Blood chemistry plays a significant role in facilitating the transport of gases between blood and alveoli. To understand something of this facilitation, we first consider a simple model for carbon dioxide transport that takes the carbon dioxide-bicarbonate chemistry into account. We assume that carbon dioxide is converted to bicarbonate via the reaction
This is the carbonic anhydrase reaction discussed in Section 16.2.3. For convenience we ignore here the intermediary H2CO3. Since the dissociation of H2CO3 into HCO-and H+ is fast, this makes no difference to the model.
Now we write conservation equations for the two chemical species CO2 and HCO-(in steady state, and ignoring diffusion within the capillary) as v^- = Dco2(°CO2Pco2 - U) + k_i[H+]V - kiU (17.11)
dU dx dV
dx where U = [CO2], V = [HCO-]. Notice that DCO2 isarate constant, similarto Dm above.
Although this is a linear problem and it can be solved exactly, it is illustrative to use an approximate, singular perturbation technique, as this technique will prove useful in the next section. First notice that we can add (17.11) and (17.12) to obtain d v-(U + V) = Dco2(OCO2PCO2 - U). (17.13)
Now we assume that V equilibrates rapidly, so that it can be taken to be in quasi-steady state. Accordingly, we set V = KcU, where Kc = k [H+]. It follows that, assuming that y, we sei v = kcu, where kc = k
This equation is identical in form to (17.6). If we take the inlet conditions to be U = U0 = aCO2P0 and V = V0 = KcU0, then the total flux Q is
Q = v(1 + KcKo2 (Po - PCO2) (1 - e-Dco2L/(v(1+Kc»), (17.15)
which is a factor of 1 + Kc larger than in (17.10). The only difference between this flux (17.15) and the original (17.8) is that the velocity v has been multiplied by the factor 1 + Kc. In other words, the conversion of carbon dioxide to bicarbonate via the carbonic anhydrase reaction effectively increases the flow rate by the factor 1 + Kc.
The equilibrium constant for the bicarbonate-carbon dioxide reaction is given by log10(k^) = -6.1. Thus (since pH = — log10[H+] with [H+] in moles per liter), at pH = 7.4, we have Kc = 20, and the improvement in carbon dioxide transport because of the carbonic anhydrase reaction is substantial.
In words, the improvement in total flux arises because the conversion of bicarbonate to carbon dioxide continually replenishes the carbon dioxide that is lost to the alveolar air. Thus, the carbon dioxide concentration in the capillary does not fall so quickly, leading to an increase in the total flux.
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