Applied Bionics and Biomechanics

Volume 2019, Article ID 4892709, 13 pages

https://doi.org/10.1155/2019/4892709

## Functional Epithelium Remodeling in Response to Applied Stress under *In Vitro* Conditions

Faculty of Technology and Metallurgy, University of Belgrade, Karnegijeva 4, Belgrade, Serbia

Correspondence should be addressed to Ivana Pajic-Lijakovic; sr.ca.gb.fmt@avi

Received 16 October 2018; Revised 14 February 2019; Accepted 21 February 2019; Published 19 May 2019

Academic Editor: Agnès Drochon

Copyright © 2019 Ivana Pajic-Lijakovic and Milan Milivojevic. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

Mathematical modeling is often used in tissue engineering in order to overcome one of its major challenges: transformation of complex biological and rheological behaviors of cells and tissue in a mathematically predictive and physically manipulative engineering process. The successive accomplishment of this task will greatly help in quantifying and optimizing clinical application of the tissue engineering products. One of the problems emerging in this area is the relation between resting and migrating cell groups, as well as between different configurations of migrating cells and viscoelasticity. A deeper comprehension of the relation between various configurations of migrating cells and viscoelasticity at the supracellular level represents the prerequisite for optimization of the performance of the artificial epithelium. Since resting and migrating cell groups have a considerable difference in stiffness, a change in their mutual volume ratio and distribution may affect the viscoelasticity of multicellular surfaces. If those cell groups are treated as different phases, then an analogous model may be applied to represent such systems. In this work, a two-step Eyring model is developed in order to demonstrate the main mechanical and biochemical factors that influence configurations of migrating cells. This model could be also used for considering the long-time cell rearrangement under various types of applied stress. The results of this theoretical analysis point out the cause-consequence relationship between the configuration of migrating cells and rheological behavior of multicellular surfaces. Configuration of migrating cells is influenced by mechanical and biochemical perturbations, difficult to measure experimentally, which lead to uncorrelated motility. Uncorrelated motility results in (1) decrease of the volume fraction of migrating cells, (2) change of their configuration, and (3) softening of multicellular surfaces.

#### 1. Introduction

One of the key challenges in tissue engineering is to consider tissue remodeling by collective cell migration in response to applied stress and simulate a tissue natural environment under *in vitro* conditions [1–3]. Deeper understanding of long-time cell rearrangement is a prerequisite in the development of functional soft tissue for potential applications in disease modeling and replacing damaged tissues [4]. The intact epithelium plays an important role in the functioning of various organs, and its ability to remodel under various stress conditions would define the level of success in tissue engineering of some organs such as the bladder and the skin.

The main goal of this contribution is to consider cell long-time rearrangement via collective cell migration under stress conditions such as (1) cell aggregate rounding after uniaxial compression between parallel plates [5, 6] and (2) cell aggregate flow subjected to one-dimensional stretching forces using micropipette aspiration [7]. In both cases, cell long-time rearrangement is influenced by external stress, locally or globally. It occurs via collective cell migration within the aggregate 3D surface region or its part driven by tissue surface tension. Consequently, induced volumetric and surface changes could be described by the Young-Laplace law [6]. These systems are analyzed from the standpoint of bionic, as the science that is formed from the combination of various natural and engineering science concepts [8]. Consequently, we discussed the fundamental interrelations between configuration changes of migrating cells and viscoelasticity of multicellular systems at the macroscopic level. Deeper understanding of the multiscale nature of viscoelasticity is necessary in designing the optimal performances of artificial epithelium.

Cell relaxations during and after applying stress occur at various time scales. The time scale of minutes corresponds to single-cell relaxation primarily by adaptation of adhesion complexes while the time scale of hours corresponds to collective cell migration. Guevorkian et al. considered the cell aggregate flow inside the pipette under pressure [7]. They indicated that the cell aggregate responds via short- and long-time pulsated contractions. Short-time contractions correspond to a few minutes and are induced by single-cell contractions. The long-time contractions correspond to tens of minutes and are induced by collective cell migration. These long-time pulsated contractions could be correlated with a change in the configuration of migrating cells. Cell aggregate compression between parallel plates also provokes the organized pattern of cell migration during aggregate rounding in order to minimize the aggregate surface free energy [5, 6, 9–12]. Pajic-Lijakovic and Milivojevic [13] modeled the experimental data of Mombach et al. [5] and pointed that aggregate shape changes take place during successive long-time relaxation cycles. These cycles have various relaxation rates per cycle. The relaxation rates per cycles are not random, but they have a tendency to gather around two or three values indicating an organized cell migration pattern. Every relaxation rate could be related to the various scenarios of cell migration. Three scenarios were considered as follows: (1) most of the cells migrate all the time, (2) some cell groups migrate while the others (at the same time) stay in the resting state, and (3) cells have successive migrating and resting periods in which most of the cells firstly migrate and then stay in the resting state. The average duration of the single relaxation cycle is about 1-2 h [13]. We correlated these scenarios with various configurations of migrating cells. Mombach et al. pointed to exponential changes in the aggregate shape from ellipsoidal to spherical [5]. Consequently, the linear nature of long-time cell rearrangement obtained experimentally at a macroscopic level has been modeled by applying the Eyring transition state theory by Marmottant et al. [6] and Pajic-Lijakovic and Milivojevic [13]. Cell surface rearrangement could be treated as a thermodynamic system close to equilibrium at the macroscopic level. However, cell surface rearrangement considered at a mesoscopic level has been treated as thermodynamic systems far from equilibrium [14]. It is in accordance with the fact that internal fluctuations, which are significant during thermodynamic system structural ordering at the mesoscopic level, could be neglected at the macroscopic level [15].

Viscoelasticity depends on the configurations of migrating cells. Migrating cell clusters are much stiffer than resting ones due to the accumulation of contractile energy. These contractions induce the generation of prestress. Lange and Fabry reported that cytoskeletal prestress causes adherent cells to stiffen [16]. Lange and Fabry reported that muscle cells can change their elastic modulus by over one order of magnitude from less than 10 kPa in a relaxed (resting) state to around 200 kPa in a fully activated (migrating) state [16]. Consequently, the multicellular surfaces could be treated as a two-phase pseudoblend from the mechanical standpoint [14]. The migrating pseudophase represents the dispersion within the resting one. The influence of configurations of migrating cells on the viscoelasticity of multicellular systems at a mesoscopic level has been discussed in the context of the mechanical coupling modes [14]. They reported that cell migration within a large number of small clusters corresponds to series mode coupling, while cell migration as monolayer sheets corresponds to parallel mode coupling. Consequently, mode coupling should be related to the biointerface size between migrating and resting cell pseudophases.

The shape of migrating cell groups could vary from small cell clusters to monolayer sheets depending on cell types and microenvironmental conditions [17–19]. Mikami et al. discussed collective cell migration of stratified epithelial cells toward the wound in the form of monolayer sheets [20]. All epithelial cells within the sheet migrate, maintaining cell-cell adhesions [21]. Migrating cell sheets slide over the surrounding cell layers in the resting state [18, 22]. The number of sheets and their sizes depend on the size, shape, and depth of injury [22]. Cell organization in the form of migrating cell sheets and their sliding over the surrounding unperturbed cell layers of epithelium pointed to the ordered lamellar structure. Friedl and Alexander considered collective cell migration during cancer invasion and metastases [17]. They concluded that some tumor types could migrate within partially connected strands through surrounding tissue while others could migrate in the forms of monolayer sheets or small cell clusters. Some other cell types could also migrate within small clusters through surrounding tissue [17].

We expand previous considerations proposed by Pajic-Lijakovic and Milivojevic [13] and formulate modified a two-step Eyring model for describing (1) resting-to-migrating cell state transition and vice versa and (2) configuration changes of migrating cells from small clusters to monolayer sheets. Obtained configuration changes of migrating cells should be related to the viscoelasticity of the multicellular surface based on mechanical coupling modes. Pajic-Lijakovic and Milivojevic [14] considered cell surface rearrangement at a mesoscopic level and proposed (1) series mode coupling for the surface parts in which cell migrates in the form of small clusters and (2) parallel mode coupling for the surface parts in which cell migrates in the form of monolayer sheets. Here, we expand this consideration obtained at the mesoscopic level to the macroscopic level by formulating mixed, series-parallel, mode coupling. Mixed mode coupling accounts for both fractions of cells (migrating and resting) coupled in series and in parallel.

#### 2. Phenomenological Background of the Model Based on Experiments of Cell Aggregate Rounding

Experimental data for the aggregate shape relaxation after uniaxial compression, considered here, shows the important feature obtained from the data fluctuations. These fluctuations clearly point to an ordered relaxation trend in the form of successive relaxation cycles. The ordered fluctuation trend in the form of long-time pulsated contractions was also obtained during cell aggregate flow inside the pipette under pressure [7]. Accordingly, the aggregate shape long-time relaxation after compression for the ^{th} cycle has been expressed by Pajic-Lijakovic and Milivojevic as (where is the deformation parameter for during the ^{th} relaxation cycle equal to , is the aggregate aspect ratio, is the initial value for the deformation parameter, and is the relaxation rate for the ^{th} cycle) [13]. The relaxation rates are not random but grouped around two or three values indicating an organized cell migrating pattern: (1) , most of the cells migrate (the volume fraction of migrating cells is ), (2) , most of the cells stay in the resting state (the volume fraction of resting cells is ), and (3) , some cell groups migrate while the others, at the same time, stay in the resting state. The relaxation rate per cycle should be related to the volume fraction and configuration of migrating cells, i.e., . However, the formulation of this relationship needs the additional surface characterization as the surface stiffness distribution and the rate of its change.

A significant difference in cell stiffness between migrating and resting cell groups indicates that volume fraction of migrating cells and their distribution could influence the long-time rheological behavior of multicellular surfaces. This aspect of cell surface rearrangement could be treated by the analogy with physics in the form of a two-phase blend composed of migrating and resting cell pseudophases. Migrating pseudophase could form various configurations: (1) small clusters, (2) monolayer sheets, and (3) mixed configurations composed of both dispersion of small clusters and lamellar structural parts [17, 21]. For mixed configurations, the volume fraction of migrating cells could be expressed as follows: where is the part of migrating cells in the form of small clusters equal to and is the part of migrating cells in the form of monolayer sheets equal to and is the number of cells in the surface region. Cell aggregate compression induces the perturbation of the aggregate surface region consisting of active cells which undergo to short- and long-time relaxations. Short-time relaxation describes relaxation of cell volumes and cell packing state within the surface region (the time scale of minutes) [5]. Long-time relaxation describes surface relaxation caused by collective cell migration (the time scale of hours). Surface tension is the main mechanism which influences the aggregate rounding. It represents the “driving force” for collective cell migration. Some of the active cells within this region become active and migrate in order to decrease the aggregate surface as well as the surface free energy. Cell packing density and cell volumes relax quickly and become constant during aggregate rounding. Long-time relaxation leads to change of the thickness and surface of the surface region while the volume remains constant. Total number of cells in the surface region consists of migrating and resting cells. The average volume fraction of resting cells is equal to where the volume fraction of resting cells is equal to , while is the volume fraction of resting cells in the surroundings of migrating cells and is the volume fraction of resting cells in the surroundings of migrating cells .

The resting pseudophase is in the surroundings of the migrating pseudophase for both configurations. Higher volume fraction corresponds to (1) higher volume fraction and (2) lower volume fractions and . When the volume fraction is (1) , it means that , and/or when the volume fraction is (2) , it means that . Consequently, we can correlate the volume fraction of the resting pseudophase with the volume fraction of the migrating pseudophase for both configurations in the form of additional condition as which offers the possibility of formulating the volume fractions of resting cells for both configurations as and , while the model parameter is equal to .

Configuration changes of a migrating cell pseudophase take place in order to minimize the interface size between the resting and migrating cell pseudophases. Higher interface size leads to intensive mechanical and biochemical perturbations caused by uncorrelated motility [23]. In the initial phase of cell rearrangement during aggregate rounding, migrating cells form small clusters . However, the increase of the volume fraction of migrating cells leads to increase of the interface size. In order to reduce the interface size as well as the surface free energy, migrating cells could form monolayer sheets instead of small clusters. Consequently, for high volume fraction of migrating cell , cell migration in the form of monolayer sheets becomes dominant, i.e., .

Cell rearrangement could be described by three variables: (1) volume fraction of resting cells , (2) volume fraction of migrating cells in the form of small clusters , and (3) volume fraction of migrating cells in the form of monolayer sheets and could be treated as a two-step process. Configuration changes of migrating cells were shown schematically in Figure 1.