Modeling of potentiometric electrode arrays for multicomponent

Anal. Chem. 1991, 63, 876-882

870

of PECL is chiefly ascribed to the following scavenging reactions:

-

+ X- SO:- + Y2X2 (X- = Br-, I-) SO4'- + NOz- + 3H20 2SOd2-+ NO,- + 2H30+ SO4'-

-

TokeCTakworyan, N. E.; Hernlngway. R. E.; Bard, A. J. J. Am. Chem. SOC. 1973,95. 6582-6589. Gleria, M.; Memmlng, R. 2. Phys. Chem. New Folge 1978, 101, 171-179. Laser, D.; Bard, A. J. J . Electrochem. Soc. 1975, 722, 632-640. White, H. S.;Bard, A. J. J . Am. Chem. Soc. 1982. 704, 6891-6895. Rublnstein, I.; Bard, A. J. J . Am. Chem. Soc. 1981, 703, 512-516. Rublnstein, I.; Martin, C. R.; Bard, A. J. Anal. Chem. 1983, 55, 1580-1582. . - - - .- - -. Ege, D.; Becker, W. G.; Bard, A. J. And. Chem. 1984, 56, 2413-2417. Bolletta, F.; Cino, M.; Balzani, V.; Serpone, N. Inorg. Chim. Act8 1982,62,207-213. Crutchley, R. J.; Lever, A. B. P. J. Am. Chem. Soc. 1980, 702, 7128-7129. Crutchley, R. J.; Lever, A. B. P. lnorg. Chem. 1982,2 f , 2276-2282. Yamazaki-Nlshida, S.; Harima, Y.; Yamashlta, K. J. Electranal. Chem. 1990,283, 455-458. Hakoila, E. Talanta 1988, 75, 55-61. Amin, D. Anelyst 1981, 106, 1217-1221. Murty, N. K.; Satyanarayana. V.; Rao, Y. P. Talanta 1977. 2 4 , 757-759. Belcher, R.; Bogdanskl, S. L.; Townshend, A. Anal. Chim. Act8 1973, 6 - .7 ,. 1-18. . . -. Belcher, R.; Bogdanski, S. L.; Knowies, D. J.; Townshend, A. Anal. Chim. Acta 1975, 77, 53-63. Evans. I. P.: Smncer. A.: Wikinson. G. J . Chem. Soc.. Dalton Trans. 1973.' 204-206. (20)Nishida, S.;Harima, Y.; Yamashita, K. Inorg. Chem. 1989, 28, 4073-4077. (21) Venturi, M.; Mulazzani, Q. G.; Ciano. M.; Hoffman, M. 2. Inorg. Chem. 1988,25, 4493-4498. (22) Crutchley, R. J.; Kress, N.; Lever, A. 8. P. J. Am. Chem. Soc. 1983, 705, 1170-1178. (23) House, D. A. Chem. Rev. 1982,62. 165-203. (24)Hap, M.; Dodsworth. E. S.; Eryavec, G.; Seymour, P.; Lever, A. B. P. Inorg. Chem. 1985,24, 1901-1906.

(8) (9)

In the present system, some amount of C1- anion is always present in the test solutions as the counterion for [Ru(bpz)3I2+, but the coexistence of C1- is not a serious problem in the determination of SzOt-. Sulfur-containing compounds, e.g., S042-,also have no influence on the ECL intensity. If metal ions or complexes such as [Fe(H20),l3+coexist in excess in the sample solutions, they affect the ECL intensities because of quenching luminescence of [ R ~ ( b p z ) ~ ](24). ~ + * According to the mechanism (reactions 1-4), ECL may be observed with powerful oxidizing agents such as SO4*- in the presence of [ R u ( b p ~ ) ~ ] In ~ +other + . words, Sn(IV), HZO2,or Fe2+/HzO2is expected to yield ECL in a similar way as in the reaction with S20Ez. However, no ECl has been observed except with S20Ez. It can be concluded that the ECL technique using the [ R u ( b p ~ ) , ] ~ + / saqueous ~ O ~ ~ system is suitable for a rapid and specific microdetermination of SzOt-. ACKNOWLEDGMENT We are grateful to Dr. and Mrs. Roy Teranishi for editing this paper. LITERATURE C I T E D (1) Gonzalez-Velasco, J. J. Phys. Chem. 1988,92, 2202-2207. (2)Gonzalez-Velasco, J.; Rubinstein, I.; Crutchley, R. J.; Lever, A. B. P.; Bard, A. J. Inorg. Chem. 1983,22, 822-825.

RECEIVED for review July 16,1990. Accepted February 7,1991.

Modeling of Potentiometric Electrode Arrays for Multicomponent Analysis Robert J. Forster, F r a n c i s Regan, a n d Dermot Diamond*

School of Chemical Sciences, Dublin City University, Dublin 9,Ireland

The use of a four-electrode array comprlslng three highly selectlve and one sparlngly selectlve electrode for the detennlnatlon of sodium, potasslum, and calclum ions In tertlary mixtures of the catlons Is descrlbed. The response surface of each electrode In the array is determined by uslng mlxed callbratlon solutlons and thls response modeled via the Nlckolskll-Elsenmann equation by uslng a variety of opthnlration procedures. The comblnatlon of highly and sparingly selective sensors offem consklerable advantages over exlsting slngbelectrode measurements, such as pOrlng of p", resulting In Improved accuracy and preclslon, and dlagnosls of electrode performance wlthout recallbratlon. The abnlty of the array to determlne sodlum, potasslum, and calclum Ions at physldoglcal levels In tertlary mixtures wlth leas than 2.8% error Is demonstrated. Unllke tradltlonal slngle-electrode measurements, the modeled array can accurately determlne low levels of lndlvldual catlons In the presence of a large and wldely varylng excess of the other two. The appllcatlon of the modeled array to the simultaneous determination of the three catlons In human plasma samples Is considered.

* To whom correspondence should be addressed. 0003-2700/91/0363-0876$02.50/0

INTRODUCTION The use of potentiometric ion-selective electrode assays in conjunction with chemometrics has received increasing attention of late (1-3). This approach has arisen since ion-selective electrodes (ISEs) respond to some degree to a range of ions. Because of this limitation, they find common application in solutions containing only low levels of interfering ions or in samples of fixed composition where it is possible to match sample and calibration matrices. To ensure that the analyte dominates the electrode response, an electrode which is very selective for the analyte ion is required in all of these situations. Hence research in this area has been dominated by the search for more selective sensors. However, through the use of sensor arrays and chemometrics, more intelligent instrumentation can be developed which permits those contributions to the total signal arising from primary and interfering ions to be decoupled, thus allowing accurate determination of individual cation activities within mixtures. In this contribution we report on the use of a computerbased array of selective and sparingly selective sensors for the determination of sodium, potassium, and calcium ions in tertiary mixtures. The response of each member of the array to changes in the concentration of these ions has been de-

0 199l American Chemical Society

ANALYTICAL CHEMISTRY, VOL. 63, NO. 9, MAY 1, 1991

termined by using calibration solutions containing the three cations and modeled via the Nickolskii-Eisenmann equation (eq l),by means of a series of iterative nonlinear optimization

techniques, where Eij is the potential of the jth electrode, measured in the ith sample, Ejo is the standard cell potential, aik and an are the activities of the primary (k) and interfering ions (1) in sample i, respectively, Kjkl is the selectivity coefficient of the jth electrode with respect to the lth interfering ion and Sj is the slope of the electrode with respect to the primary ion in the absence of interferents. The use of an array modeled by the Nickolskii-Eisenmann equation has been considered previously for the determination of sodium and potassium ions in binary mixtures using sparingly selective electrodes (4). The system was shown to be capable of circumventing some of the problems associated with sparingly selective electrodes and gave average relative errors no larger than 12% for binary solutions where the sodium ion was at physiological levels. However, the use of an array containing sparingly selective electrodes alone is likely to be restricted because of the broad range of response of these sensors. This means that in complex mixtures, at least one additional term, which may not simply add to the model, will have to be included for each likely interferent if the array response is to be accurately modeled. Failure to include likely interferents will result in inaccurate determinations where the relative concentration of interferents is variable. As well as this, the complete characterization of the response surface of an array of sparingly selective sensors will require a large number of calibration solutions. In this contribution, the use of an array containing three highly selective electrodes and a single sparingly selective electrode is examined as a means of improving accuracy and precision in multicomponent analysis using ISEs. The selective electrodes respond to only a small range of cations and hence require only a relatively simple model for accurate characterization of their response. The sparingly selective electrode responds to all three cations and hence can be modeled as a “pseudo” sodium, potassium, or calcium ion electrode. This allows polling of responses leading to increased precision and accuracy in solutions containing only these three cations. Since the sparingly selective sensor is not fully characterized in its response to ions other than sodium, potassium, and calcium, it can be used to identify the presence, but not the nature of interferents. Individual electrode performance can also be monitored, since if a particular combination of electrodes does not agree with the majority decision then the malfunctioning sensor can be identified. Furthermore, on-line diagnostic tests for probing the internal composition of the membranes are also possible ( 5 )*

EXPERIMENTAL SECTION A 16-channel National Instruments MIO-16 data acquisition card (DAS)with a sampling rate of 100 kHz installed in a Nimbus AX/2 286 (IBM AT compatible) in conjunction with impedance conversion circuitry was used to acquire the potential data in differential mode with respect to a common saturated calomel reference electrode (SCE). This, in conjunction with screening of the signal lines to ground on the DAS and software-controlled signal averaging, provided sufficiently noise-free responses. On-line data analysis and model fitting were carried out on the Nimbus AX12 286 computer using routines written in Microsoft QuickBASIC. Potentials for the manual procedures were measured by using a Corning Model 240 pH meter in the millivolt mode. All potential measurements were made at 25 f 1 “C. Electrodes. The ISE array contained three selective and one sparingly selective electrode. Tetramethyl p-tert-butylcalix[4]areneacetate (6), valinomycin, and ETH 129 (Fluka BV Ltd.) immobilized within plasticized poly(viny1 chloride) (PVC) ma-

877

Table I. Composition of Calibration Samples Used for Array Modeling sample

[Na+],mM

WI, mM

[Ca2+],mM

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24

120.3 160.4 123.2 157.9 160.3 121.9 122.7 159.5 103.2 103.3 98.7 97.8 201.4 201.3 211.4 201.4 99.2 104.6 104.5 100.1 200.3 210.6 200.4 210.6

201.3 198.7 200.0 201.8 300.6 309.2 303.2 302.5 1.02 8.1 0.9 7.9 1.0 8.3 1.2 8.3 74.3 74.9 153.8 150.3 73.2 75.4 145.8 152.7

102.6 97.9 298.8 299.8 300.9 302.3 100.4 98.6 73.3 74.3 152.1 153.2 73.4 75.1 152.4 153.2 0.6 2.3 0.6 2.1 0.5 2.1 0.5 2.1

trices, as described previously (6, 7), formed the basis of the sodium, potassium, and calcium ion selective electrodes and are labeled as Na, K, and Ca, respectively. The sparingly selective electrode was similarly constructed by using the p-tert-butylcalix[4]arene tetraoxime as the ligating agent and is labeled as SSE (8). The electrode bodies were constructed with flexible PVC tubing and were not screened for electromagnetic interference. Reagents. All solutions were prepared from Analar grade chlorides of the cations by using deionized water. Experimental Design. A three-factor, two-level factorial design comprising 24 samples, as given in Table I, was used to characterize the response surface.of each member of the array. Samples were randomly analyzed. The concentration of the primary ion was c h m n to encompass the clinical range for human serum samples (9). High concentrations of interfering ions were deliberately chosen to accurately model the response of the selective electrodes. This is necessary because the interfering ions must make a significant contribution to the observed potential if the response of the sensor to these interferents is to be accurately modeled. Plasma Samples. For cation determination within plasma samples, the array was calibrated by using the above procedure except that the calibration solutions contained a fiied background of 0.8 mM MgC12, 4.7 mM glucose, and 2.5 mM urea. Both calibration and plasma samples were similarly diluted 10-fold by adding pH 7.4 Tris buffer in 0.11 M LiCl prior to analysis (IO). The diluent dominates the ionic strength, thus giving a nearconstant cation activity. Calibration. In the calibration procedure the potential of each electrode within the array was recorded in the 24 calibration solutions. As the data were acquired, digital smoothing was carried out by averaging 10 blocks of 500 data points. The between-blocks deviation was required to be better than 0.1 mV. The raw data were stored in ASCII format to facilitate postrun analysis with applications software packages. These data were then used in conjunction with nonlinear optimization techniques to simultaneously determine the cell potential (Ejo), the slope (Sj), and selectivity coefficients (KjH). The objective function for these optimization procedures was to minimize the residual sum of squares (RSS) to less than lo-”, this being defined by 24

errj = C(Eij’- Eij)2 i-1

where Ei{ are the estimates of the potential of electrode j in sample i provided by the model and Ei. are the recorded experimental potentials. Initial estimates of and Sj were obtained from the

878


Table 11. Model Parameters (Standard Deviation Obtained by Using Five Separate Calibration Sets on a Single Array) Derived by Using Samples Given in Table I

Ejo,mV

Sj, mV/decade

j = Na+ j = K+ j = Ca2+

-70.0 (0.15) 54.0 (0.21) 54.0 (0.19)

54.9 (0.05) 53.9 (0.04) 27.1 (0.08)

j = Na+ j = K+ j = Ca2+

82.0 (0.43) 74.6 (0.54) 56.6 (0.89)

51.0 (0.09) 51.0 (0.10) 51.0 (0.11)

1 0 3 ~ ~ ~ ~ +

103~~,+

103KjC.2+

ISEs 1 4.06 (0.05) 0.01 (0.0005)

20.4 (0.9) 1 0.016 (0.0013)

17.3 (1.54) 5.09 (0.05) 1

SSEs

20

40

1 1398 (66.9) 7.54 (0.38)

60 80 100

Iteration Number

Flgure 1. Dependence of residual sum of squares (RSS) on the it-

eration number of the constrained simplex method for the case of the sparingly selective electrode modeled for sodium determination. Modeling time is indicated on the figure. intercept and slope of a plot of Eijvs log (activity of the primary ion). For each of the interfering ions, initial estimates of 3 X and 0.5 were used for the selectivity coefficients of the highly selective and sparingly selective electrodes, respectively. Several algorithms were investigated to simultaneously determine the model parameters including direct search and gradient methods ( 2 2 ) . A modified simplex procedure in which the parameters were suitably constrained typically proved efficient at satisfying the convergence criterion of minimizing the RSS to less than and was computationally simple to implement, resulting in the minimum time for model building. An example of the sequential convergence of the model estimates and experimental data as given by the residual sum of squares is shown in Figure 1, for the case of the sparingly selective electrode where sodium is the primary ion. The Davidon-Fletcher-Powell gradient method was also applied for model building and found to be superior in experiments where the model was extended to include more ions or where more solutions were used in the calibration procedure. Determination of Unknown Solutions. The modeled array was used to determine the concentrations of sodium, potassium, and calcium ions in tertiary mixtures and in plasma samples by using the following formalism. Equation 1 can be rearranged to give 10(Eij-Ejo)/sj= aik+ ~Kjklai%/zl (3) 2

For three sensors this gives rise to a system of three equations, which can be represented in matrix form as C = KA

(4)

where C is a 3 x 1 column matrix defined by the left-hand side of eq 3, K is a 3 X 3 matrix of the selectivity coefficients for the three electrodes, where the diagonal elements are unity, and A is a 3 x 1column matrix of the unknown activities of the sodium, potassium, and calcium ions, respectively. By inverting the matrix K , we obtain K-'C = A (5) where K-' is the inverse of K. The activities of the cations are then contained in the column matrix A . Since the sparingly selective electrode responds significantly to changes in the concentration of all three cations, it can be sequentially switched into this logic in place of each of the selective electrodes. This requires that the sparingly selective electrode be modeled as a pseudo sodium, potassium, and calcium ion electrode, respectively. This can be achieved by using the op-

715 (35.4) 1

5.39 (0.28)

364.2 (18.9) 185.5 (8.90) 1

timization procedures discussed previously or by rearranging eq 1in the required manner. The effect of switching in the sparingly selective electrode is to produce four combinations of three electrodes, with each combination being capable of independently predicting the activities of the sodium, potassium, and calcium ions. This gives rise to improved accuracy and precision and provides a diagnostic tool for the evaluation of individual sensor performance. Goodness of Model Fit. A measure of the model fit was obtained not only from the residual sum of squares (which is required to be less than 10-lo)but also by considering the ability of the model to estimate the concentrations of the cations in the tertiary mixtures used to construct it. Since the objective is to determine the ability of the model to estimate the concentration of the cations in the calibration solutions, and not to examine the sensor performance, the potentials obtained in the calibration procedure were used as input into the model. This procedure negelects the possibility that the ISE potential may have drifted between the time of data capture used for modeling and the time at which the goodness of model fit is investigated. Using these inputs, the model is capable of estimating the cation concentrations to within 0.1 %. The errors associated with the sodium ion highly selective electrode are relatively larger than for the potassium and calcium ion sensors. This arises since accurate modeling requires that interfering ions make a significant contribution to the overall cell potential. In the case of the sodium ISE this means having a large excess of both potassium and calcium ions. Where the sodium ion concentration is a t the physiological level, a large excess of interfering ions would require the use of impractically concentrated solutions. It is to be noted that in the case of the sparingly selective electrode, where the range of response is broader, or in other experiments where the absolute sodium ion concentration is less than the physiological range, this difficulty does not exist.

RESULTS AND DISCUSSION Model Parameters. The array examined in this contribution gives a response that is highly reproducible and exhibits a fast response time. The model parameters and the precision obtained for each sensor within the array by using five-calibration solution sets are given in Table 11. The values obtained for Kjl show that the selective electrodes show good discrimination between primary and interfering ions, while the sparingly selective electrode exhibits a broad response to both primary and interfering ions. Table I11 gives the model parameters and the precision obtained for three separate arrays by using a single-calibration solution set. From these tables it is evident that the deviation on the slopes, Sj, and selectivity coefficients, Kjkl,is relatively small, while larger deviations exist on the intra- and interarray precisions for the cell potentials Ejo. These observations suggest that the slopes and selectivity coefficients of the sensors are well defined within the limits of the investigation and remain largely constant over time and between arrays. The deviation in Ejo occurs due to slight inaccuracy in the determination of the electrode slope and, more especially, time-dependent electrode drift and the reproducibility of manufacturing these sensors. The time-dependent drift is usually monotonic and unidirectional for these sensors and may therefore be easily com-


879

Table 111. Model Parameters (Standard Deviations Obtained for Three Separate Arrays by Using a Single Calibration Solution Set)

Elo, mV

Sj,mvldecade

j = Na+ J = K+ J = Ca2+

-72.3 (5.31) 57.8 (6.12) 50.4 (5.98)

54.3 (0.65) 54.1 (0.51) 26.8 (0.92)

j = Na+ J = K+ j = Ca2+

78.6 (6.77) 70.0 (7.10) 51.4 (6.78)

52.1 (1.21) 51.2 (1.09) 48.9 (1.21)

1 0 3 ~ ~ ~ ~ +

103~~~+

103KjCa2+

ISEs

20.3 (1.1)

1

4.20 (0.11) 0.01 (0.0008)

17.88 (11.2) 5.12 (0.09)

1

0.017 (0.0011)

1

SSEs

Table IV. Electrode Characteristics as Determined by Using Manual Methods (Standard Deviations for Three Different Sensors) S, mv/

decade

103KjN1+

760 (55.8)

1

1350 (74.9) 7.3 (0.43)

367 (32.8) 186.1 (9.24)

1 5.5 (0.32)

1

1 -

N a' MANUAL A

B

9'

103~,~~2+

-2

ISEs

1

j = Na+ 54.7 (0.87) 1 12.5 (1.10) 3.16 (0.31) = K* 53.8 (0.91) 0.41 (0.03) 1 0.16 (0.02) J = Ca2+ 27.2 (1.13) 0.020 (0.002) 0.028 (0.002) 1

1

18

24

30

I

1 3 5 145 155

55Kl

SSEs i = Na'

53.8 (1.05) 1

158 (16.1)

75 (8.1) 351

pensated for. In order to circumvent inaccuracy in the determination of cation concentrations in mixtures due to drift, the modeled array response was checked by using two calibration standards prior to the analysis of unknowns. Any deviation between this response and that used to model the array was eliminated by software-controlled offsetting of the incoming potentials. The monovalent slope observed for the sparingly selective sensor when it is modeled as a pseudo calcium sensor arises because the sodium ion is present in all of the calibration solutions and, since sodium is the primary ion, this dominates the sensor response. Table IV gives the electrode characteristics evaluated by using traditional manual methods and single-ion solutions for the slope determination and for the evaluation of selectivity coefficients, the divalent ion selectivity being confirmed by using the fixed-interference (mixed solution) method (12). These data, in conjunction with Tables I1 and 111, show that the slope evaluated by the nonlinear optimization method, whilst being sub-Nernstian, does appear to follow the classical definition of ISE slope. However, there are significant differences between the selectivity coefficients obtained by the two techniques. For the sodium, potassium, and sparingly selective sensors, the selectivity coefficients determined by the nonlinear optimization procedure suggest that the sensors are less selective in the presence of interferents. In contrast, for the calcium ion sensor the selectivities estimated by both approaches are more closely related. Given the known concentration dependence of selectivity coefficients and the possibility of interactive effects occurring between ions in mixed solutions, these differences in selectivities between single and tertiary ion mixtures are not unexpected. Comparison of Selective Array a n d Arrays Containing Sparingly Selective Sensors. The ability of the modeled array combining the three selective electrodes to estimate the cation concentration in the calibration solutions was examined by using the samples given in Table I, and the results are given in Table V together with the standard deviations observed for five different calibration solution sets. These data contain an error contribution from the model itself (relatively small) as well as from experimental factors; e.g. uncorrected drift between the time of modeling and cation determination will be included. Nonetheless, total errors not exceeding 4.5% are obtained for the determination of sodium, potassium, and

.. .

15

15

25

0 5 15

25

05

I

01

04

-02

01

07

I

Ob

Figure 2. Dependence of the percent error between the relative interference concentration and the prediction obtained by (A) the mean of four combinations of three electrode arrays and (B)the manual method. Attention is drawn to the different error scales on parts A and 8.

calcium ions in these samples. Significantly, these errors are randomly distributed and are independent of sample composition, similar errors being obtained even in the determination of low concentrations of one cation in the presence a large excess of the other two (e.g. samples 17-24). The maximum relative standard deviation observed for these measurements is 3%, which shows that the array response is well defined and has acceptable precision. Table V also gives the errors obtained for the same samples where the sparingly selective electrode is included in the prediction. This gives four combinations of three electrodes, namely (Na, K, Ca), (Na, K, SSE), (Na, SSE, Ca), and (SSE, K, Ca). This table shows that improved accuracy and precision are obtained when the arithmetic mean of these four combinations (with equal weighting) is used compared to the array of three selective electrodes, with errors not exceeding 2.8%. Typically, the individual errors and standard deviations are reduced compared to the array of selective electrodes. Figure 2A shows the percent error versus log ( Q ~ / ~ & , ~ Q , # ~ / ~ ~ plot for sodium, potassium, and calcium ion determination using the mean value of the four combinations of the electrodes. This figure suggests that the errors are randomly distributed with near zero mean error. It is to be emphasized that experimental conditions were not optimized to eliminate

880


Table V. Prediction of Ion Concentrations within Calibration Solutions by Highly Selective Array and Mean of Four Combinations of Selective and Sparingly Selective Electrodes (Standard Deviations for Five Repeat Determinations)

sample 1 2 3 4 5 6 7 8 9 10 11

12

[Na'l, mM 120.3 160.4 123.3 157.9 160.3 121.9 122.7 159.5 103.2 103.2 98.7 97.8

sample [K'], mM 1 2 3 4 5 6 7 8 9 10 11 12

201.3 198.7 200.0 201.8 300.6 309.2 303.2 302.5 1.02 8.10 0.9 7.9

sample

[Caz+], mM

1

2 3 4 5 6 7 8 9 10 11 12

102.3 97.9 298.8 299.8 300.9 302.3 100.4 98.6 73.3 74.3 152.1 153.2

estimate selective array mean 117.1 167.3 127.3 151.6 158.5 118.4 124.3 162.8 105.5 100.0 96.5 99.9

(2.12) (5.01) (3.23) (4.11) (4.01) (3.22) (4.01) (4.15) (2.46) (1.83) (2.32) (2.41)

118.1 (2.14) 163.7 (3.09) 123.5 (2.55) 155.8 (3.12) 156.0 (4.03) 123.7 (1.98) 123.8 (1.78) 163.7 (2.67) 104.8 (2.33) 105.3 (2.11) 100.5 (1.99) 96.9 (0.88)

estimate selective array mean 207.7 (5.41) 195.1 (3.83) 195.7 (1.71) 193.9 (3.76) 308.7 (9.11) 322.4 (5.12) 295.8 (7.60) 316.5 (4.01) 1.03 (0.02) 8.01 (0.20) 0.89 (0.03) 7.81 (0.08)

198.9 199.9 195.7 205.8 302.8 313.3 299.1 298.2 1.0 8.0 0.9 7.9

(2.27) (2.91) (3.44) (2.94) (3.22) (4.01) (3.88) (2.88) (0.02) (0.15) (0.02) (0.08)


(3.11) 101.3 (2.87) (2.81) 96.3 (2.98) (3.44) 297.3 (6.53) (9.11) 295.5 (7.37) (8.79) 304.4 (7.45) (8.81) 297.6 (8.01) (2.85) 99.1 (2.27) (2.77) 101.2 (2.33) (2.01) 72.9 (1.78) (1.88) 72.4 (1.67) (4.65) 153.3 (3.10) (3.99) 152.5 (3.76)

90 error selective

array

mean

sample

[Na+l, mM

-2.71 4.30 3.23 -4.14 -1.10 -2.90 1.31 2.12 2.20 -3.14 -2.21 2.29

-1.82 2.04 0.20 -1.27 -2.66 1.48 0.89 2.61 1.58 1.96 1.86 -0.94

13 14 15 16 17 18 19 20 21 22 23 24

201.4 201.3 211.0 201.4 99.2 104.6 104.5 100.1 200.3 210.6 200.4 210.6

% error selective array mean

3.21 -1.82 -2.15 -3.94 2.71 4.33 -2.44 4.42 1.37 -1.11 -0.92 -0.52

-1.2 0.62 -2.13 2.00 0.73 1.32 -1.35 -1.21 0.5 -0.7 2.27 -0.02


4.32 -2.83 -4.14 3.71 3.19 -1.09 2.17 -2.82 -3.17 -4.33 2.78 2.09

-0.98 -1.63 -0.48 -1.44 1.16 -1.53 -1.34 2.61 -0.59 -2.51 0.81 -0.48

residual noise (e.g. by electrode screening), which should improve the accuracy and precision obtainable. Comparison of Array and Manual Predictions. The improvement in accuracy and precision obtained by using the array compared to the traditional manual methods was investigated by constructing calibration curves for the three selective electrodes using a pH meter in the millivolt mode. Samples in the concentration range 110-160 mM sodium ion in a background of 4.0 mM potassium ion and 1.1mM calcium ion were used for sodium ion Calibration. The potassium ISE was calibrated in the range 2-6 mM in a background of 1 4 0 mM sodium ion and 1.1 mM calcium ion. Calibration curves were similarly constructed for the calcium ISE over the range 0.5-1.5 mM in a background of 140 mM sodium ion and 4 mM potassium ion. The relevant samples from Table I were analyzed by using this method, and the results are presented in Table VI. These results show errors of approximately 5-10% are obtained where the concentrations of interfering ions in the sample and calibration standards are similar. Where the background composition deviates significantly from that used in the calibration procedure (e.g. calcium deter-


sample [K'], mM 13 14 15 16 17 18 19 20 21 22 23 24

sample 13 14 15 16 17 18 19 20 21 22 23 24

1.0 8.3 1.2 8.3 74.3 74.9 153.8 150.3 73.2 75.4 145.8 152.7

(5.38) (2.99) (5.83) (2.82) (3.21) (3.81) (1.90) (1.59) (2.92) (3.69) (6.02) (3.12)

197.2 (1.22) 203.7 (2.88) 206.4 (3.13) 204.6 (3.20) 101.0 (0.99) 103.2 (1.76) 103.0 (1.32) 99.1 (0.99) 205.8 (2.56) 206.7 (2.55) 202.5 (3.00) 215.3 (2.11)

estimate selective mean array 1.01 (0.04) 8.47 (0.27) 1.17 (0.03) 8.02 (0.21) 71.1 (2.15) 77.2 i2.36j 157.0 (3.70) 154.9 (1.91) 70.3 (2.10) 75.0 (1.31) 148.5 (3.71) 157.1 (4.71)

1.0 (0.04) 8.4 (0.18) 1.2 (0.09) 8.5 (0.12) 74.1 (1.34) 73.1 (1.67) 155.7 (1.98) 146.8 (2.01) 74.9 (1.99) 73.3 (2.87) 148.3 (2.88) 156.3 (3.01)

estimate [Ca2+l, mM selective array mean 73.4 75.1 152.4 153.2 0.6 2.3 0.6 2.1 0.5 2.1 0.5 2.1

71.1 (1.99) 76.6 (2.01) 150.8 (3.76) 160.0 (4.81) 0.57 (0.015) 2.20 (0.061) 0.58 (0.013) 2.06 (0.049) 0.54 (0.015) 2.17 (0.026) 0.48 (0.014) 2.01 (0.05)

75.2 73.6 154.3 156.9 0.6 2.3 0.5 2.1 0.5 2.1 0.5 2.0

(1.98) (1.88) (3.22) (3.56) (0.015) (0.056) (0.012) (0.04) (0.011) (0.05) (0.007) (0.028)

% error selective array mean -4.21 -3.21 -2.13 1.84 3.80 3.13 -1.91 1.71 3.21 -3.11 4.54 -3.66

-2.06 1.19 -2.15 1.57 1.85 -1.29 -1.35 -1.01 2.77 -1.84 1.06 2.23

% error

selective array mean 1.22 2.11 -1.90 -3.34 -4.43 3.10 2.09 3.09 -3.92 -0.53 1.83 2.92

-0.47 1.47 1.71 2.01 -0.28 -2.33 1.26 -2.3 2.37 -2.74 1.73 2.38


-3.08 2.07 -1.06 4.45 -3.87 -4.19 -3.27 -1.77 1.99 3.28 -3.31 -2.37

2.41 -2.04 1.25 2.37 -0.19 -0.94 -1.78 0.17 1.70 2.01 2.40 -2.37

mination in samples 21-24), unacceptably large errors occur. Figure 2B shows the dependence of the percentage error on log (ai/ C2Kjkluil)zk/zl for the determination of sodium, potassium, and calcium ions using the manual method. This figure clearly shows that, unlike the computer-based array, the accuracy in the determination of the cation concentrations is strongly dependent on the background composition. Figure 3 illustrates the correlation between the cation concentrations in the standard solutions and the predictions obtained by using (A) the mean of the four three-electrode array predicitions and (B) manual methods. This figure illustrates a recurrent problem with the calibration and subsequent analytical use of ion-selectiveelectrodes using manual methods (10, 13). In theory, if a calibrated electrode is subsequently used to determine the calibrant solution concentration, then a plot of the determined ion activity vs the real activity should yield a straight-line graph with unit slope and zero intercept. However, in practice, nonunity slopes and non-zero intercepts are obtained (IO). This problem is inherent to the manual calibration technique and has as its source the limited precision and accuracy of the single values


Table VI. Prediction of Sample Concentration Using Manual Method (Standard Deviation for Four Sets of Calibration Solutions)

sample

[Na+],mM

estimate

% error

1 2 3 4 5 6 7 8

120.3 160.4 123.2 157.9 160.3 121.9 122.7 159.5

128.9 (3.13) 170.7 (5.77) 132.3 (4.01) 167.9 (6.10) 172.7 (5.67) 133.0 (3.89) 133.8 (3.44) 172.0 (7.43)

7.21 6.44 7.43 6.38 7.76 9.11 9.09 7.83

estimate

‘3 error

sample

[K+l, mM

9 10 11 12 13 14 15 16

1.0 8.1 0.9 7.9 1.0 8.3 1.2 8.3

sample

[CaZ+],mM

estimate

% error

17 18 19 20 21 22 23 24

0.6 2.3 0.6 2.1 1.1 2.1 0.5 2.1

0.9 (0.03) 2.4 (0.11) 1.1 (0.04) 2.6 (0.13) 2.0 (0.01) 2.6 (0.14) 0.8 (0.42) 2.7 (0.12)

49.10 6.00 73.11 24.42 87.11 27.61 74.01 31.42

1.8 8.8 1.4 8.9 1.6 8.9 1.8 9.1

50 100 150 200 Standard (mM1

(0.09) (0.51) (0.07) (0.67) (0.08) (0.41) (0.08) (0.07)

50

48.68 8.16 58.88 12.61 64.5 7.89 51.66 9.10

100 150 Standard h M 1

Table VII. Prediction of Sodium, Potassium, and Calcium Concentration Using Mean of Four Combinations of Three Electrode Arrays (% Error)

sample no. 1

2 3 4 5 6 7 8 9 10 11 12 13 14

50

100 200

300

Standard ImMI

2

4 6 8 Standard (mMI

Standard ImMI

Standard ImMI

Flgure 3. Correlation between cation concentrations in the standard solutions and those gken by (A) the mean of four combinations of three electrode arrays and (B) the manual method.

used to estimate each calibration point. A similar situation also occurs when results obtained with manually calibrated ISEs are compared to a reference method. Therefore, (i) electrodes calibrated manually cannot be used outside the calibration range and (ii) the ISE response cannot be modeled by using the Nickolskii-Eisenmann equation, since the electrode slope and the cell potential cannot be accurately determined. Consequently, semiempirical methods (calibration curves, standard addition) are always used to determine un-

[Na+],mM

[K+], mM

[Ca2+],mM

125.07 (2.10) 123.81 (1.07) 124.8 (1.87) 121.4 (-0.89) 120.9 (-1.31) 121.5 (-0.81) 124.2 (1.38) 122.1 (-0.33) 201.08 (0.54) 185.82 (-2.21) 207.32 (-1.27) 51.61 (3.22) 68.2 (-2.57) 60.5 (0.85)

2.96 (-1.31) 2.94 (-1.86) 5.12 (2.40) 4.88 (-2.41) 3.09 (3.00) 2.95 (-1.66) 5.12 (2.40) 4.92 (-1.60) 20.69 (3.45) 14.88 (-0.08) 9.77 (-2.31) 0.48 (-3.42) 0.103 (3.08) 0.31 (3.33)

0.77 (2.66) 1.23 (-1.60) 0.76 (1.33) 1.23 (-1.61) 1.22 (-2.4) 0.77 (3.22) 0.75 (1.00) 1.26 (0.81) 9.88 (-1.21) 5.83 (-3.11) 5.15 (3.00) 0.21 (3.85) 0.098 (-1.97) 0.08 (3.44)

known solutions with ISEs. This problem does not arise for any of the three cations determined by using the computerbased array method. In all cases a slope of unity and zero intercept (Figure 3A) are observed; this suggests that the modeled array can effectively decouple analyte and interferent signals. In order to further investigate the prediction ability of the electrode array, tertiary solutions other than those used in the calibration procedure were analyzed, and representative results are presented in Table VII. Samples 1-8 contain cation concentrations that lie within the concentration range used for model building, while samples 9-11 and 12-14 contain cation concentrations which are above and below the calibration range, respectively. The errors observed do not exceed 4%. The errors obtained for solutions 9-14 suggest that the array response can be accurately represented by the model for a restricted concentration range outside of that used for modeling. It is significant that the modeled array can successfully decouple those signals arising from primary and interfering ions even outside the initial calibration range. This observation further suggests that the selectivity coefficients obtained by using the nonlinear regression procedure more accurately reflect sensor response than those obtained by using manual procedures. Plasma Samples. In order to investigate the ability of the modeled array to predict sodium, potassium, and calcium ions within a complex matrix, plasma samples were analyzed. This was achieved by modeling the array response as described previously, using a series of artificial plasma samples as detailed in the Experimental Design section. The results obtained by using the modeled array of three selective sensors for sodium, potassium, and calcium ion determination in real plasma samples, as compared to those obtained from a commercial analyzer (Technicon Smac 3 ion analyzer) are given in Figure 4. This figure clearly shows the ability of the array to accurately determine the cation concentrations without significant bias. This is in contrast to previous single-electrode measurements conducted by using the same sodium sensor as that employed within the array ( l o ) ,which showed acceptable accuracy but a nonunity slope in the correlation plot. This can arise if the contribution from interferents to the total signal varies between plasma samples. The fact that a high degree of accuracy in the cation determination and a unity slope are observed in Figure 4 suggests that the array of selective sensors can eliminate some of the problems associated with single-electrode measurements. It is significant to note that the results presented in Figure 4 were obtained by using the array of highly selective electrodes excluding the sparingly selective sensor. Inclusion of the SSE in the cation determination gives results that are less

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ANALYTICAL CHEMISTRY, VOL. 63,NO. 9, MAY 1, 1991

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127

131

INa'I

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Y

30

I

135 139 143 mM Technicon Smac 3

7

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29

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41

IK'I mM Technicon Smac 3

ICa*+l mM Technicon Smac 3

Flgure 4. Correlation of duplicated measurements of (A) Na+, (B) K', and (C) Ca2+using an array of selective sensors vs the Technicon Smac 3 analyzer.

accurate and show larger standard deviations than those obtained from the selective array. This result is not unexpected, since the SSE was only characterized with respect to sodium, potassium, and calcium ions. Given the sparingly selective nature of this sensor, it will respond to other components in the plasma samples that are not included in the model and cannot therefore be decoupled from the total signal. In order to achieve the improvement in accuracy and precision, which inclusion of the SSE gave in the case of tertiary solutions, the response surface of the SSE would have to be more fully characterized.

CONCLUSION The response of individual members of an array of potentiometric sensors in mixed solutions has been accurately modeled by the Nickolskii-Eisenmann equation by using nonlinear optimization techniques. The model parameters are well defined and show acceptable intra- and interarray precisions. The advantages of selective and nonselective sensors are combined within the array. Highly selective electrodes respond to only a small range of ions, are less prone to interference, and thus can be simply modeled. Sparingly selective electrodes respond to a range of ions and can therefore be modeled as pseudo selective sensors, giving an overdetermined systems, allowing prediction polling. The ability of the array of selective and sparingly selective ISEs to determine cation concentrations in tertiary mixtures to within 2.8% error where the cation concentrations are a t the physiological level has been demonstrated. The inclusion of a sparingly selective electrode can give improved precision over an array consisting only of selective sensors provided the electrode is fully characterized or the samples do not contain unmodeled interferents. The benefits of this technique over manual methods have been clearly demonstrated. Accordingly, the modeled array allows the determination of each primary ion concentration in the presence of a large excess of interferents or in samples whose background composition

varies significantly and may also be used outside the calibration range. These features are a direct consequence of the accuracy of the cell potentials, selectivity coefficients, and electrode slopes determined by the nonlinear optimization procedure, an accuracy which cannot be achieved by manual calibration procedures. In the determination of sodium, potassium, and calcium ions in human plasma, the model of the SSE response including only two interferents is inadequate and the sparingly selective sensor does not improve the precision and accuracy of the array predictions. The array of selective sensors combines the selectivity of the sensors with a model that can efficiently decouple signals arising from primary and interfering ions to provide accurate predictions even in complex media such as plasma. The inclusion of a sparingly selective sensor whose response is not fully modeled, while being incapable of providing quantitative information, can be used for diagnostic purposes such as indicating the presence of cations other than sodium, potassium, and calcium. The results obtained by using the selective array are more precise and show a direct unbiased correlation with a commercial analyzer unlike traditional single-electrode measurements. The use of computer-based modeling of sensor arrays can undoubtedly improve accuracy and precision in the determination of analytes within mixtures where the sensors are prone to interference. It can also greatly simplify calibration procedures. However, for this potential to be fully realized the interarray precision of these devices (notably in the standard cell potential Ejo)will have to be significantly improved. With present manual fabrication procedures for ISEs, sensors must be individually modeled and initial characterization of an array requires a large number of solutions. However, if identical sensors could be produced, then with the techniques described here, problems arising from interfering ions affecting the ISE response can be largely eliminated and the emphasis in ISE performance shifts from extreme selectivity toward reproducible fabrication and stability of response.

ACKNOWLEDGMENT The kind donation of plasma samples and cation analysis using the Technicon Smac 3 by Mr. Peter Gaffney, St. James Hospital, Dublin 8, is gratefully acknowledged.

LITERATURE CITED (1) Otto, M.; Thomas, J. D. R. Anal. Chem. 1985, 57, 2647. (2) Beebe, K. R.; Uerz, D.; Sandifer, J.; Kowalski, B. R. Anal. Chem. 1988, 60, 66. (3) Beebe, K. R.; Carey, W. P.; Sanchez, M. E.; Erickson, B. C.; Wllson, B. E.; Wangen, L. E.; Kowalski, B. R.; Ramos, L. S. Anal. Chem. 1986, 58, 294R. (4) Beebe, K.; Uerz, D.; Sandifer, J.; Kowalski, B. Anal. Chem. 1988, 60, 66. (5) Diamond, D.; Regan, F. Electroanalysis 1990, 2, 113. (6) Cadogan, A. M.; Diamond, D.; Smyth, M. R.; Deasy, M.; McKervey, M. A.; Harris, S. J. Analysf 1989, 114, 1551. (7) Amman, D.; Morf, W. E.; Anker, P.; Meier, P. C.; Pretsch, E.; Simon, W. IS€ Rev. 1983, 5 , 3. ( 8 ) Forster, R. J.; Cadcgan, A,; Telting-Diaz. M.; Diamond, D.; Harris, S. J.; McKervey, M. A. Sens. Actuators, in press. (9) Worth, G. J. Analyst 1988, 113, 373. (10) Telting Diaz, M.; Regan, F.; Diamond, D.; Smyth, M. R. J . pherm. Bnmed. Anal. 1990, 8, 695. (11) Press, W. H.; Flannery, B. P.; Teukoisky, S. A,; Vetterling, W. T. Numerlcal Recipes; Cambridge University Press: Cambridge, U. K., 1986. (12) Moody, G. J.; Thomas, J. D. R. Sel. Annu. Rev. Anal. Sci. 1973, 3, 59. (13) Anker, P.; Jenny, H.-6.; Wuthier, U.; Asper, R.; Ammann, D.; Simon, W. Clin. Chem. 1983, 29, 1508.

RECEIVED for review September 12,1990. Accepted January 17, 1991.

Modeling of potentiometric electrode arrays for multicomponent

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