# A Model for Oxygen Transport from Blood in Microessels to Tissue
## Governing Equations
### Mass Transfer Model  
Oxygen is mainly transported by hemoglobin Hb with n molecules bound to Hb. At equilibrium, Hill's equation is used to provide hemoglobin saturation Y as a function of partial pressure of oxygen, P\({}_{O2}\)[3], and Hill's constant n  
\[\rm Y=\frac{P_{O2}^{n}}{P_{50}^{n}+P_{O2}^{n}} \tag{9}\]  
where, at a partial pressure of oxygen equal to p\({}_{50}\), the binding sites of hemoglobin are half filled. The governing equations for dissolved oxygen balance in the core region plasma is given by  
\[\rm v_{c}\frac{\partial c_{c}}{\partial z}=D_{c}\left[\frac{1}{r}\frac{ \partial}{\partial r}\left(r\frac{\partial c_{c}}{\partial r}\right)+\frac{ \partial^{2}c_{c}}{\partial z^{2}}\right]-R_{HDO} \tag{10}\]where \(c_{c}\) and \(D_{c}\) are the free oxygen concentration in the core and the diffusion coefficient in the core region, respectively, and R is the rate of conversion to oxygenated hemoglobin. Neglecting diffusional effects as in [3], the conservation equation of oxygenated hemoglobin can be written as  
\[v_{c}\frac{\partial c_{HbO}}{\partial z}=R_{HbO} \tag{11}\]  
where \(c_{HbO}\) is the hemoglobin-bound oxygen concentration. Adding Equations (10) and (11) and using the chain rule yields [3]  
\[v_{c}\left(1+\frac{\partial c_{HbO}}{\partial c_{c}}\right)\frac{\partial c_{c }}{\partial z}=D_{c}\left[\frac{1}{r}\frac{\partial}{\partial r}\left(r\frac{ \partial c_{c}}{\partial r}\right)+\frac{\partial^{2}c_{c}}{\partial z^{2}} \right],\;m=\frac{\partial c_{HbO}}{\partial c_{c}} \tag{12}\]  
where \(D_{c}\) is estimated using Maxwell equation [3; 38]  
\[\frac{D_{c}}{D_{a}}=\frac{2D_{a}+D_{RBC}-2H_{c}(D_{a}-D_{RBC})}{2D_{a}+D_{RBC}+ H_{c}(D_{a}-D_{RBC})} \tag{13}\]  
We can also express equations using partial pressure \(P_{O_{2,c}}=c_{c}/\alpha_{a}\), where \(\alpha_{a}\) denotes oxygen solubility, to get  
\[v_{c}(1+m)\frac{\partial P_{O_{2,c}}}{\partial z}=D_{c}\left[\frac{1}{r}\frac {\partial}{\partial r}\left(r\frac{\partial\rho_{O_{2,c}}}{\partial r}\right) +\frac{\partial^{2}P_{O_{2,c}}}{\partial z^{2}}\right] \tag{14}\]  
Using [PERSON]'s equation yields [3]  
\[m=n\;p_{50}^{n}\;H\;c_{c,sat}\frac{P_{O_{2}}^{n-1}}{\left(p_{50}^{n}+p_{O_{2} }^{n}\right)^{2}},\;H=P_{O_{2,c}}/c_{c}=1/\alpha_{a} \tag{15}\]  
where \(n\) and \(p_{50}\) are equal to 2.34 and 26 mm Hg, respectively, and \(c_{c,sat}\) is the maximum concentration for oxygen (with four oxygen molecules of oxygen bound and a maximum concentration of hemoglobin in blood of 2200 \(\mu\)M) is 8800 \(\mu\)M [3]. The mass transfer of dissolved oxygen in the annular zone (cell-free layer) is governed by  
\[v_{a}\frac{\partial c_{a}}{\partial z}=D_{a}\left[\frac{1}{r}\frac{\partial} {\partial r}\left(r\frac{\partial c_{a}}{\partial r}\right)+\frac{\partial^{2 }c_{a}}{\partial z^{2}}\right] \tag{16}\]  
where \(c_{a}\) and \(D_{a}\) denote the dissolved oxygen concentration and diffusion coefficient in the annular zone, respectively. Dissolved oxygen levels can be expressed in terms of oxygen partial pressure using Henry's law [3]. The above equation yields  
\[v_{a}\frac{\partial P_{O_{2,a}}}{\partial z}=D_{a}\left[\frac{1}{r}\frac{ \partial}{\partial r}\left(r\frac{\partial P_{O_{2,a}}}{\partial r}\right)+ \frac{\partial^{2}P_{O_{2,a}}}{\partial z^{2}}\right] \tag{17}\]  
In the tissue side, the partial pressure profile in the Krogh model where mass transfer is limited to the Krogh tissue cylinder of radius \(r_{\mathrm{T}}\) surrounding the microvessel [3; 23; 25; 26; 27; 28]. The governing equation is  
\[D_{T}\left[\frac{1}{r}\frac{\partial}{\partial r}\left(r\frac{\partial c_{T}} {\partial r}\right)+\frac{\partial^{2}c_{T}}{\partial z^{2}}\right]-\Gamma=0 \tag{18}\]  
where \(c_{T}\) and \(D_{T}\) are the oxygen dissolved concentration and diffusion coefficient in the tissue side, respectively, and \(\Gamma\) is the rate of consumption of oxygen.