ISSN: 0970-938X (Print) | 0976-1683 (Electronic)

An International Journal of Medical Sciences

Research Article - Biomedical Research (2017) Volume 28, Issue 16

**Faiz-Ul-Hassan, Muhammad Adil, Ali Khaqan ^{*}, Sana Shuja, Moazzam Islam Tiwana, Qadeer-ul-Hassan, Shahzad Malik and Raja Ali Riaz**

Department of Electrical Engineering, COMSATS Institute of Information Technology, Chak Shahazad, Park Road, Islamabad, Pakistan

- *Corresponding Author:
- Ali Khaqan

Department of Electrical Engineering

COMSATS Institute of information Technology, Pakistan

**Accepted on** July 26, 2017

Diabetes is a chronic metabolic disorder affecting millions of people worldwide. Especially, type-1 diabetics have required strict glycemic control. In this paper, close loop control system is designed to normalize the high blood glucose level for diabetes patients. Glucose-insulin dynamics in blood plasma are represented by Bergman minimal mathematical model which is used as base model. The dynamical variations between different and same individual poses a major challenge in designing of controller for biological systems. The contribution of this research work lies in designing a backstepping based nonlinear controller which perfectly deals with nonlinearities present in the system. In order to visualize the robust behavior, meal and exercise are added as a disturbance factor and controller effectively track the set point value of 70 mg/dL from an initial state of hyperglycemia. The control criteria imposed on the proposed controller are hyperglycemia and hypoglycemia.

Backstepping, Blood glucose, Diabetes, Bergman model.

Diabetes mellitus (DM) also called diabetes, is prolonged metabolic disorder due to high blood sugar level. Diabetes is characterized by the failure of the human body to maintain the physiological ranges of blood glucose level [1]. This chronic condition occurs when pancreas inside the human body is not capable of producing enough insulin or cells are not in such a position to respond insulin properly produced by body [2]. There are over 220 million victims of this chronic disease in the world by a survey of World Health Organization (WHO).

Diabetes can be categorized mainly into two types; Type-1
Diabetes Mellitus (T1DM) and Type-2 Diabetes Mellitus
(T2DM). About 1.5 million people died every year between
2000-2012 as reported in [3]. 10% of diabetics have type-1
diabetes while the rest suffer from type-2 [3]. Type 1 or
insulin-dependent is the more severe form of diabetes. It
usually develops in children, teenagers or at any stage of life.
Therefore, also termed “juvenile” diabetes. Diabetics with type
1 lost their capability of insulin production and require
continuous administration of insulin *via* injection or insulin
pump. The symptoms of type 1 diabetes involve; frequent
urination, increased thirst, increased hunger and unusual
weight loss [4]. In closed loop system, insulin is delivered in
continually glucose responsive way. Closed loop Artificial
Pancreas Device (APD) systems normally combine three
functions; (1) monitoring function using Continuous Glucose
Monitor (CGM) attached wirelessly, (2) the delivery task of
hormone therapy *via* pump (3) digital controller. Integrating aforementioned three functions provides fully automated
closed loop system and these closed loop APDs are worn
externally [5]. This type of closed loop system technology
provides hands free, continuous glycemic control and eases the
life of diabetes patients. Hyperglycemia and hypoglycemia are
the two scenarios for blood glucose concentration inside the
body. Hyperglycemia occurs when the sugar level is higher
than 110 mg/dL and sugar level lower than 70 mg/dL referred
as hypoglycemia. Both of these body’s states affect the health
and life of the patient [6]. Diabetes patients manually perform
the procedure of blood glucose regulation. In such scenarios,
automatic closed loop controlling of sugar level permits the
victims to participate in daily activities with less long terms
complications. Open loop system, also defined as nonfeedback
system is such type of control system in which output
has no influence or effect on control action of input signal. An
open loop system is considered as to follow input command or
set point value regardless of final result. Closed loop system,
also termed as feedback system is type of control system which
takes into account current output value and changes it to the
desired condition. The control mechanism in such type of
systems totally dependent on output. For diabetes patients, in
an open loop system, predetermined amount of insulin is
injected to the patient according to the normal pancreatic
insulin secretion curve. In closed loop system, sensor measures
the glucose level, according to this information, control unit
will inject the required amount of insulin to maintain the
glucose level in normal range.

Automatic blood glucose regulation in diabetes patients is an extensive and dynamic field of research from decades. A number of efforts have been made by researchers and scientist to achieve the goal of maintaining high glucose level.

Bergman was the first person to provide more basic and physiologically verified mathematical model of blood glucose control inside a human body. It was a nonlinear single compartment model which simply uses the idea of remote tank for storing of insulin. Different mathematical variables were employed to symbolize the concentration of glucose, remote insulin and insulin in the blood plasma [7,8].

Farmer et al. discusses subcutaneous insulin vaccination which is an open loop system for providing the insulin to diabetics. To fulfil the basal prerequisite, mandatory task for the patient is to take the dosage early in the morning. Two considerations are important for this type of insulin injection stated as measurement of glucose level and an idea of how much food is to be eaten [9].

A prior model of regulating plasma glucose level, an algorithm of semi closed loop type has been presented in [10] for controlling the hyperglycemia more precisely. Authors presented Proportional Integral Derivative (PID) controller for the design of feedback system for glucose-insulin regulation mechanism in [11] where meal is to be taken into account as disturbance for the system.

An optimal neuro controller has been designed for T1DM diabetics. Linear Quadratic Regulator (LQR) control technique is employed to show the control performance of the system in the presence of large uncertainties. In this method, there is no need of online tuning of control variables [12].

In order to provide relaxation to diabetes patients for glucose intake, state estimation based approach has been investigated in [13] which perfectly deals with large time delays.

In [14], a closed loop control system for type-1 diabetics has been modeled using H∞ controller. Bergman minimal model is used for parametric representation of glucose insulin concentration. Uncertainties (meal, exercise) have been involved to visualize the robustness of the proposed system.

Another strategy for type-1 diabetics based on parametric programming was studied and analysed in detail for infusion of insulin to lower the high glucose level. Using this technique, inter-patient and intra-patient variability was minimized and optimal control input is obtained for insulin delivery [15].

Robust Fixed Point Transformation (RFPT) is used to make the Model Predictive Control (MPC) perfect robust for type-1 diabetics. Adaptive controller using RFPT technique deals with generic parameters of the system model without the need of system’s state estimation. Bergman model dynamics are perfectly validated via this transformation to control the blood glucose level of diabetics [16]. In [17], a meal detection based algorithm for continuous monitoring of blood glucose, is developed. It has high performance parameters and interpatient variability was also reduced. This methodology is perfect for artificial pancreas based control system.

Hyperglycemia can never be achieved through this method because bolus dose of insulin is infused for large meal inputs.

In the next section, Bergman glucose insulin mechanism is presented. Section III proposes the designed control law for insulin delivery. Section IV presents the discussion on obtained simulation results. Finally, section V concludes the research work.

Numerous efforts have been made for mathematical modeling of diabetics for glucose insulin regulation mechanism [18-20]. Dr. Bergman was one of the pioneers for developing the mathematical representation of glucose-insulin regulatory mechanism for exogenous infusion of insulin. It is also known as “minimal model” having no biological complexities.

A feedback control system which measures and regulates the
blood glucose concentration is shown in **Figure 1**. According to
block diagram, a sensor will measure the glucose level and this
signal will be fed back to the control system which will
estimate the required exogenous infusion of insulin to
normalize the glucose level. Desired amount of insulin will be
provided via mechanical pump externally.

Bergman minimal model is described as follows,

**Table 1** shows the description of parameters used by Bergman.

Parameter | Symbol | Units |
---|---|---|

Plasma glucose level | G (t) |
mg/dL |

Remote insulin | X (t) |
mU/L |

Plasma insulin level | I (t) |
mU/dL |

Glucose base level before injection | G_{b} |
mg/dL |

Insulin base level before injection | I_{b} |
μU/ml |

Input (insulin) | u (t) |
mU/min |

Glucose absorption rate to blood via food intake |
D (t) |
- |

Insulin independent constant | p_{1} |
1/min |

Decrease rate of tissue’s glucose up taking | p_{2} |
1/min |

Enhanced glucose up taking capability (insulin base) | p_{3} |
(μU/ml)/min^{2} |

Plasma insulin decay rate | n |
1/min |

Threshold value | h |
mg/dL |

Insulin secretion of β cells | γ |
μU/ml/min^{2}/(mg/dL) |

**Table 1.** Model parameters.

State space formation of system in Equations 1-3 is

Term γ (G (t)-h)^{+} represents internal regulatory system which
does not exist in diabetic patient, so it will be ignored. *D (t)* is
disturbance in the form of meal or exercise, neglects it. So,
Equation 6 in Laplace form is as follows

sI (s)=-nI (s)+u (s) → (7)

I (s)=u (s)/s+n → (8)

From (5),

sX (s)=-p_{2} X (s)+p_{3} I (s) → (9)

X (s)=p_{s} u (s)/(s+p_{2}) (s+n) → (10)

Using Equation 4,

s (G)=-p_{1} G (s)-G_{b} X (S) → (11)

G (s)=-G_{b} p_{2} u (s)/(s+p_{1}) (s+p_{2}) (s+n) → (12)

Transfer function of the overall system is given as

The main objective of this study is an appropriate infusion of
insulin to lower the high glucose level of diabetics on priority
basis. Excessive or insufficient administration of insulin may
disturb physiological ranges of glucose level inside the body
which affects the patient health. Insulin is infused *via* two
methods i.e., open loop and closed loop infusion. In open loop
method, a predetermine amount of insulin is injected manually.
Normal pancreatic insulin secretion curve is considered as reference for insulin delivery. **Figure 2 **represents the open loop
configuration for insulin delivery.

*Close loop insulin delivery*

Closed loop arrangement for insulin infusion involves a sensor
which continuously measures the level of glucose. According
to the information from sensor, controller decided the insulin
infusion rate to normalize glucose level. **Figure 1** shows close
loop insulin infusion mechanism.

*Proposed design*

This segment presents the design of controlled input using
nonlinear control technique. Since, most of the biological
systems are purely nonlinear in nature on behalf of interpatient
as well as intrapatient variability. In order to deal with these
nonlinearities, nonlinear control technique Backstepping is
used. Lyapunov function is used for stability analysis of
nonlinear system around an equilibrium point. Lyapunov is a
scalar function playing significant role in control theory.
Backstepping is Lyapunov based nonlinear control technique.
Backstepping is used for designing of stabilizing controls for a
recursive class of nonlinear systems. In this process, higher
order system is divided into sequence of lower order system
[21]. The method of designing starts with known stable system
and back out new controller which stabilizes the subsystem as
well as uncertainties of main system. Recursive process is
terminated when final control is achieved. Dynamical systems
having nth order is decomposed into nth scalar subsystem *via* Backstepping and finally the synthesis of controller is
implemented into nth iterations [22]. The state space model of
glucose insulin regulatory mechanism is

y=z_{1} → (16)

Where,

In Equations 20-22, the variables e_{1}, e_{2} and e_{3} are error
function for three states of system model, z_{d} desired level to be
tracked and a is the stabilizing function. From Equations 20-22
variables (z_{1}, z_{2} and z_{3}) can be computed as,

z_{1}=e_{1}+z_{d} → (23)

z_{2}=e_{2}+a_{1} (e_{1}) → (24)

z_{3}=e_{3}+a_{2} (e_{1}, e_{2}) → (25)

**Step-1: **Taking derivative of Equation 20 and put the value of
z_{2}, we have,

Assume the Lyapunov candidate function is

a_{1} (e_{1}), used to stabilize first error state is,

Using Equations 27 and 31 simplifies to a value for stabilizing function as

Lyapunov function simplifies to,

**Step-2: **Taking derivative of Equation 21 and put the value of
z_{3}, we have,

Again consider the Lyapunov candidate function is,

Substituting Equations 33 and 35 in Equation 37 gives,

a_{2} (e_{1}, e_{2}), used to stabilize second error state is,

Taking derivative of Equation 32 gives the value,

Using values from Equations 27, 35 and 41, stabilizing function reduces to a value,

Lyapunov function simplifies to,

**Step-3:** Taking derivative of Equation 22 and substituting
Equation 19, we have,

Lyapunov candidate function is,

Substituting Equations 33, 43 and 45 in Equation 47; the resultant is Equation 48,

Lyapunov function simplifies to,

The control input for overall system is,

The derived control input in Equation 50 track the desired level of glucose for all diabetics in acceptable physiological range. The control law perfectly compensates the interpatient and intrapatient variability by providing insulin to control the high glucose level.

Medically, there are different states of environment in which
level of glucose changes for individuals from a normal to
extreme values discussed in **Table 2**.

State no. | Glucose level | Clinical description |
---|---|---|

1 | G ≥ 150 |
Hyperglycemia |

2 | 150 ≥ G ≥ 110 |
Slight Hyperglycemia |

3 | 110 ≥ G ≥ 70 |
Normoglycemia |

4 | 40 ≥ G ≥ 70 |
Slight Hypoglycemia |

5 | 40 ≥ 0 | Hypoglycemia |

**Table 2.** States of environment.

For performance evaluation of proposed controller, parameters
value for different patient which were clinically investigated
are shown in the **Table 3** [23].

Parameter | Normal | Patient 1 | Patient 2 | Patient 3 |
---|---|---|---|---|

p1 |
0.031 | 0 | 0 | 0 |

p2 |
0.012 | 0.011 | 0.007 | 0.014 |

p3 |
4.92-6 | 5.3-6 | 2.16-6 | 9.94-6 |

γ |
0.0039 | 0.0042 | 0.0038 | 0.0046 |

n |
0.265 | 0.26 | 0.246 | 0.281 |

h |
79.035 | 80.2 | 77.578 | 82.937 |

Gb |
70 | 70 | 70 | 70 |

Ib |
7 | 7 | 7 | 7 |

G0 |
291.2 | 220 | 200 | 180 |

I0 |
364.8 | 50 | 55 | 60 |

**Table 3.** Parametric values.

This section presents the simulation results of Bergman
minimal model using MATLAB/Simulink for designing of
nonlinear Backstepping based controller. Parametric values
shown in **Table 3** are used for simulation purposes in order to
analyse the output profile for diabetes patients.

The generic profile of glucose regulation mechanism for
healthy person and diabetes patient is shown in **Figure 3**. It can
be seen from their responses that how the highly elevated level
of blood glucose for healthy person tracked the desired basal
value. High glucose level of healthy person slowly tracks the
basal value to normalize the sugar level in physiological range.
However, glucose level of diabetes patient is still very far away
from the desired basal level which demands the inclusion of
robust controller in the design [11].

The simulation setup for desired and actual output waveforms
for three different patients discussed in **Table 3 **is shown in **Figure 4**. Different clinically investigated parameters including gender, age, height and weight of patients are the main factors
to be considered while attaining the desired level of glucose via
designed control law. The controller also takes into
consideration the interpatient and intrapatient variability while
attaining the desired glucose level in normal range of 70-110
mg⁄dL.

The simulation output profile for glucose level stability using
proposed Backstepping based controller is shown in **Figure 5**.
After the infusion of exogenous insulin using the derived
control law, plasma glucose concentration variations and
tracking of basal level has been shown in simulation results. It
is evident from the obtained results that designed controller
perfectly controls the high glucose level and lowers it from
critical area into a stable range in an appropriate interval of
time. The control law of proposed controller tracks the desired
level with minimum tracking error showing the efficiency of
the design.

The robustness test of proposed system can be analysed by
adding disturbance factor *D (t)=Aexp ^{-Bt}* via controller input.
This disturbance would be in the form of meal or exercise,
because normal regulatory system does not exist in diabetics.
The controller efficiently tracks the basal value after starting from high value of blood glucose level shown in Figure 5.
Disturbance factor increases the settling time but controller
track the desired level in appropriate interval of time showing
the effectiveness of the proposed design. In order to achieve
the desired glucose level for diabetes patients, numerous
control methodologies under different control scenarios
according to patient’s conditions discussed in [13-16]. The
novelty of this work lies in designing of perfect robust real
time controller to lower the high glucose level for diabetics in
normal range. Different problems in literature review including
large time delays, input uncertainties and interpatient
variability has been considered preferably, and perfectly
minimized. The designed controller has less settling time and
minimum steady state error as compared to the other
controllers, designed and discussed in previous research work
before.

It can be observe that controlling the blood glucose level for three different patients in this study reveals that their output profile to attain the desired level is changing significantly. It depends on Lean Body Mass (LBM). LBM is primarily function of gender, height and weight of patients. For men and women, it is calculated as

**Figures 4 **and **5** show different response time for three patients
in order to track the desired basal level. LBM is one of the key
parameter responsible for patient’s such response. The other
factors are types of insulin and method through which insulin
is injected into body. Types of insulin involves rapid acting,
short acting, intermediate acting, long acting and infusion
process includes number of insulin delivery methods like
injection via syringe, insulin pens, external insulin pump,
inhalable insulin and insulin jet injectors.

The combine picture of whole scenarios of desired glucose
level tracking for three different patients in the presence of
external disturbance is shown in **Figure 5**.

Diabetes management in human regulation mechanism is
considered as one of the important and challenging task which
is addressed in this paper. Bergman minimal model is used for
dynamic modeling of controlling high glucose level for
diabetics. Feedback linearization is employed to transform the
nonlinear single compartment model into controllable
canonical form. A nonlinear control scheme, backstepping is
studied in detail and applied to explore different aspects of
glucose regulation mechanism. The designed control law
efficiently tracks the desired glucose level with minimum error
for all three patients shown in **Table 3**. Apart from designing
the control law, another important part of this research activity
is the stabilizing of glucose level in the presence of
disturbances (meal and exercise). Despite large inter-patient
variability, as shown through the different clinical parameters
of the patients, all the 3 patients achieved the desired level of
glucose.

The glucose regulatory system is a vast domain in the medical field with lot of potential for real time implementation.

With the help of National Institute of Health (NIH) Pakistan, we are going to employ and test the designed control system for critically ill patients. For future work and further analysis, disturbance parameters require comprehensive study as well as application of adaptive Backstepping for the proposed system model.

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