlates with the local divergence of the velocity field, which leads to a non-invasive estimate of the elastic modulus of the nucleus relative to the one of the cytoplasm. We also find that nuclei belonging to flocking monolayers, subjected to larger mechanical perturbations, are about two time stiffer than nuclei belonging to dynamically arrested monolayers, in agreement with atomic force microscopy results. Our results demonstrate a non-invasive route to the determination of nuclear relative stiffness for cells in a monolayer.

Biophysical studies of cancer cells' traverse-vessel behaviors under different pressures revealed cells' motion state transition

Circulating tumor cells (CTCs) survive in the bloodstream and then seed and invade to foster tumor metastasis. The arrest of cancer cells is favored by permissive flow forces and geometrical constraints. Through the use of high-throughput microfluidic devices designed to mimic capillary-sized vessels, we applied pressure differences to cancer cells (MCF-7 cell line) and recorded the cell traverse-vessel behaviors. Our results showed that cancer cells transform from a Newtonian droplet state to an adhesion/migration state when cancer cells traverse artificial vessels. To explain these phenomena, a modified Newtonian droplet model was also proposed. These phenomena and the modified model may reveal how CTCs in the blood seed and invade vessels under suitable conditions.

A high throughput microfluidic system with large ranges of applied pressures for measuring the mechanical properties of single fixed cells and differentiated cells

The mechanical properties of cells are of great significance to their normal physiological activities. The current methods used for the measurement of a cell's mechanical properties have the problems of complicated operation, low throughput, and limited measuring range. Based on micropipette technology, we designed a double-layer micro-valve-controlled microfluidic chip with a series of micropipette arrays. The chip has adjustment pressure ranges of 0.03-1 and 0.3-10 kPa and has a pressure stabilization design, which can achieve a robust measurement of a single cell's mechanical properties under a wide pressure range and is simple to operate. Using this chip, we measured the mechanical properties of the cells treated with different concentrations of paraformaldehyde (PFA) and observed that the viscoelasticity of the cells gradually increased as the PFA concentration increased. Then, this method was also used to characterize the changes in the mechanical properties of the differentiation pathways of stem cells from the apical papilla to osteogenesis.

The effects of measurement parameters on the cancerous cell nucleus characterization by atomic force microscopy in vitro

Cancer is now responsible for the major leading cause of death worldwide. It is noteworthy that lung cancer has been recognised as the highest incidence (11.6%) and mortality (18.4%) for combined sexes among a variety of cancer diseases. Therefore, it is of great value to investigate the mechanical properties of lung cancerous cells for early diagnosis. This paper focus on the influence of measurement parameters on the measured central Young's moduli of single live A549 cell in vitro based on the force spectroscopy mode of atomic force microscopy (AFM). The effects of the measurement parameters on the measured central Young's moduli were analysed by fitting the force–depth curves utilising the Sneddon model. The results revealed that the Young's moduli of A549 cells increased with the larger indentation force, higher indentation speed, less retraction time, deeper Z length and lower purity percentage of serum. The Young's moduli of cells increased first and then decreased with the increasing dwell time. Hence, this research may have potential significance to provide reference for the standardised detection of a single cancerous cell in vitro using AFM methodologies.

Cancer is now responsible for the majority leading cause of death worldwide and it is noteworthy that lung cancer has been recognised as the highest incidence (11.6%) and mortality (18.4%) for combined sexes among a variety of cancer diseases. Therefore, it is of great value to investigate the mechanical properties of lung cancerous cells for early diagnosis. This paper primarily investigated the morphological properties and the influence of measurement parameters on the measured local elastic moduli of single live A549 cell in vitro using the AFM‐based force spectroscopy mode. In practice, there are many factors for incorrect or inaccurate experimental results using AFM to measure the characteristics of live cells, such as non‐homogeneous nature of cells, probe geometry and size, mechanical analysis model, substrate stiffness and different measurement parameters. The various measurement parameters have become the huge impact factor to influence the measurement result. Hence, this research may have potential significance to provide reference for the standardised detection of a single cancerous cell in vitro using AFM methodologies.

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.

PillarX: A Microfluidic Device to Profile Circulating Tumor Cell Clusters Based on Geometry, Deformability, and Epithelial State

Circulating tumor cell (CTC) clusters are associated with increased metastatic potential and worse patient prognosis, but are rare, difficult to count, and poorly characterized biophysically. The PillarX device described here is a bimodular microfluidic device (Pillar-device and an X-magnetic device) to profile single CTCs and clusters from whole blood based on their size, deformability, and epithelial marker expression. Larger, less deformable clusters and large single cells are captured in the Pillar-device and sorted according to pillar gap sizes. Smaller, deformable clusters and single cells are subsequently captured in the X-device and separated based on epithelial marker expression using functionalized magnetic nanoparticles. Clusters of established and primary breast cancer cells with variable degrees of cohesion driven by different cell-cell adhesion protein expression are profiled in the device. Cohesive clusters exhibit a lower deformability as they travel through the pillar array, relative to less cohesive clusters, and have greater collective invasive behavior. The ability of the PillarX device to capture clusters is validated in mouse models and patients of metastatic breast cancer. Thus, this device effectively enumerates and profiles CTC clusters based on their unique geometrical, physical, and biochemical properties, and could form the basis of a novel prognostic clinical tool.

Finite Element Models of Osteocytes and Their Load-Induced Activation

PURPOSE OF REVIEW

Osteocytes are the conductors of bone adaptation and remodelling. Buried inside the calcified matrix, they sense mechanical cues and signal osteoclasts in case of low activity, and osteoblasts when stresses are high. How do osteocytes detect mechanical stress? What physical signal do they perceive? Finite element analysis is a useful tool to address these questions as it allows calculating stresses, strains and fluid flow where they cannot be measured. The purpose of this review is to evaluate the capabilities and challenges of finite element models of bone, in particular the osteocytes and load-induced activation mechanisms.

RECENT FINDINGS

High-resolution imaging and increased computational power allow ever more detailed modelling of osteocytes, either in isolation or embedded within the mineralised matrix. Over the years, homogeneous models of bone and osteocytes got replaced by heterogeneous and microstructural models, including, e.g. the lacuno-canalicular network and the cytoskeleton. The lacuno-canalicular network induces strain amplifications and the osteocyte protrusions seem to be stimulated much more than the cell body, both by strain and fluid flow. More realistic cell geometries, like minute constrictions of the canaliculi, increase this effect. Microstructural osteocyte models describe the transduction of external stimuli to the nucleus. Supracellular multiscale models (e.g. of a tunnelling osteon) allow to study differential loading of osteocytes and to distinguish between strain and fluid flow as the pivotal stimulatory cue. In the future, the finite element models may be enhanced by including chemical transport and intercellular communication between osteocytes, osteoclasts and osteoblasts.

Understanding mitochondria and the utility of optimization as a canonical framework for identifying and modeling mitochondrial pathways

The goal of this paper is to provide an overview of our current understanding of mitochondrial function as a framework to motivate the hypothesis that mitochondrial behavior is governed by optimization principles that are constrained by the laws of the physical and biological sciences. Then, mathematical optimization tools can generally be useful to model some of these processes under reasonable assumptions and limitations. We are specifically interested in optimizations via variational methods, which are briefly summarized. Within such an optimization framework, we suggest that the numerous mechanical instigators of cell and intracellular functioning can be modeled utilizing some of the principles of mechanics that govern engineered systems, as well as by the frequently observed feedback and feedforward mechanisms that coordinate the multitude of processes within cells. These mechanical aspects would need to be coupled to governing biochemical rules. Of course, biological systems are significantly more complex than engineered systems, and require considerably more experimentation to ascertain and characterize parameters and subsequent behavior. That complexity requires well-defined limitations and assumptions for any derived models. Optimality is being motivated as a framework to help us understand how cellular decisions are made, especially those that transition between physiological behaviors and dysfunctions along pathophysiological pathways. We elaborate on our interpretation of optimality and cellular decision making within the body of this paper, as we revisit these ideas in the numerous different contexts of mitochondrial functions.

Viscoelasticity, Like Forces, Plays a Role in Mechanotransduction

Viscoelasticity and its alteration in time and space has turned out to act as a key element in fundamental biological processes in living systems, such as morphogenesis and motility. Based on experimental and theoretical findings it can be proposed that viscoelasticity of cells, spheroids and tissues seems to be a collective characteristic that demands macromolecular, intracellular component and intercellular interactions. A major challenge is to couple the alterations in the macroscopic structural or material characteristics of cells, spheroids and tissues, such as cell and tissue phase transitions, to the microscopic interferences of their elements. Therefore, the biophysical technologies need to be improved, advanced and connected to classical biological assays. In this review, the viscoelastic nature of cytoskeletal, extracellular and cellular networks is presented and discussed. Viscoelasticity is conceptualized as a major contributor to cell migration and invasion and it is discussed whether it can serve as a biomarker for the cells' migratory capacity in several biological contexts. It can be hypothesized that the statistical mechanics of intra- and extracellular networks may be applied in the future as a powerful tool to explore quantitatively the biomechanical foundation of viscoelasticity over a broad range of time and length scales. Finally, the importance of the cellular viscoelasticity is illustrated in identifying and characterizing multiple disorders, such as cancer, tissue injuries, acute or chronic inflammations or fibrotic diseases.

Viscoelastic parameterization of human skin cells characterize material behavior at multiple timescales

Countless biophysical studies have sought distinct markers in the cellular mechanical response that could be linked to morphogenesis, homeostasis, and disease. Here, an iterative-fitting methodology visualizes the time-dependent viscoelastic behavior of human skin cells under physiologically relevant conditions. Past investigations often involved parameterizing elastic relationships and assuming purely Hertzian contact mechanics, which fails to properly account for the rich temporal information available. We demonstrate the performance superiority of the proposed iterative viscoelastic characterization method over standard open-search approaches. Our viscoelastic measurements revealed that 2D adherent metastatic melanoma cells exhibit reduced elasticity compared to their normal counterparts—melanocytes and fibroblasts, and are significantly less viscous than fibroblasts over timescales spanning three orders of magnitude. The measured loss angle indicates clear differential viscoelastic responses across multiple timescales between the measured cells. This method provides insight into the complex viscoelastic behavior of metastatic melanoma cells relevant to better understanding cancer metastasis and aggression.

Parvini, Cartagena and Solares introduce an iterative viscoelastic approach based on the generalized Maxwell and Kelvin-Voigt models. The results showed that metastatic melanoma cells had lower elasticity than normal fibroblasts and melanoma cells were less viscous than the fibroblasts over a large frequency range, enhancing the understanding of cellular responses at different frequencies.

Hyperbolic equations for neuronal membrane deformation waves accompanying an action potential

Experiments show that the propagation of an action potential along an axon is accompanied by mechanical deformations. We describe the mechanisms of the effect using fluid dynamic equations, Laplace's and Hook's laws for surface tension, and Lippmann's law, which relates membrane tension to membrane potential. We derived a minimal, 1-D model, which is a hyperbolic system of equations. Our model qualitatively reproduces the membrane's mechanical deformation evoked by either the propagation of an action potential or the stepwise change of membrane potential. The understanding of the relationship between electrical activity and mechanical deformation provides guidance toward non-invasive imaging of neuronal activity.

Poroelastic osmoregulation of living cell volume

Cells maintain their volume through fine intracellular osmolarity regulation. Osmotic challenges drive fluid into or out of cells causing swelling or shrinkage, respectively. The dynamics of cell volume changes depending on the rheology of the cellular constituents and on how fast the fluid permeates through the membrane and cytoplasm. We investigated whether and how poroelasticity can describe volume dynamics in response to osmotic shocks. We exposed cells to osmotic perturbations and used defocusing epifluorescence microscopy on membrane-attached fluorescent nanospheres to track volume dynamics with high spatiotemporal resolution. We found that a poroelastic model that considers both geometrical and pressurization rates captures fluid-cytoskeleton interactions, which are rate-limiting factors in controlling volume changes at short timescales. Linking cellular responses to osmotic shocks and cell mechanics through poroelasticity can predict the cell state in health, disease, or in response to novel therapeutics.

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Cell height changes can be finely captured by defocusing microscopy

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Water permeation and cellular deformability regulate dynamics of cell volume changes

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Poroelasticity describes the dynamics of cell volume changes

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The response of cell to hypo or hyperosmotic shocks are modeled by poroelasticity

Viscoelasticity Acts as a Marker for Tumor Extracellular Matrix Characteristics

Biological materials such as extracellular matrix scaffolds, cancer cells, and tissues are often assumed to respond elastically for simplicity; the viscoelastic response is quite commonly ignored. Extracellular matrix mechanics including the viscoelasticity has turned out to be a key feature of cellular behavior and the entire shape and function of healthy and diseased tissues, such as cancer. The interference of cells with their local microenvironment and the interaction among different cell types relies both on the mechanical phenotype of each involved element. However, there is still not yet clearly understood how viscoelasticity alters the functional phenotype of the tumor extracellular matrix environment. Especially the biophysical technologies are still under ongoing improvement and further development. In addition, the effect of matrix mechanics in the progression of cancer is the subject of discussion. Hence, the topic of this review is especially attractive to collect the existing endeavors to characterize the viscoelastic features of tumor extracellular matrices and to briefly highlight the present frontiers in cancer progression and escape of cancers from therapy. Finally, this review article illustrates the importance of the tumor extracellular matrix mechano-phenotype, including the phenomenon viscoelasticity in identifying, characterizing, and treating specific cancer types.

Advance of microfluidic constriction channel system of measuring single-cell cortical tension/specific capacitance of membrane and conductivity of cytoplasm

This paper reported a microfluidic platform which realized the characterization of inherent single-cell biomechanical and bioelectrical parameters simultaneously. Individual cells traveled through a constriction channel with deformation images and impedance variations captured and processed into cortical tension T_c , specific membrane capacitance C_sm , and cytoplasmic conductivity σ_cy based on an equivalent biophysical model. These properties of thousands of individual cells of K562, Jurkat, HL-60, HL-60 treated with paraformaldehyde (PA)/cytochalasin D (CD)/concanavalin A (ConA), granulocytes of Donor 1, Donor 2, and Donor 3 were quantified for the first time. Leveraging T_c , C_sm , and σ_cy , (1) high accuracies of classifying wild-type and processed HL-60 cells (e.g., 93.5% of PA treated vs. CD treated HL-60 cells) were realized, revealing the effectiveness of using these three biophysical parameters in cell-type classification; (2) low accuracies of classifying normal granulocytes from three donors (e.g., 56.4% of Donor 1 vs. 2), indicating comparable parameters for normal granulocytes. In conclusion, this platform can characterize single-cell T_c , C_sm , and σ_cy concurrently and quantify multiple parameters in single-cell analysis.

Modeling cell membrane electrodeformation by alternating electric fields

With the aim of characterizing and gaining insight into the frequency response of cells suspended in a fluid medium and deformed with a controlled alternating electric field, a continuum-based analysis is presented for modeling electrodeformation (ED) via Maxwell stress tensor (MST) calculation. Our purpose here is to apply this approach to explain the fact that the electric field anisotropy and electrical conductivity ratio Λ of the cytoplasm and the extracellular medium significantly impact the MST exerted on the cytoplasm-membrane interface. One important finding is that the modulation of electrical cues and MST force by the frequency of the applied electric field provides an extremely rich tool kit for manipulating cells. We show the extreme sensitivity of proximity-induced capacitive coupling arising concomitantly when the magnitude of the MST increases as the distance between cells is decreased and the spatial anisotropy becomes important. Moreover, our model highlights the strongly localized character of the electrostatic field effect emanating from neighboring cells and suggests the possibility of exploiting cell distribution as a powerful tool to engineer the functional performance of cell assemblies by controlling ED and capacitive coupling. We furthermore show that frequency has a significant impact on the attenuation-amplification transition of MST, suggesting that shape anisotropy has a much weaker influence on ED of the cell membrane compared to the anisotropy induced by the orientation angle itself.

Planar hydrodynamic traps and buried channels for bead and cell trapping and releasing

We present a novel concept for the controlled trapping and releasing of beads and cells in a PDMS microfluidic channel without obstacles present around the particle or in the channel. The trapping principle relies on a two-level microfluidic configuration: a top main PDMS channel interconnected to a buried glass microchannel using round vias. As the fluidic resistances rule the way the liquid flows inside the channels, particles located in the streamlines passing inside the buried level are immobilized by the round via with a smaller diameter, leaving the object motionless in the upper PDMS channel. The particle is maintained by the difference of pressure established across its interface and acts as an infinite fluidic resistance, virtually cancelling the subsequent buried fluidic path. The pressure is controlled at the outlet of the buried path and three modes of operation of a trap are defined: idle, trapping and releasing. The pressure conditions for each mode are defined based on the hydraulic-electrical circuit equivalence. The trapping of polystyrene beads in a compact array of 522 parallel traps controlled by a single pressure was demonstrated with a trapping efficiency of 94%. Pressure conditions necessary to safely trap cells in holes of different diameters were determined and demonstrated in an array of 25 traps, establishing the design and operation rules for the use of planar hydrodynamic traps for biological assays.

Brain energetics, mitochondria, and traumatic brain injury

We review current thinking about, and draw connections between, brain energetics and metabolism, and between mitochondria and traumatic brain injury. Energy is fundamental to proper brain function. Its creation in a useful form for neurons and glia, and consistently in response to the brain's high energy needs, is critical for physiological pathways. Dysfunction in the mechanisms of energy production is at the center of neurological and neuropsychiatric pathologies. We examine the connections between energetics and mitochondria - the organelle responsible for almost all the energy production in the cell - and how secondary pathologies in traumatic brain injury result from energetic dysfunction. This paper interweaves these topics, a necessity since they are closely coupled, and identifies where there exist a lack of understanding and of data. In addition to summarizing current thinking in these disciplines, our goal is to suggest a framework for the mathematical modeling of mechanisms and pathways based on optimal energetic decisions.

Numerical Modelling of Multicellular Spheroid Compression: Viscoelastic Fluid vs. Viscoelastic Solid

Parallel-plate compression of multicellular spheroids (MCSs) is a promising and popular technique to quantify the viscoelastic properties of living tissues. This work presents two different approaches to the simulation of the MCS compression based on viscoelastic solid and viscoelastic fluid models. The first one is the standard linear solid model implemented in ABAQUS/CAE. The second one is the new model for 3D viscoelastic free surface fluid flow, which combines the Oldroyd-B incompressible fluid model and the incompressible neo-Hookean solid model via incorporation of an additional elastic tensor and a dynamic equation for it. The simulation results indicate that either approach can be applied to model the MCS compression with reasonable accuracy. Future application of the viscoelastic free surface fluid model is the MCSs fusion highly-demanded in bioprinting.