Response surface calibration of 13CO2-NDIR offset values: A ‘random coefficients’ approach
Introduction
Physical processes can induce non linearities in measurements. With known underlying physical or technical laws, like that of Lambert–Beer for absorption these non linearities can be corrected. Nevertheless, frequently minor reproducible deviations persist, which are difficult to explain with physical or technical principles. This paper focuses on such deviations, specifically on those, which are additive to a scalar signal generated by the analyte and are independent of its intensity. Their impact on the zero point of the measurements should be corrected. Remaining deviations after correction can take any conceivable course and might be specific for a particular device. Hence, the impact of interference variables should be described with a general and flexible zero-point function. Polynomial functions in the interferences provide these features. For one interfering variable, denoted with x, the course of the zero point (ZP) can be described as:with z as the predicted zero point (ZP) and ki as polynomial coefficients. If the interference can be measured for each sample with negligible error then coefficients for Eq. (1) can be determined by regression using calibration samples with different amounts of the interference. Based on Eq. (1) a ZP value can be predicted from measured concentrations of the sample interference. For correction the ZP value is then subtracted from the analyte measurement, to obtain the analyte derived signal. Eq. (1) should fit the ZP course and its determination should be robust against measurement errors. This cannot be taken for granted for polynomials applied on ZP profiles with an disadvantageous shape, as demonstrated in Fig. 1.
With 11 nodes there is no chance to define and determine a polynomial, that both comes close the measurements at the nodes and shows an acceptable interpolation between these nodes. Fig. 1 also shows how a ‘ridge regression’([1], Chapter 3) improves the situation. The remaining interpolation error, however, still can be larger than the random measurement error at the nodes. The potential inefficiency of the polynomial/ridge regression approach could amount to serious difficulties for the approximation of the effects of two or more interferences. Polynomials in two or more variables will be necessary to capture a curvature in the more dimensional space, with increasing measurement effort for their determination. On the other hand, response surfaces for a particular device, collected over a longer period resemble each other, sometimes they differ just in almost constant offset value. It would be tempting to ‘calibrate’ only this small difference, however reports about corresponding efforts are rather sparse. The ‘direct standardization’ approach may come closest [2], [3]. It assumes a carefully crafted ‘master calibration set’, based on a sufficient number of calibrants, and an actual minimal calibration, just sufficient to define a transformation of actual response values such that the master calibration can be applied. It is designed for spectral data at consecutive wavelengths, where a calibration data set can be described with a response matrix. This approach is not applicable for the present case, where just one scalar value is measured per sample. As one resort we shift our focus toward the coefficients of polynomials used to describe the response surface. The variability of these coefficients over time is decomposed by ‘principal component analysis’(PCA) [4], which allows to define a subspace for the fluctuations of the polynomial coefficients. Confining on this subspace the number of coefficients, which are to be identified by regression, can be reduced. As additional ‘first order advantage’ of such a multivariate approach [5] one can identify any set of coefficients, that cannot be explained with the principal components. It then implies that the particular calibration is not consistent with earlier. It sets the base to define a ‘performance qualification’ for a ‘Good Laboratory Practice’-compliant device control [6]. Therefore, a PCR-based approach will be defined for the determination of the ZP of 13CO2/12CO2 ratios. These measurements arise with a 13C breath test, where a subject receives a 13C labeled test substrate. Metabolic processes convert the 13C label to 13CO2, which is then released with respiration. The 13CO2 / 12CO2 ratios in breath gas thereby allow a non invasive quantification of the metabolic processes involved [7]. The O2 and CO2 concentration in breath gas can vary over a wide range, especially in patient with impaired respiratory function. They receive increased amounts of O2 to assure a sufficient supply. These variable O2 and CO2 concentrations interfere with the 13CO2 / 12CO2 ratio measurements with NDIR1 devices, especially on their ZP values [8]. A corresponding correction therefore covers most of the interferences and should adapt NDIR devices to the variable conditions expected for intensive care patients [9] under respiration.
Section snippets
Material and methods
All calibration samples were based on CO2 with a 13CO2 labeling close to natural occurrence and were measured using a Helifan Ci3 / Helifan plus (Fischer Analysen Instrumente, Leipzig, Germany) equipment based on an NDIR (ABB Frankfurt, Main, Germany) device. CO2 as one interference: 22 calibrations were collected over about one year. Each calibration was based on a breath gas sample containing 2.4 to 4 vol.% CO2. CO2-devoid ambient air was added to this base sample in sequential steps to obtain
Theory/calculation
For the following lower case bold symbols denote a vector; z and denote measured and predicted values for a specific calibration set, respectively and k a set of coefficients used in Eq. (1). It defines the calibration function for a given polynomial structure and hence is used as a synonym for calibration. With specific calibration nodes and a specific polynomial structure on Eq. (1) the prediction for ZP can be written as matrix equation:Here Φ, the design matrix, reflects the
Results
An approach to correct interferences on zero point (ZP) measurements is proposed and applied on δ13C measurements2 using NDIR technology.
Discussion
We propose to reduce the effort for a response surface calibration, by including previous calibration measurements. This inclusion allows to identify a subspace for the inter-calibration variability of the coefficients, and finally the coefficients are mapped on this reduced subspace to limit the number of parameters to be identified by regression.
A first requirement for such an approach is, that the variability of the coefficients is indeed confined to a subspace, over a period of time long
Disclosure statements
J. Kappler is and W. Fabinski has been employed by ABB, a company which develops and produces NDIR Technology used in this study. H Fischer is managing director of Fischer Analysen Instrumente GmbH, which produces the system in which the NDIR module is integrated.
Acknowledgments
Ninon Nahoussi performed the measurements for two interferences in the frame of her bachelor-thesis at the Hochschule Münster, Germany.
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