Optimization of a kinetic method by response-surface methodology

Response surface methodology(RSM) is an experimental strategy that was initially developed and described by Box and Wilson (1) in 1951. It has been wi...
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Anal. Chem. 1981, 53, 1154-1159

Optimization of a Kinetic Method by Response Surface Methodology and Centrifugal Analysis and Application to the Enzymatic Measurement of Ethanol C. A. Burtis,” W. D. Bostick, J. B. Overton, and J. E. Mrochek Chemlcal Technology Dlvision, Oak Ridge National Laboratory, P.O. Box X, Oak Ridge, Tennessee 37830

Response surface methodology (RSM) offers the analytical chemlst a statistically sound, efficlent, and effective means for developlng, evaluating, and optimizlng analytlcal methods. To assist In processing the analytical workload generated by RSM, we have utilized a centrifugal analyzer. Thus, each analytical run In the Centrifugal analyzer can be considered as an individual experiment and several samples, lncludlng callbrators, synthetic and authentlc specimens, etc., can be simultaneously processed and analyzed under a single set of condltlons. Depending on the samples used, several analytlcal responses such as ilnearlty, precision, accuracy, dynamic range, speclflclty, and sensitivity can be measured for each set of experlmentai condltlons. The advantages realized In combining these two technologies are demonstrated In the optimizatlon of a klnetic method for the enzymatlc measurement of ethanol.

Response surface methodology (RSM) is an experimental strategy that was initially developed and described by Box and Wilson (1) in 1951. It has been widely used in the development of physical and chemical processes (2-8) because it provides data to (1)estimate linear, curvature, and interaction effects of the variables studied, (2) optimize or evaluate multiple responses, and (3) generate statistically valid mathematical models that can be utilized for graphic interpretation of the process under study. Analytical chemists are using RSM more frequently in method development because it provides an efficient means of developing, cooptimizing, validating, and evaluating an analytical procedure. RSM has recently been applied to the development of analytical methods to measure formaldehyde (9), serum calcium (IO),serum creatinine ( 1 0 , serum cholesterol (12), and the serum enzymes alkaline phosphatase (13, 14), aspartate aminotransferase (15, 16), alanine aminotransferase (17), creatine kinase (15, 18) and lipase (15). A disadvantage of utilizing RSM for methodology development is the considerable analytical work load that results both from the experimental design and from the number of samples that have to be analyzed under each set of experimental conditions. The number of experimental conditions is a function of the number of factors (i.e., variables such as pH, temperature, and reagent concentrations)and the number of levels of each factor. For a full-factorial experimental design, the number of sets of experimental conditions can be calculated by the following equation:

N = nLf where N is the number of sets of experimental conditions, f, the number of factors, L, the number of levels for each factor, and n, the number of times the entire design will be replicated (2, 6). If time or economic constraints dictate that a fullfactorial experimental design is impractical or impossible, the

number of experiments can be decreased by utilizing one of the fractional factorial designs, such as the Box-Behnken (19) design. The number of required experiments can also be decreased by evaluating only those factors which have the greatest effect, since such factors can usually be identified by performing screening experiments (20, 21). The application of RSM for analytical method development also requires the inclusion of a number of different samples that will be analyzed under each set of experimental conditions dictated by the experimental design. For example, several individual samples of varying composition, including standard solutions (i.e., calibrators) and synthetic and authentic samples, must be analyzed under each set of experimental conditions. In addition, numerous analytical responses, which include linearity, accuracy, precision, specificity, and sensitivity, must often be evaluated for each set of experimental conditions. Thus, the use of RSM for method development generates a considerable analytical work load. To minimize the cost and time required to process such a work load, we have used an automated centrifugal analyzer. A centrifugal analyzer is a computer-controlled, multicuvet spectrophotometer (22,23) that utilizes a centrifugal field to simultaneously initiate several individual reactions. These reactions subsequently proceed under identical conditions of time and temperature and are repetitively monitored as the cuvets spin through the analyzer’s optical system. As a result, centrifugal analyzers generate a large quantity of time-absorbance data that may either be acquired, processed, and reduced in “real time” or be stored for later processing by means of an integrated data system (24). In addition, a wide variety of computational algorithms are available (24) to calculate analytical responses from these data which, in turn, can be processed by the prerequisite RSM software. A centrifugal analyzer with its multiple-sample processing (25) and automatic data handling capabilities is thus well suited for analytical method development using RSM. To demonstrate this compatibility, we have used RSM and centrifugal analysis to optimize a kinetic method for the enzymatic measurement of ethanol. Alcohol dehydrogenase (ADH) catalyzes the following reversible reaction: ethanol

+ NAD+ + acetaldehyde + NADH + H+

(2)

where NAD+ and NADH are the oxidized and reduced forms of @-nicotinamideadenine dinucleotide. The photometric measurement of NADH, as ethanol is converted to acetaldehyde, has been used as the basis for equilibrium methods to measure ethanol in body fluids (26-29), bioprocess streams (30),and fermentation broths (31). To utilize this forward reaction for the measurement of ethanol, it is necessary to maintain an alkaline medium (Le., pH 8.0-9.0) and to include a trapping reagent such as tris(hydroxymethy1)aminomethane or semicarbazide to remove the acetaldehyde as the reaction progresses.

0003-2700/81/0353-1154$01.25/0B lQ8l American Chemical Society

ANALYTICAL CHEMISTRY, VOL. 53, NO. 8, JULY 1981

The forward reaction can also be utilized as the basis for a kinetic method for the measurement of ethanol, but the use of an enzyme as an analytical reagent to kinetically measure a substrate (32-34),requires that the ratio of the enzyme’s Michaelis constant (K,) and the concentration of the substrate ([SI) in the reaction mixture must be >IO. Under these conditions, pseudo-first-order reaction kinetics prevail, and the reaction rate is then directly proportional to the concentration of the substrate. The requirement for a K,/[S] ratio >10 places a certain limitation on this particular analytical use of an enzyme. Generally, this limitation can be circumvented (1)by using low substrate concentration, (2) by selecting and using an enzyme having a large K,, or (3) by increasing the “apparent” Michaelis constant (K,) with a competitive inhibitor, as shown in the following expression:

Kp = K,(1

+ [I]/K,]I

(3)

where K , is the Michaelis constant of the enzyme, [I] is the concentration of the inhibitor, and KI is an “inhibition” constant (35). Several enzymatic methods (30, 36-42) incorporate a competitive inhibitor, thereby increasing the apparent Michaelis constant; this results in an extension of the dynamic range of the substrate concentration over which a linear calibration curve may be obtained. In a previous study (,30),hydrazine was demonstrated to be both an effective traplping agent for acetaldehyde and also a competitive inhibitor of ADH. When hydrazine is included in an enzymatic method for measuring ethanol, it serves as a bifunctional reagent and, therefore, its use and effect must be thoroughly investigated. EXPERIMENTAL SECTION Apparatus. A centrifugal analyzer system that was developed and fabricated at the Oak Ridge National Laboratory (ORNL) was used in these studies. The integrated system consisted of a miniature centrifugal analyzer (23),an automated samplereagent loader (25),and a computerized data system. The data system was programmed by combiningassembly languagefor the acquisition and storage of‘the data and FOCAL (Digital Equipment Corp.) for data retrieval, manipulation, and computation. Modified subprograms were also added to permit program and data storage and retrieval jfrom a cassette tape unit. Direct access features of the system permit storage of data from many analyses for later recomputation and statistical anallysis. Reagents. Yeast alcolhol dehydrogenase (E.C. 1.1.1.1),as a lyophilized powder having a specific activity of 360 units/mg protein, and nicotinamidie adenine dinucleotide, hydrazine dihydrochloride, and triznia base were purchased from Sigma Chemicals. Samples. Aqueous calibrating solutions,prepared from stock solutions of absolute ethanol in distilled water, were verified gravimetrically. Synthetic samples were prepared by standard addition of ethanol stock solution into a fermentation medium containing 5% glucose and 0.5% yeast extract (Difco) in distilled water. Authentic samples were obtained from incubation of the fermentation medium with a culture of Zymomonas mobilis. Aqueous samples containing the specified quantities of various alcohols were prepared to test the specificity of this method. R S M ]EXPERIMENTS Two RSM studies, RSM-1 and RSM-8, were performed. RSM-1 was a three-factor, three-level experimental design (face-centered cube), and RSM-2 was a two-factor, five-level experimental design (star-square). The facisrs and levels used in each of the experiments are indicated in Table I. Procedure. In both RSM studies, the individual experiments were performed in a randomized sequence. The necessary combinations of reagents were prepared as dictated by the experimental design. For each experimental run (i.e,, for each rotor analyzed) alicluots of samples ,and reagents were loaded into their respective chambers in the rotor in the

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Table I. Experimental Designs, Variables, and Levels Used in RSM-1 and RSM-2 RSM-1a RSM-2‘ “star square” “face-centered cube” factor identity levels factor identity levels X1 enzyme 0.32 X1 pH 1.8 8.0 (U/mL) 0.64 0.96 8.2 X2 hydrazine 15 8.4 8.6 (mmol/L) 30 45 X2 hydrazine 10 (mmol/L) 20 30 X3 pH 8.4 40 8.7 50 9.0 a N (number of experiments) = 18 (includes four replicates of midrange experiment). N (number of experiments) = 1 2 (includes four replicates of midrange experiment ). Table 11. Loading Sequence and Samples Utilized in RSM-1 and RSM-2 RSM-2 RSM-1 cuvette content content no. H,O reference H,O reference 1 reagent blank reagent blank 2 0.125 mmol/L 0.25 mmol/L 3 ethanol ethanol 0.50 mmol/L 4 0.25 mmol/L ethanol ethanol 1.0 mmol/L 5 0.50 mmol/L ethanol ethanol 0.50 mmol/L 1.0 mmol/L 6 ethanol ethanol 0.50 mmol/L 1.0 mmol/L 7 ethanol ethanol 8 1.0 mmol/L 2.0 mmol/L ethanol ethanol 9 2.0 mmol/L 4.0 mmol/L ethanol ethanol 4.0 mmol/L 8.0 mmol/L 10 ethanol ethanol 11 8.0 mmol/L 16.0 mmol/L ethanol ethanol 16.0 mmol/L 32.0 mmol/L 12 ethanol ethanol 32.0 mmol/L 13 20.0 mmol/L ethanol 1-propanol serum blank 14 10.0 mmol/L 1-propanol serum t 1.0 15 20.0 mmol/L mmol/L ethanol 2-propanol 20 mmol/L 16 10.0 mmol/L 1-propanol 2-propanol 17 1.5 mmol/L 20.0 mmol/L acetaldehyde 1-butanol Table 111. Reaction Conditions and Data Acquisition Parameters Used in RSM-1 and RSM-2 Reaction Conditions EDTA concentration 1.5 mmol/L NAD concentration 1.5 mmol/L sample volume 1 0 PL reaction volume 125 pL wavelength 340 nm temperature 30 “C Data Acquisition Parameters delay interval 10 s observation interval 10s number of observations 50 data Doints averaaedtobservation 20

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ANALYTICAL CHEMISTRY, VOL. 53, NO. 8, JULY 1981

sequence indicated in Table 11. The reaction conditions, which were constant throughout all of the experiments, are listed in Table 111. The steps required in the routine operation of the centrifugal analyzer have been described previously (23). In each RSM study, the individual experimental runs were performed in a single afternoon of analysis which followed a morning devoted to sample and reagent preparation.

DATA ANALYSIS Two levels of data processing were required. The raw data generated by the centrifugal analyzer were acquired and processed by the minicomputer included in the centrifugal analyzer system. These data were then used as input to RSM routines in a time-shared mode on the DEC-10 computer system available at ORNL. Centrifugal Analysis. For each experimental run, transmittance data were acquired for each cuvette as indicated by the data acquisition parareters listed in Table 111. The resulting data were converted to absorbances, coded, and stored on a cassette tape. Upon command, data were recalled and processed by a variety of routines to estimate the following: (1)reaction rates, using both linear and nonlinear least-squares regression analysis (24);(2) Michaelis-Menton kinetic parameters, using the method of Wilkinson as adapted by Tiffany (43);(3) ethanol concentration, using initial rate and fixed-time kinetic routines (24),in which calibration was accomplished by either linear or nonlinear least-squares regression analysis of the calibration standards data; (4) precision by statistically processing the data obtained from the replicate samples placed in cuvettes 5-7 (Table 11, RSM-1 and RSM-2); (5) accuracy by calculating the recovery of ethanol from the serum sample spiked with a known quantity of ethanol (e.g., Table 11, RSM-1, cuvettes 14 and 15). RSM Analysis. The analytical information obtained from processing of the centrifugal analyzer data was entered as response data for each of the coded variable levels. These programs, written in BASIC, included routines to fit the response data to a polynomial model using linear regression analysis, statistically assess the goodness-of-fitof the data with the polynomial model, test the statistical significance of variable effects and their interactions, and perform canonical analysis on the resultant data. EVALUATION OF OPTIMIZED METHOD After the response surface studies were completed, the analytical performance of the optimized method was evaluated over a 10-day period, A portion of fermentation media was split into three equal volumes, which were then spiked with known quantities of ethanol, dispensed into individual vials, and stored a t -40 "C. Additional aliquots of a pool of fermentation medium of unknown ethanol concentration were also stored at -40 OC. On each day of analysis, vials from each pool were thawed and three analytical runs (one rotor per run) were performed. Within each rotor, each of the spiked samples was assayed for the ethanol content in triplicate and the unknown in duplicate. This resulted in the generation of 90 individual results for each of the spiked samples (3 replicates x 3 runs x 10 days = 90) and 60 results for the unknown sample. Calibration was accomplished by using aqueous ethanol standards. Ethanol results were calculated both on a molar and weight-to-volumepercentage basis. The percent recoveries were also calculated for the spiked samples. The resultant data were statistically evaluated after the study was completed. RESULTS AND DISCUSSION The individual experimental points obtained from the two RSM studies were used to estimate the coefficients in the

Table IV. Summary of Results Obtained from RSM-1 responsea

cali-

coefficientsb B , -ENZ

K, -1.925

bration

recov-

slope

erY

0.019*'

0.002 B,,-ENZ:ENZ -1.505 B,-HYD 15.465* -0.012" B,,-HYD:HYD 5.785 -0.004 B,-PH 4.920* -0.007*

B,,-pH :pH

B,,-ENZ:HYD B,,-ENZ:PH B,,-HYD:pH Rad

significance of regressione

0.210 3.482 4.480 4.707 0.9905 99%

-0.005 -0.008" -0.005 -0.001 0.9872

precision

-1.730* -0.369 -0.497 0.375* -0.210 0.097 0.702 0.051 0.500 0.262 0.352 0.248 -0.262 0.514* 0.287 0.068 -1.562* 0.355 0.8972 0.8799

99%

90%

9 0%

a K, = Apparent Michaelis constant; calibration slope = slope of calibration curve; recovery = recovery of known quantity of ethanol added to serum; precision = relative standard deviation of triplicate samples; see text for analytical relationships. Calculated coefficients of polyno-

mial model and variables which they represent (ENZ = ADH; HYD = hydrazine). Based on individual t tests of the null hypothesis that the coefficient = 0, * indicates those coefficients that are significantly different from 0 at the 95% confidence level. Coefficient of multiple determination (12,4 4 ) . e Regression significant at the confidence level listed. following polynomial model for each of several analytical responses: y = Bo & X I + B,1(X1)2 B2Xz B22(X2)2

+

B3X3

+

+

+

+ B33(X3I2+'B12XlX2 + B13XlX3 + B23X2X3

where y = response, X I , X2, X 3 = coded variables, Bo = constant (offset term), B1,Bz, B3 = linear effects of variables, Bll, BZ2,B33= quadratic effects of variables, and B12,B13,B23 = interaction effects of variables. The analytical responses measured include Kp,which is an index of the dynamic range of a method, the slope of the h e a r calibration curve, which is also proportional to the pseudofirst-order rate constant and is an index of analytical sensitivity, recovery of a known quantity of ethanol, which is an index of accuracy, and precision. In Table IV, the coefficients and statistical data are summarized for each of the analytical responses measured in RSM-1. When interpreting such data, the sign and magnitude of the coefficients reflect the effect that each variable (invididually) or group of variables (interactively) has on the analytical response in question. Consequently, from interpretation of these data and from the visual inspection of contour plots generated from the polynomial models, the optimum level of enzyme was estimated to be 0.75 units of ADH/mL of reaction mixture. The optimum levels of the pH and hydrazine concentration were less clearly indicated but appeared to be located between a hydrazine range of 20-40 mmol/L and a pH range of 8.2-8.6. A second RSM study was therefore performed to obtain a more exact estimate for the optimum level of both of these variables. In the second RSM study, the amount of enzyme was held constant at its determined optimum level of 0.75 units/mL. A two-factor, five-level experiment was performed in which the pH and hydrazine concentration were varied at the levels indicated in Table I. The coefficients for the analytical responses are summarized in Table v. Both pH and hydrazine had significant effects on the Kpresponse, and hydrazine also had a significant effect on the slope of the calibration curve. Significant interaction effects between hydrazine and pH were also observed for K , and calibration slope responses. The

ANALYTICAL CHEMISTRY, VOL. 53, NO. 8, JULY 1981 40

Table V. Summary of Results Obtained from RSM-2 responsea calibration recovprecicoefficientsb Kp slope erY sion Bi -PH B,,-pH:pH B,-HYD B,,-HYDZHYD BIZ-PH:HYD

4.56“C 1.87” 3.20’C 0.91 6.91’F R2d 0.9413 significance of 99%

-0.13 -0.0039 10.32 -0.21 -0.009 -10.06 0.27 -0.0040* -0.05 -0.13 -0.0005 -10.24 0.00 0.0121 10.02 0.8168 0.3476 0.5541 99% 8.3 and hydrazine concentrations >30 mmol/L. For example, if the variable levels were changed from pH 8.3 and hydrazine concentration of 30 mmol/L to 8.4 and 40 mmol/L, respectively, the contour map depicted in Figure 2 would indicate that the Kpwould increase from 28.3 to 43.2 mmol/L, an increase of 50%. Consequently, the dependability (Le., ruggedness) of the method would be diminished because its analytical performance would then be markedly influenced by the stability of the reagents and the precision and accuracy with which they could be reproducibly prepared and dispensed. This problem can be circumvented by selecting a combination of pH and hydrazine concentration such that the resultant Kpis relatively insensitive to small changes in these variables. For example, as indicated in Figure 2, any combination of the two variables between pH 8.1 to 8.3 and a hydrazine concentration of 22 to 32 mmol/L would all result in a change of less than 10% in Kp thus giving improved dependability. However, the linear dynamic range of this method would be less than at the higher levels of pH and hydrazine concentration. In Figure 4, the response surface for the slope of the calibration curve is plotted as a function of pH and hydrazine concentration. Since the magnitude of this response dictates the sensitivity and ultimately the precision of the method, Figure 4 shows that the magnitude of this response is relatively constant when the variables range from pH 8.0 and 8.3 and the hydrazine concentration is 20-35 mmol/L. We have found RSM analysis and graphical interpretation of contour plots prepared for the various analytical responses are very useful techniques in method development. With the knowledge that tradeoffs will have to be made, these techniques are capable of providing an information base from

!I 30

I

I

20 80

8.2

RESPONSE = CALIBRATION SLOPE (INITIAL RATE)

Figure 4. Contour plot of the slope of the calibration curve as a

function of the two reaction variables of pH and hydrazine concentration. (The nominal response of 100 is equlvaient to an absorbance change of 0.0662/min.) which intelligent decisions can be made so that a set of reaction conditions can be selected which will result in the best overall performance of an analytical method. For example, in Figure 2, the data indicate that by increasing the pH and hydrazine concentration, the Kp of the system and hence its linear dynamic range are increased. However, this choice of conditions would result in a strong method dependence on reagent stability and the precision and accuracy with which they can be prepared and dispensed. This would result in a decrease in the dependability of the method that ultimately would be reflected in decreased precision, especially over a long period of time. On the other hand, increased sensitivity and precision and improved dependability would be obtained by decreasing levels of the two variables, a decision that would result in a decreased linear dynamic range for the method. Consequently,in the ethanol method under development, the three responses (Le,, sensitivity, precision, and dependability) were simultaneously cooptimized using the RSM data to select the reaction conditions listed in Table VI. The final step in utilization of RSM in the development and optimization of analytical methodology is the evaluation of the resultant optimized method. We thus conducted a 10-day evaluation of the optimized ethanol method by repeatedly assaying a series of synthetic and authentic samples

RSD spike-1

spike-2 spike-3

unknown-1 a

mean concn

SD a

(%I

0.082 (wt/vol %) 17.9 (mmol/L) 0.167 (wt/vol %) 36.3 (mmol/L) 0.329 (wt/vol %) 71.5 (mmol/L) 0.377 (wt/vol %) 81.9 (mmol/L)

0.0023 0.50 0.0038 0.83 0.0077 1.67 0.0078 1.69

2.8 2.8 2.3 2.3 2.4 2.4 2.1 2.1

SD = standard deviation.

84

PH

Table VII. Precision Performance of the Optimized Ethanol Method total specimen

I

/

RSD = relative standard deviation.

interrun

variation interday

(Wa 0.0009 0.19 0.0025 0.30 0.0025 0.54 0.0043 0.93

0.0006 0.13 0.0028 0.48 0.0028 0.61 0.0042 0.91

intraday (SDIa 0.0020 0.43 0.0028 0.61 0.0067 1.45 0.0050 1.08

ANALYTICAL CHEMISTRY, VOL. 53, NO. 8, JULY 1981

Table VIII. Recovery Performance of Optimized Ethanol Method ethanol (wtlvol %) specimen spike-1 spike-2 spike-3

recovery ( a ) - -

added measured

0.08 0.16 0.32

0.082 0.167 0.329

a SD = standard deviation. deviation (%).

mean

SDa

RSDb

102.7 104.2 102.7

2.94 2.32 2.41

2.8 2.2 2.3

RSD = relative standard

for their ethanol content. The total precision and the individual error components as measured for the spiked and unknown specimens are given in Table WI. The relative standard deviations ranged from 2.1 to 2.8%, with the interrun component being responsible for most of the observed variability. Recovery data for the three spiked samples, listed in Table VIII, ranged from 102 to 104%. TJnder the optimized set of conditions, the method was very sensitive and had an absorbance change of 73.4 milliabsorbance units per minute obtained per millimole of ethanol per liter. With the centrifugal analyzer used in these studies, a signal change of this magnitude is equivalent to a signal/noiue ratio of approximately 70. Reagent costs per assay were approximately $0.01, and from 60 to 80 samloles could be assayed per hour. The evaluation data summarized in Tables VI1 and VI11 demonstrate that the reaction conditions derived from the RSM optimization studlies resulted in art analytical method that is capable of providing precise and accurate results at a very low reagent cost and at a reasonable sample throughput. Thus RSM is a very useful technique to evaluate the individual and collective effects of the method variables and to provide the data base from which an intelligent and judicious selection of an optimized set of reaction conditions could be made. The use of RSM generated a large analytical work load; however, by utilization of a centrifugal analyzer and its automatic data processing system, the work load was processed in a timely and effective manner. In conclusion, response surface methodology combined with a centrifugal analyzer provides a very efficient and effective technique to develop, evaluate, and optimize an analytical method. ACKhlO WLEDGMENT We thank Stan Demirig of the University of Houston, Steve Morgan of the University of South Caroliina, and Jim Myrick and Doug Fast, Centers for Disease Control, Atlanta, for their advice and assistance in providing much of the RSM software which was used in them studies. We also thank Ed Arcuri of ORNL for his assistance and cooperation in providing fermentation media and samples. LITEIRATURE CITED (1) Box, G. E. 0.; Wilson, K. P. J. R . Stat. Soc. 1951, 8, 13, 1-45. (2) Cochran, W. B.; Cox, Ci. M. “Experimental Designs”, 2nd ed.; Wlley: New York, 1957; Chapters 5-8. (3) Hunter, J. S. Ind. Qual. Control 1958, 15 (6),7-24. (4) Hunter, J. S. Ind. Qual Control 1958, 15 (8),8-14. ( 5 ) Hill, W. J.; Hunter, W. Ei. Technometrlcs1960, 8 , 571-590.

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RECEIVED for review January 28, 1981. Accepted April 10, 1981. Research sponsored by the Division of Biomedical and Environmental Sciences, US. Department of Energy, under Contract W-7405-eng-26 with the Union Carbide Corp. Portions of this article were previously presented a t the 33rd Annual Summer Symposium on Analytical Chemistry which was held June 5-7, 1980, at Duke University, Durham, NC.