# A Model for Oxygen Transport from Blood in Microessels to Tissue
## 4 Computational Procedure and Results
The value of \(\varphi_{\text{i}}\) is used to get the oxygen partial pressure at the two zones interface  
\[\text{P}_{\text{O}_{2}}(\text{r}_{\text{i}},\text{z})=\text{P}_{\text{O}_{2, \text{b}}}(\text{z})+\varphi_{\text{i}},\text{ }\text{P}_{\text{O}_{2,\text{b}}}(\text{z})=\text{P}_{\text{O}_{2,\text{b}}}( 0)-\text{bz},\text{ }\varphi_{\text{i}}=4\frac{\text{f}_{\text{x}=\lambda}-\text{f}_{\text{x}=1} }{\Big{(}\lambda^{2}-1\Big{)}^{2}},\text{ }\lambda=\text{R}/\text{r}_{\text{i}} \tag{47}\]  
where f is given by  
\[\text{f}(\text{x})=-\frac{\sigma^{4}}{16}\frac{\text{b}\text{A}_{ \text{a}}\text{R}^{4}}{\text{D}_{\text{a}}}\Big{[}\frac{\text{x}^{8}}{8}-\frac {5\lambda^{2}\chi^{6}}{6}+\Big{(}4\lambda^{4}+4\lambda^{2}-1\Big{)}\frac{\chi^ {4}}{\lambda}+\Big{(}-4\lambda^{4}+\lambda^{2}\Big{)}\frac{\chi^{2}}{2}\Big{]} + \tag{48}\] \[\text{K}\Big{[}\frac{\chi^{4}}{4}\Big{(}\ln\chi-\frac{1}{4}\Big{)} -\lambda^{2}\frac{\chi^{2}}{2}\Big{(}\ln\chi-\frac{1}{2}\Big{)}\Big{]}\]  
The case of metabolic consumption of oxygen in an islet of Langerhans is considered in [3]. The data, reported in [3], is a capillary blood flow rate of about 7 nL/min for a tissue perfusion rate q\({}_{\text{b}}\) (capillary blood flow rate over Krogh tissue volume) and about 4 mL/cm\({}^{3}\) min. The data used are listed in Table 1 included the relevant references. The solubilities in Table 1 are calculated using the values in mL O\({}_{2}\)/cm\({}^{3}\) mm Hg and the O\({}_{2}\) molar volume at 36.9 \({}^{\circ}\)C in [33]. Using the data in Table 1, and the above-mentioned value of \(\text{\mu}_{\text{c}}\)/\(\text{\mu}_{\text{a}}\) = 1.947 obtained for \(\sigma\) = 0.9 and H\({}_{\text{T}}\) = 34.2% (H\({}_{\text{D}}\) = 40.0%), yields the results summarized in \[\text{b}=\frac{\left[\left(\frac{\text{r}_{\text{T}}}{\text{R}}\right)^{2}-1 \right]\frac{\text{r}}{\text{\alpha}_{\text{a}}}}{(1-\sigma^{2})\overline{ \text{v}}_{\text{a}}+(1+\overline{\text{m}})\left[\frac{\text{A}_{\text{c}} \text{R}^{2}}{2}\sigma^{4}+\left(\text{v}_{\text{i}}-\text{A}_{\text{c}}\text{ R}^{2}\sigma^{2}\right)\sigma^{2}\right]} \tag{41}\]0 Using the perfusion rate of about 4 mL/cm\({}^{3}\) min used in [3] and the Krogh tissue cylinder volume as a function of R, \(\text{r}_{\text{T}}\) and L, the outer radius of the Krogh cylinder r\({}_{\text{T}}\) is found to be about 24 \(\upmu\)m. \begin{table}
\begin{tabular}{l l l l} \hline \hline
**Property** & **Symbol** & **Value** & **Reference** \\ \hline \multirow{2}{*}{\(\rm D_{a}\)} & Diffusion coefficient & \multirow{2}{*}{2.18 \(\times\) 10\({}^{-5}\) cm\({}^{2}\)/s} & [33] \\  & in plasma & & \\ \hline \multirow{2}{*}{\(\rm D_{RRC}\)} & Diffusion coefficient & \multirow{2}{*}{9.5 \(\times\) 10\({}^{-6}\) cm\({}^{2}\)/s} & [33] \\  & in RBCs & & \\ \hline \multirow{2}{*}{\(\rm D_{T}\)} & Diffusion coefficient & \multirow{2}{*}{2.41 \(\times\) 10\({}^{-5}\) cm\({}^{2}\)/s} & [33] \\  & in tissue & & \\ \hline
2 R & Capillary diameter & 10 \(\mu\)m & Table 5.1 [3] \\ \hline L & Length & 1000 \(\mu\)m & Table 5.1 [3] \\ \hline \multirow{2}{*}{\(\Gamma\)} & Rate of oxygen & \multirow{2}{*}{20 \(\mu\)M/s} & \multirow{2}{*}{Chapter 6 [3]} \\  & consumption in tissue & & \\ \hline \multirow{2}{*}{\(\rm P_{O_{2},b}(0)\)} & Average oxygen & \multirow{2}{*}{95 mm Hg} & \multirow{2}{*}{Chapter 6 [3]} \\  & partial pressure in & & \\  & blood at z = 0 & & \\ \hline \multirow{2}{*}{Q} & Volume flow rate of & \multirow{2}{*}{7 nL/min} & \multirow{2}{*}{Chapter 6 [3]} \\  & blood & & \\ \hline \multirow{2}{*}{\(\rm\alpha_{a}\)} & Oxygen solubility in & 1.110 \(\mu\)M O\({}_{2}\)/mm Hg & \\  & plasma & (2.82 \(\times\) 10\({}^{-5}\) mL O\({}_{2}\)/cm\({}^{3}\) mm Hg) & [33] \\ \hline \multirow{2}{*}{\(\rm\alpha_{T}\)} & Oxygen solubility in & 1.516 \(\mu\)M O\({}_{2}\)/mm Hg & \\  & tissue & (3.85 \(\times\) 10\({}^{-5}\) mL O\({}_{2}\)/cm\({}^{3}\) mm Hg) & [33] \\ \hline \end{tabular}
\end{table} 
Table 1: Data used for oxygen transport from blood flow in a capillary and comparison with (3).