Simulation Software for Assessment of Nonlinear and Adaptive Multivariable Control Algorithms: Glucose – Insulin Dynamics in Type 1 Diabetes

Abstract

A simulator for testing automatic control algorithms for nonlinear systems with time-varying parameters, variable time delays, and uncertainties is developed. It is based on simulation of virtual patients with Type 1 diabetes (T1D). Nonlinear models are developed to describe glucose concentration (GC) variations based on user-defined scenarios for meal consumption, insulin administration, and physical activity. They compute GC values and physiological variables, such as heart rate, skin temperature, accelerometer, and energy expenditure, that are indicative of physical activities affecting GC dynamics. This is the first simulator designed for assessment of multivariable controllers that consider supplemental physiological variables in addition to GC measurements to improve glycemic control. Virtual patients are generated from distributions of identified model parameters using clinical data. The simulator will enable testing and evaluation of new control algorithms proposed for automated insulin delivery as well as various control algorithms for nonlinear systems with uncertainties, time-varying parameters and delays.

1. Introduction

Simulation software has the potential to accelerate the development and assessment of control algorithms in complex and challenging scenarios [1]–[3]. In recent years, various novel control frameworks and algorithms are proposed to address the closed-loop stability and performance of nonlinear systems and systems with time-varying parameters [4], [5], [14]–[23], [6], [24]–[26], [7]–[13]. The efficacy of these algorithms is typically illustrated through application to tractable simulation examples, which limits the assessment of their performance and demonstration of the extent of their capabilities. New control problems may have unique sets of challenges and considerations for the theory and implementation of control systems, and simulators offering more challenging cases may assist in assessing their potential and performance.

An emerging application area for control systems deals with the regulation of biological and biomedical (physiological) processes, in particular the regulation of blood glucose concentrations in people with Type 1 diabetes (T1D) [27]–[35]. Biological systems are characterized by inherent nonlinearities, significant time delays, and stochastic disturbances that are difficult to model and degrade the realizable control performance. Despite the potential, the assessment of most control algorithms for emerging applications, such as automated drug delivery, is limited by the conventional simulation platforms that are designed primarily for evaluating feedback control algorithms. Yet, incorporating feedforward control techniques using measurable disturbances can reduce the effects of delays and enhance controller performance [36], [37]. Readily measurable supplemental physiological variables that are indicative of the presence of disturbances should be generated to progress beyond the single-input, single-output control architecture to multivariable control schemes [37], [38]. A new simulation software platform is reported in this paper to enable the assessment and expedite the development of novel multivariable control systems.

The new multivariable simulation software is developed to enable the testing and evaluation of control algorithms by using an important application, the automated insulin delivery to people with T1D. T1D is a chronic disease characterized by insulin deficiency as a result of the autoimmune-mediated destruction of insulin-producing beta cells in the pancreas. Since insulin is a vital hormone in the regulation of blood glucose concentration (BGC) levels, people with T1D must administer exogenous insulin to maintain their BGC in a safe target range. Automated insulin delivery, also called artificial pancreas (AP), is an advanced treatment approach that has the potential to improve the regulation of BGC by providing optimal insulin infusions via an insulin pump and reduce the burden on people with T1D and their care providers. The lack of sustained glycemic control within the desired BGC range increases the risk of acute and chronic long-term complications associated with the disease caused by low or high BGC (hypoglycemia and hyperglycemia). The glucose-insulin dynamics in people with T1D varies from person to person, during the course of a day for the same person, and has nonlinearities in responding to meals and physical activity (measurable disturbances). In addition, the delays in BGC inferred by continuous glucose monitoring (CGM) sensors reporting subcutaneous (interstitial) glucose concentration (8-12 minutes) and delay in insulin diffusion from infusion site to the bloodstream (20-40 minutes) add significant challenges to the control algorithms. Hence a simulator for glucose-insulin dynamics in people with T1D provides a challenging testbed for evaluating the performance of advanced control strategies and algorithms.

Several mathematical models have been developed with varying degrees of details to describe the glucose-insulin dynamics [39]–[59]. They primarily use the compartment modeling approach based on physiology. The single-compartment minimal model characterizes the glucose concentrations and the effective insulin [39], [40], [51] and it has been extended and refined over the years [41]–[43], [60], [61]. Sorenson [62] and Puckett [63] defined compartments based on organs and tissue clusters. Their models have been integrated in developing a simulator [64], [65]. The next generation of models were developed in early 2000s by using data collected from tracer-based experiments to capture the effects of carbohydrates from meals and various glucose depots in the body [46], [56]. They used several compartments representing various stages of glucose and insulin activities, and became the foundation of the next generation simulators for glucose-insulin dynamics in T1D by developing many virtual patients that span the range of actual people with T1D and enable simulations with a representative set of subjects that could be encountered in real life.

A number of advanced simulators have been developed for T1D in recent years [56], [66], [67]. The glucose insulin model (GIM) [46] provided the foundation for the University of Virginia/Padova (UVa/Padova) simulator accepted by the U.S. Food and Drug Administration to replace animal studies for AP systems that were needed before conducting human trials [66]–[68]. Hovorka’s model has also been the foundation of various simulators [57], [69], [70]. Simulators based on both models have accurate predictions of the effects of meals on glucose levels and provide a single output variable, the glucose concentration (GC). This limits their use to controllers relying on a single output measurement (GC measured by CGM sensors inserted to the subcutaneous tissue). In recent years, some of these models have been extended to capture the effects of physical exercise on BGC [56], [69], [71]–[74]. The multivariable Glucose-Insulin-Physiological Variable Simulator (mGIPsim) developed by our research group is the first simulator that includes several other “measured” variables as output variables. The additional “measured” variables generated by the simulator can be used by the control algorithm to detect and forecast the impact of measured disturbances, such as physical activities, for more durable glycemic control.

The mGIPsim software can be used to evaluate and compare the designs and computational implementations of nonlinear, adaptive, and stochastic control algorithms with realistic measured disturbances. The development of multivariable insulin dosing algorithms has been accelerated by using the mGIPsim software in our research. Currently, mGIPsim has twenty virtual subjects with T1D to recreate variants of the complex biological systems that capture a range of responses that must be optimized by the control algorithms tested. This paper describes the mGIPsim software, its user interface, and the characteristics of the various types and intensities of physical activities, in addition to the meal effects, captured by the mGIPsim software. It provides information on the inputs and outputs of the system that are exchanged with user-developed control algorithms and the simulator code for linking the model with user-developed control algorithms. It also provides references to our research results with multivariable model predictive controllers (MPC) that are tested with the simulator. The mGIPsim software has been used to assess the efficacy of several contributions, including recursively updated subspace-based system identification for GC estimation [75], unscented Kalman filters for plasma insulin concentration (PIC) estimation [37], [76], and controller performance assessment methods [77].

The simulator will be made available as a testbed to the research and education communities for testing their control algorithms for nonlinear systems with time-varying parameters, variable time delays, and uncertainties.

2. Automated Insulin Delivery for Regulating Glucose Concentration

People with T1D must administer exogenous insulin to maintain their BGC in a safe target range (BGC ∈ [70,180] mg/dL). Management of the chronic condition remains a daunting challenge for the approximately 1.25 million people with T1D in the United States and over 40 million around the world, with incidence rates rising steadily. Despite recent technological advances, the majority of the patients are not able to achieve recommended glycemic targets. The lack of durable glycemic control increases the risk of acute and chronic long-term complications associated with the disease, including hypoglycemia (low glucose concentration levels defined as BGC < 70 mg/dL), diabetic ketoacidosis (high levels of blood acids called ketones caused by insulin deficiency and the breakdown of fat for energy), and microvascular disease (damage to small blood vessels). Preventing TID-related morbidity and mortality requires novel insulin dosing algorithms that improve glycemic control. Simulation software can be harnessed to accelerate the development of therapies for T1D.

Maintaining tight glycemic control in T1D is challenging. The glucose-insulin system has significant uncertain time delays caused by the gradual absorption of insulin from the subcutaneous level to the blood and, in reverse, of glucose from the blood to the interstitial fluid. The insulin analogues used in T1D therapies gradually absorb and accumulate in the bloodstream and are metabolized at varying rates, typically reaching peak effects slower than most consumed carbohydrates. Due to the gradual absorption and utilization of insulin, prescribing insulin doses without considering the previously administered insulin lingering in the bloodstream or the subcutaneous space can cause an overcorrection and lead to hypoglycemia. In addition to the postprandial glycemic excursions, physical activity disturbances also play a major role in deteriorating glycemic control. Low- and medium-intensity physical activity causes an immediate decrease in the BGC levels as it increases the glucose uptake to and disposal from the working muscles and tissues. Physical activity is also associated with long-lasting changes in insulin sensitivity and glycemic dynamics as depleted glycogen reserves in the muscles are replenished. Appropriately quantifying the glycemic effects of the physical activity disturbances in the insulin dosing control algorithms is not possible without incorporating additional physiological measurements beyond the glucose measurements.

A common therapy for T1D involves artificially recreating the lost beta cell functionality through a CGM sensor, an insulin infusion pump, and a control algorithm that closes the loop between glucose sensing and insulin infusion. A recently commercialized automated insulin delivery system (hybrid closed-loop) necessitates manually entered meal information and user adjustments for exercise [78], [79]. It reduces the burden of managing T1D, though more sophisticated control algorithms can further automate the system to reduce the requirements for user interaction and improve glycemic control. The potential in improving the lives of people with T1D compels the development of new automated insulin delivery systems and novel control algorithms to compute the optimal amount of insulin to administer in people with T1D throughout daily life to maintain the GC in a safe target range. The development process for insulin dosing algorithms can be accelerated by using computer simulations of virtual subjects with T1D to recreate the complex biological systems in an inexpensive virtual environment where the control algorithms can be readily evaluated before migrating to a clinical setting.

Numerous physiologic and metabolic disturbances affect the glucose-insulin dynamics. Meals, exercise, stress, and sleep are some of the factors that have significant effects on the glycemic dynamics. Physical activity, in particular, is a major impediment to tight glycemic control during and after the activity period. Regulating glycemia during physical activity by relying exclusively on GC measurements is not very effective. A more comprehensive understanding of the inherent state of the subject is possible by including supplementary physiological variables that are indicative of the presence of physical activity. Multivariable AP (mAP) systems are designed to improve glucose control by using biosignals from wearable devices in addition to the glucose measurements in computing the optimal insulin doses [37], [80]. Incorporating biosignals, such as heart rate (HR), energy expenditure (EE), skin temperature (ST), and accelerometer (ACC) readings, to complement GC measurements, provides feedforward signals on the impending exercise-related glycemic disturbances. The glucose control problem is further complicated by the fluctuating insulin requirements caused by diurnal changes in the highly nonlinear and complex time-varying glucose-insulin dynamics. The unsteady time delays in the insulin absorption and utilization also necessitate control algorithms specifically designed and adapted for making insulin dosing decisions. Assessment of the new control algorithms that explicitly consider physical activity in the insulin dosing computations is limited by the conventional simulation platforms that are designed primarily for evaluating single-input, single-output control algorithms. The mGIPsim simulator was developed to enable the assessment of multi-input, single-output control algorithms and expedite the development of mAP systems.

3. Control Algorithm Assessment with Virtual Subjects

Several metabolic simulators supporting the development of control algorithms for automated insulin delivery and for in silico evaluation of their safety, efficacy, and usability are reported [29], [33], [34], [57], [58], [81]–[86]. Various proposed glucose-insulin models serve as foundations for the metabolic simulators [33], [40], [56], [62], [87]–[92]. The earlier generation of simulators involve population models for computing the general glycemic behavior without realizations of subject diversity [65], [93], [94]. The current generation of simulators use data from tracer meal protocols on a cohort of subjects to accurately measure postprandial glucose metabolism [56], [66], [67], [95], [96]. Among them, the University of Virginia (UVa)/Padova metabolic simulator is accepted by the U.S. Food and Drug Administration as a surrogate for animal studies. The simulation environment presents a sufficiently large cohort of subjects with inter-subject variability allowing for extensive and robust studies. The characteristics of subcutaneous glucose sensors and insulin infusion pumps are also involved in the simulation.

Predicting the glycemic outcomes of realistic scenarios is important to support the development and prototyping of closed-loop insulin delivery systems. Most metabolic simulators compute glycemic dynamics in response to carbohydrate consumption and a limited subset of exercises. Few glucose-insulin models are extended to incorporate a physiological variable influenced by physical activity, such as FIR or rate of oxygen consumption [70], [72], [74], [81], [97]–[100]. However, explicit consideration of physical activity is not systematically considered in the existing simulation platforms.

Integrating physical activity models with glucose-insulin dynamic models allows computing supplementary physiological variables indicative of physical activity, in addition to GC estimates. The additional physiological variables, such as HR, EE, ST, and ACC signals, are utilized in mAP systems to indicate physical activity and quantify the effects of exercise on glycemic dynamics. Multivariable simulation environments with integrated physiologic and metabolic models will thus enable in silico evaluation of next-generation mAP systems that incorporate additional physiological signals in the insulin dosing algorithms [101]–[103].

4. Beyond Glucose Outputs: The Multivariable Simulation Software

The mGIPsim software simulates virtual subjects and generates physiological variables that are measured by wearable sensors for use in mAP systems. The novel contributions of mGIPsim include the ability to describe the effects of diverse physical activities on the glucose-insulin dynamics and to generate various physiological variables, such as HR, EE, ST, and ACC, that are indicative of the presence of physical activity disturbances affecting the glucose regulation. The new simulator software, a first of its kind, is designed to assess the performance of the multi-input, single-output control architecture of the emerging mAP systems. The foundations of the simulator lie in a dynamic nonlinear integrated physiologic and metabolic model formulated to compute the glucose variations and physiological variables based on user-defined scenarios for meal consumption, insulin administration, and exercise/physical activity. Virtual patients with unique demographic profiles and T1D characteristics are generated from the distribution of identified model parameters using clinical data to represent the diversity in the population.

The foundations of the simulation software lie in Hovorka’s glucose-insulin dynamic model [56], [57] that is extended to incorporate the immediate and long-lasting effects of physical activity. The glucose-insulin dynamic model simulates the glucose variations in response to meals, administered insulin, and physical activity. The effects of physical activity are explicitly considered in the glycemic dynamics with elevated HR values during exercise causing an increase in the glucose uptake from the bloodstream to the working muscles and tissues, and an increase in the glucose disposal from the working muscles. The glucose disposal rate increases with both immediate effect and long-lasting changes in insulin sensitivity.

A dynamic physiological model computes the HR as a function of exercise intensity. The HR is characterized through three components that cumulatively contribute to the overall heart rate dynamics, including a fast component related to increased oxygen requirement of the exercising muscles, a slow component concerning the removal of accumulated lactate through supplied oxygen, and a metabolic component for the increase in demand of oxygen due to elevated core body temperature. A model to predict the EE is developed that first computes the mechanical work rate from the exercise intensity information and then translates the computed power to EE through a first-order filter. The ST is modeled by assuming a temperature gradient from the core body to the skin with physical activity as the source of increased metabolism or heat generation. A partial differential equation model of heat convection with time and distance dependencies relates the core body temperature to the ST dynamics. The physical activity effects are integrated as the source of metabolic heat generation in the core body.

A number of parameters of the models are specific to individual subjects. The identifiable model parameters are optimized to clinical experimental data. Numerous physiological variables and experimental conditions are recorded during the clinical experiments, including the timing and amount of carbohydrates consumed, physical activity type and intensity, glucose data, infused insulin information, HR, EE, ST, ACC, etc. The models are fitted to clinical experimental data and synthetic virtual subjects are generated from the joint distribution of the identified model parameters. Subspace methods and cluster analysis techniques are used to ensure the virtual subjects are unique and span the parameter space of the actual subjects.

5. Methods

This section describes the simulator structure, the clinical experiment data for model development, the models comprising the mGIPsim software, and the generation of the virtual subjects.

5.1. Simulator Structure and Environment

The multivariable simulator mGIPsim computes the transient glucose dynamics in response to meals, insulin administered (basal and bolus), and physical activity for a cohort of 20 virtual subjects with T1D. The users specify the exercise type, time, duration, and intensity, along with meal and administered insulin scenarios as inputs to the simulator, with glucose and physiological variables computed by mathematical models as the outputs. The basal and bolus insulin administration can be either open-loop (user-specified prior to starting a simulation) or closed-loop (manipulated by the control algorithm). The additional physiological variables generated include HR, EE, ST, and ACC signals. The physiological variables are employed in the control algorithms of mAP systems to manipulate insulin dosing when physical activity is detected (Figure 1). The model for meal effects is based on Hovorka’s model developed by using data from tracer experiments [56], [57]. The multivariable simulation software provides virtual subjects representative of the population of people with T1D for in silico evaluation of mAP algorithms.

Figure 1.

The input-output framework of the multivariable simulator (mGiPsim) for T1D. Meal, insulin, and physical activity are user-specified inputs to the multivariable simulator that affect the simulated glucose dynamics and physiological variables. The mGiPsim software generates physiological variables that are measured by wearable sensors for use in mAP systems. Red text (dash-dot lines) indicates new features unique to the multivariable simulation environment, blue text (dashed line) is the typical metabolic simulation environment, and green text (dotted line) is the insulin and rescue carbohydrates manipulated by the control algorithm to regulate the simulated glucose dynamics.

A graphical user interface (Figure 2) is designed to navigate and define the simulation options and inputs. The simulation time, virtual patients, control scheme, and schedules for meals, insulin, and physical activity define a simulation scenario. The insulin infusion can be determined based on provided criteria and demographic information of the virtual subjects, such as subject-specific basal infusion rate, insulin-to-carbohydrate ratio, and correction factor. The controller can manipulate either basal or bolus insulin, or both.

Figure 2.

Main menu screen of the mGIPsim software graphical user interface. Buttons for meals (carbohydrate consumption), insulin, and exercise are used to access submenus and specify the relevant information for the simulation scenarios.

5.2. Clinical Experiment Data for Development of Virtual Subjects

Clinical experiment data collected from 18 adults with T1D are used in the development of the models embedded in the simulator. The data from the 18 participants are used to identify the model parameters and generate 20 virtual subjects that span the same parameter space and mathematical relationships. Participants visited the College of Nursing at the University of Illinois at Chicago, with experiments ranging between three to six days from morning to evening on each day. The experiments were open-loop with subjects following their typical insulin therapy. The experiment protocols consisted of the Bruce treadmill test followed by two to three bouts of treadmill and stationary bike exercise sessions each lasting 30 to 40 minutes. The exercise sessions were moderate-intensity aerobic physical activity based on the maximal aerobic power determined by the Bruce treadmill test. The demographic information of participants is provided in Table 1.

Table 1.

Demographic Information and Metabolic Parameters for Participants of Clinical Experiments

Demographic variable	Mean±SD
Body weight (kg)	76.0±13.7
Age (yr)	24.2±6.1
Duration of diabetes (yr)	11.12±8.60
HbA1c(%)	7.5±1.0
Resting heart rate (BPM)	76.7±7.8
Mean glucose value (mg·dL−1)	159.5±28.3
Coefficient of variance of glucose (%)	27.1±5.7
Relative time in hypoglycemia [<70 mg·dL−1] (%)	2.6±4.7
Relative time in hyperglycemia [>180 mg·dL−1] (%)	33.0±17.6
Minimum glucose value (mg·dL−1)	87.3±19.0
Maximum glucose value (mg·dL−1)	233.3±44.0
Maximum heart rate (BPM)	194.2±10.3
Waist size (cm)	88.9±13.5
Height (cm)	170.8±6.7
Maximum VO2 (mL·kg−1·min−1)	40.5±9.4
Basal insulin rate (U·h−1)	1.0±0.34
Insulin-to-carb ratio (g CHO·U−1)	10.3±6.9
Correction factor (mg·dL−1·U−1)	43.9±15.7
Total daily basal insulin (U)	24.2±8.1
Total daily bolus insulin (U)	22.3±8.6
Max run speed in stress test (mph)	4.20±0.62
Max run grade in stress test (%)	16.00±1.55

In addition to the CGM measurements, physiological variables were collected using noninvasive wearable devices during the experiments to quantify the effects of physical activity on glycemic dynamics. Subjects wore a chest-based physiological monitoring device (BioHarness 3, Zephyr Technology Corporation, Annapolis, MD), SenseWear Pro3 Armband (BodyMedia Inc., Pittsburgh, PA), and Empatica E4 wristband (Empatica Inc., Cambridge, MA). Recorded data also included exercise information (type, duration, and intensity), carbohydrate consumption, and insulin infusion. The database constitutes a comprehensive library to characterize the glycemic dynamics and effects of physical activity in people with T1D (Figure 3). The database is used to train and validate the models employed to generate physiological signals in response to exercise and to describe the effects of physical activity on glucose variations.

Figure 3.

Summary of the CGM, HR, and Basal Insulin Data Collected During the Clinical Experiments (MED: Median, IQR: Interquartile Range)

5.3. Models

Several mathematical models are developed and used to generate various simulator output variables based on meal, insulin, and physical activity. Some model parameters are subject-specific to capture the inter-subject variability. The models are briefly described in this subsection to enhance the description of the capabilities and capacity of the multivariable simulator. A detailed description and analysis of the models is reported in a different publication [104].

5.3.1. Glycemic Model

The glycemic model simulates the glucose variations in response to meals, administered insulin, and physical activity. The effects of physical activity are explicitly considered in the glycemic dynamics described by an extended version of Hovorka’s glucose-insulin model. The original Hovorka’s model consists of a glucose subsystem, an insulin subsystem, and an insulin action subsystem. In the proposed model, physical activity instigates immediate changes in the glucose disposal and long-lasting variations in the insulin action on glucose distribution, disposal, and endogenous glucose production (Figure 4). The driving force for the changes in the glucose-insulin dynamics is the elevated HR during moderate-intensity aerobic physical activity, which increases the glucose uptake from the plasma compartment to the muscles and tissues and also increases the glucose disposal from the working muscles. The glucose disposal rate increases with both immediate effect and long-lasting changes in insulin sensitivity. The new model explicitly considering the effects of physical activity on glucose-insulin dynamics improves glucose prediction accuracy compared to the reference model. The model equations and definition of compartments are described in Appendix 1.

Figure 4.

The glucose kinetics, insulin action, and exercise subsystems including the immediate and long-lasting effects of physical activity on the glycemic dynamics. The blue circles denote the previously existing compartments in the reference glucose-insulin model [56], the orange circles are the augmented exercise compartments, and the orange dashed lines indicated the physical activity effects on glucose streams.

5.3.2. Physiological Heart Rate Model

A dynamic physiological model characterizing the HR as a function of exercise intensity is used to compute the HR in the simulator [105]. The HR during exercise is described by three different components that are combined to contribute the overall dynamics, including a fast component related to increased oxygen requirement of the exercising muscles, a slow component concerning the removal of accumulated lactate through supplied oxygen, and a metabolic component for the increase in demand of oxygen due to elevated core body temperature. A number of parameters in the physiological model are subject-specific, including demographic information such as body weight, resting HR, maximum HR, and maximum intensity achieved during the Bruce protocol test. The data spanning the exercise sessions and including a recovery period beyond the termination of exercise is considered for optimizing the model parameters to specific patients. The squared error of prediction is minimized to obtain the optimal parameters.

5.3.3. Energy Expenditure Model

Experiments are conducted for EE measurement using the COSMED K5 wearable metabolic system (COSMED srl, Rome, Italy) during treadmill and stationary bike exercises at different intensities and protocols. The data are used to develop a model to predict the EE in metabolic equivalents (MET). The mechanical work rate is computed from the exercise intensity information and the computed power is translated to the EE estimates through a first-order filter with a subject-specific time constant to capture the transient behavior of the EE dynamics.

5.3.4. Skin Temperature Model

The ST, affected by exercise, is modeled by assuming a temperature gradient from the core body to the ST with physical activity as the source of increased metabolism or heat generation in the core body. The ST depends on the exercise intensity and other factors reflecting the environment and ambient conditions (for instance, wind speed or humidity). A partial differential equation model of heat convection with time and distance dependencies relates the effect of exercise on core body temperature to the ST dynamics. The model parameters include the thermal conductivity and the distance from core body to skin [106].

5.4. Virtual Patients

A cohort of virtual subjects is derived from the clinical experiment data. The selection of the model parameters is optimized over the experimental data collected from subjects with T1D. A truncated multivariate distribution with constraints for the parameter ranges is identified for the optimized parameters of the experiment participants with T1D. The parameters of the virtual subjects are determined by sampling from the identified joint probability distribution. As there may be unrealistic realizations in the sampled parameters, principal component analysis is used to remove parameters with correlation structure not represented in the actual subjects with T1D. The virtual subjects with low probabilities or large Mahalanobis distance relative to the mean are also pruned. Similarities among virtual subjects is minimized by clustering the parameters of the virtual subjects using distance metrics and retaining only the unique virtual subjects. Maximizing the dissimilarity among the retained virtual subjects enables spanning the greatest variation of the parameter space, which increases diversity among the virtual subjects to effectively span the cohort of subjects involved in the clinical experiments.

6. Results

A cohort of 20 virtual subjects resembling in distribution and correlation the people with T1D is generated based on the data collected. The demographic information of the virtual subjects is given in Table 2. The results for optimization of the original glucose-insulin model and the extended model with explicit consideration of the effects of physical activity on the glycemic dynamics and the physiological HR model are shown in Figure 5 and Figure 6, respectively. The results correspond to a select subject participating in clinical experiments over at least three days with diverse meals and physical activities. The original Hovorka’s model and the proposed extended Hovorka’s model both consider intra-subject variability in the parameters. The original Hovorka’s model with optimized parameters is capable of describing the glucose-insulin dynamics relatively accurately, and the extended model incorporates the explicit consideration of the short- and long-term physical activity effects to improve glycemic predictions during exercise.

Table 2.

Demographic Information and Metabolic Parameters for Virtual Subjects

Demographic variable	Mean±SD
Body weight (kg)	80.3±11.8
Age (year)	26.4±4.0
Resting heart rate (BPM)	79.3±4.0
Maximum heart rate (BPM)	192.3±9.6
Waist size (cm)	92.5±11.7
Height (cm)	173.0±3.9
Maximum VO2 (mL·kg−1·min−1)	40.0±8.3
Basal insulin rate (U·h−1)	0.87±0.18
Insulin-to-carbohydrate ratio (g CHO·U−1)	11.95±2.24
Correction factor (mg·dL−1·U−1)	44.6±11.1
Total daily bolus insulin (U)	21.45±5.19
Total daily basal insulin (U)	20.8±4.29
Max run speed in stress test (mph)	4.30±0.42
Max run grade in stress test (%)	15.8±1.1
Duration of diabetes (yr)	11.72±5.90
HbA1c(%)	7.1±0.5

Figure 5.

Optimization results of the glucose-insulin model for a select subject (#2) with T1D involving diverse meals and physical activities throughout three days of clinical experiments.

Figure 6.

Optimization results for the physiological heart rate model for a select subject (#2) with T1D involving bouts of diverse exercises throughout three days of clinical experiments (treadmill – pink, bicycle – orange).

Over all 18 subjects, the predictions obtained from the original Hovorka’s model has root-mean-square error (RMSE) of 12.86±6.37 mg/dL and mean absolute error (MAE) of 10.34±5.27 mg/dL. Explicitly considering the physical activity effects in the glycemic dynamics decreases the RMSE to 9.85±5.13 mg/dL (p-value for one-sided t-test: 6.4×10⁻⁷) and MAE to 7.81±4.21 mg/dL (p-value for one-sided t-test: 1.9×10⁻⁶). The one-sided t-tests demonstrate a significant improvement (significant reduction in both RMSE and MAE) for the proposed extended Hovorka’s model relative to the original Hovorka’s model. The improvement in prediction ability is even more pronounced during physical activity. Considering the exercise time periods and the half-hour time periods succeeding the exercise, the RMSE decreases from 17.14±9.68 mg/dL for the original Hovorka’s model to 11.12±7.39 mg/dL for the proposed extended Hovorka’s model (p-value for one-sided t-test: 4.0×10⁻¹²) and the MAE decreases from 15.21±8.76 mg/dL for the original Hovorka’s model to 9.65±6.60 mg/dL for the proposed extended Hovorka’s model (p-value for one-sided t-test: 3.2×10⁻¹²). The weighted residuals for the proposed extended Hovorka’s model also demonstrate a significantly reduced prediction residual relative to the original Hovorka’s model (Figure 7), with a one-sided t-test p-value of 1.7×10⁻¹¹. The diversity of lifestyle is well characterized, and the extended model is able to track the actual glucose measurements closely, particularly in periods where exercise affects the glycemic dynamics. The characteristics of the CGM colored noise are estimated and added to the predictions. The MAE of the HR predictions for the optimized models across all subjects is 13.10±2.89 BPM during treadmill and 10.23±3.87 BPM during stationary bike exercises. The RMSE of the HR predictions compared to the actual are 16.1±4.0 and 12.47±4.6 BPM during running and cycling, respectively.

Figure 7:

Weighted Residuals for the Original Hovorka’s Model and the Proposed Extended Hovorka’s Model with Explicit Consideration of Physical Activity in the Glucose-Insulin Dynamics

Beyond the glycemic variations in response to meals, the glucose-insulin dynamics can be affected by physical activity in the proposed multivariable simulator. To elucidate this, a two-day scenario involving meals and bouts of running and biking exercise sessions is simulated (Figure 8). The exercise scenario consists of treadmill running at 9:00 (speed of 5 mph with 2% incline for 30 min) and cycling at 16:00 (60 W for 45 min) on the first day. The second day has biking at 10:30 (100 W for 30 min) and running at 17:30 (3.5 mph at 0% incline for 45 min). Modification of insulin regimens and consumption of extra carbohydrates to compensate for the glycemic responses to exercise are not considered in the simulated virtual patients to demonstrate the effects of physical activity. The effects of the bouts of exercise on the glycemic dynamics are readily observed as a rapid decrease in the glucose concentrations that occur during and subsequent to the physical activity because of the increase in glucose uptake to the working muscles and increased insulin sensitivity. These variations in the glycemic dynamics caused by physical activity are both immediate, as glucose uptake to and disposal from working muscles increase, and long-lasting, as insulin sensitivity remains elevated for a period after culmination of exercise. Figure 9 and Figure 10 show comparisons of the actual and predicted EE and ST (respectively) during bouts of running and biking exercise. The predicted EE closely follows the actual EE measurements obtained from the metabolic cart system with prediction RMSE of 0.87 MET and MAE of 0.66 MET. The ST, with a prediction RMSE of 0.49°C and MAE of 0.37°C, is relatively close to the actual measurements, though environmental factors, ambient conditions, and clothing elements are difficult to quantify in practice, which obscure the ST dynamics.

Figure 8.

Blood glucose variations in response to meals and bouts of treadmill and stationary bike exercises for all virtual subjects in mGIPsim. The solid line in the top plot represents the mean of the glucose trajectories, and the shaded area denotes the standard deviation. The dashed line in the bottom plot represents the average basal insulin and the solid bars in the bottom plot represent the average bolus insulin administered to the virtual subjects.

Figure 9.

Comparison of actual and predicted EE for (a) treadmill and (b) stationary bike exercise.

Figure 10.

Comparison of actual and predicted ST for (a) treadmill and (b) stationary bike exercise.

Figure 11 and Figure 12 illustrate the effects of different types, durations, and intensities of physical activity on the glycemic dynamics and heart rate variations. As exercise intensity or duration increases, the physical activity has a more significant immediate and more considerable long-lasting effects on glycemic dynamics. The dynamics of the HR during the recovery period after the end of exercise also changes based on the intensity of the exercise session.

Figure 11.

Mean and SD (shaded area) of the variations in glycemic dynamics for 20 virtual subjects caused by (a) different intensities of running at the same exercise duration, (b) different durations of running at the same exercise intensity, (c) different intensities of biking at the same exercise duration, and (d) different durations of biking at the same exercise intensity. Different running intensities are simulated with treadmill speeds of 3, 4, and 5 mph and an incline grade of 2% for 40 mins in duration. Different running durations are simulated with treadmill durations of 20 (cyan shaded area), 40 (cyan and yellow shaded areas), and 60 mins (cyan, yellow, and magenta shaded areas) and a speed and incline grade of 4 mph and 2%. Different biking intensities are simulated with stationary bike power of 40, 80, and 120 W for 40 mins in duration. Different biking durations are simulated with stationary bike durations of 20, 40, and 60 mins and a cycling power of 80 W. All exercise sessions are initiated at 09:00 from the same initial conditions.

Figure 12.

Mean and SD (shaded area) of the variations in heart rate dynamics for 20 virtual subjects caused by (a) different itensities of running at the same exercise duration, (b) different durations of running at the same exercise intensity, (c) different intensities of biking at the same exercise duration, and (d) different durations of biking at the same exercise intensity. Different running intensities are simulated with treadmill speeds of 3, 5, and 7 mph and an incline grade of 2% for 40 mins in duration. Different running durations are simulated with treadmill durations of 20 (cyan shaded area), 40 (cyan and yellow shaded areas), and 60 mins (cyan, yellow, and magenta shaded areas) and a speed and incline grade of 5 mph and 2%. Different biking intensities are simulated with stationary bike power of 40, 80, and 120 W for 40 mins in duration. Different biking durations are simulated with stationary bike durations of 20, 40, and 60 mins and a cycling power of 80 W. All exercise sessions are initiated at 09:00 from the same initial conditions.

7. Discussion

mGIPsim is developed to compute the glycemic variations in response to meals, infused insulin, and physical activity for a cohort of virtual subjects that are representative of the population of people with T1D. Although in silico modeling can produce credible results that can possibly substitute for preclinical and animal studies, evaluation with virtual subjects is not a replacement for in vivo clinical trials [69], [94], [107], [108]. The prerequisite simulation studies can help to identify the possible cases of AP insulin dosing algorithm shortcomings and instability prior to clinical studies. Strategies for the management of physical activity can also be investigated using the simulation environment. The mGIPsim software will further extend capabilities of in silico studies and evaluations to include the effects of physical activity on the glycose-insulin dynamics.

As the optimization results demonstrated, the architecture and models of mGIPsim are able to characterize the glycemic dynamics and heart rate variations relatively accurately. Developing models that can accurately characterize all physiological and biological phenomena in people with T1D is challenging, and it can be difficult to uniquely identify all relevant model parameters exclusively from GC measurements. Incorporating all factors that influence glycemia in the models is not practical, and some glycemic variations caused by unknown disturbances, such as stress, may be neglected in the models until such disturbances are characterized accurately [109]–[112]. Meals and physical activity are the predominant and most frequent causes of glycemic dysregulation. Simulation platforms that incorporate the effects of meals, insulin, and physical activity will enable a wider array of simulation scenarios for in silico evaluations that closely resemble the diverse causes of glycemic variations in people with T1D.

Physical activity can vary significantly based on type, duration, and intensity [113]–[116]. These factors are recognized in mGIPsim to quantify the glycemic variations in response to diverse bouts of physical activity. The variety in physical activity also instigates varied immediate changes in glucose dynamics due to increased glucose uptake to and disposal from working muscles, and distinct long-lasting effects on glycemia caused by glycogen repletion and variations in insulin sensitivity. The physical activity also generates additional physiological signals that can be used by the multivariable AP algorithms to inform the control algorithms of exercise and to quantify the effects of physical activity on the glucose-insulin dynamics considered in the closed-loop insulin dosage computation framework.

The foundations of the simulation software lie in Hovorka’s glucose-insulin dynamic model that is extended to incorporate the immediate and long-lasting effects of physical activity [56]. The state variables of the original Hovorka’s model are extended with additional new state variables related to physical activity that incorporate the effects of exercise on the glucose concentrations. The variation in the heart rate computed by a physiological model is the driving force of the new state variables describing the immediate and long-lasting effects of physical activity on glycemia. As the original Hovorka’s glucose-insulin dynamic model is widely accepted and prevalent in T1D studies [57], [108], [117], tracer studies to deduce and assess the factors involved in the regulation of postprandial glucose metabolism are beyond the scope of this work. Clinical experiments involving people with T1D conducted over multiple days comprising diverse lifestyles, meals, and physical activities facilitate the data collection and mathematical modeling. Frequent measurement of plasma glucose concentrations from blood samples is costly and impractical during bouts of exercise, therefore the CGM data is used to detect and quantify the fast-transient response of glucose dynamics to exercise. The data collected during the clinical experiments was used to identify a set of model parameters for the people with T1D participating in the studies. The identifiability of the model parameters is explored, and the precision of the parameters is validated. The joint distribution of the model parameters enabled the development of a cohort of virtual subjects that resemble the integrated metabolic and physiologic dynamics of the people with T1D.

The mGIPsim software development continues to evolve and new features will made available in future releases. A greater number of virtual subjects, including subjects from different demographic subpopulations, will be added. Hybrid gray-box modeling methods that integrate first-principles models and empirical techniques characterizing the residuals may improve the mathematical description of complex phenomena [118].

8. Conclusions

The proposed multivariable simulator, mGIPsim, quantifies the effects of meals, insulin, and physical activity on a cohort of virtual subjects with T1D. The multivariable simulation software will accelerate the development of next-generation multivariable AP systems that incorporate additional physiological signals in the insulin dosing algorithms to improve glycemic control.

A. Appendix 1

The multivariable glucose-insulin dynamics model for meals and physical activities is described for completeness. A comprehensive description of the model equations is reported elsewhere [104]. A list of the variables involved in the model is given in Table A - 1.

Table A - 1.

A list of the model variables, including the 16 state and 5 input variables, and the model parameters (* indicates subject-specific personalized parameters with respective interquartile range provided).

Symbol	Definition	Value/Range and Unit
S1(t), S2(t)	Insulin amount in two-compartment chains for subcutaneous insulin absorption	mU
I(t)	Plasma insulin concentration	mU · L−1
x1(t)	Effects of insulin on glucose distribution/transport	min−1
x2(t)	Effects of insulin on glucose disposal	min−1
x3(t)	Effects of insulin on the endogenous glucose production	min−1
Q1(t)	Glucose in the blood stream (accessible compartment)	mmol
Q2(t)	Glucose in the peripheral tissue (non-accessible compartment)	mmol
Gsub(t)	Subcutaneous glucose concentration	mmol · L−1
D1(t), D2(t)	Glucose amount in the two-compartment chains for regular meal absorption	mmol
DH1(t), DH2(t)	Glucose amount in the two-compartment chains for fast-act rescue carbohydrate absorption	mmol
E1(t)	Short-term exercise effect	BPM
TE(t)	Characteristic time for long term exercise effect	min
E2(t)	Long-term exercise effect	–
d1(t)/D1(t)	Carbohydrate intake (regular meal)	mg/mmol
d2(t)/D2(t)	Carbohydrate intake (fast-acting rescue)	mg/mmol
u1(t)	Basal insulin	mU · min−1
u2(t)	Bolus insulin	mU
HR(t)	Heart rate	BPM
k12	Transfer rate	0.066 min−1
ka,1	Deactivation rate for effect of insulin on glucose distribution/transport	0.006 min−1
ka,2	Deactivation rate for effect of insulin on glucose disposal	0.06 min−1
ka,3	Deactivation rate for effect of insulin on endogenous glucose production	0.03 min−1
SIT* = kb,1*/ka,1	Insulin sensitivities of transport	25 [21 – 29] × 10−4 min−1 per mU · L−1
SID* = kb,2*/ka,2	Insulin sensitivities of disposal	6 [4 – 7] × 10−4 min−1 per mU · L−1
SIE* = kb,3*/ka,3	Insulin sensitivities of endogenous glucose production	217 [178 – 254] × 10−4 per mU · L−1
ke*	Insulin elimination rate from plasma	0.196 [0.175 – 0.218] min−1
VI	Insulin distribution volume	0.12 (L · kg−1
VG	Glucose distribution volume	0.16 (L · kg−1
EGP0*	Extrapolated EGP at zero insulin concentration	0.0277 [0.0235 – 0.0318] mmol · kg−1 · min−1
F01*	Non-insulin-dependent glucose flux	0.0109 [0.0088 – 0.0131] mmol · kg−1 · min−1
tmax,I*	Time-to-maximum insulin absorption	69.5 [62.8 – 75.0] min
τG	Time constant for bloodstream to the interstitial tissues	15 min
a	Exercise parameter	0.77
tHR	Exercise parameter	5 min
tin	Exercise parameter	1 min
n	Exercise parameter	3
tex	Exercise parameter	200 min
c1	Exercise parameter	500 min
c2	Exercise parameter	100 min
AG*	Carbohydrate bioavailability	0.71 [0.67 – 0.76] unitless
tmax, G*	Time-of-maximum glucose appearance rate in the bloodstream (regular meal)	40.4 [37.8 – 43.4] min
β*	Exercise-induced insulin-independent glucose uptake rate	0.78 [0.57 – 1.03] mmol · min−1
α*	Exercise-induced insulin action	1.79 [1,41 – 2.19] unitless
BW	Body weight	kg
HRbase	Basal heart rate	BPM

A-1. Insulin Absorption Subsystem

The insulin absorption subsystem is a two-compartment model, represented with u(t) denoting the subcutaneous administered insulin and S₁(t) and S₂(t) as the amount of insulin in the subcutaneous tissues. These compartments regulate the insulin absorption rate that leads to the appearance of insulin I(t) [56], [57].

$$\begin{gathered} \frac{d{S}_1(t)}{dt}=u(t)-\frac{S_1(t)}{t_{\mathit{max},I}} \\ \frac{d{S}_2(t)}{dt}=\frac{S_1(t)}{t_{\mathit{max},I}}-\frac{S_2(t)}{t_{\mathit{max},I}} \\ \frac{dI(t)}{dt}=\frac{S_2(t)}{V_I.t_{\mathit{max},I}}-k_e.I(t) \end{gathered}$$ (A-1)

A-2. Insulin Action Subsystem

The insulin action subsystem translates the insulin concentration in the bloodstream to cause variations in the glucose concentration dynamics through three differential equations: the insulin action on the glucose distribution, x₁(t); the insulin action on the glucose utilization in cells by phosphorylation, x₂(t); and the inhibitory insulin action on the endogenous glucose production, x₃(t).

$$\begin{gathered} \frac{d{x}_1(t)}{dt}=-k_{a,1}.x_1(t)+k_{b,1}.I(t) \\ \frac{d{x}_2(t)}{dt}=-k_{a,2}.x_1(t)+k_{b,2}.I(t) \\ \frac{d{x}_3(t)}{dt}=-k_{a,3}.x_1(t)+k_{b,3}.I(t) \end{gathered}$$ (A-2)

A-3. Glucose Absorption Subsystem

The glucose absorption subsystem computes the gut absorption U_G(t) as the cumulative absorbed regular and fast-acting carbohydrates. The meal absorption is modeled by two compartments with concentrations D₁(t) and D₂(t), and D(t) as the amount of carbohydrate intake. Similarly, DH₁(t), DH₂(t) and DH(t) denote the two-compartment system and the absorbed fast-acting rescue carbohydrates. The time constant for the fast-acting rescue carbohydrates is assumed to be half of the time constant of the regular carbohydrates.

$$\begin{gathered} \begin{array}{c}\frac{d{D}_1(t)}{dt}=A_G.D(t)-\frac{D_1(t)}{t_{\mathit{max},G}}\\ \frac{d{D}_2(t)}{dt}=\frac{D_1(t)}{t_{\mathit{max},G}}-\frac{D_2(t)}{t_{\mathit{max},G}}\\ \frac{dD{H}_1(t)}{dt}=A_G.DH(t)-\frac{D{H}_1(t)}{t_{\mathit{max},g}}\end{array} \\ \begin{array}{c}\frac{dD{H}_2(t)}{dt}=\frac{D{H}_1(t)}{t_{\mathit{max},g}}-\frac{D{H}_2(t)}{t_{\mathit{max},g}}\\ U_G(t)=\frac{D_2(t)}{t_{\mathit{max},G}}+\frac{D{H}_2(t)}{t_{\mathit{max},g}}\end{array} \\ {t}_{\mathit{max},g}=\frac{t_{\mathit{max},G}}{2} \end{gathered}$$ (A-3)

A-4. Exercise Subsystem

The glucose concentration values decrease during exercise and in the post-recovery period following physical activity according to the types and intensities of low- and medium-intensity physical activity. High-intensity and anaerobic activities that may cause increases are not represented by these equations. The exercise subsystem describes the increased glucose consumption and insulin sensitivity as the net effect of physical activity. It is composed of three compartments driven by the elevated heart rate to describe the effects of the physical activity on glucose-insulin dynamics. The heart rate values above the resting heart rate induce the glycemic effect caused by exercise, with greater heart rate values for a particular subject signifying more intense exercise. Heart rate may vary due to other reasons, though such factors are not included in the current version of model.

The elevated heart rate above the resting heart rate, denoted ΔHR(t), drives a transient increase in E₁(t). A variable characteristic time denoted T_E(t) is proposed to represent the differing long-lasting effects of different durations and intensities of physical activity. T_E(t) influences the nonlinear prolonged effect of physical activity, denoted E₂(t).

$$\begin{gathered} \begin{array}{c}\begin{array}{c}\frac{d{E}_1(t)}{dt}=\frac{1}{t_{HR}}.[\Delta HR(t)-E_1(t)]\\ f(E_1(t))=\frac{{\left(\frac{E_1(t)}{a\cdot H{R}_{base}}\right)}^n}{1+{\left(\frac{E_1(t)}{a\cdot H{R}_{base}}\right)}^n}\end{array}\\ \frac{d{T}_E(t)}{dt}=\frac{1}{t_{ex}}\cdot [c_1\cdot f\left(E_1(t)\right)+c_2-T_E(t)]\end{array} \\ \frac{d{E}_2(t)}{dt}=-\left[\frac{f\left(E_1(t)\right)}{t_{in}}+\frac{1}{T_E(t)}\right]\cdot E_2(t)+\frac{f\left(E_1(t)\right)\cdot T_E(t)}{c_1+c_2} \end{gathered}$$ (A-4)

where t_HR denotes the time constant for the immediate effects of physical activity as the heart rate increases [97], HR_base denotes the basal or resting heart rate determined from the experimental data, c₁ and c₂ define the steady state value for T_E(t) (the variable characteristic time for the long-term effects of physical activity) depending on the exercise intensity, while t_ex defines how fast T_E(t) reaches the corresponding steady state value. Parameters a, n, and t_in specify thresholds for the initiation (in terms of intensity and time) of the prolonged effects of physical activity on insulin sensitivity.

The previously developed two-state physical activity model integrated with the minimal model motivated the proposed exercise subsystem where the physical activity effect is independent of exercise intensity [72], [97]. The proposed model for the exercise subsystem generates distinct immediate and long-lasting responses based on the intensity of the physical activity (Figure A - 1).

A-5. Glucose Kinetics Subsystem

A modified glucose kinetics subsystem is proposed to integrate the effects of physical activity on the glycemic dynamics. In this subsystem, detailed in Eq. (A-5), Q₁(t) and Q₂(t) denote the masses of glucose in the accessible and non-accessible compartments, respectively, $F_{01}^C\left(t\right)$ is the insulin-independent glucose uptake, and F_R(t) is the renal glucose clearance. The equations for $F_{01}^C\left(t\right)$, F_R(t), and the gut absorption, U_G(t), are defined similar to the reference glucose-insulin model of Hovorka. The insulin action on glucose dynamics is affected by the physical activity state variables E₁(t) and E₂(t). Physical activity increases the insulin action on glucose transfer from Q₁(t) to Q₂(t) and glucose uptake from Q₂(t).

$$\begin{array}{c}\begin{array}{c}\begin{array}{c}\frac{d{Q}_1(t)}{dt}=-(1+\alpha E_2{(t)}^2)x_1(t)Q_1(t)+k_{12}{Q}_2(t)-F_{01}^C(t)-F_R(t)+U_G(t)+EG{P}_0(1-x_3(t))\\ \frac{d{Q}_2(t)}{dt}=(1+\alpha E_2{(t)}^2)x_1(t)Q_1(t)-k_{12}{Q}_2(t)-x_2(t)Q_2(t)(1+\alpha E_2{(t)}^2)-\beta \frac{E_1(t)}{H{R}_{base}}\end{array}\\ F_{01}^C(t)=\{\begin{array}{cc}{F}_{01}& \text{if}G(t)\ge 4.5\text{mmol/L}\\ \frac{F_{01}G(t)}{4.5}& \text{otherwise}.\end{array}\end{array}\\ F_R(t)=\{\begin{array}{cc}0.003(G(t)-9)V_G& \text{if}G(t)\ge 9\text{mmol/L}\\ 0& \text{otherwise}.\end{array}\end{array}$$ (A-5)

where x₁(t), x₂(t), and x₃(t) are the state variables for insulin actions defined in Eq. (A-2), G(t) = Q₁(t)/V_G is the BGC in mmol/L. The parameters α, HR_base, k₁₂, F₀₁, EGP₀, β, and V_G are defined in Table A - 1.

A-6. Interstitial Glucose Subsystem

Eq. (A-6) expresses the relationship between the blood glucose concentration Q₁(t)/V_G and the interstitial glucose concentration G_sub(t).

$$\frac{d{G}_{sub}(t)}{dt}=\frac{1}{{\tau }_G}\left(\frac{Q_1(t)}{V_G}-G_{sub}(t)\right)$$ (A-6)

A-7. Glycemic Effects of Physical Activity

Physical activity increases glucose uptake to the working muscles and influences the insulin-mediated glucose pathways [114], [119]–[121]. Stimulated by physical activity, the insulin-independent glucose consumption from the inaccessible compartment characterizes the glycogen breakdown to glucose and the increased glucose uptake to the working muscles. Since glucose consumption is reduced when physical activity ceases, the short-term exercise effect, E₁(t), is used to define the insulin-independent glucose uptake to muscles. This new addition in the model captures the immediate effects of physical activity on glucose consumption, and the effect tends to zero immediately after physical activity ends.

Physical activity intensifies the insulin sensitivity in the proposed exercise-glucose-insulin model by manipulation of the glucose streams involved in the insulin actions. The insulin-dependent glucose transfer from the bloodstream to the intracellular distribution space, Q₁(t) to Q₂(t), and the insulin-dependent glucose disposal from the intracellular distribution space, Q₂(t), increase due to physical activity as the coefficients for the insulin actions, x₁(t) and x₂(t), are amplified by the exercise states. The change in the insulin-mediated glucose streams endures long after physical activity is terminated, and the level of the change depends on the intensity and duration of the physical activity. The model structure and the glucose streams affected by physical activity are illustrated in Figure 4. The solid lines represent the actual glucose streams while the dashed lines indicate the influence on the streams. The orange dashed lines initiating from the exercise subsystem and affecting the streams are the immediate and long-lasting glycemic effects of physical activity.

Figure A - 1.

Comparison of the effects of exercise intensity on the exercise state variables for the proposed model and the reference model [97]: (a) three different levels of heart rate represent different exercise intensities; the E₁(t) state variable to characterize the immediate effects of physical activity; and (c) the E₂(t) state variable to describe the prolonged effects of physical activity. The exercise session is conducted over the first hour followed by several hours of post-exercise recovery period.

A-8. Skin Temperature Model

The metabolic heat generated in the body during physical activity is transferred through the body to the skin surface (T_s) over distance d via conduction, where the heat dissipates from the skin to the ambient conditions and environment due to convection, evaporation, and radiation. The heat transfer in the body is given by:

$$\begin{gathered} \begin{array}{c}\begin{array}{c}\rho c_p\frac{\partial T}{\partial t}=\lambda \frac{\partial }{\partial x}\left(\frac{\partial T}{\partial x}\right)+S(x,t)\\ S(x,t)=F_{MR}\cdot MR\left(p(t)\right)\cdot X(x,\theta )\end{array}\\ X(x,\theta )={\left(\frac{x}{\theta \cdot d}\right)}^2+\frac{2x}{\theta \cdot d}\end{array} \\ \frac{dMR(t)}{dt}=\frac{{10}^6}{{\tau }_{MR}}\left(\gamma \cdot p_T(t)-MR(t)\right) \\ {p}_T(t)=\frac{p(t)}{0.8\cdot MAP} \end{gathered}$$ (A-7)

where T denotes the temperature (°C), t is the time (s), x is the distance from the core to the body tissue (m), ρ is the density (kg · m⁻³), c_p is the specific heat capacity (J · kg⁻¹ · °C⁻¹), and λ is the thermal conductivity (W · kg⁻¹ · °C⁻¹) [106]. Furthermore, p(t) is the power of the physical activity (W) and p_T(t) is the normalized power, S(x, t) is the source of metabolic heat generation (W · m⁻³) due to exercise, MR(t) is metabolic rate (W · dL⁻¹), F_MR is a subject-specific normalizing factor (unitless) for the heat source (since the same exercise power and metabolic rate may yield different heat generation for individuals), θ defines the temperature profile in the x direction with θ · d as the maximum, τ_MR is the time constant for relating the power and metabolic rate (s), and MAP is the maximum aerobic power (W) determined from the exercise protocol.

The initial condition considers the core body temperature prior to exercise as 37°C and linearly decreasing over the body tissue to the skin temperature T_s,0 before physical activity commences as

$$\begin{gathered} T(x,0)=\frac{T(d,0)-T(0,0)}{d}x+T(0,0) \\ T(d,0)=T_{s,0} \\ T(0,0)=37°\text{C} \end{gathered}$$ (A-8)

The boundary condition considers that homeostasis regulates the core body temperature at 37°C, and the heat loss from the skin surface to the ambient temperature, T_a = 25°C, can be determined from α (W · m⁻² · °C⁻¹) as

$$\begin{gathered} T(0,t)=37°\text{C} \\ -\lambda \frac{\partial T}{\partial x}=\alpha \left(T(x,t)-T_a\right) \end{gathered}$$ (A-9)

The partial differential equation is numerically solved to determine the transient temperature profile over the body and the skin temperature.