A multivariate optical computer has been constructed consisting of a spectrograph, digital micromirror device, and photomultiplier tube that is capable of determining absolute concentrations of individual components of a multivariate spectral model. We present experimental results on ternary mixtures, showing accurate quantification of chemical concentrations based on integrated intensities of fluorescence and Raman spectra measured with a single point detector. We additionally show in simulation that point measurements based on principal component spectra retain the ability to classify cancerous from noncancerous T cells.
© 2011 OSA
Raman and fluorescence spectroscopies are two tools that have long histories of being applied to biological problems. Both Raman and fluorescence spectra have been used to discriminate between cancerous and noncancerous tissues [1–3]. The techniques have also been used extensively in applications ranging from bacterial identification [4, 5], hyperspectral imaging [6, 7], and optical tomography [8, 9], to ultrasensitive analyte detection [10–12].
In many cases, the spectra themselves serve as stand-ins for more physical quantities of interest, such as concentration of a particular chemical or fluorescent probe. Even in cases where exact concentrations of known analytes is not of interest, spectra are often decomposed into a linear model using multivariate techniques such as principal components analysis (PCA) or vertex component analysis (VCA), and the concentrations of these multivariate components are used, for example, to descriminate between cells of a given type  or to false-color hyper-spectral images by presumed spectral constituency . Even ordinary least squares analyses have been shown to provide powerful insights into the chemical makeup of cells . In any multivariate technique, the original dataset contains many hundreds or thousands of individual data points for each sample which are then compressed into a relatively smaller number of highly informative points representing contributions of components of an assumed linear model to the original spectra.
In each case, the full spectral dataset can be seen not as an end in itself but as a window to a different, more informative value obtained by treating the spectra as arising from a linear model of a relatively small number of components. In cases such as hyperspectral imaging and multivariate analysis, there is a signal-to-noise price paid by dispersing the light across a CCD and recording a spectrum, rather than recording an integrated intensity value on a single pixel (as would be done in a flow cytometer or some fluorescence imaging applications). This SNR price is particularly important in Raman spectroscopy where signal strength is often the limiting factor in data acquisition speeds. In cases where the components of interest are known a priori, either through careful measurement of several pure chemicals, or through prior decomposition of similar samples, this SNR penalty can be mitigated through the use of a multivariate optical computer (MOC).
A multivariate optical computer is a device which, rather than recording spectra directly, performs dot products between spectra and arbitrary spectral functions, essentially recording their projection onto a particular multivariate axis of interest. This is accomplished by passing the light through an element whose transmission function represents the multivariate vector of interest (called the multivariate optical element), and measuring the total transmitted intensity on a point detector. Previous multivariate computers have primarily focused on utilizing intereference filters to form the multivariate optical elements [16, 17]. In these studies, having the multivariate optical element correctly model the spectral component of interest is a limiting factor , especially for sharp-featured curves typical of Raman spectra. Nevertheless, the principle of multivariate optical computing promises a significant signal to noise advantage over traditional detection of the full spectrum . This can be understood as essentially a form of the well known Felgett’s advantage. This SNR advantage could then be parlayed into significant improvements in acquisition times, particularly for Raman spectroscopy.
Previous attempts at constructing flexible multivariate elements capable of studying sharp-featured spectra have utilized pixellated liquid-crystal-based spatial light modulators to vary the transmission of each wavelength in a dispersed spectrum [20, 21]. However, spatial light modulators have several important drawbacks, most important being the requirement that they operate on linearly polarized light. In this paper we present a multivariate optical computer based upon a spectrograph and digital micromirror device (DMD) acting together as the multivariate optical element. The combination of these two pieces of equipment results in an element whose spectral throughput can be programmed by changing the displayed pattern on the DMD. We use this device to quantify the concentration of fluorescent compounds dissolved in ethanol as well as mixtures of liquid samples by their Raman spectra. Finally, we simulate the performance of such a device in discriminating between cancerous and noncancerous T-cells using the multivariate optical computer to measure principle component scores.
2. Materials and methods
2.1. Mathematical principle of multivariate optical computing
Multivariate optical computing is related to the standard linear least squares modeling of spectra. Often times spectra from complex samples are assumed to be linear superpositions of a small number of pure components, as shown in Eq. 1:Eq. 3 Eqs. 2 and 3 that letting D equal P, the measurements m can be used as a direct measure of the concentration vector c, the quantity of interest as follows,
Alternatively, rather than projecting P onto the DMD, one could equivalently project the full pseudoinverse, B, where B T = P T (PP T)−1. Thus the measurement m would identically equal c. This has one important advantage: if one is only interested in one component of the model, the concentration of component i = c i, then one need only make a single measurement, , where B i is the ith row of B. Contrast this with the case presented in Eq. 4 above, where one measurement must be made for each component of the model, regardless of whether they are of interest or not. The only downside to this technique is that P can be positive definite, while B is both positive and negative. In order to measure the projection of x onto B, then, each row of B must be split into positive and negative components, B + and B −. Thus, for each projection onto B, two measurements must be made and combined. This will be discussed further in Section 3.3.
2.2. Experimental system
The MOC system is shown schematically in Figure 1. Our system, aside from the detection arm, is a traditional Raman/fluorescence microscope. A laser illuminates a sample placed on a microscope stage, and the Stokes-shifted light is sent to the emission path by a filter cube within the microscope. Different lasers were used for fluorescence and Raman studies. The fluorescence studies were performed using a 404 nm laser beam obtained by frequency doubling a 808 nm fundamental coming from a Ti:Sapphire system (Coherent Systems, Santa Clara, CA). Raman studies were performed using a 532 nm laser system (CrystaLaser, Reno, NV). In each case, the laser was bandpass filtered and directed into the microscope with steering mirrors. A 3x telescope, composed of a 40 mm lens (L1) and a 120 mm lens (L2), expands the beam to a diameter of approximately 5 mm to more completely utilize the microscope objective’s full aperture. In the focus of the telescope we place an optical chopper, which modulates the signal at 3 kHz for the purpose of lock-in detection of our emission signal. We used an Olympus IX-71 inverted microscope equipped with a 10x objective with a numerical aperture of 0.3 (Olympus, Center Valley, PA) and beam cubes equipped with dichroic beamsplitters for 404 or 532 nm excitation (Semrock, Rochester, NY). Residual light at the excitation wavelength was further attenuated by a second bandpass filter (passbands of 415 nm-477 nm for 404 nm excitation, and 540 nm-570 nm for 532 nm excitation, both from Semrock, Rochester, NY). The emission light is then focused into a 50 micron core optical fiber using a 40 mm focal length achromatic doublet (L3) and delivered to our DMD-based detection system.
The detection system is based on a design reported by Quyen et al. . Light is coupled by optical fiber into a SpectraPro 2150i imaging spectrograph (Princeton Instruments, Trenton, NJ). However, rather than placing a CCD in the image plane of the spectrograph we reimage that plane onto the digital micromirror device (Discovery 3000, Texas Instruments, Dallas, TX), using a 75 mm achromatic doublet with a 2 inch clear aperture. Depending on whether a pixel of the DMD is in the “on” or “off” state, it reflects light either to a photomultiplier tube (H7826, Hamamatsu Photonics, Hamamatsu City, Japan) or to a beam dump, respectively. The light from the DMD is focused onto the large point detector and into the aperture of the beam dump using identical 35 mm focal length lenses (L5 and L6). The current output of the PMT is passed through a current amplifier (Model 428, Keithley, Cleveland, OH) and then to a lock-in amplifier (SR510, Stanford Research Systems, Sunnyvale, CA) that uses the optical chopper signal as a reference. Although the measurements could be performed without the lock-in amplification step, it provides highly efficient background light and noise rejection and demonstrates one benefit of using a single point detector as opposed to an array detector.
Both the DMD and lock-in are controlled by the computer using a home-built interface running in LabVIEW (National Instruments, Austin, TX). Although the DMD can only display binary patterns, because the DMD can display approximately 16500 independent patterns per second, by integrating signal over several patterns effective 8-bit greyscale patterns can be obtained at 60 frames/sec. We note that more recent generations of the DMD chip are capable of even faster operation. Accessing the highest refresh speeds possible for the DMD was accomplished using a dedicated daughter board connected to the DMD (ALP-3, DLInnovations, Austin, TX) that stores patterns in on-board RAM. Data is then fed to the DMD through an FPGA configured to act as a high speed data link.
2.3. Data processing
For the spectra shown in Figure 2, the spectra were corrected by removing a constant bias from each spectrum and then smoothed using a Whittaker smoother . The Whittaker smoother seeks to minimize a cost function composed of data fidelity (measured by sum of squared errors between the fit and the data) and data roughness (measured by a sum of the squared discrete derivative of the fit vector). The relative weight of the roughness term compared to the data fidelity term in the cost function is given by a Lagrange parameter. For this work, we used Lagrange parameters of 3 for Raman spectra, and 10 for fluorescence spectra. For representation of these curves on the digital micromirror device, the curves were normalized to their maximum value, and scaled to span a range of 0 to 255, and then compressed from doubles to 8-bit variables. The API included with the DMD (DLInnovations, Austin, TX) provides LabVIEW functions to convert the 8-bit data to instructions for displaying these patterns through rapid display of many binary patterns.
For the multicomponent experiments, the obtained voltage values were transformed into concentration predictions by multiplying the measurements by the normalized and scaled matrix PP T as discussed in section 2.1, Eq. 4. The resultant vector of concentrations must then be rescaled to account for the different normalization constants employed in generating the matrix P. If we refer to our originally measured spectral components as a matrix S, and a vector of normalization constants n, then P = n × S, where we use × to represent an element-wise multiplication (i.e. each row of S is multiplied by one element in n). Since the true concentrations we are trying to obtain areEq. 4 differ from c true by 1/n, such that
3. Results and discussion
3.1. Fluorescence experiments
Spectra of three solutions of fluorophores (perylene, coumarin 1, and coumarin 30) dissolved in ethanol are shown in Figure 2a, and were collected by turning individual columns of DMD pixels on in sequence, with each wavelength measurement consisting of an average of 10 voltage readings from the lock-in amplifier set with a time constant of 1 second, current amplifier set with a gain of 1 × 106 and a control voltage on the PMT of 0.7. The spectra have been normalized and scaled to range between 0 and 255 for display on the DMD, as discussed above. The sharp falling edge at approximately 465 nm is due to the falling edge of the bandpass filter used to reject the laser line. Laser power on the sample during these measurements was 35 microwatts.
Using these three spectra as the display patterns for the DMD, we used our multivariate computer to predict chemical concentrations of mixtures of these fluorophores in varying concentrations. We first created stock solutions of each fluorophore and from these three solutions we created two measurement solutions, one consisting only of coumarin 1 and coumarin 30 (solution A), and one consisting only of perylene and coumarin 30 (solution B), where the molar concentration of coumarin 30 is identical for each solution. For each measurement, 1 mL of solution A was pipetted out from the stock and placed in a parafilm-covered measurement chamber (A7816, Invitrogen, Carlsbad, CA), measured 5 times (placing and replacing the sample between measurements to randomize any placement-depending signal variations), and then discarded. Following this, 1 mL of solution B is added to solution A, changing the concentration of perylene and coumarin 1 within solution A, and the process was repeated for a total of 9 samples. A final 10th sample was a measurement of 1 mL of solution B. The results are shown graphically in Figure 3. The x-axis in Figure 3 represents individual samples created using the method described above, with the solid lines showing the nominal concentration of each analyte within each sample. Individual points (circles, stars and triangles) represent the mean predicted values for each analyte within each sample, with the error bars being the standard deviation across the 5 measurements. The dashed lines represent the estimated accuracy of the pipettes which limit the ultimate accuracy of our reference concentrations. Because of the huge increase in signal gained by putting photons from the entire spectrum onto a single detector, the laser had to be attenuated with an OD 2 filter to prevent saturation of the lock-in amplifier. Thus these measurements were conducted with only 350 nanowatts of power at the sample. The results clearly show the ability of the DMD-based MOC to accurately quantify concentrations of fluorophores using only a small amount of power at the sample. This could significantly aid applications where photobleaching due to high laser powers is an issue. We also anticipate that this system could be used in point-scanning hyperspectral fluorescence imaging to perform direct fluorescence unmixing and using the gain in SNR to reduce pixel dwell times and speed up overall image acquisition.
3.2. Raman experiments
Similar to the fluorescence experiments, pure spectra of toluene, benzene, and tetrahydrofuran were acquired first, shown in Figure 2b, using 40 mW of 532 nm light, using the same lock-in amplifier parameters as above. Multicomponent mixtures were also generated in an analogous manner to fluorescence samples. However, due to the hazardous nature of the chemicals used in this study, a sample chamber was constructed consisting of a coverslip cemented to a threaded brass fitting into which a teflon-taped threaded stopper could be screwed to prevent evaporation of the sample during measurement. Two solutions were created, one with only toluene and tetrahydrofuran (solution A), and one with only tetrahydrofuran and benzene (solution B), with the concentration of tetrahydrofuran kept constant across the two solutions. 1 mL of solution A was placed within the sample chamber and measured identically to the fluorescence experiments. After measurement, 100 microliters of solution B was added to the 1 mL of solution A already in the sample chamber. Due to the fact that the sample chamber can only hold approximately 1.6 mL of liquid, samples 1 through 5 were obtained by progressively adding B to A, while samples 6 through 10 were obtained by progressively adding A to B. The results are shown graphically in Figure 4, with similar labeling as in Figure 3. Once again we attenuated the laser by a factor of 100 for these measurements, resulting in a power of 400 microwatts on the sample. The system is clearly able to accurately quantify the concentration of each chemical in the mixture. The accuracy of the concentration predictions is slightly lower here than in the fluorescence experiments, but we believe that this is primarily due to the fact that the laser we used for this experiment has a slow variation in output power of approximately ±5%, which directly affects the accuracy of the predictions.
3.3. Simulation of T-cell sorting
In the above sections we created samples composed of known concentrations of pure components, and the patterns projected on the DMD represented physical spectra of these components. However, the technique is considerably more general. It is often the case that although the exact chemical constituents of a sample may not be known a priori, the spectra from many samples can be analyzed post hoc to determine meaningful regression vectors through partial least squares, principal components analysis, or other multivariate techniques. In this simulation we take Raman spectra from 90 individual T lymphocytes, evenly split between normal and neoplastic cells. The details of this dataset and its acquisition can be found in . A representative spectrum from a single cell is shown in blue in the top left panel of Figure 5. The full Raman dataset is submitted to principal components analysis. As described in , cancerous and noncancerous cells can be separated based on the first and second principal component scores. In the bottom panel of Figure 5, we plot each cell as a point in a two dimensional space whose coordinates are given by their first two principal component scores, showing the separation between cancerous and noncancerous cells. Magenta stars and green circles represent the results of the principal components decomposition using the full spectra as reported in , with the stars representing normal cells and circles representing Jurkat cells (an immortalized neoplastic T-cell line). Notice that with very few exceptions, the normal and neoplastic cells cluster into distinct regions of the principal component space.
If instead of measuring the spectra themselves we use the multivariate optical computer to return the scores directly, we can make faster measurements less sensitive to noise. The principal components model can be thought of as a linear model representing a mean centered spectral dataset. In other words,
To illustrate the robustness of this method to noise, we simulated a very weak spectrum entering our spectrometer. If we assume that the dominant noise source is shot noise, the spectrum that would be detected on a CCD, given the simulated number of photons per wavelength channel, is shown as the red curve in the upper left panel of Figure 5. Despite the fact that few photons are falling on each wavelength channel, the total number of photons across all wavelength channels is still relatively large. Therefore, our measurements will have a better signal to noise, since they integrate a large fraction of the incident photons into a single bucket. A careful comparison of the signal to noise ratios in each case is provided in Appendix A. We simulated principal component values calculated using our optical computer based on Eq. 9 and projecting the noisy data onto the first two rows of B, shown in the upper right panel of Figure 5. The calculated principal component scores are shown in the bottom panel of Figure 5 as red stars and blue circles. In the limit of no noise, the measurements from the CCD and optical computer would lie on top of each other. The added noise moves the optically computed values away from their true locations. However, as is clear from the figure, despite decreasing the number of photons by 100 times compared to the original data, with a corresponding 10-fold increase in shot noise, the estimated principal component scores are still accurate enough to permit discrimination between cancerous and noncancerous cells. This analysis also does not include the potential of lock-in amplification to improve the signal to noise ratio even further, as discussed in Appendix A. However, even in this case, the separation between the normal and neoplastic cells within the principal component space is clearly maintained. Thus, we expect such a system could be used to substantially speed up Raman-activated cell sorting or high-throughput screening.
We have reported on the construction and validation of a multivariate optical computer using a digital micromirror device and spectrometer as the multivariate optical element. By projecting grayscale patterns onto the DMD, we were able to record dot products between sample spectra and the displayed pattern that were used to quantify chemical concentrations within ternary mixtures. Although previous instruments have been constructed using multilayer stacks and liquid-crystal-based spatial light modulators, the DMD has several key advantages. Multilayer stacks offer a compact and rugged design, but suffer from being difficult to manufacture and being completely inflexible once constructed. With respect to spatial light modulator-based MOCs, because the DMD is increasingly used within consumer electronics, the cost of a DMD chip is approximately 2 orders of magnitude less expensive than a spatial light modulator while maintaining higher throughput due to the polarization and wavelength insensitivity of the device.
We further explored the possibility of separating normal from neoplastic cells based on principal component scores measured directly using the MOC. Despite reducing the number of photons measured by 100, the direct measurements of the component scores do not differ greatly from those computed from measurements of full spectra. Neoplastic and normal cells still cluster into distinct groups within the principal component space. Since our mixture experiments indicated that there is an increase in signal of approximately 100 times using the projected spectra versus a single pixel of a spectrum, and given the simulation results presented in Figure 5, we expect that our MOC system could determine if an individual cell was normal or neoplastic approximately 100 times faster than a traditional dispersive-based system (with the caveat that the relevant principal components and mean spectrum must be known a priori).
Given our results, we can also provide an order-of-magnitude estimate of detection limits for our current configuration based on the results obtained thus far. For the fluorescence experiments we were easily able to quantify micromolar concentrations of fluorophores in solution. Given that we attenuated our laser beam by 2 orders of magnitude, used a moderate objective NA of 0.3, and a control voltage on the PMT of 0.7, we estimate that there is likely 4 to 5 orders of magnitude improvement possible in our signal strength, giving us at least nanomolar sensitivity to fluorophores. Given a diffraction limited focal volume of a few femtoliters, this corresponds to nearly the single molecule limit. For Raman scattering the results are similar. We likely have 4 to 5 orders of magnitude of signal improvement, which would give us micromolar sensitivity, depending on the cross sections of the molecules of interest.
Beyond considerations of signal-to-noise, in a case where signal strength is not of primary concern, it is worth noting that the speed of the single point detectors used in this device permit detection much more rapidly than an equivalent CCD-based system. As CCDs are typically limited to a few kilohertz frame rates, while as discussed in Appendix A, the PMT used here can detect signals as fast as 600 MHz.
Extending this work to measurements of biological samples, including sorting and hyper-spectral imaging applications, are directions our group is actively pursuing.
A. Appendix: Signal to noise of multivariate optical computing measurements
Although signal to noise of a measurement is highly system dependent, we can make an order of magnitude estimate of the improvement in signal to noise of the constructed system versus other configurations or versus a traditional CCD-based spectrometer. Because the described system sees its greatest advantage when signal strengths are low, consider a signal where 10 photons are incident on a single wavelength element (pixel) of a CCD in a time of 0.1 seconds. Assuming the CCD has a quantum efficiency of 90%, the signal will produce 9 photo-electrons with a variance due to shot noise of 3. The photo-electrons are digitized by an A/D converter with a gain of 2 and a read noise of approximately 5 counts, giving a total signal of 18 counts with a noise variance of 8. Each pixel of the recorded spectrum (and thus the spectrum as a whole) would have a SNR of approximately 1.6.
Now consider a case where a multivariate optical computing system is described above, but, as in , we simply integrate the output of the single point detector over a period of 0.1 seconds. Quyen et al. accomplish this with an integrating PMT, where the electrons from the anode are stored at a capacitor for a specified integration time, and then the voltage across the capacitor is read. In this case, assuming 1024 pixels on the DMD, each receiving on average 10 photons per 0.1 seconds, with a reflectivity of the DMD of 80% and a pattern that on average reflects 50% of the photons incident on the DMD, the total number of photons incident on the photomultiplier tube would be 4096. Our detector has a peak quantum efficiency of 21% meaning that the total number of photoelectrons produced is 860, with a variance of 29 due to shot noise. The photomultiplier tube has a dark current of 3 nA, corresponding to 1.8 × 1010 electrons per second at the anode. Assuming the dominant source of noise in the PMT is thermionic emission of electrons at the cathode, and given that our detector has a gain of 0.6 million, the rate of emission of dark electrons at the cathode is 30 kHz. Thus, in 0.1 seconds, 3000 photoelectrons will be emitted with a variance of 54. Assuming the voltage reading and subsequent A/D conversion in the PMT is noiseless, the total signal will be 860 counts with a variance of 83 due to Poisson noise, giving an SNR of 10, or nearly a 10 times improvement compared to using a CCD, with the obvious caveat that full spectral information is lost.
We next consider a case where rather than use an integrating PMT, we take the current output of the PMT and send it through a lock-in amplifier. In this case we once again have 4096 photons incident on the PMT, of which 21% are detected, leading to 860 photoelectrons with a variance of 29. After a gain of 0.6 million this produces a signal current of .86 nA with a variance of .029 nA. Once again assuming that the 3nA of noise is due to a 30 kHz dark electron emission rate, the variance on the dark current is 0.054 nA, to give a signal of 0.86 nA with a variance of 0.083 nA. This is then sent through a current amplifier with a noise density of 1 nV/ with an amplification of 1 MV/A, giving a signal of 860 microvolts with a noise of 83 microvolts. The detector has a rise time of 1.5 nanoseconds, giving it a maximum response frequency of 666 MHz. Assuming the noise on our signal is white, this means that the power spectral density of the shot noise is 3.2 nV/ with an additional 1 nV/ from the amplifier. If we send the signal into a lock-in amplifier, we can reject a large fraction of this noise by placing a narrow bandwidth filter on the signal around the reference frequency. WIth a 0.1 second time constant set on the lock-in amplifier, the width of the lock-in’s filter is 2.5 Hz. Therefore, the total noise voltage leaking through the lock-in is 6.6 nV. The lock-in itself has about 11 nV of noise in this bandwidth, giving a total signal of 860 microvolts, with a noise of 17.6 nV, for an SNR of about 48000. The lock-in amplifier, by reducing the noise by a factor of close to 5000, has dramatically improved the signal to noise ratio of the measurement, enabling extremely precise measurements and thus higher accuracy when predicting concentrations of components of the spectral model.
This work was funded by NSF award IDBR 0852891. Part of this work was also funded by the Center for Biophotonics Science and Technology, a designated NSF Science and Technology Center managed by the University of California, Davis, under Cooperative Agreement No. PHY0120999. We gratefully acknowledge Agilent Technologies, Inc. for their loan of the DMD chip, and we thank Gerry Owen and Chris Coleman of Agilent for helpful discussions. We also thank James W. Chan for sharing the T-cell data utilized in this paper and Kaiqin Chu for helpful discussions.
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