2199
Anal. Chem. 1982, 5 4 , 2199-2204
Estimation of Fast Fluorescence Lifetimes with a Single Photon Counting Apparatus and the Phase Plane Method J.
Y. Jezequel, Mlchel Bouchy,
and J. C. Andre”
Le Grapp du L.A. 328 du CNRS, ENSIC, INPL,, 1, rue Grandville, 54042 Nancy Cedex, France
The phase plane deconvolutlon method allows a rapld determlnation of the lifetimes of emlttlng electronically exclted states In the case of single exponentlal decays. We demonstrate that the phase plane method Is also suitable for more complex decays, when there Is a scattered light component or the sample exhibits a decay which Is ai sum of two exponentlals. Under these conditions, this technique Is very useful for the rapid determlnatlon of the parameters of the decays; In partlcular, It Is useful with mini- or mbrocomputers.
An important problem in many scientific fields is to extract the maximum information from the result of an experimental measurement. In particular, the accurate measurement of fluorescence decays, obtained via the single photon counting technique, is crucial in the field of spectroscopy, chemical kinetics, analysis, photochemistry, and photophysics. In practice, the continuous fluorescence response function, u(t),is significantly distorted by the real time profile of the excitating flash, f ( t ) , and by the response function of the electronics, e(t). If z ( t ) is the real fluorescence time profile of the system exicited by a Dirac pulse, then we have the convolution product (eq 1) (the definitions of symbols appear in the Glossary at the end of this paper).
u(t) = f(t)*z(t)*e(t) = g(t)*z(t)
with the following initial condition: u(0) = 0. By putting
THEORY Single Exponcsntial Decay. Recently Greer, Reed, and Demas ( 3 , 4 )have described a fast and accurate deconvolution technique: the “phase plane method”. This technique allows the measurement of the lifetime, r , of emitting electronically excited states exhibiting the single exponential decay
z ( t ) = Ae+/‘
dx
(7)
sg(t) = s t0g ( x ) dx relationship 6 can be rewritten in the simpler form 1 u(t) ;su(~) = A sg(t)
+
(8)
Relationship 8 can be rewritten as -u(t) =---
sdt)
1 su(t) 7 s-d t )
+
A
(9)
Plotting u(t)/sg(t) vs. su(t)/sg(t) a t several values of the . value time t should lead to a straight line of slope - 1 / ~ The of T is then determined by use of the least-squares fitting method. Biexponential Decay. Now we show here that the same method can also be used in the case of a biexponential decay z(t) = A,e-t/Tl
+ A2e-t/rz
(10)
i.e.
z ( t ) = Z l ( t ) + zz(t)
(1)
The recovery of z(t) from experimental fluorescence time profiles and g(t) == f(t)*e(t) can be achieved by using several techniques of deconvolution as explained previously (1,2). This paper will deal with the recent phase plane technique.
& u(x) t
su(t) =
(11)
with zl(t) = Ale-t/Tl z2(t) = A2e-t/Tz
(12)
Putting Ul(t) = g(t)*z,(t) u&) = g(t)*zz(t) the total fluorescence u ( t ) is given by u ( t ) = g(t)*z(t) = u1(t) + UJt) As previously (cf. ref 6), we can write
(13) (14)
where A is a proportionality factor. We can write the convolution product as
(3)
u(t) = g(t)*z(t) = otherwise t
u ( t ) = e-t/TJ 0 g ( x ) AeZ/‘ dx
(4)
By taking the derivative of these two equations and adding them, we get
Taking the derivative of eq 4 leads to
(5) Integration on both sides of this equation leads to
knowing that
0003-2700/82/0354-2199$01.25/00 1982 American Chemical Society
ANALYTICAL CHEMISTRY, VOL. 54, NO. 13, NOVEMBER 1982
2200
and replacing du,(t)/dt and duz(t)/dt by their expressions deduced from eq 15, eq 16 is rewritten d2u(t) dt2
+ -a
du(t)
dt
+ p ~ ( t=) 7- d t + 6 g ( t )
dU-0 d t
du
u (k-1)
(18)
/
0, U < u (cf. channel j ) and if du(t)/dt < 0, U > u (cf. channel I). Note in addition that the experimental numbers of photons counted in the ith channel are different from the theoretical values U(i)and G(i) due to random errors, which can be taken into account by most of the iterative reconvolution techniques
(5,6).
s y u ( x ) d x ] dy
In the presence of this random noise, we have
0
su(i) = SU(i) sg(i) = SG(i) Using again sets of values of u ( t ) ,su(t), ssu(t), sg(t), and ssg(t) at several values of the time, t , we can get the values of the coefficients a,/3, y, and 6 by a fitting method based on the least-squares technique. Relationships (19) lead to the following expressions for the decay parameters: 71
2
= a!
+ (2- 4p)1/2
72
2
= a!
- (a2- 4 @ ) 1 / 2
Quantization of the Experimental Signal. The experimental technique of photon counting does not lead to u ( t ) and g(t) since the total time of measurement T is divided into intervals of T/n, in which n is the number of channels of the multichannel analyzer. This leads, in the absence of noise coming from the experimental technique, to a set of n discrete values (one per channel). Taking the width of one channel ( T / n )as the time unit, the numbers of counts in the ith channel are
U(i) =
si si c-1
G(i) =
u ( t ) d t for the fluorescence
1-1
g(t) dt for t h e flash
(24)
In these conditions, the integrals su(t) and sg(t) defined in eq 7 are given rigorously a t time i as i
U(j) = SU(i)
su(i) = j=l
i
sg(i) =
EGG)
j=l
= SG(i)
(25)
I t is obvious that increasing the number of channels covering the total time of measurement T reduces the quantization effect. However, this would lead to a longer counting time or a higher random noise so that the total number of channels is usually 100 to 500. We shall discuss in the following paragraph the effect of the quantization of the data and show simple methods to reduce it.
RESULTS AND DISCUSSION Effect of the Quantization of the Data on the Treatment of Single-Exponential Decays. If we use U(i) as an approximate value for u(i) and relationships 25, eq 8 can be written a t time i as 1 U(i)+ -SU(i) = A SG(i) (26) 7
There are two ways of estimating T by a least-squares method, either directly from optimization of relationship 26 or, as in ref 3 and 4, from the relationship
which allows one to visualize the result of the fitting in a convenient form because it is the equation of a straight line. T o apply and verify these methods, we have simulated a continuous excitation flash response function g(t ) by g(t) = B [ e - t / 9 - e-t/G] (28) where B = lo5, C1 = 10 channels, C, = 3 channels, and T = n = 100 channels. We have calculated the synthetic decay curves u ( t ) by integrating analytically the convolution product of g ( t ) with the single exponential function given in eq 2, assuming T = 12 channels and A = 1. We have then integrated analytically
ANALYTICAL CHEMISTRY, VOL. 54, NO. 13, NOVEMBER 1982
2201
Table I. Calculations of the Lifetime 7 in Noise-Free Single-Exponential Decays According to the Equation Used calcd value of 7 without the first calcd value n points ea o f 7 (7 = 1 2 n 7 used channels) 8
Figure 2. Visualization of the quantization effect on eq 27 (when uslng U ( i )as an approximate value for u(i)).
the functions u ( t ) and g ( t ) so as to get the simulated values of the data G(i) and U(i)according to eq 24, corresponding to the numbers of counts in the ith channel, in the absence of noise, for the excitation flash [G(i)]and the fluorescence
[Wl.
The least-squares fit of these data to eq 27 gives r = 14.34 channels instead of 12 whereas the fit o f t he same data to eq 26 gives r = 12.10. The plot of relationship 27 shows that a systematic error is made, especial1:y at low values of i (cf. plus signs on Figure 2). Indeed cancelling out the first points will lead to a better value of r; for example r = 11.99 when the first 10 points are canceled. The origin of this error lies with the data quantization effect: replacing U(i)by the correct value u(i) (i.e., when the data are not quantized) in eq 27 gives the actual value of r , 12 channels (cf. the straight line on Figure 2). This shows well the influence of ithe quantization when the data are used without care, esplecially when relationship 27 is used. Note that this problem is specific to single photon counting. In the case of datar collected by an analog tlo digital converter, the experimental data U(i)are then equal to u(i). On the other hand, the integrah su(i) and sg(i) become approximate. This case is the one effectively treated in ref 3 and 4, where su(i) and sg(i) are obtained by the classical trapezoidal rule of‘ integration. It appears that errors are very small in that case. Simple Methods to Reduce the Effect of Data Quantization. If we assume that u ( t ) is almost h e a r over two channels, there are two ways of minimizing the influence of the quantization. (i) By Use of a Better Estimation of u(i): In eq 26 or 27, if we use the values of SU(i) and SG(i)directly, it means that the integration hais been performed over ithe interval 0 to i. So, an approximation of r&), better than U(i), is (U(i) U(i + 1))/2, value obtained by linear interpolation. The equations are now
+
U ( i )+ U(i + 1) 2
1 + -SU(i) 7
= ,4SG(i)
(29)
9 26 27 29 30 32 33
12 12
12.10 14.34 12.01
10 10 2 2 2 2
11.71
11.99 12.41
11.92 11.99 12.01 12.01 11.99 11.99
Table 11. Calculations of the Lifetime 7 in Noisy Single-Exponential Decays calcd value of r without the first calcd value n points of 7 ( 7 = 1 2 ea used channels) n 7 26 27 29 30 32 33
12.3
10 10 3 3 1 1
14.7
12.2 11.8
12.2 12.7
12.2 12.3 12.2 12.1 12.2 12.1
So, to be self-consistent with this hypothesis, the sums must i.e. be performed from time 0 to (i -
SU’(i) = 5 U G ) + 1/2U(i) j=l
i-1
SG’(i) =
EGG) + l/zG(i)
j=l
and the relationships derived from eq 8 and 9 are now
1
U(i) + -SU’(i) = A SG’(i)
(32)
U(i) = -1SU’(i) +A SG’(i) r SG’(i)
(33)
7
and lead to the following optimized value of r: 7 = 11.99 and without the first two points, 7 = 11.99 from eq 32; r = 12.41 and without the first two points, r = 11.99 from eq 33, for an expected value of 12 channels. All these results are gathered in Table I. (iii)Application to Noisy Signal: To assess the influence of the quantization with the noise we have simulated noisy data by adding noise to the data, according to the following equation:
U(i)noisy = U(i)+ N[U(i)]1/2 and lead to the following optimized values of r: T = 12.01 and without the first two points, r = 12.01 from eq 29; r = 11.71 and without the first two points, T = 12.01 from eq 30, for an expected value of 12 channels. (ii) By Modification of the Integration Interval: If we approximate u(i) Iby U(i),one channel being the time unit, we can consider thLat U(i)is a good approximation of u ( t ) at time (i - l / J instead of time t = i, with the same linearity assumption.
(34)
where N is a random, normally distributed number with a mean value of zero and a standard deviation of 1,and taking a value of 2000 counts at the maximum of fluorescence. We have then calculated r by the different methods previously used for noise-free data. The results are summed up in Table 11. Application of the Phase Plane Method to More Complex Decays. In theory, the phase plane method could be applied every time there is a linear relationship, with constant parameters, between u ( t )and its derivatives and g(t) and its
2202
ANALYTICAL CHEMISTRY, VOL. 54, NO. 13, NOVEMBER 1982
Table 111. Calculations of the Lifetime r in Single-Exponential Decays When the Fluorescence Has a Scattered Light Component
U max
T(12)
free of noise 1000
12.015 11.6 11.88 11.97
10000
100000
T(2)
U max free of noise
2.002
1000 10000
1.78
100000
1.96
1.87
k(0.05) 0.0505 -0.09 0.007 0.03
A(1)
T(12)
1
12.015
k(0.20) 0.2005
1 1 1
11.6 11.88 11.97
0.05 0.15 0.19
k(0.05) 0.047
A(1)
-0.48 -0.039 0.016
1.13 1.10 1.03
1
Table IV. Calculations of the Decay Parameters in Biexponential Decays When A, = - A, U max free of
7A5) 5.02
7,(12) 11.98
Al(1) 1.006
5.6 5.2 5.07
11.07 11.71
1.09 1.06 1.02
noise 1000
10000 100000
11.9
derivatives. This occurs in particular with single exponential fluorescence decays with a scattered light component or with decays that are the sum of two exponentials [but this method cannot be used to determine experimental parameters when this is no longer true (Smoluchowski decays (7) for example)]. We have applied this method in these two important cases. (i) Measurement of the Lifetime When the Fluorescence Has a Scattered-Light Component. Relationship 6 is verified by u ( t ) . In the case of a scattered light component, the continuous fluorescence response function u’(t) is u’(t) = u ( t )
+ k g(t)
(35)
Then we can write from eq 6 and 35, with u’(0) = 0 and g(0)
A(1)
41)
A(1)
1
12.015
~(12)
1.005
1
1 1 1
11.6 11.87 11.97
0.82 0.95 0.98
1 1 1
U max
d40)
free of noise 1000
40.035 39.2 39.8 39.95
10000 100000
k(0.05) 0.05
A(1)
-0.069
0.98 0.99 0.99
1
0.014 0.04
As expected, when T decreases, the relative error on its calculated value increases for the same number of counts at the maximum, but its estimation remains good (see Table I11 when the expected values are A = 1, T = 2 channels, and k = 0.05). On the contrary, when T increases, this relative error decreases (see Table I11 when the expected values are A = 1, T = 40 channels, and k = 0.05). These results allow us to conclude that this method is suitable for the deconvolution of data represented by a single exponential impulse response with a scattered light component. (ii) Biexponential Decays. Due to the quantization of the signal, it is still necessary to find estimates of u ( t ) ,su(t), ssu(t), sg(t), and ssg(t) for the master relationship 21. We have chosen u(i)
E
[U(i)
+ U(i + 1)]/2
as in eq 29 or 30 su(i) =
i
i
j= 1
j=l
CUG), sg(i) = E G G )
as in eq 25 j=i
=o
ssu(i)
= ]=l t[i - j
+ +)
(36) By minimizing the quantization effect as done in relationships 29 and 30 and after least-square fitting, it is possible to estimate the most probable values of 7, k , and A . Table I11 gathers the results of these calculations which show the ability of this technique to determine the lifetime in these particular conditions, for data which are either free of noise or noisy (the expected values are in brackets and U max is the number of counts at the maximum of fluorescence). We can see that this method gives a good determination of 7 and A even when the estimated value of k bears an important error (this is the case for rather noisy signals with a weak scattered light component, as it can easily be understood).
using the trapezoidal rule to calculate the estimated values of ssu(i) and ssg(i). The parameters of the decay are then derived from a least-squares fitting of relationship 21. We have first studied the particular case when A2 = -Al is assumed, because it is of practical interest (excimer formation for example (8, 9)) and because it leads to a threeparameter system only instead of a four parameter one in the general case. This method is accurate when two conditions are fulfilled: T~ and r2 are sufficiently different and the data are not excessively noisy (maximum number of counts greater than
Table V. Calculations of the Decay Parameters in Biexponential Decays in the General Case (A, $ -A T~ (in channels) expected calcd
12
12 12 12 a ns =
no solution.
12.03 11.96 nsa 18.12
T?
(in channels) calcd
expected 5
5 5 5
4.95 5.03 ,sa
0.29
A,
l)
A,
expected
calcd
expected
calcd
1 1 1 1
0.988 1.01
-1 -2
-0.989 -2.01
nsa
1.34
-10 1
.sa
1.23
ANALYTICAL CHEMISTRY, VOL. 54, NO. 13, NOVEMBER 1982
IOOO), but this is also true for other methods (2). In Table IV are presented the results when r1 = 12 and 7 2 = 5 channels. This method gives similar results for other values of T~ and r2 that we have tested when 71 and 72 are different by more than two channels. In the general case when A2 # -Al, and for reasons that are not easy to understand, this method is sometimes accurate for noise-free or relatively noise-free data, but sometimes it is not accurate or suitable as illustrated in Table V. We think that this is due to the fact that eq 21 is a four-parameter one, so the minimum of the hypersurface of the sum of squared errors is very flat. As a consequence a small1 error on the values can imply an important, error for the calculated minimum bound to the accuracy of the calculator (here an Apple I1 microcomputer working with 32-bit real numbers). We are now writing new programs in which the inversion of the illconditioned least-squares matrix is not clone directly.
CONCLUSION The results presented in this paper extend and improve those published in ref 3 and 4. The phase plane method brings very significant advantages when compared to the other techniques of deconvolution or iterative reconvolution already published (see Table VI). The calculation time required by this method is especially short. This advantage, reasonable for single-exponential decays, becomes more obvious in the casle of more complex decays such as biexponential ones. The time required by the calculations is usually much shorter than the time of experiment, even when a microcomputer is used. It is possible to modify the mathematical treatment to take into account the standard deviations on the values of u(i),g(i), su(i), sg(i), ssu(i), and ssg(i). However, with these modifications, the computer program becomes much “heavier”, and it then might be more satisfactory to use another method such as the iterative reconvolution technique (5). Nevertheless the use of the technique presented here appears to be valid at the end of the calculations by the determination of the x2 on the measurements where
8
4 a 0 ri
0
G 11
m R
rlm
riri
0
E O 11 E
m m
gx
0
E
m h
c)
8 6
c & U
I
Q
p being the total number of parameters in the equation used and v(i) being the quantized convolution product of g ( t )
(known by the data G(i))with z ( t ) , the parameters having been determined using the phase plane method. In Table VI is gathered a comparison between the different techniques used for the determination of the parameters of luminescence decays obtained with a single photon counting apparatus. It clearly shows two important advantages: a short computation time and freedom from cutoff corrections as it occurs in other deconvolution techniques.
* t n A 7
Al, A, T ~ r2 ,
a,@,y, 6
T N
k P
GLOSSARY convolution product time number of channels of the multichannel analyzer proportionality factor in a single exponential decay time constant in a single exponential decay proportionality factors of the single exponential components in a biexponential decay time constants of the single exponential components in a biexponential dec?y coefficients in the master equation for biexponential decays total time of measurement random number, exhibiting a standardized normal distribution coefficient of the scattered light component in u ’( t ) total number of parameters in the equation used
.-UE
‘2 m cw
0
I
I
m m R R a,o
M
R W
8 0
R E
m
x
.-2I
c-
c a,
.r
h
8
t& &
U
3
8
m U
2203
Anal. Chem. 1982, 54, 2204-2207
continuous fluorescence response function real continuous time profile of the exciting flash continuous response function of the electronics real fluorescence continuous time profile of the system excited by a Dirac pulse continuous flash response function single exponential components in a biexponential decay components of u(t) in the biexponential decay case continuous fluorescence response function with a scattered light component integral of u(t) on (0, t) integral of g(t) on (0, t ) integral of su(t) on (0, t) integral of sg(t) on (0, t ) number of counts in the ith channel of the fluorescence histogram number of counts in the ith channel of the flash histogram sum of U(j)’son (0, i ) sum of G(j)’s on (0, i) approximate sum of U(j)’son (0, i approximate sum of G(j)’son (0, i - l/z)
quantized convolution product of g(t) with z ( t ) , the parameters having been determined by the phase plane method
LITERATURE CITED O’Connor, D.; Ware, W. R.; Andre, J. C. J . Phys. Chem. 1979, 83, 1333. McKinnon, A. E.; Szabo, A.; Miller, D. R. J . Phys. Chem. 1977, 87, 1564. Greer, J. M.; Reed, F. N.; Demas, J. N. Anal. Chem. 1981, 53,710. Demas, J. N.; Adamson, A. W. J . Phys. Chem. 1971, 75,2463. Grinvald, A.; Steinberg, I.2. Anal. Blochem. 1974, 59,583. Ware, W. R.; Doemeny, L. J.; Nemzek, T. L. J . Phys. Chem. 1973, 77,2038. Andre, J. C.;Bouchy, M.; Ware, W. R. Chem. Phys. 1979, 37, 119. Birks, J. B. “Photophysis of Aromatic Molecules”; Wiley: New York, 1970. Donner, M.; Andre, J. C.; Bouchy, M. Siochem. Siophys. Res. Commun. 1980, 97, 1183. Gafni, A.; Modlin, R. L.; Brand, L. Siophys. J . 1975, 75, 263. Isenberg, I.;Dyson, R. Blopbys. J . 1969, 9 , 1337. Valeur, B. Chem. Phys. 1978, 30, 85. Wild, U.; Holzwarth, A.; Good, H. P. Rev. Sci. Instrum. 1977, 48, 1621.
RECEIVED for review August 21,1981. Resubmitted February 16, 1982. Accepted July 7, 1982.
Structural Information from Tandem Mass Spectrometry for China White and Related Fentanyl Derivatives Michael T. Cheng, Gary H. Kruppa, and Fred W. McLafferty” Department of Chemistiy, Cornell University, Ithaca, New York 14853
Donald A. Cooper Special Testing Laboratory, U.S. Drug Enforcement Administration, McLean, Virginia 22 102
The potential of tandem mass spectrometry utilizlng colllslonally activated dlssociatlon (CAD) for molecular structure determlnation is Illustrated with a-methylfentanyl (“China Whlte”), whose complex structure required several methods for its orlglnal elucldatlon. CAD spectra of fragment ions in its electron ionization ( E I ) mass spectrum provide Information on dissociation pathways and fragment structures; the latter lnformatlon comes from both lnterpretatlon and matching against reference spectra. Although the E1 spectrum shows no odd-electron and few primary fragment Ions, Ion types most useful for such CAD studies, CAD data from the evenelectron, secondary fragment ions gave sufflcient structural Information.
Elucidation of the molecular structure of China White, an illicit narcotic implicated in drug overdose deaths, attracted unusually wide publicity (1-3). Structure 1 was assigned on the basis of mass, infrared, and nuclear magnetic resonance spectra; synthesis corroborated this identification, eliminating 3 as a candidate. The unusual narcotic activity of this compound meant that available samples contained very small concentrations, so that it would have been especially advantageous if the complete identification could have been done with a method requiring only a submicrogram sample, such as mass spectrometry. However, spectral interpretation was difficult for this polyfunctional compound; the electron ion-
C2H5CO-N(Phl
1: R’ = CH,, R2 =
R3=H,n=1 2: R ’ = R 3 = H , R2= CH,, n = 1 3: R ’ = R 2 = H , R 3 = CH,, n = 1 4: R’= RZ= R3 = H,n=l 5 : R’= R2 = R3 = H,n=O
-c
N-CH2R
6 : R = cyclopropyl 7 : R = Ph 8: R = CH,Ph 9 : PhNHCH(CH,)-
CH= CH2 10: PhNHCH,CH=
CHCH, 11: PhN(CH,)CH,CH= CH, 1 2 : PhCH,N[ CH( CH,)CH=CH,] ,
ization (EI) mass spectrum (Figure 1)“was totally unfamiliar” (1). From this spectrum the Self-Training Interpretive and Retrieval System (STIRS) ( 4 , 5) correctly identified the fiphenylethylamine moiety and the cyclic amine but gave little indication of the N-phenylpropionamide portion of the molecule. It thus appeared that these compounds might provide an interesting test of the additional structural information available from tandem mass spectrometry (MS/ MS) (6-10). In MS/MS the first mass spectrometer (MS-I) is operated as a conventional instrument, forming ions from the sample by methods such as electron and chemical ionization (E1 and CI). Ions of a specific mass separated by MS-I are then fragmented further, usually by metastable ion (MI) or colli-
0003-2700/82/0354-2204$01,25/00 1982 American Chemical Society
