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Full disclosure (if you won't be able to tell by the end of my question): I'm a biologist, not a physicist but I'm finding myself with a rather physics-intensive question that I imagine the bright minds here can help me with.

I am interested in calculating the diffusive flux rate of a gas given the following conditions:

Imagine a volume of animal tissue 100 $cm^2$ in area and 300 $μ m$ deep. On one side is seawater with a dissolved PO2 of 21 kPa and a PCO2 of 40 Pa. On the other side of the tissue is a volume of blood with PO2 of 12 kPa and a PCO2 of 300 Pa. We can treat the other four sides of the tissue as impermeable to all gases. We can also treat the seawater and blood as open systems where gases do not accumulate. Temperature is 20 °C.

So given Fick's first law of diffusion (parameterized for respiration physiology) $$ Diffusion\:rate = K \times \frac{surface\,area}{tissue\,thickness} \times (P_{O_2}\,seawater - P_{O_2}\,blood) $$

where $K$ is Krogh's diffusion coefficient. At 20 °C, $K$ for $O_2$ is 3.03 $μ mol\: O_{2}\cdot cm^{-2}\cdot\mu m\cdot kPa^{-1}\cdot hr^{-1}$ (Krogh, 1919).

Here's where things get trickier than I know how to handle myself. As $O_2$ diffuses through the tissue (from both the seawater and blood sides) the tissue is converting $O_2$ to $CO_2$ (0.8 moles of $CO_2$ produced per 1 mole $O_2$ consumed). This conversion occurs at a constant total rate of 30 $μmol\,O_2$ consumed per hour and is uniform throughout the entire tissue volume.

$CO_2$ has a much higher diffusion coefficient in tissue than $O_2$. The Krogh's diffusion coefficient for $CO_2$ at 20 °C is 108.31 $μ mol\: CO_{2}\cdot cm^{-2}\cdot\mu m\cdot kPa^{-1}\cdot hr^{-1}$.

I am trying to solve for the net flux rate of $CO_2$ into the blood that was produced from $O_2$ that was derived from the seawater.

I'm a bit lost as to how to approach this problem given the change in diffusion coefficient. If anybody could give me some pointers on how to proceed it would be very helpful.

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The problem seems to me formulated incompletely. It is unclear how the CO2 produced in the tissue would diffuse, it certainly should affect the content of both O2 and CO2 in the tissue in comparison to the case where no O2 conversion occurs. You seem to assume that CO2 produced from O2 continues to diffuse inwards (contrary to pressure gradient in the tissue) but with the diffusion coefficient value of CO2. This seems contradictory to me.

In the absence of an answer to this question, I think it is simpler to assume that CO2 produced from O2 entering the blood does not affect O2 nor CO2 diffusion rates. This would imply that a constant ratio of the O2 influx converted enters the blood converted to CO2, hence a fraction which can be obtained from your values (fraction of CO2 produced per unit of O2 gas).

On the other hand if we assume the CO2 is affected by the pressure gradient, we need to account for this, which can be done in a simple model of multilayer tissue. I can expand on this if you clarify on comments that it is the answer you are looking for.


Edit after clarification on comments (see below)

A model for the process inside the tissue is needed in order to answer the question. Assuming that Fick's law holds at infinitesimal scales one can formulate the problem as solving a system of differential equations expressing the diffusion of the different substances. Although not explicit, it should be clear that a steady state is assumed (invariable with time), hence our solution will provide the constant influx of CO2 for the constant concentrations given on the boundaries (blood and water, in and out of the boundary).

For simplification (and without affecting the solution) we will use the expression as: $$\frac{dI}{dx} = K \frac{dP}{dx}$$ where $K$ is Krogh's coefficient, and the linear diffusion intensity $I=\frac{D}{A}$ is the diffusion $D$ over the total area $A$, since these magnitudes do not change in our formulation of the problem.

We have then for O2: $$\frac{dI_{O_2}}{dx} = K_{O_2}\frac{dP_{O_2}}{dx} - \mu$$ where $\mu = 0.8 C_{O_2}$ represents the fraction of O2 converted to CO2 in the tissue per unit length, assuming as said a constant O2 consumption throughout the volume $C_{O_2}=\frac{30 {\rm \mu mol / h}}{100 {\rm cm^2} \times 300 {\rm \mu m}}$

Similarly, we have for CO2: $$\frac{dI_{{CO}_2}}{dx} = K_{{CO}_2}\frac{dP_{{CO}_2}}{dx} + \mu$$ where the term $\mu$, as above, accounts for the production of CO2 per unit length inside the tissue.

We also need expressions for the pressure change inside the tissue for each of the gases. We could formulate other criteria from biology or thermodynamics arguments, but the simplest case is when these functions are constant and equal to the pressure difference between the boundaries: $$\frac{dP_{O_2}}{dx} = const = \frac{P^{out}_{O_2} - P^{in}_{O_2}}{d_t} = \beta_{O_2} = 30 {\rm Pa/\mu m}$$ $$\frac{dP_{{CO}_2}}{dx} = const = \frac{P^{out}_{{CO}_2} - P^{in}_{{CO}_2}}{d_t} = \beta_{{CO}_2} = -0.87 {\rm Pa/\mu m}$$ where $d_t$ is the tissue thickness.

It should be noted how the sign of the $\beta$ corresponds to the convention chosen: positive for pressure gradients in the outwards direction (towards the water) and negative for pressure gradients inwards. This convention affects also the influx/outflux of the gases.

This is a system of linear differential equations fully determined due to the knowledge of the concentrations on the boundaries. There are documented methods for their solution. I am not sure if you also want the solution.


Solution and discussion

The integration of the above equations yields simply linear relations, which will have positive or negative slopes indicating the inwards/outwards direction according to the convention chosen.

$$I_{{CO}_2}(x) = \left(K_{{CO}_2}\beta_{{CO}_2} + \mu \right)x+I^o_{{CO}_2}$$

$$I_{O_2}(x) = \left(K_{O_2}\beta_{O_2} - \mu \right)x+I^o_{O_2}$$

Where the constants $I^o_{{CO}_2}$ and $I^o_{O_2}$ need to be derived from the boundary conditions. For example, assuming that the consumption of O2 in the tissue is fully provided by the blood, we find that $I_{O_2}(x=0)=I^o_{O_2} = C_{O_2}$. And assuming that CO2 entering from the water is the same as if no conversion of O2 was occurring, we get $$I_{{CO}_2}(x=d_t) = \left(K_{{CO}_2}\beta_{{CO}_2} + \mu \right)d_t+I^o_{{CO}_2} = K_{{CO}_2}\beta_{{CO}_2}d_t$$ from which we get $I^o_{{CO}_2} = -\mu d_t$.

In this picture I would not say that there is O2 flowing inwards nor CO2 going outwards, since the drift caused by the respective pressure gradient of each gas species is clearly oriented outwards in the former, inwards in the latter.

So, to answer your question about how much CO2 produced from O2 flowing inwards from the water in this case I would say none.

But the CO2 flowing inwards produced in tissue from O2 flowing outwards will by $\mu d_t = 0.24 {\rm \mu mol / {cm}^{2}h}$.

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  • $\begingroup$ rmhleo, thanks for giving this some thought. You're right, I think it is formulated incompletely but am not sure if I'm fulling understanding your suggestion. Due to the conversion of O2 to CO2 in the tissues, the PCO2 is higher in the tissue than blood. Hence why I "assume that CO2 produced from O2 continues to diffuse inwards". $\endgroup$ – CephBirk Oct 9 '17 at 21:08
  • $\begingroup$ The thing is missing here is the relation expressing what happens inside the tissue. You are saying that PCO2 inside the tissue is higher, so you are calculating it in some way, making some assumption. In my opinion, a natural assumption would be that Fick's law holds per differential unit length of tissue. This allows us to write a system of equation that describe how concentration and pressure vary across the tissue, with which one can quantify the amounts diffused. I can add this to the answer if you think that approach makes sense in your problem. $\endgroup$ – rmhleo Oct 10 '17 at 8:29
  • $\begingroup$ Yes, I think that makes sense and would love to see an answer with that assumption. $\endgroup$ – CephBirk Oct 10 '17 at 12:08
  • $\begingroup$ This is great. I would benefit greatly from the solution to these as well. It will help me follow your logic I think. $\endgroup$ – CephBirk Oct 10 '17 at 18:51
  • $\begingroup$ @CephBirk I have changed in the differential equations the term $\mu I_{O_2}$ because reading again your question I realized this was wrong. The terms as it was, implies that outer parts of the tissue consume less O2 than inner ones, since the consumption is a fraction of the O2 arriving to them. To correctly correctly represent the assumption that the consumption is the same everywhere in the tissue, the term has to be just a constant, as it stands now. This simplifies particularly the problem. $\endgroup$ – rmhleo Oct 11 '17 at 22:48

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