Robust strategies for automated AFM force curve analysis--I. Non-adhesive indentation of soft, inhomogeneous materials

The atomic force microscope (AFM) has found wide applicability as a nanoindentation tool to measure local elastic properties of soft materials. An automated approach to the processing of AFM indentation data, namely, the extraction of Young's modulus, is essential to realizing the high-throughput potential of the instrument as an elasticity probe for typical soft materials that exhibit inhomogeneity at microscopic scales. This paper focuses on Hertzian analysis techniques, which are applicable to linear elastic indentation. We compiled a series of synergistic strategies into an algorithm that overcomes many of the complications that have previously impeded efforts to automate the fitting of contact mechanics models to indentation data. AFM raster data sets containing up to 1024 individual force-displacement curves and macroscopic compression data were obtained from testing polyvinyl alcohol gels of known composition. Local elastic properties of tissue-engineered cartilage were also measured by the AFM. All AFM data sets were processed using customized software based on the algorithm, and the extracted values of Young's modulus were compared to those obtained by macroscopic testing. Accuracy of the technique was verified by the good agreement between values of Young's modulus obtained by AFM and by direct compression of the synthetic gels. Validation of robustness was achieved by successfully fitting the vastly different types of force curves generated from the indentation of tissue-engineered cartilage. For AFM indentation data that are amenable to Hertzian analysis, the method presented here minimizes subjectivity in preprocessing and allows for improved consistency and minimized user intervention. Automated, large-scale analysis of indentation data holds tremendous potential in bioengineering applications, such as high-resolution elasticity mapping of natural and artificial tissues.     

Spherical indentation method for determining the constitutive parameters of hyperelastic soft materials

A comprehensive study on the spherical indentation of hyperelastic soft materials is carried out through combined theoretical, computational, and experimental efforts. Four widely used hyperelastic constitutive models are studied, including neo-Hookean, Mooney–Rivlin, Fung, and Arruda–Boyce models. Through dimensional analysis and finite element simulations, we establish the explicit relations between the indentation loads at given indentation depths and the constitutive parameters of materials. Based on the obtained results, the applicability of Hertzian solution to the measurement of the initial shear modulus of hyperelastic materials is examined. Furthermore, from the viewpoint of inverse problems, the possibility to measure some other properties of a hyperelastic material using spherical indentation tests, e.g., locking stretch, is addressed by considering the existence, uniqueness, and stability of the solution. Experiments have been performed on polydimethylsiloxane to validate the conclusions drawn from our theoretical analysis. The results reported in this study should help identify the extent to which the mechanical properties of hyperelastic materials could be measured from spherical indentation tests.     

Elastic characterization of transversely isotropic soft materials by dynamic shear and asymmetric indentation

The mechanical characterization of soft anisotropic materials is a fundamental challenge because of difficulties in applying mechanical loads to soft matter and the need to combine information from multiple tests. A method to characterize the linear elastic properties of transversely isotropic soft materials is proposed, based on the combination of dynamic shear testing (DST) and asymmetric indentation. The procedure was demonstrated by characterizing a nearly incompressible transversely isotropic soft material. A soft gel with controlled anisotropy was obtained by polymerizing a mixture of fibrinogen and thrombin solutions in a high field magnet (B?=?11.7 T); fibrils in the resulting gel were predominantly aligned parallel to the magnetic field. Aligned fibrin gels were subject to dynamic (20-40 Hz) shear deformation in two orthogonal directions. The shear storage modulus was 1.08?±?0. 42 kPa (mean?±?std. dev.) for shear in a plane parallel to the dominant fiber direction, and 0.58?±?0.21 kPa for shear in the plane of isotropy. Gels were indented by a rectangular tip of a large aspect ratio, aligned either parallel or perpendicular to the normal to the plane of transverse isotropy. Aligned fibrin gels appeared stiffer when indented with the long axis of a rectangular tip perpendicular to the dominant fiber direction. Three-dimensional numerical simulations of asymmetric indentation were used to determine the relationship between direction-dependent differences in indentation stiffness and material parameters. This approach enables the estimation of a complete set of parameters for an incompressible, transversely isotropic, linear elastic material.     

Versatile and inexpensive Hall-Effect force sensor for mechanical characterization of soft biological materials

Mismatch of hierarchical structure and mechanical properties between tissue-engineered implants and native tissue may result in signal cues that negatively impact repair and remodeling. With bottom-up tissue engineering approaches, designing tissue components with proper microscale mechanical properties is crucial to achieve necessary macroscale properties in the final implant. However, characterizing microscale mechanical properties is challenging, and current methods do not provide the versatility and sensitivity required to measure these fragile, soft biological materials. Here, we developed a novel, highly sensitive Hall-Effect based force sensor that is capable of measuring mechanical properties of biological materials over wide force ranges (μN to N), allowing its use at all steps in layer-by-layer fabrication of engineered tissues. The force sensor design can be easily customized to measure specific force ranges, while remaining easy to fabricate using inexpensive, commercial materials. Although we used the force sensor to characterize mechanics of single-layer cell sheets and silk fibers, the design can be easily adapted for different applications spanning larger force ranges (>N). This platform is thus a novel, versatile, and practical tool for mechanically characterizing biological and biomimetic materials.     

A thermodynamically consistent constitutive equation for the elastic force-length relation of soft biological materials

Starting from the laws of thermodynamics of reversible processes, a temperature-dependent constitutive equation is derived for the elastic force-length relation of soft biological tissues. These tissues are composed of a network of fibres (mainly collagen). The equation is based on a model which uses a simplified two-dimensional representation of the alpha-helix of collagen.     

Analysis of indentation: implications for measuring mechanical properties with atomic force microscopy.

Indentation using the atomic force microscope (AFM) has potential to measure detailed micromechanical properties of soft biological samples. However, interpretation of the results is complicated by the tapered shape of the AFM probe tip, and its small size relative to the depth of indentation. Finite element models (FEMs) were used to examine effects of indentation depth, tip geometry, and material nonlinearity and heterogeneity on the finite indentation response. Widely applied infinitesimal strain models agreed with FEM results for linear elastic materials, but yielded substantial errors in the estimated properties for nonlinear elastic materials. By accounting for the indenter geometry to compute an apparent elastic modulus as a function of indentation depth, nonlinearity and heterogeneity of material properties may be identified. Furthermore, combined finite indentation and biaxial stretch may reveal the specific functional form of the constitutive law--a requirement for quantitative estimates of material constants to be extracted from AFM indentation data.     

Determination of elastic moduli of thin layers of soft material using the atomic force microscope

We address three problems that limit the use of the atomic force microscope when measuring elastic moduli of soft materials at microscopic scales. The first concerns the use of sharp cantilever tips, which typically induce local strains that far exceed the linear material regime. We show that this problem can be alleviated by using microspheres as probes, and we establish the criteria for their use. The second relates to the common use of the Hertz contact mechanics model, which leads to significant errors when applied to thin samples. We develop novel, simple to use corrections to apply for such cases. Samples that are either bonded or not bonded to a rigid substrate are considered. The third problem concerns the difficulty in establishing when contact occurs on a soft material. We obtain error estimates for the elastic modulus resulting from such uncertainty and discuss the sensitivity of the estimation methods to error in contact point. The theoretical and experimental results are compared to macroscopic measurements on poly(vinyl-alcohol) gels.     

Nanomechanics of lipid bilayers by force spectroscopy with AFM: A perspective

Lipid bilayers determine the architecture of cell membranes and regulate a myriad of distinct processes that are highly dependent on the lateral organization of the phospholipid molecules that compose the membrane. Indeed, the mechanochemical properties of the membrane are strongly correlated with the function of several membrane proteins, which demand a very specific, highly localized physicochemical environment to perform their function. Several mesoscopic techniques have been used in the past to investigate the mechanical properties of lipid membranes. However, they were restricted to the study of the ensemble properties of giant bilayers. Force spectroscopy with AFM has emerged as a powerful technique able to provide valuable insights into the nanomechanical properties of supported lipid membranes at the nanometer/nanonewton scale in a wide variety of systems. In particular, these measurements have allowed direct measurement of the molecular interactions arising between neighboring phospholipid molecules and between the lipid molecules and the surrounding solvent environment. The goal of this review is to illustrate how these novel experiments have provided a new vista on membrane mechanics in a confined area within the nanometer realm, where most of the specific molecular interactions take place. Here we report in detail the main discoveries achieved by force spectroscopy with AFM on supported lipid bilayers, and we also discuss on the exciting future perspectives offered by this growing research field.     

Species-area curve,life history strategies,and succession: a field test of relationships

Changes of species richness along temporal and environmental gradients were investigated. Two data sets were used: a successional sere of old-field plant communities in the Bohemian Karst, and a set of plant communities under various intensities of disturbance in the Krkonoe (Giant) Mts, both in Czechoslovakia. The species richness of a plant community is a spatial phenomenon, and should be described by the species-area relationship (using e.g. the power function S=c\A ^z) rather than by a single number. In the old-field succession, the number of species in very small plots (0.1×0.1 m) tends to increase with successional age while the number of species in larger plots (4×4 m) decreases from the third year of succession. The plant community under the highest rate of disturbance of the Krkonoe Mts data set shows the lowest number of species on small plots and the highest number of species on large plots. The results may be explained using the distinction between founder-and dominance-controlled communities (Yodzis 1978, 1984). In accordance with this theory, the species-area relationship within a community is shaped mainly by the type of competitive interaction and may be predicted on the basis of life-history strategies of constituent species. Disturbance causes a shift from dominance to founder control. On the landscape scale, the species-area relationship is shaped by other factors, and so it is unjustified to extrapolate the relationship outside the range in which it was originally assessed.     

Mechanical characterization of anisotropic planar biological soft tissues using finite indentation: experimental feasibility

Heart valve tissue engineering offers a promising alternative for current treatment and replacement strategies, e.g., synthetic or bioprosthetic heart valves. In vitro mechanical conditioning is an important tool for engineering strong, implantable heart valves. Detailed knowledge of the mechanical properties of the native tissue as well as the developing tissue construct is vital for a better understanding and control of the remodeling processes induced by mechanical conditioning. The nonlinear, anisotropic and inhomogeneous mechanical behavior of heart valve tissue necessitates a mechanical characterization method that is capable of dealing with these complexities. In a recent computational study we showed that one single indentation test, combining force and deformation gradient data, provides sufficient information for local characterization of nonlinear soft anisotropic tissue properties. In the current study this approach is validated in two steps. First, indentation tests with varying indenter sizes are performed on linear elastic PDMS rubbers and compared to tensile tests on the same specimen. For the second step, tissue constructs are engineered using uniaxial or equibiaxial static constrained culture conditions. Digital image correlation (DIC) is used to quantify the anisotropy in the tissue constructs. For both validation steps, material parameters are estimated by inverse fitting of a computational model to the experimental results.     

Organizing core facilities as force multipliers: strategies for research universities

Force multipliers are attributes of an organization that enable the successful completion of multiple essential missions. Core facilities play a critical role in the research enterprise and can be organized as force multipliers. Conceiving of cores in this way influences their organization, funding, and research impact. To function as a force multiplier for the research enterprise, core facilities need to do more than efficiently provide services for investigators and generate revenue to recover their service costs: they must be aligned with the strategic objectives of a research university. When core facilities are organized in this way, they can facilitate recruitment of faculty and trainees; serve to retain talented faculty; drive, acquire, and maintain cutting-edge research platforms; and promote interaction and collaboration across the institution. Most importantly, cores accelerate the discovery and sharing of knowledge that are the foundation of a modern research university. This idea has been systematically implemented through the Emory Integrated Core Facilities (cores.emory.edu), which include 16 distinct core facilities and the Division of Animal Resources. Force multiplier core facilities can significantly contribute to the many essential missions necessary for the success of the research enterprise at research universities.     

Comparison of single-phase isotropic elastic and fibril-reinforced poroelastic models for indentation of rabbit articular cartilage

Classically, single-phase isotropic elastic (IE) model has been used for in situ or in vivo indentation analysis of articular cartilage. The model significantly simplifies cartilage structure and properties. In this study, we apply a fibril-reinforced poroelastic (FRPE) model for indentation to extract more detailed information on cartilage properties. Specifically, we compare the information from short-term (instantaneous) and long-term (equilibrium) indentations, as described here by IE and FRPE models. Femoral and tibial cartilage from rabbit (age 0–18 months) knees (n=14) were tested using a plane-ended indenter (diameter=0.544 mm). Stepwise creep tests were conducted to equilibrium. Single-phase IE solution for indentation was used to derive instantaneous modulus and equilibrium (Young's) modulus for the samples. The classical and modified Hayes’ solutions were used to derive values for the indentation moduli. In the FRPE model, the indentation behavior was sample-specifically described with three material parameters, i.e. fibril network modulus, non-fibrillar matrix modulus and permeability. The instantaneous and fibril network modulus, and the equilibrium Young's modulus and non-fibrillar matrix modulus showed significant (p<0.01) linear correlations of R²=0.516 and 0.940, respectively (Hayes’ solution) and R²=0.531 and 0.960, respectively (the modified Hayes’ solution). No significant correlations were found between the non-fibrillar matrix modulus and instantaneous moduli or between the fibril network modulus and the equilibrium moduli. These results indicate that the instantaneous indentation modulus (IE model) provides information on tensile stiffness of collagen fibrils in cartilage while the equilibrium modulus (IE model) is a significant measure for stiffness of PG matrix. Thereby, this study highlights the feasibility of a simple indentation analysis.     

Unspecific membrane protein–lipid recognition: combination of AFM imaging,force spectroscopy,DSC and FRET measurements

In this work, we will describe in quantitative terms the unspecific recognition between lactose permease (LacY) of Escherichia coli, a polytopic model membrane protein, and one of the main components of the inner membrane of this bacterium. Supported lipid bilayers of 1‐palmitoyl‐2‐oleoyl‐sn‐glycero‐3‐phosphoethanolamine (POPE) and 1‐palmitoyl‐2‐oleoyl‐sn‐glycero‐3‐phosphoglycerol (POPG) (3:1, mol/mol) in the presence of Ca²⁺ display lateral phase segregation that can be distinguished by atomic force microscopy (AFM) as well as force spectroscopy. LacY shows preference for fluid (L_α) phases when it is reconstituted in POPE : POPG (3:1, mol/mol) proteoliposomes at a lipid‐to‐protein ratio of 40. When the lipid‐to‐protein ratio is decreased down to 0.5, two domains can be distinguished by AFM. While the upper domain is formed by self‐segregated units of LacY, the lower domain is constituted only by phospholipids in gel (L_β) phase. On the one hand, classical differential scanning calorimetry (DSC) measurements evidenced the segregation of a population of phospholipids and point to the existence of a boundary region at the lipid–protein interface. On the other hand, Förster Resonance Energy Transfer (FRET) measurements in solution evidenced that POPE is selectively recognized by LacY. A binary pseudophase diagram of POPE : POPG built from AFM observations enables to calculate the composition of the fluid phase where LacY is inserted. These results are consistent with a model where POPE constitutes the main component of the lipid–LacY interface segregated from the fluid bulk phase where POPG predominates. Copyright © 2015 John Wiley & Sons, Ltd.     

Extraction of quasi-linear viscoelastic parameters for lower limb soft tissues from manual indentation experiment.

A manual indentation protocol was established to assess the quasi-linear viscoelastic (QLV) properties of lower limb soft tissues. The QLV parameters were extracted using a curve-fitting procedure on the experimental indentation data. The load-indentation responses were obtained using an ultrasound indentation apparatus with a hand-held pen-sized probe. Limb soft tissues at four sites of eight normal young subjects were tested in three body postures. Four QLV model parameters were extracted from the experimental data. The initial modulus E0 ranged from 0.22 kPa to 58.4 kPa. The nonlinear factor E1 ranged from 21.7 kPa to 547 kPa. The time constant tau ranged from 0.05 s to 8.93 s. The time-dependent materials parameter alpha ranged from 0.029 to 0.277. Large variations of the parameters were noted among subjects, sites, and postures.     

The length dependence of muscle active force: considerations for parallel elastic properties.

The purpose of this study was to choose between two popular models of skeletal muscle: one with the parallel elastic component in parallel with both the contractile element and the series elastic component (model A), and the other in which it is in parallel with only the contractile element (model B). Passive and total forces were obtained at a variety of muscle lengths for the medial gastrocnemius muscle in anesthetized rats. Passive force was measured before the contraction (passive A) or was estimated for the fascicle length at which peak total force occurred (passive B). Fascicle length was measured with sonomicrometry. Active force was calculated by subtracting passive (A or B) force from peak total force at each fascicle or muscle length. Optimal length, that fascicle length at which active force is maximized, was 13.1 +/- 1.2 mm when passive A was subtracted and 14.0 +/- 1.1 mm with passive B (P < 0.01). Furthermore, the relationship between double-pulse contraction force and length was broader when calculated with passive B than with passive A. When the muscle was held at a long length, passive force decreased due to stress relaxation. This was accompanied by no change in fascicle length at the peak of the contraction and only a small corresponding decrease in peak total force. There is no explanation for the apparent increase in active force that would be obtained when subtracting passive A from the peak total force. Therefore, to calculate active force, it is appropriate to subtract passive force measured at the fascicle length corresponding to the length at which peak total force occurs, rather than passive force measured at the length at which the contraction begins.     

Evidence for a highly elastic shell-core organization of cochlear outer hair cells by local membrane indentation

Cochlear outer hair cells (OHCs) are thought to play an essential role in the high sensitivity and sharp frequency selectivity of the hearing organ by generating forces that amplify the vibrations of this organ at frequencies up to several tens of kHz. This tuning process depends on the mechanical properties of the cochlear partition, which OHC activity has been proposed to modulate on a cycle-by-cycle basis. OHCs have a specialized shell-core ultrastructure believed to be important for the mechanics of these cells and for their unique electromotility properties. Here we use atomic force microscopy to investigate the mechanical properties of isolated living OHCs and to show that indentation mechanics of their membrane is consistent with a shell-core organization. Indentations of OHCs are also found to be highly nonhysteretic at deformation rates of more than 40 microm/s, which suggests the OHC lateral wall is a highly elastic structure, with little viscous dissipation, as would appear to be required in view of the very rapid changes in shape and mechanics OHCs are believed to undergo in vivo.     

Functionalization of soft materials for cardiac repair and regeneration

Coronary artery disease is a leading cause of death in developed nations. As the disease progresses, myocardial infarction can occur leaving areas of dead tissue in the heart. To compensate, the body initiates its own repair/regenerative response in an attempt to restore function to the heart. These efforts serve as inspiration to researchers who attempt to capitalize on the natural regenerative processes to further augment repair. Thus far, researchers are exploiting these repair mechanisms in the functionalization of soft materials using a variety of growth factor-, ligand- and peptide-incorporating approaches. The goal of functionalizing soft materials is to best promote and direct the regenerative responses that are needed to restore the heart. This review summarizes the opportunities for the use of functionalized soft materials for cardiac repair and regeneration, and some of the different strategies being developed.     

Mechanical characterization of anisotropic planar biological soft tissues using large indentation: a computational feasibility study

Traditionally, the complex mechanical behavior of planar soft biological tissues is characterized by (multi)axial tensile testing. While uniaxial tests do not provide sufficient information for a full characterization of the material anisotropy, biaxial tensile tests are difficult to perform and tethering effects limit the analyses to a small central portion of the test sample. In both cases, determination of local mechanical properties is not trivial. Local mechanical characterization may be performed by indentation testing. Conventional indentation tests, however, often assume linear elastic and isotropic material properties, and therefore these tests are of limited use in characterizing the nonlinear, anisotropic material behavior typical for planar soft biological tissues. In this study, a spherical indentation experiment assuming large deformations is proposed. A finite element model of the aortic valve leaflet demonstrates that combining force and deformation gradient data, one single indentation test provides sufficient information to characterize the local material behavior. Parameter estimation is used to fit the computational model to simulated experimental data. The aortic valve leaflet is chosen as a typical example. However, the proposed method is expected to apply for the mechanical characterization of planar soft biological materials in general.     

Spherical indentation of soft matter beyond the Hertzian regime: numerical and experimental validation of hyperelastic models

The lack of practicable nonlinear elastic contact models frequently compels the inappropriate use of Hertzian models in analyzing indentation data and likely contributes to inconsistencies associated with the results of biological atomic force microscopy measurements. We derived and validated with the aid of the finite element method force-indentation relations based on a number of hyperelastic strain energy functions. The models were applied to existing data from indentation, using microspheres as indenters, of synthetic rubber-like gels, native mouse cartilage tissue, and engineered cartilage. For the biological tissues, the Fung and single-term Ogden models achieved the best fits of the data while all tested hyperelastic models produced good fits for the synthetic gels. The Hertz model proved to be acceptable for the synthetic gels at small deformations (strain < 0.05 for the samples tested), but not for the biological tissues. Although this finding supports the generally accepted view that many soft materials can be assumed to be linear elastic at small deformations, the nonlinear models facilitate analysis of intrinsically nonlinear tissues and large-strain indentation behavior.     

A theoretical model for the elastic properties of very soft tissues.

A theoretical model is developed to predict the elastic properties of very soft tissues such as glands, tumors and brain. Tissues are represented as regular arrays of polyhedral (cubic or tetrakaidecahedral) cells, surrounded by extracellular spaces of uniform width. Cells are assumed to be incompressible, with very low resistance to shear deformation. Tissue shear rigidity is assumed to result mainly from the extracellular matrix, which is treated as a compressible elastic mesh of interconnected fibers. Small-strain elastic properties of tissue are predicted using a finite-element method and an analytical method. The model can be used to estimate the compressibility of a very soft tissue based on its Young's modulus and extracellular volume fraction.