A Mathematical Model to Predict CO₂ Removal in Hollow Fiber Membrane Oxygenators

Abstract

A mathematical model has been developed to predict CO₂ removal in hollow fiber membrane oxygenators. The model is analogous to one developed previously for predicting O₂ transfer. A mass transfer correlation was determined in water for O₂ and CO₂ exchange and collapsed onto one universal curve. The correlation was used to predict CO₂ removal in blood by incorporating a ‘facilitated diffusivity’ to account for the transport of CO₂ present as bicarbonate. The diffusion of bicarbonate greatly increased the ability of the oxygenator to remove CO₂ in blood compared to water. A fiber bundle module was fabricated to test the model predictions. The fiber bundle had a length of 13 cm and a bundle thickness of 0.2 cm. The module was tested in bovine blood at flowrates of 0.75, 1.5, and 2.2 L/min and CO₂ removal rate predictions were within 9% of experimental measurements at all flowrates. The O₂ transfer rate predictions were within 10% of experimental measurements. A second module was manufactured with a bundle of length 4 cm and thickness of 1 cm. The CO₂ removal predictions were within the standard deviation of the experimental measurements.

Appendix

CO₂ Transport Equations

The following is a derivation of the convection–diffusion equations for CO₂ in a blood continuum. The flowrate of dissolved CO₂ and bound CO₂ (bicarbonate and CO₂ bound to hemoglobin) due to convection through a differential element, dA can be expressed as:

$$ C_{{\rm CO}_2}^d V_{\rm b} {\cdot}ndA+C_{{\rm CO}_2}^b V_{\rm b} {\cdot}ndA $$

(A.1)

The flowrate of dissolved (free) CO₂ and CO₂ stored as HCO₃ due to diffusion through dA is:

$$ -D_{{\rm CO}_2} \nabla C_{{\rm CO}_2}^d {\cdot}ndA-D_{{\rm HCO}_3} \nabla C_{{\rm HCO}_3}{\cdot}ndA $$

(A.2)

CO₂ bound to hemoglobin will have low diffusion capacity due to the large size of the hemoglobin molecule. The total flowrate of CO₂ through a differential volume element dV as dV → 0 is:

$$ V_{\rm b} {\cdot}\left({\nabla C_{{\rm CO}_2}^d +\nabla C_{{\rm CO}_2}^b } \right)=D_{{\rm CO}_2} \nabla^{2}C_{{\rm CO}_2}^d +D_{{\rm HCO}_3} \nabla ^{2}C_{{\rm HCO}_3} $$

(A.3)

The dissolved component of CO₂ will obey Henry’s law and can be expressed in terms of partial pressure and solubility$(\alpha_{{\rm CO}_2})$:

$$ C_{{\rm CO}_2}^d =\alpha _{{\rm CO}_2} P_{{\rm CO}_2} $$

(A.4)

Using the chain rule and rearranging gives the transport equation for CO₂ in blood:

$$ V_{\rm b} {\cdot}\left({1+\frac{1}{\alpha _{{\rm CO}_2} }\frac{\partial C_{{\rm CO}_2}^b }{\partial P_{{\rm CO}_2} }} \right)\nabla P_{{\rm CO}_2} =\nabla {\cdot}\left({D_{{\rm CO}_2} +\frac{D_{{\rm HCO}_3} }{\alpha _{{\rm CO}_2} }\frac{\partial C_{{\rm HCO}_3}}{\partial P_{{\rm CO}_2} }} \right)\nabla P_{{\rm CO}_2} $$

(A.5)

The right hand side of Eq. (A.5) is the diffusion of dissolved CO₂ and HCO₃ and the left hand side is the convection of CO₂ in all forms. The term in parenthesis on the right hand side is a facilitated diffusion coefficient:

$$ D_{\rm f}=D_{{\rm CO}_2} +\frac{D_{{\rm HCO}_3} }{\alpha _{{\rm CO}_2} }\frac{\partial C_{{\rm HCO}_3} }{\partial P_{{\rm CO}_2}} $$

(A.6)

The convective-diffusion equation for CO₂ can be obtained by dividing the right hand side of Eq. (A.5) by $1+(1/\alpha)(\partial C_{{\rm CO}_2}^b/\partial P_{{\rm CO}_2})$ assuming a constant value for $\partial C_{{\rm CO}_2}^b/\partial P_{{\rm CO}_2}$ (approximated as $\overline {\lambda _{{\rm CO}_2} }$ in the text):

$$ V_{\rm b} {\cdot}\nabla P_{{\rm CO}_2} =D_{eff\_{\rm CO}_2} \nabla ^{2}P_{{\rm CO}_2} $$

(A.7)

The convective-diffusion equation leads to the definition of the effective diffusivity used in the Sc number:

$$ D_{{\rm eff}\_{\rm CO}_2} =\frac{D_{{\rm CO}_2} +\frac{D_{{\rm HCO}_3} }{\alpha _{{\rm CO}_2} }\frac{\partial C_{{\rm HCO}_3} }{\partial P_{{\rm CO}_2} }}{1+\frac{1}{\alpha_{{\rm CO}_2} }\frac{\partial C_{{\rm CO}_2}^b }{\partial P_{{\rm CO}_2}}} $$

(A.8)

Boundary Conditions

The solution to the convective diffusion equation requires a boundary condition at the surface of the fibers. The flux across the fiber wall is due to the diffusion of dissolved CO₂ and bicarbonate ions. As the dissolved CO₂ is removed by the fibers, the bicarbonate is rapidly converted to CO₂ through the action of the enzyme carbonic anhydrase. At the fiber surface, the diffusional flux is:

$$ N_{{\rm CO}_2}=-\alpha _{{\rm CO}_2} D_{\rm f}\nabla P_{{\rm CO}_2}=k_{{\rm CO}_2} (P_{{\rm CO}_2}-P_{{\rm G}\_{\rm CO}_2}) $$

(A.9)

which results in the definition of the Sh number.