Constitutive law of healthy gallbladder walls in passive state with damage effect

Abstract

Biomechanical properties of human gallbladder (GB) wall in passive state can be valuable to diagnosis of GB diseases. In the article, an approach for identifying damage effect in GB walls during uniaxial tensile test was proposed and a strain energy function with the damage effect was devised as a constitutive law phenomenologically. Scalar damage variables were introduced respectively into the matrix and two families of fibres to assess the damage degree in GB walls. The parameters in the constitutive law with the damage effect were determined with a custom MATLAB code based on two sets of existing uniaxial tensile test data on human and porcine GB walls in passive state. It turned out that the uniaxial tensile test data for GB walls could not be fitted properly by using the existing strain energy function without the damage effect, but could be done by means of the proposed strain energy function with the damage effect involved. The stresses and Young moduli developed in two families of fibres were more than thousands higher than the stresses and Young’s moduli in the matrix. According to the damage variables estimated, the damage effect occurred in two families of fibres only. Once the damage occurs, the value of the strain energy function will decrease. The proposed constitutive laws are meaningful for finite element analysis on human GB walls.

Appendix: Custom MATLAB program for damage model

The damage model described with Eqs. (6)–(15) was encoded in MATLAB by using a main program and a user function. At first, the experimental data of two uniaxial tensile tests presented with the curves in Fig. 2c are read into the main program after the curves were digitalized by employing a digitizer. The lower and upper bounds of nine model constants are specified. To ensure a global optimization process, the lower bound should be small enough while the upper bound should be large enough. Table 4 summarizes the lower and upper bounds applied in the parameter optimization process in the paper. For the model without damage effect the lower and upper bounds of $\xi$ and $\zeta$ are 10⁸, and those of $m$ and $n$ are 1 to remove their effect on the model and restore the model represented by Eq. (1) without damage, but the bounds of the rest parameter are the same those in the model with damage.

Table 4 Summary of lower and upper bounds of nine parameters used in their optimization process

The lsqnonlin function in MATLAB was chosen to carry out the parameter optimization by minimizing the objective function Eq. (2). In the lsqnonlin function, “trust-region-reflective” optimization algorithm is implanted. In the algorithm, the objective function is approximated with a model function i.e. a quadratic function. Trust region is a subset of the region of the objective function. The minimum objective function is achieved in the trust region. In the trust region algorithm, the search step and size of trust region are decided and updated according to the ratio of the real change of the objective function to the predicted change in the objective function by the model function to ensure sufficient reduction of the objective function. Such procedures can result in the trust region may be out of one bound. Thus, the search direction should be reflected to the interior region constrained by the bounds with the law of reflection in optics on that bound. Compared with Newton method and Levenberg–Marquardt algorithm, the trust-region-reflective algorithm can ensure the optimization iteration remaining in the strict feasible region and its convergence rate is in the 2nd-order [24].

Nine internal optimization variables in the lsqnonlin function [$x_{1}$, $x_{2}$, $x_{3}$, …, $x_{9}$] were selected to represent nine parameters [$c$, $k_{1}$, $k_{2}$, $k_{3}$, $k_{4}$, $\xi$, $n$, $m$, $\zeta$] in the physical domain. However, the variables of [$x_{1}$, $x_{2}$, $x_{3}$, …, $x_{9}$] in the computational domain of the lsqnonlin function is subject to the same lower bound 0 and upper bound 1, but also the step sizes for searching the optimum solution are identical to all the variable. Thus, a transformation relationship between [$x_{1}$, $x_{2}$, $x_{3}$, …, $x_{9}$] in the computational domain and [$c$, $k_{1}$, $k_{2}$, $k_{3}$, $k_{4}$, $\xi$, $n$, $m$, $\zeta$] in the physical domain is needed. Here a linear relationship is employed and written as the followings

$$\left\{ {\begin{array}{*{20}l} {c = c_{\hbox{min} } + x_{1} \times \left( {c_{\hbox{max} } - c_{\hbox{min} } } \right)} \hfill \\ {k_{1} = k_{1\hbox{min} } + x_{2} \times \left( {k_{1\hbox{max} } - k_{1\hbox{min} } } \right)} \hfill \\ {k_{2} = k_{2\hbox{min} } + x_{3} \times \left( {k_{2\hbox{max} } - k_{2\hbox{min} } } \right)} \hfill \\ {k_{3} = k_{3\hbox{min} } + x_{4} \times \left( {k_{3\hbox{max} } - k_{3\hbox{min} } } \right)} \hfill \\ {k_{4} = k_{4\hbox{min} } + x_{5} \times \left( {k_{4\hbox{max} } - k_{4\hbox{min} } } \right)} \hfill \\ {\xi = \xi_{\hbox{min} } + x_{6} \times \left( {\xi_{\hbox{max} } - \xi_{\hbox{min} } } \right)} \hfill \\ {n = n_{\hbox{min} } + x_{7} \times \left( {n_{\hbox{max} } - n_{\hbox{min} } } \right)} \hfill \\ {m = m_{\hbox{min} } + x_{8} \times \left( {m_{\hbox{max} } - m_{\hbox{min} } } \right)} \hfill \\ {\zeta = \zeta_{\hbox{min} } + x_{9} \times \left( {\zeta_{\hbox{max} } - \zeta_{\hbox{min} } } \right)} \hfill \\ \end{array} } \right.$$

(A1)

where the lower and upper bounds of nine parameters, such as $c_{\hbox{min} }$, $c_{\hbox{max} }$, $k_{1\hbox{min} }$, $k_{1\hbox{max} }$ and so on, have been listed in Table 4. Accordingly, the step sizes in the computational domain are related to those in the counterpart in the physical domain by the following from Eq. (A1)

$$\left\{ {\begin{array}{*{20}l} {\Delta c = \Delta x_{1} \times \left( {c_{\hbox{max} } - c_{\hbox{min} } } \right)} \hfill \\ {\Delta k_{1} = \Delta x_{2} \times \left( {k_{1\hbox{max} } - k_{1\hbox{min} } } \right)} \hfill \\ {\Delta k_{2} = \Delta x_{3} \times \left( {k_{2\hbox{max} } - k_{2\hbox{min} } } \right)} \hfill \\ {\Delta k_{3} = \Delta x_{4} \times \left( {k_{3\hbox{max} } - k_{3\hbox{min} } } \right)} \hfill \\ {\Delta k_{4} = \Delta x_{5} \times \left( {k_{4\hbox{max} } - k_{4\hbox{min} } } \right)} \hfill \\ {\Delta \xi = \Delta x_{6} \times \left( {\xi_{\hbox{max} } - \xi_{\hbox{min} } } \right)} \hfill \\ {\Delta n = \Delta x_{7} \times \left( {n_{\hbox{max} } - n_{\hbox{min} } } \right)} \hfill \\ {\Delta m = \Delta x_{8} \times \left( {m_{\hbox{max} } - m_{\hbox{min} } } \right)} \hfill \\ {\Delta \zeta = \Delta x_{9} \times \left( {\zeta_{\hbox{max} } - \zeta_{\hbox{min} } } \right)}\hfill \\ \end{array} } \right.$$

(A2)

Based on Eq.(A2), even though the step sizes of the internal variables [$x_{1}$, $x_{2}$, $x_{3}$, …, $x_{9}$] are the same, i.e. $\Delta x_{1}$ = $\Delta x_{2}$ = $\Delta x_{3}$ = ··· = $\Delta x_{9}$ in the lsqnonlin function, the step sizes such as $\Delta c$, $\Delta k_{1}$, $\Delta k_{2}$, …, $\Delta \zeta$ in the physical domain still vary across the variables.

It was found that $k_{1}$($x_{2}$) and $k_{3}$($x_{4}$) vary little and affect the optimization results negligibly, but $c$($x_{1}$) changes significantly during the optimization process. Therefore $k_{1}$($x_{2}$) and $k_{3}$($x_{4}$) have to be updated by $c$($x_{1}$) after they were calculated with Eq. (A1) in the following manner

$$\left\{ {\begin{array}{*{20}c} {k_{1} \times c \Rightarrow k_{1} } \\ {k_{3} \times c \Rightarrow k_{3} } \\ \end{array} } \right.$$

(A3)

where $k_{1}$ and $k_{3}$ in the left-hand side have been determined by Eq. (A1).

Additionally, an initial nine parameters [$c_{0}$, $k_{10}$, $k_{20}$, $k_{30}$, $k_{40}$, $\xi_{0}$, $n_{0}$, $m_{0}$, $\zeta_{0}$] are generated randomly in the bounds by using rand function of MATLAB in terms of [$x_{10}$, $x_{20}$, $x_{30}$, …, $x_{90}$] to make sure a global optimization process, i.e.

$$\left\{ {\begin{array}{*{20}l} {c_{0} = c_{\hbox{min} } + x_{10} \times \left( {c_{\hbox{max} } - c_{\hbox{min} } } \right)} \hfill \\ {k_{10} = k_{1\hbox{min} } + x_{20} \times \left( {k_{1\hbox{max} } - k_{1\hbox{min} } } \right)} \hfill \\ {k_{20} = k_{2\hbox{min} } + x_{30} \times \left( {k_{2\hbox{max} } - k_{2\hbox{min} } } \right)} \hfill \\ {k_{30} = k_{3\hbox{min} } + x_{40} \times \left( {k_{3\hbox{max} } - k_{3\hbox{min} } } \right)} \hfill \\ {k_{40} = k_{4\hbox{min} } + x_{50} \times \left( {k_{4\hbox{max} } - k_{4\hbox{min} } } \right)} \hfill \\ {\xi_{0} = \xi_{\hbox{min} } + x_{60} \times \left( {\xi_{\hbox{max} } - \xi_{\hbox{min} } } \right)} \hfill \\ {n_{0} = n_{\hbox{min} } + x_{70} \times \left( {n_{\hbox{max} } - n_{\hbox{min} } } \right)} \hfill \\ {m_{0} = m_{\hbox{min} } + x_{80} \times \left( {m_{\hbox{max} } - m_{\hbox{min} } } \right)} \hfill \\ {\zeta_{0} = \zeta_{\hbox{min} } + x_{90} \times \left( {\zeta_{\hbox{max} } - \zeta_{\hbox{min} } } \right)} \hfill \\ \end{array} } \right.$$

(A3)

where $x_{10}$ = rand(1, 1), $x_{20}$ = rand(1, 1), $x_{30}$ = rand(1, 1), …, $x_{90}$ = rand(1, 1).

The option in the lsqnonlin function is as follows: MaxIter = 4000, TolFun = 10⁻⁸, TolX = 10⁻⁸, Diffminchange = 10⁻⁴, Diffmaxchange = 10⁻² and MaxFunEvals = 50,000 where MaxIter is maximum number of iterations allowed, TolFun is termination tolerance on the objective function value, TolX is termination tolerance on [$x_{1}$, $x_{2}$, $x_{3}$, …, $x_{9}$], Diffminchange and Diffmaxchange are minimum and maximum changes in variables for finite difference derivatives of the objective function, respectively; MaxFunEvals is maximum number of the objective function evaluations allowed.

The temporary nine parameters, stresses and objective function value at the experimental stretches are calculated in the user function. The user function is called repeatedly by the lsqnonlin function until a convergent optimization process arrives. The stress-stretch curves, strain energy function values, Young’s moduli, damage variables and relevant plots are figured out in the main program based on the determined nine parameters.