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CLIN. CHEM. 22/9, 1438-1450 (1976) A Propagation of Error Analysis of the Enzyme Activity Expression. A Model for Determining the Total System Random Error of a Kinetic Enzyme Analyzer Thomas 0. Tiffany,1’2 Philip C. Thayer,2 Chris M. Coelho,2 and Gilbert B. Manning2’3 We present a total system error evaluation of random error, based on a propagation of error analysis of the expression for the calculation of enzyme activity. A simple expression is derived that contains terms for photometric error, timing uncertainty, temperature-control error, sample and reagent volume errors, and pathlength error. This error expression was developed in general to provide a simple means of evaluating the magnitude of random error in an analytical system and in particular to provide an error evaluation protocol for the assessment of the error components in a prototypeMiniatureCentrifugal Analyzer system. Individual system components of error are measured. These measured error components are combined in the error expression to predict performance. Enzyme activity measurements are made to correlate with the projected error data. In conclusion, it is demonstrated that this is one method for permitting the clinical chemist and the instrument manufacturer to establish reasonable error limits. Additional Keyphrases: centrifugal analyzer source of ment #{149} analytical error #{149} enzyme #{149}variation, activity measure- Recent attention has been focused on the error components of spectrophotometric analytical instrumentation and on the establishment of acceptable error limits for such instrumentation used in the clinical laboratory (1, 2). These analytical instruments range from manually operated photometric devices to fully automated analyzers. The latter contain pipettors for sample and reagent, timing sequencers, temperature control modules, photometers, quantization modules, and data processing modules, all acting in concert to simply and rapidly produce results for the clinician. ‘Pathology Associates, Inc., Terminal Box 2687, Spokane, Wash. 99220. (To whom reprint requests should be addressed.) 2 Micro Biochemical Research Corp., Spokane, Wash. 99206. Sacred Received ‘ 1438 Heart Medical Center, May 3, 1976; accepted CLINICAL CHEMISTRY, Spokane, Wash. July 1, 1976. Vol. 22, No. 9, 1976 99204. The problem becomes how to establish and to define the acceptable limits of accuracy (systematic error) and precision (random error) for these more complex analytical systems. Photometric error and its relationship to error in kinetic analyses has been considered by several authors; Ingle (3) and Ingle and Crouch (4-6) considered the precision of normal spectrophotometric techniques and the signal-to-noise (S/N) ratio theory of fixed time spectrophotometric reaction rate measurements. Maclin (7) attempted a system analysis of the GEMSAEC Centrifugal Analyzer as a kinetic enzyme analyzer system. Maclin et al. (8) further defined the relationships between various instrumental noise sources and the performance of the instrumental system which is designed to do enzyme kinetic assays. Pardue et al. (9) developed an excellent comprehensive treatment of photometric error, which was to some extent a compendium of the other efforts. All of these efforts have produced in varying depths and usability an error analysis of spectrophotometric measurements related to different forms of kinetic analyses. Ingle and Crouch are rigorous and thorough in their treatment of photometric error. They begin from first principles to define the various photometric error components, then they methodically apply them to various types of photometric measurement techniques. They limit their discussion to photometric error. Macin extended the error estimation to the total system concept to include errors of pathlength, volume measurement, and temperature fluctuation. He further developed the idea of an error budget as it relates to system error. Pardue et al. effectively considered five categories of photometric error and combined these derived error coefficients into mathematical expressions that permit the calculation of systematic and random error components. Furthermore, the random error analysis developed by this group is useful because it allows the instrument user and manufacturer to define the instrumental nature of the photometric error. However, these authors did not consider the other error terms associated with a more complex automated kinetic photometric analyzer. Therefore this paper sets out to develop a total system error evaluation of random error based on a propagation of error analysis of the expression for the calculation of enzyme activity. The expression contains terms for photometric error, timing uncertainty, temperaturecontrol error, sample- and reagent-volume errors, and pathlength error. The error expression was developed in general to provide a simple means to evaluate the magnitude of random error in an analytical system and in particular to provide an error evaluation protocol for the assessment of the error components in a prototype miniature centrifugal analyzer (MCA) system. The error expression provides levels of sophistication for its use. It will be demonstrated to be one means, by simple experimental technique, for the clinical laboratory practitioner to measure total random error and to determine its magnitude in the various module components. It will further permit the manufacturer to dissect by methods similar to Pardue et al. the instrumental error components. Finally, such an approach is one method of permitting the clinical chemist and the instrument manufacturer to establish mutually acceptable error limits that will provide a balance between good clinical enzymology and needed economy in instrumentation. ()3u Su 2 = (U/liter) I t.A ‘Vt’ (-)#{231}--)#{231} I1\ e1(T] A (1) where LA is the absorbance change during the measurement period, tmin is the elapsed time in minutes, V is the total reaction volume, Vs is the sample volume, B is the pathlength, A isthe molar absorptivityof the monitored reaction species and K isthe temperature coefficientof the enzyme reaction.The exponential term is added in this treatiseto include the obvious temperature dependence of enzyme activity.The ad- dition of this term in no way suggests that the use of temperature coefficients to correct enzyme activity from one temperature to another is sanctioned by this author. For any given U where U = F (X, Y. . .), the basic propagation of error theory expression for the uncertainty (variance) of U in terms of all the related error source terms X, Y. . , is: . Sy2 (2) +... v4ance of enzyme activityas a function A, V8, B, and T as follows: S2=’_i_” u (MA) S2 + V, Sr2 (3) #{212}U2 1-”) Stmin2 ‘atmin, + (L)2sv2+ + time, (-)2sv2 () dU2 SB2 + aU2 (;) Solving the partial derivatives of each variable, substituting the results in expression 3, and dividing both sides by U2 results in expression 4: S2 S ..A Q .2 ‘-‘tmin 2 +%/2 min 2 S2 The relative variation (CV) standard S deviation or coefficient (4) of is defined as: CVx= A general approach to the development of a random error instrumental evaluation protocol is to perform a propagation of error analysis of the enzyme activity expression as a reasonable model to develop the error treatise (10). The formula for the calculation of enzyme activity is derived basically from Beer’s Law and is expressed as: = (;) 3U2 Sx2 + where S is the standard deviation. Therefore the formula for the enzyme activity (U), expression 1, can be operated on according to the simplified expression for the error (2) to determine the Propagation of Error Analysis of the Enzyme Activity Expression Enzyme activity 2 (5) Thus the variance or uncertainty of enzyme activity as a function of the differenterrorcontributionsof the analyticalsystem simply becomes: CVenzymeactivity = [(CVA)2 + (CVB)2 + (CVtmin)2 + (CVv5)2 + (CVv)2 + K2S2J1”2 (6) The usefulness of expression 6 is in the simplicity with which it can be used to predict the magnitude of the percent relative standard deviation of enzyme activity (CV) as a function of the magnitude of the uncertainty in each of the individual error parameters. As a simple example, if each term has an uncertainty or relative standard deviation of (0.01) or 1% relative standard deviation, the percent relative standard deviation of the measured enzyme activitywillbe: CV = 100[6(.01)2]112 = 2.45% (7) Obviously, not allthe terms willcontribute to the variance with the same magnitude and some of the terms willbe much lessthan 1%. Therefore, it isthe purpose of thisdiscussionto examine each of the error terms of expression 6 as they relateto instrumental system components, and to the MCA prototype, to make estimations of the expected magnitude of uncer- tainty from each system component, to correlate these errors with published data, and discuss the error parameters in terms of the actual enzyme analysis. It is useful to relate each term of expression 6 with an analytical system component: C V is related to the elecCLINICAL CHEMISTRY, Vol. 22, No. 9, 1976 1439 tro-optical noise of the transmission signal (T) from the analytical module and the quantization error of the analog-to-digital converter (ADC) in the computer module. CVv, CVv represent the uncertainty of volume measurement ofthe sample and reagentpipetting system.C Vtmjn istheuncertaintyoftime when theactualabsorbancemeasurements are made by the computer or other timing device. For a computer, the uncertainty is of the order of 10 ms or less. The magnitude of this error in an actual enzyme activity measurement is of the order of 4 X 10-% for a 4-mm enzyme activity assay. Therefore, this term can essentially be ignored for computerized instrumentation but not for manually timed photometer systems. CVB is the pathlength variation. This term is important for multi-cuvette centrifugalanalyzers.The K2S2 term is related to temperature control. The significance of this term will be explored in the next section. To summarize this section, the expression for the calculation of enzyme activity can serve as a good model to examine the uncertainty in the determination of enzyme activity as this uncertainty is related to the various analyticalcomponents of the instrumental system. A simple and useable expression was obtamed: CV(IU/L) = [(CV) 2 + (CVv) 2 2 + (CVv8) + K 2 + (CVB) 2 2 P/2 (8) which willbe usefulinthepredictionof the magnitude of the system error (C17) from experimentallyqf mathematicallyderivedindividualerrorcomponents. nents and to be able to predict what their worst case contribution will be to the overall noise or uncertainty of the analytical system. This is important in the quality assurance of the analytical system by the manufacturer. Furthermore it is important to the establishment of meaningful individualmodule performance characteristics that arenecessarytothefinaldefinition oftotal analytical system performance by the various society committees on instrumental standards. The individual error components will be examined in this section. A prototype Miniature Centrifugal Analyzer (11, 12) willbe used as a point of discussion. Photometric error (electrical and quantization), total volume error, sample volume error (the pipettor system), temperature uncertainty (temperature control), and pathlength error components will be considered. There are two functions of this section. First, it serves to establish individual error limits in a more rigorous manner, thereby providing criteria by which to judge the performance of analytical systems [see Pardue et al. (9)]. Second, the individual error limits can be determined for each analytical system component and these relative standarddeviationscan then be insertedinto the total error expression (expression 6) to demonstrate the system performance for the analyticalrange of conditionsproposed forthe instrument.Enzyme ac#{149} tivity assays can then be performed and the results correlatedwith the predictederror. Photometric Error Individual System Component Error Sources The usualphotometricerrortreatment has been a propagationoferroranalysisoftheBougert-Beer law. The concentrationin terms of absorbance or trans- The expression for the total system error in the measurement of enzyme activity as derived in the pre- mission error are evaluated, assuming sorbance or transmission error (13). vious sectionsimply relates the total uncertainty of enzyme activityto the relativestandard deviation (coefficient of variation) of each error component. Therefore, one can determine the magnitude of the relative standard deviation of each term (absorbance, volume, temperature, etc.) by experimental measurement. The resulting measured uncertainty of the mdividual analytical system components can then be used to predictthe overalluncertainty(aswillbe shown in thediscussion section) and toidentify critical individual errorcomponents ofthe analytical system.This isthe levelatwhich the manufacturerneeds toassessthereliability ofhisproduct (e.g., production quality control), and itisthelevelatwhich theclinical chemist can assess the reliability and continuedperformance ofthe analytical system in his laboratory. However, the enzyme values for ab- This type of treatment of photometric error has limited use for current instrumentation, which is in general not quantization error limited. However, the approach to the evaluation of photometric error as outlined by Pardue et al. (9) is extremely useful and will serve as a base of discussion. A direct result of their discussion was to demonstrate that the major error sources of a centrifugal analyzer, which has constant reference up date (minimization of source drift and dark current reading), are:transmissionerrorcaused by electrical noiseand quantizationerror caused by the analog-to-digital converter. The standard deviation of absorbance (SA) with respecttotransmission(Tr) is: 0.434 STr (1) activity error expression derived in the previous section Tr did not give a further dissection of the individual error components intotheirrespectiveinstrumentalor meHowever, this only relates the uncertainty of absorbance chanicalerrorcomponents. For example, the uncer5A as a function of the transmission error STr. Nothing taintyofabsorbancemeasurements (photometricerror) isindicatedabout the natureor magnitude ofthe error can be further broken down into electrical noise, components contained in the total transmission error quantizationerror,source flickernoise,and stray STr. Random error is expressed as: light-dark current error. It is important to be able to estimate the contribution of the actual error compoS = (5i2 + S22 + S32. . . SN2)’12 (2) 1440 CLINICAL CHEMISTRY, Vol. 22, No. 9, 1976 where the total random error is the sum of the squares of the individual error components (section 1). For this discussion, two types of error (a) electrical noise and (b) quantization error are of interest. Electrical I I 0- / Noise / Variations in photon arrival rates at low intensities (3) and shot noise-white noise (4) lead to transmittance errorsthat are proportioned to the square root of transmittance (9): = error from noise becomes, o / z 4 SNA = - / / / / / /0; (4) / I 0’ /‘\ 4/ /1 Quantization OS / ,., zv constant, SNT is the random noise or root mean square noise of the analog signal, and Tr is the transmission. __________ S & I.. I,.. I.- where B is a proportionality M$SC (S4O OITTID $40 TA MS*C 1405) / 4. 0.434 BSNT Tr112 - ‘TIU04(TIC*l. -- / 0’ and the absorbance .- x.- ‘s- / U.- (3) BTr”25N ‘ / 4- 0 Tr112SN STr / / I I/ P0WTEOS / (io’ ./ ,,./ / / S / 4/ Error / / The quantization error for a 12-bit ADC in terms of absorbance has been shown (9) to be as follows: A / #{149} I #{149}#{149}, z, i ABSORBANCE ‘5) Fig. 1. Standard deviation of absorbance (SA) vs. absorbance SQA (7.05 X 10-s) = (5) for quantization error and measured photometric error of the MCA system This was derived from the error expression for an N-bit ADC as previously described by Pardue et al. (9). Total Random Error of Absorbance (MCA) The total random error of absorbance SA for the MCA photometric system can be related to the quantizationerrorand the electrical noiseas follows: SA which becomes = 0.434 = [ [(SNA)2 BSNT (T1/2) + (SQA)2]’12 + (7.05 x 2 (6) 10_5)2]1/2 (7) T,. upon substitution of expressions 4 and 5. This type of two-term or multiple-term photometric error expression has been shown to be usefulforpredictionof photometric error over a wide range of operational absorbances, to be useful in showing what types of errors predominate for a given system at different levels of absorbance, and to be useful for establishing what can be done to minimize instrumentally the photometric error (9). Measurement of Photometric Error The test of expression 7 is how well it relates to the actual generated photometric error data and how useful it is for the prediction of what the major error components are. A first approach in its use is to assume that the total photometric error contribution is due to quantization error. Therefore expression 7 reduces to, = 3.05 X 105e23A (8) as indicated by Pardue et al. (9). Thus by simply inserting various absorbance values from 0 to 2.0 absorbance units into expression 8, the random error caused by quantization error can be obtained. This is plotted in Figure 1 (long dashed lines). The actual photometric error can be obtained by the measurement of absorbance of solutions that are relevant to the clinical laboratory. The absorbance of solutions of NADH and p-nitrophenol (PNP) at 340 nm and 405 nm, respectively, can be made. Photometric noise measurements were made by pipetting solutions of increasing concentrations of NADH in Tris buffer (0.1 mol/liter, pH 7.4) and of PNP in 0.6 mol/liter MAP buffer into a calibrated rotor. The solutions can be placed in order of increasing absorbance for convenience and with the final solution concentrations so chosen as to provide several absorbances over the range of 0-2.0 absorbance units. The absorbance data are then acquired with a tabular absorbance program that gives the mean absorbance measurements and the standard absorbance for each cuvette. These data can then be plotted as shown in Figure 1. The measured photometric error (SA) vs. absorbance as obtained from the MBRC MCA system at 340 nm and 405 nm are listed in Table 1. The data were fitted to an exponential curve, which resulted in expressions; SA = 7.1 X 105e 1.64A and = 7.7 X 10 5el69A for the 340 nm and 405 nm data, respectively. Absorbance data from the ENI SN1O9 analyzer obtained by Maclin (7) gave a similar expression of SA = 7.2 X 105e i.6OA The exponential formu- CLINICAL CHEMISTRY, Vol. 22, No. 9, 1976 1441 Table 1. Measured Photometric Precision vs. Absorbance of the MBRC MCA System at 340 nm and 405 nm 405 nm 340 nm Absorbance SA X iO 0.2273 0.5469 1.0711 1.0 0.3905 1.5 0.7361 2.8 5.3 1.0764 3.7 1.5824 10.8 1.4238 2.0616 18.2, 2.1421 A0=7.07X b=1.65 r= 0.99 = A0= 7,73x b= 1.69 r= 0.98 10 7.07 x 10’ SA X iO Absorbance e1.65A 9= 7.73 x 1.8 7.3 34.7 1O o 10e’ 00 0* 0* ci 0 0 14 Ia IS 00 ABSORBANCE- Fig. 2. Standard deviation of absorbance vs. absorbance as a function of data averaging lation of the error data as given above for the simple log data plots of the data (Figure 1) provide two types of information. First, the photometric error at any absorbance level can be readily obtained and, as will be NA05-4 PWS V-SiX 0 ,. 0.59 ojar.i tP., r- shown later, can be used to calculate the photometric contribution to the overall relative standard deviation of enzyme activity. Second, as noted by Pardue et al. (9), TUP.bI0 uM ‘A’#{176}’’ EuFF. 0.97 these data provide information about the nature of the photometric error components. As an example, the data plotted in Figure 1 reveal that the photometric error is more than quantization error. The actual data, like the data of Maclin as plotted by Pardue et al. (9), seem to show less error than predicted by the quantization error expression at higher absorbances and more error than predicted at lower absorbances. The reason for this is the dual nature (at least) of the error component of the photometric error, which is comprised of the transmission independent quantization error (Be23A) and ABSORBANCE-340 FIg. 3. Standarddeviationof absorbancevs. absorbanceof clear and turbid NADH solutions the analog error, which in turn depends on the square root of the transmission signal (Be’ 15A). A log plot of Figure 1 would reveal a slope of less than 1 for these 1442 CLINICAL CHEMISTRY, Vol. 22, No. 9, 1976 A 223 data. If the above error was quantization error, only the slope would be one, if the error were only transmission noise (analog) the slope would be 0.5 (9). Information generated in this manner is probably more useful to the manufacturer than to the clinical chemist in the laboratory. However, they do provide a definition of the limits of error faced by the instrument manufacturer in his effort to develop a reliable clinical analyzer. It is easy to say that by digitally averaging the data the error will be reduced by As shown in Figure 2, this improves the data, but digital averaging has its limits. The effort by Pardue points out that one can reduce quantization error by going to a higher resolution ADC or by imposing an auto-ranging device to permit certain portions of the transmission signal to operate over the full range of the ADC. As an example, 100-10% transmission could operate over 4096 bits, and then the 10-0% could be ranged over the same 4096 bits. Increased resolution of the ADC will not be the total answer if the analog signal is extremely noisy. Thus one NM (0) o,I,, .53 LiJ L) “ r-0999& I20 5,c-O.0t45 aD o U) so 05 <0 5 .030 0’S I I I .100 I’S ISO P-NITROPHENOLJ 13 .200 X tO MOL/LITER z.. o B 020 00) I’S I iso IS.’ Icc A, - 3.15.6/.505. A, -(0Z4XIO0 r - 0.9999 009 aD,s Q50 .05 <C 590.0. 0 0.1 [NADHJ 0.t 0.3 04 03 00 0.1 X I0 MOL/LITER Fig. 4. Optical linearity of the MCA system at 340 nm and 405 nm may decrease the noise by grounding, by changing the mode of detection to photon counting, or by other error-reducing alternatives. A combination of the above can reduce photometric error by a factor of 10, as pointed out by Pardue et al. (9). An additional error component not mentioned by Macun or Pardue is the increased error caused by turbidity of the sample. The effect of serum turbidity on photometric error is shown in Figure 3. Neglect of this component can negate the effortsto reduce quantizationand analogsignalerror components. The economic balance between further reducing photometric error and status quo can best be defined by the current level of performance of the instruments, as indicated by the enzyme activity error expression. The definition of acceptable level of performance of the instrument by the clinical chemist, based on sample throughput and acceptable error of activity measurement at critical enzyme levels, is a necessary component to the establishment of good clinical enzymology and economy in instrumentation. Systematic Photometric Table 2. LinearityStray-LightData for NADH Absorbance Measurements at 340 nm NADH concn, mmol/Iiter Absorbance 0.064 0.2037 0.2097 0.2082 0.128 0.4 192 0.4183 0.4247 0.256 0.8336 0.8356 0.8332 0.384 1.2511 1.2533 1.2574 0.512 1.6502 1.6506 1.6444 0.640 2.008 2.015 2.020 coefficient = 1. The expression for calculation V’I,A of volume: 1.lrrS 2. The error in volume caused by radius: I dV\ (A)SVr*,,j)Sr Svr (I2lTf)Sr = (B)-T7#{176}’/irr2 2r (C) SVr (/2irr) r = 3. The error in volume caused (A)Svj Svi (B)---=j ()Si11Tr2Sj irr2 S by Ieigth or linear motion: =ri 4. The total random error in volume: S, = / Sp.\1’/ (S1f +SVr2)h/2 = Error Accurate preparation of NADH and PNP solutions with absorbances over the range of 0-2.0 absorbance units provides the tools for measuring photometric error. Furthermore, the measured absorbance of these solutions can be used todeterminestraylight. A plotof the absorbancedata at405 nm and 340 nm is shown in Figure4.The measured absorbancedata and amount ofstraylightare listed in Table 2.The straylightcom- Stray-light Table 3. Random Error of Volume Caused by Mechanical-Driven Syringes 0.16% (see Appendix A) ponent at 340 nm and 405 nm was calculated to be less than 0.2%. The molar absorptivity of NADH at 340 nm was calculated to be 6.24 X 10 liter mol’ cmt, which is a 0.3% error. The method for calculation of stray light for centrifugal analyzers in general is shown in Ap- pendix A. Sample-Reagent Pipettor Error The random error caused by the pipetting and dispensing operation of a mechanically driven piston or syringe system can be determined by a propogation of error analysis of the formula for volume, as outlined in Table 3. The random error of volume displacement (sample or reagent) can be considered as being com- prised of an error term from radius variation-either of the plunger delimiter-and tioning or the barrel, whichever an uncertainty of the plunger or barrel is the tolerance in the repeatable posi(linear motion), which can be called backlash. The expression for the random error in volume (line 4, Table 3) can be used to defmethe limits of a pipetting system for a given set of tolerances. A 1OO-0’1 syringe can be chosen as an example. If the radius is 7.28 X 10-2 cm and the variation in syringe radius SR = ±3.64 X 10 cm, or 0.5%, then the standard deviation in volume (SVr) from radius error alone for pipetting a volume of 1 Ml is ±0.01 Ml, or a relative standard deviation of 1% (line 2C, Table 3). If we state that the error in linear positioning (backlash) of the syringe driving mechanism can be no greater than S1 = ±2 X 10 inch, or ±5 X 10 cm, then the standard deviation of volume caused by error in linear motion (S,1) for a volume of 1 Ml is *8.2 X 10 Ml (line 3B, Table 3). The total error from radiusand backlashuncertainty5,, is ±0.0131 Ml, or a relative standard deviation of 1.3% at 1 Ml. These are obviously very close tolerances for a pipettor system and show the stringent requirement necessary to provide a pipettor system capable of precisely dispensing the CLINICAL CHEMISTRY, Vol. 22, No. 9, 1976 1443 2 0 11J2 #{149} XAMPLFI * 00 0.5 - 5irF uJ 0 . d 111 (I 5100 > A Jo RI5..UT I- LI) VOLUME - uJ 10 > 1 -.-. -- B 0* uJ z Q0 0-- Z 05 30 M7O59 VOLUME DELIVERED -j.iL TOTAL LIJ 1._I .0 Fig. 6. Relative standard deviation (in percent) vs. volume of 00 0* lx dispensed aliquots of water 00a 1/1 ID #{176} 05 I I I I I I VOLUME -iL Fig. 5. Relative standard deviation (in percent) and absorbance of pipetted NADH solution vs. volume Temperature Uncertainty The effect of temperature on enzyme reactions has been expressed in terms of the temperature coefficient micro-scale volumesrequiredforcurrentkinetic enzyme analyzers. If one considers the error in pipetting 2, 5, or 100 Ml, the relative standard deviation drops to 0.65%, 0.26%, and 0.13%, respectively. Evaluation Several of Pipettor Error methods can be used to evaluate pipettor performance, including measurement of absorbance of pipetted optically absorbing solutions, the weighing of dispensed volumes of de-ionized water, and measurement of the radioactivity of dispensed radiolabeled materials. The first two methods have been used to measure sample pipetting precision and the precision of total volume pipetting (sample plus diluent), respectively. Concentrated solutions of accurately prepared NADH (Sigma Type lilA) in 0.1 mol/liter Tris buffer, pH 7.4, and concentrated solutions of p -nitrophenol (PNP) in 0.6 mol.liter MAP buffer were pipetted into calibrated rotors. The precision of pipetting several samples (n = 10) at various volumes in the range of 1 to 20 MI is shown in Figure 5A. The pipettorprecisionfora pipetting sample is better than 0.5% (relative standard deviation) above 5 Ml. The relative standard deviation for pipetting sample and diluent for volumes in the range of 50-75 Ml is given in Figure 6. The overall error in pipetting and dispensing total volume is less than 0.5% in this volume range. Pipettor Linearity The linearity of pipetting and dispensing increasing volumes of NADH is shown in Figure 5B. The regression line of absorbance vs. volume is y = 0.04 + 1.02x, with r = 1.00. The slope of the regression line for absorbance vs. NADH concentration yielded a molar absorptivity of 6.28 liter mmol1 cm1 compared to a literature value of 6.22, which is a 0.96% error. These data represent the systematic error of the pipettor system. 1444 CLINICAL CHEMISTRY, Vol. 22, No. 9, 1976 (Qio), the factor by which the velocity of the enzyme reaction increases on increasing the reaction temperature by 10 #{176}C (14). The temperature coefficient for enzyme catalyzed reactions usually lies between 1 and 2, depending on the enzyme and on the degree of coupling used in the enzyme activity assay. The temperature coefficient decreases for coupled enzyme catalyzed reactions. The effect of temperature on the reaction rate is generally related to the empirical Arrhenius equation: K = (9) e/RT where K is the reaction rate constant, Ea is the energy of activation of the reaction, R is the universal gas constant, and T is the temperature in K (degrees Kelvin). Enzyme activity (V) can be expressed as follows: V=K[E] (10) where K is the rate constant of the enzyme reaction, [E] is the enzyme concentration, and V is the velocity of the reaction or activity. The temperature-dependent activity becomes: (11) Ve_E/RT[E1 The type of expression is limited by the thermal denaturation of the enzyme(s) in the activity assay. However, over a limited temperature range it is useful, for this discussion, to obtain the temperature-related activity change and hence the uncertainty caused by temperature fluctuation. Henry (15) has reported temperature in enzyme activity experimentally determined correction factors for lactate dehydroge- nase, alanine aminotransferase, and aspartate aminotransferase (LDH, ALT, and AST). If one assumes an activity of 100 U/liter for each of the three enzymes at 32 #{176}C, then the activity of each enzyme can be calculated for any other temperature from 25-40 #{176}C. The resulting velocity/temperature data can be fitted to an exponential curve. The resulting expressions for LDH, ALT, and AST were: v = be.0718T, r = 0.999; V = beO649T, r = 0.999; and v = be0O629T, r = 0.999, respectively. The Qio temperature coefficients for these reactions, calculated from the relationship Qio = efiT, were found to be 2.05, 1.91, and 1.88, respectively, for the three enzymes. The use of temperature correction factors for converting a measured activity at one temperature to a calculated activity at a second temperature is not recommended because of the obvious enzymedependent nature of the factor, the reaction dependence of the factor, and the effect of temperature denaturation of the enzyme. However, the above exercise is useful in predicting the experimental magnitude of the activity variation as temperature fluctuates. The propagation of error analysis of an exponential expression, v = AeBT, results in an expression for the relative standard deviation of activity with respect to the standard deviation of temperature: (12) where S,,/V is the relative standard deviation of activity and B is the temperature coefficient of the respective enzyme reaction. For LDH, ALT, and AST the error expressions with respect to temperature become: SVLDH = (O.O7l8)ST = (O.O649)ST = (0.0629)ST VLDH 5VALT VALT SV VAST If the temperature variation (Sr) for the reaction is ±0.1 #{176}C, then the relative standard deviation of the three enzyme activity and ±0.63%. Experimental assays will be ±0.72%, ±0.65%, Measurement to a computer. 13 w lx lx u-i ci. w I- of Temperature Error This permits simulta- neous within-cuvette temperature and activity measurements to be made if so desired. A plot of temperature vs. time for several rotor transfers is shown in Figure 7. The time for temperature equilibration to be attained within the cuvette after transfer of room temperature reagents (e.g., 21 #{176}C) was approximately 150 s, and it required approximately 90 s to come within 0.1 #{176}C of equilibrium temperature for these intial experiments. A rotor modification has re- 5 TIME R0105’3g 4. 3 0 1 $ 5 0 II duced this equilibration 5 0.5144105’S Fig. 7. Temperature vs. time at 30 #{176}C for within-rotor rotor-to-rotor thermistor-measured temperature variation and time. Enzyme assays such as creatine kinase (CK), AST, and ALT require preincubation times of 150 s or longer before linearity of reaccaused by lag phase or endogenous substrate is sufficient to assure good assay conditions. However, LDH-L and hydroxybutyrate dehydrogenase do require absorbance measurements as early as 30 s after initiation of reaction. This becomes the limiting time for thermal equilibration within a cuvette. The temperature variation (ST) within a cuvette over a 10-mm reaction period after thermal equilibration was ±0.02 #{176}C. The cuvette-to-cuvette variation (ST) within a rotor was ±0.07 #{176}C. Data on equilibrium temperature variation for10 runs duringfourdays provideda mean cuvette temperature of 30.03 #{176}C and a rotor-to-rotor temperature variation of ±0.08 #{176}C. In general, the initial reaction data would not be acquired before thermal equilibration of the cuvette retion velocity action There are two levels of temperature uncertainty to be considered for a centrifugal analyzer, within-rotor variation and rotor-to-rotor temperature variation. Temperature in the rotor can be measured either by the use of a thermistor calibrated with an NBS traceable thermometer or by the use of a thermal optical solution (16). A thermistor was chosen for this work. The thermistor was placed in the cuvette during measurements and the analog signal from the thermistor circuit was interfaced 0 contents. Thus the rotor thermal conductance heatingconditionsare adjusted to assure essential equilibration of temperature of the reaction mixture by the 30-s limit. However, it is of interest to examine the effect of temperature equilibration on the variation of temperature (ST) during a reaction under “worst-case” conditions. The curves in Figure 7 show that the temperature is within 0.2 #{176}C of equilibrium at 45s after transfer, within 0.1 #{176}C at 90s, and at ±0.02 #{176}C of equilibrium by 150-180s. If an initial data point were obtained at 45 s and 11 more data points acquired at 20-s intervals thereafter, the mean temperature variation of the reaction during this 3.75-mm and characteristic reaction time would be ±0.06 #{176}C, an acceptable tem- peraturevariation. Pathlength Uncertainty The final individual error component that must be discussed is pathlength variation. For multicuvette photometers and systems with individual cuvettes such as the Du Pont aca, the cuvette variation is a random error component as wellas a systematicerrorcompo- CLINICAL CHEMISTRY, Vol. 22, No. 9, 1976 1445 nent. If the rotor is 1 cm in thickness, the rotor cannot vary more than ±0.005 cm (±0.0002 inch) to maintain a ±0.5% variation. CVB== CVT The Projection of Enzyme Activity Random Error An error expression was derived in the first section of this paper that related the relative standard deviation of enzyme activity to various random error parameters associated with a spectrophotometric kinetic analyzer. The individual error components were considered in the next section, and these were discussed with reference to a prototype centrifugal analyzer. This section will utilize the individual error components to predict the error in enzyme activity measurement by use of the Miniature Centrifugal Analyzer (MCA) for a reaction in which absorbance is decreasing (e.g., AST) and for a reaction of increasing absorbance (e.g., LDH or CK). The expression developed in the first section is CV(U/liter) [(CV)2 + (CVv1)2 Each of the individual error components was discussed, and determined experimentally for the MCA system, in the second section. The error of an enzyme activity measurement in range of 50 U/liter, 150 U/liter and 500 U/liter can be considered for an enzyme reaction that starts with a high absorbance (e.g., 1.2-1.6) and one that starts low (e.g., 0.2-0.3). A sample volume of 20 Mlin 200 Ml total reaction volume can be assumed and the enzyme reaction will be run for a total time of 4 mm. The temperature uncertainty within a rotor will be less than 0.1 #{176}C (cf. second section) and the pathlength variation will be 0.5%. The standard deviation and relative standard deviation of photometric error and sample-reagent pipetting can be obtained from Figures 1, 5, and 6, respectively. The activity error for LDH and AST at 50 U/liter can be calculated as follows. (50 U/liter) The uncertainty in photometric error: The relative standard deviation of absorbance change is C V = SA/A, where S = [(SAl)2 + (SA2)2]”2. For a downward going reaction at 340 nm, with a starting absorbance of 1.6 A, the uncertainty of absorbance is ±1.1 X 10 A. At 50 U/liter the absorbance A1 A2, because the absorbance change over a 4-mm period is 0.0622 A. Therefore, the worst-case relative standard deviation of absorbance (C V) is: CV = The remainder lows: SAV = 1.1 X iOv’ 0.0622 = of the error components 0.0033 V5 CVVT 1448 = CLINICAL CHEMISTRY, T = 0.0030 Vol. 22, No. 9, 1976 (0.0649)(.01) = 0.0250 are as fol- 0.00649 = [(0.250)2 + (.0033)2 + (0.030)2 + (.005)2 + (.00649)211/2 CVactivity = (0.0266)(100) = 2.7% The relative standard deviation for an AST sample run on the MCA system is 2.7%. The photometric error contribution is 2.5% of the total error. The above assumes the worst-case photometric measurement, which is a data-averaged two-point kinetic analysis. During a routine activity assay, error will improve because of the additional averaging provided by the fitting routine. However, this improvement is of diminishing returns and noise problems cannot be totally solved by statistical methods alone (9). Error of LDH Activity + (CVv5)2 + (CVB)2 + K2ST2I’12 Error of AST Activity = 0.005 (50 U/liter) The photometric error can be calculated in a similar manner but the standard deviation of a reaction with an initial absorbance of 0.25 A is about ±2 X 10 A or about five times lower. CVA = 2 = 0.0045 The other error terms are the same as the previous example, except temperature, which is: = [(.0045)2 + (.0033)2 + (.0030)2 + (.005)2 + (.00718)211/2 = (0.0108)(100) = 1.1% The relative standard deviation decreases more than twofold because of the fivefold reduction of photometric error. In this example, all error components were contributing significantly to the total error. The greatest contribution was due to temperature uncertainty. The random error in activity for the two enzymes are calculated for different activities, temperature uncertainty, and path length error to demonstrate the relative contributions of different errors on the overall uncertainty of enzyme activity measurement (Table 4). The most significant error contributions are photometric error for AST at low activity and temperature when controlled to ±0.2-0.3 #{176}C. It should be noted from Table 4, for example, that the establishment of stringent requirements which demand pipettors of pathlength tolerances to be better than 0.5%, while the “state of art” temperature control for most instruments remains in the range of 0.1-0.3 #{176}C will result in an insignificant improvement of system performance. Experimental Measurement of Enzyme Activity Error Verification of the above projected error estimates of enzyme activity as measured by the MCA, was first accomplished by use of premix experiments to eliminate Table 4. Effect of Various Error Components on Projected Random Error of Enzyme Activity Enzyme assay Activity, U/liter CV CVv % % CVVS* % CVB* % Calculated % relative standard ±#{176}C deviation a S1 AST LDH AST LDH AST LDH 50 50 150 150 500 500 2.5 0.5 0.8 0.2 0.3 0.1 0.3 0.3 0.3 0.3 0.3 0.3 0.3 0.3 0.3 0.3 0.3 0.3 0.5 0.5 0.5 0.5 0.5 0.5 ±0.1 ±0.1 ±0.1 ±0.1 ±0.1 ±0.1 2.7 1.1 1.1 1.0 0.9 1.0 LDH LDH 50 50 0.2 0.2 0.3 0.3 0.3 0.3 0.5 0.5 ±0.05 ±0.1 0.9 1.6 AST LDH AST AST AST 500 50 500 500 50 0.3 0.2 0.3 0.3 2.5 0.3 0.3 0.3 0.3 0.3 0.3 0.3 0.3 0.5 0.5 0.5 0.5 ± 0.05 ±0.3 ±0.2 ±0.3 ±0.3 0.8 2.3 1.5 2.1 3.2 LDH 50 0.2 0.3 0.3 1.0 ±0.1 1.4 LDH AST AST 50 50 0.2 2.5 2.5 0.5 0.3 0.5 0.5 0.3 0.5 1.0 P.O ±0.1 ±0.1 1.5 2.8 2.9 aCV(U,lite) = E(CVA)2 + (CV Vt)2 + (CV Vs)2 + (CVB)2 + K252J Parameter varied Photometric (CVA) Error J ‘ I Temperature Photometric(Sr) (CVA)and Error ‘I , Pipettor (CVv1) Pathlength (CVB) Pipettor (CVv3) 402 pipettor contributions. Second, the pipettor module was included to obtain a measurement of the total system random error. The premix experiment was performed as follows. Serum at AST and CK reagent ratios of 20 Ml to 200 Ml of reagent was chosen and 400 Ml of control sera was added to a 4 mol/liter concentration of enzyme reagent and the two were gently mixed. The resulting reaction mixture was rapidly pipetted into a rotor. The rotor was then brought to a temperature of 30 #{176}C and AST and creatine kinase activity were determined. The important element of this experiment is that all reactions in the rotor originated from a common pipetting and mixing of sample and reagent. The reactions proceed from the same initial reaction time and as long as substrate exhaustion is not allowed to occur, the precision of the in-run electro-optical rotor system can be obtained. The data are shown in Table 5. The measured CV (relative standard deviation) is listed prior to the predicted CV. These data are in excellent agreement with the previous projected random error data. The data from the total system analysis of AST activity are given in Table 6. Again there is a close correlation between predicted and measured random error Table 5. Enzyme Activity Data (Premix Experiment) sx Sample Reaction time 1. AST (SGOT), patient serum 2. AST (SGOT), turbid control serum 3. Creatine kinase (CK) (contrQl serum) a Parameters used for predicting i.A/time 285 (A) Mean activity, U/liter 0.075 105 (B) 210 (C) 450 240 5.4 (1 SD). U/liter CV, % (n = 19) 0.4 7.2 Predicted cv, 7.6 0.0268 54.6 1.6 2.9 3.4 0.0529 0.1125 0.2465 53.5 53.8 218 1.0 0.7 1.0 1.9 1.3 0.5 1.9 1.4 0.6 CV. CVactjvjty = ((CV)2 + (CVB)2+ (0.0678)2 (A) Path length variation: (measured variation) AST, CK3 (B) Std. deviation of absorbance: (data obtained from photometric error data) (C) Temperature variation: AST, = 0.4 X 103A AST2=O.6X 103A CK3=O.4X 103A AST, 2 = 0.1 #{176}C CK, 0.05#{176}C + ST2 I“ 0.79% 0.49% 2 = CLINICAL CHEMISTRY, Vol. 22, No. 9, 1976 1447 Table 6. Enzyme Activity Data (Loaded Rotor) Sample Reaction time AST (control serum) 435s Reaction conditions, a Note: Difference .A/time .1257 20-iI sample, 200-MI total; volume between premix experiment CVenzyme Parameters: SD absorbance % rel. std. dev. SD of temp. = % rel. std. dev. % rel. std. dev. = 52.4 U/liter temperature, activity = ((CVA)2+ CV = 19) 1.3% 0.5 cm; enzyme factor, Predicted cva 1.2% 3022. (CVB)2 + K2(ST)2] 0.79% activity CV enzyme activity = 1.1% = ((CV)2 + (CVB)2 + K#{176}(ST)2 + (CV)2 + (CV5)2] “ 0.6 X 103A of pathlength = 0.79% ±0.1 #{176}C of sample pipette = 0.33% of total volume = 0.30% = of AST activity. Data on the difference between pipetted and premix experiments are also provided in Table 6, and show the insignificant contribution of the pipettor error in comparison to photometric and temperature error. Error Estimation of systematic error has been included but not discussed. The individual components of systematic error (bias) are additive and can either be negative or positive. As a final approximation, if one sums the random error components with the stray light, bandwidth, and pipettor linearity errors, then for AST at borderline activities, the total error (system and random) is 5.0% while at high activities it is about 2.5% when the limits are held to the tolerances of Table 4. Summary and Conclusions The major emphasis of this paper has been on the derivation of a simple error expression that relates the random error of enzyme activity as measured by a photometric analytical system to the individual error components of that system. The expression for the calculation of enzyme activity has been the model, and the approach used by Pardue et al. (9) for the evaluation of random error components has been the guideline. The thrust of the discussion has been to break the analytical system down into its component and subcomponent parts as they relate to random error. Measurement of the magnitude of these error components enables one to establish the relationship of total error to these individual components. The relationship of total error to the variation of tolerances of each of these components under different conditions of performance therefore permits the unimpassioned rational establishment of error limits and performance characteristics for such analytical systems. The example used was the projection of random error of enzyme activity for different levels of LDH and AST at different analytical 1448 (n 0.7 U/liter 30 #{176}C; pathlength, CV enzyme activity Systematic SD) and loader rotor experiment CVenzyme Parameters: SD absorbance = 0.6 X 103A % rel. std. dev. of pathlength SD of temp. ±0.1 #{176}C S(1 Mean activity CLINICAL CHEMISTRY, Vol. 22, No. 9, 1976 = 1.2% systems tolerance limits. A prototype MCA system was used as the point of discussion. Two conclusions can be made from the discussions in this paper. First, the error expression can be thought of as a balance. On the left-hand side of the equation (balance) is the relative standard deviation of enzyme activity (random error) and on the right-hand side are the various individual components that add up to the overall error. The limit of acceptable random error (CViiiter) dictates what the limits of the individual components must combine to be, to fall within acceptable limits of tolerance. The limit of acceptable total random error must be established for the most meaningful dictation of what the limit of individual error components must be. Maclin et al. (8) have introduced the concept of error budget. This concept is reinforced by the error expression developed herein. The society committees must establish the acceptable error limits for the relative standard deviation of enzyme activity. Thus they establish the error budget. The manufacturer then must meet that error budget by establishment of proper error limits of the individual system components. The user can easily verify whether this has been achieved by simple measurement of relative standard error of enzyme actmvity at various enzyme activities. Second, temperature control will be a major error source, followed by photometric error at low enzyme activities of the aminotransferases or other reactions monitored at higher absorbances. Long-term temperature control to better than ±0.1 #{176}C for most analytical systems is difficult to achieve and will more realistically be ±0.2-0.3 #{176}C. This means a limit of relative error for enzyme activity measurements over a period of time of 0.7% to 2% without consideration of any other error components. Therefore, as an example, a requirement that the sample and reagentpipettorerrorlimitsfor random error be reduced from 0.5% to 0.25% would have an insignificant effect on the total random error (see Table 4). The establishment of error limits without careful consideration error components of the relationship of individual to the total random error can result in significant escalation of system cost without noticeable improvement in system performance. The discussion of error as developed by Macun and Pardue and their colleagues is useful for the proper es- tablishment of limits of photometric instrument performance. It is hoped that the development of a simple error expression based on the enzyme activity equation will be useful for user and instrument maker alike in the evaluation and establishment of performance limits. counting and direct current measurements for quantitative photometric methods. Anal. Chem. 44, 785 (1972). 6. Ingle, molecular Chem. spectro- J. D., Jr., and Crouch, S. R., Evaluation of quantitative absorption spectrophotometric measurements. Anal. 44, 1375 (1972). 7. Maclin, E., A systems kinetic enzyme analyzer. analysis of GEMSAEC precision used as a Clin. Chem. 17, 707 (1971). 8. Maclin, E., Rohlfing, D., and Ansour, M., Relationships between variables in instrument performance and results of kinetic enzyme assays-a system approach. Clin. C/tern. 19,832 (1973). 9. Pardue, H. L., Hewitt, T. E., and Milano, M. J., Photometric errors in equilibrium and kinetic analyses based on absorption spectroecopy. Clin. Chem. 20, 1028 (1974). P. R., Data Reduction and Error Analysis for the McGraw Hill, New York, N. Y., 1969, chap. 4. 10. Bevington, Physical Sciences, References 11. Burtis, C. A., Mailen, J. C., Johnson, W. F., et al., Development of a Miniature Fast Analyzer. Clin. Chem. 18, 753 (1972). 1. Widdowson, G. M., Performance specifications for instruments used in enzyme activity measurements. In Proceeding of the Second International Symposium on Clinical Enzymology. N. W. Tietz and A. Weinstock, Eds., In press (AACC). 12. Burtis, C. A., Johnson, W. F., Mailen, J. C., et al., Development of an analytical system based around a Miniature Fast Analyzer. Clin. C/tern. 19,895 (1973). 2. Sims, G. M., Alternative viewpoints for instruments used in enzyme activity for performance measurements. objectives Ibid. 3. Ingle, J. D., Jr., Comparison of the precision of normal and precision spectrophotometric technique. Anal. Chem. 45, 861 (1973). 4. Ingle, J. D., Jr., and Crouch, S. R., Signal-to noise ratio theory of fixed time spectrophotometric reaction rate measurements. Anal. Chem. 45, 333 (1973). 5. Ingle, J. D., Jr., and Crouch S. R., Critical comparison of photon 13. Ayres, G. H., Evaluation of accuracy in photometric analysis. (1949). 14. Dixon, M., and Webb, E. C., Enzymes, Academic Press, Inc., New York, N. Y., chap. 4. 15. Henry, R. J., Clinical Chemistry: Principles and Techniques, 1st ed., Hoeber, New York, N. V., 1964. Anal. Chem. 21,657 16. Bowie, L,, Esters, F., Bolin, J., and Gochman, N., Development of an aqueous temperature-indicating technique and its application to clinical laboratory instrumentation. Clin. Chem. 22,449 (1976). total stray light intensity = =1#{176} Appendix A. Computer Evaluation of Stray Light true reference cuvette intensity General The current generation of centrifugal analyzers, havingthe filter between therotorand photomultiplier tube, is inherently insensitive the rotor. When subsequent transmission readings on the same revolution are reduced by this amount, the result is to compensate completely for steady-state stray light entering the system below the rotor. Stray light entering the system above the rotor is, of course, additive with the source lamp and thus of no consequence. The source of stray light which is not fully compensated is that which enters the system from within the rotor (e.g., transmitted via internal reflections within the cover lens). The stray light represents identical additive terms to both the “true” sample and reference intensities, i.e., for the reference cuvette, = It + Js (1) (2) where the superscripts m, t, and s representmeasured, true, and stray light respectively. The stray light coefficient is defined as: (4) Am=ac+log(1+s)_1og(1+s) which the true absorbance has been set equal to cxc where c is the concentration of the absorbing material and a is the product of the molar absorptivity and the in pathlength. The stray light coefficient may then be determined by comparing measured absorbances with known absorbances of carefully prepared samples. Determination of a and S An iterative method of nonlinear regression is used in determining these quantities. Briefly, the method consists of estimating the quantities a and S, computing the best estimate of correction terms, iXa and itS, and finally repeating the process until a and S are determined within acceptable limits. The correction terms are obtained from a multivariant, Taylor’s series expansion of A in which only terms up to first order in LS and Za are retained. The expansion and for a sample cuvette: 15m=15t+Is The absorbance then takes the form: to effects of stray light. This situation arises from the use of self-compensating dark-current readings taken on each revolution. The dark-current readings are taken on the darkened region between two adjacent curvettes and thus are actually a measure of the ambient stray light that exists below Irm Irt takes the form: (5) AAo+Aaa+As.S, where A0 is the function A evaluated at initial values of S and a, i.e., A0 = a0c + M ln(1 + S0) - M ln(1 + SoecIM). CLINICAL CHEMISTRY, Vol. 22, No. 9, 1976 (6) 1449 Act is (Ai having the form: - Aoi)Aai z=1 Act while A8 is aS Aa c = 1L1+SoecMJ1 1, - i=1 having the form: A2 = 1 = (A (7) M [(1 + S)(1 + SoeM)] = (8) To obtain zS, za the least-squares difference between the data, A4, and A1 is minimized. The subscript i indicates that these values are to be associated with the ith value of the uniformly distributed independent variables C4. For n data points the least-squares difference is: [ha] iLSj A8Ai Agj Since this is a two dimensional matrix, it can quite easily be inverted to produce La and S. That is a - = (A - Aoj)AajAsj2 (A1 - - Aoj)AsjAsjAaj D (14) - (9) j=1 The minimum and of this function occurs when both and -- t9&x (10) -- S = >(AI - Ao)A5A02 - D (A - are exactly equal to zero. That is, when (A, an d (13) - A1)A81 0 Aoi)AaiAsiAai (15) (11) where = >Aaj2 new a52estimates (AsjAof a, ai)2 With theseD values, and S are (16) ob- 1=1 tained as follows: (A4 - A)A04 0 (12) These lead to the following equations, expressed here in matrix form: 1450 CLINICAL CHEMISTRY, Vol. 22, No. 9, 1976 SR+1 = aR + za, (17) = SR + zS. (18) The process is then repeated.