Analysis of Glucose Responsive Glucagon Therapeutics using Computational Models of the Glucoregulatory System

Abstract

Glucose‐responsive glucagon (GRG) therapeutics are a promising technology for reducing the risk of severe hypoglycemia as a complication of diabetes mellitus. Herein, the performance of candidate GRGs in the literature by modeling the kinetics of activation and connecting them as input into physiological glucoregulatory models is evaluated and projected the two distinct GRG designs, experimental results reported in Wu et al. (GRG‐I) and Webber et al. (GRG‐II) is considered. Both are evaluated using a multi‐compartmental glucoregulatory model (IMPACT) and used to compare in‐vivo experimental data of therapeutic performance in rats and mice. For GRG‐I and GRG‐II, the total integrated glucose material balances are overestimated by 41.5% ± 14% and underestimated by 24.8% ± 16% compared to in‐vivo time‐course data, respectively. These large differences to the relatively simple computational descriptions of glucagon dynamics in the model, which underscores the urgent need for improved glucagon models is attributed. Additionally, therapeutic insulin and glucagon infusion pumps are modeled for type 1 diabetes mellitus (T1DM) human subjects to extend the results to additional datasets. These observations suggest that both the representative physiological and non‐physiological models considered in this work require additional refinement to successfully describe clinical data that involve simultaneous, coupled insulin, glucose, and glucagon dynamics.

In this study, the application of the physiological model is expanded to predict blood glucose level in rodents receiving treatment with glucose‐responsive glucagon (GRG) technologies. A new element is introduced in this model for simulating insulin and glucagon infusion pumps. Such physiology‐based model is then compared with a minimal model and highlight the importance of developing a new glucagon submodel.

1. Introduction

Insulin and glucagon, as the two most important glucoregulatory hormones in the body, were discovered a century ago^{[ 1} , ^{2 ]} and have formed the basis for important therapeutics.^{[ 3} , ⁴ , ⁵ , ^{6 ]} Insulin and glucagon hormones secreted respectively by pancreatic β and α cells in the liver lead to decreased and increased blood glucose concentration, respectively.^{[ 1} , ⁷ , ⁸ , ⁹ , ^{10 ]} Apart from their separate role in regulating blood glucose concentration, they also have interconnected functions in regulating or preventing hypoglycemia or hyperglycemia events,^{[ 11} , ¹² , ¹³ , ^{14 ]} which complicate the investigation of hormonal effects on glucose metabolism. These complications can arise from the paracrine signaling pathways between alpha and beta cells, involving glucagon and Glucagon‐like peptide 1 (GLP‐1) via receptors on the beta cells. These paracrine signaling pathways can be different from one species to another, which makes translational investigation of insulin and glucagon difficult as alpha and beta cells in humans are morphologically close but distinct from rodents for example.^{[ 15 ]} Understanding such interconnected functions remains itself an experimental challenge. For example, mice with ablated alpha cells and glucagon receptor antagonist treatment still experience hyperglycemia.^{[ 16 ]} Since glucoregulation is a dynamic process and is modulated by the secretion of insulin and glucagon, and changes in blood glucose content in return, mathematical models have been developed to probe this bihormonal glucoregulation system.^{[ 17} , ¹⁸ , ^{19 ]}

In addition to experimental studies, mathematical models provided valuable directions for the logic design of therapeutic insulin and glucagon.^{[ 20} , ²¹ , ²² , ²³ , ^{24 ]} In this work, we develop a novel use of such models in evaluating the efficacy of Glucose Responsive Glucagon (GRG) therapeutics. These drugs activate or release glucagon in response to physiological glucose concentration. They are particularly promising for reducing the risk of severe hyperglycemia and hypoglycemia, with glucagon as an antagonistic hormone to insulin.

Glucose responsive therapeutics in general have been a goal of novel drug design as patients with either type 1 diabetes mellitus (T1DM) or type 2 diabetes mellitus (T2DM) experience life‐threatening hyper (>180 mg dL⁻¹) or hypoglycemia (<54 mg dL⁻¹) at least once in their lifetime.^{[ 25 ]} Glucose responsive therapeutics including molecular analogs or delivery systems sensitive to glucose concentration in the environment have provided reliable solutions for dealing with hypoglycemia events. Glucose responsive insulin (GRI) and glucagon (GRG) are two major groups of therapeutics which are being developed by adapting a close‐loop drug delivery system to reliably maintain the blood glucose concentration of diabetic patients.^{[ 26} , ²⁷ , ²⁸ , ²⁹ , ^{30 ]} The integration of closed‐loop insulin delivery with a continuous glucose monitoring system (CGMS) enhances time‐in‐range for patients and reduces hypoglycemic risks.^{[ 31} , ³² , ³³ , ^{34 ]} Advanced artificial pancreas systems now incorporate dual injection of both insulin and glucagon, using stable forms of glucagon like dasiglucagon, to maintain blood glucose levels within a normal range.

Better management of glycemia in diabetic subjects seems to be promising through glucose‐responsive release of therapeutic insulin and glucagon hormones.^{[ 35} , ^{36 ]} Insulin and glucagon are potent hormones and their controlled absorption into body is essential for the prevention of hyper‐ and hypo‐glycemia in T1D patients, respectively.^{[ 37} , ³⁸ , ³⁹ , ^{40 ]} Development of glucose‐responsive and closed‐loop technologies for the delivery of insulin and glucagon has improved the quality of life of T1D subjects by consistent regulation of their glycemia.^{[ 41} , ⁴² , ^{43 ]} By contrast to GRI having extensively evolved since its first discovery in 1979,^{[ 44 ]} GRGs as therapeutics and their delivery are still under development.^{[ 41 ]} Microneedles patches and injectable hydrogels are two examples of viable therapeutic delivery systems with glucose responsivity. Microneedle patches have shown promise for delivery of either GRG alone or both GRG and GRI concurrently.^{[ 35} , ³⁹ , ^{45 ]} Glucose responsive hydrogels were also demonstrated similarly as GRI or GRG constructs.^{[ 40} , ⁴⁶ , ^{47 ]} Despite such developments, reliable regulation of glycemia with using either GRI, GRG, or both has remained a challenge as the role of glucagon, its effect on glucose homeostasis, and its interaction with insulin is still an ongoing question.

The translation of these novel therapeutics to human clinical stages and commercial availability entails significant costs and effort. With that regard, the computational approach can offer valuable insights, potentially accelerating this translation process.^{[ 38 ]} Various in silico diabetes simulators, developed alongside CGM techniques, are able to predict patient glycemic levels more accurately.^{[ 17} , ²⁰ , ^{48 ]}These models support the control scheme of the artificial pancreas and provide better dosing strategies. We have developed an in silico computational toolbox designed to analyze glucoregulatory systems in humans, rodents, and minipigs.^{[ 21 ]} This model predicts and evaluates the translatability of new insulin therapies from animal experiments to human trials. We have demonstrated its ability to identify efficacious therapeutic parameters for glucose‐responsive insulin therapies^{[ 49 ]} and to investigate the reasons behind the gap between successful animal studies and lackluster clinical trials.^{[ 21 ]} Specifically, our comprehensive compartment model accurately captures the insulin‐glucose dynamics across rodents, minipigs, and humans.^{[ 20} , ²¹ , ²² , ³⁸ , ⁴⁹ , ^{50 ]}

In this work, we demonstrate how a glucoregulatory model can be used to analyze GRG therapeutics of different mechanistic designs,^{[ 39} , ^{40 ]} comparing as‐calculated model predictions with no adjustment to existing parameters to in‐vivo data for such therapeutics. We show approximate success in capturing the response in mice and rats, providing a foundation for computational design and optimization of next generation GRGs. However, we note several limitations that arise because the treatment of glucagon in our model^{[ 20} , ²¹ , ⁵⁰ , ^{51 ]} and others^{[ 19} , ^{48 ]} lacks key details revealed by this exercise. We further elucidate this by modeling bihormonal management of glycemia in rodents using insulin and these GRGs. Additionally, we discuss the development of insulin and glucagon infusion pumps for continuous delivery of insulin and glucagon to T1D human model and apply a non‐physiological model to fit the absorption of these therapeutics. Comparison between physiology and non‐physiology‐based models and their limitations in the prediction of blood glucose dynamics are examined. We compare our published PAMERAH and IMPACT computational platforms,^{[ 20} , ^{21 ]} and subcutaneous and intravenous injections of glucagon into T1D human subjects are finally discussed.

2. Research Methods

2.1. A physiology‐Based Model for Glucose‐Responsive Glucagon

This current work makes use of a glucoregulatory model that we have developed and utilized previously.^{[ 20} , ²¹ , ³⁸ , ^{50 ]} Unlike several semi‐empirical, control process models, our approach has been to use a physiology‐based model with compartments and vascular transport that attempts to accurately simulate the human or animal model physiology. Because of this, it is possible to expand models to a diverse set of clinical conditions, novel therapeutics and make predictions across animal models and patients.^{[ 20} , ²¹ , ³⁸ , ^{50 ]} We build on the original work of Sorensen,^{[ 51 ]} which introduced the concept of several body compartments connected by vascular transport.

In our approach,^{[ 20} , ²¹ , ³⁸ , ^{50 ]} the body compartments include the heart and lungs, brain, liver, gut, kidneys, muscles, and adipose tissues. Mass balances for glucose, insulin, and glucagon are constructed over each body compartment, assumed to be a well‐mixed continuous stirred‐tank reactor (CSTR).^{[ 52 ]} The equations, which consist of the resulting ordinary differential equations (ODEs) with sources and sink terms, are described elsewhere. In the case of the diabetic animal, we assume that pancreatic ablation was completed and approximate the physiology via the removal of all beta cell glucoregulatory effects. Scheme  1 illustrates the organ compartments and their connectivity.

Scheme 1.

Schematics of the IMPACT In‐Silico glucoregulatory model^{[ 20} , ²¹ , ³⁸ , ^{50 ]} utilized in this work to evaluate glucose responsive glucagon (GRG) therapeutics in the human body and animal models. Our glucoregulatory model is composed of different compartments or organs including the brain, heart and lungs, liver, gut, kidney, adipose tissue, and muscle. Organ systems are organized into well‐mixed compartments interconnected by vascular transport as depicted, with sources and sink terms for glucose, insulin, and glucagon described as functions of time. Glucagon submodel assumes one compartment with volume of all organs and delivery of GRG to adipose tissue. GRG‐I^{[ 39 ]} is introduced using a transdermal microneedle patch which administers the reactive particles over time. GRG‐II^{[ 40 ]} is injected into adipose tissue forming a depot that similarly releases glucagon according to the mechanism described in Section 2.2. Each symbol in this schematic is explained in Table S1 (Supporting Information).

Our previous work has focused on the analysis of data involving dynamic glucose and insulin metabolism within the glucoregulatory systems of mice, rats, mini‐pigs and humans. ^{[ 20} , ²¹ , ³⁸ , ^{50 ]} However, it is particularly relevant for this current work that we have not applied the model directly to experiments involving glucagon infusion or its resulting dynamic measurement. Glucagon is currently addressed in these models using a simple scheme originally proposed by Sorensen^{[ 51 ]} based on an empirical tertiary kinetic model described in Section 2.4 below.

2.2. Glucose‐Responsive Glucagon Delivery Mechanisms Evaluated in This Work

2.2.1. Glucose Responsive Glucagon Mechanism I (GRG‐I)

GRG therapeutics can in theory include glucagon molecules chemically modified to activate to a more potent state upon binding to glucose. Such drugs have been realized for the case of GRI,^{[ 53} , ⁵⁴ , ^{55 ]} which we have evaluated elsewhere.^{[ 49 ]} To date, however, GRGs, including those evaluated in this work, use glucose to control the delivery or release from sequestration of glucagon. Figure  1 presents two different GRG mechanisms published in the literature.

Figure 1.

Recent technologies for controlled released of glucagon in response to blood glucose concentration changes. (a) GRG‐I^{[ 39 ]} developed by using hydrogel incorporated into microneedle patches, which can be shrunken due to low glucose concentration in the environment and crosslinking between polymer chains in the hydrogel. As a result, encapsulated glucagon is released until the glucose concentration does not lead to further crosslinking. (b) ref. [39] in‐vitro experiments and the correlation between released glucagon concentration trajectories and glucose concentration. (c) GRG‐II strategy suggested by ref. [40] where glucagon and glucose oxidase enzyme is encapsulated in the hydrogel. Such design enables controlled release of glucagon from hydrogel according to glucose concentration in the environment. (d) Dependence of glucagon release trajectories on glucose concentrations for in‐vitro experiments performed by ref. [40]. Symbols in panels (b) and (d) show experimental data points and the solid lines denote the fit, and each color represents glucose concentration in the environment.

The first GRG concept evaluated in this work (GRG‐I) comes from Wu et al.^{[ 39 ]} It is based on the encapsulation of glucagon into microgel particles, which swell or contract depending on the local glucose concentration. At high glucose concentrations, the microgel swells from mono‐complexation between glucose and 4‐acrylamido 3‐fluorophenylboronic acid (AFBA) monoblocks embedded in the gel. In the swelled state, glucagon remains adsorbed within the particle. At low glucose concentrations, the AFAB blocks crosslink, and the resulting shrinkage releases the encapsulated glucagon. This then triggers an increase in hepatic glucose production (HGP) with a subsequent increase in blood glucose concentration. The prepared microgels are loaded into microneedle arrays or patches, which can enable subcutaneous infusion of the GRG microgels into the patient or animal. Figure 1a illustrates loaded microneedle arrays with microgels, the schematic of their swelling or shrinkage in response to glucose, and release of encapsulated glucagon for GRG‐I. The kinetics of glucagon release are critical to prevent hypoglycemia and avoid the onset of hyperglycemia.

The microneedle patches (Figure 1a) with either no glucagon (control) or loaded with 18% glucagon content were used to subcutaneously administer GRG into rats with T1D diabetes for in‐vivo testing of GRG performance in hypoglycemia prevention per ref. [40]. In these experiments, the diabetic rats were initially treated with patches and then overdosed with subcutaneous injection of 2 IU kg⁻¹ human recombinant insulin after 30 min.

2.2.2. Glucose Responsive Glucagon Mechanism II (GRG‐II)

A second GRG mechanism that we consider in this work (GRG‐II) was introduced by Webber et al.^{[ 40 ]} It takes advantage of a novel and reversible hydrogelation process, in which high and low glucose concentrations respectively stabilize and destabilize an encapsulating gel structure. In this design, C₁₀‐V₂A₂E₂ amphiphilic peptide chains, their self‐assembly into nanofibers, and their physical crosslinking are the basic units for hydrogelation. Glucose oxidase (GOx) is an essential reactive component that converts the glucose signal into a change in local pH while reacting it to the lactone. This change in pH makes the hydrogel structure transform from gel to sol at high and low glucose content, respectively as depicted in Figure 1c for GRG‐II.

In the GRG‐II mechanism, the enzyme GOx catalyzes the gel‐to‐sol transition by converting glucose to gluconic acid at high glucose concentration with a subsequent decrease in local pH. This disrupts the self‐assembled peptide nanofiber structure. These glucose and pH responsive hydrogels are then the dynamically active carriers for exogenous glucagon, which is controllably released as active glucagon in response to low glucose concentration for hypoglycemia prevention. Figure 1c depicts the process of reversible sol‐gel transformation and release of GRG‐II in response to rising glucose concentration.

2.3. Modeling the kinetics of in‐vitro Release of GRG in Response to Glucose Concentration

A major objective of this work is to show that GRG‐I and GRG‐II can be translated into accurate mathematical descriptions, making it possible to describe their in‐vivo performance in rodent models for comparison to published experimental results. It is then possible to extrapolate these predictions to human performance, as we have demonstrated previously for other therapeutics.^{[ 20} , ²¹ , ²² , ³⁸ , ⁴⁹ , ^{50 ]} The mathematical description can be formulated using the kinetics of release of GRG‐1 and GRG‐II in‐vitro. Various drug release models exist to describe drug release kinetics accurately, including the Higuchi model,^{[ 56 ]} Hixson‐Crowell model,^{[ 56 ]} and even complex finite‐element analysis for specific therapeutic formulation, all of which have been reviewed elsewhere.^{[ 57 ]}

We formulated a first‐order model based on the in‐vitro glucagon release rates at fixed glucose concentrations. This form is shown to describe both GRG‐I and GRG‐II well, despite the fundamental differences in their construction and mechanism. The advantage of this simple formulation is its capability to capture the essential release kinetics and compatibility with physiological glucoregulatory models. For this reason, and the complication of potential overfitting, we rejected the option of modeling these delivery devices using first principles. The in‐vitro GRG‐I and GRG‐II experiments suggest that the release of glucagon can be captured by a simple equilibrium reaction model modulated by glucose concentration in the environment. In the current scheme, the concentrations of sequestered (C_bound) and free solution glucagon (C_free) exchange rapidly according to the equilibrium:

$${\mathrm{C}}_{\mathrm{bound}}\begin{array}{c}{\mathrm{k}}_{\mathrm{f}}\\ \rightleftharpoons \\ {\mathrm{k}}_{\mathrm{b}}\end{array}{\mathrm{C}}_{\mathrm{free}}$$ (1)

Solving the mass action equations for C_free with the initial conditions C_free(t═0) = 0 and C_bound(t═0) = C_o yields:

$${\mathrm{C}}_{\mathrm{free}}=\frac{{\mathrm{C}}_{\mathrm{o}}{\mathrm{k}}_{\mathrm{f}}}{{\mathrm{k}}_{\mathrm{f}}+{\mathrm{k}}_{\mathrm{b}}}\left(1-{\mathrm{e}}^{-\left({\mathrm{k}}_{\mathrm{f}}+{\mathrm{k}}_{\mathrm{b}}\right)\mathrm{t}}\right)$$ (2)

In both GRG‐I and GRG‐II, glucose diffuses into the delivery medium and activates the glucagon for release using complex schemes for reaction and transport within the matrices. We capture this using rate constants that generally depend on the local glucose concentration:

$$k_f=f_1\left(\left[glucose\right]\right)$$ (3)

$$k_b=f_2\left(\left[glucose\right]\right)$$ (4)

For the reasons stated above, we found a semi‐empirical description of both rate constants of the form:

$$k_i=a_i{C}_{glucose}+b_i$$ (5)

Here, C_glucose is the instantaneous glucose concentration, and k_i = k_f or k_b for GRG‐I or GRG‐II. Table  1 contains the best parameters for a_i and b_i for the forward (f) and reverse (r) reactions of both GRG‐I and GRG‐II.

Table 1.

In‐vitro Glucagon release parameters for GRG‐I and GRG‐II for the forward and reverse reactions are described in Section 2.3.

	Forward		Reverse
	af [dL mg−1 min−1]	bf [min−1]	ar [dL mg−1 min−1]	br [min−1]
GRG‐I[ 39 ]	–	0.001675	2.31E‐5	−1.31E‐4
GRG‐II[ 40 ]	–	116.4	3	−82.2

As shown in Figure 1b (GRG‐I) and 1d (GRG‐II), this description, using the best‐fit parameters of Table 1, is able to capture the key dynamics of glucagon release, in agreement with in‐vitro data.

Figure 1b shows the kinetics of in‐vitro release of glucagon at different glucose concentrations in the solution over 1 h in ref. [39]. Figure 1b suggests that the maximum release of encapsulated glucagon occurs at the lowest glucose concentration of ≈ 50 mg dL⁻¹ which is below than the hypoglycemia level (<70 mg dL⁻¹) in T1D patients. Figure 1d also shows in‐vitro experimental results of ref. [40] for the kinetic of glucagon release into solution from hydrogels over 24 h at different initial glucose contents. More than half of encapsulated glucagon is released into a solution with glucose concentration of ≈50 mg dL⁻¹.

2.4. Modeling the GRG Depot in‐vivo and Glucagon Signaling

2.4.1. Modeling the Release Depot

Both GRG‐I and GRG‐II refer to microgel particles or material sections that evaluate the glucose concentration and glucagon release into adipose tissue. The injection site itself creates a depot or tissue encapsulated mass that interfaces the GRG into the tissue. For GRG‐I, this depot takes the form of a microneedle array housing the GRG microgel particles. For GRG‐II, the gel is injected subcutaneously forming the depot. Release of free glucagon (N_free = VC_free) from the depot is modeled as a first order process limited by a diffusive rate constant, k_d, where we have assumed its value is similar to insulin in dimeric form (k_d = 0.0089 min⁻¹).

$$V{\left.\frac{d{C}_{free}}{dt}\right|}_{Depot}=-k_{d}V{C}_{free}=\frac{d{N}_{free}}{dt}=-k_d{N}_{free}$$ (6)

The full set of equations interfaced to the physiological model are:

$$\frac{d{N}_{free}}{dt}=k_f\left(C_{glucose}\right)N_{bound}-k_b\left(C_{glucose}\right)N_{free}-k_d{N}_{free}$$ (7)

$$\frac{d{N}_{bound}}{dt}=-k_f\left(C_{glucose}\right)N_{bound}+k_b\left(C_{glucose}\right)N_{free}$$ (8)

where, N_bound is VC_bound and V is the depot volume.

2.4.2. Modeling Glucagon Signaling

For this current work, we did not modify the mathematical form used to link glucagon to insulin and glucose from previous studies. This will be a topic of future investigation. We call the semi‐empirical form for pancreatic glucagon release and clearance the Sorensen glucagon sub‐model as it is thus far undifferentiated from the original work of Sorensen.^{[ 51 ]} Equation 10 describes the submodel, where:

$$r_{IVGG}+k_d{N}_{free}{V}^{depot}+k_d{N}_{bound}{V}^{depot}$$ (9)

are additional GRG related terms so that the complete submodel is:

$$\begin{array}{cc}\hfill V^{\mathrm{\Gamma }}\frac{d{\mathrm{\Gamma }}^{\mathrm{N}}}{dt}& =r_{M\mathrm{\Gamma }C}(M_{P\mathrm{\Gamma }R}^G{M}_{P\mathrm{\Gamma }R}^I-{\mathrm{\Gamma }}^N+r_{IVGG}+k_d{N}_{free}{V}^{depot}\hfill \\ & +k_d{N}_{bound}{V}^{depot})\hfill \end{array}$$ (10)

Here, r_IVGG is the intravenous (IV) glucagon infusion rate upon IV delivery of glucagon treatment if relevant. The parameter V^depot denotes the total volume of the subcutaneous depot as described in the previous section. $M_{P\mathrm{\Gamma }R}^G$ and $M_{P\mathrm{\Gamma }R}^I$ respectively represent glucose and insulin cross‐reactivity on pancreatic glucagon release, and r _MΓC is metabolic rate for glucagon clearance. Γ^N and V ^Γ indicate normalized glucagon concentration and the total volume of the assumed compartment. Additional details on how these parameters were estimated in the original Sorensen glucagon submodel are found elsewhere.^{[ 51 ]}

The HGP in the liver is a glucose metabolic source incorporated into the process diagram in Scheme 1 as a mass balance equation over the liver compartment with the following equation.

$$V_L^G\frac{d{G}_L}{dt}=Q_A^G{G}_H+Q_G^G{G}_G-Q_L^G{G}_L-r_{HGU}+r_{HGP}$$ (11)

$$r_{HGP}=M_{HGP}^I{M}_{HGP}^G{M}_{HGP}^{\mathrm{\Gamma }}{r}_{HGP}^{\mathrm{B}}$$ (12)

$$M_{HGP}^{\mathrm{\Gamma }}=M_{HGP}^{\mathrm{\Gamma }0}-f_2\left(11\right)$$ (13)

$$M_{HGP}^{\mathrm{\Gamma }0}=\alpha tanh\left[\beta {\mathrm{\Gamma }}^{\mathrm{N}}\right]\alpha =2.7\beta =0.39$$ (14)

Here, the HGP rate is then correlated with a normalized glucagon concentration through the glucagon multiplier defined according to Equations 12 and 13, as per the original Sorensen glucagon submodel.

G, I, and Г superscripts indicate glucose, insulin, and glucagon components with B and A superscripts for basal and artery, respectively. Also, H, G, and L subscripts are abbreviations for heart, gut, and liver compartments respectively. The rates r_HGU and r_HGP denote hepatic glucose uptake (HGU) or production (HGP) rates in the liver. V, G, Q, and M represent volume (dL), glucose concentration (mg dL⁻¹), flow rate (dL min⁻¹), and multiplier (dimensionless). As discussed above, Γ^Nis glucagon concentration (Γ) normalized by basal glucagon concentration (Γ^B). $V_L^G$, and G_L denote liver compartment volume and glucose concentration. The term f ₂defines degradation of maximal glucagon effect on hepatic glucose production in response to a step change of initial glucagon concentration multiplier $M_{HGP}^{\mathrm{\Gamma }0}$. The original values of parameters α and β in Equation 14 in the Sorensen glucagon submodel were retained at 2.7 and 0.39. We performed a range sensitivity analysis below for α and β parameters over values of [1,4.2] and [0.1,2.1], respectively to investigate how blood glucagon concentration (from both endo/exogenous sources) modulates blood glucose concentration.

2.5. Physiology and Non‐Physiology Based Models

Non‐physiology‐based models have also been successfully utilized to describe blood glucose, insulin, and glucagon dynamics.^{[ 17} , ^{19 ]} The difference between these models, however, is the lack of one‐to‐one correspondence between physiological compartments of the glucoregulatory system. Instead, a simplified process diagram is often employed for the purpose of matching the dynamic response. Empirical constants are then fit to various data sets. Despite their simplicity, they have generally been successful in describing glucose, insulin, and glucagon dynamics. In contrast, physiology‐based models provide a more detailed description of the glucoregulatory system that is potentially applicable across a broad range of data sets or even species. Our past work has shown that they can inform the rational design of drugs, particularly for diabetes treatment (i.e., GRIs) with the promise of expediting the drug discovery process from bench to clinical practice.^{[ 49 ]} Herein, we demonstrate that our model reproduces realistic blood insulin and glucagon dynamics in a T1D human subject and compare it with the minimal model of Herrero et al.^{[ 19 ]} The latter is explained further in the Supporting Information. This current work also compares previous versions of our physiological models (PAMERAH^{[ 20 ]} and IMPACT^{[ 21 ]}) to elucidate the parametric sensitivity for this type of GRG analysis.

3. Results and Discussion

3.1. Modeling bi‐hormonal Control of Blood Glucose Dynamics in T1D Subjects

We evaluated both GRG‐I and GRG‐II using our IMPACT physiological model, comparing published in‐vivo data to model predictions based only on the parameterization of the respective therapeutic described in the method section.

3.1.1. GRG‐I Performance Test in‐Vivo Using Diabetic Rats

GRG‐I is a device that utilizes microneedle technology for glucagon delivery. The in‐vivo testing of GRG‐I proceeded as follows: initially, streptozotocin [STZ]‐induced diabetic rats with stabilized blood glucose levels at ≈300 mg dL⁻¹ were selected. They were then divided into groups receiving either glucagon‐loaded composite microneedle (cMN) patches responsive to blood glucose levels (GRG‐I) or cMN patches containing a sham buffer. The efficacy of GRG‐I in preventing hypoglycemia was evaluated by applying either GRG‐I or sham buffer cMN patches at t = 0 to diabetic rats with an initial glycemia of ≈400 mg dL⁻¹. Subsequently, an insulin challenge (2 IU kg⁻¹) was administered via subcutaneous injection of RHI at t = 30 min to induce hypoglycemia. Figure  2a illustrates the results of these two experiments. GRG‐I did not induce an initial rise in glycemia, like the sham buffer‐loaded cMN, possibly due to the initial hyperglycemia or high blood glucose concentration. Nonetheless, it exhibited a sustained effect in preventing hypoglycemia, maintaining glycemia at ≈90 mg dL⁻¹ during the final h of the experiment.

Figure 2.

Experimental and modeling data for blood glucose concentration trajectories upon subcutaneous injection of either glucagon or GRG. (a) Wu et al. experiments on T1DM rat subjects treated with either Glucagon (red circles) or GRG‐I (black circles) microneedle patches at time = 0 followed by 2 IU kg⁻¹ insulin injection at time 30 min. Red and black solid lines represent the modeling data for glucagon and GRG‐I experiments, respectively. Webber et al. experiments on T1DM mouse subjects subcutaneously dosed with (b) shame buffer (black circles), (c) 0.01 mg dasiglucagon (red circles), or (d) GRG‐II or 1% w/v hydrogel‐carried 0.01 mg dasiglucagon plus GOx/catalase (blue circles) at time = 0 followed by 2.5 IU kg⁻¹ insulin injection at time = 120 min. The corresponding modeling data are shown in solid lines in panels b to d.

3.2. Prediction of Pharmacodynamics (PD) of GRG‐I

We integrated the kinetic model for the in‐vitro release of glucagon from GRG‐I into our IMPACT, as discussed in the method section, to simulate the in‐vivo performance of GRG‐I. Following a procedure similar to the in‐vivo testing of GRG‐I, we employed a diabetic rat model in the IMPACT with an initial glycemia of 390 mg dL⁻¹. The diabetic rat was then subjected to treatment with either GRG‐I or a sham buffer at t = 0, followed by a 2 IU kg⁻¹ subcutaneous injection of RHI at t = 30 min to induce hypoglycemia.

Figure 2a illustrates that our model consistently reproduces the results of in‐vivo experiments when applying either GRG‐I or a sham buffer to diabetic rat subjects at t = 0, with no rise in glycemia observed in both experimental and model outcomes. Our calculations predicted an initial decrease in glycemia after the insulin challenge via subcutaneous delivery of RHI at t = 30 min, consistent with in‐vivo experiments. However, the predicted and experimental glycemia began to diverge after 60 min, with the final glycemia predicted to be 200 mg dL⁻¹ by the end of 3 h, compared to experimental values of 90 or 50 mg dL⁻¹ corresponding to GRG‐I or sham buffer treatment, respectively.

Our model did not accurately predict the prevention of hypoglycemia using GRG‐I treatment, and it exhibited a similar trend for both GRG‐I and sham buffer treatments. Nevertheless, despite not being specifically parameterized for the diabetic rat model in these experiments, our model was able to capture the same glycemia trend observed in in‐vivo experiments. The total glycemia area under the curve (AUC) in the case of sham buffer or GRG‐I treatment were 28470 and 34910 mg dL⁻¹ min in experiments, compared to corresponding values of 44390 mg dL⁻¹ min from predictions for both buffer and GRG‐I. These predicted AUCs were on average 41.5% ± 14% larger than the experimental AUCs.

These disparities between predictions and experimental observations likely stem from a few factors related to the modeling of the experiment. First, our model lacks a comprehensive glucagon submodel to accurately depict the secretion of glucagon and the consequent increase in glycemia following hypoglycemia induced by an insulin challenge. Another factor may be systemic deficiencies within our diabetic rat model in the IMPACT modeling framework. This necessarily results in inaccurate predictions of glycemia dynamics. Additionally, discrepancies between in‐vitro and in‐vivo releases of glucagon at the level of the devices themselves, are not easily accounted for in our model development, and require more detailed studies of device operation in‐vivo.

The first two issues will be addressed in our future research endeavors, while the third concern was explored by adjusting the pharmacokinetic (PK) parameter of GRG‐I. Since experimental data for evaluating the dissociation constant (k_d) value of GRG‐I were unavailable, the k_d value for insulin was utilized for the PK modeling of GRG‐I. Consequently, the sensitivity of glycemia or GRG‐I PD to the k_d value was investigated. Figure  3a illustrates that altering GRG‐I's PK, or in other words, device parameters like k_d, by several orders of magnitude does not enhance the predicted PD outcomes. Furthermore, it indicates that a very rapid glucagon release does induce an increase in glycemia to some degree, as anticipated, since the majority of glucagon is released within a short timeframe.

Figure 3.

Sensitivity of pharmacodynamics (PD) of GRG‐I and GRG‐II to the pharmacokinetics (PK) parameters or k_d value. The insulin k_d value of 0.0089 min⁻¹ was used for both GRG‐I and GRG‐II in our physiological model. We changed k_d values by several orders of magnitude and redid the calculations accordingly. (a) The sensitivity of diabetic rat model's glycemia or PD of GRG‐I to k_d values. (b) The same sensitivity was also performed for GRG‐II. Both panels a and b suggest that changing k_d value didn't impact glycemia in rat and mouse models respectively treated by GRG‐I and GRG‐II. Each color on the plots is representative of glycemia dynamics for each k_d values, while all other parameters and values were kept the same.

3.2.1. GRG‐II Performance in Diabetic Mice

To assess the in‐vivo performance of GRG‐II, diabetes was initially induced in healthy mouse models using STZ, as outlined in ref. [40]. The unfasted glycemia of diabetic mice was monitored over the subsequent 9–13 days to maintain it above 600 mg dL⁻¹. After a 9‐h fasting period, diabetic mice with glycemia below 450 mg mL⁻¹ were excluded from the study. This step was taken to ensure that the glycemia of diabetic mice remained within the range observed in healthy mice (≈180 mg dL⁻¹) after administration of long‐lasting insulin (detemir) 4 h before GRG delivery at t = 0. Another benefit of administering long‐lasting insulin to diabetic mice was to establish a basal level of insulin similar to that of healthy mice during GRG delivery.

The fasted diabetic mice received a subcutaneous injection of GRG at t = 0, followed by an intraperitoneal injection (i.p.) of ≈2.5 IU kg⁻¹ animal origin‐free (AOF) recombinant human insulin (Gibco) in 100 µL of saline at t = 2 h. This insulin challenge was designed to induce hypoglycemia and elucidate the role of GRG‐II in preventing it. Control experiments involved the same procedures with subcutaneous injection of either a sham buffer or dasiglucagon.

The symbols in Figures 2b and c represent the trajectories of blood glucose concentration in these in‐vivo experiments. The data indicate that the delivery of glucagon in the form of dasiglucagon or GRG‐II resulted in significant hyperglycemia, which was attenuated by the insulin challenge at t = 2 h, leading to mild hypoglycemia in all diabetic mice. However, only the diabetic mice initially treated with GRG‐II were able to recover from their hypoglycemia (reaching ≈60 mg dL⁻¹), and their glycemia gradually increased to reach 120 mg dL⁻¹ by the end of the 5‐h experiment.

3.2.2. Prediction of PD of GRG‐II

We integrated the model for in‐vitro glucagon release from GRG‐II into our in‐house IMPACT to conduct in‐silico modeling of in‐vivo GRG‐II outcomes, as previously described. These simulations were conducted following the same protocol as outlined in the in‐vivo experiments detailed above. The diabetic rat was fasted, and its initial glycemia was set at 180 mg dL⁻¹.

Our model effectively captured the dynamics of glycemia for diabetic mice treated with dasiglucagon, GRG‐II, or a sham buffer at t = 0, followed by a subcutaneous injection of regular human insulin (RHI) challenge of 2.5 IU kg⁻¹. Figure 2b demonstrates that our model accurately replicated experimental outcomes for these two phases: initial treatment with the sham buffer followed by the insulin challenge. This consistency is notable as no parameterization or fitting was employed to reproduce the glycemia of diabetic mice, and the final glycemia level was closely aligned with experimental data. In contrast, Figures 2c‐d illustrates how glucagon treatment led to discrepancies between model predictions and experimental results, as our simulations failed to capture the hyperglycemic response following glucagon treatment at t = 0. In Figure 2c, our model predicts a slight initial increase in glycemia is observed immediately after dasiglucagon delivery, while GRG‐II does not produce this initial rise in the modeling results, as depicted in Figure 2d.

The disparity between modeling outcomes and experimental findings in diabetic mice following the administration of dasiglucagon/GRG‐II at t = 0 underscores the absence of a comprehensive glucagon submodel in our IMPACT to capture glycemia elevation. Interestingly, our model successfully replicated consistent glycemia dynamics with experimental data immediately after the insulin challenge at t = 2 h in the case of dasiglucagon treatment, where glucagon release is slow and non‐responsive to glucose levels. Figure 2c illustrates that our computations fails to reproduce the recovery of mice from mild hypoglycemia at the experiment's conclusion. This observation suggests a deficiency in our model's glucagon submodel, which fails to stimulate glucose production in the liver to counteract hypoglycemia.

To quantify the disparity between experimental and modeling results, the glycemia AUC was computed. Treatment of diabetic mice with sham buffer, dasiglucagon, and GRG‐II in the experiment exhibits glycemia AUC of 34191, 50229, and 59715 mg dL⁻¹ min, respectively, compared to 32702, 37047, and 33534 mg dL⁻¹ min for the corresponding modeling outcomes. This comparison indicates that the modeling results closely underestimates the experimental glycemia AUC by an average percent error of 24.8% ± 16%. In essence, the introduction of glucagon to diabetic mice contributes to the discrepancy between simulations and experiments. This comparison also highlights the remarkable performance of our model in reproducing glycemia range, dynamics, and kinetics without requiring model parameterization.

We explored the sensitivity of our model results to k_d, a PK parameter of GRG‐II. Similar to GRG‐I, we utilized the same k_d value as insulin due to a lack of sufficient experimental data to evaluate GRG‐II k_d values. Consequently, we varied the k_d of GRG‐II by several orders of magnitude and recalculated its effects on GRG‐II PD. Figure 3b illustrates that glycemia or PD of GRG‐II remained largely unaffected by changes in the k_d value. Figures 3a and b underscore the need for further modifications to the glucagon submodel and corresponding parameters for rat/mouse subjects in our current physiological model. Additionally, GRG‐I was administered with microneedles arrays used for transdermal therapeutics delivery. In our model, we approximated transdermal delivery of glucagon via microneedles patches with subcutaneous absorption using bulk‐release into the peripheral compartment. Considering that microneedle geometry can play a significant role in both instantaneous response and effectiveness overtime, we consider the simplified release component in our model a source of deviation from the GRG‐I experiments shown in Figures 2a and 3a.

3.3. Exploring the Influence of Glucagon‐Related Parameters on Model Performance

Recognizing the limitations of our model concerning the glucagon hormone prompted an investigation into the sensitivity of the model's performance to parameters suggested by Sorensen for (HGP), which links glucagon to its PD impact. This approach involved assessing the sensitivity of glucose appearance in the liver to blood glucagon concentration, as governed by parameters α and β in Equation 14. Essentially, α and β parameters dictate the magnitude of the step change in glucagon concentration and the rate at which this change occurs.

Figure  4 presents sensitivity analysis graphs illustrating the representative range, comparing modeling data with control experiments (treatment with buffer) referenced in refs. [39] and [40] as depicted in Figure 2. Supplementary Figures S1 and S2 (Supporting Information) contain the complete set of these analyses for both control and GRG experiments. Figure 4a,b suggest that when step changes in initial glucagon concentration reach the same maximum at a faster rate, there is a corresponding increase in blood glucose concentration. Furthermore, Figures S1 and S2 (Supporting Information) indicate that the amplitude of these step changes, at the same rate, has a more pronounced effect on the level of glycemia than the pace of these changes.

Figure 4.

Blood glucose concentration modulation by parameters for glucagon‐mediated hepatic glucose production (HGP) rate. Modeled blood glucose concentration and its sensitivity to glucagon‐mediated HGP rate parameters (α, β) for (a) T1D rat subjects treated with buffer as control for GRG‐I and (b) T1D mice treated with buffer as a control for GRG‐II. (c) and (d) panels present the corresponding normalized blood glucagon sensitivity to the HGP rate parameters. Experimental and modeling data using original Sorensen parameters are shown in cyan circles and solid traces. Presentation of such range sensitivity analysis is color‐coded, where each color represents the modeling data for each pair of α and β parameters.

The corresponding normalized blood glucagon levels in Figure 4c,d show that α and β parameters do not significantly modulate normalized blood glucagon levels. This observation may align with the fact that glucagon is a potent hormone, and even minor alterations in its concentration strongly influence the glucoregulatory system. Another explanation could be attributed to the empirical form of the HGP equation. Given the complex response of glycemia levels to changes in α and β parameters in ${\mathrm{M}}_{\mathrm{HGP}}^{\mathrm{\Gamma }0}$, such observations are challenging to interpret further. Similar effects were also noted when treating rats and mice with GRG, as depicted in Figures S1 and S2. (Supporting Information)

3.4. In‐Silico Insulin and Glucagon Infusion Pumps for T1D Human Subjects

In the preceding sections, we modeled the experimental outcomes of point injections of GRG and insulin in diabetic rodent subjects. We utilized our model to investigate the PK/PD of continuous infusion of insulin and glucagon therapeutics in T1D human subjects. Clinical data from insulin and glucagon infusion pumps were digitized from literature sources.^{[ 18 ]} Subsequently, we then fit our physiological IMPACT model data with a minimal, non‐physiological model^{[ 19 ]} designed to capture the PK/PD of glucagon and insulin therapeutics. Details of this minimal model can be found in the Supporting Information.

We adapted the IMPACT^{[ 21 ]} model to design in‐silico insulin and glucagon infusion pumps for subcutaneous administration of these drugs in healthy and T1D human and rodent subjects. Non‐physiological models require training with experimental data and lack standalone platforms for predicting blood glucose, insulin, and glucagon concentration trajectories. However, our designed pump further underscores the significance of our physiological model in expediting the discovery of therapeutic insulin and glucagon.

Figure  5a‐b depict two separate experiments where either insulin or glucagon were subcutaneously injected into a T1D human subject. Simulated insulin and glucagon infusion trajectories are represented by red bars, while the black solid line denotes blood insulin or glucagon concentration in response to such infusions based on our physiological model. Subsequently, we employed a minimal, non‐physiology‐based model developed by Pau Herrero et al.^{[ 19 ]} to simulate our model‐generated results for insulin and glucagon concentrations. The corresponding modeling data are depicted by black solid lines in Figures 5a‐b, and the identified parameters of the minimal model can be found in Table  2 . Despite the imperfect fit of the IMPACT^{[ 21 ]} results, the minimal model successfully captured the dynamics of blood insulin and glucagon trajectories. It's worth noting that the insulin trajectory results are expected to be more realistic than the glucagon trajectory results, as IMPACT does not offer a comprehensive description of glucagon's role in the glucoregulatory system compared to insulin's role.

Figure 5.

Comparison between the IMPACT and minimal model^{[ 19 ]} for predicting (a) blood insulin and (b) glucagon concentration trajectories upon subcutaneous injection of insulin and glucagon, respectively. The red bar plots show either insulin or glucagon infusion rate, and the black and blue solid lines correspondingly represent results produced by IMPACT and their prediction by the minimal model.

Table 2.

Identified parameters of minimal model^{[ 19 ]} for fitting resulting blood insulin or glucagon concentration of T1D human subject treated with insulin or glucagon infusion pumps, respectively.

	Insulin infusion pump	Glucagon infusion pump
ke (min−1)	0.79	0.82
VI (mL kg−1)	23.41	22.41
tmaxI / tmaxN (min)	56.22	40.41
NI (µU mL−1) / Nb (pg mL−1)	0	98.10
R2	0.63	0.84

The comparison between these physiological and non‐physiological models underscores the importance of further developing physiology‐based compartmental models. Such models offer predictive power based on a simplified metabolic network for glucose, insulin, and glucagon. While non‐physiological models are suitable for glucose management, they lack tools for the rational design of therapeutics since their model parameters are not realistically correlated with physiology. Further comparison between IMPACT and the minimal model is discussed in the supporting information.

3.5. Subcutaneous and Intravenous Administration of Therapeutic Glucagon to T1D Human Subject via PAMERAH and IMPACT

Moreover, we conducted a comparative analysis of the performance of our two most recently published platforms, namely PAMERAH and IMPACT, in modeling therapeutic glucagon. A 1 mL subcutaneous (SC) depot containing 1 mg of therapeutic glucagon was administered to a T1D human subject using both the PAMERAH and IMPACT platforms. In another experimental setup, T1D human subjects were treated with 1 mg of glucagon administered via intravenous (IV) injection.

Figure  6a,b illustrate the PK/PD responses to either SC or IV administration of glucagon in T1D human subjects using the IMPACT with continuous injection feature. Figure 6a depicts the rapid and gradual increase in blood glucagon levels corresponding to IV and SC doses of glucagon, respectively. Figure 6b highlights the differences in PD between SC and IV administration of glucagon. IV injection results in a swift increase in blood glucose concentration, whereas SC injection leads to the gradual release of glucagon into the bloodstream via diffusion through adipose tissue, resulting in a more sustained effect on blood glucose concentration.

Figure 6.

(a) Normalized blood glucagon and (b) Blood glucose concentration trajectories generated using IMPACT after administration of either subcutaneous (black solid line) or intravenous (red trace) injection of 1 mg glucagon at the beginning.

Figure S3a,b (Supporting Information) provide a comparison of the performance of the PAMERAH and IMPACT, demonstrating that both codes generate numerically identical results. However, the current version of PAMERAH lacks the capability, unlike IMPACT, to model therapeutic glucagon injection into rodent models. In addition, therapeutic glucagon was introduced at a different time (60 min) instead of the beginning of the experiment, yielding similar results (Figure S4a,b, Supporting Information).

3.6. Future Directions for Glucagon‐Related Modification of Sorensen Model

In the original Sorensen glucagon submodel used both PAMERAH and IMPACT, the regulation of hepatic glucose production via hormonal controls was represented as source and sink terms with the response to each of the effector molecules fitted to experimental or clinical concentration‐response calibration curves. Thus, on a sub‐organ level, this approach is considered a “black box”. As discussed in the original work,^{[ 51 ]} this approach was chosen due to the lack of detailed mechanistic understanding of physiological glucoregulation at the time. An appropriate simplification was then used to both retain the physiological compartment aspect of the model as well as to accurately reflect the available data.

In the decades since, our scientific understanding of glucoregulation and the effects of hormonal signals such as insulin and glucagon have significantly advanced. From a molecular standpoint, we have gained an improved understanding of glucose transport,^{[ 58 ]} hormone receptor binding, signaling, and associated kinetics,^{[ 59 ]} as well as enzymatic processes within the cell.^{[ 60 ]} From an organ and tissue standpoint, our understanding of how signals between different bodily compartments, including which compartments and the importance of extrahepatic control of glucoregulation, have changed.^{[ 61 ]} It was also previously thought that glucagon and insulin were purely antagonistic molecules. Current understandings reflect their varying roles in physiology and pathophysiology, sometimes in synergistic manner.^{[ 62 ]}

Taken together, it is unsurprising that the Sorensen glucagon submodel, which does not consider the above factors, is not able to successfully capture the entire dynamics of a GRG administration, where both the glucagon release and kinetics significantly differ from that of native pancreatic secretion. To continue the development of a physiologically accurate model of the glucoregulatory system, not only the existing fitting functions will need to be replaced. Instead, the mechanisms of interaction between insulin, glucagon, and key organ systems will need to be reimplemented in a faithful manner to reflect the relative strength and kinetics of each interaction. Ultimately, in these physiologically based models, the combination of accurate representation of PD/PK of effector molecules and prior knowledge of biological system organization will lead to both accurate predictions and molecular insights.

4. Conclusion

We used our previously developed glucoregulatory models (IMPACT^{[ 21 ]}) to predict glucose concentration dynamics in two GRG platforms, where GRG is subcutaneously administered into either diabetic rats or mice. We left the original Sorensen glucagon submodel unchanged and added a subcutaneous injection depot after digitizing both GRG mechanisms for incorporation. Our work demonstrates that it is possible to evaluate and predict the performance of candidate GRGs in the literature by first digitizing them –constructing an accurate mass action mathematical model of their mechanistic function. The successful modeling of two distinct GRG designs, experimentally synthesized and tested by Wu et al. (GRG‐I) and Webber et al. (GRG‐II), highlights the promise of this approach. One metric for the agreement between simulation and experiment is the comparative integrated glucose material balance over the observation range, which ranged from 41.5% ± 14% more and 24.8% ± 16% less than GRG‐I and GRG‐II experimental values, respectively. We attribute the large differences to the relatively simple computational descriptions of glucagon dynamics in the literature, in comparison to insulin and glucose. This points to the need for improved descriptions of glucagon regulation in such simulations, a focus of future work. Our designed insulin and glucagon infusion pumps for continuous administration of these therapeutics to T1D subjects based on IMPACT further expand the application of such physiology‐based models.

Conflict of Interest

The authors declare no conflict of interest.

Supporting information

Supporting Information

Alizadehmojarad A. A., Yang S., Gong X., Strano M. S., Analysis of Glucose Responsive Glucagon Therapeutics using Computational Models of the Glucoregulatory System. Adv. Healthcare Mater. 2024, 13, 2401410. 10.1002/adhm.202401410

Data Availability Statement

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