(G-244) A data-driven analytical anisotropic indentation model based on finite element simulations

Introduction:: Mechanical properties are fundamental characteristics of tissues and cells which describe their ability to withstand external forces while performing physiological functions, and alteration of mechanical properties of tissues and cells are the hallmarks of many diseases. Therefore, it is critical to accurately measure mechanical properties of biological samples for diagnosis and prevention of diseases. Indentation is one of the most common methods to test mechanical properties of tissues and cells, since it has major advantages in versatility of measuring samples of different sizes and shapes and simplicity in sample preparation and data interpretation. However, traditional indentation methods lack the ability to estimate the differences in mechanical properties of tissues and cells along different directions. This anisotropy stems from the alignment of fibers in biomaterial structures, thus is of great importance and interests in many applications. Recently, the anisotropic mechanical properties have been calculated through inverse finite element analysis (FEA) based on anisotropic strain fields of samples captured through digital image correlation using spherical probes. They also could be calculated based on direction-dependent indentation forces recorded during indentations using probes with a long and short axis. These methods require complicated experimental setups and computational processes. In this study, we aim to build an accurate and ready-to-use analytical model based on anisotropic indentation along orthogonal directions using novel toroidal probes to acquire direction-dependent indentation curves. The model has been verified using FEA to simulate indentations using tips with various aspect ratios and validated through experiments using isotropic and anisotropic biological samples.


Materials and Methods:: Anisotropic probes were of a half torus shape which minimizes the volume and stress concentration. The radius of the circular cross-section was set to be 500 µm and the aspect ratio (r) was set to 1:3.25, 1:5, 1:7, and 1:10. The model was developed based on force-displacements curves obtained from finite-element simulations of indentations on incompressible transversely isotropic elastic materials with Young’s moduli range from 1 to 100kPa. The shear moduli and Poisson’s ratios were calculated based on the Young’s moduli, therefore the number of total independent variables were restricted to 2. The indentation was performed in the direction that the long axis of the probe was parallel to the out-of-plane orientation (0-deg) and perpendicular to that orientation (90-deg) using Abaqus 2020 and the indentation depth was less than 10% of the sample thickness (Fig.1). Mesh convergence was performed, and the model was validated for spherical probe on isotropic material using the Hertz contact theory.

In Experimental validation, the probes were manufactured using a 1mm-diameter stainless steel wire. A Instron EP-1000 test machine equipped with a 1 N force transducer was used (Fig.2). Chicken breast was purchased from local grocery store. Samples were dissected to 10 mm thick slices in a direction parallel to the fiber orientation. The maximum indentation force was set to 0.1 N, and the indentation speed was set as 0.1 mm/s. A total of 109 pairs of anisotropic indentations were performed on 61 pieces of chicken breast samples.


Results, Conclusions, and Discussions:: With the assumption that the tissues are transversely-isotropic, small-deformed and incompressible, the force-displacement curve for 0-degree indentation of the anisotropic sample could be described using a modified Hertzian model. The effective stiffness (NF2) is majorly determined by the in-plane elastic modulus E2, resulting in a mean absolute percentage error (MAPE) of less than 5% in each group using the Hertzian model, which is in line with previous publications. We include a correction term considering the anisotropy of the elastic moduli along 0-degree and 90-degree, thus further reduce the MAPE to less than 2%. Indentation problems on isotropic linear elastic materials has been analytically solved by Hertz more than a century ago. As one of the most used methods, Hertzian model has been demonstrated to be applicable to soft biomaterials and micro/nano indentation when the deformation is small. Notably, indentation curve in 90-degree is steeper than the curve in 0-degree. The difference between the indentation curves increases with the aspect ratio of the probe geometry and the anisotropy of the mechanical properties, and the relationship is shown in Fig.3.

Our model is first validated in experiments using an isotropic Polydimethylsiloxane (PDMS) sample. The indenters with different aspect ratio generated almost the same mechanical property (results are not significantly different) (Fig.4). The calculated Young’s modulus from our model is agreed with previous publications. For the anisotropic biomaterial (chicken breast tissue), the effective stiffness in 90-degree is higher than 0-degree (p< 0.01). The out-of-plane moduli (E2) were 24.79 ± 2.90 kPa, while the in-plane moduli were 3.98 ± 1.72 kPa, which is in line with previous publications. The curves recorded in the indentation experiments are shown in Fig.5.

Indentation can measure local mechanical properties without deforming the entire sample, which allows for measuring heterogeneity in situ. The simple sample preparation also allows easy measurement of stiffness of microscopic samples. We develop a method to extend the ability of indentation to assess anisotropic elastic moduli, which is in high demand in biomedical applications. This method may provide future researchers with an easy, cost-effective, yet reliable method to characterize mechanical anisotropy in tissues and cells.


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