A Simple and Rapid Method for Determining the

Transcription

1 Alan R. Liss, Inc. Cytometry 10: (1989) A Simple and Rapid Method for Determining the Linearity of a Flow Cytometer Amplification System C. Bruce Bagwell: David Baker, Sherry Whetstone, Mark Munson, Shelly Hitchcox, Kenneth A. Ault, and Edmund J. Lovett Maine Cytometry Research Institute, Portland, Maine Received for publication February 6, 1989; accepted June 16, 1989 We describe a simple and rapid method for determining the linearity of a flow cytometer amplification system. The method is based on a fundamental characteristic of linear amplifiers: The difference between two amplified signals increases linearly with increasing amplifier gain. Two populations of beads or cells, differing slightly in fluorescence intensity, are analyzed by the flow cytometer at increasing photomultiplier tube high-voltage settings. The distribution of the populations mean difference versus mean position is a straight line intersect- ing the origin for linear amplifiers. Although some types of nonlinearities cannot be detected with this technique, deviations from linearity indicate nonlinear components in the flow cytometer amplification system. The correlation coefficient is used to quantify degree of nonlinearity. We also describe a method for amplifier nonlinearity compensation. Key terms: Calibration, compensation, DNA analysis, quantitative immunofluorescence In 1983 Hedley et al. (5) demonstrated the feasibility of obtaining interpretable flow cytometry DNA histograms from paraffin embedded tissue. Since that landmark paper, numerous retrospective studies have confirmed the prognostic utility of DNA ploidy for many tumors (2,3). More recently, a few studies have shown that the S phase fraction has equal or greater prognostic importance than DNA ploidy in colon and breast cancer (1,6). Accurate and reproducible interpretation of ploidy and S phase fraction depends greatly on the assumption that DNA content is linearly related to the digitized fluorescence signal. When this assumption is significantly violated, it is difficult to model and interpret DNA histograms accurately. The validity of this assumption depends on such factors as stoichiometry of dye binding, cell cycle phase-specific chromatin structure effects, fluorescence absorption, and a linear amplification system. Of these factors, the linear amplification system is the most amenable to evaluation and to control. Although our interest in flow cytometer amplifier linearity arises from DNA analysis, the need to quantify the amplification characteristics of a flow cytometer is not limited to DNA applications alone. Quantitative immunofluorescence also requires instrument linearity. With a slight modification, this procedure can also be used to characterize logarithmic amplifiers (7,9). In this report we describe a rapid method to quantify accurately the degree of nonlinearity present in a flow cytometer s linear amplification system. We define the linear amplification system as including the preamplifier, amplifier, and analog-to-digital converter (ADC). The linearity of the photomultiplier tube (PMT) is not tested by this procedure. The method is based on a fundamental characteristic of linear amplification systems: The difference between any two amplified signals increases linearly with amplifier gain. This characteristic is easily demonstrated by observing the broadening of the histogram for a simple population of beads or cells with increasing linear amplifier gain. The increase in the standard deviation of the observed peak is due to the linear increase in the distance between all the points that comprise the peak. Although a test of linearity can be developed from observations of population standard deviations, a more accurate Presented in part at the Society for Analytical Cytology Meeting XIII, Breckenridge, Colorado, September Address reprint requests to Dr. C. Bruce Bagwell, Maine Cytometry Research Institute, 190 Park Avenue, Portland, ME

2 690 BAGWELL ET AL. method involves the measurement of distances between population means with slightly differing fluorescent intensities. Contained in the report is a detailed description of the method, some examples of linear and nonlinear amplifiers, a method of quantifying amplifier nonlinearity, and the mathematics associated with compensation for detected nonlinearity. MATERIALS AND METHODS Bead or Particle Mixture Two populations of either beads and/or cellular controls, slightly differing in fluorescence intensity, are combined in approximately equal proportions. The ratio of fluorescence intensity of the two populations (high-intensity population meanilow-intensity population mean) in this study is approximately 1.1, and the percent coefficients of variation are less than 1.5. Beads or cells with higher mean fluorescence ratios can be used but will result in a loss of sensitivity and a decrease in testable range. Suitable beads for linearity determination are obtained by experimentation with different lots of the same bead type. The data presented in Figure 3B were obtained with different lots of DNA-Check beads (Coulter Cytometry, Hialeah, FL). For those laboratories with tissue culture facilities, a lower cost alternative to beads is a cellular mixture stained for DNA Lymphocytes and LoVo colon adenocarcinoma cell line (4) with a DNA index of 1.1 has been used successfully (P. Wallace, personal communication). Procedure The steps to perform an amplifier linearity test are illustrated in Figure 1 and are as follows. 1. Set the flow cytometer s amplification gain to the value typically used for flow cytometry analysis. 2. Set the PMT high voltage (HV) to the lowest setting that still delineates the two peaks in the linearity mixture at the lowest channels. 3. Record the channel difference between the two peak means and the first peak mean. The mean is preferred; however, the peak mode is also acceptable. 4. Increase the HV until the first peak mean is approximately 10 channels greater than that recorded in step Repeat steps 3 and 4 until the highest intensity peak is no longer on scale. 6. Plot the difference of means (y axis) versus the first mean (x axis) for each histogram. 7. Calculate the correlation coefficient of the resulting curve. It is important to set the amplification gain to the value typically used for flow cytometry analysis, because different gain settings can have vastly different linearity characteristics. If several gain settings are typically used then each one needs to be tested. The HV range used in the linearity test procedure should fall Hv=SOO =I2 (Repeat until r2, off scale (r2-rl) H rl, r2, FLUORESCENCE (CHANNELS) FIG. 1. Data acquisition scheme for linearity determination. Two populations of beads or cellular controls with slightly differing fluorescence intensities are used. The amplification gain is set to the value normally used for flow cytometry analysis. A. The high voltage (HV) is used to set the two peaks at as low an intensity as possible but still to distinguish two separate peaks. The HV settings are for illustrative purposes only. B: The HV is increased such that the first peaks mean position increases approximately 10 channels. C: The process is repeated until the second peak is no longer on scale. within the range specified by the PMT manufacturer. If a suitable computer program is available, step 3 can be performed after all the histograms have been acquired and saved. Generally, an accuracy of one decimal digit is all that is required for mean values. The interval of 10 channels in step 4 is typical for 256 channel ADCs. This value may be scaled up or down with differing ADC resolutions. It does not matter whether the differences are plotted against the first mean, the second mean, or the average of the means.

3 TWO PEAK LINEARITY DETERMINATION 691 DIFFERENCE-MEAN RATIO-MEAN PLOT (r2-rl DIFFERENCE (R2-R1) ( r2-rl)7 (r2-rl), 1 I I rl,r17... rl" FIRST PEAK MEAN (R1) FIRST PEAK MEAN (R1) FIG. 2. Ideal linearity plots. Distance-mean plot (A): The distances between the two peak means (R2 - R1) are plotted against the first peak means (Rl). A linear amplification system will yield a locus of points that follow a line intersecting the origin. Ratio-mean plot (B): The ratio of the second peak mean position to the first is plotted against the mean value of the first peak. A horizontal line will result if the amplification system is linear. RESULTS The degree of amplification system linearity can be demonstrated by plotting either the difference between the means (R2 - R1) versus the first peak's mean (R1; see Fig. 2A) or the ratio of the second peak's mean position to the first peak's mean position (R2IR1; see Fig. 2B). If the amplification system is linear, the difference-mean plot should be a straight line intersecting the origin, and the ratio-mean plot should be a horizontal line. Deviations from these ideal relationships indicate various degrees of nonlinearity. Although both the ratio-mean and difference-mean plots convey information about the linearity of the amplification system, we will only concentrate on the analysis of difference-mean plots. Figure 3 demonstrates difference-mean linearity plots from a linear and a nonlinear flow cytometer. The correlation coefficients of Figure 3A and Figure 3B data were 0.95 and 0.73, respectively. The signal saturation type of nonlinearity observed in Figure 3B is the most typical type of nonlinearity we observed. Although a number of statistics can be employed to describe how close the observed curve is to a line with intercept 0, the method we have found to be most useful is the correlation coefficient (10). The general formula for the correlation coefficient is given by Where x = R1 - m; y = R2 - R1 - (R2- R1); R1 = mean of first peak; R2 - R1 = difference between means; = average of the first peak means; and R2 - R1 = average of the mean differences. 35'00 LINEAR AMPLIFIER r = 0.95,-:u /a DIFFERENCE (RZ-R1) ~?/rT...O < NONLINEAR AMPLIFIER r = 0.73 DIFFERENCE 6o!, FIRST PEAK MEAN (Rl) FIG. 3. Example of linearity plots. A: Difference-mean plots of a flow cytometer that has an acceptable linear amplification system (r=0.99). B: A flow cytometer with a major linearity problem (r = 0.73). The reference line in both parts is an extrapolation from the origin through a point on the curve approximately one third of the histogram range. THEORY AND COMPENSATION METHOD Assume an amplifier with a response versus signal curve as shown in Figure 4. Further assume that the

4 692 BAGWELL ET AL. By analyzing this type of amplifier in the same way we did above for the linear amplifier, we obtain RESPONSE (R) r2 rl sl s2, SIGNAL (S) FIG. 4. Amplifier response curve. The sl and s2 signals elicit rl and r2 responses. amplifier has two simultaneous input signals, sl and s2, with measured responses rl and r2, respectively. Let the signals sl and s2 be related by ~2 = k. ~l. (1) Let S1 and S2 be the set of all possible simultaneous input signals and Rl and R2 be the set of all possible responses to those signals, respectively. The differencemean plot described in this paper is the theoretical relationship between R2 - R1 and R1. If the amplifier is linear, it is described by the formu1 a R = a1. S, (2) where R is some response to signal S and a1 is the amplification factor. The difference between the two responses is given by R2 - R1 = a1. S2 - a1. S1. Substituting for S2 with equation 1 yields R2 - R1 = a1. k. S1 - a1. S1. (4) Collecting terms, R2 - Rl = (k - 1). a1. S1. (5) Since a1. S1 is R1 from equation 2, we obtain our desired relation R2 - R1 = (k - 1). R1. 16) Equation 6 predicts that if an amplifier system is linear, the plot of R2 - R1 versus Rl will be a straight line with slope k - 1 and intercept 0. Before discussing a possible method to compensate for nonlinearities, it is important to discuss some of the limitations of using R2 - R1 versus R1 to detect amplifier nonlinearities. There is a general class of amplifier nonlinearities that will not be uncovered by a difference-mean plot. A simple example is given by (3) R = al.s2 (71 R2 - R1 = (k2-1). R1. (8) In fact, amplifiers of the general form have a difference-mean equation of R = a1. S + a2 (9) R2 - R1 = (k - 1). R1. (10) The importance of these exceptions is that a linear difference-mean plot does not necessarily imply a linear amplification system; however, a nonlinear difference-mean plot does infer a nonlinear amplification system. It is also worth noting that most linear amplifiers do not have signal response curves described by equation 9. In our experience, Figure 3B represents a typical type of nonlinearity encountered with commercially available flow cytometry amplifiers presumably because of some common amplifier overloading effect. From our limited survey of different manufacturers flow cytometry systems, no particular commercial amplifier is problematic. Given the difference-mean plot shown in Figure 3B, how does one compensate for the amplifier s nonlinearity? It is recommended that if it is at all possible, identify and replace nonlinear components in the amplification system. If this procedure is not feasible, it is sometimes possible to correct mathematically for the nonlinearity. The procedure is more difficult than one would initially expect. The first step is to identify the mathematical equation that will describe the signal response curve for the amplifier. The selection of this equation can require some trial and error. One important characteristic of the equation is that it be a function with a defined inverse. This requirement becomes evident in the derivation of the difference-mean and compensation formulae. In the case of Figure 3B, we selected the following equation because of its saturation characteristics to represent the amplifier response versus signal relationship: R = a1. (1 - e? s). (11) The difference between two simultaneous signals is given by R2 - R1 = al. (e-a2 S1 - e-a2. k S1). After some algebra, we obtain (12) R2 - R1 = a1 - R1 - a1. [(a1 - Rl)/allk. (13) If we add R1 to both sides, equation 13 simplifies to R2 = a1. (1 - [(a1 - Rl)/alIk}. (14) If we use a nonlinear least-squares algorithm to solve this equation for a1 and k (8) for the observed R1 and R2 values shown in Figure 3B, we obtain a1 = k = If we substitute these values back into the theoretical difference-mean equation (see equation 13) and superimpose it on the observed data, we can visualize the

5 Rl2ial), TWO PEAK LINEARITY DETERMINATION 693 DIFFERENCE (R2-R1) a Observed ACalculated HEAN OF FIRST PEAI[ (Rl) FIG. 5. Comparison of the observed and calculated difference-mean plots. quality of our fit (see Fig. 5). A bad choice of the amplifier response versus signal equation would be evident at this point. Now that a1 and k are known, the theoretical signal response curve can be graphed (see Fig. 6). Note the differences between the true response curve shown in Figure 6 and its corresponding difference-mean plot shown in Figure 3B. The two distributions are not superimposable. The true response curve as described above approaches asymptotically a saturation response, whereas the difference-mean plot's slope can become negative. The importance of this procedure is that we eventually obtain a response versus signal curve for our amplifier without ever having to generate calibrated signals; thus there is no need for special equipment for this linearity procedure. Once the response versus signal curve is known, it is simple to compensate for nonlinearities in the amplifier. For example, if we wanted to know the actual G2-M mean/ GO-G1 mean ratio for each possible position of GO-G1, we would substitute k = 2 into equation 14 and obtain R2/R1 = a1 (1- [(a1- Rl)/al]2)/R1, (15) which simplifies to R2iRl = 2 - Rlial. (16) Thus a GO/G1 peak at channel 500 for the amplifier shown in Figure 3B would have a theoretical G2- MIGO-G1 ratio of or Another way of looking at equation 16 is that if we multiplied the observed G2-M by the factor 242. R1 ~ the result would be the expected location of G2-M for a linear amplification system or 2. Suppose we have a "flow cytometry histogram given RESPONSE SIGNAL (S) FIG. 6. Response versus signal curve for Figure 3B difference-mean plot. The calculated data points are the theoretical responses to the data presented in Figure 3B. by Y(X), where Y(X) is the number of events at channel X. If this histogram were obtained with the nonlinear amplifier described above, it can be linearized by an x axis transformation using the inverse of equation 11: Y {LN[al/(al - X)l} DISCUSSION The method described in this report assesses the linearity of a flow cytometer's preamplifier, amplifier, and ADC. The method does not test for PMT photon saturation or irregularities in the electron cascade. To test the optical and PMT portions of the entire amplification system, various calibrated neutral density filters or iilumination sources can be employed. -The

6 694 BAGWELL ET AL. method also does not detect nonlinear amplifiers with response curves described by the equation: R = a1. S" + a2, where R is the response; al, a2, n are constants, and S is the signal. These limitations in the method are mitigated somewhat by our observation that flow cytometry amplifiers generally do not have this power function characteristic and that the nonlinear component in a flow cytometer is rarely a PMT. Weaknesses in the method are, at least partially, offset by its practicality. The procedure does not require any complex equipment and can be performed in less than 20 min with simple beads or stained cells. We have observed no evidence that would suggest that the test needs to be repeated more than once a year. We encourage linearity testing if part of a flow cytometer's amplification system is serviced. How nonlinear can the difference-mean plot be before one becomes concerned? Until linearity testing is incorporated into national quality assurance programs, this question cannot be answered definitively. Our limited experience, however, suggests that the correlation coefficient of the difference-mean curve should be greater than If DNA ploidy or S phase fraction are to gain widespread clinical acceptance, it will be necessary to standardize flow cytometer performance as well as sample preparation, acquisition, analysis, and interpretation. The linearity method described in this report begins to address the general problem of flow cytometer quality control. Although it is theoretically possible to develop a compensation algorithm based on linearity curves, it is recommended that users work with vendors to correct nonlinear components. LITERATURE CITED 1. Bauer KD, Lincoln ST, Vera-Roman JM, Wallemark CB, Chmiel JS, Madurski ML, Murad T, Scarpelli DG: Prognostic implications of proliferative activity and DNA aneuploidy in colonic adenocarcinomas. Lab Invest 57:329, Kheir SM, Bines SD, Vonroenn JH, Soong S, Urist MM, Coon JS: Prognostic significance of DNA aneuploidy in stage I cutaneous melanoma. Ann Surg 207:457, Kokal W, Sheibani K, Terz J, Harada JR: Tumor DNA content in the prognosis of colorectal carcinoma. JAMA , Hay R, et al.: American Type Culture Collection Catalogue of Cell Lines and Hybridomas, Ed , p Hedley DW, Friedlander ML, Taylor IW, Rugg CA, Musgrove EA: Method for analysis of cellular DNA content of paraffinembedded pathological material using flow cytometry. J Histochem Cytochem 31:1333, McGuire WL, Dressler LG: DNA flow cytometry and prognostic factors in 1331 frozen breast cancer specimens. Cancer 61:420, Parks RD, Bigos M: Logarithmic amplifier transfer function, evaluation, and procedures for log-amp optimization and data correction. Cytometry [Suppl] 2:155A, SAS Institute Inc. SAWSTAT Guide for Personal Computers, Version 6 Edition. Cary, NC: SAS Institute Inc., Schmid I, Schmid P, Giorgi JV: Conversion of logarithmic channel numbers into relative linear fluorescence intensity. Cytometry 9:533, Zar JH: Biostatistical Analysis. Prentice-Hall, Inc., New York, 1974, p 236.