Cytometry 13:822-830 (1992)

0 1992 Wiley-Liss, Inc.

Noise, Sensitivity, and Resolution of Flow Cytometersl Harald B. Steen Department of Biophysics, Institute for Cancer Research, The Norwegian Radium Hospital, 0310 Oslo, Norway Received for publication January 8, 1992; accepted April 20, 1992

The sensitivity and resolution of flow cytometers are functions of the signal produced by a given particle as well as by the noise in the presence of which the signal is detected. The noise is primarily due to the fact that emission of light as well as its detection by photoelectric devises are stochastic processes. This fact leads to equations describing how resolution and sensitivity are limited by the magnitude of the signal, the background, and the photoelectron quantum yield of the detector. The equations are pointing

Sensitivity is a n important performance parameter of flow cytometers. It determines the smallest number of fluorescent molecules t h a t can be detected by the instrument and is a limiting factor in the measuring resolution, Sensitivity is a function of the magnitude of the signal a s well as the noise in the presence of which the signal is detected. Fluorescence sensitivity is usually given in terms of the detection Limit, that is, the smallest number of a given fluorescent molecule per cell t h a t can be detected by the instrument. The detection limit is defined as the point where the magnitude of the signal equals that of the noise. Hence, the sensitivity of a flow cytometer is limited by the noise. The noise that limits the sensitivity of flow cytometers at signal levels which approach the detection limit is primarily photon noise (2,9). This noise is a consequence of the fact that emission of light is a stochastic process, which implies that all emission of light exhibits a certain temporal fluctuation, the magnitude of which depends on the intensity. Based on this simple fact, we present equations which show how the sensitivity and resolution of flow cytometers depend on the magnitude of the signal and the noise. These equations point to a method by which the signal and the noise of any flow cytometer can be determined in absolute units, that is, in terms of the number of photons reaching the detector or the number of photoelectrons emitted in the detector. Currently, the fluorescence sensitivity of flow cytometers is determined by measuring several different samples of particles that have been calibrated with re-

to a method by which the signal and noise of a flow cytometer can be measured in absolute terms, as well as a way to determine fluorescence sensitivity without having to extrapolate to the noise level. The equations appear to be validated when applied to measuring data obtained with two different flow cytometers. 0 1992 Wiley-Liss, Inc.

Key terms: Flow cytometers, photon noise, photomultiplier tubes, background light, fluorescence detection limit

gard to the number of fluorescent molecules they carry. The detection limit is obtained by plotting the f luorescence intensity vs. the number off luorescent molecules per particle and extrapolating to the level of the background, that is, to the signal obtained from a sample of unstained particles. Since the background level is usually very low compared to the signal from the fluorescent particles, the extrapolation becomes susceptible even to small zero offsets in the signal amplifiers. This may be a t least some of the reason for the large variation in sensitivity obtained by this method for instruments of the same type (5). The present equations also provide a method for determining the sensitivity t h a t does not depend on a n extrapolation to very low signal levels, and is therefore independent of zero offsets.

THEORY Disregarding “trivial” sources of variation, like fluctuations in the excitation intensity and sample flow, the noise in a flow cytometer, at least at low signal levels, is primarily the result of the fact that emission of light is a stochastic process, which is to say that there is no time correlation between the individual photons emitted by a light source such as a fluorescing

’A short version of this work was presented at the Flow Technology Workshop, which preceeded the ISAC XV Congress, August 23-24, 1991.

823

NOISE, SENSITIVITY, AND RESOLUTION IN FCM

cell. As a consequence, if we perform a repetitive measurement of the number of photons reaching the detector from a perfectly constant light source within a given time interval, this number will vary according to a Poisson distribution (11,with a standard variation, s, given by: s =

n1’2

(1)

where n is the mean number of photons emitted within this time interval. Thus the precision of the measurement is limited by the number of photons in the signal. The relative standard variation, cv, is’: cv = n“2/n

= n-1’2

(2)

This kind of noise is called photon noise. In a flow cytometer, where other sources of noise are insignificant, the noise, N, is simply: N

~

n1/2

=

Sl/2

(3)

where S is the signal, that is, the average number of photons per cell, n, that reaches the detector. And: cv

=

N/S

= n-1’2

(4)

Eq. 4 has a fundamental implication, namely, that by measuring the variation in a signal in relative terms, i.e., the cv, we can determine the magnitude of the signal in absolute terms. The results derived in the present work are direct consequences of this basic fact. In the flow cytometer the light signal consists of two components: (1) the light pulse produced by the cell itself, containing nf photons, and (2) the background of light, nb, caused mainly by fluorescence and scattered light from the flow cell and from filters and lenses in the optical path. In flow cytometers which employ an excitation light source of constant intensity, such as a cw laser or an arc lamp running on DC current, the background has a constant intensity; nb, then, is the number of background photons reaching the detector within the time period that the detector system integrates the signal. This integration time is that of the active integrator, if such a device is being used, or the time constant of the amplifier(s) and other electronics through which the signal passes before being measured by the analog to digital converter (ADC). Hence, the total intensity is: n, + nb, which has a variation: s,

=

(n,

+ nb)l”

cv, = [(n,

(6)

Thus the precision of measurement is limited not only

cvi

[(n,

+ n,)/+,l”2/nf

(7)

=

cv,Z = (+,nf)-’ or. +enF= cv;’

(8)

Eq. 8 implies that by measuring the cv of a signal in the absence of background (see below), the magnitude of the signal can be determined on an absolute scale, the unit of which is one photoelectron. From Eqs. 7 and 8 it follows that in the absence of background the detection limit, fl, which is reached when cvp = 1, corresponds to one photoelectron per cell or particle. In the presence of background and with the same value of nf, we measure (see below) cvp = cvl. Thus, from Eq. 7 one can determine the background on the calibrated fluorescence scale: +enb = (cv:

~

CV,~)(+,IQ~ =

(cv:/cv:-1)/cv,2

(9)

Hence, +enf and +enb can be determined in absolute units just by measuring the cv of the same light pulse in the absence and presence of background, respectively. Assuming that both nf and nb are proportional to the excitation intensity, i,, i.e., that bleaching and excitation saturation are negligible, we have: n, = ai,f

’To simplify the notation, cv is not given in percent.

+ nb)8,1’’2/n&, =

where be is the photoelectron quantum efficiency, that is, the probability that a photon which reaches the photocathode of the detector will release an electron. (The multiplication of electrons, which takes place as the pulse of electrons travels down the dynode chain of the PMT, is also a stochastic process which contributes to the overall noise of the signal; see Appendix.) The pulse amplifiers and other analog electronics through which the signal is passed before it is measured may also generate noise which adds to that of the signal itself. However, proper design of the electronics can reduce this noise to negligible proportions and is therefore disregarded for this discussion. (However, see Eqs. 16-18.) Equation 7 provides a simple way to determine on an absolute scale the size of the signal, that is, the absolute number of photons or photoelectrons produced by a cell or particle. In the absence of background, i.e., for nb = 0, Eq. 7 reduces to:

(5)

The signal amplifiers of flow cytometers are normally operated in AC mode, so that only the pulsed fluorescence signal is transmitted whereas the essentially constant background is not. Hence, the relative standard variation of the fluorescence signal is: cv, = (4 + nb)l”/nf

by the magnitude of the signal from the cell, but also by the constant background that underlies the signal. But these are not the only limiting factors in the system. In the light detector, which is usually a photomultiplier tube (PMT),the photons are detected as they release photoelectrons from its light sensitive photocathode. This is also a stochastic process so that the number of photoelectrons varies according to a Poisson distribution. Hence, the relative standard variation of the signal from the PMT photocathode is given by:

(10)

and nb = ai,b

(11)

where f is the number of fluorescent molecules per cell and b the background constant, that is, the equivalent number of luminescent molecules that would give rise to the actual level of background light. The constant a

824

STEEN

-*

10

1

FIG. 1. The fluorescence detection limit, fl, a s a function of the background constant light, b, as calculated from Eq. 17. fi is given relative to its value f o r b = 0, f,.

is a function of the overlap between the fluorescence excitation spectrum of the dye and the spectrum of the excitation light, and various characteristics of the flow cytometer (Eq. 22). Introducing Eqs. 10 and 11 into Eq. 7, we have: CV:

=

(f

+

b)/+,ai,F

(12)

At the detection limit, f = fi, cvp = 1 by definition. Introducing this into Eq. 12 and solving with regard to fl, we obtain: fl

=

I1 + (1 + 4 +,ai,b)”21/2+,ai,

(13)

which, by using Eqs. 10 and 11, also can be written: fl

-

fi1 + (1

+ 4+,n,)1’z1/2+,n,

(14)

In this equation &nb is known from Eq. 9. Hence, by measuring a sample of fluorescence calibrated particles, i.e., a sample for which f is known, and determine the position of the resulting histogram peak, &nf, on the calibrated scale, i.e., in number of photoelectrons, one can calculate the fluorescence detection limit, fi. Eq. 13 confirms that the detection limit, fi, depends only on the background constant, b, and the various instrument parameters expressed as: &aix. In Figure 1 fi is plotted vs. b, according to Eq. 13. The graph illustrates the importance of the background for the sensitivity of flow cytometers. Finding the term +eaixfrom Eq. 12 with cvp = 1 and f = fl and introducing this back into Eq. 12, gives: cvg = (f/fl

+ b/fl)/ [ ( l + b/f,)(f/fl)21

1

100

10

In Figure 2 cvp is plotted, according to Eq. 15, for different values of the background constant, b. In this figure the value of the excitation intensity, i,, is chosen so that the detection limit remains the same for the different values of b (see insert). Hence, Figure 2 facilitates a comparison of the resolution (in terms of cv,) of instruments having different levels of background, but the same fluorescence detection limit. The implication of these data is that an instrument, where a relatively

10000

FIG. 2. cvp as a function of the fluorescence signal for various values of the background constant, b, plotted according to Eq. 15. The value of the excitation intensity, i,, for the different curves have been chosen so (see insert) that the fluorescence detection limit remains the same. Hence, this figure facilitates a comparison between flow cytometers having the same fluorescence sensitivity, but different values of the background constant. It is seen that instruments where a large background constant is compensated by a high excitation intensity, yields better resolution than instruments with a lower background constant and correspondingly lower excitation intensity. For example, an instrument with b = 2 should measure cells or particles having a fluorescence 100 times the fluorescence detection limit with a resolution of cv, = 5,8%,whereas a n instrument with b = 20 should measure the same sample with a cvp = 2,496.

high background constant is compensated by a correspondingly higher excitation intensity, should have a better resolution than an instrument having the same detection limit, but a lower background and excitation intensity. In the equations above, it is assumed that the cv of the signal is due to photon noise only. In the real case this is not necessarily true. In the measurement of cv, (see below) electronic noise, as well as possible fluctuations introduced by that measurement, may add to the photon noise. For example, there may be ripple on the excitation light intensity as well as fluctuations in the flow. Furthermore, a real sample is not perfectly homogenous with regard to fluorescence. Altogether such excitation intensity independent sources of noise give rise to a variation, cv,, which adds to the photon noise, so that the combined variation becomes: CVt =

(15)

1000

f/fl

(cv;

+c

v p

(16)

From Eq. 12 we have: cv; = const: i;’

(17)

Hence Eq. 16 can be written: cv,Z

=

const.i;l

+

cvz

(18)

which can be used to determine cv, by measuring cvt for various values of i,, as discussed below. Equation 13 has other interesting implications: In the usual case, where the noise is significant, that is:

NOISE, SENSITIVITY, AND RESOLUTION IN FCM

*

+eai,b 1, which is to say that the number of background photoelectrons per measurement is ? 1, Eq. 13 is reduced to: f, = [b/(Q,aix)1”2,

+,aixb

%

1

(19)

Hence, under these conditions, sensitivity, expressed as the inverse fluorescence detection limit, is proportional t o the square root of the excitation intensity. To reduce the detection limit by a factor of two, the excitation intensity has to be increased by a factor of four, and so on. In contrast, if the background can be reduced so that the average number of background photoelectrons per measurement is much less than 1, i.e., &.ai,b 4 1, Eq. 13 is reduced to:

825

chamber increases (as the inverse square root of the velocity (711, which may reduce the measuring resolution, i.e., increase cv,, if this diameter exceeds about 5% of the width of the focus, assuming a gaussian focus, and (2) that a lower flow velocity leads to longer fluorescence pulses and a correspondingly greater rate of pulse coincidences at the same count rate.

MATERIALS AND METHODS Signal Calibration According t o Eq. 8, the signal scale can be calibrated in absolute terms, i.e., in a unit which equals one photoelectron, by measuring the cv of a signal measured in the absence of background and with no other source of variation than photon noise, i.e., with cv, = 0 (Eq. 18). f, = &aiil, $,ai,b 1 (20) In order to realize this situation, one can do the followWhich is to say that if the background can be reduced ing. A light emitting diode (LED) is placed in such a to negligible proportions, the sensitivity is directly pro- position that a small fraction of its light emission portional to the excitation intensity. Hence, in this sit- reaches the fluorescence andor light scattering detecuation the advantage of increasing the excitation in- tor. Thus it may be placed somewhere in the neighbortensity is much larger than if background is hood of the flow chamber or in the light path between the detection optics and the detector, depending on the significant. The fluorescence signal is a function of the following design of the instrument. By means of a suitable pulse generator the LED is powered with pulses of approxifactors: mately the same duration as that of the fluorescence signal from a particle. With the excitation light turned rq.= const.f(NA)Tv-’i, c(h)I,(A)dh (21) off, the intensity of the pulses is adjusted to a level where NA is the effective numerical aperture of the where the cv of the resulting histogram peak is deterdetection optics, T the transmission of filters and other mined by the photon noise. In practice the cv should be components in the detection light path, v the flow ve- much larger than that of any other source of variation locity, and the integral is the overlap between the flu- in the system, say, of the order of 5%or more. Thus one orescence excitation spectrum, €(A), of the dye in ques- can measure cv,, and by means of Eq. 8 the fluorestion and the spectrum of the excitation light, I&). cence can be put on an absolute scale in units of phoHence, the constant a, as defined by Eq. 10, is: toelectrons. To the extent that +e is known, from the specifications of the PMT or from direct measurement a= const.(NA)’Tv-’ c(h)I,(h)dh (22) (which is more difficult), n, can be calculated from the Eq. 21 holds true only as long as there is no significant value of cv, (Eq. 8), and the scale can be calibrated in bleaching of the sample or fluorescence saturation. terms of the absolute number of photons. The value of Both of these effects, which will cause nf to level off +e should refer t o the wavelength of fluorescence that with increasing excitation intensity, increase roughly is to be measured and not to that of the LED. The only in proportion to the square of this intensity. The exci- purpose of the LED is to feed the detector reproducible tation intensity required to produce these effects de- pulses of light in order to release corresponding pulses pends on the dye and the dimensions of the excitation of photoelectrons. Note that the light pulses from the focus. However, as a rough guide they may become LED do not have to follow the same optical path as the significant when the excitation intensity exceeds 100 fluorescence or to have the same wavelength. mW. The background, in contrast, is likely to continue Background Determination to increase in proportion to the excitation intensity. Thus as the excitation intensity is increased, the imWith the excitation light turned on, one can then provement of the signal t o noise ratio decreases and perform the same measurement in the presence of may eventually become negative. background to determine cvl, and subsequently calcuIt is seen from Eq. 21 that an alternative t o increas- late +enb from Eq. 9. ing the excitation intensity is to reduce the flow velocNote that the measured values of cvo and cvl must be ity. This is usually simpler and less expensive than a corrected for the variation added to the signal as it larger light source. A lower flow velocity does not en- passes through the PMT, according to Eqs. A3 and A5. hance fluorescence saturation. However, it should in- Thus the measured values should be divided by (1 crease bleaching in about the same proportion as a cor- l/k)-1’2 (see Fig. 6) before they are entered into Eqs. 8 responding increase of the excitation intensity. The and 9. disadvantages of reducing the flow velocity are mainly To determine to what extent the measured cv, is limthat: (1)the diameter of the sample core in the flow ited by photon noise, one may repeat the measurement

s

s

826

STEEN

7 Flow

,,

chamber Lens

L

-

-

-I

‘Filterblack

/. Measuring slit I”Pmhole”I

FIG.3. An arc lamp-based flow cytometer of the Argus type. Only the fluorescence part of the instrument is shown (8). A semitransparent mirror is mounted in the light path between the fluorescence collecting lens and the measuring slit (“pinhole”), so as to reflect a small portion of the light from a LED into the fluorescence detector system. The “semitransparent mirror” is a microscope coverslip with

antireflection coating, so that the fraction of light reflected at an angle of incidence of around 45“ is about 2%. Hence, the loss of fluorescence caused by this device is negligible and the LED can be operated at a relatively high power where it is easier to obtain highly reproducible pulses.

with different light intensities and plot C V vs. ~ inverse intensity, that is, the inverse peak channel number. The signal intensity can be varied simply by varying the pulse amplitude of the pulse generator. Its relative value is the peak or median value of the resulting histogram peak. According to Eqs. 8 and 18, the result should be a straight line intersecting the cvg axis a t a value which equals that part of the variance which is not due to photon noise. In some cases the background contains a component of scattered light from the cell or particle that is being measured, which is due to leakage of excitation light through the emission filter(s). This component, which is usually small relative to the constant part of the background, will add to that of the fluorescence signal. In such cases &nb must be determined by measuring a nonfluorescent particle with a light scattering intensity similar to that of the cell/particle in question, and the peak or median channel number of the resulting histogram peak must be subtracted from that obtained for the fluorescent sample in order to obtain the true value of &nf. To determine the variation in the signal due to sources other than photon noise, i.e., cv,, in particular the fluorescence variation of the particle sample, the fluorescence histogram of the sample, for which cv, is to be determined, is recorded with two or three different excitation intensities, while everything else is kept the same. Different intensities are easily obtained by the use of grey filters. A reduction factor in the range 2 to 5 is suitable. The filters do not have to be calibrated since the intensity is directly proportional t o the fluorescence intensity of the particles, i.e., t o the peak or median channel number of the histogram peak. Hence, by plotting the inverse peak or median fluores-

cence channel vs. cv:, one obtains a straight line which extrapolates to cvz for cv: = 0 (Eq. 18).

Determination of Fluorescence Sensitivity The fluorescence histogram of a sample of fluorescence calibrated standard particles is recorded with the same excitation light intensity and wavelength as employed for the determination of the background, and the fluorescence detection limit calculated from Eq. 14. In the present experiments we used particles with 2,4 . lo4 FITC mol. equivalents (Flow Cytometry Standards Corp., Research Triangle Park, NC). Instruments In the present work we used two different flow cytometers: A laser-based instrument (FacScan, Becton Dickinson, San Jose, CA) and an arc lamp-based instrument which was a prototype of the Argus 100 (Skatron AS, Tranby, Norway) with specifications resembling those of the commercial version. The latter instrument was equipped with a 100 W high pressure Hg arc lamp (HBO 100Wi2, Osram, Germany) as the excitation light source. The instrument, described in more detail elsewhere (631, employs epi-illumination through a 40x/1,3 oil immersion lens, and a flow chamber of the “jet on open surface” type. A LED (HLMP1540, Hewlett Packard, Santa Clara, CAI, which was driven by a pulse generator (8015A,Hewlett Packard), was mounted immediately behind the filter block, as shown in Figure 3, so that a small portion of its light emission was reflected via a semitransparent mirror into the fluorescence detection system of the instrument. In the FacScan instrument, the LED was placed just in front of the flow chamber, right opposite the fluo-

827

NOISE, SENSITIVITY, AND RESOLUTION IN FCM 0.04

-

0.03

"9 0.02 0

0.01

d 0.00

(Signal median)-'

FIG.4. The variance, cvi, of the LED pulse in the absence of excitation light, as recorded with the Argus instrument, plotted according to Eq. 18, that is, as a function of the inverse pulse intensity. The intensity is taken to be the median channel value of the corresponding pulse histogram. The observation that the data fall on a straight line confirms the basic assumption of Eq. 2. The extrapolation of this line through the origin implies that photon noise was the only noticeable source of signal variation.

rescence detection lens. In both instruments the LED did not interfere with the normal measuring functions, except for a slight reduction of the fluorescence signal (= 3%) caused by the semitransparent mirror in the Argus prototype.

RESULTS In Figure 4 is shown the value of cv," as recorded (with the excitation light turned off) with different LED intensities, plotted according to Eq. 8. It is seen that the data fall on a straight line, which confirms the underlying assumption of the above equations, namely, that expressed by Eq. 2. The fact that this line extrapolates to the origin implies that photon noise was the only noticeable source of signal variation under these conditions. Such a result is to be expected provided the LED and the pulse generator are of reasonable quality. If the line had extrapolated to a value significantly above the origin, the most likely reason would be ambient light reaching the detector. Figure 5 shows typical histograms recorded for a LED pulsed with the same intensity in the absence and presence of excitation light, respectively. For both the Argus prototype and the FacScan, it was found that the LED signal measured in the presence of excitation light was somewhat higher than that measured with the excitation light turned off. This shift increased with the cv, that is, as the signal approached the background level. The reason must be that the baseline restore circuit in the pulse amplifier was not working ideally in the presence of the continuous stream of (small) pulses produced by the background light, with the result that the baseline was slightly raised. The value of cvl may be corrected for this increase as follows. If the mean channel value of the peak recorded in

0

50

100

150

200

250

Channel number

FIG.5. Histograms of LED pulses as recorded with one of the fluorescence detectors of the Argus instrument, with the excitation light turned off (a) and on (b),respectively. The two histograms were recorded with the same LED light intensity. The difference in the median channel is due to incomplete baseline restoration in the pulse amplifier at the low signal level of these measurements.

the presence and absence of excitation light is mch, and mcho, respectively, the measured value of cvl should be multiplied with mch,/mcho. The values of the background level, &nb, as calculated from Eq. 9 for the various fluorescence detectors of the FacScan and for various filter combinations of the Argus prototype, are shown in Tables 1 and 2, respectively. Note that the value of &nb is quite different for different excitation- and/or emission filters. In the FacScan the background constant for detector FL2, i.e., the detector typically used to detect phycoerythrin, is almost one order of magnitude larger than that of the other two detectors. The probable reason is that the raman scattering of water, which with 488 nm excitation is a t about 584 nm, falls right in the middle of the spectral range of the FL2 detector. According to Figure 1, the sensitivity of FL2 can be increased by a factor of about 2,5 if the background constant could be brought to about the same level as for the other two channels. This should be possible by the use of a narrow band blocking filter to stop the raman scattering of water. The background level for the filter combinations used in the Argus prototype for measurement of FITC and phycoerythrin conjugated antibodies, i.e., the FITC and G1 filter block, respectively, was approximately similar to that of the FL1 and FL3 detectors of the FacScan, while with shorter excitation wavelengths it was more than two orders of magnitude higher (Table 2). This is partly due t o the fact that the excitation intensity with the UV and B1 filter blocks is about one order of magnitude larger than with the FITC excitation (7). The rest of the difference is due to a much higher level of fluorescence, excited by these wavelengths, from the coverslip and immersion oil of the flow chamber and of the microscope lens used for the

828

STEEN

Table 1 Background and Fluorescence Detection Limit of FacScan Instrument" Parameter Wavelengthbmj Exc.1Em. $en,

fi (FITC mol. eqv.) f, (FITC mol. eav.)

FL1

FL2

FL3

4881515-545 595

4881563-607

48812650

48

6

826

285

T h e background, $enb, as calculated according to Eq. 9 from measurements of the cv of histograms recorded for LED pulses in the absence and presence of background, and the fluorescence detection limit, fi, as determined from measurements of fluorescence calibrated standard particles (Eq. 14) for the three different fluorescence detectors of a laser-based flow cytometer (FacScan).The unit of the background is one photoelectron (from the PMT photocathode). f, is the value of the fluorescence detection limit that would have been obtained if the background was zero.

epi illumination and fluorescence collection in this instrument. By measuring &nb with and without the flow chamber, it was found that this lens caused 50%of the background measured with the B1 filter block and 23% of that measured with the UV filter block, the remainder being due primarily to the immersion oil. If these sources of background could be reduced to the level obtained with the FITC-and G1 filter blocks, that is, if the value of the background constant, b, were the same, the detection limit would be lowered by about a factor 5. To obtain the values of the fluorescence detection limit, fi, we measured a sample of fluorescence calibrated particles. The results, as calculated from Eq. 14 are given in Tables 1and 2. The fluorescence detection limit was determined also by the conventional method, using a set of four different FITC fluorescence calibrated particles and a n unstained control (Flow Cytometry Standards). For the FacScan we obtained fl = 1,200 FITC mol. equivalents, which is significantly higher than the value obtained by the method based on the equations above, i.e., fl = 826. At least part of the reason for this difference appeared to be t h a t the unstained control particles produced a signal which was significantly above the background. This signal was also larger than t h a t obtained from a sample of much smaller (2 km diam.j unstained polystyrene particles. Thus it appears that the unstained control of the calibration particle set either had a weak fluorescence or that the band filter of the FL1 detector was not able to block out completely all scattered excitation light. Tables 1 and 2 also give the value of fi as it would have been if the background had been completely eliminated, i.e., f,. It can be seen that the background is a major factor in limiting the sensitivity of these instruments. Thus the value of fi for measurement of FITC would be reduced by a factor of 2,4 for the FacScan and 3,s for the Argus prototype if the background could be reduced to negligible proportions. From Eq. 19 it can be seen that the same increase in sensitivity would re-

quire that the excitation intensity were raised by a factor of about 6 and 12, respectively. For the filter combinations with lower excitation wavelengths (UV and B1, Table 21, the potential reduction of the fluorescence detection limit is even much larger.

DISCUSSION The present method for characterization of flow cytometers, with regard to sensitivity, resolution, and level of background light, is based on the fact t h a t emission as well a s detection of light are stochastic processes, which implies that there is a n inherent limit to the reproducibility by which the intensity of light pulses can be measured. According to the theory of probability, there is a direct correlation between the number of photons or PMT photoelectrons in a signal and its coefficient of variation, which makes i t possible to determine the signal in absolute terms. This correlation, which is indeed the basis for the present work, is confirmed by the data of Figure 4. It is generally not trivial to measure light intensity in absolute terms, except by comparison with a calibrated standard. The principle of the present method can be applied to many different types of light measurement, including continuous signals, which can be measured in absolute terms by chopping either the light itself or the signal from the detector, and compare the cv of the resulting pulses to t h a t produced by a pulsed LED. The condition for such a measurement is that the photoelectron quantum efficiency of the light detector is known. The flow cytometer appears to lend itself particularly well to the application of this method for two reasons. First, it is in itself a pulsed light source, and second, photon noise is the dominant source of signal variation, at least in situations where background and sensitivity come into question, that is, where one is approaching the detection limit. The present methods may be implemented on almost any flow cytometer simply by introducing a suitable, pulsed LED in any position where some of its emission will reach the detector(s). Note that the emission spectrum of the LED does not have to match that of the dye for which the detector is to be calibrated. The function of the LED is only to produce pulses of photoelectrons from the photocathode of the PMT, so that one can calibrate the signal scale in absolute terms, i.e., in units of photoelectrons. Once this calibration is established, the number of photons in a signal can be calculated, provided the quantum yield of photoelectrons of the PMT photocathode is known. The above equations show how important the level of background light is for the sensitivity and measuring precision of flow cytometers (Fig. 1).Hence, minimizing background should be a major concern in the design of such instruments as well a s in their daily use. The possibility to monitor the background and sensitivity of a flow cytometer in absolute terms is obviously useful in the process of design and development of such

NOISE, SENSITIVITY, AND RESOLUTION IN FCM

Table 2 Background and Fluorescence Detection Filter block Wavelength(nm) E~C.IE~.

uv

I31

366(20)/400 1800

390-4401470 1200

of

829

Argus Prototype" FITC

G1

470-4901520-560 546(12)/575 9 6 f,(FITC mol. eqv.) 1800 f,(FITC mol. eqv.) 500 "The background, +enb,as calculated according to Eq. 9 from measurements of the cv of histograms recorded for LED pulses in the absence and presence of background, and the fluorescence detection limit, fi, as determined from measurements of fluorescence calibrated standard particles (Eq. 14) for four different filter blocks, i.e., combinations of excitation-and emission filters, in an arc lamp-based flow cytometer (Argus 100).The unit of the background is one photoelectron (from the PMT photocathode). f, is the value of the fluorescence detection limit that would have been obtained if the background was zero. $enb

instruments. In particular the measurement of background may disclose weaknesses and point to ways of improvement. As noted above, bleaching and/or fluorescence saturation may become important when the excitation energy exceeds about 100 mW. Hence, these phenomena put a limit to the extent to which sensitivity can be increased just by increasing the excitation intensity. Assuming that the optics are already optimal with regard to aperture and transmission, the only way to increase sensitivity further is to reduce the background. We have been assuming above that photon noise is the primary source of noise, while electronic noise can be reduced t o negligible proportions. The insignificance of electronic noise in flow cytometers is due t o the fact that the PMT, which is itself an amplifier with a gain factor up to lo7, exhibits very little inherent noise. Hence, to minimize electronic noise one should put as much of the signal amplification as possible on the PMT by using a relatively large high voltage and correspondingly lower amplifier gains. In contrast, the PMT contributes significantly to the cv and fluorescence detection limit due to its relatively low photoelectron quantum efficiency. For most PMTs, +e is typically between 0,05 and 0,2, depending on wavelength and the type of photocathode that is being used. Thus with currently available PMTs, the transformation of the signal from light to electrons costs an increase of the cv by about a factor of 2,2 to 4,5, depending on the value of +e (Eq. 71, and a corresponding reduction of the sensitivity (Eq. 19).Other types of photoelectric detectors, such as avalanche photodiodes, can have values of +e approaching 1. However, at present this advantage is more than offset by the much higher intrinsic noise of these devices. The dark current of the PMT is an insignificant source of noise in flow cytometers. This current, which is generated almost exclusively by single electrons, thermally released from the photocathode, is of the order of 100 electrons per second for PMTs commonly used in flow cytometers (e.g., Hamamatsu R 928). For a flow cytometer with a 10 p,s pulse length, this means that the probability that such a dark current pulse will coincide with the measurement of a cell is of the order

of lop3.Even in the case that such coincidence occurs, it hardly affects the measurement since the magnitude of the dark current pulses is usually well below that of a signal corresponding t o the detection limit. Hence, the dark current is not an essential parameter in the selection of PMTs for flow cytometers. As noted above, the background, nb, is the number of background photons reaching the detector during the integration time (or time constant) of the detector system, which is to say that in instruments with a continuous excitation light source nb is proportional to this integration time. Hence, if the integration time exceeds the duration of the pulse, nb increases and sensitivity and resolution will be reduced. In contrast, an integration time much shorter than the pulse length will also reduce sensitivity and resolution because nf is reduced. Hence, to obtain optimal performance it is important that the integration time matches the pulse length. The two most common reasons for the daily variations in the performance of flow cytometers seem to be: (1)misalignment of the sample flow relative to the laser focus, and (2) variation in the background level due to impurities in the flow chamber. Misalignment is detected by reduced signal and increased cv of the fluorescence and/or light scattering of a standard sample of strongly fluorescent monodisperse particles. Variation in background can be monitored by the method outlined above. A LED driven by a stable pulse generator may be mounted in most types of instruments. If integrated into the system during the design of the instrument the device may be run by the computer through which the instrument is controlled. Thus it would be possible to determine the background level automatically in a few seconds, without interference from the operator, e.g., when the instrument was turned on. The device could also be used to check that the light detectors as well as the pulse amplifiers and other signal electronics of the instrument function properly. For the latter purpose the LED should be run with much larger pulses, so that the cv of the resulting signal becomes small, i.e., of the order of 1 percent or less. Some commercial instruments do include a LED for the latter purpose. In the present work we have been dealing exclu-

830

STEEN

sively with the measurement of fluorescence. However, the equations above and the present method of measuring background and sensitivity can be applied equally well to the light scattering detectors of the instrument. However, since the light scattering signal is not a simple function of cell or particle size (4,101,it is not straightforward to define precise criteria for the sensitivity of the light scattering detection. Nevertheless, such a device could be useful to monitor day to day performance on a relative scale. The background level in the light scattering detectors of flow cytometers is generally much larger than for the fluorescence detectors. This is especially so for the detector measuring the scattering a t low scattering angles. Even in the Argus prototype, which readily measures 0,2 pm diam. polystyrene particles, the background is several hundred PMT photoelectrons.

ACKNOWLEDGMENTS I am indebted to Tore Lindmo and Sabina Merlo for constructive criticism of the manuscript, and to Torstein Schjerven for assistance in preparing the figures.

LITERATURE CITED 1. See, e.g., Arley N, Buch K R In: Introduction to the Theory of Probability and Statistics. John Wiley & Sons, New York, 19.50. 2. Pinkel D, Steen HB: Simple methods to determine and compare the sensitivity of flow cytometers. Cytometry 3:220-223, 1982. 3. Prescott JR: A statistical model for photomultiplier single-electron statistics. Nucl Instr Meths 39:173-179, 1966. 4. Salzman GC, Singham SB, Johnston RG, Bohren CF: Light scattering and cytometry. In: Flow Cytometry and Sorting, 2nd ed, Melamed MR, Lindmo T, Mendelsohn ML (eds) Wiley-Liss, New York, 1990, pp 81-107. 5. Schwartz A, Iglesias N, Fernandez-Pepollet: Flow cytometry performance survey. Flow Cytometry Standardization Forum 2:4-6, 1990. 6. Steen HB, Lindmo T: Flow cytometry: A high resolution instrument for everyone. Science 204:403-404, 1979. 7. Steen HB: Characteristics of flow cytometers. In: Flow Cytometry and Sorting, 2nd ed, Melamed MR, Lindmo T, Mendelsohn ML (eds) Wiley-Liss, New York, 1990, pp 11-25. 8. Steen HB: Flow cytometry instrumentation. In: Particle Analysis in Oceanography, NATO AS1 series, Demers S (ed.) SpringerVerlag, Berlin, 1991, pp 3-29. 9. Ubezio P, Andreoni A: Linearity and noise sources in flow cytometry. Cytometry 6:109-115, 1985. 10. Van de Hulst HC: Light Scattering of Small Particles. Dover, New York, 1981.

1.4

1.3 N

> h

Y

\

7

1.2

v c

1.1 ~

500

250

1000

750

1250

PMT High v o l t a g e (‘4)

FIG.6. The factor by which the signal variation, cv,, is increased by the passage of the signal down the dynode chain of a PMT. The calculations are based on the specifications of the Hamamatsu type R928 PMT, which has 9 dynodes. Assuming the same voltage across all the dynode stages, the stage gain, k, of this tube vanes from 2,4 at an tube voltage of 300 V to 6,O a t 1,000 V.

cally to the total. Assuming the same value of k for all dynodes a t a given high voltage applied to the PMT, the total variation in the number of electrons reaching the anode as a result of (n, + nb)+, photoelectrons emitted from the photocathode, is therefore:

+

cvPMT2= cv2/k + cvi/k2 . . . = C V @ ~ = , ~ ~ ~ ,=’ CV,l(l H l/kd+’)/(l - lk) - 11 ~

(A2)

where d is the number of dynodes. And since l/kd == 0: cvpMT2 cv;/(k 5

-

1)

(A3)

(AH

The value of k increases with the high voltage applied to the PMT. Using the Hamamatsu R928 PMT as a typical example, k = 2,4 a t HV = 300 V and k = 6,O a t HV = 1000 V, according to the specifications of the producer. Since d = 9 for this PMT the approximation of Eq. A3 is a very good one. This variation adds to that of all other sources of variation so that the measured variation (Eq. 16) becomes: CV, = C V ~+ CV; + cvpMT2= [CV: + cv;(l - lk)~11’’2 (A41 Thus it is seen that as the signal passes through the PMT, cvp increases by a factor of (l-l/k-l’z.The value of (l-l/k)-’” is plotted as a function of the PMT high voltage in Figure 6 . This increase of cvp must be considered a lower limit since other factors, like inhomogeneities in the electron emissive material on the PMT dynodes, may raise it further (3). As seen from Eq. A2, it is the first dynode which contributes most to the total variation. Hence, in order to reduce the variation due t o the PMT it is advisable to apply a fixed and relatively high voltage to the first dynode stage, as is generally recommended by the manufacturers of such devices. In this case it may be seen that Eq. A3 becomes:

where k is the average amplification of the number of electrons at that dynode. Since the dynodes are independent in this regard, the cv of each adds quadrati-

where k, is the electron amplification factor of the first dynode.

APPENDIX Assuming that the release of secondary electrons from the dynodes of PMTs is a stochastic process, so that the number of released electrons follows the Poisson distribution, the variance of the number of electrons emitted from a dynode having received (n, + nb)& electrons is: cvp

=

(nf

+ nb)/nf+,k) = cv$k

cvPm2

=

cv&/k,(k

-

1)

(A5)