# A Model for Oxygen Transport from Blood in Microessels to Tissue
## 3 Solution  
We seek the following solution  
\[\rm{P_{O_{2},c}}=P_{O_{2},0}+\varphi_{c}-b_{c}z;\;p_{O_{2},a}=P_{O_{2},0}+ \varphi_{a}-b_{a}z;\;p_{O_{2},T}=P_{O_{2},0}+\varphi_{T}-b_{T}z \tag{25}\]  
where oxygen pressures are linear function of z, and \(\varphi_{c}\), \(\varphi_{a}\), and \(\varphi_{T}\) are functions of r. Continuity at the interfaces requires equality of the slopes: \(\rm{b_{c}}=b_{a}=b_{T}=b\), with subscripts dropped. In Equation (25), \(\varphi\) simply represents the deviation of oxygen partial pressures from the average oxygen partial pressure in blood \(\rm{p_{O_{2},b}}(z)=P_{O_{2},b}(0)-b\) z. Substituting the relevant expressions for oxygen partial pressure in Equation (31) into Equations (14) and (17) yields the following ordinary differential equations  
\[-\rm{v_{c}}(1+m)b=D_{c}\frac{1}{r}\frac{d}{dr}\bigg{(}r\frac{d\varphi_{c}}{ dr}\bigg{)} \tag{26}\]  
\[-\rm{v_{a}}b=D_{a}\frac{1}{r}\frac{d}{dr}\bigg{(}r\frac{d\varphi_{a}}{dr} \bigg{)} \tag{27}\]  
subject to the following boundary conditions  
\[\rm{at}\;r=0,\;\varphi_{c}\;is\;finite \tag{28}\]  
\[\rm{at}\;r=r_{i},\;\varphi_{c}=\varphi_{a}=\varphi_{i} \tag{29}\]  
\[\rm{at}\;r=R,\;-D_{a}\alpha_{a}\frac{d\varphi_{a}}{dr}=-D_{T}\alpha_{T}\frac{ d\varphi_{T}}{dr} \tag{30}\]  
Integrating Equations (26) and (27) using boundary conditions (28) and (29) yields  
\[\varphi_{c}-\varphi_{i}=-\frac{(1+m_{\rm{av}})b}{D_{c}}\Bigg{[}\frac{A_{c}}{1 6}\bigg{(}r^{4}-r_{i}^{4}\bigg{)}+\bigg{(}\rm{v_{i}}-A_{c}r_{i}^{2}\bigg{)} \frac{r^{2}-r_{i}^{2}}{4}\Bigg{]} \tag{31}\]  
\[\varphi_{a}-\varphi_{i}=-\frac{bA_{a}}{D_{a}}\Bigg{(}\frac{r^{4}-r_{i}^{4}}{1 6}-R^{2}\frac{r^{2}-r_{i}^{2}}{4}\Bigg{)}+K\ln\frac{r}{r_{i}} \tag{32}\]where \(\rm{m}_{\rm{av}}\) is an average value for \(\rm{m}\). Using boundary condition (30) provides a relationship between the two remaining unknowns \(\rm{b}\) and \(\rm{K}\)  
\[\rm{K}=-\frac{bA_{a}}{D_{a}}\frac{R^{4}}{4}-\frac{\Gamma}{\alpha_{a}D_{a}}\frac{ r_{T}^{2}-R^{2}}{2} \tag{33}\]  
The constant \(\rm{b}\) needed to find the drop in the partial pressure of oxygen per unit change of length in the axial direction, and a mass balance on oxygen is performed in the annular zone between \(\rm{z}\) and \(\rm{z}+\Delta z\)  
\[ \begin{split}\int_{r_{1}}^{R}2\pi r\rm{v}_{a}\big{[}\left(c_{0}+ \alpha_{a}\varphi_{a}-\alpha_{a}bz\right)\big{|}_{\rm{z}}-\left(c_{0}+\alpha_{ a}\varphi_{a}-\alpha_{a}bz\right)\big{|}_{\rm{z}+\Delta z}\big{]}d\rm{r}\\ -2\pi r_{1}\Delta zD_{\rm{c}}\alpha_{a}\left.\frac{d\varphi_{a}}{ dr}\right|_{\rm{z}=r_{i}}-\pi\big{[}r_{T}^{2}-R^{2}\big{]}\Delta z\ \Gamma=0\end{split}  \tag{34}\]  
Using Equations (31) and (32) yields after substitution into Equation (34), integration and rearrangement  
\[\rm{b}=\frac{\left(r_{T}^{2}-R^{2}\right)\Gamma/\alpha_{a}}{\left(R^{2}-r_{i} ^{2}\right)\overline{v}_{a}+\left(1+m_{\rm{av}}\right)\left[\frac{\Delta z}{2 }r_{i}^{4}+\left(v_{i}-A_{\rm{c}}r_{i}^{2}\right)r_{i}^{2}\right]} \tag{35}\]  
In order to determine \(\varphi_{i}\), we apply the overall mass balance to the annular zone between \(0\) and \(\rm{z}\)  
\[ \begin{split}\int_{r_{1}}^{R}2\pi r\rm{v}_{a}c_{0}dr& -\int_{r_{1}}^{R}2\pi r\rm{v}_{a}(c_{0}+\alpha_{a}\varphi_{a}-\alpha_{a}bz)dr \\ -2\pi r_{1}zD_{\rm{c}}\alpha_{a}\left.\frac{d\varphi_{a}}{dr} \right|_{\rm{r}=r_{i}}-\pi\Big{[}r_{T}^{2}-R^{2}\Big{]}z\ \Gamma=0\end{split}  \tag{36}\]  
Using Equation (34) reduces the above equation to  
\[\int_{r_{1}}^{R}r\,v_{a}\,\varphi_{a}dr=0 \tag{37}\]  
Substituting for the expressions of \(\rm{v}_{a}\) and \(\varphi_{a}\) given in Equations (1) and (32), and integrating gives  
\[\varphi_{i}=4\frac{f_{\rm{x}=\lambda}-f_{\rm{x}=1}}{\left(\lambda^{2}-1\right) ^{2}} \tag{38}\]  
where \(\lambda=R/r_{i}=1/\sigma\) and  
\[