An In Silico-Based Investigation on Anisotropic Hyperelastic Constitutive Models for Soft Biological Tissues

Abstract

We review twelve invariant and dispersion-type anisotropic hyperelastic constitutive models for soft biological tissues based on their fitting performance to various experimental data. To this end, we used a hybrid multi-objective optimization procedure along with a genetic algorithm to generate the initial guesses followed by a gradient-based search algorithm. The constitutive models are then fit to a set of uniaxial and biaxial tension experiments conducted on tissues with different histology. For the in silico investigation, experiments conducted on human aneurysmatic abdominal aorta, linea alba, and rectus sheath tissues are utilized. Accordingly, the models are ranked with respect to an objective normalized quality of fit metric. Finally, a detailed discussion is carried out on the fitting performance of the models. This work provides a valuable quantitative comparison of various anisotropic hyperelastic models, the findings of which can aid researchers in selecting the most suitable constitutive model for their particular analysis. The investigation reveals superior fitting performance of dispersion-type anisotropic constitutive formulations over invariant formulations.

Appendix

Root mean square error (RMSE) can be used as an alternative quality of fit metric to compare the performance of models (Tables 3, 4, 5, 6 and 7). RSME for uniaxial dataset of Cooney et al. [19], and Martins et al. [69] is

$$\begin{aligned} \begin{array}{rl} \text {RMSE} =&\sqrt{\frac{\sum \nolimits _{i=1}^{n_{UT_1}} \left( P_{11}^{UT_1}(\lambda _i) - P_{11}^{exp, UT_1}(\lambda _i)\right) ^2}{n_{UT_1}}} + \sqrt{\frac{\sum \nolimits _{i=1}^{n_{UT_2}} \left( P_{22}^{UT_2}(\lambda _i) - P_{22}^{exp, UT_2}(\lambda _i)\right) ^2}{n_{UT_2}}}\,. \end{array} \end{aligned}$$

(101)

Similarly, for the equibiaxial loading case, RMSE for the dataset of Niestrawska et al. [78] is

$$\begin{aligned} \begin{array}{rl} \text {RMSE} =&\sqrt{\frac{\sum \nolimits _{i=1}^{n_{ET}} \left( P_{11}^{ET}(\lambda _i) - P_{11}^{exp, ET}(\lambda _i)\right) ^2}{n_{ET}}} + \sqrt{\frac{\sum \nolimits _{i=1}^{n_{ET}} \left( P_{22}^{ET}(\lambda _i) - P_{22}^{exp, ET}(\lambda _i)\right) ^2}{n_{ET}}}\,. \end{array} \end{aligned}$$

(102)

Another quality of fit metric is the coefficient of determination ($R^2$). $R^2$ for uniaxial dataset of Cooney et al. [19], and Martins et al. [69] is

$$\begin{aligned} \begin{array}{rl} R^2 =&0.5 \left[ 1 - \frac{\sum \nolimits _{i=1}^{n_{UT_1}} \left( P_{11}^{UT_1}(\lambda _i) - P_{11}^{exp, UT_1}(\lambda _i)\right) ^2}{\sum \nolimits _{i=1}^{n_{UT_1}} \left( P_{11}^{UT_1}(\lambda _i) - \text {mean}(P_{11}^{exp, UT_1}(\lambda _i))\right) ^2}\right] + 0.5 \left[ 1 - \frac{\sum \nolimits _{i=1}^{n_{UT_2}} \left( P_{22}^{UT_2}(\lambda _i) - P_{22}^{exp, UT_2}(\lambda _i)\right) ^2}{\sum \nolimits _{i=1}^{n_{UT_2}} \left( P_{22}^{UT_2}(\lambda _i) - \text {mean}(P_{22}^{exp, UT_2}(\lambda _i))\right) ^2}\right] \,. \end{array} \end{aligned}$$

(103)

Similarly, for equibiaxial loading case, $R^2$ for the fitting of dataset of Niestrawska et al. [78] is

$$\begin{aligned} \begin{array}{rl} R^2 =&0.5 \left[ 1 - \frac{\sum \nolimits _{i=1}^{n_{ET}} \left( P_{11}^{ET}(\lambda _i) - P_{11}^{exp, ET}(\lambda _i)\right) ^2}{\sum \nolimits _{i=1}^{n_{ET}} \left( P_{11}^{ET}(\lambda _i) - \text {mean}(P_{11}^{exp, ET}(\lambda _i))\right) ^2}\right] + 0.5 \left[ 1 - \frac{\sum \nolimits _{i=1}^{n_{ET}} \left( P_{22}^{ET}(\lambda _i) - P_{22}^{exp, ET}(\lambda _i)\right) ^2}{\sum \nolimits _{i=1}^{n_{ET}} \left( P_{22}^{ET}(\lambda _i) - \text {mean}(P_{22}^{exp, ET}(\lambda _i))\right) ^2}\right] \,. \end{array} \end{aligned}$$

(104)

Table 3 Models sorted according to the root mean square error to the equibiaxial dataset of AAA tissue [78], uniaxial dataset of the linea alba [19] and uniaxial dataset of the rectus sheath [69]

Table 4 Models sorted according to the coefficient of determination to the equibiaxial dataset of AAA tissue [78], uniaxial dataset of the linea alba [19] and uniaxial dataset of the rectus sheath [69]

Table 5 Identified parameters and error bounds based on AAA tissue dataset

Table 6 Identified parameters and error bounds based on LA tissue dataset

Table 7 Identified parameters and error bounds based on RS tissue dataset