Abstract
Recent research has shown that hyperelastic properties of the plantar soft tissue consisting of adipose tissue and fibrous septa change from region to region. However, relatively little research has been conducted to develop analytical or computational models to describe the region-specific behavior of the plantar soft tissue. The objective of the research is to develop a region-specific constitutive model of the plantar soft tissue. Plantar soft tissue specimens were dissected from six regions [subcalcaneal (CA), sublateral (LA), subnavicular (Nav), 1st, 3rd, and 5th submetatarsal (M1, M3, M5)] from cadaveric foot samples, and a picrosirius red staining technique was used to visualize the collagen fibers in fibrous septa. The volume fractions of adipose tissue and fibrous septa and the volume fractions of the principal orientations of the fibrous septa were calculated with the intensity gradient method. Region-specific constitutive models were then developed in finite element analysis considering the microstructure of the plantar soft tissue. The hyperelastic region specific material properties of the plantar soft tissue were validated with experimental unconfined compression tests and indentation tests from the literature. The results show that the models give reasonable predictions of the stiffness of the soft tissue within a standard deviation of the tests. The region-specific constitutive models help to explain how changes in the constituents are related to mechanical behavior of the soft tissue on a region specific basis.
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This study was funded by National Nature Science Foundation of China under Grant Nos. 51505282 and 51550110233.
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Appendix
Appendix
1.1 Appendix (a)
Parametric analyses were conducted to study the influence of subimage size and collagen fibers threshold on results of the volume fractions of the principal orientations of fibrous septa. The process of image process is as following: (1) each image of the plantar soft tissue was divided into several square subimages for calculating the orientations of fibrous septa; (2) the subimages with the volume fraction of fibrous septa less than the threshold value were deleted; (3) the orientations of the fibrous septa were calculated with intensity gradient methods; (4) the volume fractions of the principal orientations of fibrous septa were calculated based on the distributions of orientations of fibrous septa.
Figures 16 and 17 show the influence of subimage size and the value of collagen fiber threshold on volume fraction results of principal orientations of fibrous septa. Subimages with a 40 pixel by 40 pixel size are used in this research. It is seen from Fig. 17 that the collagen fibers threshold has little influence on the results of volume fractions of the principal orientations of fibrous septa. The value of 15% is set as the threshold of the collagen fibers.
1.2 Appendix (b)
Error analysis of the image algorithm was conducted as the study by Karlon et al. (1998). Phantom images were generated, consisting of short line segments evenly distributed. Orientation of the short line segments follows a von Mises distribution with known mean and standard deviation (SD). The orientation of short lien segments was characterized by the image algorithm and the obtained mean and SD of orientation was compared to the specified values.
Two series of images were generated. In the 1st series, the SD of the orientation was kept constant (\(15{^{\circ }}\)). Mean orientation was varied from \(-80{^{\circ }}\) to \(80{^{\circ }}\). In the 2nd, the mean orientation was kept constant (\(0{^{\circ }}\)) and SD was varied from \(5{^{\circ }}\) to \(40{^{\circ }}\). Results are shown in Fig. 18. The obtained mean orientation is closely matching the actual mean orientation, while the obtained SD is within \(3{^{\circ }}\) of the actual.
1.3 Appendix (c)
Young’s modulus of adipose tissue was ranging from 1 kPa to 3 MPa (Comley and Fleck 2010). A parametric study was conducted to determine Young’s modulus of adipose tissue of the human plantar soft tissue.
Parametric study on Young’s modulus of adipose tissue
Six different values of Young’s modulus of adipose tissue in the range given by Comley et al. (2010) were assigned to the plantar soft tissue model in subcalcaneal region, while the rest of the parameters were kept constant. Stress versus strain relationships of plantar soft tissue model with these six different Young’s modulus of adipose tissue were compared with Lemmon’s (1997) experimental data as shown in Fig. 19, and the R-Squared values between finite element models and experimental results are calculated and summarized in Table 5. It is seen that the plantar soft tissue model with the Young’s modulus of adipose tissue of 0.03 MPa provides a reasonable prediction with R-squared value equal to 0.91.
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Ou, H., Zhan, P., Kang, L. et al. Region-specific constitutive modeling of the plantar soft tissue. Biomech Model Mechanobiol 17, 1373–1388 (2018). https://doi.org/10.1007/s10237-018-1032-9
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DOI: https://doi.org/10.1007/s10237-018-1032-9