# A Model for Oxygen Transport from Blood in Microessels to Tissue
## 4 Computational Procedure and Results
\begin{table}
\begin{tabular}{l l} \hline \hline
**Symbol** & **Value** \\ \hline \multirow{2}{*}{\(\rm v^{*}\)} & 0.149 cm/s \\ \hline \multirow{2}{*}{\(\rm r_{T}\)} & 24 \(\mu\)m \\ \hline \multirow{2}{*}{\(\rm A_{c}R^{2}\)} & \multirow{2}{*}{\(-\)0.224 cm/s} \\  & & \\ \hline \multirow{2}{*}{\(\rm A_{a}R^{2}\)} & \multirow{2}{*}{\(-\)0.436 cm/s} \\  & & \\ \hline \multirow{2}{*}{\(\rm v_{i}\)} & 0.0829 cm/s \\ \hline \multirow{2}{*}{\(\rm v_{a}\)} & 0.0415 cm/s \\ \hline \multirow{2}{*}{\(\rm b\)} & 210.88 mm Hg/cm \\ \hline \multirow{2}{*}{\(\rm K\)} & \multirow{2}{*}{\(-\)2.013 mmHg\(\rm m\)Hg} \\  & & \\ \hline \multirow{2}{*}{\(\rm D_{c}\)} & \multirow{2}{*}{\(1.60\times 10^{-5}\) cm\({}^{2}\)/s} \\  & & \\ \hline \multirow{2}{*}{\(\rm bA_{a}R^{4}\)/\(\rm D_{a}\)} & \multirow{2}{*}{\(-\)1.06 mmHg\(\rm m\) Hg} \\  & & \\ \hline \multirow{2}{*}{\(\rm bA_{c}R^{4}\)/\(\rm D_{c}\)} & \multirow{2}{*}{\(-\)0.740 mmHg\(\rm m\) Hg} \\  & & \\ \hline \multirow{2}{*}{\(\rm TR^{2}/(\rm\alpha_{T}D_{T})\)} & 0.137 mmHg\(\rm m\) Hg \\ \hline \multirow{2}{*}{\(\rm\varphi_{i}\)} & 0.0840 mmHg\(\rm m\) Hg \\ \hline \(\rm\varphi_{a}\) at R & \(-\)0.1555 mmHg\(\rm m\) Hg \\ \hline \multirow{2}{*}{\(\rm\varphi_{T}\) at \(\rm r_{T}\)} & \(-\)1.875 mmHg\(\rm m\) Hg \\ \hline \multirow{2}{*}{\(\rm P_{O_{2},b}\) at z = L} & 73.91 mmHg\(\rm m\) Hg \\ \hline \multirow{2}{*}{\(\rm m_{av}\)} & 12.33 \\ \hline \end{tabular}
\end{table} 
Table 2: Results for oxygen transport using Table 1 for \(\rm\mu_{c}/\mu_{a}\) = 1.947 and \(\sigma\) = 0.9. (HD = 40.0%). Using Equations (45) and (46) provides the values for \(\varphi\) (r = 0) and \(\varphi\) (R). Combining Equations (46) and Equation (45) (applied at r = R) gives the following expression for \(\varphi_{\mathrm{T}}\):  
\[ \begin{array}{c}\varphi_{\mathrm{T}}(\mathrm{r})=\mathrm{P}_{\mathrm{O}_{2} \mathrm{r}}(\mathrm{r},\mathrm{z})-\mathrm{P}_{\mathrm{O}_{2}\mathrm{b}}( \mathrm{z})=\varphi_{\mathrm{i}}+\frac{1}{4}\frac{\Gamma\,\mathrm{R}^{2}}{ \alpha_{\mathrm{T}}\mathrm{D}_{\mathrm{T}}}\left[\xi^{2}-1\right]-\frac{ \Gamma\,\mathrm{R}^{2}}{\alpha_{\mathrm{T}}\mathrm{D}_{\mathrm{T}}}\frac{\xi ^{2}_{\mathrm{T}}}{2}\ln(\xi)\\ -\frac{\mathrm{b}\mathrm{A}\mathrm{A}\mathrm{R}^{4}}{\mathrm{D}_{\mathrm{a}}} \left[\frac{1}{16}\left(1-\sigma^{4}\right)-\frac{1}{4}\left(1-\sigma^{2} \right)\right]+\mathrm{K}\ln\left(\frac{1}{\sigma}\right)\end{array}  \tag{49}\]  
The results obtained in Table 2 (using the data in Table 1) are used to plot the oxygen partial pressures profiles in Figure 2. As expected, the oxygen partial pressure profiles show a drop in the z direction due to consumption of oxygen in the tissue. The average oxygen partial pressure in blood is higher than the oxygen partial pressure at the outer radius of the Krogh cylinder due to mass transfer resistance to oxygen transport from blood to tissue. The results compare favorably with those in [3], as seen from Figure 2.  
The case of anoxia starts when the oxygen pressure at rT is zero at z = L (critical case):  
\[ \begin{array}{c}\mathrm{P}_{\mathrm{O}_{2}}(\mathrm{r}_{\mathrm{T}})=\mathrm{ P}_{\mathrm{O}_{2}\mathrm{b}}(0)-\mathrm{b}\mathrm{L}+\varphi_{\mathrm{i}}+ \frac{1}{4}\frac{\Gamma\,\mathrm{R}^{2}}{\alpha_{\mathrm{T}}\mathrm{D}_{ \mathrm{T}}}\left[\xi^{2}_{\mathrm{T}}-1\right]-\frac{\Gamma\,\mathrm{R}^{2}}{ \alpha_{\mathrm{T}}\mathrm{D}_{\mathrm{T}}}\frac{\xi^{2}_{\mathrm{T}}}{2}\ln( \xi_{\mathrm{T}})\\ -\frac{\mathrm{b}\mathrm{A}_{\mathrm{a}}\mathrm{R}^{4}}{\mathrm{D}_{\mathrm{a}} }\left[\frac{1}{16}\left(1-\sigma^{4}\right)-\frac{1}{4}\left(1-\sigma^{2} \right)\right]+\mathrm{K}\ln\left(\frac{1}{\sigma}\right)=0,\ \xi_{\mathrm{T}}=\mathrm{r}_{\mathrm{T}}/ \mathrm{R}\end{array}  \tag{50}\]  
In vitro experiments providing the required data for comparison with the present model are recommended for future investigations.