Computer-assisted spectrophotometry: Multicomponent analysis with a discrete fourier transform

Talmta, Vol. 37, No. 12,pp. 11834188, 1990 Printedin Great Britain

0039-9140/90$3.00+ 0.00 Pergamon Press plc

COMPUTER-ASSISTED SPECTROPHOTOMETRY: MULTICOMPONENT ANALYSIS WITH A DISCRETE FOURIER TRANSFORM MOI-LWEZD A. KORANY,* MAHMOUDA. ELSAYED,MONA M. BEDAIR and HODA MAHCOUB Department of Pharmaceutical Analytical Chemistry, Faculty of Pharmacy, University of Alexandria, Egypt EZZAT A. KOIUNY Institute of Graduate Studies and Research, UNARC, University of Alexandria, Egypt (Received 6 June 1989. Revised 16 May 1990. Accepted 14 June 1990) Summary-A computer-assisted method for analysis of multicomponent mixtures by use of conventional absorbance as well as discrete Fourier transform coefficients (combined trigonometric functions) is presented. The program can store absorbance data (A vs. A), process data by convolution with combined trigonometric functions, apply least-squares analysis and solve the resultant simultaneous linear equations, and display data on screen, printer or plotter.

There have been many attempts to devise methods of computer-assisted spectrophotometry for the analysis of complex systems,‘-17 but none of these has successfully tackled the problem of background interferences. Absorption curves have been expanded as a finite Fourier series.‘G20 If (n + 1) is an odd number, the expansion is

It has been shown previously,‘**‘9 that a correction for linear background absorption can be made by the combination of two trigonometric functions (Table 1). Thus, for discrete measurements at equally spaced wavelengths, the coefficients of cosjx, calculated from f(J) with a linear component d + mx added, are

f(l)=a,+a,cosx+a~cos2x+**~

mx,]cOS jxi/D ,f,otf(J)t+d+

- * * + a,,, cos(n/2)x + b, sin x + b2 sin 2x + . - * + b,,, sin@ /2)x

(1)

or if (n + 1) is an even number then

= aj - m/D

(4)

where D is the denominator of equation (3). If x is displaced by one interval, i.e., by 2a/(n -t I), then

f(~)=a,+a,cosx+a~cos2x+*~~ i$o [f(n), + d + mXi]cos j[xi + 2n/(n + l)]/D

* * * + a, + I)/2cos(n + 1)/2x + b, sin x +

=a;-tm/D

b2 sin 2x + * * * + b, + ,),2sin@ + 1)/2x (2)

Addition of equations (4) and (5) leads to a sum of two coefficients (aj + a;), which is directly proportional to the concentration of the pure compound and is independent of the linear component added to f(L).19 The two functions, e.g., cos x and cos[x + a/(n + I)] or the corresponding sine functions, can be linearly combined to give T’x. Thus

The calculation of the coefficients a,, a,, and b,, b2, b3,... is simplified since the trigonometric functions are mutually orthogonal. Any coefficient, t, can be calculated from a set of absorbances, measured at equally spaced wavelengths, by the following summation, in which x takes values from 0 to 27r - [2a/(n + I)], at intervals of 27r/(n + 1): a3,...

z=

i$o/(i)iTxi/ i tTxi12

i-0

where T represents cosine or sine.

(5)

t’ = ,iof(i)i

(3)

YXi/ i

(T’x,)2

i-0

(6)

where T’xi is the combined function and t’ =o’C

*Author for correspondence. 1183

(7)

MOHAMSD A. KORANY et al.

1184

Table 1. Combined trigonometric Fourier functions for n + 1 equally spaced points (generated by the computer program) (:,

[cos x, +

2;

cos(x,+

60”)]

[co52x, + co8 2(x, + 60”)]

0

1 2 3 4 5

60 120 180 240 300

1.5 0 -1.5 -1.5 0 1.5

D

3

3

n (i)

x,1 deg

[cos x, + cos(x, + 45”)]

[cos 2x, + cos 2(x, + 45’)]

[sin x, sin(x, + 45”)]

[sin 2x, sin 2(x, + 45’11

0 1 2 3 4 5 6 7

0 45 90 135 180 225 270 315

1.707 0.707 -0.707 - 1.707 - 1.707 -0.707 0.707 1.707

1.0 -1.0 -1.0 1.0 1.0 -1.0 -1.0 1.0

-0.707 -0.293 0.293 0.707 0.707 0.293 -0.293 -0.707

-1.0 1.0 1.0 -1.0 -1.0 1.0 1.0 -1.0

4

D

4

1 cm) and C is the concen-

MULTICOMPONENT SPECTROPHOTOMETRIC

ASSAY

In a multicomponent system (provided that each component obeys Beer’s law and no interaction exists between the components), the absorbance of the mixture at any wavelength is A,= Cii c1!,+ CZ~$/+ *. *+ Cin$,

(8)

where A, is the absorbance of solution i at wavelength j, Ci, is the concentration of component n in solution i, and a,,, is the At”&,value of component n at wavelength j. The Fourier coefficient of the mixture at any Iz [mean wavelengths (&ha, + &,,,)/2] is given by f$= C,lO;j+ CaOb+ ***+ CbOij

(9)

where t; is the coefficent of the combined trigonometric functions for solution i calculated at the mean wavelength (J)j, C, is the concentration of component n in solution i and CO;,is t/(1%, 1 cm) for component n at mean wavelength (A),. THE COMPUTER

PROGRAM

The program provides an interactive dialogue to control the data processing of data collected from the spectrophotometer. The program performs the following main tasks.

-0.866

[sin 2x, sin 2(x, + 60”)]

0

where o’ is t’(l%, tration.‘8~‘g

0.5 -1.0 0.5 0.5 -1.0 0.5

[sin x, sin(x, + 600)]

i.866 0.866 0 -0.866 3

4

-0.866 1.732 -0.866 -0.866 1.732 -0.866 3

4

(A) Storage of data. Wavelength and absorbance data for a compound are stored in a sequential named file on magnetic diskette. (B) Processing of data. Data stored in flies can be processed as follows:

(1) Convolute2’ the absorption data with combined trigonometric Fourier functions and output the results to a file. (2) Reduce the number of equations entered to N equations by least squares, and solve for concentrations of N compounds. (3) Solve for concentration of N compounds with N equations. The convolution process can be performed for (a) different numbers of points in a segment, (b) different orders of combined trigonometric functions and (c) different wavelength intervals. The program outputs the absorption curve convoluted with the combined trigonometric functions. These functions (Table l), with their respective divisors, are generated by the program according to the number of points entered by the user. (C) Display of data. Data are displayed on screen, printer or plotter. The plotter driver allows data to be scaled to match the dimensions of the plotting area, and specifies the labelling of the axes. A block diagram of the program is shown in Fig. 1.

Computer-assisted

spectrophotometry

1185

I

I

CHOOSE NUMBER OF POINTS LARGER TRAN NUMRER OF COMPONENTSAND APPLY LEAST-SQUARES URTROD, TREN SOLVE FOR CONCBNTRATIONS

CROOSE NDNEEB OF

POINTS EQUAL TO NUMRBR OF COMPONENTS AND SOLVE FOR CONCENTRATIONS I, I

I

I

CEOOSE NUMSBR OF POINTS EQUAL TO NUMBER OF COMPONENTS AND SOLVE FOR CONCENTRATIONS 1

I

I

CHOOSENUMRER OF POINTS LARGBRTRANNUMREROF COMPONENTSANDAPPLY LRAST-SQUARES METROD, TRRN SOLVE FOR CONCENTRATIONS

t

t +

PRINT CONCENTRATIONS OF COMPONENTS

Fig. 1. Functional block diagram of the computer program. RESULTS AND DI!3CUSSiON

The absorption spectra of the codeine phosphate, phenylephrine hydrochloride, chlorpheniramine maleate and ephedrine hydrochloride are shown in Fig. 2. The methods used to assay these components in a mixture were the Unique Absorbance Method (UAM) and Unique Fourier Function Method (UFFM), and the corresponding least-squares procedures. In UAM four absorbance readings (equal to the number of components) were determined (at different 1 values) for each component on its own. Then each mixture was measured at the selected wavelengths. In UFFM the absorption curves for different components were convoluted (Fig. 2) and the coefficients t’ = (a, + a;) were calculated by using &point T’ [ = cos x, + cos(x, + 4S”)] combined trigonometric functions (Table 1) from

&tnitipi to &,.i [268-296 nm (codeine phosphate), 260-288 nm (phenylephrine hydrochloride), 250-278 nm (chlorpheniramine maleate) and 242-270 nm (ephedrine hydrochloride)] at 4-nm intervals. For the corresponding least-squares methods, the number of coefficients must be greater than the number of components, so for A-LSM (Absorbance Least-Squares Method), 27 absorbances were measured for each component in the region 244-296 nm at 2-nm intervals. For FF-LSM (Fourier Function Least-Squares Method), 19 coefficients were calculated for each component (for 260-296 mn at 2-nm intervals). The results for four synthetic fourcomponent mixtures are given in Table 2. According to the variance-ratio test, all the methods gave F values not exceeding the theoretical value (95% confidence limits). The four methods thus gave equal reproducibility.

1186

MO-

A. KORANY et al.

(a)

0.9

(a)

a~~~~~ t

0.6

0.9 -

,/‘A

o.a-

I 1

\ \ \ \

0.7 -

a

240

250

260 270 260 Wavelength (nm)

290

0.60.5-

300 234

242

250

256

266

Wavelength

bl

274

262

290

(nm)

b) (i .I

^

O-

.C

0 +

2 (- .)

Mean

Fig. 2. Absorption curves for 60 tie/ml codeine phosphate (-), 50 &ml phenylephrine hydrochloride (---), 30 pg/ml chlorpeniramine maleate (. .*), 0.30 mg/ml ephedrine hydrochloride (- . . -), and propylparaben (-a - ) in 0.05M sulphuric acid (a), and the corresponding convoluted curves derived by using 8-point T’ [= cos x, + cos(x,+ 45”)] combined trigonometric functions at 4-nm intervals (b).

wavelength

(nml

Fig. 3. Absorption curves of 2 mg/ml gum acacia (-), 2 mg/ml gelatin (---), 41 ng/ml barium sulphate ( ..), 2pg/ml colouring matter (-. .-) in 0.05M sulphuric acid (a), and the corresponding convoluted curves derived therefrom by using I-point T’ [ = cosx, + cos(x,+ 45”)] combined trigonometric functions at 4-nm intervals (b).

Table 2. Results for assay of four-component laboratory-made mixtures by the proposed computer-assisted spectrophotometric methods Recovery, % Codeine phosphate (I)

Phenylephrine hydrochloride (II)

Chlorpeniramine maleate (III)

Ephedrine hydrochloride (IV)

Mixture*

UAM

UFFM

UAM

UFFM

UAM

UFFM

UAM

UFFM

1 2 3 4

100.0 101.4 97.8 99.1

99.4 100.5 99.2 98.7

98.6 100.1 99.9 99.6

98.7 99.8 99.7 99.9

98.4 100.1 98.7 99.5

96.8 z?:

101.2 100.8 102.6 101.9

100.4 99.1 99.7 101.1

Mean SD.

98:4

99.6 1.5

99.4 0.8 0.23 3.98

99.5 0.7

99.5 0.8 0.05 1.35

99.2 0.8

98.0 1.1 1.62 2.07

101.6 0.8

100.1 0.9 2.55 1.19

t-value? F-value? A-LSM

FF-LSM

A-LSM

FF-LSM

A-LSM

FF-LSM

A-LSM

FF-LSM

1 2 3 4

100.6 101.3 99.9 100.4

100.8 101.8 99.9 100.6

98.7 99.7 99.5 99.5

98.7 99.7 100.0 100.4

98.0 100.1 98.2 99.3

98.0 100.8 98.2 99.1

101.8 101.2 102.3 101.3

101.3 99.9 102.4 102.0

Mean S.D.

100.5 0.6 1.2 6.8

190.8 0.8

99.3 ZO

101.6 0.5 0.05 2.45

101.4 1.1 0.29 1.91

t-value F-value

:::

2:25

99.7 0.7 0.30 1.18

98.9 1.0 0.43 1.62

99.0 1.7 0.14 2.73

*Component concentrations (mg/lOO ml) in mixtures 14 were I (8,6,6,9); II (6,5,5,6); III (3,2,3,1.9); IV (30,35,30,32). tThe z- and F-values are calculated with respect to the UAM values. The corresponding theoretical values are 2.44 and 9.28, respectively.

1187

Computer-assisted spectrophotometry Table 3. Results for assay of a four-component mixture* in the presence of various additives by the proposed computer-assisted spectrophotometric method Recovery, % Codeine phosphate (I) Additive

UAM

UFFM

None

100.0 (100.6)

(1:::)

Carmoisine, 2 r8/ml

101.1 (102.5)

Barium sulphate, 150.6 (127.0) 41 lrglml 114.1 Gum acacia, (109.0) 2 mg/ml

103.5 (101.6)

123.83 (124.6)

110.89 (102.3)

Gelatine, 2 mg/ml

Ephedrine Phenylephrine Chorpheniramine hydrochloride (I) maleate (III) hydrochloride (IV) UAM

UFFM

UAM

UFFM

UAM

UFFM

98.6 (98.7)

98.7 (98.7)

98.4 (98.0)

96.8 (98.0)

101.2 (101.8)

loo.4 (101.3)

97.9 (99.4)

101.5 (99.4)

100.4 (99.9)

100.0 (101.0)

96.6 (98.6)

106.1 (106.0)

100.2 (100.2)

97.3 (98.5)

92.5 (99.2)

100.3 (99.6)

121.63 (130.2)

96.45 (98.2)

97.8 (73.9)

102.0 (100.7)

;::)

(:::;)

(1:::) 100.49 (94.5)

(::::)

(::::,

84.9 (101.1)

111.6 (118.7)

(:& 117.8 (98.1)

118.9 (105.2)

55.9 (101.4)

*The mixture contained 80 &ml (I), 60 pgjml (II), 30 &ml (III) and 0.30 mg/ml (IV). The figures in parentheses are the assay results obtained by using the same methods with the least-squares approach. Table 4. Results for assay of a four-component mixture in the presence of a preservative (propylparaben) as the fifth component, by the proposed computer-assisted spectrophotometric methods Recovery, %

Methods

Codeine phosphate (30 pg/ml)

Phenylephrine hydrochloride (20 fig/ml)

UAM UFFM A-LSM FF-LSM

97.6 112.5 99.5 100.5

99.7 97.7 97.8 101.5

Chorpheniramine Ephedrine maleate hydrochloride Propylparaben (10 &ml) (100 !&ml) (10 pg/ml)

A t-test gave 95% confidence that the four methods are of equal accuracy. To investigate matrix effects and spectral interferences, materials such as colouring matter (carmoisine), barium sulphate, (to form a turbid solution), thickening agents (gum acacia and gelatine) and a preservative (propylparaben) were added. The absorbance spectra of these additives are shown in Fig. 3. Barium sulphate at 41 pg/ml gives a significant linear spectral interference, whereas carmoisine (2 ,ug/ml) gives a negligible interference. Gelatine at 2 mg/ml absorbs strongly at wavelengths shorter than about 246 nm, then has a more or less constant absorbance at longer wavelengths. Therefore special paramaters had to be used for the multicomponent analysis when gelatine was present (FF-LSM was used with 15 coefficients calculated over the range 260-288 nm at 2-nm intervals). Gum acacia exhibits less spectral interference than gelatine. The convoluted curves derived by using combined trigonometric functions are shown in Fig. 3. The results obtained by the proposed computer-assisted spectrophotometric procedure in

96.1 104.2 99.1 98.3

93.2 99.4 97.8 104.1

100.1 102.7 99.0 98.6

the presence of various additives (Table 3), showed that neither UAM nor A-LSM can correct for systematic errors caused by addition of barium sulphate, gum acacia and gelatine. Such errors occurred with codeine phosphate, chlorpheniramine maleate and ephedrine hydrochloride. The other methods allowed correction to be made for matrix effects. The Fourier function least-squares method (FF-LSM) was usually found to be the most accurate. A different procedure was adopted in the presence of propylparaben preservative, in that the latter (Fig. 2) was treated as a fifth component. The mixture was analysed as before but with an increased number of coefficients determined. Assay results for the five-component mixture are presented in Table 4. The poorest performance was obtained with the UFFM. REFERENCES 1. S. R. Heller, R. Potenxone Jr., G. W. A. Milne and Ch. Fisk, Trends Anal. Chem., 1981,1,62; Chem. Abstr., 1982, 96, 5709e. 2. H. Gampp. M. Maecler and A. D. Zuberbilhler, Talanta, 1980, 27, 1037.

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MOHAMED A. KORANYet al.

3. A. T. Pilipenko, L. I. Savranskii and A. N. Mas’ko, Zh. Analit. Khim., 1985, 40, 232; Chetn. Abstr., 1985, 102, 142327k. 4. J. Sustek, Anal. Gem., 1974, 46, 1676. 5. A. T. Pilipenko, L. I. Savranskii, A. N. Mas’ko and V. L. Sheptun, Dokl. Akad. Nauk SSSR, 1981,260,377; Chem. Abstr., 1981, 95, 1968448. 6. W. Yang, Fenxi Huaxue, 1985, 13, 476; Chem. Abstr., 1985, 103, 152724x. 7. V. Kratochvil, A. Novak and J. Plisek, Chem. Prum., 1978, 28, 240, Chem. Abstr., 1978, 89, 164913~. 8. A. F. Vasil’ev and N. L. Aryutkina, Zavodrk. Lab., 1977, 43, 1330; Chem. Abstr., 1978, 89, 16009h. 9. N. L. Aryutkina and A. F. Vasil’ev, ibid., 1983, 49, No. 10, 53; Chem. Abstr., 1983, 99, 224404s. 10. V. V. Volkov and B. N. Grechushnikov, Zh. Prikl. Spektrosk, 1984, 40, 264; Chem. Abstr., 1984, 108, 150403f. 11. B. W. Madsen, D. Herbison-Evans and J. S. Robertson, J. Pharm. Pharmac., 1974, 26, 629. 12. B. W. Madsen and J. S. Robertson, ibid., 1974,26,682.

13. P. Arnaud, C. Metayer and N. LeGall, Lube-PharmaProbl. Tech., 1980, 2& 380. 14. J. Cheng, D. An and R. Wu, Nanjing Yaoxueyuan Xuebao, 1985, 16, No. 2, 74; Chem. Abstr., 1986, 104, 24252a. 15. T. Tajima and T. Maeda Shim&u Hyoron, 1983, 40, 139; Chem. Abstr., 1984, 100, 95865f. 16. A. E. McDowell and H. L. Pardue, J. Pharm. Sci., 1978, 67, 822. 17. A. E. McDowell, R. S. Harner and H. L. Pardue, Clin. Chem., 1976, 22, 1862; AMY. Abstr., 1978, 34, 3D53. 18. M. A. Korany, Ph.D. Thesis, University of Alexandria, 1974. 19. A. M. Wahbi, H. Abdine and M. A. Korany, Pharmazie, 1978, 33, 278. 20. I. M. Dubrovkin, Izv. Sev.-Kavk. Nauchn. Tsentra Vyssh. WC., Estestu. Nat&i, 1981, No. 1, 57. 21. A. V. Gppenheim and R. W. Schafer, Digital Signal Processing, p. 11 ff. Prentice-Hall, Englewood Cliffs, 1975.