A d . Clmnl. 1900. 58,496-499
490
0. Dimbution of Na (a) and water at mass 18 (b) In frozsnhyaated chick Intestine section. Image exposue times were as follows: Na. 5 s: H,O+. 150 s. Bar = 25 pm.
a slight decline in the water signal with time. The presputtering requirement was not limited to cultured cella alone. The tissue sections showed the same pattern. The distribution of Na from the frozen-hydrated chick intetltine section after approximately 5 min of presputtering is shown in Figure 6a. The brush border shows slightly higher intensities, The K distribution paralleled that of N a From this field of view an image of water (H,O+) was also recorded a t mass 18 (Figure 6b). Unlike the Na distribution, no higher intensities of water are observed from brush border. I n t e d u s spaces seem devoid of elements. I t is plausible that intralumenal fluid in these regions sputtered faster due to lack of cellular matrix in these regions. It should be noted that freezedrying of specimen under the instrimental parameters used is least likely. The initial chareine observed in the frozen-hydrated samplea is perhaps due-toke buildup of some f m t during sample transfer and the frozen water in the specimen since no such charging was observed for freeze-dried samples of the same thickness. I t is anticipated that presputtering of the h e n hydrated samples reduces its thickness to the level that the presence of frozen water in the sample doea not interfere with its conductivity. Thin ay~sectionsW0.5 d, although more vulnerable to ion bombardment, may reduce the presputtenq
requirement to a certain extent. I t is interesting to note that the Ca distribution observed in frozen-hydrated 3T3 cella agrees with freeze-dried samples; however, the K levels in the nuclei seem slightly elevated (9). The effect of water matrix on these signals needs further consideration and is a subject of continued research in our group. In summary, this preliminary study shows the potential of ion microscopy to perform analysis of frozen-hydrated hiological specimens a t cold temperatures. This study marks the beginning of future studies in assessing the definitive modes of sample preparation for studying the roles of diffusible ions in biological systems. Registry No. H@+, 56583-62-1;H30+.13968086;potaapium, 7440-09-7; calcium, 7440-70-2;sodium, 7440-23-5.
LITERATURECITED (1) Caatahp. R.: Sbdzbn. 0. J . Mo.asc. (Ref-&)le82. 1.395-410. (2) Mmtron. 0. H.; Skdzlan. G. Anel. Chsm. 1975. 47, 932A-943A. (3) Saa. K. M.: BMaI. K. L.; tkn!sn. 0. H. J . kliQosc. (Oxlad) 1980. 118. 408-420. (4) Chandra. S.: Mahon. G. H. In "secondary Ion Mea Spa*mmetry SIMS I V Bsnnlnghova. A,. OCBne. J.. SMmlZY. R.. Werner. H. W.. Ea.: Sphpa-Verlap: New Yolk. 1 9 W pp 489-491. (5) Roos. 0. D.; M m h n . G. H.: Sam.R. F.; Staple(i. R. C. J . W z o s C . ( O M ) 1983. 129. 221-228. 10) m l m . M. T.: Chandra. S.: Monlem. G. H. Rev. Sd. I n e m . 1985. 56. 1347-1351. (7) Chandra. S.; Monlem. G. H.; Wakoll. C. C.. submilled for pubkaG-m In J . M W . (8) Chaw&S.; Mahon. 0. H. Scbnm 9985. 228. 154S1544. (9) ChaMa. S.;Mw!nm. G. H.: C&lor. C. w.: slmm. S. E. J . cslekd. 1984. 99 (4) part 2 . 4 2 4 .
..
S u b b s h Chandra M a r k T. Bernius George H. Morrison' Department of Chemistry Cornell University Ithaca, New York 14853-1301
RFCEIVEDfor review August 5,1985. Accepted October 24, 1985. This work was funded by the National Institutes of Health under Grant No. ROlGM 24314.
Generalized Rank Annihilation Factor Analysis Sic The analytical chemist ia frequently confronted with the problem of analyzing complex mixtures of which only concentrations of a few components are of interest. In these cases,it is desirable to he able to obtain quantitative information for the analytes of interest without concern for the rent of the components in the sample. Second-order bilinear sensors, Le., 8 e m m that yield a two-dimensional data matrix of the form Mij = ~ k & x i ~ are j k , specially suited for this pupxe, and the preferred technique for qunntitation is known as rank annihilation factor analysis, RAFA ( 1 , Z ) . So far this method has been applied to excitation-emissionfluorescence (1-3). LC/UV (4), and TLC-reflectance imaging spectrophotometry (5)with good results. It is important to realize that not all two-dimensional techniques yield bilinear data arrays; e.g., 2D-NMR or MS/MS data in their raw forms are not bilinear. A limitation of rank annihilation as o r i g i d y formulated in that an iterative solution requiring many matrix diagonalizations is necegsary ( I ) . Lorber (6) has reported a noniterative solution presenting the problem as a generalized eigenvalueeigenvector equation for which a direct solution
is found hy using the singular value decomposition. With his method, to obtain the concentrations of the p analytes of interest in the sample, its bilinear spectrum and the p calibration spectra for each pure analyte must be recorded to obtain the concentrations. Analysis for each analyte requires a separate calculation. This letter presents the generalized rank annihilation method, of which Lorber's noniterative method is a particular m e , that allows simultaneous quantitation of analytes in a sample using just one bilinear calibration spectrum obtained from a mixture of standards, one standard for each analyte. Generalized rank annihilation can determine the bilinear spectrum and the relative concentration for each analyte in the unknown mixture. The calculated spectra are next matched to those of the standahls. It is then straightforward to determine the actual concentration of each analyte from its relative concentration and the concentration of the corresponding standard. The full bilinear spectrum of each analyte is not actually required for identification. One need only use a single order (e.g., only the UV spectrum in the LC/UV case) for the match. This is an unusual type of
ANALYTICAL CHEMISTRY, VOL. 58, NO. 2, FEBRUARY 1986
analysis as, in most cases, analyte concentrations are estimated one a t a time thereby precluding identification.
THEORY AND DISCUSSION Any bilinear data matrix M can be expressed as a linear combination of the n pure-component, bilinear spectra pk
are not present in the calibration N, and s is the number of common components. [3] The components in the sample data matrix M are a subset of the components present in the calibration data matrix N
{&, Pz, ..., P,, 0, ..., 0) s 1 1 (12) diagonal(t) = kl, f 2 , ..., 6, Ss+l, ..., t,+J t 2 1 (13) diagonal(@)=
n
M=
(1)
C P k p k k=l
where p k = X k Y k T and (M,)k = X i k y j k . The X k are COlUllln VeCtQrS with information in one order, e.g., excitation spectra, and the Y k T are row vectors with information in the second order, e.g., emission spectra. If we define the p k as unitary-concentration, pure-component, bilinear spectra, is the concentration of the kth compound in M. We can rewrite eq 1 in matrix notation as
M = X@YT
(2)
where x is a matrix whose columns are the n x k vectors, YT is a matrix whose rows are the n Y k T vectors, and @ is a diagonal matrix with diagonal elements that are concentrations Pk*
In general we will have two data matrices, the unknown concentrations data matrix M and the calibration data matrix N. The bilinear calibration data matrix N can similarly be represented in matrix notation as I
N = XEYT
(3)
diagonal(€) = (O,O, 0, Sr+l, Er+Z, *-,t r + s , Er+s+l, -*, Er+s+tl (15) Here, r is the number of components in the sample M that are not present in the calibration N;s is the number of common components; and t is the number of components in the calibration data matrix N that are absent in the unknown sample M. [l] FIRST C A S E ONE COMPONENT
QUANTITATION
X@ = M(YT)+
(4)
X[
(5)
M = USVT
where X and YT are the same matrices defined for eq 2, and
concentrations ( k for the calibration matrix. The matrices M and N have in common the X and YT blocks; e.g., the excitation and emission spectra are the same, differing only in their concentration matrices, @ and 6, respectively. Therefore, by solving for X in eq 2 and 3 we obtain = N(YT)’
where (YT)’represents the pseudoinverse (7) of the matrix YT. Now we right multiply eq 4 by E and eq 5 by @ and combine to get
N(YT)+@= M(YT)+E 1
Again, s is the number of common components, and t is the number of components in the calibration N that are absent from the sample M. [4] The most general case would be when there are analytes in the unknown sample that are not present in the calibration sample and vice versa diagonal(@)= { P i , Pz, ..., P r , P r + l , Pr+2, -., Pr+s, 0, .-? 01 (14)
In this case, the calibration data matrix N has just one component, &, that is also present in the sample data matrix. The solution for this case has been reported by Lorber (6) and will be included here for completeness. The first step in solving eq 7 is to apply principal components analysis (8)to the sample matrix M and then express the matrices in terms of these principal components. The principal components of M are obtained by applying singular value decomposition (7)
t: is a diagnonal matrix whose diagonal elements are the
defining Z
497
where
MV = S U M W = SV
(6)
M M W = S W (eigenequations in U space) (20)
NZ@= MZ5
(7) We only known M, N, and 6, thus we must solve for Z and @. Equation 7 is similar to the generalized eigenvalue-eigenvector problem, but it cannot be solved by standard methods since N and M are not necessarily square matrices. A solution of this equation will be discussed in the next sections, for the following different possible cases. [ 11 The calibration data matrix N has just one component that is present in the sample data matrix M
..., P n )
n 21
diagonal([) = (tl,0, ..., 0)
(8)
..., P r , P r + l t
r20 (10) tr+sl 2 1 (11) diagonal(5) = (0, 0, .-t 0, Er+l, Er+Z, Here, r is the number of components in the sample M that Pr+z,
-et
-et
The next step is to estimate the number of principal components that are significant by the use of abstract factor analysis (8)or cross validation (9,IO). In the ideal case, this number is equal to the number of components, n, in the sample mixture. The number of significant principal componenta will allow reduction to the deterministic information contained in the M matrix, with most of the random error discarded in the lesser factors. To do this, a new matrix, M, is generated from the first n “significant” columns of U, V, and the upper left corner n x n part of S
=USVT
(9)
This is the standard RAFA problem as discussed by Lorber (6). [2] The caJibration data matrix N has several components that are a subset of the components present in the sample data matrix M diagonal(@)= { P i , Pz,
(18)
MTMV = S2V (eigenequations in V space) (19)
(YT)’
diagonal(@)= {P1, Pz,
(16)
(21)
Now eq 7 can be rewritten as
NZ@= M Z € = USVTZ6
(22)
If we substitute Z = VS-lZ*, where Z* SVTZ N(VS-lZ*)@
= USVT(VS-lZ*)[
(23)
using the orthogonality properties of V,V T = f = identity matrix in the upper left n x n corner and zeros in the rest, so
NVS-lZ*@ = USIS-’ Z*[ = U ( S S - l ) Z * [
(24a)
498
*
ANALYTICAL CHEMISTRY, VOL. 58, NO. 2, FEBRUARY 1986
which reduces to
(NVS-l)Z*@ = UZ*[ (24b) Left multiplying by UT and right multiplying by 8-l gives (UWVS-l)Z*@@-1 = ( U W ) Z * l @ - l = z * x
x
&?-I
(254
or, finally,
(UTNVS-l)Z* = z * x
(25b) which is the usual eigenvalue-eigenvector equation, because the matrix (UwkvS-l)is square. The eigenvectors Z* are not perpendicular because the matrix (uwkvS-l)is not symmetric. Because the rank of N is one, there will be p 1 zero solutions for the eigenvalues hk. Therefore, the only nonzero solution will be equal to the trace of the matrix (UwkvS-').By calculating the trace of this matrix, i.e., &, the concentration of the Kth component is solved directly as P k = t k / h k . If the unknown sample does not have the component that is present in the calibration sample, we cannot expand N in terms of X and Y (eq 3), therefore eq 25b is not valid. This is an example of the fourth case introduced in the previous section, which will be considered later in this paper. In practice, a nonzero concentration value, P k , will be obtained, so the validity of eq 3 must be verified before applying eq 25b. By use of target factor analysis (8, I I ) , modified for bilinear data, it is possible to check if N is included in M (see Appendix for the details of bilinear target factor analysis). The projection matrices, UUTand VVT,should leave N unchanged
UU%VVT = N
(26) As pointed out by Lorber (6), if the calibration matrix N has more than one component, i.e., its rank is greater than one, several solutions will be obtained for the concentrations &&, ..., but there will be no way to match which concentration corresponds to which chemical component. The proposed alternative is to obtain the spectrum of all the components separately and estimate their concentration one by one. A solution to this problem is described in the next section, using the eigenvectors matrix Z in eq 7, which was defined as the pseudo-inverse of the YT,i.e., the generalized inverse of the pure component's emission spectra.
on,
[2] SECOND C A S E SIMULTANEOUS QUANTITATION OF SEVERAL COMPONENTS In this case, the calibration data matrix N has several components that are a subset of the components present in the sample data matrix M. In the first place, it is necessary to check that the components in N are a subset of the components present in the sample data matrix M, applying bilinear target factor analysis to the matrix N, i.e., eq 26 should
be true. If more than one component is represented in the calibration matrix, eq 25b has several nonzero eigenvalues. The solution is a set of eigenvalues X and their corresponding eigenvectors Z*. The eigenvectors allow us to calculate the pure spectra matrices X and YT, e.g., excitation and emission spectra
z * = SVTZ =
SVT(YT)+
YT = (VS-'Z*)+ By use of the definition of M = XPYT = USVT
X@
= M(YT)+ = U S V V S - l Z * = U Z *
(27) (28)
(29)
The eigenvalues hk are the ratio of concentrations &/& for each component, i.e., calibration/unknown. Once the pure spectra xk or YkT are obtained, it is easy to match which
concentration [ k corresponds to which ratio hk, therefore the concentrations & can be estimated by @k = &/Ah. [3] THIRD CASE: CALIBRATION AS A BASE When the sample data matrix M is a subset of the components in the calibration N, we must invert the procedure. The principal components of the matrix M do not form a basis for the representation of the matrix N, therefore eq 25b is not valid in this case. The principal components of N are estimated by N = UNSNVNT, and equations similar to eq 25b, 28, and 29 are obtained
(UNTMVNSN-l)ZN* = ZN*XN
(30)
Y T = (VNSN-'zN*)+
(31)
x@= UNzN*
(32)
The eigenvalues ( h N ) k are not defined as they were before. Now the ( X N ) k are the ratio of concentrations &/& for each component, i.e., unknown samplefcalibration. Bilinear target factor analysis can be used to test instances of the third case. The projection of the matrix M in the spaces defined by N should leave M unchanged
UNUNTMVNVNr = M
(33)
If both this test and eq 26 fail, then we are dealing with the fourth case, discussed in the next section. In practice, the third case can be solved by using principal components regression or multiple linear regression, because the spectra of all the components are known. [4] FOURTH CASE: THE GENERAL CONDITION In this case, the calibration sample will have some components that are not present in the unknown sample, and there will be some components in this unknown sample not present in the calibration sample. Projection of one matrix onto the principal components of the other matrix will change its information; eq 25b and 30 will not be valid. A solution to this problem can be obtained by using the principal components of the sum of the matrices M and N , defining W M N
+
w = UWSWVWT
(34)
(UWTMVWSw-')ZW* = Zw*Xw
(35)
YT
= (VwSw-1Zw*)+
(36)
X@ = UWZW*
(37)
The eigenvalues hk are the ratio of concentrations P k / ( l k For all the components present in both mixtures, the concentration in the unknown is P k = kk.$k/(l- hk). When one component is not present in the calibration sample, t k = 0 and hk = 1. The solution presented for this case can be applied to all the previous cases, and no testing with target factor analysis is necessary. An artifical matrix W is generated to perform the calculations. This suggests that one could instead generate the W matrix simply by making a single standard addition containing known amounts of all analytes to the unknown sample. In this way the calibration mixture is added to the unknown mixture, and the W matrix is measured directly. Quantitation by RAFA with the standard addition method (S4M) has been discussed by Lorber (14) for single analyte addition. This procedure would extend the applicability of his method to the quantitation of several analytes at a time, correcting for matrix effects, and thereby represents an extension of the generalized standard addition method (GSAM) (12, 13) to second-order tensor data. If we have several calibration matrices N1, N2, ..., N,, we can apply the method to all of them, one at a time, or we can
+ &).
499
Anal. Chem. 1988, 58,499-501
handle it as a three-way factor analysis problem, using all of the information in one calculation. We are currently working on this problem, which will be the subject of another publication. ACKNOWLEDGMENT The authors gratefully acknowledge Scott Ramos for his assistance in writing this manuscript. APPENDIX Target factor analysis (TFA) (8, 11) can be applied to bilinear data in a similar way that it is applied to one-dimensional data. For the test vectors x, or yi, TFA can be expressed as
mTxi = x, or y i T O T = yCT
(38)
where every test vector x,or y, generates a predicted target vector x, or y,. If the test vectors are present in the matrix M, i.e., if the ith component, whose spectrum is xyiTis present in M, then the predicted target vectors should be equal to the test vectors, x, = x, and y, = y,;therefore
OUTx,= xior y , T V T = or
Y T V T= Y T
(40)
Now, if N = X[YT, then
UUwVVT = (UU%)l(YTVT)
= XtYT = N
LITERATURE CITED (1) Ho, C-N.; Christian, 0. 0.; Davldson, E. R. Anal. Chem. 1978, 50, 1108-1 113. (2) Ho, C-N.; Christian, G. D.; Davldson, E. R. Anal. Chem. 1980, 52, 1071-1079. (3) Ho. C-N.; Christian, G. D.; Davldson, E. R. Anal. Chem. 1981, 5 3 , 92-98. (4) McCue. M.; Malinowski, E. R. J . Chromafogr. Sci. 1983, 21, 229-234. ( 5 ) Gianelli, M. L.; Burns, D. H.; Callls, J. B,; Chrlstlan, G. D.; Andersen, N. H. Anal. Chem. 1983, 55. 1858-1882. (6) Lorber, A. Anal. Chlm. Acta 1984, 164, 293-297. (7) Lawson, C. L.; Hanson, R. J. "Solvlng Least-Squares Problems"; Prentice-HalC Englewood Cliffs, NJ, 1974. (8) Malinowski, E. R.; Howery. D. 0. "Factor Analysis in Chemistry"; Wiley: New York, 1980. (9) Wold, S. Technomehlcs 1978, 20, 397-405. (10) Eastment, H. T.; Krranowski, W. J. Technometdcs 1982, 24, 73-77. (11) Lorber, A. Anal. Chem. 1984. 56, 1004-1010. (12) Saxberg, 6. E. H.; Kowalskl, 6. R. Anal. Chem. 1979, 57, 103 1- 1038. (13) Jochum. C.; Jochum. P.; Kowalski, 6. R. Anal. Cheh. 1981, 53, 85-92. (14) Lorber, A. Anal. Chem. 1985, 5 7 , 2397-2399.
Eugenio SBnchez Bruce R. Kowalski* Laboratory for Chemometrics Department of Chemistry, BG-10 University of Washington Seattle, Washington 98195
(41)
and this is
UU%'vOT = N
(43)
In practice, due to random noise, eq 42-44 are approximate.
(39)
By use of the definition of X and Y we can similarly write
UUTx = X
UUW = N and NVVT = N
(42)
This equation defines bilinear target factor analysis. Note that
RECEIVED for review June 3, 1985. Accepted September 3, 1985. This work was supported, in part, by the Office of Naval Research. Eugenio SBnchez is grateful to the Venezuelan Fundacidn "Gran Mariscal de Ayacucho" for the award of a scholarship.
Direct Fluorination of Silica for Use in Liquid Chromatography Sir: Silica has been studied extensively, especially for use in chromatography, because of a number of desirable properties such as rigidity and porosity (1,2). Of particular interest is the chemical modification of the silica surface to alter the surface properties. Such modifications usually take the form of attaching new chemical functionalities through the formation of chemical bonds. The preparation, characterization, and applications of bonded phases for LC have been reviewed thoroughly (3-7). The incorporation of fluorine atoms into the stationary phase is of relatively recent origin. Fluorocarbon bonded phases show unique selectivities, relative to the hydrocarbon analogues, and are useful for class separations (8-11). Generally, retention time is less when using fluorocarbon phases (12). Fluorine-containing polymers, such as Kel-F, may be used in their native form or after chemical derivatization (13, 14). The fluorine atom, because of its low polarizability, imparts unusual properties into a molecule, which affects the chromatographic properties of the surface. The direct fluorination of silica for use in liquid chromatography has not been reported. One reason for this is the extreme conditions necessary for yeplacement of -OH by -F. Fluorination of silaceous materials is possible with SF., at elevated temperature (2),but HF attacks the silica skeleton (probably the siloxane bonds). Other reactions require temperatures in excess of 700 "C (15,16). Recently, however, several new fluorinating reagents have been developed that introduce fluorine atoms under relatively 0003-2700/88/0358-0499$01.50/0
mild conditions. One such reagent is (diethy1amino)sulfur trifluoride (DAST). This is a convenient reagent for replacing primary, secondary, and tertiary hydroxyls with fluorine (17-19).This commercially available liquid can be used with ordinary glassware under moderate conditions. Results are presented here for a new LC stationary phase, formed from the reaction of DAST with a conventional silica material. EXPERIMENTAL SECTION Materials. Spherisorb silica (5 pm) and an octyl-bonded phase (5 pm) were obtained from Phase Separations, Inc. (Diethylamino)sulfur trifluoride (DAST) was obtained fram Aldrich Chemical Co. All test solutes were reagent grade or better. All mobile phases were HPLC grade and stored over molecular seives. Diglyme was dried and distilled over sodium metal. Equipment. Stationary phases were compared by using a Varian 5060 liquid chromatograph with a UV-100 absorbance detector or a Waters Model 401 refractive index detector. Flow rate was 0.5 mL/min; injection volume was 10 pL. Data were collected with an Isaac/Apple data acquisition system employing software developed in this laboratory. Preparation of Fluorinated Silica. Silica was dried under vacuum (0.5 torr) at 150 O C for 18 h. A 0.75-g sample was placed in a round-bottom flask with approximately 20 mL of dry diglyme, gently stirred, and cooled to -78 "C using dry ice/acetone. The desired amount of DAST in 50 mL of diglyme was added dropwise via a separatory funnel into the silica suspension. A nitrogen atmosphere was maintained throughout the experiment. The reaction mixture was allowed to warm to room temperature and stirred an additional hour. After the mixture was filtered, the 0 1988 Amerlcen Chemical Society
