Complete mathematical modeling method for the analysis of immunofluorescence distributions composed of negative and weakly positive cells

HD Flow Cytometry: An Improved Way to Quantify Cellular Interactions with Nanoparticles

Histogram deconvolution flow cytometry enables improved quantification of nanomaterial-cell interactions. The algorithm identifies the positive cells in highly overlapped populations and calculates the fluorescence intensity of the positive population. This technique performs better than commercially available methods with the additional benefit of visualizing the output.

Biostatistical analysis of quantitative immunofluorescence microscopy images

Semiquantitative immunofluorescence microscopy has become a key methodology in biomedical research. Typical statistical workflows are considered in the context of avoiding pseudo-replication and marginalising experimental error. However, immunofluorescence microscopy naturally generates hierarchically structured data that can be leveraged to improve statistical power and enrich biological interpretation. Herein, we describe a robust distribution fitting procedure and compare several statistical tests, outlining their potential advantages/disadvantages in the context of biological interpretation. Further, we describe tractable procedures for power analysis that incorporates the underlying distribution, sample size and number of images captured per sample. The procedures outlined have significant potential for increasing understanding of biological processes and decreasing both ethical and financial burden through experimental optimization.

Wound-healing growth factor, basic FGF, induces Erk1/2-dependent mechanical hyperalgesia

Growth factors such as nerve growth factor and glial cell line-derived neurotrophic factor are known to induce pain sensitization. However, a plethora of other growth factors is released during inflammation and tissue regeneration, and many of them are essential for wound healing. Which wound-healing factors also alter the sensitivity of nociceptive neurons is not well known. We studied the wound-healing factor, basic fibroblast growth factor (bFGF), for its role in pain sensitization. Reverse transcription polymerase chain reaction showed that the receptor of bFGF, FGFR1, is expressed in lumbar rat dorsal root ganglia (DRG). We demonstrated presence of FGFR1 protein in DRG neurons by a recently introduced quantitative automated immunofluorescent microscopic technique. FGFR1 was expressed in all lumbar DRG neurons as quantified by mixture modeling. Corroborating the mRNA and protein expression data, bFGF induced Erk1/2 phosphorylation in nociceptive neurons, which could be blocked by inhibition of FGF receptors. Furthermore, bFGF activated Erk1/2 in a dose- and time-dependent manner. Using single-cell electrophysiological recordings, we found that bFGF treatment of DRG neurons increased the current-density of NaV1.8 channels. Erk1/2 inhibitors abrogated this increase. Importantly, intradermal injection of bFGF in rats induced Erk1/2-dependent mechanical hyperalgesia.

PERSPECTIVE

Analyzing intracellular signaling dynamics in nociceptive neurons has proven to be a powerful approach to identify novel modulators of pain. In addition to describing a new sensitizing factor, our findings indicate the potential to investigate wound-healing factors for their role in nociception.

Analysis and simulation of division- and label-structured population models : a new tool to analyze proliferation assays

In most biological studies and processes, cell proliferation and population dynamics play an essential role. Due to this ubiquity, a multitude of mathematical models has been developed to describe these processes. While the simplest models only consider the size of the overall populations, others take division numbers and labeling of the cells into account. In this work, we present a modeling and computational framework for proliferating cell populations undergoing symmetric cell division, which incorporates both the discrete division number and continuous label dynamics. Thus, it allows for the consideration of division number-dependent parameters as well as the direct comparison of the model prediction with labeling experiments, e.g., performed with Carboxyfluorescein succinimidyl ester (CFSE), and can be shown to be a generalization of most existing models used to describe these data. We prove that under mild assumptions the resulting system of coupled partial differential equations (PDEs) can be decomposed into a system of ordinary differential equations (ODEs) and a set of decoupled PDEs, which drastically reduces the computational effort for simulating the model. Furthermore, the PDEs are solved analytically and the ODE system is truncated, which allows for the prediction of the label distribution of complex systems using a low-dimensional system of ODEs. In addition to modeling the label dynamics, we link the label-induced fluorescence to the measure fluorescence which includes autofluorescence. Furthermore, we provide an analytical approximation for the resulting numerically challenging convolution integral. This is illustrated by modeling and simulating a proliferating population with division number-dependent proliferation rate.

Threshold-free population analysis identifies larger DRG neurons to respond stronger to NGF stimulation

Sensory neurons in dorsal root ganglia (DRG) are highly heterogeneous in terms of cell size, protein expression, and signaling activity. To analyze their heterogeneity, threshold-based methods are commonly used, which often yield highly variable results due to the subjectivity of the individual investigator. In this work, we introduce a threshold-free analysis approach for sparse and highly heterogeneous datasets obtained from cultures of sensory neurons. This approach is based on population estimates and completely free of investigator-set parameters. With a quantitative automated microscope we measured the signaling state of single DRG neurons by immunofluorescently labeling phosphorylated, i.e., activated Erk1/2. The population density of sensory neurons with and without pain-sensitizing nerve growth factor (NGF) treatment was estimated using a kernel density estimator (KDE). By subtraction of both densities and integration of the positive part, a robust estimate for the size of the responsive subpopulations was obtained. To assure sufficiently large datasets, we determined the number of cells required for reliable estimates using a bootstrapping approach. The proposed methods were employed to analyze response kinetics and response amplitude of DRG neurons after NGF stimulation. We thereby determined the portion of NGF responsive cells on a true population basis. The analysis of the dose dependent NGF response unraveled a biphasic behavior, while the study of its time dependence showed a rapid response, which approached a steady state after less than five minutes. Analyzing two parameter correlations, we found that not only the number of responsive small-sized neurons exceeds the number of responsive large-sized neurons--which is commonly reported and could be explained by the excess of small-sized cells--but also the probability that small-sized cells respond to NGF is higher. In contrast, medium-sized and large-sized neurons showed a larger response amplitude in their mean Erk1/2 activity.

Data analysis in flow cytometry: the future just started

In the last 10 years, a tremendous progress characterized flow cytometry in its different aspects. In particular, major advances have been conducted regarding the hardware/instrumentation and reagent development, thus allowing fine cell analysis up to 20 parameters. As a result, this technology generates very complex datasets that demand for the development of optimal tools of analysis. Recently, many independent research groups approached the problem by using both supervised and unsupervised methods. In this article, we will review the new developments concerning the use of bioinformatics for polychromatic flow cytometry and propose what should be done to unravel the enormous heterogeneity of the cells we interrogate each day.

Watson's analytical approach for immunofluorescence histogram analysis

In this short note, the key points of the histogram analysis method proposed by Watson (Cytometry 2001;43:55-68) were examined. The complete relationship among the mean values of the distributions on which that method is based was derived analytically. It was shown that the symmetry assumption is not necessary, and that the assumption of equal variances of the unlabeled and labeled cell distributions cannot be verified a posteriori. Moreover, a possible shift between the unlabeled cell distribution in the test and the negative control may greatly influence the estimation of the labeled cell percentage. Finally, by using an experimental set of histograms, it was shown that the assumptions underlying Watson's analytical method are likely not to be fulfilled in real immunofluorescence data.

Sequential univariate gating approach to study the effects of erythropoietin in murine bone marrow

Analysis of multicolor flow cytometric data is traditionally based on the judgment of an expert, generally time consuming, sometimes incomplete and often subjective in nature. In this article, we investigate another statistical method using a Sequential Univariate Gating (SUG) algorithm to identify regions of interest between two groups of multivariate flow cytometric data. The metric used to differentiate between the groups of univariate distributions in SUG is the Kolmogorov-Smirnov distance (D) statistic. The performance of the algorithm is evaluated by applying it to a known three-color data set looking at activation of CD4+ and CD8+ lymphocytes with anti-CD3 antibody treatment and comparing the results to the expert analysis. The algorithm is then applied to a four-color data set used to study the effects of recombinant human erythropoietin (rHuEPO) on several murine bone marrow populations. SUG was used to identify regions of interest in the data and results compared to expert analysis and the current state-of-the-art statistical method, Frequency Difference Gating (FDG). Cluster analysis was then performed to identify subpopulations responding differently to rHuEPO. Expert analysis, SUG and FDG identified regions in the data that showed activation of CD4+ and CD8+ lymphocytes with anti-CD3 treatment. In the rHuEPO treated data sets, the expert and SUG identified a dose responsive expansion of only the erythroid precursor population. In contrast, FDG resulted in identification of regions of interest both in the erythroid precursors as well as in other bone marrow populations. Clustering within the regions of interest defined by SUG resulted in identification of four subpopulations of erythroid precursors that are morphologically distinct and show a differential response to rHuEPO treatment. Greatest expansion is seen in the basophilic and poly/orthochromic erythroblast populations with treatment. Identification of populations of interest can be performed using SUG in less subjective, time efficient, biologically interpretable manner that corroborates with the expert analysis. The results suggest that basophilic erythroblasts cells or their immediate precursors are an important target for the effects of rHuEPO in murine bone marrow. The MATLAB implementation of the method described in the article, both experimental data and other supplemental materials are freely available at http://web.mac.com/acidrap18.

Antibody arrays for quantitative immunophenotyping

Detection of multiple surface antigens expressed on living cell is an important step for cell processing and clinical diagnosis. Here we describe the preparation of antibody arrays that allow parallel detection of multiple surface antigens through affinity binding of living cells. An antibody array was fabricated by photo-assisted patterning of an alkanethiol monolayer formed on a gold-coated glass plate and subsequent immobilization of antibodies specific for cell surface antigens in an array format. We demonstrate here that rapid phenotyping can be performed on the array for both adhesion-dependent and non-dependent cells by direct cell binding assays. The density of bound cells on each antibody spot was in accordance with their contents in an original suspension. This result suggests the feasibility of the array-based method for quantitative assessment of multiple antigen expression. These findings will serve to extend the range of fundamental and clinical applications of antibody arrays.

Classifying Antibodies Using Flow Cytometry Data: Class Prediction and Class Discovery

Classifying monoclonal antibodies, based on the similarity of their binding to the proteins (antigens) on the surface of blood cells, is essential for progress in immunology, hematology and clinical medicine. The collaborative efforts of researchers from many countries have led to the classification of thousands of antibodies into 247 clusters of differentiation (CD). Classification is based on flow cytometry and biochemical data. In preliminary classifications of antibodies based on flow cytometry data, the object requiring classification (an antibody) is described by a set of random samples from unknown densities of fluorescence intensity. An individual sample is collected in the experiment, where a population of cells of a certain type is stained by the identical fluorescently marked replicates of the antibody of interest. Samples are collected for multiple cell types. The classification problems of interest include identifying new CDs (class discovery or unsupervised learning) and assigning new antibodies to the known CD clusters (class prediction or supervised learning). These problems have attracted limited attention from statisticians. We recommend a novel approach to the classification process in which a computer algorithm suggests to the analyst the subset of the "most appropriate" classifications of an antibody in class prediction problems or the "most similar" pairs/ groups of antibodies in class discovery problems. The suggested algorithm speeds up the analysis of a flow cytometry data by a factor 10-20. This allows the analyst to focus on the interpretation of the automatically suggested preliminary classification solutions and on planning the subsequent biochemical experiments.

A new spreadsheet method for the analysis of bivariate flow cytometric data

BACKGROUND

A useful application of flow cytometry is the investigation of cell receptor-ligand interactions. However such analyses are often compromised due to problems interpreting changes in ligand binding where the receptor expression is not constant. Commonly, problems are encountered due to cell treatments resulting in altered receptor expression levels, or when cell lines expressing a transfected receptor with variable expression are being compared. To overcome this limitation we have developed a Microsoft Excel spreadsheet that aims to automatically and effectively simplify flow cytometric data and perform statistical tests in order to provide a clearer graphical representation of results.

RESULTS

To demonstrate the use and advantages of this new spreadsheet method we have investigated the binding of the transmembrane adhesion receptor CD44 to its ligand hyaluronan. In the first example, phorbol ester treatment of cells results in both increased CD44 expression and increased hyaluronan binding. By applying the spreadsheet method we effectively demonstrate that this increased ligand binding results from receptor activation. In the second example we have compared AKR1 cells transfected either with wild type CD44 (WT CD44) or a mutant with a truncated cytoplasmic domain (CD44-T). These two populations do not have equivalent receptor expression levels but by using the spreadsheet method hyaluronan binding could be compared without the need to generate single cell clones or FACS sorting the cells for matching CD44 expression. By this method it was demonstrated that hyaluronan binding requires a threshold expression of CD44 and that this threshold is higher for CD44-T. However, at high CD44-T expression, binding was equivalent to WT CD44 indicating that the cytoplasmic domain has a role in presenting the receptor at the cell surface in a form required for efficient hyaluronan binding rather than modulating receptor activity.

CONCLUSION

Using the attached spreadsheets and instructions, a simple post-acquisition method for analysing bivariate flow cytometry data is provided. This method constitutes a straightforward improvement over the standard graphical output of flow cytometric data and has the significant advantage that ligand binding can be compared between cell populations irrespective of receptor expression levels.

Probability binning comparison: a metric for quantitating univariate distribution differences

BACKGROUND

Comparing distributions of data is an important goal in many applications. For example, determining whether two samples (e.g., a control and test sample) are statistically significantly different is useful to detect a response, or to provide feedback regarding instrument stability by detecting when collected data varies significantly over time.

METHODS

We apply a variant of the chi-squared statistic to comparing univariate distributions. In this variant, a control distribution is divided such that an equal number of events fall into each of the divisions, or bins. This approach is thereby a mini-max algorithm, in that it minimizes the maximum expected variance for the control distribution. The control-derived bins are then applied to test sample distributions, and a normalized chi-squared value is computed. We term this algorithm Probability Binning.

RESULTS

Using a Monte-Carlo simulation, we determined the distribution of chi-squared values obtained by comparing sets of events derived from the same distribution. Based on this distribution, we derive a conversion of any given chi-squared value into a metric that is analogous to a t-score, i.e., it can be used to estimate the probability that a test distribution is different from a control distribution. We demonstrate that this metric scales with the difference between two distributions, and can be used to rank samples according to similarity to a control. Finally, we demonstrate the applicability of this metric to ranking immunophenotyping distributions to suggest that it indeed can be used to objectively determine the relative distance of distributions compared to a single control.

CONCLUSION

Probability Binning, as shown here, provides a useful metric for determining the probability that two or more flow cytometric data distributions are different. This metric can also be used to rank distributions to identify which are most similar or dissimilar. In addition, the algorithm can be used to quantitate contamination of even highly-overlapping populations. Finally, as demonstrated in an accompanying paper, Probability Binning can be used to gate on events that represent significantly different subsets from a control sample. Published 2001 Wiley-Liss, Inc.

Probability binning comparison: a metric for quantitating multivariate distribution differences

BACKGROUND

While several algorithms for the comparison of univariate distributions arising from flow cytometric analyses have been developed and studied for many years, algorithms for comparing multivariate distributions remain elusive. Such algorithms could be useful for comparing differences between samples based on several independent measurements, rather than differences based on any single measurement. It is conceivable that distributions could be completely distinct in multivariate space, but unresolvable in any combination of univariate histograms. Multivariate comparisons could also be useful for providing feedback about instrument stability, when only subtle changes in measurements are occurring.

METHODS

We apply a variant of Probability Binning, described in the accompanying article, to multidimensional data. In this approach, hyper-rectangles of n dimensions (where n is the number of measurements being compared) comprise the bins used for the chi-squared statistic. These hyper-dimensional bins are constructed such that the control sample has the same number of events in each bin; the bins are then applied to the test samples for chi-squared calculations.

RESULTS

Using a Monte-Carlo simulation, we determined the distribution of chi-squared values obtained by comparing sets of events from the same distribution; this distribution of chi-squared values was identical as for the univariate algorithm. Hence, the same formulae can be used to construct a metric, analogous to a t-score, that estimates the probability with which distributions are distinct. As for univariate comparisons, this metric scales with the difference between two distributions, and can be used to rank samples according to similarity to a control. We apply the algorithm to multivariate immunophenotyping data, and demonstrate that it can be used to discriminate distinct samples and to rank samples according to a biologically-meaningful difference.

CONCLUSION

Probability binning, as shown here, provides a useful metric for determining the probability with which two or more multivariate distributions represent distinct sets of data. The metric can be used to identify the similarity or dissimilarity of samples. Finally, as demonstrated in the accompanying paper, the algorithm can be used to gate on events in one sample that are different from a control sample, even if those events cannot be distinguished on the basis of any combination of univariate or bivariate displays. Published 2001 Wiley-Liss, Inc.

A brief history of numbers and statistics with cytometric applications

A brief history of numbers and statistics traces the development of numbers from prehistory to completion of our current system of numeration with the introduction of the decimal fraction by Viete, Stevin, Burgi, and Galileo at the turn of the 16th century. This was followed by the development of what we now know as probability theory by Pascal, Fermat, and Huygens in the mid-17th century which arose in connection with questions in gambling with dice and can be regarded as the origin of statistics. The three main probability distributions on which statistics depend were introduced and/or formalized between the mid-17th and early 19th centuries: the binomial distribution by Pascal; the normal distribution by de Moivre, Gauss, and Laplace, and the Poisson distribution by Poisson. The formal discipline of statistics commenced with the works of Pearson, Yule, and Gosset at the turn of the 19th century when the first statistical tests were introduced. Elementary descriptions of the statistical tests most likely to be used in conjunction with cytometric data are given and it is shown how these can be applied to the analysis of difficult immunofluorescence distributions when there is overlap between the labeled and unlabeled cell populations.

Proof without prejudice revisited: immunofluorescence histogram analysis using cumulative frequency subtraction plus ratio analysis of means

BACKGROUND

Apart from the work of Lampariello and colleagues (Cytometry 15:294-301, 1994; Cytometry 32:241-254, 1998), very little analytical work has been carried out for analysis of immunofluorescence distributions containing an overlapping mixture of labeled and unlabeled cells. The methods developed tend to rely on fitting theoretical distributions to the relevant populations. However, the method described here attempts to produce an analytical solution.

METHODS

A new method for immunofluorescence histogram analysis is presented. It uses cumulative frequency distribution subtraction of the test sample from the control to predict the mean of a labeled cell component embedded within a histogram containing unlabeled cells. Ratio analysis of means (RAM) was then carried out to calculate the labeled fraction. The results were submitted to Kolmogorov-Smirnov analysis and Student's t-test for validation at a given level of probability.

RESULTS

The method was developed with a data set exhibiting a small "positive" shoulder, which was predicted to contain a labeled fraction comprising 8.0% of the total at the 99% confidence limit. It was then tested with data analyzed and published previously where the Johnson Su family of distributions was used in curve fitting.

CONCLUSIONS

There was good agreement between the known and predicted proportions of labeled cells. However, the method is dependent on the symmetry of the distributions. Some minor systematic errors were encountered due, in part, to skewed experimental distributions.

Cross-correlation for flow cytometric histogram background subtractions

Background subtraction is a widely encountered problem in flow cytometry applications for which the currently available analysis techniques are unsatisfactory. The 99% division line method, also referred to as the threshold or marker method, is widely used because it is computationally simple but it has poor accuracy and tends to underestimate the percentage of positive cells when there is overlap between histograms. Model-based approaches are preferred when there are overlapping peaks, but these methods require curve fitting and strong assumptions regarding the shape of the underlying distributions. This report assesses a mathematically rigorous, computationally facile, non-parametric technique called cross correlation for the background subtraction problem. A metric, positivity, derived from cross correlation is shown to overcome the disadvantages of both the 99% division line and model-based methods without compromise.

On the use of the Kolmogorov-Smirnov statistical test for immunofluorescence histogram comparison

BACKGROUND

The problem considered is the quantitative comparison of immunofluorescence frequency distributions in order to detect their differences of biological significance, i.e., to evaluate the potential positivity of a cell sample with respect to negative control cells. The Kolmogorov-Smirnov (KS) statistical test, proposed in the literature for this purpose, is examined and discussed through its application to a set of experimental measurements. It is shown that even differences due to the stain procedure or to instrumental biases may be considered significant by the test implemented in the standard form.

METHODS

In order to ensure valid results, it is necessary to take into account the various sources of variation in the specific experimental context. A procedure is proposed that uses the KS statistics as a reference for determining an appropriate estimate of the overall variability in the control data. This estimate is derived from the comparisons of the cumulative distributions associated with repeated measurements of the negative cell sample.

RESULTS AND CONCLUSIONS

The KS-related index thus defined provides a tool for assessing the potential positivity of a cell sample, since it allows to distinguish between statistical and biological significance of the difference between the histogram to be tested and the set of control data. In particular, if a cell sample is not included in the control variability, either a positive cell subpopulation is present, or all cells are positive. Instead, for a sample included in the control variability, the difference will be not biologically meaningful, even if statistically significant. Moreover, when a purely positive control sample is also available, it is possible to derive a measure of the precision at which a true biological positivity can be detected. Finally, since the index is not absolute, but relative to the features of the instrumentation, of the antibodies and of the fluorochromes used, it represents a quantitative measure of the stability and reproducibility of the measurement process and could be used for quality control of flow cytometric experiments in immunofluorescence.