Response of cells under various stress conditions

The response of cells under various mechanical stress conditions accounts for the interplay between various subcellular processes which influence the rate of cell spreading. Moreover, this response is a multi-scale temporal process. The cytoskeleton remodeling and the change in the state of cell adhesion contacts that lead to a cell shape change occurs on a time scale of minutes (Wottawah et al., Reference Wottawah, Schinkinger, Lincoln, Ananthakrishnan, Romeyke, Guck and Kaes2005). For example, it is known that the epithelial MCF-10A cells in the monocultured spheroid core region are rounder, while the cells in the surface region are elongated (Devanny et al., Reference Devanny, Vancura and Kaufman2021), and this time scale for changes in cell shape corresponds to cell polarization (Alert et al., Reference Alert, Blanch-Mercader and Casademunt2019) and cadherin turnover time (Lee and Wolgemuth, Reference Lee and Wolgemuth2011). A time scale of several tens of minutes corresponds to gene expression (Petrungaro et al., Reference Petrungaro, Morelli and Uriu2019) and cell persistence time (Mc Cann et al., Reference Mc Cann, Kriebel, Parent and Losert2010), while the CCM takes place on a time scale of hours. Cell division as a possible cause of cell rearrangement is neglected at the time scale of hours, because it occurs on much longer time scales, that is, days. Consequently, the movement of cells, the resulted mechanical strains (volumetric and shear), and the accumulation of normal and shear residual cell stresses occur at a time-scale of hours, while the relaxation of these stresses occurs at a time-scale of minutes (Pajic-Lijakovic and MIlivojevic, Reference Pajic-Lijakovic and Milivojevic2020b, Reference Pajic-Lijakovic and Milivojevic2020c). Cell movement induces successive short-time mechanical stress relaxation cycles under constant strain per cycle, while the strain change occurs at a time-scale of hours. Schematic presentation of short-time stress relaxation cycles caused by CCM, which results in the residual stress accumulation during movement of epithelial-like collectives, is presented in Figure 2.

Figure 2. Cell residual stress accumulation within migrating epithelial-like collectives: schematic presentation.

While mechanical stress enhances the movement of some cell types, it has no effect or reduces the movement of others. For example, the compressive stress of 773 Pa suppresses the movement of MCF-10A and MCF-7 cells (Tse et al., Reference Tse, Cheng, Tyrrell, Wilcox-Adelman, Boucher, Jain and Munn2012). This stress corresponds to the compressive stress generated within breast tissue by cell growth (Kalli and Stylianopoulos, Reference Kalli and Stylianopoulos2018). The normal mechanical stress of a several hundreds of Pa can be induced by collective movement of epithelial cells (Tambe et al. (Reference Tambe, Croutelle, Trepat, Park, Kim, Millet, Butler and Fredberg2013)). In contrast, this stress is capable of enhancing the movement of highly aggressive 4T1 and MDA-MB-231 cells, as well as, 67NR cells (Tse et al., Reference Tse, Cheng, Tyrrell, Wilcox-Adelman, Boucher, Jain and Munn2012). Riehl et al. (Reference Riehl, Kim, Lee, Duan, Yang, Donahue and Lim2020) considered and compared responses of MDA-MB-231 and MDA-MB-468 cancer cells, as well as MCF-10A epithelial cells under shear stress of 1.5 Pa. The shear stress can be induced by interstitial flow (Riehl et al., Reference Riehl, Kim, Lee, Duan, Yang, Donahue and Lim2020) or by collective movement of epithelial cells (Tambe et al. (Reference Tambe, Croutelle, Trepat, Park, Kim, Millet, Butler and Fredberg2013)). While the shear stress stimulates movement of the MDA-MB-231 cells along the flow direction, this stress has no impact on the movement of MDA-MB-468 cells, and even reduces the movement of MCF-10A epithelial cells. Higher compressive stress can suppress movement of epithelial MCF-10A cells and induce the cell jamming state transition (Grosser et al., Reference Grosser, Lippoldt, Oswald, Merkel, Sussman, Renner, Gottheil, Morawetz, Fuhs, Xie, Pawlizak, Fritsch, Wolf, Horn, Briest, Aktas, Manning and Käs2021; Pajic-Lijakovic and Milivojevic, Reference Pajic-Lijakovic and Milivojevic2021b). Consequently, the behavior of co-cultured multicellular systems should be considered in the context of cell mechano-sensitivity.

Several parameters have been discussed in the context of cell mechano-sensitivity such as: (1) single-cell stiffness (Tse et al., Reference Tse, Cheng, Tyrrell, Wilcox-Adelman, Boucher, Jain and Munn2012), (2) level of E-cadherin (Mohammed et al., Reference Mohammed, Park, Weitz and DA2021), and (3) the mechanism of cell movement (Guzman et al., Reference Guzman, Avard, Devanny, Kweon and Kaufman2020; Devanny et al., Reference Devanny, Vancura and Kaufman2021). Tse et al. (Reference Tse, Cheng, Tyrrell, Wilcox-Adelman, Boucher, Jain and Munn2012) reported that stiffer cells are less mechano-sensitive. Rudzka et al. (Reference Rudzka, Spennati, McGarry, Chim, Neilson, Ptak, Munro, Kalna, Hedley, Moralli, Green, Mason, Blyth, Mullin, Yin and Olson2019) and Riehl et al. (Reference Riehl, Kim, Bouzid and Lim2021) pointed to a correlation between cancer cell stiffness and their invasiveness. The average Young modulus decreases from 1.05 kPa for (less invasive) MCF-7 cells to 0.94 kPa for (more invasive) T47D cells and 0.62 kPa for (the most invasive) MDA-MB-231 cells (Omidvar et al., Reference Omidvar, Tafazzoli-Shadpour, Shokrgozar and Rostami2014). The stiffness of single cells is a product of cell mechanical and biochemical interactions with their surroundings which provoke various internal molecular mechanisms within single cells themselves responsible for their adaptation such as the rearrangement of their cytoskeletons and change of the strength of cell–cell adhesion contacts. Yousafzai et al. (Reference Yousafzai, Coceano, Mariutti, Ndoye, Amin, Niemela, Bonin, Scoles and Cojoc2016) found that neighboring cells significantly alter cell stiffness. The MDA-MB-231 cells become stiffer when they are in monocultured systems, while HBL-100 and MCF-7 exhibit softer character.

Mohammed et al. (Reference Mohammed, Park, Weitz and DA2021) pointed to E-cadherin-mediated cell–cell adhesion contact as the main cause of the MCF-10A epithelial cell movement reduction and the jamming state transition under the compressive stress. While MDA-MB-231 cells keep filopodia-like movement through dense adhesive surrounding such as collagen I gel, MDA-MB-468 cells prefer movement with blebs (Guzman et al., Reference Guzman, Avard, Devanny, Kweon and Kaufman2020). Enhanced motility of MDA-MB-231 cells under stress could be connected with their mechanism of movement and the adaptability of FAs, while the movement with blebs makes MDA-MB-468 cells more resistant (Riehl et al., Reference Riehl, Kim, Bouzid and Lim2021).

To understand the rearrangement of co-cultured cellular systems, in addition to cell response to various stresses it is also necessary to emphasize the role of surface tension of cell pseudo-phases (i.e., various cell types in contact) accompanied by the interfacial tension between them in cell rearrangement.

Surface tension: the driving force for segregation of co-cultured cellular systems

Tissue surface tension depends on the interplay between single-cell contractility and the state of AJs (Stirbat et al., Reference Stirbat, AMgharbel, Bodennec, Ferri, Mertani, Rieu and Delanoë-Ayari2013; Pajic-Lijakovic et al., Reference Pajic-Lijakovic, Eftimie, Milivojevic and Bordas2023c). Stronger cadherin-mediated cell–cell adhesion contacts, characteristic for the epithelial-like cells, lead to the establishment of higher tissue surface tension in comparison with the mesenchymal-like cells (Devanny et al., Reference Devanny, Vancura and Kaufman2021).

Cell contractility influences the state of AJs and FAs as well as their crosstalk (Zuidema et al., Reference Zuidema, Wang and Sonnenberg2020). Devanny et al. (Reference Devanny, Vancura and Kaufman2021) revealed that contractility plays a fundamentally different role in the cell lines in which the rearrangement is driven primarily by integrins (MDA-MB-468, along with MDA-MB-231 and MDA-MB-436 cells) versus by cadherins (MCF-10A). They emphasized that the cell contractility suppression reduces the tissue surface tension of the epithelial MCF-10A cells. The monocultured MCF-10A cell spheroids treated by 20μM blebbistatin (a molecule capable of suppressing cellular contractility) lose the smooth surface and became more irregular in shape (Devanny et al., Reference Devanny, Vancura and Kaufman2021). Intensive contractility of surface cells enhances the strength of E-cadherin-mediated adhesion contacts and on that base increases the surface tension. In contrast, enhanced contractility of mesenchymal-like cells induces an increase in cell–cell repulsions, which reduce the surface tension (Devanny et al., Reference Devanny, Vancura and Kaufman2021). The surface tension of epithelial MCF-10A spheroids is $ 45\pm 18\;\frac{mN}{m} $, while the surface tension of carcinoma MCF10DCIS.com spheroids is lower and equal to $ 21\pm 9\;\frac{mN}{m} $ (Nagle et al., Reference Nagle, Richert, Quinteros, Janel, Buysschaert, Luciani, Debost, Thevenet, Wilhelm, Prunier, Lafont, Padilla-Benavides, Boissan and Reffay2022). The surface tension of carcinoma F9 WT cell spheroids is significantly lower and equal to $ 4.74\pm 0.28\;\frac{mN}{m} $.

Besides the tissue surface tension, the segregation of co-cultured multicellular systems depends also on the interfacial tension between cell pseudo-phases.

The interfacial tension as a product of the interactions between various cell types

Interfacial tension between cell subpopulations in co-cultured systems depends on the surface tensions of the pseudo-phases in contact and the heterotypic cell–cell interactions. These heterotypic interactions account for biochemical and mechanical interactions between cells. Mechanical interactions have an impact on the stress generation at the biointerface between the cell subpopulations depending on the relative velocity between them (Pajic-Lijakovic et al., Reference Pajic-Lijakovic, Eftimie, Milivojevic and Bordas2023b). Biochemical interactions include cell signaling and gene expression which have a feedback impact on the state of cell–cell and cell-ECM adhesion contacts, migration speed and persistence. Based on our knowledge, an interfacial tension between the subpopulations has not been measured yet. This parameter could be measured by applying some noninvasive technique such as the resonant acoustic rheometry. This technique, which uses the surface capillary waves, has been used for measurement of the interfacial tension within various soft matter systems (Hobson et al., Reference Hobson, Li, Juliar, Putnam, Stegemann and Deng2021). We will provide qualitative analysis of the interfacial tension in comparison with the surface tensions of the subpopulations within the biophysical model.

Interactions between cells in co-cultured multicellular systems depend on the way of the system co-culture. Various methods have been used such as hanging drop, nonadherent surface, spinning flask, and rotating vessel methods, as well as, microfluidic, acoustic, water-in water emulsion, and 3D printing methods (Froehlich et al., Reference Froehlich, Haeger, Heger, Pastuschek, Photini, Yan, Lupp, Pfarrer, Mrowka, Ekkehard Schleußner, Markert and Schmidt2016; Raghavan et al., Reference Raghavan, Mehta, Horst, Ward, Rowley and Mehta2016; Bowers et al., Reference Bowers, Fannin, Thomas and Weis2020; Chae et al., Reference Chae, Hong, Hwangbo and Kim2021). The hanging drop method has been widely used and ensure cell aggregation within droplets under the influence of surface tension and gravitational forces, while the nonadherent surface method includes cell seeding on the nonadherent scaffolds such as polyacrylamide hydrogel or agarose hydrogel (Chae et al., Reference Chae, Hong, Hwangbo and Kim2021). Two cell types can be seeded on the same scaffold (direct co-culture) or on separate scaffolds and then cultivated together (in-direct co-culture) (Jo et al., Reference Jo, Heo, Son, Quan, Hong, Kim, Lee, Han, Park, Lee, Kwon, Park, Chae, Kim, Noh and Moon2021). The MCF-10A cells undergo the EMT when they are surrounded by the MDA-MB-231 cells within a directly co-cultured cellular system (Jo et al., Reference Jo, Heo, Son, Quan, Hong, Kim, Lee, Han, Park, Lee, Kwon, Park, Chae, Kim, Noh and Moon2021). The EMT in this case is stimulated by the inability of MCF-10A cells to establish AJs with the same type of cells (Jo et al., Reference Jo, Heo, Son, Quan, Hong, Kim, Lee, Han, Park, Lee, Kwon, Park, Chae, Kim, Noh and Moon2021). The EMT induces weakening of AJs which leads to a decrease in the surface tension of MCF-10A cell subpopulation that has a feedback impact on cell segregation. The consequence of AJs weakening within mesenchymal cell phenotype is a more intensive cell movement (Barriga and Mayor, Reference Barriga and Mayor2019). However, in in-directly co-cultured MCF-10A/MDA-MB-231 cellular systems, the MCF-10A cells keep their epithelial phenotype (Jo et al., Reference Jo, Heo, Son, Quan, Hong, Kim, Lee, Han, Park, Lee, Kwon, Park, Chae, Kim, Noh and Moon2021).

Signaling of one cell type in co-cultured cellular systems influences the movement of the other. The MCF10A cells secrete laminin-5 and fibronectin, which stimulate the FAs of cancer cells such as the MDA-MB-231 cells, which could also stimulate their movement (Nikkhah et al., Reference Nikkhah, Strobl, Schmelz, Roberts, Zhou and Agah2011). The movement of the MDA-MB-231 cells is enhanced when they are surrounded by the MCF10A cell (Lee et al., Reference Lee, Wu, Jack Rory Staunton, Ros, Longmore and Wirtz2012). Heine et al. (Reference Heine, Lippoldt, Reddy, Katira and Kaes2021) revealed that the migration of MDA-MB-231 cells is increased by the presence of the MDA-MB-436 cells. Small extracellular vesicles derived from MDA-MB-231 promote proliferation and movement in MCF7 cells (Senigagliesi et al., Reference Senigagliesi, Samperi, Cefarin, Gneo, Petrosino, Apollonio, Caponnetto, Sgarra, Collavin, Cesselli, Casalis and Parisse2022).

In the following, we briefly discuss some experimental studies that connect the surface tension and the various types of segregation phenomena that can be observed in multicellular systems.

Segregation of co-cultured multicellular systems: various scenarios

Several scenarios can be considered in the context of the rearrangement of co-cultured cellular systems: complete segregation, partial segregation, and mixed segregation, as shown in Figure 3:

Figure 3. Segregation of co-cultured breast cell spheroids: various scenarios.

The scenarios of cell segregation can result by two mechanisms: interfacial tension between the subpopulations accompanied by the interfacial tension gradients and the viscoelasticity caused by CCM. The MCF-10A cells form compact spheroids in monocultured systems, while in co-cultured cellular systems these cells perform partial or complete segregation depending on the surface tension of cancer pseudo-phase as well as the interfacial tension between them. The MDA-MB-436/MCF-10A co-cultured cellular systems, as well as the MDA-MB-231/MCF-10A co-cultured cellular systems, perform complete segregation such that MCF-10A cells, which have higher surface tension, reach out the spheroid core region, while the MDA-MB-231 (or MDA-MB-436) cells with much lower tissue surface tension cover the spheroid surface region (Carey et al., Reference Carey, Starchenko, McGregor and Reinhart-King2013; Devanny et al., Reference Devanny, Vancura and Kaufman2021). However, the MCF-10A cells perform partial segregation in a co-culture with the MDA-MB-468 cells (Devanny et al., Reference Devanny, Vancura and Kaufman2021). These results can be understood when we keep in mind that the MDA-MB-468 cells rather than the MDA-MB-436 cells and the MDA-MB-231 cells are able to establish to some extent cadherin-mediated AJs, which could be additionally stimulated by the presence of the MCF-10A cells in their surroundings (Kenny et al., Reference Kenny, Lee, Myers, Neve, Semeiks, Spellman, Lorenz, Lee, Barcellos-Hoff M, Petersen, Gray and Bissell2007). Co-culture of the MDA-MB-468 cells with other cancer cell types, such as the ZR-75-1 cells, also leads to partial segregation (Devanny et al., Reference Devanny, Vancura and Kaufman2021). The ZR-75-1 cells are capable of establishing both $ {\beta}_1 $ integ0rin and E-cadherin and on that base establish higher surface tension in comparison with MDA-MB-468 cells (Devanny et al., Reference Devanny, Vancura and Kaufman2021). The MDA-MB-468/MDA-MB-157 co-cultured cellular systems perform partial segregation. It is in accordance with the fact that the MDA-MB-157 lacks E-cadherin but can establish cell–cell adhesion contacts by using other cadherin types, and on that base establish higher surface tension (Devanny et al., Reference Devanny, Vancura and Kaufman2021). The MDA-MB-468 cells form mixed spheroids with the MDA-MB-436 cells (Devanny et al., Reference Devanny, Vancura and Kaufman2021).

Cell segregation: the mesoscopic modeling consideration

The phase model is formulated to describe the role of biophysical factors in cell segregation occurred in heterogeneous intra-tumor and inter-tumor cellular systems by considering simplified in vitro model systems such as co-cultured spheroids. The inter- and intra-tumor heterogeneity accounts for the coexistence of cell subpopulations that differ in their genetic, phenotypic, or behavioral characteristics within a given primary tumor, and between a given primary tumor and its metastasis (Martelotto et al., Reference Martelotto, Ng, Piscuoglio, Weigelt and Reis-Filho2014). The phenomenon can be caused by (1) biochemical factors such as genetic and epigenetic factors, as well as, fluctuation in signaling pathways (Marusyk et al., Reference Marusyk, Almendro and Polyak2012) and (2) physical factors such as viscoelasticity and biomechanics of cell subpopulations, as well as, the dynamics at the biointerface between them (Runel et al., Reference Runel, Lopez-Ramirez, Chlasta and Masse2021; Pajic-Lijakovic et al., Reference Pajic-Lijakovic, Eftimie, Milivojevic and Bordas2023b). The phase model, which describes the rearrangement of these three-phase cellular systems, is formulated based on the biological assumptions discussed previously in Sections “Introduction, Morphology of monocultured breast cell spheroids, CCM in co-cultured multicellular spheroids and cell residual stress accumulation” and it is briefly presented in Figure 4. Graphical presentation of the biophysical model is necessary, in order to emphasize clearly how the different components of the model discussed above interact with each other. The blue arrows indicate model parameters which are included into balances of the volume fractions of the pseudo-phases and force balances. These two parts of the model represents a system of equations. The dark red arrow points out that the cell residual stress is included into the force balances. The cell residual stress is complex parameter which consists of several contributions. The one of the contributions represents a cell residual stress generated by CCM, which is discussed from the rheological standpoint and indicated by the purple arrow. The meaning of the variables and parameters is as introduced throughout this Section “The segregation of co-cultured cellular systems: modeling consideration” and Appendices A and B. For the purpose of formulation, the phase model, it is necessary to distinguish the pseudo-phases and characterize their radial distribution within the spheroid. One phase represents the mesenchymal-like cell subpopulation, while the epithelial-like subpopulation consists of migrating and resting cell pseudo-phases. The pseudo-phases are inhomogeneously distributed within the spheroid volume. The volume fractions of these different phases (migrating epithelial $ {\phi}_{em} $, resting epithelial $ {\phi}_{rm} $, and mesenchymal $ {\phi}_c $) satisfy the following:

(1) $$ {\phi}_{em}\left(r,\tau \right)+{\phi}_{rm}\left(r,\tau \right)+{\phi}_c\left(r,\tau \right)=1 $$

where $ r $ is the radial distance in the spheroid and $ \tau $ is the long-time scale (a time scale of hours). The segregation of epithelial and mesenchymal subpopulations caused by CCM occurs on a time scale of hours $ \tau $. On this time scale, the spheroid can be treated as a canonical ensemble such that the total number of cells inside the spheroid is constant. These cell volume fractions relate to the spheroid volume through the following equations:

• • Migrating epithelial cells: $ {\phi}_{em}\left(r,\tau \right)=\frac{1}{dV}{\sum \limits}_{i=1}^{{N_r}^{\ast }}\Delta {V}_{em\;i} $, where $ dV=4{r}^2\pi dr $ is the volume increment of the spheroid, $ \Delta {V}_{em\;i} $ is the volume of the ith migrating epithelial cluster, $ {N_r}^{\ast } $ is the number of migrating epithelial clusters within the spheroid’s volume increment $ dV $.

• • Resting epithelial cells: $ {\phi}_{rm}\left(r,\tau \right)=\frac{1}{dV}{\sum \limits}_{l=1}^{{N_r}^{\ast \ast }}\Delta {V}_{er\;l} $, where $ \Delta {V}_{er\;l} $ is the volume of the lth resting epithelial cluster, $ {N_r}^{\ast \ast } $ is the number of resting epithelial clusters within the spheroid’s volume increment $ dV $.

• • Mesenchymal cells: $ {\phi}_c\left(r,\tau \right)=\frac{1}{dV}{\sum \limits}_{k=1}^{{N_r}^{\ast \ast \ast }}\Delta {V}_{ck} $, where $ \Delta {V}_{ck} $ is the volume of the kth cluster of mesenchymal cells, $ {N_r}^{\ast \ast \ast } $ is the number of mesenchymal clusters within the spheroid’s volume increment $ dV $.

Figure 4. The schematic presentation of the biophysical model.

Consequently, the spheroid volume increment can be expressed as: $ dV={\sum \limits}_{i=1}^{{N_r}^{\ast }}\Delta {V}_{em\;i}+{\sum \limits}_{l=1}^{{N_r}^{\ast \ast }}\Delta {V}_{er\;l}+{\sum \limits}_{k=1}^{{N_r}^{\ast \ast \ast }}\Delta {V}_{ck} $. Three biointerfaces are taken into consideration: (1) the “c-me” biointerface (i.e., the biointerface between migrating epithelial pseudo-phase and mesenchymal pseudo-phase), (2) the “c-re” biointerface (i.e., the biointerface between resting epithelial pseudo-phase and mesenchymal pseudo-phase), and (3) the “re-me” biointerface (i.e., the biointerface between migrating and resting epithelial pseudo-phases).

Accordingly, with the fact that only migrating epithelial cells contribute to the cell segregation, it is necessary to formulate the phase model which accounts for the jamming state transition of epithelial cells and vice versa influenced by the epithelial cell residual stress accumulation and presence of mesenchymal-like cells in their surroundings. The schematic presentation of the biophysical model is presented in Figure 4.

The model we developed consists of three interconnected parts. The main part of the model describes the dynamics of cell segregation at the spheroid level, while the dynamics at the level of cell clusters is formulated in the Appendices A and B. The main part of the model describes the interplay between the changes of the pseudo-phase local volume fractions and force balance equations for the migrating epithelial-like pseudo-phase and mesenchymal pseudo-phase depending on cell signaling, surface characteristics of the pseudo-phases, and the cell residual stress accumulation. The second part of the model formulates the normal and shear cell residual stresses accumulated within the pseudo-phases. Cell normal stress per pseudo-phase consists of isotropic and deviatoric parts. The isotropic part represents the consequence of the surface properties of the pseudo-phases, while the deviatoric part of stress is induced by CCM. Cell shear residual stress represents the product of natural and forced convections. While natural convection is caused by the interfacial tension gradient established at the biointerfaces between the pseudo-phases, the forced convection is induced by CCM (Pajic-Lijakovic et al., Reference Pajic-Lijakovic, Eftimie, Milivojevic and Bordas2023a, Reference Pajic-Lijakovic, Eftimie, Milivojevic and Bordas2023b). The normal and shear residual stress per pseudo-phases are formulated in the Appendix A. The part of cell residual stress (normal and shear) caused by CCM depends on the state of single cells and cell rearrangement. The state of single cells accounts for cell contractility and the state of AJs, while the cell rearrangement depends on cell packing density and cell velocity. These parameters influence the constitutive behaviors of migrating cell collectives. The cell residual stress caused by CCM is formulated in the Appendix B. The surface tensions of the pseudo-phases and interfacial tensions between them also depends on the strength of AJs and cell contractility. The strength of AJs and cell contractility, which represent a product of cell adaptation to microenvironmental conditions, depend on homotypic and heterotypic cell–cell and cell–matrix interactions, cell mechanotransduction, gene expression, and transport of cell metabolites through glycocalyx (Oberleithner and Hausberg, Reference Oberleithner and Hausberg2007; Oberleithner et al., Reference Oberleithner, Peters, Kusche-Vihrog, Korte, Schillers, Kliche and Oberleithner2011; Barriga and Mayor, Reference Barriga and Mayor2019; Devanny et al., Reference Devanny, Vancura and Kaufman2021). Consequently, biochemical and physical mechanisms cooperate together and influence the segregation process. Schematic presentation of the role of physical parameters in the segregation process is shown in Figure 5 and discussed in the context of formulated modeling equations.

Figure 5. Schematic presentation of the role of physical parameters in the segregation process.

The main part of the model is formulated based on modified phase model C proposed for thermodynamic systems far from equilibrium (Ala-Nissila et al., Reference Ala-Nissila, Majaniemi and Elder2004) combined with phase models for viscoelastic phase transition proposed by Tanaka (Reference Tanaka1997). Accordingly, the cellular interactions are expressed by coupling two scalar fields, that is, the volume fraction of migrating epithelial-like pseudo-phase and volume fraction of mesenchymal-like pseudo-phase. The change of the volume fraction of migrating epithelial-like pseudo-phase in the presence of mesenchymal-like pseudo-phase is expressed as:

(2)

where $ {D}_e $ is the effective dispersion coefficient of migrating epithelial-like pseudo-phase, $ F\left({\phi}_{em},{\phi}_c\right) $ is the Ginsburg-Landau free energy functional equal to $ F\left({\phi}_{em},{\phi}_c\right)={F}_e\left({\phi}_{em}\right)+{F}_c\left({\phi}_{em},{\phi}_c\right) $, $ {F}_e\left({\phi}_{em}\right) $ is the part of free energy functional which describes biochemical and mechanical interactions within the migrating epithelial-like pseudo-phase during contact with resting epithelial-like pseudo-phase, $ {F}_c\left({\phi}_{em},{\phi}_c\right) $ is the part of free energy functional which describes the contribution of mesenchymal-like pseudo-phase to the movement of epithelial cells, $ \frac{\delta F\left({\phi}_{em},{\phi}_c\right)}{\delta {\phi}_{em}} $ is the functional derivative equal to $ \frac{\delta F\left({\phi}_{em},{\phi}_c\right)}{\delta {\phi}_{em}}={\mathit{\lim}}_{\varepsilon \to 0}\frac{F\left[\left({\phi}_{em}\left({r}^{\prime },\tau \right),{\phi}_c\right)+\varepsilon \delta \left({r}^{\prime }-r\right)\right]-F\left[\Big({\phi}_{em}\left({r}^{\prime },\tau \right),{\phi}_c\right]}{\varepsilon } $, $ \varepsilon $ is an increment of $ {\phi}_{em} $. (Ala-Nissila et al., Reference Ala-Nissila, Majaniemi and Elder2004), $ {\overset{\sim }{\boldsymbol{\sigma}}}_{\boldsymbol{emR}}\left(r,\tau \right) $ represents the residual stress accumulation within migrating epithelial pseudo-phase equal to $ {\overset{\sim }{\boldsymbol{\sigma}}}_{\boldsymbol{emR}}\left(r,\tau \right)=\frac{dV}{N_r^{\ast }}{\sum \limits}_{i=1}^{N_r^{\ast }}{\overset{\sim }{\boldsymbol{\sigma}}}_{\boldsymbol{emR}\;\boldsymbol{i}}\delta \left(r-{r}_i\right) $, $ {\overset{\sim }{\boldsymbol{\sigma}}}_{\boldsymbol{emR}\;\boldsymbol{i}} $ is the residual stress accumulated within the ith migrating epithelial cluster within the spheroid volume increment $ dV $, $ \delta \left(\bullet \right) $ is the Dirac delta distribution function, and $ {\overset{\sim }{\boldsymbol{\sigma}}}_{\boldsymbol{SD}} $ is the solid stress. The residual stress accumulation within the ith migrating epithelial cluster is formulated in the Appendix A, while the part of the stress caused by CCM is formulated in the Appendix B. Note that the constitutive models in Appendices A and B, which discuss several viscoelastic regimes, depend on three other parameters: $ {n}_j $, the cell packing density of resting epithelial clusters which corresponds to the cell packing density at the jamming state; $ {n}_{mei} $, the packing density of ith migrating epithelial cell clusters; and $ {n}_c $, the cell packing density of mesenchymal cells. Since the jamming state transition induced by the cell residual stress accumulation leads to an increase in cell packing density, we have $ {n}_{mei}<{n}_j $. Moreover, since the mesenchymal-like cells keep moving and avoid cell jamming (Grosser et al., Reference Grosser, Lippoldt, Oswald, Merkel, Sussman, Renner, Gottheil, Morawetz, Fuhs, Xie, Pawlizak, Fritsch, Wolf, Horn, Briest, Aktas, Manning and Käs2021), their local packing density satisfies $ {n}_c<{n}_j $.

Returning to Eq. (2), the total stress within the migrating epithelial pseudo-phase, that is, can suppress movement by decreasing the volume fraction of migrating epithelial cells and can induce the jamming state transition. The free energy $ {F}_e\left({\phi}_{em}\right) $ has been expressed based on modified model by Cohen and Murray (Reference Cohen and Murray1981) in the form:

(3) $$ {F}_e\left({\phi}_{em}\right)=\int \left[{f}_e\left({\phi}_{em}\right)+\frac{1}{2}{k}_e^{m-r}{\left(\overrightarrow{\nabla}{\unicode{x03D5}}_{em}\right)}^2\right]{d}^3r $$

where $ {f}_e\left({\phi}_{em}\right) $ represents the energetic effect of the volumetric rearrangement of epithelial cells driven by the surface tension of epithelial cells which is simplified as: $ {f}_e\left({\phi}_{em}\right)\approx \frac{1}{2}{k}_e{\phi_{em}}^2 $, $ {k}_e $ is the parameter which accounts for interactions between migrating epithelial cells within a spheroid, and the other terms represent the gradient energy contributions, while $ {k}_e^{m-r} $ is the interaction parameter which accounts for physical interactions between migrating and resting epithelial-like pseudo-phases at the biointerface. The interfacial tension between migrating and resting epithelial-like pseudo-phases can be expressed as: $ {\gamma}_e^{m-r}\left(\tau \right){A}_e^{m-r}\left(\tau \right)=\int \frac{1}{2}{k}_e^{m-r}{\left(\overrightarrow{\nabla}{\phi}_{em}\right)}^2{d}^3r $ (where $ {A}_e^{m-r}\left(\tau \right) $ is the interfacial area between the pseudo-phases). The biointerface is finite with the thickness which is an order of magnitude larger than the size of single cells (Pajic-Lijakovic and Milivojevic, Reference Pajic-Lijakovic and Milivojevic2020a). Consequently, the gradient of the interfacial tension at the r-m biointerface influence cell movement along the interface from the resting epithelial pseudo-phase to the migrating one, which is expressed by the Marangoni flux (Pajic-Lijakovic and MIlivojevic, Reference Pajic-Lijakovic and Milivojevic2022c).

The free energy $ {F}_c\left({\phi}_{em},{\phi}_c\right) $ can be expressed as:

(4) $$ {F}_c\left({\phi}_{em},{\phi}_c\right)={\displaystyle \begin{array}{l}\int \Big[{f}_c\left({\phi}_{em},{\phi}_c\right)+\frac{1}{2}{k}_{ec}^m{\left(\overrightarrow{\nabla}{\phi}_{em}\right)}^2\\ {}+\hskip2px \frac{1}{2}{k}_{ec}^r{\left(\overrightarrow{\nabla}\Big(1-{\phi}_{em}-{\phi}_c\right)}^2\Big]{d}^3r\end{array}} $$

where $ {f}_c\left({\phi}_{em},{\phi}_c\right) $ accounts for the cumulative effects of mesenchymal-like cell signaling which influences the volumetric rearrangement of epithelial cells and the second term represents the gradient energy contribution, while $ {k}_{ec}^m $ is the interaction parameter which accounts for physical interactions between migrating epithelial-like pseudo-phase and mesenchymal-like pseudo-phase at the biointerface. The interfacial tension between migrating epithelial-like pseudo-phase and mesenchymal-like pseudo-phase can be expressed as: $ {\gamma}_{ec}^m\left(\tau \right){A}_{ec}^m\left(\tau \right)=\int \frac{1}{2}{k}_{ec}^m{\left(\overrightarrow{\nabla}{\phi}_{em}\right)}^2{d}^3r $ (where $ {A}_{ec}^m\left(\tau \right) $ is the interfacial area between these pseudo-phases), while the interfacial tension between resting epithelial-like pseudo-phase and mesenchymal-like pseudo-phase is $ {\gamma}_{ec}^r\left(\tau \right){A}_{ec}^r\left(\tau \right)=\int \frac{1}{2}{k}_{ec}^r{\left(\overrightarrow{\nabla}\left(1-{\phi}_{em}-{\phi}_c\right)\right)}^2{d}^3r $ (where $ {A}_{ec}^r\left(\tau \right) $ is the interfacial area between these pseudo-phases). The biointerface is finite and the gradient of interfacial tension established at the c-me biointerface stimulates movement of mesenchymal-like pseudo-phase along the biointerface toward to the migrating, epithelial-like pseudo-phase (Pajic-Lijakovic and Milivojevic, Reference Pajic-Lijakovic and Milivojevic2020a).

The energy $ {f}_c\left({\phi}_{em},{\phi}_c\right) $ can be expressed as:

(5) $$ {f}_c\left({\phi}_{em},{\phi}_c\right)\approx \frac{1}{2}{k}_c{\phi_{em}}^2{\phi_c}^2-\frac{1}{2}{\beta}_c{\phi_c}^2 $$

where $ {k}_c $ is the parameter which accounts for interactions between migrating epithelial cells and mesenchymal cells which arise as a product of cell signaling and $ {\beta}_c $ is the measure of cumulative effects of tractions of mesenchymal-like cells. While cell signaling is capable of enhancing the movement of both pseudo-phases (the first term of the right-hand side of Eq. (5)), the traction of mesenchymal cells can reduce movement of mesenchymal-like cells (the second term of the right-hand side of Eq. (5)). This reduction is pronounced when mesenchymal-like cells establish stronger FAs with the ECM already present within the multicellular spheroid.

The change of the volume fraction of mesenchymal-like pseudo-phase in the presence of migrating, epithelial-like pseudo-phase is expressed as:

(6) $$ \frac{\partial {\unicode{x03D5}}_c\left(r,\tau \right)}{\partial \tau }+\overrightarrow{\nabla}\left({\phi}_c{\overrightarrow{\boldsymbol{v}}}_{\boldsymbol{R}}\right)=\overrightarrow{\nabla}\left[{D}_c\left({\unicode{x03D5}}_c\overrightarrow{\mathbf{\nabla}}\frac{\delta F\left({\unicode{x03D5}}_{em},{\phi}_c\right)}{\delta {\unicode{x03D5}}_c}\right)+\overrightarrow{\nabla}\bullet \left({\overset{\sim }{\boldsymbol{\sigma}}}_{\boldsymbol{SD}}-{\overset{\sim }{\boldsymbol{\sigma}}}_{\boldsymbol{cR}}\right)\right] $$

where $ {D}_c $ is the effective dispersion coefficient which quantifies spreading of mesenchymal pseudo-phase, $ {\overrightarrow{\boldsymbol{v}}}_{\boldsymbol{R}} $ is the relative velocity between migrating epithelial and mesenchymal cell pseudo-phases equal to $ {\overrightarrow{\boldsymbol{v}}}_{\boldsymbol{R}}\left(r,\tau \right)={\overrightarrow{\boldsymbol{v}}}_{\boldsymbol{c}}\left(r,\tau \right)+{\overrightarrow{\boldsymbol{v}}}_{\boldsymbol{e}}\left(r,\tau \right) $, $ {\overrightarrow{\boldsymbol{v}}}_{\boldsymbol{c}}\left(r,\tau \right) $ is the local velocity of mesenchymal-like cells equal to $ {\overrightarrow{\boldsymbol{v}}}_{\boldsymbol{c}}\left(r,\tau \right)=\frac{dV}{N_r^{\ast \ast \ast }}{\sum}_{k=1}^{N_r^{\ast \ast \ast }}{\overrightarrow{\boldsymbol{v}}}_{\boldsymbol{c}\boldsymbol{k}}\delta \left(r-{r}_k\right) $, $ {\overrightarrow{\boldsymbol{v}}}_{\boldsymbol{ck}} $ is the average velocity of the center of mass of the kth cluster of mesenchymal cells, and $ {\overrightarrow{\boldsymbol{v}}}_{\boldsymbol{e}}\left(r,\tau \right) $ is the local velocity of epithelial-like cells equal to equal to $ {\overrightarrow{\boldsymbol{v}}}_{\boldsymbol{e}}\left(r,\tau \right)=\frac{dV}{N_r^{\ast }}{\sum}_{i=1}^{N_r^{\ast }}{\overrightarrow{\boldsymbol{v}}}_{\boldsymbol{e}\boldsymbol{i}}\delta \left(r-{r}_i\right) $, $ {\overrightarrow{\boldsymbol{v}}}_{\boldsymbol{ei}} $ is the average velocity of the center of mass of the ith cluster of epithelial cells. This formulation of the relative velocity is in accordance with the fact that the velocities $ {\overrightarrow{\boldsymbol{v}}}_{\boldsymbol{e}} $ and $ {\overrightarrow{\boldsymbol{v}}}_{\boldsymbol{c}}\left(r,\tau \right) $ have an opposite directions. While mesenchymal cells migrate from the spheroid core region toward its surface driven by the solid stress, epithelial cells migrate from the spheroid surface region toward its core region driven by the surface tension. The velocity of cancer cells $ {\overrightarrow{\boldsymbol{v}}}_{\boldsymbol{c}}\left(r,\tau \right) $ satisfies the condition $ {\overrightarrow{\boldsymbol{v}}}_{\boldsymbol{c}}\left(r,\tau \right)\ge {\overrightarrow{\boldsymbol{v}}}_{\boldsymbol{e}}\left(r,\tau \right) $ (Clark and Vignjevic, Reference Clark and Vignjevic2015), while the velocity of epithelial cells satisfies two conditions: (1) $ 0<{\overrightarrow{\boldsymbol{v}}}_{\boldsymbol{e}}\left(r,\tau \right)\le {\overrightarrow{\boldsymbol{v}}}_{\boldsymbol{c}}\left(r,\tau \right) $ for migrating cells and (2) $ {\overrightarrow{\boldsymbol{v}}}_{\boldsymbol{e}}\left(r,\tau \right)=0 $ for resting cells under jamming. Consequently, the relative velocity is $ {\overrightarrow{\boldsymbol{v}}}_{\boldsymbol{R}}\left(r,\tau \right)>{\overrightarrow{\boldsymbol{v}}}_{\boldsymbol{c}}\left(r,\tau \right) $. The first term on the right-hand side of Eq. (6) describes interactions between mesenchymal-like and migrating, epithelial-like pseudo-phases which enhance movement of mesenchymal-like cells while the second term describes an influence of the accumulated stress within the spheroid core region equal to $ {\overset{\sim }{\boldsymbol{\sigma}}}_{\boldsymbol{SD}}-{\overset{\sim }{\boldsymbol{\sigma}}}_{\boldsymbol{cR}} $ on movement of the mesenchymal-like cells (where $ {\overset{\sim }{\boldsymbol{\sigma}}}_{\boldsymbol{cR}} $ is the residual stress caused by collective movement of mesenchymal-like pseudo-phase). The stress $ {\overset{\sim }{\boldsymbol{\sigma}}}_{\boldsymbol{cR}} $ is expressed as $ {\overset{\sim }{\boldsymbol{\sigma}}}_{\boldsymbol{cR}}\left(r,\tau \right)=\frac{dV}{N_r^{\ast \ast \ast }}{\sum}_{k=1}^{N_r^{\ast \ast \ast }}{\overset{\sim }{\boldsymbol{\sigma}}}_{\boldsymbol{cR}\;\boldsymbol{k}}\delta \left(r-{r}_k\right) $, $ {\overset{\sim }{\boldsymbol{\sigma}}}_{\boldsymbol{cR}\;\boldsymbol{k}} $ is the residual stress accumulated within the kth mesenchymal cluster within the spheroid volume increment $ dV $, and $ {\overset{\sim }{\boldsymbol{\sigma}}}_{\boldsymbol{SD}} $ is the solid stress. The residual stress accumulation within the kth mesenchymal cluster is formulated in the Appendix A, while the part of the stress caused by CCM is formulated in the Appendix B. In order to understand the process of cell segregation, it is necessary to estimate change in velocity for mesenchymal cell pseudo-phase $ {\overrightarrow{\boldsymbol{v}}}_{\boldsymbol{c}}\left(r,\tau \right) $ and migrating epithelial pseudo-phase $ {\overrightarrow{\boldsymbol{v}}}_{\boldsymbol{e}}\left(r,\tau \right) $.

The force balance for the rearrangement of migrating epithelial pseudo-phase and mesenchymal pseudo-phase

CCM causes mechanical waves generation in the form of oscillatory change in cell velocity, resulted strain and corresponding cell residual stress (Serra-Picamal et al., Reference Serra-Picamal, Conte, Vincent, Anon, Tambe, Bazellieres, Butler, Fredberg and Trepat2012; Notbohm et al., Reference Notbohm, Banerjee, Utuje, Gweon, Jang, Park, Shin, Butler, Fredberg and Marchetti2016; Pajic-Lijakovic and Milivojevic, Reference Pajic-Lijakovic and Milivojevic2020c, Reference Pajic-Lijakovic and Milivojevic2022a). Oscillatory change of epithelial cell velocity (i.e., effective inertia), in the case of cell segregation, arises as the result of competition between interfacial tension force and the mixing force against viscoelastic force. The viscoelastic force is capable of suppressing movement of epithelial cells (Grosser et al., Reference Grosser, Lippoldt, Oswald, Merkel, Sussman, Renner, Gottheil, Morawetz, Fuhs, Xie, Pawlizak, Fritsch, Wolf, Horn, Briest, Aktas, Manning and Käs2021; Pajic-Lijakovic and Milivojevic, Reference Pajic-Lijakovic and Milivojevic2023). The interfacial tension force is expressed by modified model proposed by Pajic-Lijakovic and Milivojevic (Reference Pajic-Lijakovic and Milivojevic2020c) as:

(7) $$ {\phi}_{em}\left\langle {n}_{em}\right\rangle {{\overrightarrow{\boldsymbol{F}}}_{\boldsymbol{st}}}^{\boldsymbol{e}}={\phi}_{em}\left\langle {n}_{em}\right\rangle \left[-{S}_e^{r-m}\hskip0.24em {{\overrightarrow{\boldsymbol{u}}}_{\boldsymbol{e}}}^{r-m}-{S}_e^{c-m}{{\overrightarrow{\boldsymbol{u}}}_{\boldsymbol{e}}}^{c-m}\right] $$

where $ \left\langle {n}_{em}\right\rangle =\frac{{\sum \limits}_{i=1}^{{N_r}^{\ast }}{N}_{em i}}{{\sum \limits}_{i=1}^{{N_r}^{\ast }}\Delta {V}_{em\;i}} $ is the average packing density of single migrating epithelial cluster within the spheroid volume increment $ dV $, $ {N}_{emi} $ is the number of cells within the ith migrating epithelial cluster, $ {S}_e^{r-m}={\gamma_e}^r-\left({\gamma_e}^m+{\gamma_e}^{r-m}\right) $ is the spreading coefficient of migrating epithelial cells in contact with resting epithelial cells (such that $ {S}_e^{r-m}<0 $), $ {S}_e^{c-m}={\gamma}_c-\left({\gamma_e}^m+{\gamma_{ec}}^m\right) $ is the spreading coefficient of migrating epithelial cells in contact with mesenchymal-like cells (such that $ {S}_e^{c-m}<0 $), $ {{\overrightarrow{\boldsymbol{u}}}_{\boldsymbol{e}}}^{r-m} $ is the local displacement field caused by movement of epithelial clusters near the re-me biointerface, $ {{\overrightarrow{\boldsymbol{u}}}_{\boldsymbol{e}}}^{c-m} $ is the local displacement field caused by movement of epithelial clusters near the c-me biointerface. The ratios $ {X}_1=\frac{\gamma_c+{\gamma}_{ec}^m}{\gamma_e^m}<1 $ and $ {X}_2=\frac{\gamma_c+{\gamma}_{ec}^r}{\gamma_e^r}<1 $. Maximizing the ratios $ {X}_1 $ and $ {X}_2 $, such that $ {X}_1,{X}_2\to 1 $ by enhancing cell–cell adhesion contacts within the mesenchymal-like pseudo-phase can reduce the spreading of mesenchymal cells. Interfacial tension force acts to reduce the interfacial area between epithelial and mesenchymal cell subpopulations. The viscoelastic force is resistive force and acts to reduce movement of epithelial cells. The mixing force $ {{\overrightarrow{\boldsymbol{F}}}_{\boldsymbol{m}}}^{\boldsymbol{e}} $ can be expressed as: $ {{\overrightarrow{\boldsymbol{F}}}_{\boldsymbol{m}}}^{\boldsymbol{e}}=-{\phi}_{em}\overrightarrow{\nabla}\frac{\delta F\left({\phi}_{em},{\phi}_c\right)}{\delta {\phi}_{em}} $. The viscoelastic force represents a consequence of inhomogeneously distributed cell residual stress, and is expressed by Murray et al. (Reference Murray, Maini and Tranquillo1988) and Pajic-Lijakovic et al. (Reference Pajic-Lijakovic and Milivojevic2023a; Reference Pajic-Lijakovic and Milivojevic2023c) as: $ {{\overrightarrow{\boldsymbol{F}}}_{\boldsymbol{Tve}}}^{\boldsymbol{e}}=\overrightarrow{\mathbf{\nabla}}\bullet \left({\overset{\sim }{\boldsymbol{\sigma}}}_{\boldsymbol{e}\boldsymbol{mR}}+{\overset{\sim }{\boldsymbol{\sigma}}}_{\boldsymbol{SD}}\right) $. The corresponding force balance is expressed by Pajic-Lijakovic et al. (Reference Pajic-Lijakovic and Milivojevic2023a; Reference Pajic-Lijakovic and Milivojevic2023c) as:

(8)

where $ {\left\langle m\right\rangle}_e $ is the average mass of a single epithelial cell and $ {\overrightarrow{\boldsymbol{v}}}_{\boldsymbol{e}} $ is the epithelial cell velocity equal to $ {\overrightarrow{\boldsymbol{v}}}_{\boldsymbol{e}}\left(r,\tau \right)=\frac{d{\overrightarrow{\boldsymbol{u}}}_{\boldsymbol{e}}}{d\tau} $, $ \frac{D{\overrightarrow{\boldsymbol{v}}}_{\boldsymbol{e}}}{D\tau}=\frac{\partial {\overrightarrow{\boldsymbol{v}}}_{\boldsymbol{e}}}{\partial \tau }+\left({\overrightarrow{\boldsymbol{v}}}_{\boldsymbol{e}}\bullet \overrightarrow{\mathbf{\nabla}}\right){\overrightarrow{\boldsymbol{v}}}_{\boldsymbol{e}} $ is the material derivative (Bird et al., Reference Bird, Stewart and Lightfoot1960).

While the viscoelastic force reduces movement of epithelial cells, this force represents a driving force for migration of mesenchymal like cells. It is in accordance with the fact that accumulated cell stress enhances movement of cancer cells and suppresses movement of epithelial cells (Tse et al., Reference Tse, Cheng, Tyrrell, Wilcox-Adelman, Boucher, Jain and Munn2012; Riehl et al., Reference Riehl, Kim, Lee, Duan, Yang, Donahue and Lim2020, Reference Riehl, Kim, Bouzid and Lim2021). Viscoelastic force in this case is expressed by modified model proposed by Pajic-Lijakovic and Milivojevic (Reference Pajic-Lijakovic and Milivojevic2020c) as: $ {{\overrightarrow{\boldsymbol{F}}}_{\boldsymbol{Tve}}}^{\boldsymbol{c}}=\overrightarrow{\mathbf{\nabla}}\bullet \left({\overset{\sim }{\boldsymbol{\sigma}}}_{\boldsymbol{SD}}-{\overset{\sim }{\boldsymbol{\sigma}}}_{\boldsymbol{c}\boldsymbol{R}}\Big)-{\overset{\sim }{\boldsymbol{\sigma}}}_{\boldsymbol{R}\;\boldsymbol{ECM}}\right) $ (where $ {\overset{\sim }{\boldsymbol{\sigma}}}_{\boldsymbol{R}\;\boldsymbol{ECM}} $ is the residual stress accumulated within ECM). Besides the viscoelastic force, the mixing force and the interfacial tension force also drive extension of mesenchymal-like cells. The mixing force can be expressed as: $ {{\overrightarrow{\boldsymbol{F}}}_{\boldsymbol{m}}}^{\boldsymbol{c}}={\unicode{x03D5}}_c\overrightarrow{\mathbf{\nabla}}\frac{\delta {F}_c\left({\unicode{x03D5}}_{em},{\phi}_c\right)}{\delta {\unicode{x03D5}}_c} $. The interfacial tension force is equal to $ {\phi}_c\left\langle {n}_c\right\rangle $ $ {{\overrightarrow{\boldsymbol{F}}}_{\boldsymbol{st}}}^{\boldsymbol{c}}={\phi}_c\left\langle {n}_c\right\rangle \left[{S}_c^{c-m}\hskip0.24em {{\overrightarrow{\boldsymbol{u}}}_{\boldsymbol{c}}}^{c-m}+{S}_c^{c-r}{{\overrightarrow{\boldsymbol{u}}}_{\boldsymbol{c}}}^{c-r}\right] $ (where $ \left\langle {n}_c\right\rangle =\frac{{\sum \limits}_{k=1}^{{N_r}^{\ast \ast \ast }}{N}_{ck}}{{\sum \limits}_{k=1}^{{N_r}^{\ast \ast \ast }}\Delta {V}_{c\;k}} $ is the average packing density of single mesenchymal cluster within the spheroid volume increment $ dV $, $ {N}_{ck} $ is the number of cells within the kth mesenchymal cluster, $ {S}_c^{c-m}={\gamma_e}^m-\left({\gamma}_c+{\gamma_{ec}}^m\right) $ is the spreading coefficient of the mesenchymal like cells toward the migrating epithelium (such that $ {S}_c^{c-m}>0 $), $ {S}_c^{c-r}={\gamma_e}^r-\left({\gamma}_c+{\gamma_{ec}}^r\right) $ is the spreading coefficient of the mesenchymal like cells toward the resting epithelium (such that $ {S}_c^{c-r}>0 $), $ {{\overrightarrow{\boldsymbol{u}}}_{\boldsymbol{c}}}^{c-m} $ is the local displacement field caused by movement of mesenchymal-like clusters near the c-me biointerface, and $ {{\overrightarrow{\boldsymbol{u}}}_{\boldsymbol{c}}}^{c-r} $ is the local displacement field caused by movement of mesenchymal-like clusters near the c-re biointerface. The ratios $ {X}_3=\frac{\gamma_e^m+{\gamma}_{ec}^m}{\gamma_c}>1 $ and $ {X}_4=\frac{\gamma_e^r+{\gamma}_{ec}^r}{\gamma_c}>1 $ such that $ {X}_3>{X}_4 $. Minimizing the ratios $ {X}_3 $ and $ {X}_4 $ by enhancing cell–cell adhesion contacts within the mesenchymal-like pseudo-phase can reduce the spreading of mesenchymal cells. The oscillatory change of cell velocity in the case of movement of mesenchymal-like cells represents the consequence of the competition between: the viscoelastic force, mixing force, interfacial tension force, against the traction force. The traction force as a consequence of established FAs influences movement of mesenchymal cells, while the epithelial cells do not establish FAs within a spheroid (Devanny et al., Reference Devanny, Vancura and Kaufman2021). This force is capable of reducing movement of mesenchymal cells depending on the strength of FAs (Fuhrmann et al., Reference Fuhrmann, Banisadr, Beri, Tlsty and Engler2017). The traction force is expressed as: $ \rho {{\overrightarrow{\boldsymbol{F}}}_{\boldsymbol{tr}}}^{\boldsymbol{c}}=\rho k{\overrightarrow{\boldsymbol{u}}}_{\boldsymbol{ECM}} $ (where $ k $ is an elastic constant of single FA, $ \rho $ is the number density of FAs, and $ {\overrightarrow{\boldsymbol{u}}}_{\boldsymbol{ECM}} $ is the displacement field of ECM caused by movement of cancer cells) (Murray et al., Reference Murray, Maini and Tranquillo1988). The corresponding force balance can be expressed by modifying the model proposed by Pajic-Lijakovic and Milivojevic (Reference Pajic-Lijakovic and Milivojevic2020c) for 2D CCM as:

(9)

where $ {\left\langle m\right\rangle}_c $ is the average mass of a single cancer cell and $ {\overrightarrow{\boldsymbol{v}}}_{\boldsymbol{c}} $ is the epithelial cell velocity equal to $ {\overrightarrow{\boldsymbol{v}}}_{\boldsymbol{c}}\left(r,\tau \right)=\frac{d{\overrightarrow{\boldsymbol{u}}}_{\boldsymbol{c}}}{d\tau} $, $ \frac{D{\overrightarrow{\boldsymbol{v}}}_{\boldsymbol{c}}}{D\tau}=\frac{\partial {\overrightarrow{\boldsymbol{v}}}_{\boldsymbol{c}}}{\partial \tau }+\left({\overrightarrow{\boldsymbol{v}}}_{\boldsymbol{c}}\bullet \overrightarrow{\mathbf{\nabla}}\right){\overrightarrow{\boldsymbol{v}}}_{\boldsymbol{c}} $ is the material derivative (Bird et al., Reference Bird, Stewart and Lightfoot1960). While movement of mesenchymal cells corresponds to the convective regime, movement of epithelial-like cells changes from convective regime through conductive regime to the damped-conductive regime.

The efficient process of cell segregation can be postulated based on the proposed modeling consideration. In this purpose, following parameters are introduced, such as: (1) volume fraction of epithelial cells in the resting state $ {\phi}_{er}\to 0 $ and (2) the surface tension ratios $ {X}_1,{X}_2,{X}_3,{X}_4\to 1 $.

An increase in the degree of mesenchymal character of epithelial-like subpopulation leads to a decrease in the volume fraction of cells in the resting (jamming) state, corresponding surface tension, as well as, the interfacial tensions between the pseudo-phases. The surface tension of epithelial-like pseudo-phase corresponds to an order of magnitude from several $ \frac{mN}{m} $ to tens of $ \frac{mN}{m} $ (Mombach et al., Reference Mombach, Robert, Graner, Gillet, Thomas, Idiart and Rieu2005), while the surface tension of mesenchymal-like cells is significantly lower (Devanny et al., Reference Devanny, Vancura and Kaufman2021). Interfacial tensions between the subpopulations are lower than the surface tension of migrating epithelial cells, but correspond to a same order of magnitude with it. The volume fraction of epithelial-like cells in the resting state during the segregation process could be even larger than 15% of whole epithelial-like subpopulation and placed primarily within the spheroid core region. The cell normal and shear residual stress accumulation caused by CCM corresponds to several tens of Pa (Tambe et al., Reference Tambe, Croutelle, Trepat, Park, Kim, Millet, Butler and Fredberg2013) while the solid stress within the spheroid core region is significantly larger and corresponds to several kPa (Kalli and Stylianopoulos, Reference Kalli and Stylianopoulos2018).

The formulated biophysical model, pointed to the interrelation between the physical parameters responsible for the cell segregation process by accounting for the viscoelasticity of the pseudo-phases and effects along the biointerfaces between them as was presented graphically in Figure 5. Some parameters such as the residual stress accumulation caused by movement of epithelial collectives, solid stress accumulated in the spheroid core region, and tissue surface tension have been already measured, while the others such as the interfacial tension between two cell subpopulations in direct contact and interfacial tension gradient have not been measured yet. Gsell et al. (Reference Gsell, Tlili, Merkel and Lenne2023) recently confirmed cell movement along the multicellular surface in contact with liquid medium driven by the tissue surface tension gradient, that is, the Marangoni effect. However, the surface tension gradient itself has not been measured. The maximum cell residual stress caused by collective movement of epithelial monolayers corresponds to a few hundreds of Pa (Serra-Picamal et al., Reference Serra-Picamal, Conte, Vincent, Anon, Tambe, Bazellieres, Butler, Fredberg and Trepat2012; Notbohm et al., Reference Notbohm, Banerjee, Utuje, Gweon, Jang, Park, Shin, Butler, Fredberg and Marchetti2016). The solid stress corresponds to a few kPa (Kalli and Stylianopoulos, Reference Kalli and Stylianopoulos2018). Various experimental techniques have been used for the measurement the cell stress caused by CCM such as: (1) the monolayer stress microscopy (Tambe et al., Reference Tambe, Croutelle, Trepat, Park, Kim, Millet, Butler and Fredberg2013), (2) microbead/droplet-based stress sensors (Campàs et al., Reference Campàs, Mammoto, Hasso, Sperling, O’Connell, Bischof, Maas, Weitz, Mahadevan and Ingber2014; Dolega et al., Reference Dolega, Delarue, Ingremeau, Prost, Delon and Cappello2017). Some of these techniques, such as monolayer stress microscopy, are suitable for the measurement of the stress in 2D, while the others such as using droplet-based stress sensors can be used for the measurement of anisotropic normal stresses only. Development of the suitable measuring technique is a prerequisite for the improvement of our knowledge about the cell rearrangement. The tissue surface tension has been measured under equilibrium conditions only (i.e., the static tissue surface tension). Various experimental techniques have been used for the measurement of the static tissue surface tension, such as cell aggregate compression between parallel plates (Mombach et al., Reference Mombach, Robert, Graner, Gillet, Thomas, Idiart and Rieu2005; Marmottant et al., Reference Marmottant, Mgharbel, Kafer, Audren, Rieu, Vial, van der Sanden, Maree, Graner and Delanoe-Ayari2009), cell aggregate micropipette aspiration (Guevorkian et al., Reference Guevorkian, Brochard-Wyart, Gonzalez-Rodriguez, Pajic-Lijakovic and Barriga2021), and magnetic force tensiometer (Nagle et al., Reference Nagle, Richert, Quinteros, Janel, Buysschaert, Luciani, Debost, Thevenet, Wilhelm, Prunier, Lafont, Padilla-Benavides, Boissan and Reffay2022). The experimental values of the static tissue surface tension vary from a few $ \frac{mN}{m} $ to several tens of $ \frac{mN}{m} $ depending on cell type and measuring technique. However, the tissue surface tension and the interfacial tension between the two subpopulations are time dependent parameters. This modeling consideration is an effort to stimulate further biological research in this field.