¹H-NMR Spectroscopy as a New Tool in the Assessment of the Oxidative State in Edible Oils

Abstract

Within the course of lipid peroxidation, hydroperoxides are formed as primary products. They can be used as analytical markers to assess the deterioration status of oils and fats. Here a new ¹H-NMR assay to determine the hydroperoxide amount in edible oils is presented. We were able to show that the analytical performance of the method is similar to that of the commonly used peroxide value (PV) according to Wheeler. A total of 290 edible oil samples were analyzed using both methods. For some oil varieties considerable discrepancies were found between the results obtained. In the case of black seed and olive oil, two substances could be identified that cause positive (black seed oil) and negative (olive oils) deviations from the theoretical PV expected from the NMR values.

Appendix: Deming Regression and Relative Sensitivity

Assume that the relationship between the peroxide value according to Wheeler, y, and the true but unknown hydroperoxide amount z is given by a linear model of the form

$$ y = Y(z) + \eta = d + gz + \eta $$

(2)

y(z) denotes the conditional expectation of y and η represents the random error. η is supposed to follow a normal distribution with mean zero and constant variance $\sigma_{\eta }^{2}$. g and d are the model parameters. Analogously, we assume that the NMR-determined peroxide amount x is related to z by the equation

$$ x = X(z) + \varepsilon = k + lz + \varepsilon $$

(3)

where $X(z) = k + lz$ is the conditional expectation of x and the term ε designates the error. ε is supposed to be normally distributed with mean zero and constant variance $\sigma_{\varepsilon }^{2}$. For the time being, let us assume that all parameters and variances are known. Then, applying the Gaussian law of uncertainty propagation, it follows that for a given y ₀ the standard deviation of the estimate $\hat{z}(y_{0} ) = (y_{0} - d)/g$ is

$$ \sigma_{{\hat{z}(y_{0} )}} = \frac{{\sigma_{\eta } }}{g} $$

(4)

In the same way, we obtain for the standard deviation of the estimate $\hat{z}(x_{0} ) = (x_{0} - k)/l \, :$

$$ \sigma_{{\hat{z}(x_{0} )}} = \frac{{\sigma_{\varepsilon } }}{l} $$

(5)

The relative sensitivity [25] of the NMR assay with respect to the Wheeler method, RS_NMR/Wheeler, is defined as

$$ {\text{RS}}_{\text{NMR/Wheeler}} = \frac{{\sigma_{{\hat{z}(y_{0} )}} }}{{\sigma_{{\hat{z}(x_{0} )}} }} = {\frac{l}{g} \mathord{\left/ {\vphantom {\frac{l}{g} {\frac{{\sigma_{\varepsilon } }}{{\sigma_{\eta } }}}}} \right. \kern-\nulldelimiterspace} {\frac{{\sigma_{\varepsilon } }}{{\sigma_{\eta } }}}}. $$

(6)

If RS_NMR/Wheeler appreciably exceeds unity, the NMR approach would have the greater ability to detect a difference in the actual hydroperoxide amount z. The opposite conclusion holds when RS_NMR/Wheeler is clearly smaller than unity. How can we estimate RS_NMR/Wheeler? To answer this question, the Eq. (2) and (3) have to be considered. Solving the Eq. (3) for z and inserting the result into (2) gives us

$$ y = - \frac{g}{l}k + d + \frac{g}{l}x - \frac{g}{l}\varepsilon + \eta . $$

(7)

By substituting a for $- \frac{g}{l}k + d$ and b for $\frac{g}{l}$, we obtain

$$ y - \eta = a + b(x - \varepsilon ) $$

(8)

or

$$ Y = a + bX $$

(9)

with $x = X + \varepsilon$ and $y = Y + \eta$. Equation (9) connects the peroxide value y to the NMR-determined peroxide amount x and represents a so-called Deming regression model. Minimizing the sum

$$ \sum\limits_{i = 1}^{n} {\left( {(x_{i} - X_{i} )^{2} + (y_{i} - a - bX_{i} )^{2} \left( {\frac{{\sigma_{\varepsilon } }}{{\sigma_{\eta } }}} \right)^{2} } \right)} $$

(10)

for a given data set $(x_{i} , \, y_{i} )_{i = 1, \ldots ,n}$ with respect to a, b and X yields the maximum likelihood estimators for $\hat{a}$, $\hat{b}$ and $\hat{X}$:

$$ \hat{b} = \frac{{\left( {\frac{{\sigma_{\varepsilon } }}{{\sigma_{\eta } }}} \right)^{2} {\text{SS}}_{yy} - {\text{SS}}_{xx} + \sqrt {\left[ {\left( {\frac{{\sigma_{\varepsilon } }}{{\sigma_{\eta } }}} \right)^{2} {\text{SS}}_{yy} - {\text{SS}}_{xx} } \right]^{2} + 4\left( {\frac{{\sigma_{\varepsilon } }}{{\sigma_{\eta } }}} \right)^{2} {\text{SS}}_{xy}^{2} } }}{{2\left( {\frac{{\sigma_{\varepsilon } }}{{\sigma_{\eta } }}} \right)^{2} {\text{SS}}_{xy} }} $$

(11)

$$ \hat{a} = \bar{y} + \hat{b}\bar{x} $$

(12)

$$ \hat{X}_{i} = x_{i} + \frac{{\left( {\frac{{\sigma_{\varepsilon } }}{{\sigma_{\eta } }}} \right)^{2} \hat{b}}}{{1 + \left( {\frac{{\sigma_{\varepsilon } }}{{\sigma_{\eta } }}} \right)^{2} \hat{b}^{2} }}(y_{i} - \hat{a} - \hat{b}x_{i} ) $$

(13)

where ${\text{SS}}_{yy} = \sum {(y_{i} - \bar{y}_{i} )^{2} }$, ${\text{SS}}_{xx} = \sum {(x_{i} - \bar{x}_{i} )^{2} }$ and ${\text{SS}}_{xy} = \sum {(x_{i} - \bar{x}_{i} )(y_{i} - \bar{y}_{i} )} .$

With $\frac{g}{l} = b$, we can rewrite Eq. (6) as

$$ {\text{RS}}_{\text{NMR/Wheeler}} = \left( {b\frac{{\sigma_{\varepsilon } }}{{\sigma_{\eta } }}} \right)^{ - 1} $$

(14)

To compute $\hat{b}$ and to estimate the relative sensitivity, an estimate for $\sigma_{\varepsilon } /\sigma_{\eta }$ is needed. The latter can be obtained as follows: Firstly, n different mixtures of a peroxide-free and a peroxide-containing oil are prepared with mixing ratios $v_{i = 1, \ldots ,n}$. Analyzing these samples by both methods provides the x _i and y _i . Then, we regress x _i on $\nu_{i}$ and y _i on $\nu_{i}$ by simple linear least-squares fitting and calculate the corresponding residual standard deviations rsd _νx and rsd _νy. Since $\nu \propto z$, $\sigma_{\varepsilon }$ can be estimated as

$$ \hat{\sigma }_{\varepsilon } = {\text{rsd}}_{vx} = \sqrt {\frac{1}{n - 2}\sum {(x_{i} - \hat{x}(v_{i} ))^{2} } } $$

(15)

and $\sigma_{\eta }$ as

$$ \hat{\sigma }_{\eta } = {\text{rsd}}_{vy} = \sqrt {\frac{1}{n - 2}\sum {(y_{i} - \hat{y}(v_{i} ))^{2} } } $$

(16)