Elsevier

Journal of Process Control

Volume 86, February 2020, Pages 94-105
Journal of Process Control

An augmented subcutaneous type 1 diabetic patient modelling and design of adaptive glucose control

https://doi.org/10.1016/j.jprocont.2019.08.010Get rights and content

Highlights

  • A simplified control-oriented nonlinear subcutaneous glucose-insulin models for Type 1 diabetes have been proposed.

  • Parameter estimation of the proposed model is done using the data of a more complicated nonlinear model using the nonlinear least squares method.

  • A nonlinear adaptive controller is proposed that makes the uncertain system to follow a reference system.

  • The proposed adaptive controller provided an improved post-prandial glycemic control under intra-patient variability.

Abstract

In the present work, an augmented subcutaneous (SC) model of type 1 diabetic patients (T1DP) is proposed first by estimating the model parameters with the aid of nonlinear least square method using the physiological data. Next, a nonlinear adaptive controller is proposed to tackle two important issues of intra-patient variability (IPV) and uncertain meal disturbance (MD). The proposed patient model agrees quite well with the responses of one of the most popular existing nonlinear model used in the research of artificial pancreas. Further, the developed adaptive control is shown to be capable of providing desired glycemic control without feed-forward action for meal compensation or safety algorithms to avoid hypoglycemia. Due to the simple structure and capability of handling intra-patient variability of the adaptive controller, it can find immediate applicability in the development of the in-silico artificial pancreas.

Introduction

In the backdrop of metabolism, the pancreas constitutes a very vital organ that is directly responsible for maintaining the blood glucose level (BGL) within the safe range (70–180 mg/dl) [1]. The failure in the normal functioning of the β-cells of islets of Langerhans of the pancreas causes diabetes mellitus. According to the international diabetes atlas, the treatment of diabetes mellitus accounts for about 12% of the global health expenditure and is one of the most prevalent diseases with around 425 million people suffering from it [2]. Type 1 diabetes (T1D) is marked by negligible insulin-dependent glucose utilisation because of the antigen-antibody and leucocyte mediated destruction of the pancreatic β-cells, resulting in chronic hyperglycemia (BGL > 180 mg/dl) and ultimately leading to diabetic retinopathy, neuropathy and nephropathy [1], [3]. Multiple instances of prolonged hyperglycemia and severe hypoglycemia (BGL < 50 mg/dl) occur in people suffering from T1D who rely on multiple insulin dosages for blood glucose regulation [4].

Artificial pancreas (AP) provides an automated closed-loop solution to this problem by mimicking actions of the pancreas. It facilitates automated continuous insulin delivery via an insulin pump as suggested by the controller on the basis of current plasma glucose measurements facilitated by suitable sensor [5].

The models describing the dynamics of the glucose-insulin regulatory system (GIRS) of type 1 diabetic patients (T1DP) is crucial for the design and development of model-based control algorithms [6]. There exist two classes of mathematical models of GIRS of T1DP, namely, the empirical models and the physiological models, based on whether they have standard structures of time-series models or whether they model the various pharmacokinetic and physiological processes of GIRS, respectively [7]. Important works on the empirical models and design of controllers based on them were reported in [8]. The significance of various physiological parameters is lost in the empirical models because of their standard predefined structure and are completely data-driven [7]. On the other hand, the significance of important physiological parameters is preserved in the physiological models. But since the physiological models represent the various pharmacokinetic and physiological processes of GIRS, they are often complicated comprising of highly nonlinear differential equations and a large number of state variables [9]. It makes the process of designing nonlinear control algorithms based on such models extremely difficult. The physiological models of T1D can be further classified into (i) simple minimal models (representing the macroscopic functionality of GIRS) and (ii) the complicated maximal models (representing the minute-level functionality of GIRS) [10]. The most popular and the most extensively used minimal model (intravenous) was developed by Richard N. Bergman over 35 years ago [11]. Apart from this, important models of the subcutaneous T1D model include the Hovorka model [12], the UVA/Padova model [13], etc. For instance, the mathematical model used in the FDA approved UVA/Padova simulator [14] has 16 states with underlying nonlinear functions and parameters. The UVA/Padova model is also implemented in GIM simulator [15]. Likewise, the Hovorka model is represented by 8 nonlinear differential equations with some of the nonlinear functions, such as renal clearance and endogenous glucose production described by discontinuous functions. This makes the design of many nonlinear controllers such as feedback linearisation, etc. impossible without additional mathematical approximations, since, for designing such nonlinear control algorithms, the nonlinear functions of the mathematical model must be differentiable [16].

Here, in the present work, an attempt is made to address the problem and find a bridging solution to this. A nonlinear subcutaneous (SC) T1DP model is proposed by augmenting the Bergman's minimal model (BMM) with the SC glucose and insulin dynamics that is comprised of the following characteristics: (i) simple structure: it has simpler structure as compared to the UVA/Padova model [14], [3] and the Hovorka's model [12] in terms of reduced number of state variables and number as well as nature of nonlinearity, (ii) physiological significance: since the core dynamics of GIRS, i.e. the insulin action and the plasma glucose dynamics of the proposed SC T1DP model are represented by BMM, important physiological parameters like insulin sensitivity and glucose effectiveness are preserved [10], (iii) accuracy: it represents the response of the GIRS quite accurately that is statistically validated and (iv) control-oriented model: the design of nonlinear controller based on this model will be much simpler as compared to its other SC counterparts because of its simple non-linearity and structural simplicity. A similarly modified version of the BMM called the identifiable virtual patient (IVP) model was proposed in [17] using the data-set of [18]. The main differences of the proposed controller with the IVP model are as follows: (i) in this work, the meal absorption dynamics adopted from [19] which is more accurate than that of the IVP model in [17] and (ii) the SC insulin absorption dynamics of the proposed model is adopted from [12] consists of two-compartments representing the SC insulin concentrations in hexameric and monomeric forms, unlike the IVP model, where a single-compartment is utilised to represent the SC insulin dynamics, thus, providing more physiological insight into the SC dynamics. Similar works on control-oriented modelling were done recently on another biological system, like microbial fuel cell [20].

In T1D patients, two important challenges exist pertaining to uncertainty and variation in the physiological parameters of GIRS. The parameter set of a physiological model of the GIRS of T1D patients varies significantly from patients to patients within a given population which is termed as inter-patient variability. Furthermore, slow time-varying nature exists within diabetic patients, which is termed as intra-patient variability [21]. The phenomena of inter-patient variability and intra-patient variability can be addressed by designing adaptive control algorithms [22]. The application of various advanced process control methods is well summarised in [23]. In the current research, the focus is on the design of an adaptive control algorithm for AP. A critical review of some of the inherent shortcomings in the existing adaptive control methods designed in the past for this problem is presented as follows: (i) the aggressive control action (insulin infusion scheme) of the minimum variance controllers may lead to hypoglycemic events [22], (ii) linearisation of nonlinear T1D models for designing controllers result in loss of nonlinear characteristics of the original system [24], [25], [26], [27], [28], [29], [30], (iii) intra-patient variability was not explicitly taken care of in the design of adaptive controllers as reported in [22], [25], [26], [31], and moreover, these control techniques were based on time-series models that do not provide explicit information about physiologically significant parameters like insulin sensitivity, glucose effectiveness, directly, unlike the physiologically based models and (iv) presence of integral action in the model predictive control (MPC) [32] may lead to hypoglycemia during fasting due to glycemic variability as mentioned in [33].

In the present work, a nonlinear adaptive controller designed for the BMM [34] is further extended for the SC T1DP model. The reason behind this choice of the control algorithm is that it can handle uncertainty due to inter-patient as well as intra-patient variability without any sort of linear approximations, thus retaining the nonlinear characteristics of the nonlinear model. The main highlights of the proposed adaptive control technique for the proposed SC T1DP model are as follows:

  • (i)

    The time-varying adaptive control law along with the parametric updating laws facilitate an online adaptation to the parametric variations due to intra-patient variability and uncertainty in the external disturbance (meal intake).

  • (ii)

    The output of the uncertain T1DP model, i.e. BGL follows the output of the nonlinear reference system (representing the desired dynamics) despite parametric uncertainty. Unlike conventional model-based adaptive controllers [26], [32], [33] the reference system here consists of estimated parameters (in place of known/fixed parameters), since there exist no nominal/universal sets of parameters for a specific T1DP model that can represent the ideal characteristics of GIRS of diabetic subjects.

  • (iii)

    The earlier method is extended for an uncertain system with exogenous disturbance unlike the original work as reported in [34]. The current method does not require explicit information about the meal disturbance model in contrast to the controller proposed in [34].

  • (iv)

    Unlike, the control techniques in [35], [36], here the proposed feedback control law do not require prior meal announcement or meal estimation and safety algorithms like insulin-on-board (IOB) to avoid post-prandial hyperglycemia [35] and severe hypoglycemia [36], respectively. The adaptive feedback control law achieves the desired performance without any of these additional schemes.

  • (v)

    The proposed control law do not require additional feed-forward disturbance compensation [37] or IOB safety scheme to avoid severe hypoglycemia when the patient parameters vary (i.e. intra-patient variability).

  • (vi)

    Two sets of controller parameters, ci and γi facilitate a convenient way to maintain the closed-loop stability (expressed in terms of ci) and improve the transient response, respectively, by tuning their values separately.

Section snippets

Methods

A novel philosophy of closed-loop regulation of the BGL in T1DPs has been proposed in this research work as discussed below. The philosophy is a two-step strategy comprising of (i) representation of the nonlinear and complex SC dynamics of T1D/ complex physiological models of T1D (with large number of states and complex nonlinear functions) by a simple nonlinear T1D model (with less number of states and minimum non-linearity) using the input–output data-set (data of the T1D physiological

Results

The parameter estimation problem of the proposed ASMM in (5) (open-loop) and performance evaluation of the proposed adaptive controller in (20) (closed loop) are presented below.

Discussion

As depicted in Fig. 4, the predicted (simulated) plasma glucose concentration provided by the proposed ASMM in (5) fits the data set of GIM for a period of 24 h (1440 min) for a single meal and single subcutaneous insulin injection.

This result indicates the predictive ability of the proposed model to represent the nature of the response when a single meal disturbance is provided to the T1D patient in a fasting condition. The results of the parameter estimation for this scenario is provided in

Conclusion

This work discusses the modelling and adaptive controller design of T1D patients. The proposed subcutaneous patient model is simple and can capture the glucose-insulin dynamics agreeable with the existing model. The model is validated against the input–output data of the UVa Padova T1D model using GIM simulator. Further, the proposed control strategy for blood glucose regulation is achieved for the proposed augmented subcutaneous minimal model in the presence of inter-patient and intra-patient

Conflicts of interest

The authors declare no conflicts of interest.

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