Numerical Simulations as Means for Tailoring Electrically Conductive Hydrogels Towards Cartilage Tissue Engineering by Electrical Stimulation

Abstract

Cartilage regeneration is a clinical challenge. In recent years, hydrogels have emerged as implantable scaffolds in cartilage tissue engineering. Similarly, electrical stimulation has been employed to improve matrix synthesis of cartilage cells, and thus to foster engineering and regeneration of cartilage tissue. The combination of hydrogels and electrical stimulation may pave the way for new clinical treatment of cartilage lesions. To find the optimal electric properties of hydrogels, theoretical considerations and corresponding numerical simulations are needed to identify well-suited initial parameters for experimental studies. We present the theoretical analysis of a hydrogel in a frequently used electrical stimulation device for cartilage regeneration and tissue engineering. By means of equivalent circuits, finite element analysis, and uncertainty quantification, we elucidate the influence of the geometric and dielectric properties of cell-seeded hydrogels on the capacitive-coupling electrical field stimulation. Moreover, we discuss the possibility of cellular organisation inside the hydrogel due to forces generated by the external electric field. The introduced methodology is easily reusable by other researchers and allows to directly develop novel electrical stimulation study designs. Thus, this study paves the way for the design of future experimental studies using electrically conductive hydrogels and electrical stimulation for tissue engineering.

Keywords: biomaterial scaffolds; capacitive coupling; computational modelling; electrical stimulation; electrically conductive hydrogels; finite element analysis; scaffold; tissue engineering; uncertainty quantification.

Figure 1

Potential design routes for electrically conductive scaffolds to be used in cartilage tissue engineering. Possible materials for the porous, non-conductive scaffold and for the conductivity tuning using conductive fillers and dopants are shown. The dopant changes the conductivity of the scaffold by adding or removing an electron from/to the polymer, which causes a lattice distortion inducing polarons that yield increased electric conductivity [19]. Carbon nanostructures can be integrated into the scaffold network and provide a pathway for the electric current [20,21]. The final electroactive hydrogel conductivity will be strongly dependent on the degree of percolation between the conductive fillers, purity and crystalinity of the conductive polymer, doping level, redox state of the conductive filler, diffusibility of and ion mobility in the final hydrogel, hydrogel porosity, and additional factors relevant for tuning the final hydrogel conductivity.

Figure 2

General geometry of the simulation model. (a) 3D representation of the axisymmetric set-up, consisting of (b): air, culture medium/buffer (blue), plastic dish (grey), and symmetry axis (red). The voltage is applied between the top side of the upper cover slip and the bottom side of the lower cover slip. The area highlighted in green is the area for which the parallel-plate capacitor approximation was applied.

Figure 3

Uncertainty quantification (UQ) result for (a) the absolute value and (b) the phase of the impedance of the equivalent circuit (Equation (1)). The mean value is shown together with $90\%$ prediction interval for a broad frequency range.

Figure 4

First order Sobol indices for (a) the absolute value of the impedance of the equivalent circuit (Equation (1)) and (b) for the electric field in the buffer medium (Equation (2)).

Figure 5

UQ results for the TMP at the cell apex (cell top; red dot) for the benchmark model, where an elliptical cell was assumed to adhere to the bottom of the chamber, representing a 2D cell culture. The mean value is shown together with $90\%$ prediction interval for a broad frequency range (left axis). The first order Sobol indices of the uncertain parameters are shown on the right axis (lines with markers). The parameters, which have Sobol indices less than $0.1$ over the entire frequency range, are not shown for the convenience of the reader. These parameters are buffer and cytoplasm conductivity as well as buffer and cytoplasm permittivity. It turns out that the slope of the TMP is mainly defined by the cell membrane conductivity (${\sigma }_{\mathrm{m}}$) and membrane permittivity (${\epsilon }_{\mathrm{m}}$).

Figure 6

Comparison of an elliptical cell on the top surface of the hydrogel (a–c) and a spherical cell seeded on/centred in the hydrogel (d–f). Here, we only report the case where the cell is located at the hydrogel–medium interface since there is a difference between the TMP at the top and bottom of the cell to be expected (Figure S8). The result for a single cell centred in the hydrogel is shown in Figure S9. It almost perfectly resembles (f). In each part of the figure, the point on the cell membrane, where the TMP was evaluated (or could have been evaluated yielding similar results in case of the spherical cell), is indicated by a red dot. The TMP for different hydrogel conductivities is compared (a,d). In (b–f), the UQ results are shown. The mean and the $90\%$ prediction interval (left axis) are shown together with the first order Sobol indices of the uncertain parameters (right axis). Tested parameters whose Sobol index does not exceed $0.1$ over the entire frequency are not shown for the convenience of the reader. Hence, only results for membrane permittivity (${\epsilon }_{\mathrm{m}}$), cytoplasm conductivity (${\sigma }_{\mathrm{cyt}}$), buffer conductivity (${\sigma }_{\mathrm{buf}}$), hydrogel conductivity (${\sigma }_{\mathrm{hydro}}$), and permittivity (${\epsilon }_{\mathrm{hydro}}$) are shown.

Figure 7

(a) Mean value and $90\%$ prediction interval for the real part of the CM factor, $\Re \left(\mathrm{CM}\right)$. (b) Ratio between the field at the top and at the side of a cell, which is centred in the hydrogel, for different conductivities. This means that the field is always greater at the side of the cell when the ratio is less than one.