A common problem for \(^{13}CO_2\) pulse-chase studies is to measure the \(\delta^{13}C\) ratio of respired \(eCO_2\). Unfortunately, most commonly used IRGAs (e.g. LICOR-850) are optimized to detect natural abundance isotopic levels (e.g. \(^{12}CO_2\)), and thus cannot be used directly in a situation where high levels of \(^{13}C\) are expected. The usual solution to this are Picarro analyzers. However, they are very very expensive, and may not be optimized to measure \(CO_2\) concentrations per se.
There may, however, be a simple, affordable, hence scalable solution. If two IRGAs can be utilized in series or parallel to measure the same source \(eCO_2\), but be known to have two distinct overlaps with \(^{13}CO_2\) spectra, their difference will yield the true \(^{13}CO_2\) level.
The situation for a single IRGA is summarized here in equation 1: \[\tag{1} ^{IRGA1}CO_2 = \alpha^{13}CO_{2,true} + ^{12}CO_{2,true}\] where \(\alpha\) is the fraction of the true \(^{13}CO_{2}\) captured in the \(CO-2\) readout. Now this equation would be fine if we knew the true level of one or the other of the \(CO_2\) isotopic species. However, we generally do not, and especially not so in enrichment studies. Now, let’s consider a second IRGA with a different fractional capture of the true \(^{13}CO_{2}\) \(\beta\):
\[\tag{2} ^{IRGA2}CO_2 = \beta^{13}CO_{2,true} + ^{12}CO_{2,true}\] If we have designed our measurement so that we, say, pipe the outlet air straight into the inlet air of the second, with time-stamped data logs we can recreate the required \(^{13}CO_{2}\) values by differencing:
\[\tag{3} ^{IRGA1}CO_2 - ^{IRGA2}CO_2 = \alpha^{13}CO_{2,true} + ^{12}CO_{2,true} - (\beta^{13}CO_{2,true} + ^{12}CO_{2,true}) \] \[\tag{4} \frac{^{IRGA1}CO_2 - ^{IRGA2}CO_2}{\alpha-\beta} = ^{13}CO_{2,true}\] # Measurement error
\[\tag{5} ^{IRGA1}CO_2 = \alpha^{13}CO_{2,true} + ^{12}CO_{2,true} + \epsilon\] Assuming normality, we re-write as: \[\tag{6} ^{IRGA1}CO_2 \sim Normal(\alpha^{13}CO_{2,true} + ^{12}CO_{2,true},\sigma_{IRGA1}^2)\] Likewise for IRGA 2:
\[\tag{7} ^{IRGA2}CO_2 \sim Normal(\beta^{13}CO_{2,true} + ^{12}CO_{2,true},\sigma_{IRGA2}^2)\] and the distribution of their difference is:
\[\tag{8} ^{IRGA1}CO_2 - ^{IRGA2}CO_2 \sim Normal((\alpha - \beta)^{13}CO_{2,true},\sigma_{IRGA1}^2 + \sigma_{IRGA2}^2)\] Because we assume that \(\alpha\) and \(\beta\) are constants, we can invoke the symmetry of the Gaussian distribution to rewrite this as:
\[\tag{9} \frac{^{IRGA1}CO_2 - ^{IRGA2}CO_2}{(\alpha - \beta)} \sim Normal(^{13}CO_{2,true},\sigma_{IRGA1}^2 + \sigma_{IRGA2}^2)\]
Equation 9 shows clearly how we could estimate the \(^{13}CO_{2,true}\) as a latent quantity as well as the level of precision attained, which is a simple sum of the separate variance terms from each instrumental reading. Let us say this amounts to a variance of 1 ppm in each machine. Then, we have a variance of 2 ppm in our detection of \(^{13}CO_{2,true}\). In a setup where our total \(CO_2\) is perhaps around 500 ppm this yields a standard error of around 3 per mil.