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Mathematical Modelling and Simulations

Working Group 8.41

Scattering Problem in Flow Cytometry

Recently, we started to mathematical modelling of optical flow cytometry. Optical flow cytometry is characterized by light scattering on blood cells like erythrocytes. Here, information about cell properties is obtained from scattering distributions. Cell volume and geometry parameters of cell shape can be reconstructed from scattered data by solving numerically an inverse problem. For the reliable determination of  cell geometries including small uncertainties angular resolved scattering measurements are necessary. Existing reference procedures for the determination of cell volumes are intended to be extended. 

Introduction

Figure 1: Schematic diagram of the working principle of optical flow cytometry.

Optical cytometry, i. e., the measuring of biological cells by means of light scattering, is a procedure routinely used in laboratories for cell counting and cell characterization. So called flow cytometers (Fig. 1) are often used: The cell suspension to be measured flows through a narrow cuvette and is – by means of a laminarly flowing sheath fluid – hydrodynamically focused to a cross section of several micrometers extent. Flowing through the cuvette, the cell is illuminated by a beam of laser light, which is scattered by the cell and collected onto a detector by an optical system. Wavelengths are typically in the visible range (e. g., 633 nm or 488 nm). Due to the hydrodynamic focusing, the cells pass the light focus one by one.  The exact form of the scattered light field depends on the properties of the cell, such as size, shape and refractive index. The aim of this indirect measurement procedure is to infer properties of the cell from the scattered light pattern.

 

Flow cytometric measurement procedures are also used in the examination of blood samples, for instance in blood counts. The modeling of light scattering by human blood cells, such as red blood cells (erythrocytes; the most frequent type of blood cells in humans) together with the numerical solution of the inverse scattering problem serves as an important building block for optical flow cytometry, in order to draw conclusions from the measured scattering pattern on the blood cells to be characterized. In standard devices one usually employs detectors that integrate the light intensity around some few directions, meaning that, for instance, forward scatter and side scatter are measured. In order to determine cell properties such as volume and/or geometry parameters reliably and with low uncertainty, however, more information and hence an angularly resolved measurement of the scattered light is required. By doing so, we plan on extending existing reference measurement  procedures for the determination of cellular volumes.

Modelling

Maxwell's equations are the basis for the mathematical modeling of the scattering of light by blood particles. In the case of time-harmonic electromagnetic fields (monochromatic light) and homogeneous media (refractive index $n$) they simplify to the vector Helmholtz equation \[ \Delta {\bf E} + k_0^2\,n^2{\bf E} = 0 \]

for the electrical field ${\bf E}$ and accordingly for the magnetic field ${\bf H}$. The solution is sought for in the form\[ {\bf E} = {\bf E}^i + {\bf E}^s,\]

where ${\bf E}^i$ is the incident wave and  ${\bf E}^s$ is the scattered wave. In the simplest case, the cell is assumed to be a region of homogeneous refractive index. An analytical solution exists only for special cases such as the scattering of a plane wave by a homogeneous sphere (Mie scattering) or for the limiting case of small scatterers (Rayleigh scattering). For more general cases where the scattering particles or blood cells have a shape deviating from a sphere, there exist different numerical procedures such as the discrete dipole approximation (DDA). Here the scatterer's volume is discretized and the sub-volumes are approximated by point dipoles. The result is a linear system of equations for coupled discrete dipoles, which is solved numerically. As a result one computes the scattering matrix or Mueller matrix of the scattered light field for the region of interest of the solid angle, depending on the incident wave field and the refractive index of the cell. There are powerful implementations of DDA. To apply them, one needs to model the assumed shapes of the blood cells appropriately. The biconcave disk-like shape (Fig. 2) is a promising candidate for this, in order to closely resemble the real conditions. However, deformations due to hydrodynamic forces, thermical fluctuations, or pathological cell shapes (e. g., sickle cells) can make this geometric modeling much more involved. Open questions also exist concerning the wavelength dependence of the refractive index of red blood cells. There are several different literature values around, partly in contradiction to each other. Hence, it is necessary to evaluate measurements of the spectral dependence of the extinction of erythrocyte suspensions, to obtain values for the wavelength-dependent refractive index of red blood cells reliably and and as precise as possible.

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Simulations - Virtual experiments

Figure 2: Simplified biconcave disk-like shape model for erythrocytes. Axes scaled.

A powerful forward model for simulating the measurement setup of optical flow cytometry is an important tool for improving or enhancing measurement procedures. The sensitivity of scattering signals to various model parameters can be studied extensively using computer simulations. Statistical influence variables can be quantified regarding their influence on the expected scattering patterns, too, such as fluctuations of the individual cell geometries around a mean values, roughness of the cell surfaces, inhomogeneities of the intracellular refractive index (hemoglobin concentration). Fig. 3 and 4 show intensity distributions, computed with discrete dipole approximation for the shape model of Fig. 2.

Figure 3: DDA simulation data: Intensity far-field patterns around the forward axis for biconcave disk-like erythrocyte model. Incident light along $z$-axis, coming out of the image plane. From left to right Rotation around the $x$-axis: 0° (face-on illumination of disk), 46°, 90° (edge-on illumination of disk). Cell diameter 6.7 μm, wavelength 488 nm, relative refractive index 1.06+0.0011i.

Inverse problem - Inference of shape and volume

In order to infer information about the cells' shapes and volumes from the measured scattering data one needs to solve the inverse scattering problem: How must an object be composed in order to scatter light according to a given far-field distribution (Fig. 3, 4)? This inverse problem is solved as an optimization problem, i.e., by variation one tries to find that set of parameters of the mathematical cell model (geometry parameters, material properties) whose corresponding scattering pattern reproduces the measurement data best. One has to consider, that the inverse problem is generally ill-posed and without regularization, for instance via constraints for the allowed shape models or their parameter ranges and refractive index distributions, the problem will not be solvable. Using simulation data with added noise, it is possible to study the applicability of different optimization strategies such as least-squares, maximum-likelihood or Bayesian approaches.

Figure 4: Lateral light intensity distribution of a biconcave red blood cell
for two different orientations relative to the incident beam; computed with
DDA; wavelength 488 nm, relative refractive index 1.06.

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