The concentration of nanocarriers in the viable epidermis is governed by the following equation:

\[ \frac{\partial C_{NP, VE}}{\partial t} = D_{NP, VE} \nabla^2 C_{NP, VE} - \nabla \cdot (v_{is} C_{NP, VE}) - k_{rel, VE} C_{NP, VE} \tag{1} \]

For free drugs, the transport is described by:

\[ \frac{\partial C_{FD, VE}}{\partial t} = D_{FD, VE} \nabla^2 C_{FD, VE} - \nabla \cdot (v_{is} C_{FD, VE}) + k_{rel, VE} C_{NP, VE} - \frac{V_{max} C_{FD, VE}}{v_m + C_{FD, VE}} - k_{d, VE} C_{FD, VE} \tag{2} \]

The transport of nanocarriers in the papillary dermis is governed by the equation:

\[ \frac{\partial C_{NP, PD}}{\partial t} = D_{NP, PD} \nabla^2 C_{NP, PD} - \nabla \cdot (v_{is} C_{NP, PD}) - k_{rel, PD} C_{NP, PD} - Ex(C_{NP, BL}, C_{NP, PD}) - Fly C_{NP, PD} \tag{3} \]

The free drug transport in the papillary dermis is described by:

\[ \frac{\partial C_{FD, PD}}{\partial t} = D_{FD, PD} \nabla^2 C_{FD, PD} - \nabla \cdot (v_{is} C_{FD, PD}) + k_{rel, PD} C_{NP, PD} - \frac{V_{max} C_{FD, PD}}{v_m + C_{FD, PD}} - k_{AR} C_{FD, PD} + k_{DR} C_{BD, PD} - Ex(C_{FD, BL}, C_{FD, PD}) - Fly C_{FD, PD} - k_{d, PD} C_{FD, PD} \tag{4} \]