A Mathematical Model for Type 1 Diabetes Regulation Using a Smart Insulin Patch: In Silico Validation Based on Published Rat Data

Abstract

This work introduces a new mathematical model designed to describe the glucose–insulin dynamics associated with a glucose-responsive smart microneedle patch reported in the literature. The model captures the complete sequence of the patch behavior, from detecting glucose changes to controlled transdermal insulin delivery and gradually restoring blood glucose levels to the normal range. Our simulations show that the patch can effectively manage glucose not only during fasting conditions but also after single and multiple meals, restoring glucose levels to healthy levels within a short period. The model predictions are consistent with experimentally reported trends in previously published studies, which strengthens confidence in the biological realism of the proposed mechanism. Because some parameters in such systems are difficult to measure directly, we also performed a comprehensive sensitivity analysis to understand how variations in key parameters influence system stability. The results highlight the central role of the insulin release rate and the five glucose–regulation parameters examined in the sensitivity analysis, providing clear guidance on the most critical aspects of patch design for reliable performance. Overall, this study provides a simplified yet robust mathematical framework that makes the behavior of a glucose-responsive microneedle patch easy to understand and analyze. It lays the groundwork for future refinement of control strategies and optimization of patch design, improving control strategies, and developing more advanced systems that can maintain healthy glucose levels naturally and intuitively.

1. Introduction

Diabetes is an autoimmune disease characterized by impaired insulin production, leading to elevated blood glucose levels and affecting various systems in the body [1,2,3]. There are several types of diabetes, including Type 1 diabetes (T1D), Type 2 diabetes (T2D), and gestational diabetes [4,5]. Type 1 diabetes (T1D) occurs when the immune system mistakenly destroys the insulin-producing $\beta$-cells located in the islets of Langerhans in the pancreas [5,6,7,8,9]. In contrast, Type 2 diabetes (T2D) develops when the body’s cells become resistant to insulin, and the pancreas gradually produces less of it over time [6,7]. Many people can manage or improve this condition through lifestyle changes along with medication. Gestational diabetes, on the other hand, appears during pregnancy, when hormonal shifts reduce the body’s sensitivity to insulin, leading to higher blood glucose levels [6,7].

Many researchers focus on diabetes to discover more effective treatment options (for example, [10,11,12,13,14]). There are also many studies looking for new and creative ways to treat diabetes, including research on glucose-responsive microneedle-array patches reported in the literature that can sense glucose levels and release insulin automatically when the body needs it [15,16,17,18,19]. This microneedle patch shows a clear experimentally validated biological effect, because it senses when glucose levels rise and releases insulin to help bring them back to normal [20,21,22]. Ongoing research continues to advance the development of smart microneedle patches. A recent study published in 2023, Tan et al. [19], showed that the smart microneedle patch could lower blood glucose in rats with type 1 diabetes (T1D), both before and after meals, and compared its effect to traditional insulin injections (see Figure 1).

It is important to emphasize that the glucose-responsive microneedle patch considered in this study is not newly developed by the authors. Instead, the proposed model is formulated based on a biologically validated smart microneedle patch previously reported in the literature, particularly the experimental work of Tan et al. [19]. In these systems, insulin is typically encapsulated within glucose-sensitive polymeric or vesicular matrices embedded in microneedles. Elevated glucose levels trigger insulin release through mechanisms such as polymer swelling, enzymatic reactions, or changes in local pH, enabling controlled and on-demand delivery. Critical formulation factors include insulin loading capacity, glucose sensitivity threshold, polymer composition, microneedle geometry, and diffusion-controlled release kinetics, all of which influence the effectiveness and stability of the delivery system [15,19,20].

Building on recent advances in biological sciences and the emergence of smart therapeutic technologies, this work aims to develop a mathematical model that helps us understand how a smart insulin patch reported in the literature can improve blood glucose regulation. The model is designed to examine the effect of the patch under both fasting and post-meal conditions, clarify the mechanism by which this technology operates, and assess its ability to regulate glucose more efficiently and in a gentler, more responsive manner that aligns with the body’s needs. For this reason, we aim to develop a mathematical model that focuses on the dynamics of the microneedle patch’s effect on blood glucose levels and provide a theoretical demonstration that supports and reproduces the experimental outcomes reported by Tan et al. [19].

In contrast to existing glucose–insulin models that primarily describe diabetes progression or conventional insulin administration strategies, the present work introduces a mathematical framework specifically designed to capture the dynamics of glucose-responsive microneedle-based insulin delivery. Unlike traditional models, which often assume instantaneous or continuous insulin input, the proposed model explicitly incorporates glucose-triggered, delayed, and controlled insulin release through the skin. This allows the model to represent the sensing–response mechanism of smart microneedle patches and to analyze their impact on blood glucose regulation under both fasting and postprandial conditions, including multiple meal scenarios. Such features are not explicitly addressed in existing modeling approaches, thereby highlighting the novelty and relevance of the proposed framework.

Many mathematical models describe the process of diabetes [23,24,25,26,27], and others have been developed to describe the glucose–insulin regulatory dynamics in type 1 or type 2 diabetes [28,29,30,31]. In addition, several mathematical models have been proposed to simulate and evaluate different treatment strategies for diabetes, including insulin therapy, glucose-responsive systems, and drug-delivery approaches [32,33,34]. To keep pace with recent advances in diabetes treatments, new therapeutic approaches have continued to evolve, including smart microneedle patches that release insulin when glucose levels rise. These patches have shown better glucose control than traditional methods [15,19], making them a promising option for the future. Because of this progress, researchers have also started using mathematical models to understand how these patches work, especially how insulin moves from the patch into the skin and then into the bloodstream. Some mathematical models describe the movement and distribution of microneedle patches across different compartments [35,36,37]. However, the present work focuses specifically on modeling the impact of microneedle-based insulin delivery on systemic blood glucose dynamics, rather than on patch fabrication or transport mechanisms. In addition, the model reflects the microneedle patch’s ability to reduce glucose levels, consistent with the experimental trends reported by Tan et al. [19]. Moreover, we investigate the system response under multiple-meal scenarios and present simulations demonstrating a reduction in glucose levels to approximately 73 mg/dL by the end of 100 h (≈4 days).

We develop a novel mathematical framework that focuses on how a microneedle patch adopted from existing experimental studies works through the skin: how it senses rising glucose, releases insulin into the skin layer, and gradually lowers blood glucose. In addition, the model captures the natural rise in glucose after meals and how the patch responds to it (see Figure 2). The results indicate that the patch keeps glucose under control, bringing it back to normal levels after both short- and long-term meal challenges. We also compare our results with those of Tan et al. [19] at fasting and post-meal times, and observe good agreement with published experimental measurements. Since some of the parameters are unknown and based on assumptions, a sensitivity analysis is performed to examine how they influence the model behavior.

Figure 1. Schematic illustration adapted from [19] comparing conventional insulin injection and a glucose-responsive microneedle patch in a rat model: (a) blood glucose response before a meal and (b) blood glucose response after a meal. In both cases, the microneedle patch was applied approximately 6 h before or after meal intake.

Figure 1. Schematic illustration adapted from [19] comparing conventional insulin injection and a glucose-responsive microneedle patch in a rat model: (a) blood glucose response before a meal and (b) blood glucose response after a meal. In both cases, the microneedle patch was applied approximately 6 h before or after meal intake.

Figure 2. Insulin delivery mechanism of the smart microneedle patch. Insulin is first released into the skin compartment and then absorbed into the second compartment (the bloodstream), where it exerts its glucose-lowering effect. The figure also illustrates the impact of a meal on glucose dynamics in the rat.

Figure 2. Insulin delivery mechanism of the smart microneedle patch. Insulin is first released into the skin compartment and then absorbed into the second compartment (the bloodstream), where it exerts its glucose-lowering effect. The figure also illustrates the impact of a meal on glucose dynamics in the rat.

The paper is organized as follows. Section 2 outlines the mathematical framework. Section 3 analyzes the model behavior (Section 3.2), including steady-state characterization (Section 3.1) and a global sensitivity analysis (Section 4), which assesses the impact of the parameter on Glucose dynamics over 12 and 24 h. Section 5 concludes with a summary and discussion.

2. Glucose Dynamics Under Microneedle Patch Influence

In this study, we investigated a simplified mathematical model to describe the dynamics of glucose under the influence of a microneedle patch. The insulin density in the patch is denoted by S, the insulin within the skin compartment by $I_s$, the insulin in the bloodstream by $I_b$, and the glucose concentration by G. The following system of differential equations governs the temporal evolution of these variables:

$$\begin{array}{cc}\hfill {\displaystyle \frac{dS}{dt}}=& -SE,\hfill \end{array}$$

(1a)

$$\begin{array}{cc}\hfill {\displaystyle \frac{d{I}_s}{dt}}=& SE-e_s{I}_s-a_b{I}_s,\hfill \end{array}$$

(1b)

$$\begin{array}{cc}\hfill {\displaystyle \frac{d{I}_b}{dt}}=& a_b{I}_s-e_b{I}_b,\hfill \end{array}$$

(1c)

$$\begin{array}{cc}\hfill {\displaystyle \frac{dG}{dt}}=& -L+a_{G}G(1-{\displaystyle \frac{G}{m}})-e_{G}G+M.\hfill \end{array}$$

(1d)

In the following section, we describe in detail the terms that appear in model (1).

• In Equation (1a), the patch releases insulin at the rate E is defined as a piecewise function that activates only when glucose exceeds a threshold. When glucose is below 120 mg/dL [38], the patch does not release insulin $E=0$. However, once glucose is reached or higher than 120 mg/dL, the patch releases insulin, which follows a saturating Michaelis–Menten-type response. The parameter $k_s$ represents the maximum possible release rate achieved when glucose is sufficiently high [21]. In contrast, the parameter $k_m$ determines the glucose level at which the release reaches half of this maximum, analogous to a half-saturation constant in enzyme-mediated glucose-responsive systems [39]. This glucose-dependent release is represented by the following function:

  $$E=\left\{\begin{array}{cc}0,\hfill & G<120,\hfill \\ {\displaystyle \frac{k_{s}G}{k_m+G}},\hfill & G\ge 120,\hfill \end{array}\right.$$
• In Equation (1b), after the patch releases insulin at the rate $k_s$ $SE$ [40,41], the insulin enters the skin compartment, where it is cleared at the rate $e_s$ [42], and remaining insulin then moves from the skin into the bloodstream at the transfer rate $a_b$ [43].
• In Equation (1c), after insulin leaves the skin, it enters the bloodstream at the rate $a_b$ [44,45], and part of the insulin may remain in the skin and not enter the bloodstream immediately due to local degradation or tissue binding [46,47]. Thus, the insulin is naturally cleared at a rate $e_b$.
• In Equation (1d), glucose also follows logistic growth due to production from other cells in the body at a rate $a_g$ [48] approaching a carrying capacity m [49,50]. Moreover, the body still clears some glucose naturally through normal metabolism, physical activity, and hydration. These processes continue to lower glucose levels even without insulin or medication [51,52]. Thus, glucose is naturally cleared at a rate $e_g$, and it is further reduced by insulin released from the microneedle patch at the rate $n_g$ [15,49,53]. The combined glucose-dependent insulin-release term L is modeled as a piecewise stimulus that activates only when glucose exceeds a physiologically meaningful threshold. When blood glucose is below 120 mg/dL, the smart microneedle patch remains inactive and releases no insulin:

  $$L=\left\{\begin{array}{cc}0,\hfill & G<120,\hfill \\ n_{g}G{I}_b,\hfill & G\ge 120\hfill \end{array}\right.$$

We focus on two distinct states: the fasting state before the meal (Fast, $M=0$) and the post-meal state (M). The time bounds used in the definition of M are modeling assumptions introduced to represent meal events over finite time windows. These intervals do not correspond to the exact duration of eating, but rather approximate physiologically reasonable postprandial periods during which glucose enters the bloodstream. This assumption allows a simplified and tractable representation of meal-induced glucose input while preserving the essential dynamics of the system. In the case of a single meal, only one value of M is considered at 20 min:

$$M=\left\{\begin{array}{cc}0,\hfill & t<0.16,\hfill \\ 1,\hfill & 0.16\le t<0.17,\hfill \\ 0,\hfill & t\ge 0.17\hfill \end{array}\right.$$

Here, the time variable t represents a normalized (dimensionless) time obtained by scaling the physical time with respect to a characteristic time scale of the system. Accordingly, the interval $0.16\le t<0.17$ corresponds to a short postprandial window of approximately 20 min in real time. During this interval, the function M is set to $M=1$ to represent the presence of a meal and the associated glucose input. Outside this interval, $M=0$, corresponding to the fasting state. Thus, the values $t<0.16$ and $t<0.17$ are used to distinguish the pre-meal, meal, and post-meal phases within the early-time dynamics.

In the multiple meal scenario, the multiple post-meal values of (M) are introduced at 5 h, 15 h, and 22 h:

$$M=\left\{\begin{array}{cc}0,\hfill & t<5,\hfill \\ 1,\hfill & 5\le t<6,\hfill \\ 0,\hfill & 6\le t<15,\hfill \\ 1,\hfill & 15\le t<16,\hfill \\ 0,\hfill & 16\le t<22,\hfill \\ 1,\hfill & 22\le t<23,\hfill \\ 0,\hfill & t\ge 23.\hfill \end{array}\right.$$

In the multi-meal scenario, the function M represents discrete postprandial glucose inputs rather than continuous insulin release. Accordingly, $M=1$ only during the specified meal intervals (e.g., $5\le t<6$), indicating the presence of glucose intake. For $t<5$ and $6\le t<15$, no new glucose input is introduced ($M=0$). During these intervals, insulin is not actively triggered by a meal; however, the system dynamics still reflect the residual effect of previously released insulin through its interaction with glucose and other state variables.

3. Results

Now, we investigate a model (1) by analyzing its equilibrium points, which characterize the system’s long-term behavior and provide insight into its underlying stability structure. In the steady-state analysis, the absence of insulin simply indicates that the patch is inactive because glucose levels are already low, so no release is required. When glucose rises, however, the patch activates and begins delivering insulin, helping the system return to a healthier equilibrium. This contrast clearly shows how the patch distinguishes between low and high glucose states.

3.1. Steady States and Stability

We now focus only on the two steady states that can realistically occur in the body, since they are the only ones that make biological sense. We analyze the long-term behavior of model (1) by examining its steady-state dynamics.

• In the absence of insulin and in the continued presence of glucose, the system admits an equilibrium of the form $(S,I_s,I_b,G)=(0,0,0,G^{*})$, representing a physiological state in which glucose persists at a baseline level. At the same time, the microneedle patch does not produce insulin. Under the baseline parameter values listed in

  Table A1. Table summarizing the parameters that appear in the model (1).

  Param.	Description	Value	Units	Ref.
  $k_s$	The rate of insulin release from the patch	${10}^{-3}$–${10}^{-1}$ (${10}^{-3}$)	h−1	Estimation
  $k_m$	Glucose concentration producing half-maximal insulin release	126–162 (120)	mg/dL	Estimation
  $a_b$	Insulin rate moves from the skin into the bloodstream.	0.05	h−1	[54]
  m	Carrying capacity of the glucose	451–750 (600)	mg/dL	[55]
  $e_s$	The nature decay rate of the insulin in skin	${10}^{-8}$–${10}^{-4}$ (${10}^{-7}$)	h−1	Estimation
  $e_b$	The nature decay rate of the insulin in bloodstream	${10}^{-8}$–${10}^{-4}$ (${10}^{-6}$)	h−1	[54]
  $e_G$	The nature decay rate of the glucose	${10}^{-4}$–0.001 (${10}^{-3}$)	h−1	Estimation
  $a_G$	The growth rate of the glucose	${10}^{-4}$–0.05 (0.0011)	h−1	Estimation
  $n_g$	The elimination rate of the glucose via insulin	${10}^{-4}$–0.05 (${10}^{-3}$)	(μU/mL)−1 h−1	Estimation

  , this equilibrium is classified as locally asymptotically stable. The general stability properties of this state are established in Appendix B. The corresponding equilibrium glucose concentration is given by

  $$\begin{array}{c}\hfill G^{*}={\displaystyle \frac{m}{a_G}}(a_G-e_G)\mathrm{which}\mathrm{exists}\mathrm{only}\mathrm{when}(a_G>e_G).\end{array}$$
• In the presence of both insulin and glucose, the system admits a non-trivial equilibrium of the form $(S,I_s,I_b,G)=(S^{*},I_s^{*},I_b^{*},G^{*})$. Under the baseline parameter values given in Table A1, this steady state is non-zero and is classified as stable. The equilibrium components satisfy the following:

$$\begin{array}{cc}\hfill S^{*}& ={\displaystyle \frac{k_m+G^{*}}{k_s{G}^{*}}}(e_s-a_b)I_s^{*}\mathrm{which}\mathrm{exists}\mathrm{only}\mathrm{when}(e_s>a_b),\hfill \\ \hfill I_s^{*}& ={\displaystyle \frac{e_b}{a_b}}{I}_b^{*},\hfill \\ \hfill I_b^{*}& ={\displaystyle \frac{a_b}{e_b}}{I}_s^{*},\hfill \\ \hfill G^{*}& =m(1-{\displaystyle \frac{n_g{I}_b^{*}+e_G}{a_G}})\mathrm{which}\mathrm{exists}\mathrm{only}\mathrm{when}(1>{\displaystyle \frac{n_g{I}_b^{*}+e_G}{a_G}}),\hfill \end{array}$$

3.2. Numerical Results

We show how glucose behaves before the meal and after the post-meal phase to highlight how the system changes in response to eating. We assume the initial condition $G\left(0\right)=378$ mg/dL [19]. This value is chosen to represent a moderate-to-high hyperglycemic state. Clinically, postprandial glucose targets are typically below ∼180 mg/dL, whereas glucose levels exceeding 300 mg/dL are commonly regarded as markedly elevated and indicative of poor glycemic control. Moreover, severe hyperglycemia has been reported in the clinical literature at levels above 400 mg/dL. Therefore, the value 378 mg/dL (approximately 21 mmol/L) provides a physiologically relevant initial condition that is substantially above recommended targets while remaining below extreme emergency thresholds, making it suitable for evaluating the performance of the patch under pronounced hyperglycemic stress. The value $S\left(0\right)=3$ μU/mL represents the initial insulin density in the patch [19], corresponding to the amount of insulin stored in the patch before activation. In addition, $I_s\left(0\right)=0$ represents the amount of insulin in the skin compartment at the start of the simulation, indicating that no insulin has yet been delivered from the patch into the skin at time $t=0$. Similarly, $I_b\left(0\right)=0$ denotes the insulin concentration in the bloodstream at the beginning of the simulation, since insulin release from the patch is triggered only when glucose levels become elevated. Thus, the initial conditions $G\left(0\right)=378$ mg/dL, $S\left(0\right)=3$ μU/mL, $I_s\left(0\right)=0$, and $I_b\left(0\right)=0$ represent a moderate hyperglycemic state prior to patch activation.

3.2.1. Pre-Meal (Fasting, $M=0$) Phase

In Figure 3, we present the simulation of glucose levels before the meal: panel (a) shows the short-term behavior over 12 h, panel (b) illustrates the long-term behavior over 24 h, and panel (c) compares the numerical results of the glucose levels with experimental data from rats over 6 h. It can be clearly seen that glucose levels decrease when the patch is applied. This reduction occurs because the patch releases insulin when glucose levels are high, and the released insulin subsequently lowers the glucose level. The model captures the overall trend of the experimental data reported by Tan et al. [19], showing close agreement at 1 h, 2 h, and 6 h. Although the predicted value at 4 h is slightly lower than the experimental range, the deviation remains modest and biologically reasonable.

3.2.2. Post-Meal Phase

In this section, we show the post-meal simulation for the first 20 min to illustrate the immediate glucose response after food intake. Then, we illustrated the results of three-meal simulations conducted at hours 5, 15, and 22 to analyze the system’s longer-term behavior.

Single Meal Introduced Early in the Day

In Figure 4, we illustrate the simulation of glucose levels after a meal: panel (a) shows the short-term behavior over 12 h, panel (b) illustrates the long-term behavior over 24 h, and panel (c) compares the numerical results of the glucose levels with experimental data from rats over 6 h. It can be clearly seen that glucose levels decrease when the patch is applied. This reduction occurs because the patch releases insulin when glucose levels are high, and the released insulin subsequently lowers the glucose level. The model and the experimental data show the same behavior. The model fits the data of Tan et al. [19] exactly at 1 h and 2 h, and remains close at 6 h. Although the 4 h value is slightly lower than the experimental range, the deviation is modest and remains within an acceptable biological range.

Although Figure 3 and Figure 4 correspond to pre-meal and post-single-meal conditions, respectively, their trajectories appear visually similar. This observation may seem counterintuitive, as meal intake is expected to induce a transient rise in glucose levels. To clarify this behavior and avoid potential misinterpretation, we quantitatively compare the model predictions at selected time points using

Table 1. Predicted glucose levels before a meal and after a single meal at selected time points.

Time (h)	Before a Meal (mg/dL)	After a Single Meal (mg/dL)
1	350	332
2	315	308
4	260	258
6	210	208

. Table 1 reports the predicted glucose levels for both scenarios at 1, 2, 4, and 6 h. The results show that, while the meal introduces a short-lived postprandial perturbation, the glucose-responsive microneedle patch rapidly compensates for this disturbance. Consequently, glucose dynamics quickly converge to the same regulated trajectory, explaining the strong visual overlap observed in Figure 3 and Figure 4.

Multiple-Meals Scenario on Day One

In Figure 5, we show how glucose changes after three post-meal events. In panel (a), we see the short-term behavior over 50 h (≈2 days), and in panel (b), the long-term behavior over 100 h (≈4 days). When the patch is applied, glucose levels begin to fall because it releases insulin when glucose levels rise. After 5 h, the first meal causes glucose to rise, but the patch quickly responds by releasing insulin, bringing the level back to normal within about 2–3 h. The same pattern occurs at 15 h: glucose increases after the second meal, and the patch again releases insulin, returning glucose to normal within a few hours. A third rise appears at hour 22, and once more the patch lowers glucose back to normal in about 2–3 h. This third glucose peak is associated with the third meal explicitly introduced in the model through the function $M\left(t\right)$, where glucose input is activated during the interval $22\le t<23$. After this point, glucose remains stable at around 73 mg/dL for the rest of the (≈4) days.

When comparing the system under fasting with a single meal versus multiple meals, we observe that glucose levels rise after each meal, and the patch responds by releasing insulin to bring the glucose back to its normal range within a few hours. Despite the repeated meals, the system maintains stable glucose levels, demonstrating the effectiveness of the patch in controlling these frequent fluctuations.

4. Sensitivity Analysis

To better understand the relative influence of the model parameters on the system dynamics, a sensitivity analysis was carried out. By systematically varying each parameter around its baseline value, we quantify how sensitive the state variables are to small perturbations in the parameters. This approach allows us to identify which biological processes exert the strongest control over the glucose and insulin dynamics at equilibrium and during the transient phase.

To measure the influence of each model parameter on the output, an anxiety-based global sensitivity analysis was performed. For any parameter p, a sensitivity interval of $\pm \mathrm{10\%}$ around its nominal value was defined. Within this interval, $N=50$ uniformly spaced samples were generated. Let $p_{min}$ and $p_{max}$ denote the lower ($-\mathrm{10\%}$) and upper ($+\mathrm{10\%}$) bounds of the interval. The step size is computed as $\Delta p={\displaystyle \frac{p_{max}-p_{min}}{N-1}}$, and the sampled values are defined by

$$p_i=p_{min}+(i-1)\Delta p,i=1,2,\dots ,N$$

For each sampled value $p_i$, the model was simulated, and the resulting output (e.g., $G\left(12\right)$ and $G\left(24\right)$) was recorded. The variation in the output across the 50 simulations was then used to compute sensitivity indices and rank the parameters according to their influence on model behavior, following standard global sensitivity analysis methodologies. We selected the unknown parameters for the sensitivity analysis to check how much they influence the model and to ensure that the values we assumed for them are reasonable and correct.

Figure 6 presents the global sensitivity analysis of model 1 before the meal, where each parameter was perturbed by $\pm \mathrm{10\%}$ over a two-hour (12 h and 24 h). The parameters tested include the insulin release rate from the patch ($k_s$), the natural decay rate of insulin in the skin ($e_s$), the natural decay rate of glucose ($e_G$), the glucose growth rate ($a_G$), and the glucose elimination rate via insulin ($n_g$).

The results illustrate that, for most parameters, the resulting variation in the evaluated outputs ($G\left(12\right)$ and $G\left(24\right)$) is very small. This indicates that the model is relatively robust with respect to moderate uncertainties in these parameters. In contrast, the parameters $e_G$ and $a_G$ display a noticeably stronger influence. Perturbing $e_G$ and $a_G$ within the same $\pm \mathrm{10\%}$ range produces a substantially larger change in the model outputs compared with the other parameters, highlighting their dominant roles in shaping the system dynamics. This behavior, in which only a small subset of parameters contributes significantly to output variability, is consistent with findings from other biological dynamical models employing global sensitivity analysis.

Figure 7 shows the global sensitivity analysis of model (1) after the meal, where each parameter was perturbed by $\pm \mathrm{10\%}$ over a two-hour (12 h and 24 h). The parameters tested include the insulin release rate from the patch ($k_s$), the natural decay rate of insulin in the skin ($e_s$), the natural decay rate of glucose ($e_G$), the glucose growth rate ($a_G$), and the glucose elimination rate via insulin ($n_g$).

Overall, the analysis shows that most parameters induce only minimal variation in the evaluated outputs, $G\left(12\right)$ and $G\left(24\right)$. This indicates that the model is generally robust to moderate uncertainties in these parameter values. However, two parameters $e_G$ and $a_G$ display a noticeably stronger impact. Perturbing these parameters within the same $\pm \mathrm{10\%}$ interval results in substantially larger changes in the model output compared with the other parameters, underscoring their key roles in shaping the system’s behavior. sensitivity analysis.

Figure 6. The figure shows the global sensitivity analysis before the meal, based on $\pm \mathrm{10\%}$ perturbations of the parameters $k_s$, $e_s$, $n_g$, $e_G$, and $a_G$.

Figure 6. The figure shows the global sensitivity analysis before the meal, based on $\pm \mathrm{10\%}$ perturbations of the parameters $k_s$, $e_s$, $n_g$, $e_G$, and $a_G$.

Figure 7. The figure shows the global sensitivity analysis after the meal, based on $\pm \mathrm{10\%}$ perturbations of the parameters $k_s$, $e_s$, $n_g$, $e_G$, and $a_G$.

Figure 7. The figure shows the global sensitivity analysis after the meal, based on $\pm \mathrm{10\%}$ perturbations of the parameters $k_s$, $e_s$, $n_g$, $e_G$, and $a_G$.

It is also worth noting that the global sensitivity profiles were almost unchanged before and after the meal. The parameter perturbations resulted in almost the same output variations at both time points (12 h and 24 h), indicating that the model’s sensitivity structure is stable across feeding conditions. This consistency further demonstrates the robustness of the model and shows that the dominant influence of $e_G$ and $a_G$ persists whether the system is evaluated in the pre-meal or post-meal state.

The multi-meal scenario involves repeated meal inputs without altering the underlying model parameters; therefore, the parameter sensitivity structure remains unchanged, and a separate sensitivity analysis is not expected to provide additional insight.

5. Conclusions

The results of our model show that the smart microneedle patch does more than simply release insulin; it behaves like a responsive system that understands when glucose begins to rise and adjusts its release accordingly. In our simulations, even before/after one or multiple meals, glucose levels returned smoothly to the normal range without sharp spikes or drops, demonstrating the patch’s ability to maintain stable control throughout realistic daily conditions.

Another important point is the strong agreement between our simulated glucose curves and the experimental data reported [19] in previous studies during both the fasting and post-meal periods. This close match suggests that the model successfully captures the essential physiological interactions between glucose, insulin, and the skin-based delivery mechanism. It also provides confidence that the assumptions made in the model are biologically reasonable.

Because some parameters are not yet well characterized experimentally, we performed a comprehensive global sensitivity analysis to test how variations in key parameters affect model behavior. The results highlight that parameters such as the insulin release rate, the natural decay of insulin in the skin, and the glucose elimination rate have a substantial influence on the system’s stability. This insight is important for real-world development: it shows which parameters must be measured more exactly and which aspects of patch design are most critical for achieving reliable glucose control.

Future work will focus on refining the mathematical structure of the model by incorporating more realistic delay terms, improving the representation of glucose appearance after meals, and exploring alternative functional forms for insulin diffusion through the skin.

Overall, this study provides a practical mathematical framework for understanding how a glucose-responsive microneedle patch behaves in real-life scenarios. It demonstrates that such a patch can, in principle, regulate glucose effectively throughout the day, while also identifying the key factors that will need careful attention in future experimental and clinical development.