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Upon excitation with different wavelengths of light, biological tissues emit distinct but related autofluorescence signals. We used non-negative matrix factorization (NMF) to simultaneously decompose co-registered hyperspectral emission data from human retinal pigment epithelium/Bruch’s membrane specimens illuminated with 436 and 480 nm light. NMF analysis was initialized with Gaussian mixture model fits and constrained to provide identical abundance images for the two excitation wavelengths. Spectra recovered this way were smoother than those obtained separately; fluorophore abundances more clearly localized within tissue compartments. These studies provide evidence that leveraging multiple co-registered hyperspectral emission data sets is preferential for identifying biologically relevant fluorophore information.
© 2014 Optical Society of America
1. Introduction
Non-negative matrix factorization (NMF) is an unsupervised machine learning technique that has been previously applied to hyperspectral data for recovering constituent source spectra and the spatial distributions of these components [1–4]. Traditionally the NMF algorithm initialized with random spectra converges to a solution by minimizing an error criterion under either a constraint or update rule that enforces only non-negativity. Thus no information about the structure of the spectral shape of the components is typically employed in the basic NMF algorithm. However, biological fluorophores that are histologically and clinically meaningful are expected to have emission spectra with relatively smooth single peaks [5]. As a consequence, our strategy was to initialize the NMF algorithm with an estimated Gaussian mixture of components before letting it minimize the error on its own. To validate this approach, the results were compared to NMF runs initialized with random spectra.
The model tissue in these studies, the retinal pigment epithelium (RPE), is a single layer of cells in the eye. The RPE emits a strong autofluorescence signal used in diagnostics and clinical management of retinal disease. The excitation of RPE autofluorescence with multiple wavelengths gives rise to different but closely related spectral data emitted from the same cellular structures. Thus, for each single fluorophore in a tissue, the spectral responses will differ depending on the excitation wavelengths, but they are related in the simple sense that they come from the same compound. Therefore, these relationships apply to the total signal, which is the sum of these related parts, in which spectra from different excitations are recovered in related pairs from closely related groups of compounds. To exploit these hidden relationships, we studied the simultaneous decomposition of two hyperspectral data sets into major spectral signatures with identical spatial distributions and compared the results to those of factoring any single hypercube.
2. Theory: Spatially constrained simultaneous NMF of multiple related hyperspectral data sets
When tissue is excited at a given wavelength lambda (λ), hyperspectral emission data are acquired as an M × N hypercube Xλ, where M is the number of pixels per image (dimension of spatial information) and emission data are captured from each pixel at N wavelengths (dimension of spectral information). Standard NMF then factors Xλ into the product of matrices Aλ and Sλ, as in Eq. (1):
where Sλ is a K × N matrix that carries K spectra (the major recovered sources), each of which is a feature vector of emissions at N wavelengths, and Aλ is an M × K matrix that carries their spatial localizations (abundance images). K, the expected number of major spectra, is usually specified by the user depending on additional information, as described in Section 3. Intuitively, the NMF method attempts to separate the totality of emission data, which cannot be visualized, into two blocks of data, spectral and spatial. These blocks not only suggest answers to the two questions of interest — i.e., what are the strongest spectra in the total emission, and where are they coming from spatially — but are easily visualized as a set of spectra and another set of corresponding abundance images.In our formulation, an adaptation of non-negative tensor factorization (NTF) [6–8], we consider n such data sets acquired from the same tissue at excitation wavelengths λ1 … λn and assume that, based on evidence from pre-existing models or other conditions, we seek a fixed number k = K of emission spectra Sλ = [sλ1, sλ2,… sλk]T for each λ. Each sλi is thus a column vector representing the ith spectral source from excitation wavelength λ, and the elements sλi are naturally ordered by increasing peak wavelength. We further assume that these spectra are related by having the same sources for each j, where 1≤j≤k. More precisely, we assume that for each j, where 1≤j≤k, the spectral emissions sλj for all excitations λ derive from the same molecular source, which could be a single compound or a closely related family of compounds. The basis for this assumption in the present context is the known excitation/emission behavior of the RPE fluorophores, which are bisretinoid compounds. The emission peaks tend to be quite broad; likewise, the excitation spectra, while certainly showing clear maxima, also do not drop abruptly to zero [9]. For example, the well-studied bisretinoid A2E has emission maxima of comparable intensity near 600 nm for excitations at 436 and 480 nm. Hence, it must follow that, if one of two recovered signals near 600 nm had a contribution from A2E, then there would be a comparably sized contribution from A2E near 600 nm in the other. Likewise, if a family of fluorophores produced a combined signal near 600 nm for one excitation, it is reasonable that the same family would contribute to a combined signal near 600 nm for the other. Further, the order of the spectra would tend to be maintained (e.g., those excited at 480 nm would tend to be red-shifted with respect to those excited at 436 nm). In the language of the NMF decomposition, the spatial source distributions Aλ of these signals must then be constrained to be exactly the same, because they come from the same compound or family of compounds. We can thus write the factorization as in Eq. (2):
where all excitation data sets are concatenated into a single M × N × n dimensional hypercube on the left. On the right, the solution A is an M × K matrix that is the same for all λ, and the recovered sources for each excitation are also concatenated as a single K × M × n matrix. We then solve for A and [Sλ1, Sλ,… Sλn] with an appropriate standard NMF routine, such as alternating least squares, which we use in our examples. We will refer to this solution interchangeably as the concatenated NMF or the NTF solution.3. Application: Fluorophore signal recovery in the human RPE
The RPE is a monolayer of pigmented epithelial cells directly exterior to the photoreceptors of the neural retina [10]. It rests on Bruch’s membrane (BrM) [5, 11]. This five-layered extracellular matrix functions as both the substrate for RPE attachment and as a vessel wall at the inner aspect of the choroidal vasculature that nourishes the RPE and photoreceptors [5]. The RPE is considered to be central to the initiation and progression of age-related macular degeneration, a major cause of vision loss in the elderly worldwide. The RPE is responsible for generating vitamin A derivatives required for phototransduction, the initial steps of vision, through a series of biochemical reactions called the visual cycle. Byproducts of the visual cycle are thought to aggregate in the lysosomal compartment of the RPE [12] as lipofuscin, which has an intense fluorescent signal [13–15]. Because this signal comes from endogenous substances, rather than exogenously introduced fluorescent markers, it is referred to as lipofuscin autofluorescence (AF). For the purposes of this paper, it suffices to say that, first, lipofuscin AF presents a single broad emission spectrum, which is believed to be a sum total of multiple constituents. Second, knowing the true molecular constituents of this peak is considered vital to understanding the role of the RPE in health and disease [16]. The fluorescent bisretinoid A2E was long considered to be dominant in the RPE, because it was abundant and well characterized, and it was considered the major component across the entire RPE layer [17]. In 2013, imaging mass spectroscopy (IMS) revealed that A2E has strong regional variations in its tissue distribution, casting doubt on its role as a major disease initiator [18] and suggesting that additional compounds must be considered. There is extensive literature on bisretinoid biochemistry and the search for candidate RPE fluorophores [19, 20]. In addition to A2E, others reported in human tissue include DHP-A2-PE [21], A2E-DHP-PE [21], A2 GPE [22], monofuran-A2E [23, 24], monoperoxy-A2E [23, 24], iso-A2E [24], atRAL dimer [24], atRAL dimer PE [24], and atRAL dimer E [24]. Candidates localizing with lipofuscin in human RPE have also recently been identified by IMS [25]. Those in the A2E family mostly have emissions near 600 nm across a broad range of excitations, and the emissions of the atRAL family are all at about 525 nm when excited at 430 nm [12]. Thus, RPE emission signals at these wavelengths might well be combined emissions of members of one of these families. Likewise, the individual spectra retrieved by hyperspectral analysis of RPE could also be combined signals whose components are too similar to separate at the present instrumental resolutions, but these spectra could suggest what family of fluorophores is present.
At the present point in RPE research, the mixture of compounds responsible for macular RPE AF is uncertain. As noted, prior literature suggests that a major macular fluorophore is A2E, but we now know that A2E is present here in small quantities only [18, 26]. Hence, consideration must also be given to other candidates, including the known species just listed and others yet to be discovered. The goal of our studies here is not, and cannot be, to solve this detailed molecular problem. Rather, the goal is to show that NMF and NTF methods are capable of extracting plausible, abundant fluorophore signals that can guide further research with techniques that are capable of precise molecular identification, such as IMS, a newly developing and clinically important domain. Speculation on individual compounds in specific locations represented by particular signals is not yet warranted. Nevertheless, matching AF signals to families of compounds, as just discussed, might be a useful starting point for future research in this historically challenging domain.
The AF spectra of 20 flatmounts of human RPE attached to Bruch’s membrane (RPE/BrM) from donors without any retinal pathology were acquired and measured. The flatmounts were prepared as previously described [27]. From chorioretinal tissue, retina and choroid were carefully removed to prepare 20-µm-thin RPE/BrM flatmounts. During tissue preparation, images were taken at every preparation step to maintain the position of the fovea of the retina, the site of high-acuity central vision.
Three different locations on the tissue were chosen for our measurements (distances relative to the fovea): fovea, 2 mm superior (perifovea), and 10 to 12 mm superior (periphery). The RPE in these locations is distinguished by the photoreceptor population in the overlying retina: cone photoreceptors only, highest rod photoreceptor density, and highest rod/cone photoreceptor ratio, respectively. Each location from each of the 20 donors was imaged and analyzed, making 60 tissues studied in all.
Microscopy was performed using the Zeiss Axio Imager A2 microscope, equipped with a 40X oil lens (NA = 0.75) (microscope and lens: Carl Zeiss, Jena, Germany) and two filter cubes (filter 1: 436/460 nm excitation/long pass emission; filter 2: 480/510 nm excitation/long pass emission; Chroma Technology Corp., Bellows Falls, VT, US) and connected to an external mercury arc light source (Xcite 120Q, Lumen Dynamics Group, Inc., Mississauga, Ontario, Canada). For brevity, we refer to the two excitations as 436 nm and 480 nm. Thus, a total of 120 tissue data sets in all were acquired.
At each location, two hyperspectral data cubes of RPE and BrM were acquired using the two different microscope filters and a hyperspectral camera (Nuance FX, Caliper Life Sciences, Waltham, MA, US), with measurements made at 10 nm intervals between 420 and 720 nm, and 510 and 720 nm, respectively. All data were recalibrated with respect to the spectral sensitivity of the camera, which was nearly linear from 450 to 700 nm, the range of interest (Fig. 1). The data in the smaller spectral range were padded with zeroes to create hypercubes with the same spatial and spectral dimensions. Each raw data cube was saved using the integrated software (Nuance 3.0.1.2) and exported for further NMF analysis.
Fig. 1 Quantum efficiency (QE) of the Nuance camera spectral detector (arbitrary units) supplied by the manufacturer (Caliper Life Sciences). The QE is approximately linear from 400 to 700 nm, with some small shoulders, and then drops at 700 nm. Since both RPE and BrM have intrinsic autofluorescence properties, and RPE anatomically overlies BrM, the hyperspectral data cubes captured the sum of the RPE signal and a portion of that from the underlying BrM. Hence, to assist in identifying the pure RPE spectrum at each location, a pure BrM signal without overlying RPE cells was also recorded separately from areas where a few RPE cells were dislodged during preparation. For locations at which the RPE monolayer was completely intact, a pure BrM signal was separately imaged at an adjacent area.
4. Example: RPE spectra from two excitation data sets
4.1. Step 1: Pure RPE spectrum separated from underlying BrM spectrum
The emission signal from a patch of pure BrM, read in tandem with that from RPE, was subtracted to give a net signal from the RPE itself. However, the emission from BrM is also screened by overlying RPE melanin. More precisely, the average optical density of the RPE is about 0.30 DU at 500 nm, and the absorption by melanin decreases slightly with increasing wavelength [28, 29]. Hence, the two-pass absorption by RPE melanin may be approximated by that at 500 nm in our system, or about 75%, with the result that only 25% of the pure BrM signal is included in that read from intact RPE overlying BrM. The correction for this component is illustrated in Fig. 2.
Fig. 2 The pure spectrum of the RPE. Left: RGB composite AF image from a 47-year-old female donor, excitation 480 nm, perifovea. Sample raw spectral data (photon counts) were acquired in regions marked. Green mark: BrM in isolation. Red marks: RPE cells containing lipofuscin overlying BrM. Right: Separated emission curves. Green: isolated BrM. Red: RPE overlying BrM, which includes 25% of the pure BrM signal. Magenta: pure RPE signal (after subtraction of 25% of the BrM signal).
4.2. Step 2: Gaussian mixture modeling
For the pure RPE spectrum at each location for each donor (Fig. 2), Gaussian mixture modeling was applied to extract four single-peak, smooth candidates for spectral components (Fig. 3). It is important to note that we did not have prior knowledge of how many abundant signals might be present. We first observed that recovered spectra usually contained three to four peaks or shoulders, with the most consistent centered at approximately 560, 600, and 640 nm (Fig. 3, arrows), and a variable shoulder elsewhere. We then fit each spectrum with four Gaussians using a custom MATLAB program that allowed centers and amplitudes to vary for best fit. Three of these Gaussians also generally had peaks near 560, 600, and 640 nm, and the fourth Gaussian peak was variable. We also found empirically that solutions using three or five Gaussians were unsatisfactory or redundant. To rule out the possibility that these secondary peaks/shoulders were of instrumental origin, we obtained the emission spectra of pure A2E in phosphatidylcholine liposomes on BrM at both wavelengths on our system, both with and without correcting for the QE, and no secondary peaks or shoulders appeared (data not shown). As the spectrum of pure A2E is known to be smooth [12], we reasoned that any significant systematic error would have been revealed in these spectra, especially in the ranges where the signals were strong.
Fig. 3 Gaussian fits to a sample RPE spectrum. The pure RPE hyperspectral data from Fig. 2 (black line) were calibrated to the acquisition time of 18 ms and instrument gain of 3 to yield spectral intensity in units of photons/sec and then fit to the four Gaussian components of the mixture model. The arrows indicate two peaks and two slight shoulders in the original spectrum. The mixture model (sum of four Gaussians) is the solid magenta line and largely overlies the original RPE data (an overall excellent fit). Note that the centers of the Gaussians, especially Gaussian 3 (red), do not necessarily coincide with the original peaks in the RPE data. The dotted magenta lines are the 95% confidence prediction bounds of the model under the assumption of 5% random error in the original RPE spectrum.
4.3. Step 3: NMF of single excitation hyperspectral data sets
NMF was used to decompose the RPE emission hypercubes from each excitation wavelength. In addition to random initialization, we adapted the NMF technique to include a supervision step, which initialized the spectra as the four individual Gaussian curves in the mixture model, as just described. We also initialized the NMF with a fifth signal, the emission signal acquired from a region in which BrM had been completely exposed. Because BrM underlies the RPE, the total RPE emission spectrum includes this BrM signal as a component, as noted in Step 1. Thus, five spectra in all were sought by the algorithm. It should be noted that the recovered signals C1 to C5 from NMF are labeled by the MATLAB software in order of signal strength, not in order of the original Gaussians used to initialize the program. This could cause confusion. For signal analysis, however, this is a hard-coded, logical output, and so we have chosen to maintain it.
NMF initialized with random spectra recovered spectral components that were sometimes jagged, contained numerous peaks, and were not readily interpretable given the known histology of these samples. However, decomposed spectra derived from NMF, when initialized with the Gaussian spectra estimated from the mixture model fit, were generally smoother and contained fewer peaks, thus producing a solution that is more physiologically plausible.
We created two-dimensional abundance images of the spectra that could be directly compared with the original image of the tissue, and these also showed correct histological correspondence (Fig. 4). Thus, the four RPE sources all localized to areas surrounding the nuclei in a manner characteristic of lipofuscin and melanolipofuscin in organelles that are known to be autofluorescent [13, 30]. Confirmation of subcellular fluorophore attribution awaits further investigation with higher resolution microscopy techniques. A fifth spectral component representing the known emission signal for BrM corresponded to abundance images that highlighted regions of exposed BrM (Fig. 4, panels A, C).
Fig. 4 RPE flatmount, 90-year-old female donor, perifovea. A, (B), spectra recovered from 436 nm excitation; (C), (D), spectra recovered from 480 nm excitation. (A), (C), decomposed spectra from individual excitation data sets; (B), (D), spectra from simultaneous solution of both excitation data sets. The tissue image is a full 40X field. The five individual spectra in each set are labeled C1 to C5. The corresponding abundance images are also labeled C1 to C5, with false coloring to indicate the relative signal intensities. The spectra in (A), (C) have multiple subsidiary peaks, suggesting contributions from multiple sources. Those from 480 nm excitation are particularly jagged, while two signals from 436 nm excitation are nearly degenerate (C4, C5). The spectra in (B), (D) are all broad, as expected from known fluorophore data, and can be paired by shape and location. In particular, there are no degenerate solutions in the simultaneous solutions. The recovered paired spectra are smoother than either spectrum recovered separately, with more congruent shapes. The lower right element of each panel is the composite RGB image from the total AF signal for that excitation. The constrained identical abundance images on the right for each pair of spectra show precise anatomic detail and are more clearly defined than the abundances recovered individually; hence they are more consistent with well-defined species of emitter. For example, C3 is more specifically localized to isolated BrM than its counterparts in the individual cases, at both excitation wavelengths. Signals from RPE cells (C1, C2, C4, and C5) can be distinguished from each other by the relative size of the signal-poor region (blue) in the center of each hexagonal RPE cell in the concatenated solutions, whereas such distinctions between the abundances in the individual solutions are slight. C1-436 and C2-480, similarly shaped, are linked, initialized with Gaussians at 600 nm; C3 is BrM in each; C4 and C5 are linked in each; C1-480 is linked to, and red-shifted from, C2-436.
4.4. Step 4: Spatially constrained simultaneous NMF of multiple excitation hyperspectral data sets: non-negative tensor factorization (NTF)
As described above, the RPE/BrM flatmounts were excited at both 436 and 480 nm, and hyperspectral emission data were captured for both wavelengths. In Step 3, we retrieved four spectral signatures for RPE and one for BrM for each data set. We postulated that each signal found at 436 nm excitation was paired to a signal at 480 nm and that the spatial source distributions of these signals must be exactly the same, because they come from the same compound or family of compounds, as discussed in Section 2. Hence, we linked the two data sets for NMF with these constraints, as described in Section 2 with n = 2 and k = 5 in this example. Figure 4, panels B, D, shows the corresponding spectra found with simultaneous solution of the concatenated data sets. The solutions were initialized with the same Gaussians and BrM spectra used for the individual NMF solutions. The five recovered spectra for each wavelength are labeled C1 to C5 by MATLAB, and, as noted earlier, the order no longer conforms to that of the input Gaussians. In particular, the orders for the two excitation wavelengths may not be the same. However, the concatenated solutions are still internally linked and recovered in pairs, and so the pairing is always known. Thus, for example, a statement such as “C2-436 and C1-480 are linked, initialized with Gaussians at 600 nm” means that the recovered spectrum C2 in the 436 nm excitation data set is linked to C1 in the 480 nm data set, in that they were both initialized with Gaussians near 600 nm (see Fig. 4, legend, last sentence, for examples).
4.5. Results
We subjectively defined the improvement of spectral recovery by NTF compared to individual NMF as significant, moderate, or none by (a) the degree to which the spectra became less jagged (“jagged” being defined as having sharp peaks and minima); (b) the degree to which the spectra became more single-peaked; and (c) the total number of spectra that became “better” by the two previous definitions. In Fig. 4, panels C, D, among spectra recovered from 480 nm excitation, all signals except C2 are significantly improved in the NTF solution (D) over those in the individual NMF solutions (C); thus, the NTF solution is judged significantly improved from the NMF solution. Fig. 5 shows examples of moderate and no improvement.
Fig. 5 Graded improvement in spectral recovery. Top: Moderate improvement in spectral recovery with concatenated NMF. Jagged C2 and C3 are replaced by smoother C3 and C4, both significantly improved. The other signals are not improved. NTF improvement is graded moderate. Bottom: No improvement. C5, almost degenerate, regains amplitude with NTF, but C4 changes from a single peak to two. Net improvement is graded none.
Degenerate solutions, with fewer than five spectra recovered, occurred only in solving single excitation data sets. Interestingly, spectral recovery was most improved for the 480 nm excitation data sets. Precisely, for 436 nm, 30/60 (50%) showed significant improvement; 20/60 (33%) showed moderate improvement; 4/60 (7%) showed no improvement; and 6/60 (10%) became worse. For 480 nm, 54/60 (90%) showed significant improvement; 4/60 (7%) showed moderate improvement; 2/60 (3%) showed no improvement; and none became worse. In most cases, recovered signals had peaks near those of the initial Gaussians, and they generally were much broader, as would be expected from actual fluorophores. Also, in every case, a clear signal from BrM that localized correctly was recovered.
For the four RPE emission signals, the most consistent finding was a pair of signals in the 500 nm range at each excitation wavelength. Those recovered from excitation at 436 nm tended to peak between 525 and 540 nm and between 550 and 575 nm, and the corresponding signals recovered from 480 nm excitation were red-shifted by 20-30 nm (see Fig. 4, panels B, D, spectra C1, C2). Interestingly, these two signals were often detected even though only one Gaussian was used initially in this range, which reflects the power of the NMF system to find hidden structure despite the initial conditions. Secondary peaks or shoulders were commonly seen at 600 nm at both excitation wavelengths in the perifovea and periphery (Fig. 4, panels B, D, spectra C1, C2, respectively), although distinct signals at 600 nm were less common. Likewise, secondary peaks or shoulders were often seen at 640 nm at both excitation wavelengths (Fig. 4, panel D, spectrum C4; Fig. 5, lower right, spectrum C1), but distinct signals at 640 nm were uncommon. A broad, low signal around 700 nm was also fairly common (Fig. 4, panels B, D, spectrum C5), although the signal tended to increase at 700 nm, suggesting a peak further into the infrared.
In summary, a total of four RPE spectra were variably recovered at 436 nm excitation from the 60 tissues, with peaks or shoulders at about 530, 550, 600, and 640 nm; comparable signals were recovered at 550, 575, 600, and 640 nm at 480 nm excitation. The two shorter-wavelength spectra usually had distinct peaks, whereas the longer-wavelength spectra tended to be represented by secondary peaks or shoulders, reminiscent of the original pure RPE curves and suggesting that spectral separation was still incomplete and/or that more sources were present. Thus, RPE emission signals at these wavelengths might well be combined emissions of members of one of these families, whose components are too similar to separate at the present instrumental resolutions, but which could still suggest what family of fluorophores is present. For example, concerning the emission at 600 nm, A2E must be considered. However, in the macula, where A2E is present in small amounts only [18], other compounds may be contributory, such as the A2E family reviewed earlier.
5. Example: Bruch’s membrane spectra recovered from two excitation data sets
Here we focused on signals from a different tissue, isolated BrM, excited with the same two wavelengths, 436 and 480 nm, in four tissue samples. Analysis of one tissue sample is presented. The main purpose was to show that the NTF technique can be applied to a different tissue, with similar improvement in signal recovery. The NMF was initialized in each case with three Gaussians (k = 3 was found empirically to be the best choice) that were fit to the BrM total emission spectrum at each excitation (Fig. 6, BrM emissions at both excitations, Gaussians not shown), analogous to the Gaussian fits to the RPE signal (Fig. 3). In each case, three abundant spectra were recovered. All six Gaussians were then used to initialize the concatenated NMF. There was significant improvement in the quality of the recovered signals and the spatial specificity of the corresponding abundances in the concatenated solutions compared to the individual NMF solutions for each excitation data set. Thus, the three abundance images (constrained to be identical for each excitation) showed significant spatial separation of the three recovered signals, a quality that was not present in the abundance images recovered separately. This suggests that these signals are more accurate representations of separate sources, with differing localizations, than the signals retrieved from the separate NMF solutions (Fig. 7, all solutions and detailed analysis).
Fig. 6 BrM total emission spectra; perifovea of a 78-year-old female donor; excitations at 436 and 480 nm. Each signal was fit with three approximately evenly spaced Gaussians (not shown) in a manner similar to that in Fig. 3.
Fig. 7 Isolated BrM; perifovea of 78-year-old female donor. (A), (B), spectra recovered from 436 nm excitation; (C), (D), spectra recovered from 480 nm excitation. (A), (C), decomposed spectra from individual excitation data sets; (B), (D), spectra from NTF, simultaneous solution of all three concatenated excitation data sets. Three abundant signals are recovered with NMF decomposition of each individual excitation data set and with the concatenated solution. The corresponding abundance images are C1, C2, and C3 in each set. False coloring indicates the relative intensities of the signals. The lower right panel in each set is the composite RGB image from the total AF signal for that excitation.
Figure 7, panels A, B, show signal recovery with 436 nm excitation. The spectra in A are all noisy but broad, with dominant peaks. However, the signals in B, recovered by concatenation of data sets (i.e., NTF), are all smoother than their counterparts recovered from NMF of the 436-460 nm excitation data alone (C2, C3 in B are the counterparts of C3, C2 in A, respectively). The abundance images all show the intercapillary pillars of BrM as a dotted pattern, but they are better defined and more spatially differentiated by NTF than the individual NMF. The dotted pattern is due to the structure of BrM, which can be analogized to a wide deck over a blood-filled lake (choriocapillaris). The deck is supported by regularly spaced thickenings called “intercapillary pillars,” which appear as dots when viewed en face, as in this figure.
Figure 7, panels C, D contain signal recovery information for 480 nm excitation. In C, emission spectra C1 and C3 by NMF have sharp peaks, and C3 has subsidiary peaks, suggesting multiple components. Indeed, the abundance images are all quite similar, suggesting shared sources. Note that the spectra recovered by NTF in D are dramatically improved. C1 and C3 have single, broad peaks, and C2 has a smooth broad peak with a single small notch. The abundance images are all quite different, suggesting good separation of sources.
6. Discussion
As shown semi-quantitatively in Section 4.5, Results, simultaneous decomposition of multiple hyperspectral data sets constrained by common abundant sources may offer, in some cases, a superior method than standard NMF for decomposing a complex spectrum into its individual spectral signals. The greater information content, as well as strong spatial constraint, can assist in finding an improved physical solution to what is notoriously a massively underdetermined problem. A clearer outcome may also be aided with improved signal-to-noise ratio, for example in the case of BrM, where the specificity of the abundances of the signals in the two separate excitations was significantly improved by tying them together (Fig. 7, panels A, B).
This general approach has been used in biomedical imaging with multiple fluorescent dyes [7, 31], but to our knowledge it has not been previously attempted without using known spectral signatures, or with only one of several, as with Bruch's membrane in the present case. Further, when the problem is to unmix the signals from multiple known fluorescent dyes, the solution is facilitated by the fact that the individual labels presumably localize to distinct cellular structures, that is, the abundances are largely distinct. In our RPE samples, the individual fluorophores appear to localize to compartments consistent with lipofuscin and melanolipofuscin granules, and so, at least at the present level of resolution, there is scant spatial information to separate the sources, constrain the NMF solution, and assist in correct spectral resolution. Indeed, as pointed out by Neher et al. [7], the non-negativity constraint can provide a unique decomposition if (a) there is spectral separation of a precise nature, i.e., although spectra can overlap to an extent, when a given individual spectrum is compared to the group of other spectra, there must be at least one channel outside the given spectrum that is in common with the overlap of all the other spectra; and (b) there is spatial separation, i.e., the image has to contain pixels in which one source is absent and others are present in various concentration ratios. Conversely, to the extent to which these conditions do not obtain, i.e., where label distributions are similar and spectra overlap strongly, uniqueness fails. In the case of RPE or BrM spectra, neither of these conditions is met: all Gaussian candidate spectra overlap already at initialization, as do the recovered spectra in almost every case, and, as just mentioned, the abundances of the respective RPE signals are virtually identical except for magnitude. Nonetheless, in our examples of two tissue types (RPE and BrM), the recovered signals from simultaneous decomposition of multiple hyperspectral AF data sets appeared to provide consistent and better candidates for biochemical identification, further attesting to the strength of the method when applied to a historically difficult problem. Biochemical identification of the species suggested herein, of course, must await the expertise of other disciplines, including synthesis of pure candidate compounds and confirmatory experiments using a technique that provides both definitive molecular identification and spatial localization. Imaging mass spectrometry is an obvious candidate for such a technique (see Ablonczy et al. [18]); however, it cannot yet achieve the spatial resolution of our fluorescence images.
In conclusion, NTF with concatenated excitation data sets offers improved spectral recovery, even in challenging domains with unknown and overlapping spectra that are also poorly separated spatially. Thus, we submit these findings as strong support for the prediction set forth by Neher et al [7]: “The full potential of NTF is still to be explored.”
Acknowledgments
National Institutes of Health/National Eye Institute R01 EY06109 (CC)
National Institutes of Health/National Eye Institute R01 EY021470 (RTS)
National Institutes of Health/National Eye Institute R01 EY015520 (RTS)
German Research Foundation DFG # AC265/1-1 (TA)
Unrestricted funds from Research to Prevent Blindness (to the University of Alabama at Birmingham, New York University School of Medicine, and Medical University of South Carolina)
Foundation Fighting Blindness Individual Investigator Award (RTS)
National Institutes of Health/National Eye Institute R01 EY19065 (ZA)
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