Which Of The Following Measurements Is Correctly Matched With Microorganisms Of That Size?

Meas Sci Technol. Author manuscript; available in PMC 2015 May 1.

Published in final edited form as:

PMCID: PMC4265807

NIHMSID: NIHMS579925

An adaptive algorithm for tracking 3D bead displacements: awarding in biological experiments

Xinzeng Feng

^aneDepartment of Mechanical and Aerospace Engineering, Cornell Academy, Ithaca, 14853, Us

Matthew South. Hall

²Department of Biological and Environmental Engineering, Cornell Academy, Ithaca, NY 14853, U.s.

Mingming Wu

²Department of Biological and Ecology Engineering, Cornell Academy, Ithaca, NY 14853, United states of america

Chung-Yuen Hui

¹Department of Mechanical and Aerospace Engineering science, Cornell University, Ithaca, 14853, Us

Abstract

This paper presents a feature-vector-based relaxation method (FVRM) to track bead displacements within a three-dimensional (3D) volume. FVRM merges the feature vector method, a technique used in tracking bead displacements in biological gels, with the relaxation method, an algorithm employed successfully in tracking bead pairs in fluids. More specifically, FVRM evaluates the probability of a bead pairing event based on the quasi-rigidity condition between the characteristic vectors of a bead and its candidate positions within a searching domain. Computational efficiency is improved via the introduction of an adaptive searching domain size and mismatches are reduced via a two-directional matching strategy. The algorithm is validated using fake 3D dewdrop displacements caused by a strength dipole inside a linear rubberband gel. Results demonstrate a consistently high recovery ratio (above 98%) and depression mismatch ratio (below 0.1%) for tracking parameter (mean bead distance/maximum bead displacement) greater than 0.73.

Keywords: three dimensional, dewdrop tracking, two-frame, adaptive, relaxation method, feature vector

i. Introduction

Recent developments in quantitative biology, in item, the emphasis on physical forces in biological systems, demands an power to rails particles (or cells) in three dimensional (3D) infinite and time (Legant et al., 2010, Hall et al., 2013, Wu et al., 2006). Examples include microrheology (Squires and Mason, 2010), where the mechanical properties of biological fluids are extracted from the motion of tracer particles embedded inside it; hydrodynamics of living fluids (Koch and Subramanian, 2011, Wu et al., 2006) where many individual swimming bacteria are tracked to derive the mechanics of a living fluid – the bacterial suspension. More recently, tracking micrometer size particles in 3D has enabled a promising technology, 3D cellular traction force microscopy (3D TFM) (Legant et al., 2010, Hall et al., 2013). In 3D TFM, unmarried animal cells are cultured within a hydrogel, and the cellular traction force is inferred from the gel deformation that results from the release of the cellular traction (typically past administering a drug that disrupts the proper function of cytoskeletal molecules). Here, micrometer size beads with one or more fluorescence channels (Hall et al., 2013, Sabass et al., 2008) are embedded within the hydrogel, and the gel deformation is evaluated from the bead displacements. Challenges in micrometer size particle tracking in biological systems are the particle density, randomness of the particle trajectories, and constraints that limit how frequently particle positions tin can be recorded.

Existing 3D TFM methods determine the bead deportation field using either a Digital Book Correlation (DVC) (Franck et al., 2007, Franck et al., 2011, Maskarinec et al., 2009, Bay, 2008) or a ii frame particle tracking method (Legant et al., 2010, Legant et al., 2013, Hur et al., 2009, Delanoe-Ayari et al., 2010, Rieu and Delanoe-Ayari, 2012, Gjorevski and Nelson, 2012, Koch et al., 2012). The DVC algorithm finds the deportation of a sectioned volume by computing the cross correlation of ii sub-volumes in two consecutive image frames. The spatial resolution of the displacements is determined by the size of the sub-volume, and thus is limited. In add-on, the existing DVC algorithm assumes that the hydrogel deformation of the sub-book only involves rigid spatial translation and stretch, just neglects rotation (Franck et al., 2007, Franck et al., 2011, Maskarinec et al., 2009). Bold that there is no rotation of a gel sub-volume is a potential source of error because it is known that cells presume a complex geometry within a hydrogel, and many natively derived hydrogels have nonlinear material properties (Storm et al., 2005, Griffith and Swartz, 2006). In contrast to DVC, two frame particle tracking overcomes these limitations considering a correctly matched bead pair is an accurate measure of the material displacement at a particular point.

Two frame particle tracking was originally developed for measuring fluid flows in the field of fluid mechanics (Xu, 2008, Pereira et al., 2006, Ouellette, 2006, Crocker and Grier, 1996). Micrometer size beads are seeded within a fluid period, and displacements of bead pairs are used to compute the fluid velocity at the dewdrop location. Pereira et al. recently implemented a relaxation method (Baek, 1996) tracking micrometer size chaplet in 3D flow fields. They found that the relaxation method compared favorably over ii other commonly used tracking methods, the nearest neighbor and neural network method (Pereira et al., 2006). The relaxation method matches a bead pair by updating the probability of a matching bead pair iteratively based on a quasi-rigidity condition (i.e., neighboring beads take similar displacement vectors).

Current two frame particle tracking algorithms in the context of 3D cell traction microscopy include nearest neighbor (Koch et al., 2012, Crocker and Grier, 1996), auto-regression (Gjorevski and Nelson, 2012), and feature-vector-based (Legant et al., 2010, Legant et al., 2013) methods. Amid these methods, feature-vector-based methods make up one's mind the optimal lucifer by evaluating the similarity of the feature vectors of a bead (see department 2.two for the definition of characteristic vectors) with those of its candidates. They are more straightforward to implement than machine-regression methods, and yield better matching accuracy than the nearest neighbor method especially when the bead displacement is relatively big compared to the average dewdrop spacing. Legant et al. (Legant et al., 2010) beginning utilise a characteristic-vector-based tracking method to 3D traction force microscopy. Because their focus is on the biological implications of traction strength microscopy, details of their bead tracking method have non been published. From our private communication with Legant and Chen (Legant, 2013), we learnt that in their approach, characteristic vectors of a bead are formed past connecting the bead position to typically three of its nearest neighbors. A price function is then constructed by summing the Euclidian distances from the set of feature vectors in the reference frame to the fix of characteristic vectors in the deformed frame. Additional filtration and smoothing steps are practical depending on the information sets to achieve high matching ratio.

In this paper, we nowadays a 3D particle tracking method integrating the characteristic-vector-based method with the relaxation method. 2 unique aspects of the feature-vector-based relaxation method (FVRM) are its adaptive searching domain size, which is facilitated past the employ of local feature vectors; and its ii-directional matching strategy. This method is validated against the analytical solution of the bead displacements caused by the release of a known force dipole within a linear elastic gel. In the supplementary material, it is compared against a modified version of original relaxation method (MR) (Pereira et al., 2006) and a modified version of original characteristic vector method (MFV) (Legant et al., 2010).

two. Two frame particle tracking

2.one Problem statement and notation

In two frame particle tracking, the bead positions are known at the two fourth dimension points. Our goal is to track and pair the position of each dewdrop beyond fourth dimension. Let Ω denote a fixed volume in space. Suppose at time t _ane, in that location are N _ane beads A ={p ₁, p ₂,…, p _{N _one}} occupying Ω. At a later time t ₂ > t ₁, beads occupying the same volume Ω will be contained in the fix B = {q _one, q _two,…, q _{Due north _ii} }. Annotation that the sets A and B practice not necessarily incorporate the aforementioned beads or the same number of beads since some beads may enter or leave book Ω; or some beads may not be observed in one of these configurations, e.one thousand. some beads may exist too dim to be observed at time t _ii. Such chaplet volition be absent-minded in B only can belong to A.

The position of a bead is measured by its coordinate vector $\vec{p}$ with respect to a Cartesian coordinate system fixed to the laboratory. Nosotros will call (p, q) a matching pair if at time t ₂, the bead p moves to the position $\vec{q}$ occupied by bead q in B. Clearly, the number of matching pairs cannot exceed min(N ₁, N ₂). After all the matching pairs are plant, the true displacement vector of a matched bead p is given by $\vec{q} - \vec{p}$ . In the following, for arbitrary bead pair (p, q), we also call ${\vec{d}}_{pq} = \vec{q} - \vec{p}$ a displacement vector but information technology is not to be confused with the true displacement where p and q are a matching pair.

It is to be note that nosotros take assumed no stage migrate. If phase drift occurs, the observed bead positions at time t ₁ and t _ii will occupy a slightly dissimilar volume in space. We will address drift correction in Department 5.

two.two A characteristic-vector-based relaxation (FVRM) method

The bones assumption of FVRM is that deformation "preserves" local geometry effectually a dewdrop which is represented past a gear up of characteristic vectors that connect the bead with some of its neighboring chaplet. Normally the more characteristic vectors are chosen, the more authentic only less efficient the algorithm becomes. In our implementation, a number of three feature vectors are chosen for each bead at both t _ane and t _two. Specifically, the characteristic vectors of a bead p ∈ A are adamant by first finding the three nearest neighboring beads of p in A (see figure 1). Denote the set containing these 3 beads every bit Five_p . The 3 vectors that connect $\vec{p}$ with each of its three nearest neighbors in V_p are called the characteristic vectors of bead p. Note that these feature vectors are fixed at the spatial position $\vec{p}$ with respect to the laboratory frame and do not move with p. We announce $F_{p} \equiv {{\vec{ν}}_{p p^{'}}, p^{'} \in V_{p}}$ to exist the set containing these characteristic vectors. Likewise, for any bead q in B, let 5_q be the set of its iii nearest neighbors in B. The corresponding feature vectors of q are independent in $F_{p} \equiv {{\vec{ν}}_{q q^{'}}, q^{'} \in V_{q}}$ . Finally, we denote C_p to be the set containing all candidate positions of p at t ₂. Initially C_p consists of the three nearest neighbors to the spatial position $\vec{p}$ at t ₂ (q ₁, q ₂, q ₃ in figure 1). As discussed below, the fix C_p can be enlarged adaptively to include more than candidate positions.

An external file that holds a picture, illustration, etc. Object name is nihms579925f1.jpg

Overlapped bead positions at t ₁ and t _ii. p is a bead at time t ₁ occupying the spatial position $\vec{p}$ . Feature vectors ${\vec{ν}}_{p p_{i}}$ , ${\vec{ν}}_{p p_{2}}$ , ${\vec{ν}}_{p p_{3}}$ connect p with its 3 nearest neighbors at time t ₁. The candidate positions searching domain C_p includes the iii nearest beads q ₁, q _ii, q ₃ at time t ₂ to the spatial position $\vec{p}$ . Dotted arrow lines indicate the feature vectors of i of the candidates q ₃.

The FVRM is implemented in 4 steps every bit follows:

In step 1, nosotros initialize the probability of a bea p moving to a bead q. Specifically, let P_m (p, q) denote the probability of bead p moving to a bead q ∈ C_p at the k^thursday iteration. The probability of p non matching with any bead inside C_p is denoted by $P_{grand}^{*} (p)$ . By definition,

Initialization of probabilities P _thou=0 (p, q) $P_{thousand = 0}^{*} (p)$ follows

$P_{0} (p, q) = P_{0}^{*} (p) = \frac{1}{N_{p} + ane}, q \in C_{p}$

(2)

where N_p is the number of beads within C_p . Initially N_p equals three.

In step two, we update P_k (p, q) and $P_{1000}^{*} (p)$ from the k^th iteration to the (k + 1)^thursday iteration. Assume P_k (p, q) are known for all p and q ∈ C_p . We obtain P _g+ane (p, q) past first updating the pseudo-probabilities $\tilde{P}$ _k+1 (p, q), which, in the original relaxation method, is given by

${\tilde{P}}_{yard + one} (p, q) = P_{grand} (p, q) [A + B (\sum_{p^{'} \in V_{p}} \sum_{q^{'} \in C_{p^{'}}} P_{k} (p^{'}, q^{'}) Q (p, q, p^{'}, q^{'}))]$

(3)

where A = 0.three, B = 3 (Pereira et al., 2006) are prescribed constants and the weighting factor Q(p, q, p′, q′) enforces the quasi-rigidity condition, i.east. beads within a small region testify a similar design of movement (Baek, 1996). The probability of a pair is enhanced only if in that location are neighboring pairs satisfying the quasi-rigidity condition, which volition be described below.

The original relaxation method is global since all dewdrop pairs are linked to each other with equation (3); and therefore it is difficult to exist converted into an adaptive algorithm. To formulate an adaptive method, we convert (3) into a local update formula only involving P_grand (p, q) as well as the characteristic vectors of p and q

${\tilde{P}}_{k + ane} (p, q) = P_{1000} (p, q) [A + B (\sum_{p^{'} \in V_{p}} \sum_{q^{'} \in 5_{q}} {\hat{P}}_{thousand} (p^{'}, q^{'}) Q (p, q, p^{'}, q^{'}))]$

(four)

, where

${\hat{P}}_{yard} (p^{'}, q^{'}) = P_{k} (p, q) \frac{ane / c ({\vec{ν}}_{p p^{'}}, {\vec{ν}}_{q q^{'}})}{\sum_{q^{″} \in V_{q}} 1 / c ({\vec{ν}}_{p p^{'}}, {\vec{ν}}_{q q^{″}})},$

(5)

is an approximation to the probability P_thousand (p′, q′). In (5), $c (\vec{u}, \vec{ν})$ evaluates the similarity of a pair of vectors $\vec{u}$ , $\vec{ν}$ and is defined by

$c (\vec{u}, \vec{ν}) = {\begin{matrix} (\frac{| \vec{u} |}{| \vec{ν} |} + \frac{| \vec{ν} |}{| \vec{u} |} - 2) (\frac{1 - \cos 〈 \vec{u}, \vec{ν} 〉}{\cos 〈 \vec{u}, \vec{ν} 〉}) & 〈 \vec{u}, \vec{ν} 〉 < π / two \\ \infty & 〈 \vec{u}, \vec{ν} 〉 \geq π / 2 \end{matrix}$

(6)

where $〈 \vec{u}, \vec{ν} 〉$ denotes the angle between the vector pair $\vec{u}$ and $\vec{ν}$ . By definition, when characteristic vectors ${\vec{ν}}_{p p^{'}}$ and ${\vec{ν}}_{q q^{'}}$ are close in length and orientation, $c ({\vec{ν}}_{p p^{'}}, {\vec{ν}}_{q q^{'}})$ will be close to naught and therefore $\hat{P}$ _k (p′, q′) approximates to P_yard (p, q). Otherwise, $\hat{P}$ _k (p′, q′) approximates to 0.

The quasi-rigidity condition in (4) is given by

$Q (p, q, p^{'}, q^{'}) = {\begin{matrix} ane & c ({\vec{ν}}_{p p^{'}}, {\vec{ν}}_{q q^{'}}) < ω_{c} \\ 0 & otherwise \end{matrix}$

(7)

where ω_c = 0.001 is a prescribed abiding.

Finally, notation that the summation scope C_p′ in the second summation of (3) has been replaced in (4) by the fixed set V_q , which contains the neighboring beads of q. Therefore, equation (four) provides a local update of pseudo-probabilities $\tilde{P}$ _k+1 (p, q) which depends only on P₁₀₀₀ (p, q) and the characteristic vectors of p and q. These pseudo probabilities are and so normalized to yield probabilities P _k+1 (p, q), q ∈ C_P and $P_{chiliad + one}^{*} (p)$ at the (thou + ane)^th iteration and so that (ane) is satisfied. This is accomplished by

$\begin{array}{l} P_{k + 1} (p, q) = \frac{{\tilde{P}}_{k + one} (p, q)}{\sum_{q^{'} \in C_{p}} {\tilde{P}}_{thou + i} (p, q^{'}) + P_{k}^{*} (p)} \\ P_{k + 1}^{*} (p) = \frac{P_{thou}^{*} (p)}{\sum_{q^{'} \in C_{p}} {\tilde{P}}_{yard + 1} (p, q^{'}) + P_{grand}^{*} (p)} \end{array}$

(8)

Bold that these probabilities converge afterward J iterations, the match for bead p is selected by finding the bead q that gives the largest P_J (p, q) amongst q ∈ C_p . If this probability is larger than a prescribed threshold ${west}_{due south}^{h}$ , the lucifer is regarded to exist a highly confident match from A to B. If the probability is smaller than ${westward}_{s}^{h}$ merely larger than another threshold $w_{southward}^{l} < w_{s}^{h}$ , the match is regarded to be a confident friction match. In the numerical implementation, ${west}_{due south}^{h} = 0.99$ , ${west}_{south}^{50} = 0.v$ are used.

As pointed out higher up, the scheme to update the probabilities of bead p from the k^th iteration to the (chiliad + 1)^th iteration is local, and hence adaptable. For those beads with large non-matching probabilities $P_{J}^{*} (p) > 0.5$ , the candidate positions searching domain C_p is expanded adaptively to include more beads. Notation that equations (4) and (5) still remain the same, except that probabilities P _chiliad=0 (p, q), $P_{one thousand = 0}^{*} (p)$ needs to exist reinitialized using (2) and the summation in (8) will involve more than terms.

In step iii, nosotros introduce a two directional matching strategy, i.eastward., the procedure in footstep 1 and step 2 is applied to each bead q in fix B to discover its friction match in set A. This strategy is employed to handle the state of affairs when different beads at t ₁ are matched to the same bead at t _ii.

In step iv, nosotros describe the criterion to finally take a pair. Specifically, for a pair plant in both directions, it is accepted as a matching pair if both of post-obit atmospheric condition are satisfied

the pair is a highly confident lucifer from at least ane direction or the pair is a confident match from both directions;
the altitude between this pair of beads is less than the prescribed maximum displacement d_m .

In the original relaxation method (Baek, 1996, Pereira et al., 2006), the above two-directional matching strategy is non employed. Therefore different chaplet at t ₁ could be matched to the same dewdrop at t _two, which causes inconsistent bead displacements and difficulty in computing both recovery and mismatch ratios (come across adjacent section for definition). To provide a off-white comparing between the relaxation method and FVRM, we contain the original relaxation method with the two-directional matching and call it the modified relaxation (MR) method. In the supplementary textile, we present a comparing of FVRM with MR using simulated data from two cases. 1 is the Burger's vortex flow case studied previously by Pereira et al. (Pereira et al., 2006) and the other is the dipole forces case described in Department 4.

3. Tracking parameters

The performance of a tracking scheme deteriorates when the maximum dewdrop displacement | u _max| between time steps becomes comparable to the mean bead distance. To quantity this dependence, nosotros adopt the tracking parameter proposed in (Pereira et al., 2006)

where d ₀ is the mean bead distance in the observation volume Ω, i.due east.,

$d_{0} = {(\frac{3 Ω}{4 π \min ({North}_{1}, N_{2})})}^{1 / 3}$

(x)

Generally, the performance of a tracking scheme improves equally the tracking parameter increases.

To evaluate the performance of our tracking scheme, nosotros introduce two dimensionless parameters, namely: the recovery ratio η _r and the mismatch ratio η_m . Announce N_cp ≤ min(N ₁, North ₂) to be the bodily number of bead pairs in the book. Suppose a total number of K_p ≤ min(N ₁, N _two) pairs are identified past an algorithm, amidst which M_cp are correct matches while the residuum Thousand_sp =M_p − M_cp are incorrectly matched (spurious pairs). The mismatch ratio η_m is divers equally the number of spurious pairs M_sp divided by the total number of pairs found Chiliad_p , i.e.,

The recovery ratio η_r is defined equally the number of right pairs M_cp divided by the actual number of bead pairs North_cp , i.due east.,

As pointed out by Pereira et al. (Pereira et al., 2006), η_r straight measures the matching ability of a tracking scheme and η_m indicates how well a tracking scheme discriminates among candidates and dismisses unpaired beads.

4. Numerical experiments and discussion

iv.one. Tracking displacement field due to a force dipole

In cell traction experiment, certain prison cell types will polarize in space assuming an elongated shape. Forces exerted past such polarized cells are primarily exerted in the direction of elongation and can be approximated by equal and opposite forces at the ii jail cell tips. Motivated by this observation, we exam the FVRM past simulating the displacement field induced by a force dipole in a linearly rubberband gel. As long as the cell is small in comparing with its surroundings, the induced displacement field tin be modeled using the exact solution of a strength dipole in an infinite elastic solid. Let x_i, i = i,2,3 be the Cartesian coordinates of a textile point in the gel. The displacement field u_i due to a forcefulness dipole of magnitude P located at (0,0, ±a) is given by (Love, 1944)

$\begin{array}{l} {\bar{u}}_{i} & = & \bar{C} (\frac{{\bar{ten}}_{i} ({\bar{x}}_{3} + one)}{{\bar{r}}_{one}^{iii}} - \frac{{\bar{x}}_{i} ({\bar{10}}_{iii} - 1)}{{\bar{r}}_{2}^{iii}}), i = i, 2 \\ {\bar{u}}_{three} & = & \bar{C} [(\frac{three - four ν}{{\bar{r}}_{1}} + \frac{{({\bar{x}}_{3} + one)}^{2}}{{\bar{r}}_{1}^{3}}) - (\frac{3 - 4 ν}{{\bar{r}}_{ii}} + \frac{{({\bar{ten}}_{3} - i)}^{2}}{{\bar{r}}_{ii}^{3}})] \\ \bar{C} & = & \frac{(1 + ν) \bar{P}}{eight π (i - ν)}, {\bar{r}}_{one} = \sqrt{{\bar{x}}_{1}^{2} + {\bar{ten}}_{2}^{two} + {({\bar{x}}_{3} + one)}^{2}}, {\bar{r}}_{2} = \sqrt{{\bar{x}}_{ane}^{2} + {\bar{x}}_{2}^{2} + {({\bar{x}}_{three} - i)}^{2}} \end{array}$

(13)

where we take normalized all variables using

${\bar{x}}_{i} = \frac{{ten}_{i}}{a}, {\bar{u}}_{i} = \frac{u_{i}}{a}, \bar{P} = \frac{P}{Eastward a^{2}}$

(14)

to reduce the number of parameters in the simulation to two, i.east., $\bar{P}$ and ν. In equations (xiii) and (14), Eastward is the Young's modulus of the gel and ν is the Poisson's ratio. In the numerical experiment, these normalized parameters are chosen to be $\bar{P}$ =1.five, ν = 0.2. Note that the deportation field is unbounded at the 2 dipole points (r ₁ = 0, r ₂ = 0).

Simulations are carried out with Northward beads uniformly sampled in a normalized cubic domain [−two, 2] × [−2, 2] × [−2, 2] excluding the region where the magnitude of the normalized displacement vector ū exceeds 0.2, i.east., the maximum bead deportation | ū _max| ≈ 0.2. Since dewdrop displacements given past (13) get unbounded at (0, 0, ±1), the tracking performance at locations shut to (0, 0, ±1) is expected to be worse. We categorize all beads into 2 categories: I) beads with pocket-size displacements | ū | ≤ 0.3 | ū _max| where | ū | is the magnitude of the displacement vector of a particular bead; 2) beads with big displacements | ū | > 0.3 | ū _max|. Effigy 2 shows the recovery and mismatch ratios of the algorithm for beads in categories I and II respectively as well as the full population. Figure 3 shows the computation time of the algorithm. In both effigy two and 3, each indicate represents a particular dewdrop number N from 1000, 2000, 3000, 4000, 5000, 10000, 20000, 40000. For each N, three different realizations of dewdrop positions are sampled in the volume, which are used to obtain the fault bars in the figures. For each realization, the tracking parameter is calculated using in which the mean bead distance d ₀ is institute by directly averaging the distances betwixt a bead and its nearest neighbor. The average tracking parameters of the 3 realizations are used in the figures to narrate the different bead densities. For N = 5000, the average tracking parameter is 0.73 (the fourth data indicate from left to right). In real experiments, this corresponds to a bead density of 5.0 × x⁹ beads / ml. Here the maximum bead displacement is causeless to be 5μm and the mean bead altitude is estimated using (x). As shown in figure 2, for tracking parameter greater than 0.73, the overall recovery ratio is higher up 98% and the mismatch ratio is below 0.one%. Information technology takes less than ii minutes for our Matlab (The MathWorks Inc., Natick, MA) code to cease tracking for N = 5000, as shown in effigy three. Here all computations are carried out on a desktop with an AMD FX(tm)-8150 Eight-core Processor (3.6GHz) and 32GB RAM. Every bit we increment the dewdrop density, the tracking parameter falls below 0.73, causing deterioration in both recovery and mismatch ratios and increase in computation time. This is mainly due to frequent expansion of candidate searching domain and poorer tracking performance of chaplet with displacements large compared with the bead spacing.

An external file that holds a picture, illustration, etc. Object name is nihms579925f2.jpg

Dipole forces example. Plots of recovery ratio (left) and mismatch ratio (right) versus the tracking parameter for modest displacement beads (category I), large displacement chaplet (category II) and overall population (I+Ii). Points from right to left represent respectively N =grand, 2000, 3000, 4000, 5000, 10000, 20000 and 40000 beads sampled in the cube. For each N, error bars are obtained from three unlike realizations of dewdrop positions.

An external file that holds a picture, illustration, etc. Object name is nihms579925f3.jpg

Plot of the computation time of FVRM for the dipole forces case versus the tracking parameter. The Matlab code is run on a desktop with an AMD FX(tm)-8150 8-core Processor (3.6GHz) and 32GB RAM.

iv.2. Effect of observation errors

In experiments, bead positions are subjected to ascertainment errors, e.thousand., optical/electronic dissonance and image reconstruction error. We investigate this effect on the performance of our tracking scheme by introducing a random perturbation in the dewdrop positions at both time t _one and t ₂. We use the dipole forces case with N = 5000 every bit an error-free reference. The tracking parameter for this arrangement is 0.73. We apply perturbation on each bead using a compatible distribution in the interval $[- \frac{ane}{2} ò d_{0}, \frac{i}{2} ò d_{0}]$ in the x-y directions, and $[- \frac{3}{2} ò d_{0}, \frac{3}{two} ò d_{0}]$ in the z direction, where d ₀ is the mean dewdrop distance and ò is a dimensionless parameter which can be interpreted as a scaled noise aamplitude. Figure 4 shows that the performance of the tracking scheme gradually deteriorates as increasing noise undermines the validity of quasi-rigidity condition. Therefore, information technology is crucial to obtain accurate 3D dewdrop positions from raw images in club to accomplish reliable matching performance. This is facilitated by the sub-pixel localization technique introduced by Gao et al. (Gao Y, 2009). With the technique, the racket amplitude due to image reconstruction is less than 0.05 μm for a pixel size of 0.two μm. For mean bead distance d ₀ ≥ 2.5μm, the respective scaled noise aamplitude is less than 4%. For these cases, figure 4 shows that the recovery ratio is above 95% and the mismatch ratio is beneath 0.i% using FVRM.

An external file that holds a picture, illustration, etc. Object name is nihms579925f4.jpg

Result of random perturbation of bead positions on the recovery ratio η_r and mismatch ratio η_k . The tracking parameter for the error costless reference system is 0.73. For each scaled noise amplitude ò, dissonance in the x-y directions is uniformly distributed in $[- \frac{i}{2} ò d_{0}, \frac{1}{ii} ò d_{0}]$ . In the z direction, the noise amplitude is iii times larger. Fault confined are obtained from three independent realizations of perturbed dewdrop positions.

v. Drift correction

In our matching algorithm we have assumed that at that place is no migrate in the frame of reference from one time-point to the adjacent. This assumption allows us to use a stock-still coordinate arrangement to specify the position of a dewdrop. In real microscopy experiments, temperature changes in the imaging organisation and stage actuation error tin consequence in drift of the optical focal airplane and therefore the frame of reference between time-points.

Suppose A ={p ₁, p ₂,…, p _{N _one}} and B = {q _ane, q ₂,…, q _{Due north ₂}} correspond to the ii sets of chaplet observed in volume Ω at fourth dimension t ₁ and t ₂ respectively. The measured positions of beads in sets A and B are ${{\vec{p}}_{i}}$ and ${{\vec{q}}_{i} + \vec{d}}$ , where ${\vec{p}}_{i}$ , ${\vec{q}}_{i}$ are dewdrop positions with respect to a same stock-still laboratory coordinate system and $\vec{d}$ is the drift at time t ₂. If the migrate is large in comparison with hateful dewdrop distance, the above tracking scheme will interruption down. Therefore, information technology is necessary to correct the drift before feeding bead positions into the tracking algorithm.

The first pace of migrate correction is to provide an initial guess of drift. In cell experiments, most of the gel volume experiences very piddling displacement. The initial estimate of drift tin can therefore exist obtained by recording bead positions beyond time in those regions. Alternatively, the position of a characteristic fixed to the laboratory reference frame, such equally a fluorescent bead attached to the embrace glass, tin be imaged at each time-indicate to calculate the migrate. Once an initial drift ${\vec{d}}_{0}$ is obtained, the migrate is subtracted from all bead positions and the above tracking algorithm is applied to drift-corrected bead positions ${{\vec{p}}_{i}}$ and ${{\vec{q}}_{i} + \vec{d} - {\vec{d}}_{0}}$ . Every bit long every bit the initial drift approximate is sufficiently accurate, the matching algorithm should be able to match most bead pairs correctly. A correction of the migrate gauge tin can and so be obtained by averaging the matched bead displacements in the far field where displacements are expected to be minor. This process tin be performed iteratively until the drift correction is sufficiently refined.

6. Concluding remarks

To conclude, we introduce FVRM every bit an efficient and authentic tracking algorithm peculiarly suitable for biological systems where displacements are non-uniformly distributed. It extends the feature-vector-based algorithm first proposed by Legant et al. (Legant et al., 2010) past accounting for bending differences in the cost function and integrates with the relaxation method (Baek, 1996, Pereira et al., 2006) that is previously used in the fluid mechanics community. FVRM adaptively expands the size of its candidate positions searching domain and is therefore efficient specially for prison cell (Legant et al., 2010, Hall et al., 2013) and indentation experiments (Hall et al., 2012). In these experiments, bead displacements are non-uniformly distributed and most of the region experiences modest displacements.

We compare FVRM with a modified version of original relaxation method (MR) (Pereira et al., 2006) and a modified version of original characteristic vector method (MFV) (Legant et al., 2010) using faux displacement field induced by a force dipole (run across supplementary material). The results evidence that FVRM and MR perform well for large tracking parameters (> 0.73) with a high recovery ratio (above 98%) and low mismatch ratio (beneath 0.i%). For a small tracking parameter (< 0.73), MR performs slightly better in terms of recovery and mismatch ratios but its time cost increases dramatically. For instance, at tracking parameter of 0.v, FVRM is nigh 30 times faster than MR. Although MFV overall does not perform equally well equally FVRM or MR both in terms of recovery and mismatch ratios, we annotation here that MFV is a re-implementation of the original feature vector method proposed by Legant et al., and their detailed postprocessing procedure is not included in our ciphering. It is also to exist mentioned that both FVRM and MFV can be improved past using more than three feature vectors, but at the cost of greater computation fourth dimension. In real experiments, the performance of tracking algorithms is undermined by observation errors. As shown in the numerical experiment of dipole forces case, for tracking parameter greater than 0.73, the outcome of observation error using the sub-pixel localization technique (Gao Y, 2009) on the tracking operation of FVRM is minor every bit long every bit the hateful bead distance is greater than two.5 μm.

Finally, nosotros indicate out that above tracking algorithms can be naturally used to track beads with two or more fluorescence channels (Sabass et al., 2008). By tracking the beads in unlike channels separately, each channel has a lower bead density, or equivalently, larger tracking parameter which will pb to lower computation time and amend matching efficiency. Therefore a denser and more than precise sampling of displacement field could exist achieved compared to the traditional single channel approach.

Supplementary Material

mst479612supp.pdf

Acknowledgments

MW, MSH and XZF are mainly supported by the National Center for Research Resources (5R21RR025801-03) and the National Institute of General Medical Sciences (viii R21 GM103388-03) of the National Institutes of Wellness. MW thanks partial support from the Cornell Heart on the Microenvironment & Metastasis (Accolade No U54CA143876 from the National Cancer Institute), and the Cornell NanoScale Science and Engineering science and the Cornell Nanobiotechnology Center. All of us give thanks Dr. Pereira for helpful discussions on the original relaxation method.

References

BAEK SJ, L SJ. A new ii-frame particle tracking algorithm using match probability. Experiments in Fluids Experiments in Fluids. 1996;22:23–32. [Google Scholar]
BAY BK. Methods and applications of digital volume correlation. The Periodical of Strain Assay for Engineering Design. 2008;43:745–760. [Google Scholar]
CROCKER JC, GRIER DG. Methods of digital video microscopy for colloidal studies. Journal of colloid and interface science. 1996;179:298–310. [Google Scholar]
DELANOE-AYARI H, RIEU JP, SANO 1000. 4D Traction Force Microscopy Reveals Asymmetric Cortical Forces in Migrating Dictyostelium Cells. Physical Review Letters. 2010;105 [PubMed] [Google Scholar]
FRANCK C, HONG S, MASKARINEC South, TIRRELL D, RAVICHANDRAN G. Three-dimensional Full-field Measurements of Large Deformations in Soft Materials Using Confocal Microscopy and Digital Volume Correlation. Experimental Mechanics. 2007;47:427–438. [Google Scholar]
FRANCK C, MASKARINEC SA, TIRRELL DA, RAVICHANDRAN G. Three-dimensional traction forcefulness microscopy: a new tool for quantifying jail cell-matrix interactions. PloS one. 2011;6 [PMC free article] [PubMed] [Google Scholar]
GAO Y, K ML. Authentic detection and complete tracking of large populations of features in iii dimensions. Eyes express. 2009;17:4685–704. [PubMed] [Google Scholar]
GJOREVSKI N, NELSON CM. Mapping of Mechanical Strains and Stresses around Quiescent Engineered Iii-Dimensional Epithelial Tissues. Biophysical Periodical. 2012;103:152–162. [PMC free article] [PubMed] [Google Scholar]
GRIFFITH LG, SWARTZ MA. Capturing complex 3D tissue physiology in vitro. Nature reviews Molecular jail cell biology. 2006;7:211–24. [PubMed] [Google Scholar]
HALL MS, LONG R, FENG X, HUANG Y, HUI C-Y, WU Thousand. Single Prison cell Traction Microscopy within 3D Extracellular Matrices. Experimental Prison cell Inquiry. 2013 Submitted. [PMC free commodity] [PubMed] [Google Scholar]
HALL MATTHEWS, LONG R, HUI C-Y, WU Thousand. Mapping Three-Dimensional Stress and Strain Fields within a Soft Hydrogel Using a Fluorescence Microscope. Biophysical Journal. 2012;102:2241–2250. [PMC gratuitous article] [PubMed] [Google Scholar]
HUR SS, ZHAO Y, LI Y-S, BOTVINICK E, CHIEN Southward. Alive Cells Exert three-Dimensional Traction Forces on Their Substrata. Cellular and Molecular Bioengineering. 2009;two:425–436. [PMC gratis article] [PubMed] [Google Scholar]
KOCH DL, SUBRAMANIAN Yard. Collective Hydrodynamics of Swimming Microorganisms: Living Fluids. In: DAVIS SH, MOIN P, editors. Annual Review of Fluid Mechanics. Vol. 43 2011. [Google Scholar]
KOCH TM, MUNSTER S, BONAKDAR Due north, BUTLER JP, FABRY B. 3D Traction Forces in Cancer Cell Invasion. Plos Ane. 2012;vii [PMC gratis commodity] [PubMed] [Google Scholar]
LEGANT WR, CHEN CS. Private Commnuication 2013 [Google Scholar]
LEGANT WR, CHOI CK, MILLER JS, SHAO L, GAO L, BETZIG E, CHEN CS. Multidimensional traction force microscopy reveals out-of-plane rotational moments well-nigh focal adhesions. Proceedings of the National Academy of Sciences of the United states of America. 2013;110:881–6. [PMC costless article] [PubMed] [Google Scholar]
LEGANT WR, MILLER JS, BLAKELY BL, COHEN DM, GENIN GM, CHEN CS. Measurement of mechanical tractions exerted by cells in three-dimensional matrices. Nature methods. 2010;7:969–71. [PMC free commodity] [PubMed] [Google Scholar]
LOVE AEH. A treatise on the mathematical theory of elasticity. New York: Dover Publications; 1944. [Google Scholar]
MASKARINEC SA, FRANCK C, TIRRELL DA, RAVICHANDRAN G. Quantifying cellular traction forces in three dimensions. Proceedings of the National Academy of Sciences of the Us of America. 2009;104:22108. [PMC free article] [PubMed] [Google Scholar]
OUELLETTE N, Ten H, B E. A quantitative study of 3-dimensional Lagrangian particle tracking algorithms. Experiments in Fluids. 2006;40:301–313. [Google Scholar]
PEREIRA F, STÜER H, GRAFF EC, GHARIB Grand. Two-frame 3D particle tracking. Meas Sci Technol Measurement Scientific discipline and Technology. 2006;17:1680–1692. [Google Scholar]
RIEU JP, DELANOE-AYARI H. Vanquish tension forces propel Dictyostelium slugs forward. Physical Biology. 2012;9 [PubMed] [Google Scholar]
SABASS B, GARDEL ML, WATERMAN CM, SCHWARZ US. High Resolution Traction Forcefulness Microscopy Based on Experimental and Computational Advances. Biophysical Journal. 2008;94:207–220. [PMC free article] [PubMed] [Google Scholar]
SQUIRES TM, MASON TG. Fluid Mechanics of Microrheology. Annual Review of Fluid Mechanics 2010 [Google Scholar]
Tempest C, PASTORE JJ, MACKINTOSH FC, LUBENSKY TC, JANMEY PA. Nonlinear elasticity in biological gels. Nature. 2005;435:191–194. [PubMed] [Google Scholar]
WU MM, ROBERTS JW, KIM S, KOCH DL, DELISA MP. Commonage bacterial dynamics revealed using a three-dimensional population-scale defocused particle tracking technique. Applied and Environmental Microbiology. 2006;72:4987–4994. [PMC gratis article] [PubMed] [Google Scholar]
XU H. Tracking Lagrangian trajectories in position-velocity infinite. Measurement Science & Engineering. 2008;19 [Google Scholar]