Implantable bioelectronic devices with drug delivery capabilities have emerged as suitable candidates for biomedical applications focusing on localized drug delivery. These classes of miniaturized bioelectronics offer wireless operation and refillable designs that can be used for repeated animal behavioral studies without restricting their motion. The pumping mechanisms of these bioelectronic devices features soft materials, microfluidics, and electrochemical subsystems that can be scaled from behavioral studies in small animals to delivery of life-saving medication in humans. Here, we study the refillable aspect of these bioelectronic systems using an analytic model for the drug delivery time established from the ideal gas law when an initial gas volume is present in the device electrolyte reservoirs. The effect of the initial gas volume in delaying the drug delivery time is captured via a non-dimensional parameter identified as the normalized initial gas volume. An analytical solution is derived from the perturbation method, which agrees well with the numerical solution. These results have relevance in the reusability aspect of these bioelectronic systems since modifying the amount of initial gas in the device reservoirs for different experiments affects the total delivery time and can serve as a tunable parameter to ensure timely and successful delivery of the drug in the target region.
Recently developed miniaturized, refillable, and implantable bioelectronic devices with drug delivery capabilities have enabled localized drug delivery to organs/tissues without restricting the movement of small animals and avoiding excessive heat that can alter the drug properties and animal behavior [1–8]. Technologies to deliver drug locally can range from nanoparticles to implantable microneedles  to sophisticated bioelectronics with fluidic microsystems [6,8,10–12] that aim deliver the drug directly to the target area to enhance its effect. Delivering the drug locally offers certain advantages over a systemic delivery approach—where the drug must travel through the body to reach the target location—since interactions with other biological tissues and unwanted side effects can be potentially avoided [9,13,14]. Specifically, implantable bioelectronic devices have been used for localized delivery in the brain  and peripheral nerves  in the context of neuroscience research  and cancer therapeutics [16–18]. The schematic and components of a typical implantable bioelectronic device used for localized delivery studies is shown in Fig. 1. This design features two refillable reservoirs: (1) electrolyte and (2) drug. In the partially filled electrolyte reservoir, a set of interdigitated electrodes—in direct contact with the liquid electrolyte—initiate an electrochemical reaction (e.g., water hydrolysis) to generate gas when electric current flows through them. The gas generation (illustrated via bubble formation in Fig. 1(b)) pressurizes the electrolyte reservoir, which is only partially filled with the liquid electrolyte leaving an initial gas volume V0 to deform a flexible membrane; the deformation transitions from bending dominated (small deformation) to stretching dominated (finite deformation) [19,20], where the flexible membrane adopts the shape of a spherical cap (like the drug reservoir). As the flexible membrane deforms, the drug (located on top of the flexible membrane in the drug reservoir) is pushed through a network of microfluidic channels into the target location inside the body [21–23]. Let V denote the volume of gas due to electrochemical reaction such that the total volume of gas in the electrolyte reservoir is V + V0.
The mechanics of drug delivery in these microsystems is influenced by environmental, microfluidic, flexural, electrochemical, and geometrical parameters totaling approximately 12 individual parameters (ten are shown in Fig. 1, the remaining two are flexural parameters listed below). These parameters include the initial environmental pressure at the target organ/tissue P0 that varies depending on the internal pressure of the organ/tissue (e.g., eye, blood, brain, lungs) , the microchannel length L and cross-sectional dimensions a and b , the drug viscosity μ, the flexible membrane thickness h and radius R0 as well as flexural parameters such as the Young’s modulus E, Poisson’s ratio v, and the constitutive stress–strain relationship σ − ɛ [20,26,27]. Inside the partially filled electrolyte reservoir, the liquid electrolyte temperature T and electrical current i flowing in the electrodes influence the gas generation dynamics . Most importantly, for the analysis in this article, the presence of an initial gas volume V0 when the electrolyte reservoir is only partially filled with the liquid electrolyte (shown in blue in Fig. 1(a)). This initial gas volume V0 and volume of the liquid electrolyte add to the constant volume of the electrolyte reservoir.
This electrochemical actuation principle and bioelectronic device form factors used in small animal experiments can be scaled up for drug delivery studies in larger animals, and in the future, human applications. During experimental trials, controlling the parameters that influence the drug delivery process yield insights of the total delivery time and volume temporal profile, and both are keys to ensure timely and successful delivery. To that end, a scaling law that combines all the environmental, microfluidic, flexural, electrochemical, and geometrical terms into two nondimensional parameters (one for the environmental pressure and the other for the microfluidic resistance) has been previously established  to model and optimize the drug delivery process when the flexible membrane is modeled as a Marlow hyperelastic material  under finite deformation. The scaling law, and analytical solution obtained using the perturbation method, neglects the effect of the initial gas volume present at the beginning of the delivery, i.e., the electrolyte chamber is completely full of the liquid electrolyte . However, when considering the refillable capabilities of these devices [4,5], the presence of an initial gas volume V0 in the electrolyte chamber must be accounted for as it affects the drug delivery. For a partially full electrolyte chamber (Fig. 1(a)), the presence of an initial gas volume V0 will slow down the drug delivery process. Therefore, studying the effect of V0 in the total delivery time can yield an additional parameter that can be tuned to control the drug delivery process on clinically relevant settings.
Here, we propose a modified scaling law that accounts for the effects of the initial gas volume and obtain a new analytical solution using the perturbation method , validated by the numerical results, that can be used to control the drug delivery process using three nondimensional parameters. The drug volume temporal profile is used to determine the drug delivery time for bioelectronic devices of different sizes that correspond to representative applications in small animals, large animals, and humans.
The organization of this article is as follows. First, the Mooney Rivlin  hyperelastic strain energy potential is used for the flexible membrane mechanics analysis in Sec. 2. Then, scaling law of drug delivery that accounts for the effect of the initial gas volume in the electrolyte chamber is derived and solved analytically using the perturbation method in Sec. 3. The validation and the parametric study of the drug delivery time for different initial gas volumes is discussed in Sec. 4, where designs sizes for small animals, large animals, and human applications are presented. Finally, alternative material models are presented to highlight the sensitivity of the analytical model to the mechanics of the flexible membrane.
2 Mechanics of the Flexible Membrane—Mooney Rivlin Solid
The function of the flexible membrane is to respond to an asymmetric pressure stimulus from inside the device electrolyte reservoir, which results in hydraulic power to pump the drug into the target location through the microchannels. This pumping mechanism is achieved by a pressure difference P − Pdrug, where P is the gas pressure generated from the electrochemical reaction and Pdrug is the pressure from the drug on top of the flexible membrane. Before the drug delivery process begins, the pressures are at equilibrium, i.e., P = Pdrug = P0.
For H ≫ h or equivalently drug volume as shown in Eq. (4), the effect of bending deformation (particularly near the clamped edge of the flexible membrane) is negligible such that the deformed shape is a spherical cap, characterized by the spherical angle ϕ that tracks the deformation of the membrane from flat into the shape of a spherical cap as shown in Fig. 2(a).
Together, Eqs. (6) and (10) are the analytic, but parametric, equations for the function f(V) for a Mooney Rivlin incompressible solid.
Figure 2(a) shows the vertical displacement of a flexible membrane with dimensions R0 = 5 mm and h = 0.15 mm to illustrate the deformation (from flat to spherical cap) during the drug delivery process. The function f(V) obtained from finite element analysis (FEA) agrees well (Fig. 2(b)) with the analytical, but parametric, equations for volume and pressure in Eqs. (6) and (10) and validates the analytic model for the Mooney Rivlin hyperelastic potential.
3 Mechanics of Drug Delivery—Effect of Initial Gas Volume
It includes the initial gas volume V0, which was previously unaccounted.
The three nondimensional parameters in Eqs. (20)–(22) combine all the relevant parameters (∼12) of the bioelectronic device shown in Fig. 1 and can be analyzed separately to understand their effect in the drug delivery process.
It degenerates to the analytic model proposed in Ref.  for V0 = 0. Equation (25) includes two terms; the first term, which is inversely proportional to electrical current i (and temperature T) but independent of microchannel sizes (L and a) and drug viscosity (μ), represents the time required for the flexible membrane to overcome the external environmental pressure P0 and deform to the desired volume V given by the drug reservoir shaped like a spherical cap. The last term on the right-hand side, which is linearly proportional to drug viscosity (μ) (and microchannel sizes L and a−4) but independent of electrical current i (and temperature T), is the time required for the total drug volume to travel through the microchannels. The two terms in Eq. (25) occur simultaneously, i.e., as the flexible membrane deforms is pushing the drug through the microchannels, to give the total drug delivery time.
4 Results and Discussion
The volume temporal profile provides valuable insights to understand, and optimize, the drug delivery process. This temporal quantity is affected by the three non-dimensional parameters , , and , which must be carefully selected to ensure timely and safe (i.e., no damage to surrounding tissues) delivery. The scalability of the microsystem design allows us to analyze (non-dimensionally) the effect of (by fixing ) in the context of the drug volume temporal profiles and apply these findings to bioelectronic devices intended for small or large animal models, and ultimately humans. The following terminology is adopted in this section:
The term “numerical” is used for solutions of the ODE (Eq. (19)) described in Sec. 3 when the non-dimensional function is obtained from FEA using the Mooney Rivlin hyperelastic potential.
The term “analytical” is used for the analytical solutions in Eqs. (25) and (26) described in Sec. 3 when the non-dimensional function is obtained analytically for stretching-dominated deformation using the Mooney Rivlin hyperelastic potential as shown in Fig. 2.
For a representative bioelectronic device with a target delivery volume of V = 125 μl, flexible membrane of radius R0 = 5 mm, thickness h = 0.15 mm, Young’s modulus E = 15 MPa, and initial environmental pressure P0 = 101.3 kPa, the non-dimensional environmental pressure and microfluidic resistance are calculated as and , respectively (all the parameters used to generate the volume temporal profiles are listed in Table 1). For this example, the target delivery volume of the flexible membrane is large enough to satisfy the stretching-dominated deformation condition in Eq. (4). Figure 3 shows the effect of increasing the initial gas volume in delaying the delivery of the drug. When the solutions show excellent agreement on the entire delivery range. As increases to 0.623, 1.256, and 1.880 the agreement remains over the entire volume temporal profile except at the beginning of the drug delivery process for times (Fig. 3(b)). Figure 3(b) shows that at the beginning of the drug delivery process, all the analytical solutions are stacked with each other up to because initially (when gas formation starts) the external pressure P0 exerted on the flexible membrane dominates Eq. (25), and it is independent of the initial gas volume V0. Once sufficient gas has been generated from the electrochemical reaction, the dominant factor becomes the mechanics of the flexible membrane, i.e., f(V)(V + V0) dominates in Eq. (25), where the presence of V0 results in different total delivery times showed by the curves in Fig. 3(a). Figure 4 shows that, for a fixed volume (e.g., ), the total delivery time is approximately linearly related to , which is consistent with Eq. (24) for small . The difference between the numerical and analytical solutions is less than 1%.
|Parameter||Small devices||Medium devices||Large devices||Units|
|T||310 (core body temperature)||K|
|Parameter||Small devices||Medium devices||Large devices||Units|
|T||310 (core body temperature)||K|
Figures 3 and 4 show that the presence of a non-zero only delays the drug delivery (i.e., increase the total drug delivery time). Therefore, when and are fixed, the solution (electrolyte reservoir is filled) can be taken as the fastest delivery time achievable with the bioelectronic device. For example, if the drug must be delivered between two and four minutes, the limit can be designed to deliver the drug in two minutes and a non-zero can be used to deliver the drug in four minutes with the same bioelectronic device. Once the bioelectronic device is fabricated and ready to be used in animal experiments, only a few parameters can be changed during or before the drug delivery such as the electrical current i and the initial gas volume V0. The electrolyte reservoir in Fig. 1 is refillable which allows the bioelectronic device to be used repeatedly. Therefore, by partially filling the electrolyte reservoir, the presence of an initial gas volume V0 can be used to control the drug delivery time without having to fabricate a new device with a different set of materials.
4.1 Design Scalability for Small Animals, Large Animals, and Humans.
An advantage of these type of electrochemical bioelectronic devices is that the geometry can be scaled to tailored applications in small and large animals as well as humans by changing the three non-dimensional parameters in Eqs. (20)–(22) . Figure 5 shows the dimensional volume temporal profiles for three representative device sizes with a target drug volume of 3 μl, 125 μl, and 1000 μl for experiments in small and large animals where each configuration has a unique set of non-dimensional parameters based on the typical experimental parameters listed in Table 1. In all cases, the flexible membrane is modeled as a Mooney Rivlin incompressible solid with stretching-dominated deformation as explained in Sec. 2. The temporal profiles during delivery are very similar, only the magnitude of the target drug volume is different as shown in Fig. 5. To scale up the bioelectronic device from small animals to humans, the geometrical parameters such as R0 and the microfluidic cross-section a increase to accommodate larger volumes of drug inside the device reservoirs and deliver them in a timely fashion. Also, larger device geometries can accommodate different power strategies to deliver higher electrical currents to the interdigitated electrodes for a speedy drug delivery process. All these changes result in a unique combination of the three non-dimensional parameters. For the non-dimensional parameter specifically, its value can be changed relatively easy as compared to the other two non-dimensional parameters since it is only related to the amount of initial gas present before the delivery.
4.2 Radius of Flexible Membrane for Expedited Delivery.
Therefore, for a fixed drug volume V, increasing R0 can significantly decrease the delivery time before approaching the asymptote. However, the overall device size must be carefully considered when designing these bioelectronic systems as a large R0 can lead to bulky devices that pose bio-integration challenges in device securing or implantation strategies for epidermal or implantable applications, respectively. Figure 6 demonstrates that for a device with a drug volume V = 125 μl, h = 0.15 mm and R0 = 5 − 10 mm, the total delivery time for R0 = 5 mm is approximately t = 260 s. When R0 = 10 mm, the delivery time is substantially reduced (by ∼74%) to t = 68 s, beyond this point further increasing the radius results in almost negligible improvement as it approaches the asymptote value given in Eq. (26).
4.3 Alternative Material Models.
Figure 7 shows the influence of the material model in the drug delivery process where the corresponding normalized are shown in Fig. 7(a) for the linear elastic relation between the true stress and logarithmic strain and the Marlow hyperelastic potential for a flexible membrane with E = 15 MPa, R0 = 5 mm, and h = 0.15 mm. All curves are the same for small drug volume ). For , the curves are separated due to nonlinear deformation. The non-dimensional volume temporal profiles are shown in Fig. 7(b) where the numerical and analytical solutions agree well for a representative combination of the non-dimensional parameters , , . The results show a clear distinction in the volume temporal profiles and drug delivery time, which is related to the curves. For a non-dimensional volume of (corresponding to 125 μl of drug), the Marlow model yields ∼ which is equivalent to approximately t = 116 s, while the linear elastic relation between the true stress and logarithmic strain predicts almost ten times longer at a delivery time of t = 965 s.
In summary, this works presents the analytical model for drug delivery time using bioelectronic devices while accounting for the effect of the initial gas volume present in the electrochemical reservoir before the delivery process. The drug delivery process is controlled by three non-dimensional parameters: the normalized environmental pressure, the initial gas volume, and the microfluidic resistance which can be scaled from small animal studies to large animal and human trials following unique combinations of the non-dimensional parameters. The analytic model, detailed analysis of the non-dimensional parameters related to the initial gas volume, and volume temporal profile results are important to guide parametric optimization of these devices for refillable/repeated studies and obtain the estimated total delivery time for localized drug delivery studies.
R.A. and J.L.C. acknowledge support from the National Science Foundation Graduate Research Fellowship (NSF Grant No. DGE-1842165). R.A. also acknowledges funding from the Ford Foundation Predoctoral Fellowship. J.L.C., A.V.G., Y.Z., and J.A.R. acknowledge the support of the HEAL Initiative of the National Institutes of Health under award number UG3DA050303. The conclusions, opinions and other statements in this publication are the authors’ and not necessarily those of the sponsoring institution.
Conflict of Interest
There are no conflicts of interest.
Data Availability Statement
The datasets generated and supporting the findings of this article are obtainable from the corresponding author upon reasonable request. The authors attest that all data for this study are included in the paper. No data, models, or code were generated or used for this paper.