## Spectral Imaging with Linear Unmixing

In a typical spectral imaging experiment based on fluorescence, there are usually several fluorophores present in the specimen, each one labeling a different structure. Throughout an image of this specimen, the fluorophores are found either separately or as a mixture depending upon their spatial distribution within the targeted organelles or macromolecules. The purpose of linear unmixing analysis is to determine the relative contribution from each fluorophore for every pixel of the image. This tutorial explores how individual fluorophores can be identified within a complex mixture in a triple-labeled specimen.

The tutorial initializes with a complex spectral data set appearing in the window graph and an image of the triple-stained specimen placed in the upper right-hand corner. In order to operate the tutorial, click in the specimen image window and drag the bulls-eye cursor to a point of interest. As the cursor is translated across the image, the spectral profiles of the various probes are altered to reflect their concentrations at that particular region (denoted by the progress bars in the lower portion of the tutorial window. The white curve is the added spectra, whereas the red curve represents EGFP, the green curve represents Alexa Fluor 488, and the blue curve represents Alexa Fluor 514. Click on the red **Auto** button adjacent to a selected fluorophore to relocate the cursor to a region of the specimen containing 100 percent of that fluorophore.

The fundamental concept underlying linear unmixing calculations is relatively simple. Each pixel in the spectral image is categorized as representing a mixture of fluorophore signals (intensities) when the measured spectrum (**I(λ)**) can be deconvolved into the proportion, weight, or concentration (**C**) of each individual fluorophore reference spectrum (**R(λ)**) when the values are summed. Thus, each reference spectrum of a pure fluorophore is described as **R _{i}(λ)** where i = 1,2,3.....n represents the index of the fluorophore (

**C**). For a particular number of fluorophores (

_{i}**n**), this relationship can be represented as:

Or more simply:

In practice, the signal intensity for each pixel (**I**) in the spectral image is determined and recorded during acquisition of the lambda stack and the reference spectra for the known fluorophores are measured independently in separate control specimens labeled with only a single fluorophore using identical sample preparation techniques and instrument settings. The overall spectral contributions from the various fluorophores in the specimen can then be determined as a simple linear algebra matrix exercise by calculating their individual contributions to each point in the measured spectrum, as described in the equations above. For many of the commercially available linear unmixing software packages, the solution is obtained by inputting reference spectral profiles and using an inverse least squares fitting approach that minimizes the square difference between the measured and the calculated spectra.

In order to ensure the best chances to obtain successful results when applying linear unmixing algorithms, several experimental criteria must be met. One of the most important considerations is to ensure that the number of spectral detection channels is at least equal to the number of fluorophores present in the specimen. Failure to meet this specification can result in multiple solutions to the spectral separation calculation and a unique result may not be possible. Another critical requirement for linear unmixing is that all fluorophores present in the specimen must be considered in the calculations or the results may be skewed towards the dominant (most concentrated) fluorophore at the expense of less concentrated species. Ironically, including spectra in the calculations that do not match any of the fluorophores in the lambda stack will not affect linear unmixing results (a zero contribution will be assigned to the missing fluorophore). Finally, autofluorescence and/or high background levels should also be defined spectrally (if possible) and treated as an additional fluorophore in order to achieve optimum results. Optionally, an error term can also be calculated and output as an error residuals image.

The linearity involved in adding fluorophore spectra is illustrated in Figure 1 for a mixture of two different, but highly overlapping hypothetical fluorophores having emission maxima residing in the yellow-orange (Fluorophore 1) and orange-red (Fluorophore 2) spectral regions. The black curves in Figure 1(a) through 1(c) represent the summed spectra of the two fluorophores at different concentrations: Figure 1(a) 1 to 1; Figure 1(b) 0.5 to 1; and Figure 1(c) 1 to 0.5. Although the spectra presented in Figure 1 represent examples of only three fluorophore combinations, the summed spectrum can readily be predicted for every possible combination of these two fluorophores simply by adding the intensities as a function of concentration. Note that the peak of the summed spectra changes with the proportions of the component fluorophores such that the maximum is 594 nanometers in Figure 1(a), 598 nanometers in Figure 1(b), and 589 nanometers in Figure 1(c). It should be emphasized that linear unmixing takes advantage of the entire spectral curve(s), not just the peak positions. Robust algorithms, such as those used in spectral karyotyping and confocal microscopy, also handle minute spectral shifts by sophisticated curve analysis and correction.

When analyzing the spectral content of a specimen labeled with two fluorophores, similar to that presented in Figure 1, the simplest approach is to match the summed spectrum from any particular pixel with all possible sum combinations residing in a spectral reference library. As an example, if the measured summed spectrum was a very close match to the black curve presented in Figure 1(a), it would indicate that the pixel contains a 50-percent contribution from each of the fluorophores and that they are evenly mixed in the specimen (at least for that pixel). Similarly, if the summed spectrum matches the black curve in Figure 1(b), one could assume that the pixel contains 66 percent of Fluorophore 2 and 33 percent of fluorophore 1. Thus, it can be summarized that linear unmixing operates by comparing a matrix representing the summed spectra measured in an image against a reference library of predicted spectra according to the best-fit parameters applied by the software. Once the spectral contribution from each fluorophore has been determined, the lambda stack can be segregated into individual images for each fluorophore.

### Contributing Authors

**Jeffrey M. Larson** and **Stanley A. Schwartz** - Nikon Instruments, Inc., 1300 Walt Whitman Road, Melville, New York, 11747.

**Adam M. Rainey** and **Michael W. Davidson** - National High Magnetic Field Laboratory, 1800 East Paul Dirac Dr., The Florida State University, Tallahassee, Florida, 32310.