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TECHNICAL PAPERS: Soft Tissue

Robust Strategies for Automated AFM Force Curve Analysis—I. Non-adhesive Indentation of Soft, Inhomogeneous Materials

[+] Author and Article Information
David C. Lin

Laboratory of Integrative and Medical Biophysics, National Institutes of Health, 9 Memorial Drive, Bldg. 9 Rm. 1E118, Bethesda, MD 20892lindavid@mail.nih.gov

Emilios K. Dimitriadis

National Institute of Biomedical Imaging and Bioengineering, National Institutes of Health, 13 South Drive, Bldg. 13 Rm. 3N17, Bethesda, MD 20892dimitria@helix.nih.gov

Ferenc Horkay

Laboratory of Integrative and Medical Biophysics, National Institutes of Health, 13 South Drive, Bldg. 13 Rm. 3W16, Bethesda, MD 20892horkay@helix.nih.gov

J Biomech Eng 129(3), 430-440 (Nov 15, 2006) (11 pages) doi:10.1115/1.2720924 History: Received June 12, 2006; Revised November 15, 2006

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.

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Copyright © 2007 by American Society of Mechanical Engineers
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Figures

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Figure 8

Plot of the first derivative (∂d∕∂w) for a simulated curve (indicated by +) that obeys the Hertz equation and for the same curve after performing a smoothing operation using splines (indicated by ◻). Predicted contact points (indicated by 엯) are from fits to the region of interest in the range 10–40% of the maximum value of ∂d∕∂w (solid line for the simulated data and dotted line for the smoothed data). Note the deviation in the behavior of the derivative after smoothing and the concomitant degradation in the fit, resulting in an early predicted point of contact. The inset shows the simulated d vs. w data (indicated by ◇; range of d: −122to−59nm) and the smoothed fit (solid line).

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Figure 9

Error in force as a function of the ratio of equivalent deflection to tip radius

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Figure 1

Blunt and sharp tips of the same tip angle. Both conical and pyramidal tips are shown. Radius a is the contact radius and c is the indentation radius. The blunt tips transition at radius or half-width b to a round tip with radius R. The incline angle of the faces of the pyramid is represented by θ.

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Figure 2

(a)–(c): Force curves (open diamonds), best-fit curves (solid lines) and corresponding plots of the MSE as a function of position of the assumed contact point. Three separate indentations of cartilage specimens are presented. The portion of each curve bracketed by the ◻ symbols is the linearly elastic region identified during data preprocessing. For ease of visualization, curves were shifted vertically with no effect on the solutions. When the contact point (indicated by ●) lies within the range of the retained data, the MSE plot is unimodal (curves “a” and “b”). When the contact point lies outside the range of the retained data, the MSE plot does not have a global minimum; in this example (curve “c”), tip-sample interactions (not adhesive in nature) obscured the location of the contact point by distorting the initial contact portion of the force curve. The inset in the large plot shows the first, second, and third derivatives of deflection d with respect to w for curve “c.” Each derivative is normalized such that the difference between the maximum and minimum value is one. The cutoff point (f″=0 and f‴>0) is indicated by ∎. (d): A representative force-displacement curve showing significant adhesive interactions.

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Figure 3

Flowchart representation of the algorithm

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Figure 4

Results of macroscopic compression and AFM indentation tests on PVA gels. Due to the large difference in sample size between AFM and macroscopic measurements, error bars show standard deviation rather than standard error. Inset shows data for the 3% gel in larger scale. Large strain AFM values were obtained by including points up to maximum indentation. Small strain values were obtained by truncating the data at ∼20–25% strain.

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Figure 5

Sample plot of Young’s modulus of tissue-engineered cartilage as a function of position along a line. Points were spaced ∼2.5μm apart. Regions of very low stiffness most likely are chondrocytes. Insets show sample curve fits.

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Figure 6

Simulated force-indentation curves for tips of various geometries. Forces are normalized against material properties of the indented half space.

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Figure 7

Simulated force-indentation curves for blunt conical tips of various tip radii. The curves are nearly parallel beyond a certain indentation depth and at small radii, are virtually indistinguishable from one another.

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