statistical and sampling issues when using multiple particle...

Statistical and sampling issues when using multiple particle tracking

Thierry Savin and Patrick S. Doyle*Chemical Engineering Department, Massachusetts Institute of Technology, 77 Massachusetts Avenue,

Cambridge, Massachusetts 02139, USA�Received 14 December 2006; published 8 August 2007�

Video microscopy can be used to simultaneously track several microparticles embedded in a complexmaterial. 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 evaluatestructural and/or mechanical properties of the assessed material. However, the sampling of measured displace-ments 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 trackingtechnique, we derive estimators for the mean and variance of the probes’ dynamics that are independent of thepeculiar statistical characteristics. We expose stringent tests of these estimators using simulated and experi-mental complex systems with a known heterogeneous structure. Up to a certain fundamental limitation, whichwe characterize through a material degree of sampling by the embedded probe tracking, these estimators can beapplied to quantify the heterogeneity of a material, providing an original and intelligible kind of information oncomplex fluid properties. More generally, we show that the precise assessment of the statistics in the multipleparticle tracking output sample of observations is essential in order to provide accurate unbiasedmeasurements.

DOI: 10.1103/PhysRevE.76.021501 PACS number�s�: 83.10.Pp, 05.40.�a, 07.05.Kf, 83.80.Kn

INTRODUCTION

Soft matter scientists have expressed an increasing inter-est in resolving the structure of complex fluids at a micron-size length scale. This interest is justified by the implicationof the spatial micro-organization of the components of acomplex material in its transport, rheological, and opticalproperties at equilibrium �see Ref. �1� and referencestherein�, as well as in its eventual kinetics of formation �2,3�.This microscale characterization has also been shown to beparticularly important in biological applications. From inter-and intracellular transport phenomena to molecular motoractivity, the driving �or resisting� forces occur on micron andsubmicron length scales, and cannot be inferred from bulkmeasurements �4,5�. Also, in the biology framework, it hasbeen reported that the fate and behavior of single cells de-pend on their local level of confinement �6� or the stiffness oftheir microenvironment �7�. More strikingly, it has been re-cently pointed out that, in general, systems with heteroge-neous features can lead to apparently peculiar measurements,themselves leading to incorrect interpretations �8�. Thus, ac-cess to some measure of the spatial distribution of a givenmaterial property �pore size, viscoelasticity, charge density,etc.� provides fundamental information for the understandingof a plethora of phenomena involved in complex fluid sci-ence.

Consequently, the range of techniques available toachieve these goals has been broadened over recent decades.Direct observations of the structural elements and their orga-nization are made possible thanks to a wide range of micros-copy techniques. Light, fluorescence, electron, and atomic

force microscopy are able to report a spatial mapping of thecomplex fluid structure at various length scales. Microscopytechniques provide us with two-dimensional pictures, andhigh-order measurements of the spatial distribution of struc-tures can eventually be made from the micrographs by usingimage postprocessing techniques �9,10�. Three-dimensionalinformation can be obtained from modified versions of theprevious techniques �confocal and differential interferencecontrast for optical microscopy, quick-freeze deep-etchsample preparation for electron microscopy, etc.�. Scatteringtechniques such as neutron, x-ray, and light scattering, how-ever, spatially average the structural features in the sampleand thus lose some information on the eventual heterogene-ity.

Rheology of a complex fluid is intimately related to itsnetwork microstructure �11�. Compared to the morphologicalmeasurements described above, inferring structural informa-tion from rheological data is, however, a complicated inverseproblem �see, for example, �12� for an application of rheom-etry to extract information on network heterogeneity�. Directmicrorheological mapping of a material is possible by scan-ning its surface with an atomic force microscope �see Ref.�13� and references therein�, and has been addressed recentlyusing a passive microrheology technique �14,15�. Passivemicrorheology measures the response of a material to thethermal motions of micron-sized probes. Under certain con-ditions, it reports either the local mechanical properties orthe microstructure of the material in which the probes areembedded. Video multiple particle tracking is a simple andinexpensive microrheology technique that, to date, providesthe highest throughput of spatial microrheological samplingof a material. It is a passive technique, where the thermalfluctuations of about a hundred particles dispersed in thematerial can be tracked simultaneously �see Ref. �16� for adetailed description of the technique�. Measurements areusually fast, and the statistical errors are typically considered

*Author to whom correspondence should be addressed. Email ad-dress: [email protected]

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small because of the large amount of data collected. Its sim-plicity and availability have made it very popular, and in thelast few years, investigation of microenvironments usingvideo multiple particle tracking has been performed on vari-ous materials, such as actin systems, agarose gels, cells, andDNA solutions �17–20�. In recent studies, it has been used toextract the pore size distribution in cross-linked actin net-works �21� and to characterize anisotropic gels of alignedDNA �22�. We also recall at this point that the material het-erogeneity can be a problem for microrheologists trying tocalculate bulk properties �23�. A method called two-pointmicrorheology that tentatively overcomes this problem hasrecently been introduced using the cross correlation of thepaired motions of the probes �24,25�. However, local one-point microrheology from video multiple particle tracking, asused in this paper, remains the only available method thatcan assess heterogeneity of a sample.

In this regard, the complex materials investigated withmicrorheology are often known to be heterogeneous, suchthat, depending on where the probe is located in the sample,its motion will exhibit different dynamics �26�. In parallel,due to the finite imaging volume, the amount of data col-lected from a given particle is limited by the duration of itsresidence in the volume of observation. The complexitycomes from the fact that the duration of the measured par-ticle trajectory depends also on the local material properties.For example, a model heterogeneous system can let someparticles travel throughout its porous structure, but willtightly trap a subpopulation of particles in its smaller pores�such dynamically bimodal behavior has been observed inactin gels �27��. The trapped particles will be tracked for thewhole acquisition time, whereas free particles will leave thevolume of observation and/or enter—possibly severaltimes—during the same acquisition time. As a result, numer-ous short trajectories will be extracted in regions of loosemeshing where particles are free to move, whereas only afew long trajectories will be extracted from the signal oftrapped particles, even if the material hypothetically exhibitsthe same amount of small and large pores. This toy modelexample gives an immediate sense of the peculiar statisticalsampling of a heterogeneous material using this technique.The sampling depends on both the size of the imaging vol-ume and the probed material’s structure.

Ultimately, the distribution of material properties needs tobe characterized independently of the measurement tech-nique. The distribution of a given property can be quantifiedby its mean, variance, skewness, kurtosis, and other higher-order moments or functions of moments, such as the non-Gaussian parameters �28�. The latter have already been usedto quantify dynamical heterogeneity in colloidal systems�29,30�. Through the dynamics of the probes, the materialheterogeneity is indirectly, but almost uniquely, quantified�see subsequent sections�. For example, the mean of theprobes’ individual mean-squared displacements, that is, theensemble-averaged mean-squared displacement, is usuallycalculated by dividing in time individual trajectories into dis-placements and accounting for all displacements in the sameway, disregarding the trajectory they were extracted from �asimultaneous time and ensemble average is thus performed�.But the ensemble-averaged mean-squared displacement has

also been calculated as the center of the individual mean-squared displacement distribution obtained from prior time-averaging calculations on the individual trajectories �31�.Similar calculations based on time-averaged estimationsfrom each probes’ trajectory were used in Refs. �14,32� toperform qualitative analyses that circumvent, to a certain ex-tent, the statistical limitations described in the previous para-graph.

In the current literature, we found only two methods toassess a material’s heterogeneity using multiple particletracking. In the first kind of heterogeneity study, bin partitionanalysis based on percentile calculations is used to improvestatistical accuracy. This analysis was successful in assessinga number of mechanical mechanisms in the cellular machin-ery �2,3,19,20,33�. Nevertheless, it remains overall a qualita-tive perspective, since it compares results in complex sys-tems with measurements in homogeneous glycerol solutions.It has, however, the advantage of being able to provide asomewhat surveyable degree of heterogeneity, allowing, forinstance, the ordering of materials from the most heteroge-neous to the least. In a second attempt, hypothesis testingbased on the F ratio of paired mean-squared displacementswas performed to classify particles’ dynamics into statisti-cally distinguishable groups �15�. Although it is not the moreconvenient way to quantify the heterogeneity �i.e., with asingle number�, this classification allowed the authors to mapthe locations of given microenvironments in an agarose gelsample. Following this idea, it is actually possible to testhomogeneity of multiple dynamics at once using a singlestatistical test �see �34� for a review of available statisticalmethods�.

In this paper, we develop a way to rigorously calculate thefirst two moments of the probe particle dynamics in a het-erogeneous system. Using a mathematical formulation forthe peculiar statistical sampling obtained from multiple par-ticle tracking output, we derive estimators of these two mo-ments �mean and variance� that are independent of the sam-pling design, up to a certain fundamental limitation of thetechnique that we call the material’s detectability. In the firstpart, the theoretical approach is discussed and we introducean important factor, called the degree of sampling, that quan-tifies the level of detectability of certain probe dynamics bythe technique. From this approach, estimators for the twofirst moments of individual mean-squared displacements arederived. Section III provides stringent testing of these esti-mators on simulated and experimental systems of increasingcomplexity that cover a wide range of actual scenarios.

I. THEORY

A. Sampling design of multiple particle tracking

Let �� ,x� be the value of a material property � at thelocation x= �x ,y ,z��V in the material, evaluated at the fre-quency �. We write as P��v�=V−1�V��v−�� ,��d� theprobability density function of this material property in avolume V of medium, where � designates the Dirac deltafunction. A schematic of such a heterogeneous fluid is pic-tured in Fig. 1. An indicator of heterogeneity of the material

THIERRY SAVIN AND PATRICK S. DOYLE PHYSICAL REVIEW E 76, 021501 �2007�

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in this volume is then given by the moments about the meanE�� of P�� :�r��=E[��−E��r], whereE�¯� designates the expectation value.

In a passive microrheology experiment such as the mul-tiple particle tracking technique used in this paper, a measureof the local material property is made through the thermalmotion of a micron-sized particle embedded in the material.A thermally fluctuating particle following the trajectory x�t�exhibits dynamics that, at a time scale �=�−1, can be quan-tified by the distribution of the displacement dt��=x�t+��−x�t�, given the position x�t� from which the step displace-ment is taken. Usually, a measure of the local material prop-erty is made by calculating the mean-squared displacementmt�� ,��=E�dt��2 �x�t�=��=�� 2Pdt��x�t�� d�, where theintegral runs over all possible displacements. This materialproperty is defined at a location x�t�, within a minimumlength scale given by the extent dt�� of the trajectory �i.e.,the amplitude of the deformation applied to the material�,and over which the material is assumed homogeneous. Fromthe stationary assumption, this property is independent of t,and we will write it m�� ,��. Hence the distribution that isaccessible with the technique is Pm��x�m ��=��m−m�� ,�� and the central moments �or moments about themean� of Pm��

u �m�=V−1�VPm��x�m ��d�, obtained with uni-form spatial distribution of positions in the volume V�Px��=V−1�, can be used to quantify the heterogeneity ofthe material through the probes’ heterogeneous dynamics�we will expose some limitations to this idea in the nextsections�. Such moments can be written in terms of the rawmoments,

E�m��r� = 0

+�

mrPm��u �m�dm

= V−1V � 2Pd��x��d��r

d� . �1�

We note at this point that, since the quantity to estimateinvolves two integrations, there will be two levels of sam-pling when performing the experiments. A first level of sam-pling is obtained in the spatial integration �outer integral inEq. �1�� since, in practice, only a finite number of locationswill be investigated. A second level of sampling needs to becharacterized, as the number of sample displacements ob-tained per given sample location will be eventually small,and will thus require a binning of the volume of observation.It is then possible to gather observed displacements bygrouping their corresponding locations into spatial bins, andthen calculate the inner integral of Eq. �1� in each bin.

In the multiple particle tracking technique, trajectories ofthe probes’ Brownian motion are obtained by processingmovies acquired with video microscopy. The sampling of thematerial is limited in space by the volume of observationVb�V �the camera field of view in the plane and the trackingdepth in the direction perpendicular to the plane� which con-tains Nb�t� probe particles at a given time t �see Fig. 1�. Alimitation in time is naturally given by the duration of acqui-sition Tb=nb�t, where �t is the time interval between two

successive movie frames, and nb+1 is the number of frames.The output of the tracking is a set of N probe trajectories�xi�t��1iN, each trajectory being sampled every �t. We willwrite as X={�xi,j =xi�j�t+ ti��Vb�0jni

}1iN the sampleof observed positions, where Ti=ni�t is the duration of thetrajectory i, and ti is the time of first observation. Severaltrajectories can eventually correspond to a single particle thatleaves and comes back in the volume of observation, suchthat in general Nb�t�N and TiTb. For a lag time �=n�tand for trajectories such that Ti� �i.e., nin�, we can ex-tract qi�n�=ni+1−n overlapping �hence a priori nonindepen-dent; see the next section� displacements from the trajectoryi, and we obtain the sample D�n�={�di,j�n�=xi,j+n−1

−xi,j−1�1jqi�n�}i�I�n� of observed displacements, whereI�n�= �i :nin� is the set of indices corresponding totrajectories longer than �. The corresponding set ofpositions associated with D�n� will be written X�n�={�xi,j−1�1jqi�n�}i�I�n� �45�.

As noted earlier, in order to estimate Eq. �1� the volumeVb must be subdivided into M cells �or bins� of volume�Vk�1kM, with Vb=�k=1

M Vk. To each bin corresponds a con-stant �assuming homogeneity on the bin size scale� value ofthe inner integral, which we will call mk��=E�m�� ,�� Vk�=m�� ,�k�, where �k�Vk is a hypothetical location ofVk. Suppose that it is possible to determine mk�� using anestimator with arbitrary certainty, g(Dk

v�n�), such that

objective

side view

bottom viewνmax

νmin

x

yz

xb

yb

zb

⇓

FIG. 1. �Color online� Schematic of a heterogeneous system asseen through a microscope. Different colors in the material corre-spond to different values of a certain material property �. The ma-terial is held in a chamber of observation �side view� and randomlydispersed particles �represented by the dots� are visible only whenmoving in a certain imaging volume Vb=xb�yb�zb �dashedframe� whose smallest dimension zb in the z direction determinesthe limitations in sampling. The bottom view pictures a typicalsnapshot at a given time t.

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E�g(Dkv�n�)�=mk�� from any subset Dk

v�n� of D�n� contain-ing only displacements �di,j�n�� for which the correspondingpositions �xi,j−1� are in Vk. By making such an assumption,we aim at characterizing the first level of sampling only,discarding the second level, which will be discussed later inthe text. The estimators of the spatial moments of m�� arethen

E�m�� = �k=1

M

�kg„Dkv�n�… ,

�2�m�� =

�k=1

M

�k g„Dkv�n�… − �

k=1

M

�kg„Dkv�n�…�2

�k=1

M

�k�1 − �k�

, �2�

where �k=Vk /Vb, as obtained from the usual sample estima-tors of the weighted mean and variance with uncorrelatedsamples �35�. The difficulty is to build the set �Vk�1kM

such that, on the one hand the statistical accuracy withineach cell is high enough �obtained by increasing Vk, which inturn will decrease M�; on the other hand, the number of binsis large enough such that moments are accurately estimatedand such that the assumption of uniformity within each clus-ter is valid by sampling the space at a fine enough scale�large M�. In other words, there is a compromise betweengaining accuracy in estimating the inner integral in Eq. �1�and accuracy in the outer integral calculation. For generality,it is convenient to have a formula independent of the choiceof �Vk�1kM. This can be achieved under some reasonableassumptions. We first assume that each trajectory is entirelycontained in one of the bins �Vk�1kM. We can then write

g„Dkv�n�… = �

i�Ik

ni�−1 �i�Ik

nig„Di�n�… ,

where Di�n� is the subset of D�n� corresponding to trajectoryi, and Ik is the set of indices corresponding to trajectoriescontained in Vk. Next we assume that ��i�Ik

ni� /Vk=Cbnb isindependent of k, where Cb is the uniform density of particlein the material �recall that �i�Ik

ni is the total number ofobserved positions in Vk�. We get

E�m�� = �i=1

N

pig„Di�n�… ,

�2�m�� =

�i=1

N

pi g„Di�n�… − �i=1

N

pig„Di�n�…�2

�k=1

M

�k�1 − �k�

, �3�

where pi=ni /�i=1N ni. The denominator of �2�m�� is a cor-

recting factor for the bias, and approaches 1 for M 1. Asmentioned earlier, the use of Eq. �3� for the sample varianceassumes that the probes’ motions are mutually independent�or at least, uncorrelated�. This is an acceptable approxima-

tion, since the residual correlation, measured for example intwo-point microrheology techniques, is a higher-order hydro-dynamic effect that vanishes with increasing interparticledistance. At this point, it is also possible to characterize theuncertainty of these estimators solely due to the first level offinite spatial sampling. Using again a common formula forthe variance of the sample mean and of the sample variance�35�, we have

�2†E�m��‡ = �2�m��k=1

M

�k2,

�2†�2�m��‡ = ��4�m�� − �2�m��2��k=1

M

�k2 + O�

k=1

M

�k2� .

�4�

The calculation of the quantity �k=1M �k

2 requires knowl-edge of the individual volumes Vk for all 1kM. How-ever, at an instant t of the measurement, the number ofsample locations observed in the volume Vb is Nb�t�. We canthen argue for this calculation that the uncertainty in the firstlevel of spatial sampling is proportional to E�Nb�t��−1=Nb

−1.Thus we will write �k=1

M �k2�Nb

−1 when estimating the abovequantities.

As pointed out earlier in the text, only trajectories forwhich Ti� will be counted in the displacements’ sample.For a trajectory shorter than �, the corresponding Di�n� isempty and no estimator g(Di�n�) can be computed. The du-ration of a trajectory Ti is in fact an observation of anotherrandom variable whose distribution depends, among otherfactors, on the dynamics of the corresponding particle.Hence, a particle that is likely to travel across the volume ofobservation �say its smallest dimension zb; see Fig. 1� over alag time � will not be tracked for a sufficient time to performany computation from its trajectory. The corresponding ma-terial in which probes undergo such dynamics will not bedetectable by the technique. A proposed quantitative indica-tor of this effect is the degree of sampling, which we defineas

��n� =

�i�I�n�

ni

�i=1

N

ni

. �5�

It is the ratio of the number of positions for which a displace-ment at lag n has been measured to the total number ofobserved positions �over which the sum runs in the estima-tors Eq. �3�, where this effect has been discarded by theinitial assumptions�. Limiting behaviors of this indicator arereached when all positions are associated with a displace-ment ��n�=1 and the sampling is at its best�, and when nosample displacement could be calculated ��n�=0 and thematerial is totally transparent to the technique�. Furthermore,we will show in the following sections that this degree ofsampling � is a good measure to characterize the effect of thevolume of observation on the quality of the estimators sub-sequently derived.


021501-4

To conclude this section, we notice that this choice of bins�Vk�1kM is not unique. For example, bins of identical shapeand size could be chosen to divide Vb independently of thesample X. Their size and number would then need to beadjusted for each lag time to reach the best statistical accu-racy. The latter might be challenging to evaluate on the fly.The bins could also be constructed randomly by Voronoitessellation from a subset of X. The choice made here, rely-ing on the assumption that a particle probes a unique mate-rial property along its path, is a more natural procedure in theparticle tracking framework �the case where a single, freelydiffusing particle encounters liquids of different viscositiesalong its trajectory has been recently discussed �36��.

In the next two sections, we investigate the second levelof statistics involved in the evaluation of g(Di�n�).

B. Characterization of the sample of displacements

For simplicity, we will consider only one-dimensionalrandom walks in the x direction. The output of a multipleparticle tracking experiment is a list of one-dimensionaloverlapping displacements as defined earlier, D�n�={�di,j�n�=xi,j+n−1−xi,j−1�1jqi�n�}i�I�n�. Here we look at thecharacteristics of this sample. If all processes xi,j are as-sumed stationary with respect to the time j, as well as inde-pendent of one another �see discussion in the previous sec-tion�, then we can immediately write the following second-order characterization:

E�di,j�n�� = 0,

E�di,j2 �n�� = mi�n� ,

E�di,j�n�di�,j��n��/E�di,j2 �n�� = �ii��n,i

�1��j − j�� ,

where �ii� designates the Kronecker delta and where the cor-relation coefficient �n,i

�1��h� can be expressed in terms of themean-squared displacement mi�n� by

�n,i�1��h� =

�mi�h + n� − mi�h�� + �mi�h − n� − mi�h��2mi�n�

.

Under the assumption that all displacements are Gaussiandistributed, this second-order characterization is sufficient toknow all other moments of the displacement samples. Forexample, an interesting characteristic when calculating themean-squared displacement to some power is the correlationcoefficient of the squared displacements,

E�di,j2 �n�di�,j�

2 �n�� − E�di,j2 �n��E�di�,j�

2 �n��

E�di,j4 �n�� − E�di,j

2 �n��2 = �ii��n,i�2��j − j�� ,

where it can be shown that �n,i�2��h�= ��n,i

�1��h��2. We show typi-cal values of �n

�1��h� in Fig. 2 for various dynamics m�n�. Weobserve that in general the overlapping displacements arecorrelated up to a nonuniversal lag h. More interestingly, wesee on this figure that even if the displacements are not over-lapping �hn�, anticorrelations can be observed. In this re-gard, only the Newtonian dynamics �pure diffusion, m� �n� in

Fig. 2�a�� exhibit uncorrelated successive displacements. Wecan conclude then that the observations of the squared dis-placement will also exhibit correlations in a nonuniversalway.

This will present a problem when applying usual estima-tor formulas involving squared sums of observations. A com-mon way to decorrelate observations is to perform a block-average transformation �37�. We define the s-sized blockaverage of the rth power displacement in the following way:

dri,j�n� =

1

s�l=0

s−1

di,sj−lr �n� .

The characteristic of these new observations can also be cal-culated:

E�dri,j�n�� = E�di,j

r �n��

and

(a)

(b)

(c)

(d)

100 103

0.7

0.3

1

mm

mm

hh

hh

ss

ss

ρ(1)n (h) ρ1

n(1)m(n)

nnn

nnn

nnn

nnn

−1 1

FIG. 2. �Color online� Autocorrelation coefficients of the dis-placements and the block-averaged displacements. Different dy-namics m�n� are investigated: �a� is a purely Newtonian fluidm�n�� n�, �b� and �c� are 0.7 and 0.3 power-law dynamics, m�n�� n�0.7 and m�n�� n�0.3, respectively, and �d� is a Voigt fluid modelwith relaxation time of 10, m�n��1−e−�n�/10. The plots on the sec-ond column give the displacements’ autocorrelation coefficient�n

�1��h� as a function of �n ,h� for the corresponding dynamics shownin the first column. The third column of plots shows the autocorre-lation coefficient of two successive boxed averaged displacements�1n

�1� as a function of �n ,s� �see text for notations�.


021501-5

E�dri,j�n�dri�,j�

�n�� − E�dri,j�n��E�dri�,j�

�n��

E�dri,j2 �n�� − E�dri,j

�n��2 = �ii��rn,i�j − j�� ,

where we compute the block-averaged transform of the dis-placement autocorrelation coefficient,

�rn,i�h� =

�k=−�s−1�

s−1

�1 − �k�/s��n,i�r��h�s + k�

�k=−�s−1�

s−1

�1 − �k�/s��n,i�r��k�

.

We plot in Fig. 2 the correlation �1n,i�1� of successive block-

averaged displacements as a function of s, and we see thatthe choice s=n decorrelates successive block-averaged dis-placements. This apparent decorrelation is effectively asmoothing of the raw displacements correlation, that can beobserved for higher order correlation coefficient ��rn

�h� is

also almost zero for r�1 and for h1, even though it is notthe case for the corresponding �n

�r��h�; data not shown�. Amore general definition of the block-averaged rth power dis-placement is given by

dri,j�n� =

1

si�n� �l=0

si�n�−1

di,si�n�j−lr ,

where si�n�=n if qi�n�n, si�n�=qi�n� otherwise. Foreach trajectory, we obtain q� i

�n� block-averaged displace-

ments with q� i�n� equal to the biggest integer smaller

than qi�n� /si�n�. It forms a sample Dr�n�={�dri,j

�n��1jq�i�n�}i�I�n� of mutually uncorrelated observa-

tions. Note that this choice of si�n� does not lead to a signifi-cant loss of statistics as compared to calculations made withnonoverlapping displacements. In the latter, approximatelyni /n displacements can be extracted from trajectory i, com-parable to q� i

�n�.

C. The estimators

We define the weighted sample mean

dp�n� = �i�I�n�

�1jqi�n�

wi�n�di,jp �n� ,

with wi�n�= �ni /qi�n�� /�i�I�n�ni over the sample D�n�. Simi-larly, we will write

drp�n� = �

i�I�n��

1jq� i�n�

w�i�n�dri,j

p �n� ,

with w�i�n�= �ni /q� i

�n�� /�i�I�n�ni over the sample Dr�n�. In

general, drp�n��drp�n� unless p=1, in which case the two

terms differ only by the number of sample displacements lostwhen taking the integer part of qi�n� /si�n�. However, wehave E�dr�n��=E�dr�n��.

We define the estimator M1�n� of M1�n�=E�m�n�� by

M1�n� = d2�n� = �i�I�n�

wi�n� �1jqi�n�

di,j2 �n� . �6�

With pi�=ni /�i�I�n�ni, the second-level expectation is

E�M1�n��Vk�1kM� = �i�I�n�

ni�−1 �i�I�n�

nimi�n�

= �i�I�n�

pi�mi�n� ,

and is unbiased in the first level for ��n�=1 �see Eq. �3� with

pi�= pi�. If the system is homogeneous, however, M1�n� isunbiased for any value of ��n�, as will be shown in the nextsection.

To calculate the variance of M1�n�, we note that there aretwo independent contributions in the uncertainty of the esti-mator, each contribution coming from the two different lev-els of sampling mentioned earlier in the text. To see this, letus briefly derive a simplified result. We consider a set of Nunbiased estimators �ai�1iN of a corresponding set of val-ues ��i�1iN, the latter forming a sample of independentobservations of a random variable �. Similar to the currentstudy, we have here two levels of statistical uncertainty whenestimating some moments of �: one level is affecting theaccuracy of estimating each �i with ai, and the other is com-ing from the limited number of observations �i. In the firstlevel, we have E�ai ��i�=�i and we define the variance of theestimator ai by �2�ai ��i�=E�ai

2 ��i�−E�ai ��i�2=�i /ni whereni is the size of the sample used to calculate ai. It can beshown that �i=O�ni

0� in most cases �38�. We can calculatethe mean and variance of the quantity a=N−1�1iNai that isan estimator of E��. We use the iterated expectation to write

E�a� = E†E�a��i�1iN�‡ = E N−1 �1iN

�i� = E�� ,

which shows that the estimator is unbiased. The same way,we find

E†E�a2��i�1iN�‡ = E N−2 �1iN

�i/ni + N−2 �1iN

�i�2�= E��/n�/N + �2��/N + E��2,

such that finally the variance of the estimator a,

�2�a� = �2��/N + E��/n�/N ,

is the sum of two terms. The first term comes from the levelof sampling of the random variable � with the N observa-tions �i and is proportional to N−1, and the second term in-cludes the uncertainties from the estimation of each �i by aiand is of order �Nn�−1, where Nn is the total number of initialobservations. In the current study, the first term correspondsto the expression given by Eq. �4� for the first level of spatialsampling. The second term can be calculated by assumingthe system homogeneous �that is, taking �2��=0 in thedemonstration presented above�, which means that all initialobservations are taken from the same distribution. Note alsothat from intermediate calculations in the above demonstra-tion, we get


021501-6

�2�a��i�1iN� = E�a2��i�1iN� − E�a��i�1iN�2

= N−2 �1iN

�i/ni �7�

of order �Nn�−1. This result will be useful in the followingderivations.

We follow this idea to calculate the variance of M1�n�. Weuse the approximation d2�n��d2�n�, the mean of the sampleD2�n� of uncorrelated observations. In that case, it is possible

to show that the variance �2�M1�n��h obtained when all ob-servations in D2�n� are identically distributed �assumption ofhomogeneity� is equal to �2�d2i,j

��i�I�n�q� i�n�w�

i�n�2, which is

estimated by

�2�M1�n��h =

�d22�n� − d2�n�2� �

i�I�n�q� i

�n�w�i�n�2

�i�I�n�

q� i�n�w

i�n��1 − w�

i�n��

,

so that the total expression is �see Eq. �4��

�2�M1�n�� =M2�n�

Nb+ �2�M1�n��h, �8�

where we have used Eq. �4� with M2�n� an estimator ofM2�n�=�2�m�n��.

To calculate M2�n�, we assume that all displacements areGaussian distributed. In that case, we find that

M2�n� = d4�n�/3 − M1�n�2 �9�

is almost unbiased. Indeed, for the first term in M2�n�, wehave

E�d4�n��Vk�1kM� = 3

�i�I�n�

nimi�n�2

�i�I�n�

ni

since E�d4�=3E�d2�2 for a zero-mean Gaussian random vari-able d. The expectation of the second term can be written

E�M1�n�2��Vk�1kM� = E�M1�n��Vk�1kM�2

+ �2�M1�n��Vk�1kM� ,

where the second term is of order �i�I�n�q� i�n�w�

i�n�2 from Eq.

�7�, that is, of the order of the inverse total number of dis-placements �i�I�n�ni in D�n�. This second term can thus beneglected, and we finally get

E�M2�n��Vk�1kM� = �i�I�n�

pi� mi�n� − �i�I�n�

pi�mi�n��2

+ O� �i�I�n�

ni�−1� ,

which is the estimate given in Eq. �3� for the first level of

uncertainty when ��n�=1. To estimate �2�M2�n�� we follow

the same line of reasoning as for the variance of M1�n�. We

used the software SYMSS �39�, relying on the computer alge-bra package MATHEMATICA �Wolfram Research�, to calculate

the variance �2�M2�n��h under the homogeneity assumption,and found

�2�M2�h = d42 − d4

2

9+ 4d2

2�d2

2 − d2

2� +

4d2

3�d2 d4 − d2 d4��

� �i�I�n�

ni�−1+ O� �

i�I�n�ni�−2� ,

where we have dropped the dependency in n for concisenessin the notation �also, we used d2 d4=�i�I�n��1jq�

i�n�w� i

d2i,jd4i,j

�. The first-level term given by

Eq. �4� requires the estimation of �4�m�n�� the other term

�2�m�n�� is estimated by M2�n��, which is obtained under theGaussian assumption

�4�m� = d8/105 − 4d2 d6/15 + 2d22d4 − 3d2

4,

so that finally

�2�M2� = ��4�m� − M22�Nb

−1 + �2�M2�h. �10�

II. METHODS

A. Simulations

We used Brownian dynamics simulations to test the va-lidity of the estimator. Simulations are convenient as theyprovide both freedom of design and a well-defined system.The input parameters are easily changed and well controlledin a wide range of values. However, we must first ensure thevalidity of this range as compared to real experimental de-signs.

1. Scaling and range of parameters

The smallest dimension of the volume of observationgiven in the z direction is called zb �see Fig. 1�. Its valuedepends on the depth of field of the multiple particle trackingtechnique as well as the tracking parameters. But typical val-ues range from 1 to 10 �m �see Sec. II B�. The concentra-tion of probe particles is chosen such that minimal interac-tion between particles is expected. Given the magnification,between 10 and 100 particles can be tracked simultaneously�10�Nb�100�. The video rate is usually not greater than100 Hz; thus the time interval between consecutive frames is�t�0.01s. The duration of the movie is limited by the num-ber of frames storable in memory, but typically Tb=1000�t.The smallest viscosity encountered in typical applications isthat of water �=10−3 Pa s, and at T=25 °C with smallesttrackable particle radius a=0.05 �m, we get that �=6�a��1 cP �m. Throughout the following sections, we will usequantities made dimensionless with the distance zb and thetime �t. Hence, in the following we will designate the di-mensionless quantities using a tilde, such that, for example,m=m /zb

2 designates the dimensionless mean-squared dis-placement. However, to avoid redundancy, the dimensionless


021501-7

lag time and acquisition duration are written n= �=� /�t and

nb= Tb=Tb /�t, respectively, as already introduced earlier.

2. Brownian dynamics simulations

A Brownian dynamics simulation was employed to createparticle trajectories �40�. An explicit first-order time-steppingalgorithm was used to advance the position x j�t� of a particlej at time t,

x j�t + �t� = x j�t� + x j�t��t ,

where �t is the time step and x j�t� satisfies the followingstochastic differential equation:

x j�t� �1

� jF j„x j�t�… +�2kBT

� j�tdW j ,

obtained by assuming the drag on the particle to be Stokesianand by neglecting any other hydrodynamic interactions.Also, F j�x�=−kj�x−c j� is the Hookean linear force law ap-plied to the particle by the medium from the fixed centerposition c j �F j�x�=0 when simulating a Newtonian fluid� andW j is a Wiener process that satisfies �dW j�=0 and�dW jdW j�=� where � is the unit second-order tensor �40�.The model fluid is then characterized by the properties � j= �� j ,kj�, and dynamics by m��=2kBT�1−e−kj��/�j� /kj ineach direction. Nb trajectories of duration Tb were simulated.The effect of the finiteness of the imaging volume is takeninto account only in the z direction of the motion x�t�= (x�t� ,y�t� ,z�t�), where only positions verifying 0z�t��zb can be observed. In practice, the Nb force centers�cj�1jNb

are uniformly distributed in an interval �0,zb� inthe z direction. About these centers, Nb initial positions�zj�0��1jNb

are randomly chosen from the equilibrium dis-

tribution Pzj�0��z��e−kj�z − cj�2/�2kBT�. This ensures that the sys-

tem is at thermal equilibrium at t=0. The trajectories are thensimulated starting from these positions. For each trajectoryx j�t�, with 1 jNb, the first z position that lies outside thebox interval �0,zb� is translated by a length zb to fall back inthe observable interval. Accompanying this translation, anindex of trajectory is incremented and assigned to the frag-ment of trajectory starting from this translated point. Theprocess is repeated over time to obtain fragments of theoriginal x j�t� as divided by these periodic boundary condi-tions in the z direction. After performing this transformationto the Nb initially simulated trajectories, all the fragments arereindexed to obtain the sample �xi�t��1iN of N tracks. Byalways keeping all successive positions in the volume ofobservation, the condition of constant particle density,�i=1

N Ti=NbTb, is satisfied for all time. For all sets of simula-tion, �t=10−1�t. We verified that our results did not appre-ciably change for smaller values of �t. Allowing the densityNb�t� to fluctuate by observing only a subvolume of height zb

in an initial box of height larger than zb is a more detailedmodel of real multiple particle particle tracking experiments.But again, by doing so, we could not observe significantchange in the results, whereas the computation time is in-creased.

In the following, the friction coefficient and spring con-

stant are made dimensionless by using �=� / �kBT�t /zb2� and

k=k / �kBT /zb2�, respectively. From the previous discussion

about the range of parameters met in experiments, we con-

clude that typically ��10.

B. Experiments

Simple bimodal heterogeneous systems were made bycreating gel features in the field of view of the multiple par-ticle tracking setup. We used microscope projection photoli-thography �41� to create well-defined regions of gel withembedded beads, within a region of purely viscous, unpoly-merized material also populated with the probe particles.

We used a solution of 10% �v/v� poly�ethylene gly-col��700� diacrylate �PEG-DA, Sigma-Aldrich�, 0.5% �v/v�solutions of Darocur 1173 �Sigma Aldrich� initiator, 20%�v/v� ethanol �95% grade, Pharmco�, and 20% �v/v� 5�TBE buffer, in which 2a=0.518 �m diameter carboxylate-modified yellow-green particles �Polysciences� at a volumefraction �=2.48�10−3% were thoroughly dispersed by20 min sonication. The sample was injected in a150-�m-high, custom built chamber of observation sealedwith vacuum grease, and mounted on an inverted microscope�Zeiss�. Four photomasks with unique pinholes of diameter470, 700, 800, and 900 �m were designed in AUTOCAD 2005

and printed using a high-resolution printer at CAD Art Ser-vices �Poway, CA�. The masks were then inserted into thefield-stop of the microscope. A 100 W HBO mercury lampserved as the source of uv light. A filter set providing wideuv excitation �11000v2, uv, Chroma� was used to select lightof the desired wavelength and a VS25 shutter system �Unib-litz� driven by a computer-controlled VMM-D1 shutterdriver provided specified pulses of uv light. The oligomersolutions mixed with probed particles were exposed for atime of 1.5 s to photopolymerize poles in the chamber byusing the 10� objective �numerical aperture �NA�=0.25�. Ithas been shown that the poles are not cylinders of uniformradius because of the shape of the light beam �42�. However,they can be considered as straight cylinders over a lengthscale of 60 �m around the focal plane of the beam. At thisaltitude in the chamber, their diameters are respectively2Rgel=110±4, 170±2, 200±2, and 230±3 �m �this leads toa magnification of approximately 1/4 from the field-stopplane to the objective focal plane�. With a 20� objective�NA=0.5�, we recorded the motion of the particles in a vol-ume of observation placed such that, within this imagingvolume, the poles are straight cylinders at all altitudes z�Vb. The movies were acquired at a rate of ten frames persecond �i.e., �t=0.1 s� and for a duration of 2000 frames.We verified that the tracking dynamic errors caused by thevideo interlacing do not affect the validity of the results pre-sented in this study �43�. We also verified that such an expo-sure time of the solution using the fluorescein-matching fluo-rescent filter does not induce any photopolymerization. Themovies were analyzed offline using the IDL language �Re-search Systems� tracking package �16�. We find that we track�Nb��250 particles in a field of view of dimensions 300


021501-8

�300 �m2, with a standard deviation of less than 5% of thismean �Nb�. Using �Nb�=Cbxbybzb with Cb=� / ��2a�3 /6�and xb=yb=300 �m, we can conclude that, for these experi-ments, using the 20� objective, we have zb=8.2±0.4 �m.Note that this length scale is smaller than the length scale ofcurvature of the poles’ profile in the z direction, ensuring thatthe polymerized structures are seen as straight cylinders inthe experiments.

III. RESULTS AND DISCUSSION

We investigated the quality of the estimator for two dif-ferent dynamics. The simple diffusive dynamics in Newton-ian liquids allow us to scan the parameter space �n , m� using

the relation m�n�=2�n� / � for varying �. In the single-relaxation-time Voigt fluid, the space �n , m� is mapped by

changing � and k using the following dynamical relation:

m�n�=2�1−e−�n�k/�� / k. Admittedly, these two kinds of dy-namics do not cover all possible multiple-relaxation-timeviscoelastic dynamics often encountered in microrheometrymeasurements on complex systems. However, they aresimple models that are easily implemented with Browniandynamics simulations, and they exhibit sufficiently differentdynamical characteristics to capture the effect of probe dy-namics on the quality of the estimators, and more generallyon the statistics of the obtained sample.

Before analyzing heterogeneous fluids having these twodynamics, it is instructive to evaluate and characterize theformula in homogeneous fluids.

A. Homogeneous fluids

We performed simulations of homogeneous fluids using

��rN= ��r=10r/2 , k=0��0r11 to scan the parameter space

�n , m� using Newtonian dynamics m�n�=2�n� / �r with differ-

ent friction coefficients. Also, we used ��rV= ��r=10r/2 , kr

=10r/2 /5��0r11 to sample the parameter space using Voigt

dynamics with a dimensionless relaxation time at �r / kr=5:m�n�=2�1−e−�n�/5� / kr. These parameters allow us to investi-gate �n , m� around m=1 where the main effects of the finitevolume of observation are expected to occur. But their rangealso report mean-squared displacements sufficiently low, m�1, so that the limit in which the volume of observation isinfinite is approached. In the simulation method, the latterlimit has been separately simulated by discarding the laststep consisting of the trajectories subdivision.

For each of the 24 types of fluid �12 Newtonian and 12Voigt�, simulations were repeated 100 times to obtain 100observations of the various random parameters of interest.These include the largest lag time nmax for which it is pos-sible to calculate the estimators �chosen such that there re-mains at least two indices in the set I�nmax��, the degree of

sampling ��n�, and the estimators M1�n� and M2�n�. Fromthe 100 simulations performed under identical conditions,these observations are calculated at common values of lagtime from one simulation to another, so that statistics �meanand standard deviation� for each measure are calculated for agiven n.

Figure 3 gives an overview of the results obtained forhomogeneous fluids with Nb=100 and for two acquisitiontimes nb=1000 and 100. In particular, it shows the mappingof the degree of sampling in the parameter space �n , m� forthe two kinds of dynamics. In this figure, each solid blackline represents the relation m�n� used in the simulations tomap the space, m�n�� n� for �a� and m�n��1−e−�n�/5 for �b�,and their end points �the maximum n at which they aredrawn� is at E�nmax�.

First, we see that for mean-squared displacements abovethe observation limit m=1, the degree of sampling vanishesfor both kinds of dynamics. This result is not surprising, as

100 101 102 10310-4

10-3

10-2

10-1

100

101

100 101 102 100 101 102 10310-4

10-3

10-2

10-1

100

101

100 101 1020.0

0.2

0.4

0.6

0.8

1.0

(a) (b)

nb = 1000nb = 1000 nb = 100nb = 100

nnnn

mm θ(n)

m(n) ∝ |n| m(n) ∝ 1 − e−|n|/5

FIG. 3. �Color online� Degree of sampling ��n� in the parameter space �n , m� scanned by two different kinds of dynamics m�n�: �a� is apurely Newtonian fluid, for which m�n�� n�, and �b� is a Voigt fluid with relaxation time 5, that is, m�n��1−e−�n�/5. For each fluid type, twoacquisition times were simulated, nb=1000 on the left and nb=100 on the right. The dashed line represents the observation limit m=1 due

to the volume of observation. The connected squares represent the limit above which the relative bias of M1, defined by Eq. �11�, is greaterthan 5%.


021501-9

for such values of mean-squared displacement, the probesare more likely to cross the entire volume of observationover a time less than the lag time. Hence a particle in the boxat a given time t is likely to be outside the volume of obser-vation at a time t+�, and thus its observed trajectory cannotbe used to extract a displacement at lag time �.

More strikingly, we see that the degree of sampling in amaterial where probes have a mean-squared displacement mat a lag time n depends on the entire dynamics relation m�n�and not only on the position �n , m� in the parameter space�compare the different mappings of Figs. 3�a� and 3�b��. Thisdependency may seem surprising, in particular for the Voigtfluid where some values of �n , m� are undetectable ��n��0� even though m�1. To provide some insights into thiseffect, we show in Fig. 4 typical trajectory durations ni in aselection ��r

N�r��0,3,6� of Newtonian dynamics and��r

V�r��0,3,6� of Voigt dynamics, as well as the resulting varia-tion ��n� of the degree of sampling with the lag time n, forNb=100 and nb=100. Figure 4�c� was obtained by gatheringtogether on the same line the trajectories coming from thesame initial simulated trajectory, subsequently chopped whentaking the effect of a finite volume of observation into ac-

count. Thus, in these plots, a line corresponds to one of theNb simulated particles, and the chopping of this line occurs atthe successive absolute times when the corresponding par-ticle crosses one of the observation volume’s z boundaries.From r=0 to 6, we observe that trajectories are becominglonger, or, equivalently, that the particles are leaving the vol-ume of observation less frequently: from a given position inthe volume of observation, slower particles �lower m at con-stant n� will not be as likely to reach a given boundary thanfaster particles �higher value of m for the same n�. Conse-quently, the degree of sampling at a given lag time n willincrease from faster �higher m� to slower �lower m� dynam-ics, as seen in Fig. 4�b�. Also, for each value of r, since theVoigt dynamics are “slower” than the corresponding New-tonian dynamics, we also verify in Fig. 4�b� that the curvefor the degree of sampling in Newtonian fluids falls belowthe corresponding curve for Voigt fluids with same value ofr. The peculiarity of the detectability is, however, visible bycomparing the Newtonian dynamics �6

N, m�n�=2�n� /103, andthe Voigt dynamics �3

V, m�n�= �1−e−�n�/5� /101/2. At a lag timen=100, both exhibit similar values of m �see Fig. 4�a��whereas the corresponding degrees of sampling are remark-ably different, as seen in Fig. 4�b�. Thus the degree of sam-pling, and hence the detectability, does not depend only onthe particular value of m, but rather on the function m�n� thatdescribes the dynamics of the probes in the entire range oflag times n. The undetectability of certain fluids with mul-tiple particle tracking is an important limitation of the tech-nique. But the dynamic dependency of this detectability is aparticularly strong weakness that prevents some universalityin judging a priori the feasibility of an experiment.

We also report in Fig. 3 the relative bias of the first esti-

mator M1�n�, defined as follows:

b�M1�n�� = E�M1�n��/M1�n� − 1, �11�

where the above expected value is calculated from the 100experiments repeated for each kind of fluid. For these homo-geneous fluids, a bias of less than 5% is measured for almostthe entire observed parameter space. In particular, it is notaffected by the drop in the degree of sampling of the mate-rial. In a wide part where the fluid is almost totally undetect-able ��1�, but where the mean-squared displacement canbe reported, the bias is negligible. This is not surprising forhomogeneous fluids, since any observed displacement willreport a valid measurement of the entire fluid. However, wewill show in the next part how the bias is greatly affected bythe nonuniform detectability of heterogeneous fluids. We

also observed that the estimator M2�n� is unbiased for thiskind of fluid. We defined, for this section only, the bias of

M2�n� relative to M12�n� by

b��M2�n�� = E�M2�n��/M12�n� ,

and we saw that the trends followed by this bias �not re-ported here� are very similar to the one followed by

b�M1�n��. That is, M2�n� is unbiased for almost all the ob-servable parameter space �the ensemble of points �n , m� forwhich ��0�. Finally, we observe that these qualitative be-

10-2

10-1

100

101

102

1 50100 101 1020.0

0.5

1.0

0.0

n

mνN0

νN0

νN0

νN3

νN3

νN3

νN6

νN6

νN6

νV0

νV0

νV0

νV3

νV3

νV3

νV6

νV6

νV6

(a)

(b)

(c)

t/∆t

θ(n)

FIG. 4. �Color online� Trajectory statistics. As shown in �a�, theparameter space �n , m� is sampled using purely viscous dynamicswith material properties ��r

N�r��0,3,6� and Voigt dynamics with relax-ation time 5, ��r

V�r��0,3,6� �see the labels on the figure�. In �a�, thedashed line represents the observation limit m=1 due to the volumeof observation. �b� shows the corresponding variations of the degreeof sampling ��n� as a function of n for each fluid. �c� representstypical trajectory durations: a single dot means that the particle wasobservable for only one frame, and lines indicate trajectories of atleast �t long. Each line in these plots refers to the original simu-lated trajectory that has been fragmented to take limited observationvolume into account �see Sec. II A 2�, and we present results for thetwo kinds of fluid from top to bottom with increasing values of theindex r.


021501-10

haviors are not significantly changed when decreasing nbfrom 1000 to 100 �see Fig. 3� or when decreasing Nb from100 to 10 �not shown�.

B. Heterogeneous fluids

We use now the two model fluids, Newtonian and Voigt,mentioned in the previous part to build heterogeneous fluids.Rather than trying to cover a wide variety of heterogeneousfluids, we will use a canonical model of a heterogeneousfluid to point out the influence of a finite volume of obser-vation on measurements. The study of these simple systems,performed using both simulations and experiments, will befollowed by a model fluid with a more complex heteroge-neous nature, which will illustrate the power of the formula.

1. Bimodal fluids

A canonical model for a heterogeneous fluid is given by abalanced bimodal system, where the fluid is composed ofhalf of one kind and half of another kind. More precisely, wedefine a set of balanced bimodal Newtonian fluids where halfof the volume of observation is occupied with a liquid of

friction �r=10r/2, and the other half exhibits a friction �r+1=10�r+1�/2, where 0r10. The resulting equally likely

probe dynamics are written m�1��n�=2�n� / �r and m�2��n�=2�n� / �r+1, and the heterogeneous fluid will be denominated�r

N+�r+1N . Similarly, a set of bimodal Voigt fluids is built from

a balanced mixture of ��r=10r/2 , kr=10r/2 /5� on one side and

��r+1=10�r+1�/2 , kr+1=10�r+1�/2 /5� on the other side with 0r10, and will lead to the two equally likely dynamics

m�1��n�=2�1−e−�n�/5� / kr and m�2��n�=2�1−e−�n�/5� / kr+1. Thelatter fluid will be referred to as �r

V+�r+1V . Looking at Fig. 3,

we see that the described bimodal fluids are made of a bal-anced composite of two successive dynamics represented by

the black solid lines. The estimators M1�n� and M2�n� arebuilt to evaluate

M1�n� =m�1��n� + m�2��n�

2,

M2�n� =�m�1��n� − m�2��n��2

4

in these models.In the simulation method, these fluids are obtained by

simulating 50 initial trajectories in one homogenous compo-nent of the fluid, and then 50 trajectories in the other com-ponent. The effect of the limited volume of observation istaken into account in the same way as described in Sec.II A 2. In practice, an experiment in such a hypothetical fluidshould be simulated by assigning random initial positions foreach trajectory, uniformly distributed through the heteroge-neous fluid. Hence, by exactly splitting the trajectory set inhalf, we discard in this part the random spatial sampling. It isaimed at isolating a specifically strong limitation of themethod: the effect of detectability on the bias of the estima-tors. In the experiments described in the next section, thespatial sampling is naturally performed.

Referring again to Fig. 3, we observe that, in general,along the lines representing the two dynamics involved in thebimodal fluid, the drop in the degree of sampling is reachedat different values of n and m. The dynamics exhibitinghigher m, that is, m�1�, will eventually become undetectableat smaller lag time, and the tracking technique will then onlyprobe the remaining observable side of the heterogeneousfluid at larger lag time. Consequently, the estimators of themean and variance of the probes’ dynamics will be biased forthese large lag times.

To be more precise, we want to compare the detectabilitymap � as scanned by dynamics in the fluid of type 1, forwhich the mean-squared displacement is m�1��n�, with a mapof the estimators’ bias as scanned by the composite dynamics

described by M1�n�. To perform this comparison quantita-tively, we juxtapose in Figs. 5�a� and 5�b� the behavior of thelevels of constant value of �, with levels of constant bias of

M1�n� and M2�n� in the �n , m� space. The bias is defined herefor both estimators using Eq. �11�,

b�Mi�n�� = E�Mi�n��/Mi�n� − 1 for i = 1,2,

and is visible in Figs. 5�a� and 5�b� by comparing the theo-

retical composite dynamics M1�n� �black solid lines in these

figures� with the estimator M1�n� �dash-dotted lines�. Note

how M1�n� tends to m�2��n� �see labeled dashed line� as thedynamics m�1��n� of fluid 1 becomes less detectable, as ex-plained above.

More specifically, we used the result for the degree ofsampling obtained in homogenous fluid of kind 1 to uniquelyread the coordinates �n� , m�� on the corresponding curvem�= m�1��n�� at which detectability reaches a given value ��the triangles in Figs. 5�a� and 5�b� are at �n� , m�� for �=0.5�. Then, at the same lag time n�, we read the bias of the

estimators at the point (n� ,M1�n��) �the diamonds in Figs.

5�a� and 5�b� are at (n� ,M1�n��) for �=0.5�. Next, by varyingr, we obtained a curve of constant degree of sampling �thesolid lines connecting the triangles in Figs. 5�a� and 5�b� areextracted from the contour plots given in Fig. 3� for the fluidof kind 1, and a corresponding set of bias values on the

associated composite dynamics M1�n�. The mean of this setof bias values was calculated for a given degree of sampling,and is reported in Figs. 5�c� and 5�d� for varying values of �,the error bars being given by the standard deviation of thesame set. If these error bars are small, it means that to agiven value of detectability of fluid 1 corresponds a well-defined value of bias in the estimator.

To compare with other methods of calculation of the mo-ments of dynamics, we also reported in Figs. 5�c� and 5�d�calculations of the mean and variance of prior time averagedindividual mean-squared displacements extracted from eachtrajectory �14,31,32�. The latter methods completely disre-gard eventual differences in trajectory durations �hence anapparent statistical heterogeneity in the accuracy of the indi-vidual mean-squared displacement estimations�. For theseestimates, we repeated the study described in the previous


021501-11

paragraph and the results are given by the open symbols inFigs. 5�c� and 5�d�.

In this figure, we see that as � increases the bias of eachestimator derived in this study vanishes. This is not the casefor the calculations with prior time averages on individualtrajectories as described above. Also, we remark that the er-ror bars on each point are small compared with the ones onthe open symbols, indicating that the degree of samplingdefined by Eq. �5� in the first section is well correlated withthe bias in the estimators. Note finally that the variation ofthe biases with the degree of sampling is fairly independent

of the nature of the dynamics involved, Newtonian or Voigt.The last two plots, Figs. 5�e� and 5�f�, show the results cor-responding to balanced bimodal fluids with greater differ-

ences in the dynamics: �rN+�r+2

N designates a fluid where �r

=10r/2 is mixed with �r+2=10r/2+1 for the Newtonian dynam-

ics, and �rV+�r+2

V is for a bimodal fluid with ��r=10r/2 , kr

=10r/2 /5� and ��r+2=10r/2+1 , kr+2=10r/2+1 /5� for the Voigtdynamics. We observe that the same conclusions still hold,although the reported biases are significantly higher.

In the remaining, we will use an experimental unbalancedbimodal system and also simulate a complex and randomheterogeneous fluid. To show the strength of the estimators,we will place these studies in conditions where the fluid isalmost entirely detectable.

2. Experimental bimodal fluid

As explained in Sec. II, the experimental bimodal fluid iscomposed of a region of viscous non-cross-linked oligomersolution, and a cylindrical region of known radius where thematerial is a stiff gel. In Figs. 6�a�–6�d�, we show the trajec-tories extracted from these experiments, where we see thatimmobile particles are found in the circular region at thecenter, with varying radius. Around the gelled area, the par-ticles’ motion is not constrained. By performing multiple par-

100 101 102 10310-2

10-1

100

0 0.2 0.4 0.6 0.8 1-40

-20

0

20

40

0 0.2 0.4 0.6 0.8 1-40

-20

0

20

40

100 101 102 103

0 0.2 0.4 0.6 0.8 1

0 0.2 0.4 0.6 0.8 1

m

nn

b[ M

(n)]

(%)

b[ M

(n)]

(%)

θ

θ

θ

θ

νNr + ν

Nr+1

νNr + ν

Nr+2

νVr + ν

Vr+1

νVr + ν

Vr+2

(a) (b)

(c) (d)

(e) (f)

m(1

)

m(2

)

m(1)

m(2)

FIG. 5. �Color online� Bias as a function of the degree of sam-pling �. The first column of plots �a�, �c�, and �e� corresponds to amapping of the parameter space �n , m� using Newtonian dynamicsand the second column is for a Voigt fluid. �a� and �b� show therelation between the degree of sampling of the homogeneous fluid 1�the line connecting the triangles corresponds to a constant �=0.5�with the bias �the line connecting the diamonds is a constant level

of bias for M1�n�� in the bimodal mixtures �rN+�r+1

N �a� and �rV

+�r+1V �b�. Refer to the text for further explanations on how these

lines are built. The composite fluid �M1�n� is given by the blacksolid line� is made of a balanced mixture of two successive dashed

lines, m�1��n� and m�2��n�. The estimator M1�n� is given by the

dash-dotted line. �c� and �d� show the bias in M1 �squares� and M2

�circles� as a function of �. �e� and �f� show the variation of the biasfor a bimodal fluid with greater difference in the dynamics �r

N

+�r+2N and �r

V+�r+2V , respectively. In �c�–�f�, the open symbols are

obtained using a prior time averaging for the corresponding estima-tors �see text�. Note that the open circles, corresponding to an esti-

mation of M2 with prior time averages on individual tracks, lieabove 100%, outside the range of bias shown here.

0.0 0.1 0.2 0.3 0.4 0.50.0

0.2

0.4

0.6

0.8

1.0

1.2

M2(

10)/ M

1(1

0)2

p(2) = πR2gel/(xbyb)

(a) (b) (c) (d)

(e)

FIG. 6. �Color online� �a�–�d� are the measured trajectories in aheterogeneous unbalanced bimodal model experimental fluid. Eachimage has dimensions xb=yb=300 �m. The centered circular gelledregion, where the particles are trapped, have radii Rgel=55, 85, 100,and 115 �m, respectively, from �a� to �d�. �e� is the squared coef-ficient of variation of the probes’ dynamics in these fluids. The opensquares are estimates of the measure �2�m�n�� /E�m�n��2 using

M2�n� /M1�n�2 for the four bimodal fluids created by photopolymer-ization at n=10. The filled square was obtained in the homogeneoussystem Rgel=0. The solid line is the theoretical curve, Eq. �12�. Thecircles are the calculations of the squared coefficient of variationusing prior time averages on individual tracks �as explained in theprevious section about the simulated bimodal fluids; see also Fig.5�.


021501-12

ticle tracking on a homogeneous gel system, we observedthat the motion of the beads in the gel is below spatial reso-lution, such that the effective mean-squared displacement isconstant and measured at m��=5�10−4 �m2 �44�. In theother kind, particles are freely diffusing and we found a dif-fusion coefficient D=0.25±0.005 �m2 s−1. At unit distancezb=8.2 �m and unit time �t=0.1 s �see Sec. II�, we have anunbalanced bimodal fluid with m�1��n�=2�n� /103.44 m�2��n�=2/105.43 at all observed lag times n. Each fluid is observedwith respective probabilities p�1�=1− p�2� and p�2�

=�Rgel2 / �xbyb�. In that case, we can show that the squared

coefficient of variation is given by

�2�m��E�m��2 =

M2�n�M1�n�2 =

p�2�

1 − p�2� + O m�2�

m�1�� . �12�

We estimate the above quantity using M2�n� /M1�n�2. The lagtime n is chosen such that the degree of sampling of fluid 1is closer to 1. From Fig. 3, we evaluate that � significantlydrops at about n=100, in accordance with what we observedin experiments on the homogeneous fluid of kind 1 �data not

shown�. We evaluated in Fig. 6�e� the ratio M2�n� /M1�n�2 atn=10 to discard any bias from an eventual undetectability offluid 1. At this lag time, we found an excellent agreementbetween the estimator of the squared coefficient of variationand the theoretical value given by Eq. �12� for the wholerange of reported p�2�. On Fig. 6�e�, the error bars on theexperimental points �open squares� were calculated usingEqs. �8� and �10�, whose validity is demonstrated in the nextsection. In contrast with the accuracy of the estimators de-rived in this paper, the evaluation of the squared coefficientof variation, using estimators of �2�m�� and E�m�� cal-culated with a prior time average on individual trajectory�see description above�, reports poor estimates �see thecircles in Fig. 6�. This illustrates the importance of the char-acterization and understanding of the sampling design inmultiple particle tracking measurements.

3. Random fluid

To conclude the characterization of the estimators, wesimulated a random heterogeneous fluid the following way.Each probe undergoes purely Voigt dynamics, m�n�=2�1−e−�n�k/�� / k, where the parameters � and k are independentlydistributed according to the following uniform and Gammaprobability density functions: P��x�= �105−104�−1 for 104

x105 �0 otherwise�, and Pk�x�=4xe−2x/104/108 for 0

x �0 otherwise�. In particular, random values of k are gen-

erated using the expression k=−ln�U1U2��104/2, where 0�U1 ,U2�1 are random variates drawn from the uniformstandard distribution. We simulated 300 initial trajectories ina volume of observation whose z dimension is 3zb, but onlypositions in �zb ,2zb� were counted to build the trajectory.Hence we let Nb�t� fluctuate in time around a constant valuenear 100 �see Sec. II�. Thus this simulates more realisticexperimental measurements. We compare in Fig. 7 the esti-

mators M1 and M2, and the theoretical results for M1 and M2,respectively, whose expressions are given below:

M1�n� =2n

9 � 104 ln20 + n

2 + n� ,

M2�n� =16n

9 � 108 ln �1 + n�n+1�1 + n/20�n+20

�1 + n/2�n+2�1 + n/10�n+10� − M1�n�2.

As seen in Fig. 7, we again find a good agreement betweenthe estimators and their theoretically expected values overthe range of lag times n investigated here. Small deviationsare visible at high lag times in Fig. 7�b�, where eventuallyundetectability of the faster dynamics in the fluid, for whichthe mean-squared displacement approaches the volume sizelimit zb

2, can alter the bias of the estimators.

10-5

10-4

10-3

10-2

10-1

100

104 105

102

104

100 101 102 103

10-9

10-8

10-7

10-6

nm

(n), M

1(n)

˜ξ

˜

k

M2(n

)

(a)

(b)

FIG. 7. �Color online� Estimators M1 and M2 calculated for arandom fluid. For the random fluid simulated here, the parameterspace �n , m� is sampled with purely Voigt dynamics for which the

drag coefficient � is uniformly distributed between 104 and 105, and

the spring constant k is Gamma distributed with mean 104 and a

shape parameter of 2 �� and k are statistically independent�. Theresulting probability density of �n , m� is given �a�, where the occur-rence is color coded from light to dark for increasing likelihood.The inset of �a� displays the probability density P�,k following thesame color-code convention. In �a�, the solid line indicates the mean

M1 and the squares report the corresponding estimate M1. �b� com-

pares the results of M2 �circles� obtained by the simulations with

the theoretical values M2 �solid line�. Error bars for both M1 and

M2 were calculated using Eqs. �8� and �10�, respectively �see Fig.8�.


021501-13

For this advanced example, we plot in Fig. 8 the estima-

tors �2�M1� �Eq. �8�� and �2�M2� �Eq. �10�� for the variances

of M1 and M2, respectively. In this figure, we also report the

corresponding estimates of �2�M1� and �2�M2� obtainedfrom the set of 100 simulations performed for the same ran-dom fluid �solid lines�. We see on this figure a good agree-ment between the two estimates, ensuring the validity of the

expressions of �2�M1� and �2�M2� derived in Sec. I C. Smalldifferences, although almost consistently within the errorbars reported in the figure �these error bars were obtained bydirect calculations of the sample variance of the sets of ob-servations�, could come from residual correlations in thesample between block-averaged displacements �see Fig. 2�,but also from statistical uncertainty in estimating �2�M�from the finite-sized set of 100 observations. The accuracy of

�2�M� is not critical, however, these estimators being used to

indicate error bars on M1 and M2 �as performed in Figs. 6

and 7�. Similar agreements between �2�M� and �2�M� havebeen observed for all simulation results reported throughoutthis paper.

IV. CONCLUSIONS

Multiple particle tracking experiments return a list ofBrownian probe trajectories. From this output, a sample ofdisplacements is extracted to calculate statistical quantities

revealing the properties of the probed material. We outlinedhere that the sample of displacements presents several pecu-liarities in a spatially heterogeneous system. If not carefullytaken into account, these peculiarities can bias classical esti-mations of the aforementioned statistical quantities. We pre-sented a description of the sampling design of the multipleparticle tracking measurements. This description allowed usto derive the main results of this paper: Eq. �6� estimates theensemble-averaged mean-squared displacement, a widelyused quantity to characterize the spatially averaged mechani-cal property of the material, and Eq. �8� is used to evaluatethe error bars affecting the latter quantity; Eq. �9� gives theensemble spatial variance of the individual probes’ mean-squared displacements, which is a valid and intelligible indi-cator of the material’s heterogeneity, and Eq. �10� can thenbe used to calculate its error bars. We expect that these for-mulas will become standard routines to analyze multiple par-ticle tracking data. Moreover, if a material is found to beheterogeneous, we showed an important limitation of thetechnique coming from undersampling regions in the mate-rial where the thermal motion of the embedded probes isallowed to exhibit fluctuations larger than the tracking depth.We called this effect detectability, quantified through the de-gree of sampling, and showed its influence when attemptingto calculate material properties with video microscopy par-ticle tracking. For example, when particles are tracked toassess the micromechanical mapping of live cells, an analy-sis of individual mean-squared displacements indicates a sig-nificant heterogeneity of the cytoplasm �19�. Some probescan reach low-frequency fluctuations of amplitudes close tothe depth of tracking of the setup �see results in Ref. �19�obtained with high microscope magnification, leading tosmall zb�1�m�. Thus in this important application of thevideo particle tracking technique, special attention should bepaid when tentatively applying the estimators derived in thispaper.

Overall, this study shows that great care should be allo-cated to understanding the experimental output of the tech-nique when calculating statistical quantities. Notably, weprovided here an initial pathway to further this understand-ing.

ACKNOWLEDGMENTS

The authors thank P. T. Underhill and K. Titievsky forinsightful discussions, and K. M. Schultz for her help withexperiments. This material is based upon work supported bythe National Science Foundation under Grants No. 0239012and No. 0304128 �NSF-NIRT�.

�1� J. Quintanilla, Polym. Eng. Sci. 39, 559 �1999�.�2� Y. Tseng, K. M. An, and D. Wirtz, J. Biol. Chem. 277, 18143

�2002�.�3� Y. Tseng and D. Wirtz, Phys. Rev. Lett. 93, 258104 �2004�.�4� K. Luby-Phelps, P. E. Castle, D. L. Taylor, and F. Lanni, Proc.

Natl. Acad. Sci. U.S.A. 84, 4910 �1987�.�5� K. Luby-Phelps, F. Lanni, and D. L. Taylor, Annu. Rev. Bio-

phys. Biophys. Chem. 17, 369 �1988�.�6� C. S. Chen, M. Mrksich, S. Huang, G. M. Whitesides, and D.

E. Ingber, Science 276, 1425 �1997�.

100 101 102 10310-3

10-2

10-1

100

101

n

µ2[ M

]/M

2,E

[µ2[ M

]]/M

2

FIG. 8. �Color online� Estimators �2�M1� �squares� and �2�M2��circles� calculated for the random fluid in Fig. 7 and scaled by thesquared mean M1

2=E�m�2 and the squared variance M22=�2�m�2,

respectively. The associated solid lines are the correspondingly

scaled estimates of �2�M1� and �2�M2� obtained from repeating thesimulation 100 times �the vertical lines are error bars�.


021501-14

�7� C.-M. Lo, H.-B. Wang, M. Dembo, and Y.-l. Wang, Biophys. J.79, 144 �2000�.

�8� S. Gheorghiu and M.-O. Coppens, Proc. Natl. Acad. Sci.U.S.A. 101, 15852 �2004�.

�9� N. D. Gershon, K. R. Porter, and B. L. Trus, Proc. Natl. Acad.Sci. U.S.A. 82, 5030 �1985�.

�10� N. Pernodet, M. Maaloum, and B. Tinland, Electrophoresis 18,55 �1997�.

�11� W. W. Graessley, Polymeric Liquids & Networks: Structureand Properties �Garland Science, New York, 2003�.

�12� A. R. Kannurpatti, K. J. Anderson, J. W. Anseth, and C. N.Bowman, J. Polym. Sci., Part B: Polym. Phys. 35, 2297�1997�.

�13� F. C. MacKintosh and C. F. Schmidt, Curr. Opin. Colloid In-terface Sci. 4, 300 �1999�.

�14� J. Apgar, Y. Tseng, E. Fedorov, M. B. Herwig, S. C. Almo, andD. Wirtz, Biophys. J. 79, 1095 �2000�.

�15� M. T. Valentine, P. D. Kaplan, D. Thota, J. C. Crocker, T.Gisler, R. K. Prud’homme, M. Beck, and D. A. Weitz, Phys.Rev. E 64, 061506 �2001�.

�16� J. C. Crocker and D. G. Grier, J. Colloid Interface Sci. 179,298 �1996�.

�17� I. M. Tolić-Nørrelykke, E.-L. Munteanu, G. Thon, L. Odder-shede, and K. Berg-Sørensen, Phys. Rev. Lett. 93, 078102�2004�.

�18� D. T. Chen, E. R. Weeks, J. C. Crocker, M. F. Islam, R. Verma,J. Gruber, A. J. Levine, T. C. Lubensky, and A. G. Yodh, Phys.Rev. Lett. 90, 108301 �2003�.

�19� Y. Tseng, T. P. Kole, and D. Wirtz, Biophys. J. 83, 3162�2002�.

�20� A. Goodman, Y. Tseng, and D. Wirtz, J. Mol. Biol. 323, 199�2002�.

�21� J. H. Shin, M. L. Gardel, L. Mahadevan, P. Matsudaira, and D.A. Weitz, Proc. Natl. Acad. Sci. U.S.A. 101, 9636 �2004�.

�22� I. A. Hasnain and A. M. Donald, Phys. Rev. E 73, 031901�2006�.

�23� T. A. Waigh, Rep. Prog. Phys. 68, 685 �2005�.�24� J. C. Crocker, M. T. Valentine, E. R. Weeks, T. Gisler, P. D.

Kaplan, A. G. Yodh, and D. A. Weitz, Phys. Rev. Lett. 85, 888�2000�.

�25� A. J. Levine and T. C. Lubensky, Phys. Rev. Lett. 85, 1774�2000�.

�26� M. J. Solomon and Q. Lu, Curr. Opin. Colloid Interface Sci. 6,430 �2001�.

�27� I. Y. Wong, M. L. Gardel, D. R. Reichman, E. R. Weeks, M. T.Valentine, A. R. Bausch, and D. A. Weitz, Phys. Rev. Lett. 92,178101 �2004�.

�28� A. Rahman, Phys. Rev. 136, A405 �1964�.�29� W. K. Kegel and A. van Blaaderen, Science 287, 290 �2000�.�30� E. R. Weeks, J. C. Crocker, A. C. Levitt, A. Schofield, and D.

A. Weitz, Science 287, 627 �2000�.�31� R. Tharmann, M. M. A. E. Claessens, and A. R. Bausch, Bio-

phys. J. 90, 2622 �2006�.�32� Y. Tseng and D. Wirtz, Biophys. J. 81, 1643 �2001�.�33� P. Panorchan, D. Wirtz, and Y. Tseng, Phys. Rev. E 70, 041906

�2004�.�34� D. D. Boos and C. Brownie, Stat. Sci. 19, 571 �2004�.�35� A. Stuart and J. K. Ord, Kendall’s Advanced Theory of Statis-

tics, 6th ed. �Hodder Arnold, London, 1994�, Vol. 1.�36� D. Montiel, H. Cang, and H. Yang, J. Phys. Chem. B 110,

19763 �2006�.�37� H. Flyvbjerg and H. G. Petersen, J. Chem. Phys. 91, 461

�1989�.�38� R. Zwanzig and N. K. Ailawadi, Phys. Rev. 182, 280 �1969�.�39� D. R. Bellhouse, Am. Stat. 55, 352 �2001�; software may be

obtained from the American Statistician Software archive athttp://lib.stat.cmu.edu/

�40� H. C. Öttinger, Stochastic Processes in Polymeric Fluids:Tools and Examples for Developing Simulation Algorithms�Springer-Verlag, New York, 1996�.

�41� J. C. Love, D. B. Wolfe, H. O. Jacobs, and G. M. Whitesides,Langmuir 17, 6005 �2001�.

�42� D. Dendukuri, D. C. Pregibon, J. Collins, T. A. Hatton, and P.S. Doyle, Nat. Mater. 5, 365369 �2006�.

�43� T. Savin and P. S. Doyle, Phys. Rev. E 71, 041106 �2005�.�44� T. Savin and P. S. Doyle, Biophys. J. 88, 623 �2005�.�45� See EPAPS Document No. E-PLEEE8-75-142705 for a

summary of the main notations used in this paper. For moreinformation on EPAPS, see http://www.aip.org/pubservs/epaps.html.


021501-15

statistical and sampling issues when using multiple particle...

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