## Abstract

We report on a new method based on speckle size analysis and devoted to particle aggregation measurements. The experimental measurements give the speckle size variation during a salt aggregation process of polystyrene microspheres. The measurements are taken at a fixed monomer concentration, varying the salt concentration. Moreover, we applied this technique to follow blood platelet aggregation, usually monitored with a visible light transmittance photometer (aggregometer). Aggregation process was induced by ADP (adenosine diphosphate) addition, then we measured the speckle size variation versus time at two different ADP concentrations.

©2004 Optical Society of America

## 1. Introduction

Numerous works about colloidal aggregation were produced in the last few years. These researches used different techniques as low angle static light scattering. This well-known method is particularly interesting and gives information about reaction kinetics (associated evolution of clusters sizes) and fractal morphology of the aggregates [1,2]. Dynamics Light Scattering (DLS) is another interesting method because of its ability to provide data on the size and shape of particles [3,4] from dynamical speckle analysis [5].

Speckle or laser light-induced granularity can be observed either in free space (objective speckle) or on the image plane of a diffuse object illuminated with a coherent source (subjective speckle) [6]. The speckle effect through a coherent light source-illuminated medium results from random fluctuations of refractive index. Photons travel along random optical paths; the random dephasing of associated wavelets produce random interferences that induce a statistical distribution of light intensity.

Speckle parameters (size, contrast, intensity, polarization,…) can bring information on scattering media. For example, spatial characteristics such as the speckle size can be used to measure the roughness of surfaces [7,8,9] or to determine the thickness of a Teflon plate [10].

The study reported here was aimed at following the kinetics of aggregation process from speckle size measurements.

Thus, we first investigated polystyrene-microsphere aggregation induced by different salt additions and characterized the two limiting regimes of aggregation kinetics [2]: the slow aggregation regime that it is named Reaction-Limited Cluster Aggregation (RLCA) and the fast aggregation regime that it is named Diffusion Limited Cluster Aggregation (DLCA). The sticking probability between particles determines the behavior as follows: a sticking probability equal to one leads to the DLCA, whereas values noticeably smaller than 1 give rise to RLCA. The chosen salt concentration assures fast or slow aggregation.

Secondly we realized a biomedical application for blood platelet aggregation monitoring. Platelet aggregation is of paramount importance to stop a hemorrhage after a vessel injury. So the clinical evaluation of the status of aggregation platelet is necessary in the diagnosis of imbalance. A visible light transmittance photometer (aggregometer) is classically used to follow the aggregation process because of the changes induced in scattering characteristics, and then in transmitted light [11]. Nevertheless this method can be limited by the photometer geometry or multiple scattering [12]. This led us to propose, here, a new method based on speckle size measurement to assess the aggregation process.

## 2. Methodology and speckle size measurement

In a previous study [13], we showed the relation between the speckle properties and the scattering of media. We measured the effect of the particle size into the media and that of the scattering coefficient, *μ _{s}*, on the speckle size. In the previous investigations about the speckle
produced in transmission by scattering fluid media, we followed its evolution from size measurement versus the optical thickness,

*ls*=

*Lμ*(

_{s}*L*is the thickness of the slab), and versus the polystyrene-microsphere size in the media. The different optical thicknesses were obtained by varying the diffuser number,

*N*, since

*μ*is proportional to

_{s}*N*[14] for a given particle size. At weak scattering the speckle size decreases linearly with scattering increase. Moreover, for the biggest particles and a fixed optical thickness, the speckle size increases concomitantly with the particle diameter or diffuser size,

*d*. This variation was measured for diameters within 0.53 and 6.36 μm.

During an aggregation process, the two parameters *N* and *d* vary, and then both produce an increase of the speckle size: the sticking between particle induces an increase of *d* and a decrease of *N*.

To estimate the speckle size, we calculated the normalized autocovariance function of the intensity speckle pattern obtained in the observation plane (*x*,*y*). This function corresponds to the normalized autocorrelation function of the intensity; it has a zero base and its width provides a reasonable measurement of the “average width” of a speckle [6]. If *I* (*x*
_{1}, *y*
_{1}) and
*I*(*x*
_{2},*x*
_{2})are the intensities of two points in the observation plane (*x*,*y*), the intensity autocorrelation function is defined by Eq. (1) [6]:

where Δ*x* = *x*
_{1} - *x*
_{2} and Δ*y* = *y*
_{1}-*y*
_{2}. ⟨ ⟩ corresponds to a spatial average. If *x*
_{2} = 0, *y*
_{2} = 0, *x*
_{1} = *x* and *y*
_{1} = *y*, we can write:

The normalized autocovariance function of the intensity, *c _{I}* (

*x*,

*y*), is given by Eq. (3):

As implied by the Wiener-Khintchine theorem, the autocorrelation function of the intensity is given by the Inverse Fourier Transform (*FT*
^{-1}) of the Power Spectral Density (PSD) of the intensity:

FT is the Fourier Transform.

*c _{I}* (

*x*,

*y*) calculated from the intensity distribution measured of the speckle is:

*c _{I}* (

*x*, 0) and

*c*(0,

_{I}*y*) are the horizontal and the vertical profile of

*c*(

_{I}*x*,

*y*), respectively. Let us term

*dx*the width of

*c*(

_{I}*x*,0) so that

*c*(

_{I}*dx*/2,0) = 0.5 and

*dy*the width of

*c*(0,

_{I}*y*) such as

*c*(0,

_{I}*dy*/2) = 0.5.

Figure 1 illustrates the experimental set up. The 7-mW HeNe laser used emits a 1.12-mm-wide polarized (linear) beam at *I _{0}*/

*e*with

^{2}*I*being the maximum laser intensity, at the 632.8 nm wavelength with a coherence length of about 20 cm. The light intensity is monitored by a half wave plate

_{0}*L*associated to a polarizer P1. The length

*L*of the fluid sample is 1 cm.

A CCD camera records the medium-produced speckle field. The CCD imager contains 788×268 pixels that are 8 μm×8 μm in size. When a newly acquired image is digitized, the analog-to-digital converter assigns an intensity value (gray level) in the range of 1 to 1024 (10-bit precision).

To record the speckle, one should be aware of the Brownian motion of particles in the medium. These movement induce a random agitation of the speckle (“boiling speckle”). Consequently, the image acquisition time must be very short. So, the time exposure of our CCD camera could vary from 60.10^{-3} to 10^{-4} s. However, in order to keep a correct signal-to-noise ratio, in our experiments image acquisition took 0.001 s.

Another consideration of importance in recording the speckle data is the speckle size on the CCD array: the speckles must be large as compared to the pixel size so as to resolve variations in speckle intensity [15]. Moreover, each image needs to contain many speckles for meaningful statistical evaluation. These conditions were met by choosing the distance *D* (*D* = 36 cm) between the sample and the CCD camera.

In order not to record the directly transmitted laser beam the CCD camera was positioned at an angle *θ* with the optical axis (Fig. 1). In our previous study [13] we showed that contrarily to *dx*, when *θ*< 10° dy variation versus *θ* is small. In our experiments we have *θ*= 5°.

## 3. Polystyrene microsphere aggregation

We will first illustrate on polystyrene microsphere aggregation how this process can be monitored by speckle size measurements. The polystyrene microspheres used had a diameter of 0.535 μm and were from Polyscience Inc.; they were provided to us in a deionized water solution and their number per cubed meter, *N _{0}*, is known. The undiluted solution contains

*N*

_{0}=3.147 * 10

^{17}m

^{-3}. By dilution in deionized water, we adjusted the parameter

*N*and so the scattering coefficient.

*μ*determined by Beer-Lambert Law application:

^{s}*μ*= 6.23

_{s}*c*, where

*c*is the concentration of the initial solution. Furthermore, we employed deionized water to avoid the possible effect of particle aggregation due to electrostatic interaction. The refractive indices were 1.59 and 1.33 for the spheres and medium, respectively.

As the scattering coefficient must be high enough to get a small initial speckle size and allow it to greatly vary during aggregation. It is worth recalling that the speckle size decreases when the scattering coefficient is increasing. So the density of particles was chosen so that *μ _{s}* = 4 cm

^{-1}, and the initial speckle size measured about 46 μm. For our experimental configuration, the speckle size cannot exceed about 220 μm.

*dy*220 μm when

*μ*0 cm

_{s}^{-1}and this speckle size limit depends on the intensity distribution of the laser beam size [13]. The salt used to induce aggregation was MgCl

_{2}. Figure 2 illustrates

*dy*evolution versus the time,

*t*, for several salt concentrations.

These three curves show clearly different behaviors. For the strongest salt concentration (300 mM), *dy* increases a lot after the salt addition. Speckle size varies from 46 to 126 μm, which corresponds to a fast aggregation. After this first variation, *dy* evolution is insignificant. At the weakest salt addition (3 mM), there is no strong variation. The small speckle size evolution corresponds to a slow aggregation. For the intermediary salt addition (45 mM), we can observed a strong variation of *dy* next the salt addition and afterward we observe a weak variation: the aggregation process is fast further to the salt addition, then slows down.

Finally these curves show that speckle size measurement can allow one to characterize an aggregation process and distinguish between a RLCA and a DLCA (weak and strong salt concentrations, respectively).

## 4. Blood platelets aggregation

In this section we will apply such a speckle size measurement to the monitoring of a blood platelet aggregation process. The samples, platelet suspensions in plasma, were prepared from whole blood after centrifugation to remove the larger red and white cells. To induce the aggregation, we added ADP (ADP denotes adenosine diphosphate) to the samples previously heated at 37°C. The measured scattering coefficient was 11 cm^{-1} and the speckle size before aggregation *dy* = 100 μm. Large diffusers induce a large speckle size [13], so compared to the polystyrene microsphere, *dy* is higher even though *μs* is higher (platelet size is about 4 μm and polystyrene microsphere size is 0.535 μm).

Figure 3 depicts *dy* versus *t* at two ADP concentrations: 25 μM and 10 μM.

It shows that, at the higher ADP concentration, the speckle size is strongly increased for one minute after ADP addition; this corresponds to a fast aggregation (elevation of *dy* from 100 to 135 μm). Afterwards, one should note the slow increase of the speckle size concomitant with the slowing down of the aggregation process.

At the weaker ADP concentration, despite the first fast aggregation the speckle size extends only to 111 μm; this goes along with the formation of smaller aggregate. Moreover, after the first minute, there is a reduction of the speckle size followed with a stabilization at 107 μm after *t* = 1.5 min.

So, the obtained curves show clearly different behavior according to the ADP concentration.

## 5. Conclusion

In this study we experimentally showed that aggregation process can be monitored by speckle size measurement because of the changes undergone by scatters-size and -density. So such a speckle size measurement allowed us to characterize the fast and the slow aggregation processes, respectively denoted DLCA and RLCA, during a salt-induced aggregation of polystyrene microspheres.

Our method was also applied to the monitoring of blood platelet aggregation further to ADP addition. Among the two ADP concentrations used, the higher one permitted us to demonstrate that aggregation was fast during the first minute after ADP addition, and then slow. The lower ADP concentration induced a different aggregation; it produced smaller aggregates before a stabilization of the process.

Aggregometers are currently used in medicine to monitor platelet aggregation, but according to some investigations like the ones by P. Latimer [12] the effects of the photometers geometry (angle of the incident laser beam, detector size) and multiple scattering may spoil the analysis. Thus, speckle measurement may be an alternative method.

Finally the real-time, cheap method presented here sounds very promising for getting data on blood platelet aggregation and in general on the aggregate systems.

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