Regulation of Migration of Chemotactic Tumor Cells by the Spatial Distribution of the Collagen Fibers' Orientation
RRegulation of Migration of Chemotactic Tumor Cells by the SpatialDistribution of the Collagen Fibers’ Orientation
Youness Azimzade, Abbas Ali Saberi, , † and Muhammad Sahimi , ‡ Department of Physics, University of Tehran, Tehran 14395-547, Iran Mork Family Department of Chemical Engineering and Materials Science, University ofSouthern California, Los Angeles, California 90089-1211, USA
Collagen fibers, an important component of the extracellular matrix (ECM), can both in-hibit and promote cellular migration.
In-vitro studies have revealed that the fibers’ orienta-tions are crucial to cellular invasion, while in-vivo investigations have led to the developmentof tumor-associated collagen signatures (TACS) as an important prognostic factor. Studyingbiophysical regulation of cell invasion and the effect of the fibers’ oritentation not only deepensour understanding of the phenomenon, but also helps classifying the TACSs precisely, which iscurrently lacking. We present a stochastic model for random/chemotactic migration of cells infibrous ECM, and study the role of the various factors in it. The model provides a framework,for the first time to our knowledge, for quantitative classification of the TACSs, and reproducesquantitatively recent experimental data for cell motility. It also indicates that the spatial dis-tribution of the fibers’ orientations and extended correlations between them, hitherto ignored,as well as dynamics of cellular motion all contribute to regulation of the cells’ invasion length,which represents a measure of metastatic risk. Although the fibers’ orientations trivially affectrandomly moving cells, their effect on chemotactic cells is completely nontrivial and unexplored,which we study in this paper. 1 a r X i v : . [ q - b i o . CB ] J un . INTRODUCTION Up to 90 percent of cancer-associated mortality is attributed to metastasis, but despite thisfact, metastasis has remained one of the least understood aspects of the disease [1]. Duringmetastasis, cells disseminate from the initial tumor, intravasate into the the surrounding vessels,and colonize within a new host tissue [2]. Despite development of many prognostic measuresfor evaluating the metastatic risk, there is still intensive ongoing research for gaining deeperunderstanding of the phenomenon and evaluating accurately its risk, in order to minimizetreatment failure and costs [3].Cell migration plays a crucial role in metastasis [4], as the physical translocation duringmetastasis happens through cellular migration that is either directed or random [5]. If thebasic machinery of cell mobility is activated, but without guiding principle, the cells migraterandomly. In the presence of external/internal guidance cues, however, the cells undergo di-rected migration [5]. When soluble chemotactic agents, such as chemokine and growth factor,represent the external cue, cancer cells may climb the gradient and undergo chemotaxis in orderto metastasize [6]. As a result, during metastasis, cancer cells migrate randomly or are directeduntil they reach blood vessels and enter its stream. Cellular migration in vivo happens within aheterogeneous environment that is composed mainly of extracellular matrix (ECM), which cansignificantly alter cellular migration [7]. Many models, such as random [8] and persistent walks[9], as well as other types of models [10-12] have been developed to describe and/or simulatecellular migration.The ECM provides the environment that supports cell maintenance [13] and it influences[14] cellular migration through its physical properties [14], such as confinement [15-20], fibertopography [21-26], and bulk characteristics [27-36]. In particular, orientation of the ECM’sfibrils affects cells’ direction of migration [37-41]. Alignment of the fibrillar matrix, both in vitro and in vivo controls migration and promotes directional cell migration [42,43], and reorientscell motility without altering its overall magnitude [44,45]. Moreover, the fibers can eitherimpede tumor invasion by acting as a barrier against migration [46-48], or facilitate it byproviding high-speed “highways” [49] based on their orientation. Experimental investigationshave studied the effect of the orientation of the fibers on cellular invasion [50-52]. They haveindicated larger invasion extent along the direction of aligned ECM’s fibers for both random and2irected migration, but they still fail to provide a quantitative understanding of such regulation.Physics-based models based on percolation [53,54] have also been used to simulate the ECMstructure.The profound effect of fiber orientation in vivo has led to the emergence of a prognosticfactor, known as tumor-associated collagen signature (TACS), which predicts the behavior oftumor based on the the type and structure of the ECM alignment [3,55]. According to thisapproach, there are three types of signatures [56]: TACS-1, representing dense wavy collagenfibers (CFs); TACS-2, which is indicative of linear CFs parallel to the tumor’s border, andTACS-3, identified by the presence of linear CFs perpendicular to the tumor’s border. Screeningof the TACS is a clinical prognostic tool, and TACS-3 could indicate poor survival rate, hencesuggesting that quantifying CFs alignment may be an independent prognostic marker [3,55,57].But, despite the significance of the classification as a strong prognostic factor, as well as aquantitative approach to study alignment of the ECM fibers [58], it remains qualitative [55]because, for example, it is not clear what angle between the fibers and the tumors’ borderconstitutes the ”dividing angle” between the TACS-2 and TACS-3, and how the transitionbetween the two occurs. Thus, fiber classification should be addressed by development ofquantitative understanding of the effect of the orientations of the fibers on cell migration. Inthis paper we describe a new model, and utilize analytical arguments and numerical simulationto study the effect of the ECM’s fibers’ structure on cell migration, which provide quantitativeunderstating of the ECM’s fibers dividing angle.The rest of this paper is organized as follows. In Sec. II we describe the details of the modelthat we use in our study. The results are presented and discussed in Sec. III, while the paperis summarized in Sec. IV.
II. THE MODEL
Physical cues of the cellular environment, such as organization of the ECM components,affect the cells and their motility, and force transduction [59]. Collagen alignment regulatesmigration by directing cellular protrusions along aligned fibers [60]. The alignment also pro-motes directed migration by a combination of traction forces and contact guidance mechanisms[61]. Fibrillar topographical cues in the form of one-dimensional (1D) nanofibres guide cellmigration in vivo [62]. As such, the cells migrate directionally along oriented fibers [21]. The3ize of the CFs and the structure of the ECM exhibit wide diversity, however. Experimentalstudies [44,45] have attempted to mimic the observed structures [50-52], and have indicatedthat the fibers have an average length of 20 µ m, with their orientations following a Gaussiandistribution. In this paper we rely on such data as the basis of our model. ( a ) ( b )( c ) ( d ) Figure 1: (a) A cell and the medium into which it migrates from a torn boundary at (0,0)(middle left). The fibers are shown by the solid lines. Once one step of migration takes place,the cell will be in one of the nearest neighbor units and continues to migrate according to thesame probabilities. The green dashed line shows a possible random trajectory for the migration.Sample trajectories are for ( σ, ¯ θ ) (b) ( π/ , π/ π/ , π/ pi/ , × µ m units,corresponding to the length of a fiber, and assign a direction to each unit along the fiber insideit; see Fig. 1(a). In principle, the size of the fibers can be different for different tissues, but thequalitative aspects of the results would be the same, if we use different fiber size for different4issues. It would also not be difficult to use different fiber size for different tissues. Note alsothat the exact length of the fibers does not play an important direct role in the simulations.What is significant is the existence of the fibers in the ECM that provide a medium for thecells to advance.The ECM is, of course, three dimensional. There is also extensive evidence that extrapola-tion from 2D is far from straightforward [5]. But, as emphasized by others [21], the fact is thatthe fibers have a 1D structure. As such, we argue that migration happens on 1D structuresembedded in a higher-dimensional space. Given this assumption, then, regardless of dimen-sions of the space in which the fibers are embedded, our results should be valid. In fact, in theexperiments with which we compare our results (see below), the cellular medium is quasi-2D.Migration happens on the 1D collagen fibers, and the height (thickness) of the medium is sosmall compared to the other dimensions that the cellular medium is essentially 2D. If we extendthe cellular environment in our model in the third direction z by keeping the same configurationof the fibers around the x axis (shown in Fig. 1), the result for the motility would remain thesame.The heterogeneous cellular environment is not completely random, but contains extendedcorrelations [63] the existence of which has been confirmed by in-vivo studies [55], although thestructure of the correlation function has not been characterized yet. To generate a distributionof the fibers with spatial correlations, we use the fractional Brownian motion (FBM) [64]according to which for the fiber orientations at x and x’ one has, (cid:104) ( θ x − θ x (cid:48) ) (cid:105) ∝ | x − x (cid:48) | H .Here, 0 < H < H > . H →
1, whereas negative correlations arerepresented by
H < . H →
0. Note that wedo not claim that the FBM represents the actual type of the correlations, rather we use it as atypical stochastic functions that produces extended correlations. At the same time, the FBMhas found many applications in biological phenomena [66-70]. Note that the model that we useis not for migration in 2D substrates. Instead, it imitates migration on collagen fibers, whichfor convenience is considered to be two dimensional. In a 3D model one would have 20 × × µ m cubes, in which we assign θ as the angle with the normal to the tumor’s border plain, anda second angle φ for the orientation in the tumor’s border plain.As Fig. 1(a) indicates, the tumor’s boundary is at x = 0. Then, if a cell is in a unit with5 fiber orientation θ with respect to the y axis [vertical axis in Fig. 1(a)], the probabilities ofmigrating in the x and y directions are, respectively, sin θ and cos θ . In the corresponding3D model we set the probabilities to be sin θ , cos θ cos φ , and cos θ sin φ for x , y and z directions, respectively, where φ desribes the direction in the plane parallel to tumor’s border.The probabilities provide us with a means of understanding the effect on cell migration of thefibers’ alignment and their distribution. Note that the orientations lead to cellular alignmentthat causes directed migration [44,45]. We assume that the migration probabilities capturethe effects of both the ECM’s and the cellular alignment. Then, the orientations are selectedaccording to the FBM with a standard deviation of σ around ¯ θ .To study chemotaxis we use the normalized barrier [71] or the Keller-Segel [72] model,according to which for a cell moving in a 1D medium in a constant external chemical potentialgradient, the probabilities of moving right and left are r = p and l = 1 − p , where | p − / | ( | p − / | in 2D) is the strength of chemotaxis that is regulated by the gradient strength andthe cells’ ability to detect and respond to it. Extending the model to 2D with no chemtaxis inthe y direction leads to r = p and l = u = d = (1 − p ) /
3, in which u and d are the probabilitiesof moving up and down. Coupling between the effect of chemotaxis and the ECM alignment isimplemented by considering r ∝ p sin θ , l ∝ (1 / − p ) sin θ , and u = d ∝ (1 / − p ) cos θ ,which after normalization lead to, r = 3 p sin θS , (1) l = sin θ (1 − p ) S , (2) u = d = cos θ (1 − p ) S , (3)with the drift velocity v x given by, v x = δ ( r − l ) τ = δ sin θ (4 p − D , (4)and v y = 0, where S = 2(1 − p ) cos θ + (1 + 2 p ) sin θ and D = Sτ , with δ = 20 µ m beingthe jump’s length (the units’ size), and τ is the duration of a single jump. Though we do notpresent results in which time is explicitly present, consistent with the experiments [44,45] oncells’ velocity, we set τ = 1 hour in the simulations, and consider the drift only in the positive x direction, whereas in the y direction only diffusion with no drift occurs.6 II. RESULTS AND DISCUSSIONS
We first consider the cells to be initially at x = 0, the tumor boundary, moving for alarge number of time steps in the cellular medium with the FBM disribution of the fibers’orientations with given σ and ¯ θ . At each time step a cell moves according to the probabilitiesgiven by Eqs. (1) - (3). It is free to move anywhere, except crossing the x = 0 line. Examplesof the trajectories of the cells are presented in Figs. 1(b)-1(d).We then check if the model reproduces recent experimental data for cell motility in var-ious directions. Defining the directional motilities by µ x = (cid:104) x (cid:105) /t and µ y = (cid:104) y (cid:105) /t , re-cent experimental studies on the ECM’s fiber alignment with ¯ θ = π/ σ = 0 . π re-ported that [44,45], µ x /µ y ≈
5. Figure 2(a) presents the simulation results with the sameparameters. We find that after long enough times, (cid:104) x (cid:105) ≈ (cid:104) y (cid:105) ≈
90 and, therefore, µ x /µ y = (cid:104) x (cid:105) / (cid:104) y (cid:105) = 420 / ≈ .
67, in excellent agreement with the experimental data.Since our model is 2D, but the experiments had been carried out in seemingly 3D media,the agreement may seem to be fortuitous. Note, however, that the experiments were actuallycarried out in a quasi-two-dimensional cellular medium, and although migration occurs on thecollagen fibers embedeed in 3D space, the evironment is limited in third direction z . But, evenif the cellular medium were truly 3D, (cid:104) x (cid:105) should remain unchanged, because in the 3D modelwe define the motility in the border plain by µ r = µ y + µ z . In that case we would still have µ x /µ r ≈
5, hence indicating that our result, at least for the uncorrelated cellular media, wouldnot change if we use a fully 3D model.More generally, however, it is important to understand how the spatial distribution of thefibers’ orientations affect the extent of cell invasion. Since migration perpendicular to the tumorboundary at x = 0 plays the main role in metastasis, we take the net mean displacement, (cid:104) x (cid:105) / , as the extent or length of the invasion, which is indicative of the metastatic risk.Consider, first, the non-chemotactic case with p = 1 /
4. The simulations indicate that as σ ,the standard deviation of the orientations’ distribution, increases in an uncorrelated medium, (cid:104) x (cid:105) /t converges to the same value of about 0.5 for all ¯ θ , which is expected for a randomwalk in a homogeneous medium; see Fig. 2(b). Precisely the same behavior develops for thechemotactic case, p > / σ , the dependence on ¯ θ of all7igure 2: (a) Evolution of (cid:104) x (cid:105) , (cid:104) y (cid:105) , and (cid:104) R (cid:105) = (cid:104) x (cid:105) + (cid:104) y (cid:105) . (b) Effect of σ of the orientationdistribution. (c) Dependence of the motility µ x on σ and ¯ θ for p = 1 /
4, defining the crossingpoint θ ∗ . (d) Same as (c) but with p = 0 .
8. (e) Effect of directionality, represented by theprobability p . (f) Same as in (e) but in a correlated cellular medium with σ = 0 . θ = π/ (cid:104) x (cid:105) /t exhibits sigmoidal behavior with a common crossing point at a specific orientation θ ∗ ,defined by (cid:104) x ( θ ∗ ) (cid:105) = (cid:104) x ( π/ (cid:105) /
2. Note that although the shape of (cid:104) x (¯ θ ) (cid:105) may vary in variouslimits, it still is an important property for quantitative understanding of the effect of the fibers’alignment. Figure 2(c) shows that in random cell advancement the transition is at ¯ θ ∗ = π/ θ < π/ θ > π/ θ ∗ may be considered, but our approach allows us to classify thefibers based on any reasonable definition of θ ∗ . Note that we do not claim, at this point, thatthe value of θ ∗ is universal, independent of the type of fibers’ oritentation distribution, or thedimensionality of the cellular medium. The important point that we would like to emphasizeis rather the existence of such a critical angle. We will return to this point shortly.Consider, next, chemotactic migration with p > / (cid:104) x (cid:105) (cid:54) = µ x t . Thus, we focuson (cid:104) x (cid:105) /t , which is a measure of mobility (“diffusivity”) of the cells, and study its dependenceon σ and ¯ θ . We first note that chemotactic migration does have a transition point θ ∗ ; seeFig. 2(d) that presents the results for p = 0 .
8. But, before analyzing the characteristics of θ ∗ , let us consider the effect of the directionality of cell migration on (cid:104) x (cid:105) , which is regulatedby p . Using the expressions for the probabiliies of motion and the drift velocities, we obtain, (cid:104) x (cid:105) = µ x t + v x t = 2 (cid:104) sin θ (cid:105) (1 − p ) t/ t (cid:104) δ sin θ (4 p − /D (cid:105) ] .To understand the effect of p we first considered uncorrelated cellular media and varied both p and ¯ θ . For a given ¯ θ (cid:104) x (cid:105) /t increases with increasing p ; see Fig. 2(e). Depending on p , thecorrelations may increase or decrease (cid:104) x (cid:105) /t ; see Fig. 2(f). Thus, the correlations are indeedrelevant and regulate (cid:104) x (cid:105) . Moreover, regardless of the correlations’ type ( H < . H > . (cid:104) x (cid:105) and θ ∗ significantly.To study the effect of σ , we computed (cid:104) x (cid:105) for p = 0 . θ . As Fig.3(a) shows, for large σ all (cid:104) x (cid:105) /t converge to the same eventual value. But, more interestinglyand contrary to the limit p = 1 /
4, the dependence of (cid:104) x (cid:105) /t on σ is not trivial. It initiallydecreases and then increases. Such non-monotonic behavior may be understood by noting thatthe average probability (cid:104) r (cid:105) of moving in the positive x direction does not vary monotonicallywith σ . Figure 3(a) also indicates that, although all the (cid:104) x (cid:105) /t eventually converge to the samevalue, the shape of their variations depends on the value of σ .One goal of this paper is to understand the effect of the orientations’ correlations on theresults, and as Fig. 3(b), which presents the dependence of (cid:104) x (cid:105) /t on σ for various Hurstexponents H in the limits ¯ θ = 0 and p = 0 .
9, indicates, the effect is completely nontrivial. AsFig. 3(b) indicates, Not only is the growth of (cid:104) x (cid:105) /t with σ completely different from thosein Fig. 3(a), it also indicates that the cells advance more slowly in a cellular environment9igure 3: Dependence of (cid:104) x (cid:105) /t on (a) σ and ¯ θ for p = 0 .
9. Increasing σ decreases the differencesbetween the results for various ¯ θ ; (b) σ and the Hurst exponent H for ¯ θ = 0, showing that cellsmove slower in environments with positive correlations ( H > . σ and H for ¯ θ = π/ p = 0 .
75. The correlations affect the mobility, but the nonmonotonic behavior of (cid:104) x (cid:105) /t remains unchanged, and (d) H for p = 0 . H > . p = 0 . θ = π/
8. In this case, the correlations give rise once again to nonmonotonic dependenceof (cid:104) x (cid:105) /t on σ , hence playing a major role in regulating (cid:104) x (cid:105) . As Fig. 3(d) indicates, thecorrelations may increase or decrease (cid:104) x (cid:105) /t , depending on σ and ¯ θ .As discussed earlier, the limits p = 1 / θ ∗ = π/ θ ∗ as the dividing angle at which a transition fromlow-risk (TACS2) to high-risk (TACS3) metastasis occurs are important. As Fig. 4(a) shows,for every p in a noncorrelated medium θ ∗ vanishes with increaing σ , hence indicating that in acellular medium with a rather wide σ , the risk of metastasis increases significantly for almost10igure 4: Dependence of θ ∗ on (a) σ and p , showing that varying p slightly changes the magni-tude of θ ∗ and its dependence on σ , but the main qualitative behavior remains almost the samefor all p ; (b) σ and H for p = 0 .
95; (c) p and σ , indicting that directionality of the motion mayincrease or decrease θ ∗ , and (d) p for various H .all the fiber directions and migration modes. This provides a framework for classification of theTACSs. Equally importantly, Fig. 4(b) indicates that the correlations have a nontrivial effecton θ ∗ . Figures 4(a) and 4(b) both indicate that the transition orientation θ ∗ is not universaland does depend on the details of the oritentation distribution and other parameters.Since cells may migrate in various modes, the question of how they alter θ ∗ is also important.As Fig. 4(c) indicates, directionality of the tumor advancement initially increases and thendecreases θ ∗ . The significance of these results is in demonstrating that, while for randommotion of the cells the fibers’ alignment is the main contributing factor to the magnitude ofthe invasion length, one needs a more precise and better defined framework for chemotaxticmigration, as the physical parameters that affect the phenomenon, such as the extent and typeof the correlations between the fibers’ orientations and the migration mode, play major roles.11igure 5: Dependence of (cid:104) x (cid:105) /t on σ and p for (a) ¯ θ = 0, and (b) ¯ θ = π/ p on the mobility (cid:104) x (cid:105) /t . Figures 5(a) and5(b) present, respectively, the results for ¯ θ = 0 and π/
8. Thus, even a small change in themean angle of the fibers’ orientations gives rise to remarkable changes in the mobility. Inparticular, decreasing the probability p of moving forward for ¯ θ = π/ µ x with σ . IV. SUMMARY AND CONCLUSIONS
We described what we believe is the first model that introduces a dividing angle for clas-sification of tumor-associated collagen signature (TACS), demonstrating how various physicalfeatures, such as the spatial distribution of the extre-cellumar matrix’s fibers’ orientations andthe cells’ migration dynamics regulate the cell invasion length and, therefore, the metastaticrisk. The distribution of the orientations of the fibers plays a crucial role, and may promoteor inhibit cell migration. The cells’ migration mode, ranging from random walks to entirelybiased walks, also affects the invasion length. Thus, the three factors should be consideredtogether for complete classification of the TACSs. This may explain why classification of tumorenvironment based solely on the fibers’ alignment has not proven to be fruitful for all the cases.We emphasize that cell migration is a complex process regulated by biochemical communi-cations between the cells and various constituents of the host tissue, as well as the biophysicalinteractions. Our goal in this paper was to investigate the effect of physical interactions. Gain-ing a comprehensive understanding of the ECM regulation of cell migration and the metastasisshould integrate all the chemical and physical aspects.12 [email protected] ‡ [email protected][1] C.L. Chaer and R.A. Weinberg, A perspective on cancer cell metastasis, Science , 1559 (2011).[2] C.A. Klein, The metastasis cascade, Science , 1785 (2008).[3] A. Case et al. , Identification of prognostic collagen signatures and potential thera-peutic stromal targets in canine mammary gland carcinoma, PLOS one , e0180448(2017).[4] F. van Zijl, G. Krupitza, and W. Mikulits, Initial steps of metastasis: cell invasionand endothelial transmigration, Mutat. Res. , 23 (2011).[5] R.J. Petrie, A.D. Doyle, and K.M. Yamada, Random versus directionally persistentcell migration, Nat. Rev. Mol. Cell Biol. , 538 (2009).[6] E.T. Roussos, J.S. Condeelis, and A. Patsialou, Chemotaxis in cancer, Nat. Rev.Cancer , 573 (2011).[7] W.J. Polacheck, I.K. Zervantonakis, and R.D. Kamm, Tumor cell migration in com-plex microenvironments, Cellular Mol. Life Sci. , 1335 (2013).[8] P.-H. Wu, A. Giri, S.X. Sun, and D. Wirtz, Three-dimensional cell migration doesnot follow a random walk, Proc. Natl. Acad. Sci. USA , 3949 (2014).[9] P.-H. Wu, A. Giri, and D. Wirtz, Statistical analysis of cell migration in 3D usingthe anisotropic persistent random walk model. Nat. Protoc. , 517 (2015).[10] S. Evje, An integrative multiphase model for cancer cell migration under influenceof physical cues from the microenvironment, Chem. Eng. Sci. , 240 (2017).[11] C. Deroulers, M. Aubert, M. Badoual, and B. Grammaticos, Modeling tumor cellmigration: From microscopic to macroscopic models, Phys. Rev. E , 031917(2009). 1312] A.J. Loosley, X.M. O’Brien, J.S. Reichner, and J.X. Tang, Describing directionalcell migration with a characteristic directionality time, PLOS One , e0127425(2015).[13] C. Frantz, K.M. Stewart, and V.M. Weaver, The extracellular matrix at a glance,J. Cell Sci. , 4195 (2010).[14] G. Charras and E. Sahai, Physical influences of the extracellular environment oncell migration, Nat. Rev. Mol. Cell Biol. , 813 (2014).[15] K. Wolf et al. , Physical limits of cell migration: control by ECM space and nucleardeformation and tuning by proteolysis and traction force, J. Cell Biol. , 1069(2013).[16] Y.-J. Liu, M. Le Berre, F. Lautenschlaeger, P. Maiuri, A. Callan-Jones, M. Heuz´e,T. Takaki, R. Voituriez, and M. Piel, Confinement and low adhesion induce fastamoeboid migration of slow mesenchymal cells, Cell , 659 (2015).[17] A. Rahman-Zaman, S. Shan, and C.A. Reinhart-King, Cell migration in microfabri-cated 3D collagen microtracks is mediated through the prometastatic protein girdin,Cell. Mol. Bioeng. , 1 (2018).[18] A. Rahman, S.P. Carey, C.M. Kraning-Rush, Z.E. Goldblatt, F. Bordeleau, M.C. Lampi, D. Y. Lin, A.J. Garc´ıa, and C.A. Reinhart-King, Vinculin regulatesdirectionality and cell polarity in two- and three-dimensional matrix and three-dimensional microtrack migration, Mol. Biol. Cell , 1431 (2016).[19] S.P. Carey, A. Rahman, C.M. Kraning-Rush, B. Romero, S. Somasegar, O.M. Torre,R.M. Williams, and C.A. Reinhart-King, Comparative mechanisms of cancer cellmigration through 3D matrix and physiological microtracks, Am. J. Physiol. CellPhysiol. , C436 (2015).[20] C.M. Kraning-Rush, S.P. Carey, M.C. Lampi, and C.A. Reinhart-King, Microfab-ricated collagen tracks facilitate single cell metastatic invasion in 3D, Integr. Biol.(Camb.) , 606 (2013). 1421] A.D. Doyle, F.W. Wang, K. Matsumoto, and K.M. Yamada, One-dimensional to-pography underlies three-dimensional fibrillar cell migration, J. Cell Biol. , 481(2009).[22] C.M. Kraning-Rush and C.A. Reinhart-King, Controlling matrix stiffness and to-pography for the study of tumor cell migration, Cell Adh. Migr. , 274 (2012).[23] L.A. Hapach, J.A. VanderBurgh, J.P. Miller, and C.A. Reinhart-King, Manipula-tion of in vitro collagen matrix architecture for scaffolds of improved physiologicalrelevance, Phys. Biol. , 061002 (2015).[24] M.C. Lampi, M. Guvendiren, J.A. Burdick, and C.A. Reinhart-King, Photopat-terned hydrogels to investigate the endothelial cell response to matrix stiffness het-erogeneity, ACS Biomater. Sci. Eng. , 3007 (2017).[25] F. Bordeleau, L.N. Tang, and C.A. Reinhart-King, Topographical guidance of 3Dtumor cell migration at an interface of collagen densities, Phys. Biol. , 065004(2013).[26] S. Rhee, J.L. Puetzer, B.N. Mason, C.A. Reinhart-King, and L.J. Bonassar, 3Dbioprinting of spatially heterogeneous collagen constructs for cartilage tissue engi-neering, ACS Biomater. Sci. Eng. , 1800 (2016).[27] K.R. Levental et al. , Matrix crosslinking forces tumor progression by enhancingintegrin signaling, Cell , 891 (2009).[28] M.J. Paszek et al. , Tensional homeostasis and the malignant phenotype, CancerCell , 241 (2005).[29] T.A. Ulrich, E.M. de Juan Pardo, and S. Kumar, The mechanical rigidity of theextracellular matrix regulates the structure, motility, and proliferation of gliomacells, Cancer Res. , 4167 (2009).[30] A. Pathak and S. Kumar, Independent regulation of tumor cell migration by matrixstiffness and confinement, Proc. Natl. Acad. Sci. USA , 10334 (2012).1531] F. Bordeleau et al. , Matrix stiffening promotes a tumor vasculature phenotype,Proc. Natl. Acad. Sci. USA , 492 (2017).[32] M.C. Lampi, C.J. Faber, J. Huynh, F. Bordeleau, M. R. Zanotelli, and C.A. Reinhart-King, Simvastatin ameliorates matrix stiffness-mediated endothelial monolayer dis-ruption, PLOS One , e0147033 (2016).[33] S. Lin, L.A. Hapach, C. Reinhart-King, and L. Gu, Towards tuning the mechani-cal properties of three-dimensional collagen scaffolds using a coupled fiber-matrixmodel, Materials , 5376 (2015).[34] F. Bordeleau, J.P. Califano, Y.L.N. Abril, B.N. Mason, D.J. LaValley, S.J. Shin,R.S. Weiss, and C.A. Reinhart-King, Tissue stiffness regulates serine/arginine-richprotein-mediated splicing of the extra domain B-fibronectin isoform in tumors, Proc.Natl. Acad. Sci. USA , 8314 (2015).[35] B.N. Mason, A. Starchenko, R.M. Williams, L.J. Bonassar, and C.A. Reinhart-King, Tuning three-dimensional collagen matrix stiffness independently of collagenconcentration modulates endothelial cell behavior, Acta Biomater. , 4635 (2013).[36] J.C. Kohn, D.W. Zhou, F. Bordeleau, A.L. Zhou, B.N. Mason, M.J. Mitchell, M.R.King, and C.A. Reinhart-King, Cooperative effects of matrix stiffness and fluidshear stress on endothelial cell behavior, Biophys. J. , 471 (2015).[37] A. Wood, Contact guidance on microfabricated substrata: the response of teleost finmesenchyme cells to repeating topographical patterns, J. Cell Sci. , 667 (1988).[38] A. Webb, P. Clark, J. Skepper, A. Compston, and A. Wood, Guidance of oligoden-drocytes and their progenitors by substratum topography, J. Cell Sci. , 2747(1995).[39] N. Gomez, S. Chen, and C.E. Schmidt, Polarization of hippocampal neurons withcompetitive surface stimuli: contact guidance cues are preferred over chemical lig-ands, J. Roy. Soc. Interface , 223 (2007).1640] A.I. Teixeira, G.A. Abrams, P.J. Bertics, C.J. Murphy, and P.F. Nealey, Epithelialcontact guidance on well-defined micro- and nanostructured substrates, J. Cell Sci. , 1881 (2003).[41] W. Loesberg, J. Te Riet, F. van Delft, P. Scon, C. Figdor, S. Speller, J. van Loon,X. Walboomers, and J. Jansen, The threshold at which substrate nanogroove di-mensions may influence fibroblast alignment and adhesion, Biomaterials , 3944(2007).[42] N. Nakatsuji and K.E. Johnson, Ectodermal fragments from normal frog gastrulaecondition substrata to support normal and hybrid mesodermal cell migration invitro, J. Cell Sci. , 49 (1984).[43] N. Nakatsuji and K.E. Johnson, Experimental manipulation of a contact guidancesystem in amphibian gastrulation by mechanical tension, Nature , 453 (1984).[44] A. Ray, Z.M. Slama, R.K. Morford, S.A. Madden, and P.P. Provenzano, Enhanceddirectional migration of cancer stem cells in 3D aligned collagen matrices, Biophys.J. , 1023 (2017).[45] A. Ray, R. Morford, N. Ghaderi, D. Odde, and P. Provenzano, Dynamics of 3Dcarcinoma cell invasion into aligned collagen Integ. Biol. (Camb.) , 100 (2018).[46] M. Grossman, N. Ben-Chetrit, A. Zhuravlev, R. Afik, E. Bassat, I. Solomonov, Y.Yarden, and I. Sagi, Tumor cell invasion can be blocked by modulators of collagenfibril alignment that control assembly of the extracellular matrix, Cancer Res. ,4249 (2016).[47] A. Parekh and A.M. Weaver, Regulation of cancer invasiveness by the physicalextracellular matrix environment, Cell Adh. Migr. , 288 (2009).[48] P.P. Provenzano, D.R. Inman, K.W. Eliceiri, S.M. Trier, and P.J. Keely, Contactguidance mediated three-dimensional cell migration is regulated by Rho/ROCK-dependent matrix reorganization, Biophys. J. , 5374 (2008).1749] M. Sidani, J. Wycko, C. Xue, J. E. Segall, and J. Condeelis, Probing the microen-vironment of mammary tumors using multiphoton microscopy, J. Mammary GlandBiol. Neoplasia , 151 (2006).[50] W. Han et al. , Oriented collagen fibers direct tumor cell intravasation, Proc. Natl.Acad. Sci. USA , 11208 (2016).[51] D. Truong, J. Puleo, A. Llave, G. Mouneimne, R.D. Kamm, and M. Nikkhah, Breastcancer cell invasion into a three dimensional tumor-stroma microenvironment, Sci.Rep. , 34094 (2016).[52] S.P. Carey, Z.E. Goldblatt, K.E. Martin, B. Romero, R.M. Williams, and C.A.Reinhart-King, Local extracellular matrix alignment directs cellular protrusion dy-namics and migration through Rac1 and FAK, Integr. Biol. , 821 (2016).[53] A.L. Bauer, T.L. Jackson, and Y. Jiang, Topography of extracellular matrix medi-ates vascular morphogenesis and migration speeds in angiogenesis, PLOS Comput.Biol. , e1000445 (2009).[54] V. Gorshkov, V. Privman, and S. Libert, Lattice percolation approach to 3D mod-eling of tissue aging, Physica A , 207 (2016).[55] M.W. Conklin, J.C. Eickho, K.M. Riching, C.A. Pehlke, K.W. Eliceiri, P.P. Proven-zano, A. Friedl, and P.J. Keely, Aligned collagen is a prognostic signature for survivalin human breast carcinoma. Am. J. Pathol. , 1221 (2011).[56] V. Pavithra, R. Sowmya, S.V. Rao, S. Patil, D. Augustine, V. Haragannavar, andS. Nambiar, Tumor-associated collagen signatures: An insight, World J. Dent. ,224 (2017).[57] A. Brabrand, I.I. Kariuki, M.J. Engstrom, O.A. Haugen, L.A. Dyrnes, B.O. Asvold,M.B. Lilledahl, and A.M. Bonfin, Alterations in collagen fibre patterns in breastcancer. A premise for tumour invasiveness? APMIS , 1 (2015).[58] J.S. Bredfeldt, Y. Liu, M.W. Conklin, P.J. Keely, T.R. Mackie, and K.W. Eliceiri,Automated quantification of aligned collagen for human breast carcinoma prognosis,18. Pathol. Informatics , 28 (2014).[59] A.J. Ford and P. Rajagopalan, Extracellular matrix remodeling in 3D: implicationsin tissue homeostasis and disease progression, Nanomed. Nanobiotechnol. (4),(2018).[60] K.M. Riching et al. , 3D collagen alignment limits protrusions to enhance breastcancer cell persistence, Biophys. J. , 2546 (2014).[61] P. Friedl and K. Wolf, Tube travel: The role of proteases in individual and collectivecancer cell invasion, Cancer Res. , 7247 (2008).[62] E. Schnell, K. Klinkhammer, S. Balzer, G. Brook, D. Klee, P. Dalton, and J. Mey,Guidance of glial cell migration and axonal growth on electrospun nanofibers of poly-epsilon-caprolactone and a collagen/poly-epsilon-caprolactone blend, Biomaterials , 3012 (2007).[63] A.A. Saberi, Recent advances in percolation theory and its applications, Phys. Rep. , 1 (2015).[64] T.H. Keitt, Spectral representation of neutral landscapes, Landscape Ecol. , 479(2000).[65] B.B. Mandelbrot and J.W. van Ness, Fractional Brownian motion, fractional Guas-sian noise, and their applications. SIAM Rev. , 422 (1968).[66] V. Tejedor, O. ´Benichou, R. Voituriez, R. Jungmann, F. Simmel, C. Selhuber-Unkel, L. B. Oddershede, and R. Metzler, Quantitative analysis of single particletrajectories: Mean maximal excursion method, Biophys. J. , 1364 (2010).[67] S. Weber, A.J. Spakowitz, and J.A. Theriot, Bacterial chromosomal loci move sub-diffusively through a viscoelastic cytoplasm, Phys. Rev. Lett. , 238102 (2010).[68] G.R. Kneller, K. Baczynski, and M. Pasenkiewicz-Gierula, Consistent picture of lat-eral subdiffusion in lipid bilayers: Molecular dynamics simulation and exact results,J. Chem. Phys. , 141105 (2011). 1969] F. Ghasemi, J. Peinke, M. Sahimi, and M.R. Rahimi Tabar, Regeneration of stochas-tic processes: An inverse method. Europ. Phys. J. B , 411 (2005).[70] F. Ghasemi, M., Sahimi, J., Peinke, and M.R. Rahimi Tabar, Analysis of non-stationary data for heart-rate fluctuations in terms of drift and diffusion coefficients.J. Biol. Phys. , 117 (2006).[71] H.G. Othmer and A. Stevens, Aggregation, blowup, and collapse: The ABC’s oftaxis in reinforced random walks, SIAM J. Appl. Math. , 1044 (1997).[72] E.F. Keller and L.A. Segel, Model for chemotaxis, J. Theor. Biol.30