Elasticity Models for the Spherical Indentation of Gels and Soft Biological Tissues
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1060-LL05-07
Elasticity Models for the Spherical Indentation of Gels and Soft Biological Tissues David C. Lin, Emilios K. Dimitriadis, and Ferenc Horkay National Institutes of Health, Bethesda, MD, 20892
ABSTRACT AFM micro- or nanoindentation is a powerful technique for mapping the elasticity of materials at high resolution. When applied to soft matter, however, its accuracy is equivocal. The sources of the uncertainty can be methodological or analytical in nature. In this paper, we address the lack of practicable nonlinear elastic contact models, which frequently compels the use of Hertzian models in analyzing force curves. We derive and compare approximate forceindentation relations based on a number of hyperelastic general strain energy functions. These models were applied to existing data from the spherical indentation of native mouse cartilage tissue as well as chemically crosslinked poly(vinyl alcohol) gels. For the biological tissue, the Fung and single-term Ogden models were found to provide the best fit of the data while the Mooney-Rivlin and van der Waals models were most suitable for the synthetic gels. The other models (neo-Hookean, two-term reduced polynomial, Fung, van der Waals, and Hertz) were effective to varying degrees. The Hertz model proved to be acceptable for the synthetic gels at small strains ( 2 mm thick) for macroscopic displacement-controlled compression and AFM nanoindentation, respectively. Sixty-micrometer thick cartilage samples were transversely sectioned from the femoral heads of one-day old wild-type mice using a microtome. Samples were lightly fixed in formaldehyde and frozen in embedding matrix prior to sectioning. General-purpose silicon nitride tips with 5.5 µm glass (for the synthetic gels) or 5 µm polystyrene (for cartilage) beads attached were used for the AFM measurements, performed using a commercial AFM (Bioscope I with Nanoscope IV controller, Veeco). The spring constant of the cantilever was measured by the thermal tune method while bead diameters were measured from images acquired during the attachment process. A raster scanning approach (“force-volume”) was applied to automatically perform indentations over an area of ~ 30×30 µm, at a resolution of 16×16 (256 total indentations) for the hydrogel and 32×32 (1024 indentations) for the cartilage. Code written in Matlab was used to automatically process each dataset and extract values of Young’s modulus. For the cartilage, height images were used to determine whether each measurement location corresponded to the extracellular matrix or to the cells.
RESULTS AND DISCUSSION Ten random indentations were selected from the force-volume scan of the poly(vinyl alcohol) gel for processing using the different force-indentation equations in Table I. From the scan of the cartilage, ten indentations each of the matrix and cells were chosen. Mean square errors and extracted Young’s moduli are listed in Table II. Two representative fits for the synthetic gel are shown in Figure 1. Likewise, representative fits for the cartilage ext
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