## Abstract

In vitro tissue engineering is emerging as a potential tool to meet the high demand for replacement tissue, caused by the increased incidence of tissue degeneration and damage. A key challenge in this field is ensuring that the mechanical properties of the engineered tissue are appropriate for the in vivo environment. Achieving this goal will require detailed understanding of the interplay between cell proliferation, extracellular matrix (ECM) deposition and scaffold degradation. In this paper, we use a mathematical model (based upon a multiphase continuum framework) to investigate the interplay between tissue growth and scaffold degradation during tissue construct evolution in vitro. Our model accommodates a cell population and culture medium, modelled as viscous fluids, together with a porous scaffold and ECM deposited by the cells, represented as rigid porous materials. We focus on tissue growth within a perfusion bioreactor system, and investigate how the predicted tissue composition is altered under the influence of (1) differential interactions between cells and the supporting scaffold and their associated ECM, (2) scaffold degradation, and (3) mechanotransduction-regulated cell proliferation and ECM deposition. Numerical simulation of the model equations reveals that scaffold heterogeneity typical of that obtained from \(\mu \)CT scans of tissue engineering scaffolds can lead to significant variation in the flow-induced mechanical stimuli experienced by cells seeded in the scaffold. This leads to strong heterogeneity in the deposition of ECM. Furthermore, preferential adherence of cells to the ECM in favour of the artificial scaffold appears to have no significant influence on the eventual construct composition; adherence of cells to these supporting structures does, however, lead to cell and ECM distributions which mimic and exaggerate the heterogeneity of the underlying scaffold. Such phenomena have important ramifications for the mechanical integrity of engineered tissue constructs and their suitability for implantation in vivo.

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## Acknowledgments

This work was supported by funding from the EPSRC in the form of a Ph.D studentship (RDO) and an Advanced Research Fellowship (SLW). It was also supported by the EPSRC/BBSRC-funded OCISB project BB/D020190/1 (JMO), and based on work supported in part by Award KUK-013-04 made by King Abdullah University of Science and Technology (HMB). We are also grateful to E. Baas, ISTM, Keele University for the provision of experimental data.

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R. D. O’Dea and J. M. Osborne are the ioint first authors.

## Appendix: Model derivation

### Appendix: Model derivation

We consider a bioreactor of length \(L^*\) and width \(h^*\), modelled as a two-dimensional channel containing a mixture of four interacting phases, representing cells, culture medium, PLLA scaffold and ECM and denote these via a subscript \(i=n,\ w,\ s,\ e\), respectively. The viscosity of any fluid phase is denoted \(\mu ^*_i\), and the typical timescale for tissue growth (comprising both cell proliferation and ECM deposition) is denoted \(K^*\). Asterisks distinguish dimensional quantities from their dimensionless equivalents.

We introduce a Cartesian coordinate system \(L^*\varvec{x}=L^*(x,y)\) and time \(K^*t\) and the channel occupies the dimensionless region \(0\le x \le 1,\,0\le y\le h=h^*/L^*\). The volume fraction of each phase is denoted \(\theta _i\), while the dimensionless volume-averaged velocities, pressures and stress tensors of the each phase are denoted \(K^*L^*\varvec{u}_i=K^*L^*(u_i,v_i),\,K^*\mu _w^*p_i\) and \(K^*\mu _w^*\varvec{\sigma }^i\). Tissue growth, scaffold degradation and ECM deposition are captured via material transfer functions \(K^*S_i\). We assume that all dimensionless dependent variables are functions of \(\varvec{x}\) and \(t\).

The model is constructed by considering mass and momentum balances for each phase, assuming that each phase is incompressible, with equal density, and neglecting inertial effects; the equations governing the \(i\mathrm{th }\) phase (with volume fraction \(\theta _i\)) are as follows [see Lemon et al. (2006), O’Dea et al. (2010), Osborne et al. (2010)]:

Additional conservation conditions may be obtained by summing over all phases and exploiting the no-voids condition \(\sum _i \theta _i=1\).

In Eq. (21) \(K^* S_i\) is the net material production term associated with phase \(i\) (mass conservation demands that \(\sum S_i=0\)); in (22), \(K^*\mu _w^*/L^* \mathbf F ^{ij}\) is the interphase force exerted by phase \(j\) on phase \(i\), obeying \(\mathbf F ^{ij}=-\mathbf F ^{ji}\). These interphase forces comprise interphase viscous drag (with drag coefficient \(\mu _w^*/L^{*2} k\)) and active forces, the latter being embodied within extra pressures which arise due to cell–cell, cell–ECM and cell–scaffold interactions; interactions between the culture medium and scaffold phases are assumed to involve only viscous drag. The mechanics of this four phase formulation is simplified by lumping the scaffold and ECM components into a single ‘substrate’ phase, denoted \(\theta _S=\theta _s+\theta _e\), and modelled as a rigid porous material. For notational convenience, in this Appendix, we employ the subscript \(S\) to denote the substrate, in preference to \(\Theta \). Separate mass conservation equations are nevertheless employed for \(\theta _s\) and \(\theta _e\) to track their individual evolution.

The cell population and culture medium are represented as distinct viscous fluids, modelled by standard viscous stress tensors; the rigidity of the substrate implies \(\varvec{u}_S \!=\!\varvec{0}\). These constitutive assumptions are embodied in the following equations.

wherein \(\mu _i\) are the dimensionless viscosities of each phase, and \(\Sigma _{n}\) and \(\psi _{nS}\) are defined

In Eq. (26) \(\nu ,\ \chi ,\ \delta _a,\ \delta _b>0\) dictate the cells’ tendency to aggregate, their affinity for the scaffold/ECM and the strength of cell–cell/cell–scaffold repulsion. In a more general formulation, the coefficient of viscous drag \(k\) between two phases \(i\) and \(j\) varies depending upon the phases under consideration and may depend upon their respective volume fractions or other state variables. A suitable representation is to replace \(k\) by, say, \(k_{ij}(\theta _i,\theta _j)\) (obeying \(k_{ij}=k_{ji}\)). Full details and discussion of the above choice of interphase interaction terms may be found in Lemon et al. (2006).

Figure 9 depicts the two-dimensional model of the bioreactor, together with appropriate boundary conditions. These correspond to no-slip and no-penetration of cells or culture medium through the channel walls, a pressure-driven flow imposed via up- and downstream pressures \(K^*\mu _w^* P_U\) and \(K^*\mu _w^* P_D\), partitioned normal stress conditions and fully-developed flow at \(x=0,1\).

We now simplify the two-dimensional equations by considering the limit for which the aspect ratio of the bioreactor is asymptotically small \((h\ll 1)\). We remark that, since the culture medium volume fraction may be eliminated via \(\theta _w=1-\theta _n-\theta _s-\theta _e\) and the substrate is rigid, we need consider momentum conservation equations for the fluid (cell and culture medium) phases, only.

Following O’Dea et al. (2010), the reduced model is obtained by rescaling according to:

and averaging across the channel in the transverse direction (imposing the boundary conditions at \(y=0,h\) depicted in Fig. 9). We find that the pressure and the volume fraction of each phase are functions of \(x\) and \(t\) only and the flow of cells and culture medium is unidirectional at leading order \((\overline{v}_i=0)\). Expressions for the averaged axial velocities \(\langle \overline{u}_w \rangle \) and \(\langle \overline{u}_n \rangle \) are obtained from the remaining momentum equations, on substitution of which into the (averaged) mass conservation equations (dropping the overbars), we obtain the following system of coupled partial differential equations for the volume fractions \(\theta _e(x,t),\,\theta _s(x,t),\ \theta _n(x,t)\) and the culture medium pressure, \(p_w(x,t)\):

in which \(\mu _n\) is the relative viscosity of the cell and culture medium phases. The extra pressures \(\Sigma _n\) and \(\psi _{nS}\) are scaled according to Eq. (27) so that these interactions are retained at leading order, which implies \((\nu ,\delta _a,\chi ,\delta _b)=(\bar{\nu },\bar{\delta }_a,\bar{\chi },\bar{\delta }_b)/h^2\); the remaining parameters are \(\mathcal O (1)\). Equations (28)–(31) embody conservation of mass for the ECM, PLLA scaffold and cell phases, and the multiphase mixture. Dropping the subscripts on the interaction functions \(\Sigma _n\) and \(\psi _{nS}\) gives the equations stated in the main text.

Under the rescaling, (27), the boundary conditions shown in Fig. 9 become:

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O’Dea, R.D., Osborne, J.M., El Haj, A.J. *et al.* The interplay between tissue growth and scaffold degradation in engineered tissue constructs.
*J. Math. Biol.* **67**, 1199–1225 (2013). https://doi.org/10.1007/s00285-012-0587-9

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DOI: https://doi.org/10.1007/s00285-012-0587-9