## Classification

# of variables | # of modes | # of jumps |
---|---|---|

4 | 2-3 | 1-4 |

Type | Continuous dynamics | Guards & Invariants | Resets |
---|---|---|---|

hybrid | non-polynomial | linear polynomial | identity |

## Download

Flow* Strategy I | glucose_control_I |

Flow* Strategy II | glucose_control_II |

Flow* Strategy III | glucose_control_III |

## Model description

We study the “minimal model” defined in [1-3] for the dynamics of glucose and insulin interaction in the blood system. It is described by the following ODE.

wherein are constant parameters whose typical values are min, min and min U L. The values of and are the basal values of plasma glucose concentration and free plasma insulin concentration respectively, and they are given by mmol L, U L. The constant is the insulin distribution volume, and the value of denotes the fractional disappearance rate of insulin. We take their values as L and min. The remaining parameters are variables, we give their explanations as below.

the difference of plasma glucose concentration | |

the free plasma insulin concentration | |

the insulin concentration in an interstitial chamber | |

the rate of infusion of exogeneous glucose | |

the rate of infusion of exogeneous insulin |

We use the formulation given in [4] for the rate of insulin infusion, and consider three different strategies for the insulin delivery rate .

*Strategy I* is proposed in [5].

The rate is U h when mmol L,

and it is U h when mmol L.

If is in the range from to mmol L, we use U h which is a linear transition between the rates and mmol L.

*Strategy II* is taken from [6], it is similar to Strategy I but considers the rates and mmol L.

That is, we set U h when mmol L,

and U h when mmol L.

For the case that is between and mmol L, is defined by .

In order to better stabilize the glucose level, a more sophisticated control strategy is presented in [6]. We call it *Strategy III*. The controller reads the value of in the beginning of every hours and do the following job.

If mmol L, then we use the input rate U min.

Otherwise is set to be the linear form U min.

## Reachability settings

We consider the initial condition , , and under all control strategies.

## Results

An over-approximation computed by Flow* for Strategy I is shown below. The time horizon is :

Similarly, we show the figures of the overapproximations computed by Flow* for Strategy II and Strategy III.

## References

[1] R. Bergman, Y. Ider, C. Bowden, C. Cobelli.*Quantitative estimation of insulin sensitivity.*In The American Journal of Physiology, volume 236, pages 667-677, The American Physiological Society, 1979. [2] R. Bergman, L. Phillips, C. Cobelli.

*Physiologic evaluation of factors controlling glucose tolerance in man: measurement of insulin sensitivity and -cell glucose sensitivity from the response to intravenous glucose.*In The Journal of Clinical Investigation, volume 68, pages 1456-1467, The American Society for Clinical Investigation, 1981. [3] R. Bergman, D. Finegood, M. Ader.

*Assessment of insulin sensitivity in vivo.*In Endocrine Reviews, volume 6, pages 45-86, Endocrine Society, HighWire Press, 1985. [4] M. Fisher.

*A semiclosed-loop algorithm for the control of blood glucose levels in diabetics.*In IEEE transactions on biomedical engineering, volume 38(1), pages 57-61, IEEE,

1991. [5] D. Chisholm, E. Kraegen, D. Bell, D. Chipps.

*A semiclosed loop computer-assisted insulin infusion system.*In The Medical journal of Australia, volume 141, pages 784–789, Australasian Medical Publishing Company Proprietary Limited, 1984. [6] S. Furler, E. Kraegen, R. Smallwood, D. Chisholm.

*Blood glucose control by intermittent loop closure in the basal mode: computer simulation studies with a diabetic model.*In Diabetes Care, volume 8, pages 553–561, American Diabetes Association, 1985.