Particle sizing by multiangle light-scattering data using the high-order neural networks

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Abstract

The problem of retrieval of size and refractive index of a spherical particle by angular dependence of scattered light in scanning flow cytometry is considered. For its solution, the high-order neural networks are used. We restricted the range of angles available for measurement from 10° to 60°. The retrieval errors of characteristics of nonabsorbing particles were investigated at the ranges of the radius and relative refractive index 0.6–10.6 μm, and 1.02–1.38, respectively.

Introduction

The problem of retrieval of particle parameters by angular distribution of light scattered by a single particle is actual for different applications, in particular for flow cytometry [1], [2], [3], [4]. Especially, this problem is urgent for the scanning flow cytometry [4]. Nonspherical and spherical particles are the objects of investigation by this method.

There are many approaches to solve this problem. In paper [5], the solution of the inverse problem of scattering for a homogeneous spherical particle is proposed. To determine parameters of a particle, it is necessary to calculate integrals from the measured angular distribution of scattered light multiplied on oscillating functions of the angle in limits from 0° to180°.

The trial-and-error method is widely used [6]. It consists of a multiple solution of a direct problem and selection of particle parameters so that the calculation results as much as possible coincided with the experimentally measured angular distribution of scattered light. The method requires considerable time even for modern computers. To decrease the time of analysis, the empirical techniques for solving an inverse problem of scattering in flow cytometry were offered [4], [7], [8], [9].

Recently, for a solution of the inverse problem of scattering by spherical and nonspherical particles, the neural network method is used [10], [11], [12], [13], [14]. The neural network presents a set of elementary calculators called neurons [15], [16]. Depending on a solved problem, the neurons fulfill different operations. For example, neuron-summator adds the input signals and converts the output signal, according to nonlinear function f(x). The last is a neuron activation function. Usually the S-shaped function of neuron activation is applied. The single-level and multilevel neural networks are used [15], [16]. The arbitrary functions of many variables with a required accuracy can be approximated by the multilevel neural networks of such type [17], [18], [19], [20]. The neural networks, where the neurons fulfill more complicate operations, are applied. For example, in neural networks with the radial basis function (RBF networks), the output signal exponentially decreases with the distance in the Machalanobis metric. Such neural networks are widely used in the pattern-recognition theory. RBF networks are optimal for approximation of functions entered with errors [21].

To solve correctly the problem by neural networks, it is necessary to choose neural network coefficients adequately. This problem is solved on a level of training of the neural network. As a rule, the training of the neural network demands a lot of time. It consists of determining a global extremum of function of many variables. It is a very complicate problem. Due to simplicity and possibility to implement parallel calculations fulfilled by neurons of the network, the trained neural network solves the problem very fastly.

The neural networks with radial basis function are used for identification of particles in flow cytometry [22], [23]. In papers [10], [11], two-level neural networks with radial basis function were proposed to determine parameters of nonabsorbing particles at the range of radius from 0.5 to 1.5 μm and relative refractive index in the range from 1.12 to 1.29. For retrieval of particle parameters, it is necessary to know intensity of scattered light in the range from 0° to 180°.

The neural networks with sigmoidal function of neuron activation were used for retrieval of a mean size of particles and variance of size distribution function [12]. Multilevel neural networks with a symmetric function of neuron activation were applied to determine characteristics of spherical absorbing particles with radius from 0.5 to 5.0 μm by intensity of scattered light in the range from 0° up to 180° [13]. Such neural networks were used for retrieval of volume, refractive index, aspect ratio and orientation of spheroidal particles [14].

The experimental equipment usually allows to measure intensity of scattered light in a limited range of angles. In the scanning flow cytometer, characteristics of scattered light can be measured in the range of angles from 10° to 60°–70° [4], [24].

In the present paper, the problem of retrieval of size and refractive index of spherical particle by intensity of scattered light in the range of angles from 10° to 60° is considered. The ranges of the radius and the refractive index are 0.6–10.6 μm and 1.02–1.38, respectively. In these regions, there are values of parameters of many biological cells. Many of the cells have more complex structure. Their shape cannot be always considered as spherical. Some of them, for example lymphocytes, contain eccentric nuclei. The creation of neural networks for different types of cells has to take into account the specificity of the cell. We consider here only spherical cells in a wide range of optical parameters. This is a step for further investigation of more complex cells.

Section snippets

Forming and training of a neural network

Let us consider a nonabsorbing spherical particle with a relative refractive index n and radius R. The refractive index n varies in the range from 1.02 to 1.38, radius varies in the range from 0.6 to 10.6 μm. The particle is illuminated by a plane wave along z-axis. It is necessary to retrieve R and n by intensity of light Ij=I(θj), scattered in the directions θj (j=1,2,,Nθ, Nθ is the amount of angles). Angle θ varies in the range from 10° to 60°. To model scattered light, the Mie theory is

Neural network testing

To test the neural network, we multiply retrieved particle parameters at the presence of the measuring error. A multiplicative error of measurement was modeled in the same manner as described in the previous section. We multiply determined particle parameters on different examples, and by the obtained results, we calculated the root-mean-square deviation of the retrieval error of radius σR and refractive index σn.

Data of Fig. 8 illustrate sensitivity of the single-level and two-level neural

Conclusion

High-order neural networks for retrieval of radius R in the range from 0.6 to 10.6 microns and refractive index n in the range from 1.02 to 1.38 of spherical nonabsorbing particles by the relative intensity of scattered light in the range of angles from 10° to 60° are developed.

The possibilities of retrieval of particle parameters by multilevel neural networks are analyzed at changes in the amount of neurons and other structural parameters of the neural network.

Retrieval errors of R and n are

Acknowledgement

This research is partially sponsored by NATO's Scientific Affairs Division in the framework of the Science for Peace Programme.

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