Improved Compensation in Flow Cytometry by Multi-Variable Optimization

Abstract

Conventional compensation of flow cytometry (FMC) data of an N-stained sample requires additional data sets, of N single-stained control samples, to estimate the spillover coefficients. Single-stained controls however are the least rigorous controls because any of the multi-stained controls are closer to the N-stained sample. In this paper a new, optimization based, compensation method has been developed that is able to use not only single- but also multi-stained controls to improve estimates of the spillover coefficients. The method is demonstrated on a data set from 5-stained dentritic cells (DCs) with 5 single-stained and 8 multi-stained controls. This approach is practical and leads to significant improvements in FCM compensation.

Introduction

Every fluorescent molecule emits light with a spectrum characteristic of that molecule. These emission spectra overlap, in some cases very significantly. Optical filters are used to limit the range of frequencies measured by a given detector. However, when two or more fluorochromes are used, the spectral overlap often makes it impossible for optical filters to isolate light from a given fluorochrome. As a result, light emitted from one fluorochrome appears in a detector intended for another. This is referred to as spillover and it is represented by a constant known as the spillover coefficient. Spillover can be corrected mathematically by using a method called compensation. Compensation removes the signal of any given fluorochrome from all detectors except the one devoted to measuring that fluorochrome. Several compensation methods have been developed, as reviewed by Roederer(1). The most useful method to date, the multicolor method, we refer to as conventional compensation.

Conventional compensation of an N-stained sample has a simple way of estimating the spillover coefficients by the direct analysis of N controls from single-stained samples. By using these spillover coefficients one can construct a set of N linear algebraic equations, providing relationships between the measured intensities of the N-stained sample and the unknown compensated intensities. Finally, by means of standard matrix inversion, one can get the result: the N compensated intensities.

However it was long recognized that single-stained controls are the least rigorous controls because any of the multi-stained controls are closer to the N-stained sample. For example the most rigorous controls, the (N−1)-stained controls, have been utilized to discriminate positive from negative events in N-stained FCM data. This is called fluorescence-minus-one (FMO) gating(1).

The method, described in this paper, can utilize any number and type of controls to compensate the N-stained data. From these controls at least 1 and at most N−1 stains are missing. The detector designated to measure the photons from the j-th dye actually measures photons from three sources: (1) primarily from the j-th dye, (2) spillover photons from other dyes, and (3) photons from autofluorescence. Naturally if the j-th dye is missing, photons are coming to the detector only from source (2) and (3). The controls are utilized to estimate the spillover coefficients by means of an optimization technique. Once spillover coefficients are determined, the compensated intensities are calculated similarly to conventional compensation.

The optimization is based on the recognition that, by using the above mentioned linear algebraic calculations, one can compensate every control as well. These compensations are considered to be correct if the compensated intensities at the missing dyes are close to zero. The spillover coefficients are optimally selected when the compensation of each control is correct. The new compensation method is demonstrated on a data set from 5-stained dentritic cells (DCs) with 5 single-stained and 8 multi-stained controls.

Materials and Methods

Experiments

Experiments have been performed on DCs by using a BD LSR II flow cytometer. Table 1 lists the utilized 5 fluorochromes and the characteristics of the respective band pass filters. 5-stained samples were prepared as follows. DCs were fixed with 1.5% paraformadehyde for 15 minutes at room temperature, then washed twice, and permeabilized with 100% methanol. During the permeabilization, the cells were stained with appropriate concentration of Alexa-350 and Alexa-750. After washing the cells twice, the cells were stained with the other three dyes. Single and multi-stained controls were prepared similarly by using the appropriate stains. Each control sample can be specified by a 5-element vector, v. For example vector v=(00101) specifies a control sample where the DCs were stained only by the 3^rd and 5^th dyes. Table 2 lists the v vectors that specify the control samples. Every experiment utilized the same type of cells, dyes from the same supplier, and exactly the same settings of the flow cytometer, i.e. fluorescent intensities were measured by 5 detectors and using the same 5 band pass filters.

Table 1.

Fluorochromes and band pass filters utilized in the FCM experiments on DCs

Fluorochrome’s code number	1	2	3	4	5
Fluorochrome’s name*	FITC	Alexa-350	APC	Alexa-750	PE
Band pass filter characteristics (center wavelength/band width in nm)	530/30	450/50	660/20	780/60	575/30
Detector’s code number	1	2	3	4	5

^*The full names and catalog numbers of the reagents used are:

CD45-FITC (Beckman Coulter, Cat# PN IM0782U)

Alexa Fluor 350 carboxylic acid, succinimidyl ester (Invitrogen, Cat# A10168)

CD45-APC (Beckman Coulter, Cat# PN IM2473U)

Alexa Fluor 750 carboxylic acid, succinimidyl ester (Invitrogen, Cat# A20011)

PE anti-human CD45 (BioLegend, Cat# 304008)

The spectra of the fluorochromes can be viewed at http://www.invitrogen.com/site/us/en/home/support/Research-Tools/Fluorescence-SpectraViewer.html

Table 2.

List of v vectors specifying the controls to the 5-stained experiment with DCs

1-stained	2-stained	3-stained	4-stained
(1,0,0,0,0)	(0,1,0,1,0)	(1,0,1,0,1)	(0,1,1,1,1)
(0,1,0,0,0)		(1,1,0,1,0)	(1,0,1,1,1)
(0,0,1,0,0)			(1,1,0,1,1)
(0,0,0,1,0)			(1,1,1,0,1)
(0,0,0,0,1)			(1,1,1,1,0)

Theory

Data Sets

In this section we describe step-by-step the method of compensating FCM data of cells stained by N fluorescent dyes. The data set belonging to any of the control experiments D^(v) is a matrix, where the i-th row lists the fluorescence intensities measured by N(=5) detectors on the i-th cell. The number of rows, n^(v), is the number of cells measured in this data set. The average intensity measured by the j-th detector is:

$$\langle D_j^{(v)}\rangle =\frac{{\displaystyle \sum _{i=1}^{n^{(v)}}{D}_{\mathit{\text{ij}}}^{(v)}}}{n^{(v)}}$$ (1)

Although the above defined vector v specifies any of the control samples, for convenience we use the following alternative notations for single-stained controls: D^(1K), where the cells are stained only by the K-th dye. The data matrices belonging to the N-stained sample and the unstained sample are marked by D^(N) and D⁽⁰⁾, respectively.

Determining Spillover Coefficients in Conventional Compensation

The K-th detector is designated to measure photons of the K-th dye. However, the spectrum of the K-th dye is usually broad and other detectors may also detect photons emitted by the K-th dye. Let $G_j^{(K)}\text{ and }{G}_K^{(K)}$ denote fluorescent intensities from the K-th dye that are detected by the j-th and K-th detector, respectively. The ratio of these intensities defines the spillover coefficient, $S_j^{(K)}$, i.e.:

$$S_j^{(K)}=G_j^{(K)}/G_K^{(K)}$$ (2)

In conventional compensation the spillover coefficients, $S_j^{0(K)}$ are usually calculated by using every single-stained control as follows(1,2):

$$S_j^{0(K)}=\frac{\langle D_j^{(1,K)}\rangle -\langle D_j^{(0)}\rangle }{\langle D_K^{(1,K)}\rangle -\langle D_K^{(0)}\rangle }$$ (3)

Calculating Compensation by Linear Algebra

Once the spillover coefficients are available the following linear algebraic method is utilized to calculate the compensated intensities.

Let us consider the intensities measured on the i-th cell of the N-stained sample. The intensity measured by the j-th detector, $D_{\mathit{\text{ij}}}^{(N)}$ is originated primarily from the j-th dye. The detected intensity of this dye at the j-th detector is $C_{\mathit{\text{ij}}}^{(N)}$, but signals from other dyes may spill over into the j-th detector too. Compensation of flow cytometry data means the determination of every $C_{\mathit{\text{ij}}}^{(N)}$ intensity by using both the detected data set, D^(N) and the spillover coefficients, S̿. If the intensity of the K-th dye (that is detected by the K-th detector) is $C_{\mathit{\text{iK}}}^{(N)}$ the intensity that is spilled over to the j-th detector is $S_j^{(K)}{C}_{\mathit{\text{iK}}}^{(N)}$. Thus the overall intensity measured by the j-th detector is:

$$D_{\mathit{\text{ij}}}^{(N)}\cong S_j^{(1)}{C}_{i1}^{(N)}+S_j^{(2)}{C}_{i2}^{(N)}+\dots +C_{\mathit{\text{ij}}}^{(N)}+\dots +S_j^{(N)}{C}_{\mathit{\text{iN}}}^{(N)}+\langle D_j^{(0)}\rangle$$ (4)

where $\langle D_j^{(0)}\rangle$ is the average intensity of the cells’ auto-fluorescence that falls into the j-th detector. After averaging Eq.4 over the cells we get

$$\langle D_j^{(N)}\rangle =S_j^{(1)}\langle C_1^{(N)}\rangle +S_j^{(2)}\langle C_2^{(N)}\rangle +\dots +\langle C_j^{(N)}\rangle +\dots +S_j^{(N)}\langle C_N^{(N)}\rangle +\langle D_j^{(0)}\rangle$$ (5)

Similar equations hold for the average intensities measured by the remaining N−1 detectors. This set of N linear algebraic equations can be written by using matrix notations:

$$\overline{\langle D^{(N)}\rangle }-\overline{\langle D^{(0)}\rangle }=\overline{\langle C^{(N)}\rangle }\overline{\overline{S}}$$ (6)

where $\overline{\langle D^{(N)}\rangle }=(\langle D_1^{(N)}\rangle ,\langle D_2^{(N)}\rangle ,\cdots ,\langle D_N^{(N)}\rangle )\text{ and }\overline{\langle C^{(N)}\rangle }=(\langle C_1^{(N)}\rangle ,\langle C_2^{(N)}\rangle ,\cdots ,\langle C_N^{(N)}\rangle )$ are N-element row vectors of the average measured intensities and average compensated intensities, respectively. S̿ is an N×N square matrix of the spillover coefficients, where the element of the K-th row and j-th column is $S_j^{(K)}$. One can calculate the average compensated intensities, $\overline{\langle C^{(N)}\rangle }$, by solving the above set of equations (Eq.6):

$$\overline{\langle C^{(N)}\rangle }=(\overline{\langle D^{(N)}\rangle }-\overline{\langle D^{(0)}\rangle }){\overline{\overline{S}}}^{-1}$$ (7)

where S̿⁻¹ is the inverse of the spillover matrix. In this work the inversion has been performed by using the standard technique of LU decomposition(3).

Compensating the Controls

Each control sample is specified by a v vector of N components (see Table 2). Each component refers to one of the dye of the N-stained sample. The value of a vector component is 1 or 0 if the respective dye is present or missing from the control. There is always one or more 0’s in every v vector because certain dyes are always missing from any of the control samples.

By using the same inverse spillover matrix as in Eq.7 one can calculate the average compensated intensities, $\overline{\langle C^{(v)}\rangle }$ for any of the controls:

$$\overline{\langle C^{(v)}\rangle }=(\overline{\langle D^{(v)}\rangle }-\overline{\langle D^{(0)}\rangle }){\overline{\overline{S}}}^{-1}$$ (8)

where the components of the $\overline{\langle C^{(v)}\rangle }$ vector are $(\langle C_1^{(v)}\rangle ,\langle C_2^{(v)}\rangle ,\cdots ,\langle C_N^{(v)}\rangle )$ and the j-th component of this vector, $\langle C_j^{(v)}\rangle$ is the average intensity of the j-th dye that is detected at the j-th detector. Since certain dyes are missing from every control sample in the case of correct compensation it is expected that the respective components of the $\overline{\langle C^{(v)}\rangle }$ vectors are close to zero.

Based on the above criterion one can check the goodness of conventional compensation, i.e. when $\overline{\overline{S}}=\overline{\overline{S^0}}$. By using the definition of $\overline{\overline{S^0}}$ (see Eq.3) we could solve the linear algebraic equations, $\overline{\langle D^{(v)}\rangle }-\overline{\langle D^{(0)}\rangle }=\overline{\langle C^{(v)}\rangle }\overline{\overline{S^0}}$, for single-stained controls and obtained: $\langle C_j^{(1,K)}\rangle =0$ for every j except for j=K we get $\langle C_K^{(1,K)}\rangle =\langle D_K^{(1,K)}\rangle -\langle D_K^{(0)}\rangle$. Thus the compensation of every single-stained control is correct. However, as we will point out in the Results and Discussions, the compensations of multi-stained controls are not so accurate.

Determining Spillover Coefficients by Optimization

Conventionally the spillover coefficients are determined from the single-stained controls by Eq.3. These spillover coefficients result in correct compensations of the single-stained controls, but not so correct compensations of the multi-stained controls (see Results and Discussions). Since single-stained samples are the least rigorous controls, this uneven nature of the compensation, based on the $S_j^{0(K)}$ spillover coefficients, is not satisfactory. We need refined values of the spillover coefficients, $S_j^{(K)}$, that result in an overall better compensation. An overall better compensation can be achieved by a spillover matrix S̿ that minimizes the following objective function:

$$F_{\mathit{\text{obj}}}\left(\overline{\overline{S}}\right)={\displaystyle \sum _{\{v\}}\left[{\displaystyle \sum _{\begin{array}{c}\hfill i=1\hfill \\ \hfill v_i\ne 1\hfill \end{array}}^N{\langle C_i^{(v)}\rangle }^2}\right]}$$ (9)

where the right hand side of the equation characterizes the overall deviation from the correct compensation for all the control samples. As an example let us assume that N=3, and the available controls are specified by the following v vectors: (1,0,0), (0,1,0), (0,0,1), (1,1,0),(1,0,1). In this case the objective function is:

$${\displaystyle \sum _{\{v\}}\left[{\displaystyle \sum _{\underset{v_i\ne 1}{i=1}}^3{\langle C_i^{(v)}\rangle }^2}\right]}=\left[{\langle C_2^{(1,0,0)}\rangle }^2+{\langle C_3^{(1,0,0)}\rangle }^2\right]+\left[{\langle C_1^{(0,1,0)}\rangle }^2+{\langle C_3^{(0,1,0)}\rangle }^2\right]+\left[{\langle C_1^{(0,0,1)}\rangle }^2+{\langle C_2^{(0,0,1)}\rangle }^2\right]+\left[{\langle C_3^{(1,1,0)}\rangle }^2\right]+\left[{\langle C_2^{(1,0,1)}\rangle }^2\right]$$ (10)

The local minimum of the objective function F_obj(S̿) is searched around the initial estimate of the spillover coefficients, $S_j^{(0,K)}$ by means of the standard Powell method(3). This multi-variable optimization takes place for the N(N−1) off-diagonal elements of the spillover matrix, S̿, while the N diagonal elements remain equal to 1.

Results and Discussions

In this section the control FCM data sets (from DCs) are compensated by both the conventional and the optimization based methods, and then comparison is made between their performances.

Figure 1a shows the average intensities, $\langle D_j^{(v)}\rangle$ ’s, measured for different control samples at different detectors. The control sample and the detector is specified by the following notation: (v)_ j, where v vector specifies the control sample and j index specifies the detector. The v vectors, specifying all the controls of the basic 5-stained experiment, are listed in Table 2. Out of the 5 stains at least 1 and at most 4 stains are missing in these controls. The zero components of each v vector refer to the missing dyes. In Figure 1a only those averages are plotted that belong to missing dyes. For example in the case of control sample (1,0,1,0,1) only the average intensities measured at the 2^nd and 4^th detector are plotted. The sum of the squares of the plotted averages is: $F={\displaystyle \sum _v{\displaystyle \sum _{\begin{array}{c}\hfill \{j\}\hfill \\ \hfill v_j\ne 1\hfill \end{array}}^5{\langle D_j^{(v)}\rangle }^2}}=31,666,000$ and the root of the mean of these squares is: $|\mathrm{\Delta }|=\sqrt{F/32}=994.8$, where 32 is the number of plotted columns. In the absence of autofluorescence and spillover all these averages, $\langle D_j^{(v)}\rangle$ ’s would be equal to zero and the respective F value would be equal to zero too. In Figure 1a the error bars show the standard deviations of each plotted data. The average of the standard deviations is about 150.

Figure 1. Steps of compensation of the FCM data of 5-stained dentritic cells.

a) The column plot shows 32 averages, $\langle D_j^{(v)}\rangle$ and standard deviations of fluorescence intensities detected in 13 control samples. The control samples of the 5-stained sample are specified by v vectors that are listed in Table 2. At the abscissa each column is marked by a (v)_ j symbol, where vector v and subscript j specify the control experiment and the detector where the intensity was measured, respectively. Only those averages are plotted that belong to missing dyes, i.e. the j-th component of the v vector is zero. After correct compensation all these averages should approach zero. At the abscissa symbols referring to single-stained controls are framed by red. b) Result of conventional compensation. $\langle C_j^{(v)}(\overline{\overline{S^0}})\rangle$ is the compensated average intensity of the j-th stain that is detected at the j-th detector in control sample v. In conventional compensation the spillover matrix, $\overline{\overline{S^0}}$ is determined from Eq.3 by using data from single-stained controls. c) Result of optimization based compensation. $\langle C_j^{(v)}(\overline{\overline{S}})\rangle$ is the compensated average intensity of the j-th stain that is detected at the j-th detector in control sample v. For the compensation the spillover matrix, S̿ was determined by minimizing the objective function in Eq.9. d) Result of optimization based compensation using a modified objective function (in Eq.9 performing the first summation over all the multi-stained controls while disregarding the single-stained controls). $\langle C_j^{(v)}(\overline{\overline{S}})\rangle$ is the compensated average intensity of the j-th stain that is detected at the j-th detector in control sample v. For the compensation the spillover matrix, S̿ was determined by minimizing the modified objective function.

In every subfigure F is the sum of the squares of the plotted averages. $|\mathrm{\Delta }|=\sqrt{\frac{F}{32}}$ is the root of the mean of the squares of the plotted averages.

Conventional Compensation

In order to perform conventional compensation, we determine from Eq.3 the spillover coefficients, $S_j^{(0,K)}$ by using the single-stained controls.

After the inversion of $\overline{\overline{S^0}}$ spillover matrix we use Eq.8 to calculate the vector of the average compensated intensities, $\overline{\langle C^{(v)}\rangle }$ for every control. The components of the $\overline{\langle C^{(v)}\rangle }$ vector are $(\langle C_1^{(v)}\rangle ,\langle C_2^{(v)}\rangle ,\cdots ,\langle C_N^{(v)}\rangle )$ and the j-th component of this vector, $\langle C_j^{(v)}\rangle$ is the average intensity of the j-th dye that is detected at the j-th detector. Since certain dyes are missing from every control, in the case of correct compensation it is expected that the respective components of the $\overline{\langle C^{(v)}\rangle }$ vectors are close to zero. In Figure 1b only those components of the $\overline{\langle C^{(v)}\rangle }$ vectors are shown that belong to the missing dyes. We can see that the compensation is correct for single-stained controls (framed by red at the abscissa), i.e.: $\langle C_j^{(v)}\rangle =0$, while for multiple-stained controls $\langle C_j^{(v)}\rangle$ ’s may deviate considerably from zero. The sum of the squares of the plotted averages, $\langle C_j^{(v)}\rangle$, is: $F={\displaystyle \sum _v{\displaystyle \sum _{\begin{array}{c}\hfill \{j\}\hfill \\ \hfill v_j\ne 1\hfill \end{array}}^5{\langle C_j^{(v)}\rangle }^2}}=56,504$. The actual value of F measures the overall deviation from perfect compensation. The root of the mean of these squares is: $|\mathrm{\Delta }|=\sqrt{F/32}=42$ which is about 28% of the average standard deviation of the uncompensated data. This is a considerable reduction of the F value relative to the uncompensated case. But it remains disturbing that the compensation of the multi-stained controls remains not so good. Finally it is important to note that in this case the definition of F and the definition of the objective function, F_obj, coincide (see Eq.9).

Compensations by Optimization

The conventional method results in correct compensation of the single-stained controls but inaccurate compensation for multi-stained controls. One can improve the overall compensation of the controls by finding refined spillover coefficients, $S_j^{(K)}$, that minimize the objective function in Eq.9. The initial values of the spillover coefficients are given by $\overline{\overline{S^0}}$ matrix. At these initial values the value of the objective function is $F_{\mathit{\text{obj}}}\left(\overline{\overline{S^0}}\right)=F=56,504$, while at the optimal values of the spillover coefficients the objective function reduces to $F_{\mathit{\text{obj}}}\left(\overline{\overline{S}}\right)=F=18,238$. By using the optimized spillover coefficients from Eq.8 we calculate the vector of the average compensated intensities, $\overline{\langle C^{(v)}\rangle }$ for every control. In Figure 1c components of the $\overline{\langle C^{(v)}\rangle }$ vectors are shown that belong to the missing dyes. The compensation is perfect for none of these components, i.e. $\langle C_j^{(v)}\rangle \ne 0$, but 1) the deviation from perfect compensation is homogeneously distributed along the controls, and 2) the root of the mean of the squares of the deviations |Δ|= 23.9 is considerably smaller than for conventional compensation. The significant reduction of the F and |Δ| values show that the optimization based compensation is overall better than the conventional compensation.

In the above example the deviation from perfect compensation was similar for single-stained and multi-stained controls. By means of the optimization based method one can emphasize the importance of the multi-stained controls and make their compensation perfect. In order to do this we slightly modify the objective function in Eq.9 performing the first summation over all the multi-stained controls while disregarding the single-stained controls. The spillover coefficients obtained from this optimization result in uneven compensation of the controls: perfect for multi-stained controls and not so good for single-stained controls (see Figure 1d). The F and |Δ| values of this optimization based compensation are similar to that of the conventional compensation. Conventional compensation gave an uneven compensation too but in an opposite sense: perfect for single-stained and not so good for multi-stained controls.

An executable file containing the compensation program is available at the authors' website (www.tsb.mssm.edu/Primeportal).

Conclusions

Conventional compensation of flow cytometry data of an N-stained sample requires additional data sets, of N single-stained control samples, to estimate the spillover coefficients. Single-stained controls however are the least rigorous controls because any of the multi-stained controls are closer to the N-stained sample. In this paper a new, optimization based, compensation method has been developed that is able to use not only single- but also multi-stained controls to improve estimates of the spillover coefficients. The method is demonstrated on a data set from 5-stained dentritic cells with 5 single-stained and 8 multi-stained controls. This approach is practical and leads to significant improvements in FCM compensation.