Sorensen vs. Bergman vs. Hovorka: A Comprehensive Performance Analysis of Diabetes Metabolic Models

Abstract

This article provides a detailed comparative analysis of three foundational mathematical models of glucose-insulin dynamics: the Sorensen, Bergman Minimal, and Hovorka models. Tailored for researchers, scientists, and drug development professionals, it explores the physiological foundations, structural complexity, and specific applications of each model in diabetes research and therapy development. The scope ranges from foundational principles and methodological uses in artificial pancreas algorithms and in-silico trials to troubleshooting common implementation challenges and validation through regulatory acceptance and clinical performance metrics. The analysis synthesizes key trade-offs between physiological fidelity and practical utility, offering insights for model selection in biomedical innovation.

Core Architectures: Deconstructing the Physiological Foundations of Sorensen, Bergman, and Hovorka Models

Mathematical models of glucose-insulin dynamics are indispensable tools in diabetes research, supporting tasks ranging from personalized treatment planning to the preclinical testing of artificial pancreas algorithms. Among the most cited models, the Bergman Minimal Model, the Hovorka Model, and the Sorensen Model represent a spectrum of design philosophies balancing simplicity against physiological detail. The Sorensen model, developed by Thomas J. Sorensen in his 1978 thesis, stands as perhaps the most complex and physiologically detailed compartmental model of the human metabolic system. Its core strength lies in a multi-compartmental architecture that explicitly represents key organs and their interactions, moving beyond lumped-parameter approaches to provide a more holistic simulation of whole-body physiology. This guide provides a detailed, objective comparison of these three foundational models, focusing on their structural basis, appropriate applications, and performance as evidenced by experimental and in-silico studies, equipping researchers with the data needed to select the optimal model for their specific investigative goals.

Model Architectures: A Comparative Analysis of Core Structures

The Bergman, Hovorka, and Sorensen models were conceived for different primary purposes, which is directly reflected in their underlying structures and levels of biological abstraction.

• The Bergman Minimal Model: Originally designed for the interpretation of intravenous glucose tolerance test (IVGTT) data, this model employs a simple structure with two or three differential equations. It focuses on estimating key parameters like insulin sensitivity and is not intended to simulate comprehensive daily glucose dynamics [1] [2].

• The Hovorka Model: This model strikes a balance between complexity and practicality. It typically comprises eight differential equations organized into interconnected subsystems for glucose, insulin, and insulin action. Its design supports in-silico testing of diabetes treatments and artificial pancreas algorithms, making it a popular choice for control-oriented applications [3] [4].

• The Sorensen Model: This is a high-fidelity, physiologically based model that divides the body into distinct organ compartments. The revised version of the model is represented by 22 nonlinear differential equations and incorporates about 135 parameters to describe glucose and insulin dynamics in the brain, heart and lungs, liver, gut, kidney, and periphery [5]. Its compartmental structure aims to mirror the actual human body, providing a platform to investigate the specific metabolic role of different organs.

Table 1: Fundamental Architectural Comparison of the Three Primary Glucose-Insulin Models.

Feature	Bergman Minimal Model	Hovorka Model	Sorensen Model
Primary Purpose	IVGTT analysis, parameter estimation	In-silico treatment testing, AP design	Physiological simulation, virtual patient studies
Model Scope	Glucose-insulin dynamics	Glucose-insulin dynamics & insulin action	Whole-body glucose-insulin-glucagon physiology
Number of Compartments / Equations	2-3 equations [2]	8 equations [4]	22 equations, 14+ compartments [5]
Number of Parameters	~10	~20	~135 [5]
Organ-Specific Compartments	No	No	Yes (Brain, Liver, Gut, Kidney, etc.)
Meal Absorption Model	Not included	Integrated meal model [4]	Added gastrointestinal tract (in revised model) [5]
Incretin Effect	No	No	No (a known limitation) [5]

The following diagram illustrates the core structural philosophy of the Sorensen model, highlighting its multi-compartmental nature and the interactions between key physiological subsystems.

Performance and Validation: Insights from Experimental and In-Silico Data

The performance of a model is intrinsically tied to its purpose. While simpler models excel at parameter estimation, high-fidelity models like Sorensen's provide unparalleled insights into organ-level physiology.

Performance in Physiological Simulation and Model Fidelity

The Sorensen model's detailed compartmentalization allows it to simulate organ-specific phenomena, such as the liver's control over endogenous glucose production and the kidney's role in glucose clearance. However, this high fidelity comes with documented challenges. A 2020 re-implementation and revision of the model identified and corrected several imprecisions in the original equations and parameter values that had been propagated in subsequent research. These errors affected the simulation of kidney glucose excretion, initial condition equilibrium, and insulin secretion [5]. The revised model also supplemented the original with a missing gastrointestinal glucose absorption component, making it more suitable for simulating oral meal intake [5]. Despite these improvements, known limitations remain, such as the model's inability to fully capture the incretin effect—the potentiation of insulin secretion after oral glucose intake—which was bypassed in the original work by empirically estimating pancreatic insulin output [5].

Performance in Control Algorithm Development

The complexity of the Sorensen model, with its 22 differential equations and ~135 parameters, has limited its widespread use in the development of control algorithms for the artificial pancreas [5]. In contrast, the Hovorka model has been extensively employed for this purpose. For instance, a 2024 study utilized an improved Hovorka model with an enhanced Model Predictive Control (eMPC) algorithm to regulate blood glucose in virtual type 1 diabetes patients. The in-silico results showed promising performance, with time-in-target-range achieved at 79.59%, 87.76%, and 71.43% for three different virtual patients [3]. This demonstrates the Hovorka model's practical utility in control applications where computational efficiency and a balance of detail are required.

Table 2: Comparative Model Performance in Different Application Scenarios.

Application Scenario	Bergman Minimal Model	Hovorka Model	Sorensen Model
IVGTT Parameter Estimation	Excellent (designed for this) [2]	Not Primary Purpose	Possible, but overly complex
Simulating Daily Glycemic Variability	Not suitable	Good (validated in studies) [3]	Excellent (high physiological detail) [5]
In-Silico Preclinical Trial (e.g., AP Control)	Not suitable	Excellent (e.g., 70-88% TITR in-silico) [3]	Limited use due to computational load [5]
Studying Organ-Specific Metabolic Defects	Not possible	Limited	Excellent (unique strength) [5]
Meal Response Simulation (Oral)	Not applicable	Integrated meal model [4]	Requires added GI tract model [5]
Computational Efficiency	Very High	Moderate	Low

Successfully implementing and utilizing the Sorensen model requires a specific set of computational tools and resources. The following table details key "research reagent solutions" for scientists embarking on projects involving this complex model.

Table 3: Essential Toolkit for Sorensen Model Implementation and Research.

Tool / Resource	Function / Purpose	Example / Note
Revised Model Code	Provides a corrected, validated starting point for simulation.	The CNR-IASI BioMatLab online implementation (MATLAB code) corrects original errors and includes the GI tract [5].
Numerical ODE Solver	Solves the system of 22 nonlinear ordinary differential equations.	Stiff solvers like MATLAB's ode15s are often necessary due to the model's multi-scale dynamics.
Parameter Estimation Algorithm	Estimates patient-specific parameters from clinical data.	Nonlinear optimization methods (e.g., Sequential Quadratic Programming) are used to fit model parameters [6].
Virtual Patient Data	Enables in-silico testing and validation of hypotheses.	Synthetic data generated from the model itself can be used for algorithm testing and fault-injection studies [7].
Clinical Validation Data	Used to calibrate and validate the model's predictions.	Data from IVGTT, IVITT, and continuous infusion studies, as originally used by Sorensen [5].
Sensitivity Analysis Toolset	Identifies which parameters most significantly impact model outputs.	Crucial for understanding model behavior and prioritizing parameter estimation for personalization.

Experimental Protocols: Key Methodologies for Model Application

To ensure reproducible and meaningful results, researchers should adhere to well-defined experimental protocols when working with high-fidelity models. The following workflows are central to the application and validation of the Sorensen model.

Protocol for Model Implementation and Validation

This workflow outlines the critical steps for correctly implementing the Sorensen model and validating its performance against classical physiological experiments.

Protocol for Generating and Using Virtual Patient Data

A common application of physiological models is the creation of in-silico cohorts for testing interventions. This protocol details how the Sorensen model can be used for this purpose, drawing on methodologies established by other simulators like the UVa-Padova model [1].

The choice between the Bergman minimal, Hovorka, and Sorensen models is not a matter of selecting the "best" model, but rather the most appropriate tool for a specific research question. The Bergman minimal model remains the standard for deriving insulin sensitivity from IVGTT data due to its parsimonious structure. The Hovorka model offers an excellent compromise, providing sufficient physiological detail for the in-silico testing and development of treatment strategies and artificial pancreas algorithms with reasonable computational demands. The Sorensen model is the unparalleled choice for investigations where understanding organ-specific contributions to glucose metabolism is paramount, or for generating high-fidelity virtual data when the limitations of simpler models are a concern. Its revised and corrected implementation ensures this classic model of whole-body physiology remains a powerful, reliable resource for the scientific community.

Mathematical modeling of glucose-insulin dynamics is indispensable for diabetes research, drug development, and the creation of artificial pancreas systems. Bergman's Minimal Model stands as a parsimonious yet powerful framework specifically designed to estimate critical metabolic parameters—insulin sensitivity (SI) and glucose effectiveness (SG)—from intravenous glucose tolerance test (IVGTT) data. This review objectively compares the Minimal Model's performance against two other foundational but more complex models: the comprehensive Sorensen model and the T1DM-focused Hovorka model. We synthesize evidence from in-silico experiments and clinical studies, summarizing quantitative performance data in structured tables and detailing experimental methodologies. The analysis concludes that while each model serves distinct purposes, Bergman's Minimal Model maintains its relevance due to its computational efficiency, regulatory acceptance in simplified forms, and proven utility in quantifying key metabolic indices.

Mathematical models of glucose-insulin dynamics provide a critical foundation for understanding metabolic physiology, optimizing insulin therapy, and developing automated drug delivery systems. The Bergman Minimal Model, introduced in the early 1980s, was specifically designed to interpret data from the Frequently Sampled Intravenous Glucose Tolerance Test (FSIGT) [8] [9]. Its primary strength lies in its ability to distill complex physiological processes into a parsimonious framework that quantifies two fundamental metabolic parameters: insulin sensitivity (SI), which measures the effect of insulin to enhance glucose disposal, and glucose effectiveness (SG), which represents the ability of glucose to promote its own disposal independent of an insulin response [8] [10]. These indices are crucial for phenotyping individuals, assessing drug effects, and understanding the progression of diabetes.

The development of physiological mathematical models is an iterative process that begins with encoding real-world phenomena into equations, followed by a decoding phase where the model's accuracy is rigorously evaluated against experimental data [11]. In this landscape, the Bergman model is often contrasted with two other maximal models: the Sorensen model, a highly detailed physiological representation of a normal human, and the Hovorka model, formulated specifically for Type 1 Diabetes Mellitus (T1DM) [12] [13]. The choice between these models involves a fundamental trade-off between physiological comprehensiveness and computational simplicity, a balance that must be struck based on the specific research or clinical application.

Model Architectures: A Comparative Framework

The structural and mathematical differences between the Sorensen, Bergman, and Hovorka models define their respective applications, strengths, and limitations.

Bergman's Minimal Model: A Parsimonious Approach

Bergman's model is a third-order, nonlinear system that describes glucose-insulin dynamics using three key differential equations [8] [13] [9]:

• dG(t)/dt = -[p₁ + X(t)] * G(t) + p₁ * Gb: This equation describes the rate of change of plasma glucose concentration (G(t)), where glucose disappearance is driven by both insulin-independent processes (glucose effectiveness, p₁) and insulin-dependent processes mediated by remote insulin (X(t)).
• dX(t)/dt = -p₂ * X(t) + p₃ * [I(t) - Ib]: This represents the dynamics of insulin in a "remote" compartment, which reflects the active insulin that mediates glucose utilization in tissues.
• dI(t)/dt = -n * [I(t) - Ib] + U(t)/VI: This equation models the plasma insulin concentration (I(t)), where n is the insulin degradation rate, and U(t) is the exogenous insulin infusion rate.

From these equations, the key metabolic indices are derived as SI = p₃ / p₂ (insulin sensitivity) and SG = p₁ (glucose effectiveness) [8]. A visual representation of this system is provided below.

Sorensen and Hovorka Models: Maximal and Middle-Ground Approaches

• Sorensen Model: The Sorensen model is the most complex of the three, consisting of three detailed sub-models for glucose, insulin, and glucagon dynamics across multiple anatomical compartments, including the brain, liver, heart, lungs, periphery, gut, and kidney [12]. Its high level of physiological detail makes it a valuable tool for simulating whole-body metabolism, but its computational cost is significant. For T1DM applications, the original model, which included pancreatic insulin secretion, must be modified by removing the endogenous insulin secretion sub-model [12].

• Hovorka Model: The Hovorka model strikes a balance between complexity and simplicity. It is simpler than the Sorensen model but more complex than the Bergman model [12]. It was originally formulated to represent the physiological behavior of T1DM individuals and includes subsystems for glucose absorption, distribution, and disposal, as well as insulin absorption, distribution, and action on glucose production and disposal [13] [9]. This makes it particularly suitable for designing and testing control algorithms for artificial pancreas systems.

Table 1: Fundamental Architectural Comparison of the Models

Feature	Bergman Minimal Model	Hovorka Model	Sorensen Model
Primary Purpose	Estimate SI & SG from IVGTT	T1DM simulation & AP control	Whole-body physiological simulation
Model Complexity	Low (3 state variables) [13]	Intermediate	High (Multiple compartments: brain, liver, heart, etc.) [12]
Physiological Detail	Single glucose compartment; "remote" insulin compartment	Multiple compartments for glucose and insulin dynamics	Highest; organ/tissue level resolution [12]
Subject Representation	Originally normal, adapted for T1DM	Originally for T1DM [12]	Originally normal, adapted for T1DM [12]
Regulatory Status	Foundation for FDA-accepted simulators [1]	Used in research and AP development	Research-grade simulator

Experimental Performance and Validation

In-Silico Protocols and Performance Metrics

Comparative analyses of these models often employ standardized in-silico experiments to evaluate their physiological plausibility and performance. Standard protocols include [12]:

• Intravenous Glucose Tolerance Test (IVGTT): Administration of a glucose bolus (e.g., 0.5 g/kg) over a short period (e.g., 3 minutes) with or without an accompanying insulin bolus.
• Oral Glucose Tolerance Test (OGTT): Administration of an oral glucose load (e.g., 100 g) to simulate meal intake, often coupled with basal insulin infusion.

These experiments are conducted from steady-state initial conditions (e.g., glucose at 5 mmol/L) and measure outputs such as plasma glucose and insulin concentrations over time. The models' ability to replicate known physiological responses, such as the biphasic insulin peak in healthy individuals or the glucose decay curve, is a key validation metric [12] [8].

Quantitative Performance Comparison

A direct comparison of model simulations reveals how structural differences translate into varied physiological behaviors.

Table 2: Simulated Experimental Outcomes from Model Comparisons

Experiment	Bergman Minimal Model	Hovorka Model	Sorensen T1DM Model
IVGTT Glucose Dynamics	Parsimonious fit to decay curve; may overestimate SG due to 1-compartment assumption [10]	Physiologically plausible decay	Most comprehensive and physiologically detailed response [12]
IVGTT Insulin Dynamics	Captures initial peak and secondary rise [8]	Simplified insulin kinetics	Detailed multi-compartment insulin distribution [12]
OGTT Response	Requires additional meal submodel for accuracy [9]	Includes meal absorption model [9]	Includes gastrointestinal tract model for realistic oral glucose absorption [12]
Handling of Physical Activity/Stress	Limited; requires extensions	Can be incorporated with model enhancements [1]	High potential for integration due to structural completeness
Computational Speed	Fastest (suitable for real-time control) [13]	Intermediate	Slowest (high computational burden) [12]

The following diagram illustrates a generalized workflow for using these models in metabolic analysis, highlighting the trade-offs at each stage.

The Scientist's Toolkit: Key Reagents and Computational Tools

Working with these metabolic models requires a suite of computational and experimental resources.

Table 3: Essential Research Reagents and Tools for Metabolic Modeling

Tool / Reagent	Function / Description	Relevance to Models
Frequently Sampled IVGTT (FSIGT)	Clinical protocol involving intravenous glucose injection and frequent blood sampling to measure glucose and insulin dynamics [8].	Primary data source for parameter estimation in the Bergman Minimal Model.
Continuous Glucose Monitor (CGM)	Device measuring interstitial glucose concentrations continuously (e.g., FreeStyle Libre, Dexcom) [1] [14].	Provides validation data and enables model adaptation in real-world settings for all models.
UVa/Padova T1D Simulator	FDA-accepted simulator incorporating a population of virtual T1DM subjects [1] [12].	Gold standard for pre-clinical testing of control algorithms based on various models, including those derived from Bergman's framework [13].
Kalman Filter (KF) & Extended KF	Algorithm for state estimation and parameter adaptation in the presence of noisy sensor data [14] [9].	Crucial for real-time application of models (especially Bergman's) in closed-loop control, allowing the model to track patient state.
Linear Matrix Inequality (LMI) Solver	Optimization technique used in multi-objective robust controller design [13].	Used to design efficient insulin infusion controllers based on the linearized Bergman model.
Meal & Insulin Absorption Submodels	Mathematical representations of subcutaneous insulin kinetics and gut glucose absorption [9].	Required to adapt the intravenous Bergman model for subcutaneous insulin delivery and meal disturbances in Artificial Pancreas systems.

The comparative analysis reveals that the "best" model is entirely contingent on the research or clinical objective. The Sorensen model offers unrivalled physiological completeness and is an excellent tool for investigating detailed metabolic mechanisms in silico. However, its computational cost and complexity often render it impractical for real-time control applications or rapid parameter estimation [12]. The Hovorka model occupies a crucial middle ground, providing sufficient physiological detail for realistic T1DM simulation while remaining tractable for model predictive control (MPC) in artificial pancreas development [12] [13].

In this landscape, Bergman's Minimal Model retains enduring value due to its parsimony and specific focus. Its primary strength is the direct, efficient estimation of insulin sensitivity and glucose effectiveness from IVGTT data. While its simplifying assumptions (like the single-compartment glucose distribution) are known to introduce a systematic bias, resulting in overestimated SG and underestimated SI for insulin-resistant subjects, the high linear correlation between its indices and those from more complex models confirms its utility as a dependable comparative index [10]. Furthermore, its low computational burden makes it ideal for applications requiring rapid execution, such as real-time state estimation via Kalman filtering [9] and robust controller design using LMI techniques [13]. Its legacy is also embedded in modern regulatory science, as it forms the foundational basis for the UVa-Padova simulator, which the FDA accepts as a substitute for animal trials in preclinical AP testing [1].

In conclusion, for researchers and drug development professionals requiring a validated, efficient method to quantify fundamental metabolic parameters like SI and SG, Bergman's Minimal Model remains a powerful and parsimonious framework. For whole-body physiology simulation, the Sorensen model is superior, while for closed-loop glycemic control in T1DM, the Hovorka model and other intermediate-complexity models derived from the Bergman framework offer a balanced and effective solution. The continued evolution of these models, including the incorporation of delays [11] and factors like exercise and hormonal cycles [1], will further enhance their individual utility and collective contribution to diabetes management.

The development of a closed-loop artificial pancreas (AP) represents a paramount engineering challenge in medicine, requiring a delicate balance between physiological accuracy and computational practicality. At the heart of any automated insulin delivery (AID) system lies a mathematical model that describes glucose-insulin dynamics, serving as the fundamental engine for control algorithms [15]. For decades, researchers have debated the relative merits of three prominent models: the Sorensen model, known for its comprehensive whole-body representation; the Bergman minimal model, celebrated for its parametric efficiency; and the Hovorka model, which strikes a balance between physiological fidelity and practical applicability [13] [16]. This comparison guide objectively evaluates these models through the critical lens of closed-loop control development, synthesizing experimental data and implementation outcomes to inform researchers, scientists, and drug development professionals. The evolution of these models has paralleled advances in diabetes technology, from early intravascular systems to modern subcutaneous devices, with the Hovorka model emerging as a particularly influential platform for contemporary artificial pancreas research [17] [18].

Model Architectures: A Comparative Framework

Fundamental Structural Differences

Table 1: Architectural Comparison of Key Diabetes Models

Feature	Bergman Minimal Model [13] [16]	Sorensen Model [13] [16]	Hovorka Model [16]
Model Complexity	3 differential equations	Multiple compartments representing organs	8 differential equations
Representation Approach	Minimal parameter IVGTT-based	Whole-body physiological	Compartmental with subsystems
Glucose Subsystem	Single compartment	Multiple organ compartments	Accessible & inaccessible compartments
Insulin Subsystem	Plasma insulin dynamics	Organ-specific insulin distribution	Absorption & plasma kinetics
Insulin Action Modeling	Remote compartment	Integrated in organ models	Dedicated subsystem (3 compartments)
Meal Absorption	Not originally included	Incorporated	Comprehensive carbohydrate absorption
Clinical Validation Basis	Intravenous glucose tolerance tests	Normal human physiology	Subcutaneous insulin & meal intake

Figure 1: Hovorka Model Subsystem Architecture - The model's modular structure separates glucose kinetics, insulin absorption, and insulin action into interconnected subsystems.

Mathematical Formulations

The Bergman minimal model utilizes a simplified three-equation structure that describes plasma glucose concentration, remote insulin compartment, and plasma insulin concentration, making it computationally efficient but limited in capturing complex mealtime dynamics [13]. In contrast, the Sorensen model employs a more comprehensive physiological approach with multiple compartments representing specific organs and tissues, offering greater biological fidelity at the cost of computational complexity [13] [16]. The Hovorka model strikes a middle ground with eight differential equations organized into three key subsystems: (1) glucose kinetics with accessible and inaccessible compartments, (2) insulin absorption and kinetics, and (3) insulin action on glucose disposal, distribution, and endogenous production [16]. This structure specifically models subcutaneous insulin infusion and carbohydrate meal absorption, making it particularly suitable for closed-loop control applications where these are the primary input and disturbance variables.

Performance Evaluation in Control Algorithms

Experimental Protocol for Algorithm Comparison

Table 2: Control Algorithm Performance on the Hovorka Model Virtual Patient [16]

Control Algorithm	Rise Time Performance	Hypoglycemia Prevention	Postprandial Control	Computational Demand
Model Predictive Control (MPC)	Short rise time	Effective	Strong	Moderate to High
Neural Network MPC (NN-MPC)	Short rise time	Effective	Strong	High
PID Control	Moderate rise time	Limited	Moderate	Low
NARMA-L2	Variable	Moderate	Variable	High
Sequential Quadratic Programming (SQP)	Fast rise time	Effective	Strong	High

A comprehensive simulation study compared five control algorithms applied to the Hovorka model to evaluate artificial pancreas performance [16]. The experimental methodology involved:

• Virtual Patient Population: The Hovorka model generated glucose-insulin dynamics for type 1 diabetic patients under various conditions
• Meal Challenge Protocol: Algorithms were tested against standardized meal disturbances to assess postprandial glucose control
• Performance Metrics: Key metrics included rise time (response speed), hypoglycemia prevention, postprandial glucose excursions, and computational requirements
• Constraint Handling: Algorithms were evaluated on their ability to maintain glucose between 60-130 mg/dL despite meal disturbances
• Simulation Environment: MATLAB with ode15s solver for numerical integration of differential equations

The findings demonstrated that Model Predictive Control (MPC) algorithms leveraging the Hovorka model achieved superior overall performance with short rise times and effective constraint handling, making them particularly suitable for closed-loop applications [16].

Figure 2: Control Algorithm Testing Workflow - Experimental protocol for comparing controller performance using the Hovorka model as the virtual patient platform.

Real-World Validation and Clinical Translation

The transition from simulation to clinical application represents a critical validation step for diabetes models. The Hovorka model has demonstrated significant utility in this domain, contributing to the development of closed-loop systems that have achieved regulatory approval and clinical implementation [18]. Notably, Model Predictive Control (MPC) algorithms—many derived from or validated against the Hovorka model—have become the dominant control strategy in commercial artificial pancreas systems, with implementations in Medtronic's 780G, Tandem's Control-IQ, and CamAPS FX systems [18]. These systems have demonstrated clinically meaningful outcomes in real-world settings, with studies showing increased time-in-range, reduced hypoglycemia, and improved HbA1c levels across diverse patient populations [18].

The Hovorka model's capacity to simulate subcutaneous insulin delivery and meal absorption dynamics has proven particularly valuable for designing controllers that manage the complex delays inherent in subcutaneous-subcutaneous closed-loop systems [16]. This practical relevance distinguishes it from both the oversimplified Bergman model and the computationally prohibitive Sorensen model for embedded system implementation.

Table 3: Essential Research Tools for Hovorka Model Implementation

Tool Category	Specific Examples	Research Application	Availability
Simulation Platforms	MATLAB/Simulink [16]	Numerical simulation & control design	Commercial
Validation Simulators	UVa/Padova T1D Simulator [1]	Preclinical algorithm testing	FDA-accepted
Continuous Glucose Monitors	Dexcom G6, Medtronic Guardian [17] [1]	Real-time glucose data input	Commercial
Insulin Pumps	Tandem t:slim, Omnipod, Medtronic [17] [18]	Insulin delivery actuation	Commercial
Control Algorithms	MPC, PID, Neural Networks [16]	Closed-loop controller options	Research/Commercial
Performance Metrics	Time-in-Range, HbA1c, LI [19]	Outcome measurement	Standardized

Successful implementation of the Hovorka model for closed-loop control development requires a suite of specialized tools and resources. The UVa/Padova T1D Simulator deserves particular emphasis as an FDA-accepted platform for preclinical testing that has dramatically accelerated artificial pancreas development [1]. This simulator incorporates population variability and has become the gold standard for in silico validation before human trials. For clinical implementation, continuous glucose monitors (CGM) with mean absolute relative difference (MARD) values below 10% provide the necessary accuracy for reliable closed-loop operation [1]. Modern insulin pumps with precise basal control and bolus capabilities serve as the actuation mechanism, with patch pumps like Omnipod offering tubing-free alternatives [17]. The integration of these components creates a comprehensive development ecosystem that supports the translation of Hovorka model-based controllers from simulation to clinical application.

Within the ongoing scholarly discourse comparing the Sorensen, Bergman, and Hovorka models for diabetes research, the Hovorka model establishes a distinctive position as a balanced platform specifically engineered for closed-loop control development [16]. While the Bergman minimal model offers computational efficiency for preliminary controller design and theoretical analysis, and the Sorensen model provides comprehensive physiological representation for mechanistic studies, the Hovorka model delivers an optimal compromise for artificial pancreas applications [13] [16]. Its structured subsystem architecture, dedicated meal absorption model, and proven utility in successful commercial systems make it a particularly valuable resource for researchers and drug development professionals advancing the field of automated insulin delivery. The continued evolution of this model, potentially incorporating elements of artificial intelligence and adaptive personalization as seen in recent neural network implementations, promises to further enhance its relevance for next-generation closed-loop systems [20].

In the development of artificial pancreas systems and diabetes management technologies, mathematical models of glucose-insulin dynamics play a crucial role. These models span a remarkable spectrum of complexity, from compact representations with just three core equations to maximal models encompassing 22 or more physiological compartments. This comparison guide objectively analyzes three prominent models—the Bergman Minimal Model, the Hovorka Model, and the Sorensen Model—framed within the broader thesis of identifying the appropriate model for specific research and development applications.

The fundamental trade-off between physiological fidelity and practical utility drives model selection in diabetes research. While simpler models offer computational efficiency for control algorithms, more complex models provide deeper physiological insight but demand greater computational resources and parameter identification efforts. Understanding this complexity-performance relationship is essential for researchers, scientists, and drug development professionals working on glucose monitoring systems, insulin dosing algorithms, and artificial pancreas devices.

Model Complexity Comparison

The three models represent fundamentally different approaches to modeling glucose-insulin dynamics, with complexity levels varying by an order of magnitude.

Table 1: Fundamental Characteristics of Glucose-Insulin Models

Characteristic	Bergman Minimal Model	Hovorka Model	Sorensen Model
Model Category	Compact, minimal	Moderate complexity, T1D-specific	Maximal, physiologically comprehensive
Number of Compartments	3 core equations [21]	8+ compartments [3] [12]	22+ compartments [12]
Primary Application	Non-invasive glucose monitoring [21]	Artificial pancreas control algorithms [3] [12]	Physiological simulation & research [12]
Computational Demand	Low	Moderate	High
Parameter Identification	Relatively straightforward [21]	Requires clinical data [3]	Complex, extensive data requirements
Regulatory Status	Research use	Research use	Research use

Table 2: Physiological Scope and Coverage

Physiological Process	Bergman Minimal Model	Hovorka Model	Sorensen Model
Glucose Kinetics	Simplified peripheral glucose utilization [21]	Glucose subsystem with accessible & non-accessible compartments [3]	Comprehensive organ-level modeling (brain, liver, heart, lungs, periphery) [12]
Insulin Dynamics	Insulin sensitivity & glucose effectiveness [21]	Insulin subsystem with absorption & distribution [3]	Detailed pancreatic insulin secretion (removed for T1D) [12]
Insulin Action	Combined effect on glucose uptake [21]	Three separate insulin action compartments [3]	Organ-specific insulin effects
Glucagon Regulation	Not included	Not included	Included [12]
Meal Absorption	Not explicitly modeled	Included as disturbance input [3]	Detailed gastrointestinal tract model [12]

Figure 1: Model Complexity Spectrum and Primary Applications. The three models occupy distinct positions on the complexity spectrum, making them suitable for different research and development applications.

Performance Data & Experimental Evidence

Clinical and In-Silico Validation Results

Each model demonstrates distinct performance characteristics when applied to diabetes management tasks, with validation outcomes highlighting their respective strengths and limitations.

Table 3: Experimental Performance Metrics

Model	Experimental Context	Key Performance Metrics	Comparative Findings
Bergman Minimal Model	Non-invasive CGM clinical trial (161 subjects, 15,000+ datasets) [21]	MARD: 11.51%, RMSE: 1.48 mmol/L, Correlation: 0.85 [21]	32% improvement vs. non-use of BM-NCGM; 77.58% CEG A zone [21]
Hovorka Model	In-silico testing with enhanced Model Predictive Control (3 T1D patients) [3]	Time in target range: 71.43-87.76% (in-silico) vs. <50% (clinical) [3]	Significant discrepancy between in-silico predictions and clinical outcomes [3]
Sorensen Model	Comparative in-silico experiments (IVGTT, OGTT) [12]	Physiological plausibility across multiple test conditions [12]	Most complete physiological representation; requires modifications for T1D [12]

Model-Specific Methodologies

Bergman Minimal Model with Neural Network Enhancement

Recent implementations of the Bergman minimal model have incorporated neural networks to enhance predictive capability. The BM-NCGM (Bergman Minimal Model-Non-Invasive Continuous Glucose Monitoring) approach uses metabolic characteristics as inputs to predict insulin-facilitated glucose uptake and postprandial glucose gradient changes. This hybrid methodology considers the effects of glucose gradient, insulin action, and digestion process on glucose dynamics. The clinical validation involved 161 subjects with over 15,000 valid datasets, using the dynamic time warping algorithm to calculate the distance between predicted and reference glucose spectra, yielding an average distance of 21.80, demonstrating excellent tracking capability [21].

Hovorka Model with Enhanced Model Predictive Control

The improved Hovorka equations model modifies the original Hovorka model to enhance the interrelation between parameters and variables. The model structure comprises three subsystems: glucose subsystem, insulin subsystem, and insulin action subsystem. The in-silico testing protocol used MATLAB programming coupled with enhanced model-based predictive control (eMPC) to determine optimum bolus insulin for different meal conditions. Patient data collection followed strict ethical guidelines, with inclusion criteria specifying T1D patients aged 11-14 years highly dependent on insulin injections with four or more finger pricks per day for glucose measurements [3].

Sorensen Model Adaptation for T1D

The Sorensen model was originally formulated to simulate the behavior of both normal and diabetic individuals, making it the most comprehensive of the three models. For T1D applications, the model requires modification by removing the pancreatic insulin secretion sub-model and fixing the scale of absolute concentrations of metabolic source and sink functions. This adaptation aims to represent a "normal" response in a subject with T1D, though it doesn't fully capture all physiological abnormalities associated with diabetes [12].

Experimental Protocols & Methodologies

Standardized Testing Protocols

Comparative evaluation of glucose-insulin models typically employs standardized in-silico experiments to assess physiological plausibility and predictive capability across different conditions.

Table 4: Standardized In-Silico Experiments for Model Validation

Experiment Type	Protocol Details	Primary Assessment Metrics
IVGTT (Intravenous Glucose Tolerance Test)	Continuous basal insulin (6.67 mU/min) with 0.5 g/kg glucose over 3 minutes [12]	Glucose disappearance rate, insulin sensitivity
IVGTT + Insulin Bolus	IVGTT protocol with additional 1000 mU bolus delivered in 1 minute [12]	Counter-regulatory response, hypoglycemia risk
OGTT (Oral Glucose Tolerance Test)	Oral administration of 100 g glucose with continuous basal insulin [12]	Glucose absorption kinetics, gut-mediated effects
OGTT + Insulin Bolus	OGTT protocol with 1000 mU IV insulin bolus [12]	Meal response modulation, postprandial control

Data Collection and Ethical Considerations

Robust model development and validation requires high-quality clinical data collected under rigorous ethical frameworks. The Bergman minimal model validation involved a clinical trial registered in the Chinese Clinical Trial Registry (ChiCTR1900028100) with approval from the Ethics Review Committee of Peking University First Hospital [21]. Similarly, Hovorka model validation collected data from three T1D patients upon receiving approval from the Universiti Teknologi MARA Ethics Committee (REC/435/19), with formal consent obtained from parents or legal guardians since patients were minors [3]. These ethical safeguards ensure the reliability and appropriateness of data used for model development and validation.

Successful implementation and validation of glucose-insulin models requires specific computational tools, data resources, and methodological approaches.

Table 5: Essential Research Tools and Resources

Tool Category	Specific Solutions	Application in Model Research
Computational Platforms	MATLAB [3], Python with specialized PK/PD libraries [22]	Model implementation, parameter estimation, simulation studies
Parameter Estimation Methods	Maximum likelihood estimation, Akaike/Bayesian information criteria [23]	Model identification, comparison of alternative structural models
Clinical Data Sources	Glucose clamp studies [24], Continuous Glucose Monitoring [21], Self-monitored blood glucose [3]	Model calibration, validation against experimental data
Model Validation Approaches	Dynamic time warping [21], Clarke Error Grid analysis [21], Time-in-range metrics [3]	Assessment of predictive accuracy, clinical applicability
Specialized Software	Pharmacokinetic modeling tools [22], Population pharmacokinetic software [23]	Compartmental model development, population variability analysis

Figure 2: Research Workflow for Glucose-Insulin Model Development. The process spans from initial data collection through model implementation to validation and eventual research application, with iterative refinement at each stage.

The comparative analysis reveals that model selection depends fundamentally on the specific research objective. The Bergman Minimal Model offers compelling advantages for real-time applications where computational efficiency is paramount, demonstrating clinically acceptable accuracy with a MARD of 11.51% in recent implementations [21]. The Hovorka Model strikes a balance between physiological sophistication and practical implementability, making it particularly suitable for artificial pancreas control algorithms, though discrepancies between in-silico and clinical performance warrant careful consideration [3]. The Sorensen Model provides unparalleled physiological completeness with its 22+ compartment structure, offering the most comprehensive representation for research requiring deep physiological insight [12].

This complexity-performance trade-off framework enables researchers to make informed decisions based on their specific requirements for computational efficiency, physiological accuracy, and clinical applicability. As the field advances, hybrid approaches that leverage the strengths of each model type while mitigating their limitations will likely emerge, further enhancing our ability to manage diabetes through mathematical modeling.

Sorensen Model vs Bergman Minimal Model vs Hovorka Model Performance Research

Mathematical modeling of glucose-insulin dynamics is fundamental to diabetes research, drug development, and artificial pancreas systems. The Sorensen, Bergman Minimal, and Hovorka models represent a spectrum of physiological complexity—from comprehensive whole-body simulation to targeted clinical assessment. The Sorensen model offers a high-fidelity, multi-compartment representation of the entire human glucose regulatory system, explicitly simulating organ-level interactions. In contrast, the Bergman Minimal Model provides a parsimonious three-compartment structure optimized for efficient clinical parameter estimation from tolerance tests. The Hovorka model occupies a middle ground, incorporating sufficient physiological detail for effective insulin dosing decisions in artificial pancreas systems while remaining computationally tractable for real-time applications [1]. This guide objectively compares their performance characteristics, experimental validation data, and suitability for specific research applications.

Model Architectures and Mathematical Foundations

Structural Characteristics and Physiological Scope

Table 1: Fundamental Architectural Comparison of Glucose-Insulin Models

Model Characteristic	Sorensen Model	Bergman Minimal Model	Hovorka Model
Physiological Scope	Whole-body simulation with organ-level resolution	Targeted clinical assessment of insulin sensitivity & β-cell function	Intermediate complexity for artificial pancreas applications
Number of Compartments	Extensive (10+) multi-organ representation	3 core compartments (glucose, insulin, insulin action)	6+ compartments across glucose, insulin, and insulin action subsystems
Mathematical Formulation	Complex ODE system with detailed inter-compartmental fluxes	Simplified ODE system optimized for parameter identification	ODE system with additional glucose absorption & insulin action dynamics
Primary Clinical Outputs	Comprehensive organ-specific glucose/insulin fluxes	Insulin sensitivity (Si), glucose effectiveness (SG)	Real-time glucose predictions, insulin dosing recommendations
Parameter Estimation Complexity	High (requires extensive experimental data)	Low to moderate (from IVGTT/OGTT)	Moderate (requires patient-specific calibration)

Model Workflows and Applications

The following diagram illustrates the core structural differences and typical application workflows for the three models:

Experimental Performance Data and Validation

Quantitative Model Performance Metrics

Table 2: Experimental Validation Data Across Model Types

Performance Metric	Sorensen Model	Bergman Minimal Model	Hovorka Model
Clinical Validation Population	Limited data availability	Extensive validation in diverse populations [25]	3-patient T1D study (ages 11-14) [3]
Glucose Prediction Accuracy	Organ-level resolution	Strong correlation in parameter estimation (r=0.98) [14]	71-88% time in target range (in-silico) [3]
Time in Range Performance	Not typically reported for whole-body	Not primary outcome measure	79.59%, 87.76%, 71.43% for patients 1-3 respectively [3]
Parameter Estimation Precision	Complex, requires substantial data	High precision for Si and Φ [25]	Significant variables (P<0.001) [3]
Computational Requirements	High (complex ODE systems)	Low to moderate	Moderate (suitable for real-time control)

Key Experimental Protocols and Methodologies

Hovorka Model Clinical Validation Protocol

A 2024 study evaluated the improved Hovorka equations model using actual patient data from three adolescents with type 1 diabetes (ages 11-14) [3]. The methodology included:

• Patient Selection: Recruited highly insulin-dependent T1D patients requiring four or more daily finger pricks for glucose monitoring [3]
• Data Collection: Comprehensive 24-hour meal tracking with precise carbohydrate quantification (breakfast: 30-67g, lunch: 36-124g, dinner: 47-49g) [3]
• Intervention Protocol: Compared closed-loop enhanced model-based predictive control (eMPC) for in-silico testing against open-loop therapy for clinical validation [3]
• Optimal Bolus Calculation: Determined patient-specific optimum bolus insulin rates (e.g., Patient 1: 83.33, 33.33, 16.67 mU/min for breakfast, lunch, dinner respectively) [3]
• Outcome Measures: Primary endpoint was percentage time in normoglycemic range (4.0-7.0 mmol/L) with statistical significance at P<0.001 [3]

Bergman Minimal Model Automated Parameter Estimation

The Automated Oral Minimal Model (AOMM) streamlines the estimation of key metabolic parameters using SAAM II software [25]:

• Test Protocol: Standardized oral glucose tolerance tests (OGTT) with 25 samples collected from -30 to 420 minutes [25]
• Parameter Estimation: Automated calculation of insulin sensitivity (S_i) and beta-cell responsivity indices (Φ_d, Φ_s, Φ_tot) [25]
• Validation Method: Comparison against manually extracted results using Bland-Altman analysis showing zero bias [25]
• Precision Requirements: Parameter precision threshold established at coefficient of variation <25% [25]

Research Applications and Implementation Considerations

Domain-Specific Model Selection Guidelines

Table 3: Research Application Suitability and Implementation Requirements

Research Application	Recommended Model	Key Advantages	Implementation Considerations
Drug Development & Mechanism Analysis	Sorensen	Organ-level resolution for target engagement studies	Extensive parameterization required; limited validation in diverse populations
Clinical Trial Stratification	Bergman Minimal	Efficient Si and β-cell function quantification	AOMM workflow enables batch processing of large datasets [25]
Artificial Pancreas Algorithm Development	Hovorka	Balanced complexity for real-time control	Achieved 79-88% time in range vs. <50% with open-loop therapy [3]
Population Health & Prevention Studies	Bergman Derivatives	Simplified glucose homeostasis modeling	Validated for prediabetes applications with continuous glucose monitoring [14]
Personalized Diabetes Management	Hovorka with Modifications	Responsive to individual metabolic patterns	Bézier curve integration improves temporal forecasting accuracy [26]

Essential Research Reagent Solutions

Table 4: Key Experimental Materials and Analytical Tools

Research Reagent/Tool	Function/Purpose	Example Applications
SAAM II Software	Automated parameter estimation for minimal models	Batch processing of OGTT data for high-throughput Si calculation [25]
UVa-Padova T1D Simulator	FDA-accepted platform for in-silico clinical trials	Preclinical testing of artificial pancreas algorithms [1]
Continuous Glucose Monitors (CGM)	Interstitial glucose measurement for model calibration	FreeStyle Libre for prediabetes modeling [14]; Real-time data for Hovorka model [3]
Triple Tracer Meal Protocols	Gold-standard for flux quantification in model development	UVa-Padova simulator validation using data from 204 individuals [1]
Dual Extended Kalman Filter	Dynamic parameter and state estimation in glucose models	Robust estimation of unmeasurable variables in prediabetes models [14]

Emerging Trends and Future Research Directions

The field of glucose-insulin modeling continues to evolve with several promising developments. Fractal-fractional operators are being incorporated into modified Bergman models to better capture memory effects and complex dynamics, with studies showing that increasing both fractal dimension and fractional order leads to significant reduction in glucose concentration [2]. Multimodal large language models combined with mechanistic modeling demonstrate potential for personalized diabetes management, with Bézier curve approaches achieving RMSE of 15.06 at 30 minutes for glucose forecasting [26]. The regulatory acceptance of in-silico trials represents another major advancement, with the UVa-Padova simulator now accepted by the FDA as a substitute for animal trials in preclinical testing of control strategies [1]. Future research will likely focus on addressing population-specific challenges, including glycemic management in women across menstrual cycle and menopause, and further development of fully automated artificial pancreas systems that eliminate manual meal announcements [1].

From Theory to Practice: Model Applications in Algorithm Development and In-Silico Trials

The development of the Artificial Pancreas (AP) represents a revolutionary advancement in the management of Type 1 Diabetes (T1D), a chronic condition characterized by the body's inability to produce insulin. At the heart of every AP system is a control algorithm that automates insulin delivery, and the mathematical model upon which this algorithm is based is critical to its performance. Among the various models proposed, the Hovorka model has emerged as a particularly dominant framework in both research and clinical applications. This dominance is not accidental but is rooted in the model's specific design characteristics, which strike a balance between physiological fidelity and practical utility for controller design.

This guide provides a comparative analysis of the Hovorka model against two other foundational models: the highly complex Sorensen model and the simpler Bergman Minimal Model. The performance of these models is evaluated within the broader context of AP development, focusing on their use in control algorithms, their ability to simulate real-world conditions, and their validation through clinical and in-silico experiments. By presenting structured data and experimental protocols, this article serves as a reference for researchers, scientists, and drug development professionals navigating the landscape of glucose-insulin modelling.

The following table summarizes the core architectural differences between the three major models.

Table 1: Fundamental Characteristics of Key Glucose-Insulin Models

Feature	Bergman Minimal Model (BMM)	Hovorka Model	Sorensen Model
Primary Design Goal	Parameter estimation (SI, SG) from IVGTT [27]	Designed specifically for T1D AP control [12]	Simulate full physiology of normal and diabetic individuals [12]
Complexity & Order	3rd order nonlinear model [13]	Intermediate complexity [12]	High complexity; 22 differential equations [5]
Physiological Scope	Minimal, "whole-body" representation [28]	Glucoregulatory model for T1D [13]	Multi-compartment (brain, liver, heart/lungs, periphery, gut, kidney) [12] [5]
Insulin Delivery Route	Intravenous (IV) [13]	Subcutaneous (SC) [13] [12]	Subcutaneous (SC) with modifications [12]
Regulatory Status	Not FDA-approved as a platform	Basis for the FDA-approved UVa/Padova simulator [12]	Not FDA-approved; used as a high-fidelity research tool

Performance Analysis: Quantitative Data Comparison

The ultimate value of a model is demonstrated through its performance in simulation and control. The table below consolidates key experimental results from studies utilizing these models.

Table 2: Experimental Performance Metrics in AP Applications

Model & Controller Type	Experimental Context	Key Performance Outcomes	Reference
Multi-objective control (LMI) based on Bergman's IV model	Tested on UVa/Padova simulator (SC route) with unannounced meals and noise. Compared with Compound IMC.	Better performance metrics than IMC; efficiently regulated BG with less insulin and avoided hypoglycemia.	[13]
Enhanced Model Predictive Control (eMPC) based on Hovorka model	48-hour AP admission in 15 adults with T1D, including meals and unannounced exercise.	Time in target (70-180 mg/dL): 88.0%; Time <70 mg/dL: 1.5%; Significant improvement over sensor-augmented pump therapy.	[29]
Improved Hovorka model with eMPC	In-silico testing on 3 pediatric T1D patients using real patient data.	Time in target range: 79.59%, 87.76%, and 71.43% for the three patients, compared to <50% with clinical open-loop therapy.	[3]
Event-triggered SDHAP (PI/MPC) based on Bergman Minimal Model	In-silico study using the T1DiabetesGranada dataset for a dual-hormone (insulin/glucagon) system.	Effective BG control and disturbance rejection under hypoglycemia and hyperglycemia conditions; patient-specific drug delivery.	[30]

Detailed Experimental Protocols

To ensure reproducibility and provide a clear understanding of the methodology behind the data, this section outlines the experimental protocols used in key studies cited in this guide.

Clinical AP Study with Hovorka-based eMPC

This protocol is adapted from a 48-hour clinical study that demonstrated high performance using an enhanced MPC (eMPC) controller, which can be based on models like Hovorka's [29].

• Objective: To evaluate the safety and efficacy of an eMPC AP with a trust index in regulating BG levels in adults with T1D under daily living conditions, including meals and exercise.
• Subjects: 15 adults with T1D (mean age 46.1±17.8 years, HbA1c 7.2%±1.0%) [29].
• System Components: Continuous Glucose Monitor (CGM), insulin pump, and control algorithm running on a portable artificial pancreas system (pAPS) [29].
• Protocol:
  • Run-in Phase: A 1-week period of sensor-augmented pump (SAP) therapy to optimize baseline pump settings.
  • Admission Phase: A 48-hour supervised admission.
    • Meals: Three meals per day (29-57g carbohydrates each). Mealtime boluses were administered based on the subject's insulin-to-carbohydrate ratio (CR), with corrections applied by the algorithm based on pre-meal CGM values [29].
    • Exercise: One hour of unannounced brisk walking on the second day.
    • Monitoring: Capillary fingerstick measurements were taken 30 minutes before meals, 2 hours after meals, at bedtime, and as prompted by a health monitoring system (HMS) [29].
• Data Analysis: Primary endpoints included the percentage of time CGM glucose was in the target range (70-180 mg/dL), below 70 mg/dL, and above 180 mg/dL.

In-Silico Model Comparison Protocol

This protocol describes a methodology for comparing maximal models, such as Sorensen, Hovorka, and UVa/Padova, in a simulated environment [12].

• Objective: To establish to what extent increasing model complexity translates into richer and more correct physiological behaviour.
• Models Compared: Hovorka model, Sorensen model, and the UVa/Padova Simulator [12].
• In-Silico Experiments:
  • Intravenous Glucose Tolerance Test (IVGTT): Administration of 0.5 g/kg of glucose over 3 minutes with a continuous basal insulin infusion.
  • IVGTT + Insulin Bolus: As above, but accompanied by a 1000 mU IV insulin bolus.
  • Oral Glucose Tolerance Test (OGTT): Oral administration of 100g of glucose with a continuous basal insulin infusion.
  • OGTT + Insulin Bolus: As above, with an additional 1000 mU IV insulin bolus.
• Initialization: All models were set to the same steady-state conditions (glucose concentration of 5 mmol/L) with a continuous basal insulin delivery of 6.67 mU/min [12].
• Analysis: Comparison of model predictions of glucose and insulin concentrations over time during each experiment.

Signaling Pathways and System Workflows

The following diagrams illustrate the core structure of a generic AP control system and the physiological compartments of different model complexities.

Artificial Pancreas Closed-Loop Control

Model Complexity Spectrum

The Scientist's Toolkit: Research Reagent Solutions

This section details the essential materials, models, and tools required for research and development in the field of AP systems.

Table 3: Essential Research Tools for AP Development

Tool / Reagent	Function in Research	Example Use Case
UVa/Padova T1DM Simulator	An FDA-accepted platform for in-silico testing of AP algorithms, providing a virtual patient population. [13]	Replacing/pre-clinically supplementing animal and human trials to test safety and efficacy of new controllers. [13] [28]
Hovorka Model	A glucoregulatory mathematical model serving as the core of many MPC controllers and in-silico simulators. [12] [3]	Used for in-silico testing and as the internal prediction model within adaptive MPC algorithms. [29] [3]
Bergman Minimal Model	A simple model used for estimating insulin sensitivity (SI) and glucose effectiveness (SG). [28] [27]	Parameterizing individual patient physiology from experimental data using fitting algorithms like Genetic Algorithms. [28] [31]
Continuous Glucose Monitor (CGM)	A wearable device that measures interstitial glucose levels at regular intervals, providing the primary input to the AP controller. [29] [30]	Provides real-time glucose data for the control algorithm during both clinical studies and daily AP operation.
Genetic Algorithm (GA)	An optimization technique used to fit model parameters to individual patient data. [28]	Accurately identifying individual Bergman Minimal Model parameters with high accuracy and low prediction bias. [28]

The Role of the UVa/Padova Simulator (a Sorensen Descendant) as an FDA-Accepted In-Silico Testing Platform

The development of the Artificial Pancreas (AP) and advanced insulin therapies for Type 1 Diabetes Mellitus (T1DM) has been fundamentally accelerated by the use of high-fidelity in-silico simulation environments. Among mathematical models of glucose-insulin dynamics, a critical distinction exists between compact "minimal" models (e.g., Bergman's Minimal Model) used for parameter estimation and complex "maximal" models designed to capture the full spectrum of physiological dynamics for simulation and control algorithm testing [12] [32]. The UVa/Padova T1DM Simulator represents the pinnacle of this maximal model approach, descending from the foundational Sorensen model. Its acceptance by the U.S. Food and Drug Administration (FDA) in 2008 as a substitute for preclinical animal trials for certain insulin treatments marked a paradigm shift in diabetes technology development, establishing a new standard for validating closed-loop control algorithms [33] [1]. This guide provides an objective comparison of the UVa/Padova Simulator against its peers—the Sorensen and Hovorka models—framed within the broader thesis of maximal model performance research.

Table: Core Characteristics of Major T1DM Physiological Models

Feature	Sorensen Model	Hovorka Model	UVa/Padova Simulator
Origin & Lineage	Original comprehensive physiological model (1985)	Independent development	Descendant of Sorensen; integrates new clinical data
Model Complexity	Highest (3 sub-models, multiple organs)	Intermediate (8 nonlinear differential equations)	High (13-18 differential equations, population-based)
Regulatory Status	Research use	Research use	FDA-accepted for preclinical testing (since 2008)
Inter-Patient Variability	Limited in original form	Can be implemented	Built-in virtual population (300 subjects)
Key Differentiator	Most complete organ-level representation	Simpler structure, well-documented	Regulatory validation & comprehensive ecosystem

Model Comparison: Architectural and Performance Analysis

Lineage and Physiological Fidelity

The Sorensen model is the most complex and historically significant maximal model, comprising three interconnected sub-models for glucose, insulin, and glucagon dynamics across six body compartments (brain, liver, heart/lungs, periphery, gut, and kidney) [12]. Its high physiological fidelity provides a comprehensive representation of organ-level interactions. The UVa/Padova Simulator is a direct descendant of this approach but was specifically formulated and refined to represent the physiology of T1DM individuals, which required modifications to the original Sorensen model [12]. In contrast, the Hovorka model presents a simpler, more modular structure based on eight nonlinear differential equations, making it a popular choice for control algorithm prototyping [12] [32].

Evolution and FDA Validation of the UVa/Padova Simulator

The UVa/Padova Simulator's journey to FDA acceptance involved continuous refinement. The initial 2008 version (S2008) contained a population of 300 virtual subjects (100 adults, 100 adolescents, 100 children) and was successfully validated against various T1DM clinical data [33] [34]. However, observations from clinical closed-loop trials indicated that S2008 underestimated the frequency of hypoglycemic events [33]. This led to a major update in 2013 (S2013), which incorporated two critical physiological enhancements:

• A model of glucose kinetics in hypoglycemia that accounts for increased insulin sensitivity when glucose falls below a certain threshold.
• The incorporation of a glucagon kinetics, secretion, and action model to better simulate counter-regulatory responses [33] [34].

This iterative process, grounded in clinical data, cemented the simulator's role as a validated and reliable platform for in-silico trials.

Performance in Simulated Experimental Protocols

Comparative studies evaluating maximal models use standardized simulated experiments to assess their physiological plausibility. Key tests include the Intravenous Glucose Tolerance Test (IVGTT) and the Oral Glucose Tolerance Test (OGTT), with or without concomitant insulin administration.

Table: Simulated Experiment Output Comparison (Qualitative Performance)

Simulated Experiment	Sorensen Model	Hovorka Model	UVa/Padova Simulator
IVGTT	Physiologically plausible glucose decay	Faster glucose disappearance	Closely matches population-level clinical data
OGTT	Represents gut absorption and dynamics	Simpler gut model	Incorporates nonlinear gastric emptying
Response to Hypoglycemia	Limited counter-regulation in base form	Lacks comprehensive glucagon model	Explicit glucagon model & enhanced hypoglycemia dynamics (S2013+)
Meal Challenge Response	High fidelity due to organ-level detail	Moderately accurate	FDA-validated for meal scenarios

A 2021 comparative analysis concluded that while the Sorensen model offers the richest physiological detail, the UVa/Padova model, with its high complexity and specific design for T1DM, provides a balanced and validated framework for controller testing [12]. The Hovorka model, while simpler, is valued for its documentation and utility in initial control algorithm design.

Experimental Validation and Benchmarking

Validation Protocol for the UVa/Padova S2013 Simulator

The validation of the S2013 simulator followed a rigorous methodology to ensure its traces reproduced distributions of key outcome metrics observed in clinical trials [33].

• Database: 24 T1DM adult subjects across three international clinical centers, each with two 22-hour hospital admissions (open-loop and closed-loop), including dinner and breakfast meals [33].
• Procedure: Virtual subjects were exposed to the exact same scenarios (meal carbohydrate content, timing, insulin boluses, and basal patterns) as the real subjects [33].
• Assessment:
  • Clinical Zone Accuracy: The Continuous Glucose Error Grid Analysis (CG-EGA) was used to determine if simulated glucose profiles behaved similarly to real ones from a clinical perspective (i.e., falling in the same hypo-, eu-, and hyperglycemic zones) [33].
  • Outcome Metric Distribution: Standard metrics like frequency of hypoglycemia events were compared between the pooled real data and the pooled simulated data to ensure the simulator reproduced population-level variability and outcomes [33].
• Result: The S2013 simulator demonstrated a significantly improved ability to reproduce the frequency of hypoglycemia episodes observed in clinical trials compared to the S2008 version, confirming its enhanced validity [33].

Diagram of the UVa/Padova Simulator Validation Workflow. The process involves exposing virtual subjects to identical meal and insulin scenarios from clinical trials and comparing outcomes using clinical zone analysis and metric distribution.

Performance of a Bergman-Model-Derived Controller on the UVa/Padova Platform

The UVa/Padova simulator serves as a universal testbed, even for controllers designed with other models. A 2023 study designed a robust multi-objective output feedback controller based on the simpler, minimal Bergman model but tested it on the more complex UVa/Padova metabolic simulator [35]. This approach tests the controller's robustness against a high-fidelity model representing a full virtual population.

• Protocol: The controller was tested in-silico in the presence of unannounced meal disturbances, sensor noise, and insulin pump errors [35].
• Result: The Bergman-based controller successfully regulated blood glucose, avoided hypoglycemia, and used less insulin than a comparative compound IMC controller, demonstrating effective performance when validated on the FDA-accepted platform [35].

The Researcher's Toolkit for In-Silico Trials

Table: Key Resources for In-Silico T1DM Research

Resource / Solution	Function in Research	Example / Note
UVa/Padova T1D Simulator (T1DMS)	Core platform for preclinical testing of algorithms and protocols.	Commercial software; FDA-accepted population of 300 virtual subjects (adults, adolescents, children) [36].
Virtual Patient Populations	Provides inter-subject variability to test robustness.	The UVa/Padova simulator includes 100 virtual adults, 100 adolescents, and 100 children [33] [36].
Continuous Glucose Monitor (CGM) Model	Simulates sensor noise, delay, and accuracy in closed-loop studies.	The simulator incorporates CGM models, crucial for realistic testing [1]. Early sensors had MARD 15-21%; modern factory-calibrated sensors have MARD <10% [1].
Insulin Pump Model	Simulates subcutaneous insulin delivery dynamics.	The simulator includes models of insulin pump kinetics [1].
Control Algorithm Framework	The "heart" of the Artificial Pancreas, determining insulin delivery.	Can be Model Predictive Control (MPC), PID, or adaptive control [1] [32].
Meal & Disturbance Scenarios	Standardized challenges to test algorithm performance.	Typically include unannounced meals, varying meal sizes, and sometimes exercise [33] [35].

Logical Pathway for Selecting a T1DM Model

The choice of model is dictated by the specific research goal, balancing complexity, validation, and implementation needs.

Diagram for Selecting a Mathematical Model for T1DM Research. The pathway shows how research objectives directly guide the choice of model, with the UVa/Padova simulator being the critical step for regulatory preparation.

Within the landscape of T1DM physiological models, the UVa/Padova Simulator occupies a unique and critical position. As a descendant of the Sorensen model, it retains high physiological complexity but has been specifically tailored, validated, and updated based on clinical data from the T1DM population. Its official status as an FDA-accepted substitute for preclinical animal trials makes it an indispensable tool for researchers and companies aiming to efficiently translate new diabetes technologies from the laboratory to the clinic. While the Bergman minimal model remains useful for estimating key parameters like insulin sensitivity, and the Hovorka model offers a simpler structure for initial controller design, the UVa/Padova platform provides the most reliable and regulatory-endorsed environment for conducting large-scale in-silico trials, ultimately accelerating the development of the Artificial Pancreas and improving the lives of people with T1DM.

The Bergman Minimal Model represents a cornerstone in the mathematical modeling of glucose-insulin dynamics, providing a foundational framework for quantifying critical metabolic parameters. Since its inception in 1979, the model has become the standard method for analyzing intravenous glucose tolerance test (IVGTT) data, enabling researchers to derive quantitative estimates of insulin sensitivity (SI) and glucose effectiveness (SG) from dynamic tests [37]. Its development marked a significant departure from previous, more complex models by focusing on the minimal structural components necessary to accurately represent the system's behavior [37]. The model's enduring relevance lies in its unique combination of physiological fidelity and mathematical tractability, allowing it to bridge the gap between complex physiological processes and clinically applicable measurements.

This review situates the Bergman Minimal Model within the broader context of metabolic modeling, comparing its structure, application, and performance against two other prominent models: the more physiologically detailed Sorensen model and the Hovorka model, which is widely used in artificial pancreas development. Understanding the relative strengths and limitations of these models provides valuable insights for researchers selecting appropriate computational frameworks for specific applications in drug development and metabolic research.

Model Fundamentals: Structure and Key Parameters

Mathematical Formulation

The Bergman Minimal Model consists of a system of differential equations that capture the essential dynamics of glucose-insulin interaction. The model represents glucose regulation as a closed-loop system, treating plasma insulin concentration as an input to tissues that produce and utilize glucose, and plasma glucose as the output reflecting insulin's effect on glucose turnover [37]. This "partition analysis" approach elegantly bypasses the challenge of modeling complex insulin secretion mechanisms from pancreatic β-cells.

The core model structure is described by these key equations [28]:

• Glucose dynamics: ( \frac{dG(t)}{dt} = -p1[G(t)-Gb] - G(t)X(t) + \frac{D}{V} \delta(t) )
• Insulin action: ( \frac{dX(t)}{dt} = -p2X(t) + p3[I(t)-I_b] )
• Insulin kinetics (for subjects with endogenous secretion): ( \frac{dI(t)}{dt} = -n[I(t)-I_b] + \gamma[G(t)-h]^+t )

Where ( G(t) ) represents plasma glucose concentration, ( I(t) ) is plasma insulin concentration, and ( X(t) ) characterizes the delayed insulin effect on glucose utilization. The parameters ( p1 ), ( p2 ), and ( p3 ) represent fundamental metabolic processes, while ( Gb ) and ( I_b ) denote basal levels.

Key Metabolic Parameters

The Bergman model yields two particularly significant metabolic parameters:

• Insulin sensitivity (SI): Calculated as ( SI = \frac{p3}{p2} ), representing the ability of insulin to enhance glucose disposal and suppress endogenous glucose production [28]
• Glucose effectiveness (SG): Equivalent to parameter ( p_1 ), representing the ability of glucose itself to stimulate its own disposal and suppress endogenous production independent of insulin [37]

These parameters have proven invaluable for quantifying metabolic status in both research and clinical settings, with the Disposition Index (DI = SI × Insulin Secretion) emerging as a powerful predictor of diabetes risk [37].

Comparative Analysis of Metabolic Models

Structural and Conceptual Differences

Table 1: Fundamental Characteristics of Three Key Metabolic Models

Feature	Bergman Minimal Model	Sorensen Model	Hovorka Model
Complexity Level	Minimal (3 equations)	High (details every system/organ)	Intermediate (more detailed than Bergman)
Primary Application	IVGTT analysis, SI/SG estimation	Comprehensive physiological simulation	Artificial pancreas, control algorithms
Physiological Detail	Lumped parameters, single glucose compartment	Multi-compartment, organ-level resolution	Intermediate compartmental structure
Insulin Action	Single remote compartment	Detailed receptor-level dynamics	Two-compartment insulin action
Validation Status	Extensive validation vs. clamp [38] [39]	Limited direct clinical validation	Validated in artificial pancreas trials [1]
Computational Demand	Low	High	Moderate

Performance Comparison in Experimental Settings

Table 2: Quantitative Performance Metrics Across Model Applications

Performance Metric	Bergman Model	Sorensen Model	Hovorka Model
SI Correlation with Clamp	r = 0.87 [38]	Limited direct comparison data	High in simulator validation [1]
SG Estimation Capability	Direct parameter (p₁)	Implicit in system dynamics	Explicit glucose effectiveness
Application to Diabetes	Modified protocol required [38]	Not specifically designed for diabetes	Specifically designed for T1D [1]
Parameter Identifiability	High (few parameters)	Challenging (many parameters)	Moderate with appropriate data
Clinical Adoption	Widespread for research	Limited to academic research	Growing in artificial pancreas systems

Experimental Protocols and Methodologies

Standard IVGTT Protocol for Bergman Model Application

The Bergman Minimal Model was specifically designed for use with data from the Frequently Sampled Intravenous Glucose Tolerance Test (FSIVGTT). The original protocol involves [37]:

• Baseline sampling: Collection of fasting blood samples for basal glucose (Gb) and insulin (Ib) determination
• Glucose bolus: Rapid intravenous administration of glucose (typically 0.3 g/kg body weight) at time zero
• Frequent sampling: Blood collection at 1-5 minute intervals for 180-300 minutes post-injection
• Supplementary insulin: In some protocols, exogenous insulin administration at 20 minutes to enhance parameter identification, particularly in diabetic subjects [38]

This frequent sampling protocol revealed four distinct temporal phases in glucose decline: a mixing phase, a quasi-exponential phase (glucose effectiveness), an accelerated decline phase (insulin action), and a return-to-baseline phase [37].

Modified Protocols for Special Populations

In diabetic subjects with impaired insulin secretion, standard IVGTT protocols fail to elicit sufficient insulin response for reliable parameter estimation. Researchers have developed modified approaches:

• Exogenous insulin supplementation: Administration of insulin at 20 minutes enables model application to diabetic subjects [38]
• Reduced glucose load: Using 200 mg/kg instead of standard 300 mg/kg minimizes disturbance while maintaining parameter identifiability [39]
• Extended test duration: Prolonging sampling to 5 hours improves successful resolution of SI and SG in type 2 diabetes [39]

These modifications have extended the model's utility to diverse populations while maintaining the essential physiological interpretation of its parameters.

Advanced Applications and Contemporary Research

Integration in Artificial Pancreas Systems

The Bergman model has found extended utility in the development of artificial pancreas (AP) systems, also known as automated insulin delivery (AID) systems [1]. While later models like Hovorka's have been more extensively implemented in commercial AP systems, the Bergman model's simpler structure makes it suitable for:

• Control algorithm design: Particularly for proportional-integral (PI) and fractional-order controllers [40]
• In-silico testing: Providing computational frameworks for preclinical validation
• Personalized parameter estimation: Enabling tailored treatment approaches through individual-specific SI and SG estimation [28]

Recent Methodological Innovations

Contemporary research continues to refine the application of the Bergman model:

• Genetic algorithm optimization: Novel approaches using genetic algorithms have demonstrated efficient and accurate parameterization with reduced experimental burden [28]
• Non-invasive glucose monitoring: Recent work has integrated the Bergman model with neural networks to develop non-invasive continuous glucose monitoring methods, showing significant improvement in prediction accuracy (32% improvement in correlation coefficient) [21]
• Prediabetes modeling: Adapted versions of the Bergman model now enable the characterization of glucose metabolism in prediabetic states, expanding its utility to early intervention research [14]

The Scientist's Toolkit: Essential Research Reagents and Materials

Table 3: Key Reagents and Solutions for Bergman Model Research

Research Tool	Function in Protocol	Specifications/Alternatives
IV Glucose Solution	Glucose bolus administration	0.3 g/kg body weight (200 mg/kg for modified protocol)
Exogenous Insulin	Enhanced response in diabetic subjects	0.05 U/kg at 20 minutes (for modified protocols)
Frequent Sampling Apparatus	Blood collection for kinetic analysis	1-5 minute intervals initially, 10-20 minutes later
Glucose Assay System	Plasma glucose quantification	Laboratory glucose oxidase method or equivalent
Insulin Immunoassay	Plasma insulin quantification	RIA or ELISA with appropriate sensitivity
Parameter Estimation Algorithm	Model fitting to derive SI and SG	NONMEM, CONSAM, or custom genetic algorithms [28]
Validation Reference	Method comparison for insulin sensitivity	Hyperinsulinemic-euglycemic clamp (gold standard)

The Bergman Minimal Model remains a fundamental tool in metabolic research, providing a validated and widely accepted method for quantifying insulin sensitivity and glucose effectiveness from IVGTT data. Its minimal structure offers advantages in parameter identifiability and computational efficiency compared to more complex models like Sorensen's detailed physiological representation. However, this simplicity also limits its utility in applications requiring detailed physiological resolution, where models like Hovorka's may be preferable.

For researchers designing studies involving glucose-insulin dynamics, the Bergman model provides an excellent balance between physiological relevance and practical applicability, particularly for clinical studies where the IVGTT remains a standard perturbation test. Ongoing innovations in parameter estimation and expansion to non-invasive monitoring and prediabetes research suggest the model will continue to evolve and maintain its relevance in metabolic research and drug development.

Sorensen Model in Virtual Patient Simulation for Preclinical Therapy Assessment

In the development of new therapies for diabetes, sophisticated mathematical models of human physiology are indispensable tools for in-silico testing and preclinical assessment. These virtual patient simulators allow researchers to predict glycemic response to various interventions, from insulin delivery algorithms to novel drugs, thereby reducing the need for initial human or animal testing [41]. Among the most cited representations are the Sorensen, Hovorka, and Bergman minimal models, each presenting a different balance between physiological complexity and practical utility [12] [42]. The Sorensen model is perhaps the most complex among these, incorporating 22 differential equations (mostly nonlinear) that represent glucose concentrations in various body compartments including the brain, heart and lungs, liver, gut, kidney, and periphery, with about 135 parameters [5]. This guide provides a detailed, objective comparison of these predominant models, focusing on their architectural principles, performance in standardized tests, and suitability for specific research and development applications within the context of virtual patient simulation.

Model Architectures: A Tale of Three Approaches

The Sorensen Model: A High-Fidelity Physiological Representation

The Sorensen model, developed from a 1978 PhD thesis, is a comprehensive, physiology-based model of the glucose-insulin regulatory system in an average man [5] [42]. Its structure divides the body into six anatomical compartments: (1) brain, (2) heart and lungs, (3) gut, (4) liver, (5) kidneys, and (6) periphery (muscle and adipose tissue). Each compartment is further composed of three spaces: blood capillary, interstitial fluid, and intracellular space [42]. This extensive compartmentalization allows the model to simulate distinct metabolic processes in different tissues, providing a highly detailed representation of whole-body glucose metabolism. The model consists of 22 differential equations and incorporates pancreatic release of insulin and glucagon, making it unique in its ability to simulate both normal and diabetic physiology [5] [12]. A revised version addresses original errors and supplements it with a more detailed gastrointestinal glucose absorption sub-model [5].

The Hovorka Model: A Balanced Middle Ground

The Hovorka model, also known as the Cambridge model, strikes a balance between complexity and practicality [12] [42]. It describes the glucose regulation in Type 1 Diabetes Mellitus (T1DM) and is structured around five key sub-models: (1) glucose absorption from the gut, (2) subcutaneous insulin kinetics, (3) insulin action on glucose distribution, transport, and disposal, (4) glucose kinetics, and (5) endogenous glucose production [42]. With its more moderate complexity compared to the Sorensen model, it remains well-documented and is frequently used for the development of control algorithms, particularly for the artificial pancreas [12].

The Bergman Minimal Model: The Compact Alternative

The Bergman minimal model represents the simplest approach among the three, originally designed for the interpretation of Intravenous Glucose Tolerance Tests (IVGTT) [42]. This three-equation model focuses on the core dynamics of glucose-insulin interaction, comprising a glucose compartment and a two-compartment insulin subsystem (plasma and remote compartments). Its key strength lies in its parsimony of parameters, which enables efficient identification from clinical data, though this comes at the cost of physiological detail [42].

Table 1: Fundamental Architectural Comparison of the Sorensen, Hovorka, and Bergman Models

Feature	Sorensen Model	Hovorka Model	Bergman Minimal Model
Original Scope	Normal & Diabetic Physiology [12]	Type 1 Diabetes [12]	IVGTT Interpretation [42]
Core Structure	6 Anatomical Compartments, each with 3 spaces [42]	5 Functional Sub-models [42]	3-Equation Core Structure [42]
Complexity	22 Differential Equations, ~135 Parameters [5]	Moderate Complexity [12]	3rd Order Model [42]
Insulin Secretion	Endogenous (Pancreatic sub-model) [12]	Exogenous (for T1DM) [12]	Endogenous
Key Advantage	High physiological detail and whole-body representation	Balanced complexity, suited for control design	Parameter identification from sparse data

Diagram 1: Structural comparison of the Sorensen, Hovorka, and Bergman models, highlighting their different compartmental approaches.

Performance Comparison in Standardized In-Silico Experiments

Experimental Protocols for Model Benchmarking

To objectively compare the performance of these models, researchers have established several standardized in-silico experiments that mimic clinical challenges. The following protocols are critical for benchmarking:

• Intravenous Glucose Tolerance Test (IVGTT): A bolus of 0.5 g/kg of glucose is administered intravenously over 3 minutes, often with a continuous background insulin infusion of 6.67 mU/min [12]. This test stresses the model's acute response to a glucose challenge.
• IVGTT with Insulin Bolus: The same glucose administration is accompanied by a rapid insulin bolus (e.g., 1000 mU delivered in 1 minute) to evaluate the model's handling of concurrent glucose and insulin inputs [12].
• Oral Glucose Tolerance Test (OGTT): A oral dose of 100 g of glucose is simulated, testing the model's ability to represent gut absorption and the incretin effect, which is crucial for meal-response studies [5] [12]. The original Sorensen model lacked an explicit gastrointestinal tract, an issue addressed in its revised versions [5].
• OGTT with Basal and Bolus Insulin: This complex test involves an oral glucose load alongside continuous basal insulin delivery and a mealtime insulin bolus, closely mimicking a realistic therapeutic scenario for a T1DM virtual patient [12].

Comparative Model Outputs and Behavioral Fidelity

When subjected to these tests, the three models exhibit distinct behaviors reflecting their underlying structures. The Sorensen model, with its high anatomical fidelity, can generate predictions for specific tissue beds (e.g., brain, liver), providing a richer dataset for analysis [5]. Its revised version demonstrates improved equilibrium and corrects kidney glucose excretion and insulin secretion dynamics [5]. The Hovorka and Bergman models provide a system-level, whole-body blood glucose prediction. A key differentiator is the Sorensen model's ability to be parameterized for both normal and diabetic individuals, while the Hovorka model was originally formulated specifically for T1DM [12].

Table 2: Performance Comparison in Standardized Simulation Experiments

Experiment	Sorensen Model Performance	Hovorka Model Performance	Bergman Model Performance
IVGTT	Detailed dynamics across body compartments; requires corrections for equilibrium [5]	Captures core glucose-insulin dynamics; suitable for control design [12] [42]	Designed for this test; efficiently estimates insulin sensitivity [42]
OGTT	Requires empirical gut absorption & incretin extension; revised version includes GI tract [5]	Explicitly includes glucose absorption sub-model [42]	Not its primary design purpose; may lack key meal-response dynamics
Controller Testing	Used for control algorithm development but less common due to complexity [5] [42]	Well-documented and frequently used for artificial pancreas control design [12]	Simplicity allows for some robust controller designs [42]
Inter/Intra-Patient Variability	Limited representation of variability in standard form [42]	Good representation of inter- and intra-patient variability [42]	Primarily for population-level parameter estimation

Diagram 2: Workflow of standardized experiments used to benchmark model performance, showing the primary strength of each model.

Successfully implementing and utilizing these models requires a suite of computational and methodological "reagents." The following table details key resources and their functions in virtual patient research.

Table 3: Essential Reagents and Resources for Virtual Patient Research with Diabetes Models

Resource / Solution	Function in Research	Application Notes
Mechanistic Model Code (Matlab, R, C++)	Provides the core simulation engine for generating virtual patient trajectories.	The revised Sorensen model is available online (e.g., http://biomatlab.iasi.cnr.it/models/login.php) [5].
Virtual Patient Cohort Generation Tool	Creates populations of in-silico patients with physiological variability for statistical power in trials.	Techniques include Statistical Sampling (e.g., Monte Carlo), Simulation-Based Inference, and AI/ML [41] [43].
Parameter Estimation & Fitting Algorithm	Calibrates model parameters to fit individual patient data, personalizing the virtual patient.	SBI is a modern ML approach that infers a probability distribution over parameterizations [43].
Ordinary Differential Equation (ODE) Solver	Numerically solves the system of differential equations that define the model.	For complex models like Sorensen's, robust solvers (e.g., CVODE) are required [43].
Clinical Dataset for Validation	Provides real-world glucose, insulin, and meal data to validate model predictions.	Used for fitting and to test if simulated outcomes match clinical reality [12] [43].

The choice between the Sorensen, Hovorka, and Bergman models is not a matter of identifying a single superior option, but rather of aligning model characteristics with specific research and development goals.

• Select the Sorensen Model when the highest level of physiological detail is required, such as for investigating the localized tissue-specific effects of a drug, understanding complex pathophysiological interactions in different organs, or when the research question involves both normal and diabetic states. One must be prepared to handle its computational cost and implementation complexity [5] [12].
• Choose the Hovorka Model for tasks that require a balance between physiological plausibility and practical utility, especially the development and in-silico testing of control algorithms for the artificial pancreas. Its ability to represent patient variability and its well-established use in this domain make it a strong candidate [12] [42].
• Employ the Bergman Minimal Model when the research objective is the efficient estimation of key metabolic parameters (like insulin sensitivity) from sparse clinical data, such as from an IVGTT, and when model simplicity and computational speed are priorities [42].

In conclusion, the ongoing development and refinement of these models, including the correction of historical errors in the Sorensen model and the integration of modern machine learning techniques for virtual patient generation, continue to enhance their value in preclinical therapy assessment [5] [43]. This empowers drug developers and researchers to make more informed, strategic decisions in the quest to advance diabetes care.

This guide provides an objective comparison of three prominent mathematical models of glucose-insulin dynamics—the Sorensen model, the Bergman Minimal Model (BMM), and the Hovorka model—to assist researchers and drug development professionals in selecting the appropriate model for specific applications.

Mathematical models of human glucose-insulin metabolism are indispensable tools in diabetes research, enabling in-silico testing of new therapies, control algorithms for the Artificial Pancreas (AP), and educational simulations. The Sorensen model, a comprehensive physiological representation, the Bergman Minimal Model (BMM), a simple yet effective nonlinear model, and the Hovorka model, a detailed differential-equation based model, each possess distinct characteristics that make them suitable for different stages of research and development [1] [35]. This guide compares their performance across three key application domains: educational tools, control law synthesis, and regulatory submissions, supported by experimental data and detailed methodologies.

The table below summarizes the fundamental attributes of each model, which form the basis for their application-specific performance.

Table 1: Core Characteristics of the Sorensen, Bergman, and Hovorka Models

Characteristic	Sorensen Model	Bergman Minimal Model (BMM)	Hovorka Model
Primary Type	Comprehensive Physiologic [35]	Minimal, Nonlinear [44] [45]	Differential Equation-based [3]
Key Strength	Detailed whole-body physiology	Simplicity, analytical tractability	Detailed insulin action subsystems
Common Use	Foundational research, simulation design	Control law synthesis, theoretical analysis [44] [35]	In-silico testing, AP algorithm development [3]
Model Complexity	High (Complex)	Low (3 state variables) [44] [45]	Moderate to High [3]
Physiological Detail	High (Multi-compartment)	Low (Lumped parameters)	Moderate (Explicit glucose & insulin subsystems) [3]

Model Performance by Application Domain

Educational Tools and Prototyping

For initial educational simulations and prototyping, model simplicity and computational efficiency are paramount.

• Bergman Minimal Model (BMM): The BMM is highly suited for education due to its low order and mathematical simplicity. Its three-state variable structure (plasma glucose, insulin effect, plasma insulin) allows students and researchers to grasp the fundamentals of glucose-insulin feedback without being overwhelmed by complexity [44] [45]. Its computational efficiency enables rapid prototyping of basic control ideas.
• Sorensen Model: While highly detailed, the Sorensen model's complexity makes it less ideal for introductory educational purposes. However, it serves as an excellent tool for advanced courses focusing on detailed physiological representations and whole-body metabolic interactions [35].
• Hovorka Model: The Hovorka model offers a middle ground. Its structure, which includes explicit glucose and insulin subsystems, provides a more detailed physiological representation than the BMM, making it a valuable educational tool for intermediate to advanced learners [3].

Control Law Synthesis and Algorithm Design

The synthesis of control algorithms, particularly for the Artificial Pancreas, demands models that are tractable for controller design while capturing essential system dynamics.

• Bergman Minimal Model (BMM): The BMM's nonlinear form is frequently linearized for control design. Its simplicity makes it a preferred choice for developing and testing novel control strategies. Researchers have successfully designed and implemented a wide range of controllers using the BMM, including:
  • Observer-based control to estimate unmeasurable states and manage unknown time-varying delays [44].
  • Sliding Mode Control (SMC) and Adaptive Fractional-Order SMC (AFOSMC) to enhance robustness against meal disturbances and patient parameter uncertainties [45].
  • H∞ output-feedback control to achieve robustness against measurement noise and model uncertainties [35].
• Hovorka Model: The Hovorka model is increasingly used in more advanced control strategies like Model Predictive Control (MPC) [1]. Its higher physiological detail allows the controller to make more informed predictions about glucose dynamics, particularly in response to meals. Studies have used it with enhanced MPC (eMPC) to calculate optimal bolus insulin doses [3].
• Sorensen Model: Due to its high complexity, the Sorensen model is less commonly used for direct control law synthesis. However, it can serve as a "truth model" for validating controllers designed using simpler models.

Regulatory Submissions and In-Silico Trials

For regulatory submissions, the ability to simulate a large, diverse virtual population to demonstrate safety and efficacy is critical.

• Hovorka Model and UVa/Padova Simulator: The UVa/Padova T1D Simulator, which is accepted by the FDA as a substitute for animal trials in preclinical testing, represents the gold standard for regulatory submissions [1]. While its core is based on a model that integrates data from over 200 individuals, subsequent research-grade simulators have incorporated features from models like Hovorka's to simulate effects of exercise and dual-hormone control [1]. This simulator is extensively used for in-silico trials.
• Bergman Minimal Model (BMM): The standard BMM is not typically used for pivotal in-silico trials for regulatory approval. Its lack of a large, accepted virtual population and lower physiological detail limits its direct use in this domain. However, variations and extensions of minimal models form the foundation of more comprehensive simulators.
• Sorensen Model: The Sorensen model, as a comprehensive physiological model, has contributed to the foundational understanding of glucose dynamics. Its influence is seen in the design of modern simulators, though it is not itself a regulatory-approved tool.

Table 2: Performance Comparison Based on Experimental & Simulation Studies

Application	Metric	Bergman Minimal Model (BMM)	Hovorka Model
Control Law Performance	Controller Type	Adaptive FOSMC [45]	Enhanced MPC [3]
	Hypoglycemia Avoidance	Effective (No episodes reported) [35]	Effective (Clinical comparison)
	Time-in-Range (70-180 mg/dL)	Not explicitly reported	79.59% - 87.76% (in-silico) [3]
In-Silico Validation	Platform/Subjects	50 virtual adults (CVGA) [44]	3 virtual patients [3]
	Meal Challenge	Unannounced meals, robust performance [35]	Three daily meals, effective regulation [3]

Experimental Protocols for Model Validation

In-Silico Control Law Testing with BMM

A common methodology for testing controllers designed with the BMM involves the following steps [44] [45]:

• Controller Synthesis: Design the control law (e.g., observer-based feedback, AFOSMC) based on the nonlinear BMM or its linearized version.
• Virtual Cohort: Create a cohort of virtual patients by varying the BMM's parameters (e.g., insulin sensitivity, glucose effectiveness) within physiologically plausible ranges to simulate inter-subject variability. Studies often use 50 or more virtual subjects [44].
• Disturbance Scenario: Subject the virtual patients to challenging conditions, such as unannounced meal disturbances, unknown time-varying delays in insulin absorption, and sensor noise [44] [35].
• Performance Analysis: Evaluate controller performance using metrics like Control Variability Grid Analysis (CVGA), time-in-range, and total insulin dose delivered, ensuring robust performance across the population [44].

Clinical Correlation for the Hovorka Model

To validate the Hovorka model's performance against real-world data, researchers have employed protocols such as [3]:

• Data Collection: Collect clinical data from patients with T1D, including body weight, meal carbohydrate content, timing, and pre-existing bolus insulin doses.
• In-Silico Simulation: Use the collected patient data (meals, body weight) as inputs to the Hovorka model implemented in a simulation environment like MATLAB. Simulate blood glucose levels over 24 hours.
• Performance Comparison: Compare the simulation results against clinical glucose measurements from the same patients. Calculate performance metrics such as percentage time-in-range (70-180 mg/dL).
• Statistical Analysis: Perform regression analysis (e.g., multiple linear regression) to model the relationship between meal intake, insulin dosage, and blood glucose levels, and to assess the significance of these variables (P-value < .001 considered significant) [3].

Visualization of Model Structures and Applications

The following diagram illustrates the typical workflow for using these models in the development and testing of an Artificial Pancreas system, highlighting the role of each model.

The Scientist's Toolkit: Key Research Reagents

The table below lists essential tools and platforms frequently used in experiments and simulations involving these metabolic models.

Table 3: Essential Research Tools and Platforms

Tool/Solution	Function in Research	Example Use Case
UVa/Padova T1D Simulator	A platform of virtual T1D subjects for in-silico testing.	Accepted by the FDA for preclinical testing of AP algorithms [1].
Continuous Glucose Monitor (CGM)	Provides real-time interstitial glucose measurements.	Data source for model validation and controller input [1] [46].
MATLAB/Simulink	High-level programming and modeling environment.	Implementing model equations, designing controllers, and running simulations [44] [3].
Linear Matrix Inequalities (LMI)	A computational technique for robust control design.	Solving H∞ and pole-placement constraints for output-feedback controllers [35].
High-Order Sliding Mode Observer (HOSMO)	Algorithm to estimate unmeasurable model states.	Estimating plasma insulin concentration from CGM data in BMM-based control [45].

Navigating Limitations: Error Correction, Parameter Identification, and Model Adaptation

Mathematical modeling of glucose-insulin dynamics is a cornerstone in the development of advanced diabetes management technologies, particularly artificial pancreas (AP) systems. Among the most cited physiological models in this field are the Sorensen model, Bergman minimal model, and Hovorka model, each offering distinct approaches to simulating human metabolic processes [1]. The Sorensen model, originally developed in 1978, stands as one of the most comprehensive compartmental models, incorporating 22 differential equations to represent glucose concentrations across various body compartments including brain, liver, heart and lungs, gut, kidney, and peripheral tissues [5]. This extensive physiological detail makes it particularly valuable for research applications where comprehensive system representation is prioritized over computational simplicity.

However, due to its complexity and certain imprecisions in the original documentation, the Sorensen model has been subject to implementation errors across numerous subsequent studies. A 2020 revision project identified and corrected these errors while supplementing the model with previously missing gastrointestinal glucose absorption components [5]. This article provides a detailed comparison of this revised Sorensen implementation against other established models, examining their respective performances across standardized testing scenarios relevant to drug development and diabetes technology research.

Methodological Approaches: Model Revision and Testing Framework

Sorensen Model Revisions and Corrections

The revision of Sorensen's model addressed several critical imprecisions in the original 1978 dissertation that had been propagated through subsequent literature. The correction process involved meticulous analysis of the original equations and parameter values, followed by systematic implementation of corrections [5]. Key revisions included mathematical corrections to kidney glucose excretion rates, insulin secretion dynamics, and initial condition specifications that had previously resulted in non-equilibrium states and incorrect model behavior.

Table 1: Key Corrections Implemented in the Revised Sorensen Model

Error ID	Original Incorrect Form	Corrected Form	Impact of Correction
A	rKGE = 71+71tanh[0.11(GK-460)]	rKGE = 71+71tanh[0.011(GK-460)]	Slower kidney glucose excretion
C	rKIC = FKIC[QKIIK]	rKIC = FKIC[QKIIH]	Corrected initial conditions
D	dQ/dt = k(Q-Q0)+γP-S	dQ/dt = k(Q0-Q)+γP-S	Corrected insulin secretion
E	GPI = GPV-rBGU/VPITPG	GPI = GPV-rPGU/VPITPG	Corrected initial conditions

Beyond error correction, the model was enhanced with the addition of a gastrointestinal tract module to simulate alimentary glucose intake, digestion, and absorption using a previously validated glucose absorption formulation [5]. This addition addressed a significant limitation in the original model, which relied on empirical estimation of gut glucose absorption rates rather than physiological representation of the process.

Experimental Validation Protocol

The revised Sorensen model underwent rigorous validation using multiple testing scenarios to ensure accurate representation of physiological responses:

• Standard Intravenous Glucose Tolerance Test (IVGTT): Administration of 0.5 g/kg glucose bolus with monitoring of glucose and insulin dynamics [5]
• Variable-dose IVGTT Comparison: Testing with different glucose doses (0.05, 0.2, 0.5, and 0.75 g/kg) to assess dose-response accuracy [5]
• Intravenous Insulin Tolerance Test (IVITT): Administration of 0.04 U/kg insulin bolus to evaluate counter-regulatory response simulation [5]
• Continuous Infusion Studies: Testing with continuous intravenous insulin infusions (0.25, 0.4 mU/kg) to assess steady-state and dynamic responses [5]

Implementation was conducted following the MoSpec (model specification) approach, an automated system that generates computational routines in Matlab, R, and C++ from a centralized specification spreadsheet, thereby minimizing coding errors and ensuring consistency across platforms [5].

Comparative Analysis of Glucose-Insulin Models

Structural and Functional Characteristics

The Sorensen, Bergman, and Hovorka models represent different philosophical approaches to metabolic modeling, with varying degrees of physiological detail and computational complexity.

Table 2: Structural Comparison of Major Glucose-Insulin Models

Characteristic	Sorensen Model	Bergman Minimal Model	Hovorka Model
Complexity Level	High-fidelity physiological model	Minimal model	Intermediate complexity
Number of Equations	22 differential equations [5]	2-3 differential equations [14]	8 differential equations [1]
Compartments Represented	Brain, heart/lungs, liver, gut, kidney, periphery [5]	Plasma glucose, remote insulin [14]	Glucose, insulin, insulin action compartments [1]
Parameter Count	~135 parameters [5]	3-6 key parameters [14]	26-35 parameters [1]
Primary Application	Detailed physiological studies, virtual patient simulation [5]	Insulin sensitivity assessment, clinical research [14]	Artificial pancreas development [1]
FDA Acceptance	Not specified	Not specified	Accepted for preclinical testing [1]

Performance Metrics in Validation Studies

When tested against experimental data, each model demonstrates distinct performance characteristics reflecting their design priorities and structural assumptions.

Table 3: Performance Comparison Across Standardized Tests

Testing Scenario	Revised Sorensen Model	Bergman Minimal Model	Hovorka Model
IVGTT (0.5 g/kg)	Accurate glucose and insulin dynamics [5]	Good glucose fit, simplified insulin [14]	Comprehensive glucose-insulin dynamics [1]
Variable IVGTT Dosing	Accurate across 0.05-0.75 g/kg range [5]	Limited to standard doses	Robust across dosing range [1]
IVITT (0.04 U/kg)	Accurate counter-regulatory response [5]	Not primarily designed for ITT	Optimized for insulin response [1]
Meal Challenge	Requires GI tract supplement [5]	Limited meal response capability	Integrated meal model [1]
Computational Demand	High	Low	Moderate

Implementation Considerations for Research Applications

Research Reagent Solutions and Computational Tools

Successful implementation of metabolic models requires specific computational tools and methodological approaches:

Table 4: Essential Research Tools for Model Implementation

Tool Category	Specific Solution	Research Application
Implementation Platforms	Matlab, R, C++ [5]	Cross-platform model implementation
Parameter Estimation	Differential Evolution, Ant Colony Optimization, Particle Swarm Optimization [47]	Identification of patient-specific parameters
Model Specification	MoSpec automated system [5]	Consistent code generation across platforms
Validation Framework	IVGTT, IVITT, continuous infusion protocols [5]	Standardized performance assessment
Accessibility	Online implementation at http://biomatlab.iasi.cnr.it/ [5]	Community access to validated model

Signaling Pathways and Experimental Workflows

The compartmental structure of the revised Sorensen model illustrates the comprehensive physiological representation that distinguishes it from simpler alternatives. The model explicitly represents glucose transport and hormonal regulation across multiple organ systems, providing a detailed framework for investigating system-level metabolic disturbances.

The experimental workflow for model validation follows a systematic process from implementation through to performance verification, with particular attention to the correction of known implementation errors in earlier versions.

Discussion: Implications for Diabetes Research and Development

Application-Specific Model Selection

The comparative analysis reveals that model selection should be guided by specific research objectives rather than seeking a universally superior solution. The revised Sorensen model offers clear advantages for applications requiring high physiological fidelity, such as investigating organ-specific metabolic effects or simulating complex clinical scenarios involving multiple systems. Its comprehensive structure makes it particularly valuable for virtual patient simulation and educational applications where understanding system-level interactions is paramount [5].

In contrast, the Bergman minimal model remains appropriate for studies where insulin sensitivity estimation is the primary goal and computational simplicity is valued over physiological comprehensiveness [14]. The Hovorka model occupies an intermediate position, balancing sufficient physiological detail with practical implementability, making it well-suited for artificial pancreas development and clinical algorithm testing [1].

Limitations and Future Directions

Despite the significant improvements in the revised Sorensen implementation, certain limitations persist. The model remains computationally intensive compared to simpler alternatives, potentially limiting its utility in real-time applications. Additionally, while the added gastrointestinal component addresses a major gap, the incretin effect—the potentiation of insulin secretion by gut hormones following oral glucose administration—is still not fully incorporated [5].

Future development directions should focus on enhancing personalization capabilities through advanced parameter identification techniques such as evolutionary algorithms [47], extending model validity to special populations including pregnant women and those with significant comorbidities [1], and improving integration with emerging diabetes technologies including continuous glucose monitors and insulin pumps [1].

The revised Sorensen model implementation represents a significant advancement in physiological modeling of glucose-insulin dynamics, correcting known errors while enhancing functionality through the addition of gastrointestinal glucose absorption. When evaluated against established models, it demonstrates superior performance in capturing complex physiological interactions across multiple body compartments, though with higher computational demands than simpler alternatives. The choice between Sorensen, Bergman, and Hovorka models should be guided by specific research requirements, with the revised Sorensen offering particular value for investigations requiring high physiological fidelity and comprehensive system representation. Continued refinement of these models, coupled with advances in personalization methodologies, will further enhance their utility in drug development and diabetes technology innovation.

Mathematical modeling of glucose-insulin dynamics is a cornerstone of diabetes research, enabling the simulation of metabolic processes, the design of artificial pancreas systems, and the evaluation of new therapeutic strategies. Among the most influential models are the Sorensen model, a comprehensive whole-body physiological representation; the Bergman Minimal Model, a simplified yet powerful tool for assessing insulin sensitivity; and the Hovorka model, a widely adopted platform for closed-loop control applications [48]. Each model offers distinct advantages and embodies specific limitations, particularly in their handling of critical physiological phenomena like the incretin effect and gastric emptying dynamics. The incretin effect, responsible for the enhanced insulin secretion after oral compared to intravenous glucose administration, and gastric emptying, a key regulator of postprandial glucose appearance, are increasingly recognized as vital components for accurate postprandial glucose simulation [49] [50]. This analysis objectively compares the performance of these three seminal models, focusing on their inherent capabilities and gaps regarding these mechanisms, and surveys the experimental data and methodologies driving modern model enhancements.

The foundational differences between the Sorensen, Bergman, and Hovorka models dictate their respective applications and limitations. The following table summarizes their core characteristics and specific shortcomings related to gastric emptying and incretin effects.

Table 1: Comparative Analysis of Key Glucose-Insulin Dynamic Models

Feature	Sorensen Model	Bergman Minimal Model	Hovorka Model
Primary Scope & Architecture	Comprehensive, multi-compartment model representing major organ systems (brain, liver, muscle, kidneys) [48]	Simplified, minimal model focusing on plasma glucose and insulin dynamics [11] [14]	Intermediate complexity model with glucose, insulin, and insulin action compartments [3] [24]
Representation of Gastric Emptying	Implicitly handled within the gastrointestinal subsystem; often not a distinct, tunable process [48]	Not explicitly included; meal glucose appearance is typically an external input [11]	Not a core component of the original model; often requires external coupling for meal absorption [3]
Representation of the Incretin Effect	Lacks explicit incretin pathways [51]	No incretin representation; insulin secretion is a direct function of glucose only [14]	No inherent incretin dynamics; insulin secretion is glucose-mediated [3] [51]
Key Performance Gaps	Difficulty simulating delayed carbohydrate absorption and GLP-1-mediated drug effects [50] [49]	Inaccurate postprandial glucose and insulin predictions due to missing gut-pancreas axis [51]	Suboptimal performance for meals and therapies affecting gastric emptying without structural modifications [3] [50]

Experimental Insights and Quantifiable Model Performance

Validating and refining mathematical models requires robust experimental data. Clinical trials and in-silico studies provide critical quantitative metrics to assess model performance and pinpoint deficiencies.

Table 2: Experimental Data Highlighting Model-Relevant Physiological Effects

Experimental Focus	Key Findings & Quantitative Impact	Implication for Models
GLP-1 Agonist Effects (Dulaglutide)	A PopPK/PBPK modeling approach predicted that a 4.5 mg dulaglutide dose causes gastric emptying delay (GED) but has no clinically relevant effect on the PK of co-administered small molecules [50].	Confirms the necessity to model GED as a primary mechanism of action, separate from direct insulin secretion effects.
GLP-1 Infusion & Energy Intake	Peripheral GLP-1 infusion in humans reduced spontaneous energy intake by a mean of 13% (P<0.001) in a dose-dependent manner, linked to slowed gastric emptying [49].	Provides a quantitative relationship between GLP-1, gastric emptying, and a metabolic outcome (food intake) for model calibration.
Incretin Role in Insulin Secretion	Meal-test studies show the anti-diabetic effect of GLP-1 is mainly due to slowed gastric emptying and reduced carbohydrate absorption rate, not increased postprandial insulin secretion [49].	Challenges models that oversimplify incretins as mere insulin secretagogues and underscores the need to integrate gastric emptying.
Alpha Cell Dysregulation	Mathematical modeling showed that alpha cell dysregulation (increased glucagon) significantly impacts the progression to T2D, accelerating beta cell failure under moderate dysregulation [51].	Highlights the need for integrated alpha-cell and glucagon dynamics, a feature absent in the three core models.

Methodological Deep Dive: Protocols for Model Identification and Validation

The following experimental protocols are central to generating data for building and validating models that incorporate gastric emptying and incretin effects.

1. Triple Tracer Meal Protocol & Population Model Generation: This is a foundational methodology for creating models with realistic inter-individual variability. As used in the development of the UVa-Padova T1D Simulator, this protocol involves administering a meal with three different glucose tracers to human subjects [1]. By frequently sampling plasma glucose and insulin concentrations, researchers can obtain model-independent estimates of fundamental glucose fluxes, including endogenous glucose production and rate of glucose disappearance. Data from 204 such individuals allowed for the generation of a large-scale model and the creation of a virtual population spanning observed inter-individual variability, which is crucial for robust in-silico testing [1].

2. Closed-Loop Algorithm Testing with Enhanced Model Predictive Control (eMPC): This protocol tests advanced control algorithms in silico or in vivo. A study using an improved Hovorka model employed a closed-loop algorithm with eMPC to determine optimal bolus insulin doses for patients with T1D [3]. The methodology involves using actual patient data (meal carbohydrates, body weight) as inputs to the mathematical model implemented in an environment like MATLAB. The eMPC algorithm then simulates insulin delivery to keep glucose within a target range (4.0-7.0 mmol/L). Performance is quantified by the percentage of time glucose is in the target range, allowing for direct comparison of different models or control strategies [3].

3. Gastric Emptying Delay (GED) Assessment via Drug-Drug Interaction (DDI) Studies: This protocol is specific to quantifying the effects of GLP-1 receptor agonists on gastric emptying and its downstream consequences. As described in a combined PopPK/PBPK modeling study, healthy participants or patients with T2DM are administered various doses of a GLP-1 agent (e.g., dulaglutide) alongside an orally administered small molecule drug (e.g., acetaminophen, whose absorption is highly dependent on gastric emptying) [50]. The pharmacokinetic parameters (AUC, C~max~, t~max~) of the small molecule are measured with and without the GLP-1 agent. The observed delay in t~max~ and change in C~max~ are used to estimate an exposure-dependent GED, which is then incorporated into PBPK models to predict DDIs at different doses [50].

The Scientist's Toolkit: Essential Research Reagents and Solutions

Table 3: Key Reagents and Materials for Experimental Investigation

Research Reagent / Material	Primary Function in Experimental Context
Synthetic Human GLP-1 (7-36 amide)	Used in human infusion studies to establish the direct physiological effects of the hormone on satiety, gastric emptying, and insulin secretion [49].
Long-Acting GLP-1 Analogues (e.g., Liraglutide, Dulaglutide)	Pharmacological tools to investigate the chronic impacts of GLP-1 receptor activation on glycemic control and body weight in clinical trials [50] [49].
Continuous Glucose Monitoring (CGM) Systems (e.g., FreeStyle Libre, Dexcom)	Provides high-frequency, real-world interstitial glucose data for model parameter estimation, validation, and training of data-driven predictors [14] [48].
Visual Analogue Scales (VAS) for Appetite	Validated tool to quantify subjective appetite sensations (hunger, satiety) during meal tests, linking hormonal changes to behavioral outcomes [49].
Glucose Clamp Setup	The gold-standard method for assessing insulin sensitivity and beta-cell function, providing critical data for parameterizing PK-PD models of insulin action [24].

Visualizing Workflows and Physiological Pathways

GLP-1 Mechanism and Model Gap Analysis

Diagram Title: GLP-1 Mechanisms and Model Gaps

Integrated Workflow for Model Enhancement

Diagram Title: Model Enhancement Workflow

The comparative analysis reveals that while the Sorensen, Bergman, and Hovorka models have profoundly advanced diabetes research and therapy development, their inability to explicitly simulate gastric emptying and the incretin effect constitutes a significant performance gap. This limitation impedes the accurate prediction of postprandial glucose dynamics, particularly in the context of modern therapies like GLP-1 receptor agonists. Future research must focus on the structural enhancement of these models, integrating these physiological mechanisms using quantitative data from clinical infusion studies, DDI trials, and advanced in-silico validation platforms. Success in this endeavor will yield more robust and predictive tools, accelerating the development of next-generation automated insulin delivery systems and personalized therapeutics for diabetes and related metabolic disorders.

In the field of metabolic disease modeling, particularly for diabetes, a fundamental trade-off exists between the physiological fidelity of a model and the practical identifiability of its parameters from real-world data. Model-based approaches for drug development and treatment optimization require mathematical models that accurately represent human glucose-insulin dynamics while remaining practically estimable from clinically feasible data. High-fidelity models with numerous parameters often face severe identifiability issues when calibrated with sparse, noisy data collected in routine clinical practice or free-living conditions [52]. This challenge forms the core dilemma in selecting among three prominent models: the detailed Sorensen model, the intermediate-complexity Hovorka model, and the simplified Bergman minimal model.

The parameter identifiability problem refers to the inability to uniquely determine parameter values from available observation data, which can result in non-robust predictive performance and unreliable scientific insights [53] [52]. As dynamical systems are increasingly learned from data rather than prescribed manually, identifiability has become a paramount consideration—without it, no guarantees can be given about model behavior under new conditions or about possible control mechanisms to steer the system [53]. This comparative analysis examines how three established models navigate this critical challenge while maintaining sufficient utility for research and clinical applications.

Model Specifications and Theoretical Foundations

Bergman Minimal Model

The Bergman minimal model represents a simplified yet physiologically relevant representation of glucose-insulin dynamics, characterized by its mathematical tractability and efficiency in parameter estimation [14]. The model employs two quasi-linear differential equations: one representing insulin kinetics in plasma and a second representing the effects of insulin and glucose itself on restoration of the glucose after perturbation [1]. Its simplified structure makes it particularly suitable for control applications where computational efficiency is paramount [54].

Key Features:

• Low parameter count (typically 3-6 key parameters)
• Control-oriented structure favoring algorithm implementation
• Proven utility in insulin sensitivity assessment from IVGTT
• Balance between physiological interpretability and mathematical simplicity

Hovorka Model

The Hovorka model offers intermediate complexity, structuring glucose-insulin dynamics into multiple compartments including glucose absorption, plasma glucose, insulin absorption, and plasma insulin [55]. This model expands upon minimal models by incorporating additional physiological detail while maintaining computational feasibility for clinical applications.

Key Features:

• Moderate parameter count (approximately 20-30 parameters)
• Compartmental structure capturing key physiological processes
• Demonstrated utility in artificial pancreas applications
• Balance between physiological detail and identifiability

Sorensen Model

The Sorensen model represents one of the most comprehensive physiological models of glucose-insulin regulation, incorporating 22 differential equations (mostly nonlinear) that represent glucose concentrations in various body compartments including brain, heart and lungs, liver, gut, kidney, and periphery [56]. With approximately 135 parameters (including initial conditions), this model aims for high physiological fidelity but faces significant identifiability challenges [56].

Key Features:

• High parameter count (∼135 parameters including initial conditions)
• Multi-compartment structure representing organ-specific dynamics
• Detailed representation of physiological mechanisms
• Primarily used for virtual patient simulation rather than individual patient parameter estimation

Table 1: Comparative Specifications of Diabetes Metabolic Models

Characteristic	Bergman Minimal Model	Hovorka Model	Sorensen Model
Model Complexity	Low	Medium	High
Number of Compartments	2-3	4-6	22
Parameter Count	3-6	20-30	~135
Primary Strength	Parameter identifiability	Balance of fidelity and identifiability	Physiological completeness
Primary Limitation	Simplified physiology	Limited organ-specific dynamics	Severe identifiability challenges
Computational Demand	Low	Moderate	High
Clinical Utility	Insulin sensitivity assessment; bolus calculators	Artificial pancreas systems; in-silico trials	Virtual patient simulation; research
Data Requirements	Sparse measurements feasible	Moderate data requirements	Extensive data required

Comparative Identifiability Analysis

Theoretical Identifiability Framework

Identifiability analysis examines whether model parameters can be uniquely determined from observed data, with both structural (theoretical) and practical (estimation) dimensions [53]. Structural identifiability concerns the model structure itself—whether parameters can theoretically be identified given perfect, noise-free data. Practical identifiability addresses whether parameters can be reliably estimated given real-world data constraints including measurement noise, sparsity, and limited observation periods [53] [52].

For linear ordinary differential equation systems, a key theoretical result states that "almost all linear ODEs are identifiable from a single solution trajectory" [53]. However, this result assumes dense matrices and does not hold for sparse systems, which naturally arise in biological contexts where not all variables interact with all others [53]. This has profound implications for physiological models like Sorensen's, where sparsity patterns reflect known biological constraints but simultaneously introduce identifiability challenges.

Practical Identifiability Assessment

In practical applications, identifiability becomes severely constrained by data limitations. Real-world clinical settings typically provide sparse, noisy measurements—sometimes as few as 10-15 glucose measurements per day in ICU settings [55]—which are insufficient to resolve complex models with numerous parameters. This sparsity induces practical unidentifiability, where multiple parameter combinations can equally explain the observed data [52].

The Bergman minimal model demonstrates superior practical identifiability under sparse data conditions due to its parsimonious parameterization [14]. Studies implementing the model with continuous glucose monitoring (CGM) data have shown robust parameter estimation with as few as 6-8 data points per day [14]. The Hovorka model achieves moderate identifiability but often requires Bayesian estimation methods or population approaches to constrain parameter space [55]. The Sorensen model faces severe practical identifiability challenges, with attempts to estimate all parameters from individual patient data typically resulting in ill-conditioned estimation problems and parameter non-identifiability [56].

Diagram 1: Model selection trade-off between physiological fidelity and parameter identifiability under sparse data constraints.

Experimental Protocols and Performance Validation

In-Silico Validation Methodologies

The development of the University of Virginia-University of Padova (UVa-Padova) T1D Simulator represents a landmark achievement in diabetes model validation, providing a platform for in-silico testing of control algorithms [1]. This simulator, accepted by the FDA in 2008 as a substitute for animal trials in preclinical testing, incorporates inter-subject variability through a cohort of virtual subjects generated using joint multivariate probability distributions of model parameters [1]. The simulator has evolved through iterative refinements, initially focusing on meal effects and gradually incorporating additional physiological factors such as circadian changes in insulin sensitivity and dawn phenomenon [1].

Standardized in-silico testing protocols typically involve:

• 30-virtual-subject cohorts representing population variability
• 24-48 hour simulation periods with 3-4 standardized meals
• Performance metrics: time-in-range (70-180 mg/dL), hypoglycemia events, hyperglycemia exposure
• Controller performance comparison against standard therapy baselines

Clinical Validation Studies

Clinical validation approaches vary significantly based on model complexity and application context. For the Bergman minimal model, validation typically employs intravenous glucose tolerance tests (IVGTT) or oral glucose tolerance tests (OGTT) with frequent sampling (10-20 measurements over 2-4 hours) [14]. The Hovorka model has been validated through artificial pancreas clinical trials involving continuous glucose monitoring and insulin pump data over several days [1] [55]. The Sorensen model validation relies primarily on physiological plausibility assessments rather than individual patient parameter estimation, given its structural complexity [56].

Recent advances have explored using continuous glucose monitoring (CGM) as a minimally invasive alternative to plasma glucose measurements for model calibration [57]. Studies simultaneously measuring interstitial glucose using CGM and plasma glucose during OGTT demonstrate that CGM glucose may improve practical identifiability of model parameters compared to plasma glucose, despite the inherent time lag between compartments [57].

Table 2: Experimental Performance Metrics Across Model Types

Performance Metric	Bergman Minimal Model	Hovorka Model	Sorensen Model
Time-in-Range (TIR) Achievement	65-75%	75-85%	Not directly applicable
Hypoglycemia Prevention	Moderate	Good	Not directly tested
Meal Response Accuracy	Limited	Good	Excellent
Exercise Response Capture	Poor	Moderate	Good
Computational Speed	Seconds	Minutes	Hours-Days
Individualization Capacity	High	Moderate	Low
CGM Data Compatibility	Excellent	Good	Poor

Advanced Methodologies for Enhanced Identifiability

Stochastic and Reduced-Order Modeling Approaches

To address identifiability challenges with sparse data, researchers have developed innovative modeling approaches that sacrifice some physiological fidelity for improved parameter estimation. Linear stochastic differential equation models approximate complex nonlinear dynamics while explicitly accounting for uncertainty through stochastic terms [52] [55]. These models represent mean BG behavior through drift terms and capture oscillatory dynamics through diffusion terms, resulting in improved identifiability from sparse clinical data [52].

The Minimal Stochastic Glucose (MSG) model exemplifies this approach, designed specifically for glycemic management in ICU settings where data sparsity and nonstationary patient responses present significant challenges [55]. This model structure mitigates parameter identifiability issues that typically occur when unmeasured system components (e.g., interstitial insulin) are included as states in dynamical system models [55].

Hybrid and Fractional-Order Approaches

Recent research has explored fractional-order calculus as an alternative to traditional integer-order differential equations for representing glucose-insulin dynamics [54]. Fractional derivatives with power-law kernels can model systems with memory effects and may provide better representation of physiological processes with varying time scales [54]. However, these approaches introduce additional mathematical complexity and may exacerbate identifiability challenges despite their theoretical advantages in capturing physiological realism.

Hybrid methodologies that combine mechanistic modeling with data-driven approaches have shown promise in balancing fidelity and identifiability [52] [57]. These approaches use mechanistic models to capture known physiology while employing machine learning techniques to model residual variations and unmeasured influences, potentially offering a path forward for utilizing complex models like Sorensen's in practical applications.

The Scientist's Toolkit: Essential Research Reagents

Table 3: Essential Research Tools and Computational Resources

Tool/Resource	Function	Application Context
UVa-Padova Simulator	Preclinical testing of control algorithms	Regulatory submission; algorithm development
CGM Devices (Medtronic, Dexcom, Abbott)	Continuous interstitial glucose monitoring	Model calibration; free-living validation
Dual Extended Kalman Filter	Simultaneous parameter and state estimation	Personalized model calibration from sparse data
Linear Quadratic Gaussian Control	Optimal control under uncertainty	Insulin dosing algorithms; clinical decision support
Fractional-Order Calculus	Modeling memory effects in physiology	Enhanced physiological representation
Stochastic Differential Equations	Representing uncertain dynamics	Robust control under physiological variability
Model Predictive Control	Constrained optimization for drug dosing	Artificial pancreas systems; bolus calculators

The fundamental trade-off between physiological fidelity and parameter identifiability remains a central consideration in metabolic modeling for drug development and treatment optimization. The Bergman minimal model offers superior practical identifiability from sparse clinical data but sacrifices physiological completeness. The Sorensen model provides exceptional physiological fidelity but faces severe identifiability challenges that limit its practical utility for individual patient modeling. The Hovorka model represents a middle ground, balancing sufficient physiological detail with reasonable identifiability characteristics.

Emerging approaches including stochastic modeling, fractional-order calculus, and hybrid mechanistic-machine learning methods offer promising avenues for navigating this fidelity-identifiability trade-off. Furthermore, the increasing availability of continuous glucose monitoring data provides opportunities for enhancing parameter identifiability across all model classes. Researchers must carefully select models based on their specific application context, data availability, and precision requirements, recognizing that the optimal choice represents a calculated compromise between biological realism and practical estimability.

Mathematical models of glucose-insulin dynamics are indispensable tools in diabetes research, enabling the simulation of metabolic processes, prediction of disease progression, and development of automated treatment systems. The Bergman Minimal Model, the Sorensen model, and the Hovorka model represent three foundational approaches with distinct structures, complexities, and applications. These models vary significantly in their physiological detail, parameter requirements, and adaptability to different diabetic populations. The Bergman Minimal Model offers a simplified three-equation structure that captures essential glucose-insulin interactions, making it suitable for basic research and controller design [13] [58]. In contrast, the Sorensen model provides a comprehensive organ-based compartmental approach with up to 28 differential equations, offering detailed physiological representation but requiring extensive parameterization [59]. The Hovorka model strikes a balance with its modular structure focusing on glucose absorption, insulin kinetics, and insulin action subsystems [1].

Adapting these models for specific populations—Type 1 Diabetes (T1DM), Type 2 Diabetes (T2DM), and prediabetes—requires careful modification to account for distinct pathophysiological mechanisms. T1DM adaptations typically focus on replacing endogenous insulin secretion with exogenous insulin delivery, while T2DM modifications must address insulin resistance and beta-cell dysfunction [1] [60]. Prediabetes models aim to capture the early metabolic dysregulation that precedes overt diabetes [14]. This review systematically compares how these three fundamental models have been modified for different diabetic populations, providing researchers with a structured analysis of adaptation methodologies, validation approaches, and performance characteristics across specific use cases and patient groups.

Model Structures and Theoretical Foundations

Bergman Minimal Model

The Bergman Minimal Model employs a simplified three-compartment structure described by ordinary differential equations representing plasma glucose concentration, remote insulin concentration, and plasma insulin concentration [13] [58]. Its mathematical representation follows:

• Glucose dynamics: ẋG(t) = -p₁xG(t) - Gb - xG(t)xRI(t) + d(t) [13]
• Remote insulin dynamics: ẋRI(t) = -p₂xRI(t) + p₃xI(t) - Ib [13]
• Plasma insulin dynamics: ẋI(t) = -p₄xI(t) - Ib + u(t) [13]

Where xG is plasma glucose concentration, xRI is remote insulin concentration, xI is plasma insulin concentration, Gb and Ib are basal levels, d(t) represents glucose appearance from meals, and u(t) is the insulin delivery rate [13]. The model's key parameters (p₁-p₄) represent insulin-independent glucose uptake, decrease in tissue glucose uptake ability, insulin-dependent glucose uptake enhancement, and insulin degradation rate respectively [13]. This parsimonious structure enables efficient parameter identification and controller design but sacrifices physiological detail, making it most suitable for applications where computational efficiency outweighs the need for comprehensive physiological representation.

Sorensen Model

The Sorensen model employs a detailed physiological structure with compartmental representations of key organs including brain, heart, lungs, liver, and peripheral tissues [58] [59]. This organ-based approach originally comprised 19 differential equations [58], with extensions expanding it to 28 dimensions [59]. The model simulates solute transport through blood flow connections between compartments, with mass balance equations quantifying concentration changes for glucose, insulin, glucagon, and incretins [59]. Its state-space representation includes variables for glucose concentrations across different vascular and tissue compartments (GBV, GBI, GH, GL, GK, GPV, GPI, GG, GPN), insulin concentrations (IB, IH, IL, IK, IPV, IPI, IG, I_PN), and related metabolic functions [59]. This comprehensive structure enables detailed physiological simulation but requires extensive parameterization with 46 parameters for hemodynamical processes and 67 parameters for metabolic rates [59], making it computationally intensive and challenging to personalize for individual patients.

Hovorka Model

The Hovorka model employs a modular structure with three primary subsystems: glucose absorption, insulin kinetics, and insulin action [1] [58]. Unlike the organ-based Sorensen approach, the Hovorka model focuses on functional compartments representing distinct physiological processes. The glucose subsystem models transport between plasma and tissues, while the insulin subsystem represents subcutaneous insulin absorption and plasma kinetics [1]. The insulin action subsystem quantifies the effects of insulin on glucose disposal, endogenous production, and transport [1]. This modular design facilitates integration with artificial pancreas systems, as individual components can be modified without overhauling the entire model structure. The model has been incorporated into the University of Virginia-University of Padova (UVa-Padova) T1D Simulator, which was accepted by the FDA in 2008 as a substitute for animal trials in preclinical testing of control strategies [1]. This regulatory acceptance highlights the model's robustness and establishes it as a benchmark for closed-loop system development.

Table 1: Fundamental Characteristics of Core Diabetes Models

Characteristic	Bergman Minimal Model	Sorensen Model	Hovorka Model
Model Structure	3-compartment ODE	Organ-based compartments (19-28 ODEs)	Modular subsystems (glucose, insulin, insulin action)
Primary Applications	Basic research, controller design	Physiological simulation, pathophysiology studies	Artificial pancreas, clinical trials
Parameter Complexity	Low (4 key parameters)	High (46+67 parameters)	Moderate
Computational Demand	Low	High	Moderate
Regulatory Status	Research use	Research use	FDA-accepted for preclinical trials

Model Adaptations for Specific Populations

Type 1 Diabetes (T1DM) Adaptations

T1DM model adaptations primarily address the absence of endogenous insulin secretion, necessitating exogenous insulin delivery. The UVa-Padova T1D Simulator, which incorporates a modified Hovorka model, represents the gold standard for T1DM simulation. This simulator substitutes the endogenous insulin secretion subsystem with exogenous insulin delivery and incorporates inter-individual variability through parameters derived from clinical data of 204 individuals [1]. The resulting T1DM simulation model includes 13 differential equations with 35 parameters (26 free and 9 derived from steady-state constraints) and has been expanded to include 100 adult, 100 adolescent, and 100 children virtual subjects [1]. This population variability enables robust testing of control algorithms across different age groups.

For the Bergman model, T1DM adaptation typically involves setting endogenous insulin production to zero and adding an external insulin delivery term u(t) [13]. This simplified approach enables controller design, as demonstrated in artificial pancreas systems where Bergman-based controllers successfully regulated blood glucose in the UVa-Padova simulator environment despite being designed for intravenous insulin delivery but tested with subcutaneous delivery [13]. The Sorensen model requires more extensive modification for T1DM, including disabling pancreatic insulin secretion functions and incorporating subcutaneous insulin absorption kinetics. These adaptations make the Sorensen model valuable for studying metabolic differences between healthy and T1DM states but computationally challenging for real-time control applications.

Type 2 Diabetes (T2DM) Adaptations

T2DM model adaptations must address insulin resistance and progressive beta-cell dysfunction, creating more complex modification requirements. The Beta-cell-Insulin-Glucose (BIG) model, an extension of the Bergman Minimal Model, incorporates beta-cell mass dynamics with a slow, glucose-dependent feedback loop to regulate beta-cell mass [60]. This addition enables the model to simulate long-term disease progression and treatment effects, including lifestyle, pharmacologic, and insulin therapies [60]. The BIG model has been further extended to incorporate metformin dynamics by explicitly including metformin as a continuous-state variable that affects insulin sensitivity, hepatic glucose production, and beta-cell function [60].

The Sorensen model has been adapted for T2DM through individual parameterization of mathematical functions representing impaired metabolic rates [59]. This methodology involves fitting specific functions to clinical data from T2DM patients, particularly those representing metabolic processes significantly contributing to hyperglycemia [59]. The adapted model successfully simulates graded intravenous glucose tests and oral glucose tolerance tests at different doses, demonstrating its ability to capture T2DM pathophysiology [59]. The Hovorka model requires modifications to represent insulin resistance and beta-cell dysfunction for T2DM applications, including parameter adjustments to decrease insulin sensitivity and modify beta-cell response characteristics [60]. These adaptations enable the simulation of T2DM-specific metabolic abnormalities but may sacrifice some physiological detail compared to the comprehensive Sorensen approach.

Prediabetes Adaptations

Prediabetes modeling requires capturing early metabolic dysregulation before overt diabetes development. A novel mathematical model based on Bergman's minimal model with parametric adjustments has been developed specifically for prediabetes [14]. This model incorporates four components: (1) plasma glucose concentration, (2) insulin-induced glucose reduction, (3) plasma insulin concentration, and (4) interstitial glucose concentration [14]. The model uses a Dual Extended Kalman Filter for dynamic parameter estimation and accounts for parametric variability, enabling accurate representation of the subtle metabolic alterations characteristic of prediabetes [14].

Validation against 311 days of continuous glucose monitoring data from 43 participants (14 healthy and 29 at risk) demonstrated strong agreement with experimental data (r = 0.98, p < 0.01) [14]. This prediabetes-specific model represents one of the first tailored to this transitional metabolic state, addressing a critical gap in preventive strategy development. The Sorensen and Hovorka models can be adapted for prediabetes by adjusting parameters to reflect intermediate states of insulin resistance and beta-cell dysfunction, but their complexity may be unnecessary for early metabolic changes that prediabetes models aim to capture.

Table 2: Population-Specific Model Modifications and Applications

Population	Bergman Model Adaptations	Sorensen Model Adaptations	Hovorka Model Adaptations
T1DM	Endogenous insulin set to zero; external insulin delivery added [13]	Pancreatic insulin secretion disabled; subcutaneous insulin absorption added [59]	Endogenous insulin subsystem replaced with exogenous delivery; UVa-Padova simulator [1]
T2DM	BIG model extension with beta-cell mass dynamics; metformin integration [60]	Individual parameterization of impaired metabolic functions [59]	Parameter adjustments for insulin resistance and beta-cell dysfunction [60]
Prediabetes	Parametric adjustments with Dual Extended Kalman Filter [14]	Limited research on specific adaptations	Limited research on specific adaptations
Validation Metrics	Controller performance in artificial pancreas [13]	OGTT simulation accuracy [59]	FDA acceptance as preclinical tool [1]

Experimental Validation and Performance Comparison

In Silico Trial Methodologies

The UVa-Padova T1D Simulator represents the most rigorously validated experimental platform for diabetes model testing, having received FDA acceptance as a substitute for animal trials in preclinical testing of control strategies [1]. This simulator employs a comprehensive methodology beginning with model parameterization based on dynamic profiling of carbohydrate metabolism in over 200 adults [1]. The simulation approach involves generating virtual subject cohorts that span observed inter-individual variability in the general population with diabetes, enabling realistic assessment of treatment performance [1]. Simulation scenarios typically include standardized meal challenges, unannounced meals, and various disturbance conditions to test robustness [1] [13].

For T2DM and prediabetes models, validation typically follows different methodologies. The extended Sorensen model for T2DM was validated through oral glucose tolerance tests at different doses, with comparison between simulations and clinical data showing acceptable description of blood glucose dynamics in T2DM [59]. The prediabetes model validation utilized continuous glucose monitoring data from FreeStyle Libre sensors, with parameter estimation using the Levenberg-Marquardt algorithm to minimize estimation error [14]. These methodological differences reflect the distinct application priorities: T1DM models emphasize controller robustness for artificial pancreas systems, while T2DM and prediabetes models focus on accurate representation of metabolic pathophysiology.

Performance Metrics and Comparative Outcomes

Performance evaluation of adapted diabetes models employs standardized metrics relevant to each population's clinical needs. For T1DM artificial pancreas applications, the key metrics include:

• Time in Range (TIR): Percentage of time glucose remains between 70-180 mg/dL
• HbA1c reductions: Typically 0.2% to 0.5% compared with open-loop therapy [1]
• Hypoglycemia avoidance: Minimal time below 70 mg/dL
• Postprandial glucose control: Peak reduction after meals

Bergman-based controllers have demonstrated effective performance in these metrics, with one multi-objective output feedback controller achieving robust glucose regulation with lesser insulin and hypoglycemia avoidance compared to compound IMC controllers [13]. The Hovorka-model-based artificial pancreas systems have shown significant improvements in time-in-range metrics across multiple commercial systems (Medtronic 670G/770G/780G, Tandem t:slim X2, OmniPod 5) [18].

For T2DM models, performance focuses on accurate pathophysiological representation rather than control performance. The adapted Sorensen model successfully simulated scheduled graded intravenous glucose tests and oral glucose tolerance tests at different doses, with statistical analysis confirming acceptable agreement with clinical data [59]. The prediabetes model demonstrated exceptionally high correlation with experimental data (r = 0.98, p < 0.01) across 311 simulations [14], indicating strong predictive capability for early metabolic dysregulation.

Diagram 1: Model Adaptation Workflow (63 characters)

Successful adaptation of diabetes models requires specialized computational tools and experimental resources. The UVa-Padova T1D Simulator represents the most critical resource, providing a validated platform for in silico testing accepted by regulatory agencies [1]. This simulator includes 100 adult, 100 adolescent, and 100 children virtual subjects with realistic inter-subject variability, enabling comprehensive assessment of population-wide performance [1]. Continuous Glucose Monitoring (CGM) systems with mean absolute relative differences (MARD) ≤10% provide essential input data, with factory-calibrated sensors (FreeStyle Libre, Dexcom, Medtronic) enabling reliable glucose monitoring without frequent calibrations [1] [14].

For parameter estimation, the Dual Extended Kalman Filter (DEKF) implementation enables dynamic parameter estimation while accounting for parametric variability, as demonstrated in prediabetes modeling [14]. The Levenberg-Marquardt algorithm provides effective error minimization for model fitting to clinical data [14]. MATLAB Simulink with ODE solvers (ode45) serves as the primary computational environment for model implementation and simulation [13] [59]. These tools collectively provide the foundation for robust model adaptation and validation across different diabetic populations.

Table 3: Essential Research Resources for Diabetes Model Development

Resource Category	Specific Tools	Primary Applications	Key Features
Simulation Platforms	UVa-Padova T1D Simulator [1]	Preclinical testing of control algorithms	FDA-accepted; 300 virtual subjects (adults, adolescents, children)
Parameter Estimation	Dual Extended Kalman Filter [14]	Dynamic parameter estimation	Accounts for parametric variability
Optimization Algorithms	Levenberg-Marquardt [14]	Error minimization in model fitting	Robust convergence for nonlinear systems
Clinical Data Collection	CGM Systems (FreeStyle Libre) [14]	Continuous glucose monitoring	Factory-calibrated; MARD <10%
Computational Environment	MATLAB Simulink [13] [59]	Model implementation and simulation	ODE solvers (ode45)

The adaptation of mathematical models for specific diabetic populations requires careful consideration of model characteristics relative to research objectives. The Bergman Minimal Model offers computational efficiency and straightforward parameter estimation, making it ideal for controller design and preliminary investigations. The Sorensen Model provides comprehensive physiological representation, enabling detailed pathophysiological studies but requiring extensive parameterization. The Hovorka Model strikes an effective balance with its modular structure, supporting artificial pancreas development and regulatory submissions through the UVa-Padova simulator.

Population-specific adaptations follow distinct pathways: T1DM focuses on exogenous insulin delivery, T2DM requires representation of insulin resistance and beta-cell dysfunction, and prediabetes demands sensitive detection of early metabolic alterations. The continued refinement of these models, incorporating factors like physical activity, stress, menstrual cycle variations, and therapeutic interventions, will enhance their clinical relevance. As diabetes modeling evolves, the strategic selection and adaptation of these foundational models will remain essential for advancing both basic understanding and clinical management of different diabetic populations.

Mathematical modeling of the glucose-insulin system is fundamental to advancing diabetes research, drug development, and artificial pancreas technologies. Researchers face a fundamental trade-off: choosing between physiologically detailed models that offer comprehensive biological insight and simpler, computationally efficient models that enable rapid simulation and parameter estimation. This guide objectively compares three established mathematical models—the Sorensen, Hovorka, and Bergman Minimal models—focusing on their computational performance and applicability in different research contexts. The Sorensen model represents the high-complexity end of the spectrum, the Bergman Minimal model occupies the low-complexity end, and the Hovorka model strikes a middle ground, often described as a "maximal" model with intermediate complexity [61] [1]. Understanding their performance trade-offs is crucial for selecting the appropriate tool for specific research objectives, whether for detailed physiological investigation, control algorithm development, or high-throughput in silico trials.

The table below summarizes the core architectural and physiological characteristics of the three models, which directly influence their computational demands.

Table 1: Fundamental Characteristics of the Glucose-Insulin Models

Characteristic	Sorensen Model	Hovorka Model	Bergman Minimal Model
Physiological Scope	Whole-body physiology (brain, liver, heart/lungs, periphery, gut, kidney) [61]	Glucose metabolism for Type 1 Diabetes [61]	Plasma glucose-insulin dynamics [1]
Model Complexity	High-complexity "maximal" model [61]	Intermediate-complexity "maximal" model [61]	Low-complexity "compact" model [61] [1]
Number of Compartments	3 sub-models (glucose, insulin, glucagon) across multiple organs [61]	Not explicitly stated, but significantly fewer than Sorensen	2-3 core compartments [1] [25]
Primary Application	Investigation of detailed physiological behaviors [61]	Development of control algorithms [61]	Rapid parameter estimation (e.g., insulin sensitivity) [1] [25]
Regulatory Status	Research tool	Research tool	Foundation for FDA-accepted simulators [1]

Quantitative Performance and Experimental Data

The structural differences between the models lead to direct trade-offs between simulation speed and physiological richness. The following table synthesizes quantitative and qualitative performance metrics based on comparative studies.

Table 2: Computational Performance and Physiological Fidelity Comparison

Performance Metric	Sorensen Model	Hovorka Model	Bergman Minimal Model
Computational Speed	Slowest (high computational load) [61]	Intermediate [61]	Fastest (optimized for parameter estimation) [25]
Physiological Detail	Highest (most complete representation) [61]	Intermediate (well-documented T1D physiology) [61]	Lowest (focus on core dynamics) [1]
Parameter Identifiability	Challenging (many parameters, requires extensive data) [61]	More robust than Sorensen, used for control [61]	High (designed for reliable estimation from clinical tests) [52] [25]
Inter-individual Variability	Can be implemented but computationally intensive [61]	Supports personalization for AP algorithms [61]	Foundation for population simulators (e.g., UVa-Padova) [1]
Ideal Use Case	Simulating complex organ-specific phenomena [61]	In-silico testing of Artificial Pancreas systems [61] [1]	Large-scale cohort studies and rapid insulin sensitivity screening [25]

Experimental Protocols for Model Comparison

The performance data in Table 2 is derived from standardized in-silico experiments that stress-test the models under controlled conditions. The primary methodologies include:

• Intravenous Glucose Tolerance Test (IVGTT): A continuous basal insulin infusion (e.g., 6.67 mU/min) is administered alongside an intravenous glucose bolus (e.g., 0.5 g/kg) over a short period (e.g., 3 minutes). This tests the model's ability to capture rapid plasma glucose and insulin dynamics [61].
• IVGTT with Insulin Bolus: The standard IVGTT protocol is combined with a large insulin bolus (e.g., 1000 mU delivered in 1 minute) to further challenge the model's response to extreme hormonal perturbations [61].
• Oral Glucose Tolerance Test (OGTT): An oral glucose load (e.g., 100 g over 1 minute) is simulated, often with concurrent basal insulin delivery. This tests the model's capability to represent delayed intestinal glucose absorption and the incretin effect, which is more critical for the Sorensen model with its gastrointestinal tract compartment [61].

Decision Framework and Research Toolkit

Selecting the optimal model depends on the researcher's primary objective. The workflow below outlines the key decision points.

The Scientist's Toolkit: Essential Research Reagents

Table 3: Key Tools and Solutions for Glucose-Insulin Modeling Research

Tool/Solution	Function in Research	Example Use Case
UVa-Padova T1D Simulator	A population of in-silico virtual subjects replacing animal trials for preclinical testing [1].	Testing safety and efficacy of new control algorithms across a diverse virtual population.
SAAM II Software	A validated software tool for performing minimal model analysis and parameter estimation [25].	Automating the calculation of insulin sensitivity (Si) and beta-cell responsivity (Φ) from OGTT data.
Automated Oral Minimal Model (AOMM)	A tool that streamlines and automates the entire Oral Minimal Model workflow [25].	High-throughput batch processing of large clinical trial datasets for metabolic parameter extraction.
Continuous Glucose Monitor (CGM)	Provides real-time, dense glucose concentration data for model personalization and validation [1] [62].	Collecting interstitial glucose measurements to fit and validate patient-specific model parameters.
Insulin Pump (CSII)	Enforces controlled subcutaneous insulin delivery for experimental input or closed-loop control [1] [62].	Acting as the actuator in an Artificial Pancreas system driven by a model-based control algorithm.

The choice between the Sorensen, Hovorka, and Bergman Minimal models is not a question of which is universally superior, but which is optimal for a specific research task. The Sorensen model offers the highest physiological fidelity for investigating complex, organ-level questions but at the greatest computational cost. The Bergman Minimal model provides the fastest computational performance for high-throughput parameter estimation and large-scale studies. The Hovorka model effectively bridges this gap, offering sufficient physiological detail for developing and testing artificial pancreas algorithms with robust computational performance. By aligning research goals with the inherent strengths and limitations of each model, scientists can dramatically improve the efficiency and effectiveness of their diabetes research and drug development workflows.

Benchmarking Performance: Validation, Regulatory Status, and Comparative Efficacy

The development of effective treatments and management strategies for diabetes relies heavily on mathematical models that can accurately simulate glucose-insulin dynamics. Among the most prominent models used in research are the Sorensen model, Bergman Minimal Model, and Hovorka model, each offering different approaches to representing the complex physiological processes involved in glucose regulation [12] [16] [63]. These models serve as critical tools for in-silico testing of closed-loop insulin delivery systems (the artificial pancreas), drug development, and understanding fundamental pathophysiology [63] [64].

The Intravenous Glucose Tolerance Test (IVGTT) and Oral Glucose Tolerance Test (OGTT) represent two fundamental experimental protocols used to perturb the glucose-insulin system and reveal its dynamic characteristics [65] [66]. IVGTT involves direct intravenous administration of glucose, allowing assessment of acute insulin response and insulin sensitivity without the confounding variables of gastrointestinal absorption [65] [67]. OGTT, which involves oral glucose administration, provides a more physiological assessment that incorporates incretin effects and other gut-mediated processes [68] [66]. This comparison guide objectively evaluates the performance of three key mathematical models when simulating these important experimental protocols, providing researchers with quantitative data to inform model selection for specific applications.

The three models represent different philosophical approaches to modeling glucose-insulin dynamics, with varying levels of physiological detail and mathematical complexity.

Bergman Minimal Model (Minimal Model)

The Bergman Minimal Model (MINMOD) is a compactly parameterized model originally developed to estimate insulin sensitivity and glucose effectiveness from IVGTT data [16] [67]. It consists of three differential equations representing glucose kinetics, insulin kinetics, and insulin action [67] [69]. Its simplicity enables relatively straightforward parameter identification, but this comes at the cost of physiological comprehensiveness. The model has known limitations, including potential artifacts in insulin sensitivity estimation when acute insulin response is high [67].

Hovorka Model

The Hovorka model strikes a balance between complexity and practicality, with eight differential equations organizing the glucose-insulin system into three main subsystems: glucose, insulin, and insulin action [16]. The glucose subsystem includes two compartments representing accessible (plasma) and inaccessible glucose, while the insulin action subsystem represents the effect of insulin on glucose transport, glucose disposal, and endogenous glucose production [16]. This model was specifically formulated for type 1 diabetes applications and has seen substantial use in artificial pancreas development [12] [16].

Sorensen Model

The Sorensen model is the most comprehensive and physiologically detailed of the three compared systems. It is a compartmental model that explicitly represents key organ systems including the brain, liver, heart and lungs, periphery, gut, and kidneys [12] [69]. Originally developed to simulate both normal and diabetic physiology, it offers unparalleled physiological fidelity but requires identification of numerous parameters [12]. The model's complexity makes it computationally demanding but highly valuable for applications requiring detailed physiological insight.

Table 1: Fundamental Characteristics of the Compared Models

Characteristic	Bergman Minimal Model	Hovorka Model	Sorensen Model
Original Purpose	Estimate insulin sensitivity from IVGTT	Type 1 diabetes simulation and control	Simulate both normal and diabetic physiology
Model Complexity	Low (3 differential equations)	Medium (8 differential equations)	High (19+ differential equations)
Physiological Resolution	Single glucose compartment	Two glucose compartments	Multiple organ-system compartments
Primary Applications	Insulin sensitivity assessment; research	Artificial pancreas development; control algorithms	Detailed physiological studies; therapy design
Regulatory Status	Research use	Research use	Research use

Comparative Performance in Simulated Experiments

IVGTT Simulation Protocols and Outcomes

The Intravenous Glucose Tolerance Test provides a standardized method for assessing first-phase insulin secretion and insulin sensitivity without the confounding factors of gastric emptying or incretin effects [65]. In simulation studies, the three models exhibit markedly different behaviors when subjected to IVGTT protocols.

A comparative study implemented an IVGTT with a continuous basal insulin infusion of 6.67 mU/min alongside 0.5 g/kg of glucose administered over 3 minutes [12]. The Sorensen model demonstrated the most detailed response capture, reflecting its multi-compartment structure, while the Hovorka model provided a balanced representation suitable for control applications. The Minimal Model, while computationally efficient, showed limitations in capturing the full dynamics, particularly in representing the distribution phases [12] [67].

A critical finding from simulation studies is that the Minimal Model may artifactually interpret strong insulin secretion as weak insulin action [67]. This occurs because glucose rises rapidly at the start of the IVGTT and reaches levels largely independent of insulin sensitivity, while insulin during this period is determined by the acute insulin response (AIR). The Minimal Model effectively interprets this combination as low insulin sensitivity even when actual insulin sensitivity is unchanged [67]. This limitation is particularly relevant when comparing populations with different characteristic AIR levels, such as Black versus White individuals, where the model may underestimate insulin sensitivity in groups with higher AIR [67].

Table 2: IVGTT Simulation Performance Across Models

Performance Metric	Bergman Minimal Model	Hovorka Model	Sorensen Model
Acute Insulin Response Capture	Limited; potential artifact [67]	Moderate	Comprehensive
Glucose Disposition Accuracy	Good for simple kinetics	Very good	Excellent
Parameter Identifiability	High	Moderate	Challenging
Computational Demand	Low	Moderate	High
Known Limitations	May underestimate SI with high AIR [67]	Less physiological detail than Sorensen	Requires modification for T1DM [12]

OGTT Simulation Protocols and Outcomes

The Oral Glucose Tolerance Test introduces additional complexity through gastrointestinal glucose absorption and incretin effects, presenting different challenges for mathematical modeling [68] [66]. Simulation protocols typically involve administration of 75-100g of glucose orally, with subsequent monitoring of glucose and insulin dynamics [12] [68].

In comparative simulations using OGTT protocols with 100g of glucose administered over 1 minute alongside continuous basal insulin infusion, the Sorensen model demonstrated superior capability in capturing the prolonged dynamics of glucose absorption and disposition, owing to its explicit gastrointestinal compartment and detailed organ-system representation [12]. The Hovorka model, with its dedicated meal absorption model, performed well for control-oriented applications [69]. The Minimal Model requires significant modifications to handle OGTT protocols effectively, as its original formulation was designed for intravenous challenges [67].

Validation studies comparing OGTT-based indices to gold standard measures have found that certain derived parameters (AUC (I0-30)/AUC (G0-30), I30/G30, first-phase Stumvoll) show high correlation with clamp-derived measures of insulin secretion [66]. The more complex models (Sorensen and Hovorka) can better simulate these indices under varying physiological conditions, making them valuable for pre-clinical drug development [68].

Figure 1: OGTT Simulation Workflow - The diagram illustrates the key physiological processes involved in OGTT simulations, highlighting points where model differences emerge.

Implications for Artificial Pancreas Development

The choice of simulation model has profound implications for artificial pancreas development, where in-silico testing provides a critical pathway for regulatory approval and algorithm refinement [64]. The FDA has accepted specific simulation platforms based on modified meal models for proof-of-concept testing of closed-loop algorithms, highlighting the importance of validated models in the device development pipeline [64].

Control algorithm studies demonstrate that success in simulation does not always translate to clinical effectiveness, and model choice significantly impacts predicted controller performance [69]. For instance, PID control strategies that appear effective in simpler models may prove inadequate when tested against more physiologically comprehensive models or in clinical settings [69]. The Sorensen and Hovorka models have shown particular utility for testing advanced control strategies like Model Predictive Control (MPC) due to their ability to represent critical nonlinearities and delays in the glucose-insulin system [16] [63].

Table 3: Model Selection Guidelines for Specific Applications

Research Application	Recommended Model	Rationale
Insulin Sensitivity Estimation	Bergman Minimal Model	Purpose-built for IVGTT-derived SI estimation
Artificial Pancreas Algorithm Development	Hovorka Model	Balanced complexity; control-relevant formulation
Physiological Mechanism Studies	Sorensen Model	Comprehensive organ-system resolution
Drug Development & Testing	Hovorka or Sorensen Models	Adequate dynamics for pharmacodynamic assessment
Educational Applications	Bergman or Hovorka Models	Manageable complexity with physiological relevance

Successful implementation of in-silico comparative studies requires specific methodological tools and computational resources. The following toolkit outlines essential components for researchers undertaking IVGTT and OGTT simulation studies.

Table 4: Essential Research Resources for In-Silico Glucose-Insulin Studies

Tool Category	Specific Solution	Function & Application
Simulation Software	MATLAB/Simulink	Implementation and simulation of model equations; control algorithm design
Parameter Estimation	ADAPT	System identification and parameter estimation from experimental data
Closed-Loop Testing	JDRF Accepted Simulator [64]	Regulatory-accepted platform for artificial pancreas algorithm validation
Clinical Data	Wynn Database [65]	Large-scale IVGTT datasets for model validation and parameter identification
Model Validation Metrics	Control Variability Grid Analysis (CVGA) [64]	Standardized assessment of glycemic control performance

The comparative analysis of the Bergman Minimal, Hovorka, and Sorensen models reveals a classic trade-off in mathematical modeling: balancing physiological comprehensiveness against practical identifiability and computational burden. The Bergman Minimal Model remains valuable for specific applications like insulin sensitivity estimation from IVGTT data, despite its limitations in interpreting high acute insulin responses. The Hovorka model offers an effective compromise for control-oriented applications, particularly in artificial pancreas development. The Sorensen model provides the most physiologically comprehensive platform for mechanistic studies and detailed physiological investigation.

Model selection should be guided by research objectives rather than presumed superiority. Simpler models suffice for parameter estimation-focused studies, while complex physiological investigations necessitate more comprehensive models. Future model development should address identified limitations while maintaining validation against both controlled experiments and clinical data, ensuring these important tools continue to advance diabetes research and therapeutic development.

Table of Contents

• Introduction
• Model Architectures and Physiological Basis
• Comparative Performance Analysis
• Experimental Protocols for Model Validation
• The Scientist's Toolkit: Essential Research Reagents and Materials
• Conclusion

Mathematical models of glucose-insulin dynamics are indispensable tools in diabetes research, aiding in everything from understanding fundamental physiology to developing automated insulin delivery systems. The Sorensen model, Bergman Minimal Model, and Hovorka model represent different philosophical approaches to this challenge, each making distinct trade-offs between physiological detail and practical utility. This guide provides a structured, objective comparison of these three prominent models, focusing on their core architecture, experimental validation, and physiological plausibility. The assessment is framed within ongoing research efforts to refine these models, such as correcting implementation errors in the Sorensen model and extending the Hovorka model with improved equations [5] [3]. The performance of algorithms based on these models is critical for advancing treatments, particularly for Type 1 Diabetes (T1D) [70] [1].

Model Architectures and Physiological Basis

The fundamental difference between the models lies in their conceptual design: the Sorensen model is a maximal model aiming for a comprehensive physiological representation, the Bergman model is a minimal model focused on parameter estimation from data, and the Hovorka model strikes a balance as an intermediate-complexity model well-suited for simulation and control.

The diagram below illustrates the core structural relationships and typical applications of each model.

Key Model Characteristics

Feature	Sorensen Model	Bergman Minimal Model	Hovorka Model
Modeling Philosophy	Maximal, physiologically based	Minimal, data-driven	Intermediate, control-oriented
Complexity	22+ differential equations; ~135 parameters [5]	3 differential equations [71] [72]	8+ differential equations (3 main subsystems) [3]
Compartments	Brain, heart/lungs, liver, gut, kidney, periphery [5]	Plasma glucose, remote insulin, plasma insulin [72]	Glucose (2 compartments), insulin, insulin action [3]
Key Outputs	Organ-specific glucose/insulin concentrations	Insulin sensitivity (SI), Glucose effectiveness (SG) [71] [28]	Blood glucose level, insulin kinetics
Primary Application	Detailed physiological studies [5]	Metabolic parameter estimation from IVGTT [71] [72]	Artificial pancreas (AP) algorithms & in-silico testing [70] [1] [3]

Comparative Performance Analysis

A model's performance is multi-faceted, evaluated based on its ability to replicate known physiology, its clinical validation, and its computational efficiency for specific tasks like controller design.

Quantitative Performance and Validation Data

Performance Metric	Sorensen Model	Bergman Minimal Model	Hovorka Model
Physiological Plausibility	High (explicit organ-level dynamics) [5]	Low (lumped-parameter, "remote" compartment) [72]	Medium-High (captures key gluco-regulatory processes) [1]
Clinical Validation	Good fit for IVGTT, IVITT; requires empirical adjustment for meals [5]	SI validated against glucose clamp [71]	Foundation of FDA-approved simulator [1]
Handling of Meals	Requires added gastrointestinal sub-model [5]	Not designed for oral meals (IVGTT only)	Explicitly includes meal absorption [1]
Inter-individual Variability	Represents a single "average" virtual patient [5]	Parameters (e.g., SI) vary between individuals [28]	100+ virtual adult, adolescent, and pediatric subjects in simulator [1]
Reported Simulation Error	-	Mean RMSE ~15 mg/dL (rat data) [28]	Time-in-target: 72-88% (in-silico, improved equations) [3]

Experimental Protocols for Model Validation

The credibility of model predictions rests on rigorous experimental validation. The protocols below are standardized tests used to parameterize and validate these models.

Intravenous Glucose Tolerance Test (IVGTT)

The IVGTT is a cornerstone experiment for validating model dynamics, particularly for the Bergman Minimal Model [71] [72].

• Purpose: To assess the body's ability to clear a glucose bolus and the corresponding insulin response, enabling estimation of insulin sensitivity (SI) and glucose effectiveness (SG).
• Procedure:
  • After an overnight fast, a standardized intravenous (IV) bolus of glucose (e.g., 0.3-0.5 g/kg body weight) is administered over a short period.
  • Blood samples are collected frequently (e.g., every 1-10 minutes) for 3 hours to measure plasma glucose and insulin concentrations.
  • The resulting time-course data is fitted to the model equations using optimization algorithms (e.g., genetic algorithms, nonlinear least squares) to estimate patient-specific parameters [28].

Meal Tolerance Test (MTT)

The MTT assesses a model's ability to handle the complex physiology of oral glucose intake.

• Purpose: To validate model predictions of postprandial glucose dynamics, including the roles of gastric emptying, gut-derived incretin hormones, and carbohydrate absorption.
• Procedure:
  • After an overnight fast, a standardized meal with a known carbohydrate content (e.g., 50-100g) is consumed.
  • Blood samples are collected frequently over 4-6 hours to measure glucose, insulin, and sometimes incretin hormones.
  • For complex models like Sorensen's, this test revealed the need for an explicit gastrointestinal tract sub-model to accurately simulate glucose appearance [5].

Euglycemic-Hyperinsulinemic Clamp

The clamp is considered the "gold standard" for independent validation of a key parameter, insulin sensitivity.

• Purpose: To directly measure insulin sensitivity in vivo for comparison against model-predicted SI values.
• Procedure:
  • A constant IV insulin infusion is started to create a steady-state of hyperinsulinemia.
  • A variable IV glucose infusion is simultaneously adjusted to "clamp" blood glucose concentration at a basal, euglycemic level.
  • The glucose infusion rate (GIR) required to maintain euglycemia is a direct measure of whole-body insulin sensitivity. The Bergman Minimal Model's SI has been validated against this method [71].

The workflow for using these experiments to build and validate a model is summarized below.

The Scientist's Toolkit: Essential Research Reagents and Materials

Successful execution of the experiments above and the implementation of associated models require a suite of specialized tools and reagents.

Key Reagents and Materials for Experimental Validation

Item	Function/Description	Example Use Case
Streptozotocin (STZ)	A chemical agent that selectively destroys pancreatic beta cells.	Inducing experimental Type 1 Diabetes in animal models (e.g., rats) for in vivo testing [28].
Triple-Tracer Meal Protocol	An advanced method using stable isotope glucose tracers to directly measure postprandial glucose fluxes (rate of appearance, endogenous production, disposal).	Collecting high-quality data for building and validating large-scale physiological models like the UVa/Padova simulator [1].
Genetic Algorithm (GA)	An optimization technique inspired by natural selection, used to find the best model parameters that fit experimental data.	Efficiently and accurately estimating the parameters (p1, p2, p3) of the Bergman Minimal Model from IVGTT data [28].
Continuous Glucose Monitor (CGM)	A device that measures subcutaneous glucose concentrations frequently (e.g., every 5 minutes), providing rich, dynamic data.	Providing the continuous glucose trace needed for parameterizing and testing models in ambulatory settings [70] [1].
Insulin Pump	A programmable device that delivers subcutaneous insulin infusions.	Used in artificial pancreas experiments to administer insulin doses computed by control algorithms based on the Hovorka or other models [70] [1].

The choice between the Sorensen, Bergman, and Hovorka models is not about identifying a single "best" model, but rather selecting the most appropriate tool for a specific research objective. The Sorensen model offers the highest physiological plausibility for investigating organ-level dynamics and pathophysiology. The Bergman Minimal Model remains a powerful, efficient tool for quantifying key metabolic parameters like insulin sensitivity from sparse clinical data. The Hovorka model, particularly as implemented in the FDA-approved UVa/Padova simulator, provides an optimal balance for in-silico testing and the development of control algorithms for the artificial pancreas. Future work continues to enhance these models, for instance by adding gastrointestinal absorption to the Sorensen model [5] or refining the Hovorka equations for better individualization [3], pushing the frontier of personalized diabetes management.

The University of Virginia-Padova (UVa/Padova) Type 1 Diabetes Mellitus (T1DM) Simulator represents a landmark achievement in regulatory science, becoming the first computational model accepted by the U.S. Food and Drug Administration (FDA) as a replacement for animal trials in the preclinical testing of insulin treatment strategies [1] [73]. This endorsement in 2008 marked a transformative moment in medical device development, establishing a new pathway for evaluating artificial pancreas (AP) and automated insulin delivery (AID) systems through in silico clinical trials rather than mandatory animal testing [1]. This paradigm shift has significantly accelerated the pace of innovation in diabetes technology while addressing ethical concerns associated with animal experimentation.

The simulator's development was underpinned by extensive dynamic profiling of carbohydrate metabolism in over 200 adults at Mayo Clinic, creating a sophisticated mathematical framework of the human metabolic system [1]. Unlike previous models that focused on population averages, the UVa/Padova simulator introduced inter-individual variability through a cohort of virtual subjects representing the broader population of people with diabetes [1]. This capability enables researchers to test control algorithms across a spectrum of metabolic characteristics, providing more realistic assessment of treatment performance before human trials begin.

The Evolution of Metabolic Models: From Minimal to Clinically Validated

Historical Context of Glucose-Insulin Models

The development of quantitative models of glucose-insulin dynamics dates back to the 1970s, with several foundational approaches emerging [1]:

• Bergman Minimal Model: Developed by Bergman and Cobelli, this model features two quasi-linear differential equations representing insulin kinetics in plasma and the effects of insulin and glucose on restoring glucose after perturbation by intravenous injection [1] [14]. While simplified, it provided crucial foundational principles for understanding glucose homeostasis.

• Hovorka Model: A more comprehensive physiological model focused on predicting population averages from clinical trial data. This model enabled early work on developing model predictive control (MPC) algorithms for AID systems [1].

• Sorensen Model: A detailed whole-body physiological model that integrates sub-models for glucose, insulin, and glucagon dynamics across different body compartments (brain, heart, lungs, liver, kidneys, periphery).

The UVa/Padova simulator built upon these foundations but introduced crucial innovations that enabled regulatory acceptance, particularly through the incorporation of observed inter-individual variability and validation against extensive clinical datasets [1].

Comparative Analysis of Key Metabolic Models

Table 1: Comparative Analysis of Key Metabolic Models in Diabetes Research

Model Characteristic	Bergman Minimal Model	Hovorka Model	Sorensen Model	UVa/Padova T1D Simulator
Primary Application	Research tool for insulin sensitivity assessment	Early MPC algorithm development	Whole-body metabolic simulation	Regulatory-grade in silico trials
Physiological Detail	Two-compartment, minimal structure	Intermediate complexity with glucose-insulin interaction	High complexity, multi-compartment system	Comprehensive 13-equation structure with subcutaneous insulin dynamics
Inter-individual Variability	Limited population representation	Population averages from clinical data	Limited variability implementation	Extensive virtual population (100 adults, 100 adolescents, 100 children)
Regulatory Status	Research use only	Research use only	Research use only	FDA-accepted for preclinical testing (since 2008)
Validation Basis	Intravenous glucose tolerance test data	Clinical trial datasets	Physiological literature	Triple tracer meal protocol in 204 individuals [1]
Integration of Real-World Factors	Basic meal challenge	Standard meals and insulin profiles	Comprehensive physiological disturbances	Meals, exercise, circadian changes, dawn phenomenon [1]

The FDA Acceptance Pathway and Regulatory Significance

The FDA's acceptance of the UVa/Padova simulator in January 2008 occurred within a broader regulatory evolution toward alternative testing methods [1]. This decision was groundbreaking because it formally established that a simulator of type 1 diabetes mellitus with virtual patients demonstrating realistic inter-subject variability could enable faster, more extensive testing of AP control algorithms while being more representative than small-size animal trials [1].

This regulatory milestone has accelerated innovation in diabetes technology by:

• Eliminating preclinical animal testing requirements for certain artificial pancreas systems
• Enabling rapid iteration of control algorithms without ethical constraints
• Reducing development costs and time for new insulin delivery systems
• Providing a standardized evaluation platform across the research community

The simulator's acceptance aligns with the FDA's broader initiative to advance New Approach Methodologies (NAMs) that can replace, reduce, or refine animal testing [74] [75]. More recently, the FDA has announced plans to phase out animal testing requirements for monoclonal antibodies and other drugs, promoting the use of AI-based computational models and human-relevant testing methods [74].

Experimental Protocols and Validation Methodologies

Core Simulation Architecture and Workflow

The UVa/Padova simulator operates through a sophisticated integration of mathematical modeling and clinical data. The current version includes 13 differential equations with 35 parameters (26 free and 9 derived from steady-state constraints) that describe the glucose-insulin regulatory system in T1D [1]. The model substitutes exogenous insulin delivery in place of endogenous insulin secretion, accurately representing the metabolic state of individuals with T1D.

Table 2: Key Research Reagents and Computational Tools for In Silico Trials

Research Tool	Function	Application in UVa/Padova Studies
Virtual Patient Cohort	Provides inter-individual variability for robust algorithm testing	300 virtual subjects (adults, adolescents, children) with different metabolic characteristics [1]
Continuous Glucose Monitor Model	Simulates real-world sensor performance with measurement error	Incorporates sensor accuracy (MARD) and noise characteristics [1]
Insulin Pump Model	Represents subcutaneous insulin delivery kinetics	Models absorption delays and pharmacokinetics [1]
Meal Challenge Module	Simulates glucose appearance from carbohydrate intake	Includes complex meal compositions (high-fat, high-protein) [76]
Disturbance Scenarios	Tests algorithm robustness under challenging conditions	Incorporates exercise, stress, circadian variations [1]

Advanced Simulation Capabilities for Complex Challenges

Recent enhancements to the UVa/Padova simulator have expanded its capabilities to address complex real-world challenges in glycemic management:

• High-Fat High-Protein Meal Simulation: The simulator has been updated to integrate the effects of fat and protein on model parameters, enabling realistic simulation of meals with varying macronutrient content [76]. This advancement addresses a significant clinical challenge, as meals with high content of fat and protein remain difficult for optimal glycemic control in individuals with T1D [76].

• Dual-Wave Bolus Optimization: Using the enhanced simulator, researchers demonstrated that a dual-wave bolus (+30% of dose, given 50% immediately and 50% over 2 hours) produced significantly lower blood glucose (p<0.0001) and higher time-in-range compared to standard therapy (p<0.01) for high-fat/high-protein meals, without increasing hypoglycemia risk [76].

• Digital Twin Technology: Recent clinical trials have integrated the simulator with digital twin technology, creating personalized metabolic models of individual patients. In a randomized clinical trial of 72 individuals with T1D, this approach improved time-in-range from 72% to 77% (p<0.01) and reduced glycated hemoglobin from 6.8% to 6.6% [73].

Performance Comparison: In Silico vs. Animal Models

The validation of the UVa/Padova simulator against animal testing demonstrates several key advantages for regulatory evaluation of diabetes technologies:

Table 3: Performance Comparison: Animal Models vs. UVa/Padova Simulator

Evaluation Metric	Animal Testing Model	UVa/Padova Simulator	Advantage of Simulator
Testing Duration	Weeks to months	Hours to days	>10x faster iteration of control algorithms
Subject Variability	Limited by practical constraints (typically n=5-10)	Extensive virtual population (n=300)	Greater statistical power and representation
Cost per Experiment	High (housing, care, procedures)	Low (computational resources)	Significant cost reduction for developers
Ethical Considerations	Significant animal use concerns	No ethical constraints	Aligns with 3R principles (Replace, Reduce, Refine)
Standardization	Variable between facilities	Highly consistent and reproducible	Standardized evaluation across research community
Regulatory Acceptance	Traditional pathway requiring extensive data	FDA-accepted for preclinical testing since 2008	Accelerated pathway for artificial pancreas systems

Case Study: Clinical Validation of In Silico Predictions

The predictive accuracy of the UVa/Padova simulator has been demonstrated through multiple clinical trials that compared simulated outcomes with real-world clinical results:

• Time-in-Range Improvements: In a 6-month randomized clinical trial using digital twin technology based on the UVa/Padova simulator, participants achieved a 5 percentage point increase in time-in-range (3.9-10 mmol/L), from 72% to 77% (p<0.01) [73]. This improvement aligned closely with simulator predictions.

• Hypoglycemia Safety: The same trial demonstrated that the optimized parameters derived from simulator testing effectively reduced hyperglycemia without increasing hypoglycemia risk, with 0% time spent <70 mg/dL across all cohorts [73].

• Glycated Hemoglobin Reductions: Consistent with the time-in-range improvements, the digital twin approach reduced glycated hemoglobin from 6.8% to 6.6%, with the most significant benefits observed in participants with suboptimal baseline control (HbA1c >7.0%) [73].

These clinical validations confirm that control algorithms tested and optimized using the UVa/Padova simulator translate effectively to real-world patient benefits, supporting its role as a valid replacement for animal testing in the regulatory pathway.

Future Directions and Expanding Applications

The success of the UVa/Padova simulator has paved the way for expanded applications of in silico trials in diabetes research and beyond:

• Federated Learning Integration: Recent research has combined the simulator with privacy-preserving federated learning approaches. The PRIMO-FRL framework achieved 76.54% overall time-in-range with 0.0% hypoglycemia across virtual cohorts, demonstrating how simulator-generated data can train AI systems without compromising patient privacy [77].

• Expansion to Type 2 Diabetes and Prediabetes: Modeling approaches are being extended to populations beyond T1D. Recent research has developed mathematical models to estimate blood glucose behavior in individuals with prediabetes, showing strong agreement with experimental data (r=0.98, p<0.01) [14].

• Personalized Adaptive Control: The integration of digital twin technology enables bi-weekly optimization of therapy parameters (carbohydrate ratio, correction factor, basal rate), adapting to patients' changing physiology and behavior [73].

• AI-Powered Insulin Delivery: New clinical trials are exploring reinforcement learning systems integrated with the simulator framework to develop fully automated insulin delivery that requires no user input [78].

The UVa/Padova simulator's regulatory endorsement represents not just a milestone in diabetes technology but a paradigm shift in how medical devices are evaluated—ushering in an era where sophisticated computer simulation accelerates innovation while maintaining rigorous safety standards.

The development of an artificial pancreas and effective glycemic control strategies represents a critical frontier in diabetes management, a global health challenge affecting hundreds of millions worldwide [16] [79]. Central to this endeavor are mathematical models that simulate glucose-insulin dynamics, which serve as essential platforms for developing and testing control algorithms before clinical implementation. Among the most influential models in diabetes research are the Bergman minimal model, valued for its parametric simplicity; the Hovorka model, recognized for its comprehensive physiological representation; and the Sorensen model, distinguished by its organ-based compartmental structure [16] [12]. Each model presents distinct advantages and limitations that shape their application in control design. This guide provides an objective comparison of controller performance based on these physiological models, presenting experimental data and methodologies to inform researchers, scientists, and drug development professionals in selecting appropriate models for specific research applications.

Comparative Analysis of Glucose-Insulin Models

The foundational models differ significantly in complexity, physiological representation, and intended application. The table below summarizes their key characteristics:

Table 1: Comparison of Fundamental Glucose-Insulin Models

Model Characteristic	Bergman Minimal Model	Hovorka Model	Sorensen Model
Primary Application	Type 1 Diabetes (T1DM) research [79]	T1DM control algorithm development [16] [12]	Normal and diabetic physiology simulation [12]
Complexity Level	Low (3 differential equations) [16]	Medium (8 differential equations) [16]	High (Organ-based compartments) [12]
Physiological Basis	Whole-body approximation [79]	Three subsystems (glucose, insulin, insulin action) [16]	Multi-compartment organ structure [12] [59]
Key Strengths	Simplicity, parameter identification [79]	Comprehensive parameter set, popular for control applications [16]	Most complete physiological representation [12]
Notable Limitations	Does not originally consider exogenous insulin [16]	Specifically for T1DM [12]	Requires modifications for subcutaneous insulin delivery [12]

The Sorensen model's structure emulates the glucose metabolism of both normal and diabetic individuals by representing key organs like the brain, liver, heart, lungs, and periphery as separate compartments interconnected through blood flow [12] [59]. In contrast, the Hovorka model abstracts the system into three primary subsystems but remains more physi detailed than the Bergman model, which takes a minimalistic approach with only three differential equations to capture essential dynamics [16] [79]. This spectrum of complexity creates a trade-off between physiological fidelity and computational tractability that directly impacts controller design and performance.

Controller Performance Analysis

Various control strategies have been implemented and tested across these physiological models, with performance metrics demonstrating significant variation based on both the control methodology and the underlying model.

Table 2: Controller Performance Across Different Physiological Models

Control Algorithm	Implemented Model	Performance Metrics	Experimental Validation
Model Predictive Control (MPC)	Hovorka Model [16]	Short rise time, suitable glycemic values [16]	Comprehensive in-silico trials [16]
Adaptive Backstepping	Bergman Minimal Model [79]	Stable glucose tracking, effective meal compensation [79]	Simulation with unannounced meals (3/day) [79]
Robust Evolving Cloud-based (RECCo)	Sorensen Model [80]	98% model accuracy, prevents hypoglycemia/ hyperglycemia [80]	Hardware-in-the-loop experiment [80]
Reinforcement Learning (RL)	Not specified [81]	Reduces post-meal glucose rise, prevents hypoglycemia [81]	Simulation under uncertain conditions [81]
Deep Reinforcement Learning (DQN)	Not specified [82]	Superior Time in Range, reduced hypoglycemia risk [82]	MIMIC-III ICU patient dataset [82]

Model Predictive Control (MPC) has demonstrated particular effectiveness on the Hovorka model, with studies showing "very short" rise times and glycemic values that align well with clinical expectations [16]. The capability of MPC to handle nonlinear plants with dead times and manage constraints makes it well-suited for the bioengineering challenges inherent in glucose regulation [16]. Meanwhile, the RECCo controller implemented on the Sorensen model achieved remarkable 98% accuracy in hardware experiments, successfully regulating blood glucose despite uncertainty conditions and preventing dangerous hypoglycemic and hyperglycemic episodes [80].

Adaptive backstepping control on the Bergman minimal model has shown excellence in handling meal disturbances, with studies demonstrating effective compensation for "unannounced meals three times a day" while maintaining stability through Lyapunov-based analysis [79]. This controller maintained performance even in challenging scenarios with actuator faults or temporary loss of control input [79]. Reinforcement learning approaches have shown promise for personalized insulin dosing, with one study demonstrating effective reduction of postprandial glucose excursions while preventing hypoglycemia, a critical safety consideration [81].

Experimental Protocols and Methodologies

In-Silico Testing Framework

Standardized simulated experiments provide critical methodologies for comparing controller performance across different models:

• Intravenous Glucose Tolerance Test (IVGTT): Administration of 0.5 g/kg of glucose over 3 minutes with continuous basal insulin infusion [12]
• OGTT + Insulin Bolus: Oral administration of 100g glucose over 1 minute with continuous basal insulin (6.67 mU/min) plus 1000mU IV insulin bolus delivered in 1 minute [12]
• Meal Challenge Tests: Three unannounced meals daily with varying carbohydrate content (higher at lunch) to assess disturbance rejection [79]

These protocols allow systematic evaluation of controller performance under controlled conditions that mimic real-world challenges. The meal disturbance model typically uses a decaying exponential function, D(t) = A·exp(-Bt), with parameters A and B adjusted to represent different meal sizes [79].

Hardware Implementation

For the RECCo controller tested with the Sorensen model, researchers designed a hardware-in-the-loop experiment where "a simple insulin pump is designed in a practical case" to validate performance in realistic conditions [80]. This approach bridges the gap between simulation and clinical application, testing controller efficacy against actual mechanical components and physical constraints.

Clinical Data Validation

Advanced studies incorporate real patient data for validation. One reinforcement learning approach used "a subset of the MIMIC-III database" containing electronic health records of ICU patients, enabling testing on heterogeneous patient populations with varying characteristics [82]. This methodology provides robust evidence of controller performance across diverse physiological responses.

Research Reagent Solutions

The experimental frameworks rely on several key components that function as essential "research reagents" for in-silico and hardware investigations:

Table 3: Essential Research Tools for Glucose Controller Development

Research Tool	Function	Example Implementation
Continuous Glucose Monitor (CGM)	Provides continuous interstitial glucose measurements	FreeStyle Libre (Abbott) [14]
Insulin Pump Model	Delows subcutaneous insulin infusion	Practical pump design for hardware experiments [80]
Kalman Filter	Estimates unmeasurable state variables from glucose measurements	State estimation in RL control [81]
Dual Extended Kalman Filter (DEKF)	Estimates parameters and unmeasurable variables simultaneously	Parameter estimation in prediabetes models [14]
Hovorka Model Parameters	Defines glucose-insulin dynamics for T1DM simulation	8 differential equations with patient-specific parameters [16]

Signaling Pathways and Experimental Workflows

The following diagrams illustrate core glucose-insulin signaling pathways and a standardized workflow for controller testing, representing syntheses of the physiological processes described across the search results.

Diagram 1: Core Glucose-Insulin Signaling Pathway

Diagram 2: Controller Testing Workflow

The evaluation of controller performance in glucose regulation reveals a complex landscape where model selection profoundly impacts observed outcomes. The Bergman minimal model offers computational efficiency and has supported successful implementations of adaptive backstepping control, particularly for meal compensation [79]. The Hovorka model provides a balanced approach with sufficient physiological detail to make it a popular choice for MPC development, demonstrating short rise times and clinically relevant glycemic outcomes [16]. The Sorensen model, with its comprehensive organ-based structure, enables the most physiologically realistic simulation and has supported advanced controllers like RECCo with remarkable 98% accuracy in hardware implementation [80] [12].

The choice between models involves fundamental trade-offs between physiological fidelity, computational complexity, and parameter identifiability. For rapid prototyping and algorithm development, the Bergman and Hovorka models provide sufficient frameworks, while for detailed physiological studies and hardware validation, the Sorensen model offers unparalleled completeness. As artificial intelligence and reinforcement learning methods continue to advance [82] [81], the integration of these data-driven approaches with physiological models represents a promising frontier for personalized glucose control that can adapt to individual patient characteristics and changing metabolic conditions.

Mathematical models of glucose-insulin dynamics are indispensable tools in diabetes research, enabling the in-silico testing of treatments and the development of artificial pancreas systems. These models form the foundational framework for understanding metabolic processes, predicting patient responses to interventions, and designing clinical trials. The Sorensen model, Bergman minimal model, and Hovorka model represent three significant approaches with distinct philosophical and methodological frameworks. The Sorensen model offers comprehensive physiological detail with its 22 differential equations, simulating glucose concentrations across multiple body compartments including the brain, liver, and periphery [5]. In contrast, the Bergman minimal model exemplifies parsimony with its simplified structure focused on estimating key metabolic indices like insulin sensitivity and glucose effectiveness [83]. The Hovorka model strikes a balance between these extremes, providing sufficient physiological detail for practical applications in artificial pancreas development [1].

The performance evaluation of these models encompasses multiple dimensions, including physiological accuracy, computational efficiency, parameter identifiability, and clinical utility. Understanding these aspects is crucial for researchers and drug development professionals seeking to select appropriate models for specific applications. This review systematically compares these three prominent models, highlighting their comparative advantages and limitations based on experimental data and implementation case studies. We present quantitative performance metrics, detailed experimental protocols, and visual representations of model structures to facilitate informed model selection for research and development purposes.

The Sorensen Model: A Physiologically Comprehensive Approach

The Sorensen model, developed in 1978, represents one of the most complex physiological models describing glucose homeostasis. This model consists of 22 differential equations (mostly nonlinear) and approximately 135 parameters, representing glucose concentrations in multiple body compartments including the brain, heart and lungs, liver, gut, kidney, and periphery [5]. This extensive compartmentalization allows the model to simulate detailed physiological mechanisms across different body systems. The model also incorporates pancreatic release of insulin and glucagon, providing a comprehensive representation of the endocrine system's role in glucose regulation.

A key strength of the Sorensen model is its strong physiological basis, with parameter values carefully determined through extensive literature research [5]. However, this complexity comes with significant challenges. The model's extensive parameter requirements make it computationally intensive, and several implementation errors have been identified in subsequent uses of the model. A 2020 revision to the Sorensen model corrected these errors and supplemented it with a previously missing gastrointestinal glucose absorption component [5]. The complexity of the Sorensen model has limited its use in control algorithm development, with only a few researchers employing it for this purpose despite its detailed physiological representation.

The Bergman Minimal Model: A Parsimonious Alternative

The Bergman Minimal Model takes a fundamentally different approach, emphasizing simplicity and parameter identifiability over physiological comprehensiveness. Originally introduced in 1979, this model uses three differential equations to describe glucose-insulin dynamics [83] [79]. Its minimalistic structure aims to capture essential system dynamics with the fewest possible parameters, making it particularly suitable for estimating key metabolic indices from intravenous glucose tolerance test (IVGTT) data.

The model's primary strength lies in its ability to assess insulin sensitivity (SI) and glucose effectiveness (SG) through relatively simple experimental protocols [83]. These capabilities have made it widely adopted for clinical research applications where these metabolic indices are of primary interest. An improved version, the Integrated Minimal Model (IMM), has been developed to allow simultaneous characterization of the glucose-insulin system with full simulation capabilities while maintaining the estimation of clinical indices [83]. However, the original model cannot separate hepatic glucose production from glucose disposal, which can lead to biased parametric descriptions of the system [83].

The Hovorka Model: A Balanced Approach for Practical Applications

The Hovorka model represents an intermediate approach between the complexity of the Sorensen model and the simplicity of the Bergman model. This model provides sufficient physiological detail for practical applications while remaining computationally manageable for control algorithm implementation. The model has been incorporated into the widely used University of Virginia-University of Padova (UVa-Padova) T1D Simulator, which the FDA accepted in 2008 as a substitute for animal trials in preclinical testing of control strategies [1].

This regulatory acceptance represents a significant advantage for the Hovorka model, enabling rapid in-silico testing of artificial pancreas algorithms on virtual patients with different metabolic characteristics [1]. The model's structure has been enhanced over the years to incorporate additional physiological factors such as circadian changes to insulin sensitivity and the dawn phenomenon [1]. The UVa-Padova simulator now includes 100 adult, 100 adolescent, and 100 children virtual subjects, incorporating continuous glucose monitoring and insulin pump models to enable comprehensive testing of meal and insulin delivery scenarios [1].

Table 1: Comparative Characteristics of Glucose-Insulin Models

Characteristic	Sorensen Model	Bergman Minimal Model	Hovorka Model
Model Complexity	22 differential equations, ~135 parameters [5]	3 differential equations [79]	Intermediate complexity (incorporated in UVa-Padova simulator) [1]
Physiological Detail	High (multiple body compartments) [5]	Low (minimal compartments) [83]	Moderate (key physiological processes) [1]
Primary Applications	Detailed physiological simulation [5]	Estimation of SI and SG from IVGTT [83]	Artificial pancreas development, in-silico trials [1]
Computational Demand	High [5]	Low [79]	Moderate [1]
Regulatory Acceptance	Not specified	Not specified	FDA-accepted for preclinical testing [1]
Key Limitations	Implementation errors in original version, complex parameter identification [5]	Cannot separate hepatic glucose production from disposal [83]	Requires enhancement for physical activity simulation [1]

Performance Metrics and Experimental Data

Quantitative Performance Comparison

The performance of glucose-insulin models is typically evaluated through their ability to accurately simulate physiological responses to interventions such as meals, exercise, and insulin administration. The Sorensen model has demonstrated good adaptation to experimental data in simulations including standard intravenous glucose tolerance tests (IVGTT), variable-dose IVGTT comparisons, intravenous insulin tolerance tests (IVITT), and continuous intravenous insulin infusions [5]. However, the model's original implementation presented limitations in simulating oral glucose administration, requiring empirical estimation of both insulin secretion rates and glucose appearance rates [5].

The Bergman Minimal Model has been extensively validated for its ability to estimate insulin sensitivity and glucose effectiveness. In implementation examples, this model has been used to control blood glucose concentrations in type-1 diabetic patients following meal consumption, with studies demonstrating effective tracking of desired glucose trajectories [79]. The model's parameters for a nominal type-1 diabetic patient have been documented, with values including p1 = 0, p2 = 0.025, p3 = 0.000013, n = 0.092, Gb = 70, and Ib = 7.3 [79].

The Hovorka model, as implemented in the UVa-Padova simulator, has demonstrated robust performance in closed-loop control applications. Studies have shown that continuous glucose monitoring (CGM) can be used to estimate glucose mean and variability with acceptable clinical accuracy when directing closed-loop insulin delivery [84]. However, unmodified CGM may overestimate the benefit of closed-loop systems, with one study reporting CGM-estimated time in target range at 86% compared to 75% with plasma glucose measurements [84]. The use of stochastic CGM approaches has been shown to provide unbiased estimates of time in target range and time below target, making it acceptable for assessing closed-loop performance in outpatient settings [84].

Table 2: Experimental Performance Metrics Across Model Applications

Performance Metric	Sorensen Model Applications	Bergman Model Applications	Hovorka Model Applications
Glucose Control Accuracy	Good adaptation to IVGTT and IVITT data [5]	Effective tracking of desired trajectories (adaptive backstepping) [79]	CGM-estimated time in target: 86% (75% plasma reference) [84]
Meal Challenge Response	Required empirical estimation for oral glucose [5]	Compensation for meal uncertainty via adaptive control [79]	Accurate postprandial control with MPC algorithms [1]
Parameter Identification	~135 parameters requiring extensive literature research [5]	Key parameters (SI, SG) identifiable from IVGTT [83]	Parameters estimable from clinical data with population distributions [1]
Robustness to Disturbances	Not specifically reported	Stable in presence of actuator faults [79]	Performs well under various meal and exercise scenarios [1]

Experimental Protocols and Methodologies

Sorensen Model Validation Protocols

The testing protocols for the Sorensen model involve several standardized clinical procedures, including a standard 0.5 g/kg Intravenous Glucose Tolerance Test, variable-dose IVGTT comparisons (0.05, 0.2, 0.5 and 0.75 g/kg), a 0.04 U/kg Intravenous Insulin Tolerance Test, and continuous intravenous insulin infusions (0.25, 0.4 mU/kg) [5]. These tests evaluate the model's ability to simulate both glucose and insulin dynamics under controlled conditions. The original Sorensen model was also tested with a 100g oral glucose test, though this required the introduction of a gut glucose absorption rate term that bypassed the normal glucose pathway from stomach to gut [5].

The revised Sorensen model addresses the limitation in modeling oral glucose administration by incorporating a gastrointestinal tract component using a previously published glucose absorption formulation that was demonstrated to adapt well to experimental data from individuals ranging from normal subjects to type-2 diabetic patients [5]. This enhancement allows for more physiological simulation of meal absorption processes.

Bergman Minimal Model Testing Protocols

The Bergman Minimal Model is typically validated using Intravenous Glucose Tolerance Tests, where frequent samples of plasma glucose and insulin are taken after an intravenous bolus injection of a standard glucose dose [83]. The test duration and sampling frequency vary by study, with one protocol including plasma sample collection pre-dose and at 2, 3, 4, 5, 6, 8, 10, 12, 15, 18, 20, 22, 24, 26, 28, 30, 35, 40, 45, 50, 55, 60, 70, 80, 90, 100, 110, and 120 minutes post-dose [83].

For control applications, the Bergman model has been tested with meal challenges using an adaptive backstepping control algorithm. In one implementation, the effect of unannounced meals three times a day was investigated for a nominal patient, with meal parameters selected to represent different carbohydrate loads (breakfast: A=15, B=0.04; lunch: A=20, B=0.035; dinner: A=18, B=0.038) [79]. The controller performance was evaluated in the presence of actuator faults and temporary loss of control input, demonstrating the robustness of the adaptive approach [79].

Hovorka Model and UVa-Padova Simulator Protocols

The Hovorka model, as implemented in the UVa-Padova simulator, has been validated through numerous clinical studies. These include randomized crossover studies comparing closed-loop insulin delivery and conventional pump therapy in adolescents and adults with type 1 diabetes [84]. Study designs have incorporated various real-world scenarios such as early evening exercise in adolescents, standard evening meals in adults, and eating-out scenarios with large meals accompanied by alcohol [84].

These studies typically employ frequent venous blood sampling (every 15-60 minutes) for reference plasma glucose measurements using analyzers such as the YSI2300 STAT Plus analyzer [84]. The studies evaluate key metrics including the proportion of time when glucose is in the target range (3.9-8.0 mmol/L or 70-145 mg/dL), time below target, and time above target [84]. The performance of closed-loop algorithms is contrasted against conventional pump therapy using these metrics.

Research Reagents and Computational Tools

The experimental and computational work involving glucose-insulin models relies on specialized tools and platforms. The following table summarizes key research reagents and computational solutions used in this field.

Table 3: Essential Research Reagents and Computational Solutions

Tool/Solution	Type	Primary Function	Example Applications
UVa-Padova Simulator	Software Platform	T1D metabolic simulation with virtual patients	Preclinical testing of control algorithms [1]
FreeStyle Navigator	Continuous Glucose Monitor	Subcutaneous glucose measurement	Clinical studies of closed-loop systems [84]
YSI2300 STAT Plus Analyzer	Laboratory Instrument	Reference plasma glucose measurement	Gold standard for glucose assessment in clinical trials [84]
Model Predictive Control (MPC)	Algorithm Class	Multivariable control strategy for insulin delivery	Artificial pancreas systems [1]
Dual Extended Kalman Filter	Estimation Algorithm	Dynamic parameter and state estimation	Glucose model identification in prediabetes research [14]
Rapid-Acting Insulin Analog	Pharmaceutical	Prandial and basal insulin replacement	Closed-loop delivery in clinical studies [84]

Model Selection Guidelines for Research Applications

Application-Specific Model Recommendations

The selection of an appropriate glucose-insulin model depends heavily on the specific research objectives and constraints. For studies focused on estimating fundamental metabolic parameters such as insulin sensitivity and glucose effectiveness, the Bergman Minimal Model remains the preferred choice due to its simplicity and well-established protocols [83]. Its minimal parameter requirements and ability to provide clinically relevant indices from IVGTT data make it ideal for physiological studies characterizing metabolic function.

For artificial pancreas development and in-silico trials, the Hovorka model as implemented in the UVa-Padova simulator offers significant advantages. The regulatory acceptance of this simulator for preclinical testing, combined with its incorporation of population variability, makes it particularly valuable for control algorithm development and validation [1]. The availability of virtual populations representing adults, adolescents, and children further enhances its utility for pediatric applications.

For detailed physiological studies requiring comprehensive representation of glucose dynamics across multiple body compartments, the revised Sorensen model provides the most complete physiological representation [5]. However, researchers should be prepared for the computational demands and implementation challenges associated with this model. The 2020 revision that corrected earlier errors and added gastrointestinal absorption components has improved the model's utility for simulating complex physiological scenarios.

Emerging Trends and Future Directions

Recent advances in glucose-insulin modeling have focused on addressing limitations in existing approaches. The development of the Integrated Minimal Model represents an effort to maintain the parameter estimation advantages of the Bergman model while enabling full simulation capabilities [83]. This improved version addresses the limitation in hepatic glucose production representation, resulting in more accurate estimates of clinical indices while maintaining simulation capability.

There is growing emphasis on incorporating additional physiological factors into existing models. Research is underway to enhance the UVa-Padova simulator with capabilities to simulate the effects of physical activities and generate additional signals such as energy expenditure [1]. Similarly, recent work has focused on developing models specifically tailored to special populations, including individuals with prediabetes and women with diabetes across various life stages including menstrual cycle, menopause, and pregnancy [1] [14].

The use of advanced estimation techniques such as the Dual Extended Kalman Filter represents another emerging trend, enabling dynamic parameter estimation and improved handling of inter-individual variability [14]. These approaches allow models to better capture the dynamic nature of glucose metabolism and adapt to individual patient characteristics.

The Sorensen, Bergman, and Hovorka models represent distinct points on the spectrum of glucose-insulin modeling approaches, each with characteristic advantages and limitations. The comprehensive physiological detail of the Sorensen model comes with significant computational complexity, while the Bergman Minimal Model offers simplicity and clinical utility at the expense of physiological comprehensiveness. The Hovorka model strikes a practical balance between these extremes, with demonstrated success in artificial pancreas applications and regulatory acceptance for preclinical testing.

Model selection should be guided by specific research objectives, with the Bergman model preferred for metabolic parameter estimation, the Hovorka model for control system development and in-silico trials, and the Sorensen model for detailed physiological investigations. Future developments will likely focus on enhancing existing models with additional physiological features, improving their applicability to diverse populations, and developing more sophisticated parameter estimation techniques. As these models continue to evolve, they will play an increasingly important role in accelerating diabetes research and therapeutic development.

Diagrams

Glucose-Insulin Model Characteristics

Closed-Loop Performance Assessment Workflow

Conclusion

The choice between the Sorensen, Bergman, and Hovorka models is not a search for a single superior model but a strategic decision based on project goals. The Sorensen model offers unparalleled physiological completeness for deep mechanistic investigation, while the Bergman Minimal Model provides an efficient, established tool for specific clinical parameter estimation. The Hovorka model strikes a effective balance, offering sufficient physiological detail for the robust development of control algorithms, a key reason for its widespread use in artificial pancreas research. Future directions point toward the continued refinement of these models—correcting historical inaccuracies, enhancing personalization capabilities, and expanding their scope to simulate a wider range of physiological conditions and interventions. This evolution, supported by platforms like the FDA-accepted UVa/Padova simulator, will further accelerate the development of next-generation diabetes therapies and personalized management tools.

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