## 3 Results
### Dependence on Perfusion  
The solution of Equation (12) with boundary conditions Equations (13) and (14) is a quadratic function. Thus, in the active (dermal) layer  
Figure 3: The dependence of the epidermal temperature drop \(\Delta T_{e}\) as a function of the passive layer thickness \(E\) for two dermal temperatures/ambient temperature gradients: (\(T_{d}-T_{\infty}\)) 15 \({}^{\circ}\)C (solid red) and 10 \({}^{\circ}\)C (dashed blue).  
\[T\ =\ -\frac{Qz^{2}}{2k_{d}}+C_{1}z+C_{2} \tag{23}\]  
From the boundary conditions on the lower boundary Equation (13), we can find that \(C_{1}\ =\ -\rho_{b}C_{b}\omega_{b}\big{(}T_{c}-T_{p}\big{)}/k_{d}\) and \(C_{2}\ =\ T_{p}\).  
Then, substituting Equation (23) into the boundary condition on the upper boundary Equation (14), we can write  
\[T|_{z\ =\ D}\ =\ T_{d}\ =\ T_{p}-\frac{\rho_{b}C_{b}\omega_{b}\big{(}T_{c}-T_{p} \big{)}L}{k_{d}}D-\frac{Q}{2k_{d}}D^{2} \tag{24}\]  
\[-k_{d}\frac{dT}{dz}\bigg{|}_{z\ =\ D}\ =\ h(T_{skin}-T_{\infty})\ =\ QD+\rho_{b}C_{b}\omega_{b}\big{(}T_{c}-T_{p}\big{)}L \tag{25}\]  
The skin temperature \(T_{skin}\) can be linked to the dermal temperature \(T_{d}\) using Equation (20). In particular, Equation (25) can be transformed into  
\[T_{d}\ =\ T_{\infty}+\frac{QD+\rho_{b}C_{b}\omega_{b}\big{(}T_{c}-T_{p} \big{)}L}{h}(1+hE/k_{e}) \tag{26}\]  
We can solve Equations (24) and (26) for \(T_{p}\)  
\[T_{p}\ =\ T_{c}-\frac{T_{c}-T_{\infty}-\frac{QD}{h}\left(1+\frac{hE}{k_{e}} \right)-\frac{Q}{2k_{d}}D^{2}}{1+\omega_{b}/\bar{\omega}} \tag{27}\]  
Here \(\widetilde{\omega}^{-1}\ =\ \frac{\rho_{b}C_{b}L}{h}\Big{(}1+\frac{hD}{k_{d}}+ \frac{hE}{k_{e}}\Big{)}\).  
However, \(T_{p}\) is not observable, so it does not have a high practical value. By substituting \(T_{p}\) into Equation (25), we can obtain an explicit expression for the observable \(T_{skin}\).  
\[T_{skin}\ =\ T_{\infty}+\frac{QD+\rho_{b}C_{b}L\omega_{b}\frac{T_{c}-T_{ \infty}-\frac{QD}{h}\left(1+\frac{hE}{k_{e}}\right)-\frac{Q}{2k_{d}}D^{2}}{1+ \omega_{b}/\bar{\omega}}}{h} \tag{28}\]  
This expression can be estimated in two important limits:  
* In the case of very low perfusion (\(\omega_{b}\)\(\rightarrow\)0), we have \(T_{skin}\ =\ T_{\infty}+\frac{QD}{h}\)
* In the case of very high perfusion (\(\omega_{b}\)\(\rightarrow\)\(\infty\)), we have \(T_{skin}\ =\ T_{c}-(T_{c}-T_{\infty})\Big{(}\frac{hD}{k_{d}}+\frac{hE}{k_{e}} \Big{)}\)  
We can estimate factors in Equations (27) and (28). Using realistic assumptions, we can get \(\frac{QD}{h}\ =\ 0.033\ ^{\circ}\)C, \(\frac{QD^{2}}{2k_{d}}\ =\ 7.5\times 10^{-4}\ ^{\circ}\)C, \(\frac{hD}{k_{d}}\ =\ 0.045\), \(\frac{hE}{k_{e}}\ =\ 0.0057\), 0.086, and 0.11 for the dry hairy skin, dry glabrous epidermis, and 2 mm callus, respectively.  
Based on Equation (20), we can write an explicit expression for the temperature drop across the skin (_Delta_):  
\[Delta\ =\ (T_{c}-T_{\infty})\frac{1+\omega_{b}/\ \widetilde{\Omega}}{1+ \omega_{b}/\bar{\omega}} \tag{29}\]  
Here \(\widetilde{\Omega}^{-1}\ =\ \rho_{b}C_{b}L\Big{(}\frac{D}{k_{d}}+\frac{E}{k_{e}} \Big{)}\). The dependence of temperature drop across the skin (_Delta_) on perfusion for several ambient temperatures is depicted in Figure 4. To maintain this temperature drop constant, the body needs to adjust the perfusion rate in the broad range. The calculated perfusion rate to keep _Delta_ at 3.5 \({}^{\circ}\)C as a function of the ambient temperature, \(T_{\infty}\) is shown on insert in Figure 4.  
Knowing _Delta_, the perfusion rate can be explicitly expressed  
\[\omega_{b}\ =\ (\frac{T_{c}-T_{\infty}}{Delta}-1)\ /(\frac{1}{\widetilde{ \omega}}-\frac{T_{c}-T_{\infty}}{Delta}\frac{1}{\widetilde{\Omega}}) \tag{30}\]