Analysis of Multicomponent Fluorescence Data 1. M. Warner, G. D. Christian,* and E. R. Davldson Department of Chemistry, University of Washington, Seattle, Wash. 98 195
J. B. Callis Department of Pathology, University of Washington, Seattle, Wash. 98 195
A method for the analysis of a fluorescent sample containing multlple components Is presented which utilizes the experimental “Emlsslon-Excltatlon Matrlx”, M. The elements MI,of thls matrix represent the fluorescent intensity measured at wavelength XI for excltatlon at X I . The number of Independently emlttlng components may be obtalned by a determlnatlon of the number of non-zero elgenvaluesof the matrlces MMTor MTM. If only two components are present a range of posslble excltatlon and emlsslon spectra for each may be obtalned through a linear transformatlon of the elgenvectors of MMTand MTM.Analysls of real and slmulated multicomponent data using the above strategles Illustrates the potential of the techniques. Also, examples are glven of how nolse affects the outcome and how spectral overlaps lead to partial amblgultles In the derived spectra.
In using fluorescence as an analytical technique, one is often confronted by a sample containing an unspecified number of fluorescent species, some or all of which are compounds with unknown spectral properties. If the components are well separated spectrally it is sometimes possible by trial and error to selectively excite one or more of the components, and to make reasonable interpretations of the data in terms of known fluorescence spectra. Some excellent examples of how an experienced spectroscopist can use this technique of selective excitation are given in the monograph by C. A. Parker (I).His examples illustrate the potential of fluorescence for multicomponent analysis. In this paper we provide a systematic procedure for the analysis of a multicomponent fluorescent or phosphorescent sample. Our analysis requires the experimental determination of the “Emission-Excitation Matrix”, M, whose elements Mij represent the fluorescent intensity measured a t wavelength A, for excitation a t A,. If the wavelengths are properly sequenced, then a row of M represents a fluorescence spectrum taken a t a particular wavelength of excitation, while a column represents a fluorescence excitation spectrum at a particular monitoring wavelength. We shall show that this matrix can be used to provide information about (a) the number of independent components contributing to the spectra and (b) the possibile excitation and emission spectra of each component in the case where only two are present. The work reported here represents a logical extension of the early studies of Weber (2) on multicomponent fluorescent systems, and has certain features in common with the technique of principal component analysis which is now finding application in a number of areas of analytical chemistry (3-5). The algorithms for spectral decomposition we have developed should prove particularly valuable now that automated instrumentation exists which can acquire the necessary fluorescence data in a very short period of time (6-8).
THEORY Eigenanalysis Method. For a sample containing a single 564
ANALYTICAL CHEMISTRY, VOL. 49, NO. 4, APRIL 1977
emitting species, we shall assume that the elements of the emission-excitation matrix are given to an adequate approximation (9) by:
MI, = 2.303 @$o( A, 1 (A, 1bc Y ( A, 1K (A, )
(1)
where Zo(A,) is the intensity of the monochromatic exciting light incident on the sample in units of quanta/s; the expression 2.303~(A,)bcrepresents the optical density of the sample and is the product of the compound’s molar extinction coefficient €(A,), the pathlength b , and the concentration of the emitting species c; @pf is the quantum efficiency of the fluorescence; y(A,) is the fraction of fluorescence photons emitted at wavelength A,; and K(X,) expresses the wavelength dependence of the sensitivity of the analyzing system, including geometrical factors, transmission of the monochromator’s grating, quantum efficiency of the detector, etc. In deriving Equation 1 we have assumed the optical densities are low enough so that the condition 2.3034 A,)bc 0 and V’ > 0. Proof that K exists depends upon the equality
UVT = UKIV(KT)-l]T
(7)
i.e., if
M = U P then also
M = U’V’T
(9)
where
U’ = UK and
V’ = V(KT)-l In practice, because of the noise in experimental data, we have found it necessary to relax the constraints on U’ and V’ slightly, so they are allowed to have some elements slightly below zero. For rank two,
+ + K ~ z u , 5 -T
u,(’)’ = K l l ~ , ( l ) Kzlu, (2) I-T U ,(’)’ u,(’)’
K12uL( l )
(’)
(12) (13)
= ( K Z ~ U ,-( K12~,(~))/(KllK22 ~’ - K12K21) I-7’’ (14)
UJ(*)’=
(KllfJ,(2)- K21Uj(l))/(KllK22- K12K21) 1 -T’ (15)
where 7’ and T‘ are estimates of maximum contribution of noise to the spectral vectors and (K11K22 - K12K21) is the determinant of K , which we can represent by IKI . Clearly one can assume K11 = K22 = 1 without loss of generality. Then because u(l) 1 0 and u(1) 1.0 (except for noise contributions) and u(2) and u ( 2 ) have some positive elements, Equations 12-15 can be rewritten for IKI > 0 as:
Then T’JK(is adjusted (using T = 4 T’IKJ)to the smallest positive value for which the upper bounds to K12 and Kzl are not less than the lower bounds. There are no solutions to Equations 12-15 for IKI I 0. Hence Equations 16 and 17 define the permissible range of Kl2 and K21. Clearly, as additional data are added to M, these bounds tend to become tighter. For large enough ranges of spectral frequencies, a nearly unique answer for x and y becomes probable. The problem of making U and V positive is different from the standard oblique rotation problem of factor analysis because the transformations of U and V are not done independently. If the matrix K is used to transform U, then the matrix (KT)-l must be used to transform V. For more than two components, IKI (KT)-l is not linear in the elements of K so the inequalities become much more difficult to solve. Resolution of Two Ideal Components. In the preceding sections we have indicated how the emission-excitation matrix may be decomposed into its individual components. In this section we present an analysis of how the spectral properties of the individual components affect the outcome of the analysis. This will be accomplished by working “backwards” from the true spectra to the eigenvalues, eigenvectors, and spectral vectors. The insight gained by this approach should prove useful in predicting the success of a proposed separation of a particular mixture of components as well as emphasizing the strengths and limitations of the analysis. As before, the discussion will be confined to the case of two components. ANALYTICAL CHEMISTRY, VOL. 49, NO. 4, APRIL 1977
565
First let us discuss how the true spectra determine the eigenvalues and eigenvectors of MMT and MTM. Suppose the true emission and excitation vectors are normalized so that the scalar products xTx = 1and yTy = 1.The matrix M then has the form M = P , x ( l ) y ( l )+ P2x(2)y(2) (18)
E2
=
P12P22 (P1
w ( l ) = (PlP2
(19)
+ P2)2
MMT = Xj3YTYBXT= X/3Tj3XT
(23)
where
T=
I[:
(34)
- x(2))/(1
- ,2)1/2
From Equations 32 and 33 it is clear that the second eigenvalue becomes smaller as the spectral overlap becomes greater. Since noise introduces other extraneous non-zero eigenvalues into MMT, the existence of two components can only be detected with certainty if ( 2 is larger than the eigenvalues arising from noise. If the extraneous eigenvalues are as large as t, detection of two components requires 52 >> t, or for similar components (1
- s2)(1- t 2 ) >> 4 1 + P 2 ) 2 / P 1 2 P 2 2
(35)
while for nonoverlapping components only Pz2 >> t is needed. For simplicity, consider two components whose spectra can be approximated as single Gaussian peaks in both emission and excitation, then
where 6k is a measure of the width of the peak, and value of X when x ( ~is) a maximum. For closely spaced peaks of equal width,
Xk
is the
(37)
Since the eigenvectors w of MMT must span the same space as the x vectors,
w = a l x ( l ) + u ~ x (=~Xu )
- P)
+ P12)x(1) + (PIP2 + P,2)x(Z)
w ( 2 )= ( x ( 1 )
It will be useful to define the scalar products of the emission and excitation spectra a s s = x(1)Tx(2)and t = y ( l ) ~ (These ~ ) ~ . quantities provide a measure of the overlap of the two component spectra and are directly related to the cosine of the angle between the vectors. Because all entries in X and Y are positive, zero overlap implies that x ( I ) and x @ ) are disjoint (i.e., not simultaneously non-zero a t any wavelength). Similarly, small overlap implies that the main peaks in the spectra are disjoint. In terms of the above,
(1 - s ” ( 1
and
where
and
+ P2I2
The corresponding eigenvectors are
which can be more conveniently expressed as
M = Xj3YT
(P1
51
and
and
(25)
where
Under these circumstances, Equation 35 becomes (26)
Thus Equation 6 may be rewritten as
XrrMMTw= 5XTw
(27)
or where (29) In the case where the spectra of the two components are quite different, s and t will be small, and to the first approximation 51
Assuming that
#
02,
= P12; t 2 = P22
(30)
(39) Clearly, detection of two similar but distinct components is a strong function of the difference of their wavelength maxima, and their relative concentrations. Although not shown by Equation 39, when 1 1 = X 2 and X I = X2, a separation can still be made as long as 61 # 62 (so that s and t are sufficiently less than unity). Ambiguity of Derived Spectra. Let us now evaluate how the degree of overlap of the true spectra of the two components affects the degree of ambiguity (Equations 16 and 17) in the attempt to reconstruct x and y from the eigenvectors of M‘rM and MMT. Since the approximate vectors (uh’and u k ’ ) span the same space as the correct spectra ( x and y ) , we may write: u (1)‘ = x (1) - a21x(2) 2 0 .(2)’
the eigenvectors are
= U I 2 X ( 1 ) + ,(a 2 0
(40)
and
-
+
~ 1 2 ~ 21 1 )0
+ y ( 2 ) ) / ( 1+
~ 1 2 ~ 22 1 )0
~ ( 1 )= ‘ ( ~ ( 1 ) a12y(2))/(1 ~ ( 2 )= ’ (azly(1)
If the spectra of the two components are quite similar, then s and t will be close to unity. Consequently, 566
ANALYTICAL CHEMISTRY, VOL. 49, NO. 4, APRIL 1977
(41)
Because of the inequality signs in the right hand side of Equations 40 and 41, a12 and a21 will generally have a range of values so that u’and u’ are ambiguous. The bounds for a 12
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567
Figure 3. (A) True (Mf)(Mr)T -t (Mr)(Mf)T.(B) True (Mr)(Mr)' Figure 2. (A) Simulated noise data. (B) True noise matrix
Table I. A. Simulated Noise Data Component , ';A: 1 2
nm 500 440
A,m,B", nm
u , nm
s /t
416 356
20 20
0.11/0.11
476 416
20 20
0.11/0.11
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516 456
and a21 can be written as (using the fact that x and y are non-negative)
and
From these equations, we can immediately see that a21 will be restricted to equal zero if and only if x l = 0 for some i such that x L ( 2 # ) 0 and y,@) = 0 for some j such that y,(l) # 0. Similarly, a12 will be restricted to equal zero if and only if xL(2= ) 0 for some i such that # 0 and y I (l) = 0 for some j such that yl # 0. The above reasoning is quite consistent with the generally appreciated notions that to obtain an unambiguous fluorescence spectrum of one component of a mixture we will have t o excite it in a region where it alone absorbs, and to obtain an unambiguous excitation spectrum of a particular component we will have to observe its emission in a region where it alone emits. The conditions for ~2~ and a12 to become bounded by zero from above or below is most easily expressed in terms of the spectral range A over which the spectrum is non-zero. Thus if- A ( x ) is the range of wavelengths for which x is non-zero, and A ( x ) is the complementary range for which x is zero, one can write A(x(1))n A ( X ( ~ as ) ) the range over which x(l)is zero and x ( L )is non-zero. Then, for example, 0 2 a21 if h(x('))n A ( x ( ~ ) ) tf fl, where fl is the empty set. A pair of spettra, x(l)and x(?) are well-separated if A(x(l))n A ( X ( ~ )and ) A ( X ( ~ )n ) A(x(')) are both non-empty. In Figure 1,we have summarized the expected ambiguities in the sixteen possibile combinations of spectral range inter568
ANALYTICAL CHEMISTRY, VOL.
49, NO. 4, APRIL 1977
sections in emission and excitation. For each combination, we have listed the range of values which a12 and a21 may assume, and which of the derived spectra, u(l)',uQ)',u ( l ) ' ,u ( ~ ) will ' , be ambiguous. We have also included in Figure 1 the implications on the elements of the transformation matrix, K, based on the spectral overlaps. We cannot predict which element of K is ambiguous because of the arbitrariness in sign convention of the second eigenvector, u ( ~and ) u ( ~ )In. seven of the sixteen combinations, one is afforded at least two completely unambiguous "fingerprint spectra", and in only one case do none of the spectra derived from extreme values of K12 and KP1 correspond to correct spectra. By examination of a wider range of wavelengths it should be possible to convert any ambiguous case into a less ambiguous (or unambiguous) one.
EXPERIMENTAL Apparatus. Computerized Fluorometer. For the study of the fluorescence of a mixture of free base and zinc octaethylporphins, the fluorometer of Gouterman et al. was employed (12).This system has been recently interfaced to a PDPS/E minicomputer which controls the scanning of the monochromators with stepping motors, and acquires fluorescence intensities through digitization of the output of the analog photomultiplier signal. The data acquisition, display and formatting routines were written using the high level language UW/ Focal. Scanning was accomplished by holding the excitation wavelength constant while acquiring a 50-point emission spectrum, then changing the excitation wavelength and recording another fluorescence spectrum until 2500 data points were acquired. The data were then formatted and transmitted via phone line t o a remote computer (CDC 6400). Conventional Fluorometry. The emission spectra of the five aromatic hydrocarbons, anthracene, chrysene, fluoranthene, perylene, and pyrene were obtained on a Baird Atomic SF/100 fluorometer. Absorption Spectra. The absorption spectra of the five aromatic hydrocarbons listed above were obtained on a Cary, model 14, spectrometer. Reagents. The zinc-octaethylporphin compound was synthesized and provided by the research group of Martin Gouterman, Department of Chemistry, University of Washington. Data Analysis. Real and simulated data were analyzed with a Fortran IV coded program, sIM-1, (Spectral Interpretation Model, version I), written for the CDC-6400 computer and based on the algorithm presented in the introductory part of this paper. RESULTS Noise Characterization through Simulated Spectra. In order to accurately determine the number of independently emitting species, it is important to understand how various sources of noise in a fluoroescence experiment affect the
+EXCITATION SPECTRlIa
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+EXCITATION
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596
WAVELENGTH.Nfl
Figure 4. (A) Isometric projection of simulated two component mixture. (8) First eigenvectors. (C) Second eigenvectors. (D) Derived spectra of component 1 (ABS. INT. = 100 photons). (E) Derived spectra of component 2 (ABS. INT. = 20 photons).(F) Resynthesized data containing only the first two components
magnitudes of the eigenvalues of MMT and MTM. We will confine the discussion to the three most important sources of error: 1)photoelectron statistical noise in the signal, 2) dark current and its statistical fluctuations, and 3) scattered excited light and its statistical fluctuations. The observed fluorescence matrix Mobsdcan then be represented as Mobsd = Mf + Md + Ms + Mr (43) where Mf is the noise-free fluorescence matrix, Md is a matrix of constant intensities representing the average value of the
dark current, Ms represents the noise-free scattered exciting light contribution, and Mr is the noise matrix, the i,jth element of which is represented by a random value taken from a Poisson distribution characterized by a mean and variance equal to the sum of the elements Mf,, M$ and MYl~Additional noise caused by digitization is included in Mr. Figure 2A shows a typical fluorescence data matrix simulated according to Equation 43. Here, M‘ is an “idealized” two-component fluorescence mixture, each having a single Gaussian peak in both excitation and emission. The excitation ANALYTICAL CHEMISTRY, VOL. 49, NO. 4, APRIL 1977
569
Table 111. Spectral Ambiguity of Binary Mixtures of Aromatic Hydrocarbons
Table 11. Non-Zero Spectral Range of Five Aromatic Hydrocarbons Aromatic hydrocarbons Anthracene Chrysene Fluoranthene Perylene Pyrene
Emission range, nm
Excitation range, nm
362-502 354-534 370-546 418-538 354-526
290-382 290-362 290-394 350-450 290-366
Mixture
wavelength extends from 320 to 516 nm in 4-nm increments. Similarly, the emission wavelengths extend from 400 to 596 nm in 4-nm increments. Table IA summarizes the input parameters of each component and shows the calculated values of s and t . The maximum intensities of both components were set equal to 100 photons. The dark current Md was characterized by a constant intensity of 10 photons while Ms was set a t a maximum intensity of 50 photons a t 400 nm in emission which decreased linearly to a maximum intensity of 40 photons a t 520 nm. A random number following a Poisson distribution was then generated and truncated to the nearest integer and used as M:Pd. Figure 2B shows the noise matrix Mr for this example. Since Md is a matrix of constant intensity which will accumulate in a single eigenvalue, it is desirable to remove Md before eigenanalysis. In actual experimental data, Md may easily be removed from Mobsdif one can find a submatrix of Mobsdwhere Mobsd = Md. (Alternatively, one can perform a separate measurement of Md.)The average value of Md over the submatrix may then be subtracted from Mobsd,In this simulated analysis, Md was accurately known and, therefore, the correct value of 10 photons was subtracted. As discussed earlier it is important to note from Equation 5 that scattered exciting light does not have a simple bilinear form and must be removed by a stripping technique. If we suppose that a standard scattered light matrix Mss has been determined, and the dark current contribution to it removed, and that there exists a spectral region where
Mobsd -Md
I.LM~~
(44)
then is easily determined, and pMss can be subtracted from the whole matrix. Again, in this simulation we know the correct value of lrMSsand can accurately remove the scattered exciting light signal. After the scattered exciting light and dark current contributions were removed the stripped, observed matrix Ms,Obsd was used in the eigenanalysis algorithm described earlier. The first step of the analysis is the formation of the product matrix (Ms,obsd)(Ms,obsd)T or (Ms@bsd)T(Ms,obsd). The former can be expressed as (Ms,obsd)(Ms,obsd)T = (Mf)(MqT (M')(Mr)T+ (Mr)(Mf)T+ (Mr)(Mr)T. Magar (13) has suggested that (a) the variance contributions of (Mq(Mr)Tand (Mr)(MqTare negligible compared to (Mr)(Mr)T and (b) (Mr)(Mr)Tis diagonal. If this were true, one could remove the "unique" variance due to noise by an iterative procedure. Unfortunately, the assumptions in his prescription are not generally valid for shot noise limited signals. The matrix elements of (Mq(MqTwill be of the order of N m 2 where N is the number of elements in a column of Mf and m is of the order of the mean value of all elements of Mf. The matrix elements of (Mf)(Mr)Tand (M')(MqT will be of the order of N1/2m3/2,while those of (Mr)(Mr)T will be of the order of N m for the diagonal elements, and N1I2mfor the off-diagonal elements. Thus (Mf)(Mr)Tand (Mr)(Mf)Tare small compared to (Mr)(Mr)T only when N >> m. Since N will not likely exceed 50 (2500 data points) and m is typically
+
570
ANALYTICAL CHEMISTRY, VOL. 49, NO. 4, APRIL 1977
slt
values
0.0810.21 0.000610.24 0.000610.40 0.04/0.88 0.4110.87 0.2310.92 0.8310.30 0.5410.90 0.5810.31 0.5010.48
Anthracenelperylene Perylenelpyrene Perylenelchrysene Perylenelfluoranthene Anthracenelpyrene Anthracenelchrysene Anthracenelfluoranthene Pyrenelchrysene Pyrenelfluoranthene Chrysenelfluoranthene
Spectral overlap type
I I I I1 I1 I11 I1 I11 I1 I1
+
100-10 000 photons, (Mf)(Mr)T (Mr)(Mf)Tis much larger than (Mr)(Mr)T.Consequently, Magar's analysis does not usually apply to fluorescence data. Figure 2B shows the true noise matrix for this simulation. Figures 3A and 3B are plots of (Mf)(Mr)T (Mr)(Mf)Tand (Mr)(Mr)T.These figures confirm the validity of our conclusions. One notes that even in this case where N = 50 and m = 100, the magnitude of (Mf)(Mr)T (Mr)(Mf)Tis much greater than (Mr)(Mr)T. From the eigenvectors of Ms,obsd(Ms,ohsd)T and (Ms,ohsd)TMs,obsd the best least squares fit of rank two,
+
+
MU" = ~ ( 1 ) u ( 1+ ) ~ ( 2 ) ~ ( 2 ) (45) was constructed and the least squares estimate of the residual noise Mres= Ms,ohsd- Mu"was formed. This decomposition of Ms,Ohsd leads to Ms,ohsd(Ms.ohsd)TI Muv(Muv)T + Mres(Muv)T
+ MUV(MreS)T + Mres(Mres)T(46) which is similar in form to the true decomposition into input spectra and noise. Within computational error (Muv)(Mres)T (Mres)(MUV)T is identically zero. This is merely a consequence of the least squares character of the eigenanalysis approach which forces this matrix product to be zero. It is clear from this result that the least-squares approach cannot be expected to provide a correct separation of the spectrum and noise. Consequently, the Mu"matrix has noise induced errors of the same magnitude as the elements of Mr or MreS.One can expect, however, that the norms of the matrices IIMrII and IIMresI/are comparable so that /lMresI/gives an indication of the error /IMf - MuvlI which is the error of greatest concern. Since Mf M' = Muv Mres,then
+
+
JJMf - M""/J
+
JIMres- M' I1 < IIMresII+ IIM'II
(47)
or
IIMf - Mu"//< 2//MreSll
(48)
6
if it is true that 1 M'II = I)MresI/. The norm, II Mres11, is the for this case of only two components. As another example, the maximum intensity of the second component was reduced from 100to 20 photons and no stray light was introduced. Figure 4A shows an isometric projection of the emission-excitation matrix for this simulated mixture. Not surprisingly, the spectra appear to be very noisy, and the minor component is almost completely obscured in the noise. The first part of SIM-1 finds the eigenvalues of MMT and MTM according to Equations 6. For this example, the square roots of the first four eigenvalues were 903.1,180.1,68.2,and 61.0. Clearly, the first two eigenvalues are reasonably greater than the third and one can conclude that only two components are present even though component 2 appears to be barely
+EXCITRTION SPECTRUM
+EXCITATION SPECTRUM
-EMISSION SPECTRUM
-EMISSION SPECTRUM
WRVELENGTH,NN +EXCITATION SPECTRUM
-EMISSION SPECTRUM
B
WAVELENGTH .NM +EXCITATION SPECTRUM
+EXCITRTION SPECTRUM
-EMISSION SPECTRUM
-EMISSION SPECTRUM
+EXCITATION SPECTRUN
-EMISSION SPECTRUM
C Figure 5. Ambiguities of binary mixtures of aromatic hydrocarbons. (A) Anthracene (left) and perylene (right). (B) Anthracene (left) and pyrene (right). (C) Fluoranthene (left) and pyrene (right)
ANALYTICAL CHEMISTRY, VOL. 49, NO. 4, APRIL 1977
571
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.
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550 574
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EMISSION WAVELENGTH,NM +EXCITATION SPECTRUfl -EflISSION SPECTRUfl
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+EXCITATION SPECTRUfl -EflISSION SPECTRUM 1001
h
Figure 6. (A) Contour plot of data of experimentally acquired twoComponent mixture of ZnOEP and H20EP.(B) Resolved two-dimensional spectra of component 1 (ZnOEP). (C) Resolved two-dimensional spectra of component 2 (H20EP)
above noise in Figure 4A. The derived eigenvectors are plotted in Figures 4B and 4C. The small negative entries in u ( l )and u ( 1 ) are a consequence of noise in the spectral data. The negative entries in u(2)and u ( 2 ) can be largely attributed to their required orthogonal relationship to u ( l )and u ( l ) . 572
ANALYTICAL CHEMISTRY, VOL. 49,
NO. 4, APRIL 1977
In the final part of SIM-1, the spectral vectors u(l)‘,u ( ~ )u(l)‘, ’, and u ( ~ )are ’ generated from the eigenvectors according to Equations 12-17, and are plotted in Figures 4D and 4E. Ambiguity limits result from the range of values of Kl2 and K Z I which fulfill Equations 16 and 17 with T’IKI = 0.019 and T = 17.2. One notes that most of the negative entries are not perceptible on a relative scale of 100. As we would expect, the second component is much noisier since its signal was significantly affected by the dark current noise level and photon noise from the much more intense first component. Based on spectral overlap of the digitized spectra, Figure 1would predict a type I spectral overlap and consequently no ambiguities. Here, however, the severe noise level caused deviations from predicted ambiguities. The variation in ambiguity as a function of wavelength results from the extent of overlap of the two components. Although the transformation elements K12 and Kzl may often have large ambiguities, there are regions of small ambiguities due to the eigenvector associated with the other component having values close to zero in these regions. Finally, we have used the spectral vectors to regenerate the excitation-emission matrix (Figure 4F). The resynthesized matrix is less noisy because much of the noise is contained in the eigenvectors following the second. Simulated Spectra. In the next series of trials, we simulated ten “two-component” emission-excitation matrices using the five aromatic hydrocarbons, anthracene, pyrene, perylene, chrysene, and fluoranthene. Since we desired to study the inherent ambiguities of our transformation and to compare the results with the predictions of Figure 1, the simulated spectra were synthesized without noise or truncation. Figures 5A-C show the correct spectra (solid lines) of selected compounds together with the observed range in ambiguity (given as error bars). Tables I1 and I11 give the non-zero spectral range of each compound, the calculated values of s and t and a summary of the results of the simulation. We find no ambiguities in the combinations anthracene1 perylene, perylenelchrysene, and perylenelpyrene. In the case of anthracenelperylene, we note that the emission spectrum of perylene extends about 40 nm (-506 to 545 nm) beyond the emission spectrum of anthracene. The excitation spectrum of anthracene extends about 56 nm (-290-346 nm) outside the excitation range of perylene. This is in accordance with a type Ispectral overlap and one would predict no ambiguities as verified in Figure 5A. Similarly, we can examine the regions of overlap in the perylenelpyrene and perylene/chrysene pair and find that both are also type I overlaps. In the other seven cases, we note some ambiguities. For example, in the pyrene/chrysene mixture, the emission spectrum of chrysene completely overlaps the emission spectrum of pyrene, while the excitation spectrum of pyrene overlaps the excitation of chrysene. This corresponds to a type I11 overlap and, as predicted, one of the K elements is unique while the other is intermediate. Analyses of Experimental Fluorescence Data. Figure 6A shows the observed fluorescence matrix for a mixture of free-base octaethylporphin (H20EP) and zinc octaethylporphin (ZnOEP). These data were taken with the computer driven fluorometer described in the Experimental section. In addition to the fluorescence components, one also observes a stray light component which is a maximum at A,, = Aem. T o remove the latter, a fluorescence matrix of the pure solvent was determined. After removal of the estimated dark current, a multiple of this matrix was subtracted from the original fluorescence matrix to eliminate the stray light contribution. With experimental data, we have found that it is difficult to find the correct multiple for elimination of the scattered light contribution. Apparently, this is due to differences in noise
patterns and possible differences in scattering phenomena from the pure solvent. However, for analysis purposes, it suffices to subtract enough of the scattered exciting light matrix so that the residual scattered light is below the noise level. Analysis of the pretreated data matrix showed the presence of two components which are displayed in Figures 6B and 6C. One spectrum from each component is unambiguous (within experimental error). For the first component, the excitation spectrum was unambiguous, and identical to that of ZnOEP. For the second component, the emission spectrum was unambiguous, and identical to that for H20EP. With knowledge of the actual spectra and Figure 1,we would have predicted a type I1 spectral overlap. This is in agreement with the results shown. The square roots of the first four eigenvalues of MMT were 2790.2, 809.1, 167.8, and 87.6 so IIMresl/I= 167.8. Much of the noise is associated with the scattered light region. For this example T’lKl was 0.010 and 5” was 27.9.
DISCUSSION We have shown that it is possible to analyze the EmissionExcitation matrix to determine (a) the number of independent components contributing to the emission, and (b) the possible spectra of each, if only one or two components are present. Furthermore, a separation of the components can be made even if the spectra severely overlap and the signal/noise is very poor. Recently several authors (14-1 7 ) have asserted that the use of “selective excitation” to separate fluorescent components is very inadequate for severely overlapping components, and that more sophisticated approaches will be required. Thus the techniques of derivative spectroscopy (16,17) and time (14) and phase (15) resolved spectroscopy have been introduced for multicomponent analysis. However, all of these techniques require “a priori” knowledge of the components to work, and all break down in the presence of background fluorescence without alerting the analyst. In contrast, the use of the fluorescence-excitation spectra is of a higher order of generality. The detection of multiple components is accomplished without making assumptions about the individual spectra. Moreover, for two components, fingerprint spectra are produced. Thus, if a library of known spectra is available, a search may be initiated, and comparisons made. In this manner, quantitative as well as qualitative analysis may be accomplished. One possible objection to the algorithms presented here is the huge amount of data they require. Admittedly, with a manually controlled fluorescence spectrophotometer, the analysis can be both time consuming and tedious. A computer controlled fluorometer such as we have used here would of course alleviate much of the tedium of collecting the data; however, it would do little to speed up the data acquisition time, especially if the signal-to-noise ratio is photon limited.
In the case where most of the samples are expected to have the same components, a computer controlled system could be programmed to quickly examine a carefully selected subset of wavelength pairs. If these are consistent, no further data would be required, and the analysis would be finished at that point. If the data were not consistent, then the full fluorescence matrix could be determined to uncover the reasons for the discrepancy. The attractiveness of the selective excitation technique is greatly enhanced by the recent development of the video fluorometer (6, 7). This instrument simultaneously aquires up to 241 fluorescence spectra, excited a t up to 241 different exciting wavelengths in 16.7 ms.
ACKNOWLEDGMENT The authors gratefully acknowledge the technical assistance of C. Connell, F. Kampas, and M. Gouterman in the use of their computerized fluorometer which allowed the acquisition of experimental data for analysis. The zinc-octaethylporphin compound was kindly provided by M. Gouterman. Finally, we thank J. Van Zee for his invaluable assistance in transmitting data to the larger computer and writing the u W / FOCAL language.
LITERATURE CITED (1) C. A. Parker, “Photoluminescence of Solutions with Applications to Photochemistry and Analytical Chemistry”, American Elsevier, New York, N.Y., 1968, p 440. (2) G. Weber, Nature (London), 190, 27 (1961). (3) R. W. Rozett and E. M. Petersen. Anal. Chem., 48, 817 (1976). (4) J. T. Bulmer and H. F. Shuruell, J. Phys. Chem., 77, 256 (1973). (5) G. Wernimont, Anal. Chem., 39, 554 (1967). (6) I. M. Warner, J. B. Callis, E. R. Davidson, M. Gouterman, and G. D. Christian, Anal. Lett., 8, 665 (1975). (7) I. M. Warner, J. B. Callis, E. R. Davidson, and G. D. Christian, Clin. Chem. ( Winston-Salem, N.C.), 22, 1485 (1976). (8) G. H. Haugen, B. A. Raby, and L. P. Rigdon, Chem. Instrum., 6, 205 (1975). (9) J. W. Longworth. “Creation and Detection of the Excited State”, Vol. 1, A. A. Lamola, Ed., Marcel Dekker, Inc., New York, N.Y., 1971. 10) G. A. Crosby, J. N. Demas, and J. B. Callis, “Absolute Quantum Efficiencies”, in National Bureau of Standards Special Publication 378, “Accuracy in Spectrometry and Luminescence Measurements”, Proceedings of a Conference held at NBS, Gaithersburg, Md., 1972. 11) R . Courant and and D. Hilbert, “Methods of Mathematical Physics”, Interscience Publishers, Inc., New York, N.Y., 1953. 12) M. Gouterman, L. K. Hanson, G. E. Kahiil, J. W. Buckler, K. Rohbock, and D. Dolphin, J. Am. Chem. Soc., 97, 3142 (1975). 13) M. E. Magar, “Membrane Conformation: A Factor Analysis Approach”, in “The Physical Principles of Neuronal and Organismic Behavior”, M. Conrad and M. E. Magar, Ed., Gordon and Breach Science Publishers,New York, N.Y., 1973, p 189. 14) P. A. St. John and J. D. Winefordner, Anal. Chem., 39, 500 (1967). 15) J. J. Mousa and J. D. Winefordner, Anal. Chem., 48, 1195 (1974). 16) T. C. O’Haver and W. M. Parks, Anal. Chem., 46, 1886 (1974). 17) T. C. O’Haver and G. L. Green, Anal. Chem., 48, 312 (1976).
RECEIVEDfor review September 16,1976. Accepted December 20, 1976. This research was supported in part by an American Chemical Society, Division of Analytical Chemistry, summer fellowship to I.M.W., sponsored by the Carle Instrument Company. Later phases of the work were supported by NIH grant IROlGM22311-01 MCHB.
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