# Altered Mechano-Electrochemical Behavior of Articular Cartilage in Populations with Obesity
## 2 Material and Methods
### Flow Equations  
Outgoing and/or incoming water and ion fluxes are considered in the mathematical formulation through the mass balance equation for the total mixture and the charge balance for each ion. The main fluxes present in the model are water flux (\(\mathbf{J}^{\mathrm{w}}\)), cation flux (\(\mathbf{J}^{+}\)) and anion flux (\(\mathbf{J}^{-}\)). The flux equations are detailed below.  
Mass balance of the mixture:  
\[\
abla\cdot\mathbf{v}^{\mathbf{s}}+\
abla\cdot\mathbf{J}^{\mathrm{w}}=0 \tag{1}\]  
Charge balance of ions:  
\[\frac{\partial\left(\Phi^{\mathrm{w}}c^{+}\right)}{\partial t}+\
abla\cdot \underbrace{\mathbf{J}^{+}}_{\text{diffusion}}+\
abla\cdot\underbrace{ \left(\[PERSON]^{\mathrm{w}}c^{+}\mathbf{v}^{\mathbf{s}}\right)}_{\text{convection}}=0, \tag{2}\]  
\[\frac{\partial\left(\[PERSON]^{\mathrm{w}}c^{-}\right)}{\partial t}+\
abla\cdot \underbrace{\mathbf{J}^{-}}_{\text{diffusion}}+\
abla\cdot\underbrace{ \left(\[PERSON]^{\mathrm{w}}c^{-}\mathbf{v}^{\mathbf{s}}\right)}_{\text{convection}}=0, \tag{3}\]  
where \(\mathbf{v}^{\mathbf{s}}=\frac{\partial\mathbf{u}^{\mathbf{s}}}{\partial t}\) is the velocity of each point of the solid matrix; \(c^{+}\) and \(c^{-}\) are cation and anion concentrations, respectively; and \(\Phi^{\mathrm{w}}\) is the porosity of the tissue [26]. Hereafter, the fluxes can be mathematically expressed as a function of the electrochemical potentials:  
\[\mathbf{J}^{\mathrm{w}}=-\frac{R\ T\Phi^{\mathrm{w}}}{\alpha}\left(\
abla \varepsilon^{\mathrm{w}}+\frac{c^{+}}{\varepsilon^{+}}\
abla\varepsilon^{+}+ \frac{c^{-}}{\varepsilon^{-}}\
abla\varepsilon^{-}\right) \tag{4}\]  
\[\mathbf{J}^{+}=-\frac{R\ T\[PERSON]^{\mathrm{w}}c^{+}}{\alpha}\
abla\varepsilon^{ \mathrm{w}}-\left[\frac{\[PERSON]^{\mathrm{w}}c^{+}D^{+}}{\varepsilon^{+}}+\frac{ R\ T\[PERSON]^{\mathrm{w}}\left(c^{+}\right)^{2}}{\alpha\varepsilon^{+}}\right] \
abla\varepsilon^{+}-\frac{R\ T\[PERSON]^{\mathrm{w}}c^{+}c^{-}}{\alpha \varepsilon^{+}}\
abla\varepsilon^{-}, \tag{5}\]  
\[\mathbf{J}^{-}=-\frac{R\ T\[PERSON]^{\mathrm{w}}c^{-}}{\alpha}\
abla\varepsilon^{ \mathrm{w}}-\left[\frac{\[PERSON]^{\mathrm{w}}c^{-}D^{-}}{\varepsilon^{-}}+\frac{ R\ T\[PERSON]^{\mathrm{w}}\left(c^{-}\right)^{2}}{\alpha\varepsilon^{-}}\right] \
abla\varepsilon^{-}-\frac{R\ T\[PERSON]^{\mathrm{w}}c^{+}c^{-}}{\alpha \varepsilon^{+}}\
abla\varepsilon^{+}, \tag{6}\]  
where \(\alpha\) is the drag coefficient between the solid and the water phases; \(R\) is the universal gas constant; \(T\) is the absolute temperature; and \(D^{+}\) and \(D^{-}\) are the cation and anion diffusivities, respectively [27]. The electrochemical potentials are defined as follows:  
\[\varepsilon^{\mathrm{w}}=\frac{P}{R\ T}-\Phi\left(c^{+}+c^{-}\right)+\frac{B_ {\mathrm{w}}}{R\ T}\theta, \tag{7}\]  
\[\varepsilon^{+}=\gamma^{+}c^{+}exp\left(\frac{F_{\mathrm{c}}\psi}{R\ T} \right), \tag{8}\]  
\[\varepsilon^{-}=\gamma^{-}c^{-}exp\left(-\frac{F_{\mathrm{c}}\psi}{R\ T}\right) \tag{9}\]  
where \(\Phi\) represents the osmotic coefficient, \(B_{\mathrm{w}}\) is the fluid-solid coupling coefficient, \(P\) is the fluid pressure, \(\theta=div\ \mathbf{u}^{\mathbf{s}}\) is the expansion of the solid matrix related to the infinitesimal strain tensor of the solid matrix, \(F_{\mathrm{c}}\) is the Faraday constant and \(\psi\) the electrical potential. \(\gamma^{+}\) and \(\gamma^{-}\) refer to the cation and anion activity coefficients, respectively.