Quantized sampled-data static output feedback control of the glucose–insulin system

Abstract

In this paper, the plasma glucose regulation problem for type 2 diabetic patients is studied. A nonlinear time-delay model of the glucose–insulin regulatory system is exploited to design a quantized sampled-data static output feedback control, using only glucose measurements. It is shown that the proposed control law achieves semiglobal practical stability of the related quantized sampled-data closed-loop glucose–insulin system with arbitrary small steady-state tracking error. The controller involves past values of the glucose, which may not be available in the buffer, for instance because of non-uniform sampling. Such a drawback is overcome by means of spline interpolation. Furthermore, quantization in both input and output channels are taken into account. A pre-clinical validation, concerning the performances of the proposed glucose regulator, is carried out by means of a well-known simulator of diabetic patients, broadly accepted for testing insulin infusion therapies. The simulation results pave the way for further clinical evaluation.

Introduction

Diabetes Mellitus (DM) is a chronic condition arising due to high levels of blood glucose concentrations generated by a disruption of the physiological glucose control exerted by the insulin hormone. The most common type of diabetes is Type 2 DM (T2DM), which is related to an inadequate production of insulin and/or to insulin resistance, i.e. the inability of the body to fully respond to insulin. T2DM accounts for about 415 million patients worldwide of all cases of diabetes, with an important draining of the national healthcare budgets (Ogurtsova et al., 2017). The problems related to DM disease have been also amplified by the recent pandemic COVID-19. Although the risk of contracting SARS-COV-2 does not increases in T2DM patients, viral infections can cause more severe symptoms and complications in people with diabetes, who are also at increased risk due to concomitant immunosuppression following solid organ transplantation (Kaiser et al., 2020, Schofield et al., 2020, Weinrauch et al., 2018). Infection with SARS-COV-2 is also generally associated with an increased insulin requirement and, as an acute viral infection, has been linked to the rapid development of transient insulin resistance (Guo et al., 2020, Schofield et al., 2020, Sestan et al., 2018). These new factors increase more and more the necessity of providing more efficient technologies for solving the blood glucose regulation problem for DM patients.

The Artificial Pancreas (AP) is a set of technologies combining control systems, actuators, and sensors, which are used in order to develop a proposed plasma glucose control therapy. In the literature concerning AP, many results are given for Type 1 DM (T1DM), i.e. for diabetic patients who totally lack of a pancreatic endogenous insulin release (see among the others, Beneyto et al., 2020, Bertachi et al., 2018, Colmegna et al., 2018, Gondhalekar et al., 2018, Gondhalekar et al., 2013, Goodwin and Seron, 2019, Herrero et al., 2017, van Heusden et al., 2012, Hovorka, 2011, Kovàcs, 2017a, Kovàcs, 2017b, Kovàcs et al., 2019, Kovatchev et al., 2016, Lunze et al., 2013, Magni et al., 2009, Messori et al., 2018, Turksoy et al., 2017, Zavitsanou et al., 2015 and references therein). In the present contribution, a quantized sampled-data glucose regulator for T2DM patients is provided. A model-based approach is used to design the glucose regulator. Model-based approaches are usually counterposed to model-less approaches (aka empirical approaches), where the control law is synthesized on the basis of the available data and the corresponding input/output relationship, without attempting to build any glucose–insulin model: examples can be found in look-up tables, rule-based control or more recently on neuro-fuzzy and PID regulators (Dalla Man, Raimondo et al., 2007, Huyett et al., 2015, Laxminarayan et al., 2012, Ruiz et al., 2012, Steil, 2013). On the other hand, a model-based approach aims at designing the control law for the plant of the glucose–insulin system. Advantages are evident, since most sophisticated control strategies can be applied: we may cite, among the others, observer-based approaches to cope with the lack of real-time insulin measurements (Palumbo, Pepe, Panunzi, & De Gaetano, 2012), optimal control achieving the best chosen performance (Chee, Savkin, Fernando, & Nahavandi, 2005), robust control to cope with model uncertainties (Kovàcs, 2017a, Kovàcs et al., 2013), etc. Clearly, a model-based approach requires the use of a sufficiently reliable model of the glucose–insulin system: on one hand, it is required to be physiologically meaningful to mimic the insulin-dependent glucose homeostasis at least at a macroscopic level; on the other hand, it is required to be a “minimal” model, i.e. identifiable according to a minimal set of parameters, by means of non-invasive experimental procedures (many details on the comparison of the above mentioned two routes to the AP can be found in Chee & Fernando, 2007 and references therein). The proposed control law is designed by means of a nonlinear Delay Differential Equation (DDE) model (Palumbo et al., 2007, Panunzi et al., 2007). Motivations for the choice of the DDE model are below reported. First, it is nowadays established that DDE models best represent the pancreatic insulin response to circulating glucose, which cannot be neglected for T2DM (Makroglou et al., 2006, Palumbo et al., 2013). Second, its parameters can be correctly identified according to standard glucose perturbation procedures and, indeed, the model has been clinically validated by means of the Intra-Venous Glucose Tolerance Test (IVGTT) in different studies (Panunzi et al., 2010, Panunzi et al., 2007). Third, it is mathematically coherent since its qualitative behavior well mimics the correct functioning of the glucose–insulin system. Besides, regards to the glucose control perspective, the chosen DDE model is “simple” enough to allow the building of a theoretical control strategy that ensures specific issues by design, e.g. the stability properties of the proposed closed-loop glucose–insulin system (i.e. of the closed-loop system described by the nonlinear DDE model and the proposed regulator). Finally, the chosen DDE model has been successfully exploited in many different AP architectures for T2DM patients (Borri et al., 2021, Borri et al., 2017, Di Ferdinando et al., 2017, Di Ferdinando, Pepe, Palumbo, Panunzi and De Gaetano, 2020, Palumbo et al., 2012, Palumbo et al., 2014) and Pepe, Palumbo, Panunzi, and De Gaetano (2017). Similarly to Chee et al., 2005, Di Ferdinando, Pepe, Palumbo, Panunzi and De Gaetano, 2020, Dua et al., 2006, Kovàcs et al., 2013, Palumbo et al., 2012, Palumbo et al., 2014 and RuizVelazquez, Femat, and CamposDelgado (2004), the intra-venous route is here considered, providing insulin appearance in circulating blood without delays. Although of limited application, the intra-venous route is straightforwardly applicable to problems of glycemia stabilization in critically ill subjects, such as in intensive care units (den Berghe, 2003).

In the context of sampled-data control theory (see, for instance, Clarke et al., 1997, Di Ferdinando, Pepe and Borri, 2020, Fridman, 2014, Grune and Nesic, 2003, Hetel et al., 2017, Karafyllis and Krstic, 2012, Naghshtabrizi et al., 2006, Omran et al., 2014, Pepe, 2014 and references therein), in Di Ferdinando et al. (2017) and Pepe et al. (2017) sampled-data static-state feedback controllers for T2DM patients are provided. The control laws proposed in Borri et al., 2021, Di Ferdinando et al., 2017 and Pepe et al. (2017) exploit both measurements of glucose and insulin concentrations. Since insulin measurements are time-consuming, cumbersome to achieve and, in no way, exploitable for real-time applications, their use in the design of control laws makes these therapies just a proof-of-concept. The sampled-data dynamic output feedback controller for T2DM patients proposed in Di Ferdinando, Pepe, Palumbo, Panunzi and De Gaetano (2020) overcomes the aforementioned drawback by exploiting only glucose measurements. At each sampling instant, such control law requires both the current glucose measurement and a suitable past value of the glucose concentration. In Di Ferdinando, Pepe, Palumbo, Panunzi and De Gaetano (2020), the theoretical results are provided assuming the availability in the buffer of the delayed values of the glucose and problems related to the non-availability in the buffer of such measurements are not taken into account. Moreover, these theoretical results are valid only if a suitable assumption involving control and model parameters is satisfied. These drawbacks will be overcome by the present methodology taking also into account quantization in the input/output channels which, in Di Ferdinando, Pepe, Palumbo, Panunzi and De Gaetano (2020), has not been addressed at all.

To our best knowledge, quantized sampled-data glucose control strategies have never been provided in the literature concerning the plasma glucose regulation problem for T1DM and T2DM patients. In this paper, we fill this gap by providing a quantized sampled-data static output feedback controller for the plasma glucose regulation problem in T2DM patients. In the context of the sampled-data stabilization, theoretical results are provided for the following aspects:

• problems related to the non-availability in the buffer of the past values of the glucose required by the proposed controller are taken into account, and spline approximation methodologies are used in order to provide an approximation of the suitable needed past values of the glucose (Di Ferdinando, Pepe & Borri, 2020);

• sampling and quantization of the glucose measurements, as well as of the signal related to the exogenous intra-venous insulin delivery rate, is considered;

• no assumptions concerning the control and model parameters are introduced.

The theory concerning the stabilization in the sample-and-hold sense (Clarke et al., 1997, Di Ferdinando and Pepe, 2017, Di Ferdinando and Pepe, 2019, Di Ferdinando, Pepe and Borri, 2020, Pepe, 2014, Pepe, 2016, Pepe, 2017) is used to provide the results. In particular, the results provided in Di Ferdinando, Pepe and Borri (2020), which address the quantization in the input/output channels, as well as the problems related to the non-availability in the buffer of the value of the system variables at some past times, are used in order to prove that the provided quantized sampled-data glucose control law ensures the semiglobal practical stability of the closed-loop GI system, with arbitrary small steady-state tracking error. The improvements of the proposed controller with respect to the one in Di Ferdinando, Pepe, Palumbo, Panunzi and De Gaetano (2020) are mainly two: (i) new theoretical results are provided taking into account sampling and quantization in the input/output channels, as well as problems related to the necessity for the controller of the knowledge of the past glucose values (see Remark 1 in Di Ferdinando, Pepe, Palumbo, Panunzi & De Gaetano, 2020); (ii) the proposed controller is much simpler to implement since no assumption is introduced on the control and model parameters involved in the controller (see Assumption 2 in Di Ferdinando, Pepe, Palumbo, Panunzi & De Gaetano, 2020) and no dynamics are involved (see (5) in Di Ferdinando, Pepe, Palumbo, Panunzi & De Gaetano, 2020). Summarizing, the main contributions provided in this paper, with respect to the literature concerning the feedback glucose control for T2DM patients, are the following:

• differently from model less-approaches (see, for instance, Ekram, Sun, Vahidi, Kwokand, & Gopaluni, 2012), here, theoretical results concerning the stability property of the GI system in closed-loop with the proposed quantized sampled-data glucose regulator are provided and analytically proved by exploiting a model-based approach;

• in the context of the model-based approaches for the design of feedback glucose controllers for T2DM patients (see, for instance, Borri et al., 2021, Borri et al., 2017, Di Ferdinando et al., 2017; Di Ferdinando, Pepe, Palumbo, Panunzi and De Gaetano, 2020, López-Palau et al., 2018, Palumbo et al., 2012, Palumbo et al., 2014, Pepe et al., 2017), to our best knowledge, this is the first time in the literature that a fully digital glucose control strategy, which takes simultaneously into account the sampling (non-necessarily uniform) as well as the quantization (non-necessarily uniform) in the input/output channels, is designed and theoretically validated by proving the stability property of the related quantized sampled-data closed-loop GI system (i.e. of the closed-loop system described by the nonlinear DDE model exploited for the design of the regulator and the proposed quantized sampled-data glucose control algorithm).

A pre-clinical validation of the provided glucose regulator is performed by the use of the comprehensive mathematical model provided by Dalla Man, Rizza, Rizza, and Cobelli (2007), which allows dealing with T2DM patients, and provides the bases for the UVA/Padova Type 1 Diabetes Simulator (Kovatchev, Breton, Dalla Man, & Cobelli, 2008) accepted by the Food and Drug Administration (FDA) for testing insulin infusion therapies for diabetic patients and very used in the literature concerning the validation of glucose control laws (see, for instance, Di Ferdinando, Pepe, Palumbo, Panunzi and De Gaetano, 2020, Gondhalekar et al., 2018, Lunze et al., 2013, Magni et al., 2009, Messori et al., 2018, Palumbo et al., 2014, Turksoy et al., 2014, Zavitsanou et al., 2015). Similarly to the virtual environment used in Di Ferdinando, Pepe, Palumbo, Panunzi and De Gaetano (2020) and Palumbo et al. (2014), firstly the parameters of the control law are designed upon the DDE model, tailored for the T2DM average virtual patient provided in Dalla Man, Rizza and Cobelli (2007). Then, a population of virtual patients is generated by exploiting the comprehensive model (Dalla Man, Rizza & Cobelli, 2007), and simulations are performed by keeping the parameters of the regulator fixed for any virtual patient. A deep data analysis of the simulation results have been carried out according to the ones proposed in Chassin et al., 2004, van Heusden et al., 2012 and Maahs et al. (2016).

Preliminary results concerning the control law proposed in this paper have been published in Di Ferdinando, Pepe, Di Gennaro and Palumbo (2020). With respect to Di Ferdinando, Pepe, Di Gennaro and Palumbo (2020), here a significant theoretical improvement is provided by introducing quantization in the input/output channels and by carrying out a massive campaign of simulations of the proposed fully digital glucose control law involving a population of 10,000 virtual patients sampled from the model of an average T2DM virtual patient provided in Dalla Man, Rizza and Cobelli (2007).

The rest of the paper is organized as follows: in Section 2 the compact GI model exploited for the design of the proposed quantized sampled-data glucose regulator is presented; in Section 3 the proposed quantized sampled-data glucose regulator is introduced and the semiglobal practical stability property (with arbitrarily small steady-state tracking error) of the related quantized sampled-data closed-loop GI system is proved; in Section 4, a detailed description concerning the practical implementation of the proposed glucose control strategy is provided and the pre-clinical validation of the regulator is performed; the conclusions are provided in Section 5.

Notation

$\mathbb{N}$ denotes the set of integer numbers in $[0,+\infty )$, $\mathbb{R}$ denotes the set of real numbers, ${\mathbb{R}}^{\star }$ denotes the extended real line $[-\infty ,+\infty ]$, ${\mathbb{R}}^{+}$ denotes the set of nonnegative reals $[0,+\infty )$. The symbol $\left|\cdot \right|$ stands for the Euclidean norm of a real vector, or the induced Euclidean norm of a matrix. For a positive integer $n$, for a positive scalar $\Delta$, a Lebesgue measurable function $f:[-\Delta ,0]\to {\mathbb{R}}^n$ is said to be essentially bounded if $\mathrm{ess}{sup}_{t\in [-\Delta ,0]}\left|f\left(t\right)\right|<+\infty$, where $\mathrm{ess}{sup}_{t\in [-\Delta ,0]}\left|f\left(t\right)\right|=inf\{a\in {\mathbb{R}}^{\star }:\lambda \left(\{t\in [-\Delta ,0]:\left|f\left(t\right)\right|>a\}\right)=0\}$, $\lambda$ denoting the Lebesgue measure. The essential supremum norm of an essentially bounded function is indicated with the symbol ${\Vert \cdot \Vert }_{\infty }$. For a positive integer $n$, for a positive real $\Delta$ (maximum involved time-delay): ${\mathcal{C}}^n$ and $W_n^{1,\infty }$ denote the space of the continuous functions mapping $[-\Delta ,0]$ into ${\mathbb{R}}^n$ and the space of the absolutely continuous functions, with essentially bounded derivative, mapping $[-\Delta ,0]$ into ${\mathbb{R}}^n$, respectively. For a positive scalar $p$, for $\varphi \in {\mathcal{C}}^n$, ${\mathcal{C}}_p^n\left(\varphi \right)=\{\psi \in {\mathcal{C}}^n:{\Vert \psi -\varphi \Vert }_{\infty }\le p\}$. The symbol ${\mathcal{C}}_p^n$ denotes ${\mathcal{C}}_p^n\left(0\right)$. For a continuous function $x:[-\Delta ,c)\to {\mathbb{R}}^n$, with $0<c\le +\infty$, for any real $t\in [0,c)$, $x_t$ is the function in ${\mathcal{C}}^n$ defined as $x_t\left(\tau \right)=x(t+\tau )$, $\tau \in [-\Delta ,0]$. For a positive integer $n$, for $\mathbb{S}={\mathbb{R}}^n$ (or ${\mathbb{R}}^{+}$): $C^1(\mathbb{S};{\mathbb{R}}^{+})$ denotes the space of the continuous functions from $\mathbb{S}$ to ${\mathbb{R}}^{+}$, admitting continuous (partial) derivatives; $C_L^1(\mathbb{S};{\mathbb{R}}^{+})$ denotes the subset of the functions in $C^1(\mathbb{S};{\mathbb{R}}^{+})$ admitting locally Lipschitz (partial) derivatives. Let us here recall that a continuous function $\gamma :{\mathbb{R}}^{+}\to {\mathbb{R}}^{+}$ is: of class ${\mathcal{P}}_0$ if $\gamma \left(0\right)=0$; of class $\mathcal{P}$ if it is of class ${\mathcal{P}}_0$ and $\gamma \left(s\right)>0$, $s>0$; of class $\mathcal{K}$ if it is of class $\mathcal{P}$ and strictly increasing; of class ${\mathcal{K}}_{\infty }$ if it is of class $\mathcal{K}$ and unbounded. The symbol $I_d$ denotes the identity function in ${\mathbb{R}}^{+}$. For a given positive integer $n$, for a symmetric, positive definite matrix $P\in {\mathbb{R}}^{n\times n}$, ${\lambda }_{max}\left(P\right)$ and ${\lambda }_{min}\left(P\right)$ denote the maximum and the minimum eigenvalue of $P$, respectively. The symbol $\circ$ denotes composition (of functions). For positive integers $n$, $m$, for a map $f:{\mathcal{C}}^n\times {\mathbb{R}}^m\to {\mathbb{R}}^n$, and for a locally Lipschitz functional $V:{\mathcal{C}}^n\to {\mathbb{R}}^{+}$, the derivative in Driver’s form (see Pepe, 2007 and the references therein) $D^{+}V:{\mathcal{C}}^n\times {\mathbb{R}}^m\to {\mathbb{R}}^{\star }$, of the functional $V$, is defined, for $\varphi \in {\mathcal{C}}^n$, $u\in {\mathbb{R}}^m$, as

$$D^{+}V(\varphi ,u)=\underset{h\to 0^{+}}{lim\; sup}\frac{V\left({\varphi }_{h,u}\right)-V\left(\varphi \right)}{h},$$

where, for $0\le h<\Delta$, ${\varphi }_{h,u}\in {\mathcal{C}}^n$ is defined, for $s\in [-\Delta ,0]$, as

$${\varphi }_{h,u}\left(s\right)=\left\{\begin{array}{cc}\varphi (s+h),\hfill & s\in [-\Delta ,-h),\hfill \\ \varphi \left(0\right)+(s+h)f(\varphi ,u),\hfill & s\in [-h,0].\hfill \end{array}\right.$$