Establishing System Linearity
The easiest way to establish the linearity of a flow cytometer is to analyze particles that fluoresce or scatter light with known relative intensities. Absolute intensities of the particles need not be known; however, relative intensities must be known with an accuracy acceptable for the application for which the flow cytometer is being tested to run. The range of particle intensities is chosen to span the full histogram scale. By far the most common calibrators are fluorescent particles, because applications that require the linearity of the flow cytometer to be known are fluorescence based (e.g., quantitation of DNA, RNA, protein, and antigen, and determination of antigen density). The many types of fluorescence "standard" particles of biological or manufactured origin each have advantages and disadvantages.
The biological particles used most commonly to determine the linearity of a flow cytometer are nucleated erythrocytes, typically of chicken or trout origin. Owing to their uniformity, significant difference in chromatin content from human cells, and widespread availability (Vindelov et al., 1983), chicken and trout erythrocytes are used extensively to verify the linearity of flow cytometers for DNA quantitation applications (Vindelov and Christensen, 1990). When used in standardizations, the particles are stained with the same DNA dye used in the DNA quantitation application and then run on the flow cytometer. The positions in the histogram of the populations of singlet, doublet, triplet, and quadruplet particle clusters are noted. The goal is to determine if the flow cytometer is both linear and proportional; thus, the expected result is for the doublet, triplet, and quadruplet particles to be two, three, and four times, respectively, the singlet peak location. Doing a linear regression of channel position versus number of particles in the multiplet would reveal any nonlinearity and/or offset (Ubezio and Andreoni, 1985).
A disadvantage of nucleated erythrocytes as linearity calibrators is the difference in size among the singlet, doublet, triplet, and quadruplet clusters. As the particle clusters increase in size, the pulse width of the larger clusters would exceed that of a singlet particle, leading to a corresponding reduction in pulse bandwidth. A single particle with the same fluorescence intensity as a quadruplet cluster would have a higher pulse bandwidth than the cluster and thus may not be measured as having as much fluorescence intensity as an equivalent quadruplet cluster.
The difference in bandwidth requirements between a singlet and a multiplet increases as the excitation beam size along the axis of particle travel decreases, because for narrower beams particle size is more a determinant of pulse width (Leary et al., 1979). Thus, a flow cytometer may seem linear when tested with particle systems using singlets, doublets, etc.; but the system may not be linear with a different set of particles. For example, the results obtained with a set of smalldiameter particles may be different from the results obtained with a set of largerdiameter particles. The dimmer singlets require a wider bandwidth, but the smallsignal bandwidth may be wide enough to accommodate the singlets. Quadruplets, which require a narrower bandwidth, may be accommodated by the narrower largesignal bandwidth, whereas a singlet particle of the same fluorescence intensity may require a bandwidth exceeding the largesignal bandwidth. Thus, the flow cytometer may seem linear with the wider brighter particles when it may not be when using smaller particles of the same brightness. The important factor is the pulse width. If the pulse widths of singlet particles are significantly less than pulse widths of multiplets, then investigators need to consider the issue of amplifier bandwidth when interpreting results of linearity tests using singlet and multiplet biological particles.
Another method of measuring the linearity of a linear amplifier system is to analyze two types of particles with different fluorescence intensities and measure the difference in pulse amplitudes at different PMT voltage settings or laser powers. This method has been used for biological particles (Vindelov and Chistensen, 1990) and microspheres (Bagwell et al., 1989). The difference in the means of the two particle populations M1 and M2 , where M2 is the particle population with the higher mean (M2 > M1), is plotted against the mean of the dimmer particle population. The relationship is given by (M2 M1) = k x M1, where k is a constant related to the relative intensities of the two particles. Thus, a plot of (M2  M1) versus M1 is expected to be linear and to intercept the origin (Bagwell et al., 1989; Vindelov and Chrisensen, 1990). If the resulting plot is not linear, then the instrument response is not linear and the instrument needs to be serviced. Unfortunately, this test for linearity is not complete, because a linear plot does not guarantee that the system is linear. Some nonlinear amplifiers, such as amplifiers that have an exponential transfer function (y = axb) will give a linear plot (Bagwell et al., 1989). With this caveat in mind, this method can be used to prove nonlinearity but not linearity.
Manufactured fluorescent particles (i.e., fluorochromeconjugated plastic beads) can be used in the same way as biological particles by locating the singlet, doublet, triplet, and quadruplet peak positions, but it is also possible to obtain manufactured particles of identical sizes with variable fluorescence intensities. Using plastic beads of the same size in multiple intensities avoids the problems of the influence of bandwidth in the measurement of system linearity and the need to use the particles in pairs. For a linear amplifier, a plot of the mean channel of each of the particle populations versus the relative or calibrated intensities should be a straight line. Linear regression analysis of the data can be done to determine the degree of correlation and the existence of an offset. If the linear amplifiers have no detectable offset, then the regression line should intercept the origin as well. If the plot is linear, then the amplifiers are linear. If the line does
Flow Cytometry Instrumentation not pass through the origin, then there is an offset in the linear amplifier or amplifiers. The amplifiers would be linear but not proportional. As long as the manufactured particles are approximately the same size as or smaller than the particles to be analyzed, then a flow cytometer that is determined to be linear with the manufactured particles should be linear for the application to be run. The converse is true as well: i.e., a flow cytometer that gives a nonlinear response with standard particles will also be nonlinear with actual samples.
It is important to determine the linearity at the gain setting (and attenuation setting, if applicable) at which the application will be run, because the bandwidth of the amplifiers is different for each gain setting. In general, though, if the bandwidth at the highest gain setting is adequate then it probably will be adequate at the lower gain settings. However, because most linear amplifiers in flow cytometers are multistage, it is not easy to predict what the bandwidth would be at a given gain setting. Thus, it is better to test for linearity at the gain at which the application is to be analyzed.
Although the discussion so far has focused on testing of linear amplifiers, similar methods can be applied to the testing of logarithmic amplifiers. Because of the wide dynamic range of logarithmic amplifiers, manufactured particles are favored over biological particles, as the manufactured particles can be fabricated to cover up to a 10,000:1 range commonly found in logarithmic amplifiers used in flow cytometers. If the scale of the logarithmic histogram is presented in linear units (e.g., 0.1 to 1000, 1 to 10,000) then the analysis techniques for linear amplifiers can be used to analyze the data. However, if the scale is in histogram channel units (e.g., 0 to 1024, 0 to 256), then the analysis techniques need to be changed to reflect the fact that the channel units represent the logarithm of the input values. For more detailed discussion of data analysis techniques, see Chapter 10.
Two approaches are used specifically to evaluate the logarithmic transform function of logarithmic amplifiers. The first uses two types of microspheres with differing known relative fluorescence intensities. The difference between the mean intensities of the two populations is measured in histogram channel units at different PMT highvoltage settings. Initial highvoltage settings are chosen so that signals Establishin and from the pair of microsphere populations are Maintaining located near the bottom of the histogram, and System Linearity the voltage is raised until the signals are near the top. The intermediate voltage settings are chosen to cover the whole histogram range uniformly. Because the histogram channel units represent the logarithm of the input signal, taking the difference in the logarithmic domain is equivalent to dividing the means in the linear domain. Thus, taking the difference between the means is equivalent to determining the relative intensity of the one bead versus the other. As the relative intensities of the beads are invariant, this difference should remain constant regardless of the highvoltage setting. Any deviation would indicate a deviation from an ideal logarithmic transformation (Schmid et al., 1988).
The second approach in evaluating the linearity of logarithmic amplifiers is to use a series of microspheres of known relative intensities. A plot of mean histogram channel versus the logarithm of relative microsphere intensity shows the relationship between microsphere intensity and histogram channel value (Muirhead et al., 1983; Parks et al., 1988; Schwarz et al., 1996). Because the histogram channel value represents the logarithm of the input signal, this plot is in reality a loglog plot (see Fig. 1.4.6). A straight line model of the loglog plot that is based on the relationship y = a(xb) used to fit the data, where y is the measured signal intensity, x is the actual particle intensity, b is the linearity factor, and the offset is zero (see line A in Fig. 1.4.6). Taking the logarithm of both sides of the equation gives log(y) = b x log(x) + log(a), which is the equation of a line in a loglog plot. The relationship between the histogram channel and log(y) is given by histogram channel = cpd x log(y), where y is scaled from 1 to the maximum intensity value (e.g., 10,000 for a fourdecade logarithmic amplifier) and cpd is channels per decade. The slope of the line in Figure 1.4.6 is the product of the channels per decade (cpd) and linearity factor (b). The linearity factor will typically be unity but could be different because of nonlinearities in the linear amplifiers preceding the logarithmic amplifiers. If b is greater than or less than unity, then the effective dynamic range will decrease or increase, respectively. The slope of the line in the loglog plot represents the effective channels per decade, and power functionrelated nonlinearities before the logarithmic amplifier are masked.
The last two methods involving pairs or series of microspheres can be used to evaluate the overall accuracy of the logarithmic transformation of a logarithmic amplifier and any linear amplifiers that precede it. Neither of
Figure 1.4.6 The loglog plot version of Figure 1.4.1 is commonly used to standardize logarithmic amplifiers. Curve A illustrates a proportional response. Curve B show the effect on an otherwise linear amplifier response of an offset from noise (optical and/or electronic). Curve C illustrates a nonlinearity at the high end of the histogram scale.
Figure 1.4.6 The loglog plot version of Figure 1.4.1 is commonly used to standardize logarithmic amplifiers. Curve A illustrates a proportional response. Curve B show the effect on an otherwise linear amplifier response of an offset from noise (optical and/or electronic). Curve C illustrates a nonlinearity at the high end of the histogram scale.
these methods, however, will determine the magnitude of the offset (if present). An offset may be apparent as a deviation from the ideal logarithmic transformation at the lower end of the histogram, but the two methods do not provide enough information to determine if the deviation is due to an inaccurate logarithmic transformation or to an offset (see Fig. 1.4.6, curves B and C). Although available from the data, the offset information is either hidden or ignored by these methods of data analysis.
One way to detect and quantitate the offset is to use a set of multiple microspheres, as in the second method mentioned above that uses a series of particles, but to perform a linear regression of the converted (linear) intensities calculated from the histogram channel values versus the relative microsphere intensities, instead of a linear regression of loglog data. This method using a series of particles or particle multiplets is equivalent to that described above for linear amplifiers and will provide the same information about the degree of correlation and the magnitude of the offset. A high degree of correlation confirms an accurate logarithmic transformation. An alternative is adding the offset to the earlier equation, resulting in:
histogram channel = cpd x log(axb + c)
A nonlinear leastsquares regression is needed to fit the four parameters. Because there are four variables in the new equation, four or more fluorescent microsphere populations must be included in the test mixture. Either the linear regression or the nonlinear regression analysis would provide more complete characterization of the logarithmic transformation of the logarithmic amplifiers in a flow cytometer.
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