Adaptive and Personalized Plasma Insulin Concentration Estimation for Artificial Pancreas Systems

Abstract

Background:

The artificial pancreas (AP) system, a technology that automatically administers exogenous insulin in people with type 1 diabetes mellitus (T1DM) to regulate their blood glucose concentrations, necessitates the estimation of the amount of active insulin already present in the body to avoid overdosing.

Method:

An adaptive and personalized plasma insulin concentration (PIC) estimator is designed in this work to accurately quantify the insulin present in the bloodstream. The proposed PIC estimation approach incorporates Hovorka’s glucose-insulin model with the unscented Kalman filtering algorithm. Methods for the personalized initialization of the time-varying model parameters to individual patients for improved estimator convergence are developed. Data from 20 three-days-long closed-loop clinical experiments conducted involving subjects with T1DM are used to evaluate the proposed PIC estimation approach.

Results:

The proposed methods are applied to the clinical data containing significant disturbances, such as unannounced meals and exercise, and the results demonstrate the accurate real-time estimation of the PIC with the root mean square error of 7.15 and 9.25 mU/L for the optimization-based fitted parameters and partial least squares regression-based testing parameters, respectively.

Conclusions:

The accurate real-time estimation of PIC will benefit the AP systems by preventing overdelivery of insulin when significant insulin is present in the bloodstream.

Type 1 diabetes mellitus (T1DM) is a chronic condition characterized by the autoimmune destruction of pancreatic beta cells, resulting in the inability of the pancreas to produce the insulin necessary for maintaining euglycemia. Without the normal glucose homeostasis mechanism, individuals living with T1DM depend on exogenous insulin administered through either multiple daily injections or continuous subcutaneous insulin infusion (CSII). Insulin infusion pumps and automated control algorithms have enabled automated insulin delivery, the artificial pancreas (AP) technology, where control algorithms calculate the required insulin dose based on continuous glucose monitoring (CGM) measurements and control criteria.^1-17 Regardless of the control algorithms, AP systems require a safety constraint to moderate the potential aggressive control actions (insulin overdosing) to minimize the risk of hypoglycemia and significantly improve the performance of the AP.

The primary limiting factor in determining the amount of active insulin present in the body is the lack of sensors capable of nonintrusive and real-time measurement of insulin concentrations in the bloodstream. Estimating the amount of available insulin in the body is challenging because of the inter- and intrapatient variability attributed to physiological differences and metabolic changes throughout the course of the day. Despite the lack of direct measurement, maintaining the blood glucose concentration (BGC) in the target range requires AP systems that are cognizant of the quantity of insulin previously administrated, which if not appropriately incorporated into the control algorithm may cause overcorrection for the postprandial rise in BGC. Such excessive dosing in either the bolus or basal insulin administered through CSII pumps can potentially lead to hypoglycemia. Hence, in addition to the current and target BGC, a constraint expressing an approximation of the insulin present in the body, such as the conventional insulin on board (IOB) estimates, is needed for insulin-dosing calculations.

The IOB is an estimate of the amount of insulin that is present in the blood and the interstitial fluid cavity. It is typically determined through the approximation of the insulin decay curves, which represent the amount of insulin still remaining in the body due to the prior insulin infusions. Static approximations of the insulin action curves are typically utilized in insulin pumps, with IOB calculations primarily relying on basic insulin decay profiles. Furthermore, significant time-varying delays induced by the absorption and utilization of the subcutaneously administrated insulin as well as diurnal variations in the metabolic state of individuals have significant effects on the IOB. Therefore, the insulin action curves for IOB calculations, usually involving static models with basal and bolus insulin as inputs and active insulin as the output, are not accurate enough to be used in an AP control system. Regardless of the sophistication of the IOB calculation, the information obtained from insulin action curves is usually an approximation of the active insulin in the body and is not a direct estimate of the concentration of insulin in the bloodstream. Other approaches to determine the bloodstream insulin information involve estimating plasma insulin concentrations (PICs) and calculating the amount of subcutaneously administered insulin present through insulin absorption models.^1,2,8,18-24

Accurate estimates of PIC can be obtained by using CGM measurements with adaptive observers designed for simultaneous state and parameter estimation based on reliable glucose-insulin models. Several methods have been proposed for real-time estimation of PIC.^8,25-33 In one study, an extended Kalman filter is used to compute the real-time estimates of PIC from CGM data based on Hovorka’s glucose-insulin model with various time-varying model parameters considered as extended states in the original model.²⁵ The proposed method is tested using an in-silico study of 100 patients with T1DM and clinical data from 12 patients on CSII therapy to demonstrate the estimation results. An unscented Kalman filter (UKF) is designed to estimate the current PIC based on the measurement of the plasma glucose using a discretized version of Bergman’s minimal model with BGC as an output.²⁶ Clinical data from an intravenous glucose tolerance test (IVGTT) are used to evaluate the estimation results. The UKF approach is also applied to the extended Bergman’s model to simultaneously estimate parameters and states.²⁷ An estimator incorporating error feedback is also proposed based on the measured and predicted BGC using Bergman’s third-order nonlinear model designed to tolerate measurement noise as well as discretization errors by means of the $H_{\infty }$ criterion.²⁸ The estimator is tested by using synthetic patient data produced using IVGTT. A particle filter approach using Bergman’s minimal model is proposed to estimate PIC and the dataset from an IVGTT is used to evaluate the performance.²⁹ A nonlinear observer is designed on the basis of an aggregate model where Bergman’s nonlinear model is expanded in terms of local linear models weighted by normalized radial basis functions.³⁰ The performance of the nonlinear observer is assessed using the data set from IVGTT as well.

In our previous studies, the design of adaptive and personalized PIC estimators that directly take into account the inter- and intrasubject variabilities in glucose-insulin dynamics is investigated using three different estimation techniques, including continuous-discrete extended Kalman filter (CDEKF), UKF, and moving horizon estimation (MHE).^31-33 The results are based on clinical experiments conducted with adolescents at the Yale Children’s Diabetes Clinic (New Haven, CT) involving 13 datasets from subjects with T1DM. Five-hour-long euglycemic clamps were employed on two separate occasions. One clamp was performed with the insulin infusion site warming device (IISWD) and the other without the IISWD. All subjects received insulin Aspart at 0.2 U/kg bolus relative to body weight at the start with or without the IISWD, while the basal infusion of insulin via the insulin pump was suspended. Subsequent to the bolus insulin injection, a variable flow rate of fluid with 20% dextrose was infused and adjusted every five minutes based on bedside measurements of plasma glucose to maintain the BGC within the desired range of 90 to 100 mg/dL throughout the study. Although, the three estimation techniques showed good performance for the clinical data,³³ the MHE had the highest computation time and the CDEKF approach was likely to result in instability or poor performance at certain sampling instances due to the linear approximation of the nonlinear model around the operating point. The UKF provided good PIC estimation and was relatively tractable for on-line real-time applications.

The results from the previous study are limited to the five-hour euglycemic clamp study on adolescent subjects. The above considerations provide a strong motivation for evaluating the proposed UKF-based PIC estimation algorithm in a more realistic setting. Therefore, this study analyzes the performance of the proposed individualized PIC estimation algorithm using 20 clinical datasets from closed-loop experiments conducted continuously over 60 hours involving young adults with T1DM. The diversity of the subjects and the length of the clinical experiments allow for a more comprehensive and critical evaluation of the performance of the PIC estimation method that will be incorporated into the AP system. The significant variability in data for various subjects is due to the different meals (amount and type of carbohydrate intake), varied daily basal rates, varied bolus insulin infusions, physical activity levels, and sleep characteristics. The discrete sampling nature of the CGM measurement output, the lack of knowledge on the exact time and amount of meals, the time-varying nature of human physiology, the unmeasured disturbances caused by exercise and sleep, the intersubject variability, the constraints on the state variables, and the rate of change of the model parameters are some of challenges addressed in this work. The PIC estimator is individualized using readily available demographic information, such as body weight, height, body mass index (BMI), and total daily insulin dose. The PIC estimation results are compared against those obtained through the conventional IOB curves to demonstrate the merits of the proposed individualized PIC estimator.

Methods

Subjects and Clinical Study Experiments

The subjects involved in this study were recruited by the Kovler Diabetes Center, University of Chicago Medical Center (Chicago, IL) and were scheduled for a visit at the University of Chicago General Clinical Research Center. The subjects included healthy, physically active young adults between the ages of 18-35 years with T1D. All subjects used CSII pump therapy. Each patient’s visit was approximately 60 hours long during the closed-loop experiment. The subjects’ own insulin type and pumps were used during the experiments. Subjects were provided a total of 13 meals and snacks during the three days of the closed-loop experiment. Each subject participated in exercise bouts of 20-min sessions for aerobic and resistance (days 1 and 3) or interval (day 2) exercise before and after lunch, respectively. An intravenous catheter was placed in the dominant arm for blood sampling every 30 mins, which was later analyzed to quantify the PIC in the blood samples. These PIC values are used to evaluate the accuracy of the proposed PIC estimation algorithm. Overall there are 20 clinical closed-loop experiments. Table 1 shows the characteristics of the participants of the closed-loop studies.

Table 1.

Demographic Information of the Clinical Subjects.

Demographic criteria	Mean ± standard deviation
Age (year)	24.56 ± 1.62
Body weight (kg)	77.43 ± 9.58
Height (cm)	175.70 ± 9.19
Body mass index (kgm-2)	25.09 ± 3.17
Total daily insulin dose [basal] (U)	50.05 [29.49] ± 20.77 [11.87]
Duration of time with diabetes (year)	14.99 ± 3.74
Waist (cm)	88.19 ± 0.61

Glucose-Insulin Dynamic Model

Several models are proposed in literature to provide a mathematical representation of the glucose-insulin system. Hovorka’s model,³⁴ a widely utilized physiological model for describing the insulin action and the glucose kinetics system, is used in this study for designing the UKF estimator. A brief description of Hovorka’s model is provided for completeness. The model consists of nine state variables and various differential equations that describe the glucose-insulin dynamics, the subsystem pertaining to the BGC dynamics, the subsystem concerning the subcutaneous insulin infusion, and the subsystem for the glucose transport from plasma to interstitial tissues. The BGC dynamics are described using a two-compartment model. The two state variables $Q_1\left(t\right)$ and $Q_2\left(t\right)$ denote the mass of glucose in the accessible and nonaccessible compartments, respectively. As precise information about the time and quantity of meals is difficult to ascertain, meal information is considered as an unknown disturbance characterized by the gut absorption rate, $U_{\mathrm{G}}\left(t\right)$. To capture the meal intake, the time-varying gut absorption rate parameter in the dynamics of the state variable $Q_1\left(t\right)$ is determined by the PIC estimator to automatically quantify the effects of the consumed carbohydrates. Another two-compartment model, with state variables $S_1\left(t\right)$ and $S_2\left(t\right)$, defines the absorption rate of subcutaneously administered insulin. The PIC, denoted $I\left(t\right)$, is represented by a first-order differential equation. When considering the subsystem that describes the subcutaneous insulin infusion and PIC, including the dynamics of the state variables $S_1\left(t\right)$, $S_2\left(t\right)$ and $I\left(t\right)$, it is readily shown that the parameters $t_{\max ,I}$ and $k_{\mathrm{e}}$ have direct effect on the PIC. Therefore, in this study the parameters for the time-to-maximum of the absorption of subcutaneously injected insulin ($t_{\max ,I}$) and insulin elimination from plasma ($k_{\mathrm{e}}$) are considered uncertain parameters that are estimated by a PIC estimator over time. The measure of insulin action on glucose kinetics is calculated through three variables: the influence on glucose transport and distribution $x_1\left(t\right)$; the utilization and phosphorylation of glucose in adipose tissue $x_2\left(t\right)$; and the endogenous glucose production in the liver $x_3\left(t\right)$. The relationship between BGC and subcutaneous glucose reported by CGM measurements is considered a first-order dynamic equation. A detailed description of the model and the PIC estimator are provided in Appendix A and Appendix B, respectively, as well as in Hajizadeh et al³³ and Hovorka et al.³⁴

PIC Estimator Individualization

The time-varying model parameters $t_{\max ,I}$ and $k_{\mathrm{e}}$ have significant effect on the estimation of the PIC values, and their appropriate initialization is crucial to improving the performance and convergence of the PIC estimator. Because the values of these parameters differ from one patient to another, individualizing the initialization of the PIC estimators to each patient is necessary. The most accurate approach for estimating these parameters is to optimize them by fitting the insulin compartmental model to the clinically sampled PIC data for each subject. However, this approach has several limitations. The main obstacle to the direct optimization of the $t_{\max ,I}$ and $k_{\mathrm{e}}$ parameters over PIC measurements acquired from blood samples is that accurate, reliable, and nonintrusive sensors for the real-time measurement of PIC are not available, as the PIC measurements need to be evaluated through expensive and time-consuming laboratory analysis involving specific assays.

To address the issue of the appropriate personalized initialization of the parameters $t_{\max ,I}$ and $k_{\mathrm{e}}$ without utilizing the PIC measurements attained from impractical blood samples, an approach based on partial least squares (PLS) regression is proposed in this study to obtain a good initial guess of these parameters (ie, a close approximation to the true parameter). In this work, the accuracy of predicting the initial values for the parameters $t_{\max ,I}$ and $k_{\mathrm{e}}$ using partial least squares regression requires the true value of these temporal parameters. As the values of the parameters $t_{\max ,I}$ and $k_{\mathrm{e}}$ are never directly measured, the initial values of these parameters are determined by optimizing the parametrized insulin compartment model over the observed PIC measurements from each experiment to yield the parameter values that best fit the data. The proposed estimators are then evaluated based on either the optimized values of the parameters $t_{\max ,I}$ and $k_{\mathrm{e}}$, called fitting parameters, or the more suitable regression-based estimates, called testing parameters. The testing parameters are computed using an individualized PLS regression approach and cross-validation. The PLS regression technique is briefly reviewed here for completeness, and a description of the independent variables for predicting the initial values of the parameters is provided.

Partial Least Squares Models

The demographic information of subjects, such as weight, height, BMI, age, total daily insulin and duration with T1DM, is easily attainable. Hence, this demographic information can be exploited to personalize the initialization of the PIC estimator. To this end, the readily available demographic information is used to identify a relationship between the demographic variables and the model parameters to be individually initialized to each patient. Data-driven approaches, such as PLS regression models, are widely used when the exact underlying mathematical relationship between two sets of data, demographic inputs (matrix $V$) and the output parameters (matrix $Z$), is not explicitly and mathematically formalized through fundamental physiological models. PLS is a multivariate regression method for modeling the relationship between two groups of data (explaining the variations in $Z$ by using $V$) consisting of numerous noisy and correlated variables while appropriately handling potentially incomplete measurements with missing data.³⁵ The PLS-based regression relationship is derived from data collected from several experiments. The latent variables of the PLS model are identified to maximize the prediction performance of the model. This is achieved by finding components that maximize the covariance between the independent ($V$) and the dependent variables ($Z$). The $V$ matrix consists of the demographic information of the patients, such as bodyweight, height, BMI, total daily insulin dose, and the $Z$ matrix is defined to be the two critical parameters of interest $t_{\max ,I}$ and $k_{\mathrm{e}}$ that were determined for each patient using real PIC measurements. In addition to the parameters $t_{\max ,I}$ and $k_{\mathrm{e}}$, the initial condition for $U_{\mathrm{G}}$ should also be defined. The value for $U_{\mathrm{G}}$ is obtained from a steady-state assumption at initialization. Following the initialization of the parameter values using the PLS regression models, the proposed estimation technique can rapidly converge to provide good estimates of the PIC in a timely manner by using only the CGM measurements and infused insulin data.

Results

Individualized Parameter Estimation Results

The results for the individualization of the PIC estimators by predicting the initial values for the time-varying parameters using PLS regression models are first presented. In Table 2, the demographic information used as the independent variables and the average of relative error percentage (AREP) for each time-varying model parameter based on leave-one-out cross-validation (LOOCV) are presented.

Table 2.

Input Variables and Accuracy of the PLS Models for Individualization of PIC Estimators.

Parameter	Input variables	AREP
$t_{\max ,I}$	Height, BMI, total daily insulin, and waist	$21.71$
$k_{\mathrm{e}}$	Weight, average basal rate, and total daily bolus insulin	$11.85$

The AREP for the $t_{\max ,I}$ and $k_{\mathrm{e}}$ parameters is $21.71\%$ and $11.85\%$, respectively, demonstrating the excellent capability for predicting the initial value of these model parameters by relying only on the readily available demographic information for each subject. The performance of PLS models is also evaluated using the leave-p-out cross-validation (LPOCV) technique, (considering $p$ to be $5$, representing $80\%$ of the samples for training the PLS models and $20\%$ of the samples for testing the identified models). The AREP for the $t_{\max ,I}$ and $k_{\mathrm{e}}$ parameters for LPOCV is $25.69\%$ and $12.53\%$, respectively, which demonstrates the ability of PLS models to provide satisfactory results for predicting the initial value for the model parameters. It is clear that using a greater number of experiments in training the PLS models improves the accuracy of the models.

The use of the demographic information for training the PLS models to provide initial values of the PIC estimators gives good predictions for the initialization of the time-varying model parameters. Table C1 in Appendix C presents the coefficient matrices of the regression model for predicting the $t_{\max ,I}$ and $k_{\mathrm{e}}$ parameters.

Plasma Insulin Concentration Estimation Results

The PIC values are estimated using the UKF algorithm with the initialization of the model parameters either through the optimization-based fitting parameters or the PLS-regression-based testing parameters. The root mean square error (RMSE) and mean absolute error (MAE) of estimating the PIC values based on different sets of individualized initial parameters are computed. The average values for the RMSE and MAE in PIC estimation using Hovorka’s model without any feedback correction through the CGM measurements for various parameter estimation approaches are presented in Table 3.

Table 3.

RMSE and MAE Values for Estimating the PIC Using Hovorka’s Model With Various Estimates of the Parameters and Without Feedback Correction.

Performance index	Nominal parameter values	PLS-regression-based testing parameters	Optimization-based fitted parameters
RMSE	11.38	9.53	7.53
MAE	8.39	6.95	5.02
P valuea	4.24 × 10−5	3.20 × 10−3	—
P valueb	—	7.60 × 10−2	4.24 × 10−5

^aP value for statistical significance in relation to the optimization-based fitted parameters.

^bP value for statistical significance in relation to the nominal parameter values.

The RMSE and MAE values for PIC estimation without individualization of the time-varying model parameters are greater than the approaches involving personalized parameters. This demonstrates that the nominal model parameters are not sufficient and that the individualization of the parameters for each patient is important in reducing the estimation error. The PLS-regression-based parameters do not significantly improve the PIC estimates (P value = $7.60\times {10}^{-2}$), perhaps because certain characteristics of the subjects are not captured in the demographic information used in the PLS regression models. The most accurate PIC estimation results are obtained by the fitting parameters determined by matching the insulin dynamic model to the PIC data using optimization. The optimized parameters provide better results in comparison to the PLS-regression-based parameters (P value = $3.20\times {10}^{-3}$). However, the PIC data are seldom available. Hence, the best practical PIC estimation results are based on the testing parameters obtained by PLS regression models, as evident by the low AREP values for the PLS models reported in Table 2. The PLS-based regression parameters provide the best practical initialization of the time-varying parameters as the actual PIC measurements are often not available and thus optimization-based fitting of the parameters is typically infeasible.

The results based on UKF for feedback correction through the available CGM measurements are also evaluated and presented in Table 4. The UKF-based approach is also personalized through various sets of model parameters that are determined by different initialization methods of the time-varying parameters. Once again, accurate results are obtained for PIC estimation through the optimization-based fitting parameters, while the initialization of the time-varying parameters through the use of the PLS regression models provides better results than the initialization using the nominal model parameters (P value = $2.55\times {10}^{-2}$). Determining the time-varying parameters through optimizing the parameter values to available PIC measurements provides better estimates compared to the PLS-regression-based PIC estimates (P value = $2.84\times {10}^{-3}$), though optimization with actual PIC values is not always a practical solution as PIC measurements are not available. Furthermore, employing the UKF for closed-loop feedback also improves the results compared to using the model in an open-loop manner (ie, no feedback).

Table 4.

RMSE and MAE Values for Estimating the PIC Using Hovorka’s Model With Various Estimates of the Parameters and With UKF-Based Feedback Correction.

Performance index	Nominal parameter values	PLS-regression-based testing parameters	Optimization-based fitted parameters
RMSE	11.41	9.25	7.15
MAE	8.58	6.72	4.77
P valuea	9.70 × 10−6	2.84 × 10−3	—
P valueb	—	2.55 × 10−2	9.70 × 10−6

^aP value for statistical significance in relation to the optimization-based fitted parameters.

^bP value for statistical significance in relation to the nominal parameter values.

Figure 1 shows the results for the optimization-based fitting parameters and PLS-regression-model-based testing parameters for one clinical experiment. The proposed estimation algorithm is able to accurately quantify the PIC values and closely track the corresponding CGM measurements. Overall, by comparing the presented results in Tables 3 and 4, the requirement for developing an adaptive and individualized PIC estimator is demonstrated.

Figure 1.

(A) Comparison of estimated and measured CGM data based on UKF. Black line: estimated CGM based on fitting parameters; green line: estimated CGM based on testing parameters obtained by PLS model; filled red circle: measured CGM. (B) Comparison of estimated and measured PIC data based on UKF. Black line: estimated PIC based on fitting parameters; green line: estimated PIC based on testing parameters obtained by PLS model; filled red circle: measured PIC. (C) Basal (blue line) and bolus insulin (black bar).

Figure 2 shows the time-varying parameters, $t_{\max ,I}$ and $k_{\mathrm{e}}$, for select random subjects to represent the variations in the trajectories of the estimated parameters over time. The changes in the values are relative to the subject-specific initial starting conditions for the initialized parameters. The average change in the $t_{\max ,I}$ $t_{\max ,I}$ parameter over the duration of the experiments is $2.27\pm 3.30$ and $2.72\pm 3.76$ min for the PLS regression based testing and optimization based fitting parameters, respectively. Similarly, the average change in the $k_{\mathrm{e}}$ parameter over the duration of the experiments is $1.06\times {10}^{-4}\pm 2.66\times {10}^{-4}$ and $1.13\times {10}^{-4}\pm 3.51\times {10}^{-4}mi{n}^{-1}$ for the PLS regression based testing and optimization based fitting parameters, respectively. The estimated parameter values drift to various levels to better characterize the inter- and intra-subject variability in the glycemic dynamics. The variability in the parameters of the glucose—insulin dynamic models can be diverse and may be attributed to many unknown or only approximately known phenomena. As these phenomena affect the glycemic dynamics, the estimation of the time-varying parameters is necessary to accurately characterize the glucose—insulin dynamics and to improve the plasma insulin concentration estimates.

Figure 2.

Time-varying trajectory profiles for the $t_{\max ,I}$ and $k_{\mathrm{e}}$ parameters estimated using the proposed approach (red/blue/green/brown/purple lines are values for five randomly selected subjects for optimization based fitted parameters [dashed line] and PLS regression based testing parameters [solid line]).

Discussion

A major drawback of using standard IOB curves is the inability to accurately characterize the temporal dynamics of active insulin in the body. Figure 3 shows the estimated onboard insulin based on various IOB curves¹⁸ along with the PIC estimates from the proposed approach for one clinical experiment. Since the units of the values computed by the IOB curves and the PIC estimates are not consistent, both the IOB and PIC estimates results are scaled. The IOB trajectories do not correspond well with the PIC estimates, as exemplified by the fact that the maximum PIC estimates do not coincide with the largest values predicted by IOB curves. This inherent limitation is due to the static IOB curves that do not characterize the temporal dynamics of the absorption of subcutaneously injected insulin.

Figure 3.

Comparison of scaled estimated PIC and different IOB curves results for one experiment.

As evident in the clinical results, individuation of the PIC estimators is important for the accurate estimation of the PIC. If the initial values for the time-varying model parameters are not specified appropriately, the performance of the PIC estimators may be compromised. Furthermore, temporal variability within the subjects is accounted for by the on-line estimation of the $t_{\max ,I}$ and $k_{\mathrm{e}}$ parameters as well as the $U_{\mathrm{G}}$ parameter representing the gut absorption. Therefore, the adaptation and individualization of the algorithm have significant effects on the PIC estimator. The unique properties of the proposed estimation technique that improve the PIC estimation results are: (1) the capability for individualization through PLS regression models incorporating demographic information; and (2) the ability to capture variations in the glucose-insulin dynamics through the effective on-line estimation of the time-varying model parameters.

Nevertheless, the variations in the insulin-glucose dynamics because of the differences in daily activities or the metabolic state of the individuals may also affect other parameters of the model. The proposed PIC estimation approach can also take into account the effects of these other factors by incorporating additional model parameters as extended states to the original model. In this work, a technique to personalize two model parameters in the glucose-insulin dynamic model is presented. Individualizing other model parameters could also affect the accuracy of the model, potentially resulting in better tracking of the CGM data, and consequently yielding better estimates of the PIC values and the time-varying parameters.

In studies investigating the AP system, various types of controllers are applied to regulate BGC, including the proportional-integral-derivative (PID) control and the model predictive control (MPC) algorithm. In all AP control systems, insulin infusion should be limited for safety considerations through the use of information on the available infused insulin present in the body that will gradually affect the BGC. One of the advantages of the MPC control algorithm is the convenience of incorporating additional constraints into the optimization problem formulation.

Another important consideration for AP systems is the on-line computational tractability of the control and estimation algorithms. Although more sophisticated estimation algorithms can be employed to estimate the PIC values, the approaches generally require greater computational effort. On the other hand, decreasing the computational complexity may adversely affect the prediction accuracy as these are typically competing criteria and the tradeoffs between them should be effectively balanced. For on-line real-time applications, it is necessary to consider the accuracy of the estimated PIC values along with the associated computational burden, especially when implementing the algorithms on computationally constrained wearable or implantable hardware.

Conclusions

Accurate information on the amount of active insulin present in the body is necessary for AP systems to avoid overdelivery of insulin. In comparison to conventional IOB curves, the PIC provides a better representation of the active insulin present in the body. Although the PIC cannot be measured in real time, it can be readily estimated through algorithms based on glucose-insulin dynamic models. The proposed method utilizes CGM and infused insulin data to compute the PIC estimates with time-varying model parameters incorporated as augmented states to capture the temporal dynamics in patients. The proposed method would benefit AP systems by providing PIC estimates in real time.

Appendix A

Plasma Insulin Concentration Estimator

The glucose-insulin dynamic model can be written in the form

$$\frac{dX\left(t\right)}{dt}=f\left(X\left(t\right),u\left(t\right)\right)$$

$$y\left(t\right)=h\left(X\left(t\right)\right)$$

where

$$X\left(t\right)={\left[\begin{array}{l}{S}_1\left(t\right)S_2\left(t\right)I\left(t\right)x_1\left(t\right)x_2\left(t\right)\\ x_3\left(t\right)Q_1\left(t\right)Q_2\left(t\right)G_{\mathrm{sub}}\left(t\right)\end{array}\right]}^T$$

denotes the vector of state variables, $h\left(X\left(t\right)\right)$ denotes the measurement function and $y\left(t\right)$ as the subcutaneous glucose output measurement given by $G_{\mathrm{sub}}\left(t\right)$. When considering the subsystem that describes the subcutaneous insulin infusion and PIC, including the dynamics of the state variables $S_1\left(t\right)$, $S_2\left(t\right)$ and $I\left(t\right)$, it is readily shown that the parameters $t_{\max ,I}$ and $k_{\mathrm{e}}$ have direct effect on the PIC. Therefore, in this study the parameters for the time-to-maximum of the absorption of subcutaneously injected insulin ($t_{\max ,I}$) and insulin elimination from plasma ($k_{\mathrm{e}}$) are considered uncertain parameters that are included as extended states in the model. Furthermore, as precise information about the time and quantity of meals is difficult to ascertain, meal information is considered as an unknown disturbance to be characterized by the gut absorption rate $U_{\mathrm{G}}\left(t\right)$, which is also included as an extended state in the model to capture the effect of meal intake. Hence, the time-varying model parameters $t_{\max ,I}\left(t\right)$ and $k_{\mathrm{e}}\left(t\right)$ that have an effect on the PIC and the parameter $U_{\mathrm{G}}\left(t\right)$ pertaining to the carbohydrate intake information are all employed to develop an augmented state vector

$$X\prime \left(t\right)={\left[X{\left(t\right)}^T{t}_{\max ,I}\left(t\right)k_{\mathrm{e}}\left(t\right)U_{\mathrm{G}}\left(t\right)\right]}^T\in {\mathrm{ℝ}}^n$$

The dynamics of the augmented system, after incorporating uncertainty in the dynamics of the system (referred to as process noise) and measurement noise in the discrete-time sampled outputs,³⁶ can be expressed as

$$\frac{dX\prime \left(t\right)}{dt}=f^{\prime }\left(X\prime \left(t\right),u\left(t\right)\right)+G\left(t\right)\omega \left(t\right),\omega \left(t\right)\sim N\left(0,Q\left(t\right)\right)$$

$$y_k=h^{\prime }\left({X\prime }_k\right)+{\nu }_k,{\nu }_k\sim N\left(0,R\left(t\right)\right)$$

where $\omega \left(t\right)$ and ${\nu }_k$ represent the process and observation noise vectors, respectively, $Q\left(t\right)$ and $R\left(t\right)$ denote the covariance and variance of the process and measurement noise, respectively, and $X\prime \left(t\right)$ denotes the augmented state vector including the uncertain model parameters to be simultaneously estimated. To design a state estimator for the augmented system, the augmented system should be observable. The observability of the system has been proven in previous studies.^25,31-33 With observability established, we now briefly describe the observer used in this work, with specific algorithm equations detailed in Hajizadeh et al.³³

In the UKF algorithm outlined in Appendix B, the unscented transformation (UT) method is employed for calculating the statistics of a random variable that undergoes a nonlinear transformation^37-39 such as the augmented Hovorka’s glucose-insulin dynamic model.

The UKF algorithm can handle the nonlinear dynamics of the glucose-insulin model, is robust to noise, and has the ability to compensate for deviations and converge to the true value of the augmented states through the Kalman-gain-based correction term added to the estimation.³³

The state variables in the augmented Hovorka’s model represent a physiological process based on first principles, and the state variables should be maintained within a physically realizable range. For example, a negative value for the PIC due to measurement noise and system uncertainty is not physically possible. Therefore, we employ constraints in the UKF algorithm to ensure the augmented state estimates correspond with the physical definitions

$${X\prime }_{\min }\le {\widehat{X\prime }}_k\le {X\prime }_{\max }$$

where ${X\prime }_{\min }$ and ${X\prime }_{\max }$ denote the minimum and maximum allowable values for the augmented state vector, respectively. In addition, maximum rates of change constraints are defined for the parameters $t_{\max ,I}$ and $k_{\mathrm{e}}$ to avoid sudden changes in the parameter values due to measurement noise or unknown disturbances that may result in inappropriate corrections:

$$\left|{\widehat{t}}_{\max ,I,k}-{\widehat{t}}_{\max ,I,k-1}\right|\le {\delta }_1$$

$$\left|{\widehat{k}}_{\mathrm{e},k}-{\widehat{k}}_{\mathrm{e},k-1}\right|\le {\delta }_2$$

where ${\delta }_1$ and ${\delta }_2$ are the constraints on the rate of change of the estimated parameters ${\widehat{t}}_{\max ,I,k}$ and ${\widehat{k}}_{\mathrm{e,k}}$ between consecutive sampling instances k and k-1, respectively.

Appendix B

Unscented Kalman Filter

The advantages of the UKF algorithm over regular linear or extended Kalman filters are that: (1) it can readily handle the nonlinear dynamics of the glucose-insulin model; (2) it is robust to noise, because it takes the process and measurement uncertainties into consideration; and (3) it has the ability to compensate for deviations and converge to the true value of the augmented states through the Kalman-gain-based correction term added to the estimation.

The UT characterizes the mean and covariance estimates with a minimal set of sample points called sigma points. Let the set of sigma points at sampling instance $k$ be denoted

$${X\prime }_{i,k},i\in \left\{0,\dots ,2n\right\}$$

with each point being associated with a corresponding weight $w_i$. In the UKF approach, both the sigma points and the weights are determined deterministically via specific criteria and equations. The sigma points are propagated through the nonlinear state dynamics of the glucose-insulin function, which yields propagated states ${X\prime }_{i,k|k-1}$ and the corresponding mean representing the prior estimates ${\widehat{X}\prime }_{k|k-1}$ approximated by the weighted average of the transformed points as

$${\widehat{X}}_{k|k-1}={\displaystyle \sum }_{i=0}^{2n}{w}_i{X}_{i,k|k-1}$$

where

$${\displaystyle \sum }_{i=0}^{2n}{w}_i=1$$

and the covariance of the prior state estimates $P_{x,k|k-1}$ is computed by the weighted outer product of the transformed points as

$$P_{x,k|k-1}={\displaystyle \sum }_{i=0}^{2n}{w}_i\left({X\prime }_{i,k|k-1}-{\widehat{X\prime }}_{k|k-1}\right){\left({X\prime }_{i,k|k-1}-{\widehat{X\prime }}_{k|k-1}\right)}^T$$

The sigma points are similarly propagated through the measurement function as

$$y_{i,k|k-1}=h^{\prime }\left({X\prime }_{i,k|k-1}\right)$$

and the estimated prior CGM output ${\widehat{y}}_{k|k-1}$ is approximated by the weighted average of the transformed points as

$${\widehat{y}}_{k|k-1}={\displaystyle \sum }_{i=0}^{2n}{w}_i{y}_{i,k|k-1}$$

as well as the estimated covariance matrices

$$P_y={\displaystyle \sum }_{i=0}^{2n}{w}_i\left(y_{i,k|k-1}-{\widehat{y}}_{k|k-1}\right){\left(y_{i,k|k-1}-{\widehat{y}}_{k|k-1}\right)}^T$$

$$P_{xy}={\displaystyle \sum }_{i=0}^{2n}{w}_i\left({X\prime }_{i,k|k-1}-{\widehat{X\prime }}_{k|k-1}\right){\left(y_{i,k|k-1}-{\widehat{y}}_{k|k-1}\right)}^T$$

The Kalman gain $K_k$ and posterior updates for the augmented state estimate ${\widehat{X\prime }}_{k|k}$ as well as the posterior error covariance matrix $P_{x,k|k}$ of the augmented state estimate are given by the standard Kalman update equations

$$K_k=P_{xy}{P}_y^{-1}$$

$${\widehat{X\prime }}_{k|k}={\widehat{X\prime }}_{k|k-1}+K_k\left(y_k-{\widehat{y}}_{k|k-1}\right)$$

$$P_{x,k|k}=P_{x,k|k-1}-K_k{P}_y{K}_k^T$$

Appendix C

PLS regression models

The PLS regression model can be describe by the following general equation:

$$Z^{\prime }=\left(\beta \times V^{\prime }\right)\times Z_s+Z_{avg}$$

where $\beta$ is the PLS model coefficients, $V^{\prime }$ is the scaled input values (the selected demographic information), $Z_s$ is the standard deviations of output values ($t_{\max ,I}$ and $k_{\mathrm{e}}$ parameters, $Z_{avg}$ is the average values of output values and $Z^{\prime }$is the output values. For predicting the initial values of the $t_{\max ,I}$ and $k_{\mathrm{e}}$ parameters, the PLS model coefficient matrices $\beta$ are presented in Table C1.

Table C1.

Matrices of the PLS Regression Model Coefficients for Predicting the $t_{\max ,I}$ and $k_{\mathrm{e}}$ Parameters.

Parameter	PLS model coefficients $\beta$
$t_{\max ,I}$	$\left[\begin{array}{cc}-0.930& \begin{array}{cc}-0.969& \begin{array}{cc}0.302& 1.915\end{array}\end{array}\end{array}\right]$
$k_{\mathrm{e}}$	$\left[\begin{array}{cc}0.118& \begin{array}{cc}-0.120& -0.492\end{array}\end{array}\right]$

Appendix D

Detailed Results of Plasma Insulin Concentration Estimation Over All the Experiments

The RMSE and MAE indices for the 20 clinical closed-loop data sets are provided to demonstrate the performance of PIC estimators across all subjects.

Table D1.

RMSE and MAE Values for PIC Estimates Using Hovorka’s Model With Various Estimates of the Parameters and With UKF-Based Feedback Correction for All Clinical Subjects

CL data	Nominal parameter values		PLS-regression-based testing parameters		Optimization-based fitted parameters
	RMSE	MAE	RMSE	MAE	RMSE	MAE
1	19.29	15.95	11.08	7.59	10.40	6.65
2	13.35	9.57	8.51	5.21	7.98	4.82
3	12.35	8.64	8.46	5.96	8.49	5.98
4	8.88	5.85	5.42	3.69	4.34	3.08
5	7.47	5.67	5.14	3.68	5.09	3.42
6	11.43	7.58	6.94	5.24	6.48	4.47
7	10.90	7.58	14.06	8.81	9.20	5.86
8	15.34	10.96	15.32	9.40	13.09	7.77
9	10.24	7.24	8.74	6.02	9.63	5.85
10	8.86	6.75	6.23	3.99	4.42	2.80
11	11.36	9.80	8.41	7.25	4.19	2.86
12	9.54	7.19	6.41	4.36	5.71	3.65
13	4.59	3.16	10.74	9.53	4.14	2.95
14	14.32	11.65	9.60	7.19	7.79	5.67
15	11.31	8.65	10.71	7.43	10.87	6.87
16	10.61	8.27	15.19	13.94	6.09	4.57
17	9.41	6.58	11.42	7.38	9.11	6.38
18	12.05	9.37	8.28	6.22	7.95	5.80
19	17.02	14.04	8.98	7.49	3.11	2.30
20	9.80	7.10	5.35	3.82	5.14	3.73

Figure D1 shows the mean and standard deviations of estimated PICs, as well as the actual measured PICs, for all 20 clinical closed-loop data sets.

Figure D1.

Mean and standard deviation of estimated PIC values and measured samples of PIC for all 20 clinical closed-loop data sets. Different colors are used in asterisks to denote data from various subjects.