A Computational Glucoregulatory Model of Liver and Glucagon for the Evaluation of Therapeutics

Abstract

Computational models of the glucoregulatory system constitute a powerful tool for preclinical evaluation and mechanistic insight into therapeutics. However, in the case of diabetes, there is a dearth of physiological models capable of accurately describing the hormone glucagon, which is important for the study and design of new classes of therapeutics, such as glucose-responsive glucagon (GRG). In this work, we construct a physiological compartment model, IMPACT 2.0, which integrates a refined liver submodel and explicit whole-body glucagon kinetics. Key mechanistic enhancements include glucose transporter dynamics, receptor binding, and hepatic glycogen metabolism, allowing for the improved prediction of glucose excursions in response to both insulin- and glucagon-based therapeutics. Model validation against experimental data from healthy and diabetic rats demonstrated accurate glucose predictions following insulin and glucagon administration. Sensitivity analysis was used to evaluate our model’s identifiability in the case of insulin or glucagon subcutaneous injections. By comparing diabetic and healthy model fits, we found that 16 of the 37 fitting parameters were significantly different between the health states. Additionally, we applied IMPACT 2.0 to evaluate a recently developed GRG based on controlled release via a microneedle patch, illustrating its utility in mechanistic drug design and bridging in vitro characterization with physiological outcomes. By offering a physiologically detailed and validated framework for glucagon and liver metabolism, IMPACT 2.0 is an improved pharmacokinetic and pharmacodynamic model that will be valuable for accelerating drug discovery, optimizing GRG formulations, and informing the design of closed-loop insulin and glucagon therapeutics.

Diabetes mellitus is a prevalent chronic disease worldwide, imposing substantial societal and economic burdens. The dysregulation of the glucoregulatory system is responsible for the disease effects, specifically insulin production in Type 1 diabetes mellitus (T1D) and insulin sensitivity in type 2 diabetes mellitus (T2D). As a result, the physiological and pathophysiological roles of key glucoregulatory hormones, insulin and glucagon, were extensively studied for decades. Traditionally, insulin is viewed as a blood-glucose-lowering hormone, while glucagon counteracts this effect by raising blood glucose levels. Although these hormones have been effectively utilized for glucose management, their inherently narrow therapeutic windowheavily affected by factors such as diet, physical activity, and comorbiditiesposes significant risks of morbidity and mortality. , Developing effective diabetes treatments is challenging due to the complexity of glucose regulation and the high costs of clinical trials. While preclinical animal models are more affordable than human trials, species differences can impact their translatability. Murine models are more cost-effective but less translatable, whereas porcine, canine, and primate models offer better translatability but are more challenging to implement. In silico glucoregulatory models offer a promising alternative for predicting therapeutic effects and trial outcomes, facilitating faster and more cost-effective drug development. While promising, no model currently provides a comprehensive physiological description of the body, coupled with a mechanistic liver component. Especially in the case of glucagon, an increasingly important therapeutic target, we found that such a description is necessary and address these limitations in this work.

Multiple advanced therapeutic approaches have emerged in recent years to mitigate the therapeutically related risks in T1D therapy. Continuous glucose monitoring (CGM) and artificial pancreas were developed to manage treatment with minimal patient intervention. , These solutions are especially useful in T1D, where precise insulin and meal management are involved. − An important diabetic care metric of these management strategies is called “time in range” (TIR), a means to quantify the success of short time-scale mitigation of glucose level excursions outside the target 70–180 mg/dL. Medicinally, therapeutics that modulate effect in response to glucose concentration changes to increase TIR have been of significant interest. , Insulin analogues can be conjugated to glucose-responsive moieties to significantly extend the therapeutic window in vivo. − The integration of glucose-responsive materials with controlled release technologies has achieved a similar effect. − In addition to insulin-based therapeutics, glucagon has recently been explored for combination therapy to reduce fatal hypoglycemic risk. , The counteracting glucagon has been historically underutilized due to its instability in formulations and its blood glucose increasing effect. , However, since the FDA approval of the ultrastable glucagon analogue, dasiglucagon in 2021, interest in glucagon therapeutics has grown as a means to reduce hypoglycemic risk specifically associated with insulin treatment, , specifically glucose-responsive glucagon (GRG) to address the above therapeutic design considerations. −

Over the years, a number of diabetes simulators were developed. Notably, the UVA/Padova type 1 diabetes simulator was the first one approved by the FDA, providing a framework for designing and testing closed-loop hormone controllers. In 2004, Hovorka et al. developed a relatively simple yet powerful predictive control model designed to prevent hypoglycemia for T1D patients. In contrast to these two computational models focusing on simplified control algorithms to describe glucose metabolism, the Sorensen model involved a whole-body compartment network based on the physiology of both healthy and diabetic subjects. , Building upon this foundation, our group developed and refined a glucoregulatory compartment model based on the Sorensen approach. We initially incorporated a tissue-implantable insulin sensor and tested a computational framework applicable for designing and simulating the glucose-responsive insulins (GRIs). , Next, we further extended our physiological model to be applicable not only to humans but also to rodents (PAMERAH) and minipigs (IMPACT 1.0). The minipig model further enabled us to investigate the mechanisms underlying the clinical trial failure of GRI MK-2640. Ultimately, we demonstrated that this computational approach allowed for a quantitative evaluation of different GRI designs within their respective chemical design spaces.

While these models accurately captured glucose-insulin dynamics across various species, it was evident that the Sorensen model’s treatment of glucagon was insufficient to describe the detailed pharmacokinetics (PK) and pharmacodynamics (PD) of exogenous glucagon therapeutics. Traditionally, computational compartment models of glucoregulation have involved only insulin and few compartments, simplifying complex physiology in a manner that allowed for model fits to assays and data available at the time. − Sufficient for specific CGM and glucose controller applications, these approaches are common today. These methods often involved choosing an appropriate mathematical functional form as the best estimate of observed glucose dynamics. While the Sorensen model introduced extensive physiological detail for insulin, its glucagon treatment involved only a single ordinary differential equation (ODE), and the level of detail for hepatic glucose uptake and release did not extend to the cellular level. Liu and Tang (2008) introduced a model where systemic transport was simplified, but specific detail was paid to active components within the hepatocyte. Important considerations included insulin- and glucagon-receptor binding, transmembrane glucose transport, and glycogen synthesis. Other examples include König et al. (2012), where extensive detail of cellular metabolic pathways were the focus, and Schaller et al. (2013), where intercompartmental biological signals were considered. When selecting model components for this work, we prioritized physiological accuracy, component significance, and balance between model complexity and identifiability.

In this work, we construct IMPACT 2.0, a multiscale glucoregulatory model of the liver, with the goal of improving the description of glucagon and insulin dynamics in compartment models. The liver is modeled by incorporating insulin, glucose, and glucagon-receptor binding, along with downstream signaling pathways that regulate glycogen synthase (GS) and glycogen phosphorylase (GP) activity. The model also accounts for endocytosis, degradation, and conversion of glucose to lactate. This work significantly expands upon our previous efforts with a particular emphasis on establishing a physiologically accurate treatment of glucagon, similar to existing models of insulin. Glucagon is treated as a whole-body circulating species, and the liver submodel has been reconstructed to incorporate physiological details down to the cellular and protein levels for both insulin and glucagon. This represents a significant improvement over our previous physiological model, where glucagon dynamics were described using a single-compartment subsystem with fitted response functions. Key biochemical mechanisms were also incorporated, including glucose transporter dynamics within hepatocytes, receptor binding and degradation pathways, and well-characterized feedback relationships among key modeled analytes. The identifiability, or the ability to uniquely estimate model parameters from observed data, of the 42-parameter liver submodel was assessed using sensitivity analysis (SA), allowing for the determination of parameter significance in response to subcutaneous insulin or glucagon administration. When integrated into our existing physiological compartment model, the liver submodel accurately captured literature-reported experimental glucose dynamics in both healthy and diabetic rats. As a final validation, the model was applied to simulate a hydrogel-based GRG candidate tested in rats, demonstrating its utility in therapeutic simulations. The advancements presented in this work enhance the physiological accuracy of glucoregulatory models and enable a more comprehensive description of glucagon dynamics, improving their applicability in drug development and glucose regulation studies.

Research Design and Methods

Compartment Model Structure

The overall structure of our physiological compartment model of the glucoregulatory system has been detailed in previous work and is summarized briefly here (terminology used in this work is defined in aggregate in Table ). The organism was divided into seven well-mixed compartments that can be considered continuous stirred-tank reactors (Figure ). These major organ systems are considered to cover the majority of the volume of distribution of the three key analytes in our model: glucose, insulin, and glucagon. Where needed, compartments can be separated into subdivisions (i.e., vascular and interstitial) to introduce transport barriers between convective pumping and tissue uptake. As an example, generic compartment ODEs that contain both vascular (subscript v) and interstitial (subscript i) subcompartments for a species concentration C would be described by the equations:

$$V_{\mathrm{C}-\mathrm{v}}\frac{\mathrm{d}{C}_{\mathrm{v}}}{\mathrm{d}t}=Q(C_{\mathrm{h}}-C_{\mathrm{v}})-\frac{V_i(C_{\mathrm{v}}-C_{\mathrm{i}})}{T}+R_{\mathrm{v}}$$ 1

$$V_{\mathrm{C}-\mathrm{i}}\frac{\mathrm{d}{C}_{\mathrm{i}}}{\mathrm{d}t}=\frac{V_{\mathrm{i}}(C_{\mathrm{v}}-C_{\mathrm{i}})}{T}+R_{\mathrm{i}}$$ 2

where V denotes compartment volume, C _h represents the species concentration in the heart and lungs compartment (which can be another inflow compartment), Q denotes the blood flow rate through the organ, T is the characteristic transport time between the vascular and interstitial subcompartments, R represents any compartment specific sources or sinks of the species described. For organs with rapid transcapillary equilibrium, interstitial subcompartments can be removed.

1. Terminology and Definitions.

PKs	the body’s effects on an administered therapeutic, including transport (absorption, distribution, and excretion) and conversion (metabolism)
PDs	a drug’s effects on the body related to how drug concentration at the site of action is correlated to the observed physiological response (e.g., glucose uptake or release)
INS/I	insulin both native and injected were assumed to perform similarly
GLN/N	glucagon both native and injected were assumed to perform similarly
GLY/Y	glycogen
GRG/GRI	glucose-responsive glucagon/glucose-responsive insulin
sub-model	set of subcompartments that describes the PK and PD within a specific bodily organ provided with only arterial input and venous output
sub-compartment	a compartment component within an organ that is connected via transport to others (e.g., vascular, interstitial, and cellular)
subsystem	an aggregate term for all model differential equations that tracks a particular analyte (e.g., glucose, insulin, glucagon, and glycogen)
subcutaneous (SC)	used as in the context of injections of therapeutics into the adipose interstitial tissue, creating a fluid volume from which drug uptake occurs
GS/GP	GS and GP

1.

Block diagram of model compartments with definitions. A block diagram is displayed with compartments within the model organized spatially in a physiological manner, where each organ is interconnected via arterial supply (red) and venous drainage (blue). The liver is the only compartment with both central and gut arterial flow. The abbreviation C was used for insulin and glucagon as the two species were considered identically within the model. PV and PI are abbreviated forms of peripheral vascular and peripheral interstitial subcompartments, respectively. The first capital letter of subscripts denotes each organ submodel. Designations and abbreviation of the organ submodels were also shown. Subcutaneous injections were performed via a subcompartment connected only to the adipose interstitial subcompartment.

Other key features of the model include: (1) interstitial transport of glucose within the brain, (2) treatment of subcutaneous injections as separate compartments, when introduced, connected to the interstitial adipose tissue, and (3) description of the pancreas within the liver compartment, capturing its ability to modulate key hormones, including insulin and glucagon. The entire ODE system was simultaneously solved using numerical methods in MATLAB (ode15s, MathWorks, Inc., Natick, MA) (see Supporting Information for details on numerical optimization).

IMPACT 2.0 Treatment of Glucagon and the Liver Compartment

Previous versions of our model retained the Sorensen approach to glucagon, where the entire body’s glucagon was treated as a single compartment. The effect of glucagon on hepatic glucose dynamics was then fitted by using experimental data via the tanh functional form following normalization. This allowed for the collation of data from multiple studies to generate the responsivity curves of the glucagon concentration as a function of glucose or insulin levels.

IMPACT 2.0 revises the treatment of glucagon as an independent circulating species similar to that of insulin. Due to the lack of specific experimental data, we reasonably assume that the volume of distribution of glucagon is similar to that of insulin. Similarly, glucagon subcutaneous injection was implemented with the same structure as insulin without the hexamer-dimer interconversion. Transport coefficients such as injection absorption and subcompartment transport were left as fitting parameters due to lack of specific experimental data. The liver submodel in IMPACT 2.0 was designed to take into account important physiological complexity without over parametrization (the full set of liver submodel equations can be found in Supporting Information).

One of our initial considerations for the liver submodel was a two-compartment approach, consisting of vascular and cellular components connected by a transport coefficient. However, we quickly found that model fitting did not converge satisfactorily on the animal data sets, regardless of the algorithm or initial conditions used. Specifically, the model returned to equilibrium too quickly compared to the animal data (Supporting Information, Figure S2). This suggests that a third subcompartment is necessary to introduce an appropriate delay in transport, which aligns with the reasoning that molecules require time to pass through the interstitium between blood vessels and cells. Thus, the liver submodel contains the exchange of all 3 species of interest transported between vascular, interstitial, and intracellular subcompartments (Figure A).

2.

Liver submodel and the hepatic control of glucose. (A) A schematic is shown of the components of the liver submodel, including the subcompartments: vascular, interstitial, and intracellular. Within each subcompartment, transport mechanisms, degradation, and biochemical effects that were considered within IMPACT 2.0 are diagrammed. (B) A mechanistic diagram of hepatocyte that includes the transport of glucose, and the interconversion between glucose and glycogen (solid lines). Regulatory effects of glucose, glucagon, and insulin on the interconversion between glucose and glycogen were shown (dashed lines), with (+) or (−) detailing either a positive or negative effect on the depicted type of action.

Glucagon and Insulin Liver Model Components

Within the liver submodel, both insulin and glucagon are treated identically as receptor-binding effector molecules. They extravasate from the blood into the interstitial and bind to surface receptors, which then result in both subsequent biochemical signaling effects as well as their degradation. The glucagon equations are detailed below for the reader:

$$V_{\mathrm{N}-\mathrm{L}\mathrm{V}}\frac{\mathrm{d}{N}_{\mathrm{L}\mathrm{V}}}{\mathrm{d}t}=Q_{\mathrm{H}}^{\mathrm{N}}{N}_{\mathrm{H}}+Q_{\mathrm{G}}^{\mathrm{N}}{N}_{\mathrm{G}}-Q_{\mathrm{L}\mathrm{V}}^{\mathrm{G}}{N}_{\mathrm{L}\mathrm{V}}-\frac{V_{\mathrm{N}-\mathrm{L}\mathrm{I}}}{T_{\mathrm{N}-\mathrm{L}}}(N_{\mathrm{L}\mathrm{V}}-N_{\mathrm{L}\mathrm{I}})+r_{\mathrm{P}\mathrm{N}\mathrm{R}}$$ 3

$$V_{\mathrm{N}-\mathrm{L}\mathrm{I}}\frac{\mathrm{d}{N}_{\mathrm{L}\mathrm{I}}}{\mathrm{d}t}=\frac{V_{\mathrm{N}-\mathrm{L}\mathrm{I}}}{T_{\mathrm{N}-\mathrm{L}}}(N_{\mathrm{L}\mathrm{V}}-N_{\mathrm{L}\mathrm{I}})-k_{\mathrm{a}}^{\mathrm{N}}{G}_{\mathrm{L}\mathrm{I}}(R_{\mathrm{N}}^0-B_{\mathrm{N}})$$ 4

$$V_{\mathrm{N}-\mathrm{L}\mathrm{C}}\frac{\mathrm{d}{B}_{\mathrm{N}}}{\mathrm{d}t}=k_{\mathrm{a}}^{\mathrm{N}}{G}_{\mathrm{L}\mathrm{I}}(R_{\mathrm{N}}^0-B_{\mathrm{N}})-k_{\mathrm{deg}}^{\mathrm{N}}{B}_{\mathrm{N}}$$ 5

where compartment designations are LV = liver vascular, LI = liver interstitial, LC = liver cellular, H = heart and lung, Ggut. N designates glucagon, where glucagon bound (B) is tracked as an intracellular concentration. Q is blood flow; T is the interstitial time constant; and r _PNR is the rate of pancreatic glucagon release (which is vascularly attached to the liver). The receptor-binding process was enabled by a single binding coefficient k _a and the conservation of a total number of receptors R _N . We assume here that the number of receptors does not change within the course of an experiment for a specific fitted physiological state. Finally, the bound ligand was assumed to degrade at a first order rate of k _deg . Compared to previous models, the implementation of receptor binding as a major path of ligand degradation is more congruent with current understanding. The bound species concentration was then eventually used to correlate with downstream signaling effects. While being well aware that receptor binding and signaling is much more complex (including multiple cascading phosphorylation of enyzmes, multiple receptor interactions, unbinding, and recycling), we chose to limit the complexity of the model as to describe the PD phenomena given the data sets available and over-parametrization concerns.

Glucose Transport and Control Model Components

Similar to the hormones, glucose within the liver was also modeled as 3 compartments with transport in between. Similarly, the movement between vascular and interstitial subcompartments was accomplished via an interstitial time constant. Glucose moves from the interstitial space into the hepatocyte via the process of facilitated diffusion through the GLUT2 membrane transporter. It is trivial to show that the flux of glucose into the hepatocyte is correlated with the concentration across the membrane and the association constant with the membrane transporter in the following relationship (see Supporting Information for derivation):

$$J_{\mathrm{G}\mathrm{L}\mathrm{U}\mathrm{T}2}={n_{\mathrm{c}\mathrm{e}\mathrm{l}\mathrm{l}}J}_{\mathrm{t}\mathrm{m}}(\frac{G_{\mathrm{L}\mathrm{I}}}{G_{\mathrm{L}\mathrm{I}}+K_{\mathrm{m}‐\mathrm{G}\mathrm{L}\mathrm{U}\mathrm{T}2}}-\frac{G_{\mathrm{L}\mathrm{C}}}{G_{\mathrm{L}\mathrm{C}}+K_{\mathrm{m}‐\mathrm{G}\mathrm{L}\mathrm{U}\mathrm{T}2}})$$ 6

where K _m‑GLUT2 is the transporter associated coefficient. As the model considers all intracellular transport within the liver, the total flux is then the number of cells (n _cell) multiplied to the flux coefficient per cell (J _tm). Within the cell, two major processes were considered: glucose–glycogen interconversion and the glucose conversion to lactate, the latter of which we choose to model as a first order process controlled by receptor bound insulin. The liver glucose ODEs are then the following:

$$V_{\mathrm{G}-\mathrm{L}\mathrm{V}}\frac{\mathrm{d}{G}_{\mathrm{L}\mathrm{V}}}{\mathrm{d}t}=Q_{\mathrm{H}}^{\mathrm{G}}{G}_{\mathrm{H}}+Q_{\mathrm{G}}^{\mathrm{G}}{G}_{\mathrm{G}}-Q_{\mathrm{L}\mathrm{V}}^{\mathrm{G}}{G}_{\mathrm{L}\mathrm{V}}-\frac{V_{\mathrm{G}-\mathrm{L}\mathrm{I}}}{T_{\mathrm{G}-\mathrm{L}}}(G_{\mathrm{L}\mathrm{V}}-G_{\mathrm{L}\mathrm{I}})$$ 7

$$V_{\mathrm{G}-\mathrm{L}\mathrm{I}}\frac{\mathrm{d}{G}_{\mathrm{L}\mathrm{I}}}{\mathrm{d}t}=\frac{V_{\mathrm{G}-\mathrm{L}\mathrm{I}}}{T_{\mathrm{G}-\mathrm{L}}}(G_{\mathrm{L}\mathrm{V}}-G_{\mathrm{L}\mathrm{I}})-J_{\mathrm{G}\mathrm{L}\mathrm{U}\mathrm{T}2}$$ 8

$$V_{\mathrm{G}-\mathrm{L}\mathrm{C}}\frac{\mathrm{d}{G}_{\mathrm{L}\mathrm{C}}}{\mathrm{d}t}=J_{\mathrm{G}\mathrm{L}\mathrm{U}\mathrm{T}2}+n_{\mathrm{G}\mathrm{P}}{r}_{\mathrm{G}\mathrm{P}}-n_{\mathrm{G}\mathrm{S}}{r}_{\mathrm{G}\mathrm{S}}-k_{\mathrm{deg}‐\mathrm{l}\mathrm{a}\mathrm{c}}{B}_{\mathrm{I}}{G}_{\mathrm{L}\mathrm{C}}$$ 9

where k _deg‑lac controls the rate of lactate conversion and B _I is the receptor bound insulin conversion. In terms of glucose–glycogen interconversion, we choose to use the rate-limiting enzymes GS for the forward reaction and GP for the release of glucose. Thus, the total conversion rates for all liver cells within the model were the product of the individual enzyme production rates (r _GP, r _GS) and the total number of enzymatic units within the liver (n _GP, n _GS). Finally, a single ODE was used to track the amount of glycogen (species Y) within the liver, where the initialization amount to represent the organism’s fed state was a fitting parameter.

$$V_{\mathrm{Y}‐\mathrm{L}\mathrm{C}}\frac{\mathrm{d}Y}{\mathrm{d}t}=n_{\mathrm{G}\mathrm{S}}{r}_{\mathrm{G}\mathrm{S}}-n_{\mathrm{G}\mathrm{P}}{r}_{\mathrm{G}\mathrm{P}}$$ 10

Within the hepatocyte, a complex network of factors exists that influence the glucose–glycogen interconversion process. The major contributions of glucose, bound insulin, and bound glucagon were represented in IMPACT 2.0 as mathematical controls diagrammed in (Figure B), where (+) or (−) shows the correlation of signal concentration on the process direction. A decision was made here to model the phosphorylation of the GS and GP enzymes, as a control function multiplied to a Michaelis–Menten form of enzymatic rate was not able to capture the dynamics of this process (see Supporting Information for details). Biochemically within the cell, receptor binding results in a phosphorylation cascade that eventually achieves enzymatic modulation. Here, we choose to include only the last step of phosphorylation of the effector enzymes GS and GP. Thus, the GS equations consist of 2 ODEs for each phosphorylation state (the phosphorylated stated labeled with subscript p) and 1 equation for the rate calculation below:

$$\frac{\mathrm{d}\mathrm{G}{\mathrm{S}}_{\mathrm{p}}}{\mathrm{d}t}=k_{1\mathrm{s}}(\mathrm{G}\mathrm{S})f(B_{\mathrm{I}},B_{\mathrm{N}})-k{2}_{\mathrm{G}\mathrm{S}}(\mathrm{G}{\mathrm{S}}_{\mathrm{P}})B_{\mathrm{I}}$$ 11

$$f(B_{\mathrm{I}},B_{\mathrm{N}})=(1-\frac{{B_{\mathrm{I}}}^{n_{\mathrm{g}\mathrm{s}‐\mathrm{i}\mathrm{n}\mathrm{s}}}}{k{m}_{\mathrm{g}\mathrm{s}‐\mathrm{i}\mathrm{n}\mathrm{s}}+{B_{\mathrm{I}}}^{n_{\mathrm{g}\mathrm{s}‐\mathrm{i}\mathrm{n}\mathrm{s}}}})\left(\frac{k{3}_{\mathrm{g}\mathrm{s}‐\mathrm{g}\mathrm{l}\mathrm{n}}{B_{\mathrm{N}}}^{n_{\mathrm{g}\mathrm{s}‐\mathrm{g}\mathrm{l}\mathrm{n}}}}{k{m}_{\mathrm{g}\mathrm{s}‐\mathrm{g}\mathrm{l}\mathrm{n}}+{B_{\mathrm{N}}}^{n_{\mathrm{g}\mathrm{s}‐\mathrm{g}\mathrm{l}\mathrm{n}}}}\right)$$

$$\frac{\mathrm{d}\mathrm{G}\mathrm{S}}{\mathrm{d}t}=k{2}_{\mathrm{G}\mathrm{S}}(\mathrm{G}{\mathrm{S}}_{\mathrm{P}})B_{\mathrm{I}}-k_{1\mathrm{s}}(\mathrm{G}\mathrm{S})f(B_{\mathrm{I}},B_{\mathrm{N}})$$ 12

$$r_{\mathrm{G}\mathrm{S}}=\frac{V_{\max ‐\mathrm{G}\mathrm{S}}(\mathrm{G}\mathrm{S})G_{\mathrm{L}\mathrm{C}}}{(k{m}_{\mathrm{G}\mathrm{S}}+G_{\mathrm{L}\mathrm{C}})(1+\frac{k{m}_{\mathrm{g}\mathrm{l}\mathrm{c}‐\mathrm{g}\mathrm{s}‐\mathrm{a}\mathrm{c}\mathrm{t}}}{{G_{\mathrm{L}\mathrm{C}}}^{n_{\mathrm{g}\mathrm{s}‐\mathrm{g}\mathrm{l}\mathrm{c}}}})}$$ 13

The increase of GS phosphorylation was controlled by both insulin and glucagon binding in the functional form f(B _I, B _N), where the action of each signal was modeled via a Hill equation-like process. Dephosphorylation was first order and correlated with bound insulin only. For the rate of glucose production, only the contribution of the dephosphorylated GS was considered as a simplification. For the calculation of r _GS, the rate of glycogen synthesis, we assume a Michaelis–Menten form with the inclusion of allosteric promotion of glycogen synthesis via glucose to simulate the glucose-6-phosphate activation of GS.

Similarly for GP, we include the receptor binding actions of insulin and glucagon as well as glucose concentration in the control of glycogen synthesis in the following equations:

$$\frac{\mathrm{d}\mathrm{G}{\mathrm{P}}_{\mathrm{P}}}{\mathrm{d}t}=k{1}_{\mathrm{G}\mathrm{P}}(\mathrm{G}\mathrm{P})B_{\mathrm{N}}-k{2}_{\mathrm{G}\mathrm{P}}(\mathrm{G}{\mathrm{P}}_{\mathrm{p}})f_{\mathrm{g}\mathrm{l}\mathrm{n}}{f}_{\mathrm{g}\mathrm{l}\mathrm{c}}$$ 14

$$f_{\mathrm{g}\mathrm{l}\mathrm{n}}=\frac{1}{1+\left(\frac{B_{\mathrm{N}}}{k{d}_{\mathrm{g}\mathrm{p}‐\mathrm{g}\mathrm{l}\mathrm{n}}}\right)}$$

$$f_{\mathrm{g}\mathrm{l}\mathrm{c}}=1+{\left(\frac{G_{\mathrm{L}\mathrm{C}}}{k{d}_{\mathrm{g}\mathrm{p}‐\mathrm{g}\mathrm{l}\mathrm{c}}}\right)}^{n_{\mathrm{g}\mathrm{p}‐\mathrm{g}\mathrm{l}\mathrm{c}}}$$

$$\frac{\mathrm{d}\mathrm{G}\mathrm{P}}{\mathrm{d}t}=k{2}_{\mathrm{G}\mathrm{P}}(\mathrm{G}{\mathrm{P}}_{\mathrm{p}})f_{\mathrm{g}\mathrm{l}\mathrm{n}}{f}_{\mathrm{g}\mathrm{l}\mathrm{c}}-k{1}_{\mathrm{G}\mathrm{P}}(\mathrm{G}\mathrm{P})B_{\mathrm{N}}$$ 15

$$r_{\mathrm{G}\mathrm{P}}=\frac{V_{\max ‐\mathrm{G}\mathrm{P}}(\mathrm{G}{\mathrm{P}}_{\mathrm{p}})Y_{\mathrm{s}0}}{k{m}_{\mathrm{G}\mathrm{P}‐\mathrm{e}\mathrm{f}\mathrm{f}}+Y_{\mathrm{s}0}}$$ 16

$$k{m}_{\mathrm{G}\mathrm{P}‐\mathrm{e}\mathrm{f}\mathrm{f}}=k{m}_{\mathrm{G}\mathrm{P}}(1+\frac{G_{\mathrm{L}\mathrm{C}}^{n_{\mathrm{g}\mathrm{p}‐\mathrm{g}\mathrm{l}\mathrm{c}2}}}{k{m}_{\mathrm{g}\mathrm{p}‐\mathrm{g}\mathrm{l}\mathrm{c}}})$$

The GP enzyme is active in the phosphorylated form, which converts glycogen to glucose. Again, the hill-like equation functional form was used to modulate control effects. GP phosphorylation is increased by glucagon and decreased by glucose level increases. Finally, glucose also binds to the GP enzyme itself, which has been shown to effectively shift the enzymatic km. (A complete set of equations for the liver submodel is detailed in Supporting Information.)

Model Fitting Outcomes

Taken as a whole, following the removal of the empirical Sorensen liver submodel and introduction of the 3 subcompartment liver submodel system above, 19 additional model parameters were introduced. A comprehensive evaluation of model parameters determined that 37 parameters need to be fitted from the experimental data, either due to lack of clinical measurements or the ability to determine a reasonable estimate (see Supporting Information, Table S1, for parameter designation, values, and sources). Values of newly introduced parameters in the table served as a means to determine appropriate ranges in the fitting process. Fitting was primarily performed using the pattern search function in Matlab via the nups-mads algorithm (see Supporting Information for more information on model fitting algorithm parameters, and Figure S1 for comparison of algorithmic outcomes).

We chose rats as the target organism due to experimental data availability. Approaches in this work can be readily adapted to fit data from other organisms. Data for glucose measurements after subcutaneous injection of glucagon in healthy and diabetic rats was digitized from literature. , Data for glucose measurements after subcutaneous injection of insulin was obtained from our previously published work, Yang et al. 2020. These male Lewis rats were rendered diabetic via intraperitoneal streptozocin injections, with a diabetic state confirmed by blood glucose data.

Figure details the outcome of all of the model fits. For healthy, there was a single glucagon injection (3.13 × 10⁶ pg) and 6 insulin injections (1, 2, 3, 4, 6, and 8 μg). For diabetics, there was a single glucagon injection (2.7 × 10⁶ pg) and 8 insulin injections (3.3, 5, 10, 15, 20, 35, 50, and 60 μg). All IMPACT 2.0 simulations in this work assume a 300 g rat. The model fit loss function was the sum of the mean squared error between the mean value of each experimental data set time point and the model predictions, with the diabetic and healthy animals fitted separately. As a whole the model was able to capture all fitting data sets well, with an average per data point residual of 5 mg/dl for healthy, 10 mg/dl for diabetic, corresponding to 5% and 2.5% accuracy, respectively. We can conclude that the model has a high level of fidelity in predicting injection concentrations of insulin or glucagon near the fitted values.

3.

Model fits to experimental rat data. Model fitting was performed on experimental data of SC injections of insulin (at t = 0 min) into healthy and diabetic rats (n = number of animal subjects) with time-dependent blood glucose concentration changes measured afterward. Glucagon data sets were extracted from literature for heathy and diabetic rats separately. Error bars represent experimental data standard deviation, with sample number detailed above. Solid lines overlaid represented model fits to data sets of the corresponding color.

Of note, during the fitting process, particular attention was paid to the hill coefficients within the control of phosphorylation. These parameters were assumed to be integers between 1 and 4, representing the number of concurrent discrete molecules that can bind to affect a process. Following successful model fits of both healthy and diabetic states, it was discovered that 4/5 of the values were close to 1 and setting these values to 1 did not significantly affect fitting outcomes. On the remaining hill coefficient, n _gp‑glc, had a best fit value of 3. This outcome can be interpreted as 3 glucose molecules are needed to activate the dephosphorylation of GP, a known biophysical finding determined via experiment.

Discussion on PK Fitting

We defined PD data sets as those that injected insulin or glucagon and then measured subsequent glucose concentration changes, while a PK data set is the injection of a therapeutic and then measuring plasma concentrations of that therapeutic over time post injection. While not included in a part of the final model, one of our efforts also involved simultaneously fitting glucagon PK data in addition to insulin and glucagon PD data. The goal would have been to validate the model fidelity via glucagon injection measurements. Unfortunately, nearly all studies reported glucose concentration measurements following glucagon injection, not glucagon PK. This was especially true for rats, since the existing clinical trial pharmacokinetic (PK) data used for therapeutic approval pertains only to human patients. We attribute the paucity of such data to the availability of glucose measurements as compared to that of direct glucagon measurement assays. In the above study, the PK data was measured using liquid chromatography with tandem mass spectrometry. Other common assays include commercially available ELISA kits.

Of the glucagon PK data sets available in the literature, none were found for rats. We chose a dog study as an initial attempt (see Supporting Information, Figure S3). To appropriately scale injection amounts, we used the calculation of body surface area as a comparison between the model organisms. An injection concentration was chosen such that the PK and PD studies had similar orders of magnitude. While performing these model fits, it was discovered that while the PD data can be effectively captured, the PK data had significant differences with the model predictions, with predicted plasma GLN concentrations much higher than what was measured. We attribute this outcome to the difference between animal models and the shorter half-life of GLN, resulting in its various sinks within the body having more complex and significant effects. Finally, as glucagon PK/PD data sets become available for animal models, improving the injection submodel would be one of the first steps of future work.

Results and Discussion

As a demonstration of the appropriate treatment of species and compartments, example injections of experimental concentrations of insulin and glucagon are modeled in healthy rats in Figure . SC injection of both glucagon (left, 2.7 × 10⁶ pg) and insulin (right, 10 μg) was performed following a standard equilibration of the model to ensure equilibrium. Time-dependent concentrations of glucagon, insulin, and glucose were plotted (Figure , rows). Following glucagon injection, glucagon concentration was expectedly highest closest to the injection compartment within the adipose interstitium. As expected, blood glucose concentrations increased in response, with liver cellular glucose having the highest increase due to liver cells being the location of the direct conversion between glycogen and glucose. Insulin concentrations exhibited a delayed increase as the body glucose concentrations rose to levels necessitating a response. Similarly, insulin injection expectedly showed the opposite effect as glucagon, where glucose concentrations decrease following injections, triggering a glucagon release. Of note, in both injection cases, liver cellular glucose increases. This is reasonable, as both conversion to and from glycogen involve transient buildup within the hepatocytes.

4.

Model validation injections. Example subcutaneous injections were performed as a means to validate model fits, including glucagon (left, 2.7 × 10⁶ pg) and insulin (right, 10 μg) at time 0 min. The 3 rows of plots correspond to glucagon, insulin, and glucose concentrations in time, respectively, during the 200 min following injection. Particular compartments were displayed including: adipose interstitialdirectly connected to the injection compartment, renalwhere significant loss occurs, and liver cellularwhere glycogen storage happens.

Interestingly, we find that a single parameter set does not fit both diabetic and healthy control data sets, assuming that diabetes is simplified as a loss of insulin secretion. From a physiological standpoint, it was reasonable that the two states required different models, as the diabetic state also is a result of significant physiological modifications. Accordingly, each was fit separately, generating distinct parameter sets for comparison. Additionally, the differences in model equilibrium would not allow IMPACT 2.0 to accomplish a simultaneous fit, in contrast with the Sorensen approach with analyte normalization and concentration set points. To further study the differences between health states from a model point of view, fitted model coefficients were compared (Figure A). For ease of visualization across multiple orders of magnitude, fitted parameters for diabetic and healthy models were normalized to their means using the equation:

$$N_{\mathrm{p}\mathrm{a}\mathrm{r}\mathrm{a}\mathrm{m}}=(P_x-\frac{(P_{\mathrm{D}}+P_{\mathrm{H}})}{2})/\frac{(P_{\mathrm{D}}+P_{\mathrm{H}})}{2}$$ 17

where P _x is the parameter value to be normalized, P _D is the diabetic model fit for the same parameter, and P _H is the healthy model fit for the same parameter. A value higher in the diabetic fit shows a bar toward the right (red) and healthy bars toward the left (blue). Values less than 10% normalized fold change were not displayed and were interpreted as a minimal change, leaving 16 of the 37 fitting parameters displayed.

5.

Analysis of model fit parameters. The fitted parameters, a total of 37, were compared for diabetic and healthy rats (A). For ease of visualization across multiple orders of magnitude, fitted parameters for diabetic and healthy models were normalized to their means. Values less than 10% normalized fold change were not displayed. (B) Sensitivity indices were calculated for the same narrowed set of parameters for the model using T1D rat parameters subjected to SC injection of glucagon (B, top) or insulin (B, below), comparing Sobol (blue) and FAST (red) algorithms. Parameters definitions for ones with largest change: km _gs‑gln – GS; km _gp‑glc – GP; R _I liver insulin receptor concentration; k _deg liver insulin bound insulin degradation rate; k _a insulin receptor association constant; k _tmliver transport coefficient of glucose into hepatocytes; k _N‑sinkabsorption rate of glucagon SC injection depot (see Supporting Information or all parameter definitions).

Several observations can be made from the fitting parameter comparison (parameter designations noted in parentheses). The diabetic state has increased glucose transport across the hepatocyte membrane (P101), as well as increased insulin receptor association (P120) and degradation (P122). The diabetic GP has an increased k _m (P149), entailing a decreased ability to bind to and break down glycogen. The healthy state has an increased number of glucagon receptors (P124) and an increased number of glucagon sinks (P131). Additionally, the effectiveness of GS was decreased through its k _m (P140) and allosteric glucose binding (P141). Taken together, the fitting of the model predicted that the diabetic state has an increased need for insulin and that the glucose to glycogen interconversion was biased toward glycogen synthesis. It is also well-known that glucose transporter levels are modulated by diseased states.

Model Identification Using SA

To test the identifiability of our newly developed model, we applied two global SA (GSA) algorithms, including Sobol SA and Fourier amplitude sensitivity testing (FAST). , In the case of our model, with many parameters and temporal responses to drug injections, GSA can be computationally expensive. Thus, we first narrowed down the 37 model fitting parameters to the 11 most significantly different between healthy and T1D diabetic rat fits in Figure A. Sobol and FAST were performed and compared using the GSA toolbox in MATLAB.

SA is used to study the significance of model parameters, or variance of model predictions, thus assessing the reasonableness of model fits. In this case, the focus is on the response of the model system to an SC injection of interest. The model response was computed using a diabetic rat model treated with a SC injection of either 0.3 × 10⁶ pg glucagon or 3 μg insulin, starting with an initial blood glucose concentration of 411 mg/dL. The blood glucose concentration trajectories over the course of 5 h for these treatments using fitted parameters are presented in Figure S8. Sensitivity is evaluated using a cost function, where the blood glucose trajectory, G _H(t, p ₁, p ₂, p ₃, ..., p _k) at time t with a set of p _i parameters on the sampling space, is compared with the blood glucose concentration G _H(t), the fit outcome. The absolute value of this comparison is then used as the cost function in the GSA algorithms. The following equation describes such cost function C _SA:

$$C_{\mathrm{S}\mathrm{A}}=\sum _t|G_{\mathrm{H}}(t,p_1,p_2,p_3,...,p_{\mathrm{k}})-G_{\mathrm{H}}(t)|$$ 18

Variance-based GSA algorithms first define a parameter space using Monte Carlo sampling, where each sample consists of a set of selected parameters with values normalized in the [0,1] range. The unconditional variance of the model response across all samples is then computed. This represents the total variability of the model output without fixing any specific parameter values, capturing the general spread of the model response over all input variations. Next, the conditional variance of the response for each parameter value is computed. This measures how much the output varies when a particular parameter is fixed at a specific value while the other parameters are allowed to vary. The sensitivity index (SI) is determined as the ratio of the conditional variance to the unconditional variance, indicating the model’s sensitivity to each parameter. A higher SI suggests that variations in the parameter have a significant impact on the model response, while a lower SI suggests minimal impact. The computed SIs using these approaches can have high variance if a low number of Monte Carlo samples were used. We chose the number of samples to be 10⁶ after testing the stabilization of SIs by repeating SA for different numbers of samples between 10³ and 10⁶ (see Supporting Information for additional SA details and Figure S4 for SA analysis injection baselines).

The importance of each fitting parameter on the model response varies depending on the system’s state, including factors such as the species of interest, the health condition of the subject, the type of insulin or glucagon treatment, and the administration sites of therapeutics. These variations are expected, as the set of ODEs are solved simultaneously, with each equation having a different weight and impact on the model response. Our SA assesses this importance by computing first-order SIs in the case of T1D diabetic rats treated with insulin or glucagon injections into adipose tissue compartments. Figure B presents the first-order SIs of the 11 chosen fitting parameters for modeling diabetic rats treated with SC injections of glucagon (top) and insulin (bottom). As a sanity check for the SA calculations, a comparison of the SIs from the Sobol and FAST algorithms was shown, illustrating the consistency of the SIs between these two methods. We also found that the SIs differ when different injections are performed, which reasonably engage different components of the model.

In the case of SC glucagon injection in to the T1D rat, the top panel of Figure B suggests that the most important parameter is ${k_{\mathrm{t}\mathrm{m}}=n_{\mathrm{c}\mathrm{e}\mathrm{l}\mathrm{l}}J}_{\mathrm{t}\mathrm{m}}$ with a SI of ∼0.6. This parameter, which appears in eq , describes the transport of glucose across hepatocytes via GLUT2 transporters. This finding aligns with the condition of the T1D rat model, which lacks insulin secretion, and the fact that GLUT2 transporters regulate glucose uptake or release by hepatocytes independently of insulin. , The next two important factors include k _N‑sink and k _mgp‑glc with an SI of ∼0.2. k_N‑sink is a multiplication factor that modulates the rate at which injected glucagon is absorbed in the T1D rat model, thus directly affecting the result of glucagon injection. The k _mgp‑glc parameter modulates glycogen breakdown as shown in eqs and and is proportional to glucose production by hepatocytes.

SC insulin injection in the T1D rat model leads to a different set of sensitive parameters. Here, k _deg with a SI of ∼0.7 is the kinetic constant for total degradation of bound insulin in the liver. GS activity in the phosphorylated or active state is modulated by total bound glucagon (B _N) and insulin (B _I) concentrations in the liver, which, respectively, lead to a decrease and increase in glycogen synthesis shown in eq . Therefore, k _deg directly affects the total bound insulin concentration and the effects of insulin on glucose from the SC administration. k _mgs‑gln is one of the kinetic constants that changes GS phosphorylation, which, with both bound glucagon and insulin, modulates GS activity. Its significance with SI ∼ 0.2 can be due to it being directly multiplied to bound insulin in eq . The second reason is that T1D subjects’ liver is capable of producing glucagon, also present in eq . Thus, k _mgs‑gln regulates glucagon response following insulin injection.

Prediction of Microneedle Patch GRG Therapeutic Effect

Previously, our work in 2024 employed IMPACT 1.0 in the evaluation of published GRG constructs with laboratory in vitro and in vivo data. As the current version of IMPACT 2.0 was specifically fit to rat physiology, we chose the Toronto GRG from GhavamiNejad et al. 2019 as an example therapeutic where the experimental data is species consistent. Briefly, the Toronto GRG construct consists of a microneedle patch with embedded microgel particles (Figure A). Each hydrogel particle was constructed with polymers that bind to glucose and self-cross-link when glucose concentration is low, resulting in shrinking. When loaded with glucagon, the change in the hydrogel geometry results in glucagon release.

6.

Prediction of cMN patch GRG experimental response to insulin challenge in STZ-induced diabetic rats. (A) A schematic illustrates a GRG device incorporating hydrogel-bound glucagon into a cMN patch, which releases glucagon in response to low local glucose concentrations via polymer cross-linking and shrinkage (A) reproduced with permission from Alizadehmojarad et al. Copyright 2024 John Wiley and Sons. GhavamiNejad et al. experimentally evaluated the ability of the cMN patch to prevent hypoglycemia in T1D rats over a 3 h period. Two groups of rats were tested: one treated with a glucagon-loaded cMN patch (yellow circles) and a control group with a sham patch (purple marks). The patches were applied at t = 0 min, followed by a 2 IU/kg subcutaneous recombinant insulin injection at t = 30 min to induce hypoglycemia. The glucagon-loaded cMN patch successfully maintained blood glucose levels above 90 mg/dL for at least 2.5 h post insulin injection, whereas the sham-treated group experienced severe hypoglycemia (black dotted line). (B) Model implementation of GhavamiNejad et al.’s experiment was performed for both the glucagon-loaded patch (blue line) and sham patch (orange line) conditions. Model agreement was further examined by adjusting insulin injection amounts to simulate the experimental effect of the challenge injection (C) and by modifying glucagon loading to account for differences in glucagon response across organisms (D).

The prediction capability of IMPACT 2.0 regarding the physiological hepatic response of STZ-induced diabetic rats was explored by recreating the experiment of GhavamiNejad et al. in the model, with parameters and formulation outlined by Alizadehmojarad et al., where the GRG construct was represented by first order reversible kinetics, with coefficients dependent on the local glucose concentration. The experiment consisted of the composite microneedle (cMN) GRG patch administration to the rodent at t = 0, followed by an insulin challenge of 2 IU/kg at t = 30 min to induce hypoglycemia. The goal of the experiment was to assess the device’s ability to prevent hypoglycemia following an insulin challenge. First, the experiment was modeled using the same insulin dose and patch drug loading, which was estimated from the in vitro leaching experiments (Figure B). To better match experimental values, we next increased the insulin dosage in the modeled experiment such that the model’s prediction of the insulin response at the lowest glucose level more closely resembles that of the experimental data (Figure C). Lastly, both insulin dosage and the amount of glucagon loading were adjusted in an attempt to further improve the agreement between model predictions and the in vivo experimental data (Figure D).

For a GRG in which effectiveness is modulated by glucose concentration, an initial release of glucagon is expected upon implantation, which would trigger a corresponding increase in blood glucose. This increase immediately following time = 0 min and before insulin injection was observed in the model prediction but not in the experimental data. Then it follows that an increase in glucagon loading in the GRG patch would show an increase in glucose rise. To the contrary, there was no initial blood glucose response increase as a consequence of the GRG administration in the experimental data. We attribute this difference to experimental drug losses in the injection or uptake process of the in vivo experiment. It is also very likely that the dynamics of glucagon release in the in vitro fitted model do not capture the patch’s actual in vivo behavior. Additionally, increasing insulin dosage results in further disagreement between the model and experimental data in both the control- and loaded-patch results. Similarly, the rise in blood glucose concentration following insulin injection occurred significantly earlier in the model prediction. Both of these differences suggest the disparity of glucagon sinks between the experimental animal and IMPACT2.0. Finally, the model required significantly more insulin to produce the same glucose decrease (approximately 4 IU/kg, compared to 2 IU/kg as indicated in the experiment) to induce hypoglycemia, suggesting a lower sensitivity to insulin than observed in the test subjects (Figure D).

Conclusions and Outlook

In this work, we developed an in silico physiological model of the glucoregulatory system built upon a compartment transport framework. The model integrates detailed glucagon PK/PD alongside insulin, allowing for the prediction of outcomes involving both therapeutic administrations. A key advancement is the liver submodel, which significantly improves upon our previous work by capturing both multiscale transport of key species (glucose, insulin, and glucagon) and the subcellular interconversion between glucose and glycogen. The inclusion of major signaling controls and feedback mechanisms provides a robust framework for in-depth glucoregulation analysis.

In constructing IMPACT 2.0, we prioritized physiological accuracy in hepatic glucose regulation by incorporating: (1) GLUT2-mediated glucose transport, (2) receptor binding and ligand degradation, (3) rate-limiting enzyme phosphorylation, and (4) key regulatory mechanisms within the phosphorylation cascade. The model successfully balances physiological simplification with sufficient mechanistic detail, allowing it to capture glucose dynamics in response to insulin and glucagon injections in both healthy and diabetic individuals. Additionally, by examining parameter differences between healthy and diabetic model fits, we gained insight into the underlying physiological changes.

To assess the parameter significance, Sobol and FAST SA tools were applied to a T1D rat model treated with SC injections of glucagon or insulin. Results indicated that for glucagon injections, the parameter associated with GLUT2 transporters had the highest SI, whereas for insulin injections, the total bound insulin concentration in the liver was the most influential parameter. Furthermore, we demonstrated IMPACT 2.0’s ability to simulate and predict the effects of a GRG. Model predictions revealed significant differences between in vivo and in vitro outcomes, underscoring the value of simulation tools in drug design and validation.

IMPACT 2.0 builds upon a modular physiological framework developed through successive iterations of our earlier models, PAMERAH and IMPACT 1.0. These models represent a stepwise advancement in physiological compartment modeling, with each version expanding capabilities while preserving physiological fidelity. PAMERAH extended our 2017 human models by incorporating the corresponding rodent models to support translational studies. IMPACT 1.0 further advanced the framework by adapting it for minipigs and applying it to simulate the first clinical trial of GRI, MK-2640. IMPACT 2.0 introduced a significantly refined liver module and added a physiological glucagon component to complement the existing insulin module. Importantly, these updates enable the framework to support glucagon-based therapeutics while maintaining insulin modeling capabilities.

Throughout the IMPACT 2.0 development process, we identified several components that require further investigation. One key aspect is the injection submodel, where we currently assume a similar mechanistic treatment for insulin and glucagon. Differences between model predictions and PK measurements suggest that improvements in the injection submodel could enhance predictive accuracy. However, refining these submodels would require high-quality and high-quantity PK data, along with detailed mechanistic studies to account for formulation-specific effects and uptake dynamics. Future iterations should also differentiate between the SC, IV, and IP administration routes to enhance accuracy. Additionally, the nutrient uptake was not explicitly modeled in IMPACT 2.0. Our previous work incorporated glucose inputs similar to OGTT or IVGTT. However, to better simulate real-world patient risk and metabolic dynamics, future iterations should include gut physiology, nutrient transport, and metabolism, enabling more comprehensive modeling of day-to-day glucose fluctuations in response to meals.

Expanding IMPACT 2.0 to model therapeutic outcomes across different speciesincluding dogs, minipigs, and humansis a key goal. While this work focuses on rats, the framework is designed for scalability. The detailed incorporation of glucagon, on equal footing with insulin, presents exciting opportunities to study their interplay. However, achieving accurate cross-species predictions will require high-quality data sets, particularly for glucagon therapeutics. Finally, as shown in our GRG example, care must be taken to develop drug construct-specific therapeutic models as an important component to accurately capture their in vivo performance. With these considerations, IMPACT 2.0 will become an important in silico tool for improving therapeutic development cycles and eventual patient safety.

The Supporting Information is available free of charge at https://pubs.acs.org/doi/10.1021/acsptsci.5c00177.

• Hepatocyte membrane GLUT2 glucose transporter equations; outlines of the numerical optimization and model fitting approach; SA methodology; comparison of different model fitting algorithms; effect of the number of liver subcompartments on model fit; PK fitting attempts and outcomes; standard injection baselines used in SA; alternative implementation of a control function for hormonal glucose regulation; comprehensive table of parameter designations, definitions, values, and references; and the full set of liver submodel equations (PDF)

X.G. and M.S.S. conceived the mathematical model. X.G. generated code, analyzed data, and wrote the manuscript. A.A.A. aided with SA and associated text. S.Y. edited and wrote manuscript sections. M.M. performed the Toronto GRG analysis. All authors commented on, edited, and have given approval to the final version of the manuscript.

This work was supported by the National Institutes of Health (1R01DK127761) and the Helmsley Charitable Trust (#1902-03727 to MPI team M.S.S., A.C., and M.A.W.). The authors also acknowledge a grant from the Helmsley Charitable Trust to M.S.S. at MIT, no. 2202-05781, for support of IMPACT 2.0 development and refinement.

The authors declare no competing financial interest.