A microstructurally inspired constitutive model for skin mechanics

Abstract

This study investigates the link between the mechanical properties of skin and its microstructural characteristics. Rat back skin samples from different locations, orientations, and sexes were collected and subjected to quasi-static uniaxial tensile tests. Stress–stretch behavior at low stress ranges and rupture data at high stress ranges were collected. The influence of location, orientation, and sex on skin mechanical properties was examined by comparing the mechanical parameters (i.e., initial slope, maximum slope, ultimate tensile strength, rupture stretch, and toughness) evaluated from the tensile testing data. Location and orientation were both found to have a significant effect on the mechanical properties. Collagen structural data (i.e., fiber orientation distribution, relative content, and fiber straightness) were evaluated using histology images. It was found that the rat lower (caudal) back had higher relative collagen content when compared to the upper (cranial) back. A microstructurally based constitutive model was proposed to describe the mechanical behavior of preconditioned rat back skin. The constitutive model incorporated the distribution of collagen fiber bundle orientations and relative collagen content measured from histology, and showed good agreement with the tensile test data. The influence of location and orientation was also evident in the optimized constitutive parameters. This study was a comprehensive investigation that combines skin mechanical behavior, micro-structure, and constitutive modeling.

Appendices

Appendix 1: Influence of storage condition on skin mechanical properties

To investigate the influence of storage condition on skin mechanical properties, three different storage protocols were developed as follows, Protocol I: samples were stored at 4 °C in Hank’s Balance Salted Solution (HBSS) for 24 h before testing. Protocol II: samples were flash frozen in isopentane that was cooled down by dry ice (− 78.5 °C). The flash frozen samples were then stored in a − 80 °C chamber for up to 2 weeks, and thawed at 4 °C for 6 h before testing. Protocol III: flash freezing and storage temperature were the same as Protocol II, but samples were thawed at 4 °C for 24 h before testing. Control: samples were tested fresh, within 2 h of excision. Skin samples obtained from each animal in Group I were randomized into the four groups (three storage protocols plus control), while skin samples obtained from animals in Group II were stored using Protocol II.

After the mechanical testing, a one-way ANOVA was carried out to examine the influence of storage condition on skin mechanical parameters. While the averaged stress–stretch behavior difference between the control group and each storage protocol was evaluated using normalized root-mean-square deviation (RMSD), which was calculated as

$$e_{2} = \frac{{\sqrt {\frac{{\mathop \sum \nolimits_{j = 1}^{n} \left( {\lambda^{\text{prtcl}} - \lambda^{\text{cntrl}} } \right)_{j}^{2} }}{n}} }}{{ \mathop {\hbox{max} }\limits_{1 \le j \le n} \left( {\lambda^{\text{cntrl}} } \right)_{j} }},$$

(5)

where $\lambda^{\text{prtcl}}$ and $\lambda^{\text{cntrl}}$ represented stretch ratios of a sample from a storage protocol and a control sample, respectively, at 20 (i.e., n = 20) different evenly spaced stress values along the stress–stretch curve.

The five mechanical parameters of Group I samples were evaluated and grouped based on storage protocol (see Fig. 8). The storage protocol did not have a significant effect on initial slope, maximum slope, UTS, rupture strain, or toughness (p > 0.05). Normalized RMSD between the control group (fresh samples) and each of the other three storage protocols were calculated to further compare the influence of the storage protocol on the nonlinear mechanical behavior of skin, see Table 2. Results showed that Protocol II had the lowest total normalized RMSD (7.11%), followed by Protocol III (13.75%) and Protocol I (16.97%).

Fig. 8

Inflluence of storage condition on initial slope, maximum slope, UTS, rupture strain, and toughness. Values shown in the bar graph are means and standard deviations. Values shown below each bar graph are p values of one-way ANOVA comparing the mechanical parameter difference between four different storage conditions

Table 2 The normalized RMSD between each storage protocol and the control group for cyclic loading stress–stretch curves and rupture loading stress–stretch curves

Since the influence of different storage conditions on the mechanical properties of skin was not significant, samples from Group I and Group II animals were put together to study the influence of location, orientation, and sex on skin mechanical properties.

Appendix 2: Parameter sensitivity

Best-fit values for the four constitutive parameters (i.e., $c^{gm}$, $c_{1}$, $c_{2}$, and $G_{q}$) were obtained from the estimation process. We, then, investigated the sensitivity of the model to each one of these parameter for one representative sample. Starting from the best-fit constitutive parameters of the representative sample UL22, we changed the value of one parameter at a time, namely for the constitutive parameters $c^{gm}$, $c_{1}$, $c_{2}$ in the range 20–180% of the reference value, and the kinematic parameter $G_{q}$ in the range 0.92–1. We, then, estimated the theoretical stress–stretch curve employing each new set of modified parameters, chosen considering one parameter varying from the reference value, as described, and all the other parameters set to reference values. Finally, the difference between the curves generated with the modified parameter and the original curve is measured using normalized RMSD, described as

$$e_{3} = \frac{{\mathop \sum \nolimits_{i = 1}^{N} \left( {\lambda_{\text{ref}} - \lambda_{\bmod } } \right)_{i}^{2} }}{{\left[ {\mathop {\hbox{max} }\limits_{1 \le i \le N} \left( {\lambda_{\text{ref}} } \right)_{i} } \right]^{2} }},$$

(6)

where $\lambda_{\text{ref}}$ and $\lambda_{\bmod }$ represent the stretch values at a certain stress level in the curve generated with reference parameter values and the curve generated with one modified parameter, respectively. Stretch values at 30 evenly spaced stress values were considered, i.e., $N = 30.$

The stress–stretch curve generated by the constitutive model, using reference parameter values for the representative sample (i.e., $c^{gm} = 0.44 \,{\text{MPa}}$, $c_{1} = 59 \,{\text{MPa}}$, $c_{2} = 32$, $G_{q} = 0.96$) are shown as thick dashed lines in Fig. 9a–e. The stress–stretch curves generated by the constitutive model with one modified parameter, using values that range from 20 to 180% (or 0.92–1) of the reference value, are shown as solid lines in lighter to darker gray in Fig. 9a–e. The normalized RMSDs, as described by Eq. (6), are shown in Fig. 10a–d. When $c^{gm}$ was varied, as shown in Fig. 9a, the change of the stress–stretch is very small, suggesting the model is not sensitive to this parameter. However, when only the portion of the curve for stretches < 1.02 is considered, the normalized RMSD could reach as high as 46.7%, as shown in Figs. 9b and 10a, suggesting the low stretch potion of the curve is significantly sensitive to $c^{gm}$. When $c_{1}$, $c_{2}$, and $G_{q}$ were varied, as shown in Figs. 9c–e and 10b–d, the normalized RMSD could reach as high as 27.4%, 19.0%, and 167.4%, respectively. This suggest that the model is significantly sensitive to these parameters, especially $G_{q}$. It is also interesting to notice that since $c_{2}$ and $G_{q}$ exist in the exponential term of the stress–stretch equation, the normalized RMSD is not symmetrical with respect to the reference values, compared to the symmetrical changes showed for $c^{gm}$ and $c_{1}$.

Fig. 9

Simulated stress–stretch curve when the value of a parameter is changed for the reference sample. For a$c^{gm}$, c$c_{1}$, and d$c_{2}$, the parameter values is changed from 20 to 180% of the reference value (i.e., $c^{gm} = 0.44\,{\text{MPa}}$, $c_{1} = 59\,{\text{MPa}}$, $c_{2} = 32$). For e$G_{q}$ (reference value $G_{q} = 0.96$), the parameter value is changed from 0.92 to 1. The simulated stress–stretch curves between stretch range 1–1.02 when $c_{gm}$ is changed is shown in b

Fig. 10

Normalized RMSD between the simulated stress–stretch curve with one altered parameter value and the simulated stress–stretch curve with reference parameter values. For a$c^{gm}$ (between stretch range 1–1.02), b$c_{1}$, and c$c_{2}$, the parameter values is changed from 20 to 180% of the reference value (i.e., $c^{gm} = 0.44 \,{\text{MPa}}$, $c_{1} = 59 \,{\text{MPa}}$, $c_{2} = 32$). For d$G_{q}$ (reference value $G_{q} = 0.96$), the parameter value is changed from 0.92 to 1

Appendix 3: Location, orientation and sex effect

Mechanical parameters evaluated from rupture stress–stretch curves were grouped based on three different factors, i.e., location (upper vs. lower back), orientation (axial vs. transverse direction), and sex (male vs. female). Group median and quartiles based on different factors are shown in box plot in Fig. 11.

Fig. 11

Influence of location, orientation, and sex on a initial slope, b maximum slope, c rupture strain, d UTS, and e toughness. Quartiles of each parameter based on the grouping factor were shown in the figure. On each box, the central mark indicates the median, and the bottom and top edges of the box indicate the 25th and 75th percentile, respectively. The Whiskers extent to the most extreme data points not considered outliers, and the outliers are plotted individually using the ‘+’ symbol. P values of the three-way ANOVA are shown below the bar graph for comparison within each factor, i.e., upper versus lower, axial versus transverse, male versus female. Statistical significance is marked by the asterisk (p < 0.05)