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Letters |
1
Dept. of Forensic Med., Tokai Univ. School of Med., Isehara, Kanagawa 259–11, Japan,
2
School of Computer Sci., Carnegie Mellon Univ., Pittsburgh, PA 15213,
3
Dept. of Clin. Chem., Royal Hallamshire Hosp., Sheffield S10 2JF, UK.
a Author for correspondence.
To the Editor:
CO-oximeters are specialized spectrophotometers that automatically determine hemoglobin (Hb) derivatives by measuring absorbance at selected wavelengths (1). We believe that a good understanding of the relevant theory may allow users to avoid many pitfalls during operation of these instruments. The mathematical basis of their operation has not, however, been fully explained by the manufacturers apart from Instrumentation Laboratory (Lexington, MA) at the introduction of their first CO-OximeterTM (2). Here, we discuss what mathematical methods for data processing might be used in commercial CO-oximeters, particularly in those models that use an "overdetermined" system.
CO-oximeters depend on the observation that Hb solutions obey the
Lambert–Beer Law; thus, the absorbance measured at a given wavelength
is the sum of the absorbance of each Hb derivative at the same
wavelength (2). When we measure n wavelengths
to determine the m Hb derivatives
i, we get n equations:
 | (1) |
j,
Ci is the concentration of derivative
i, and l is the pathlength.
ji is the molar
absorptivity at wavelength j for derivative
i.
When n = m, we can solve Eq. 1
to get
Ci. This is termed an "exactly
determined" system (3) and has been implemented in the
IL 482 CO-Oximeter (Instrumentation Laboratory), the IL 282 (its
predecessor), and the Radiometer OSM3 HemoximeterTM
(Radiometer, Copenhagen, Denmark). The IL 482 uses four wavelengths for
four Hb derivatives, whereas the OSM3 uses six wavelengths for six
unknowns: five Hb derivatives plus one for noise (attributed to
"turbidity"). The report by Steinke and Shepherd (4)
illustrates how the full exposition of the algorithms used in specific
instruments is useful not only to users but also to manufacturers.
After measuring absorptive spectra of Hb derivatives at three different
temperatures, Steinke and Shepherd applied their data to simulate the
effect of temperature variation on the accuracy of the IL 482,
according to the published mathematical formula. They found that
significant errors occurred if the apparatus was not
temperature-controlled. The marketed IL 482, however, precisely
controls the temperature of the measuring cell.
When n > m, the set of equations for Eq. 1
has been described as an "overdetermined" system (3).
Overdetermined systems have been utilized in the Corning 270
CO-oximeter (Ciba Corning Diagnostic Corp., Medfield, MA), the Corning
2500 (its predecessor), and the AVL 912 CO-OxyliteTM (AVL
Scientific Corp., Roswell, GA). The Corning 270 uses 7 wavelengths
for five Hb derivatives, whereas the AVL 912 uses 17 wavelengths for
five Hb derivatives.
We suspect that these CO-oximeters with overdetermined systems use the
least-squares method for data reduction. In such cases, errors
dj are defined as the difference
between observed value Aj and the predicted
values
 | (2) |
 | (3) |
 |
j is the weighting factor,
 |
 |
 |
S/Ck = 0 (for 1
k m) hold true. By solving these
simultaneous equations, we can get
Ci:
 | (4) |
 |
According to standard mathematical textbooks, the usual way to fix the
weighting factor j is to use the inverse
of the standard error 
j of absorbance
Aj at wavelength
j (j =
1/j for 1 j
n), if we can determine j.
Although we cannot determine the standard error for blood samples
because of the heterogeneity in their content of Hb derivatives, we can
determine the standard error for any (essentially) pure Hb derivative
by experiment. We therefore estimate the standard errors
j for a particular blood sample by assuming
that j at a given wavelength is the total of
the standard error due to each Hb derivative at the same wavelength:
 | (5) |
ji(Ci0)
(for 1 i m, 1
j n) is the standard error at wavelength
j for a pure Hb derivative
i having the concentration
Ci0, and
Ci is the concentration of the Hb
derivative i in the blood sample. The problem
with these formulas is that they include
Ci, which is the very unknown to be
determined. This problem can be avoided by using the following
algorithms:
1) Fix all the initial weighting factors as 1
[j (1) = 1, for 1
j n].
2) Calculate the provisional concentration
Ci(1) from the measured absorbance
Aj (1 j
n) by using Eq. 4
.
3) Calculate the provisional standard error
j(1) from
Ci(1) by using Eq. 5
.
4) Replace j(1) with
j(2) = 1/j(1).
5) Calculate the next provisional concentration Ci(2).
6) Calculate the next provisional standard error
j(2) from
Ci(2).
7) Repeat 4), 5), and 6) until j and
Ci converge.
Clearly, there may be other possibilities.
Brown (2) attributed a primary cause of the measurement
error to the drift in the wavelength of the emitted light
("wavelength shift"). This suggests that the standard error
j may be linear with respect to the slope
(first derivative) of the absorbance spectrum
dA/d (where =
j). If we accept this assumption, we notice
that Eq. 5
overestimates the standard error j
for a blood sample because the standard error of each Hb derivative in
the sample could cancel out each other (the slope of the absorbance
spectrum for the mixture of Hb derivatives is the total of the slope
for each Hb derivative in the mixture only when the directions of the
slope for all Hb derivatives are the same). On the other hand, we can
measure absorbance Aj accurately when
Aj falls in a given range (the
background noise due to any turbidity produces large errors when
Aj is too small, and the upper limit
for Aj depends on the linearity of the
photooptical system). Considering all of these factors, we suggest that
the standard error is a function of the absorbance and of its first
derivative: j =
F(Aj,
dA/d) (where =
j). However, speculation as to the form of
the function F is beyond our ability.
We ask the two companies who use overdetermined systems (Corning and
AVL) to comment on this approach—in particular, as to whether the
least-squares method is, in fact, used for processing an overdetermined
data set. If it is, we hope the manufacturers would explain how they
fixed their weighting factors j, i.e.,
how they have weighted the redundant wavelengths to achieve more
accurate results.
References
Chiron Diagnostics Corp., Medfield, MA 02052
a Author for correspondence.
To the Editor:
We have read the letter of Yukawa et al., and believe that it clearly addresses some often misunderstood aspects of CO-oximetry. We offer what we believe to be some additional perspectives on the topic.
The technique of "least-squares" analysis of error is firmly grounded in the technical and analytical literature and can be a powerful tool in the simultaneous multicomponent analysis used in CO-oximetry. We believe, however, that the rationale for wavelength selection and the use of weighting factors is not quite so clear. In that light we offer the following discussion based on our experience over the several generations of CO-oximeters that Chiron Diagnostics (formerly Ciba Corning Diagnostics) has developed and manufactured.
Wavelength selection for a determined system is a conceptually simple process. One chooses a combination of absorption maxima and isosbestic points, with a total number of wavelengths equal to the number of components of interest. After experimental determination of the absorptivities, one can set up a matrix for solving for the various fractions of a test specimen.
The process for an overdetermined system is similar, except that one decides on an arbitrary number of additional wavelengths, then tests the system in the presence of typical interferents using the least-squares approach. When one has a choice of wavelengths, the process is repeated multiple times to determine the best set of wavelengths to combine measurement of fractions and minimization of the effects of interferents. As a part of the process, one might decide, on the basis of the results of the experimental findings, that either different wavelengths or a greater or lesser number are more robust than those originally selected.
The seven wavelengths used in the M270 CO-oximeter as used by Yukawa et al. have performed quite well for the vast majority of blood samples. However, after years of operation with many hundreds of field placements, it became evident that a small number of blood samples from apparently normal donors have significant deviations in their absorbance spectra in the region between 580 and 610 nm. These deviations resulted in small changes in the reported Hb fractions as well. Although this region of the spectrum would appear to be useful for differentiating MetHb from the other fractions, it introduced unwanted variability in the results. Avoiding this variable region of the Hb spectrum is one of the reasons that led us to select a different set of 10 wavelengths for the new 800 Series CO-oximeter.
Although it is customary to use the standard error or variance of observations as weighting factors for least-squares analysis, it is often difficult to measure or estimate appropriate weighting factors to be used in real applications. Such is the case with CO-oximetry. As Yukawa et al. suggest, the standard error cannot be determined before the measurement because it depends on the actual concentrations of the components being measured. As they also suggest, the standard error could be estimated from measurements on donor samples and then applied to unknown samples in some iterative scheme. However, such an approach can account for only systematic errors in the measurement of absorbances and the variability in wavelength.
Our experience shows that for the measurement of typical blood samples, i.e., samples that are fresh and free of interfering substances and contamination, weighting factors of 1.0 yield both precise and accurate results. The systematic errors that affect all measurements in some measurable or estimatable manner have been minimized. It is the errors resulting from exceptional conditions such as interfering substances and improper preanalytical sample handling that cause the majority of inaccuracy and imprecision. Thus, our approach has been to select wavelengths that avoid many common interferents and minimize susceptibility to minor variations in wavelength.
Of course, it is not possible to avoid all interferents, so detection and correction algorithms have been devised for some known interferents such as lipid, SulfHb, CNMetHb, and methylene blue. For samples that contain uncharacterized interferents, the overdetermined measurement affords a "quality-of-fit index" that is used to detect the presence of unknown interferents and flag the samples with the message, "If blood, question data."
Yukawa et al. have clearly outlined the mathematics behind CO-oximetry measurements with overdetermined systems. However, we feel that the wider system issues of wavelength selection and detection of atypical samples also play an equally vital role in the performance of CO-oximeters. Hopefully this discussion has helped to enlighten users in regard to the characteristics and analytical limitations of present-day CO-oximeters.
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