A Phase-field model of cell motility in biodegradable hydrogel scaffolds for tissue engineering applications

Abstract

Tissue engineering aims at producing in the laboratory new biological tissues by combining cells, scaffold materials, and biochemistry. Recent successes in the field are promising, but sustainability and reproducibility issues still limit the large-scale applications of this technique. The present work addresses the development of a computational model that describes cell motility in biodegradable hydrogel scaffolds. The goal is to support the understanding and the control of the first stage of tissue formation, when cells seeded into the scaffold congregate to form clusters, necessary precursors of tissue blocks. Cellular migration is treated as an advective/diffusive process modeled via the phase-field approach. The chemo-biological mechanisms incorporated in the modeling framework are: (i) the natural degradation of the hydrogel; (ii) chemotaxis induced by cell-cell signaling pathways; (iii) nutrient diffusion through the construct and its consumption by cells. Each cell initially moves following a random path, subsequently switching to a directed chemically-driven motion when the presence of other cells is sensed in its neighborhood. Numerical results highlight the role of the interplay between nutrient availability in the construct, chemoattractant production, scaffold degradation and cell motility, showing that the proposed model could pave the way towards efficient computationally-aided tools for the optimization of neotissue mass production.

Appendix A: Notations and analytic expression of the sigmoid function

Throughout the paper, reference has frequently been made to the sigmoid (also termed logistic) function $\mathcal {S}(x)$, which is often employed to describe a smooth transition between a lower and an upper bound value. In detail, if x is a real-valued scalar variable, symbol $\mathcal {S}(x; \, A_-,A_+,x_0,\kappa _{\text {A}})$ can be used to synthetically denote a hyperbolic-tangent-type sigmoid function of x, parametrized by the asymptotes $y=A_{\pm }$ for $x \rightarrow \pm \infty$ and by parameters $x_0$, $\kappa _{\text {A}}$, i.e.:

$$\begin{aligned}{} & {} \mathcal {S}(x; \, A_-,A_+,x_0,\kappa _{\text {A}}) \nonumber \\{} & {} \quad = \frac{A_+ + A_-}{2} + \frac{A_+ - A_-}{2}\,\tanh \bigl ( \kappa _{\text {A}} (x-x_0) \bigr ) \end{aligned}$$

(54)

Specifically, $\kappa _{\text {A}} > 0$ is a parameter regulating the steepness of the variation between $A_+$ and $A_-$, and $x_0$ is the sigmoid center, namely the value of x such that $\mathcal {S}(x_0) = (A_+ + A_-)/2$ (see Fig. 11).

In the main text, the asymptotes always correspond to the values $y=0$ and $y=1$. As such, the following simplified notation is introduced:

$$\begin{aligned} \mathcal {S}_+(x; \, x_0, \kappa _{\text {A}})= & {} \mathcal {S}(x; \, 0,1,x_0,\kappa _{\text {A}}) \nonumber \\= & {} \frac{1}{2} \biggl (1+ \tanh \bigl (\kappa _{\text {A}} (x-x_0) \bigr )\biggr ) \end{aligned}$$

(55)

$$\begin{aligned} \mathcal {S}_-(x; \, x_0, \kappa _{\text {A}})= & {} \mathcal {S}(x; \, 1,0,x_0,\kappa _{\text {A}}) \nonumber \\= & {} \frac{1}{2} \biggl (1 - \tanh \bigl ( \kappa _{\text {A}} (x-x_0) \bigr )\biggr ) \end{aligned}$$

(56)

where the subscript $+$ (respectively, −) suggests that the sigmoid is an increasing (resp., decreasing) function of x between 0 and 1.

Finally, note that if a real function F assumes a constant value $\tilde{F} \ne 0$ only in a bounded interval $[x_1,x_2]$ of real numbers, it can be approximately described in a smooth way through the following function:

$$\begin{aligned} F = \tilde{F} \, \min \{ \mathcal {S}_+(x; \, x_1, \kappa _{\text {A}}), \mathcal {S}_-(x; \, x_2, \kappa _{\text {A}}) \}. \end{aligned}$$

(57)