Fractional Electrodamage in A549 Human Lung Cancer Cells

Abstract

Fractional electrodamage in A549 human lung cancer cells was analyzed by introducing a non-integer order parameter to model the influence of electrical stimulation on cellular behavior. Numerical simulations were conducted to evaluate the conversion of electrical energy to heat within A549 cancer cells, emphasizing the electrocapacitive effects and electrical conductivity in modulating dielectric properties. Using the Riemann–Liouville fractional calculus framework, experimental results were accurately fitted, demonstrating the non-integer nature of electrodamage processes. The study identified a strong dependency of electrical behavior on frequency, revealing a critical role of fractional dynamics in the dielectric breakdown and susceptibility of A549 cells to voltage changes. These findings advance our understanding of cellular responses to electrical fields and provide insights into applications in cancer diagnostics, monitoring, and potential therapeutic treatments.

1. Introduction

Important applications in the treatment of cancer can be addressed through the study of consequences of exceeding cancer cell membrane potentials [1]. The concept of electrical power dissipation in biological cells, leading to heat generation, has been explored as an alternative for cancer treatment.

Data from several studies suggest that at the beginning of this decade, lung, prostate, and breast cancer were a globally leading cause of death [2]. Among the most-encountered forms of lung cancer are non-small cell lung carcinomas [3]. Carcinoma human alveolar basal epithelial cells (A549 cancer cells) have emerged as a prominent in vitro model for studying cancer. The key characteristics of A549 cells that make them suitable for this study include their established use as an in vitro model for non-small cell lung carcinoma and their distinct electrochemical properties, such as membrane capacitance and resistance. These properties allow analysis of the interactions in cells with electrical stimuli, making them ideal for exploring electrocapacitive effects and fractional electrodamage mechanisms.

Apoptosis results when electrical pulses that are administered to cancer cells provoke thermal damage to internal structures and cell membranes [4]; it is known as inhibition of proliferation on biological cells [5]. Healthy cells that are in the surrounding media are less sensitive to applied electrical pulses, avoiding significant damage.

For the purpose of acquiring knowledge about the electrical properties and electrochemical behavior of A549 cancer cells, researchers have used electrochemical impedance spectroscopy (EIS) [6]. As a method for observing and determining how cells react to various stimuli in diverse disciplines, such as medicine [7] and biology [8], EIS is widely used in these disciplines. In the context of cancer research, EIS represents a prominent option for exploring electrical features of A549 cancer cells such as resistance [9] and membrane capacitance [10]. Cyclic voltammetry (CV) is commonly considered as an essential instrument in electrochemistry. This approach is often used to explore and evaluate electrode performance while also offering insights into electron transport mechanisms. Electrical impulse transmission over biological cells membranes is responsible for electrical signal movement that comes from outside [11]. External stimulation and signal transduction in cells are caused by electrical signal propagation over cells, resulting in a change in the electrical potential in the cell membrane [12]. A capacitive effect can be observed in cells, due to the existence of cell membranes, and they can be considered as natural capacitors [13], due to their composition. Proteins and lipids are the main components in cell membranes, and electrical charges on opposite sides are segregated by these membranes [14]. The cells membranes work as a natural energy store of an electrical charge on cells surface when a change on voltage exists. This dynamic electrical energy store is provoked by a capacitive effect on cell membranes. These capacitive features have an important role in biological processes such as cellular communication and neural excitability [15]. Cells response to an external electrical signal depends on the nature of stimulation, for instance, the response time of nervous cells is measured in the order of milliseconds [16], and it can be faster than other cells. Parameters such as propagation distance synaptic cell connectivity and ion channel density have an important relationship with the cellular response time and bioelectrical reactions [16]. A frequency-dependent electrical signal changes biological processes in cells and tissues in response to stimulation. Slow muscular contractions can be registered at low frequencies, whereas faster contractions can be observed at higher frequencies [17]. Signal transmission in neurons is influenced by frequency dependence and neurotransmitter release. It is known that frequency dependence can influence and contribute to biological functions but its ability to control cellular responses and physiological systems based on the electrical signals properties is still limited [17].

In this direction, anomalous replication of cancer cells can be described by fractional differential equations, since fractional modeling provides a more accurate description for representation of dynamic processes. Fractional calculus can be used for modeling complex interactions between cancer cells replication and the growth rate. Cellular proliferation can be described by fractional calculus and modeling anomalous diffusion and movements patterns in cancer cells. Localized damage due to light exposition in cancer cells can be described by fractional photodamage as a technique for determining it. Deeper knowledge about fractional photodamage in A549 cancer cells has prominent importance in several areas of health such as cancer treatment optimization. On the other hand, fractional electrodamage can be used as a therapeutic approach that uses electrical currents to selectively eliminate cancer cells. Fractional electrodamage refers to the analysis of electrical damage in biological cells using fractional calculus; it incorporates non-integer order parameters to model accurately dynamic processes. In comparison with conventional electrodamage approaches, fractional electrodamage considers the frequency-dependent and non-linear electrical behavior of cells, providing a representation of electrocapacitive effects and dielectric breakdown mechanisms.

It is remarkable that the electrical response of cells can be regulated by electrical pulse duration, repetition rate, and strength [18].

Figure 1 shows a roadmap of progress in the field of studies related to fractional electrodamage in A549 cancer cells. From Figure 1 can be seen chronological progression from the initial investigation of electric fields and cancer cell behavior to more refined techniques for characterizing cancer cells, ultimately culminating in the specialized study of cancerous cell differentiation based on their biophysical phenotype.

Past research has highlighted the role of material properties in enhancing therapeutic outcomes, particularly in targeting cancer cells while preserving healthy tissue [25]. These studies align with the present research by emphasizing the importance of precision and innovation in therapeutic approaches [26]. Incorporating these perspectives enriches the understanding of fractional electrodamage as a non-invasive, targeted method for cancer diagnostics and treatment, bridging materials science and biomedical applications. The present research surpasses prior studies by leveraging fractional calculus to model frequency-dependent dielectric breakdown mechanisms with unprecedented precision. It integrates experimental validation with the Riemann–Liouville framework, offering a significant advancement over conventional models in capturing the non-linear electrical behavior of A549 cancer cells.

In view of all these considerations, the main purpose of this study is to analyze physical mechanisms responsible for the fractional electrodamage exhibited by A549 cancer cells. The primary objective of studying fractional electrodamage in A549 human lung cancer cells is to analyze the physical mechanisms underlying the fractional dynamics of electrodamage induced by electrical stimulation. This has the purpose of enhancing the conception of the electrocapacitive properties of cancer cells and their susceptibility to voltage changes.

This study applies fractional calculus to model electrodamage in A549 lung cancer cells, emphasizing frequency-dependent dielectric breakdown. Using the Riemann–Liouville theory, it highlights the cancer cells’ susceptibility to voltage changes and advances in non-invasive diagnostic and therapeutic methods, such as electrochemical hyperthermia.

We considered A549 cancer cell activities in the setting of lung carcinoma, underlying the response of these cells to electrical fields, enhancing the knowledge about electrocapacitive changes that can be related to cancer progression. It is considered that studies related to fractional electrodamage in cancer cells have prominent applications on biotechnology, drug delivery systems, and development of bioelectronic devices. It allows development of new therapeutic strategies such as pulsed electric current applications for inducing damage and inhibiting the growth of cancer cells. It highlighted the potential of monitoring cellular changes induced by electrical influence for assessing cancer cells response and different strategies in treatments.

2. Materials and Methods

2.1. Fractional Electrodamage in A549 Cancer Cells

With the purpose of calculating the impedance of the biological samples in a culture medium, we used the following equations [27]:

(1)$R_i={\rho }_i{\displaystyle \frac{l_i}{A}}.$

(2)$C_i={\displaystyle \frac{{\epsilon }_0{\epsilon }_{ri}A}{l_i}}.$

where ${\rho }_i$ is the resistivity of the material, ${\epsilon }_{ri}$ is the relative permittivity, ${\epsilon }_0$ is the permittivity of the vacuum, A is the cross sectional area of the used Metrohm DS 220 AT electrode (the equipment was sourced from the Metrohm brand, Herisau, Switzerland), and the thickness of the culture medium drop is represented by $l_i$.

Considering that $Z_i$ is the impedance of the drop and $R_i$ is the resistance, we have [27]

(3)$Z_i={\displaystyle \frac{R_i\left(1-j\omega C_i{R}_i\right)}{1+{\omega }^2{C}_i^2{R}_i^2}}.$

where $C_i$ is the capacitance in the system, $j$ is the imaginary part, and $\omega$ is angular frequency.

On the other hand, with the purpose of modeling the numerical behavior of impedance and capacitance as a function of applied frequency for different concentration of cells we employed the Cole–Cole model. The total impedance can be expressed as follows [28]:

(4)$Z_T=R_s+{\displaystyle \frac{R_p}{1+{\left(j\omega R_p{C}_p\right)}^n}}.$

where $R_p$ represents the resistance in the cells, $C_p$ is the total capacitance on them, $R_s$ denotes the total series resistance, and $n$ represents the power order that most accurately fits the model obtained.

The numerical capacitance was calculated employing the Giaver theory. It describes the impedance between adherent cells and an electrode. The numerical capacitance can be considered as follows [29]:

(5)$C_{mem}={\displaystyle \frac{{\epsilon }_0{\epsilon }_{ri}}{d_{mem}}}.$

where $C_{mem}$$\left[{\mathrm{Fm}}^{-2}\right]$ represents the capacitance of the cancer cells membrane per unit area. The permittivity of the vacuum is ${\epsilon }_0$, ${\epsilon }_{mem}$ is the relative permittivity of the surrounding media, and $d_{mem}$ represents the thickness of the cell membrane [29].

The imaginary part of the impedance can be described as follows [30]:

(6)$\underset{\omega \to 0}{\lim }\left[\omega \mathrm{Im}\left(Z\right)\right]={\displaystyle \frac{1}{C_{mem}}}.$

On the other hand, the real part was numerically calculated considering [30]:

(7)$\mathrm{Re}\left(Z\right)={\displaystyle \frac{{\rho }_i\left(0\right)L_{BULK}}{{\rho }_i{l}_i}}.$

where ${\rho }_i\left(0\right)$ denotes the resistivity at the center area of the cells, where the potential was applied, while $L_{BULK}$ represents the distance half between two cancer cells [30].

With the intention of determining the electrodamage induced in A549 cancer cells, we used fractional calculus. To evaluate fractional electrodamage in A549 cancer cells, we introduce the concept of fractional calculus theory to effectively describe this phenomenon.

Fractional calculus and mathematical modeling techniques play a crucial role in this research because it provides a framework to describe the dynamic and non-linear behavior of A549 cancer cells under electrical stimulation. They enable simulations of electrical impedance and capacitance as functions of frequency, allowing for a better understanding of fractional electrodamage and its implications in cancer diagnostics and therapies.

Following the principles of the Riemann–Liouville fractional calculus operator theory, the integral of a function $f(t)$ of order $\alpha$ is defined as follows [31]:

(8)${\displaystyle \frac{d^{-\alpha }f(t)}{d{t}^{-\alpha }}}=t_0{D}_t^{-\alpha }f(t)={\displaystyle \frac{1}{\Gamma (\alpha )}}{\displaystyle \underset{\alpha }{\overset{t}{\int }}{\left(t-x\right)}^{-\alpha -1}}F(x)dx.$

here, D in this context represents the antiderivative of $\alpha$ order, which is derived from the Cauchy formula. The parameter $\alpha$ signifies the order of the fractional derivative, and x indicates the elongation during the electrochemical changes in the A549 cancer cells. Additionally, t denotes the duration related to the experiment. For the purposes of this study, $\alpha =0.8$ was selected as it provided the best fit to the experimental data.

In order to model interactions between historical and non-local properties in a process, we have employed the Caputo derivative formula. All these mathematical expressions are applied to calculate temperatures using a fractional approach, as explained in the Supplementary Material S1.

2.2. Cell Culture

A549 epithelial cells grown in culture bottles were detached using 2 mL of 0.05 g/L trypsin and 0.05 g/L EDTA solution (Sigma Aldrich, St. Louis, MO, USA). After, cell suspension was transferred to 15 mL conical tube and centrifuged at 1500 rpm during 5 min. The cell pellet was resuspended with 1 mL fresh culture medium and viable cells were quantified in a Neubauer chamber with 0.4% trypan blue solution. Then, the cell suspension was diluted with fresh medium to analyze three different concentrations of cells (125,000; 250,000; and 500,000 cells/60 µL). In order to observe the cells’ morphology, the actin cytoskeleton was labeled with rhodamine-phalloidin, and nuclei were stained with DAPI (4′,6-diamidino-2-phenylindole). A confocal system coupled to an inverted microscope (LSM710 NLO Zeiss, Jena, Germany) was employed.

2.3. Electrochemical Impedance Analysis of A549 Cancer Cells: Frequency Response Study

Electrochemical investigations were carried out on biological samples of A549 cancer cells at a constant room temperature of 25 °C by triplicate. The cells contained in a drop of 60 µL were deposited in a Metrohm DS 220 AT electrode to measure their electrical response in an AUTOLAB 302N PGSTAT potentiostat (Metrohm, Herisau, Switzerland). An Ag/AgCl reference electrode and a platinum counter electrode were employed for experimentation [32]. The supporting electrolyte used was a 0.5 M KOH solution, which was pre-treated by degassing in an ultrasonic bath for 15 min and subsequent bubbling with nitrogen gas for 10 min. Data analyses were performed by using MATLAB software (R2023a) to extract meaningful information. From Figure 2a is possible to observe a representative photo of the studied A549 cancer cells located in the electrode. From Figure 2b is possible to visualize the connections of the Metrohm DS 220 AT electrode: A Metrohm electrode, specifically the DS 220 AT model, was used.

2.4. Apoptosis and Cell Viability in A549 Cells Through Electrical Stimulation and Flow Cytometry Analysis

Apoptosis and cell viability experiments were undertaken in order to guarantee changes in the cellular structure as part of the experimental procedure in our model. A549 cells were cultured in F-12 medium supplemented with 8% fetal bovine serum (FBS), 1% penicillin-streptomycin, and maintained at 37 °C in a humidified atmosphere containing 5% CO₂. Cells were subcultured every 3–4 days to maintain exponential growth. An electrical circuit for inducing electrical current in a suspension of 2.5 × 10⁵ A549 cells prepared in a 96-well plate with 50 μL of culture media. The electrodamage stimulus was maintained for different time periods (0, 5, 10, 20, 30, 60, and 120 s). Following treatment, cells were immediately analyzed for membrane integrity and cell viability. Membrane integrity was evaluated using a trypan blue exclusion assay. Cells were mixed with an equal volume of 0.3% trypan blue solution and incubated for 5 min at room temperature. Cells with compromised membrane integrity were identified by their ability to uptake the trypan blue stain, whereas viable cells remained unstained. Photographs of stained cells were taken to document the results. Apoptosis and cell viability were assessed using flow cytometry with annexin V-FITC and propidium iodide (PI) staining. Following electrodamage treatment, cells were resuspended in annexin V binding buffer and stained with annexin V-FITC and PI according to the manufacturer’s instructions (Cat. No. 640914, Biolegend, San Diego, CA, USA). The stained cells were then analyzed using a flow cytometer (FACSAria II, BD Bioscience, San Jose, CA, USA). Data were collected and analyzed using the FlowJo v.10 software to determine the percentage of live, early apoptotic, and necrotic cells. Live cells were identified as annexin V and PI negative (Q4), early apoptotic cells as annexin V positive and PI negative (Q1), and necrotic cells as annexin V and PI positive (Q2). The results are presented as the percentage of 10,000 cells collected. All experiments were performed in duplicate, and data were expressed as mean ± standard deviation. Comparisons between experimental conditions were performed using one-way ANOVA followed by Tukey’s post hoc test. A p-value of <0.05 was considered statistically significant. GraphPad Prism software v.8 was used for all statistical analyses and graph generation.

To ensure the observed effects were specifically attributable to electrodamage, experiments were conducted with negative controls, including samples that did not receive electrical stimuli. These controls allowed for a baseline comparison, verifying those changes in cell viability, membrane integrity, and other parameters were a direct result of the applied electrical stimulation.

Figure 3 illustrates the sample in contact with the electrodes.

3. Results

Figure 4 shows a confocal image of studied A549 cancer cells. From the images can be seen A549 cells with actin filaments distributed longitudinally in the cytoplasm. In order to contrast, the nuclei of cells were stained with DAPI (4′,6-diamidino-2-phenylindole) and were detected in blue emission. The purpose of using fluorescence was to observe the complex details and structure of the cellular components. Figure 5 combines red fluorescence for the actin cytoskeleton and blue fluorescence for the cell nuclei, providing a comprehensive and visually striking representation of the cellular morphology.

For measuring the electrical response as a function of frequency in the A549 cancer cells, they were prepared under controlled conditions in order to ensure a uniform cell population. Later, the cells were exposed to electrical impulses. We induced a range of electric frequency parameters that were known to provoke electrodamage with voltage signals around 200 mV [33]. According to the literature it is sufficient to disrupt the cell membrane, inducing changes without causing cell death [33].

Figure 5a, represents the experimental impedance of A549 cancer cells for different cells concentrations. It must be mentioned that this measurement was conducted for three different cell concentrations: 126,000 cells, 250,000 cells, and 480,000 cells. Figure 5b,c presents the numerical and experimental calculation of the real and imaginary parts of impedance as a function of frequency of the samples; a numerical simulation for the ideal behavior is included. Using fractional calculus for the simulation of electrical impedance seems to be necessary for accurate modeling, it is well known that cells exhibit non-linear behavior that fractional models can solve effectively. On the other hand, fractional calculus allows a precise characterization of the frequency-dependent development of impedance and capacitance. Fractional calculus enables a better simulation of differences between different concentrations of cells, due to its better accuracy and reliability.

The real and imaginary parts of the impedance of A549 cancer cells at different electrical frequencies are plotted in Figure 5b,c, which suggests that the cells have capacitive properties, primarily associated with their cell membranes. Different fractional calculus theories, such as Riemann–Liouville, Caputo, and Grünwald–Letnikov, were considered with the purpose of determining the best fit with the experimental results. We used the Riemann–Liouville fractional-order derivative because it provides a precise framework for modeling the dynamic, non-linear, and frequency-dependent behavior of A549 cancer cells under electrical stimulation. It enables accurate fitting of experimental data, particularly in capturing the electrocapacitive effects and dielectric breakdown mechanisms, outperforming other fractional calculus theories.

MATLAB was used to perform all numerical simulations presented in the document, including those corresponding to Figure 5.

For further investigation of the electrochemical behavior of the cells, a sample was heuristically diluted in order to easily visualize the capacitive effect in the samples. Figure 6a shows electrical impedance measurements and Figure 6b shows the electrical capacitance. In Figure 6a the experimental determination of the impedance is represented by the black line. A numerical estimation was performed, depicted by the blue line. The fractional behavior was determined, and it is shown by the red dotted line at a value of $\alpha =0.60$. We employed the Grünwald–Letnikov method, illustrated by the green dotted line with $ƛ=0.75$. Additionally, the Riemann–Liouville (R-L) parameter was used, indicated by the yellow dot at $\hslash =0.80$, and the Caputo parameter is shown by the brown dotted line. It was observed that the Riemann–Liouville fractional calculus theory provided the best fit to the experimental results. This implies that, among the considered fractional calculus approaches, the Riemann–Liouville theory most accurately describes the behavior of capacitance of A549 cancer cells based on the experimental data.

The changes in impedance and capacitance with varying cell concentrations are primarily due to the increased number of cell membranes acting as capacitive elements and the altered distribution for current flow in the suspension. A capacitance behavior is a measure related to the ability of a system to store electrical charge, and its decrease with electrical frequency. These effects are more pronounced as the concentration increases, showing a clear trend that can be explained by the cumulative impact of the cell’s electrical properties and their interactions within the culture medium. For example, when the concentration of cancer cells was quadrupled to 500,000 cells, the close distribution of them reduced the resistance path making easier the current flow.

Experimentation indicates that damage to A549 human lung cancer cells by modulated voltage easily occurs at a higher frequency range since cells show an adverse response or suffer damage under voltage modulated applications. Identification of this fractional capacitive condition provides valuable information for understanding the frequency limits that affect the viability or behavior of A549 cells in the context of electrical experimentation.

The error in the numerical fit relative to the fractional fit to describe the electrical behavior of the sample refers to the discrepancies between the fractional model predictions and the actual values measured during experiments.

The challenges encountered in the study included ensuring the accuracy of experimental measurements and achieving reliable simulations of fractional electrodamage. These were addressed by using validated experimental setups, such as electrochemical impedance spectroscopy, and employing robust mathematical modeling techniques like the Riemann–Liouville fractional calculus theory. Additionally, repeated trials and statistical analyses ensured the reproducibility and reliability of the results.

The calculated error bar was ±2.5%, this shows the extent of the deviation between model predictions and experimental observations. As it can be considered, the percentage error is low; this indicates a close correspondence between the fractional model and the experimental data.

Figure 7 represents the heating induced by electrical signals in A549 cancer cells as a function of electrical frequency and voltage change for different cancer cells concentrations. The x-axis displays the electrical frequency ranging from 0 to 100 KHz, while the y-axis shows the voltage change from 100 mV to 200 mV. The color intensity in the heat map indicates the level of electrodamage, with darker colors representing higher damage levels and lighter colors indicating lower damage levels. The distribution of the electrodamage is linear, reflecting a proportional increase in damage with increasing frequency and voltage change. This visualization helps to identify the conditions under which the cells experience the most significant electrodamage.

In Figure 7, each subplot represents the relationship between electrical frequency (x-axis) and voltage change (y-axis) for a specific cell concentration. The z-axis represents the electrical damage caused. The color intensity on the surface of each subplot indicates the level of electrical damage. Each subplot is labeled with the corresponding cell concentration in the title, and the axes are labeled with the respective units.

Figure 8 provides a visual representation of how electrical frequency and voltage change affect the electrodamage for different cell concentrations. As the frequency increases, the membrane resistance decreases; it makes the cell more permeable to electrical currents. Higher frequencies and voltages both increase the electrodamage in cells.

Figure 9a represents the conversion of electrical energy to heat within cancer cells. For calculating the heat distribution within the A549 cancer cells and to evaluate processes in which electrical energy is converted into heat inside A549 cancer cells, we used the following formulas [34]:

(9)$Q={{\displaystyle \underset{0}{\overset{pw}{\int }}\left(I_0{e}^{\raisebox{1ex}{$-t$}\!\left/ \!\raisebox{-1ex}{$RC$}\right.}\right)}}^{2}Rdt={\displaystyle \frac{C{R}^2{I}_0^2}{2}}\left(1-e^{{\displaystyle \raisebox{1ex}{$-2pw$}\!\left/ \!\raisebox{-1ex}{$RC$}\right.}}\right)={\displaystyle \frac{C{V}_0^2}{2}}\left(1-e^{{\displaystyle \raisebox{1ex}{$-2pw$}\!\left/ \!\raisebox{-1ex}{$RC$}\right.}}\right).$

(10)$Z=Z_0\left(1+\phi \left(T-T_0\right)\right).$

where Q represents the heat generated in joules (J), t is the time exposition in seconds, R denotes the cancer cells’ resistance at the experimental stage, pw is the electrical signal pulsed width, $I_0$ and $V_0$ are the peak current and voltage, C is the cancer cells capacitance, Z represents the cancer cells impedance, $Z_0$ denotes the cancer cells impedance at temperature $T_0$, $\phi$ is the temperature coefficient of resistivity in A549 cancer cells, and $\left(T-T_0\right)$ represents the change in temperature at the experiment stage. We decided to modify Equation (6), changing impedance instead of resistance, because electrical impedance was in function of electrical frequency.

Figure 9b describes the relationship between voltage applied and heat that originates inside cancer cells, which is governed by Joule law. The electrical resistance of the cells is considered. Where the change in voltage, $\left(T-T_0\right)$, was from 100 mV to 200 mV, the time, t, was considered from 0 to 30 s, the R resistance of A549 cells was 110 ohms, the capacitance C was 10 nanofarads, the peak current, $I_0$, was 3 mA, the pulsed width, pw, was considered as 10 milliseconds, the peak voltage, $V_0$, was 150 mV, and the temperature coefficient of resistivity, $\phi$, was 0.0040 ohms [34].

The error bar associated with the measurement of the change in temperature and electrical conversion into heat in A549 cancer cells during the experimentation was calculated at about ±5%. It is important to mention that all the experiments reported in this article were performed 10 times, in order to guarantee repeatability and reliability in the measurements.

In order to experimentally confirm the electrodamage effect on the membrane integrity and cell viability of A549 cells, cells in suspension were exposed to electrical current. Trypan blue staining allowed determination of the effect of stimulus on membrane integrity.

A significant reduction in membrane integrity was observed after 20 s of treatment, which was evidenced by the increase in the number of cells that were capable of uptaking the stain as can be observed in Figure 10.

Since a significant reduction in cell viability was noted after 30, 60, and 120 s of treatment, the fate of these cells was analyzed through an apoptosis assay, which identified apoptotic and necrotic cells.

The results are presented as the percentage of 10,000 cells collected in gate corresponding to A549 cells (Supplementary Material S2). This assay revealed a significant increase in apoptosis and cellular necrosis after 60 and 120 s, as can be observed in Figure 4, indicated that cell death was in progress. These findings highlight the impact of electrodamage on the viability and membrane integrity of A549 cells, with significant effects observed at specific time points, showing the induction of early apoptosis and the cell death by a necrosis process.

The observed effects of fractional electrodamage on A549 cells included significant reductions in cell viability, alterations in membrane integrity, and morphological changes indicative of apoptosis and necrosis. The experiments showed a strong dependence of electrodamage on electrical frequency and voltage, with higher frequencies and voltages causing more pronounced effects. These findings highlight the susceptibility of A549 cells to fractional electrodamage, providing critical insights into their metabolic and structural responses to electrical stimulation.

4. Discussion

Electrical behavior in suspended cells exhibits a clear frequency dependence, with reported capacitance inversely proportional to the applied frequency [35]. The cells can influence the ability of a biological system to store electrical charge due to their electrical and morphological properties [36]. The variation in capacitance at different frequencies reflects dynamic interactions between cancer cells and electrodes, offering a valuable perspective on their electrochemical behavior [37]. The culture medium influenced the electrical response. It affected electrical conductivity, dielectric properties, ionic composition, and the interaction between cells and the medium [38]. Changes in the medium due to cell growth, ion concentrations, and cell viability impacted the electrical response [39]. The potential impact of discovering the dominant fractional electrical response in cell measurements is significant. It enhanced our ability to monitor and assess the effects of different treatments or environmental conditions on cells [40]. The cells possess mechanisms to convert an electrical stimulus into a cellular response, involving substantial changes in the cell membrane. For example, allowing the entry of ions, such as sodium, which is responsible for changing the electrical potential in the region where the stimulus is received [41]. Various biological functions in cells are activated when the propagation of a voltage signal passes through their membrane [42]. An example of this is cell communication, which occurs when cells are able to transmit information through electrical signals to other cells that are around; this is essential for the coordination of functions in tissues and organs [43]. The permeability of cell membranes also facilitates ionic regulation, playing a crucial role in neurotransmitter secretion and cellular excitability [44]. Another event that occurs when an external voltage signal is applied to the cells is that it is responsible for using this signal to regulate the internal balance between ions and nutrients, thus, helping the control, cell survival, and adaptation of cells to their environment [45]. The maximum that a cell can withstand depends on the type of cell it is, as well as its physiological state. There is no single value that is applicable to all cells since they have different electrical properties and tolerances depending on the cell type. However, cell membrane potentials are of the order of 100 millivolts (mV), but if they exceed 200 mV, a dielectric breakdown within the biological cell could be reported, thus, forming arcs inside the cell [33]. In brain neurons, the resting membrane potential of these neurons usually registers in the range of −60 to −70 mV [46]. Drastic changes in voltage inside the cells could cause irreparable damage, such as undesirable cellular responses like apoptosis and cell lysis [47]. The electric current used to produce a killing effect on a cancer cell can be described by controlled application of electric current through electrodes placed near or within an affected tissue with cancer cells [48]. Electrochemical damage or electroporation involves cell membrane disruption caused by high voltage application [49]. It causes pores in the cell membrane and eventually cell death when the voltage is sufficiently high [49]. However, the electroporation can be reversible according to the magnitude and duration of the electrical stimulation in some cases when the voltage is moderate and a short exposition the damage is reversible [50]. On the other hand, prolonged exposures to high voltage cause an irreversible damage to cells, generating apoptosis [51]. Furthermore, this can be achieved by a pulsed electric current of low intensity and high frequency applied through the electrodes. The pulsed electric current causes a phenomenon called electroporation in the cell membrane of the cancer cell. During electroporation, pores in the cell membrane are temporarily opened, allowing molecules to enter that would not normally be able to enter [47]. The combination of the pulsed electric current and therapeutic agents inside the cancer cell can damage it and cause its death or inhibition of growth, thus, contributing to the destruction of the cancer. Electrical power dissipation in a cell occurs when an electrical current flows through the cell due to an applied potential difference. For instance, biological cells convert electrical energy into other types of energy such as heat during power dissipation [52]. It can be explained by the result of electrical resistance of structural components in cells such as ion channels and cell membranes. Because of the intrinsic resistance of the cell, a part of the electrical energy flowing through it is converted to heat [53]. Some other mechanisms for generation of heat in cells occur in the electrical signals propagated in nerve cells [54]. The voltage threshold for damage varies with frequency; at lower frequencies, higher voltage is required to provoke damage due to greater membrane resistance [55]. At lower frequency the cell membrane resists electrical current more, so it is necessary to have higher voltage application to provoke damage [55]. A critical frequency was identified around 90,000 Hz; here the membrane is susceptible to dielectric breakdown. In this study, the primary type of cellular damage observed is dielectric breakdown, where the cell membrane loses its integrity due to high voltage and frequency, leading to cell death [56,57]. Other potential types of damage include plasmolysis, which would require specific staining techniques and microscopy to identify [57]. To determine different types of damage, various assays such as live staining, electron microscopy, and biochemical markers for cell integrity and apoptosis would be needed [58]. In areas such as medicine, cells biology, and biotechnology, information that we have described is crucial for developing therapeutic approaches, diagnostics, and research.

This study advances existing work by focusing on the fractional electrodamage mechanism, explaining its frequency-dependent nature and the relevance of fractional modeling in predicting cellular responses to electrical stimulation. Different fractional calculus theories, such as Riemann–Liouville, Caputo, and Grünwald–Letnikov, were considered with the purpose of determining the best fit with the experimental results. It was observed that the Riemann–Liouville fractional calculus theory provided the best fit to the experimental data, suggesting a better fit in modeling the fractional electrodamage processes. This finding contrasts with past published studies where alternative approaches and tests were applied. Results did not achieve the same level of precision in capturing the frequency-dependent electrocapacitive behavior of A549 cancer cells.

The proposed exploration method leverages fractional calculus to model the electrodamage of A549 cancer cells, providing a better accuracy in capturing frequency-dependent and non-linear electrical behaviors. This method enables more precise simulations of electrocapacitive responses and dielectric breakdown, facilitating real-time monitoring.

The potential advantages of using fractional electrodamage over existing cancer treatment modalities include its ability to precisely target cancer cells by leveraging frequency-dependent electrical stimulation, minimizing damage to surrounding healthy tissue. The fractional approach enables a more accurate modeling of cellular responses. It offers real-time monitoring of cellular changes and provides a non-invasive platform for combining with other anticancer therapies.

The experiments were conducted at a fixed temperature of 25 °C and within specific voltage and frequency range to ensure controlled and reproducible conditions. While these parameters optimize experimental accuracy, they limit the direct applicability of findings to varied physiological or clinical environments where temperature and electrical conditions may fluctuate. Future studies should explore a broader range of conditions to enhance generalizability. In future research, nonlinearities in biological responses, such as thresholds for apoptosis or necrosis, will be considered to enhance the analysis of electrodamage. Although statistical tools like ANOVA were employed in this study to analyze experimental data, incorporating more robust statistical methods, such as effect size or regression analysis, could offer deeper insights into the relationships between voltage, frequency, and cellular damage. These approaches would enhance the interpretation of the results by quantifying the strength and nature of these interactions, providing a more comprehensive understanding of the underlying mechanisms.

To enhance its biological relevance, future analyses could explore how this heat generation correlates with known thermal thresholds for cancer cell death. Establishing these correlations would provide a clearer understanding of the role of heat in the electrodamage process.

This will address the current simplification of linear damage trends and provide a more accurate representation of the relationship between voltage, frequency, and biological outcomes.

5. Conclusions

This study demonstrates that fractional electrodamage effectively models the dynamic electrical behavior of A549 cancer cells, highlighting the frequency-dependent nature of dielectric breakdown. The use of the Riemann–Liouville fractional calculus theory offers unparalleled precision in simulating electrocapacitive responses, enabling the identification of key thresholds for cellular damage. These findings open new opportunities for advancing cancer diagnostics and therapies, particularly through non-invasive, targeted techniques such as electrochemical hyperthermia and electroporation. The integration of experimental data with robust mathematical modeling underscores the potential of fractional approaches to revolutionize cancer treatment by minimizing harm to healthy tissues while enhancing therapeutic efficacy. The novel aspects of this work include the application of fractional calculus to model electrodamage in A549 lung cancer cells, revealing a frequency-dependent dielectric breakdown mechanism. This study incorporates non-integer order mathematical modeling with experimental validations, pointing out the superior accuracy of the Riemann–Liouville theory compared to other fractional approaches. This work not only advances our understanding of cell–electrical interactions but also sets the stage for innovative applications in biotechnology and precision medicine.

Author Contributions

H.M.-A.: writing—original draft, writing—review and editing, investigation; J.A.A.-M.: writing—review and editing, investigation; M.A.M.-P.: writing—review and editing, investigation; A.V.-F.: writing—review and editing, investigation; M.T.-V.: writing—review and editing, investigation; B.E.G.-P.: writing—review and editing, investigation; C.T.-T.: writing—review and editing, investigation, conceptualization. All authors have read and agreed to the published version of the manuscript.

Data Availability Statement

Data and materials are available upon reasonable request to Carlos Torres-Torres ([email protected]).

Acknowledgments

The authors kindly acknowledge to Instituto Politécnico Nacional, Universidad Autónoma del Estado de Hidalgo, Escuela Superior Tepeji del Río, and Consejo Nacional de Humanidades Ciencias y Tecnologías. The authors are also thankful to the Central Microscopy facilities of the CNMN-IPN.

Conflicts of Interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Footnotes

Disclaimer/Publisher’s Note: The statements, opinions and data contained in all publications are solely those of the individual author(s) and contributor(s) and not of MDPI and/or the editor(s). MDPI and/or the editor(s) disclaim responsibility for any injury to people or property resulting from any ideas, methods, instructions or products referred to in the content.

Figures

Figure 1. Roadmap describing the progress of the topic of study based on representative works. Refs. [19,20,21,22,23,24] for 2015–2021, respectively.

Figure 2. (a) A549 cancer cells measured and deposited in the Metrohm DS 220 AT electrode; cancer cells were integrated into the drop using an electronic pipette, allowing precise control over the placement of cells. (b) Scheme of the experimental setup.

Figure 3. A549 cancer cells with an electrical current induced by metallic electrodes located along a diameter of the well.

Figure 4. Confocal image of the representative A549 human lung epithelial cancer cells studied; (a) nuclei of individual cells in blue; (b) typical actin filaments were stained with rhodamine-phalloidin (red), and its longitudinal distributions are show; (c) merge of two tracks (red and blue). All figures at 40X.

Figure 5. (a) Numerical and experimental results of electrical impedance as a function of frequency for varying cell concentrations in a drop, with clear labels highlighting the trends. (b) Real part of the measured electrical impedance with distinct legends emphasizing differences across frequencies. (c) Imaginary part of the measured electrical impedance, with annotations to aid in interpreting the observed variations.

Figure 6. Comparison of experimental and numerical data for: (a) electrical impedance of the studied samples, and (b) capacitance of A549 cancer cells.

Figure 7. Heat map showing the electrodamage in A549 cancer cells as a function of electrical frequency (0–100 KHz) and voltage change (100–200 mV), for different cells concentration. The color intensity represents the level of electrodamage, with darker colors indicating higher damage levels. This linear distribution illustrates how electrodamage increases proportionally with higher frequencies and greater voltage changes. (a) Shows electrodamage as a function of applied voltage and electrical frequency, with a maximum electrodamage of 250 kHz/mV for 125,000 cells. (b) Displays a maximum electrodamage of 500 kHz/mV for 250,000 cells. (c) Exhibits a maximum electrodamage of 1000 kHz/mV for 500,000 cells.

Figure 8. Variation in electrodamage as a function of electrical frequency and voltage change for different concentrations of cancerous cells.

Figure 9. Change in temperature and electrical conversion into heat within A549 cancer cells. (a) Change in temperature representation inside A549 cancer cells. (b) Voltage and converted heat magnitude for electrical energy when it is converted into heat within biological cells.

Figure 10. Effect of electrodamage on membrane integrity and viability of A549 cells. Blue cells in photographs represent the uptake of trypan blue staining by cells with compromised membrane integrity. Dot blots show the results for the apoptosis assay. The results are presented as the percentage of 10,000 cells. Live cells are found in quadrant 4 (Q4; Annexin V and PI negative), early apoptotic cells are indicated in Q1 (Annexin V positive and PI negative), dead cells by necrosis are demonstrated in Q2 (Annexin V and PI positive). The colors represent the collection of cells with the same intensity detected during flow cytometry, with blue representing low intensity (single cells) and red representing high intensity. Statistical analysis was conducted, with graphs representing the mean of two independent experiments with standard deviation. Comparisons between experimental conditions were performed using one-way ANOVA followed by Tukey’s post hoc test (* indicates a p-value of <0.05 vs. time 0 living cells condition; ** indicates a p-value of <0.001 vs. time 0 early apoptosis condition; *** indicates a p-value of <0.001 vs. time 0 necrosis condition). Scale bars = 20 µm.

Supplementary Materials

The following supporting information can be downloaded at: https://www.mdpi.com/article/10.3390/fractalfract9010034/s1. Supplementary Material S1: Fractional Description; Supplementary Material S2: Gating Strategy for the Apoptosis Assay. References [31,57,59] are cited in Supplementary Materials.

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