Numerical simulation of PDRN iontophoresis: comparison with passive diffusion and key parameter analysis

Abstract

Polydeoxyribonucleotide (PDRN) is a bio-derived therapeutic agent known to exhibit regenerative, angiogenic, and anti-inflammatory effects. However, its relatively large molecular size limits penetration across the skin barrier, thereby reducing the feasibility of noninvasive transdermal delivery. Iontophoresis is a promising approach for enhancing the transport of charged macromolecules, yet quantitative understanding of PDRN iontophoresis remains limited. Here, we present an in silico framework for cathodal PDRN iontophoresis based on the Nernst–Planck equation, focusing on passive diffusion and electromigration. The PDRN diffusion coefficient was determined from Franz diffusion cell experiments using the lag-time method, and the effective charge was estimated from prior experimental reports on dsDNA. A three-layer skin model comprising the stratum corneum, epidermis, and dermis was employed, and PDRN transport was analyzed as a function of applied current density, initial PDRN concentration, delivery time, and PDRN diffusion coefficient. Transport was quantified using molar flux and cumulative permeation at the epidermis–dermis interface and mid-dermis. The results clarify parameter-dependent regulation of delivery and reveal a depth-dependent shift in the dominant transport mechanism, highlighting the importance of iontophoresis for macromolecular delivery. Using calibrated outputs, we assessed agreement with experimental permeation data using the root-mean-square deviation (RMSD), mean absolute percentage error (MAPE), and the coefficient of determination (R²). Overall, this work provides a computational baseline to predict PDRN iontophoresis outcomes and proposes practical iontophoresis strategies for PDRN to guide device design.

1. Introduction

Polydeoxyribonucleotide (PDRN) is a bio-derived DNA polymer with molecular weights ranging from 50 to 1,500 kDa and has been reported to promote tissue regeneration, wound healing, angiogenesis, anti-ischaemic responses, and anti-inflammatory effects (Squadrito et al. 2017). However, owing to its high molecular weight, PDRN shows limited penetration across the stratum corneum via passive diffusion (Prausnitz et al. 2004; Prausnitz and Langer 2008; Bird and Ravindra 2020; Ramadon et al. 2022; Fujii et al. 2023). Therefore, PDRN is currently administered primarily via injection for clinical use. However, due to their invasive nature, injection procedures inherently breach the skin barrier and pose substantial risks of infection, including skin and soft-tissue infections such as abscess or cellulitis (Wheeler et al. 2025). Similar concerns also apply to microneedle-based approaches, which have recently attracted attention as minimally invasive transdermal delivery methods (Jung and Jin 2021).

To overcome the limitations, several non-invasive strategies have been proposed. Chemical enhancers can improve drug permeation rates but often require extensive safety validation (Whitehead et al. 2008; Gupta et al. 2019). Other approaches using external energy, such as electricity (Kalia et al. 2004; Ramadon et al. 2022), ultrasound (Park et al. 2019; Bok et al. 2020; Li et al. 2024), light (Sun et al. 2023; Yin et al. 2023) have also been explored. However, ultrasound- and light-based methods typically require external equipment, which can increase device size and system complexity (Park et al. 2019; Sun et al. 2023; Yin et al. 2023; Li et al. 2024). In contrast, iontophoretic drug delivery employs electrical energy and can be implemented in compact, integrated patches powered by miniaturised electronic circuits (Van Der Geest et al. 1996; Bird and Ravindra 2020; Bakshi et al. 2020).

Therefore, iontophoresis is gaining increasing attention as a noninvasive strategy to enhance transdermal drug transport by applying small electrical current (Morrow et al. 2007; Prausnitz and Langer 2008; Ramadon et al. 2022). In the skin, the anions are mobilised to the anode by a repel pulse from the cathode, and vice versa for the cation (Dhote et al. 2012; Bakshi et al. 2020). Recent studies indicate that even high-molecular-weight therapeutics can penetrate skin primarily through relatively large appendageal pores such as sweat glands and hair follicles (Hinsberg et al. 1995; Wang et al. 2007; Bakshi et al. 2020; Hasan et al. 2022). However, because these openings occupy only a small fraction of the skin surface area (typically on the order of ~0.1% overall), passive delivery remains inefficient (Abbasi and Heath 2025). To overcome this limitation, iontophoresis can enhance transport by preferentially driving flux through these low-resistance pathways (Han et al. 2026).

Consistent with this trend, interest in PDRN iontophoresis and iontophoretic device development has been increasing, and electrode-array designs for iontophoretic drug delivery represent a meaningful direction for advancing transdermal delivery systems (Moarefian et al. 2020; Bok et al. 2023; Han et al. 2026; Baniya et al. 2025). However, iontophoresis involves numerous controllable parameters, including current density, drug concentration, delivery time, and electrode spacing, which makes it impractical to identify optimal conditions and predict outcomes solely through experiments (Vemulapalli et al. 2008; Dhote et al. 2012; Bora and Dasgupta 2022). Therefore, there is a strong need for simulation-based guidance that can predict PDRN iontophoretic transport under a range of delivery conditions and inform experimental design.

In this study, we simulate cathodal iontophoresis using the Nernst–Planck equation (Imanidis and Luetolf 2006; Lu et al. 2010; Filipovic et al. 2016; Pontrelli et al. 2017; Noor et al. 2021; Bora and Dasgupta 2022). To enhance the validity of the simulations, key material properties for the drug and skin were obtained from experiments or the literature. Specifically, a Franz diffusion cell experiment was performed to measure passive diffusion of PDRN (Ng et al. 2010; Salamanca et al. 2018; Pulsoni et al. 2022), and the time-lag method was applied to estimate the effective diffusion coefficient in skin (Pillai and Panchagnula 2003; Kharis Nugroho et al. 2004; Mitragotri et al. 2011). In addition, the effective charge of PDRN was estimated based on prior literature, enabling extraction of a critical parameter required for iontophoretic simulations (Keyser et al. 2006). Using these inputs, we investigated transdermal transport as a function of four key parameters—applied current density, initial PDRN concentration, delivery time, and PDRN diffusion coefficient—to clarify the role of each parameter in iontophoretic delivery and to provide practical guidelines for experimental design (Vemulapalli et al. 2008; Jiang et al. 2021). The novelty of this study is the development of a PDRN-specific numerical model that not only predicts cumulative delivery but also resolves the relative contributions of diffusion and electromigration to identify depth-dependent parameter sensitivity during cathodal iontophoresis.

2. Materials and methods

2.1. Governing equations

First, the electrical potential, a key variable distinguishing passive diffusion from iontophoresis, was calculated using the electrostatic Laplace equation shown below (Ciuculete and Morega 2015; Pontrelli et al. 2017; Filipovic et al. 2017).

$$\nabla \bullet (\sigma \nabla \phi )=0$$ (1)

where $\sigma$ (S m⁻¹) is the electrical conductivity, and $\phi$ (V) is the electrical potential.

The electric field and current density were then computed using the following equations (Patriciu et al. 2005; Pontrelli et al. 2017; Filipovic et al. 2017).

$$\vec{E}=-\nabla \phi$$ (2)

$$\vec{j}=\sigma \vec{E}$$ (3)

where $\vec{E}$ (V m⁻¹) and $\vec{j}$ (A m⁻²) represent the electric field and the current density, respectively.

In this study, to evaluate the molar flux of PDRN during iontophoresis, we employed the Nernst–Planck equation without electro-osmosis, consistent with previous work (Filipovic et al. 2017). Because PDRN is a negatively charged DNA-derived macromolecule, its iontophoretic transport cannot be described by passive diffusion alone. Therefore, the Nernst–Planck equation was used to account for concentration-gradient-driven diffusion and electric-field-driven electromigration. Accordingly, the one-dimensional (1D) form of the Nernst–Planck equation used in this study is expressed as follows (Pontrelli et al. 2017; Filipovic et al. 2017; Bora and Dasgupta 2022; Lee et al. 2022).

$$J_i=-D_i\frac{{\partial C}_i}{\partial x}-\frac{z_i{D}_{i}F}{\mathit{RT}}{C}_i\frac{\partial \phi }{\partial x}$$ (4)

where $J_i$ (mol m⁻² s⁻¹) is the molar flux of species $i$ , $D_i$ (m² s⁻¹) is the diffusion coefficient of species $i$ , $C_i$ (mol m⁻³) is the concentration of species $i$ , $z_i$ is the effective charge of species $i$ , $F$ (C mol⁻¹) is the Faraday constant, $R$ (J mol⁻¹ K⁻¹) is the universal gas constant, $T$ (K) is the absolute temperature, and $x$ (m) is the distance.

The Nernst–Planck equation can be decomposed into two terms. The first is the diffusion term, driven by the diffusion coefficient and the concentration gradient. The second is the electromigration term, representing the motion of charged molecules under an electric field generated by an applied potential. When either $z_i$ or $\partial \phi /\partial x$ is zero, the Nernst–Planck equation becomes independent of the applied current. Thus, in Equation (4), the electromigration term can be neglected, and the expression reduces to the diffusion term only, which is equivalent to Fick’s first law (Pontrelli et al. 2017; Filipovic et al. 2017; Bora and Dasgupta 2022).

$$J_{\mathit{id}}=-D_i\frac{\partial C_i}{\partial x}$$ (5)

Conversely, when $\partial C_i/\partial x$ is zero, Equation (4) simplifies such that $J_{\mathit{id}}$ reduces to the electromigration term alone, with the diffusion term neglected (Pontrelli et al. 2017; Filipovic et al. 2017; Bora and Dasgupta 2022).

$$J_{\mathit{ie}}=-\frac{z_i{D}_{i}F{C}_i}{\mathit{RT}}\frac{\partial \phi }{\partial x}$$ (6)

On the other hand, the diffusion coefficient and electric mobility are proportionality constants relating the flux to the concentration gradient and the potential gradient, respectively. This relationship is associated with the Nernst–Einstein equation. The electric mobility of species $i$ is given by (Pontrelli et al. 2017; Filipovic et al. 2017)

$$u_i=\frac{z_i{D}_{i}F}{\mathit{RT}}$$ (7)

In addition, to quantify the cumulative amount of PDRN permeated at a specified depth, the molar flow rate was first computed using the surface integral below. Because the analysis is based on a two-dimensional (2D) axisymmetric model, the integral is evaluated over the rotated surface area generated by revolving the cross-sectional line (Pontrelli et al. 2017; Bora and Dasgupta 2022).

$$\frac{\mathit{dQ}}{\mathit{dt}}={\int }_{\Gamma }N(r,s)2\pi r \mathit{ds}$$ (8)

where $\mathit{dQ}/\mathit{dt}$ (mol s⁻¹) is the molar flow rate, $\mathrm{\Gamma }$ denotes the interface line in the 2D axisymmetric cross-section over which the surface integral is evaluated, $N(x)$ (mol m⁻² s⁻¹) is the molar flux, $r$ (m) is the radial coordinate measured from the axis of revolution, $\mathit{ds}$ (m) is the differential line element along the cross-section, and $2\pi \mathit{r\; ds}(m^2)$ is the corresponding differential surface area element generated by revolution.

The cumulative permeated amount was then obtained by integrating $\mathit{dQ}/\mathit{dt}$ over the total delivery time $t$ (Pontrelli et al. 2017; Bora and Dasgupta 2022).

$$Q={\int }_0^t\left(\frac{\mathit{dQ}}{\mathit{dt}}\right)\mathit{dt}={\int }_0^t{\int }_{\Gamma }N(r,s)2\pi \mathit{r\; ds}\mathit{dt}$$ (9)

The governing equation of drug concentration in 2D axisymmetric model used in this study is expressed as (Pontrelli et al. 2017; Filipovic et al. 2017; Bora and Dasgupta 2022; Lee et al. 2022):

$$\frac{\partial C}{\partial t}+\nabla \bullet (-D\nabla C-\frac{\mathit{zDF}}{\mathit{RT}}C\nabla \phi )=0$$ (10)

where $C$ (mol m⁻³) is the drug concentration, $D$ (m² s⁻¹) is the diffusion coefficient, and $z$ is the effective charge.

When an electric current flows through a medium, it generates heat, a phenomenon known as Joule heating. To incorporate this effect, the energy equation governing heat transfer is written as (Patriciu et al. 2005; Shafirstein and Moros 2011; Noor et al. 2021):

$$\frac{\partial }{\partial t}\left(\rho (e+\frac{v^2}{2})\right)=\nabla \bullet (k_{\mathit{eff}}\nabla T)+S_h$$ (11)

where $\rho$ (kg m⁻³) is the density, $e$ (J kg⁻¹) is the specific internal energy, $v^2/2$ (J kg⁻¹) is the specific kinetic energy, $k_{\mathit{eff}}$ (W m⁻¹ K⁻¹) is the effective thermal conductivity, and $S_h$ (W m⁻³) is the volumetric heat source term.

Because $e\approx C_{p}T$ and the flow velocity is negligible under iontophoresis and electric-field-driven transport, the kinetic-energy term $\partial /\partial t(\rho v^2/2)$ can be neglected. Accordingly, the governing equation solved in the simulations is (Patriciu et al. 2005; Noor et al. 2021)

$$\rho C_p\frac{\partial T}{\partial t}=\nabla \bullet (k_{\mathit{eff}}\nabla T)+S_h$$ (12)

where $C_p$ (J kg⁻¹ K⁻¹) is the specific heat capacity.

The source term used to model Joule heating is given by (Patriciu et al. 2005; Noor et al. 2021):

$$S_h=\vec{J} \bullet \vec{E}=\sigma {|\nabla \phi |}^2$$ (13)

2.2. Numerical model and mesh generation

The numerical model used in the simulations was constructed using the ANSYS SpaceClaim module. Among the electrodes comprising the array device shown in Figure 1a, simulations were performed to evaluate the performance of a single-electrode model shown in Figure 1b.

Figure 1.

Schematic overview of the iontophoretic device configurations and simulation model: (a) electrode array applied to facial skin, (b) single-electrode configuration applied to skin (c) PDRN iontophoresis and 2D axisymmetric model used in the simulation, and (d) mesh refinement near the anode and cathode.

The human skin geometry was designed to reflect the thickness of the malar and lower cheek regions reported in the literature (Lee et al. 2002; Chopra et al. 2015). Because a 2D axisymmetric analysis was conducted, only the right-hand side of Figure 1c was modelled. However, the dermis thickness was set to 7 mm, rather than the commonly used value of approximately 1–2 mm, in order to examine the maximum delivery depth of PDRN by intentionally extending the computational domain. Accordingly, the ‘mid-dermis’ described later does not refer to the midpoint of the 7 mm dermis layer. Instead, it denotes 0.6 mm, which corresponds to the midpoint of a 1.2 mm dermis thickness representative of the malar and lower cheek regions. The electrode spacing and hydrogel thickness were set to 20 mm and 3 mm (Han et al., 2026). Table 1 summarises the model parameters and their corresponding values (Lee et al. 2002; Chopra et al. 2015).

Table 1.

Geometrical parameters of the drug layer and human skin used in numerical simulations.

	Cell zone	Parameter	Length(mm)
Material	pH 7.0 buffer gel	X-length	1.25
		Y-length	3
	PDRN hydrogel	X-length	2.5
		Y-length	3
Skin	Stratum corneum	X-length	40
		Y-length	0.01
	Epidermis	X-length	40
		Y-length	0.04
	Dermis	X-length	40
		Y-length	7

Figure 1d shows the mesh generated near the pH 7.0 buffer gel and the PDRN hydrogel regions. To avoid excessive numerical concentration at sharp corners, the model edges were rounded. Meshing was performed using Ansys Meshing with the automatic (PrimeMesh) method. Mesh sizing was applied such that smaller elements were used in the stratum corneum and epidermis, whereas larger elements were used in the remaining regions. This strategy increased mesh resolution in the stratum corneum and epidermis, which are critical to the analysis. The final model consisted of a total of 322,981 mesh elements.

2.3. Numerical solution

In this study, ANSYS Fluent 2024 R2 was used to perform numerical simulations. Material properties, including the physical and electrical properties of human skin, were obtained from the literature, and the values listed Table 2 were used (Miklavč et al. 2006; Shafirstein and Moros 2011; Moscicka-Studzinska and Ciach 2012; Ciuculete and Morega 2015; Filipovic et al. 2017; Noor et al. 2021; Newell and Zhan 2024; Mohizin and Sung 2025; Bhuimali et al. 2025; G et al. 2025). The convergence criteria values were set to 10⁻⁶, and all governing equations are discretized using the second-order upwind scheme.

Table 2.

Material properties of the drug layer and human skin used in the numerical simulations.

	Cell zone	Property	Value	Unit
Material	Common value	Temperature (initial condition)	24	℃
	pH 7.0 buffer gel	Electrical conductivity	1.6	S m−1
		Specific heat capacity	4100	J kg−1 K−1
		Thermal conductivity	0.6	W m−1 K−1
	PDRN hydrogel	Electrical conductivity	1.15	S m−1
		Specific heat capacity	4180	J kg−1 K−1
		Thermal conductivity	0.6	W m−1 K−1
Skin	Common value	Diffusion coefficient	1.55 × 10−10	m2 s−1
		Temperature (initial condition)	31	℃
	Stratum corneum	Electrical conductivity	0.0005	S m−1
		Specific heat capacity	1880	J kg−1 K−1
		Thermal conductivity	0.213	W m−1 K−1
	Epidermis	Electrical conductivity	0.026	S m−1
		Specific heat capacity	4400	J kg−1 K−1
		Thermal conductivity	0.62	W m−1 K−1
	Dermis	Electrical conductivity	0.227	S m−1
		Specific heat capacity	4184	J kg−1 K−1
		Thermal conductivity	0.582	W m−1 K−1

2.4. Determination of the effective diffusion coefficient of PDRN—Franz diffusion cell experiment and time-lag analysis

To estimate the diffusion coefficient—one of the key parameters for simulating the transdermal delivery of PDRN (Derbiome, SHEBAH BIOTECH Inc.)—a Franz diffusion cell experiment was conducted. Based on the experimental results, the time-lag method was applied to extract the effective diffusion coefficient of PDRN in skin.

The Franz diffusion cell experiment is a passive diffusion test performed using a Franz diffusion cell, as shown in Figure 2a. A micropig skin membrane (Apures; thickness: 600 μm, dimensions: 30 × 30 mm) with a stratum corneum, epidermis, and dermis layer was placed between the donor and receptor chambers and secured using a pinch clamp. Micropig dorsal skin was used as an ex vivo human-skin substitute because porcine/minipig skin shows comparable permeability characteristics to human skin and is suitable for reproducible Franz diffusion cell experiments (Qvist et al. 2000). The receptor chamber was filled with phosphate-buffered saline (PBS, pH 7.2) and a stirring bar, whereas the donor chamber was loaded with a PDRN solution (10 mg mL⁻¹), which was prepared by dissolving PDRN in DIW under stirring at 50 °C. At predetermined time points, receptor samples were collected through the sampling port. To allow bubbles formed in the receptor chamber to escape through the sampling port, the apparatus was slightly tilted. The donor chamber was sealed with parafilm to prevent evaporation of the PDRN solution. To maintain the receptor volume and temperature, an equal volume of PBS at the same temperature was replenished immediately after each sampling.

Figure 2.

Evaluation of the effective diffusion coefficient of PDRN in micropig skin. (a) Schematic illustration of the Franz diffusion cell. (b) Cumulative amount of permeated PDRN over time and its linear regression (R² = 0.9844). Experimental data in a and b are presented as mean ± SD (n = 3).

A standard curve was constructed from the absorbance of dsDNA at 260 nm at various concentrations. Using this calibration, the cumulative permeated amount $Q_n$ and the cumulative permeated amount per unit area $Q_n/A$ were calculated as follows:

$$Q_n=V_r{C}_n+\sum _{i=1}^{n-1}{V}_s{C}_i$$ (14)

where $V_r$ (mL) is the total volume of the receptor chamber, $C_n$ (μg mL⁻¹) is the receptor concentration at time point $n$ , $V_s$ (mL) is the sampling volume, and $C_i$ (μg mL⁻¹) is the receptor concentration measured at sampling time point $i$ .

When $Q_n/A$ was plotted as a function of time, the linear region (excluding the initial non-steady-state period) was fitted by linear regression to obtain a first-order relationship with slope $J_{\mathit{ss}}$ and x-intercept $t_{\mathit{lag}}$ , as shown in Figure 2b and expressed by:

$$\frac{Q_n}{A}=J_{\mathit{ss}}(t-t_{\mathit{lag}})$$ (15)

where $A$ (cm²) is the permeation area, $J_{\mathit{ss}}$ (μg cm⁻² h⁻¹) is the steady-state flux, and $t_{\mathit{lag}}$ (h) is lag time.

Using the time-lag method (Pillai and Panchagnula 2003; Kharis Nugroho et al. 2004; Mitragotri et al. 2011), the effective diffusion coefficient can be derived as:

$$t_{\mathit{lag}}=x^2/6D$$ (16)

where $x$ (m) is the membrane thickness and $D$ (m² s⁻¹) is the effective diffusion coefficient in skin.

Table 3 lists the experimental parameters and intermediate results used to calculate the effective diffusion coefficient. From this analysis, the effective diffusion coefficient of PDRN in skin was determined to be 1.55 × 10⁻¹⁰ m² s⁻¹, and this value was used as the PDRN diffusion coefficient in the simulations as shown in Table 2.

Table 3.

Experimental parameters and calculated results from the Franz diffusion cell experiment used to determine the effective diffusion coefficient of PDRN in skin.

Parameter	Value	Unit
Temperature of the donor PDRN solution	24.6	℃
Temperature of the receptor PBS	31	℃
Total receptor volume ( $V_r$ )	9	mL
Sampling volume ( $V_s$ )	1	mL
Permeation area ( $A$ )	2.27	cm2
Skin membrane thickness ( $x$ )	600	μm
Slope $(J_{\mathit{ss}})$	0.837	μg cm−2 h−1
x-intercept ( $t_{\mathit{lag}}$ )	0.108	h
Effective diffusion coefficient of PDRN	1.55 $\times$ 10−10	m2 s−1

2.5. Estimation of the effective charge of PDRN—effective charge of dsDNA per base pair

To define the electromigration term, the effective charge of PDRN was estimated based on a previous experimental study that quantified the effective charge of double-stranded DNA (dsDNA) (Keyser et al. 2006). Because this value reflects experimentally derived physical parameters that account for reduced electrostatic forces due to counterion condensation as well as electroosmotic effects, it can serve as a reasonable approximation for modelling the electromigration behaviour of PDRN. The study reported an effective charge of dsDNA of $0.50\pm 0.05e^{-}$ per base pair, which was applied to the 150-bp PDRN considered in this work. Based on the report that the effective charge is not strongly dependent on DNA length or salt concentration, the effective charge of an entire PDRN molecule was calculated as follows and used as the effective charge input to the Nernst–Planck equation:

$$Q_{\mathit{eff}}=N_{\mathit{bp}}\times q_{\mathit{eff}}=150\times 0.50e^{-}=75e^{-}$$ (17)

where $N_{\mathit{bp}}$ is the number of base pair and $q_{\mathit{eff}}$ is the effective charge per base pair of dsDNA.

2.6. Calibration

Unlike simulations, in vitro iontophoretic experiments are influenced by multiple factors that are difficult to parameterise explicitly. For example, prolonged hydration reduces stratum corneum resistance and alters the effective diffusion geometry, sustaining flux beyond predictions of a time-invariant model. In addition, the net negative charge of the skin at physiological pH induces electroosmotic flow from anode to cathode, which may hinder anionic transport during cathodal delivery (Guy et al. 2000; Pikal 2001; Kalia et al. 2004; Sahraoui et al. 2024). However, the magnitude of electro-osmosis is highly sensitive to the state of fixed charges (pH), electrolyte composition, and time. Therefore, it requires additional experiments or independent parameter estimation that reflects skin condition and formulation-specific factors. Because of these factors, implementing a fully time-dependent simulation is challenging. Therefore, we derived simple calibration parameters. These parameters account for geometric differences between the previously reported experimental setup and our model (concentric annulus vs. rectangular geometry), as well as unit conversion from mol to μg (Han et al. 2026). They also capture additional discrepancies between simulations and experiments.

As distinct from cases that can be corrected using a single scaling factor, delivery time acts as a governing variable in time-dependent simulations, implicitly modulating all of the aforementioned effects. Consequently, as delivery time increases, the discrepancy between simulation and experiment tends to become more pronounced. Therefore, the calibration procedure was divided into two cases, depending on the varying parameter: initial PDRN concentration shown in Equation (18) and delivery time shown in Equation (19). First, the results obtained by varying the initial PDRN concentration were corrected using a single scaling factor:

$$Q_C=k_C{n}_{\mathit{sim}}$$ (18)

where $Q_C$ (μg) is the cumulated mass calibrated for changes in the initial PDRN concentration, $k_C$ (1.95 × 10⁶ μg mol⁻¹) is the concentration calibration factor, and $n_{\mathit{sim}}$ (mol) is the simulated cumulative amount of permeated PDRN.

Additionally, we corrected the results for changes in delivery time using a calibration function that follows a Korsmeyer–Peppas–type power-law form, which has been widely used to empirically fit nonlinear time-course behaviour in drug release and permeation (Rinaki et al. 2003). Specifically, the calibrated cumulative mass is defined as:

$$Q_{\mathit{CT}}=k_{\mathit{CT}}{(n_{\mathit{sim}})}^p$$ (19)

where $Q_{\mathit{CT}}$ (μg) is the cumulated mass calibrated for changes in delivery time, $k_{\mathit{CT}}$ (803 μg mol⁻¹) and $p$ (0.183) are time-dependent calibration parameters estimated by least-squares fitting to the reference dataset. Taking the natural logarithm yields the following linear relationship:

$$\ln {(Q}_{\mathit{CT}})=\ln {(k}_{\mathit{CT}})+\mathit{pln}(n_{\mathit{sim}})$$ (20)

Accordingly, $p$ is obtained as the slope of the least-squares regression in log-log space, yielding 0.183. We then multiply this calibration factor by the result of Equation (9) to map our simulations to experimentally reported cumulative mass values, thereby enabling extrapolation to conditions beyond the experimentally validated range (Han et al. 2026).

3. Results

We performed numerical simulations to examine PDRN molar concentration contours while varying key iontophoretic parameters. The four primary parameters considered were the applied current density, initial PDRN concentration, delivery time, and the PDRN diffusion coefficient. This analysis enables prediction of the penetration depth and lateral spread of PDRN within the skin under each condition. For each case, the PDRN molar flux was evaluated at two representative depths: the epidermis–dermis interface (skin depth of 0.05 mm) and the mid-dermis (skin depth of 0.65 mm). By comparing fluxes at these depths, we assessed how transport varies with depth. In particular, under iontophoretic conditions, the relative contributions of passive diffusion and electromigration to the total flux were quantified, providing insight into the transport rate of PDRN at each depth. In addition, the cumulative permeated amount of PDRN at the epidermis–dermis interface and across the mid-dermis plane was compared between passive diffusion and iontophoresis, allowing evaluation of depth-dependent delivery efficiency under different iontophoretic conditions. Finally, by calibrating the cumulative permeated amount of PDRN, we compare our simulation outcomes with experimental measurements and predict permeation behaviour in conditions that were not experimentally evaluated.

The baseline conditions for the parameter sweeps are summarised in Table 4, and Figures 3–6 present the delivery results obtained by varying one parameter at a time while keeping other parameters fixed at their respective baseline values. The molecular weight distribution of the PDRN used in this study was approximately 30–200 kDa. Therefore, when the initial PDRN concentration of 0.11 mol m⁻³ is converted to a mass concentration, it corresponds to approximately 3.3–22 mg mL⁻¹. This range is higher than the approximately 1 mg mL⁻¹ used in previously reported local PDRN delivery studies (Lazzarotto et al. 2004; Valdatta et al. 2004), while falling within or below the range of 20 mg mL⁻¹ used in a related preclinical PDRN iontophoresis study (Han et al., 2026). Therefore, considering the representative molecular weight range of PDRN, 0.11 mol m⁻³ was selected as a reasonable baseline and upper-bound concentration in this study.

Table 4.

Baseline values of key parameters used in parametric studies.

Figure	Varying parameter	Fixed value	unit
3	Applied current density	0.5	mA cm−2
4	Initial PDRN concentration	0.11	mol m−3
5	Delivery time	60	min
6	PDRN diffusion coefficient	1.55 $\times$ 10−10	m2 s−1

Figure 3.

PDRN molar concentration profiles and molar fluxes of passive diffusion (diff), electromigration (elec) and total flux (total) at the epidermis-dermis interface and the mid-dermis under applied current densities of (a) 0, (b) 0.25, (c) 0.5, (d) 0.75, and (e) 1 mA cm⁻². Cumulative permeated PDRN (f) at the epidermis-dermis interface and (g) in the mid-dermis plane. Maximum values in the computational domain at the end of the simulation: (h) temperature, electric potential, current density magnitude for each applied current density.

Figure 4.

PDRN molar concentration profiles and molar fluxes of passive diffusion (diff), electromigration (elec) and total flux (total) at the epidermis-dermis interface and the mid-dermis for initial PDRN concentrations of (a) 0.014, (b) 0.028, (c) 0.055 and (d) 0.11 mol m⁻³. Cumulative permeated PDRN (e) at the epidermis-dermis interface and (f) in the mid-dermis plane.

Figure 5.

PDRN molar concentration profiles and molar fluxes of passive diffusion (diff), electromigration (elec) and total flux (total) at the epidermis-dermis interface and the mid-dermis for delivery times of (a) 15, (b) 30, (c) 45 and (d) 60 min. Cumulative permeated PDRN (e) at the epidermis-dermis interface and (f) in the mid-dermis plane.

Figure 6.

PDRN molar concentration profiles and molar fluxes of passive diffusion (diff), electromigration (elec) and total flux (total) at the epidermis-dermis interface and the mid-dermis for PDRN diffusion coefficients of (a) 1 × 10⁻¹¹, (b) 1.55 × 10⁻¹⁰, (c) 1 × 10⁻⁰⁹ and (d) 1 × 10⁻⁰⁸ m² s⁻¹. Cumulative permeated PDRN (e) at the epidermis-dermis interface and (f) in the mid-dermis plane.

3.1. Varying parameter: applied current density

First, we analysed how changes in the applied current density—one of the key parameters in iontophoresis—affect the iontophoretic delivery of PDRN. The difference in PDRN transport between passive diffusion and iontophoresis is readily observed by comparing Figure 3a with Figure 3b–e. For example, when comparing the no-current case shown in Figure 3a with the baseline current density of 0.5 mA cm⁻² shown in Figure 3c, the PDRN molar flux at the epidermis–dermis interface is more than 10³ times higher.

This trend is also evident in Figure 3f and g, which show the cumulative permeated amount of PDRN at the epidermis–dermis interface and at the mid-dermis, respectively, as a function of the applied current density. In Figure 3f, iontophoresis at 0.5 mA cm⁻² yields a cumulative permeated amount that is 2 × 10³ times higher than that of passive diffusion (0 mA cm⁻²). Notably, in Figure 3g, the cumulative permeated amount at the mid-dermis is approximately 1.2 × 10⁴ times higher than that under passive diffusion.

At first glance, these results may suggest that higher current densities are advantageous for PDRN delivery. However, the current density cannot be increased indiscriminately, because Figure 3h shows that once the applied current density exceeds 0.5 mA cm⁻², Joule heating and the potential difference increase sharply, which may raise safety concerns such as changes in skin pH (Patriciu et al. 2005; Han et al., 2026). This behaviour corresponds to the commonly cited maximum safe current density (MSCD), and for cathodic iontophoresis, the use of current densities up to 0.5 mA cm⁻² is generally recommended (Khan et al. 2011; Dhote et al. 2012). Therefore, considering both delivery performance and safety constraints, 0.5 mA cm⁻² shown in Figure 3c was selected as the operating current density to maximise the efficiency of PDRN iontophoresis.

3.2. Varying parameter: initial PDRN concentration

Figure 4a–4f presents the simulation results obtained by varying the concentration of the PDRN hydrogel applied to the cathodic drug reservoir. Overall, the trends are similar to those observed in Figure 3. However, the changes in PDRN molar flux with increasing initial PDRN concentration are relatively modest at both the epidermis-dermis interface and the mid-dermis. This indicates that, compared with the strong influence of current density, the effect of the initial PDRN concentration is minor throughout the skin domain.

Notably, in Figure 4f, increasing the initial PDRN concentration from 0.014 to 0.11 mol m⁻³ increases the iontophoretic delivery at the mid-dermis by only 1.4 times. This suggests that under iontophoretic conditions, increasing the initial PDRN concentration has a negligible impact on PDRN delivery to the mid-dermis. Moreover, 0.11 mol m⁻³ corresponds to a relatively high donor concentration of approximately 3.3–22 mg mL⁻¹, based on the molecular weight distribution of the PDRN used in this study. Therefore, considering the practical trade-off in which higher viscosity generally reduces diffusivity, using concentrations higher than this level is not desirable.

3.3. Varying parameter: delivery time

Figure 5a–f shows the simulation results obtained by varying the total delivery time after the electric potential was applied. A key observation is the depth-dependent rate of change in the cumulative permeated amount of PDRN. As shown in Figure 5e, the cumulative permeated PDRN at the epidermis-dermis interface increases by approximately 2 × 10² times with increasing delivery time. In contrast, at the mid-dermis, the cumulative permeated PDRN increases much more substantially, reaching approximately 9.5 × 10⁴ times the initial value, as shown in Figure 5f. This behaviour is consistent with the trend that the influence of the electromigration term diminishes with increasing distance from the electrodes, whereas the contribution of the diffusion term becomes more pronounced.

As indicated by Equations (4–6), delivery time affects transport primarily through the passive-diffusion component, whereas applied current density and initial PDRN concentration scale more strongly with the electromigration component. Because passive diffusion dominates near the mid-dermis, delivery time has a greater impact on PDRN delivery at the mid-dermis. These results suggest that increasing the delivery time has minimal influence on the initial penetration through the stratum corneum and epidermis. But it also plays a critical role in post-epidermal transport, particularly diffusion within deeper skin layers. This conclusion is also supported by comparing the cumulative permeated amounts in Figure 5e and f. Therefore, optimising the delivery time according to the target delivery depth can improve the overall efficiency of PDRN delivery.

However, the delivery time cannot be increased indefinitely to improve transport performance. Longer iontophoresis increases the likelihood of side effects, including tingling, itching, erythema, and oedema, and can also lead to larger pH changes (Li et al. 2005). Although the use of a pH 7.0 buffer can mitigate these effects, adverse reactions may still arise when iontophoresis is applied for longer than 60 min (Guffey and Guffey 1999). Therefore, we set the maximum current application time to 60 min.

3.4. Varying parameter: PDRN diffusion coefficient

Finally, Figure 6a–f compares the simulation results obtained by varying the PDRN diffusion coefficient. Interestingly, this parameter exhibits trends that differ from those observed for the other variables. In the molar-flux plots of Figure 6a and d, the relative dominance of the diffusion and electromigration contributions differs markedly from the other cases. Specifically, in Figure 6a, the passive-diffusion term dominates over the electromigration term at all locations, whereas in Figure 6d, the electromigration term accounts for nearly the entire flux at all locations.

The behaviour in Figure 6a can be explained by extremely low diffusion coefficient, which makes skin permeation of PDRN nearly impossible. As a result, the drug accumulates near the concentration boundary layer, leading to a situation in which $\nabla C$ becomes large relative to $C$ in Equation 4. Consequently, the passive-diffusion term becomes dominant compared with the electromigration term. Conversely, in Figure 6d, the high diffusion coefficient enables rapid permeation and spreading of PDRN within the skin, resulting in a regime in which $C$ becomes large relative to $\nabla C$ , thereby increasing the relative contribution of electromigration.

In addition, the diffusion coefficient produced the largest overall impact among the investigated parameters. While the difference between 1 × 10⁻¹¹ and 1.55 × 10⁻¹⁰ m² s⁻¹ in F igure 6e is notable, corresponding to an 10⁵ times increase in PDRN delivery, the effect is even more pronounced at the mid-dermis in Figure 6f, where the increase reaches 5.4 × 10¹⁰ times. The cumulative permeated amount at the mid-dermis increased dramatically as the diffusion coefficient increased, and under iontophoretic conditions, the maximum value was on the order of 10²⁷ times the minimum value at the mid-dermis. Such a disparity is not realistically achievable by varying the other parameters. As shown in Equation (4), this is because the diffusion coefficient, unlike the other variables, is directly proportional to both the diffusion and electromigration terms. Although the diffusion coefficient was swept over a wider range than the other parameters, the value used in Figure 6d is within a realistic range for smaller molecules, making the result meaningful (Filippin et al. 2024). Therefore, maximising the diffusion coefficient is a critical challenge for enhancing the efficiency of PDRN iontophoresis.

3.5. Calibration and performance summary

To enable quantitative comparison with previously reported experimental data (Han et al., 2026), we introduced calibration parameters. These parameters represent factors such as geometric discrepancies, unit conversion, and residual mismatches between simulations and experiments. We compared the experimental measurements with the simulated results extracted at the epidermis–dermis interface, beyond the stratum corneum. Equations (18) and (19) describe the calibration procedures for variations in the initial PDRN concentration and delivery time, respectively. Using the calibrated outputs, we quantified the agreement with the reference dataset by calculating RMSD, MAPE, and $R^2$ . These metrics were used to assess how closely the calibrated simulations reproduced the reference results.

However, because the available experimental data were insufficient, we could not establish additional calibration functions for variations in the applied current density and the PDRN diffusion coefficient. Accordingly, for these parameter sweeps, we applied the concentration-based calibration constant $k_C$ obtained in Equation (18), which already incorporates geometry-related scaling, unit conversion, and the systematic simulation–experiment offset. This approach aligns the overall scale for current-density and diffusion-coefficient variations, allowing a first-order projection of trends associated with these parameters.

First, Figure 7a compares the calibrated simulation results and the experimental data as a function of the initial PDRN concentration. The simulations reproduced the experimental trend with high fidelity, yielding an RMSD of 23.34 μg, a MAPE of 6.741%, and an $R^2$ of 0.9821. Figure 7b presents the time-dependent permeation results, including the simulation calibrated using $Q_C$ , the simulation calibrated using $Q_{\mathit{CT}}$ , and the experimental measurements. The $Q_C$ -based calibration showed a substantial discrepancy from the experimental time course, because the simulations did not explicitly capture time-evolving factors present in the experiments. Therefore, we compared the results using the power-law calibration $Q_{\mathit{CT}}$ and the best agreement was obtained at $p=0.183$ , for which the RMSD, MAPE, and $R^2$ were 1.46 μg, 2.019%, and 0.9971, respectively.

Figure 7.

Cumulative amount of permeated PDRN: (a) simulated and experimental cumulative permeation with the concentration calibration Q_C​ as a function of the initial PDRN concentration; (b) simulated and experimental cumulative permeation with Q_C and the time-dependent calibration Q_CT​ (with different p values) as a function of delivery time; (c) simulated cumulative permeation with Q_C​ as a function of applied current density; and (d) simulated cumulative permeation with Q_C​ as a function of the diffusion coefficient. Experimental data in a and b are presented as mean ± SD (n = 5).

Figure 7c and d show the projected trends for variations in the applied current density and the PDRN diffusion coefficient, respectively, inferred using the $Q_C$ scaling. Because corresponding experimental measurements were not available for these parameter sweeps, these results should be interpreted as first-order estimates. Notably, when compared with Figure 7a, which was calibrated using the same approach, both sweeps suggest a substantially larger increase in the cumulative permeated amount. In addition, the predicted total PDRN delivery amounts for each parameter condition are summarised in Table 5. Future experiments under matched conditions will be used to directly assess the accuracy of these projections and refine the calibration if needed.

Table 5.

Predicted cumulative PDRN delivery under varying parameter conditions after model calibration.

Varied parameter	Tested range	Applied calibration factor	Predicted cumulative PDRN delivery (μg)
Applied current density	0–0.5 mA cm−2	QC	0.14–283
Initial PDRN concentration	0.014–0.11 mol m−3	QC	36–850
Delivery time	5–60 min	QCT	18–230
PDRN diffusion coefficient	1 × 10−11–1 × 10−8 m2 s−1	QC	0.0026–3.1 × 104

4. Discussion

This study provides a numerical framework for analysing PDRN iontophoresis by separating the diffusion and electromigration terms using the Nernst–Planck equation. Because PDRN is a high-molecular-weight DNA-derived therapeutic agent, its passive penetration across the stratum corneum is intrinsically limited. Previous studies on transdermal delivery have emphasised that the stratum corneum is the main barrier to hydrophilic and macromolecular drugs, and that physical enhancement strategies such as iontophoresis are required to improve their transport (Prausnitz et al. 2004; Prausnitz and Langer 2008; Ramadon et al. 2022). In this context, the present model helps clarify how each operating or material parameter contributes to PDRN transport and provides design guidance for iontophoresis-assisted transdermal delivery. By separating diffusion and electromigration, the model clarifies not only whether iontophoresis enhances PDRN delivery, but also how the dominant transport mechanism and parameter sensitivity change with delivery depth.

The most important mechanistic finding is the depth-dependent transition in the dominant transport mechanism. At the epidermis–dermis interface, iontophoresis markedly enhanced PDRN delivery compared with passive diffusion. Under the baseline current density of 0.5 mA cm⁻², the cumulative permeated amount was approximately 2 × 10³ times higher at the epidermis–dermis interface and approximately 1.2 × 10⁴ times higher at the mid-dermis compared with passive diffusion. These results indicate that electromigration plays a major role in overcoming the initial skin barrier. However, comparison of the molar fluxes at the epidermis–dermis interface and mid-dermis showed that the contribution of electromigration decreased with increasing distance from the electrodes. At the mid-dermis, the diffusion term became the dominant contributor to the PDRN molar flux even under iontophoretic conditions. Therefore, iontophoresis primarily enhances the initial penetration of PDRN through the stratum corneum and epidermis, whereas delivery into deeper dermal regions requires designs that improve diffusion-driven transport and effective molecular mobility rather than relying solely on electromigration.

The parameter-sweep results provide practical guidance for optimising PDRN iontophoresis protocols. The applied current density was the most important external driving parameter for initial skin entry. However, its practical use is limited by safety-related factors such as Joule heating, potential difference, pH variation, skin irritation, and patient discomfort. Therefore 0.5 mA cm⁻² was considered a practical upper current density condition for cathodal PDRN iontophoresis under the present operating conditions. Increasing the current density beyond this level is unlikely to be an appropriate optimisation strategy unless additional safety validation is performed.

In contrast, increasing the initial PDRN concentration alone was not an efficient strategy for deep dermal delivery. Although donor concentration determines the amount of PDRN available in the reservoir, the mid-dermis delivery increased by only 1.4 times across the tested concentration range. This limited improvement indicates that deep transport is constrained more by skin transport resistance and effective PDRN mobility than by donor loading itself. Moreover, further increasing PDRN concentration may increase hydrogel viscosity and reduce effective diffusivity. Therefore, donor concentration should be selected within a practical formulation range rather than simply increased to maximise loading.

Delivery time should be optimised according to the target delivery depth. The time-dependent increase was much larger at the mid-dermis than at the epidermis–dermis interface, indicating that longer application mainly supports post-epidermal diffusion into deeper skin layers rather than the initial penetration step. Therefore, superficial delivery can be optimised primarily by selecting an appropriate current density within the safe range, whereas deeper dermal delivery requires sufficient application time. However, prolonged iontophoresis may increase discomfort, erythema, oedema, and pH changes (Guffey and Guffey 1999; Li et al. 2005). Thus, delivery duration should be balanced with safety and patient compliance, and 60 min is suggested as a practical upper duration under the present operating conditions.

Among all investigated parameters, the PDRN diffusion coefficient was the most influential factor for increasing the overall delivered amount. Its effect was particularly pronounced at the mid-dermis, where increasing the diffusion coefficient produced a 5.4 × 10¹⁰-fold increase in cumulative delivery. This strong dependence arises because the diffusion coefficient directly affects both the diffusion and electromigration terms in the Nernst–Planck equation. Therefore, future optimisation of PDRN iontophoresis should prioritise strategies that increase the effective mobility of PDRN, for example by balancing crosslinking density and mesh size, reducing formulation viscosity, and limiting excessive polymer–PDRN interactions while maintaining sufficient mechanical stability. Taken together, effective PDRN iontophoresis should be designed by jointly optimising current density, delivery time, donor concentration, and PDRN diffusivity, rather than by increasing a single parameter independently.

The calibrated model reproduced the previously reported concentration- and time-dependent permeation trends with high agreement, with R² values of 0.9821, and 0.9971, respectively. These results support the ability of the model to capture the major experimental trends. However, the current-density and diffusion-coefficient sweeps should be interpreted as first-order projections because corresponding experimental datasets were not available for direct calibration. Therefore, the model is most reliable for comparing relative parameter sensitivity and identifying design priorities, rather than for claiming exact delivered doses under all possible experimental conditions. Accordingly, its predictions should be interpreted primarily within the tested and calibrated parameter ranges and should be further validated when applied to different formulations, skin sites, electrode configurations, or conditions in which electroosmosis may play a substantial role.

Several limitations should be considered. First, electroosmosis was not explicitly included, although electroosmotic flow can influence iontophoretic transport depending on skin pH, electrolyte composition, and fixed-charge conditions (Guy et al. 2000; Pikal 2001; Sahraoui et al. 2024). Second, the effective charge of PDRN was estimated from dsDNA literature, although PDRN has a molecular-weight distribution and may not behave as a single uniform species. Third, the diffusion coefficient was assumed to be constant across the skin layers and concentration range, although it may vary with skin hydration, hydrogel viscosity, and formulation composition. Fourth, the skin was modelled as a homogeneous multilayer structure and did not explicitly resolve appendageal pathways such as hair follicles and sweat glands, which may contribute to macromolecular iontophoresis (Hinsberg et al. 1995; Hasan et al. 2022). Finally, the model evaluates transport behaviour rather than biological efficacy; therefore, simulated permeation should not be interpreted as a direct therapeutic endpoint.

These findings are also relevant to clinical translation. In practice, iontophoresis-assisted transdermal delivery must satisfy multiple constraints, including sufficient drug delivery, skin safety, thermal effects, pH stability, patient comfort, device size, and reproducibility across different skin sites and individuals. Directly optimising all of these variables through experiments would require a large number of repeated ex vivo, animal, and clinical tests. Computational modelling can help address this challenge by narrowing the experimental design space, identifying the most influential parameters, and excluding conditions that are unlikely to provide meaningful delivery improvement or may raise safety concerns. Thus, the proposed model can serve as a pre-validation tool for designing improved PDRN iontophoresis protocols and for supporting the stepwise translation of this technology toward clinically applicable non-invasive transdermal delivery systems.

5. Conclusion

In this study, we developed a numerical framework for analysing cathodal PDRN iontophoresis using the Nernst–Planck equation. By separating passive diffusion and electromigration, the model enabled quantitative evaluation of how applied current density, initial PDRN concentration, delivery time, and PDRN diffusion coefficient affect transdermal PDRN transport. The effective diffusion coefficient was determined from Franz diffusion cell experiments, and the simulated cumulative permeation was calibrated against previously reported experimental data.

The results demonstrate that iontophoresis substantially enhances PDRN delivery compared with passive diffusion and that the dominant transport mechanism depends on delivery depth. Electromigration plays a major role in the initial penetration across superficial skin layers, whereas deeper dermal delivery becomes increasingly dependent on passive diffusion and effective PDRN mobility. Among the tested parameters, current density was the dominant external parameter for initial delivery enhancement, while the diffusion coefficient was the most influential material parameter for deep transport. In contrast, increasing the initial PDRN concentration alone had a relatively limited effect on mid-dermis delivery.

Based on the parameter analysis, an applied current density of 0.5 mA cm⁻², an initial PDRN concentration of 0.11 mol m⁻³, and a delivery time of 60 min were identified as practical baseline conditions for cathodal PDRN iontophoresis under the present model assumptions. These values should be adjusted depending on the target tissue depth, formulation properties, electrode geometry, and safety requirements.

Overall, this work provides a physics-based simulation strategy for predicting and optimising PDRN iontophoresis. The framework can be used as a pre-validation tool to compare candidate delivery conditions, identify key design parameters, and reduce repeated experimental tuning during the development of non-invasive transdermal PDRN delivery systems.

Funding Statement

This work was supported by the IITP (Institute of Information & Communications Technology Planning & Evaluation)-ITRC (Information Technology Research Centre)((IITP-2026-RS-2023-00260091,10) grant funded by the Korea government (Ministry of Science and ICT) and National R&D Program through the National Research Foundation of Korea (NRF) funded by the MSIT (grant nos. RS-2024-00411764, RS-2025-24533288, RS-2025-16072886, 2021M3H2A1038042)).

Disclosure statement

No potential conflict of interest was reported by the author(s).

Data availability statement

The data that support the findings of this study are available from the corresponding author upon reasonable request.

Use of AI tools

We partially used ChatGPT (OpenAI) to edit the English language and improve the readability of this manuscript, and the authors take full responsibility for the content and interpretation of the final version.