an experimental design approach to selecting the optimum

8/14/2019 An Experimental Design Approach to Selecting the Optimum

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c o m p l e t e s e p a r a t i o n o f t h e w a t e r - s o l u b l e v it a m i n s in ar e a s o n a b l e a n a l y s is t i m e b y s y s t e m a t i c s t u d y o f t h e f a c -t o r s i n fl u e n c in g t h e c h r o m a t o g r a p h i c b e h a v i o r o f t h eW S V s . W e p r o c e e d e d b y a s e q u e n t ia l a p p r o a c h u s in ge x p e r i m e n t a l d e s i gn m e t h o d o l o g y , w h i c h e n a b l e d m o d -e l in g a n d o p t i m i z a t i o n o f t h e c h r o m a t o g r a p h i c s e p a r a -t i o n o f W S V s .T h e f a c t o r s e v a l u a t e d w e r e t h e c o n c e n t r a t io n o f m e t h a -n o l a s o r g a n i c m o d i f i e r , t h e c o n c e n t r a t i o n o f h e x a n e s u l -f o n a t e a s i o n - p a i r r e a g e n t , m o b i l e p h a s e p H a n d t h ec o n c e n t r a t i o n o f t r ie t h a n o l a m i n e ( T E t O H A ) a s c o m -p e t i t i v e b a s e . T h e r e l a t i v e i m p o r t a n c e o f th e f o u r m o b i l ep h a s e p a r a m e t e r s a n d t h e i r i n t e r a c t i o n s t o t h e c h r o m a -t o g r a p h ic b e h a v i o r o f e a c h c o m p o u n d w e r e e x a m i n e db y a p p l i c a t i o n o f a D r a p e r - L i n s m a l l c o m p o s i t e d e s i g n[ 2 , 3 ] . R e t e n t i o n t i m e s a n d p e a k - w i d t h s a t h a l f - h e i g h tw e r e m e a s u r e d a n d q u a d r a t i c p o ly n o m i a l m o d e l s e n-ab l e d f i t ti ng o f t he da t a .A f u ll s e c o n d - o r d e r p o l y n o m i a l m o d e l is o n e o f t h e m o s tu s e f u l m o d e l s f o r a p p r o x i m a t i n g a r e g i o n o f a m u l t i-f a c t o r r e s p o n s e s u r f a c e . A r a t i o n a l i z a t i o n o f t h e w i d e -s p r e a d u s e o f f u l l s e c o n d - o r d e r p o l y n o m i a l m o d e l s i st h a t t h e y r e p r e s e n t a t r u n c a t e d T a y l o r s e r i e s e x p a n s i o no f a n y c o n t i n u o u s f u n c t i o n , a n d s u c h m o d e l s w o u l dt h e r e f o r e b e e x p e c t e d t o p r o v i d e a r e a s o n a b l y g o o d a p -p r o x i m a t i o n o f t h e t r u e r e s p o n s e s u r f a c e o v e r a l o c a l r e -g i o n o f e x p e r i m e n t s p a c e [ 4] . F o r f o u r f a c t o rs , t h e q u a d -r a t i c m o d e l i s o f t h e f o r m :

Y = B 0 + B1X1 + B2X2 + B3X3 + B4X4 +B12X1 X 2 + B I 3 X 1 X3 + B14X1X4 + B23X2X 3 +B24X2X~ + B34X3X4 + Bl lX1 2 + B22X22 + B33X32+ B 44 X4 + E R R O R

w h e r e Y is th e m e a s u r e d r e s p o n s e v a r i a b le , X1-X4 a r et h e i n d e p e n d e n t c h r o m a t o g r a p h i c v a r i ab l e s , B O is th e i n -t e r c e p t . ( o r c o n s t a n t t e r m ) , B1-B4 are t he f i r s t o rder co -ef f i c i en t s , B12-B34 are t he i n t e r ac t i on coef f i c i en t s andB l l - B 4 4 a r e t h e ( p u r e ) s e c o n d - o r d e r c o e f f i c i e n t s .I n t h e g e n e r a l c a s e , c o n s i d e r t h e s i t u a t i o n i n w h i c h a r e -s p o n s e y d e p e n d s o n k f a c to r s , c o d e d a s xl , x2 ...... Xk. W ec a n a p p r o x i m a t e t h e r e s p o n s e f u n c t i o n o v e r a l i m i t e de x p e r i m e n t a l r e g i o n b y a q u a d r a t i c r e s p o n s e - s u r f a c em o d e l o f t h e f o r m :

k k kY=13o + ZI3~x~ + E [ ~ ii x ~ 1- E E ~ i j x i x j 1 -~i l i l i>j l

T h e q u a d r a t i c m o d e l h a s a c o n s t a n t t e r m , k f i r s t - o r d e rt e r m s , k q u a d r a t i c t e r m s , k(k - 1) / 2 i n t e r ac t i on t e rmsa n d t h u s a to t a l o f p = 89 + 1 ) ( k + 2 ) p a r a m e t e r s . T h es p e c i f ic c h o i c e o f d e s i g n t o o b t a i n t h e d a t a f o r s u c h am o d e l f i tt in g w o u l d d e p e n d o n t h e r e l a t iv e i m p o r t a n c et o t h e e x p e r i m e n t e r o f v a r i o u s d e s i g n f e a t u r e s [ 5 ] . I nt h is w o r k w e u s e d a n a p p r o p r i a t e s m a l l e x p e r i m e n t a ld e s i g n o f c o m p o s i t e f o r m f o r e s ti m a t i n g t h e8 9 p a r a m e t e r s , s u c h t h a t t h e n u m b e r o f r u n si s a s l it t le i n e x c e s s o f p a s p o s s ib l e . C o m p o s i t e d e s i g n f o rf i tt i n g s e c o n d - o r d e r s u r f a c e s w e r e i n it ia l ly s u g g e s t e d b yB o x a n d W i l s o n [ 6] ( w h o a d d e d s t a r d e s i g n s t o 2 k f a c t o -r i a ls ) a n d f o l l o w e d u p b y B o x a n d H u n t e r [ 7] .

I n g e n e r a l t h e c o m p o s i t e d e s i g n c o n s i s t s o f a 2 k fa c t o r i a lo r a 2 k -q f r a c ti o n a l f a c t o r i a l p o r t i o n ( c a l l e d a c u b e ) , w i thruns sele cte d f ro m the 2 k run s (Xl , x2 .. .. . Xk) = (1, 1,..., 1)usua l l y o f r eso l u t i on V o r h i gher , p l u s 2k ax i a l po i n t s a t ad i s t ance (x f rom t he o r i g i n , p l u s no cen t e r po i n t s (0 ,0 . .. . 0 ). C o m p o s i t e d e s i g n a r e e x t r e m e l y u s e f u l f o r s e -q u e n t i a l e x p e r i m e n t a t i o n i n w h i c h t h e c u b e p o r t i o n e n -a b l e s e s t im a t i o n o f t h e f i r s t - o r d e r e f f e c t s a n d i n t e r a c -t i o n e f fe c t s , a n d t h e a d d i t i o n o f s t a r p o i n t s e n a b l e ss e c o n d - o r d e r te r m s t o b e a d d e d t o th e m o d e l a n d t o b ee s t i m a t e d . A g e o m e t r i c r e p r e s e n t a t i o n o f a t h r e e - f a c t o rD r a p e r - L i n s m a l l c o m p o s i t e d e s i g n i s d e p i c t e d i n F i g -ure 1 .

xperimentalSamples Chem icals and SolventsT h e r e f e r e n c e s u b s t a n c e s , a s c o r b i c a c i d ( v i t a m i n C ) ,t h i a m i n e m o n o n i t r a t e ( v i ta m i n B 1 ) , p y r i d o x i n e h y d r o -ch l o r i de (v i t am i n B6), n i co t i n am i de (v i t ami n B3) and r i -b o f l a v i n p h o s p h a t e ( v i t a m i n B 2 ) , w e r e s u p p l i e d b y F e d -e r a ( B r u s s e l s ) . C h e m i c a l s a n d s o l v e n t s w e r e o f p r oa n a l y si s q u a li ty : h e x a n e s u l f o n i c a c i d N a s a l t . H 2 0 , 9 8 % ,w a s f r o m J a n s s e n C h i m i c a ( B e e r s e , B e l g i u m ) . T r ie t h a -n o l a m i n e , p h o s p h o r i c a c i d, a c e t i c a c i d, d i m e t h y l s u l f o x -i de , p o t a s s iu m d i h y d r o g e n p h o s p h a t e a n d s o d i u m h y -d r o x id e s o l u t io n w e r e s u p p l ie d b y M e r c k ( D a r m s t a d t ,G e r m a n y ) a n d m e t h a n o l w a s f r o m L a b - S c a n ( D u b l i n ,I r el a n d) . T h e w a t e r u s e d f o r H P L C w a s p r e p a r e d w i t h aM i l li - Q s y s t e m ( M i l li p o r e , M i l f o r d , M A , U S A ) .Apparatus and ColumnL i q u i d c h r o m a t o g r a p h y w a s p e r f o r m e d w i t h a W a t e r sm o d e l 6 0 0 m u l t i - s o l v e n t d e l i v e r y s y s t e m ( M i l l i p o r e -W a t e rs , M i l fo r d , M A , U S A ) e n a b l in g u s e o f a m i x e d m o -b i l e p h a s e f r o m f o u r r e s e r v o i r s c o n t i n u o u s l y d e g a s s e dw i t h h e l i u m a t a f l o w r a t e o f 2 5 m L m i n - 1. T h e p u m p h a da b u i l t - i n ' s i l k ' f e a t u r e , a p u m p t r a n s d u c e r f e e d - b a c kl o o p e n a b l in g t h e f l o w t o b e c o n s t a n t l y m o n i t o r e d a n da d j u s t e d f o r p u l s e - f r e e f lo w . A W a t e r s 9 9 6 p h o t o - d i o d e -a r r ay d e t e c t o r w a s l in k e d t o a n N E C P o w e r m a t e 3 8 6 /3 3 id a t a s t a t i o n . A M a r a t h o n a u t o s a m p l e r ( S p a r k H o l l a n d ,E m m e n T h e N e t h e r l a n d s ) w a s p r o v i d e d w i t h a 2 0 ~ t Ll o o p . T h e 2 5 0 x 4 .6 m m i .d . A l l t i m a c o l u m n ( A l l t e c h )w i t h p a c k e d w i t h C 8 - b o n d e d 5 - ~ m s p h e r i c a l s i l i c a . T h ef l o w r a t e w a s k e p t c o n s t a n t a t 1 m L m i n - 1. T w o -d i m e n s i o n a l c h r o m a t o g r a m s w e r e r e c o r d e d a t a w a v e -l eng t h o f 270 nm.Mobi le Phase Com posi t ionF o u r s e p a r a t e a q u e o u s s o l u t io n s , e a c h c o n t a i n i n g a c o m -p o n e n t o f t h e m o b i l e p h a s e , w e r e s t o r e d i n f o u r s o l v e n tr e s e r v o i r s A , B , C a n d D . R e s e r v o i r A c o n t a i n e d 1 00 m Mt r i e t h a n o l a m i n e ; r e s e r v o i r B 5 0 m M h e x a n e s u l f o n a t e ;r e s e r v o i r C 5 0 % (v/v) m e t h a n o l - w a t e r ; a n d r e s e r v o i r D5 0 m M p o t a ss i u m d i h y d r o g e n p h o s p h a t e ( K H 2 P O 4 ) .T h e p H o f e a ch s o l u t i o n w a s p r e v i o u s ly a d j u s t e d w i t h

63 5 Chromatographia Vot. 44, No. 11/12, June 1997 Original


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T a b l e I I . A p p l i e d D r a p e r - L i n s m a l l c o m p o s i t e d e s i g n ( i n c o d e d u n i t s) .R u n T r i e t h a n o l a m i n e H e x a n e s u l f o n a t e M e t h a n o l P H( m M ) ( m M ) ( v / v

1 0 0 0 02 1 1 1 - I3 1 1 - 1 - 14 1 -1 1 15 - 1 1 - 1 16 1 - 1 - 1 17 - 1 - 1 1 - 18 -1 1 1 19 - 1 - 1 - 1 - 110 -1.41 0 0 011 1.41 0 0 012 0 -1.41 0 013 0 1.41 0 014 0 0 -1.41 015 0 0 1.41 016 0 0 0 -1 .4117 0 0 0 1.4118Low leve l ( -1 ) 15 2 .5 15 3Cen t ra l leve l (0 ) 20 5 20 4

Hig h leve l (+1) 25 7 .5 25 5L e v e l ( - c0 12.93 1.46 12.93 2.54Le vel (+ct) 27.07 8.54 27.07 5.41

T a b l e H I . M e a s u r e d v a l u e s f o r t h e r e s p o n s e v a r i a b l e r e t e n t i o n t i m e ( m i n ) .R u n B 1 B 2 B 3 B 6 C S a c c h a r i n E l l 0

1 9.222 7.348 5.960 6.817 2.818 4.407 9.8972 9.910 5.130 4.813 6.460 3.013 3.915 5.7323 19.713 11.432 6.232 10.997 3.165 5.528 14.4124 4.930 5.997 5.097 4.297 2.513 4.287 8.6205 17.463 10.045 7.478 7.447 2.378 4.550 12.1676 8.913 14.812 7.562 5.947 2.528 6.163 26.9137 4.813 6.208 4.525 4.813 3.008 4.543 8.9108 7.628 4.582 5.065 4.762 2.365 3.442 4.7929 7.568 15.433 5.700 7.568 3.183 6.798 27.31510 9.690 7.037 5.928 7.013 2.797 4.205 8.92711 9.120 7.987 6.170 6.853 2.820 4.567 11.21712 4.808 10.332 6.065 4.957 2.882 5.650 19.68313 15.233 6.463 5.997 8.333 2.780 3.930 7.34714 17.652 15.562 8.112 10.868 2.912 5.710 22.29315 6.542 4.654 4.728 5.175 2.745 3.657 5.49016 8.937 8.410 4.727 7.020 3.110 5.730 11.14717 8.475 6.832 6.015 4.925 2.382 4.298 9.86518 9.313 7.438 5.982 6.863 2.827 4.397 10.218

D a t a n a l y s i sT h e s t a t is t i c a l a n a l y s i s o f d a t a a n d m u l t i p l e r e g r e s s i o nw e r e p e r f o r m e d b y t h e s o f t w a r e S t a t g ra p h i c s ve r s io n7 .0 . ( M a n u g i s t i c s , I n c . , 2 1 1 5 E a s t J e f f e r s o n S t r e e t ,R o c k v i l l e , M a r y l a n d , U S A , 1 9 93 ). T h e p r e d i c t i o n s o f r e -t e n t i o n t i m e s a n d p e a k - w i d t h a t h a l f - h e i g h t f o r v a r i o u sm o b i l e p h a s e c o n d i t i o n s w e r e c a l c u l a t e d b y m e a n s o f as p r e a d s h e e t p r o g r a m ( E x c e l v e r s i o n 5.0 ; M i c r o s o f t ) .T h i s s o f t w a r e w a s a l s o u s e d t o d r a w c o n t o u r p l o t s o fm i n i m u m e f fe c t iv e r e s o l u t i o n a n d m a x i m u m r e t e n t i o nt i m e s a s f u n c t i o n s o f m o b i l e p h a s e p a r a m e t e r s .

R e s u l t s a n d D i s c u s s i o n

E s t i m a t i o n o f t h e M a i n E f f e c t s a n d I n t e r a c t io nE f f e c t sT h e c l a ss i c al d e f i n i t i o n o f t h e e f f e c t o f a n i n d i v i d u a l m o -b i l e p h a s e p a r a m e t e r o n t h e r e s p o n s e v a r i a b l e i s t h em e a n ( a v e r a g e ) r e s p o n s e v a l u e a t i ts h ig h l e v e l s ( + ) m i -n u s t h e m e a n r e s p o n s e v a l u e a t i t s l o w l e v e l s ( - ) . T h e e f -f e ct s o f p a r a m e t e r i n t e r a c t i o n s a r e o b t a i n e d b y s u b t r a c t -i n g th e m e a n v a l u e o f t h e i r p o s i t i v e p r o d u c t s a n d t h e i r

6 7 C h r o m a t o g r a p h i a V o l . 4 4 , N o . 1 1/ 12 , J u n e 1 9 97 O r i g i n a l


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Table V. Measu red values for the response va riable peak-w idth at half-height (min).Run B1 B B3 B6 C Sacchar in E l l0

1 0.255 0.222 0.149 0.158 0.095 0.116 0.4472 0.267 0.169 0.144 0.151 0.085 0.109 0.2153 0.366 0.265 0.147 0.269 0.129 0.136 0.4334 0.138 0.178 0.131 0.126 0.083 0.112 0.2735 0.365 0.339 0.180 0.200 0.086 0.120 0.3586 0.219 0.390 0.180 0.166 0.083 0.144 0.6637 0.166 0.188 0.118 0.166 0.078 0.119 0.3158 0.205 0.148 0.130 0.140 0.083 0.094 0.1689 0.284 0.449 0.141 0.284 * 0.155 0.75210 0.312 0.211 0.148 0.161 0.099 0.112 0.34111 0.251 0.224 0.161 0.162 0.113 0.120 0.46012 0.135 0.283 0.151 0.130 0.132 0.137 0.47713 0.323 0.202 0.181 0.191 0.107 0.107 0.26514 * 0.385 0.198 0.2 57 0.125 0.137 0.59715 0.182 * * 0.136 0.100 0.103 0.18916 0.248 0.252 0.118 0.158 0.134 0.138 0.48617 0.214 0.207 0.151 0.154 0.082 0.115 0.45918 0.268 0.224 0.150 0.160 0.097 0.115 0.459

*Not m easured.

n e g a t i v e p r o d u c t s . A n e s t i m a t e d e f f e c t (h ) c a n b e c o n -s i d e r e d a s s i g n i fi c a n t if t h e v a l u e z e r o i s n o t i n c l u d e d i nt h e 9 5 % c o n f i d e n c e i n t e r v a l , h + 2 o ( h ) ( i. e. t h e e s t i-m a t e d e f f e c t + t w i c e it s s t a n d a r d e r r o r ) , o f t h e e s t i m a t e de f f e c t c a l c u l a t e d b y a S t u d e n t t - t e s t. [8 , 9 ] . I f t h e c o n f i -d e n c e i n t e r v a l c o n t a i n s z e r o , th e e s t i m a t e d e f f e c t i s n o ts i g n if i c an t . C a l c u l a t i o n o f s t a n d a r d e r r o r f o r e f f e c t s o ri n t e r a c ti o n s u s i n g r e p l i c a t e d r u n s a n d f o r h ig h e r o r d e ri n t e r a c t i o n s ( i n t h e c a s e o f n o r e p l i c a t e s ) i s d i s c u s s e d b yB o x e t a l . [ 10 ] .I n r e g r e s s i o n a n a l y s is f a c t o r a n d i n t e r a c t i o n e f f e c t s a r ej u s t h a l f t h e v a l u e s o f c la s s ic a l f a c t o r a n d i n t e r a c t i o n e f -f e c ts . T h i s c a n b e e x p l a i n e d b y t h e c l a ss i ca l d e f i n i t i o n o ff a c t o r e f f e c t w h i c h is s im p l y t h e d i f f e r e n c e in a v e r a g e r e -s p o n s e . T h i s d e f i n i t i o n i n v o l v e s a c h a n g e i n r e s p o n s eo n l y , a b a r e B y. T h e c h a n g e i n f a c t o r f r o m l o w t o h i g hl e v e l , 5 x is i g n o r e d . T h i s w a s t h e c a s e i n e a r l y t y p e s o f d e -s i g n e d e x p e r i m e n t s w h e r e r e s e a r c h e r s ( f o r e x a m p l e ina g r i c u l t u r e ) u s u a l l y u s e d n o m i n a l o r o r d i n a l v a r i a b l e s ,n o t v a r i a b l e s e x p r e s s e d o n i n t e r v a l o r r a t i o s c a l e s . I nm o d e r n r e s e a r c h u s i n g i n t e r v a l a n d r a t i o s c a l e s t h e 8 xs h o u l d n o t u s u a l l y b e i g n o r e d . B e c a u s e t h e v a r i a b l e X lc h a n g e d f r o m a c o d e d l e v e l o f - 1 t o a c o d e d l e v e l o f + 1 ,8 x l = 2 . T h u s t h e f a c t o r e f f e c t -- 5y1 Sxl p e r t w o c o d e du n i t s a n d t h e f a c t o r e f f e c t p e r o n e c o d e d u n i t f r o m Xl = 0t o X l = 1 ( a s m e a s u r e d b y r e g r e s s i o n a n a l y s is ) is sm a l l e rb y 8 9 t h a n t h e r e s u l t o b t a i n e d f r o m t h e c l a s s ic a l a p -p r o a c h .T h e m e t h o d o f c a lc u l a t i o n o f t h e m a i n e f f e c t a n d i n t e r-a c t i o n e f f e c t s , w i t h t h e i r s t a n d a r d e r r o r , is g i v e n b y B o xet a l . [11] .

nal ys i s o f Var i anceT h e t h e o r y o f a n a l y s i s o f v a r i a n c e i s w e l l d is c u s s e d b yA l l u s e t a l . [ 12 , 1 3 ] A n a l y s i s o f v a r i a n c e c a n h e l p d e t e r -m i n a t i o n a n d c o n f i r m a t i o n o f w h i c h f a c t o r s a r e s ig n i fi -

c a n t o r n o t . S i g n i f ic a n t p a r a m e t e r s a r e t h o s e f o r w h i c ht h e F i s h e r v a r i a n c e r a t i o ( F - r a t i o ) is l a r g e a n d e x p r e s s e db y a P - v a l u e ( s i g n i f i c a n c e l e v e l ) l e s s t h a n 0 . 0 5 .R e g r e s s io n M o d e l in gT h e r e t e n t i o n t i m e a n d t h e p e a k - w i d t h a t h a l f - h e ig h ta r e m o d e l e d . M u l t i p l e r e g r e s s i o n m o d e l s e x p r e s s t h em a t h e m a t i c a l r e l a t i o n s h i p b e t w e e n t h e m e a s u r e d r e -s p o n s e v a r i a b l e s a n d t h e i n d e p e n d e n t v a r i a b l e s . I f a fu l lf a c t o r i a l d e s i g n i s u s e d t o f i t a l i n e a r q u a d r a t i c m o d e l ,t h r e e l e v e l s f o r e a c h f a c t o r w i l l b e n e c e s s a r y . F o r f o u rf a c to r s , t h e n u m b e r o f e x p e r i m e n t s i s t o o g r e a t ( 8 1, ;3 4) . I n t h i s w o r k , h o w e v e r , w e u s e d a s m a l l c o m p o s i t ed e s i g n w i t h 1 8 r u n s o n l y to f i t s e c o n d - o r d e r r e g r e s s i o ne q u a t i o n s . W e c o m b i n e d a t w o - l e v e l f r a c t i o n a l f a c t o r i a ld e s i g n ( o r c u b e p o r t i o n ) 2 4 .1 ( e i g h t r u n s ) , w i t h a s t a r d e -s i g n - t h e a x i a l d i s t a n c e c t = 1 . 4 1 4 ( e i g h t a x i a l p o i n t s )p l u s t w o c e n t e r p o i n t s. S m a l l d e s i g n s m i g h t b e a p p r o p r i -a t e w h e n e x p e r i m e n t a t i o n i s e x p e n s i v e , d i f fi c u l t, o rt i m e - c o n s u m i n g . T a b l e s V a n d V I l i s t t h e r e g r e s s i o ne q u a t i o n c h a r a c t e r i s t ic s c a l c u l a t e d f o r r e t e n t i o n t i m ea n d p e a k - w i d t h a t h a l f -h e i g h t , r e s p e c t iv e l y , o f t h e d i f f e r-e n t c o m p o u n d s ; t h e i n t e r c e p t s a n d r e g r e s s i o n c o e f fi -c i e n t s a r e a l s o i n c l u d e d i n t h e t a b l e s .M o d e l d e q u a c y C h e c k i n gT h e p r i m a r y d i a g n o s ti c t o o l f o r c h e c k i n g t h e a d e q u a c yo f t h e m o d e l s c a l c u l a t e d i s r e s id u a l a n a ly s is . T h e r e s u l t sr e v e a l e d a n o r m a l d i s t r i b u t i o n o f r e s id u a l s . T h e s u m o fs q u a r e s ( S S Q ) o f a b s o l u t e d e v i a t io n s b e t w e e n p r e d i c t e da n d e x p e r i m e n t a l d a t a , a n d t h e a v e r a g e r e l a t i v e d e v ia -t i on b e t w e e n c a lc u l a t ed a n d e x p e r i m e n t a l d a t a ( A R D ) ,a s il l u s tr a t e d i n T a b l e V I I, p r o v i d e s o m e i d e a o f th e a c -c u r a c y o f t h e m o d e ls . T h e l a r g e S S Q v a l u e s f o r r e t e n t i o nt im e s a r e l a r g e ly b e c a u s e o f t h e m a g n i t u d e o f t h e p a -r a m e t e r .

Or iginal Ch rom atog raph ia Vol . 44, No. 11/12, June 1997 6 8


6/11

T a b l e V . R e g r e s s i o n e q u a t i o n s o b t a i n e d f o r r e t e n t i o n t i m e , tR ( th e e s t i m a t ed c o e f f ic i e n ts a r e s h o w n a n d t h e i r s t a n d a r d e r r o r s ) .C o m p o u n d M o d e l e q u a t i o n f o r r e t e n t i o n t i m e ( tR )V i t a m i n B 1

V i t a m i n B 2

V i t a m i n B 6

V i t a m i n B 3V i t a m i n C

S a c c h a r i n

E l 1 0

tR = ( 17 .1 + 4 .61 ) + ( 5 .22 + 0 .536) B + ( - 1 .93 + 0 .388) C + ( - 0 .129 + 0 .019) B C ( 0 .047 + 0 .009) C 2 +( - 0 .29 9 0 .094) B D ( 1 .188 + 0 .506) DtR = (54.75 _+ 1.55) + (0.041 + 0.014) A +' ( -2 .4 98 0 .190 ) B + ( -3. 12 + 0.138) C + ( -0.4 2 5: 0 .068) D +( 0 .063 + 0 .007) B C (0 .068 5: 0 .013) B ~ + (0.051 + 0.003) C 2tR = (19.9 9 + 1.929) + (1.915 0 .415) B + ( -1. 405 + 0.183) C + (0.063 -+ 0.01 4) A B ( - 0 .016 -+ 0 .004) A C( - 0 .064 0 .010) B C + ( - 0 . 3 4 8 0 . 0 7 7 ) B D (0 .176 0 .023) CD ( 0 .024 5 : 0 .004) C z + ( - 0 .302 + 0 .053) D 2tR = ( -1 .89 7 + 1.819) + (0.024 + 0.065) C + (4.44 + 0.721) D + ( -0.057-+0.016) CD +( - 0 . 352 -+ 0 .080) D 2tR = (4.16 5: 0 .256 ) + ( -0 .0 41 0 .008) B + ( - -0 .039 _+0.009) C + ( -0 .09 3 -+ 0.101) D + (0.001 _+ 0.000) A B(0 .007 5: 0 .002) CD ( -0 .0 43 5:0.011) D 2tR = ( 16 .53 5 : 1 .371) + ( - 0 .748 + 0 .171) B + ( - 0 .233 + 0 .032) C + ( - 2 .461 + 0 .603) D + ( 0 .014 + 0 .006) B C( 0 .024 + 0 .012) B2 + ( 0 .262 + 0 .075) D 2tR = ( 100 .45 + 5 .2 ) + ( - 8 .457 5 : 0 .660) B + ( - 5 .20 0 .479) C +( 0 .206 + 0 .023) B C (0 .259 _+0.046) B z +( 0 .072 + 0 .012) C ~

T a b l e V I . R e g r e s s i o n e q u a t i o n s o b t a i n e d f o r p e a k - w i d t h a t h a l f -h e i g h t, W ~ ( t h e e s t i m a t e d c o e f f i ci e n ts a r e s h o w n a n d t h e i r s t a n d a r d e r r o rs ) .C o m p o u n d M o d e l e q u a t i o n f o r p e a k - w i d t h a t h a l f- h e i g h t (W , A)V i t a m i n B 1V i t a m i n B 2

V i t a m i n B 6

V i t a m i n B 3S a c c h a r i n

E l 1 0

W,A = ( 0 .403 + 0 .054) + ( - 0 .002 + 0 .002) A + ( 0 .022 5 : 0 .003) B + ( - 0 .011 + 0 .002) CW,A = ( 1 .278 5 : 0 .179) + ( - 0 .051 + 0 .015) B + ( - 0 .07 1 + 0 .017) C +(0.001 + 0.000) C 2W,,~ = (0.885 5: 0 .122) + ( -0 .03 6 _+ 0.013) B + ( -0.0 53 + 0.011 ) C +(-0.0 03 _+ 0.001 ) A D ( 0 .002 + 0 .001) CD (0 .001 + 0.00 0) C 2W ~ = ( 0 .227 + 0 .022) + ( - 0 .00 4 + 0 .001) CW ~ = ( 0 .292 5 : 0 .022) + ( - 0 .0 04 +_ 0 .000) B + ( - 0 .003 + 0 .000) C +( 0 .004 0 .001) D 2W ~ = ( 1 .62 + 0 .157) + ( -0 .121 + 0 .029) B + ( - 0 .051 + 0 .004) C + ( 0 .004 + 0 .001) B C

0 . 0 0 2 + 0 . 0 0 1 ) B C

0 .002 + 0 .001) A B

- 0 .0 42 + 0 .011) D +

T a b l e V I I . A c c u r a c y o f m o d e l d e s c ri p ti o n s .P a r a m e t e r B 1 B z B 3 B 6 C S a c c h a r i n E l 1 0t R ( m i n ) A R D ( ) 4 . 1 0 1 .6 5 2 . 5 0 2 . 3 6 0 . 7 1 2 . 9 3 4 . I 6SS Q 4.831 0.554 0.671 0.524 0.011 0.494 8.005WI /2 A R D ( ) 8 .22 7 .02 7 .71 6 .34 - 2 .25 9 .47SS Q 0.011 0.008 0.005 0.004 - 0 .0002 0.035

6 9 C h r o m a t o g r a p h i a V o l. 4 4 , N o . 1 1 /1 2 , J u n e 1 9 97 O r i g i n a l


7/11

V i t a m i n B 1

1 6 ~ - i . . . . . i . . . . . . . . . E . . . . i . . . . . . . . . .

f i ! i i ! i . . . . ~ .L ; : : : : : :.o B ' . ' : : : : :0 : : : : ; : :

0 4 B 1 2 1 6 2 0 2 4p r e d i c t e d

V i t a m i n B Z

1 64

m

i i in0 8

:.:~.:....:viv:.:i.:...:.i...::.~:....:~,4 G B 1 0 1 2 1 4 1 6

p P e d i c t e d

Figure 2

V i t a m i n B 3

5 i . . . . . . . . . . . . . . . . - i . . . . . . . . . . . . . . . . . i

o s . 5 . . . . . . . . : . . . / . . . ~ . . . . . . . . . : : . . . . . . . . ?

4 5 :;4 , 5 5 . 5 6 . 5 7 . 5 8 . 5

p r ' e d c t e dV i t a m i n C

3 . 3

3 , 1

Lal~ . . 7o

2 . 5

Z . 3 2 . 5 2 . 7 Z . ? 3 . I 3 . 3p r e d i c t " e d

S a c c h a r i n7 4 : C . . . : . : . i . : . . . . . . { . : . : . : . . . ; . . . . . : . . :

6 4 i . . . . . . . i . . . . . . . . ~ . . . . . . . . . . . . . . . i~ i i iIll : : : : :J] : : : :

4 . 4 ~ . . . . . . 9 . . . . . . " . . . . . . . . . . . . . . . . . . !: . r _ ~ i i i

" ..vv-'i':': :i' :':':';'v :'i43 . 3 4 , 3 5 . 3 6 , 3 7 . 3

p r e d t c t e d

Diagno stic plots of observed retention tim es against predicted retention times.

V i t a m i n B 6

~ . . . . . . . . . . . . . . . . . . . . . . / . . . . l" o i i ! ! i

4 . 2 6 . 2 8 . 2 1 0 . Z 1 Z . 2p r - e d i c t e d

S u n s e t y e l l o w F C F3 . . . . . . : . . . . . . . . . . . : . . . . . . : . . . . . : . . . . . .

z s . . . . . . . . . . . . . . . . . . ..... :

9 |i i i E i : :o ~ o i . . . . ~ , " ' ~ . . .. . . . . .. . . . .. .

0 5 1 { } t 5 2 0 2 5 3 { ]l o p e d i c t e d

N o v a l u e o f A R D w a s r e c o r d e d fo r v i ta m i n C b e c a u s en o p a r a m e t e r s s e e m t o i n f l u e n c e i t s p e a k - w i d t h i n t h ep a r a m e t e r s p a c e e x a m i n e d . T h e a v e r a g e p e a k - w i d t h a tha l f -he ight was c ons ide r e d in pr e dic t ions for v i tamin C .T h e A R D va lue s fa l l w i th in the in te r va l 0 .71 --4.16 % forr e t e n t i o n t i m e s a n d 2 . 2 5 - 9 . 4 7 % f o r t h e p e a k - w i d t h a th a l f - h ei g h t . T h e s e i n t e r v a l s d e m o n s t r a t e d t h e a d e q u a c yof the mode l s for f i t t ing the data .D i a g n o s t i c p l o t s o f o b s e r v e d a g a i n st p r e d i c t ed r e t e n t i o nt ime s for v i tamins B t , B 2 , B 3 , B 6 and C , and for s ac c har ina n d E l 1 0 a r e r e p r o d u c e d i n F i g u r e 2, w h i c h d e m o n -s tr a te s the powe r fu l ab i l i ty o f the quad r at i c mo de l s to f i tthe data .

O p t i m i z at i on a n d R u g g e d n e s s o f th e M e t h o d sT w o r e s p o n s e c r i t e r i a w e r e c h o s e n f o r o p t i m i z a t i o n o ft h e m o b i l e p h a s e p a r a m e t e r s f o r th e s e p a r a t i o n o f c o m -p o u n d s: th e m i n i m u m e f f e c t i v e r e s o l u t i o n ( R s m i n ) a n dthe max imum r e te nt ion t ime . ( tR max) . T he quadr at i cm o d e l s o b t a i n e d e n a b l e d t h e p r e d i c t i o n o f r e t e n t i o nt ime s and the pe ak-width a t ha l f -he ight for a l l the c om-p o u n d s u n d e r s tu d y. T h e s p r e a d s h e e t p r o g r a m E x c e l 5 . 0was us e d for c a lc u la t ion w i th var ious s e t s o f c ondi t ionsi m p o s e d o n t h e p a r a m e t e r s , a t f i x e d m o b i l e p h a s e p H .T h e t h r e e m o b i l e p h a s e p a r a m e t e r s w e r e c o m b i n e d w i t hi n c r e m e n t s o f tw o u n i t s w i t h i n t h ei r e x a m i n e d b o u n d a -r ie s, r e s u l ting in 225 c om binat ions . T h e r e te n t ion t ime s

Original Chromatographia Vol.44, No. 11/12, June 1997 64


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t a i n e d u s in g t h e m o b i l e p h a s e c o m b i n a t i o n g i v in g o pt i -Tab le V IIl . Exa mp les of the values obtaine d for minim um effective resolution (hexanesulfonate 7.5 mM, pH 3.0).T E t O H A( m M )

Min imu m effective resolution, Rs min, for the metha nol conce ntrations ( v/v :14 15 16 17 18 19 20 21 22 23 24 25 26 27

13 1.33 2.15 1.38 0.40 0.49 1.26 1.88 1.66 1.51 1.44 1.49 0.97 0.51 0.5214 1.76 1.96 0.93 0.03 0.90 1.65 1.78 1.56 1.40 1.33 1.37 1.20 0.31 0.4615 2.18 1.51 0.50 0.45 1.30 1.98 1.69 1.46 1.29 1.22 1.25 1.41 0.11 0.4016 2.11 1.06 0.07 0.86 1.69 1.89 1.60 1.36 1.19 1.10 1.13 1.28 0.07 0.3517 1.66 0.63 0.35 1.26 2.08 1.80 1.50 1.26 1.08 0.99 1.01 1.15 0.25 0.3018 1.22 0.20 0.77 1.66 2.06 1.72 1.41 1.16 0.97 0.88 0.89 1.02 0.42 0.2619 0.79 0.22 1.17 2.04 1.98 1.63 1.31 1.06 0.87 0.76 0.77 0.90 0.59 0.2120 0.37 0.63 1.57 2.29 1.90 1.54 1.22 0.96 0.76 0.65 0.64 0.77 0.75 0.1721 0.05 1.03 1.95 2.21 1.82 1.45 1.13 0.86 0.65 0.53 0.52 0.64 0.90 0.1222 0.46 1.43 2.33 2.13 1.73 1.36 1.03 0.76 0.55 0.42 0.40 0.51 0.77 0.0823 0.86 1.82 2.48 2.06 1.65 1.28 0.94 0.66 0.44 0.31 0.28 0.38 0.63 0.0524 1.26 2.20 2.41 1.98 1.57 1.19 0.85 0.56 0.33 0.19 0.16 0.25 0.50 0.0125 1.65 2.58 2.33 1.90 1.49 1.10 0.75 0.46 0.23 0.08 0.04 0.13 0.36 0.0326 2.03 2.71 2.26 1.82 1.40 1.01 0.66 0.36 0.12 0.03 0.08 0.00 0.23 0.0627 2.41 2.64 2.19 1.75 1.32 0.92 0.56 0.26 0.01 0.15 0.20 0.13 0.09 0.10

a n d p e a k - w i d t h a t h a lf - h e i g h t w e r e p r e d i c t e d f o r e a c hc o m p o u n d . T h e c o m p o u n d s w e r e t h e n s o r t e d in a s ce n d -i n g o r d e r o f e l u ti o n a n d t h e r e s o l u t i o n b e t w e e n s u c c e s -s i v e p e a k s w a s c a l c u l a t e d . S u b s e q u e n t l y , t h e m i n i m u me f f ec t iv e r e s o lu t i o n a n d m a x i m u m r e t e n t i o n t i m e s w e r es e l e c t e d f o r e a c h p a r a m e t e r c o m b i n a t i o n . I n a d d i t i o np r e d i c t i o n s a t a f i x e d h e x a n e s u l f o n a t e c o n c e n t r a t i o n o f7 .5 m M a n d p H = 3.0 . w e r e p e r f o r m e d w i t h o n e u n i t in -c r e m e n t . A n e x a m p l e o f t h e v a lu e s o f R S m i n o b t a i n e da r e s h o w n i n T a b l e V I I I . C o n t o u r p l o t s w e r e c o n -s t r u c t e d f o r t h e m i n i m u m e f f e c t iv e r e s o l u ti o n , R s r a in ,a n d t h e m a x i m u m r e t e n t i o n t im e s w e r e p r e d i ct e d . T h ec o m m o n s e c t i on s f or m o b i l e p h a s e p a r a m e t e r s w h e r em i n i m u m r u g g e d r e s o l u t i o n R s r a i n > 1 .5 ( s e e re f . 1 4)a n d t R m a x w a s m i n i m u m w e r e s e l e c t e d f r o m th e c o n -t o u r p lo ts . T h e c o m m o n s e c ti o ns s h o w o p t i m u m m o b i l ep h a s e c o m b i n a t i o n s f o r c h r o m a t o g r a p h i c s e p a r a ti o n o ft h e c o m p o u n d s e x a m i n e d .R e s o l u t i o n w a s c a l c u l a t e d u s i n g t h e f o r m u l a R s = [ 1.1 8( tR 2 - tR 1 ) ] / [W , A (2 ) + W , A (1 ) ] .A c c o r d i n g t o t h e c o n t o u r p l o t ( F i g u r e 3a ) o f m i n i m u me f f e c t i v e r e s o l u t i o n a s a f u n c t i o n o f m e t h a n o l c o n t e n ta n d t r i e t h a n o l a m i n e c o n c e n t r a t i o n a t a f i xe d p H o f 3 .0w h e n t h e h e x a n e s u l f o n a t e c o n c e n t r a t i o n w a s 1 . 5 r a M ,t h e r e a r e t h r e e d o m a i n s t h a t f u r n i s h o p t i m u m r e s ul ts .N e a r t h e r i g h t si d e t h e r e i s a d o m a i n w i th R s r a i n b e -t w e e n 2 .0 0 a n d 2 .5 0 a n d a n o t h e r w i t h R s r a i n b e t w e e n1 .5 0 a n d 2 .0 0 ; t h e t h i r d o p t i m u m d o m a i n , i n t h e m i d d l e ,h a s a R s m i n o f 1 . 50 - 2 .0 0 . T h e l a s t is p r e f e r a b l e b e c a u s ea l t h o u g h t h e s e p a r a t i o n q u a l i t y i s s l i g ht l y l o w e r i t i sm o r e r u g g e d . I t s h o u l d b e s t r e s s e d t h a t h e r e t h e r u g g e d -n e s s i s a m e a s u r e o f a s y s t e m ' s l a c k o f s e n s i t i v i t y t o s m a l lc h a n g e s i n o p e r a t i n g c o n d i t io n s . T h e d o m a i n s o n t h er i g h t a r e s e n s i t i v e t o s m a l l f l u c tu a t i o n s i n m e t h a n o l c o n -t e nt . T h e c o n t o u r p l o t o f d e p e n d e n c e o f m i n i m u m e f fe c -t i v e r e s o l u t i o n o n m e t h a n o l a n d t r i e t h a n o l a m i n e c o n -c e n t r a t i o n s a t p H 3 . 0 w h e n t h e h e x a n e s u l f o n a t e c o n c e n -t r a t i o n i s 3.5 m M i s s h o w n i n F i g u re 3 b . T h e d o m a i n o n

t h e r i g ht g i v e s h i gh q u a l i t y s e p a r a t i o n w i t h R s m i n b e -t w e e n 1 .5 a n d 2 . 0 0; t h i s d o m a i n i s, h o w e v e r , h i g h ly p r o n et o v a r i a ti o n o n s m a l l c h a n g e s i n m e t h o d c o n d i t i o n s a n di s, t h e r e f o r e , n o t r u g g e d . F i g u r e 3 c s h o w s t h e c o n t o u rp l o t o f m i n i m u m e f f e c t iv e r e s o l u t i o n a s a f u n c t i o n o f t h ec o n c e n t r a t i o n s o f m e t h a n o l a n d t r i e t h a n o l a m i n e a tp H 3 .0 w h e n t h e c o n c e n t r a t i o n o f h e x a n e s u l f o n a t e i s5 .5 r aM ; n o o p t i m u m d o m a i n w i t h R s m i n > 1 .5 w a s d e -t e c t e d .A p l o t o f t h e d e p e n d e n c e o f m i n i m u m e f f e c t i v e r e s o l u -t i on in h e x a n e s u l f o n a t e a n d t r i e t h a n o l a m i n e c o n c e n t r a -t i on w h e n t h e m e t h a n o l c o n t e n t i s 1 4 v / v ( F i g u r e 3 e )r e v e a l e d t w o o p t i m u m d o m a i n s w i t h R s m i n o f 2- 2 .5a n d 1 . 5 - 2 , o n t h e r i g h t o f t h e p l o t . T h e s e r e g i o n s a r e n o tr o b u s t , h o w e v e r . E x a m i n a t i o n o f t h e c o n t o u r p l o t o fm i n i m u m e f f e c t iv e r e s o l u t i o n a g a i n s t h e x a n e s u l f o n a t ea n d t r i e t h a n o l a m i n e c o n c e n t r a t i o n f o r a m e t h a n o l c o n -t e n t o f 1 8 v / v ( F i g u re 3 f ) s h o w s t w o o p t i m u m z o n e so n e o n t h e r i g h t w i t h R s m i n = 2 . 0 0 - 2 . 5 0 a n d 1 . 5 0 - 2 . 0 0a n d a n o t h e r o n t h e l e f t w i t h R s m i n = 1 . 5 0 -2 . 0 0 ; a g a i nn e i t h e r o f t h e t w o z o n e s i s r u g g e d .T h e m o s t o p t i m u m a n d r u g g e d c o n d i ti o n s w e r e r e -v e a l e d b y e x p lo r i n g t h e r e s p o n s e s u r f a c e o f m i n i m u me f f e c ti v e r e s o l u t i o n f o r a h e x a n e s u l f o n a t e c o n c e n t r a -t i o n o f 7 .5 m M a n d a f i x e d p H o f 3 .0 , a s r e p r o d u c e d i nF i g u re 3 d. T h e s a m e m o b i l e p h a s e p a r a m e t e r s w e r e e x -p l o r e d w i t h o n e u n i t i n c r e m e n t s , a s s h o w n i n F i g u r e 4 .T w o d o m a i n s s h o w r e g i o n s w i t h R s m i n . b e t w e e n 2 .5 0a n d 3 . 00 a n d b e t w e e n 2 . 00 a n d 2 . 50 ( F i g u r e 4 , u p p e rp l o t ) . T h e s e d o m a i n s p r o v i d e m a r k e d l y b e t t e r r u g g e d -n e ss . C o m b i n a t i o n o f th i s c o n t o u r p l o t w i t h t h a t o f m a x i -m u m r e t e n t i o n t i m e ( F i g u r e 4 , l o w e r ) f o r t h e s a m e e l u -e n t c o m p o s i t i o n , e n a b l e s s e l e c t i o n o f t h e b e s t c o m p r o -m i s e b e t w e e n r u g g e d m o b i l e p h a s e c o n d i t i o n s t h a t e n -s u r e c o m p l e t e s e p a r a t i o n o f a ll t h e c o m p o u n d s u n d e rs t ud y , i n a r e a s o n a b l e a n a l y s i s t i m e .F i g ur e 5 s h o w s t w o e x a m p l e s o f t h e c h r o m a t o g r a m s o b -

64 1 Ch rom atog rap h ia Vol . 44 , No . 11 /12 , Jun e 1997 Orig inal


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FigureCon tour plots of minimum effective resolution as a function of mobile phase compositions at pH 3: (a) hexane-sulfonate 1.5 mM; (b) hexanesulfonate 3.5 mM; (c) hexanesulfonate 5.5 mM; (d) hex anesulfonate 7.5 mM; (e)methanol 14 v/v); (f) methanol 18 v/v).

Original Chr oma togra phia Vol. 44, No. 11/12, June 1997 64


10/11

FigureContou r plots of minimum effective resolution ( lef t) and maxim um retentio n time (r ight) as functions of the concen trations of methano l andtr iethanolamine in the mobile phase (pH 3; hexanesulfonate 7.5 mM).

0 . 2 0 - -

0 . ? . 0 -0 . 1 6 -0 . 1 4 -0 . 3 . 2 -

~ 0 . 1 0 -0 . 0 8 .w

0 . 0 4 -O. ~;2 -0 . 00

0 . 0 00 . 2 0 -0 . ; . 8 -0. . I . 6-o . 142

0 . 1 2 -~ r

0 0 8

0 ' . L?b -0 , 0 4 -0 . 0 2 -0 , 0 0 -

0 . 0 (

.)

g

,~ - A 9 ,TI ' ' ' : I . . . . I ' '5.00 ~.0.00 15.00~iJ.n ..I ;eJ

b)

m

S. 00 10.00 15.00l, i .z~ CaJ

Figure 5Chrom atogram s showing separatio n of the water-soluble vitamins in (a) s tandard solution and (b) e ffervescenttablet. Column: Allt ima 250 x 4.6 mm i.d. 5 pm Cs-bon ded spherical silica. Mobile phase: isocratic; methan ol,20 v v; hexanesulfon ate, 7.5 mM; tr iet hanolamin e 15 mM; pH 3 (KH2PO4 buffer) ; flow 1 m L min -1. PD A de-tection at 270 nm. Orde r of elution: ascorbic acid, saccharin, nicotinamide, r iboflavin phosphate, sun set yellowFCF, pyr idoxine hydrochlor ide and th iamine n i t r a te .

64 Chr oma togra phia Vol . 44 , No. 11112, June 1997 Or ig in a l


11/11

m u m s e p a r a t i o n o f th e c o m p o u n d s e x a m i n e d . W h e n t h ec h r o m a t o g r a m s w e r e r e c o r d e d 2 m o n t h s l a te r u n d e r t h es a m e c o n d i t i o n s , i .e . m o b i l e p h a s e m e t h a n o l 2 0 % , h e x -a n e s u l f o n ic a c id 7 .5 m M , t r i e t h a n o l a m i n e 15 m M a t p H3 .0 0 ( b u f f e r K H 2 P O 4 5 0 m M ) , t h e r e t e n t i o n t i m e s w e r ea g o o d m a t c h w i t h t h e p r e v i o u s r e s u lt s . T h i s sh o w s t h a tt h e s t a b i l it y o f t h e c h r o m a t o g r a p h i c s y s t e m i s g o o d .

o n c l u s i o nT h e e x p e r i m e n t a l d e s i g n m e t h o d o l o g y [ 15 - 18 ] f o l l o w e di n t hi s w o r k e n a b l e d m o d e l i n g a n d o p t i m i z a ti o n o f t h ec h r o m a t o g r a p h i c s e p a r a t i o n o f t h e W S V s a n d t h e e x -c i p i e n t m a t e r i a l s s a c c h a r i n a n d t h e d y e S u n s e t y e l l o w( E l l 0 ) . T h e D r a p e r - L i n s m a l l co m p o s i t e d e s ig n e n a b l e dc o n s i d e r a b le r e d u c t i o n o f t h e n u m b e r o f e x p e r i m e n t sw i t h o u t l o ss o f i n f o r m a t i o n a n d a c c u r a cy . T h e d o m i n a n tp a r a m e t e r s a f f e c ti n g r e t e n t i o n t i m e a n d p e a k - w i d t h a th a l f - h e i g h t w e r e f i rs t i d e n t i f i e d a n d s u b s e q u e n t l y f i t t e dt o s e c o n d - o r d e r p o l y n o m i a l m o d e l s ( c a l c u l a te d b y m u l -t i p l e r e g r e s s i o n ) , w i t h g o o d s t a t i s t i c a l r e l i a b il i t y .T h e a c c u r a c y o f t h e m o d e l s w a s a s s e s s e d b y e v a l u a t i o no f t w o c r i te r i a : t h e s u m o f s q u a r e s ( S S Q ) o f t h e a b s o l u t ed e v i a t i o n s b e t w e e n p r e d i c t e d a n d e x p e r i m e n t a l d a t a ,a n d t h e a v e r a g e r e l a t i v e d e v i a t i o n b e t w e e n c a l c u l a t e da n d e x p e ri m e n t a l d a t a ( A R D ) . T h e A R D ( % ) b e t w e e nm e a s u r e d a n d p r e d i c t e d v a l u e s w e r e 0 . 71 - 4 .1 6 % f o r r e -t e n t i o n t i m e s a n d 2 . 2 5 -9 . 47 % f o r p e a k - w i d t h a t h a lf -h e i g h t . P r e d i c t i o n o f r e t e n t i o n t i m e s a n d p e a k - w i d t h s a th a l f - h e i g h t w e r e c a l c u l a t e d b y u s e o f a s p r e a d s h e e t f o r2 2 5 d i f f e r e n t m o b i l e p h a s e c o m b i n a t i o n s w i t h i n t h e p a -r a m e t e r b o u n d a r i e s e x a m i n e d .T h e m i n i m u m e f f ec t iv e r e s o l u t io n ( R s m i n ) a n d t h em a x i m u m r e t e n t i o n t i m e ( t R m a x ) w e r e m o n i t o r e d a so b j e c t i v e c r i te r i a . T h e r e s u l t i n g c o n t o u r p l o t s f u r n i s h e do p t i m u m a n d r u g g e d c o m b i n a t i o n s o f m o b i l e p h a s e p a -r a m e t e r s t h a t e n s u r e d b a s e l i n e - r e s o l v e d p e a k s w i t h i n ar e a s o n a b l e a n a l y s i s t i m e . T h e b e s t c o n d i t i o n s c a n b e s e -l e c t e d a t a g l a n c e b y c o n s i d e r i n g th e t h r e s h o l d m i n i m u me f f e c t i v e r e s o l u t i o n R s r a i n > 1 .5 a n d t h e m i n i m u ma n a l y s i s t i m e ( i . e. t R m a x h a s t o b e m i n i m i z e d ) .

Re f e r e n c e s[1] A. A lber t , E . P . Ser jean t , The Determination of IonizationConstants, A L abora tory Manual , 3rd edn, Chapman an dHall, USA, 1984, p. 167.[2] N. R. Draper, D. J . K. Lin, Technometrics 27, 173 (1985).[3] N. R. Draper, D. J . K. Lin, Technometrics 32, 187 (1990).[4] S. N. Derning, S. L. Mo rga n, Experimental D esign: A Che-mom etrics App roach , Elsevier, Am sterdam, 1993, p. 246.[5] G. E. P. Box, N R. Drap er, Empirical Model Building andResponse Surfaces , John W iley and Sons, Inc. , New York,1987, pp. 502-525.[6] G . E . P . B o x , K . B . W i l s o n , J. Ro y. Statist. Soc. Ser. B 13, 1(1951).[7] G. E. P. Box, W. G. Hunter, J . S . Hu nter, An n. M ath. Stat. 28,195 (1957).[8] D. L. M assart, B. G. M. Vandeginste, S. N. De mi ng , Y. M ichotte,L . Kau frnan , in Chemometrics: A Text Boo k , Elsevier,Am sterdam, 1988, p. 101.[9] G Sado, M. C. Sado, Les Plans d'Exp6rience; de l'Experi-mentation ~ l 'Assurance de Qualit6 , Afnor Technique,Paris, 1991, p. 70.[10] G.E .P . Box , W. G. Hun ter , Z S . Hun ter , in Statistics for Ex-perimenters, An Introduction to D esign, Da ta A nalysis andModel Building , John Wiley and Sons, Inc., New York, 1978,

p. 327.[11] G. E. P. Box , W . G. Hunter, J . S . H un ter in Statistics for Ex-perimenters, An Introduction to Design, Da ta A nalysis andModel Building, John W iley and Sons, Inc. New York, 1978,p. 309.[12] M. A . A l lus , R . G. Brere ton , G. N ick less , Chemometr. Intell.Lab. Syst. 3, 215 (1988).[13] M. A . A l lus , R . G. Brere ton , G. N ick less , Chemometr. Intell.Lab. Syst. 6, 65 (1989).[14] L. R . Snyder, J . Glajch, J . Kirk land, Practical HPL C MethodDevelop ment, John Wiley and Sons, Inc., New York, 1988, p.17.[15] E. Morgan , K . W Bur ton , P . A . Church , Chemometr. Intell.Lab. Syst. 5, 283 (1989).[16] J.O. De Beer, C. V. Vandenbroucke, D. L. Massart, J. Pharm.Biomed. Anal. 12, 1379 (1994).[17] J. (9. De Beer, C. V. Vandenbroucke, D. L. Massart, B. M. DeSpiegeleer, J. Pharm. Biomed. Anal. 14, 525 (1996).[18] E Rouber ty , J .Fourn ier , Chromatograph ia . 41 ,553 (1995).

Received: Nov 21,1996Revised manuscriptsreceived: Feb 11 and Ap r 8,1997Accepted: Ap r 25, 1997

Origina l Chro mato graphia Vol . 44, No. 11/12, June 1997 6

an experimental design approach to selecting the optimum

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