Simulation of Steatosis Zonation in Liver Lobule—A Continuummechanical Bi-Scale, Tri-Phasic, Multi-Component Approach

Abstract

The human liver is an important metabolic organ which regulates metabolism of the body in a complex time depending and non-linear coupled function-perfusion-mechanism. Harmful microstructure failure strongly affects the viability of the organ. The excessive accumulation of fat in the liver tissue, known as a fatty liver, is one of the most common liver micro structure failures, especially in western countries. The growing fat has a high impact on the blood perfusion and thus on the functionality of the organ. This interaction between perfusion, growth of fat and functionality on the hepatic microcirculation is poorly understood and many biological aspects of the liver are still subject of discussion. The presented computational model consists of a bi-scale, tri-phasic, multi-component approach based on the theory of porous media. The model includes the stress and strain state of the liver tissue, the transverse isotropic blood perfusion in the sinusoidal micro perfusion system. Furthermore, we describe the glucose metabolism in a two-scale PDE-ODE approach whereas the fat metabolism is included via phenomenological functions. Different inflow boundary conditions are tested against the influence on fat deposition and zonation in the liver lobules. With this example we can discuss biological assumptions and get a better understanding of the coupled function-perfusion ability of the liver.

Appendix

1.1 Perfusion-Model with a Multi-phasic Approach

The perfusion of the blood through the liver lobules is an important part to depict realistic descriptions for the viability of the organ. For that we use a homogenized approach on the mesoscopic scale, see Fig. 14. We consider three phases: the liver tissue $\varphi ^{\mathbf {{S}}}$, fat tissue $\varphi ^{\mathbf {{TG}}}$ and fluid phase $\varphi ^{\mathbf {{L}}}$.

Fig. 14

Numerical implementation: perfusion approach with permeability in dependence of vessel distribution and preferred flow direction. Homogenization of the real structure into a smeared model (TPM)

The phases are assumed as mutually immiscible materials $\varphi ^{\pmb {\alpha }}$ with a heterogeneous arrangement in the overall volume. Each phase consists of a carrier phase $\varphi ^{\alpha }$, namely a solvent, and small miscible microscopic components $\varphi ^{\alpha \beta }$, called solutes in the solvent. The TPM is an approach, which is composed of the mixture theory (Greve [7] and Hutter et al. [9]) and the concept of volume fractions (de Boer [2, 3] and Ehlers [5]). The saturation condition completes the approach.

The overall structure is a mixture of all included components, so the whole body $\varphi$ can be decomposed by

$$\begin{aligned} \varphi = \displaystyle \sum \limits _{{\varvec{\alpha }}=1}^\kappa \varphi ^{{\varvec{\alpha }}} := \displaystyle \sum \limits _{\alpha =1}^\kappa [\displaystyle \sum \limits _{\beta =1}^{\nu -1} (\varphi ^{\alpha \beta }) + \varphi ^{\alpha } ]. \end{aligned}$$

(3)

To account the contribution of different phases we use the volume fraction $\mathrm{n}^{\pmb {\alpha }}$ expressed by the ratio of partial volume ${\mathrm{dv}}^{\pmb {\alpha }}$ to the overall mixture volume $\mathrm{dv}$

$$\begin{aligned} \mathrm{n}^{\pmb {\alpha }}= \displaystyle \frac{{\mathrm{dv}}^{\pmb {\alpha }}}{\mathrm{dv}}. \end{aligned}$$

(4)

In view of the volume fractions, the saturations condition has to be fulfilled with

$$\begin{aligned} \displaystyle \sum \limits _{{\varvec{\alpha }}=1}^\kappa \mathrm{n}^{\pmb {\alpha }}\,\, \mathrm {mit} \quad \kappa \, \in \, \left\{ \pmb {\mathrm {S,TG,L}}\right\} . \end{aligned}$$

(5)

For an effective connection between the mixture theory and the concept of volume fractions we consider the density $\rho ^{\pmb {\alpha }}$ with

$$\begin{aligned} \mathrm {m} = \displaystyle \sum \limits _{{\varvec{\alpha }}=1}^\kappa \mathrm {m}^{{\varvec{\alpha }}} = \displaystyle \int \limits _{\mathrm {Bs}} \displaystyle \sum \limits _{{\varvec{\alpha }}=1}^\kappa \rho ^{{\varvec{\alpha }}} \mathrm{dv}\,. \end{aligned}$$

(6)

where $\mathrm {m}$ is the mass of the mixture and $\mathrm {m}^{{\varvec{\alpha }}}$ is the partial mass. Thereby $\ \mathrm {\rho ^{{\varvec{\alpha }}}}$ describes the partial density, which is derived by the ratio of mass $\mathrm {dm}^{{\varvec{\alpha }}}$ to the volume $\mathrm{dv}$ of the structure

$$\begin{aligned} \mathrm { \rho ^{{\varvec{\alpha }}}=\displaystyle \frac{dm^{{\varvec{\alpha }}}}{dv} }\,. \end{aligned}$$

(7)

The true density $\rho ^{{\varvec{\alpha }}\mathrm {R}}$ of the phases follows with

$$\begin{aligned} \mathrm { \rho ^{{\varvec{\alpha }}R}=\displaystyle \frac{dm^{{\varvec{\alpha }}}}{dv^{{\varvec{\alpha }}}} }\,. \end{aligned}$$

(8)

Including the given Eqs. (3) and (8) we get a connection between the density and the volume fractions with

$$\begin{aligned} \mathrm { \rho ^{{\varvec{\alpha }}}=\displaystyle \frac{dm^{{\varvec{\alpha }}}}{dv} =\displaystyle \frac{dm^{{\varvec{\alpha }}}}{dv^{{\varvec{\alpha }}} / n^{{\varvec{\alpha }}}} =n^{{\varvec{\alpha }}} \ \rho ^{{\varvec{\alpha }}R} }\,. \end{aligned}$$

(9)

The concentration $\mathrm{c}^{\alpha \beta }$ of the microscopic components which are included in the phases are described by the ratio of the number of moles $\mathrm{dn}^{\beta }_{\mathrm {mol}}$ and the partial volume ${\mathrm{dv}}^{\pmb {\alpha }}$ with

$$\begin{aligned} \mathrm{c}^{\alpha \beta }=\displaystyle \frac{\mathrm{dn}^{\beta }_{\mathrm {mol}}}{{\mathrm{dv}}^{\pmb {\alpha }}}\,. \end{aligned}$$

(10)

The partial molar density of the microscopic components ${\rho ^{\alpha \beta }}$ is decomposed by the volume fraction $\mathrm{n}^{\pmb {\alpha }}$, the concentration $\mathrm{c}^{\alpha \beta }$ and the molecular weight $\mathrm {M}^{\beta }_\mathrm{mol}$ of the constituent

$$\begin{aligned} {\rho ^{\alpha \beta }}=\mathrm{n}^{\pmb {\alpha }}\mathrm{c}^{\alpha \beta }\mathrm {M}^{\beta }_\mathrm{mol}\,. \end{aligned}$$

(11)

Whereas, the molecular weight of the constituent is calculated by the fraction of the mass $\mathrm {dm}^{\beta }$ of the component and the number of moles $\mathrm{dn}^{\beta }_{\mathrm {mol}}$

$$\begin{aligned} \mathrm {M}^{\beta }_\mathrm{mol}=\displaystyle \frac{\mathrm {dm}^{\beta }}{\mathrm{dn}^{\beta }_{\mathrm {mol}}}\,. \end{aligned}$$

(12)

1.2 Balance Equations in the Framework of TPM

The balance equations for the porous media contain the description for each phase $\varphi ^{\pmb {\alpha }}$ analogous to a one phase continuum. The multiphase approach incorporates the chemical and physical interactions of the phases $\varphi ^{\pmb {\alpha }}$ with the interaction forces $\hat{\mathbf {p}}^{{\varvec{\alpha }}}$ and the mass exchange ${\hat{\rho }}^{{\varvec{\alpha }}}$ following the metaphysical principles of Truesdell [15]. The local statements of the balance equation of mass, momentum and moment of momentum follow with

$$\begin{aligned} \begin{array}{lcl} \displaystyle {\sum \limits _{{\varvec{\alpha }}={1}}^{\kappa }}\,[\displaystyle {\frac{\partial \mathrm{n}^{\pmb {\alpha }}}{\partial \mathrm{t}}}+{\mathrm{div}}(\mathrm{n}^{\pmb {\alpha }}\,{\mathbf {{x}}}^{\prime }_{\varvec{\alpha }})\,-\,\frac{1}{{\rho }^{{\varvec{\alpha }}{\mathrm{R}}}}\,{\hat{\rho }^{\varvec{\alpha }}}\,]\, &{} = &{} \,\mathrm {0} \\ \displaystyle {\sum \limits _{{\varvec{\alpha }}=\mathrm {1}}^{\kappa }}\,[\mathrm{div}\>{\mathbf {T}}^{{\varvec{\alpha }}}\, + \,\rho ^{\pmb {\alpha }}\,(\mathbf {{b}}^{{\varvec{\alpha }}}\,-\,{\mathbf {{x}}}^{\prime \prime }_{\pmb {\alpha }})\,+ \,\hat{\mathbf {p}}^{{\varvec{\alpha }}}\, - \,{\hat{\rho }}^{{\varvec{\alpha }}}\,{\mathbf {{x}}}^{\prime }_{\varvec{\alpha }}\,]\, &{} = &{} \,\pmb {\mathrm {0}} \\ \displaystyle {\sum \limits _{{\varvec{\alpha }}=\mathrm {1}}^{\kappa }}\,[\mathbf {{T}}^{{\varvec{\alpha }}}\,-\,(\mathbf {{T}}^{{\varvec{\alpha }}})^{\mathrm {T}}\,]\, &{} = &{} \,\pmb {\mathrm {0}}\, \end{array} \end{aligned}$$

(13)

The balance equation of mass applies the time derivative $\partial \mathrm {t}$ of the volume fraction $\mathrm{n}^{\pmb {\alpha }}$. Furthermore, “div” describes the spatial divergence operator, ${\mathbf {{x}}}^{\prime }_{\varvec{\alpha }}$ the velocity of the phases and ${\rho }^{{\varvec{\alpha }}{\mathrm{R}}}$ the true density. $\mathbf {{T}}^{{\varvec{\alpha }}}$ is the Cauchy stress tensor, $\mathbf {{b}}^{{\varvec{\alpha }}}$ is the specific volume force and ${\mathbf {{x}}}^{\prime \prime }_{\pmb {\alpha }}$ describes acceleration. Truesdell introduced the metaphysical principles in [15] with fundamental formulations for the mixture bodies which lead to the restrictions

$$\begin{aligned} \begin{array}{lcl} \displaystyle {\sum \limits _{{\varvec{\alpha }}=\mathrm {1}}^{\kappa }}\,{\hat{\rho }}^{{\varvec{\alpha }}}\, &{} = &{} \,\mathrm {0} \\ \displaystyle {\sum \limits _{{\varvec{\alpha }}=\mathrm {1}}^{\kappa }}\,\hat{\mathbf {p}}^{{\varvec{\alpha }}}\, &{} = &{} \,\pmb {\mathrm {0}}\,. \end{array} \end{aligned}$$

(14)

1.3 Assumptions for the Perfusion Model

In this approach we apply three main phases for the mixture body. Thus, the volume counts the main phases $\varphi ^{\pmb {\alpha }}$ with ${\varvec{\alpha }}\, \in \,$ S (liver tissue), TG (fat tissue), L (blood). We assume miscible concentrations included in the main phases which are important for the metabolism processes. The liver tissue is described by two components, the hepatocytes which include glycogen $\varphi ^{{\mathbf {{S}}\beta }}\,\in \,\{\mathrm {Gy}\}$ as an internal concentration and fat as the second component which includes triglyceride $\varphi ^{{\mathbf {{TG}}\beta }}\,\in \,\{\mathrm {TG}\}$. Both internal miscible components are results of the metabolism. Furthermore, the blood phase includes external solutes which are important metabolites coming from the intestines. Focusing on the glucose and fat metabolism we apply external concentrations for glucose, FFA and lactate $\varphi ^{{\mathbf {{L}}\beta }}\,\in \,\{\mathrm {Gu,Lc,FFA}\}$. Since the overall solutes $\varphi ^{\alpha \beta }$ are negligibly small in contrast to the phases $\varphi ^{\pmb {\alpha }}$ we do not take the volume fraction of the concentrations into account. So, the volume of the main phases is nearly the same as the volume of the carrier phases and we can simplify $\varphi ^{\pmb {\alpha }}\,\cong \,\varphi ^{\alpha }$. Thus, we summarize the description for the mixture body with $\kappa \,=\,\mathrm {3}$

$$\begin{aligned} \varphi ^{\alpha }=\{\mathrm {S,TG,L\} } =\alpha _{\mathrm {i}}\,\mathrm {\vert i=1 \ldots 3} \end{aligned}$$

(15)

and $\mathrm {(\nu -1)}$ the microscopic components

$$\begin{aligned} \begin{array}{lclcl} \varphi ^{\mathrm {S\beta }}\, &{} = &{} \,\{\mathrm {Gy}\}\, &{} = &{} \,{\mathrm {\beta _i}}\,{\mathrm {\vert i=1}} \\ \varphi ^{\mathrm {TG\beta }}\, &{} = &{} \,\{\mathrm {TG}\}\, &{} = &{} \,{\mathrm {\beta _i}}\,{\mathrm {\vert i=1}} \\ \varphi ^{\mathrm {L\beta }}\, &{} = &{} \,\{\mathrm {Gu,Lc,FFA}\}\, &{} = &{} \,{\mathrm {\beta _i}}\,{\mathrm {\vert i=1 \ldots 3}}\, \end{array} \end{aligned}$$

(16)

The overall volume $\mathrm {v}$ can be calculated by the volume fractions $\mathrm {n}^{\alpha }$ of the phases

$$\begin{aligned} \mathrm {v}= \displaystyle \sum \limits _{\alpha =1}^\kappa \,\mathrm {dv}^{{\alpha }}= \displaystyle \int \limits _{\mathrm {Bs}} \displaystyle \sum \limits _{\alpha =1}^\kappa \,\mathrm {dv}^{{\alpha }}= \displaystyle \int \limits _{\mathrm {Bs}} \displaystyle \sum \limits _{\alpha =1}^\kappa \,\mathrm {n}^{\alpha }\,\mathrm{dv}\,\, \mathrm {mit} \quad \kappa \, \in \, \{\mathrm {S,TG,L\}} \end{aligned}$$

(17)

The basis of the TPM applies superimposed continua with interactions and independent motion functions for the included phases. De Boer [4] and Ehlers [5] give an explanation of the kinematics of TPM. We assume a Lagrange description of the motion for the liver tissue ${\varvec{\chi }}_{\mathrm {S}}(\pmb {X}_{\mathrm {S}},\,\mathrm {t})$ with $\pmb {X}_{\mathrm {S}}$ describing the reference configuration of the liver cells and $\mathrm {t}$ describing the time. As the fat tissue is connected to the liver cells we use the same motion function with ${\varvec{\chi }}_{\mathrm {S}}\,=\,{\varvec{\chi }}_{\mathrm {TG}}$. Additionally, the internal concentrations which are exclusively present in the cells of the liver have the same motion function as the liver and fat tissue. So we extend ${\varvec{\chi }}_{\mathrm {S}}\,=\,{\varvec{\chi }}_{\mathrm {TG}}\,=\,{\varvec{\chi }}_{\mathrm {S}\beta }=\,{\varvec{\chi }}_{\mathrm {TG}\beta }$. As a consequence of the identical motion function one velocity follows for the liver tissue, fat tissue and the internal concentration (glycogen, triglyceride) with $\mathbf {{x}}^{\prime }_{\mathrm {S}}\,=\,\mathbf {{x}}^{\prime }_{\mathrm {TG}}\,=\,\mathbf {{x}}^{\prime }_{\mathrm {S}\beta }\,=\,\mathbf {{x}}^{\prime }_{\mathrm {TG}\beta }$. Beside the fixed solid fraction the porous body contains the fluid phase which represents the blood flow. For the kinematics of the blood flow we use a modified Eulerian description with respect to the solid phase. We apply independent motion functions for the blood phase ${\varvec{\chi }}_{\mathrm {L}}(\pmb {X}_{\mathrm {L}},\mathrm {t})$ and the external concentration: glucose $\varvec{\chi }_{\mathrm {L,Gu}}(\pmb {X}_{\mathrm {LGu}},\mathrm {t})$, lactate $\varvec{\chi }_{\mathrm {L,Lc}}(\pmb {X}_{\mathrm {LLc}},\mathrm {t})$ and FFA $\varvec{\chi }_{\mathrm {L,FFA}}(\pmb {X}_{\mathrm {LFFA}},\mathrm {t})$, which are included in the blood phase. The velocities follow with $\pmb {\mathrm {x}}^{\prime }_{\mathrm {L}}\,=\,\pmb {\mathrm {x}}^{\prime }_{\mathrm {L}}(\pmb {X}_{\mathrm {L}},\,\mathrm {t})$ for the main phase of the fluid and $\mathbf {{x}}^{\prime }_{\mathrm {L,Gu}}\,=\,\mathbf {{x}}^{\prime }_{\mathrm {L,Gu}}(\pmb {X}_{\mathrm {LGu}},\,\mathrm {t})$ for glucose, $\mathbf {{x}}^{\prime }_{\mathrm {L,Lc}}\,=\,\mathbf {{x}}^{\prime }_{\mathrm {L,Lc}}(\pmb {X}_{\mathrm {LLc}},\,\mathrm {t})$ for lactate and $\mathbf {{x}}^{\prime }_{\mathrm {L,FFA}}\,=\,\mathbf {{x}}^{\prime }_{\mathrm {L,FFA}}(\pmb {X}_{\mathrm {LFFA}},\,\mathrm {t})$ for FFA.

The blood flow in the liver lobules mainly depends on the vascular system, which is designed by the sinusoidal arrangement. The sinusoids guide the blood from the portal triads to the central vein and lead to an anisotropic diffusivity. We introduced an approach for the anisotropic perfusion in Ricken et al. [13]. This includes an ansatz for the filter velocity $\mathrm {n}^{\mathrm {L}}\,\mathbf {{w}}_{\mathrm {LS}}$ with the volume fraction of the fluid $\mathrm {n}^{\mathrm {L}}$ and the seepage velocity $\mathbf {{w}}_{\mathrm {LS}}$. The seepage velocity defines the difference in velocity of the fluid and solid phase $\mathbf {{w}}_{\mathrm {LS}}\,=\,\mathbf {{x}}^{\prime }_{\mathrm {L}}\,-\,\mathbf {{x}}^{\prime }_{\mathrm {S}}$.

$$\begin{aligned} \,\mathrm {n}^{\mathrm {L}}\,\mathbf {{w}}_{\mathrm {LS}}\,=\,\mathrm {k}_{\mathrm {0S}}^{\mathrm {S}}\,(\frac{\mathrm {n}^{\mathrm {L}}}{\mathrm {1}\,-\,\mathrm {n}^{\mathrm {S}}_{\mathrm {0S}}})^{\mathrm {m}}\,\, \mathbf {{M}}^*\,[\,-\mathrm {grad}\lambda \,+\,\rho ^{\mathrm {LR}}\,\mathbf {{b}}] \end{aligned}$$

(18)

It depends on the deformation $(\frac{\mathrm {n}^{\mathrm {L}}}{\mathrm {1}\,-\,\mathrm {n}^{\mathrm {S}}_{\mathrm {0S}}})^{\mathrm {m}}$ (see Eipper [6]) which includes a dimensionless material parameter $\mathrm {m}$, controlling the isotropic dependency of the permeability. Furthermore, it depends on the Darcy permeability with $\mathrm {k}_{\mathrm {0S}}^{\mathrm {S}}\,[\frac{\mathrm {m}^{4}}{\mathrm {Ns}}]$ and on the transverse isotropic permeability relation $\mathbf {{M}}^*$ which includes the alignment of the sinusoids (for further information see Ricken et al. [12, 13]).

1.4 Field Equations and Constitutive Modeling

Summarizing, we consider a quasi-static, isothermal, tri-phasic porous model with microscopic substances. The model takes into account an incompressible solid phase $\varphi ^{\mathrm {S}}$ and fat phase $\varphi ^{\mathrm {TG}}$ which are derived by the same motion function and an incompressible fluid phase $\varphi ^{\mathrm {L}}$. We assume mass exchange between the fat and fluid phase. All phases include substances $\varphi ^{\alpha \beta }$ which are calculated by the microscopic model and allow phase transition to build up the metabolism. The solid phases $\varphi ^{\mathrm {S}}$ and fat phase $\varphi ^{\mathrm {TG}}$ include the internal concentrations $\varphi ^{\mathrm {S\beta }}$ with $\varphi ^{\mathrm {S\beta }}\,\in \,\{\mathrm {Gy}\}$ and $\varphi ^{\mathrm {TG\beta }}$ with $\varphi ^{\mathrm {TG\beta }}\,\in \,\{\mathrm {TG}\}$. The external concentrations $\varphi ^{\mathrm {L\beta }}$ with $\varphi ^{\mathrm {L\beta }}\,\in \,\{\mathrm {Gu,Lc,FFA}\}$ are included in the fluid phase. For the calculation of the presented model we use the following independent relations. On the one hand we use the local form of the balance equation of mass and momentum for each component with

$$\begin{aligned} \begin{array}{lcl} (\mathrm {n}^{\alpha })^{\prime }_{\alpha }\,+\,\mathrm {n}^{\alpha }\,\mathrm {div}\>{\mathbf {x}}^{\prime }_{\alpha }\,&{}=&{}\,\,\displaystyle {\frac{\mathrm {1}}{\rho ^{\alpha {\mathrm {R}}}}}\,\hat{\rho }^{\alpha }\\ {\mathrm{div}\>{\mathbf {T}}^{\alpha }}\, + \,\rho ^{\alpha }\,\mathbf {{b}}^{\alpha }\,+ {\hat{\mathbf {p}}}^{\alpha }&{}=&{}\,\hat{\rho }^{\alpha }\,{\mathbf {x}}^{\prime }_{\alpha }\, \end{array} \end{aligned}$$

(19)

for the main phases and

$$\begin{aligned} \begin{array}{lcl} (\mathrm {n}^{\alpha })^{\prime }_{\alpha \beta }\,\mathrm{c}^{\alpha \beta }\,\mathrm {M}^{\beta }_\mathrm{mol}\,+\,\mathrm {n}^{\alpha }\,(\mathrm {c}^{\alpha \beta })^{\prime }_{\alpha \beta }\,\mathrm {M}^{\beta }_\mathrm{mol}\,+\,\mathrm {n}^{\alpha }\,\mathrm{c}^{\alpha \beta }\,\mathrm {M}^{\beta }_\mathrm{mol}\,\mathrm {div}\>\pmb {\mathrm {x}}^{\prime }_{\alpha \beta }\,&{}=&{}\,\hat{\rho }^{\alpha \beta }\\ {\mathrm {div}}\>{\mathbf {T}}^{\alpha \beta }\, + \,{\rho ^{\alpha \beta }}\,\mathbf {{b}}^{\alpha }\,+ \,\hat{\mathbf {p}}^{\alpha \beta }\,&{}=&{}\,\hat{\rho }^{\alpha \beta }\,{\mathbf {x}}^{\prime }_{\alpha }\end{array} \end{aligned}$$

(20)

for the included concentrations. The field equations include the interaction terms between the main phases $\hat{\rho }^{\alpha }$ and the miscible substances $\hat{\rho }^{\alpha \beta }$. On the other hand we consider the physical constraint condition derived by the assumptions of the porous medium with the saturation condition

$$\begin{aligned} \mathrm {n}^{\mathrm {S}}\,+\,\mathrm {n}^{\mathrm {TG}}\,+\,\mathrm {n}^{\mathrm {L}}\,=\,\mathrm {1}, \end{aligned}$$

(21)

the conditions for the mass exchange between the components with

$$\begin{aligned} \hat{\rho }^{\mathrm {S}}\,+\,\hat{\rho }^{\mathrm {TG}},+\,\hat{\rho }^{\mathrm {L}}\,+\,\hat{\rho }^{\alpha \beta }=\,\mathrm {0}\,, \end{aligned}$$

(22)

and the interaction forces

(23)

Beside the field equations we need constitutive relations to complete the calculation concept of the saturated porous body. The constitutive equations are derived by the evaluation of the entropy inequality

$$\begin{aligned} \sum \limits _{{\varvec{\alpha }}=\mathrm {1}}^{\kappa }\,[-\rho ^{\pmb {\alpha }}\,(\psi ^{{\varvec{\alpha }}})^{\prime }_{{\varvec{\alpha }}}\,-\,{\hat{\rho }}^{{\varvec{\alpha }}}\,(\,\psi ^{{\varvec{\alpha }}}\,-\,\frac{\mathrm {1}}{\mathrm {2}}\,{{\mathbf {x}}}^{\prime }_{\pmb {\alpha }}\,\cdot \,{{\mathbf {x}}}^{\prime }_{\pmb {\alpha }})+\,\mathbf {{T}}^{{\varvec{\alpha }}}\cdot \pmb {\mathrm {D}}_{{\varvec{\alpha }}}\,-\,\hat{\mathbf {p}}^{{\varvec{\alpha }}}\,\cdot \,{{\mathbf {x}}}^{\prime }_{\pmb {\alpha }}]\, \ge \mathrm {0}, \end{aligned}$$

(24)

The constitutive equations follow with

$$\begin{aligned} \begin{array}{lcl} \pmb {\mathrm {T}}^{\mathrm {S}}\,+\,\pmb {\mathrm {T}}^{\mathrm {S}\beta }\,+\,\pmb {\mathrm {T}}^{\mathrm {TG}}\,+\,\pmb {\mathrm {T}}^{\mathrm {TG}\beta }\,&{}=&{}\,\mathrm {2}\,\rho ^{\mathrm {S}}\,\pmb {\mathrm {F}}_{\mathrm {S}}\,\displaystyle {\frac{\partial \psi ^{\mathrm {S}}}{\partial \pmb {\mathrm {C}}_{\mathrm {S}}}}\,\pmb {\mathrm {F}}_{\mathrm {S}}^{\mathrm {T}}\,+\,\mathrm {2}\,\rho ^{\mathrm {TG}}\,\pmb {\mathrm {F}}_{\mathrm {S}}\,\displaystyle {\frac{\partial \psi ^{\mathrm {TG}}}{\partial \pmb {\mathrm {C}}_{\mathrm {S}}}}\,\pmb {\mathrm {F}}_{\mathrm {S}}^{\mathrm {T}}\\ &{} &{}\,-\,\,(\mathrm {n}^{\mathrm {S}}\,+\mathrm {n}^{\mathrm {TG}})\,\lambda \,\pmb {\mathrm {I}}\\ \pmb {\mathrm {T}}^{\mathrm {L}}\,&{}=&{}\,-\,\lambda \,\mathrm {n}^{\mathrm {L}}\,\pmb {\mathrm {I}}\,+\,{\rho ^{\alpha \beta }}\,\mathrm{c}^{\alpha \beta }\,\displaystyle {\frac{\partial \psi ^{\alpha \beta }}{\partial \mathrm{c}^{\alpha \beta }}}\,\pmb {\mathrm {I}}\\ \pmb {\mathrm {T}}^{\mathrm {L}\beta }\,&{}=&{}\,-{\rho ^{\alpha \beta }}\,\mathrm {c}^\mathrm{L\beta }\,\displaystyle {\frac{\partial \psi ^{\alpha \beta }}{\partial \mathrm{c}^{\alpha \beta }}}\,\pmb {\mathrm {I}}. \end{array} \end{aligned}$$

(25)

for the description of the stresses.

We postulate a hyperelastic material description of the solid following the Hooke’s law with the Neo-Hookean Helmholtz free energy function

$$\begin{aligned} \begin{array}{lcl} \psi ^{\mathrm {S}}\,&{}=&{}\,\displaystyle {\frac{\mathrm {1}}{\rho ^{\mathrm {S}}_{\mathrm {0S}}}}\,\left[ \lambda ^{\mathrm {S}}\,\frac{\mathrm {1}}{\mathrm {2}}\,(\mathrm {ln}\mathrm {J}_{\mathrm {S}})\,-\,\mu ^{\mathrm {S}}\,\mathrm {ln}\mathrm {J}_{\mathrm {S}}\,+\,\frac{\mathrm {1}}{\mathrm {2}}\,\mu ^{\mathrm {S}}\,(\mathrm {tr}\,\pmb {\mathrm {C}}_{\mathrm {S}}\,-\,\mathrm {3})\right] \\ \psi ^{\mathrm {TG}}\,&{}=&{}\,\displaystyle {\frac{\mathrm {1}}{\rho ^{\mathrm {TG}}_{\mathrm {0S}}}}\,\left[ \lambda ^{\mathrm {TG}}\,\frac{\mathrm {1}}{\mathrm {2}}\,(\mathrm {ln}\mathrm {J}_{\mathrm {S}})\,-\,\mu ^{\mathrm {TG}}\,\mathrm {ln}\mathrm {J}_{\mathrm {S}}\,+\,\frac{\mathrm {1}}{\mathrm {2}}\,\mu ^{\mathrm {TG}}\,(\mathrm {tr}\,\pmb {\mathrm {C}}_{\mathrm {S}}\,-\,\mathrm {3})\right] \end{array} \end{aligned}$$

(26)

Table 1 Material parameters of liver lobule

including the Lamé constants $\lambda ^{\mathrm {S}}$ and $\mu ^{\mathrm {S}}$ for the liver solid and $\lambda ^{\mathrm {TG}}$ and $\mu ^{\mathrm {TG}}$ for the liver fat (Table 1). Moreover we describe the free energy function for the concentrations $\psi ^{\alpha \beta }$ with the general gas constant $\mathrm {R}\,[\frac{\mathrm {J}}{\mathrm {mol}\,\mathrm {K}}]$, the temperature of the mixture $\theta \,[\mathrm {K}]$, the molecular weight of the constituent $\mathrm {M}^{\beta }_\mathrm{mol}\,[\frac{\mathrm {g}}{\mathrm {mol}}]$ and the reference chemical potential $\mu ^{\alpha \beta }_{\mathrm {0}}\,[\frac{\mathrm {J}}{\mathrm {mol}}]$

$$\begin{aligned} \psi ^{\alpha \beta }\,=\displaystyle {\frac{\mathrm {1}}{\mathrm{c}^{\alpha \beta }}}\,\left[ \frac{\mathrm {R}\,\theta }{\mathrm {M}^{\beta }_\mathrm{mol}}\,(\mathrm {ln}(\displaystyle {\frac{\mathrm{c}^{\alpha \beta }}{\mathrm {c}^{\alpha \beta }_{\mathrm {0}}}})\,-\mathrm {1})\,+\,\mu ^{\alpha \beta }_{\mathrm {0}}\,\right] . \end{aligned}$$

(27)