MXPA97010090A - Determination of component concentrations taking into account medic errors - Google Patents

Abstract

The present invention relates to apparatuses and methods for determining the concentration of sample components of a sample by an analytical technique that produces a spectrum that can be written as Y (w) = P (w) C. The apparatuses and methods of the invention they respond to experimental errors that cause distortions in the observed spectrum, and that consequently result in inaccurate determinations of the concentrations of the components of the sample. The invention considers such errors by modeling the total experimental error as the sum of one or more types of errors that can be written as S K. The spectrum is then modeled as Y = PC + S K. Using the observed spectrum, known values for P , and a mathematical model for S, this equation can be solved for the best adjusted value of the concentrations of the components of the sample, C and the magnitudes of the errors, K. The method can be used for any error that can be molded in the above manner, such as a displacement includes a constant displacement as well as a linear displacement through the entire spectrum. The devices and methods are used advantageously in absorbance spectroscopy and chromatography

Description

DETERMINATION OF COMPONENT CONCENTRATIONS TAKING INTO ACCOUNT MEASUREMENT ERRORS BACKGROUND FIELD OF THE INVENTION The present invention relates to the field of analytical spectrometry and, in particular, to apparatus and correction methods for determining concentrations of analytical sample components that consider instrumental error.

Summary of the Related Art A wide variety of analytical techniques has been developed over the years to detect and determine the concentrations of the components of a sample. Some techniques are completely spectrophotometric, such as ultraviolet-visible-infrared absorbance spectroscopy, (UV-VIS-IR) and NMR spectroscopy. Other techniques, such as column chromatography are not spectrophotometric per se, but frequently use spectrophotometric techniques to detect the presence of compounds. Each of these techniques generally provides a "spectrum" in which a dependent variable, typically the intensity of some quantity (eg, absorbance), is plotted against a dependent variable (e.g., wavelength). The relative concentrations of the components of the sample are determined by obtaining the better adjustment to the experimental spectrum by varying the relative spectral contribution of each component. This requires knowledge of the spectral characteristics of the individual components. Due to instrumental errors and / or deviations in the experimental conditions, small displacements of the independent variable frequently occur which can lead to large displacements in the relative concentrations of the components of the sample determined from the measured spectrum. To date, the prior art is devoid of adequate methods for determining displacement and correcting it. According to these, new methods that respond to displacements of the independent variable in analytical spectrophotometric techniques are desirable.

SUMMARY OF THE INVENTION The present invention provides apparatus and methods for the precise determination of the concentrations of components of the sample. The apparatuses and methods of the invention advantageously correct experimental errors (including errors induced by the instrument) that otherwise introduce errors in the concentrations of the components of the sample. The present invention compensates a wide variety of phenomena exogenous to the analytical sample that contribute to and manifest themselves in the observed analytical spectrum, from which the concentrations of the components of the sample are determined.

The present invention can be used advantageously in analytical spectrophotometric techniques, such as UV-VIS-IR spectroscopy and NMR spectroscopy. The present invention can also be used in analytical techniques that are not spectrophotometric per se, but which incorporate spectrophotometric detection and / or produce a "spectrum-like" graph (i.e., a graph that resembles a spectrum). An example of a technique that produces a "spectrum-like" graph is column chromatography, in which materials are detected by absorbance spectroscopy at one (or more) wavelength, producing a "spectrum-like" graph "which shows the intensity of absorbance extracted as a function of time. As used herein, the term "spectrum" encompasses the traditional spectrophotometric spectrum in which a spectral intensity is plotted against the frequency of radiation, wavelength, or wave number (or some equivalent thereof), as well as graphics. similar to spectra "produced in techniques such as column chromatography. Under ideal conditions, an analytical spectrum can be described by the equation Y (?) = P (?) "C, (i) where? Is a functional parameter (eg, frequency, wavelength, wave number, or time) ), Y is a vector whose elements are the spectral intensities, C is a vector whose elements are the concentrations of the components of the sample, and P is a matrix whose elements are a measure of the magnitude of the contribution of each component of the sample for the spectral intensity in each value from ?. The elements of P are known quantities and can be, for example, the UV-VIS-IR extinction coefficients of each component of the sample at each wavelength?. In practice, when one wishes to use such an analytical spectrum to determine the concentrations of the components of the sample that provide an increase to the observed spectrum, Yobs, equation (i) can be used to obtain a better estimate of C fit when using Yobs instead of Y. Due to instrumental errors and other experimental errors, however, equation (i) frequently does not describe the observed spectrum well. In this case, equation (i) can be modified to incorporate a term representing the experimental error: Y = P • C + dY, (ii) where dY is the deviation induced by the experimental error of the observed spectrum of the ideal. In the present invention, dY is written as: dY =? K, (iii) where K is a scalar that represents the magnitude of the error y? is a vector whose elements are the relative errors of each value of?. Equation (ii) is then written as Y = P • C +? K (iv) The apparatuses and methods of the present invention can be used to correct experimental errors whenever the observed spectrum can be written in the form of equation (iv), that is, each time the experimental error can be written as in the equation (iii). Equation (iv) is easily solved for the concentrations of the components of the best fitted sample, C, and the magnitude of the error, K, for the spectrum observed, Yobs, (using Yobs instead of Y in equation (iv)) as described more fully below. In one aspect of the invention, the apparatus and methods model the error, dY, as a displacement of the entire spectrum by an amount d ?. In this aspect of the invention: dY = _dY d? = Y 'd? . (v) d? In this aspect of the invention, the observed spectrum is estimated as: Yobs = P C + Y 'd? . (saw) Depending on the model chosen for the displacement, d? can it be a scalar (that is, the same for everything?) or a vector whose elements vary with? In one embodiment of this aspect of the invention, using an appropriate model for d ?, equation (vi) takes the form of equation (iv), which is then solved for the best adjusted values of C and K. In another embodiment of this aspect of the invention, the apparatuses and methods respond the displacement due to the experimental error when adjusting the complete spectrum (to produce an adjusted spectrum, Yadj) using an average compensated d? "of previously determined values of displacement d ?. The complete spectrum is then adjusted by d? Yadj = Yobs (? + D?). (vii) When 3? is small, Y 'd? it is a good estimate of the displacement in the spectrum, which is then calculated preferably from: Yadj = Yobs + Y 'd? . (viii) On the other hand, when d? is large, it is preferable to answer the displacement using equation (vii) directly. In any case, in this embodiment of the invention, Yadj is used in any equation (i) or (vi) to obtain the best adjusted value of C. In another embodiment of this aspect of the invention, the Y form in the equation (i) is used in equation (v) to obtain an expression of dY in terms of the derivative of P with respect to?: dY = dY d? = d_Y_C d? = dP • C d? , (ix) d? d? and the estimated spectrum is: Yest = P • C + P '• C d? . (x) Equation (x) reduces to the shape of equation (iv) when d? is modeled appropriately and can then be solved for C. In another embodiment of this aspect of the invention, the apparatuses and methods of the present invention respond to the displacement by adjusting P using some compensated average of previously determined values of d ?. An adjusted value of P that responds to the displacement is given by: Padj = P (? + D?). (xi) Yes d? is large, equation (xi) is preferably used directly. Yes d? is small, however, (dP / d?) d? is a good estimate of the change induced by the displacement in P, and Padj is then obtained preferably from: Padj = P + dP d? . (XÜ) d? In any case, Padj is then used in any equation (i) or (x) to obtain the best adjusted value of C. In this aspect of the invention, the displacement can be modeled in a number of ways. In one modality, displacement is modeled by being constant across the entire spectrum. In this model d? = S, a scalar. In another modality, the displacement is modeled by varying linearly around a central value of?,? C, and is given by the vector whose elements are d? I = (? I -? C) M, where M is the magnitude of the displacement. In this mode, d? It is a vector. When M > 0 the full spectrum is amplified around? C; when M < 0, the full spectrum is compressed around? C. Still in another modality, the experimental displacement is modeled as a linear combination of the two previous models. As described more fully below, each of these models can respond to errors in the observed spectrum that are manifested by shifts in?. The apparatuses and methods of the present invention incorporate the above equations to compensate for the experimental displacement in the observed spectrum and therefore produce more accurate measurements of the concentrations of the components of the sample. In a particularly preferred embodiment of the present invention, the apparatuses and methods of the invention employ UV-VIS-IR absorbance spectroscopy. Co-oximeters, which measure the relative concentrations of blood components, are an example of a modality. In this modality, Y is the spectrum of Absorbance A, P is the matrix of the extinction coefficients, E, and the independent variable,?, is the wavelength,?. In another preferred embodiment, the apparatuses and methods of the invention employ chromatographic means. In this mode,? is the time in which the components of the sample are extracted, and is the absorbance at the wavelength in which the components are detected, and P is a matrix of the relative absorbances of each sample component as a function of time of extraction.

DETAILED DESCRIPTION OF THE PREFERRED MODALITIES The apparatuses and methods of the present invention are useful for correcting a wide variety of instrumental and other experimental errors in a wide variety of spectrophotometric and analytical techniques. In general, the apparatuses and methods of the present invention can be used in conjunction with any spectrophotometric and / or analytical technique that produces a spectrum that can be described by the equation. Y (?) = P (?) • C, (1) where Y is a vector whose "m" components Yi are the spectrum intensities in each of the "m" values of the independent variable? (? I), C is a vector whose "n" elements are the concentrations of the components of the sample that contribute to the measured response Y, and P is an "mxn" matrix whose elements Pij relate the contribution of the component Cj to the intensity Yi. "m" is an integer and equal to the number of values in which Y is measured, "n" is also an integer and equal to number of sample components that contribute to Y. In UV-VIS-IR spectroscopy, for example, Yi is the absorbance of the sample at the wavelength? i, and the Pij are the extinction coefficients of the absorbent "j" in the wavelength? i. In column chromatography, as another example, Yi is the intensity of the absorbance over time? I, and the Pij are the extinction coefficients of the absorber "j" at the wavelength monitored in the extraction time? I under the particular conditions of extraction (for example, the identity of the shock absorber and force, pH, temperature, etc.). In practice, when one wishes to use such an analytical spectrum to determine the best estimate of the concentrations of the components of the sample, equation (1) can be used by inserting Yobs by Y and solving for the best adjusted value of C. As it becomes clearer later, it will be appreciated by those skilled in the art that the present apparatuses and methods are ideally suited for analytical techniques in which the identity of the "n" components of the sample and the matrix elements of P are known and we wish to determine the best estimated concentrations of the components of the sample from the observed (measured) spectrum Yobs. The apparatuses and methods of the present invention can be advantageously employed in analytical spectrophotometric techniques such as UV-VIS-IR and NMR spectroscopy. The present methods can also be used in analytical techniques that are not spectrophotometric per se, but which incorporate detection spectrophotometric and / or produce a "spectrum-like" graph (ie, a graph that resembles a spectrum). An example of a technique that produces a "spectrum-like" graph is column chromatography, in which materials are detected by spectroscopy at one (or more) wavelength, producing a "spectrum-like" graph. which shows the intensity of absorbance extracted as a function of time. As used herein, the term "spectrum" encompasses the traditional spectrophotometric spectrum in which a spectral intensity is plotted against the frequency of radiation, wavelength, or wave number (or some equivalent thereof), as well as graphics. similar to spectra "produced in techniques such as column chromatography. It is often the case that errors in the observed spectrum are introduced due to an experimental technique and / or instrument performance less than ideal. As used herein, the term "experimental error" means any error (e.g., arising from less than ideal instrument performance or suboptimal experimental technique) that results in a deviation of the measured spectrum from the theoretical ideal. These errors translate into errors in the concentrations of the measured sample components. These errors in the observed spectrum can be considered by rewriting equation (1) as Yobs = P • C + dY, (2) Where dY is the deviation induced by the experimental error in Yobs of the ideal. The present apparatus and methods correct such errors by incorporating a mathematical model for dY: dY =? • K, (3) where K is a dimension vector "r" whose elements are the magnitudes of the errors for each of the types V of modeled error as it contributes to the spectrum, and? is a matrix "m x r" composed of "r" vectors each of whose elements "m" are the relative errors in each value of? for each type of error modeled as it contributes to the spectrum. Where only one type of error is modeled r = 1, and K is a scalar and? It is a vector. Equation (3) is written as: Yobs = P C +? K (4) In the broadest aspect, the apparatuses and methods of the present invention incorporate corrections for instrumental errors and / or other experimental errors induced in the measured spectrum each time the spectrum can be written in the form of equation (4). In this aspect, therefore, the invention comprises an improved apparatus and methods for determining the concentrations of the components of the sample using an analytical technique that produces a spectrum that can be estimated as equation (1), the improvement comprising correcting the error experimental when modeling the experimental error as types of errors "r" given by the product? • K, where K is a vector whose elements "r" are the magnitudes of each of the "r" types of experimental errors and? is a matrix "m x r" whose elements are the relative errors in each value of? for each type of experimental error, add the product? • K to the estimated spectrum as in equation (4), and solve for the best adjusted values of C and K.

As used in the present invention r > 1; preferably r =. 10; and most preferably 1 < r < 3. N > 1 and preferably 1 < n < 20. M > n + r and preferably m is approximately twice n + r. Equation (4) is easily solved for the best adjusted values of C and K by least squares analysis. To do so, equation (4) is "collapsed" by defining an increased matrix "m x (n + r)", P ?, which has the form: and an increased vector, C «, of the dimension" n + r ", which has the form: Equation (4) then becomes: Yobs = P? • C? - (7) The least squares solution for equation (7) is given by: C? = P? ' Yobs, (8) where P? 1 = (P? 1 P?) "1 D P? E (9) is the least squares transformation matrix. For example, Noble and Daniel, Applied Linear Algebra, pp. 57-65 (Prentice-Hall, Inc., N.J., 1977).

CK is easily determined from equation (8) using standard algorithms. See, for example, Press et al. , Numeric! Recipes: The Art of Scientific Computing (Cambridge University Press, Cambridge 1986). The first "n" elements of CK are the best adjusted values of "n" concentrations of components of the sample and the remaining "r" elements are the best adjusted magnitudes of the errors. In another aspect of the invention, the apparatuses and methods model the error, dY, as a displacement of the complete spectrum by an amount d ?. In this aspect of the invention: dY = _dY d? = Y 'd? . (10) d? In one embodiment of this aspect of the invention, the spectrum is estimated by substituting equation (10) in equation (2) to obtain Yobs = P • C + Y 'd? . (eleven ) The term Y'd? it must be able to be written in the form of equation (3) and, consequently, equation (1 1) in the form of equation (4). As will be shown later, this can be so uniform when d? by itself it is different from a scalar. In this case, d? it is decomposed into a scalar K and a vector or matrix and the vector or matrix combined with Y 'to form a product matrix? Equation (11) is then solved in the same manner described above for equation (4). Specific examples of this method are presented in more detail below.

In another embodiment of this aspect of the invention, displacement due to experimental error is considered by adjusting the full spectrum using some compensated average of the displacement magnitude, d ?, obtained, for example, from previously determined displacement values, d ?. The full spectrum is corrected by the experimental error when displacing it by a quantity d ?: Yadj = Yobs (? + D >). (12) When d? is large in relation to the resolution of the measured spectrum, equation (12) is preferably used directly to obtain the corrected and adjusted spectrum, Yadj. When d? is small in relation to the measured spectrum, Y 'd? it is a good estimate of the displacement in the spectrum, which is then preferably calculated from: Yadj = Yobs + Y 'd ^. (13) In any case, however, Yadj is then used in any equation (1), (1 1) or (19) (see below) instead of Yobs and the solved equations to obtain the best value adjusted for C that is corrected for the displacement. d? is any suitable scalar representing the displacement in?. In a preferred embodiment, d? is the previously calculated value of d ?, or an average over the last "k" measurements, where "k" is 2 or more. In a preferred embodiment, K is 5 or more. In another preferred embodiment, k = 8. This method compensates each of the last measurements "k" in the same way. Alternatively, a filter can be used to provide greater compensation to the most recent values of d ?. In this mode, each of the last "k" values are compensated by a factor "? i". d? is then represented by a vector, d? °, where Each of the "k ° elements d? i ° is a previously determined value of d ?, such as d? i ° is the most recently determined value of d? and d? k ° is the oldest value, d? is then obtained from the equation: d ^ =? t d? °, (14) where? t is the transpose of the vecotr? whose "k" values "? i" are chosen so that? i > ? 2 > ... > ? k and: k S, w¡ = 1. (1 5) i = 1? It can be determined in any suitable way. To compensate equal the last "k" values of d ?, each? i is 1 / k. Alternatively, when it is desired to give greater compensation to the most recently determined values of d ?, a function such as: k ¡= _ and J_, (16) ai ¿-faii = 1 ok ¡= / r, (17) a ' ¿Sa 'i = 1 can be used, when "a" is a real number greater than 0. Preferably, "a" is greater than 1. Other suitable compensation functions are well known to those skilled in the art.

In another embodiment of this aspect of the invention, the Y-form in equation (1) is used in equation (10) to obtain an expression of dY in terms of the derivative for P with respect to?: dY = dY d? = dY • C d? = dP • C d? = P 'Cd? , (18) d? d? and the best adjusted value of C corrected for displacement is obtained from: Yobs = P • C + P '• C d? . (19) Equation (19) reduces to the shape of equation (4) when d? is modeled appropriately and an estimated value for C, Cest, is used in the expression P '' C. Cest can be obtained, for example from the solution to equation (1): where P * = (Pt P) -1 Pt (21) is the least squares transformation matrix and Yobs is the observed (or measured) spectrum. With this estimate, equation (19) is then solved for the best adjusted value of C and the magnitude of the displacement in equation (4) is resolved in the same way, as described above. In another embodiment of this aspect of the invention, the displacement is considered by adjusting P using some compensated average of the previously determined values of d ?. An adjusted value of P that considers the displacement is then given by: Padj = P (? + D?). (22) Yes d? is large, equation (21) preferably is used directly. Yes d? is small, however, P'd? is a good estimate of the change induced by the displacement in P, and Padj is then preferably obtained from: Padj = P + P 'd ?. (2. 3) In any case, Padju is then used in either equation (1) or (19) instead of P to obtain a value for C that is corrected by the displacement. In any case, Padj is used in any of the equations (1), (1 1), and (19) to determine the magnitude of the displacement, d? and a corrected value of C. Displacement can be modeled in a variety of ways. For example, in one modality, the displacement in? it is modeled to be constant across the full spectrum. In this mode d? is defined as a scalar S, which can be positive or negative, d? is given by some compensated average, S, of S. In this modality the equation (1 1) takes the form: Yobs = P C + Y 'S. (24) In this modality, the displacement can be considered to obtain more precise values of C when solving equation (24) in the way described to solve equation (4), supra, where? = Y 'and K = S. Alternatively, the entire spectrum can be displaced by an amount S, preferably as in equation (12) when S is large: Yadj = Yobs (? + S). (25) or as in equation (13) when S is small: Yadj = Yobs + Y'S. (26) In any case, however, Yadj is then used in any of equations (1), (24) or (27) (see below) instead of Yobs to obtain S and a better adjusted value of C that is corrected by the displacement. Alternatively, the displacement is considered using the derivative of the matrix P, as in equation (19): Yobs = P C + P 'C S. (27) Equation (27) is solved in the same way as equation (19) to obtain the magnitude of the displacement S and a value of C that is corrected by the displacement. Or, if a reasonable estimate of S is available, equation (22) can be used to correct the displacement by adjusting the matrix P, using: Padj = P (? + S). (28) if S is large, or: Padj = P + P 'S. (29) if S is small. In any case, Padj is then used in any of the equations (1), (24), and (27) instead of P to obtain a value for C that is corrected for the displacement. If any of equations (24) or (27) is used, a new value for the displacement magnitude, S, is also obtained. In another modality to model the experimental error that arises from a shift in the spectrum, the displacement is modeled as a compression / increase of the spectrum around a central value of?,? C. The change in? due to the increase or compression in this model is given by d? i = (? i -? c) M, (30) where? i is the "th" component of the vector? whose elements are the values of? to which the measurements are taken, and M is the increase / compression factor. You see that in this model d? It is a vector. Yes M >; 0, then the scale of the independent variable? increases. If M < 0, then the scale is compressed. It is seen from equation (30) that the change in? for which this method compensates is directly proportional not only to the increase / compression factor M, but also to the distance from the central value of?,? c. Thus, the greater the distance from the central value, c, the greater the change. In this mode, the equation (1 1) takes the form: Yobs = P C + Y '? M. (31) where? is a diagonal matrix whose diagonal elements? ¡¡are (? i -? c). Equation (31) is the same as equation (4), with? = Y '? and K = VI, and can be solved in the same way. Alternatively, if a reasonable value M is available (as described in equations (14) - (17) in the associated text), the spectrum can be displaced by a quantity? M, preferably as in equation (12) when M is large: Yadj, i = Yobs (? I + (? I -? C) M), (32) or as in equation (13) when M is small : Yadj, i = Yobs + Y '? M. (33) In any case, however, Yadj is used in any of equations (1), (31), or (34) (see below) instead of Yobs to obtain M and a better adjusted value of C that is corrected by the displacement type increase / compression. Alternatively, the displacement type increase / compression is considered using the derivative of the matrix P, as in equation (19): Yobs = P C +? P 'C M. (3. 4) Equation (34) is solved in the same way as equation (19) (when estimating C in the expression? P 'C, which is equivalent to? In equation (4)) to obtain M and a value of C that is corrected for displacement type increase / compression. Or, if a reasonable estimate of M is available (as described in equations (14) - (17) and associated text), equation (34) can be used to correct the displacement type increase / compression by adjusting the matrix P, using: Padj, i = P (? I + (? I -? C) M). (35) if M is large, or: Padj = P +? P 'C. (36) if M is small. In any case, Padj is then used in any of the equations (1), (31), or (34) instead of P to obtain a better value adjusted for C that is corrected for the displacement of type increase / compression. If any of equations (31) or (34) is used, a new value of M is also obtained. In yet another aspect of the invention, the apparatuses and methods incorporate the above techniques to compensate for both displacement and augmentation / compression. In this aspect of the invention, the displacement, d ?, is given by d? i = S + (? i -? c) M. (37) Using this expression for d? in the equation (1 1) produces: Yobs = P C + Y'S +? Y'M. (38) Equation (36) can be solved in the same way as equation (4) when defining? as the matrix [Y ',? Y '] and K as a vector [S, M]. Whereas the previous two modalities illustrated models for equation (4) in which? and K where a vector and a scalar, respectively, this modality illustrates a situation in which? and K are a matrix and a vector, respectively. In an alternative embodiment of this aspect of the invention, a compensated average of the scalars S and M is used to precalculate a fitted spectrum, Yadj, using equation (12) where d? is S + (? i -? c) M: Yadj = Yobs (? i + S + (? i -? c) M) (39) which is used if the sum S + (? i -? c) l \? "is large, or: Yadj = Y + Y'S +? Y'M (40) when the sum S + (? i -? c) M is small In any case, Yadj is used in any of the equations ( 1), (38) or (41) instead of Yobs to obtain new values for S and M and a better value adjusted for C that is corrected by both types of displacement error In another embodiment of this aspect of the invention, the change in? due to the combined effects of displacement and increase / compression can be considered when calculating the derivative of the matrix P. Using the previously reported results, equations (27) and (34), the estimated spectrum is: Yobs = P C + P 'CS +? • P 'CM. (41) Equation (41) is analogous to equation (38) and can be solved in the same way as equation (4) by defining the matrix? like [P 'C ,? P 'C] and the vector K as [S, M]. Alternatively and / or in addition, an adjusted matrix P, Padj, can be calculated using previously determined values of S and M. Using equation (22), the adjusted matrix P is given by: Padj, i = P (? i + S + (? i -? c) M) (42) when the sum S + (? i -? c) M is large and by equation (23): Padj = P + P 'C +? P 'C (43) when S + (? I -? C) M is small. In any case, Padj can be used instead of P in any of equations (1), (38), or (41) to obtain a better adjusted value of C that is corrected for both types of displacement. If any of equations (38) or (41) is used, new values for S and M are also obtained. In a particularly preferred embodiment of the present invention, the above methods are applied to analytical absorbance spectroscopy. In this embodiment of the invention, equation (1) is the well-known Beer-Lambert law: Aobs (?) = E (?) C, (44) where Aobs (?) Is the absorbance spectrum measured as a function of the wavelength?, E (?) is the matrix of wavelength-dependent extinction coefficients for the absorbents, and C is the concentration of absorbents. A is a vector whose "m" elements Ai are the absorbencies at "m" discrete wavelengths? ¡, E is an "mxn" matrix whose elements Ejj are the extinction coefficients of the component "j" to the length of wave? ¡, and C is a vector, each of whose "n" elements Cj is the concentration of the absorbent "j". From here, it is seen that in this particularly preferred embodiment, the following correspondences exist:? =? (45a) Yobs = Aobs (45b) Y '= dA (45c) d? P = E (45d) P '= dP (45e) d? ? ¡= (? ¡-? C) (45f) These definitions are then used in equations (1) to (43) to measure the magnitude of the experimental error and calculate a value of C that is corrected for the error, as described above. This modality is particularly useful, for example, in analytical UV-VIS-IR absorbance spectrophotometry (for example, as conducted with Ciba-Corning Diagnostics 800 Series Co-oximeters) used to determine the concentrations of blood components in samples of blood. The main components of the blood that ordinarily will be used in the present methods are reduced hemoglobin (HHb), oxyhemoglobin (O2Hb), carboxyhemoglobin (COHb), methaemoglobin (MetHb), sulfahemoglobin (SHb) and lipid. The Wavelength-dependent extinction coefficients for blood components based on hemoglobin for use in E can be measured directly (preferably in a highly calibrated spectrophotometer) or obtained from the literature. For example, Zijlstra et al. , Clin. Chem. 37 (9), 1633-1638 (1991). The lipid spectrum can be measured from an intravenous fat emulsion (for example, the commercially available intralipid lipid product) in an aqueous dispersion of about 10% by weight. In a particularly preferred aspect of the invention, the inventive method is used to correct the instrumental wavelength deviation observed in UV-VIS-I absorbance spectroscopy R. In another embodiment, the above methods are used to correct the experimental error in column chromatography. In this modality, the independent variable? is equal to the extraction time, t. The Yobs spectrum is the magnitude of the signal to detect the presence of the components of the sample in the eluate (for example, absorbance, refractive index). The shape of the extraction profile measured under controlled and standard conditions provides the elements of P. In this modality, a displacement in the extraction time, dt, is equivalent to the displacement in the independent variable, d ?. Such displacement may arise due to deviations from standard parameter conditions such as flow velocity, solvent strength and column temperature. The change in extraction time can be modeled as a combination of scalar displacement and linear displacement, as described in equations 837) - (43) and associated text. For example, a change in the flow velocity of the column affects the extraction time for the non-retained components and will change the extraction time for the retained components in the approximate proportion to their initial extraction times minus the extraction time for the components. components not retained. As a result, all peaks will be displaced by a fixed delay plus one proportional to the difference in extraction time.

All previous mathematical manipulations can be conducted using standard program packages, such as MATHCAD (MathSoft, Cambridge, MA). In all the methods of the present invention, the spectrum of a sample is generated from which the concentrations of the components of the sample can be determined. The concentrations of the components of the sample are then determined by employing one or more of the previously described methods. The apparatuses according to the invention incorporate the methods according to the invention to more precisely determine the concentrations of the components of the sample. According to this, the apparatus of the invention comprises a means for generating a spectrum (as defined above) of an analytical sample, a means for detecting the spectrum, a means for recording the spectrum, and a means for manipulating the spectrum according to any of the methods described herein. There are innumerable means to generate, detect and record a spectrum and are well known to those skilled in the art. For example, Hobart H. Willard et al. , Instrumental Methods of Analysis (7th ed., Wadsworth Pub. Co., Belmont, CA, 1988). The means for manipulating the spectrum comprises any computer means that can run programs that encompass one or more of the above methods to determine the concentrations of the corrected sample components for the experimental error from the measured spectrum. Of course, the practitioner will appreciate that the apparatus needs not to be an integrated unit. In a preferred embodiment, the apparatus is a spectrophotometer capable of generating a sample spectrum in the UV, VIS, or IR regions of the electromagnetic spectrum. In another preferred embodiment, the apparatus comprises column chromatography equipment. The following examples are provided to illustrate certain embodiments of the invention and are not intended, nor should they be considered, to limit the invention in any way.

EXAMPLES All the manipulations described herein are performed using the MATHCAD program from MathSoft (Cambridge, MA). Matrices of extinction coefficients owned by Ciba-Corning Diagnostic were used.

EXAMPLE 1 Estimation of wavelength shift using the derivative of the absorbance spectrum The absorbance spectrum of eleven blood samples was measured at a resolution of 1 nm and the fractions of HHb, O2Hb, COHb and MetHb were measured. calculated using a least squares solution of equation (42). The results were as follows: 0.5 -0.5 0.2 1.2 0.4 0 0.8 0.4 2.2 0.2 -0.4 93.3 100.4 98.7 97.4 98.4 100 98 98.4 97.1 94.4 99.6 5.1 -0.2 -0.1 0 0 -.03 -0.1 0.2 0.6 4.8 0.4. (to) 1 .2 0.4 1.1 1.3 1.2 0.4 1 .3 1 0.1 0.6 0.3 where each column is a sample and the rows are the fractional concentrations of HHb, O2Hb, COHb and MetHb, respectively. The data were displaced by 0.1 nm by adjusting the spectrum measured with a cubic tab and displacing the adjusted spectrum. The fractional concentrations were then determined by solving equation (42) for C using the least squares method. The result was: -0.3 -1.3 -0.5 0.4 -0.4 -0.8 0.1 -0.4 1.5 -0.5 -1.2 93.7 101 99.3 98 99 100.3 98.6 98.9 97.6 94.9 100.2 6.1 0.7 0.8 1 0.9 0.6 0.8 1.1 1.5 5.8 1.4 • (b) 0. 5 -0.3 0.4 0.7 0.5 -0.4 0.6 0.3 -0.6 -0.1 -0.4 The difference between the original data (a) and the displaced data (b) was determined to be: '-0.8 -0.8 -0.8 -0.8 -0.8 -0.8 -0.8 -0.8 -0.8 -0.8 -0.8 0.4 0.6 0.6 0.5 0.6 0.6 0.6 0.6 0.5 0.4 0.6 1 0.9 0.9 0.9 0.9 0.9 0.9 0.9 0.9 1 1. (c) 0. 7 -0.7 -0.7 -0.7 -0.7 -0.7 -0.7 -0.7 -0.7 -0.7 -0.8 As can be seen from the matrix (c), the relative change in the fractional concentrations due to the frequency shift as small as 0.1 nm can be quite pronounced, particularly for the components that are present in relatively small concentrations. Using the derivatives of the absorption data spectrum, the displacements and concentrations were calculated in a sample by the base of the sample using equation (22). The results for the displacement were: (0.1 1 0.04 0.07 0.14 0.09 0.15 0.1 1 0.1 1 0.17 0.09 0.05), (d) and the results for the concentrations were: 0.6 -1 0 1 .5 0.3 0.3 0.9 0.5 2.8 0.2 -0.7 93.2 100.7 98.9 97.2 98.5 99.7 97.9 98.3 96.7 94.4 99.9 5 0.3 0.2 -0.3 0.1 -0.8 -0.2 0.2 0 4.8 0.9 (e ) 1 .2 0 0.9 1 .6 1 .2 0.7 1 .4 1 .1 0.6 0.5 -0.1 J The difference between matrices (a) and (e) was calculated to be: 0.1 -0.5 -0.2 0.3 -0.1 0.4 0.1 0.1 0.5 0 -0.4 -0.1 0.3 0.2 -0.2 0 -0.3 -0.1 -0.1 -0.4 0 0.3 - 0.1 0.6 0.3 -0.3 0.1 -0.4 -0.1 -0. 1 -0.6 0.1 0.5 (0 0. 1 -0.4 -0.2 0.3 -0.1 1 0.4 0. 1 0.1 0.5 0 -0.4 By comparing the matrices 8c) and (f) it is seen that the differences of the original unshifted spectrum are much smaller when one considers the displacement. The average displacement of the matrix (d) is 0.104 nm. Note that even when the individual estimates of S can vary as much as 50%, the average value is exceptionally good. Using the average displacement of 0.104 nm compensates the displacement in the spectrum of each sample, the fractional concentrations were calculated using equation (24): 0.5 -0.5 0.3 1.3 0.4 0 0.9 0.4 2.3 0.3 -0.3 93.2 100.4 98.7 97.4 98.4 100 98 98.3 97.1 94.4 99.6 5 -0.3 -0.1 0 -0.1 -0.1 -0.1 0.2 0.6 4.7 0.4 (9) 1. 2 0.4 1.1 1.4 1.3 0.4 1.3 1 0.1 0.6 0.3 and the difference between the original fractional concentrations (matrix (a)) and the fractional concentrations calculated using the average over 11 displacement values, S, (matrix (h)), was determined to be: 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -0. 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 () 0 0 0 0 0 0 0 0 0 0 0 Given that the average value of the displacement was a very good estimate of the current displacement, it is not surprising that the calculated estimates of the concentrations are coupled to the concentrations of the data without shifting as well.

Example 2 Estimation of the wavelength shift using the derivative of the extinction coefficient matrix Using the same data that resulted in the matrix (a) in Example 1 and shifting the spectrum by 0.1 nm, we compensated the displacement by the method of calculate the derivative of the matrix of extinction coefficients as described in equations (18) - (27) and the associated text using estimated values of the concentrations of the components obtained in solving equation (42). The estimated fractional concentrations were determined as: r0.6 -1 .3 0.0 1 .5 0.3 0.4 0.9 0.5 2.7 0.2 -0.8 93.2 100.8 98.9 97.6 98.4 99.7 97.9 98.4 96.7 94.4 99.9 5.0 0.3 0.2 -0.3 0.1 -0.7 -0.2 0.1 0.0 4.9 0.8 (i) 1 .3 0.0 0.9 1 .6 1 .1 0.8 1 .4 1.1 0.6 0.6 0.0 which differ from the original concentrations of the matrix (a) as follows: O.1 -0.5 -0.2 0.3 -0.1 0.4 0.1 0.1 0.5 0 -0.4 -0.1 0.4 0.2 -0.2 0 -0.3 -0.1 0 -0.4 0 0.3 -0 . 1 0.5 0.3 -0.3 0. 1 -0.4 -0.1 -0.1 -0.6 0.1 0.4 (j) 0.1 -0.4 -0.2 0.3 -0. 1 0.4 0. 1 0.1 0.5 0 -0.3 The displacements of the wavelength, S, were determined to be: (0.1 0.04 0.07 0. 13 0.09 0.15 0.1 1 0.1 1 0.16 0.09 0.05 0.03). (k) The average displacement, S, is 0.097 nm. Using the estimated average displacement, S, and solving for concentrations C using equation (74) produced the following concentrations: 0.5 -0.6 0.2 1 .3 0.4 -0.1 0.9 0.4 2.3 0.3 -0.4 93.2 100.4 98.7 97.3 98.4 100 97.9 98.4 97 94.3 99.6 5.1 -0.2 -0.1 0 0 -0.3 -0.2 0.3 0.6 4.5 0.5. (I) 1 .2 0.4 1 .1 1 .4 1 .3 0.3 1 .3 1 0.1 0.6 0.3 which resulted in the following differences of the original concentrations in the matrix (a): 0 0 0 0 0 0 0.1 0 0 0 0 -0.1 0 0 -0.1 0 0 -0.1 0 -0.1 -0.1 0 0.1 0 0 0 0 0.1 -0.1 0 0 0 0 (m) 0 0 0 0.1 0 0 0.1 0 0 0.1 0.00 Example 3 Averaging S over eight previous measurements Using the data from Example 1 and focusing on the ninth sample, one can average over the first eight samples to obtain a value of S. The original fractional values for the ninth sample (obtained from the unexpanded spectrum ) were: 2. 2 97.1 0.6 (o) 0.1 and the fractional concentrations after displacing the spectrum by 0.1 nm were: The average displacement of the first eight samples was 0.096. The fractional concentrations and the wavelength shift were calculated using the derivatives of the extinction coefficients. The fractional concentrations obtained were: 2.3 97 0.6 (q) 0.1 and the displacement of the calculated wavelength was 0.06. The estimated displacement for the first sample was 0.1. Continuing to average the displacement over the last 8 samples, the new average displacement S is given by 0.096 x 8 - 0.1 + 0.06 = 0.103 7 8 using this revised figure for S, the adjusted value for the matrix of extinction coefficients Eadj, was calculated and used to obtain a revised set of fractional concentrations: which, for this sample, differs insignificantly from the previous estimate.

Claims (10)

4 CLAIMS

1. An improved method to determine the concentrations of one or more components of an analytical sample of an observed spectrum estimated by Yobs (?) = P (?) C, where Yobs is a vector whose "m" elements are the magnitudes of the observed spectrum in each value of an independent variable?, C is the vector whose "n" elements are the estimated concentrations of "n" components that contribute to the measured spectrum, and P is an "mxn" matrix whose elements are the magnitudes of the contribution of the spectrum of each of the "n" components in each of the "m" values of the independent variable?, where the method comprises generating a sample spectrum from which the concentrations of the components of the sample are determined, and to determine the concentrations of the sample from the spectrum, including the improvement to correct the experimental error when modeling the experimental error as "r" types of errors given by the product? K, where K is a vector whose "r" elements are the magnitudes of each of the "r" types of experimental errors and? is a matrix "m x r", whose elements are the relative errors in each value of? for each type of experimental error, add the product? K to the spectrum estimated as Yobs (?) = P (?) C +? K and solve for the best adjusted values of C and K, where "n" and "r" are integers each greater than or equal to 1 and "m" is an integer at least "n + r".

2. The method according to claim 1, wherein the best adjusted values of C and K are determined by least squares analysis.

3. The method according to claim 2, wherein r is 1 and the experimental error is modeled as a displacement in? for the quantity d ?, and? = Y '(dYobs) / (d?).

4. The method according to claim 2, wherein r is 1 and the experimental error is modeled as a displacement in? for a quantity d? = P 'C, P' = dP d?

5. An improved method to determine the concentrations of one or more components of an analytical sample, whose observed spectrum are estimated by the equation Yobs (?) = P (?) C, where Yobs is a vector whose "m" elements are the magnitudes of the spectrum observed in each value of an independent variable?, C is the vector whose "n" elements are the estimated concentrations of "n" components that contribute to the measured spectrum, and P is an "mxn" matrix whose elements are the magnitudes of the contribution to the spectrum of each of the "n" components in each of the "m" values of the independent variable?, wherein the method comprises generating a sample spectrum from which the concentrations of the sample component are determined, and determining the concentrations of the components of the sample from the spectrum, comprising the improvement correcting the experimental error by modeling the experimental error as a displacement of the spectrum by an amount d ?, estimating a displacement, d ?, calculate an adjusted spectrum, Yadj, using an equation selected from the group consisting of Yadj = Yobs (? + d?) and Yadj = Yobs + Y '- d ?, and determine the concentrations, C, when solving an equation selected from the group consisting of Yadj = PC and Yadj = PC +? K for the best adjusted value of C, where? Take the shape? = Y '= (d Yobs) / (d?) = P 'C, where P '= dP d? "m" and "n" are integers, "n" is greater than or equal to 1, and "m" is at least "n".

6. A method according to claim 5, wherein d? is average compensated of "k" previous values of d ?, d? * is a vector of length "k" whose elements are previously determined values of d ?, and w satisfies the equation: k C Í = 1. i = 1

7. The method according to claim 6, wherein each element of w is selected from the group consisting of 1 / k, k i = 1 and k w¡ = J_ < _1_ . a¡ L-, a¡ i = 1 where "a" is any real number greater than 1 and "i" is an integer from 1 to "k".

8. An improved method to determine the concentrations of one or more components of an analytical sample whose observed spectrum is estimated by the equation Yobs (?) = P (?) C, where Yobs is a vector whose "m" elements are the magnitudes of the spectrum observed in each value of an independent variable?, C is the vector whose "n" elements are the estimated concentrations of "n" components that contribute to the measured spectrum , and P is an "mxn" matrix whose elements are the magnitudes of the contribution to the spectrum of each of the components "n" in each of the "m" values of the independent variable?, where the method comprises generating a sample spectrum from which the concentrations of the components of the sample are determined, and determine the concentrations of the components of the sample from the spectrum, including the improvement to correct the experimental error when modeling the experimental error as a displacement in? for a quantity d ?, estimate a displacement, d ?, calculate an adjusted P matrix, Padj, using an equation selected from the group consisting of Padj = Pobs (w + d?) and Padj = Pobs + P 'd? and determine the concentrations, C, by solving an equation selected from the group consisting of Yobs = Padj C and Yobs (?) = Padj (?) C +? K, for the best adjusted value of C, where? Take the shape of? = Y '= (dPobs) / (d?) Or? = P '• C, where P' = dP d? and "n" is greater than or equal to 1 and "m" is at least "n".

9. A method according to claim 8, wherein the compensated average of d ?, d? , is the average compensated for k previous values of d ?, where k is greater than or equal to 1, and is determined from the equation d? = wt d? *, where wt is the transpose of a vector w of length "k" whose elements are the relative ones to be applied to each one of the values k previous to d? * is a vector of length "k" whose elements are the previously determined values of d ?, and w satisfies the equation: k = 1. 1 = 1

10. The method according to claim 9, wherein each element of w is selected from the group consisting of 1 / k, k. = J_ / ^ _1_. a i * -f a i i = 1 i = 1 where "a" is any real number greater than 1 and "i" is an integer of 1 to "k" eleven . The method according to any of claims 3, 4, 5, or 9, wherein the displacement in? is modeled as a constant across the full spectrum and equal to a scalar d? = S. 12. The method according to any of claims 3, 4, 6, or 9, wherein the displacement in? is modeled as d? ¡= (w¡ - wc) M, where d? ¡is the displacement in the ¡és, ¿value of?,? ¡¡¡¡¡¡¡¡¡¡¡¡¡¡¡¡¡Esirno value of?,? c is a constant value of?, and M, which is any real number, is the magnitude of the error. 13. The method according to claim 1, wherein Y is an absorbance spectrum of the sample, P is the matrix of coefficients of 0 Extinction for the absorbent components of the sample, and? is the frequency at which the absorbance is measured. The method according to claim 12, wherein Y is an absorbance spectrum of the sample, P is the matrix of extinction coefficients for the absorbent components of the sample, and? is the frequency at which the absorbance is measured. 15. An apparatus for determining the concentration of the components in a sample comprising a means for generating a spectrum of the sample; a means to detect the spectrum; a means to record the spectrum; and a means for determining the concentrations of the components of the sample from the spectrum according to the method of claim 13. 16. An apparatus for determining the concentration of the components in a sample comprising a means for generating a spectrum of the sample; a means to detect the spectrum; a means to record the spectrum; and a means for determining the concentrations of the components of the sample from the spectrum according to the method of claim 14. 17. The apparatus according to claim 15, wherein the apparatus comprises a spectrophotometer that generates a spectrum of the sample in a region selected from the group consisting of the UV, VIS, and IR of the electromagnetic spectrum. 18. The apparatus according to claim 16, wherein the apparatus comprises a spectrophotometer that generates a spectrum of the sample in a region selected from the group consisting of the UV, VIS, and IR regions of the electromagnetic spectrum.