APPLICATION OF OPTIMAL CONTROL STRATEGIES FOR PHYSIOLOGICAL MODEL OF TYPE 1 DIABETES-T1D

1Laboratory Analysis, Modeling and Simulation LAMS. Department of Mathematics and Computer Science, Faculty of Sciences Ben M’Sik, Hassan II University of Casablanca, Morocco 2Endocrinology, Diabetology and Metabolic Diseases Department, Ibn Rochd University Hospital of Casablanca, Morocco 3Mathematical Engineering Team (INMA), Department of Mathematics, Faculty of Sciences El Jadida, Chouaib Doukkali University, El Jadida, Morocco


INTRODUCTION
To entirely understand diabetes in first, we explain the mechanism of the use of the body's energy from food. In fact, during digestion, a large part of the supply transforms into glucose ( Figure. 1a) When glucose enters the bloodstream, it moves to cells that need energy and support for absorption. The presence of glucose in the blood stimulates the production of insulin by the pancreas (Figure. 1b). Therefore, the insulin is considered as the primordial key in bringing glucose into the cells where the mechanism of glucose transformation and storage takes place. When this regulatory system does not work or when in other words enough insulin or use it properly. Then we can talk about diabetes. The rest of the paper is organized as follows. In section 2, we present our description of diabetes. Section 3 gives a general description of our proposed method and the description of the carbohydrate metabolism model. In section 4, we illustrate by an example and numerical simulations. Finaly, in section 5, we provide the conclusion.

DESCRIPTION OF DIABETES
Diabetes is then a disorder of the assimilation, the use and storage of sugar brought by the diet. This result by a high level of glucose within the blood called glycemia.
While fasting, the glycemia of a normal person is included between 0.72 and 1.26. When the glycemia is inferior than 0.72 g/l or superior than 1.26 g/l, we are then respectively speaking hypoglycemia or After a meal, glycemia may increase slightly and then return to normal state.
Diabetes is then hyperglycemia when the glycemia is higher or equal to 1.26 while fasting and higher than 2 g/l during a normal day. The organ that is responsible of the regulation of the glycemia is called the pancreas.  The role of these two hormones in the regulation of glucose is as follows ( Figure. 2b): • When the Glycemia is superior than 1.26 g/l or hyperglycemia, the β cells will secrete insulin. Having a hypoglycemic effect, that is to say an effect to lower the glycemia.
• When the Glycemia is inferior than 0.72g/L or hypoglycemia, the alpha cells will secrete glucagon: Having a hyperglycemic effect, that is to say an effect which will increase the glycemia.  to insulin. Here we may say that diabetes is precisely a disease that comes from a flaw in the production of insulin. There are two main types diabetes, type 1 diabetes T1D which affected 10% of the diabetics, and type 2 diabetes T2D which is the most common type, affects 85%, while the remaining 5% are affected by an other type.
The type 1 diabetes, also called DID, is usually found in children, adolescents and adults for some unknown reasons. This type of diabetes is characterized by an absence of β -cells in the islet of Langerhans, thing that makes it possible to say that this type of diabetes is an autoimmune disease, it means that the immune system starts to fight its owncells.
Concerning people with this type of diabetes. The body does not langer manufacture any more insulin so the only current treatment is the contribution of this one, either in the form of injection with a syringe or a pen or with a pump portable or implantable treatment device intended to administer the insulin continuously.
For people with type 1 diabetes that's mean the tack of insulin, the glucose cannot enter cells so it accumulated within the blood which causes an abnormal increase of the level of glucose in the blood. where case the insulin injection allows the glucose to enter cells and be used as energy.
Traitement D1D. During the treatment, it is the patient who controls his disease. However, the patient relies on the subcutaneous administration of insulin, called insulin therapy, associated with a balanced diet and regular physical activity. This treatment must be for life. And The patient must himself determine the dose of insulin to be injected ( Figure. 3a)according to the measurements of his glucose level In fact, the injectable insulin replaces the insulin that should be manufactured by the body, so the doctor offers an insulin regimen adapted to the patient's glycemic profile Slow insulin is used to regulate the glycemia throughout the day, while the fast insulin is used to regulate the glycemia after meals The injected insulin is divided into "necessary to live" which is the elementary insulin, and the insulin "necessary to eat" is the bolus "roundmass" of insulin. Yet, the issue here is how to properly manage basal doses of insulin.
The    Figure. 4) is a one compartment model, meaning that the body is described as a compartment tank with a basal concentration of glucose and insulin. The minimal model actually contains two minimal models. One describing glucose kinetics, how blood glucose concentration reacts to blood insulin concentration and one describing the insulin kinetics, how blood insulin concentration reacts to blood glucose concentration. The two models respectively take insulin and glucose data as an input.

FIGURE 4. Schema of Bergman model
The Bergman model. Minimal model involves two physiological compartments : a glucose compartment and a plasma insulin compartment, which is assumed acting through a remote form to influence net glucose uptake. The glucose-insulin systems is given as follows.
and the problem is to minimize the objective functional : The study will be focused on examining the glucose insulin model oscillation related to food intake, and to examine their ability to simulate homeostatic glucose control.
The Pontryagin's Maximum Principle provides necessary conditions for an optimal control problem. This principle converts into a problem of minimizing the Hamiltonian, such that ∂ I we obtain the following theorem: Theorem 3.1. Given optimal control u * and solutions G * , X * and I * of the corresponding state system, there exists adjoins satisfying the following equations: with transversally conditions λ 1 t f = η, λ 2 t f = 0 and λ 3 t f = 0. Moreover, the optimal control is given by: The optimal control u can be solve from the optimality condition,

NUMERICAL SIMULATIONS
In this section we will present a numerical simulation to illustrate the contribution of our method Note that it is always difficult to assign a set of general settings for patients with diabetes 1 types and representing varying clinic cases. However, since the main purpose of this study is to use the optimal control theory to reduce the rate of glucose in blood, the values of the parameters found in [7] are kept and it is stated that the stability properties of the model (S 1 ) are stored for these parameters.
The glucose minimal model is used to interpret the glucose kinetics of an IVGTT. These interpretations are based on parameter estimation, using measured blood insulin concentrations during the test as input to the model. Bergman et al. [11].
The basal values measured are G b = 92 mg/dL and I b = 7.3 mU/dL. By inserting the insulin data as input, and using Matlab/Simulink they derive the following parameters:

CONCLUSION
In this work, we proposed an effective strategy to reduce or to decrease the glycimia. The optimal control theory has been applied in the context of Bergman's model, and that includes one control representing the effort reducing the glucose by digestion, lost and . By using the Pontryagins maximum principle, the explicit expression of the optimal controls was obtained.
The numerical simulation of both the systems i.e with control and without control, shows that this strategy helps to reduce the glycemia.
In morocco, we don't currently have available application adapted to the Moroccan context.
So, in future work and in collaboration with the endocrinology department of the University Hospital Center UHC-Casablanca, we will try to implement our model with parameters adapted to Moroccan habits (family meal,... ) using android or smartphone.

CONFLICT OF INTERESTS
The author(s) declare that there is no conflict of interests.