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Mathematical modelling of glucose-insulin system is very important in medicine as a necessary tool to understand the homeostatic control of human body. It can also be used to design clinical trials and in the evaluation of the diabetes prevention. In the last three decades so much work has been done in this direction. One of the most notable models is the global six compartment-mathematical model with 22 ordinary differential equations due to John Thomas Sorensen. This paper proposes a more simplified three compartment-mathematical model with only 6 ordinary differential equations by introducing a tissue compartment comprising kidney, gut, brain and periphery. For model parameter identification, we use inverse problems technique to solve a specific optimal control problem where data are obtained by solving the global model of John Thomas Sorensen. Numerical results show that the proposed model is adaptable to data and can be used to adjust diabetes mellitus type I or type II for diabetic patients.

It is common knowledge that lifestyle factors largely influence our health. These factors are diet, physical activity, smoking and psychological stress. The lifestyle changes have the influence on metabolism of some systems of human body including glucose-insulin system, for example disordered glucoregulation. Therefore two variables that have a bearing on glucose homeostasis are affected, those are: pancreas beta cell response to glucose and sensitivity of body to insulin. The key organs that control blood glucose are pancreas and liver. The key hormones are insulin and glucagon. Large-scale clinical trials have demonstrated the benefits of tight control of glucose-insulin system, minimizing disease complications and improving quality of life [

Since the 1960s, mathematical models have been developed to describe glucose-insulin dynamics [

The rest of the paper is organised as follows. In Section 2, we set mathematical model equations. The Section 3 deals with qualitative study. Estimation of model parameters is presented in Section 4 while Section 5 focuses on concluding remarks.

The Sorensen model [

The metabolic sources and sinks in the glucose-insulin mathematical model are from the physiologic processes that happen at a constant rate. The mathematical nomenclature is defined in

Taking into account the exchanges illustrated in

{ V H G d G H ( t ) d t = Q L G G L ( t ) + γ T G ( G T ( t ) ) α − Q H G G H ( t ) − R H G , V L G d G L ( t ) d t = Q A G G H ( t ) − Q L G G L ( t ) + R L G , V T G d G T ( t ) d t = Q P G ( G H ( t ) − G T ( t ) ) − R T G , V H I d I H ( t ) d t = Q L I I L ( t ) + γ T I ( I T ( t ) ) β − Q H I I H ( t ) , V L I d I L ( t ) d t = Q A I I H ( t ) − Q L I I L ( t ) + R P I R − R L I , V T I d I T ( t ) d t = Q P I ( I H ( t ) − I T ( t ) ) − R T I . (1)

Variable | Description |
---|---|

V | Volume (L) |

G | Glucose concentration (mg/dL) |

I | Insulin concentration (mU/dL) |

Q | Vascular blood flow (dL/min) |

Subscript | Description |

H | Heart and Lungs |

L | Liver |

T | Tissues |

A | Hepatic Artery |

PIR | Peripheral insulin release |

Superscript | Description |

G | Glucose |

I | Insulin |

Constant to be estimated | Description |

R | Metabolic source |

or sink rate (mg/min or mU/min) | |

γ | Vascular blood flow rate (dL/min) |

α | Parameter |

β | Parameter |

Let X e = ( G H e , I H e , G L e , I L e , G T e , I T e ) ′ be the steady state vector where X' denotes the transpose of X. In order to analyse the steady state, we need to solve the following system:

{ Q L G G L e + γ T G ( G T e ) α − Q H G G H e = R H G , Q A G G H e − Q L G G L e = − R L G , G H e − G T e = R T G Q P G , Q L I I L e + γ T I ( I T e ) β − Q H I I H e = 0 , Q A I I H e − Q L I I L e = R L I − R P I R , I H e − I T e = R T I Q P I . (2)

Note that the first three equations and the last three equations of (2) form the glucose model and insulin model respectively. From the glucose model we get G H e and G L e as functions of G T e

G H e = G T e + R T G Q P G and G L e = 1 Q L G [ − R L G + Q A G ( G T e + R T G Q P G ) ] , (3)

and the glucose model becomes

{ Q L G G L e + γ T G ( G T e ) α − Q H G G H e = R H G , G H e = G T e + R T G Q P G , G L e = 1 Q L G [ − R L G + Q A G ( G T e + R T G Q P G ) ] .

In the same way, from the insulin model we get I H e and I L e as functions of I T e

I H e = I T e + R T I Q P I and I L e = 1 Q L I [ Q A I ( I T e + R T I Q P I ) + R P I R − R L I ] , (4)

and the insulin model can be rewritten as follows

{ Q L I I L e + γ T I ( I T e ) β − Q H I I H e = 0 , I H e = I T e + R T I Q P I , I L e = 1 Q L I [ Q A I ( I T e + R T I Q P I ) + R P I R − R L I ] .

Let J G = ∂ G ∂ X ( X e ) and J I = ∂ I ∂ X ( X e ) be Jacobian matrices of glucose model and insulin model respectively where all derivatives are evaluated at the equilibrium point X e . After some algebraic calculations we get

J G = ( − Q H G Q L G α γ T G ( G T e ) α − 1 Q A G − Q L G 0 Q P G 0 − Q P G ) and J I = ( − Q H I Q L I β γ T I ( I T e ) β − 1 Q A I − Q L I 0 Q P I 0 − Q P I ) . (5)

The behaviour of the mathematical model (1) near the steady state can be analysed by the nature of the real parts of the eigenvalues of matrices J G and J I . The proof of the theorem below will use the following proposition due to Routh-Hurwitz [

Proposition 1. Let a 1 , a 2 and a 3 be positive real numbers. The roots of the polynomial

λ 3 + a 1 λ 2 + a 2 λ + a 3 = 0

have negative real parts when a 1 a 2 > a 3 .

Theorem 2.

The system (1) is asymptotically stable if the following conditions are satisfied:

1)

α γ T G ( G T e ) α − 1 < Q H G − Q A G , (6)

2)

α γ T G ( G T e ) α − 1 < Q L G Q P G ( Q H G − Q A G ) + Q H G + Q L G , (7)

3)

α γ T G ( G T e ) α − 1 < 1 [ Q P G ( Q P G + Q H G ) ] [ 2 Q P G Q L G Q H G − Q L G Q A G Q H G + ( Q H G ) 2 ( Q P G + Q L G ) + ( Q L G ) 2 ( Q P G + Q H G − Q A G ) + ( Q P G ) 2 ( Q H G + Q L G ) ] , (8)

4)

β γ T I ( I T e ) β − 1 < Q H I − Q A I , (9)

5)

β γ T I ( I T e ) β − 1 < Q L I Q P I ( Q H I − Q A I ) + Q H I + Q L I , (10)

6)

β γ T I ( I T e ) β − 1 < 1 [ Q P I ( Q P I + Q H I ) ] [ 2 Q P I Q L I Q H I − Q L I Q A I Q H I + ( Q H I ) 2 ( Q P I + Q L I ) + ( Q L I ) 2 ( Q P I + Q H I − Q A I ) + ( Q P I ) 2 ( Q H I + Q L I ) ] . (11)

Proof.

The system (1) is asymptotically stable if and only if J G and J I are stability matrix; that is, every eigenvalue of J G and J I has a negative real part. The characteristic equation associated to J G is P G ( λ ) = 0 given by

| − Q H G − λ Q L G α γ T G ( G T e ) α − 1 Q A G − Q L G − λ 0 Q P G 0 − Q P G − λ | = 0.

That is

λ 3 + ( Q H G + Q L G + Q P G ) λ 2 + [ Q P G ( Q H G + Q L G − α γ T G ( G T e ) α − 1 ) + Q L G ( Q H G − Q A G ) ] λ + Q P G Q L G ( Q H G − Q A G − α γ T G ( G T e ) α − 1 ) = 0. (12)

Similarly P I ( λ ) = 0 is equivalent to

λ 3 + ( Q H I + Q L I + Q P I ) λ 2 + [ Q P I ( Q H I + Q L I − α γ T I ( I T e ) α − 1 ) + Q L I ( Q H I − Q A I ) ] λ + Q P I Q L I ( Q H I − Q A I − α γ T I ( I T e ) α − 1 ) = 0. (13)

Using Proposition 1, we need to verify the following requirements: all the coefficients of (12) are positive and the product of coefficients of second and third terms of (12) is strictly greater than its fourth term.

Indeed, since all vascular blood flow rates are positive, then the relation Q H G + Q L G + Q P G > 0 is obvious. The next requirement is

Q P G Q L G ( Q H G − Q A G − α γ T G ( G T e ) α − 1 ) > 0 ,

which after some calculations is equivalent to

α γ T G ( G T e ) α − 1 < Q H G − Q A G .

Similarly, the requirement

Q P G ( Q H G + Q L G − α γ T G ( G T e ) α − 1 ) + Q L G ( Q H G − Q A G ) > 0 ,

is equivalent to

α γ T G ( G T e ) α − 1 < Q L G Q P G ( Q H G − Q A G ) + Q H G + Q L G .

The last requirement

[ Q P G ( Q H G + Q L G − α γ T G ( G T e ) α − 1 ) + Q L G ( Q H G − Q A G ) ] ( Q H G + Q L G + Q P G ) > Q P G Q L G ( Q H G − Q A G − α γ T G ( G T e ) α − 1 ) , (14)

yields after calculations (8).

The requirements (9), (10) and (11) are obtained in a similar way by considering the insulin model.,

The nonlinear system (1) can be represented in the following compact form

X ˙ ( t ) = f ( X ( t ) , μ ) , (15)

where μ is the vector of parameters to be estimated. That is

μ = ( γ T G , α , R H G , β , γ T I , R L G , R T G , R T I ) ′ .

The mathematical model requires parameter identification which can be carried out by setting the following optimal control problem. We determine the control μ such that the cost functional

J ( μ ) = ∫ 0 t f q G ( G H ( t ) − G H o b s ) 2 + q I ( I H ( t ) − I H o b s ) 2 d t , (16)

is minimized under the restriction of the model Equation (15). The positive scalar coefficients q G and q I determine how much weight is associated to each term in the integrand. Superscript “obs” refers to the observed state to which the system is transferred by the control. In order to obtain the observed data, we solve the global model of [

For computational purposes we discretize the system (15) using N linear B-splines. Let us consider

B N = { ψ j N , j = 1 , ⋯ , N } , (17)

a linear B-splines basis functions on the uniform grid

Ω N = { t k = k T N , k = 0 , ⋯ , N } , (18)

such that

ψ i N ( t k ) = δ i k ,

where δ i k is Kronecker symbol. Let us introduce the vector space W N whose

the basis is B N . It follows that dim W N = N and W n ⊂ W n + 1 , n = 1 , ⋯ , N . We assume that functions appearing in the system (15) are continuous on [ 0, T ] . Let us denote W = C 0 ( 0 , T ) and consider the interpolation operator

Π N : W → W N ,

satisfying ∀ ϕ ∈ W

Π N ϕ ( t k ) = ϕ ( t k ) , k = 1 , ⋯ , N .

Therefore, in this setting we are looking for the solution X N ∈ W N of the following discrete problem

X ˙ N ( t ) = f ( X N ( t ) , μ N ) , such that X N ( 0 ) = X 0 N , (19)

where the control is

μ N = ( γ T G , N , α N , R H G , N , β N , γ T I , N , R L G , N , R T G , N , R T I , N ) ∈ ( W N ) 8 .

The corresponding discrete optimal control problem (16) is to minimize

∑ k = 1 N ( q G ( G H N ( t k ) − G H o b s ) 2 + q I ( I H N ( t k ) − I H o b s ) 2 ) h with h = T N (20)

with respect to (19). In compact form the problem (20) can be rewritten as follows

min μ N J N ( μ N ) = ∑ k = 1 N h Y k T R Y k (21)

subject to

{ X ˙ N ( t ) = f ( X N ( t ) , μ N ) X N ( 0 ) = X 0 N (22)

where Y k , k = 1 , ⋯ , N is the following matrix

( ( G H N ( t k ) − G H o b s ) , ( I H N ( t k ) − I H o b s ) ) ′ ,

and R is the matrix defined by

R = ( q G 0 0 q I ) .

The numerical computations have been carried out using a collection of MaTlaB routines [

Physiological and dynamical conditions for glucose and insulin impose the need for relatively simple models that should be able to describe as accurately as possible the mechanical behavior of glucose-insulin system. In this work we have proposed a three compartmental mathematical model that describes the variation of glucose and insulin for human being. The modelling technique is used to

Parameter | Value |
---|---|

γ T G | 5.1575 |

0.0354 | |

β | 1.0422 |

γ T I | 2.1056 |

R H G | 9.9966 |

R L G | 1042.1509 |

R L I | 7.6076 |

R T G | 40.8085 |

R T I | 1.0013 |

provide interesting answers to the question of determining the global mathematical model with lower number of equations for glucose-insulin system. Numerical results show that the proposed model is adaptable to data. In fact

The authors declare no conflicts of interest regarding the publication of this paper.

Ntaganda, J.M., Minani, F., Banzi, W., Mpinganzima, L., Niyobuhungiro, J., Gahutu, J.B., Rutaganda, E., Kambutse, I. and Dusabejambo, V. (2018) Simplified Mathematical Model of Glucose-Insulin System. American Journal of Computational Mathematics, 8, 233-244. https://doi.org/10.4236/ajcm.2018.83019