Signal generation and mixing electronics for frequency-domain lifetime and spectral fluorometry
The present invention additionally comprises a method and apparatus for generating and mixing signals for frequency-domain lifetime and spectral fluorometry. The present invention comprises a plurality of signal generators that generate a plurality of signals where the signal generators modulate the amplitude and/or the frequency of the signals. The present invention uses one of these signals to drive an excitation signal that the present invention then directs and transmits at a target mixture, which absorbs the energy from the excitation signal. The property of fluorescence causes the target mixture to emit an emitted signal that the present invention detects with a signal detector. The present invention uses a plurality of mixers to produce a processor reference signal and a data signal. The present invention then uses a processor to compare the processor reference signal with the data signal by analyzing the differences in the phase and the differences in the amplitude between the two signals. The processor then extracts the fluorescence lifetime and fluorescence spectrum of the emitted signal from the phase and amplitude information using a chemometric analysis.
Latest Systems & Processes Engineering Corporation Patents:
- Method and system for measuring optical scattering characteristics
- Method and system for measuring optical scattering characteristics
- Signal generation and mixing electronics for frequency-domain lifetime and spectral fluorometry
- Chemometric analysis for extraction of individual fluorescence spectrum and lifetimes from a target mixture
- Precision optical displacement measurement system
The U.S. Government has a paid-up license to certain technologies disclosed in this invention and the right in limited circumstances to require the patent owner to license others on reasonable terms as provided for by the terms of the following contracts: NAS1-20426 and NAS1-20162 awarded by NASA; DAAH01-91-R198 awarded by US Army Missile Command; DAAL06-92-C-0014 and DAAD07-91-C-0127 awarded by US Army White Sands Missile Range; DAA15-93-C-0034 and DAAM01094-C-0033 awarded by US Army Chemical and Biological Defense Agency; and F41624-95-C-6010 and F41627-97-C-6029 awarded by US Air Force Armstrong Laboratory.
BACKGROUND OF THE INVENTION1. Field of the Invention
The present invention generally relates to fluorescence measurements. More specifically, the present invention relates to frequency domain measurements of the fluorescence lifetime and the fluorescence spectrum.
2. Description of the Related Art
The process of fluorescence occurs when a substance, such as a molecule, absorbs light at one wavelength (or energy), and then emits light at a longer wavelength (or lower energy). A slight time delay occurs from when the substance absorbs light and when the substance re-emits light at the longer wavelengths. This time delay is known as the fluorescence lifetime.
A group of horizontal lines 20, 22, and 24 appear in each of the potential energy wells 10 and 12. Each horizontal line represents the individual vibrational energy states possible for the diatomic molecule. Quantum mechanics requires that the vibrational frequency of the spring be within certain values. In other words, the spring may oscillate at frequency ‘a’, and the spring may oscillate at frequency ‘b’, but it is a physical impossibility for the spring to oscillate at any frequency between ‘a’ and ‘b’. In
The different potential energy wells 10 and 12 represent another type of energy in the diatomic molecule: electronic energy. Different electronic energies occur when the molecule absorbs energy in such a fashion that it causes an electron to improve to a higher energy configuration within the molecule. The classical example of the energy absorption is the ‘changing the spring’ that connects the two atoms. If we “add” electronic energy to the potential energy well 10, the energy raises the potential energy well 10 to the energy level of potential energy well 12.
With this background, we can use
In practice, we typically probe many different types of molecules at once with the excitation light pulse.
I(t)=I0·e−(t−5
I(t)=I0·e−(t−t
where t0 is the time of the excitation pule, I0 is the initial fluorescence and I(t) is the observed fluorescence intensity as a function of time. emission of a photon is the process of fluorescence. The process of emitting the photon has an associated time delay, which is the fluorescence lifetime.
In the cases with multiple molecular types (where more than one lifetime decay is presented within a sample of a target mixture), the exponential decay seen in
The present invention is a system for chemometric analysis for the extraction of the individual fluorescence spectrum and fluorescence lifetime from a target mixture. The present invention comprises a processor with an apparatus for generating an excitation signal to transmit at a target mixture and an apparatus for detecting the emitted signal from the target mixture. One embodiment of the present invention uses a processor that comprises a computer that extracts the individual fluorescence spectrum and lifetime measurements from the frequency and wavelength data acquired from the emitted signal. The present invention first determines the G and S matrices from the frequency and wavelength vectors. A renormalization of w−1Sx to Gx occurs next due to the differences in units. Next, the present invention determines the initial U, V, g and w−1s approximation. The present invention uses an iterative solution that first requires the initialization of the decision variables including η. The iterative solution compares the decision variables for convergence to see if further approximation determinations of the U, V, g and w−1s are necessary. When the solution converges, the present invention then determines the reduced best fit error for the analysis of the individual fluorescence lifetime and the fluorescence spectrum. And finally, the system of the present invention extracts individual component fluorescence lifetime and fluorescence spectra from the emitted signal of the target mixture.
The present invention additionally includes a method and apparatus for generating and mixing signals for frequency-domain lifetime and spectral fluorometry. The present invention comprises a signal generator that generates a driving/references signal that modulates the amplitude and/or the frequency of the driving/reference signal over time and a signal generator that generates the mixing signal that modulates the amplitude and/or the frequency of the mixing signal over time. The driving/reference signal generator drives an excitation signal generator that generates the excitation signal. The present invention then directs and transmits the excitation signal at a target mixture, which absorbs the energy from the excitation signal. The property of fluorescence causes the target mixture to emit an emitted signal that the present invention detects with a signal detector. The present invention uses a mixer that mixes the emitted signal with the driving/reference signal to produce a processor reference signal. Another mixer mixes the emitted signal with the mixing signal to produce a data signal. The present invention then uses a processor to compare the processor reference signal with the data signal by analyzing the differences in the phase and the differences in the amplitude between the two signals. The processor then extracts the fluorescence lifetime and fluorescence spectrum of the emitted signal from the phase and amplitude information using a chemometric analysis.
To further aid in understanding the invention, the attached drawings help illustrate specific features of the invention and the following is a brief description of the attached drawings:
The present invention is a system for chemometric analysis for the extraction of the individual component fluorescence spectra and fluorescence lifetimes from a target mixture. Additionally, the present invention is a method and apparatus for generating and mixing signals for frequency-domain lifetime and spectral fluorometry. This disclosure describes numerous specific details that include specific processes, structures, and circuits in order to provide a thorough understanding of the present invention. For example, the present invention describes the extraction of the fluorescence spectra and fluorescence lifetimes from a target mixture. However, the practice of the present invention includes the extraction or analysis of other physical properties of a target mixture other than the previously described ones. One skilled in the art will appreciate that one may practice the present invention without these specific details. Additionally, this disclosure does not describe some well known processes such as Fourier transforms, eigenequations, eigenfunctions, eigenvalues, transforms, or best fit analysis in detail in order not to obscure the present invention.
The present invention utilizes a novel system that uses a chemometric analysis to determine the fluorescence lifetimes and the fluorescence spectra of a target mixture. Instead of irradiating the target sample with a single short pulse of light (photon counting) as other prior art systems, the present invention continuously irradiates the target mixture with a light source whose amplitude modulation frequency is stepped with time.
Referring back to
Coupled to the driving/reference signal generator 71 is a filter 82 and an amp 86. The filter 82 filters out any unwanted components from the driving/reference signal, and the amp 86 provides additional gain to the signal if necessary. Coupled to the mixing signal generator 73 is a filter 84 and an amp 88. The filter 82 filters out any unwanted components from the mixing signal, and the amp 86 provides additional gain to the signal if necessary. The present invention uses the driving reference signal 90 as a phase and amplitude reference signal on the receiver component 56 (of FIG. 6). And, the present invention sends the mixing signal 92 to the receiver component 56 where it is split.
The present invention uses the driving/reference signal 90 to directly drive and modulate an excitation signal generator 58 (of
The present invention comprises several techniques or alternative embodiments to “target” the target mixture. One embodiment of the present invention uses a target compartment for holding the target mixture within a chamber. A container may contain the target mixture, or the mixture may be free floating, in a fluid state, within the chamber. Another embodiment of the present invention uses an optical probe that combines the signal generator 58 and the detector 60 into a single probe. A user of the present invention would then probe or place the optical probe into the target mixture. And, another embodiment of the present invention may use a laser to target a free floating target mixture, such as the emission from a car (at a distance). One skilled in the art will appreciate that other embodiments of targeting the target mixture are possible after reviewing the disclosure of the present invention.
If we apply a ‘monochromatic’, short light pulse to the molecules of a fluorescent material at a time t0, ideally represented by:
E(t)=E0·δ(t−t0) (1)
where E0 is the total energy in the pulse and E(t) is the intensity of the incident pulse as a function of time, we find that the fluorescent intensity at a fixed wavelength has the form:
I(t)=I0 e−(t−ω)κ (2)
where
I0=α·E0 (3)
and α has appropriate units. Thus, by measuring the decay of the fluorescent intensity with time and fitting a straight line (linear least squares fit) to a graph of log(I(t)) vs time, we can determine the lifetime of a fluorophore. However, if more than one fluorophore is present, the intensity variation with time depends on the concentrations, Cn, and lifetimes, τn, of the individual fluorophores:
Now the intensity decay curve must be fit by a non-linear least squares process, which requires initial guesses for the number of fluorophores, r, their concentrations, and their lifetimes. In general, convergence of the solution cannot be guaranteed as it depends on the accuracy of the initial guesses.
Weber showed that decay constants and relative intensities of an arbitrary number of independence components in a heterogeneous fluorescence emission could, in principle, be determined by measuring phase shift and relative modulation of the total fluorescence at an equal number of frequencies. See Weber, G., Resolution of the Fluorescence Lifetimes in a Heterogeneous System by Phase and Modulation Measurements, J. Phys. Chem., 85, 949-953 (1981). At each frequency, the phase and modulation measurements yield the complex Fourier transform of the fluorescence impulse response, G+iS. The moments of a distribution of lifetimes are found as linear combination of the G's and S's and Prony's method is used to obtain lifetimes and concentrations of the components from the moments. Due to its sensitivity to noise, this method is really not practicable for more than binary mixtures.
Lackowicz and his coworkers developed an analysis of frequency domain phase and modulation data based on a weighted non-linear least square analysis. See Lackowicz, J. R., et al., Analysis of Fluorescence Decay Kinetics from Variable-Frequency Phase Shift and Modulation Data, Biophys. J., 46, 463-477 (1984). By taking data at several emission wavelengths, they were able to resolve binary mixtures whose components differed in lifetime by 30% and ternary mixtures the ratio of whose range of lifetimes spanned a decade. Best fit is determined by minimizing a reduced chi-squared function. Poor initial guesses can still produce unpredictable results.
Working with an Excitation-Emission-Frequency Array, McGown and her coworkers employed a principal components analysis followed by a non-linear least squares minimization on an over determined set of data to extract the spectra and lifetimes of components in a mixture. See Burdick, D. S., et al., Resolution of Multicomponent Fluorescent Mixtures by Analysis of the Excitation-Emission-Frequency Array, J. Chemometrics, 4, 15-28,(1990). In addition to requiring a non-linear minimization, a disadvantage of this method is that it requires evaluation over a large (3 dimensional) parameter space.
The present invention utilizes a novel system that uses a chemometric analysis for measuring the fluorescence lifetime and the fluorescence spectrum of a target mixture. Instead of irradiating the target sample with a single short pulse of light (photon counting) as other prior art systems, the present invention continuously irradiates the target mixture with a light source whose amplitude (intensity) and phase (frequency) are modulated (varied) with time. In evaluating experimental phase resolved fluorescent spectroscopy (PRFS) data, the variables sought are the concentrations, ci, the lifetimes, τi, and the emission spectra, αi(λ), of the individual fluorescent contributors. The variables that we may control include the emission wavelength, λ, the modulation frequency, ω, and the detector reference phase angle ΦR. We first consider an amplitude modulated excitation:
E(t)−E0[1+Me sin (ωt)], t>0 (5)
where ω is the modulation angular frequency, Me is the modulation amplitude and E0 is the average amplitude of the excitation.
At steady state, the modulated part of the fluorescent intensity may be given by
Iλ(t)p=E0MeA sin (ωt−Φ) (6)
where A is the amplitude response at the fluorescent wavelength and Φ is the phase lag at this wavelength and frequency.
We originally detected this signal (the emitted signal) by correlating it with a reference signal (the driving/reference signal) by mixing the two signals with the mixing signal and integrating them over some multiple of a period. The correlated signal could therefore be expressed as:
where r is the number of components present (or fluorophores) and ΦR is the phase of the reference signal relative to the backscattered excitation signal. We observe that Sλ(ω,ΦR) may be written as:
Therefore, as an alternative to Eqn. (8), we may determine G(ω,λ) and S(ω,λ) to within a common normalization factor, E0Me/2,for any fixed ω and λ by making measurements at two different, but known, values of ΦR. In the presence of noise, more reference phase values would be used in the measurements and fit in a least squares manner to G and S.
We then find that for each value of ω and λ (emission wavelength) we have (modulo the normalization factor):
with tan φt=ωτi, where i runs over the number of fluorophores present and assumed to be acting in an independent (uncorrelated) fashion.
The preferred embodiment of the present invention allows us to determine G and S by mixing the emitted signals from the mixture down to an intermediate frequency of 10 kHz, sample and digitize it at 50 kHz, Fourier transforming the result, and picking off the signal at 10 kHz. When this is done the mixed signal has the form:
Iλ(t)=E0Me[G sin (ωHt)−S cos (ωHt)] (11)
From which we see that G and S are proportional to the Fourier transform of Equation (11).
In the case of noiseless data, where r fluorophores are present, it is necessary to make measurements at r values of λ and r values of ω to obtain:
G−cc·A and S=cs·A, (12)
where
G=[Gjk]
S=[Sjk]
cc=[cos2 φr(ωj)=[ccη]
cs=[cos φi(ωj) sin φi(ωj)]=[csji]
A=[εi(λk)]=[αik] (13)
where the quantities defined in boldface are matrices. The matrix indices i, j, and k run over the components, the modulation frequencies, and the emission wavelengths, respectively.
If no noise is present, we may take the indices i, j, and k above to run from 1 to r, the number of components. As the fluorophores are assumed to be independent and as the frequencies {ωj} may be chosen arbitrarily, all the matrices defined above have true inverses. Thus, from Eqns. (12) and (13) we find:
cc−1G=A (14)
and
S=cs·cc−1·G
or
S·G−1·cc=cs
Now, we note that:
cs = [cosφi(ωj)sinφi(ωj) (15)
= w·cc·T
By substitution from tan φi=ωτi, we have:
cs = [(cosφi(ωj))2ωjτi] (16)
= w·cc·T
where w=diag[ωi] and T=diag[τi]. Hence, we find
S·G−1·cc=w·cc·T (17)
or
w−1·S·G−1·cc=cc·T
This equation is in the form of a standard eigenvalue/eigenfunction equation, where the τi are the eigenvalues and the columns of cc are the eigenfuctions. We may also write this in the form:
cc−1·w−1·S·G−1·cc=T tm (18)
From which we have:
Since φi(ωj) is known once τi is know, we may determine cc or cs and solve the equation:
G=cc·A or S=cs·A (21)
for A with a standard LU decomposition, which gives us
A=T−1·ccpi·w−1S (22)
where ccpi is the pseudo inverse of cc.
Another, equivalent, way of writing the eigenfunction equations proceeds from the observation that:
G·A−1=C (23)
and
S=cs·A=W·cc·T·A (24)
so that
w−1·S·A−1=cc·T=G·A−1·T (25)
or
G−1·w−1S·A−1=A−1·T G−1·w−1·S·A−1=A−1·T (26)
Solving this eigenequation determines τi a the eigenvalues and A−1 as the eigenvectors. However, as the eigenvectors are normalized, A may be better determined as above if information about the component concentrations is required.
Once the spectrum and lifetime of each component is known, as may use the information to identify the fluorophore. The concentration of each fluorophore may then be determined by comparing its spectrum against its standard spectrum. We find:
where αi(λ) is the standard spectral value of the ith component at wavelength λ and τi is its lifetime.
The number of components in a mixture may not be known a priori, and the method requires the number of frequencies and wavelengths employed to at least equal the number of components. Furthermore, some degree of noise will be present in the experimental data. Therefore, it is useful to determine a means of solving for lifetimes and spectra when the number of frequencies and the number of emission wavelengths are unequal and when each is greater than the number of independent components present in the mixture.
In the following, we shall therefore assume that r components are present in a mixture, for which we have taken data a N frequencies and M emission wavelengths. We assume that N,M≦r and that N≠M, in general.
From Eqns. (12) and (16), we therefore have:
G=cc·A (28)
(N×M)−(N×r)−(r×M)
and
S=w·cc·T·A (29)
(N×M)·(N×N)·(N×r)·(r×r)−(r×M)
where the number of rows and columns of each matrix is indicated below it. In the noise free case, all the matrices are of rank r. In the presence of noise, G and S will be of rank N or M, whichever is smaller.
In the absence of noise, all matrices are of rank r. We manipulate Eqns. (28) and (29) slightly, to write:
G=cc·I·A
and
w−1·S=cc·T·A (30)
where I is the r×r identity matrix.
We may now make a singular value decomposition on the first of Eqns. (30):
G=U·C1·VT (31)
(N×M)·(N×r)·(r×r)·(r×M)
where C1 is the diagonal matrix of singular values of G and U and V are each column orthonormal, i.e.:
UT·U=VT·V=I. (32)
From Eqns. (31) and (32), we have:
C1=UT·G·V (33)
We define:
C2=UT·w−1·S·V (34)
where C2 is not diagonal, in general, but is r×r.
It may easily be shown, by a process that is similar to Gram-Schmidt orthogonalization, that while, in general, neither matrix is invertible, cc has a left inverse and A has a right inverse:
ccli·cc=I and A·Ari=I (35)
From Eqns. (30) and (31), we may therefore write:
cc=U·Pu and A=Pv·VT (36)
with
PU=C1·VT·Ari and Pv=ccli·U·C1.
where PU and Pv are each r×r.
We may interpret the first of Eqns. (36) in terms of its column vectors as the representation of the r vectors of cc in the orthonomral basis of the r vectors of U:
where cj cj is the jth column of cc, Ui Ui is the ith column of U and Pij is the corresponding element of Pu. From this expression and the linear independence of the cj cj and Ui Uj, we see that Pu is invertible. Linear independence of the cj cj requires that:
From Eqn. (36), we see that Eqn. (37) implies:
From the linear independence of the Ui Uj, we must have
Thus, if the condition of Eqn (37) is to be satisfied, the system of Eqns. (39) must have only the trivial solution. This is equivalent to the requirement that
det [PU]=0.
Therefore, PU has an inverse. A similar argument shows that Pv is also invertible.
From Eqns. (30), (33) and (34), we have:
C2=PU·T·Pv and C1=Pu·I·Pv. (40)
Therefore, we may write:
C1−1·C2=Pv−1·T·Pv (41)
or
C−1·C2·Pv−1=Pv−1·T. (42)
Eqn. (42) is the eigenvalue equation for the matrix, C1−1·C2, with the eigenvalues equal to the diagonal elements of T, and eigenvectors proportional to the column vectors of Pv−1. The spectrum, A, may be determined from Eqns. (13) and (30) since cc is determined once the lifetimes are known.
A second method for solving Eqns. (30) is to use Eqn. (31) to write the pseudo inverse of G:
Gpi=V·C1−1·UT (43)
so that we have:
G·Gpi=U·UT and Gpi·G=V·VT,
where U·UT is and an N×N matrix and V·VT is an M×M matrix, and each of these matrices is of rank r.
From Eqns. (30) and (35), we may write:
Gpi=V·C1−1·UT (44)
Substituting this into the first of Eqns. (30), we have:
Gpi·w−1·S·Ari=V·VT·Ari·T (45)
From Eqns. (30) and (36) we have:
w−1·S=U·Pu·T·Pv·T. (46)
From Eqn. (32), we therefore find:
w−1·S=w−1·S·V·VT.
Substituting this into Eqn. (45), we therefore have:
Gpi·w−1·S·V·VT·Ari=V·VT·Ari·T (47)
Eqn. (47) is the eigenvalue equation for the matrix, Gpi·w−1·S, with eigenvalues equal to the diagonal elements of T and eigenvectors proportional to the columns of V·VT·Ari. The spectra, A, may be found as described above.
When determining the lifetimes, we can renormalize Eqn. (47) by using
pseidoinv(g) w−1S evec=evec T (48)
where evec=V·VT·Ari for T=diag(τ1, . . . , τr)
For the case when noise is present, let us assume that r components are actually present, and let us take N<M. The experimentally determined matrices, Gx and Sx, are therefore of rank N. We wish to determine the N×M matrices, G and S, of rank r that minimize some suitably defined error.
We shall define the Euclidean norm of a matrix, M, to be given by:
The error we define will depend on our knowledge of the statistics of the experimental data. We shall assume that the errors at each data point are uncorrelated with those at our data points and that the standard deviations from the sample means are the same for all data points. This is tantamount to the assumption that the statistics are independent of the modulation frequency and the emission wavelength for the ranges of the variables explored. Previous experience with experimental data tends to bear this assumption out.
Given these assumptions, it is reasonable to define the error, χ2, in terms of the equally weighted distance of the experimental points from their analytical counterparts, i.e.:
χ2=∥Gx−G∥2+∥w−1·Sx−w−1·S∥2 (50)
where G satisfies Eqn. (31) and where, from Eqns. (40) and (46), we see that S satisfies:
w−1·S=U·C2·VT (51)
Therefore, from Eqns. (31), (50) and (51), we must find U, V, C1 and C2 to minimize:
χ2=∥Gx−U·C1·VT∥2+∥w−1·Sx−U·C2·VT∥2 (52)
Minimizing Eqn. (52) will yield a least squares fit for the data.
For Eqn. (52) to be a minimum, its partial derivatives with respect to each free variable must vanish. Applying this requirement with respect to the variables of C1, we find:
C1=UT·Gx·V (53)
or, from Eqn. (31),
G=U·UT·Gx·VT=Qu·Gx·Qv (54)
where
Qu=U·UT and Qv=V·VT (55)
Proceeding similarly for the variables of C2, we find:
C2=UT·w−1·Sx·V (56)
or w−1·S=Qu·w−1·Sx·Qv (57)
or =U·UT·w−1·S·Vx·VT (58)
Eqns. (54) and (57) determine C1 and C2 once U and V are determined. We now proceed to determine these variables.
By substituting for G and w−1S from Eqns. (54) and (57) into Eqn. (50), we find that minimizing that expression implies that we must maximize:
η=∥Qn·Gx·Qv∥2+∥Qu+w−1·Sx·Qv∥2 (59)
or
η=Tr(ggT)+TR|(w−1S)(w−1S)T] (60)
Observing that:
Qu=QuT, Qu·Qu=Qu, (61)
and Qv=QvT, Qv·Qv=Qv, (62)
we may rewrite Eqn. (59) in the alternative forms:
η=Tr(Mv·U·UT)
η=Tr(MV·U·UT) (63)
or η=Tr(Mu·V·VT) (64)
where Mv=Gx·Qv·GxT+(w−1·Sx)·Qv·(w−1·Sx)T (65)
and MU=GxT·QU·Gx+(w−1·Sx)r·QU·(w−1·Sx) (66)
We note that Mv is an N×N matrix, MU is an M×M matrix, and both matrices are symmetric.
If we assume that V is known, we may determine U by requiring that it be chosen to maximize Eqn. (63) subject to the constraint imposed by Eqn. (32). We may solve for U by means of Lagrange multipliers.
Define: η1=η−Tr[Lv·(UT·U−I)] (67)
where Lv is a symmetric r×r matrix formed from the multipliers.
We may now extremize η′ with U unconstrained to find:
Mv·U=U·Lv (68)
Since it is symmetric, Lv may be diagonalized by a similarity transform with an appropriate orthogonal matrix. Therefore, we may write:
Mv·U=U·O·D·O−1 (69)
where D is an r×r diagonal matrix and O is an r×r orthogonal matrix.
We may then write:
(Mv=U·O=U·O·D (70)
Substituting Eqn. (70) into Eqn. (63) we find:
η=Tr(Mv·U·O·O−1·UT)
=Tr[(U·O)·D·(U·O)T]
=Tr[D·(U·O)·(U·O)]
or, finally,
η=Tr(D) (70)
Thus, to maximize η we choose U to maximize D as determined by Eqn. (70). We may easily do this by observing that Eqn. (70) has the form of an eigenvalue equation. The matrix D yields r of the N possible eigenvalues of the matrix Mv. We therefore solve the eigenvalue equation for Mv:
Mv·Û=Û·E (71)
taking the largest r eigenvalues of E for D and, without loss of generality, choosing U to equal the corresponding eigenvectors of U.
A similar procedure may be used to determine V from Eqn. (64) if U is known using the relationship MU·V=LU to find LU.
As the frequencies in w are large, we renormalize ∥w−1·Sx∥ to equal ∥Gx∥ so as to avoid skewing the results in favor of the Gx data 138. This amounts to a resealing rescaling of the units for the frequencies and the lifetimes. This resealing rescaling of units is compensated once the lifetimes are found, so they are expressed in seconds.
We begin the iterative approach by determining the initial approximations for U, V, g and w−1s approximations 140, 142, and 144. The initial iteration begins by choosing the r columns of U1 to be the eigenvectors corresponding to the r largest eigenvalues of the matrix
Gx·GxT+(w−1·Sx)·(w−1·Sx)T (73)
or [G·GT+(w−1·S)·(w−1·S)T]·U=UL (74)
where L=diag(λ1, . . . , λT) for λ1> . . . λT. Next, U is used to form the matrix MU1 in accordance with Eqn. (66). We can determine V1 from MU1 by solving the eigenfunction equation MU
We next initialize the decision variables 146 to produce the solution in a reasonable time, with a reasonable accuracy, and a reasonable number of iterations. We begin the iterative solution by comparing the decision variables 148 for convergence to see if further determinations for U, V,g and w−1s are necessary as above. In the iterative solution, we proceed to maximize η, given by Eqn. (59), and recognizing that:
0≦η≦∥Gx∥2+∥w−1·Sx∥2=N (72)
Proceeding in this way, we form a sequence of η values {ηi}. We see easily that the sequence is monotonic non-decreasing. By Eqn. (72) and the Bolzano-Weierstrass theorem, the sequence has at least one limit point in the interval [0,N]. The monotonic nature of the sequence guarantees that there can be cut one limit point. Therefore the sequence must converge.
The values of U and V corresponding to the limit of the sequence are then used to form G and w−1S in accordance with Eqns. (54) and (57). Eqn. (47) may subsequently be solved for the lifetimes and spectra as described previously 160, 162, and 164.
In the preceding, we assumed an arbitrary but fixed value, r, for the number of fluorescing components present. Fits to the data may be found for different values of r. The question then arises as to the value of r that best fits the experimental data. Two qualitative criteria may be stated: (1) the spectra should look reasonable; and (2) the lifetime should be positive real numbers.
Somewhat more quantitatively, we recognize that χ2, suitably renormalized for frequency as described above, represents our error for a fixed value of r. However, the value of χ2 depends on the number of free parameters available for the fit, which, in turn, depends on r. Instead, we attempt to remove this dependence by dividing by the difference between the number of experimental data points and the number of free parameters present:
where χ2R is the reduced error team, Nx is the number of experimental data points and Nf is the number of free parameters.
Since we collect an amplitude and a phase value for each emission wavelength and each modulation frequency, we have:
Nx=2NwNλ (76)
where Nw is the number of frequencies and Nλ is the number of wavelengths used.
From Eqns. (54), (57) and (68), we see that the free parameters being fit to the experimental data are the elements of U, V, LV and LU subject to the constraints of Eqns. (32). For r components, we therefore find that:
We evaluate χ2R for each r, and choose the value of r corresponding to the smallest value of χ2R to represent the number of components which best fits the data 158.
If two or more lifetimes are the same, their components will not be separable by this method, nor will they be separable by the method of PREEMSs. Their contributions will remain combined as that of a single component with the common lifetime and a spectrum given by the combined spectrum appropriate to their concentrations.
When the solution converges, a determiner 186 determines the reduced best fit error for the analysis of the individual fluorescence lifetimes and fluorescence spectra. The present invention then uses an extractor 188 that extracts the individual fluorescence lifetimes from the target mixture, and an extractor 190 that extracts the individual fluorescence spectra.
The present invention overcomes the limitations of the prior art systems by utilizing a novel technique to measure the fluorescence lifetime. Instead of irradiating the target sample with a single short pulse of light, the present invention continuously irradiates the target sample with a light source whose amplitude and phase are modulated with time. This technique allows the present invention to use a chemometric analysis to automatically extract the lifetimes from the ‘phase delay’ and ‘intensity vs. time’ characteristics of the emitted light.
The present invention is a system for chemometric analysis for the extraction of the individual fluorescence spectrum and fluorescence lifetime from a target mixture. The present invention comprises a processor with an apparatus for generating an excitation signal to transmit at a target mixture and an apparatus for detecting the emitted signal from the target mixture. The present invention extracts the individual fluorescence spectrum and fluorescence lifetime measurements from the frequency and wavelength data acquired from the emitted signal. The present invention uses an iterative solution that first requires the initialization of several decision variables and the initial approximation determination of intermediate matrices. The iterative solution compares the decision variables for convergence to see if further approximation determinations are necessary. If the solution converges, the present invention then determines the reduced best fit error for the analysis of the individual fluorescence lifetime and the fluorescence spectrum before extracting the individual fluorescence lifetime and fluorescence spectrum from the emitted signal of the target mixture.
The present invention additionally comprises a method and apparatus for generating and mixing signals for frequency-domain lifetime and spectral fluorometry. The present invention comprises a plurality of signal generators that generate a plurality of signals where the signal generators modulate the amplitude and/or the frequency of the signals. The present invention uses one of these signals to drive an excitation signal that the present invention then directs and transmits at a target mixture, which absorbs the energy from the excitation signal. The property of fluorescence causes the target mixture to emit an emitted signal that the present invention detects with a signal detector. The present invention uses a plurality of mixers to produce a processor reference signal and a data signal. The present invention then uses a processor to compare the processor reference signal with the data signal by analyzing the differences in the phase and the differences in the amplitude between the two signals. The processor then extracts the fluorescence lifetime and fluorescence spectrum of the emitted signal from the phase and amplitude information using a chemometric analysis.
Other embodiments of the invention will be apparent to those skilled in the art after considering this specification or practicing the disclosed invention. The specification and example above are exemplary only, with the true scope of the invention being indicated by the following claims.
Claims
1. An apparatus for fluorescence lifetime and spectral measurements, comprising:
- a driving/reference signal generator that generates a driving/reference signal, said driving/reference signal is amplitude and/or frequency modulated over time;
- a mixing signal generator that generates a mixing signal, said mixing signal is amplitude and/or frequency modulated over time;
- an excitation signal generator that generates an excitation signal, the driving/reference signal drives said excitation signal generator;
- a signal detector that detects the emitted signal;
- a mixer that mixes the emitted mixing signal with the driving/reference signal and produces the processor reference signal;
- a mixer that mixes the emitted signals with the mixing signal and produces the data signal; and
- a processor that extracts the fluorescence lifetime and fluorescence spectrum of the emitted signal from the comparison of the processor reference signal with the data signal using a chemometric analysis.
2. The apparatus of claim 1 wherein the driving/reference signal and the mixing signal vary by an adjustable offset frequency.
3. The apparatus of claim 1 wherein said chemometric analysis extracts the fluorescence lifetime of the emitted signal from the phase difference between the processor reference signal and the data signal.
4. The apparatus of claim 1 wherein said chemometric analysis extracts the fluorescence spectrum of the emitted signal from the amplitude difference between the processor reference signal and the data signal.
5. The apparatus of claim 1 wherein said chemometric analysis further comprises a converging iterative solution.
6. A system for fluorescence lifetime and spectral measurements, comprising:
- means for generating a driving/reference signal, said driving/reference signal means modulates the amplitude and/or the frequency of the driving/reference signal over time;
- means for generating a mixing signal, said mixing signal means modulates the amplitude and/or the frequency of the mixing signal over time,
- means for generating an excitation signal, the driving/reference signal drives said excitation signal means;
- means for detecting the emitted signal;
- means for mixing the emitted mixing signal with the driving/reference signal to produce the processor reference signal;
- means for mixing the emitted signal with the mixing signal to produce the data signal; and
- a processor that extracts the fluorescence lifetime and fluorescence spectrum of the emitted signal from the comparison of the processor reference signal with the data signal using a chemometric analysis.
7. The system of claim 6 wherein the driving/reference signal and the mixing signal vary by an adjustable offset frequency.
8. The system of claim 6 wherein said chemometric analysis extracts the fluorescence lifetime of the emitted signal from the phase difference between the processor reference signal and the data signal.
9. The system of claim 6 wherein said chemometric analysis extracts the fluorescence spectrum of the emitted signal from the amplitude difference between the processor reference signal and the data signal.
10. The system of claim 6 wherein said chemometric analysis further comprises a converging iterative solution.
11. A method for forming the fluorescence lifetime and the fluorescence spectrum, comprising the following steps:
- generating a driving/reference signal and modulating the amplitude and/or the frequency of the driving/reference signal over time;
- generating a mixing signal and modulating the amplitude and/or the frequency of the mixing signal over time;
- generating an excitation signal form the driving/reference signal;
- detecting the emitted signal,
- mixing the emitted mixing signal with the driving/reference signal and producing the processor reference signal;
- mixing the emitted signal with the mixing signal producing the data signal; and
- extracting the fluorescence lifetime and fluorescence spectrum of the emitted signal from the comparison of the processor reference signal with the data signal to measure using a chemometric analysis.
12. The method of claim 11 wherein the driving/reference signal and the mixing signal vary by an adjustable offset frequency.
13. The method of claim 11 wherein said chemometric analysis extracts the fluorescence lifetime of the emitted signal from the phase difference between the processor reference signal and the data signal.
14. The method of claim 11 wherein said chemometric analysis extracts the fluorescence spectrum of the emitted signal from the amplitude difference between the processor reference signal and the data signal.
15. The method of claim 11 wherein said chemometric analysis further comprises a converging iterative solution.
16. A method of producing an apparatus for fluorescence lifetime and spectral measurements, comprising:
- providing a driving/reference signal generator that generates a driving/reference signal, said driving/reference signal is amplitude and/or frequency modulated over time;
- providing a mixing signal generator that generates a mixing signal, said mixing signal is amplitude and/or frequency modulated over time;
- coupling an excitation signal generator that generates an excitation signal and a reference signal to said driving/reference generator;
- providing a signal detector that detects the emitted signal;
- coupling a first mixer to said excitation signal generator, said mixer mixes the emitted mixing signal with the driving/reference signal to produce the processor reference signal,
- coupling a second mixer to said mixing signal generator, said mixer mixes the emitted signal with the mixing signal to produce the data signal; and
- coupling a processor to said first mixer and said second mixer, said processor extracts the fluorescence lifetime and fluorescence spectrum of the emitted signal from the comparison of the processor reference signal with the data signal using a chemometric analysis.
17. The method of claim 16 wherein the driving/reference signal and the mixing signal vary by an adjustable offset frequency.
18. The method of claim 16 wherein said chemometric analysis extracts the fluorescence lifetime of the emitted signal from the phase difference between the processor reference signal and the data signal.
19. The method of claim 16 wherein said chemometric analysis extracts the fluorescence spectrum of the emitted signal from the amplitude difference between the processor reference signal and the data signal.
20. The method of claim 16 wherein said chemometric analysis further comprises a converging iterative solution.
21. A program storage device readable by a computer, tangibly embodying a program of instructions executable by the computer to perform method steps for a method for measuring the fluorescence lifetime and the fluorescence spectrum, comprising the following method steps:
- generating a driving/reference signal and modulating the amplitude and/or the frequency of the driving/reference signal over time;
- generating a mixing signal and modulating the amplitude and/or the frequency of the mixing signal over time;
- generating an excitation signal from the driving/reference signal;
- detecting the emitted signal;
- mixing the emitted mixing signal with the driving/reference signal and producing the processor reference signal;
- mixing the emitted signal with the mixing signal producing the data signal; and
- extracting the fluorescence lifetime and fluorescence spectrum of the emitted signal from the comparison of the processor reference signal with the data signal to measure using a chemometric analysis.
22. The program storage device of claim 21 wherein the driving/reference signal and the mixing signal vary by an adjustable offset frequency.
23. The program storage device of claim 21 wherein said chemometric analysis extracts the fluorescence lifetime of the emitted signal from the phase difference between the processor reference signal and the data signal.
24. The program storage device of claim 21 wherein said chemometric analysis extracts the fluorescence spectrum of the emitted signal from the amplitude difference between the processor reference signal and the data signal.
25. The program storage device of claim 21 wherein said chemometric analysis further comprises a converging iterative solution.
5196709 | March 23, 1993 | Berndt et al. |
- Weber, G., Resolution of the Florescence Lifetimes in a Heterogeneous System by Phase and Modulation Measurements, J. Phys. Chem., 85, (1981) pp. 949-953.
- Lackowicz, J.R., Analysis of Fluorescence Decay Kinetics from Variable-Frequency Phase Shift and Modulation Data, Biophys. J., 46 (1984) pp. 463-477.
- Burdick, D.S. et al., Resolution of Multicomponent Fluorescent Mixtures by Analysis of the Excitation-Emission-Frequency Array, J. Chemometrics, 4 (1990) pp. 15-28.
Type: Grant
Filed: Nov 9, 2001
Date of Patent: Apr 3, 2007
Assignee: Systems & Processes Engineering Corporation (Austin, TX)
Inventors: Tommy Clay Cruce (Leander, TX), William H. Hallidy (Austin, TX), Robert C. Chin (Austin, TX)
Primary Examiner: Scott J. Sugarman
Assistant Examiner: Richard Hanig
Application Number: 10/035,461
International Classification: G01N 21/64 (20060101);