Glucose-insulin models appeared in the literature are varying in complexity. Hence, their use in control theory is not trivial. The paper presents an optimal controller design framework to investigate the type 1 diabetes from control theory point of view. Starting from a recently published glucose-insulin model a Quasi Model with favorable control properties is developed minimizing the physiological states to be taken into account. The purpose of the Quasi Model is not to model the glucose-glucagon-insulin interaction precisely, but only to grasp the characteristic behavior such that the designed controller can successfully regulate the unbalanced system. Different optimal control strategies (pole-placement, LQ, Minimax control) are designed on the Quasi Model, and the obtained controllers' applicability is investigated on two more sophisticated type 1 diabetic models using two absorption scenarios. The developed framework could help researchers engaging the control problem of diabetes.

According to the data provided by the World Health Organization (WHO), diabetes mellitus is predicted to be the “disease of the future” especially in the developing countries (due to the stress and the unhealthy lifestyle). The diabetic population (in 2000, being estimated 171 million people) is predicted to be doubled by 2030 (366 million worldwide) [

The normal blood glucose concentration level in the human body varies in a narrow range (70–120 mL/dL). If for some reasons the human body is unable to control the normal glucose-insulin interaction (e.g., the glucose concentration level is constantly out of the above-mentioned range), diabetes is diagnosed. Type 1 (also known as insulin-dependent diabetes mellitus) is one of the four classified types of this disease (type 2, gestational-diabetes, and other types, like genetic deflections, are the other three categories of diabetes) and is characterized by complete pancreatic

From an engineering point of view, the treatment of diabetes mellitus can be represented by an outer control loop, to replace or artificially regulate the partially or totally deficient blood-glucose-control system of the human body. The quest for artificial pancreas can be structured in three different tasks: glucose sensor, insulin pump, and control algorithm problem [

To design an appropriate control, an adequate model is necessary. In the last few decades many scientists have tried to create mathematical models describing the human blood glucose system. A brief overview can be found here [

Due to the fact that models of diabetic systems are imprecise by nature, the modeling of the glucose-insulin system and controlling its behavior are two tightly connected questions; hence the problems could not be discussed separately. Regarding the applied control strategies the palette is very wide [

In case of type 1 diabetes mellitus, the insulin secreted by the pancreas is insufficient, therefore, external insulin needs to be injected, whereas glucose intake can be regarded as disturbance to the system. Therefore, external automatic regulator needs to be applied in order to restore balance (Figure

Diagram of the closed-loop system.

The aim of this paper is to present a controller design framework to investigate the type 1 diabetes from control theory point of view. A Quasi Model (a type 1 diabetes mellitus (T1DM) linear model) with favorable control properties is developed starting from the Liu-Tang model [

The paper is organized as follows: in Section

The kernel of the controller design framework is represented by the created Quasi Model, which is summarized below. In order to develop this simple, but useful model, physiological and mathematical considerations are taken into account. The chapter also briefly summarizes the optimal control methods which are used to demonstrate the utility of the developed framework.

In contrast with the previous models, like Bergman et al. [

The model can be naturally divided into three subsystems: the transition subsystem of glucagon and insulin, the receptor binding subsystem, and the glucose subsystem. Here, only main parameters and variables are explained, and detailed description and parameters can be found in [

The first two equations denote concentrations of glucagon (

The receptor binding system is captured by four equations:

Finally, the glucose system is represented by two equations:

Hormones of the pancreas have a cardinal role in blood glucose regulation and homeostatic stability, since negative feedback of glucagon and insulin through blood glucose level assures controllability (in medical sense):

In order to analyze the model in a quantitative manner, a physiologically correct input has to be defined for

Observing Figure

Open-loop simulation results of the original Liu-Tang model.

The original Liu-Tang model is only capable of describing the healthy human blood glucose system, although a type 1 diabetic model is required for controller design. Therefore, the model is reparameterized in order to describe type 1 diabetes mellitus [

In case of type 1 diabetes mellitus, insulin secretion of the pancreas becomes insufficient to regulate blood glucose. Equation (

Open-loop simulation results of the modified Liu-Tang model (type 1 diabetic).

The Liu-Tang model is a nonlinear system, and controlling a nonlinear system is not a trivial task. Controlling linear systems however has a vast theoretical background, and there is a wide range of tools available to implement a proper controller. Therefore, especially in our current case when the system is needed to be maintained in a certain steady state, linearization and creating controller for this linearized system are good approach. Linearizing the Liu-Tang model, one may face some serious problems.

Elements of the system matrix vary in a wide range, since it spans to

Control properties of the full rank system are not perfect, since the rank of the controllability matrix is 6, whereas the rank of the observability matrix is 5. Therefore, the system is neither controllable, nor observable.

Model reduction results in a transformed system where state variables do not have any physiological meaning. Therefore, their application cannot be carried out since measurements will not have any connection to actual state variables.

Simply selecting state variables from the full-rank system do not take interconnections into account. For instance, selecting plasma insulin, glucagon, and glucose the system matrix is

Therefore, another solution has to be found.

In order to avoid the above-mentioned problems, a simple, but useful linear model can be created. The goals are as follows:

elements of the system matrix should be from a narrow range,

controllability and observability,

state variables should have physiological meaning,

interconnections should be taken into account.

In order to create a physiologically plausible and useful model, three state variables should be considered: the two control hormones, insulin (

The developed Quasi Model is a simple linear system with one control input (intravenous insulin), one output (plasma glucose), and one disturbance (glucose intake):

The main contribution of this modeling technique which helps us overcome the above-mentioned problems is that we do not derive the simplified model directly from a complex nonlinear model, but we define the structure of the model based on theoretical and control theoretic aspects, and we use the behavioral simulation of the more complex Liu-Tang model for parameter tuning of the system matrix

In order to check the physiologically correct behavior of the model, glucose absorption input based on experiments presented in [^{2}) =

For our case this situation can be regarded as a worst-case disturbance (= large meal intake) as meal intake is not directly incorporated in the model, but treated as a disturbance, which makes the control task harder. This method is used for long time in control theory, and the bigger the meal intake (disturbance) is, the harder the control task becomes.

It has to be remarked that the system is linearized in steady state

In case of the healthy system, regulation mechanism is intact and it functions properly: there is enough insulin to decrease the elevated glucose level and insulin receptors are not insensitive to insulin.

The system matrix is

It should be emphasized here that the aim of this linear model is not to describe the accurate operation of the human blood glucose regulation system. The main contribution of the Quasi Model is to get a simple model which holds the characteristics of the original system, which yields only qualitative, but not quantitative description. This model can be used for controller designs, which can be tested on the clinically more reliable models. Hence, the Quasi Model is only a tool for control engineers, and it cannot be directly used in clinical aspects.

One good property of the system matrix (

In case of type 1 diabetes mellitus, regulation mechanism does not function properly: there is no insulin to decrease the elevated glucose level; however, insulin receptors are not insensitive to insulin. We can get a system matrix for the type 1 diabetes mellitus by canceling the terms

Therefore, the system matrix is

This section presents optimal control methods used to test the utility of the Quasi Model defined in the previous section. The purpose of the developed framework is to help control engineers create and test control laws for blood glucose regulation system. In this section, several examples are elaborated.

In order to check the applicability of the Quasi Model, different feedback controllers are designed and checked later on clinically more comprehensive models. First we design a feedback control with pole-placement, which is one of the simplest feedback design techniques. Then we design an observer, since in practice only the blood glucose concentration can be measured, and we need all the state variables available for feedback. Next, LQ control and

General feedback scenarios mean that the input signal is calculated from the state variables as

The feedback matrix (

By setting the desired poles bigger than the original ones, four scenarios are observed and the feedback matrices are

Closed-loop simulation results in case of Pole-Placement Control.

Control Input (Intravenous Insulin)

Output (Plasma Glucose)

This technique is not as explicit as the upcoming LQ and Minimax design techniques in the means of the resulting signals; however it is not negligible since the engineer can freely define the closed-loop model.

In the general feedback scenario we need the values of the state variables, however in practice only the blood glucose concentration can be measured. To overcome this problem we design an observer that estimates the state variables from the measured blood glucose concentration and the insulin input defined by the controller.

Let us consider the observer in the form

Consequently, the observer can be designed in three steps:

Using the general form of a dynamic LTI (linear time invariant) system

Considering the Quasi Model the disturbance (glucose) should be overweighted in the discussion of

Now, the problem is to find a control

Consequently, the general closed-loop system consists of four blocks (Figure

Type 1 Diabetic Model:

Absorption Model:

Observer:

Feedback Gain:

Structure of the closed-loop system.

The Controller consists of two subsystems: the observer and the feedback gain, which can be either the pole-placement, LQ Control or the minimax control feedback matrix.

In order to observe the performance and robustness of the designed controller, several different input data should be used on the reparameterized Liu-Tang model and modified Sorensen model [

The required input of the model is glucose absorbed from the gut, so providing plausible data is not a trivial task. One possibility is to use a static absorption profile like the one presented in [

Based on theoretical models of absorption [

The meal simulation model of Dalla Man et al. [

The key issue of the model is the rate of gastric emptying (

Change of gastric emptying rate (

In our case three carbohydrate intake absorption scenarios are considered (Figure

Absorption scenarios taken into account based on the meal model of Dalla Man et al. [

Absorption scenarios taken into account based on the use of the Weibull-curve.

All three meal scenarios through static and dynamic meal absorption were tested on both the reparameterized Liu-Tang model and the modified version of the Sorensen-model [

Figures

Comparison of pole-placement, LQ, and minimax control methods on the modified Liu-Tang model and modified Sorensen-model for the three dynamic absorption scenarios taken into account.

Glucose and control input variation for 30 g CHO absorption scenario

Glucose and control input variation for 60 g CHO absorption scenario

Glucose and control input variation for 100 g CHO absorption scenario

Comparison of pole-placement, LQ, and minimax control methods on the modified Liu-Tang model and modified Sorensen model for the three static absorption scenarios taken into account.

Glucose and control input variation for 30 g CHO dynamic absorption scenario

Glucose and control input variation for 60 g CHO dynamic absorption scenario

Glucose and control input variation for 100 g CHO dynamic absorption scenario

For the considered absorption scenarios results can be seen in Figures

The widely known “two-hump behavior” can be observed only on the Liu-Tang model. Due to the differences in the considered models, the glucose peaks also differ. This can be explained with the fact that the Liu-Tang model was created on the Korach-André et al. [

We should emphasize again that our goal was not to create the precise model of glucose absorption with the Weibull-curve, but to generate plausible absorption curves based on Korach-André et al. [

Regarding the applied control methods, it can be seen that the Minimax control gives better results, especially by means of maximal plasma glucose concentration, proving control theory results [

The hypoglycemia trend observed in Figures

It has to be remarked that the controller is developed for the Quasi Model, which is only a rough approximation of the type 1 diabetic system; however, when the controller is applied to the modified Liu-Tang model and the modified version of the Sorensen-model the closed-loop system produces the desired behavior.

An optimal controller design framework to investigate T1DM from control theory point of view was developed. First we created the structure of a linear Quasi Model (both healthy and type 1 diabetic versions) based on theoretical and control aspects, and then the parameters were tuned using Monte Carlo algorithm. The Liu-Tang model was used for validation in parameter tuning. Next, optimal control techniques (pole-placement, LQ, and Minimax methods) were applied to the type 1 diabetic version of the Quasi Model, proving that it can be successfully and easily used for controller design. Validation of the developed controllers, as well as demonstration of the performance and robustness of the developed framework was emphasized on different absorption scenarios and use of two more sophisticated glucose-insulin models: the reparameterized Liu-Tang model and the modified Sorensen-model. Consequently, the developed framework could help researchers engaging the control problem of diabetes as well as in physiological control education.

Further research directions could focus on applying modern control methods to the Quasi Model (e.g., modern robust control) or testing the developed framework on other well-known and sophisticated type 1 diabetic models (e.g., [

The research is partially funded by Hungarian National Scientific Research Foundation, Grants nos. OTKA T69055 and 82066. It is connected to the scientific program of the “Development of quality-oriented and harmonized R+D+I strategy and functional model at BME” project, supported by the New Széchenyi Plan (Project ID: TÁMOP-4.2.1/B-09/1/KMR-2010-0002). The authors address special thanks to Dr. Zsuzsanna Almássy from Heim Pál Hospital Budapest, Pediatric Department for her advices on diabetics questions. Moreover, the authors gratefully thank the important and constructive comments of the unknown reviewers.

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