Optical tracking methods are increasingly employed to characterize the size of nanoparticles in suspensions. However, the sufficient separation of different particle populations in polydisperse suspension is still difficult. In this work, Nanosight measurements of welldefined particle populations and MonteCarlo simulations showed that the analysis of polydisperse particle dispersion could be improved with mathematical methods. Logarithmic transform of measured hydrodynamic diameters led to improved comparability between different modal values of multimodal size distributions. Furthermore, an automatic cluster analysis of transformed particle diameters could uncover otherwise hidden particle populations. In summary, the combination of logarithmically transformed hydrodynamic particle diameters with cluster analysis markedly improved the interpretability of multimodal particle size distributions as delivered by particle tracking measurements.
It has often been shown that the size of nanoparticles determines, among other factors, its biologic or even toxic effects [
During the past 5 years, the Nanoparticle Tracking Analysis (NTA) became increasingly important in nanotoxicology to describe the size distribution of nanoparticle suspensions [
The purpose of this paper is to demonstrate these features by means of MonteCarlo simulations of polydisperse suspensions. We furthermore will show methods useful for the analysis and improved interpretation of polydisperse particle size distributions (PSDs). Therefore, different proportions of sizedefined particle populations were simulated, and size distributions were analysed. To effectively attenuate the broadening effect and to increase the interpretability of polydisperse distribution, calculated particle diameters will be logarithmised to normalize their variance. This means that the logarithm of each diameter is calculated and used for further analysis. To exploit the logarithmised diameters, a cluster analysis will be used. The efficacy of these procedures to improve the analysis of size distributions will be validated by simulations and verified by experimental results.
To track the Brownian motion of NP in suspension, we used a Nanosight instrument (LM10) which combines a conventional light microscope and a laser illumination device. The laser light is guided approximately perpendicular to the optical axis and scattered by NPs which, therefore, can be viewed according to Huygens principle. A CCD camera captures the diffraction patterns of diffusing particles at 30 frames per second. Then, the Nanosight software detects the center of each single diffraction pattern and measures the length of the trajectory [
Particle size measurements were carried out with welldefined polystyrene standard particles sized 50 nm (Thermo Scientific, 3050A NIST), 100 nm (KiskerBiotech, PPs0.1), 150 nm (Thermo Scientific, 3150A NIST), and 200 nm (KiskerBiotech, PPs0.2). Particle suspensions were adequately diluted with particlefree, doubledistilled H_{2}O to obtain 30–50 particles within one field of view, pipetted on the stage of a Nanosight LM10 laser device (530 nm) and viewed with an intensified CCD camera (AndorDL658MOEM) mounted on a Nanosight LM10. Tracking data were recorded for 160 s using the NTA Software version 2.2. All measurements were repeated at least three fold. In mixed particle populations camera settings were adapted such that the smallest particles could be recognized by the software. The numerical composition of mixed particle suspensions was studied by scanning electron microscopy (SEM). The aqueous suspensions were identical to those measured in Nanosight experiments but were at least 1000fold less diluted. Suspension was dried on Thermanox slices and sputtered with a thin layer of gold (40 nm, Sputter Coater S150B, Edwards, North Walsham, UK). Scanning electron microscopic examinations were done with the Gemini DSM 982 (ZEISS, Oberkochen, Germany, m 15 kV). At least 1000 particles from 10 different images were evaluated.
NTA measurements strongly depend on instrument and software settings such as camera gain, threshold mode and value, background subtraction, expected particle size, and other parameters. Also the homogeneity of the suspension influences size, distribution, and results are prone to be biased towards larger particles [
To eliminate these sources of error, “semireal” tracking data of a polydisperse suspension were generated from monodisperse tracking data. For this purpose, tracks from real monodisperse particle suspensions were gathered within one data file using predefined proportions (see Figure
Generation of virtual polydisperse suspensions.
To analyse the properties of PSDs, a MonteCarlo simulation of the Brownian motion of single particles with a specific diffusion coefficient was applied. According to Michalet [
Using the socalled transformation method [
Procedure for simulating tracks with a total length
Our approach for simulating size distributions of polydisperse suspensions is based on the procedure for monodisperse samples. For this purpose, we defined
Finite Mixture Densities Models assume that a population is mixed of
Normally, PSDs are computed on the basis of untransformed diameter data. To show that the resolution of single particle populations in conventionally calculated PSDs are limited, we used a MonteCarlo simulation of a polydisperse suspension containing 50 nm, 100 nm, 150 nm, and 200 nm particles. Table
Composition of the polydisperse simulation shown in Figure
Diameter  50 nm  100 nm  150 nm  200 nm 


Number 





Proportion  15%  30%  5%  50%  100% 
Simulated PSD of a polydisperse suspension.
Standard deviation depending on the mean size diameter of simulated (circles) and measured data (triangles).
A further disadvantage of the PSD in Figure
It is a known property of the logarithm that it reduces heteroscedasticity of random variables, so it stabilizes the variance. For this reason, the diameter data were logarithmised to gain more comparability between two particle populations in the PSD of polydisperse suspensions. The logarithmised version of Figure
Logarithmised PSD (black) and the results of the cluster analysis, showing five different clusters indicated by different colours. Clusters with a difference of the means less than 7 nm were merged.
Despite the variance stabilizing transform, the 150 nm particle population remains hidden behind the 200 nm population. To uncover such hidden populations and determine quantitative values for the population ratios, an MCLUST cluster analysis was performed. It must be emphasized that the logarithmised diameters are better suited for a cluster analysis, because the variance between particle populations can be assumed as equal.
The result of such a cluster analysis is shown as colored PSDs in Figure
Results of the cluster analysis of a simulated polydisperse suspension as shown in Figure

1  2  3  4  5  6 

Cluster mean  49  71  99  144  199  284 
Proportion  14.5%  1.5%  28.1%  7%  47%  1.7% 
For the validation of our method described in Section
Results of the cluster analysis of the virtual polydisperse suspension.

1  2  3  4  5 

Cluster mean  52  98  133  183  277 
Proportion  14.9%  27.1%  9.3%  47.4%  1.1% 
The PSDs of the monodisperse polystyrene particle with a diameter of 50 nm (a), 100 nm (b), 150 nm (c), and 200 nm (d) as measured with the Nanosight system.
Cluster analysis of a virtual polydisperse suspension.
To verify the method experimentally, we prepared a defined suspension of 100 nm and 150 nm polystyrene particles with equal number concentrations of both particle types. Therefore, an SEM analysis of the mixed suspension was carried out, a representative micrograph of which is shown in Figure
NTA PSD of a mixed suspension of 100 nm and 150 nm polystyrene particles (a) and typical SEM picture of the same suspension (b). Cluster analysis of polydisperse polystyrene suspension. Means of the two main clusters are 104 nm and 144 nm. Two false clusters were also detected (c). Simulation of polydisperse polystyrene suspensions with particle diameters of 101 nm and 134 nm (d).
The untransformed PSD is shown in Figure
A fundamental assumption of the proposed method is that the standard deviations of the subpopulations in polydisperse suspension are equal after logarithmic transformation. If this is correct, the logarithmic transform is a good method to reduce the impact of the particle size on the broadening effect of the PSD. Another factor influencing the broadening of the PSD is the mean number of steps contributing to the analysed tracks because the standard deviation of
Figure
The standard deviation of a PSD in dependence of the number of steps per track of untransformed (a) and logarithmised data (b).
One may object that the different mean step lengths of different particle populations in a polydisperse suspension lead to different SDs. This would contradict the assumption of equal SDs and, consequently, would impair the cluster analysis. To avoid this difficulty, we removed all tracks with a length below 10 steps in the examples of this work, thus avoiding to track “random noise” in particle videos. The remaining tracks of the 100 nm, 150 nm, and 200 nm particle suspensions had a mean track lengths of 56, 37, and 65 steps, respectively. Regardless of some variations in SD which might be due to this disparity, the cluster analysis worked convincingly well with this data. Nevertheless, a cluster analysis may fail if data with larger variation in SD are being processed.
Accepting the above limitations, the methodological improvements suggested in this study may be helpful to analyse more complex and multimodal systems. For example, the apparent size of nanoparticles suspended in biological fluids such as serum or cell culture fluid increases due to agglomeration and/or the formation of a protein corona [
We presented a method for improving polydisperse particle size distributions based on the logarithmic transform of the estimated diameters to reduce the heteroscedasticity, which is partly due to the constant coefficient of variation of the diameter data. Transformed data were then subjected to a cluster analysis, that shows the ability to uncover hidden populations. Calculated cluster proportions were validated and verified by MonteCarlo simulations of polydisperse suspensions, NTA measurements, and SEM images. The procedure appears helpful to correctly interpret the composition of polydisperse particle suspensions. The novel method for MonteCarlo simulation of polydisperse suspension and the concept of virtual polydisperse suspensions seems to be useful for further investigations of the properties of polydisperse size distributions.
The authors declare that they do not have a direct financial relation with the trademarks mentioned in their work that might lead to a conflict of interest.
This paper was supported by Grants of the German Federal Ministry of Education and Research (BMBF, NanoGEM Project, FKZ 03X0105G, and 03X0105H).