Finite-time control of plasma glucose in insulin therapies for diabetes
Abstract
To study the finite-time control of plasma glucose for diabetic patients with impulsive injections of insulin, we propose an impulsive differential equation model with initial and boundary conditions. The goal of glucose control is supposed to be achieved if the system has a solution, otherwise the goal cannot be achieved. By constructing two comparison systems and using a comparison principle, several conditions under which the system has a solution are obtained. Furthermore, some numerical simulations are given. The results show that a relatively higher initial insulin level is beneficial for the glucose control. For a predefined finite time, injection dose and injection period are two important adjustable factors which can guarantee the achievement of the control goal.
Keywords
Finite-time control Insulin therapy Impulsive injection Comparison principle1 Introduction
Diabetes mellitus, as a common chronic disease, is characterized by high concentration level of plasma glucose due to the body’s inability to produce insulin or the ineffective utilization of the insulin produced. Diabetes mellitus is generally classified into three main types: type 1 diabetes, commonly seen in children and young adolescents, is caused by the malfunction of the body’s immune system and almost no insulin is produced from the pancreas; type 2 diabetes, commonly seen in adults, is thought to be caused by the dysfunction of the glucose–insulin regulatory system such as insulin resistance, so that body cells cannot absorb glucose timely by utilizing insulin; and gestational diabetes, which is first detected during pregnancy for glucose intolerance of various degrees. In recent decades, the number of diabetic patients increased rapidly all over the world and diabetes has come to be seen as an epidemic disease worldwide. Long-term complications of this disease, including possible blindness, amputation and kidney failure, affect hundreds of millions of people around the world. Based on such a reality, many researchers are motivated to study the pathogenesis and therapy of diabetes and other problems associated with it.
Current therapies for diabetic patients include taking medications, insulin supplementation and dietary adjustment. The subcutaneous injection of insulin is usually carried out through a syringe or insulin pump. Insulin pump is a medical device to administrate insulin or its analogs. In clinical practice, an insulin pump is popularly used in the therapy for both type 1 and type 2 diabetes and it can greatly relieve the pain of diabetic patients [1, 2, 3, 4, 5, 6, 7, 8, 9]. Even so, the life style of the patients is still seriously affected, for example, patients have to inject insulin manually before meals to avoid hyperglycemia, and the injection dose needs to be carefully computed according to the carbohydrate to be ingested [4, 5]. So in recent years, researchers have been attempting to solve technological problem to develop an artificial pancreas [10, 11], which can substitute the endocrine functionality of a real and healthy pancreas [4, 12, 13].
In order to carefully choose the correct dose of insulin and the right time of injection, several mathematical models have been constructed and studied. However, most of these studies focused on the change of the glucose concentration after a sufficiently long time. But in some clinical situations, glucose concentration needs to be controlled under a certain level in a finite time. For example, critically ill patients whose plasma glucose is in an extremely high level need to drop their glucose concentration in a relatively short time. In this paper, we propose an impulsive differential equation model for insulin injection with finite-time control. Through qualitative analysis, we shall give the conditions under which the glucose concentration can be controlled in a range predefined.
The rest of the paper is organized as follows. In Sect. 2, a mathematical model of impulsive injection of insulin with initial and boundary value conditions is formulated. In Sect. 3, the existence of the solution is discussed by comparison theorem. Finally, some discussions and numerical simulations are provided in Sect. 4.
2 Model formulation
Mathematical models of insulin therapy for diabetes have played an important role in understanding the pharmacological mechanism of insulin in treating diabetes mellitus, and they are also applied to anticipate the efficacy of different therapeutic schedules. In order to design effective control strategies for plasma glucose, researchers have formulated many models to simulate interactions of glucose and insulin. For example, the glucose–insulin regulatory system of normal people was studied in [14, 15, 16, 17, 18], and the insulin sensitivity is considered as a whole in [19, 20, 21, 22]. In order to determine the main cause of the sustained oscillation of the endocrine metabolic system, delay differential equation models are proposed in [23, 24, 25, 26]. Doran et al. [27] formulated a mathematical model to study the insulin therapy for critically ill patients in ICU and insulin infusion was considered. Noticed that the insulin injection is a relatively transient behavior in the whole course of therapy, so it can be see as a pulse. Impulsive differential equation models have a big advantage in describing such kind of behaviors. For now, several impulsive differential equation models have been proposed to study the impulsive injection of insulin in diabetes treatments [4, 9, 12].
Most of the studies in the literature focused on the dynamical behaviors in infinite time, corresponding to the long-term control of the plasma glucose level. However, in clinic practice, when the glucose concentration is at a level much higher than normal, the life of the patient will be threatened in a very short time. So how to lower the glucose concentration in a given time is considered by the doctor. That is to say, the glucose concentration is required to be reduced to a normal level in a given time which is finite. As far as we know, there is very little study about the finite-time control of the glucose concentration. Based on our previous work about glucose control, in this paper we construct a new mathematical model for the finite-time control of plasma glucose, and qualitatively analyze conditions under which the goal of control can be achieved.
3 Finite-time control of glucose
In this section, we consider the finite-time control of glucose by analyzing the system (2). Firstly, we formulate two comparison systems for variable I, study the relationship between the comparison systems and system (2) and get some basic properties. Then we give the upper and lower bounds of the two comparison systems. The conditions under which the system (2) has a solution are finally obtained.
3.1 Preliminary
3.2 The upper and lower bounds of the comparison systems
In the following, we give the upper and lower bounds of the comparison systems (4) and (5). These results will be applied to discuss the existence of the solution of system (2) with initial and boundary value problem.
Proposition 1
If\(I_{0}\geq\frac{\sigma}{1-\exp(-d_{i}\tau)}\), then the system (4) satisfies\(I_{1}(t)\geq I_{1}(T)\triangleq I_{0}\exp(-d_{i} T)+\frac{\sigma[\exp(-d_{i}\tau)-\exp(-d_{i} T)]}{1-\exp (-d_{i}\tau)}\). If\(I_{0}< \frac{\sigma}{1-\exp(-d_{i}\tau)}\), then the system (4) satisfies\(I_{1}(t)\geq I_{1}(\tau)\triangleq I_{0}\exp(-d_{i} \tau)\).
Proof
If \(I_{0}\geq\frac{\sigma}{1-\exp(-d_{i}\tau)}\), \(I_{1}(t)\) is monotonic decreasing on \((k\tau, (k+1)\tau]\) and \(\{I_{1}((k+1)\tau)\}\), \(k=0,1,2,\ldots,p-1\) is a monotonic decreasing sequence, then \(I_{1}(t)\geq I_{1}(p\tau)=I_{1}(T)=I_{0}\exp(-d_{i} T)+\frac{\sigma[\exp(-d_{i}\tau )-\exp(-d_{i} T)]}{1-\exp(-d_{i}\tau)}\).
Conversely, if \(I_{0}< \frac{\sigma}{1-\exp(-d_{i}\tau)}\), \(I_{1}(t)\) is monotonic decreasing on \((k\tau, (k+1)\tau]\) and \(\{I_{1}((k+1)\tau)\}\), \(k=0,1,2,\ldots,p-1\) is a monotonic increasing sequence, then \(I_{1}(t)\geq I_{1}(\tau)=I_{0}\exp(-d_{i} \tau)\). That completes the proof. □
Proposition 2
Proof
If \(I_{0}\geq\frac{\sigma_{1}}{d_{i}}\) and \(I_{0}-\frac{\sigma_{1}}{d_{i}}\leq \frac{\sigma}{1-\exp(-d_{i}\tau)}\), then \(I_{2}(t)\) is monotonic decreasing on \((k\tau, (k+1)\tau]\), \(\{I_{2}((k+1)\tau)\}\) and \(\{I_{2}((k+1)\tau^{+})\}\), \(k=0,1,2,\ldots,p-1\), are monotonic increasing sequences, thus we get \(I_{2}(t)\geq I_{2}(\tau)\) and \(I_{2}(t)\leq I_{2}((p-1)\tau^{+})=I_{2}[(T-\tau )^{+}]\) for \(t\in[0,T]\). That completes the proof. □
3.3 Existence of solution of system (2)
Now we discuss the existence of the solution of the system (2) for the finite-time glucose control.
Theorem 3.1
Proof
Theorem 3.2
Proof
By a similar discussion to Theorem 3.1, we can also get \(G(T)\leq G_{U}\). That completes the proof. □
Theorem 3.3
Proof
Because \(I_{0}\geq\frac{\sigma_{1}}{d_{i}}\) and \(I_{0}\leq\frac {\sigma}{1-\exp(-d_{i}\tau)}\), we have \(I_{0}-\frac{\sigma_{1}}{d_{i}}\leq\frac {\sigma}{1-\exp(-d_{i}\tau)}\).
By a similar discussion to Theorem 3.2, we can also get \(G(T)\geq G_{L}\). That completes the proof. □
4 Numerical simulation and discussion
In this paper, we build an impulsive differential equation model with initial and boundary value conditions to study the finite-time control of glucose for the insulin therapy of diabetics. Compared with system (2.2) proposed in [4], our new model concentrates on the finite-time control of glucose under periodic injection of insulin. This is meaningful for some clinical situations, for example, seriously ill patients whose plasma glucose level is extremely high and needs to be controlled under certain level in a relatively short time. By applying comparison theorem, we obtain several results which guarantee that the glucose concentration drops to a safe level in a finite time. This potentially contributes to the insulin therapy for diabetics in the clinic.
According to the relationship of initial insulin level (\(I_{0}\)), injection dose (σ) and injection period (τ), we obtain sufficient conditions for the existence of solution of the system (2) (see Theorem 3.1, Theorem 3.2, Theorem 3.3). For every case, we give the upper and lower bounds of the glucose concentration. In clinic, we are more interested in the upper bound of the glucose level. The upper bound in Theorem 3.2 is the same as in Theorem 3.1, so there are two types of upper bound (see inequality (22) and (30)). Obviously, both \(f_{1}(0,T)=(\frac{n+I_{1}(0)}{n+I_{1}(T)})^{\frac{am}{d_{i}}}\) and \(f_{1}(\tau,T)=(\frac{n+I_{1}(\tau)}{n+I_{1}(T)})^{\frac{am}{d_{i}}}\) decrease monotonically with σ, so the upper bound of glucose level is a monotonically decreasing function of σ. That is to say, the upper bound of the glucose concentration we obtained in Sect. 3 will decrease with the increase of the injection dose.
However, the upper bound of the glucose concentration in \([0,T]\) is not necessarily the glucose concentration at time T (i.e. G(T)). In order to evaluate the control effect when the control objective (\(G_{L}< G(T)\leq G_{U}\)) is achieved, we perform a series of numerical simulations to explore factors that affect the control effect.
Model parameter values from [4]
Parameters | Values | Units | Parameters | Values | Units |
---|---|---|---|---|---|
\(G_{\mathrm{in}}\) | 216 | mg/min | m | 900 | mg/min |
b | 100 | mg/min | n | 80 | mg |
\(\sigma_{2}\) | 5 × 10^{−6} | min^{−1} | \(\sigma_{1}\) | 6.27 | mU/min |
a | 3 × 10^{−5} | mg^{−1} | \(\alpha_{1}\) | 105 | mg |
c | 40 | mg/min | \(d_{i}\) | 0.08 | min^{−1} |
\(G_{U}\) | 190 | mg/dl | \(G_{L}\) | 60 | mg/dl |
Notes
Acknowledgements
This work is supported by the National Natural Science Foundation of China (11501489, 11371306, 11671346, 11601466 and 11701495), Nanhu Scholars Program of XYNU, Nanhu Scholars Program for Young Scholars of XYNU, Foundation and frontier project of Henan Province (152300410019) and Youth Teacher Foundation of XYNU (2016GGJJ-14, 2011079).
Authors’ contributions
All authors contributed equally to this work. All authors read and approved the final manuscript.
Competing interests
The authors declare that they have no competing interests.
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