Background

Introduction

Flow cytometry clustering for several hundred million cells has long been hampered by software limitations. Julia allows us to go beyond these limits. Through the high-performance GigaSOM.jl package, we gear up for huge-scale flow cytometry analysis, softening these limitations with an innovative approach on the existing algorithm, by implementing parallel computing with stable and robust mathematical models that would run with an HPC cluster, allowing us to run cytometry datasets as big as 500 million cells.

Multidimensional Data

Multidimensional data is a term used when combining four, five (or more) characteristics simultaneously to create a data space where each measured cell parameter becomes a dimension in the multidimensional space.
The data generated by Flow Cytometry, Mass Cytometry or RNA-seq are good examples of multidimensional datasets, as they combine all information from all characteristics to create a multidimensional data space that preserves the integrity of the relationships between each of the parameters.[1]

Flow and Mass Cytometry

The use of flow cytometry has grown substantially in the past decade, mainly due to the development of smaller, user-friendly and less expensive instruments, but also to the increase of clinical applications, like cell counting, cell sorting, detection of biomarkers or protein engineering. Flow cytometry is an immunophenotyping technique used to identify and quantify the cells of the immune system by analysing their physical and chemical characteristics in a fluid. These cells are stained with specific, fluorescently labelled antibodies and then analysed with a flow cytometer, where the fluorescence intensity is measured using lasers and photodetectors. [2] More recently, a variation of flow cytometry called mass cytometry (CyTOF) was introduced, in which antibodies are labelled with heavy metal ion tags rather than fluorochromes, breaking the limit of multiplexing capability of FACS (fluorescence-activated cell sorting) and allowing the simultaneous quantification of 40+ protein parameters within each single cell. The ability of flow cytometry and mass cytometry to analyse individual cells at high-throughput scales makes them ideal for multi-parameter cell analysis and high-speed sorting. [3]

General workflow of cell analysis in a mass cytometer. The cells are introduced into the nebulizer via a narrow capillary. As the cells exit from the nebulizer they are converted to a fine spray of droplets, which are then carried into the plasma where they are completely atomized and ionized. The resulting ion cloud is filtered and selected for positive ions of mass range 80–200 and measured in a TOF chamber. The data are then converted to .fcs format and analyzed using traditional flow cytometry software. [4]

Self-organising maps (SOMs)

Self-organising maps (also referred to as SOMs or Kohonen maps) are artificial neural networks introduced by Teuvo Kohonen in the 1980s. Despite of their age, SOMs are still widely used as an easy and robust unsupervised learning technique for analysis and visualisation of high-dimensional data. The SOM algorithm maps high-dimensional vectors into a lower-dimensional grid. Most often the target grid is two-dimensional, resulting into intuitively interpretable maps. After initializing a SOM grid of size n*n, each node is initialized with a random sample (row) from the dataset (training data). For each input vector (row) in the training data the distance to each node in the grid is calculated, using Chebyshev distance or Euclidean distance equations, where the closest node is called BMU (best matching unit). The row is subsequently assigned to the BMU making it move closer to the input data, influenced by the learning rate and neighborhood Gaussian function, whilst the neighborhood nodes are also adjusted closer to the BMU. This training step is repeated for each row in the complete dataset. After each iteration (epoch) the radius of the neighborhood function is reduced. After n epochs, clusters of nodes should have formed and as a final step, consensus cluster is used to reduce the data (SOM nodes) into m clusters. [5]

Example of a self-organizing network with five cluster units, Yi, and seven input units, Xi. The five cluster units are arranged in a linear array. [6]

Implementation

The high-performance GigaSOM.jl package enables the analysis and clustering of huge-scale flow cytometry data because it is HPC-ready and written in Julia, prepared to handle very large datasets. Also, the GigaSOM.jl package, provides training and visualization functions for Kohonen's self-organizing maps for Julia. Training functions are implemented in pure Julia, without depending on additional binary libraries. The SOM algorithm maps high-dimensional vectors into a lower-dimensional grid. Most often, the target grid is two-dimensional, resulting into intuitively interpretable maps. The general idea is to receive huge-scale .fcs data files as an input, load and transform them accordingly to enable the analysis and clustering of these data and automatically determine the required number of cell populations, and their sensitivity and specificity using parallel computing. In order to achieve this, GigaSOM.jl implementes a batch SOM algorithm, in which the weight vectors are only updated at the end of each epoch, and the BMU is calculated using the weight vectors from the previous epoch. Since the weight updates are not recursive, the order of the inputs does not affect the final result. In addition, there is no learning rate coefficient, which reduces the data dependency. This means that, compared to the original one, the batch algorithm has faster convergence, requires less computing, and it is the only one suitable for parallel computing. Parallel computing is the last step of our package implementation and two possible approaches exist for this kind of computation: the Model Parallelism and the Data Parallelism. In the Model Parallelism approach, the algorithm sends the same data to all the processes, and then each process is responsible for estimating different parameters and exchange their estimates with each other to come up with the right estimate for all the parameters. In the Data Parallelism approach, the algorithm distributes the data between different processes, and then each process independently tries to estimate the same parameters and then exchange their estimates with each other to come up with the right estimate. On this project, we use the Data Parallelism approaches because our nodes grid is too small for the Model Parallelism approach.

Data Parallelism vs Model Parallelism [7]