Statistical and sampling issues when using multiple particle tracking

Video microscopy can be used to simultaneously track several microparticles embedded in a complex material. The trajectories are used to extract a sample of displacements at random locations in the material. From this sample, averaged quantities characterizing the dynamics of the probes are calculated to evaluate structural and/or mechanical properties of the assessed material. However, the sampling of measured displacements in heterogeneous systems is singular because the volume of observation with video microscopy is finite. By carefully characterizing the sampling design in the experimental output of the multiple particle tracking technique, we derive estimators for the mean and variance of the probes’ dynamics that are independent of the peculiar statistical characteristics. We expose stringent tests of these estimators using simulated and experimental complex systems with a known heterogeneous structure. Up to a certain fundamental limitation, which we characterize through a material degree of sampling by the embedded probe tracking, these estimators can be applied to quantify the heterogeneity of a material, providing an original and intelligible kind of information on complex fluid properties. More generally, we show that the precise assessment of the statistics in the multiple particle tracking output sample of observations is essential in order to provide accurate unbiased measurements.

Figure 1

(Color online) Schematic of a heterogeneous system as seen through a microscope. Different colors in the material correspond to different values of a certain material property $\nu$. The material is held in a chamber of observation (side view) and randomly dispersed particles (represented by the dots) are visible only when moving in a certain imaging volume $V_{\mathrm{b}}=x_{\mathrm{b}}\times y_{\mathrm{b}}\times z_{\mathrm{b}}$ (dashed frame) whose smallest dimension $z_{\mathrm{b}}$ in the $z$ direction determines the limitations in sampling. The bottom view pictures a typical snapshot at a given time $t$.

Figure 2

(Color online) Autocorrelation coefficients of the displacements and the block-averaged displacements. Different dynamics $m\left(n\right)$ are investigated: (a) is a purely Newtonian fluid $m\left(n\right)\propto \mid n\mid$, (b) and (c) are 0.7 and 0.3 power-law dynamics, $m\left(n\right)\propto {\mid n\mid }^{0.7}$ and $m\left(n\right)\propto {\mid n\mid }^{0.3}$, respectively, and (d) is a Voigt fluid model with relaxation time of 10, $m\left(n\right)\propto 1-e^{-\mid n\mid ∕10}$. The plots on the second column give the displacements’ autocorrelation coefficient ${\rho }_n^{\left(1\right)}\left(h\right)$ as a function of $(n,h)$ for the corresponding dynamics shown in the first column. The third column of plots shows the autocorrelation coefficient of two successive boxed averaged displacements ${\underset{̱}{{\rho }_1}}_n\left(1\right)$ as a function of $(n,s)$ (see text for notations).

Figure 3

(Color online) Degree of sampling $\theta \left(n\right)$ in the parameter space $(n,\tilde{m})$ scanned by two different kinds of dynamics $\tilde{m}\left(n\right)$: (a) is a purely Newtonian fluid, for which $\tilde{m}\left(n\right)\propto \mid n\mid$, and (b) is a Voigt fluid with relaxation time 5, that is, $\tilde{m}\left(n\right)\propto 1-e^{-\mid n\mid ∕5}$. For each fluid type, two acquisition times were simulated, $n_{\mathrm{b}}=1000$ on the left and $n_{\mathrm{b}}=100$ on the right. The dashed line represents the observation limit $\tilde{m}=1$ due to the volume of observation. The connected squares represent the limit above which the relative bias of ${\widehat{M}}_1$, defined by Eq. (11), is greater than 5%.

Figure 4

(Color online) Trajectory statistics. As shown in (a), the parameter space $(n,\tilde{m})$ is sampled using purely viscous dynamics with material properties ${\left\{{\nu }_r^{\mathrm{N}}\right\}}_{r∊\{0,3,6\}}$ and Voigt dynamics with relaxation time 5, ${\left\{{\nu }_r^{\mathrm{V}}\right\}}_{r∊\{0,3,6\}}$ (see the labels on the figure). In (a), the dashed line represents the observation limit $\tilde{m}=1$ due to the volume of observation. (b) shows the corresponding variations of the degree of sampling $\theta \left(n\right)$ as a function of $n$ for each fluid. (c) represents typical trajectory durations: a single dot means that the particle was observable for only one frame, and lines indicate trajectories of at least $\Delta t$ long. Each line in these plots refers to the original simulated trajectory that has been fragmented to take limited observation volume into account (see Sec. 2A2), and we present results for the two kinds of fluid from top to bottom with increasing values of the index $r$.

Figure 5

(Color online) Bias as a function of the degree of sampling $\theta$. The first column of plots (a), (c), and (e) corresponds to a mapping of the parameter space $(n,\tilde{m})$ using Newtonian dynamics and the second column is for a Voigt fluid. (a) and (b) show the relation between the degree of sampling of the homogeneous fluid 1 (the line connecting the triangles corresponds to a constant $\theta =0.5$) with the bias [the line connecting the diamonds is a constant level of bias for ${\widehat{M}}_1\left(n\right)$] in the bimodal mixtures ${\nu }_r^{\mathrm{N}}+{\nu }_{r+1}^{\mathrm{N}}$ (a) and ${\nu }_r^{\mathrm{V}}+{\nu }_{r+1}^{\mathrm{V}}$ (b). Refer to the text for further explanations on how these lines are built. The composite fluid [${\tilde{M}}_1\left(n\right)$ is given by the black solid line] is made of a balanced mixture of two successive dashed lines, ${\tilde{m}}^{\left(1\right)}\left(n\right)$ and ${\tilde{m}}^{\left(2\right)}\left(n\right)$. The estimator ${\widehat{M}}_1\left(n\right)$ is given by the dash-dotted line. (c) and (d) show the bias in ${\widehat{M}}_1$ (squares) and ${\widehat{M}}_2$ (circles) as a function of $\theta$. (e) and (f) show the variation of the bias for a bimodal fluid with greater difference in the dynamics ${\nu }_r^{\mathrm{N}}+{\nu }_{r+2}^{\mathrm{N}}$ and ${\nu }_r^{\mathrm{V}}+{\nu }_{r+2}^{\mathrm{V}}$, respectively. In (c)–(f), the open symbols are obtained using a prior time averaging for the corresponding estimators (see text). Note that the open circles, corresponding to an estimation of ${\tilde{M}}_2$ with prior time averages on individual tracks, lie above 100%, outside the range of bias shown here.

Figure 6

(Color online) (a)–(d) are the measured trajectories in a heterogeneous unbalanced bimodal model experimental fluid. Each image has dimensions $x_{\mathrm{b}}=y_{\mathrm{b}}=300\mu \mathrm{m}$. The centered circular gelled region, where the particles are trapped, have radii $R_{\text{gel}}=55$, 85, 100, and $115\mu \mathrm{m}$, respectively, from (a) to (d). (e) is the squared coefficient of variation of the probes’ dynamics in these fluids. The open squares are estimates of the measure ${\mu }_2\left[\tilde{m}\left(n\right)\right]∕\mathbb{E}{\left[\tilde{m}\left(n\right)\right]}^2$ using ${\widehat{M}}_2\left(n\right)∕{\widehat{M}}_1{\left(n\right)}^2$ for the four bimodal fluids created by photopolymerization at $n=10$. The filled square was obtained in the homogeneous system $R_{\text{gel}}=0$. The solid line is the theoretical curve, Eq. (12). The circles are the calculations of the squared coefficient of variation using prior time averages on individual tracks (as explained in the previous section about the simulated bimodal fluids; see also Fig. 5).

Figure 7

(Color online) Estimators ${\widehat{M}}_1$ and ${\widehat{M}}_2$ calculated for a random fluid. For the random fluid simulated here, the parameter space $(n,\tilde{m})$ is sampled with purely Voigt dynamics for which the drag coefficient $\tilde{\xi }$ is uniformly distributed between ${10}^4$ and ${10}^5$, and the spring constant $\tilde{k}$ is Gamma distributed with mean ${10}^4$ and a shape parameter of 2 ($\tilde{\xi }$ and $\tilde{k}$ are statistically independent). The resulting probability density of $(n,\tilde{m})$ is given (a), where the occurrence is color coded from light to dark for increasing likelihood. The inset of (a) displays the probability density $P_{\tilde{\xi },\tilde{k}}$ following the same color-code convention. In (a), the solid line indicates the mean ${\tilde{M}}_1$ and the squares report the corresponding estimate ${\widehat{M}}_1$. (b) compares the results of ${\widehat{M}}_2$ (circles) obtained by the simulations with the theoretical values ${\tilde{M}}_2$ (solid line). Error bars for both ${\widehat{M}}_1$ and ${\widehat{M}}_2$ were calculated using Eqs. (8, 10), respectively (see Fig. 8).

Figure 8

(Color online) Estimators ${\widehat{\mu }}_2\left[{\widehat{M}}_1\right]$ (squares) and ${\widehat{\mu }}_2\left[{\widehat{M}}_2\right]$ (circles) calculated for the random fluid in Fig. 7 and scaled by the squared mean $M_1^2=\mathbb{E}{\left[m\right]}^2$ and the squared variance $M_2^2={\mu }_2{\left[m\right]}^2$, respectively. The associated solid lines are the correspondingly scaled estimates of ${\mu }_2\left[{\widehat{M}}_1\right]$ and ${\mu }_2\left[{\widehat{M}}_2\right]$ obtained from repeating the simulation 100 times (the vertical lines are error bars).