Phys. Rev. E, 2017: Decorrelation correction for nanoparticle tracking analysis of dilute polydisperse suspensions in bulk flow
Citation: Hartman J, Kirby BJ. Decorrelation correction for nanoparticle tracking analysis of dilute polydisperse suspensions in bulk flow, Phys. Rev. E, 95, 033305 (2017). doi
Abstract: Nanoparticle tracking analysis, a multiprobe single particle tracking technique, is a widely used method to quickly determine the concentration and size distribution of colloidal particle suspensions. Many popular tools remove nonBrownian components of particle motion by subtracting the ensembleaverage displacement at each time step, which is termed dedrifting. Though critical for accurate size measurements, dedrifting is shown here to introduce significant biasing error and can fundamentally limit the dynamic range of particle size that can be measured for dilute heterogeneous suspensions such as biological extracellular vesicles. We report a more accurate estimate of particle meansquare displacement, which we call decorrelation analysis, that accounts for correlations between individual and ensemble particle motion, which are spuriously introduced by dedrifting. Particle tracking simulation and experimental results show that this approach more accurately determines particle diameters for lowconcentration polydisperse suspensions when compared with standard dedrifting techniques.
Figures:


Methods to generate PSDs used in this work. Specified PSD is a continuous probability density function generated from the
summation of two lognormally distributed random variables with specified parameters and therefore is not associated with a set of displacements
or any calculation of MSD. Sampled PSD contains the particle diameter values that were sampled from specified PSD in SPT simulations
and therefore is associated with the discrete MSD values used to generate particle trajectories. No correction implies that observed particle
trajectories were not modified in any manner. Perfect correction only considers particles’ Brownian motion and therefore represents ideal values
given instrument broadening. Monodisperse correction utilizes the MSD correction factor from [26]. Both raw dedrifting and smooth dedrifting
are presented in order to illustrate the effect of applying linear weighting functions to estimated nonBrownian motion. Accordingly, both raw
DA and smooth DA forms are presented for clarity. Although only the x component of displacement is shown, the y component is equivalently
determined and r2 = x2 + y2 is used in calculations of MSD (column 3). Individual particle MSDs were converted into suspension PSDs
by using kmeans clustering to initialize Gaussian mixture model fitting of multivariate distribution parameters, as described in Sec. II.
A


Dedrifting causes particle number and compositiondependent errors related to the correlation between estimated drift and particle
displacement. (a)–(c) Pr is the measured probability density distribution normalized by the perfect correction. Particle size distributions
represent MCaveraged (1000 iterations) distribution parameters measured from different total number particles per time step Nt or per window
Nw. The magnitude of peak shift was inversely related to the number of particles defined by the ensemble and the direction of shift depended
on the suspension composition. Particles were sampled equally from lognormal distributions with standard deviation = 8 nm and applied
constant drift (1 um/s). (d)–(f) Here R^xy is the unbiased zerolag crosscorrelation function (9) normalized by the observed value for each
peak. Data points represent the MC average ±2std in the case of Nt = 4, 2 × 105 displacements per particle and no applied drift. Data points
(circles) indicate whether the spurious peak shift was dominated by residual correlation (less than 1) or introduced correlation (greater than 1).
The legend indicates vectors being assessed for correlation.


Spurious shifts from simulated particle motion are exactly
predicted analytically. Results from SPT simulations of bimodal
PSDs with (a) 100 and 400nmdiam peak means and (b) 200 and
400nmdiam peak means are shown. The y axes are a measure of
the spurious peak shifting caused by dedrifting where values greater
than 1 correspond to an underestimated diameter (add < aBr), values
equal to 1 have no bias (add = aBr), and values less than 1 correspond
to an overestimated diameter (add > aBr). Lines represent the result
of Eq. (10) given the relative number of each subpopulation that
was simulated. Data points represent the average diameter ratio of
the indicated size subpopulation (open square or closed diamond)
calculated from ten simulations in which particles were tracked
for 1e4 displacements with no bulk flow. Line color and style
represent five cases corresponding to the number of small to large
particles in infinity:0 (red solid line), 3:1 (green dotted line), 1:1 (cyan
dotdashed line), and 1:3 (blue dashed line) ratios. The inset in
(b) zooms in on the case N = 8 in order to better visualize the
agreement between analytical and simulation results. Analytically
calculated bias and true bias calculated from simulated trajectories
are in perfect agreement, demonstrating that Eq. (10) predicts
compositiondependent error introduced by dedrifting.


Accuracy of a single large particle in a bath of small
particles. Results are from SPT simulations of N = 1 bath particles
and a single large particle over a range of diameters zeta = abig/abath.
Plots show (a) a broad range and (b) a practical range of zeta for vesicles. The y axes are the dedrifted diameter ratio zeta^dd = addbig/abath, while zeta indicates the true diameter ratio, thus the dedrifted ratio pproaches the known true ratio as N approaches inf. Whether values approach zeta from inf+ or inf− indicates the dominant source of error.


Simulations and experiments demonstrate that DA is more accurate than dedrifting for constant, uniform bulk flow measurement
conditions. Plots show the percent error of distribution parameters obtained from analysis of SPT simulations using the (a) raw estimated drift
and (b) smooth estimated drift, as well as from a particle tracking experiment using the (c) raw estimated drift and (d) smooth estimated drift.
Marker shape represents size mode, marker color represents the approach used for determining particle diameters, and marker size indicates
the target number of particles in each time step, 6 (large), 12, or 18 (small). (a) and (b) For simulations, particle diameters were sampled from
bimodal 100 and 400nm lognormal distributions (std = 10 nm) and prescribed trajectories with a random number of time steps (between 100
and 1000) in the presence of 65um/s bulk fluid flow (∼10 s to traverse NanoSight field of view). Plotted values represent averages from
100 simulations. (c) and (d) For experimental data, 100 and 400nmdiam polystyrene bead standards were tracked in quiescent fluid and the number of particles in each time step was controlled in postprocessing. In both simulations and experiments, decorrelating after dedrifting obtained the lowest percent error of peak mean of all approaches for a given number of particles in each time step.


A raw approach determined more accurate PSDs than a smooth approach in oscillatory bulk flow conditions. Plots show
percent error of (a) distribution parameters obtained from DA of
SPT simulations and (b) data from a particle tracking experiment in
which uniform bulk flows with constant and oscillatory components
were present. The frequency spectrum of applied simulated bulk
flow was matched to the frequency spectrum of the experimental
ensembleaverage displacement time series using Fouriertransform
analysis. Marker shape represents size mode, marker color represents
the approach used for determining particle diameters, and, for
simulations only, marker size indicates the target number of particles
in each time step, 6 (large), 12, or 18 (small). In both simulations and
experiments, raw DA bulk flow removal obtained the lowest percent error of peak mean of all approaches for a given number of particles
in each time step. Smoothing the estimated drift time series removed
true oscillatory drift such that subtracting the smooth estimated drift
did not remove highfrequency bulk flow components, spuriously
underestimating inferred particle diameters.


Simulationsdemonstrate that DA is more accurate than dedrifting for a broad monomodal PSD. Plots show MCaveraged binned histograms of particle diameter obtained from analysis of SPT simulations using (a) raw estimated drift and (b) smooth estimated
drift (w = 3). Particle diameters were sampled from a lognormal PSD (open circles) defined by f(x;mean = 165 nm,std = 75 nm)
and prescribed trajectories with a random number of time steps
between 1000 and 2000 that began at random time steps such that
Nt = 4 particles. Decorrelation analysis [Eq. (12), green asterisks] is compared to standard dedrifting [Eq. (8), red marks] and to standard
dedrifting with a monodisperse correction factor (blue marks) [26,31]. Plotted values represent averages from 100 simulations. The DA histogram values were nearly identical to binned sampled
diameters, demonstrating high measurement accuracy regardless of particle diameter. On the other hand, current dedrifting methods both spuriously shift and spuriously narrow the PSD.
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