Evaluating the electrical stimulation of bone cells based on an induced transmembrane potential model and intracellular calcium levels

Abstract

Electrical stimulation holds promise for enhancing bone healing; however, it has not yet seen widespread clinical adoption. A significant obstacle is the limited understanding of the biological mechanisms involved and the electric field parameters required to trigger them. It has been observed that the intracellular calcium ion concentration increases upon electrical stimulation, possibly via the activation of voltage-gated calcium channels. In this work, we introduced a digital twin framework to rationally choose stimulation parameters. We aimed to induce a transmembrane potential sufficient to activate the voltage-gated calcium channels. We focused on kilohertz-frequency stimulation, which offers advantages for clinical translation, and applied electrical stimuli using a well-established direct-contact stimulation chamber. By combining this with live-cell calcium imaging, we observed the immediate stimulation effect. We found that a stimulation at $1$  kHz or $100$  kHz, adjusted to induce a transmembrane potential of about $10$  mV, did not alter the intracellular calcium concentration. In contrast, direct current stimulation at $5$  V consistently increased intracellular calcium concentrations. However, our results indicate that this effect is not caused by the electric field itself but by electrochemical by-products leading to local changes in the pH value. We further confirmed that chemical stimulation could reproduce the effect. The presented workflow enables researchers to distinguish between purely electrical and mixed electrochemical stimulation. It is easily transferable and contributes to a more precise understanding of the effects of electrical stimulation.

Introduction

Improving the bone healing process is of high clinical relevance, for example, for non-union fractures or critical-size defects. ¹ Initiated by Yasuda in the early 1950s, electrical stimulation has been considered a biophysical therapy for bone injuries. However, electrical stimulation has not yet become a standard clinical treatment option for bone regeneration because of insufficient evidence of its effectiveness. ² It has been hypothesised that conflicting observations are linked to the unknown local electric field and insufficient technical documentation of in vivo experiments. ³ Nonetheless, many studies have reported a positive effect of electric fields on cells in cell culture experiments. ⁴

A major limitation of most in vitro studies is the absence of a clearly defined electrical stimulation-related objective. Such a goal could be, for instance, a target electric field strength at the site of injury ⁵ or an induced transmembrane potential (iTMP) deemed sufficient to activate voltage-gated membrane channels. However, clear activation thresholds have not yet been established. As a result, Verma et al., for example, focused on maximising the electric field strength at the injury site, comparing it to field strengths commonly used in neuromodulation, given the limited knowledge of target field strengths for bone healing. ⁶

Calcium ions ( ${\mathrm{Ca}}^{2+}$ ) have emerged as a focal point in unravelling the intricate mechanisms by which electrical stimulation elicits biological responses in bone-forming cells. Voltage-gated calcium channels have been reported to open in response to electrical stimulation ⁷ and have therefore been considered potential targets for optimising electronic bone growth stimulators. ⁶ Many publications show that electrical stimulation elevates the intracellular calcium ion concentration.^8–11

${\mathrm{Ca}}^{2+}$ is an important extracellular and intracellular messenger. Increased extracellular ${\mathrm{Ca}}^{2+}$ levels have been shown to lead to increased intracellular ${\mathrm{Ca}}^{2+}$ concentrations and osteoblast proliferation.^12,13 The extracellular and intracellular ${\mathrm{Ca}}^{2+}$ levels are connected by an interplay of membrane channels, calcium-sensitive receptors, and calcium-sensing proteins such as calmodulin, which activate downstream signalling pathways that ultimately enhance osteogenic differentiation.^14–16

The membrane channels, which regulate the inflow of ${\mathrm{Ca}}^{2+}$ ions from the extracellular space, can be classified into mechanosensitive cation and voltage-gated calcium channels. Stretch-sensitive transient receptor potential channels have been demonstrated to mediate the effects of electrical stimulation, as evidenced by their significantly upregulated expression following seven hours of direct current (DC) stimulation. ¹⁷ Furthermore, the mechanosensitive piezo channels have been shown to be involved in rejuvenating and enhancing osteogenic differentiation of aged bone marrow-derived stem cells through electrical stimulation. ¹⁸ Voltage-gated calcium channels can be divided into two functional types: the high-voltage and the low-voltage activated channels. ¹⁹ High-voltage-activated channels require a strong membrane depolarisation. For example, in mouse or rat osteosarcoma cells, high-voltage activated calcium channels begin to conduct inward currents at membrane potentials near −30 mV to $0$  mV. ²⁰ Low-voltage activated ${\mathrm{Ca}}^{2+}$ channels are activated at membrane potentials around $-50$  mV. ²¹

To establish the iTMP that needs to be induced by electrical stimulation to sufficiently depolarise the cells, it is important to know the resting membrane potential. It has been reported to be $-60$  mV in MG-63 cells. ²² Thus, a depolarisation of $10$  mV (i.e. an iTMP of $10$  mV) should be sufficient to open low-voltage calcium channels and induce an increase in intracellular ${\mathrm{Ca}}^{2+}$ . Osteoblasts and osteoblast-like cells such as MG-63 were reported to transcribe various low-voltage channel subunits. ²³ Notably, the ${\mathrm{Ca}}_v$ 3.1– ${\mathrm{Ca}}_v$ 3.3 channels are defined by a short transient opening slightly above the resting potential.^19,24 Previous work by our group identified the ${\mathrm{Ca}}_v$ 3.1 to be expressed in MG-63 cells. ²⁵

Optimising the electrical stimulation parameters with respect to the iTMP is not straightforward. An analytical expression can be used to estimate the iTMP for a spherical cell in a spatially homogeneous electric field (as, e.g. generated by a parallel-plate capacitor). ²⁶ These assumptions generally do not hold true for in vitro electrical stimulation with adherent cells. ²⁷ Instead, numerical methods must be used to obtain a reliable estimate of the extracellular field ²⁸ and compute the iTMP based on realistic cell geometries. ²⁹ The numerical models can account for mild electrochemical reactions but must be continuously calibrated by impedance measurements.^30,31 This approach can be considered a digital twin because a validated model is updated by monitoring data to yield reliable predictions of the electric field, thereby enabling direct control of the electrical stimulation experiment. However, the digital twin workflow is not applicable in the presence of strong, irreversible electrochemical reactions, which can occur during DC stimulation. ²⁸

In recent years, various stimulation devices have been proposed. A stimulation chamber originally designed to promote osteogenic differentiation ³² has become a popular choice for electrical stimulation in 6-well plates due to its simple and reproducible fabrication.^{28,31,33–35} It supports stimulation with both DC and alternating current (AC) signals. There is no consensus on an optimal frequency for bone healing via electrical stimulation. Reported frequencies range from a few Hz to $60$  kHz. ³ In previous research, we found that biphasic pulses at $20$  Hz and 6 mA induced electrochemical processes at the electrode-electrolyte interface. ³¹ These electrochemical processes complicated the interpretation of results, as electrical and electrochemical effects on the cells could not be clearly distinguished. Conclusions about the impact of electrical stimulation could only be drawn by conducting separate chemical stimulations. Similar observations have been made in the field of neuromodulation, where it has been shown that typical charge-balanced stimulation waveforms lead to significant production of reactive oxygen species (ROS). ³⁶ Given that most studies neglect the chemical aspects of electrical stimulation, experimental protocols should be designed to detect or minimise electrochemical reactions, thereby enabling the isolation of purely electrical effects or gaining insight into the importance of ROS.

Therefore, we have investigated the underlying biological mechanisms by examining the immediate effect of solely electrical stimulation on osteoblast-like MG-63 cells in this work. We focused on the well-accepted hypothesis that immediate calcium events are triggered by the electrical stimulation when the iTMP is sufficiently large. We have employed MG-63 osteoblasts due to their well-characterised calcium responses and reproducible resting membrane potentials. This stability allowed us to define induced transmembrane potentials ( $10$  mV) necessary for hypothesis-driven electrical stimulation. While primary osteoblasts more closely represent native bone, their resting potentials vary between donors, making them unsuitable for the precise electrical parameters required in this study.

We have integrated the aforementioned stimulation chamber with live-cell imaging of intracellular ${\mathrm{Ca}}^{2+}$ to observe the effect of electrical stimulation in real time. Going beyond the state-of-the-art, we have used the resulting imaging data to extend the previously introduced digital twin of the stimulation chamber ²⁸ and achieved direct control of the electric field reaching the cells by coupling macro- and cellular-scale models. The macroscale model provides the boundary conditions for the cellular-scale model. The cellular-scale model is a fine-grained numerical representation of cells based on 3D confocal fluorescence microscopy images used to obtain cell-specific iTMP and the local electric field distribution. At the macroscale, the electric potential in the cell culture was obtained based on thorough validation and calibration. By combining these simulations with the voltage and current measurements as well as live-cell imaging during electrical stimulation, we are able to monitor the iTMP during the experiment and rationally select stimulation parameters that induce transmembrane potentials potentially strong enough to open voltage-gated calcium channels.

Materials and methods

Cell culture

Osteoblast-like cells, MG-63 (CRL1427™, ATCC), were cultured under standard cell culture conditions at $37$ °C and $5$ % ${\mathrm{CO}}_2$ in DMEM (Gibco, Thermo Fisher Scientific) with $10$ % foetal bovine serum (FBS Superior; Sigma Life Science, Thermo Fisher Scientific) and $1$ % gentamicin ( $5$  mg/mL, Ratiopharm). The cells were used from passages 5 to 30. ³⁷ For electrical stimulation, $2\times {10}^4$  cells/cm² were seeded in six-well glass bottom plates (Cellvis, Mountain View) containing 3 mL DMEM without pyruvate (Gibco, $10\%$ FBS, $1\%$ gentamicin). The cells were incubated for two hours to let them adhere to the surface. Afterwards, the staining for intracellular ${\mathrm{Ca}}^{2+}$ ions was performed. During live-cell imaging, the cells were kept at $37$ °C and $5$ % ${\mathrm{CO}}_2$ .

Staining and live-cell imaging

After cell adhesion, the intracellular ${\mathrm{Ca}}^{2+}$ ion concentration was detected via Ca²⁺-sensitive dye Calbryte™ 520 AM (AAT Bioquest). Cells were stained with 5µM Calbryte™ 520 AM, mixed with $1\%$ PowerLoad (Invitrogen, Thermo Fischer Scientific) in hypotonic HEPES buffer for $30$ min. ³¹

After incubation, the cells were washed with phosphate-buffered saline and covered with $3$ mL FluoroBrite™ DMEM (Gibco, Thermo Fischer Scientific). This medium is optimised for fluorescence imaging ³⁸ and was supplemented with $4$ mM Glutamax, $10\%$ FBS (FBS Superior; Sigma Life Science, Thermo Fisher Scientific) and $1\%$ gentamicin (5 mg/mL, Ratiopharm). For control experiments, $2.5$ mM of the calcium ion chelator 1,2-bis(o-aminophenoxy)ethane-N,N,N′,N′-tetraacetic acid (BAPTA Tetrapotassium Salt, cell impermeant, Invitrogen) was added to the medium to eliminate extracellular calcium ions. Additionally, $50$ µM verapamil, $1$ µM thapsigargin (both Sigma-Aldrich), or 5mM N-acetylcysteine (NAC, abcam) was added to investigate the contributions of L-type calcium channels, endoplasmic reticulum calcium stores, and ROS, respectively.

The samples were directly put under the live-cell imaging incubator hood at $37$ °C, $5\%$ ${\mathrm{CO}}_2$ . The fluorescence was imaged using the confocal laser scanning microscope (LSM 780, Carl Zeiss Microscopy) with a $40\times$ /1.2 water immersion objective (Carl Zeiss Microscopy) and the Zen software (Zen 2.3 SP1 FP3 black, Carl Zeiss Microscopy). The frame size was $1024\times 1024$ pixels (212.55 µm × 212.55 µm) per image. Images were acquired every 5 s for 96 cycles (eight minutes) in the centre of the well.

During the image acquisition of unstimulated controls, the electrodes were still in contact with the cell culture media. Electrical stimulation, hydrogen peroxide ( $H_2{O}_2$ ), or sodium hydroxide (NaOH) was applied for stimulation at minute three of the image acquisition time course. Data on calcium levels of single cells were obtained from $n=3$ to $7$ independent biological experiments per stimulation condition, each conducted on separate days using independently cultured MG-63 cell populations. In every experiment, more than ten individual cells were analysed.

For the simulation of the iTMP at a cellular scale, 3D images of the cells were required. To obtain these images, MG-63 cells were stained with the red fluorescent cell linker PKH26 (Sigma-Aldrich, Merck KGaA) following the manufacturer’s protocol. Afterwards, the cells were seeded into six-well glass-bottom plates (Cellvis, Mountain View, USA) and incubated for two hours. Following adhesion, the cells were stained with Hoechst H33342 $10$ µg/mL in PBS (Life Technologies, Thermo Fisher Scientific) and calcein-AM (2 mM, LIVE/DEAD™ viability and cytotoxicity kit, Invitrogen) for five minutes. After washing, DMEM FluoroBright was added to the wells, and z-stacks of the cells were acquired at $37$ °C and $5\%$ ${\mathrm{CO}}_2$ using an LSM 780 (Carl Zeiss Microscopy) with a 40×/1.2 water immersion objective (Carl Zeiss Microscopy) and the ZEN software (version 2.3 SP1 FP3 Black; Carl Zeiss Microscopy). Z-stacks were recorded with a step size of $1$ µm, optimised for each image to a total height of around $20$ µm. The frame size was $1024\times 1024$ pixels (212.55 µm × 212.55 µm) per image.

pH and hydrogen peroxide measurement

pH was measured before and after electrical stimulation using a MicroFET pH Probe (Sentron Europe BV). For this, 100µL of medium was collected from three distinct spots (cathode, centre, anode) immediately after 5 min DC stimulation. $H_2{O}_2$ was quantified in DMEM FluoroBrite using the fluorometric hydrogen peroxide assay kit (Sigma-Aldrich, Merck KGaA) according to the manufacturer’s protocol. Fluorescence was measured with the fluorescence reader (ex/em: $540$ / $590$ nm; infiniteM200, Tecan Group Ltd.) after stimulation with 5V for $5$ min. $H_2{O}_2$ concentration was determined using the standard curve.

Electrical stimulation

Direct contact electrical stimulation was applied through a stimulation chamber consisting of six L-shaped platinum electrode pairs, which can be inserted into a six-well plate.^31,32 The entire setup is seen in Figure 1. The electrodes are submerged in the cell culture media with the horizontal part just above the bottom of a well. Each electrode pair was connected to a circuit board to connect and stimulate each well individually. A function generator ( $60$ MHz arbitrary waveform generator 4115, PeakTech GmbH) was used to generate AC (sinusoidal waveform) and DC signals. The signal was amplified 10 times by a PA1011 Power Amplifier (Rigol Technologies, Inc.). The applied voltage was measured with an oscilloscope (digital oscilloscope DS1054, Rigol Technologies Inc.). Currents were calculated by measuring the voltage per well across a shunt resistor with a resistance of 47 Ω. Measurements were saved every minute using a Python script ^* . This procedure ensured that each experiment and stimulation was documented and that biological results could be directly associated with the applied stimuli. The recorded measurements served as input for simulating the electric field corresponding to each stimulation scheme.

Figure 1.

Overview of the electrical stimulation setup. (A) Stimulation set up under the microscope. (B) Stimulation chamber (top view) with two L-shaped platinum wires in each well (diameter: 1 mm, distance: 26 mm, horizontal length: 22 mm, vertical length: $19$ mm). (C) Schematic of electrode position and imaging position in the well.

Data analysis of calcium levels

To ensure standardised analysis of the fluorescence signals from individual cells over time, the pre-trained deep learning segmentation algorithm Cellpose (v.3.1.1.1) with cyto3 model ³⁹ was utilised. The segmentations were then manually reviewed and adjusted as necessary to separate touching cells or to merge fragmented regions of the same cell into a single object (see Figure 2). Manual corrections were performed using Napari (v.0.5.6). ⁴⁰ Subsequently, cell segmentation masks were linked across time points using u-segment3D (v.0.1.1). ⁴¹ In this way, 2D masks belonging to the same cell had the same index at all time points.

Figure 2.

Data processing steps. Workflow for automated segmentation and cell tracking mask for fluorescence analysis. The video was segmented frame-by-frame using the Cellpose3 model. Manual correction was occasionally required due to unclear boundaries between touching cells and irregular cell shapes. Quantitative evaluation of the Cellpose3 model using manually corrected masks as ground truth yielded an accuracy of $0.74$ at an overlap threshold of $\tau =0.5$ . Scale bar: $20$ µm. The values were exported as CSV files and analysed using R. The mean fluorescence intensity (MFI) values of the intracellular ${\mathrm{Ca}}^{2+}$ (i ${\mathrm{Ca}}^{2+}$ ) ions were normalised to the minimum fluorescence of each cell. Finally, peaks were detected for each cell using our custom peak detection algorithm within the timeframe before the onset of electrical stimulation (no ES) and during electrical stimulation (ES). Orange dashed lines: detected intensity peaks; purple dotted lines: start and end of each peak; blue dashed line: baseline used for peak detection; red dash-dotted lines: end of non-ES and ES periods for statistical analyses.

The fluorescence intensity data obtained from this semi-automated image analysis pipeline were then processed using R (v.4.4.1) ⁴² and RStudio (2024.04.2, Posit Software). In the R script, the fluorescence intensities of each recording were normalised first. Debleaching was not necessary. Next, the maximum peak of a Gaussian kernel density estimate of the intensity values over time was calculated for each individual cell to serve as the threshold for peak detection. The detected peaks were then analysed to determine their widths and intensities. The R script is available in our data repository. Figure 2 shows an example of this data analysis.

Accelerating and improving segmentation in calcium imaging with deep learning

To support future automation and reduce the need for manual corrections, we trained a dedicated segmentation model using the InstanSeg framework, ⁴³ with manually corrected masks serving as the ground truth. A total of $3404$ images from 36 calcium imaging videos were used for training, along with $192$ validation and $304$ testing images. Details of the model architecture and training parameters can be found in Supplemental Table S1, the segmentation performance is summarised in Supplemental Table S2, and an example is shown in Supplemental Figure S1.

Statistical analysis

The output from the single-cell calcium analysis script was imported into a separate R script. Data from all experiments were combined into a single table, with each entry annotated according to the respective experimental conditions. The output from the single-cell calcium analysis script was imported into a dedicated R script for further evaluation. For each oscillating cell, we analysed calcium signal characteristics – including number, intensity, duration, and frequency of the peaks – by comparing the same cell during the initial unstimulated period (first three minutes) with the equivalent time interval during subsequent control or stimulation phase. This cell-wise comparison allowed us to precisely evaluate potential effects of electrical stimulation on occasionally occurring spontaneous oscillations. Statistical analyses were performed using the Kruskal-Wallis test for non-normally distributed samples, followed by Dunn’s test for multiple comparisons, with p-values adjusted for multiple testing using the Bonferroni method. The same tests were used to analyse the chemical stimulations.

Construction of the image-based digital twin

To monitor and control the electrical stimulation, we further advanced the methodology proposed in our earlier research. ²⁸ The digital twin construction process involved the following preparatory steps prior to simulation:

1. Electrochemical characterisation of the stimulation setup using electrochemical impedance spectroscopy (EIS).

2. Extraction of equivalent circuit parameters by fitting the impedance data with ImpedanceFitter. ⁴⁴ Some parameters are related to the electrochemical interface, while others can be directly attributed to the electrode geometry and conductivity of the cell culture medium.

3. Validation of the macroscopic numerical model of the chamber by comparing the numerically estimated medium resistance with the resistance values obtained from EIS measurements.

4. Acquisition of fluorescent confocal images of the cells in the chamber.

5. Generation of a finite element mesh from the 3D images.

During the electrical stimulation, the following steps were carried out:

1. Recording the applied current and voltage.

2. Comparison of measured and predicted current from the impedance model using the parameters obtained before stimulation for quality control.

3. Calculation of the voltage drop across the culture medium, which is used as a boundary condition for the macroscale numerical model.

4. Extraction of the electric potentials on the boundaries of the microscope field of view, which is used as a boundary condition in the cell-scale model.

5. Calculation of the electric field distribution and the iTMP in the cell-scale model.

6. Adjustment of the applied voltage or current to reach a desired iTMP or local electric field.

Electrochemical impedance spectroscopy

We conducted EIS without cells, as previous work demonstrated no detectable difference between the impedance spectrum with or without cells. ²⁸ All six wells were filled with $3$ mL of conductivity standards or FluoroBrite DMEM at 37 °C and characterised by conducting EIS using a Reference 620 potentiostat (GAMRY instruments, Warminster, PA, USA). This procedure ensured that all wells had similar electrochemical properties. Additionally, EIS measurements were conducted on a single well at varying input voltages, ranging from 10 mV_rms to 1V_rms.

The conductivity of the cell culture medium FluoroBrite DMEM at 37°C was determined by measuring the impedance with the BDS1309 sample cell and the NEISYS potentiostat (Novocontrol, Montabaur, Germany). The impedance spectra were fitted to an equivalent circuit model comprising a constant-phase element in series with a resistor and analysed using the open-source package ImpedanceFitter. ⁴⁴ The conductivity was computed from the fitted resistance using the known cell constant of the BDS1309 measurement chamber.

Electric field simulation

The electro-quasistatic field equation

$$\nabla \cdot [\left(\sigma +j\omega \epsilon \right)\nabla \varphi ]=0$$ (1)

was solved, where $\sigma$ is the conductivity, $\epsilon ={\epsilon }_r{\epsilon }_0$ is the permittivity with ${\epsilon }_0=8.854\times {10}^{-12}$ F/m, and $\omega$ is the angular frequency. ⁴⁵

Equation (1) was solved using the finite element method (FEM) with second-order curved elements in the open-source software NGSolve. ⁴⁶

We adopted the macro-scale model of the stimulation chamber described by Zimmermann et al. and its associated uncertainty quantification to account for the impact of input parameter uncertainties on the numerical results. In this model, the geometry consists of a single domain (the medium), therefore, solving Laplace’s equation is sufficient. The only modification was the use of $3$ mL of culture medium in our setup. A $90\%$ prediction interval for the chamber medium resistance and the electric field was estimated based on the most influential parameters: electrode spacing, horizontal electrode length, and culture medium volume, assuming a 2 % variation in medium conductivity as described previously. ²⁸

At the micro-scale, two modelling approaches were considered. The single-shell model includes only the cell and its membrane, whereas the double-shell model comprises cytoplasm, cell membrane, nuclear envelope, and nucleoplasm. As the cell membrane (thickness: $7$ nm) and nuclear envelope (thickness: $40$ nm) are significantly thinner than other model components, they were modelled using a thin-layer approximation.^29,47

The dielectric properties used in the model are summarised in Supplemental Table S3. Values for the membrane, cytoplasm, nuclear envelope, nucleoplasm, and extracellular medium were taken from the literature. ⁴⁸ The conductivity of the extracellular medium σ = 1.892 S/m was derived from impedance spectroscopy measurements at 37°C.

To ensure mesh convergence, we applied adaptive mesh refinement to the base cellular geometry using the Zienkiewicz-Zhu error estimator. ⁴⁹ The refinement process was performed until the relative difference in the magnitude and phase of the impedance fell below $0.01\%$ .

The iTMP was evaluated as

$$\mathrm{iTMP}=\left|{\varphi }_o-{\varphi }_i\right|,$$ (2)

where ${\varphi }_o$ and ${\varphi }_i$ denote the electric potentials outside and inside the cell membrane, respectively.

As equation (1) is a linear partial differential equation, the iTMP, electric field, and current scale linearly with the applied voltage difference $U$ . In all numerical experiments, we used the reference voltage U₀ = 1 V. The resulting reference values for observables such as the voltage drop across the image boundaries, iTMP, and the electric field, can be scaled to match the experimental conditions via

$$X_1=\frac{U_1\cdot X_0}{U_0}=\frac{U_1}{1V}\cdot X_0,$$ (3)

where $X_0$ and $X_1$ are the reference and scaled values of the respective observable.

To validate the macroscopic model, the resistance $R$ of the cell culture medium is computed using

$$R=\frac{U}{I}=\frac{{\sigma }_0}{\sigma \left(T\right)}\cdot \frac{U_0}{I_0},$$ (4)

and compared with experimental measurements. ²⁸ Here, ${\sigma }_0=1$ S/m denotes the reference conductivity, $\sigma \left(T\right)$ the temperature-dependent conductivity, and $I_0$ , $U_0$ the reference current and voltage, respectively.

The numerical computations were performed using a workstation equipped with Intel Xeon CPUs (24 cores total) and $256$ GB RAM and a NVIDIA RTX A6000 GPU ( $48$ GB VRAM).

Mesh creation from 3D fluorescent images

The simulation of the iTMP at a cellular scale was achieved using 3D images of the cells. We designed an automated process for image segmentation and mesh generation to enable fine-grained numerical simulations of cells (Figure 3(A)). First, the pre-trained deep learning segmentation algorithm Cellpose (cyto model) ⁵⁰ was used to segment each slice of the z-stack images individually. Next, 3D consensus cell segmentations were generated from the 2D segmented stacks using u-segment3D. ⁴¹ Cellpose segmented the weak intensity signals which did not belong to the cells. This resulted in oversized segmentations. To correct these, an additional 3D Otsu-foreground mask was applied. Further post-processing steps were required to enable mesh generation. This included removing border-touching cells and converting background pixels from intensity zero to intensity one, thereby treating the background as a single object. The cell regions were then relabelled using consecutive integer values starting from two. Padding the image with layers of zero intensity created an artificial background that framed the cells and ensured correct meshing of the bounding box by the CGAL algorithm. ⁵¹ We used CGAL to generate a surface mesh from a 3D labelled image, where each voxel was assigned a label corresponding to the object it represented (e.g. cell or nucleus). This resulted in a multi-domain geometry, with each subdomain representing an individual cell. Finally, a finite volume mesh was generated from the CGAL surface mesh using Netgen (v.6.2.2406). ⁵²

Figure 3.

Automated process for image segmentation and mesh generation. (A) Workflow for generating the mesh shown in Panel (C). The z-stack images were segmented slice-by-slice using Cellpose. ⁵⁰ The segmented slices were reconstructed into a 3D image and then combined with an Otsu-based ⁵³ foreground mask. Post-processing involved cleaning the 3D volume by removing edge-touching cells and padding, which was then used for mesh generation. (B) Comparison of mesh generation using CGAL (v.6.0.1) ⁵¹ with and without polyline features. Enabling polylines improves boundary precision. (C) From left to right: 3D view of the original image, the Cellpose segmentation, its combination with the Otsu-foreground mask, and the final mesh. (D) Illustration of the importance of using Otsu-foreground masks, especially in the uppermost slices of the z-stack. The enhanced-brightness image was used solely for visualisation purposes to illustrate how Cellpose performs predictions and was not utilised for segmentation. The length of the scale bar is $20$ µm.

Nuclei were optionally incorporated into the geometric model by merging 3D cell and nuclear label images, based on two criteria: (1) a nucleus was considered valid only if its voxels were fully enclosed within a single cell, and (2) cells were retained only if they contained at least one valid nucleus. Each cell and its corresponding nucleus were then assigned identical labels to facilitate the assignment of material properties. See the example in Figure 3, where no cells were removed.

Results

Establishing the digital twin of the stimulation chamber

The digital twin of the stimulation chamber at the macroscale was established in two steps. First, the conductivity of the cell culture medium was measured, and it was verified whether the macroscale model correctly predicted the resistance of the cell culture medium. Second, it was assessed whether the equivalent circuit could accurately predict the measured current.

The EIS spectra of the medium were obtained in a separate measurement chamber for liquid samples, and the conductivity was calculated from the fitted resistance to be $1.829$ S/m. Knowing the conductivity allowed us to numerically estimate the expected resistance in the stimulation chamber. Considering the uncertainty of the measured conductivity and geometric uncertainties, the uncertainty quantification analysis of the macroscopic volume conductor model yielded a resistance interval of 142 Ω $142$ to $164$ Ω. We acquired EIS spectra at $1$ V_rms and 0.1 V_rms, and fitted the parameters of the equivalent circuit model described by Zimmermann et al. to obtain a parametric impedance model including the resistance of the cell culture medium. ²⁸ The fitted resistance of $150$ Ω lay inside the resistance interval predicted by the numerical model. This validated the geometry of the numerical model and the conductivity value of the medium.

Next, we included the electrochemical interface impedance by considering the parametric equivalent circuit model calibrated by EIS. Furthermore, we added the shunt resistor, which was not present in the EIS measurements, to this model. We evaluated this parametric model at $1$ kHz and 100 kHz with an applied voltage amplitude of $10$ V. In all cases, the models obtained at the two EIS amplitudes were in good agreement and predicted currents with a relative deviation between the models of less than $5\%$ (see Supplemental Table S4). Compared to the measured current, the model obtained at 1V_rms EIS amplitude deviated by less than $5\%$ , whereas the model obtained at $0.1$ V_rms had one outlier with a relative difference of about $10\%$ . We therefore selected the impedance model obtained at $1$ V_rms as it provided a more accurate representation of the system (see an example in Supplemental Figure S2(A)). In this way, we established the digital twin, which permits the prediction of the stimulation electric field for an applied stimulation voltage and can be calibrated by matching the observed stimulation current.

The established digital twin enabled us to reliably estimate the voltage drop across the medium (excluding the electrochemical interface layer) at 1 kHz and 100 kHz (Supplemental Table S5 and Figure 4). The mean and standard deviation of the predicted voltage drop were computed from repeated measurements at each measurement condition to define boundary conditions for the macroscale numerical simulations (see Supplemental Table S5 and Supplemental Figure S3). The electric field strength in the centre of the well is approximately $300$ V/m at an applied voltage amplitude of $10$ V (Figure 5(A)). We evaluated the solution in the frequency domain; the time course of the electric field strength is a sine wave (see Figure 4). The field is generally uniform across the medium, with localised non-uniformities observed only near the electrode surfaces, where the cells were not seeded.

Figure 4.

Electrical stimulation during calcium imaging using alternating current. (A) Input voltage measurements and calculated voltage drop across the medium during stimulation with $20$ V_pp at $1$ kHz and 100 kHz. (B) Simulated electric field strength in the culture medium. The solid line indicates the mean value, and the shaded region represents the prediction interval. (C) Intracellular ${\mathrm{Ca}}^{2+}$ (i ${\mathrm{Ca}}^{2+}$ ) levels over time. Note that the stimulation began three minutes after the start of the image acquisition, as indicated by the vertical white line. MFI = mean fluorescence intensity.

Figure 5.

Results of macroscale and microscale numerical simulations at $10$ V and 100 kHz. (A) Voltage distribution (left) and electric field strength (right) in the central transverse cross-section of the stimulation chamber relative to the electrode pair. The voltage drop across the field of view (highlighted by the black rectangle) was extracted from the macroscale model and applied as a Dirichlet boundary condition for the microscale simulations. (B) Electric field distribution inside and around the cells, evaluated in the plane indicated by the dashed line in panel C. Microscale simulations are shown for models including (left) and excluding (right) the nucleus. (C) Absolute values of the iTMP for cells located at the centre of the chamber, corresponding to different cell distributions and spatial morphologies.

The successfully established digital twin is only valid for AC stimulations. DC stimulation can cause strong electrochemical reactions, which lead to a time-dependent change in the current response. Therefore, we measured the current amplitude over a time of five minutes while maintaining a constant voltage (Supplemental Table S6 and Supplemental Figure S2(B)). When applying 3 V and 5V, a significant decrease in current was observed after one minute, indicating electrochemical reactions at the electrode-electrolyte interface. After one minute, the current level reached a steady state, but the steady-state current varied considerably between different wells. This indicates that the underlying electrochemical reactions cannot be reliably described by the equivalent circuit model used for the DC case, and that a different modelling approach would be required to establish a digital twin. To estimate the voltage drop across the medium, we assumed that the medium resistance remains constant under DC stimulation and applied Ohm’s law. For each voltage condition (at 3V and 5V), the voltage drop across the medium was calculated at the beginning of the experiment and after the current had reached steady state. As in the AC case, the mean and standard deviation of the calculated voltage drops were determined from repeated measurements (see Supplemental Table S6). These values were used as boundary conditions in subsequent macroscopic numerical simulations (see Supplemental Table S6). The electric field in the centre of the well depended nonlinearly on the applied DC voltage. At $3$ V DC, the electric field decreased from $20$ V/m to approximately $10$ V/m after one minute and remained stable thereafter. The electric field at 5 V DC followed a similar course, decreasing from about $68$ V/m to $60$ V/m after one minute (see Figure 6).

Figure 6.

Electrical stimulation during calcium imaging using DC. (A) Current in the medium during stimulation experiments with 3V and 5V, respectively. (B) Simulated electric field strength in the medium. The mean is shown as a solid line. The shaded area represents the prediction interval. (C) Intracellular ${\mathrm{Ca}}^{2+}$ (i ${\mathrm{Ca}}^{2+}$ ) levels over the course of the experiments. Note that stimulation started at three minutes, as indicated by a thin vertical white line.

Estimating the induced transmembrane potential

The electric field in the cell culture medium is reliably computed in the macroscale numerical model. To connect this model to the cellular scale, we pursued a multiscale approach. We assumed that the cells perturb the electric field only locally. Because the cells are seeded sparsely on the bottom of the well, we assumed that the electric potential at the outer edges of the microscope’s field of view is independent of the presence of the cells. Accordingly, we extracted the solution of the macroscale numerical model on these surfaces and imposed it as a Dirichlet boundary condition in the cell-scale model based on the dimensions of the imaged z-stack.

Three z-stacks of MG-63 cells located at the centre of the well were segmented. These z-stacks were acquired prior to the stimulation experiments. Each 3D-image has a width of $212.15$ µm in x- and y-direction, with z-stack heights of 25.86 µm, 21.00 µm, and $18.90$ µm, respectively, to fully capture the entire cell layer. We established an automated segmentation pipeline based on the deep learning segmentation tool Cellpose ⁵⁰ and multi-Otsu’s thresholding ⁵³ that accurately captured the morphology of individual cells (see Figure 3 and the overlay of the segmentation mask on the original image in Supplemental Figure S4). More details are provided in the Methods section. The segmentation masks were meshed using CGAL. ⁵¹ We ensured that the resulting mesh preserves both the surface area and volume of each cell in the segmentation masks. Relative differences in surface area were less than $2\%$ , and relative volume differences were less than $1\%$ (see Supplemental Table S7). Furthermore, the mesh maintained the orientation and spatial position of the cells as observed in the original microscopic image (Figure 3(C)). It also accurately reproduced the boundaries of the field of view. We found that polyline features need to be enabled in CGAL to ensure that mesh points align precisely with the bounding box edges (see Figure 3).

We evaluated the simulations with and without modelling the cell nucleus under DC stimulation, as well as under AC stimulation at $1$ kHz and 100 kHz. A baseline voltage of $1$ V was applied in all cases, resulting in a voltage drop of approximately $7.88$ mV across the faces of the image box. Based on this voltage drop, we calculated the iTMP, electric field, and the impedance magnitude and phase (see Supplemental Table S8). To obtain actual values corresponding to experimental conditions, these baseline results were linearly scaled according to the applied voltage using equation (3). This approach eliminates the need to re-run simulations as long as only the applied voltage changes.

Based on the baseline simulations, both models, that is, with and without nuclei, produced virtually identical iTMP and impedances across all conditions. Nonetheless, we always included the nucleus, as it provides a more detailed understanding of the electric field distribution within individual cells. For example, under a $10$ V applied voltage at $1$ kHz, the intracellular electric field strength and its spatial distribution differ between the two models (Figure 5(B)). In the model including the nucleus, the field ranges from $0.05$ V/m to $0.35$ V/m, whereas in the model without the nucleus, it ranges from $0.1$ V/m to $0.25$ V/m.

In any case, the extracellular field reaches significantly higher values, up to $700$ V/m near the triple point, where the membrane, medium, and insulator meet. Our results illustrate that the electric field strength at the boundary of the modelling domain remains consistent with the macroscale value of $300$ V/m, despite the presence of the cells (compare the electric field shown in Figures 4(B) and 5(B)). This finding confirms the validity of our multiscale modelling assumption.

Using an applied voltage of $10$ V, the simulations yielded an iTMP close to $10$ mV for cells located at the centre of the chamber, depending on their specific cell morphology (see Figure 5(C)). This estimated iTMP aligns with the target values aimed for in the stimulation experiments. An overview of the resulting values is provided in Table 1.

Table 1.

Comparison of voltage drop and iTMP.

Experiment	Mean voltage drop / V		Voltage drop across 3D image / mV		iTMP / mV
$20$  Vpp, $1$  kHz	7.67 ± 0.08		60.44 ± 0.63		9.66 ± 0.10
$20$  Vpp, $100$  kHz	7.64 ± 0.07		60.20 ± 0.55		9.63 ± 0.10
	Initial	After $1$  min	Initial	After $1$  min	Initial	After $1$  min
DC, 3 V	0.50 ± 0.03	0.28 ± 0.03	3.94 ± 0.24	2.20 ± 0.24	0.70 ± 0.04	0.40 ± 0.04
DC, $5$  V	1.73 ± 0.03	1.54 ± 0.03	13.63 ± 0.24	12.14 ± 0.24	2.40 ± 0.04	2.14 ± 0.05

Intracellular ${\mathrm{Ca}}^{2+}$ oscillations without electrical stimulation

We tested the normal calcium dynamics in the absence of electrical stimulation but in the presence of the electrodes. The MG-63 osteoblasts were stained with a calcium-sensitive dye and imaged for eight minutes. Using the manually corrected masks as ground truth, we evaluated cell-tracking segmentation and found that the Cellpose3 model achieved an accuracy of $0.74$ at an overlap threshold of $\tau =0.5$ (see equation (S1) in Supplemental material). This indicates a reliable segmentation performance with a relatively low number of false or missed detections.

Intracellular ${\mathrm{Ca}}^{2+}$ levels fluctuated periodically in some cells (see Figure 7(C), cell 3), which exhibited strong, recurring peaks of high fluorescence. Not all cells in the field of view exhibited strong and well-defined oscillations: some did not display any oscillations at all (see, e.g. Figure 7(C), cell 11), while others showed slower changes in mean fluorescence intensity and generally lower fluorescence intensity levels. In control experiments, a median of $53.8\%$ of cells exhibited calcium oscillations. The oscillations occurred at a median frequency of $7.6$ mHz with an interquartile range (IQR) of $1.8$ mHz. To evaluate the influence of live cell imaging on the intracellular ${\mathrm{Ca}}^{2+}$ levels, we compared the first three minutes of observation with the following three minutes. The number of peaks, their intensity, and duration did not change over the time span (see Table 2). However, the percentages of oscillating cells and the frequencies varied between experiments, expressed by the relatively high IQR.

Figure 7.

Intracellular calcium ion oscillations in control cells. (A) Time series of live-cell imaging showing intracellular ${\mathrm{Ca}}^{2+}$ (i ${\mathrm{Ca}}^{2+}$ ) levels. ${\mathrm{Ca}}^{2+}$ ions depicted in green; arrows highlight elevated ${\mathrm{Ca}}^{2+}$ levels in selected cells. Scale bar: $50$ µm. (B) Normalised fluorescence intensity over time for individual cells under control conditions and after addition of the ${\mathrm{Ca}}^{2+}$ chelator BAPTA to the culture medium. (C) Heatmap of fluorescence intensity for each cell over the full 8-min observation period for control and BAPTA experiments. MFI = mean fluorescence intensity; arb. units = arbitrary units.

Table 2.

Analysis of intracellular ${\mathrm{Ca}}^{2+}$ oscillations.

Parameter	Control	Control BAPTA	$20$  Vpp, $1$  kHz	$20$  Vpp, $100$  kHz
Oscillating cells / %	53.8 (24.1)	0.0 (0.0)*	32.6 (17.6)	45.0 (12.1)
Frequency / mHz	7.6 (1.8)	n/a	6.4 (1.0)	7.6 (2.0)
$\Delta$ peak number	0.0 (0.0)	0.0 (0.0)	0.0 (0.0)	0.0 (0.0)
$\Delta$ peak intensity / arb. units	−0.14 (0.15)	n/a	−0.13 (0.12)	−0.09 (0.19)
$\Delta$ peak duration / s	5.0 (13.1)	n/a	−5.0 (19.9)	10.0 (11.2)
$n$	7	3	4	4

Median with interquartile range in $(\dots )$ . $\Delta$  = values during minutes three to six minus the values during minutes 0–three of the observation time. $n=3$ to $7$ biological replicates with $>10$ cells per experiment. Kruskal-Wallis test followed by Dunn’s post hoc test for multiple comparisons.*: $p\le 0.05$ compared to control. n/a indicates that analysis was not possible.

To examine the origin of the intracellular ${\mathrm{Ca}}^{2+}$ peaks, we added the ${\mathrm{Ca}}^{2+}$ chelator BAPTA to the cell culture medium. BAPTA binds free calcium ions in the extracellular space, thereby reducing their availability to the cells. Its addition terminated the intracellular ${\mathrm{Ca}}^{2+}$ oscillations, and the percentage of oscillating cells declined to zero (see Table 2). This decrease was significant compared to the control.

Intracellular ${\mathrm{Ca}}^{2+}$ levels during AC electrical stimulation

We applied sinusoidal waves with an amplitude of $10$ V, which was just below the amplifier’s limit. The digital twin workflow predicted an iTMP of $10$ mV at the cells situated in the centre of the well at $10$ V amplitude. Therefore, we expected the activation of low-voltage calcium channels. To prevent electrochemical reactions at the electrodes, we initially selected a frequency of $1$ kHz. The stimulation was started after three minutes of observation. Due to the shunt resistor used for current measurement and the contribution of the electrochemical interface, the voltage drop across the cell culture medium (i.e. the effective stimulation amplitude) differed from the applied voltage. Using the validated impedance model, we calculated the voltage drop across the cell culture medium to have an amplitude of 7.67 ± 0.08 V (see Table 1). The impact of the electrode-electrolyte interface impedance on the voltage drop across the medium was small at the chosen frequencies. Thus, the stimulation was indeed dominated by the electrical rather than the electrochemical properties of the setup.

We did not detect a significant effect of the stimulation on the intracellular $C{a}^{2+}$ levels or oscillations (see Table 2 and Supplemental Figure S5). The percentage of oscillating cells and their oscillation frequencies were comparable to those in the control group. Furthermore, the electrical stimulation did not affect the number, intensity, or duration of calcium peaks of individual cells. As in the control group, there was substantial variability among oscillating cells. In addition, we conducted the same experiments at 100 kHz, which did neither substantially alter the stimulation current nor the voltage amplitudes. Consistently, the higher frequency also had no detectable effect on intracellular $C{a}^{2+}$ levels or oscillatory behaviour (Table 2).

Influence of DC electrical stimulation on intracellular ${\mathrm{Ca}}^{2+}$ levels

We could not observe an effect of AC stimulation with sinusoidal waves in the kilohertz range on intracellular ${\mathrm{Ca}}^{2+}$ levels. To explore potential reasons for previous reports of elevated intracellular ${\mathrm{Ca}}^{2+}$ levels in response to electrical stimulation,^8,10,54 we tested the effect of DC stimulation. We applied a constant voltage of $3$ V for five minutes. The applied current decreased over the stimulation time, and thereby the estimated electric field strength dropped from $17$ V/m to 10V/m (see Supplemental Figure 6(A) and (B)). Apart from some endogenous intracellular ${\mathrm{Ca}}^{2+}$ oscillations, no increase in basal intracellular ${\mathrm{Ca}}^{2+}$ levels could be detected (Figure 6(c) left). Therefore, we increased the applied voltage to 5 V. The resulting voltage drop in the cell culture media was initially around 1.73 V and decreased to $1.54$ V over the time of the electrical stimulation (see Table 1). The corresponding mean electric field strength in this part of the well initially ranged between $66$ V/m and 71 V/m, decreasing to a range of $58$ V/m to $63$ V/m after one minute (Supplemental Table S9). While current, voltage, and electric field strength were highest right after the onset of electrical stimulation, the cells did not show an immediate calcium response. After a median duration of $3.1$ min (IQR $0.9$ min) of electrical stimulation, the intracellular ${\mathrm{Ca}}^{2+}$ levels of all cells in the field of view increased dramatically (see Figure 6(C) right). To investigate whether this effect depends on extracellular calcium availability, we repeated the experiment with $2.5$ mM BAPTA added to the culture medium. The stimulation remained within the same range as in the condition without BAPTA, and the estimated voltage drop across the medium was unchanged (Supplemental Table S6). All cells again showed a pronounced increase in intracellular ${\mathrm{Ca}}^{2+}$ , with a slightly delayed onset (median $3.4$ min, IQR $0.7$ min), although the difference in timing was not statistically significant (see Supplemental Figure S5(F)). Inhibition of ROS with NAC and blockage of the L-type channels using verapamil resulted in values comparable to the DC-only condition. We repeated the experiment with thapsigargin added to the culture medium. All cells again showed a pronounced increase in intracellular ${\mathrm{Ca}}^{2+}$ ; however, the onset of the calcium rise occurred earlier (see Supplemental Figures S5(F) and S6(A)–(D)), consistent with the blockade of SERCA pumps by thapsigargin. ⁵⁵

The applied voltage of 5V resulted in an initial iTMP of $2.4$ mV, which decreased to $2.14$ mV after one minute (see Table 1). We were unable to attribute the observed increase in intracellular ${\mathrm{Ca}}^{2+}$ concentration to this relatively low iTMP. Therefore, we hypothesised that the electrochemical by-products generated at the electrodes were responsible for the observed elevation in intracellular ${\mathrm{Ca}}^{2+}$ . During stimulation with 5V, the concentration of hydrogen peroxide ( $H_2{O}_2$ ) in the culture medium increased from $0.52$ µm to $2.67$ µm (see Supplemental Figure S7). To test whether $H_2{O}_2$ alone could induce a calcium response, we added it directly to the culture medium after three minutes of observation in accordance with the electrical stimulation experiments. Concentrations corresponding to those measured during electrical stimulation ( $\approx 3$ µM) or even 100-fold higher ( $0.3$ mM) did not elicit any calcium responses (see Figure 8(B)). To address further electrochemical effects, we measured other potential by-products of electrical stimulation, including pH, temperature, and nitrite/nitrate levels. During 5 V DC stimulation, pH changes were most pronounced, with an extracellular increase of $\approx 0.5$ units (see Supplemental Figure S7) in the centre of the well, while temperature changes were minimal ( $0.1$ °C) and levels of nitrite and nitrate remained below the detection limit. To test whether the pH change alone could induce calcium responses, we chemically mimicked the pH shift by adding a base, which reproduced the calcium increase seen during 5V DC electrical stimulation (see Figure 8(a)), even in the presence of BAPTA to chelate extracellular ${\mathrm{Ca}}^{2+}$ (see Supplemental Figure S6(E)).

Figure 8.

Time series of live-cell imaging showing intracellular ${\mathrm{Ca}}^{2+}$ levels under different chemical conditions shown as heatmaps over the full $8$ min observation period, with treatment starting at $3$ min (vertical white line). (A) Effect of adjusting the medium pH to the values measured after 5V DC stimulation, which induces ${\mathrm{Ca}}^{2+}$ mobilisation with a time course similar to DC stimulation. (B) Effect of adding $H_2{O}_2$ at $0.3$ mM, which does not induce changes in intracellular ${\mathrm{Ca}}^{2+}$ .

Discussion

Electrical stimulation has been considered for bone regeneration for more than 60 years. ³ Despite these research efforts, electrical stimulation has not yet found widespread application in the clinical treatment of bone fractures and non-unions. Two reasons have contributed to this situation: (1) the underlying biological mechanisms of interaction remain poorly understood; and (2) limited documentation of electrical stimulation experiments has hampered retrospective analyses. As a result, various stimulation approaches have been successfully tested in vitro and small animal models in vivo, but a reliable translation to clinical stimulation devices remains elusive.^3,6 Knowing the exact mechanism of interaction and required stimulation parameters to trigger this mechanism would pave the way for rationally designed stimulation protocols. Currently, two approaches are being pursued separately to reveal the exact effect of electrical stimulation. (1) Numerical models are used to estimate the electric field cells are exposed to,^6,28,34 and more detailed models allow estimation of single-cell stimulation target values such as the iTMP.^29,56 (2) Wet-lab experiments have employed channel blockers ⁵⁷ and live-cell imaging^8,10,54 to assess the effects of electrical stimulation on intracellular ${\mathrm{Ca}}^{2+}$ levels and their origin. In this work, we combined these numerical and experimental approaches to test the hypothesis that voltage-gated calcium channels in MG-63 osteoblasts can be triggered by electrical stimulation.^7,9–11

Digital twin framework

It has been shown that estimates of the stimulating electric field in the literature are often drastically overestimated due to inappropriate or inaccurate model assumptions.^27,28,58 Furthermore, the electrochemical interface of the electrode must be taken into account, making an experimental calibration of the numerical method essential.^28,30 The combination of experimental and numerical methods is commonly referred to as the digital twin approach and has provided reliable estimates of the actual field strength experienced by the cells. In this work, we extended this approach by integrating a model of the effect of electrical stimulation at the cellular scale. The macroscale model determined the boundary conditions for a selected sub-volume containing cells. We defined this sub-volume as the field of view of a 3D fluorescent image of cells with stained membranes and nuclei.

The digital twin should react adaptively to input data. Thus, we aimed to automate the creation of the numerical model from fluorescent microscopy images. The first step is instance segmentation to identify individual cells and nuclei. The pre-trained deep learning segmentation model Cellpose, combined with advanced classical image processing techniques, enabled automated and accurate segmentation. Retraining the deep learning model was not required for our dataset. However, since deep learning segmentation models have been shown to generalise well on fluorescence microscopy data, ³⁹ we expect our approach to be readily translatable to images of other cell types. This automated workflow enables fast model generation and is compatible with live-cell imaging experiments. Generating a numerical model from the 3D image and computing the corresponding electric field required approximately ten minutes on a workstation for our specific dataset. Thus, future studies may go beyond retrospective analysis and build as well as calibrate the multiscale digital twin during the stimulation session. This would pave the way for real-time control of electrical stimulation and adaptive adjustment of stimulation parameters based on live-cell imaging data. To substantially reduce computational load and minimise manual post-processing effort in the segmentation of calcium imaging data, we explored the novel deep learning InstanSeg model. ⁴³ On our dataset, InstanSeg demonstrated a significantly faster inference time, approximately 0.2 seconds per slice, compared to 3.4 seconds for Cellpose3. This speed advantage could enable the application of our workflow on highly motile cells. For the slowly migrating bone cells studied here, this benefit is not critical, but it becomes relevant when stimulating other, more motile cell types.

Combined cytoplasmic and nuclear imaging enables the construction of more complex models. However, we did not observe significant differences in iTMP and impedance between the configurations studied. This suggests that modelling the cell without the nucleus is sufficient to estimate these observables. Nevertheless, the model including the nucleus provides more detailed insights into the electric field distribution within individual cells and may become relevant for other stimulation parameters or more complex cell geometries. Creating a multi-domain mesh enables the assignment of distinct material properties, for example, to individual cells and/or nuclei (see Supplemental Figure S8). This allows the numerical model to incorporate mixed cell populations or data from live/dead assays. ⁵⁹ Similarly, the digital twin approach is not limited to the osteoblast-like cells investigated in this study.

While the cell culture model and intracellular ${\mathrm{Ca}}^{2+}$ staining are well established, the iTMP model relies on assumptions and requires further validation. The iTMP could be validated using a potentiometric fluorescent dye. ²⁹ We tested several voltage-sensitive probes for this purpose. ${\mathrm{DiBAC}}_4$ (3) exhibited low fluorescence intensity and a slow response, whereas FluoVolt™, which is expected to react rapidly, did not respond to increased extracellular KCl in positive control experiments. This indicated that staining protocols require further optimisation. Furthermore, special equipment is required to resolve the fast temporal changes as induced by alternating current stimulation at $1$ kHz. ⁶⁰ To gain deeper insights into the membrane potential and its modulation by electrical stimulation, we plan to apply the patch clamp technique. This method will not only allow direct measurements of membrane potential changes but also enable the calibration of membrane potential-sensitive dye intensities against defined voltage values.

Furthermore, microelectrode arrays could be used to measure the local voltage distribution in the presence of cells, which would allow direct comparison with the numerical models. ³⁰ Importantly, the microelectrodes must be compatible with live-cell fluorescence imaging. An indirect validation might be achieved by detecting electrically induced calcium events, synchronised in multiple cells.

Endogenous $C{a}^{2+}$ oscillations

To separate the effect of electrical stimulation on calcium levels in MG-63 osteoblasts, we first investigated intracellular ${\mathrm{Ca}}^{2+}$ concentrations in the absence of electrical stimulation using live-cell imaging. The setup was adapted from previous experiments ³¹ to allow for simultaneous electrical stimulation and calcium imaging. We optimised the staining protocol and observed endogenous, spontaneous intracellular ${\mathrm{Ca}}^{2+}$ oscillations. To our knowledge, this is the first report of such oscillations in MG-63 osteoblasts. In contrast, calcium oscillations have previously been described in other bone-related cells, such as foetal human osteoblasts, human mesenchymal stem cells, and ex vivo chick bone osteoblasts.^61–64 We observed oscillations at a median frequency of approximately $7$ mHz. Human osteoblasts have been shown to oscillate at frequencies of $4.2$ to $10$ mHz, ⁶⁴ which is consistent with our findings. Sun et al. demonstrated that mesenchymal stem cells exhibit spontaneous intracellular ${\mathrm{Ca}}^{2+}$ oscillations that were faster and more regular than those in osteoblasts. Upon osteogenic differentiation, the frequency of these oscillations decreased. Other studies have likewise reported an initial reduction in intracellular ${\mathrm{Ca}}^{2+}$ oscillations during osteogenic differentiation. ⁶⁵ This oscillatory behaviour has been associated with mechanical stress on the cells, such as shear stress or osmotic swelling caused by the staining procedure.^64,66 A similar mechanism might have been present in our experiments, where washing steps with the staining solution and medium exchange could have induced shear or osmotic stress, triggering the observed oscillations. In such cases, the influx of extracellular ${\mathrm{Ca}}^{2+}$ observed in response to mechanical stress appears to be primarily mediated by stretch-activated cation channels, such as TRPM7, ⁶⁶ rather than by voltage-gated calcium channels. The fact that we did not see changes in intracellular ${\mathrm{Ca}}^{2+}$ oscillations indicates that there was no cell deformation by electrical stimulation.

We added the calcium chelator BAPTA to elucidate the role of extracellular ${\mathrm{Ca}}^{2+}$ ions. Indeed, the addition of BAPTA inhibited spontaneous oscillations, indicating that extracellular ${\mathrm{Ca}}^{2+}$ plays a critical role in generating these spikes. This observation is consistent with previous findings, which show that the use of the calcium chelator EGTA in the buffer suppresses intracellular ${\mathrm{Ca}}^{2+}$ oscillations. ⁶⁴

MG-63 osteoblastic cells, just like primary osteoblasts, possess gap junctions ⁶⁷ and are therefore capable of transmitting not only ions such as ${\mathrm{Ca}}^{2+}$ to neighbouring cells but also electrical signals. ⁶⁸ In our experiments, we did not observe any propagation of intracellular ${\mathrm{Ca}}^{2+}$ signals between cells. It is possible that no gap junctions have formed due to the short incubation time. Previous studies have shown that gap-junction communication is critical for mediating endogenous intracellular ${\mathrm{Ca}}^{2+}$ oscillations in intact embryonic chick calvarial bone explants. ⁶⁹ Notably, pharmacological inhibition of gap junctions significantly reduced osteocyte oscillatory activity, while osteoblast activity remained unaffected. ⁶⁹ Thus, gap junctions may represent potential targets for electrical stimulation. However, in this study, we focussed on the direct effects on individual cells, while future work could explore responses within multicellular networks.

Electrical stimulation experiments

We investigated the effect of electrical stimulation on intracellular ${\mathrm{Ca}}^{2+}$ concentration, as previous studies have reported increased intracellular ${\mathrm{Ca}}^{2+}$ levels following both DC and AC electrical stimulation using various stimulation setups.^8,10,31,54 Additionally, previous work from our group has consistently shown that MG-63 cells exhibit robust calcium responses to ATP stimulation across passages 5–25,^9,25,37,70 highlighting the reliability and physiological responsiveness of this cell line. In our study, we integrated simultaneous intracellular ${\mathrm{Ca}}^{2+}$ live-cell imaging, real-time electrical measurements, and numerical simulations to precisely characterise the stimulation experienced by the cells and their corresponding response. Our hypothesis was that changes in the intracellular ${\mathrm{Ca}}^{2+}$ concentration would be observable and spatiotemporally correlated with the applied electrical stimulation. However, neither stimulation at $1$ kHz nor at 100 kHz led to such an effect. This result was somewhat surprising, as the applied electric field strength was around $260$ V/m. Also, the estimated iTMP reached up to $9.6$ mV; a value expected to trigger low-voltage-activated calcium channels. Two explanations may account for the lack of activation. First, the iTMP is not evenly distributed along the cell membrane and is highest at the cell edges. As a result, it may not affect membrane regions that contain a higher density of calcium channels. Second, the activation thresholds of the calcium channels may be higher than the achieved iTMP. In theory, iTMP scales linearly with the applied stimulation voltage. However, we already applied an amplitude of $10$ V, which approached the upper limit of the amplifier’s linear range. Applying even higher voltages could also pose a safety risk for the experimenter. Therefore, the only way to determine whether a higher iTMP could trigger calcium events is to develop a stimulation chamber that delivers higher local field strengths while remaining compatible with live-cell imaging. This will be the focus of future research.

To evaluate whether cells can elicit a response, we applied a 5V DC ‘high-intensity’ condition. This extreme stimulation was intended to investigate whether intracellular calcium mobilisation can occur in principle, while acknowledging that such a condition may partially compromise cell viability and does not necessarily reflect physiologically intended responses. DC stimulation with 5V led to an increase in intracellular ${\mathrm{Ca}}^{2+}$ after several minutes of stimulation. The same effect was observed slightly delayed when BAPTA was added to the cell culture medium. This indicates that the effect was independent of the extracellular ${\mathrm{Ca}}^{2+}$ availability. However, the observed increase in intracellular ${\mathrm{Ca}}^{2+}$ levels was unexpected, as the estimated local electric field and iTMP were approximately four–five times lower than those achieved during AC stimulation. Consequently, the DC electrical stimulation should not have been sufficient to directly open voltage-gated calcium channels. Given the relatively long delay between the onset of electrical stimulation and the rise in intracellular ${\mathrm{Ca}}^{2+}$ , we hypothesised that the effect was primarily mediated by faradic by-products, particularly local pH shifts. Measurements confirmed an increase in extracellular pH during stimulation, indicating that DC electrical stimulation was accompanied by pronounced electrochemical reactions (see Supplemental Figure S7). Moreover, experimental modulation of the medium’s pH reproduced similar increases in intracellular fluorescence, supporting the notion that extracellular pH elevation contributed to the observed rise in intracellular ${\mathrm{Ca}}^{2+}$ levels, as pH shifts are known to significantly elevate intracellular calcium concentrations, indicating that extracellular alkalinisation can contribute to ${\mathrm{Ca}}^{2+}$ dynamics in our system.^55,71,72 Control experiments altering osmolality within the same range as during alkalinisation showed no effect on intracellular ${\mathrm{Ca}}^{2+}$ , supporting that the response is not due to osmotic changes (see Supplemental Figure S6(F)). In the presence of thapsigargin, which blocks SERCA-mediated calcium re-uptake, the calcium rise at 5V DC occurs earlier (Supplemental Figure S5(F)), consistent with a contribution from intracellular stores. This suggests that the response is not exclusively mediated by calcium influx through classical channels, but is likely a cumulative effect of multiple factors, including local pH shifts, mild cell depolarisation, and ROS formation. However, our main focus was on AC-induced transmembrane potential changes and their potential effects on calcium signalling. DC stimulation was included for comparison, showing that moderate DC (e.g. $3$ V) does not open calcium channels, while extreme conditions (5V) induce additional effects due to strong electrochemical perturbations.

Our findings highlight the importance of a comprehensive electrochemical characterisation of electrical stimulation experiments, with particular attention to faradic by-products. In our case, the increase in intracellular ${\mathrm{Ca}}^{2+}$ occurred with a temporal delay and was not a direct consequence of electrical stimulation, but rather the result of secondary effects. This finding sheds new light on the observations made by Khatib et al., ¹⁰ who reported an increase in intracellular ${\mathrm{Ca}}^{2+}$ approximately 20 min after the onset of DC electrical stimulation. Similar to our study, the addition of EGTA, as a chelator of extracellular ${\mathrm{Ca}}^{2+}$ ions, resulted in a slightly delayed and reduced intracellular ${\mathrm{Ca}}^{2+}$ increase. Consistent with our results, a chemically mediated mechanism, blocking voltage-gated calcium channels with verapamil had no effect on the observed intracellular ${\mathrm{Ca}}^{2+}$ elevation. Compared to our findings, the intracellular ${\mathrm{Ca}}^{2+}$ increase in Khatib’s study occurred considerably later, which may be attributed to their use of agar-salt bridges rather than bare platinum wires in direct contact with the cell culture. Thereby, the transport of by-products from the electrodes to the cells is reduced. ⁷³

Workflow for adaptive electrical stimulation in cell cultures

We aimed to streamline the calcium imaging analysis pipeline and ensure it can be readily applied to other experimental settings. All image segmentation, intensity measurements, peak visualisation, and statistical analyses were performed using open-source software and are freely accessible. This approach ensures transparency, reproducibility, and broad usability for future studies involving calcium imaging data. We found that InstanSeg, as a segmentation network, performs significantly faster than other deep learning networks. Training it on segmentation masks from our dataset yielded high accuracy. Thanks to its speed and precision, InstanSeg may enable real-time analysis, thereby helping to close the digital twin loop – for example, by adapting electrical stimulation in response to detected calcium events. We propose a three-phase workflow for developing and implementing a digital twin model for cell culture systems (Supplemental Figure S9). Phase one involves data acquisition and characterisation of the stimulation setup and cell culture conditions using EIS and voltage/current measurements to assess electrode properties. 3D imaging and morphology analysis of cellular and tissue architecture are integrated into the digital twin. Phase two focuses on parameter extraction, model calibration, and optimisation of the digital twin model. Phase 3 envisions real-time operation and closed-loop control, integrating live-cell imaging and analysis (e.g. calcium dynamics, morphology, cell tracking, and viability assessment) alongside live electrical monitoring of current responses, impedance changes, and boundary conditions. These feedback signals are used to dynamically adapt the digital twin and control stimulation parameters.

Conclusion

We demonstrated a digital twin approach that integrated live-cell imaging, electrical recordings, and numerical models. This approach enabled a rational and systematic analysis and control of the applied electric field and of the resulting induced transmembrane potential in MG-63 osteoblasts.

In summary, our results demonstrate that cellular responses to electrical stimulation strongly depend on the underlying electrochemical environment. DC stimulation above a specific threshold triggered electrochemical reactions such as electrolysis, leading to a pH shift that modulated intracellular calcium levels. AC stimulation, which presumingly changed the transmembrane potentials, had no effect on intracellular calcium. This indicates that mere electrical activation with the tested stimulation parameters is insufficient to trigger calcium signalling. In contrast, chemical stimulation by faradic by-products generated during DC stimulation did lead to a pronounced calcium response. Understanding these distinct mechanisms is crucial for the meaningful interpretation of electrostimulation experiments and for designing more targeted and physiologically relevant stimulation strategies. It may also help explain the limited translation of in vitro findings to clinical applications.

Our open-source workflow provides a basis for determining optimal stimulation parameters in a systematic and reproducible manner (see Supplemental Figure S9). It offers the potential to adapt stimulation protocols in real time based on feedback from live-cell imaging.