# Three-dimensional cell migration does not follow a random walk

^{a}Department of Chemical and Biomolecular Engineering,^{b}Johns Hopkins Physical Sciences–Oncology Center, and^{c}Department of Mechanical Engineering, The Johns Hopkins University, Baltimore, MD 21218; and^{d}Departments of Oncology and Pathology and Sidney Kimmel Comprehensive Cancer Center, The Johns Hopkins University School of Medicine, Baltimore, MD 21205

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Edited by Robert H. Austin, Princeton University, Princeton, NJ, and approved February 5, 2014 (received for review October 27, 2013)

## Significance

The motility of cells in the absence of gradients has long been described in terms of random walks. Most of what we know about eukaryotic cell migration has stemmed from well-controlled studies of cell migration on flat dishes. However, cells in vivo often move through 3D environments. Despite this difference, cell speed and persistence are typically extracted from fits using the same persistence random walk (PRW) model. This paper shows that the assumptions of the PRW model are erroneous for 3D cell migration. We introduce and validate a new model of 3D cell migration that takes into account cell heterogeneity and the anisotropic movements induced by local remodeling of the 3D matrix.

## Abstract

Cell migration through 3D extracellular matrices is critical to the normal development of tissues and organs and in disease processes, yet adequate analytical tools to characterize 3D migration are lacking. Here, we quantified the migration patterns of individual fibrosarcoma cells on 2D substrates and in 3D collagen matrices and found that 3D migration does not follow a random walk. Both 2D and 3D migration features a non-Gaussian, exponential mean cell velocity distribution, which we show is primarily a result of cell-to-cell variations. Unlike in the 2D case, 3D cell migration is anisotropic: velocity profiles display different speed and self-correlation processes in different directions, rendering the classical persistent random walk (PRW) model of cell migration inadequate. By incorporating cell heterogeneity and local anisotropy to the PRW model, we predict 3D cell motility over a wide range of matrix densities, which identifies density-independent emerging migratory properties. This analysis also reveals the unexpected robust relation between cell speed and persistence of migration over a wide range of matrix densities.

Random walks are ubiquitous in biology (1). In particular, the motility of bacteria and eukaryotic cells in the absence of symmetry-breaking gradients has long been described in terms of random walk statistics. Eukaryotic cell migration is a complex process that is a tightly regulated and critical to the normal development of organs and tissues (2⇓–4). Cell migration is activated in a wide range of human diseases, including cancer metastasis (5, 6), immunological responses (7), and wound healing (8). Most of what we know about eukaryotic cell migration at a mechanistic molecular level has stemmed from well-controlled studies of cell migration on flat dishes (i.e., 2D environment). However, cell migration in vivo often forces cells to remodel, exert pulling forces on, and move through a 3D collagen I-rich matrix. Recent work has demonstrated that mechanisms of 3D migration are often different from their 2D counterparts (9⇓⇓⇓⇓⇓–15). Migration on 2D dishes, which induces a basal-apical polarization of the cell, is driven by actomyosin contractility of stress fibers between large focal adhesions and the formation of a wide lamellipodium terminated by thin filopodial protrusions at the leading cellular edge (4, 16). The same cells in collagen-rich 3D matrix do not display a lamellipodium or filopodia. Instead, they display highly dendritic pseudopodial protrusions controlled by distinct proteins that rely both on acto-myosin contractility and microtubule assembly/disassembly dynamics (11, 17). 3D cell migration depends on the expression of metalloproteinases (MMPs), which are dispensable in 2D migration, and physical properties of the 3D matrix (18), such as pore size (6). Recent work has also shown how cancer cells in 3D can alternate between a mesenchymal and an amoeboid migratory phenotype depending on the physical properties of the matrix (19, 20) and MMP inhibition (17), phenomena that do not occur in the 2D case. Finally, through the zyxin/α-actinin/p130Cas module, 3D cancer cell migration features tight molecular control of the temporal and spatial patterns of movements in the matrix, which do not exist in conventional 2D migration (10).

However, despite these important differences, cell speed and persistence of migration in 2D and 3D microenvironments are typically extracted from fits of the mean squared displacements (MSDs) using the same persistence random walk (PRW) model (21⇓⇓⇓–25). Fits of MSDs, however, do not rigorously test several key underlying assumptions of the PRW model, including a Gaussian distribution of velocities, a single-exponential decay of the velocity correlation function, an isotropic velocity field, and a flat distribution of angles between cell movements at long time scales (1). This paper shows that incorporating cell heterogeneity into the PRW model is sufficient to fully explain statistical descriptors of 2D migration. In contrast, we show that the assumptions of the PRW model are quantitatively and qualitatively erroneous for 3D cell migration: cancer cell migration in 3D matrix does not follow a random walk. We introduce and validate a model of 3D cell migration that takes into account cell heterogeneity and the anisotropic movements induced by local remodeling of the 3D matrix.

## Results

### PRW Model.

The PRW model of cell motility is derived from a stochastic differential equation describing the motion of a self-propelled cellwhere is time, is the cell velocity, is the persistent time, is the cell speed, and is the random vector of a Wiener process (23). A main characteristic of this model is that the MSD is given bywhere is the dimension of the extracellular space (which can be 1D, 2D, and 3D) (10, 17, 26, 27), and *τ* is the time lag between positions of the cell. The autocorrelation function of the cell velocity vector for the PRW model exhibits a single exponential decaywhere *D* is the cell diffusivity. In 2D, the velocity direction is described by an angle with respect to a laboratory frame, *θ*. The change in angle over a small time interval, *dθ*, is a random variable given by a uniform distribution with a peak near *dθ* = 0. Typically, Eq. **2** is used to fit measured MSD data. The statistics of *dθ* and the time lag dependence of the velocity autocorrelation function (Eq. **3**) are generally not examined in details.

### Rigorous Test of the PRW Model of Cell Migration.

Using live-cell microscopy, we measured the spontaneous displacements of individual, low-density, human, WT fibrosarcoma HT1080 cells—a cell model used extensively in cell migration studies—on 2D collagen-coated substrates and inside 2 mg/mL collagen matrices in the absence of symmetry-breaking directional (chemotactic, galvanotactic, durotactic, etc.) gradients. Type I collagen was chosen because it is by far the most abundant protein of the extracellular matrix in fibrous connective tissues from which malignant mesenchymal tumors are derived and disseminate (6). Cell movements were recorded at a rate of 30 frames/h for >8 h, corresponding to ∼2.5 decades in time scales (Fig. 1 *A* and *B*). Trajectories of cells in 2D and 3D conditions readily showed distinct patterns (Fig. 1*B*). The trajectories of cell migration in 3D displayed a more linear morphology compared with trajectories of cells in 2D conditions. Importantly, we verified that the instantaneous speed of cells (averaged distance traveled every 2 min) was independent of time over the entire observation time period, which indicated that cells displayed a steady motility behavior both in the 2D and 3D cases (Fig. 1 *C* and *D*). Cells displayed a significantly lower speed in 3D matrices than cells on 2D flat substrates at both a short time scale (*τ* = 2 min) and a long time scale (*τ* = 60 min) (Fig. 1*E*). Accordingly, the MSDs of cells on 2D substrates were significantly higher than of cells in 3D matrices at any given time lag between 2 min and 8 h, indicating that 2D cell motility is faster than 3D cell motility (Fig. 1*F*). At short time scales (*τ* < 1 h), both MSD profiles in 2D and 3D displayed an exponent α > 1 (measured from a fit of MSD ∼ *τ*^{α}), indicating that cell motility was directional (superdiffusive) (Fig. 1*F*).

The PRW model was introduced close to 30 years ago and has been used ubiquitously to describe and analyze the random migration of cells on substrates (21⇓⇓–24) and, more recently, cell migration in 3D matrices (25). The MSD for the PRW model is given in Eq. **2**. If we include the observational error in the measurements, the MSD is then given byHere, 4*σ*^{2} is the noise (error) in the position of the cell (*SI Text*). The PRW model provides an overall good fit to MSDs of individual cells (*R*^{2} value: 0.88–0.98) for both 2D and 3D migration. This model also seemed to perfectly describe cell MSDs at the cell population level in both 2D and 3D environments (*R*^{2} value: ∼1; Fig. 1*F*). Therefore, one could conclude that the PRW model explains 2D and 3D cell migration.

However, the PRW model has a number of underlying assumptions, such as a Gaussian distribution of velocity at all time scales, exponential decay for the autocorrelation correlation function, and isotropic cell movements. There is a practical challenge to test these assumptions for individual cells: the inherently limited resolution of measuring these statistical profiles at the single-cell level. Indeed, the resolution is mainly determined by the sample size of measured cell velocities, which is naturally restricted by two intrinsic limits: (*i*) the sampling rate and (*ii*) the observation time. The limited sampling rate is due to the fact that cell velocity becomes difficult to define clearly at high camera frame rates when apparent cell migration is mostly due to subcellular movements and fast irregular changes of cell morphology without real cell translocation. The total observation time period is also intrinsically limited by the time between cell divisions, which is ∼16–24 h for HT-1080 cells. As a consequence, determination of whether the distribution of cell velocities is Gaussian or exponential for a single cell is prone to error. Hence, beyond fits of individual MSDs, fully validating the PRW model at single-cell level is inherently difficult.

An alternative solution has been to obtain statistical characteristics of cell motility from population-averaged profiles to reach the required resolution (19, 28, 29). The underlying assumption of this method is that individual cells have equally probable motile behavior, a notion we test rigorously in this paper.

### Fundamental Statistical Differences Between 2D and 3D Migration.

A first implication of the excellent fit between measured MSDs and MSDs predicted by the PRW model (Fig. 1*F*) is that the autocorrelation function (ACF) of cell velocity, both in 2D and 3D environments, should decay as a single exponential with a relaxation time equal to the persistence time *P*. We found that the decay of the ACF did not follow a single-exponential relaxation. Rather, ACF profiles followed a two-step process characterized by two characteristic time scales. For 2D migration, we observed a slower-than-predicted decrease of the ACF at long time scales, > 30 min (Fig. 2*A*, blue curve). Such a two-step profile for the ACF has previously been observed in 2D migration (29). The slower-than-expected decrease of the ACF was even more pronounced in 3D motility (Fig. 2*A*, red curve). To be noted, ACF at one-frame lag was not adopted since it would be corrupted by observation noise (Fig. S1).

A second implication of the goodness of fits between measured MSDs and MSDs predicted by the PRW model (Fig. 1*F*) is that the distribution of cell velocities should follow Gaussian statistics. Instead, ensemble-averaged results showed that cell displacements followed an exponential distribution at all probed time scales (Fig. 2 *B* and *C*), not only in the 2D case (blue curves) as previously observed (19, 29), but also for in the 3D case (red curves). Importantly, we found that this exponential distribution of cell velocity was independent of the method of tracking of cell movements (*SI Text* and Fig. S2).

A third implication of the excellent fits between measured and predicted MSDs (Fig. 1*F*) is that the angular distribution of cell movements should flatten over time. We measured angular displacements *dθ* during cell migration and computed their distribution (Fig. 2*D*). We found that the distribution in *dθ* at different time scales in 3D showed profiles fundamentally different from those in 2D. For 2D motility, the distribution in *dθ* was elevated at small angles, corresponding to cells moving persistently at short time scales, becoming a uniform distribution at long time scales. This result is predicted by the conventional PRW model (*SI Text*). However, the high probability to observe small *dθ* values observed during 3D motility at short time scales did not disappear over time (Fig. 2*E*). Instead of the expected flattening of the distribution of angles between cell movements over time, the probability to observe large angular displacements progressively increased around 180°, corresponding to cells moving in the exact opposite direction to the direction of movements separated by long time lags. This result indicates that the probability of observing cells moving back into the 1D tunnel-like tracks in the 3D matrix formed by the cells during their initial exploration of the matrix increased at long time scales.

Based on this result, we studied whether the magnitude of velocity was spatially anisotropic. First, we identified and then realigned the primary direction of migration () of individual cells using the singular-vector decomposition method (SVD; Fig. 2*F*, *Inset*). for each cell is an estimate of the primary direction of migration equal to the principal axis of all instantaneous velocities of that cell. We measured the magnitude of the velocity (at a 2-min time lag) at different orientations relative to . This analysis indicated that cells in 3D matrix displayed a higher velocity along their primary migration axis, which include both and −, compared with the velocity along the direction of migration perpendicular to (Fig. 2*F* and Fig. S3). In sum, when analyzed through their individual or ensemble-averaged MSD profiles, cell motility patterns in 2D and 3D seem to be quantitatively different, but qualitatively similar. However, good fits of MSDs constitute a weak test for models of cell migration and comprehensive statistical analysis reveals instead that cell motility patterns in 2D and 3D environments are qualitatively different. Cells migrating in a 3D matrix display qualitatively different angular displacement distributions from their 2D counterparts and, unlike in 2D migration, display an anisotropic velocity.

### Cell Heterogeneity Alone Explains the Non-Gaussian Velocity Distribution in 2D.

Accumulating evidence suggests a strong correlation between cell phenotypic heterogeneity and clinical outcomes, particularly in cancer. We hypothesized that the non-Gaussian nature of the velocity distribution could stem from cell heterogeneity. Therefore, we assessed the degree of migratory heterogeneity in 2D and 3D environments. Here we found that, despite the homogeneous environment of 2D substrates, individual HT-1080 cells already displayed significantly different motility profiles from each other. A one-way ANOVA test of velocities of different pairs of individual cells evaluated at a time lag of 2 min showed that more than 50% of paired cells had different mean velocities with *P* < 0.05 (Fig. S4*A*). Similar results were obtained for cell motility in 3D matrices (Fig. S4*B*).

We first described the motility of individual cells using the PRW model by simulating cell trajectories using experimentally measured paired values of persistent time *P* and speed *S* for each individual cell (Fig. 3*B*; see details about simulations in *SI Experimental Procedures*). For the sake of comparison, we also simulated cell trajectories using the same *P* and *S* derived from population-averaged MSDs to model trajectories (Fig. 3*A*). Ensemble-averages MSDs (Fig. 3*C*), ACFs (Fig. 3*D*), velocity distributions (Fig. 3*E*), and anisotropic maps (Fig. 3*F*) of these two sets of simulated trajectories were then computed and compared. Although MSD profiles predicted by both approaches were in good agreement with the experimental results (Fig. 3*C*), ACFs obtained from the PRW model that included single-cell distribution provided better fits when including cell heterogeneity (Fig. 3 *E* and *D*). Remarkably, when incorporating cell heterogeneity, the PRW model correctly predicted the exponential distribution of cell velocities (Fig. 3*E*). Both approaches also correctly predicted the distributions of angular displacements (Fig. 3 *G* and *H*).

Together, our results indicate that the simple PRW model, when it includes cell heterogeneity, captures essential statistical characteristics of cell migration, at least on 2D substrates. In contrast, 3D migration using the PRW model, even when incorporating cell heterogeneity, yielded trajectories and associated statistical characteristics that were qualitatively distinct from experimental results (Fig. 4 *A* and *E–H*). This result suggests that, unlike the 2D case, the PRW model, even when including cell heterogeneity, does not explain qualitatively or quantitatively cell migration in 3D matrix.

### Anisotropic PRW Model Fully Describes 3D Migration.

In the conventional PRW model, the velocity of cells is presumed to be spatially isotropic. However, an important characteristic of 3D cell migration is its highly anisotropic velocity profile (Fig. 1*B*). SVD analysis of cell velocities identified primary and nonprimary directions of migration (Fig. 2*F*). We extracted the MSDs and ACFs of individual cells along these two directions and found that cell migration is a self-correlative process and that MSDs in each direction are well described by the PRW model (Fig. S3 and *SI Experimental Procedures*). Hence, we extended the PRW model to the anisotropic PRW model (APRW), which incorporates different persistent times and speeds in the primary (*P*_{p}, *S*_{p}) and nonprimary (*P*_{np}*, S*_{np}) directions of migration and found that in these different directions cells followed PRW statistics. R^{2} values derived from fitting APRW models into primary and nonprimary directions of migration were >0.95, which suggests that the APRW model describes 3D migration.

To test the APRW model, we simulated 3D cell migration trajectories with experimentally measured single-cell values of *P* and *S* (Fig. 4 *A* and *B* and *SI Experimental Procedures*). MSD profiles obtained from the PRW model, that does not acknowledge anisotropy, and the APRW model that does acknowledge anisotropy, both fitted well the experimental MSDs (Fig. 4*C*). However, we already know that a good fit of MSDs is a weak test of models of cell migration (Fig. 1). The two-step decay of the ACF and the exponential velocity distribution were qualitatively and quantitatively better predicted by the APRW model than the PRW model (Fig. 4 *D* and *E*). Moreover, the observed anisotropic velocity profiles and distributions of angular displacements, which were inaccurately anticipated by the PRW model, were correctly predicted by the APRW model (Fig. 4 *F–H*). Together, our results indicate that the APRW model successfully describes the heterogeneous and anisotropic motility patterns of migratory cells in 3D matrix.

### Diffusive Patterns and Effects of Collagen Density.

We demonstrated that the APRW model properly characterizes cell motility in 3D matrix at a fixed concentration collagen I. As a more comprehensive test of the APRW model, we next investigated how statistical characteristics of 3D cell migration were modulated by changes in collagen density (Fig. 5). MSDs, displacement distributions, autocorrelation functions, and angular distribution were well fitted over a wide range of collagen density with the APRW model of 3D migration (Fig. 5*A* and Fig. S5). We note the great improvement of the fits of anisotropic profiles of velocity and angular displacement distributions compared with the PRW model and PRW model that takes into account cell heterogeneity.

Cells in a 3D collagen I matrix moved most persistently at a concentration of 1 mg/mL; the mean persistent time along the primary migration direction decreased with increasing collagen density (Fig. 5 *B* and *C*). Cell migration in 1 mg/mL collagen matrices also showed the highest diffusivity, measured as *D*_{tot} = ∼ *MSD*_{long} _{times}/4*τ*: the mean cell diffusivity decreased monotonically with collagen concentration before plateauing at 4 mg/mL (Fig. 5 *D* and *E*). The ratio of diffusivities along the primary and nonprimary migration directions (which we call the anisotropic index *ϕ*) also depended on collagen concentration (Fig. 5*F*). In sum, these results show that the mean values of descriptors of 3D cell migration, including persistent time, diffusivity, and anisotropic index, are tightly regulated by collagen density and that the APRW model describes well 3D migration over a wide range of collagen concentrations.

### Cell Diffusive Patterns and Searching Strategies in 3D.

We next identified functional relationships among the different descriptors of 3D cell migration through systematic correlative analysis. We found that some of these cell motility descriptors were correlated with each other. For example, the persistence time and diffusivity were highly correlated for cell motilities in 1 and 4 mg/mL matrices with correlation coefficients of 0.81 and 0.70, respectively (Fig. 6*A*). The extent of interdependence among the five major motility descriptors, including total diffusivity (*D*_{tot}), persistence time and diffusivity along the primary axis (*P*_{p} and *D*_{p}), nonprimary axis of migration (*P*_{np} and *D*_{np}), the anisotropic index *ϕ*and their mutual-correlation profiles, were evaluated through heat maps (Fig. 6*B*) and correlation network diagrams (Fig. 6*C*) as a function of collagen density.

Some expected correlations among descriptors of migration were observed, such as the high correlation between total diffusivity and primary or nonprimary diffusivity (e.g., see thick lines between *D*_{tot} and *D*_{p} and, to a lesser extent, between *D*_{tot} and *D*_{np}; Fig. 6*C*), because total diffusivity is a weighted combination of both. These constitute positive controls. However, our analysis revealed a strong dependency between a priori independent variables, including the persistent time and diffusivity along the principal axis of migration (*P*_{p} and *D*_{p}; Fig. 6*C*) and a strong association between the primary diffusivity and nonprimary diffusivity (*D*_{p} and *D*_{np}; Fig. 6*C*) across a wide range of conditions.

These results suggest the existence of underlying constrains for cell migration set by common robust molecular pathways that regulate 3D cell motility, independently of changes in collagen density. Moreover, because persistent time and speed are correlated across a wide range of conditions, they are not controlled by purely stochastic processes. We also found that the relation of the anisotropic index *ϕ* with other migration descriptors (i.e., the high connectivity of the blue dot with other dots in the hexagon network; Fig. 6*C*) changed qualitatively with collagen concentration. Indeed, the anisotropic index was negatively correlated with diffusivity along the minor axis of migration at the 1 mg/mL condition while strongly positively correlated with total diffusivity at 6 mg/mL These findings suggest that highly motile cells in low-density and high-density collagen are mechanistically distinct. Collectively, these analyses indicate that strategies for matrix exploration by cells are tightly regulated by collagen density.

## Discussion

This work shows that the traditional PRW model, which is ubiquitously used to parameterize cell migration into speed and persistence time (20), is based on assumptions that are not met in the patho-physiological relevant case of a 3D extracellular matrix. Because the underlying assumptions of the PRW model—exponential velocity autocorrelation, Gaussian distribution of velocities, and flat angular distributions at long time scales—cannot be rigorously tested at the single-cell level because of limited (meaningful) spatial resolution and a time of collection limited by cell division, we and others (19, 28, 29) have used population-averaged values of these parameters. These discrepancies in the 2D case have typically been accounted for by adding new fitting parameters (28, 29). Here we show that, in 2D migration, cell heterogeneity is sufficient to understand the population-averaged non-Gaussian (exponential) distribution of velocities and the nonexponential decay of the velocity autocorrelation. Therefore, the PRW model qualitatively holds in 2D migration.

However, the PRW model, even when incorporating cell heterogeneity, does not properly describe 3D migration even qualitatively (Figs. 4 and 5). This result indicates that cell migration patterns in a 3D matrix do not follow persistent random walks. Any random walk model of cell migration would predict that the distribution of angles between cell movements flattens at time scales longer than the persistence time. Instead, we find that the probability of complete 180° turnabout in the 3D matrix increases with time (Fig. 2*E*). Rather than the angular distribution becoming flat at long time scales, it peaks at 0° and 180° and is the minimum at 90° (Fig. 2*E* and Fig. S5). One of the possible reasons for this unexpected polarized distribution is due to the fact that fibrosarcoma cells create tube-like microchannels of diameter approximately equal to their nucleus by locally digesting the collagen matrix at their leading edge, mainly because of the surface-bound metalloproteinase MT1-MMP (19, 20). These microchannels can greatly reduce the number of possible cell movements. In contrast, there are no local obstacles that would restrict the reorientation of migratory cells in the 2D case.

SVD analysis of cell trajectories (Fig. 2*F*) shows that, unlike its 2D counterpart, 3D migration is anisotropic (Fig. S3*A*). A complete parametrization of 3D migration at the single-cell level, therefore, requires not only speed and persistence time, but also an anisotropy index (Fig. 5*F*). This unique model of 3D cell migration correctly predicts the nonexponential profile of the velocity autocorrelation function and the complex time-dependent angular distribution of cell movements in 3D matrix.

When matrix density is changed, a myriad of microstructural parameters of the matrix are changed accordingly, including ligand density, pore size of the matrix, fiber thickness, fiber alignment, and matrix stiffness. However, despite this wide range of conditions, we found that cell diffusivity in 3D matrices consistently predicts its persistence two parameters of cell migration that are a priori independent (Fig. 6 *B* and *C*).

## Experimental Procedures

Human fibrosarcoma cells (HT1080; ATCC) migrating on 2D petri dishes and in 3D collagen I matrix were imaged every 2 min for more than 8 h. The trajectories of cells were tracked using Metamorph (Molecular Devices) software. Analysis of cell trajectories and simulations were conducted using custom software in MATLAB (Mathwork). Detailed experimental and computational procedures for cell culture, embedding cells in 3D collagen I matrices, cell tracking, statistical profiling of cell trajectories, characterizing the PRW model, details about the new model for 3D cell migration, analysis of angular displacement of PRW model, computer simulations of cell trajectories, and statistics are given in *SI Experimental Procedures*.

## Acknowledgments

This work was supported by National Institutes of Health Grants R01CA174388 and U54CA143868.

## Footnotes

↵

^{1}P.-H.W. and A.G. contributed equally to this work.- ↵
^{2}To whom correspondence may be addressed. E-mail: pwu{at}jhu.edu and wirtz{at}jhu.edu.

Author contributions: P.-H.W., S.X.S., and D.W. designed research; P.-H.W. and A.G. performed research; P.-H.W. and A.G. contributed new reagents/analytic tools; P.-H.W., A.G., and S.X.S. analyzed data; and P.-H.W., A.G., S.X.S., and D.W. wrote the paper.

The authors declare no conflict of interest.

This article is a PNAS Direct Submission.

This article contains supporting information online at www.pnas.org/lookup/suppl/doi:10.1073/pnas.1318967111/-/DCSupplemental.

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