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

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## Abstract

**:**

## 1. Introduction

^{−}, NO

_{3}

^{−}) or large polyelectrolyte molecules (e.g., sodium polystyrenesulfonate (PSS))—are a key ingredient for adjusting the conductivity of the hydrogel [15,19].

_{4}has been suggested for CTE [25]. To fabricate hydrogels that mimic the complexity of articular cartilage, 3D printing techniques have been identified as a promising route [26,27]. The interaction of the scaffolds with the chondrocytes depends on the structure of the scaffold. For example, in the case of fibrous collagen scaffolds, the scaffold fibres should have a smaller diameter than the cells to ensure a spherical cell shape [28] that is beneficial for chondrogenesis [29]. Spherical chondrocytes have occured dominantly in sponge-like collagen scaffolds [28]. Printed scaffolds allow to include tailored macroscopic porosity in the range of several hundred microns as well as microporosity by the hydrogel itself. In addition, hydrogel porosity can be tuned by, e.g., freeze-drying techniques to optimize the microporosity towards increased cell–material interaction [30,31]. Furthermore, recent research has demonstrated the 3D printing of electroactive hydrogels with tuneable conductivity [7,27]. Both the conductivity of about $1.5{\mathrm{S}\mathrm{m}}^{-1}$, which is commonly assumed for a cell culture medium [32], and $1{\mathrm{S}\mathrm{m}}^{-1}$, which is approximately the conductivity of bovine articular cartilage [33], are included in the feasible range. Hence, established 3D printing techniques can be employed to manufacture hydrogels that mimic the electrical properties of the cellular ECM environment and of common in vitro cell culture conditions. Unlike the conductivity, the permittivity of the scaffolds has usually not been investigated and it is not clear if it can be tailored as well.

## 2. Results and Discussion

#### 2.1. Validation of the Modelling Approach

#### 2.2. A Numerical Model for Cell-Laden Electrically Conductive Hydrogels

- ${\sigma}_{\mathrm{hydro}}>{\sigma}_{\mathrm{buf}}\Rightarrow {E}_{i}<{E}_{o}$.
- ${\sigma}_{\mathrm{hydro}}<{\sigma}_{\mathrm{buf}}\Rightarrow {E}_{i}>{E}_{o}$.
- ${\sigma}_{\mathrm{hydro}}={\sigma}_{\mathrm{buf}}\Rightarrow {E}_{i}={E}_{o}$.

^{2+}when triggered, which has been linked to the effect of ES [37]. However, Xu et al. studied chondrocytes seeded on cover slips, which corresponds to a 2D cell culture, i.e., the benchmark problem of an elliptical cell adhering to the bottom of the ES chamber (Figure 5). Voltage-gated channels are activated by changes in the TMP. In general, a change in the TMP ranging between ${10}^{0}$ mV [43] and 10

^{2}mV [40] has been estimated to be sufficient for a significant effect. While we found an increased TMP at $60$ kHz for the benchmark model, but not for cells in contact with hydrogels, an effect on cells inside a hydrogel due to a change in the TMP seems unlikely. Frequencies, for which we found the TMP to reach biologically relevant values in the $\mathrm{mV}$ range, have yet not been considered in CTE. They could be tried in the future when conducting experiments with electrically conductive hydrogels. A frequency of about $5$ MHz could be optimal to verify the hypothesis that an induced change in the TMP causes the biological effect referring to the reported increased ECM synthesis [12,13,14], improved re-differentation of de-differentiated cells [9,10], and the enhanced chondrogenic differentiation [11].

#### 2.3. Theoretical Considerations Regarding Cellular Organisation

## 3. Materials and Methods

#### 3.1. Geometric Modelling and Equivalent Circuits

#### 3.2. Finite Element Analysis and Uncertainty Quantification

^{®}, V5.3a employing second-order Lagrange elements and a direct solver.

^{3}[86]. Thus, we considered a broad span from the cell culture medium case to the cartilage case in our UQ analysis of the hydrogel model (see Table 1).

## 4. Conclusions

- Development of chemically and electrically stable low-conductivity hydrogels (<$1.5{\mathrm{S}\mathrm{m}}^{-1}$) to increase the electrical field strength acting on the chondrocytes and thus increased change in their TMP. Here, low-impedance insulation of the electrodes from the cell culture medium becomes highly relevant to ensure an electrically efficient solution. Possible hydrogel materials could be, for example, solely ionically conductive hydrogels [7] or a suitable hydrogel functionalised with, for example, polypyrrole [25] or reduced graphene oxide [24].
- Usage of high-conductivity hydrogels (>$1.5{\mathrm{S}\mathrm{m}}^{-1}$) together with non-uniform, strong electric fields much greater than $1{\mathrm{V}\mathrm{m}}^{-1}$ to exploit the attraction of chondrocytes by low-field regions. This could be used to influence cellular ingrowth inside hydrogels and increase efficiency of initial cell seeding. Such high conductivities might only be reached by the use of, for example, highly conductive carbon nanotubes [20], reduced graphene oxide [21,24], or well-percolating doped conductive polymer networks made from, among others, polypyrrole or polyaniline [15,19].

## Supplementary Materials

## Author Contributions

## Funding

## Conflicts of Interest

## Abbreviations

UQ | Uncertainty Quantification |

FEA | Finite Element Analysis |

ECM | Extracellular Matrix |

TMP | Transmembrane Potential |

CM | Clausius-Mossotti |

MC | Monte Carlo |

PC | Polynomial Chaos |

TE | Tissue Engineering |

CTE | Cartilage Tissue Engineering |

ES | Electrical Stimulation |

GO | Graphene Oxide |

## Appendix A

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**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 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.

**Table 1.**Assumptions for the dielectric properties of electrically conductive hydrogels with an exemplary single cell model [32]. The uniform distribution is denoted by $\mathcal{U}$.

Domain (Subscript) | Conductivity $\mathit{\sigma}$[S m${}^{-1}$] | Permittivity $\mathit{\epsilon}$ |
---|---|---|

Hydrogel (hydro) | $\mathcal{U}(0.1,2.0)$ | $\mathcal{U}(60,1\xb7{10}^{3})$ |

Buffer medium (buf) | $\mathcal{U}(0.5,1.5)$ | $\mathcal{U}(60,80)$ (benchmark) or 80 |

Membrane (m) | $\mathcal{U}(0,5\xb7{10}^{-5})$ | $\mathcal{U}(5,15)$ |

Cytoplasm (cyt) | $\mathcal{U}(0.1,1.0)$ | 60 |

**Table 2.**Parameters for the numerical model based on [32].

Domain | Subscript | Electrical Conductivity [S m${}^{-1}$] | Rel. Permittivity |
---|---|---|---|

Insulator/Lid | ins | 0 | $2.6$ |

Cover slip | cs | 0 | 4 |

Culture medium | buf | $1.5$ | 80 |

Cytoplasm | cyt | $1.5$ | 60 |

Cell membrane | m | 0 | $11.3$ |

**Table 3.**Assumptions for the dielectric properties of eukaryotic cells as reported in [65]. The uniform distribution is denoted by $\mathcal{U}$.

Domain (Subscript) | Conductivity $\mathit{\sigma}$[S m${}^{-1}$] | Permittivity $\mathit{\epsilon}$ |
---|---|---|

Membrane (m) | $\mathcal{U}(8\xb7{10}^{-8},5.6\xb7{10}^{-5})$ | $\mathcal{U}(1.4,16.8)$ |

Cytoplasm (cyt) | $\mathcal{U}(0.033,1.1)$ | $\mathcal{U}(60,77)$ |

Nuclear envelope (ne) | $\mathcal{U}(8.3\xb7{10}^{-5},7\xb7{10}^{-3})$ | $\mathcal{U}(6.8,100)$ |

Nucleoplasm (np) | $\mathcal{U}(0.25,2.2)$ | $\mathcal{U}(32,300)$ |

**Table 4.**Assumptions for the geometric properties of eukaryotic cells as reported in [65]. Note that instead of the explicit nucleus radius ${R}_{\mathrm{n}}$, a scale parameter was introduced such that ${R}_{\mathrm{n}}=\mathrm{scale}\phantom{\rule{0.166667em}{0ex}}\xb7\phantom{\rule{0.166667em}{0ex}}{R}_{\mathrm{c}}$. This ensures that the nucleus radius is always less than the cell radius. The uniform distribution is denoted by $\mathcal{U}$.

Parameter | Symbol | Probability Distribution |
---|---|---|

Cell radius | ${R}_{\mathrm{c}}$ | $\mathcal{U}(3.5,10.5)\phantom{\rule{3.33333pt}{0ex}}\mathsf{\mu}\mathrm{m}$ |

Membrane thickness | ${d}_{\mathrm{m}}$ | $\mathcal{U}(3.5,10.5)\phantom{\rule{3.33333pt}{0ex}}$nm |

scale | $\mathrm{scale}$ | $\mathcal{U}(0.28,0.84)$ |

Nuclear envelope thickness | ${d}_{\mathrm{n}}$ | $\mathcal{U}(20,60)\phantom{\rule{3.33333pt}{0ex}}$nm |

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**MDPI and ACS Style**

Zimmermann, J.; Distler, T.; Boccaccini, A.R.; van Rienen, U. Numerical Simulations as Means for Tailoring Electrically Conductive Hydrogels towards Cartilage Tissue Engineering by Electrical Stimulation. *Molecules* **2020**, *25*, 4750.
https://doi.org/10.3390/molecules25204750

**AMA Style**

Zimmermann J, Distler T, Boccaccini AR, van Rienen U. Numerical Simulations as Means for Tailoring Electrically Conductive Hydrogels towards Cartilage Tissue Engineering by Electrical Stimulation. *Molecules*. 2020; 25(20):4750.
https://doi.org/10.3390/molecules25204750

**Chicago/Turabian Style**

Zimmermann, Julius, Thomas Distler, Aldo R. Boccaccini, and Ursula van Rienen. 2020. "Numerical Simulations as Means for Tailoring Electrically Conductive Hydrogels towards Cartilage Tissue Engineering by Electrical Stimulation" *Molecules* 25, no. 20: 4750.
https://doi.org/10.3390/molecules25204750