ABSTRACT
This study discusses the application of a hybrid experimental-numerical approach to analyze nano-indentation curves of a biological membrane acquired with an Atomic Force Microscope. The proposed procedure combines experimental measurements, FEM analysis and numerical optimization and is completely general. Variations of estimated Young modulus of the membrane are determined when attributing different constitutive laws to the sample and in the case of progressive blunting of the AFM tip during the measurement. Since traditional analysis of Atomic Force Microscope indentation curves relies on an inappropriate application of the classical Hertz theory, a comparison between the hybrid approach and the Hertzian model in the determination of the elastic properties of the sample is presented. In particular, it is found that large errors occur in the derivation of the Young modulus when the Hertzian model is used for the analyis of experimental data.
INTRODUCTION
Many physiological and patho-physiological processes alter the mechanical properties of the biological tissues they affect. It is well known that aging causes deterioration in mechanical strength of human tissues and that muscles get harder with weight training. Correlation between tissue structural behavior and pathologies manifests itself already at the cellular level, as observed in the case of inflammations, some forms of cancer and cardiac diseases. Mechanical characterization of cells and biological membranes "in vitro", i.e. in their physiological environment, can be performed thanks to the recent developments in the field of nanotechnology of instruments with nanoscale resolution: these allow to apply and detect forces and displacements with picoNewton (pN) and nanometer sensitivity, respectively. The Atomic Force Microscope (AFM) has recently emerged as a powerful tool to investigate the elastic properties of biological specimens. The AFM was designed to provide high resolution images of the surfaces of non-conductive samples and consists of a very sharp tip mounted at the end of a cantilever that scans the surface of the sample. The 3D topography of the selected area is reconstructed by recording the minute deflections of the cantilever during the scanning procedure. Soon after its invention, the AFM was also used as a nanoindenter to measure the mechanical properties of a sample with a nanometric resolution. In this operating mode, the deflection of the cantilever, is monitored as a function of the indentation depth of the tip into the sample and the instrument registers a force-indentation curve.
Traditional analysis of AFM indentation curves relies on an inappropriate application of the classical Hertz theory [1,2] with its hypotheseis of linear elastic material properties, infinitesimal strains and infinite sample thickness and dimensions. None of these assumptions is likely to be valid when a biological membrane is indented with an AFM. Most biological materials exhibit nonlinear constitutive behavior. Furthermore, the AFM probe induces large deformations during the indentation process and the half-space assumption cannot be adapted to thin bio membranes. Previous studies used Finite Element Modeling to simulate AFM indentation curves and evaluated the effect of indentation depth, tip geometry and material nonlinearity on the finite indentation response [3,4]. The current trend in the interpretation of AFM data is to describe mechanical behavior of cell membranes by means of hyperelastic constitutive relationships and to extract values of elastic properties of the specimen with the aid of FEM analysis [5,6]. This work proposes the application of a hybrid procedure which combines experimental measurements, FEM analysis and optimization algorithms to analyze AFM indentation curves. The proposed methodology is applied to the determination of the mechanical properties of a biological membrane. In particular, the membrane analyzed in the paper is the Zona Pellucida (ZP), which is the extracellular coat that surrounds mammalian oocyte. The limit of applying the Hertzian model to extract the elastic properties of a sample are put in evidence as well as the errors incurred in the derivation of the Young modulus. The hybrid procedure allows to take into consideration all the parameters involved in the experiment, such as radius of curvature of the tip, the thickness of the membrane, the constitutive law of the sample. The variations of the estimated Young modulus of the ZP membrane are determined in the case of attribution of different constitutive laws to the sample and in the case of blunting of the AFM tip during the measurement.
FINITE ELEMENT ANALYSIS
The AFM nanoindentation experiments conducted on the membrane were simulated with the ABAQUS® Version 6.7 commercial finite element software [7]. For that purpose, an axisymmetric FE model was developed: the model includes a rigid blunt-conical indenter (tip radius of 10nm and half-open angle of 20°) pressing against a soft layer adherent on a rigid substrate. The Young modulus of the silicon-nitride AFM tip is 300 GPa. The biomembrane was modeled as an incompressible hyperelastic slab with diameter 60 ^m and thickness 10 ^m.
Figure 1. Finite element model simulating the nanoindentation process. The deformation field corresponding to 100nm indentation is shown in the figure.
Figure 1 shows the finite element model with the rigid indenter and the membrane: the deformation field corresponding to 100nm indentation is presented. The mesh of the membrane included 69716 four-node bilinear, hybrid CAX4H elements with constant pressure and 70562 nodes. The hybrid pressure-displacement formulation implemented in the chosen element type allowed incompressible behavior to be modeled.
Convergence analysis was carried out in order to have mesh independent solutions. The mesh was properly refined in the contact region between the AFM tip and the membrane: the element size is 0.06 nm where such a value allowed to reach a good compromise between convergence of nonlinear analysis and computation time.
The penetration of the indenter was simulated by progressively increasing the value of the force applied to the AFM tip in the vertical direction: the load transferred by the rigid blunt-conical tip to the membrane generates a state of compression in the soft material of the slab. The bottom and side of the slab were fixed in space, and both the rigid blunt-cone and axis of symmetry for the slab are permitted to move only in the vertical direction. Finite element analysis accounted for geometric non-linearity (i.e. large deformations) and the automatic time stepping option was selected to facilitate the convergence of nonlinear analysis. The contact between the indenter and the membrane was assumed to be frictionless; the "hard contact" (i.e. no force is exchanged before surfaces come in contact) option available in ABAQUS was chosen.
HYPERELASTIC MODELS
Three different hyperelastic constitutive models were considered in this study in order to describe the structural behavior the ZP membrane: (i) Two-parameter Mooney-Rivlin (MR); (ii) Neo-Hookean (NH); (iii) Arruda-Boyce Eight-chain (8chain) model (AB).
The two-parameter MR constitutive law [8-10] is a very classical phenomenological model described by the following strain energy density function:
where Q0 and CM are the MR constants given in input to ABAQUS as material properties. Strain invariants are defined, respectively, as
where [C] is the Cauchy-Green strain tensor. The corresponding uniaxial stress (ct) – stretch (X) equation can be derived as:
The shear modulus
is defined as
while the Young modulus is equal to:
The NH model [9-11] was selected in this study because it is based on the statistical thermodynamics of cross-linked polymer chains. Although, this model is not phenomenological, it can be however derived from the two-parameter MR model by setting CM=0. Consequently, only one material parameter must be given in input to ABAQUS. The shear modulus |NH is defined as | nh=2Ci0 while the Young modulus is equal to:
The AB model [12] was previously used in literature to describe the mechanical behavior of biospecimens including filamentous collagen networks [13,14] and monolayers of endothelial cells [6]. This model relies on the statistical mechanics of a material with a cubic representative volume element containing eight chains along the diagonal directions. The strain hardening behavior of an incompressible material is predicted by using two constants: the shear modulus | 8cham and the distensibility XL where the latter corresponds to the limiting network stretch. The strain energy function can be expressed as Eq. (5):
The corresponding uniaxial stress (ct) – stretch (X) equation is:
The Arruda-Boyce model is activated in ABAQUS by giving in input the values of
as material parameters.
The Young modulus is defined as
FORMULATION OF THE INVERSE PROBLEM
In order to extract more realistically the hyperelastic properties of the ZP membrane from the above described FE model which accounts for non-linearity of finite indentation process and material non-linearity, a hybrid procedure combining experimental measurements, FE analysis and nonlinear optimization was utilized. Displacement values measured experimentally are compared with the corresponding results of FE analysis. This leads to formulate an optimization problem including the unknown material properties as design variables. The optimization problem describing the inverse problem of material characterization can be stated as follows:
where Q is the error functional to be minimized. The design vector X(Xi,X2,.. .,XNMP) includes the unknown number of material properties (NMP) to be determined that can vary between the lower and upper bounds.
In Eq. (8), 5FEMj and 0 , respectively, are the displacement values for the j-th load step computed with FE analysis and those measured experimentally with AFM. The number of control locations Ncnt is equal to the number of load steps to complete nonlinear FE analysis. Nanoindentation values measured experimentally can be taken as target values in the identification problem because the execution of AFM measurements does not require any a priori knowledge of material properties. Conversely, the "correct material properties", i.e. the actual material properties, must be given in input to the FE model to obtain the force-indentation curve that matches the F-8 curve determined experimentally. Theoretically, the functional error Q computed at the optimum design (i.e. the target material properties) will be equal to 0. However, since target values are measured experimentally, there will be a residual deviation between the force-indentation curve reconstructed numerically and the actual F-8 curve measured with AFM.
The suitability of the optimization-based for mechanical characterization problems of nonlinear materials is well documented in literature [15,16].
The inverse problem (8) was solved with the Sequential Quadratic Programming (SQP) method, a gradient-based optimization algorithm that has the property of global convergence. SQP satisfies the necessary Kuhn-Tucker optimality conditions regardless of the initial point from which the optimization process is started [17]. SQP is universally considered as the most efficient gradient-based optimization method. The powerful SQP optimization routine implemented in the commercial general mathematics software MATLAB® Version 7.0 was utilized [18]. The finite element solver of ABAQUS was interfaced with the SQP optimization routine of MATLAB that processed the results of FE analysis, compared the numerical F-8 curve with the experimental data, computed the error functional Q, and perturbed the material parameters for the subsequent design cycles.
RESULTS AND DISCUSSION
The elastic properties of the ZP membranes extracted from mature oocytes were evaluated applying both the Hertz model and the hybrid procedure described previously. An indentation range of 100nm was considered, as in this interval the hypothesis of infinitesimal strain could still be considered valid.
Firstly, the indentation curves were examined with the modified Hertzian model for the conical indenter and, from the analysis of the experimental data recorded on 50 different points of each sample, the following Young modulus was derived: Enertz=18.5kPa ± 1.58kPa.
Figure 2. Force-indentation curves acquired experimentally on ZP of mature oocytes (symbol) and the corresponding numerical curves obtained with the hybrid procedure (solid line). Three different constitutive models for the membrane were considered: a) Neo-Hookean model; b) Two-parameter Mooney Rivlin model; c) Eight chain model, Arruda-Boyce model.
The elastic parameters of the ZP membrane were then extracted applying the hybrid procedure. This approach is completely general, and therefore the derived elastic properties can be considered more reliable. Figure 2 shows a comparison between the force-indentation curves acquired experimentally on the sample and the corresponding curves obtained with the optimization algorithm. Three different constitutive models for the membrane were considered: the NH, the MR and the AB. The corresponding Young modulus of the sample, calculated from Eqs. (3), (4) and (7), respectively for the Neo-Hookean, Mooney-Rivlin and Arruda-Boyce models, are equal to ENH=9.73kPa, EMR=8.88kPa and EAB=6.4kPa. It can be observed that these values are considerably smaller (two or three times) than the Young modulus extracted with the Hertzian model. Hence, large errors are incurred in the estimation of elastic modulus when using linear elastic model to fit the data even for small indentation ranges. This result is consistent with the findings of other authors who conducted FE studies on the indentation of materials with an hyperelastic mechanical behavior [5,6]. The fit between the experimental data and the model was evaluated by means of the coefficient of correlation R2. Table 1 summarizes the R2 values calculated for the three hyperelastic laws and for the Hertzian model. It was found that the best fit of the experimental curve is obtained with the Arruda Boyce model. Therefore, it can be concluded that this constitutive law is the most appropriate to describe the mechanical behavior of the ZP membrane.
Table 1. Fitting parameters and Young modulus of ZP isolated from mature oocyte attributing linear elastic and hyperelastic constitutive behavior to the membrane. R2: correlation coefficient
|
Model |
Fitting parameters |
Young modulus E (kPa) |
R2 |
|
Hertzian |
 |
18.5 |
0.993 |
|
Neo-Hookean |
 |
9.73 |
0.996 |
|
Mooney-Rivlin |
 |
8.88 |
0.997 |
|
Arruda-Boyce |
 |
6.4 |
0.998 |
After having evaluated how the Young modulus varies when attributing different constitutive laws to the samples, it was considered the influence of the variation of the radius of the tip in the determination of the Young modulus. The tip may be blunted, for example, during a scan of the sample. In Figure 3 it is shown the variation of the derived shear modulus with the radius of curvature of the tip, in the case of the Arruda Boyce hyperelastic constitutive behavior. It can be seen that when the radius of the tip increases from 10nm to 50nm, the corresponding shear modulus is halved. Therefore, in case of blunting of a tip during an AFM measurement, a considerable error can be induced in the estimation of the Young modulus.
Figure 3. Variation of the shear modulus with the radius of curvature of the tip when attributing the Arruda-Boyce hyperelastic constitutive law to the membrane.
CONCLUSIONS
This study proposed the application of a hybrid procedure which combines experimental measurements, FEM analysis and nonlinear optimization to analyze the nanoindentation curves on a biological membrane acquired with an AFM. A comparison with the Hertzian model in the determination of the elastic properties of the sample is presented. In particular, it is found that large errors occur in the in the derivation of the Young modulus when applying the Hertzian model to analyze the experimental data. The hybrid procedure is completely general as it takes into consideration all the parameters involved in the experiment, such as radius of curvature of the tip, the thickness of the membrane, the constitutive law of the sample. The variations of the estimated Young modulus of the membrane are determined when attributing different constitutive laws to the sample and in the case of blunting of the tip during the measurement.
