Modeling function–perfusion behavior in liver lobules including tissue, blood, glucose, lactate and glycogen by use of a coupled two-scale PDE–ODE approach

Abstract

This study focuses on a two-scale, continuum multicomponent model for the description of blood perfusion and cell metabolism in the liver. The model accounts for a spatial and time depending hydro-diffusion–advection–reaction description. We consider a solid-phase (tissue) containing glycogen and a fluid-phase (blood) containing glucose as well as lactate. The five-component model is enhanced by a two-scale approach including a macroscale (sinusoidal level) and a microscale (cell level). The perfusion on the macroscale within the lobules is described by a homogenized multiphasic approach based on the theory of porous media (mixture theory combined with the concept of volume fraction). On macro level, we recall the basic mixture model, the governing equations as well as the constitutive framework including the solid (tissue) stress, blood pressure and solutes chemical potential. In view of the transport phenomena, we discuss the blood flow including transverse isotropic permeability, as well as the transport of solute concentrations including diffusion and advection. The continuum multicomponent model on the macroscale finally leads to a coupled system of partial differential equations (PDE). In contrast, the hepatic metabolism on the microscale (cell level) was modeled via a coupled system of ordinary differential equations (ODE). Again, we recall the constitutive relations for cell metabolism level. A finite element implementation of this framework is used to provide an illustrative example, describing the spatial and time-depending perfusion–metabolism processes in liver lobules that integrates perfusion and metabolism of the liver.

Appendix

1.1 Kinematics

The saturated porous solid is treated as an immiscible mixture of all constituents $\varphi ^{\varvec{\alpha }}$ with particles $X_{{\varvec{\alpha }}}$ with an own independent motion function

$$\begin{aligned} \mathbf x ={\varvec{\chi }}_{\small {\varvec{\alpha }}}(\mathbf{X}_{\varvec{\alpha }},\mathrm{t})\>,\quad \mathbf{X}_{\varvec{\alpha }}={\varvec{\chi }}_{\small {\varvec{\alpha }}}^{\mathrm{-1}}(\mathbf{x},\mathrm{t})\>. \end{aligned}$$

(40)

where (40)$_1$ represents the Lagrange description of motion. The function ${\varvec{\chi }}_{\small {\varvec{\alpha }}}$ is postulated to be unique and uniquely invertible at any time t. The existence of a function inverse to (40)$_1$ leads to the Euler description of motion, see (40)$_2$. A mathematical condition, which is necessary and sufficient for the existence of equation (40)$_2$, is given if the Jacobian $\mathrm J_{\varvec{\alpha }}=\mathrm{det}\>\mathbf{F}_{\varvec{\alpha }}$ differs from zero. Therein, $\mathbf{F}_{\varvec{\alpha }}$ is the deformation gradient. The tensor $\mathbf{F}_{\varvec{\alpha }}$ and its inverse $\mathbf{F}_{\varvec{\alpha }}^{\mathrm{-1}}$ are defined as $\mathbf{F}_{\varvec{\alpha }}=\partial \mathbf{x}/\partial \mathbf{X}_{\varvec{\alpha }}=\mathrm{Grad}_{{\varvec{\alpha }}}\>{\varvec{\chi }}_{\small {\varvec{\alpha }}}$ and $\mathbf{F}_{\varvec{\alpha }}^{\mathrm{-1}}=\partial \mathbf{X}_{\varvec{\alpha }}/\partial \mathbf{x}= \mathrm{grad}\>\mathbf{X}_{\varvec{\alpha }}$. The differential operator “$\mathrm{Grad}_{{\varvec{\alpha }}}$” is referring to a partial differentiation with respect to the reference position $\mathbf{X}_{\varvec{\alpha }}$ of the constituent $\varphi ^{\varvec{\alpha }}$ and the differential operator $``\mathrm{grad}$” referring to the spatial point $\mathbf{x}$. During the deformation process, $\mathbf{F}_{\varvec{\alpha }}$ is restricted to $\mathrm{det}\>\mathbf{F}_{\varvec{\alpha }}> \mathrm 0$. The spatial velocity gradient $\mathbf{L}_{\varvec{\alpha }}= ( \, \mathrm{Grad}_{\varvec{\alpha }}\>\mathbf{x}^\prime _{\varvec{\alpha }}\, ) \, \mathbf{F}_{\varvec{\alpha }}^\mathrm{-1} = \mathrm{grad}\>\mathbf{x}^\prime _{\varvec{\alpha }}$, where $( \mathbf{F}_{\varvec{\alpha }})^\prime _{\varvec{\alpha }}= \partial \mathbf{x}^\prime _{\varvec{\alpha }}/\partial \mathbf{X}_{\varvec{\alpha }}= \mathrm{Grad}_{\varvec{\alpha }}\>\mathbf{x}^\prime _{\varvec{\alpha }}$ denotes the material velocity gradient, can be additively decomposed into a symmetric part $\mathbf{D}_{\varvec{\alpha }}= ( \, {\mathbf{L}_{{\varvec{\alpha }}}} \, + \, {\mathbf{L}_{{\varvec{\alpha }}}^\mathrm{T}} \, )\,/\, 2$ and a skew-symmetric part ${\mathbf{W}_{{\varvec{\alpha }}}} = ( \, {\mathbf{L}_{{\varvec{\alpha }}}} \, - \, {\mathbf{L}_{{\varvec{\alpha }}}^\mathrm{T}} \, ) \,/\,2$ with $\mathbf{L}_{\varvec{\alpha }}= \mathbf{D}_{\varvec{\alpha }}+ \mathbf{W}_{\varvec{\alpha }}$.

With the Lagrange description of motion (40)$_1$, the velocity and acceleration fields of the constituents $\varphi ^{\varvec{\alpha }}$ are defined as material time derivatives of the motion function (40)$_1$, see Fig. 19,

$$\begin{aligned} \mathbf{x}^{\prime }_{{\varvec{\alpha }}}=\frac{\partial {\varvec{\chi }}_{\small {\varvec{\alpha }}}(\mathbf{X}_{\varvec{\alpha }},\mathrm{t})}{\partial \mathrm{t}}\>,\quad {\mathbf{x}}''_{\varvec{\alpha }}=\frac{\partial ^{\mathrm{2}}{\varvec{\chi }}_{\small {\varvec{\alpha }}}(\mathbf{X}_{\varvec{\alpha }},\mathrm{t})}{\partial \mathrm{t}^{\mathrm{2}}}\>. \end{aligned}$$

(41)

For scalar fields depending on ${\mathbf{x}}$ and t, the material time derivatives are defined as

$$\begin{aligned} (...)^\prime _{\varvec{\alpha }}=\partial (...)/\partial \mathrm{t}+\mathrm{grad}\>(...)\cdot {\mathbf{x}}'_{\varvec{\alpha }}, \end{aligned}$$

(42)

see, e.g., de Boer (2000).

Fig. 19

Motion of solid and fluid particle in a fluid saturated porous solid

1.2 Constitutive theory

With respect to the assumptions made in Sect. 3, the material time derivative of the saturation condition (24) is an equation in excess that restricts the motion of the incompressible constituents. Therefore, the set of unknown field quantities must be extended by a scalar, namely the Lagrange multiplier $\bar{\lambda }$, which is understood as an indeterminate reaction force due to the saturation condition. Thus, the set of unknown field quantities is:

$$\begin{aligned} {\mathcal {U}}=\left\{ \,\varvec{\chi }_{\varvec{\alpha }},\>{\mathrm{c}}^{\alpha \beta },\>\bar{\lambda }\,\right\} \end{aligned}$$

(43)

Considering that the external acceleration $\mathbf{b}$, i.e., the acceleration of gravity, is known, the remaining quantities

$$\begin{aligned} {\mathcal {C}}=\left\{ \,\mathbf{T}^{\varvec{\alpha }},\>\hat{\mathbf{p}}^\mathrm{F},\>\hat{\mathbf{p}}^{\alpha \beta },\>{\hat{\mathrm{c}}}^{\alpha \beta }\,\right\} \end{aligned}$$

(44)

in the set of field equations, see Sect. 4, require constitutive relations in order to close the system of equations. In view of the treatment of the entropy inequality in analogy to Coleman and Noll (1963) the following set of free but not overall independent process variables is chosen:

$$\begin{aligned}&\mathcal {P}=\left\{ \,\mathbf{C}_\mathrm{S},{\mathrm{n}}^{\mathrm{S}},{\mathrm{n}}^{\mathrm{F}},\mathbf{w}_\mathrm{FS},\mathbf{w}_{\mathrm{F} \beta S},{\mathrm{c}}^{\mathrm{S}\beta },\mathrm{c}^{\mathrm{F}\beta },\mathrm{grad}\mathrm{n}^\mathrm{F},\right. \nonumber \\&\quad \left. \mathrm{grad}\mathrm{c}^\mathrm{S\beta },\mathrm{grad}\mathrm{c}^{\mathrm{F}\beta }\right\} \end{aligned}$$

(45)

In (45) the right Cauchy-Green deformation tensor $\mathbf{C}_\mathrm{S}=\mathbf{F}_\mathrm{S}^\mathrm{T}\,\mathbf{F}_\mathrm{S}$ is considered. The entropy inequality for the mixture body

$$\begin{aligned}&\displaystyle \sum _{{\varvec{\alpha }}}^{\mathrm{S,F,\alpha \beta }}\left\{ \,-\,\rho ^{\varvec{\alpha }}\,(\psi ^{\varvec{\alpha }})^\prime _{\varvec{\alpha }}-\hat{\rho }^{\varvec{\alpha }}\,(\,\psi ^{\varvec{\alpha }}-\frac{1}{2}\,\mathbf{x}^\prime _{\varvec{\alpha }}\cdot \mathbf{x}^\prime _{\varvec{\alpha }}\,)\right. \nonumber \\&\quad \left. +\mathbf{T}^{\varvec{\alpha }}\cdot \mathbf{D}_{\varvec{\alpha }}+\hat{\mathrm{e}}^{\varvec{\alpha }}-\hat{\mathbf{p}}^{\varvec{\alpha }}\cdot \mathbf{x}^\prime _{\varvec{\alpha }}\,\right\} \ge \mathrm{0} \end{aligned}$$

(46)

will be rearranged to ensure no neglecting of dependencies which can influence constitutive modeling. In order to keep the complexity of the evaluation in a justifiable scope, the dependency of the Helmholtz free energies $\psi ^{\varvec{\alpha }}$ on the process variables $\mathcal {P}$ will be restricted as follows:

$$\begin{aligned} \psi ^\mathrm{S}=\psi ^\mathrm{S}\,\left\{ \,\mathbf{C}_\mathrm{S}\,\right\} \>,\quad \psi ^\mathrm{F}=\psi ^\mathrm{F}\,\{\,-\,\}\>, \quad \psi ^{\alpha \beta }=\psi ^{\alpha \beta }\,\{\,{{\mathrm{c}}^{\alpha \beta }}\,\}\>.\nonumber \\ \end{aligned}$$

(47)

For further investigations, the relations $\rho ^{\varvec{\alpha }}\,(\psi ^{\varvec{\alpha }})^\prime _{\varvec{\alpha }}$ will be replaced by

$$\begin{aligned} \begin{array}{llllll} \rho ^{\mathrm{S}}\,(\psi ^{\mathrm{S}})^{\prime }_{\mathrm{S}}&{}=&{}\displaystyle 2\,{\mathrm{n}}^{\mathrm{S}}\,\rho ^{\mathrm{SR}}\,\mathbf{F}_{\mathrm{S}}\,\frac{\partial \psi ^{\mathrm{S}}}{\partial \mathbf{C}_{\mathrm{S}}}\,\mathbf{F}_{\mathrm{S}}^{\mathrm{T}}\cdot \mathbf{D}_{\mathrm{S}},\\ \rho ^{\mathrm{F}}\,(\psi ^{\mathrm{F}})^{\prime }_{\mathrm{F}}&{} = &{} 0,\\ \rho ^{\alpha \beta }\,(\psi ^{\alpha \beta })^{\prime }_{\beta }&{}=&{}\displaystyle {\rho ^{\alpha \beta }\,\frac{\partial \psi ^{\alpha \beta }}{\partial {\mathrm{c}}^{\alpha \beta }}\,(\mathrm{c}^{\alpha \beta })^{\prime }_{\beta }}\>. \end{array} \end{aligned}$$

(48)

For saturated porous media consisting of incompressible constituents the saturation condition (20) is a constraint with respect to the overall volumetric deformation. Therefore, the saturation condition must be considered in view of the evaluation of the entropy inequality. Here, the material time derivative of the saturation condition following the motion of the solid phase with

$$\begin{aligned} (\mathrm{n}^{\mathrm{S}})^{\prime }_{\mathrm{S}}+ (\mathrm{n}^{\mathrm{F}})^{\prime }_{\mathrm{S}}= (\mathrm{n}^{\mathrm{S}})^{\prime }_{\mathrm{S}}+ (\mathrm{n}^{\mathrm{F}})^{\prime }_{\mathrm{F}}- \mathrm{grad}\, {\mathrm{n}}^{\mathrm{F}}\cdot \mathbf{w}_{\mathrm{FS}}= \mathrm 0 \end{aligned}$$

(49)

will be considered, where (42) and the difference velocity

$$\begin{aligned} \mathbf{w}_{\mathrm{FS}}= \mathbf{x}^{\prime }_{\mathrm{F}}- \mathbf{x}^{\prime }_{\mathrm{S}}\end{aligned}$$

(50)

are used. Applying the identity

$$\begin{aligned} \begin{array}{lll} (\mathrm{n}^{\alpha })^{\prime }_{\alpha }&{} = &{} \displaystyle {\frac{{\mathrm{n}}^{\alpha }}{\rho ^{\alpha }}} \, (\rho ^{\alpha })^{\prime }_{\alpha }- \displaystyle {\frac{{\mathrm{n}}^{\alpha }}{\rho ^{\alpha \mathrm{R}}}} \, (\rho ^{\alpha })^{\prime }_{\alpha }\\ &{} = &{} - \, {\mathrm{n}}^{\alpha }\, ( \, \mathbf{D}_{\alpha }\, \cdot \, \mathbf{I}\, ) -\displaystyle {\frac{{\mathrm{n}}^{\alpha }}{\rho ^{\alpha \mathrm{R}}}}\,\underbrace{(\rho ^{\alpha \mathrm{R}})^{\prime }_{\alpha }}_\mathrm{=\,0},\\ &{} = &{} - \, {\mathrm{n}}^{\alpha }\, ( \, \mathbf{D}_{\alpha }\, \cdot \, \mathbf{I}) \end{array} \end{aligned}$$

(51)

the rate of the saturation condition (49) can be rearranged to

$$\begin{aligned} {\mathrm{n}}^{\mathrm{S}}\, ( \, \mathbf{D}_{\mathrm{S}}\cdot \mathbf{I}\, ) + {\mathrm{n}}^{\mathrm{F}}\, ( \, \mathbf{D}_{\mathrm{F}}\cdot \, \mathbf{I}\, ) + \mathrm{grad}\, {\mathrm{n}}^{\mathrm{F}}\, \cdot \, \mathbf{w}_{\mathrm{FS}}= \mathrm 0. \end{aligned}$$

(52)

In view of the constitutive modeling, we use the concept of Lagrange multipliers. Using the identity $\mathrm{div}\>\mathbf{x}^{\prime }_{\alpha }=\mathbf{D}_{\alpha }\cdot \mathbf{I}$, we add (21)$_\mathrm{1,2}$ to the entropy inequality (46), multiplied by the Lagrange multiplier $\bar{\lambda }$, cf. de Boer (1996). The interconnection between the spatial velocity deformation gradients $\mathbf{D}_{{\varvec{\alpha }}}$ with the volume fractions and their material time derivative ${\mathrm{n}}^{\alpha }$ and $(\mathrm{n}^{\alpha })^{\prime }_{\alpha }$ as well as the mass supplies $\hat{\rho }^{\alpha }$ will be considered with the balance equations of mass (14)$_1$ multiplied with a respective Lagrange multiplier:

$$\begin{aligned}&\bar{\lambda }^\mathrm{S}\,\left[ \,(\mathrm{n}^{\mathrm{S}})^{\prime }_{\mathrm{S}}+{\mathrm{n}}^{\mathrm{S}}\>(\mathbf{D}_{\mathrm{S}}\cdot \mathbf{I})\,\right] =\mathrm{0}\nonumber \\&\bar{\lambda }^\mathrm{F}\,\left[ \,(\mathrm{n}^{\mathrm{F}})^{\prime }_{\mathrm{F}}+{\mathrm{n}}^{\mathrm{F}}\>(\mathbf{D}_{\mathrm{F}}\cdot \mathbf{I})\,\right] =\mathrm{0}\nonumber \\&{\bar{\lambda }}^{\mathrm{S}\beta }\,\left[ \,(\mathrm{n}^{\mathrm{S}})^{\prime }_{\mathrm{S}}\,{\mathrm{c}}^{\mathrm{S}\beta }+{\mathrm{n}}^{\mathrm{S}}\,({\mathrm{c}}^{\mathrm{S} \beta })^{\prime }_\mathrm{S}+{\mathrm{n}}^{\mathrm{S}}\,{\mathrm{c}}^{\mathrm{S}\beta }\,\mathbf{D}_{\mathrm{S}}\cdot \mathbf{I}-\hat{\mathrm{c}}^{\mathrm{S} \beta }\right] =\mathrm{0}\nonumber \\&\bar{\lambda }^\mathrm{F\beta }\,\left[ \,(\mathrm{n}^{\mathrm{F}})^{\prime }_{\mathrm{F}}\,\mathrm{c}^{\mathrm{F}\beta }+{\mathrm{n}}^{\mathrm{F}}\,(\mathrm{c}^{\mathrm{F} \beta })^{\prime }_{\beta }+\mathrm{grad}\>{\mathrm{n}^{\mathrm{F}}}(\mathbf{w}_{\mathrm{F}\beta \mathrm{S}}-\mathbf{w}_{\mathrm{FS}})\,\mathrm{c}^{\mathrm{F}\beta }\right. \nonumber \\&\quad \left. +\,{\mathrm{n}}^{\mathrm{F}}\,\mathrm{c}^{\mathrm{F}\beta }\,\mathbf{D}_{\mathrm{F} \beta }\cdot \mathbf{I}-{\hat{\mathrm{c}}}^{\mathrm{F} \beta }\right] =\mathrm{0}\nonumber \\&\quad -\,\bar{\lambda }\,\left[ \,-{\mathrm{n}}^{\mathrm{S}}\,\mathbf{D}_{\mathrm{S}}\cdot \mathbf{I}-{\mathrm{n}}^{\mathrm{F}}\,\mathbf{D}_{\mathrm{F}}\cdot \mathbf{I}-\mathrm{grad}\>{\mathrm{n}^{\mathrm{F}}}\,\mathbf{w}_{\mathrm{FS}}\right] =\mathrm{0} \end{aligned}$$

(53)

The entropy inequality (46), supplemented with the material time derivative of the saturation condition (53)$_5$ and the balance equations of mass (53)$_{1-4}$, all multiplied with the corresponding Lagrange multiplier, reads

$$\begin{aligned}&\mathbf{D}_{\mathrm{S}}\>\cdot \left\{ \,\displaystyle {\mathbf{T}^{\mathrm{S}}+\mathbf{T}^\mathrm{S \beta }-\mathrm{2}\,{{\mathrm{n}}^{\mathrm{S}}\,\rho ^{\mathrm{SR}}}\,{\mathbf{F}_{\mathrm{S}}}\,{\frac{\partial \psi ^{\mathrm{S}}}{\partial \mathbf{C}_{\mathrm{S}}}}\,{\mathbf{F}_{\mathrm{S}}^{\mathrm{T}}}}\right. \nonumber \\&\left. \quad \,+\displaystyle {\bar{\lambda }\,{\mathrm{n}}^{\mathrm{S}}\,\mathbf{I}\,+\bar{\lambda }^\mathrm{S}\,{\mathrm{n}}^{\mathrm{S}}\,\mathbf{I}\,+{\bar{\lambda }}^{\mathrm{S}\beta }{\mathrm{c}}^{\mathrm{S}\beta }\,{\mathrm{n}}^{\mathrm{S}}\,\mathbf{I}}\,\right\} \,\nonumber \\&\quad +\,\mathbf{D}_{\mathrm{F}}\>\cdot \left\{ \,\displaystyle {\mathbf{T}^{\mathrm{F}}}+\displaystyle {\bar{\lambda }\,{\mathrm{n}}^{\mathrm{F}}\,\mathbf{I}}+\displaystyle {\bar{\lambda }^\mathrm{F}\,{\mathrm{n}}^{\mathrm{F}}\,\mathbf{I}}\,\right\} \,\nonumber \\&\quad +\,\displaystyle {\mathbf{D}_{\mathrm{F} \beta }}\>\cdot \left\{ \,\displaystyle {\mathbf{T}^\mathrm{F \beta }+\displaystyle {\bar{\lambda }^\mathrm{F\beta }\,{\mathrm{n}}^{\mathrm{F}}\,\mathrm{c}^{\mathrm{F}\beta }\,\mathbf{I}}}\,\right\} \,\nonumber \\&\quad -\,(\mathrm{n}^{\mathrm{S}})^{\prime }_{\mathrm{S}}\left\{ \, -\bar{\lambda }^\mathrm{S}-{\bar{\lambda }}^{\mathrm{S}\beta }\,{\mathrm{c}}^{\mathrm{S}\beta }\right\} \,-(\mathrm{n}^{\mathrm{F}})^{\prime }_{\mathrm{F}}\left\{ \, -\bar{\lambda }^\mathrm{F}-\bar{\lambda }^\mathrm{F\beta }\,\mathrm{c}^{\mathrm{F}\beta }\right\} \,\nonumber \\&\quad -\,\displaystyle {(\mathrm{c}^{\mathrm{F}\beta })^{\prime }_{\beta }}\left\{ \,\displaystyle {\rho ^{\mathrm{F} \beta }\,\frac{\partial \psi ^{\mathrm{F} \beta }}{\partial \mathrm{c}^{\mathrm{F}\beta }}-\bar{\lambda }^\mathrm{F\beta }\,{\mathrm{n}}^{\mathrm{F}}} \right\} \,\nonumber \\&\quad -\,\displaystyle {({\mathrm{c}}^{\mathrm{S}\beta })^{\prime }_{\beta }}\left\{ \,\displaystyle {\rho ^{\mathrm{S} \beta }\,\frac{\partial \psi ^{\mathrm{S} \beta }}{\partial {\mathrm{c}}^{\mathrm{S}\beta }}-{\bar{\lambda }}^{\mathrm{S}\beta }\,{\mathrm{n}}^{\mathrm{S}}} \right\} \,\nonumber \\&\quad -\,\hat{\rho }^{\mathrm{S} \beta }\displaystyle {\left\{ \psi ^{\mathrm{S} \beta }-\frac{\mathrm{1}}{\mathrm{2}}\,\mathbf{x}^{\prime }_{\mathrm{S}}\cdot \mathbf{x}^{\prime }_{\mathrm{S}}+\frac{{\bar{\lambda }}^{\mathrm{S}\beta }}{\mathrm{M}^{\beta }_\mathrm{mol}}\right\} }\,\nonumber \\&\quad -\,\hat{\rho }^{\mathrm{F} \beta }\displaystyle {\left\{ \psi ^{\mathrm{F} \beta }-\frac{\mathrm{1}}{\mathrm{2}}\,\mathbf{x}^{\prime }_\mathrm{F \beta }\cdot \mathbf{x}^{\prime }_\mathrm{F \beta }+\frac{\bar{\lambda }^\mathrm{F\beta }}{\mathrm{M}^{\beta }_\mathrm{mol}}\right\} }\,\nonumber \\&\quad -\,\displaystyle {\mathbf{w}_{\mathrm{FS}}\>\cdot }\left\{ \,\displaystyle {{\hat{\mathbf{p}}}^{\mathrm{F}}-\bar{\lambda }\,\,\mathrm{grad}\>{\mathrm{n}^{\mathrm{F}}}+\bar{\lambda }^\mathrm{F\beta }\,\mathrm{c}^{\mathrm{F}\beta }\,\mathrm{grad}\>{\mathrm{n}^{\mathrm{F}}}}\,\right\} \,\nonumber \\&\quad -\,\displaystyle {\mathbf{w}_{\mathrm{F}\beta \mathrm{S}}\>\cdot }\left\{ \,\displaystyle {{\hat{\mathbf{p}}}^{\mathrm{F}\beta }-\bar{\lambda }^\mathrm{F\beta }\,\mathrm{c}^{\mathrm{F}\beta }\,\mathrm{grad}\>{\mathrm{n}^{\mathrm{F}}}}\,\right\} \,\ge 0\>, \end{aligned}$$

(54)

where use has been made of $\sum \rho ^{\alpha \beta }= 0$ and ${\hat{\mathbf{p}}}^{\mathrm{S}}=-{\hat{\mathbf{p}}}^{\mathrm{F}}-{\hat{\mathbf{p}}}^{\mathrm{S}\beta }-{\hat{\mathbf{p}}}^{\mathrm{F}\beta }$, see (19). The inequality must hold for fixed values of the process variables, see (45), and for arbitrary values of the so-called free-available quantities $\mathcal {A}=\displaystyle {\{\mathbf{D}_{{\varvec{\alpha }}},\>(\mathrm{n}^{\alpha })^{\prime }_{\alpha },\>(\mathrm{c}^{\alpha \beta })^{\prime }_{\alpha \beta }\}}$ which contains selective derivatives of the values contained in $\mathcal {P}$. Thus, the entropy inequality (54) can be satisfied if the following structure is obtained:

$$\begin{aligned}&\displaystyle {\mathbf{D}_{\mathrm{S}}\cdot \left\{ \underbrace{(\ldots )}_{=\,\mathbf{0}}\right\} +\mathbf{D}_{\mathrm{F}}\cdot \left\{ \underbrace{(\ldots )}_{=\,\mathbf{0}}\right\} +\mathbf{D}_{\mathrm{F} \beta }\cdot \left\{ \underbrace{(\ldots )}_{=\,\mathbf{0}}\right\} }\,\nonumber \\&-(\mathrm{n}^{\mathrm{S}})^{\prime }_{\mathrm{S}}\,\left\{ \underbrace{(\ldots )}_{=\,\mathrm{0}}\right\} -(\mathrm{n}^{\mathrm{F}})^{\prime }_{\mathrm{F}}\,\left\{ \underbrace{(\ldots )}_{=\,\mathrm{0}}\right\} \,\\&-\displaystyle {(\mathrm{c}^{\mathrm{S} \beta })^{\prime }_\mathrm{S\beta }\,\left\{ \underbrace{(\ldots )}_{=\,\mathrm{0}}\right\} \,-(\mathrm{c}^{\mathrm{F} \beta })^{\prime }_\mathrm{F\beta }\,\{\underbrace{(\ldots )}_{=\,\mathrm{0}}\}\,+}\,\,\displaystyle {\underbrace{\mathcal {D}is}_{\ge {\mathrm{\,0}}}\,}\,\ge 0\>,\nonumber \end{aligned}$$

(55)

where the dissipation mechanism

$$\begin{aligned} \mathcal {D}is&= -\hat{\rho }^{\mathrm{S} \beta }\displaystyle {\left\{ \psi ^{\mathrm{S} \beta }-\frac{\mathrm{1}}{\mathrm{2}}\,\mathbf{x}^{\prime }_{\mathrm{S}}\cdot \mathbf{x}^{\prime }_{\mathrm{S}}+\frac{{{\bar{\lambda }}^{\mathrm{S}\beta }}}{\mathrm{M}^{\beta }_\mathrm{mol}}\right\} }\,\nonumber \\&-\hat{\rho }^{\mathrm{F} \beta }\displaystyle {\left\{ \psi ^{\mathrm{F} \beta }-\frac{\mathrm{1}}{\mathrm{2}}\,\mathbf{x}^{\prime }_\mathrm{F \beta }\cdot \mathbf{x}^{\prime }_\mathrm{F \beta }+\frac{\bar{\lambda }^\mathrm{F\beta }}{\mathrm{M}^{\beta }_\mathrm{mol}}\right\} }\,\\&-\displaystyle {\mathbf{w}_{\mathrm{FS}}\>\cdot }\left\{ \,\displaystyle {{\hat{\mathbf{p}}}^{\mathrm{F}}-\bar{\lambda }\,\,\mathrm{grad}\>{\mathrm{n}^{\mathrm{F}}}+\bar{\lambda }^\mathrm{F\beta }\,\mathrm{c}^{\mathrm{F}\beta }\,\mathrm{grad}\>{\mathrm{n}^{\mathrm{F}}}}\,\right\} \,\nonumber \\&-\displaystyle {\mathbf{w}_{\mathrm{F}\beta \mathrm{S}}\>\cdot }\{\,\displaystyle {{\hat{\mathbf{p}}}^{\mathrm{F}\beta }-\bar{\lambda }^\mathrm{F\beta }\,\mathrm{c}^{\mathrm{F}\beta }\,\mathrm{grad}\>{\mathrm{n}^{\mathrm{F}}}\,\}}\,\ge \,0\nonumber \end{aligned}$$

(56)

must hold. Considering the aforementioned remarks, we obtain necessary and sufficient conditions for the unrestricted validity of the second law of thermodynamics. The Lagrange multipliers associated with the solutes can be identified from (55), terms 4–7, as

$$\begin{aligned} \begin{array}{ll} \bar{\lambda }^{\alpha }&{} = -\bar{\lambda }^{\alpha \beta }\,{\mathrm{c}}^{\alpha \beta },\\ \bar{\lambda }^{\alpha \beta }&{} = \displaystyle \frac{\rho ^{\alpha \beta }}{{\mathrm{n}}^{\alpha }}\, \frac{\partial \psi ^{\alpha \beta }}{\partial {\mathrm{c}}^{\alpha \beta }}= {\mathrm{c}}^{\alpha \beta }\,{\mathrm{M}}^{\beta }_\mathrm{mol}\, \frac{\partial \psi ^{\alpha \beta }}{\partial {\mathrm{c}}^{\alpha \beta }}. \end{array} \end{aligned}$$

(57)

Thus, we obtain from (55), terms 1–3 for the Cauchy stress tensors

$$\begin{aligned} \mathbf{T}^\mathbf{S}&= \mathbf{T}^{\mathrm{S}}+ \mathbf{T}^\mathrm{S\beta }\nonumber \\&= \!\displaystyle {-\bar{\lambda }\,{\mathrm{n}}^{\mathrm{S}}\,\mathbf{I}\,\underbrace{-\bar{\lambda }^\mathrm{S}\,{\mathrm{n}}^{\mathrm{S}}\,\mathbf{I}\,\!-\!{\bar{\lambda }}^{\mathrm{S}\beta }{\mathrm{c}}^{\mathrm{S}\beta }\,{\mathrm{n}}^{\mathrm{S}}\,\mathbf{I}}_\mathrm{= 0}\!+\!2\,\rho ^{\mathrm{S}}\,\mathbf{F}_{\mathrm{S}}\,{\displaystyle {\frac{\partial \psi ^{\mathrm{S}}}{\partial \mathbf{C}_{\mathrm{S}}}}}\,\mathbf{F}_{\mathrm{S}}^{\mathrm{T}}\,}\nonumber \\&= -\,\bar{\lambda }\,{\mathrm{n}}^{\mathrm{S}}\,\mathbf{I} +\underbrace{2\,\rho ^{\mathrm{S}}\,\mathbf{F}_{\mathrm{S}}\,{\displaystyle {\frac{\partial \psi ^{\mathrm{S}}}{\partial \mathbf{C}_{\mathrm{S}}}}}\,\mathbf{F}_{\mathrm{S}}^{\mathrm{T}}}_{\mathbf{T}^\mathbf{S}_\mathrm{E}} \>, \nonumber \\ \mathbf{T}^{\mathrm{F}}&= +\displaystyle {\bar{\lambda }\,{\mathrm{n}}^{\mathrm{F}}\,\mathbf{I}}+\displaystyle \bar{\lambda }^\mathrm{F}\,{\mathrm{n}}^{\mathrm{F}}\,\mathbf{I}\nonumber \\&= -\,\bar{\lambda }\,{\mathrm{n}}^{\mathrm{F}}\,\mathbf{I}+ {\mathrm{n}}^{\mathrm{F}}\, (\mathrm{c}^{\mathrm{F}\beta })^2 \mathrm M^{\beta }_\mathrm{mol}\,\displaystyle {\frac{\partial \psi ^{\mathrm{F} \beta }}{\partial \mathrm{c}^{\mathrm{F}\beta }}} \,\mathbf{I}\>,\nonumber \\ \mathbf{T}^\mathrm{F \beta }&= -\bar{\lambda }^\mathrm{F\beta }\,{\mathrm{n}}^{\mathrm{F}}\,\mathrm{c}^{\mathrm{F}\beta }\,\mathbf{I}\nonumber \\&= -\,{\mathrm{n}}^{\mathrm{F}}\,(\mathrm{c}^{\mathrm{F}\beta })^2\,\mathrm M^{\beta }_\mathrm{mol}\, \displaystyle \frac{\partial \psi ^{\mathrm{F} \beta }}{\partial \mathrm{c}^{\mathrm{F}\beta }}\,\mathbf{I}\nonumber \\&= -{\mathrm{n}}^{\mathrm{F}}\,\mu ^{\mathrm{F} \beta }\,\mathbf{I}\>, \end{aligned}$$

(58)

using (57)$_\mathrm{1, 2}$ and the abbreviations $\mathbf{T}^\mathbf{S} = \mathbf{T}^{\mathrm{S}}+ \mathbf{T}^\mathrm{S\beta }$ for the overall solid stress and $\mu ^{\mathrm{F} \beta }=(\mathrm{c}^{\mathrm{F}\beta })^2\,\mathrm M^{\beta }_\mathrm{mol}\, (\partial \psi ^{\mathrm{F} \beta }/ \partial \mathrm{c}^{\mathrm{F}\beta })$ for the chemical potential. Moreover, the Lagrange multiplier $\bar{\lambda }$ is understood as the reaction force between the solid and fluid phase, which can be identify as the pore pressure $\lambda$ with $\bar{\lambda }=\lambda$, cf. eg. Bluhm (2002). In view of the dissipative mechanism (56), the following approaches for the interaction forces ${\hat{\mathbf{p}}}^{\mathrm{F}}$ and ${\hat{\mathbf{p}}}^{\mathrm{F}\beta }$ and for the mass supply $\rho ^{\alpha \beta }$are postulated:

$$\begin{aligned} {\hat{\mathbf{p}}}^{\mathrm{F}}&= \,\bar{\lambda }\,\mathrm{grad}\>{\mathrm{n}^{\mathrm{F}}}-\bar{\lambda }^\mathrm{F\beta }\,\mathrm{c}^{\mathrm{F}\beta }\,\mathrm{grad}\>{\mathrm{n}^{\mathrm{F}}}-\alpha _{\mathbf{w}_{\mathrm{FS}}}\,\mathbf{w}_{\mathrm{FS}}\\&+\alpha _{\mathbf{w}_{\mathrm{FF}\beta }}\,\mathbf{w}_{\mathrm{FF}\beta }\>,\\ {\hat{\mathbf{p}}}^{\mathrm{F}\beta }&= \,\bar{\lambda }^\mathrm{F\beta }\, \mathrm{c}^{\mathrm{F}\beta }\,\mathrm{grad}\>{\mathrm{n}^{\mathrm{F}}}- \alpha _{\mathbf{w}_{\beta \mathrm{S}}}\,\mathbf{w}_{\beta \mathrm{S}}-\alpha _{\mathbf{w}_{\mathrm{FF}\beta }}\,\mathbf{w}_{\mathrm{FF}\beta }\>,\\ \hat{\rho }^\mathrm{FLc}&= \,-\delta ^\mathrm{FLc}(\Psi ^\mathrm{FLc}-\Psi ^\mathrm{SGy})\>,\\ \hat{\rho }^\mathrm{FGu}&= \,-\delta ^\mathrm{FGu}(\Psi ^\mathrm{FGu}-\Psi ^\mathrm{SGy})\>, \end{aligned}$$

(59)

with the abbreveation $\Psi ^{\alpha \beta }=\psi ^{\alpha \beta }-\displaystyle \mathrm{{1}}/{2} \, {\mathbf{x}^{\prime }_{\alpha \beta }}\cdot {\mathbf{x}^{\prime }_{\alpha \beta }}+ \displaystyle {\bar{\lambda }^{\alpha \beta }}/{{\mathrm{M}}^{\beta }_\mathrm{mol}}$. The factors connected to the relative permeabilities $\alpha _{\mathbf{w}_{\mathrm{FS}}}$, $\alpha _{\mathbf{w}_{\mathrm{FF}\beta }}$, $\alpha _{\mathbf{w}_{\beta \mathrm{S}}}$ and connected to the mass supply terms $\delta ^\mathrm{FLc}$, $\delta ^\mathrm{FGu}$ are material parameters depending on the set $\mathcal {U}$ given in (43) and are restricted to positive values ($\ge 0$).

Regarding the liver lobe level (microperfusion scale), we receive constitutive restrictions with respect to the concentration exchange of lactate and glucose from (59)$_{3,4}$ with

$$\begin{aligned} \hat{\rho }^\mathrm{FLc}&= \mathrm{M}^\mathrm{Lc}_\mathrm{mol}\,\hat{{\mathrm{c}}}^\mathrm{FLc}= -\delta ^\mathrm{FLc}(\Psi ^\mathrm{FLc}-\Psi ^\mathrm{SGy})\>,\\ \hat{\rho }^\mathrm{FGu}&= \mathrm{M}^\mathrm{Gu}_\mathrm{mol}\,\hat{{\mathrm{c}}}^\mathrm{FGu}= -\delta ^\mathrm{FGu}(\Psi ^\mathrm{FGu}-\Psi ^\mathrm{SGy})\>, \end{aligned}$$

(60)

where $\Psi ^{\alpha \beta }=\psi ^{\alpha \beta }-\mathrm{{1}}/2 \, \mathbf{x}^{\prime }_{\alpha \beta }\cdot \mathbf{x}^{\prime }_{\alpha \beta }+ (\bar{\lambda }^{\alpha \beta }/{\mathrm{M}}^{\beta }_\mathrm{mol})$. We assume now that the concentration exchange depends neither on kinetic energy $\mathrm 1/2 \,\, \mathbf{x}^{\prime }_{\alpha \beta }\cdot \mathbf{x}^{\prime }_{\alpha \beta }$ nor on the pressure state $\bar{\lambda }^{\alpha \beta }$. Thus, the concentration exchange rates reduce with (18) to

$$\begin{aligned} \hat{{\mathrm{c}}}^\mathrm{FLc}&= \displaystyle \frac{\hat{\rho }^\mathrm{FLc}}{\mathrm{M}^\mathrm{Lc}_\mathrm{mol}} = \displaystyle -\frac{\delta ^\mathrm{FLc}}{\mathrm{M}^\mathrm{Lc}_\mathrm{mol}}(\psi ^\mathrm{FLc}-\psi ^\mathrm{SGy})\>,\nonumber \\ \hat{{\mathrm{c}}}^\mathrm{FGu}&= \displaystyle \frac{\hat{\rho }^\mathrm{FGu}}{\mathrm{M}^\mathrm{Gu}_\mathrm{mol}} = \displaystyle -\frac{\delta ^\mathrm{FGu}}{\mathrm{M}^\mathrm{Gu}_\mathrm{mol}}(\psi ^\mathrm{FGu}-\psi ^\mathrm{SGy})\>,\\ \hat{{\mathrm{c}}}^\mathrm{SGy}&= -\hat{{\mathrm{c}}}^\mathrm{FLc}-\hat{{\mathrm{c}}}^\mathrm{FGu}.\nonumber \end{aligned}$$

(61)

1.3 Derivation of field equations

Considering that the solid and fluid carrier phases are treated to be material incompressible ($(\rho ^{\alpha \mathrm{R}})^{\prime }_{\alpha }=0$), see Sec. 3, leads to the simplification

$$\begin{aligned} (\rho ^{\alpha })^{\prime }_{\alpha }=({\mathrm{n}}^{\alpha }\,\rho ^{\alpha \mathrm{R}})'_S=(\mathrm{n}^{\alpha })^{\prime }_{\alpha }\,\rho ^{\alpha R} + {\mathrm{n}}^{\alpha }\,\underbrace{(\rho ^{\alpha R})}_{=0} = (\mathrm{n}^{\alpha })^{\prime }_{\alpha }\,\rho ^{\alpha R}.\nonumber \\ \end{aligned}$$

(62)

Incorporating additionally that mass exchanges are only acts between the solute ($\hat{\rho }^{\alpha }=0$), the balance equation of mass (14)$_1$ reduce for the solid and fluid carrier phases to

$$\begin{aligned} (\mathrm{n}^{\mathrm{S}})^{\prime }_{\mathrm{S}}+{\mathrm{n}}^{\mathrm{S}}\>\mathrm{div}\>\mathbf{x}^{\prime }_{\mathrm{S}}=\mathrm 0\,,\quad (\mathrm{n}^{\mathrm{F}})^{\prime }_{\mathrm{F}}+{\mathrm{n}}^{\mathrm{F}}\>\mathrm{div}\>\mathbf{x}^{\prime }_{\mathrm{F}}=\mathrm 0. \end{aligned}$$

(63)

In addition, with (11) the mass for the solutes is given with

$$\begin{aligned} \begin{array}{lll} {\mathrm{m}}^{\beta }(\mathbf{x},\mathrm{t}) &{} = &{}\displaystyle \int _\mathrm{M_{{\varvec{\beta }}}}\,\mathrm d{\mathrm{m}}^{\beta }\\ &{} = &{}\displaystyle \int _{\mathrm{B}_{\varvec{\alpha }}} {\mathrm{c}}^{\alpha \beta }\,{\mathrm{M}}^{\beta }_\mathrm{mol}\,{\mathrm{d}\mathrm{v}}^{{\alpha }}\\ &{} = &{}\displaystyle \int _{\mathrm{B}_{\varvec{\alpha }}} {\mathrm{n}}^{\alpha }\,{\mathrm{c}}^{\alpha \beta }\,{\mathrm{M}}^{\beta }_\mathrm{mol}\,{\mathrm{d}\mathrm{v}}, \end{array} \end{aligned}$$

(64)

which directly leads to the solute balance equation

$$\begin{aligned} \left( {\mathrm{n}}^{\alpha }\,{\mathrm{c}}^{\alpha \beta }\,{\mathrm{M}}^{\beta }_\mathrm{mol}\right) '_{\alpha \beta }+{\mathrm{n}}^{\alpha }\,{\mathrm{c}}^{\alpha \beta }\,{\mathrm{M}}^{\beta }_\mathrm{mol}\>\mathrm{div}\>\mathbf{x}^{\prime }_{\alpha \beta }=\hat{\rho }^{\alpha \beta }. \end{aligned}$$

(65)

Further, we consider that ${\mathrm{M}}^{\beta }_\mathrm{mol}$ is a constant of the substance $\varphi ^{\beta }$ and applies for the time derivatives for scalar fields (42). Thus, the solute balance equation of mass (65) yields with (13) to

$$\begin{aligned} (\mathrm{n}^{\alpha })^{\prime }_\mathrm{S}\,{\mathrm{c}}^{\alpha \beta }+{\mathrm{n}}^{\alpha }\,(\mathrm{c}^{\alpha \beta })^{\prime }_\mathrm{S}+ \mathrm{div}\left( \mathbf{j}^{\alpha \beta }\right) + {\mathrm{n}}^{\alpha }\,{\mathrm{c}}^{\alpha \beta }\,\mathrm{div}\>\mathbf{x}^{\prime }_{\mathrm{S}}= {\hat{\mathrm{c}}}^{\alpha \beta }.\nonumber \\ \end{aligned}$$

(66)

Herein,

$$\begin{aligned} \mathbf{j}^{\alpha \beta }\!=\! {\mathrm{c}}^{\alpha \beta }\,{\mathrm{n}}^{\alpha }\,\mathbf{w}_{\alpha \beta \mathrm{S}}\!=\! -\mathrm{D}_{\alpha \beta }\,[-{\mathrm{n}}^{\alpha }\,\mathrm{grad}{\mathrm{c}}^{\alpha \beta }]\!+\!\hat{\mathrm{D}}_{\alpha \beta }\,\mathbf{w}_{\alpha \mathrm{S}}\end{aligned}$$

(67)

denotes the molar flux, with $\mathrm{D}_\mathrm{\alpha \beta }$ as the diffusion coefficient (Fick’s part), $\hat{\mathrm{D}}_{\alpha \beta }$ as the advective coefficient (advective part) and $\mathbf{w}_{\alpha \beta \mathrm{S}}=\mathbf{x}^{\prime }_{\alpha \beta }-\mathbf{x}^{\prime }_{\mathrm{S}}$ as the velocity of the solute $\varphi ^{\alpha \beta }$ solved in the carrier phase $\varphi ^{\alpha }$ relative to the solid velocity.

Fig. 20

Developing different angel $\theta _{\mathrm{ap}}^\mathrm{t}$ in the $\mathbf a_\mathrm{0}\times \mathbf p_\mathrm{0}$ plain spanned by the pressure gradient $\mathbf p_\mathrm{0}$ and preferred flow direction $\mathbf a_\mathrm{0}$ or $\mathbf a_\mathrm{0}'$

1.4 Evolutionary approach for preferred flow direction

The stationary solution for the anisotropic flow is achieved following the phenomenological approach that sinusoids tend to be oriented in the direction of the pressure gradient $\mathbf{\mathbf p_\mathrm{0}} =\Vert \mathrm{Grad}\,\bar{\lambda }\Vert$ with $\mathbf{|\mathbf p_\mathrm{0}|} = \mathrm{1}$; see Fig. 20. Thus, starting from an arbitrary preferred flow direction $\mathbf{a}_\mathrm{0}$ with $| \mathbf{a}_\mathrm{0} | = \mathrm 1$ an updated preferred flow direction $\mathbf{a}_\mathrm{0}'$ is calculated with the relation

$$\begin{aligned} \mathbf{a}_\mathrm{0}' = \mathbf{a}_\mathrm{0} + \triangle \mathbf{a}_\mathrm{0}' \quad \mathrm{with} \quad \mathbf{a}_\mathrm{0} \cdot \triangle \mathbf{a}_\mathrm{0} = \mathrm 0 \end{aligned}$$

(68)

where $\triangle \mathbf{a}_\mathrm{0}'$ denotes the incremental update of the preferred flow direction $\mathbf{a}_\mathrm{0}$Werner et al. (2012). The incremental update can be expressed by a rigid body rotation of $\mathbf a_\mathrm{0}$ by $\triangle \mathbf a_\mathrm{0}= \varvec{\omega }\times \mathbf a_\mathrm{0}\,$ where $\varvec{\omega }$ denotes the angular velocity of reorientation. Since $\varvec{\omega }$ is perpendicular to the plane spanned by $\mathbf a_\mathrm{0}$ and $\mathbf p_\mathrm{0}$ the condition ${\varvec{\omega }} = \frac{\delta _\mathrm{t}}{\delta _\mathrm{d}} ( \mathbf a_\mathrm{0}\times \mathbf p_\mathrm{0})\,$ must hold with $\delta _\mathrm{t}$ denoting a virtual damping coefficient with respect to the time-dependent model and $\delta _\mathrm{d} = |\mathbf a_\mathrm{0}-\mathbf p_\mathrm{0}|$ weighting the distance between the preferred flow direction and the pressure gradient. Thus, the incremental update reads

$$\begin{aligned} \triangle \mathbf a_\mathrm{0}= \frac{\delta _\mathrm{t}}{\delta _\mathrm{d}} ( [\mathbf a_\mathrm{0}\times \mathbf p_\mathrm{0}] \times \mathbf a_\mathrm{0}). \end{aligned}$$

(69)

1.5 Supplemental information model reduction

${V}_\mathrm{liver}$	1.5 l	liver volume in full kinetic model
${V}_\mathrm{sim}$		liver volume for simulation in liter
${V}_\mathrm{f}$	$\displaystyle \frac{{V}_\mathrm{sim}}{{V}_\mathrm{liver}}$	Relative volume for adaptions of model fluxes and properties like the glycogen storage capacity to the actual liver volume
${C}_\mathrm{HGU}, {C}_\mathrm{GLY}$		Coefficients for polynomial approximation of quasi-steady state
${m}_\mathrm{bw}$	75 kg	Body weight used for conversion of fluxes given per body weight in concentration changes
${k}_\mathrm{z} = {k}_\mathrm{x}$	0.05 mM	Saturation parameter accounting for substrate limitation: HGU and glycolysis are limited by the available $\varphi ^{\mathrm{FGu}}$. HGP and gluconeogenesis are limited by available gluconeogenic substrate (handles the border conditions for integration in the polynoms)

$$\begin{aligned} {x}&= \varphi ^{\mathrm{FGu}}\,y=\displaystyle \frac{\varphi ^{\mathrm{SGy}}}{V_f},\,z=\varphi ^{\mathrm{FLc}}\\ C_\mathrm{HGU}&= \begin{pmatrix} 0.002037960420379\\ -0.000367490977632\\ -0.069301032419012\\ -0.000002823120484\\ 0.011282864074433\\ 3.740276159346358\\ -0.000000181515364\\ 0.000157328485157\\ -0.100050917438436\\ -18.414978834613287\\ \end{pmatrix}\\ {C}_\mathrm{GLY}&= \begin{pmatrix} -0.013260401508103\\ -0.000078240970095\\ 0.478235644004833\\ 0.000002861605817\\ 0.000932752106971\\ -2.492569641130055\\ 0.000000166945924\\ -0.000125285017396\\ 0.015354944655784\\ -4.975026288067225\\ \end{pmatrix} \\ \end{aligned}$$

$$\begin{aligned}&{p}_{\mathrm{HGU}}(\varphi ^{\mathrm{FGu}},\varphi ^{\mathrm{SGy}},\varphi ^{\mathrm{FLc}})\nonumber \\&\quad =\displaystyle \frac{{V}_{\mathrm{f}}\,{{m}}_{\mathrm{bw}}}{60\cdot 10^{-3}}\cdot ({C}_{\mathrm{HGU}}[1]\cdot {{x}}^3+{{C}}_{\mathrm{HGU}}[2]\cdot {x}^2\mathrm{y}\nonumber \\&\quad +\,{C}_{\mathrm{HGU}}[3]\cdot {x}^2 +{C}_{\mathrm{HGU}}[4]\cdot {xy}^2+{C}_{\mathrm{HGU}}[5]\cdot {xy}\nonumber \\&\quad +\,{C}_{\mathrm{HGU}}[6]\cdot {x}+{{C}}_{\mathrm{HGU}}[7]\cdot {y}^3 +{C}_{\mathrm{HGU}}[8]\cdot {y}^2\nonumber \\&\quad +\,{C}_{\mathrm{HGU}}[9]\cdot {y} +{C}_{\mathrm{HGU}}[10])\end{aligned}$$

(70)

$$\begin{aligned}&{v}_{\mathrm{HGU}}(\varphi ^{\mathrm{FGu}},\varphi ^{\mathrm{SGy}},\varphi ^{\mathrm{FLc}})\nonumber \\&\quad ={p}_{\mathrm{HGU}}(\varphi ^{\mathrm{FGu}},\varphi ^{\mathrm{SGy}},\varphi ^{\mathrm{FLc}})\cdot {\left\{ \begin{array}{ll} \displaystyle \frac{{z}}{{z}+{k}_{\mathrm{z}}} \quad \forall \,{p}_{\mathrm{HGU}}<0\\ \displaystyle \frac{{x}}{{ x+k}_{\mathrm{x}}} \quad \forall \,{p}_{\mathrm{HGU}}\ge 0\\ \end{array}\right. }\end{aligned}$$

(71)

$$\begin{aligned}&{p}_{\mathrm{GLY}}(\varphi ^{\mathrm{FGu}},\varphi ^{\mathrm{SGy}},\varphi ^{\mathrm{FLc}})\nonumber \\&\quad =\displaystyle \frac{{V}_\mathrm{f}\,\mathrm{m}_\mathrm{bw}}{60\cdot 10^{-3}}\cdot ({C}_\mathrm{GLY}[1]\cdot \mathrm{x}^3+\mathrm{C}_\mathrm{GLY}[2]\cdot { x}^{2}{ y+C}_\mathrm{GLY}[3]\cdot {x}^2\nonumber \\&\quad +\,{ C}_\mathrm{GLY}[4]\cdot {xy}^{2}+{C}_\mathrm{GLY}[5]\cdot {xy+C}_\mathrm{GLY}[6]\cdot {x+C}_\mathrm{GLY}[7]\cdot { y}^3\nonumber \\&\quad +\,{C}_\mathrm{GLY}[8]\cdot {y}^{2}+{C}_\mathrm{GLY}[9]\cdot {y+C}_\mathrm{GLY}[10])\end{aligned}$$

(72)

$$\begin{aligned}&{v}_\mathrm{GLY}(\varphi ^{\mathrm{FGu}},\varphi ^{\mathrm{SGy}},\varphi ^{\mathrm{FLc}})\nonumber \\&\quad ={p}_\mathrm{GLY}(\varphi ^{\mathrm{FGu}},\varphi ^{\mathrm{SGy}},\varphi ^{\mathrm{FLc}})\cdot {\left\{ \begin{array}{ll} \displaystyle \frac{{z}}{{z+k}_\mathrm{z}} \quad \forall \,{p}_\mathrm{GLY}<0\\ \displaystyle \frac{{x}}{{x+k}_\mathrm{x}} \quad \forall \,{p}_\mathrm{GLY}\ge 0\\ \end{array}\right. }\end{aligned}$$

(73)

$$\begin{aligned}&{ v}_\mathrm{GS}(\varphi ^{\mathrm{FGu}},\varphi ^{\mathrm{SGy}},\varphi ^{\mathrm{FLc}})\nonumber \\&\quad ={v}_\mathrm{HGU}(\varphi ^{\mathrm{FGu}},\varphi ^{\mathrm{SGy}},\varphi ^{\mathrm{FLc}})-{\mathrm{v}}_\mathrm{GLY}(\varphi ^{\mathrm{FGu}},\varphi ^{\mathrm{SGy}},\varphi ^{\mathrm{FLc}})\end{aligned}$$

(74)

$$\begin{aligned}&\displaystyle \frac{\mathrm{d}\,\varphi ^{\mathrm{FGu}}}{\mathrm{dt}}=\mathrm{-v}_{\mathrm{HGU}}(\varphi ^{\mathrm{FGu}}, \varphi ^{\mathrm{SGy}}, \varphi ^{\mathrm{FLc}})= \hat{{\mathrm{c}}}^\mathrm{FGu}, \nonumber \\&\displaystyle \frac{\mathrm{d}\,\varphi ^{\mathrm{SGy}}}{\mathrm{dt}}=\mathrm{-v}_{\mathrm{HGU}}(\varphi ^{\mathrm{FGu}}, \varphi ^{\mathrm{SGy}}, \varphi ^{\mathrm{FLc}})= \hat{{\mathrm{c}}}^\mathrm{FGu}, \\&\displaystyle \frac{\mathrm{d}\,\varphi ^{\mathrm{FLc}}}{\mathrm{dt}}= 2\cdot \mathrm{v}_{\mathrm{GLY}}(\varphi ^{\mathrm{FGu}}, \varphi ^{\mathrm{SGy}}, \varphi ^{\mathrm{FLc}})= \hat{{\mathrm{c}}}^\mathrm{FLc}.\nonumber \end{aligned}$$

(75)