Isotope dilution

Tutorial example

Isotope dilution is analogous to the mark and recapture method, commonly used in ecology to estimate population size.

For instance, consider the determination of the number of fish (n_A) in a lake. For the purpose of this example, assume all fish native to the lake are blue. On their first visit to the lake, an ecologist adds five yellow fish (n_B = 5). On their second visit, the ecologist captures a number of fish according to a sampling plan and observes that the ratio of blue-to-yellow (i.e. native-to-marked) fish is 10:1. The number of fish native to the lake can be calculated using the following equation:

${\displaystyle n_{\mathrm{A}}=n_{\mathrm{B}}\times \frac{10}{1}=50}$

This is a simplified view of isotope dilution but it illustrates the method's salient features. A more complex situation arises when the distinction between marked and unmarked fish becomes fuzzy. This can occur, for example, when the lake already contains a small number of marked fish from previous field experiments; and vice versa, where the amount of marked fish added contains a small number of unmarked fish. In a laboratory setting, an unknown (the "lake") may contain a quantity of a compound that is naturally present in major ("blue") and minor ("yellow") isotopic forms. A standard that is enriched in the minor isotopic form may then be added to the unknown, which can be subsequently analyzed. Keeping to the fish analogy, the following expression can be employed:

${\displaystyle n_{\mathrm{A}}=n_{\mathrm{B}}\times \frac{R_{\mathrm{B}}-R_{\mathrm{A}\mathrm{B}}}{R_{\mathrm{A}\mathrm{B}}-R_{\mathrm{A}}}\times \frac{1+R_{\mathrm{A}}}{1+R_{\mathrm{B}}}}$

where, as indicated above, n_A and n_B represent the number of fish in the lake and the number of fish added to the lake, respectively; R_A is the ratio of the native-to-marked fish in the lake prior to the addition of marked fish; R_B is the ratio of the native-to-marked fish in the amount of marked fish added to the lake; finally, R_AB is the ratio of the native-to-marked fish captured during the second visit.

Applications

Isotope dilution is almost exclusively employed with mass spectrometry in applications where high-accuracy is demanded. For example, all National Metrology Institutes rely significantly on isotope dilution when producing certified reference materials. In addition to high-precision analysis, isotope dilution is applied when low recovery of the analyte is encountered. In addition to the use of stable isotopes, radioactive isotopes can be employed in isotope dilution which is often encountered in biomedical applications, for example, in estimating the volume of blood.

Single dilution method

Consider a natural analyte rich in isotope ⁱA (denoted as A), and the same analyte, enriched in isotope ^jA (denoted as B). Then, the obtained mixture is analyzed for the isotopic composition of the analyte, R_AB = n(ⁱA)_AB/n(^jA)_AB. If the amount of the isotopically enriched substance (n_B) is known, the amount of substance in the sample (n_A) can be obtained:

${\displaystyle n_{\mathrm{A}}=n_{\mathrm{B}}\frac{R_{\mathrm{B}}-R_{\mathrm{A}\mathrm{B}}}{R_{\mathrm{A}\mathrm{B}}-R_{\mathrm{A}}}\times \frac{x{(}^j\mathrm{A}{)}_{\mathrm{B}}}{x{(}^j\mathrm{A}{)}_{\mathrm{A}}}}$

Here, R_A is the isotope amount ratio of the natural analyte, R_A = n(ⁱA)_A/n(^jA)_A, R_B is the isotope amount ratio of the isotopically enriched analyte, R_B = n(ⁱA)_B/n(^jA)_B, R_AB is the isotope amount ratio of the resulting mixture, x(^jA)_A is the isotopic abundance of the minor isotope in the natural analyte, and x(^jA)_B is the isotopic abundance of the major isotope in the isotopically enriched analyte.

For elements with only two stable isotopes, such as boron, chlorine, or silver, the above single dilution equation simplifies to the following:

${\displaystyle n_{\mathrm{A}}=n_{\mathrm{B}}\frac{R_{\mathrm{B}}-R_{\mathrm{A}\mathrm{B}}}{R_{\mathrm{A}\mathrm{B}}-R_{\mathrm{A}}}\times \frac{1+R_{\mathrm{A}}}{1+R_{\mathrm{B}}}}$

In a typical gas chromatography analysis, isotopic dilution can decrease the uncertainty of the measurement results from 5% to 1%. It can also be used in mass spectrometry (commonly referred to as isotopic dilution mass spectrometry or IDMS), in which the isotopic ratio can be determined with precision typically better than 0.25%.

Optimum composition of the blend

In a simplified manner, the uncertainty of the measurement results is largely determined from the measurement of R_AB:

${\displaystyle u(n_{\mathrm{A}}{)}^2\propto {\left(\frac{\mathrm{\partial }{n}_{\mathrm{A}}}{\mathrm{\partial }{R}_{\mathrm{A}\mathrm{B}}}\right)}^{2}u(R_{\mathrm{A}\mathrm{B}}{)}^2=n_{\mathrm{A}}^2\frac{(R_{\mathrm{A}}-R_{\mathrm{B}}{)}^2}{(R_{\mathrm{A}}-R_{\mathrm{A}\mathrm{B}}{)}^2(R_{\mathrm{A}\mathrm{B}}-R_{\mathrm{B}}{)}^2}u(R_{\mathrm{A}\mathrm{B}}{)}^2}$

From here, we obtain the relative uncertainty of n_A, u_r(n_A) = u(n_A)/n_A:

${\displaystyle u_{\mathrm{r}}(n_{\mathrm{A}}{)}^2\propto \frac{(R_{\mathrm{A}}-R_{\mathrm{B}}{)}^2}{(R_{\mathrm{A}}-R_{\mathrm{A}\mathrm{B}}{)}^2(R_{\mathrm{A}\mathrm{B}}-R_{\mathrm{B}}{)}^2}u(R_{\mathrm{A}\mathrm{B}}{)}^2}$

The lowest relative uncertainty of n_A corresponds to the condition when the first derivative with respect to R_AB equals zero. In addition, it is common in mass spectrometry that u(R_AB)/R_AB is constant and therefore we can replace u(R_AB) with R_AB. These ideas combine to give

${\displaystyle u_{\mathrm{r}}(n_{\mathrm{A}}{)}_{\mathrm{m}\mathrm{i}\mathrm{n}}\mapsto \mathrm{\partial }\left(\frac{(R_{\mathrm{A}}-R_{\mathrm{B}})}{(R_{\mathrm{A}}-R_{\mathrm{A}\mathrm{B}})(R_{\mathrm{A}\mathrm{B}}-R_{\mathrm{B}})}{R}_{\mathrm{A}\mathrm{B}}\right)/\mathrm{\partial }{R}_{\mathrm{A}\mathrm{B}}=0}$

Solving this equation leads to the optimum composition of the blend AB, i.e., the geometric mean between the isotopic compositions of standard (A) and spike (B):

${\displaystyle R_{\mathrm{A}\mathrm{B}}=\sqrt{R_{\mathrm{A}}{R}_{\mathrm{B}}}}$

This simplified equation was first proposed by De Bievre and Debus numerically and later by Komori et al. and by Riepe and Kaiser analytically. It has been noted that this simple expression is only a general approximation and it does not hold, for example, in the presence of Poisson statistics or in the presence of strong isotope signal ratio correlation.

Double dilution method

The single dilution method requires the knowledge of the isotopic composition of the isotopically enriched analyte (R_B) and the amount of the enriched analyte added (n_B). Both of these variables are hard to establish since isotopically enriched substances are generally available in small quantities of questionable purity. As a result, before isotope dilution is performed on the sample, the amount of the enriched analyte is ascertained beforehand using isotope dilution. This preparatory step is called the reverse isotope dilution and it involves a standard of natural isotopic-composition analyte (denoted as A*). First proposed in the 1940s and further developed in the 1950s, reverse isotope dilution remains an effective means of characterizing a labeled material.

Reverse isotope dilution analysis of the enriched analyte:

${\displaystyle n_{\mathrm{B}}=n_{\mathrm{A}\ast }\frac{R_{\mathrm{A}\ast }-R_{\mathrm{A}\ast \mathrm{B}}}{R_{\mathrm{A}\ast \mathrm{B}}-R_{\mathrm{B}}}\times \frac{x{(}^j\mathrm{A}{)}_{\mathrm{A}\ast }}{x{(}^j\mathrm{A}{)}_{\mathrm{B}}}}$

Isotope dilution analysis of the analyte:

${\displaystyle n_{\mathrm{A}}=n_{\mathrm{B}}\frac{R_{\mathrm{B}}-R_{\mathrm{A}\mathrm{B}}}{R_{\mathrm{A}\mathrm{B}}-R_{\mathrm{A}}}\times \frac{x{(}^j\mathrm{A}{)}_{\mathrm{B}}}{x{(}^j\mathrm{A}{)}_{\mathrm{A}}}}$

Since isotopic composition of A and A* are identical, combining these two expressions eliminates the need to measure the amount of the added enriched standard (n_B):

${\displaystyle n_{\mathrm{A}}=n_{\mathrm{A}\ast }\frac{R_{\mathrm{A}\ast }-R_{\mathrm{A}\ast \mathrm{B}}}{R_{\mathrm{A}\ast \mathrm{B}}-R_{\mathrm{B}}}\times \frac{R_{\mathrm{B}}-R_{\mathrm{A}\mathrm{B}}}{R_{\mathrm{A}\mathrm{B}}-R_{\mathrm{A}}}}$

Double dilution method can be designed such that the isotopic composition of the two blends, A+B and A*+B, is identical, i.e., R_AB = R_A*B. This condition of exact-matching double isotope dilution simplifies the above equation significantly:

${\displaystyle n_{\mathrm{A}}=n_{\mathrm{A}\ast }(R_{\mathrm{A}\ast \mathrm{B}}=R_{\mathrm{A}\mathrm{B}}\wedge R_{\mathrm{A}\ast }=R_{\mathrm{A}})}$

Triple dilution method

To avoid contamination of the mass spectrometer with the isotopically enriched spike, an additional blend of the primary standard (A*) and the spike (B) can be measured instead of measuring the enriched spike (B) directly. This approach was first put forward in the 1970s and developed in 2002.

Calculations using calibration curve

Many analysts do not employ analytical equations for isotope dilution analysis. Instead, they rely on building a calibration curve from mixtures of the natural primary standard (A*) and the isotopically enriched standard (the spike, B). Calibration curves are obtained by plotting measured isotope ratios in the prepared blends against the known ratio of the sample mass to the mass of the spike solution in each blend. Isotope dilution calibration plots sometimes show nonlinear relationships and in practice polynomial fitting is often performed to empirically describe such curves.

When calibration plots are markedly nonlinear, one can bypass the empirical polynomial fitting and employ the ratio of two linear functions (known as Padé approximant) which is shown to describe the curvature of isotope dilution curves exactly.