Theoretical model and experimental study of red blood cell (RBC) deformation in microchannels

The motion and deformation of red blood cells (RBCs) flowing in a microchannel were studied using a theoretical model and a novel automated rheoscope. The theoretical model was developed to predict the cells deformation under shear as a function of the cells geometry and mechanical properties. Fluid dynamics and membrane mechanics are incorporated, calculating the traction and deformation in an iterative manner. The model was utilized to evaluate the effect of different biophysical parameters, such as: inner cell viscosity, membrane shear modulus and surface to volume ratio on deformation measurements. The experimental system enables the measurement of individual RBCs velocity and their deformation at defined planes within the microchannel. Good agreement was observed between the simulation results, the rheoscope measurements and published ektacytometry results. The theoretical model results imply that such deformability measuring techniques are weakly influenced by changes in the inner viscosity of the cell or the ambient fluid viscosity. However, these measurements are highly sensitive to RBC shear modulus. The shear modulus, estimated by the model and the rheoscope measurements, falls between the values obtained by micropipette aspiration and laser trapping. The study demonstrates the integration of a theoretical model with a microfabricated device in order to achieve a better understanding of RBC mechanics and their measurement using microfluidic shear assays. The system and the model have the potential of serving as quantitative clinical tools for diagnosing deformability disorders in RBCs.     

High-resolution spatial mapping of shear properties in cartilage

Structural properties of articular cartilage such as proteoglycan content, collagen content and collagen alignment are known to vary over length scales as small as a few microns (Bullough and Goodfellow, 1968; Bi et al., 2006). Characterizing the resulting variation in mechanical properties is critical for understanding how the inhomogeneous architecture of this tissue gives rise to its function. Previous studies have measured the depth-dependent shear modulus of articular cartilage using methods such as particle image velocimetry (PIV) that rely on cells and cell nuclei as fiducial markers to track tissue deformation (Buckley et al., 2008; Wong et al., 2008a). However, such techniques are limited by the density of trackable markers, which may be too low to take full advantage of optical microscopy. This limitation leads to noise in the acquired data, which is often exacerbated when the data is manipulated. In this study, we report on two techniques for increasing the accuracy of tissue deformation measurements. In the first technique, deformations were tracked in a grid that was photobleached on each tissue sample (Bruehlmann et al., 2004). In the second, a numerical technique was implemented that allowed for accurate differentiation of optical displacement measurements by minimizing the propagated experimental error while ensuring that truncation error associated with local averaging of the data remained small. To test their efficacy, we employed these techniques to compare the depth-dependent shear moduli of neonatal bovine and adult human articular cartilage. Using a photobleached grid and numerical optimization to gather and analyze data led to results consistent with those reported previously (Buckley et al., 2008; Wong et al., 2008a), but with increased spatial resolution and characteristic coefficients of variation that were reduced up to a factor of 3. This increased resolution allowed us to determine that the shear modulus of neonatal bovine and adult human tissue both exhibit a global minimum at a depth z of around 100 μm and plateau at large depths. The consistency of the depth dependence of |G*|(Z) for adult human and neonatal bovine tissue suggests a functional advantage resulting from this behavior.     

Quantifying the contribution of actin networks to the elastic strength of fibroblasts

The structural models created to understand the cytoskeletal mechanics of cells in suspension are described here. Suspended cells can be deformed by well-defined surface stresses in an Optical Stretcher [Guck, J., Ananthakrishnan, R., Mahmood, H., Moon, T.J., Cunningham, C.C., K?s, J., 2001. The optical stretcher: a novel laser tool to micromanipulate cells. Biophys. J. 81(2), 767-784], a two-beam optical trap designed for the contact-free deformation of cells. Suspended cells have a well-defined cytoskeleton, displaying a radially symmetric actin cortical network underlying the cell membrane with no actin stress fibers, and microtubules and intermediate filaments in the interior. Based on experimental data using suspended fibroblasts, we create two structural models: a thick shell actin cortex model that describes cell deformation for a localized stress distribution on these cells and a three-layered model that considers the entire cytoskeleton when a broad stress distribution is applied. Applying the models to data, we obtain a (actin) cortical shear moduli G of approximately 220 Pa for normal fibroblasts and approximately 185 Pa for malignantly transformed fibroblasts. Additionally, modeling the cortex as a transiently crosslinked isotropic actin network, we show that actin and its crosslinkers must be co-localized into a tight shell to achieve these cortical strengths. The similar moduli values and cortical actin and crosslinker densities but different deformabilities of the normal and cancerous cells suggest that a cell's structural strength is not solely determined by cytoskeletal composition but equally importantly by (actin) cytoskeletal architecture via differing cortical thicknesses. We also find that although the interior structural elements (microtubules, nucleus) contribute to the deformed cell's exact shape via their loose coupling to the cortex, it is the outer actin cortical shell (and its thickness) that mainly determines the cell's structural response.     

Cell poking. Determination of the elastic area compressibility modulus of the erythrocyte membrane

Cell poking, a new method for measuring mechanical properties of single cells was used to determine the elastic area compressibility modulus of osmotically swollen human erythrocytes. With this method we determined the force required to indent cells attached to a glass coverslip (Petersen, N.O., W. B. McConnaughey , and E. L. Elson , 1982, Proc. Natl. Acad. Sci. USA, 79:5327. Forces on the order of one millidyne and indentations on the order of one micron were detected. An analysis of these data in terms of a simplified mechanical model yielded the elastic area compressibility modulus. This analysis used a variational approach to minimize the isothermal elastic potential energy density function given by E. A. Evans and R. Skalak (Mechanics and Thermodynamics of Biomembranes, 1980, CRC Press, Boca Raton , FL). Measurements on swollen erythrocytes gave a range of values, depending in part on the osmotic conditions, of 17.9 +/- 8.2 to 34.8 +/- 12.0 mdyn /micron for the elastic area compressibility modulus at 25 degrees C. Fractional area expansion greater than 2.6 +/- 0.8% produced rapid cell lysis. These values were not corrected for the reversible movement of water across the cell membrane in response to hydrostatic pressure gradients. Our results agree reasonably with those obtained by Evans et al. (Evans, E.A., R. Waugh , and L. Melnick , 1976, Biophys. J., 16:585-595.) using micropipette aspiration under similar conditions.     

The stress-free shape of the red blood cell membrane.

The two main proposals found in the literature for the stress-free shape of the red cell membrane are (a) the bioconcave shape and (b) the sphere of the same surface area. These possibilities are evaluated in this paper using theoretical modeling of equilibrium membrane shapes according to Zarda et al. (1977. J. Biomech. 10:211-221) and by comparison to experiments on red cells whose membrane shear modulus has been increased by treatment with diamide. Neither proposal is found to be compatible with all the experimental behaviour of native red cells. Neither proposal is found to be compatible with all the experimental behaviour of native red cells. To account for this discrepancy we propose that either the shear modulus of the native membrane is dependent on the membrane strain or that the bending stiffness is higher than estimated by Evans (1980. Biophys. J. 30:265-286). These studies suggest that the bioconcave disk is the more likely possibility for the stress-free shape.     

On the calculation of the elastic modulus of a biofilm streamer

Biofilm mechanical properties are essential in quantifying the rate of microbial detachment, a key process in determining the function and structure of biofilm systems. Although properties such as biofilm elastic moduli, yield stress and cohesive strength have been studied before, a wide range of values for the biofilm Young's modulus that differ by several orders of magnitude are reported in the literature. In this article, we use experimental data reported in Stoodley et al. [Stoodley et al., Biotechnol Bioeng (1999): 65(1):83-92] and present a methodology for the calculation of Young's modulus, which partially explains the large difference between the values reported in the literature.     

An extended relationship for the characterization of Young's modulus and Poisson's ratio of tunable polyacrylamide gels

Substrates with tunable mechanical properties are crucial for the study of cellular processes, and polyacrylamide gels (PAGs) are frequently used in this context. Several experimental techniques have been proposed to obtain the mechanical properties of PAGs. However, the range of the considered Poisson's ratio values remains quite large and no attempt has been made to propose an analytical relationship allowing the estimation of PAG Young's modulus when both bis-acrylamide and acrylamide concentrations are known. In order to complete the actual knowledge on the mechanical properties of PAGs, we took benefit of our original method based on the micropipette aspiration technique (Boudou et al., J. Biomech. 2006) for characterizing gels made with concentrations in the range 0.02% < or =[Bis]< or =0.20% and 3% < or =[Acry]< or =10%. We found that the PAGs Young's modulus varies nonlinearly with the acrylamide amount. Moreover, our study validates the quasi-incompressibility hypothesis usually made in studies using PAGs (mean Poisson's ratio of 0.480+/-0.012). More generally, and in agreement with data published by other groups, we propose an original nonlinear mathematical relationship allowing the computation of Young's modulus of PAG for any given acrylamide and bis-acrylamide amounts taken in the range of values we considered.     

Vibration and length-dependent flexural rigidity of protein microtubules using higher order shear deformation theory

Microtubules are hollow cylindrical filaments of the eukaryotic cytoskeleton characterized by extremely low shear modulus. A remarkable controversy has occurred in the literature, regarding the length dependence of flexural rigidity of microtubules predicted by the classical elastic beam model. In this study, a higher order shear deformable beam model for microtubules is employed to study unexplained length-dependent flexural rigidity and Young’s modulus of microtubules reported in the literature. The formulation allows for warping of the cross-section of the microtubule and eliminates the need for using arbitrary shear correction coefficients as in other theories. It is showed that vibration frequencies predicted by the present parabolic shear deformation theory (PSDT) are much lower than that given by the approximate isotropic beam model for shorter microtubules, although the two models give almost identical results for sufficiently long microtubules. It is confirmed that transverse shearing and the warping of the cross-section of microtubules are mainly responsible for the length-dependent flexural rigidity of an isolated microtubule reported in the literature, which cannot be explained by the widely used Euler-Bernoulli beam model. Indeed, the length-dependent flexural rigidity predicted by the present model is found to be in qualitative agreement with the existing experimental data ( [Kurachi et al., 1995] and [Pampaloni et al., 2006]). These results recommend that the parabolic shear deformation-beam theory offers a unified simple 1D model, which can capture the length dependence of flexural rigidity and be applied to various static and dynamic problems of microtubule mechanics.     

Axisymmetric optical-trap measurement of red blood cell membrane elasticity

The elastic properties of the cell membrane play a crucial role in determining the equilibrium shape of the cell, as well as its response to the external forces it experiences in its physiological environment. Red blood cells are a favored system for studying membrane properties because of their simple structure: a lipid bilayer coupled to a membrane cytoskeleton and no cytoplasmic cytoskeleton. An optical trap is used to stretch a red blood cell, fixed to a glass surface, along its symmetry axis by pulling on a micron-sized latex bead that is bound at the center of the exposed cell dimple. The system, at equilibrium, shows Hookean behavior with a spring constant of 1.5×10(-6)?N/m over a 1-2 μm range of extension. This choice of simple experimental geometry preserves the axial symmetry of the native cell throughout the stretch, probes membrane deformations in the small-extension regime, and facilitates theoretical analysis. The axisymmetry makes the experiment amenable to simulation using a simple model that makes no a priori assumption on the relative importance of shear and bending in membrane deformations. We use an iterative relaxation algorithm to solve for the geometrical configuration of the membrane at mechanical equilibrium for a range of applied forces. We obtain estimates for the out-of-plane membrane bending modulus B≈1×10(-19)?Nm and an upper limit to the in-plane shear modulus H<2×10(-6)?N/m. The partial agreement of these results with other published values may serve to highlight the dependence of the cell's resistance to deformation on the scale and geometry of the deformation.     

The density and strength of proteoglycan-proteoglycan interaction sites in concentrated solutions.

Rheological flow properties of link-stable and link-free proteoglycan (PG) aggregates in concentrated solutions were measured using a cone-on-plate viscometer. A second-order constitutive model, based upon the statistical-network theories of Lodge, [Rheol. Acta 7, 379-392 (1968)] and De Kee and Carreau [J. Non-Newtonian Fluid Mech. 6, 127-143 (1979)], was developed to describe the measured steady and transient flow responses exhibited by the PG solutions. Our measurements confirmed previous experimental findings that the complex shear modulus of PG solutions depends on the frequency of the imposed small-amplitude oscillatory shear, and the apparent viscosity and primary normal-stress difference depend nonlinearly on the shear rate under steady-shear flow conditions [Mow et al., J. Biomechanics 17, 325-338 (1984b); Hardingham et al., J. orthop. Res. 5, 36-46 (1987)]. In the present study, we found that PG solutions exhibit pronounced stress overshoot responses and large hysteresis loop effects. These transient responses were shown to be sensitive to acceleration strain (i.e. the second rate of strain) as well as PG structure (i.e. link-protein stabilization). The model parameters were determined by curvefitting of the second-order constitutive model and experimental data from steady, oscillatory and transient shear flow measurements. Using this network model, we calculated the density of the idealized interaction sites existing in the PG network, and the average strength of these interaction sites. The results indicate that link-protein stabilization of PG aggregates does not change the density of interaction sites formed in the PG network, rather, it increases the average strength of these interaction sites.     

Deformation properties of articular chondrocytes: a critique of three separate techniques

This paper presents a series of techniques, which examine the deformation characteristics of bovine articular chondrocytes. The direct contact approach employs well established methodology, involving AFM and micropipette aspiration, to yield structural properties of local regions of isolated chondrocytes. The former technique yields a non-linear response with increased structural stiffness in a central location on a projected image of the chondrocyte. A simple viscoelastic model can be used with data from the micropipette aspiration technique to yield a mean value of Young's modulus, which is similar to that recently reported (Jones et al., 1999). An indirect approach is also described, involving the response of chondrocytes seeded within compressed agarose constructs. For 1% agarose constructs, the resulting cell strain, yields a gross cell modulus of 2.7 kPa. The study highlights the difficulties in establishing unique mechanical parameters, which reflect the deformation behaviour of articular chondrocytes.     

Deduction of intrinsic mechanical properties of the erythrocyte membrane from observations of tank-treading in the rheoscope

Measurements of the dimensions and membrane rotational frequency of individual erythrocytes steadily tank-treading in a Rheoscope are used to deduce the surface shear viscosity (eta m) and the shear elastic modulus (mu m) of the membrane. Previously published algorithms (Trans-Son-Tay et al., Biophys. J. 46: 65, 1984, and 51: 915, 1987) plus an assumed area-conserving membrane velocity field (Secomb and Skalak, Q. J. Mech. Appl. Math. XXXV 2: 233, 1982) are applied to calculate eta m as a function of the second invariant of the surface strain rate and mu m as a function of the second invariant of membrane strain. The results indicate density-related increases in membrane stiffness and viscosity, shear-thinning viscous behavior, and strain-stiffening elastic behavior.     

Nanoindentation measurements of biomechanical properties in mature and newly formed bone tissue surrounding an implant

The characterization of the biomechanical properties of newly formed bone tissue around implants is important to understand the osseointegration process. The objective of this study is to investigate the evolution of the hardness and indentation modulus of newly formed bone tissue as a function of healing time. To do so, a nanoindentation device is employed following a multimodality approach using histological analysis. Coin-shaped implants were placed in vivo at a distance of 200 μm from the cortical bone surface, leading to an initially empty cavity of 200 μm * 4.4 mm. Three New Zealand White rabbits were sacrificed after 4, 7, and 13 weeks of healing time. The bone samples were embedded and analyzed using histological analyses, allowing to distinguish mature and newly formed bone tissue. The bone mechanical properties were then measured in mature and newly formed bone tissue. The results are within the range of hardness and apparent Young's modulus values reported in previous literature. One-way ANOVA test revealed a significant effect of healing time on the indentation modulus (p < 0.001, F = 111.24) and hardness (p < 0.02, F = 3.47) of bone tissue. A Tukey-Kramer analysis revealed that the biomechanical properties of newly formed bone tissue (4 weeks) were significantly different from those of mature bone tissue. The comparison with the results obtained in Mathieu et al. (2011, "Micro-Brillouin Scattering Measurements in Mature and Newly Formed Bone Tissue Surrounding an Implant," J. Biomech. Eng., 133, 021006). shows that bone mass density increases by approximately 13.5% between newly formed bone (7 weeks) and mature bone tissue.     

Effect of hydroperoxides on red blood cell membrane mechanical properties

We investigate the effect of oxidative stress on red blood cell membrane mechanical properties in vitro using detailed analysis of the membrane thermal fluctuation spectrum. Two different oxidants, the cytosol-soluble hydrogen peroxide and the membrane-soluble cumene hydroperoxide, are used, and their effects on the membrane bending elastic modulus, surface tension, strength of confinement due to the membrane skeleton, and 2D shear elastic modulus are measured. We find that both oxidants alter significantly the membrane elastic properties, but their effects differ qualitatively and quantitatively. While hydrogen peroxide mainly affects the elasticity of the membrane protein skeleton (increasing the membrane shear modulus), cumene hydroperoxide has an impact on both membrane skeleton and lipid bilayer mechanical properties, as can be seen from the increased values of the shear and bending elastic moduli. The biologically important implication of these results is that the effects of oxidative stress on the biophysical properties, and hence the physiological functions, of the cell membrane depend on the nature of the oxidative agent. Thermal fluctuation spectroscopy provides a means of characterizing these different effects, potentially in a clinical milieu.     

Microstructural analysis of deformation-induced hypoxic damage in skeletal muscle

Deep pressure ulcers are caused by sustained mechanical loading and involve skeletal muscle tissue injury. The exact underlying mechanisms are unclear, and the prevalence is high. Our hypothesis is that the aetiology is dominated by cellular deformation (Bouten et al. in Ann Biomed Eng 29:153-163, 2001; Breuls et al. in Ann Biomed Eng 31:1357-1364, 2003; Stekelenburg et al. in J App Physiol 100(6):1946-1954, 2006) and deformation-induced ischaemia. The experimental observation that mechanical compression induced a pattern of interspersed healthy and dead cells in skeletal muscle (Stekelenburg et al. in J App Physiol 100(6):1946-1954, 2006) strongly suggests to take into account the muscle microstructure in studying damage development. The present paper describes a computational model for deformation-induced hypoxic damage in skeletal muscle tissue. Dead cells stop consuming oxygen and are assumed to decrease in stiffness due to loss of structure. The questions addressed are if these two consequences of cell death influence the development of cell injury in the remaining cells. The results show that weakening of dead cells indeed affects the damage accumulation in other cells. Further, the fact that cells stop consuming oxygen after they have died, delays cell death of other cells.     

Passive mechanical behavior of human neutrophils: power-law fluid.

The mechanical behavior of the neutrophil plays an important role in both the microcirculation and the immune system. Several laboratories in the past have developed mechanical models to describe different aspects of neutrophil deformability. In this study, the passive mechanical properties of normal human neutrophils have been further characterized. The cellular mechanical properties were assessed by single cell micropipette aspiration at fixed aspiration pressures. A numerical simulation was developed to interpret the experiments in terms of cell mechanical properties based on the Newtonian liquid drop model (Yeung and Evans, Biophys. J., 56: 139-149, 1989). The cytoplasmic viscosity was determined as a function of the ratio of the initial cell size to the pipette radius, the cortical tension, aspiration pressure, and the whole cell aspiration time. The cortical tension of passive neutrophils was measured to be about 2.7 x 10(-5) N/m. The apparent viscosity of neutrophil cytoplasm was found to depend on aspiration pressure, and ranged from approximately 500 Pa.s at an aspiration pressure of 98 Pa (1.0 cm H2O) to approximately 50 Pa.s at 882 Pa (9.0 cm H2O) when tested with a 4.0-micron pipette. These data provide the first documentation that the neutrophil cytoplasm exhibits non-Newtonian behavior. To further characterize the non-Newtonian behavior of human neutrophils, a mean shear rate gamma m was estimated based on the numerical simulation. The apparent cytoplasmic viscosity appears to decrease as the mean shear rate increases. The dependence of cytoplasmic viscosity on the mean shear rate can be approximated as a power-law relationship described by mu = mu c(gamma m/gamma c)-b, where mu is the cytoplasmic viscosity, gamma m is the mean shear rate, mu c is the characteristic viscosity at characteristic shear rate gamma c, and b is a material coefficient. When gamma c was set to 1 s-1, the material coefficients for passive neutrophils were determined to be mu c = 130 +/- 23 Pa.s and b = 0.52 +/- 0.09 for normal neutrophils. The power-law approximation has a remarkable ability to reconcile discrepancies among published values of the cytoplasmic viscosity measured using different techniques, even though these values differ by nearly two orders of magnitude. Thus, the power-law fluid model is a promising candidate for describing the passive mechanical behavior of human neutrophils in large deformation. It can also account for some discrepancies between cellular behavior in single-cell micromechanical experiments and predictions based on the assumption that the cytoplasm is a simple Newtonian fluid.     

结合AFM探测副伤寒沙门氏茵B感染红细胞膜的生物力学特性

采用原子力显微镜和衍射显微术,在纳米精确尺度探测副伤寒沙门氏菌B（Sp B）感染宿主红细胞（RBC）膜微观结构和力学特性,涉及细胞的形变、膜面内剪切模量和弯曲模量。结合这两种单分子测量技术,利用相关的数学模型表述RBC膜对菌体印B的入侵非常敏感。实验结果显示,不同感染期间的SpB寄生菌体,能够引起宿主RBC膜结构改变,形变能力降低,膜剪切模量和弯曲模量显著增加。这些力学特性的变化影响RBC的输氧和循环功能。实验结果表明,印B具有独特的鞭毛调控系统,入侵的毒性菌体寄生蛋白与血影蛋白网络中的运输蛋白有特异结合位点,导致RBC膜骨架网络、波动力学和细胞内、外基质都产生应激反应,这有可能为理解勋曰感染RBC的发病机理和寄生途径提供一些新的实验思路和分析依据。     

A model of non-uniform lung parenchyma distortion

A finite element model of mammalian lung parenchyma is used to study the effect of large non-uniform distortions on lung elastic behaviour. The non-uniform distortion is a uni-axial stretch from an initial state of uniform pressure expansion. For small distortions, the parenchymal properties are linearly isotropic and described by two elastic moduli. However, for large distortions, the parenchyma has anisotropic non-linear elastic properties described by five independent elastic moduli dependent on the degree of distortion; they are computed for a range of distortions and initial pressures. Ez, the Young's modulus in the direction of stretch, increases significantly with distortion (epsilon(z)) while Ex, the Young's modulus in the plane perpendicular to the stretch, is approximately constant. The greater the initial pressure, the bigger the difference between the two moduli at larger distortion strains. The shear modulus G(xz) is approximately independent of degree of distortion except at the highest initial pressure. The Poisson's ratio, nu(xz) is approximately constant with distortion strain for lower initial pressures, but increases significantly with epsilon(z) at higher pressures. Model predictions of the relation between G(xz) and initial uniform inflation pressure show a good correlation with reported experimental data for small distortion strains in a range of species. The model also exhibits similar behaviour to the experimentally measured uni-axial large deformations of a tri-axially pre-loaded block of parenchyma (Hoppin et al., 1975, Journal of Applied Physiology 39, 742-751).     

低强度激光对人红细胞生物物理特性的影响

研究不同功率的低强度He-Ne激光对正常人体红细胞流变学特性影响。以正常人体红细胞为研究对象,测量了低强度He-Ne激光在不同照射时间、不同功率条件下红细胞的变形、取向、膜流动性、膜的微粘度和渗透脆性的变化情况。结果表明:照射后红细胞的变形性和膜流动性增强、渗透脆性下降。照射对红细胞流变学特性影响显著,其中激光能量为0.24 J、照射血样为2 mL时取得的照射效果最佳。     

Computer simulation of a model network for the erythrocyte cytoskeleton.

The geometry and mechanical properties of the human erythrocyte membrane cytoskeleton are investigated by a computer simulation in which the cytoskeleton is represented by a network of polymer chains. Four elastic moduli as well as the area and thickness are predicted for the chain network as a function of temperature and the number of segments in each chain. Comparisons are made with mean field arguments to examine the importance of steric interactions in determining network properties. Applied to the red blood cell, the simulation predicts that in the bilayer plane the membrane cytoskeleton has a shear modulus of 10 +/- 2 x 10(-6) J/m2 and an areal compression modulus of 17 +/- 2 x 10(-6) J/m2. The volume compression modulus and the transverse Young's modulus of the cytoskeleton are predicted to be 1.2 +/- 0.1 x 10(3) J/m3 and 2.0 +/- 0.1 x 10(3) J/m3, respectively. Elements of the cytoskeleton are predicted to have a mean displacement from the bilayer plane of 15 nm. The simulation agrees with some, but not all, of the shear modulus measurements. The other predicted moduli have not been measured.