Glycemic control


# of variables # of modes # of jumps
4 2-3 1-4
Type Continuous dynamics Guards & Invariants Resets
hybrid non-polynomial linear polynomial identity


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.

     \[ \left\{ \begin{array}{lcl} \dot{G} & = & -p_1 \cdot G - X\cdot(G + G_B) + P \\ \dot{X} & = & -p_2\cdot X + p_3\cdot I \\ \dot{I} & = & -n\cdot (I + I_B) + \frac{u}{V_I} \end{array} \right. \]

wherein  p_1,p_2,p_3 are constant parameters whose typical values are  p_1 = 0 min^{-1},  p_2 = 0.025 min^{-1} and  p_3 = 0.013 min^{-2} U^{-1} L. The values of G_B and I_B are the basal values of plasma glucose concentration and free plasma insulin concentration respectively, and they are given by G_B = 4.5 mmol L^{-1}, I_B = 0.015 U L^{-1}. The constant V_I is the insulin distribution volume, and the value of n denotes the fractional disappearance rate of insulin. We take their values as V_I = 12 L and n = \frac{5}{54} min^{-1}. The remaining parameters are variables, we give their explanations as below.

G the difference of plasma glucose concentration
I the free plasma insulin concentration
X the insulin concentration in an interstitial chamber
P the rate of infusion of exogeneous glucose
u the rate of infusion of exogeneous insulin

We use the formulation P = 0.5 \cdot \exp(-0.05\cdot t) given in [4] for the rate of insulin infusion, and consider three different strategies for the insulin delivery rate u.

Strategy I is proposed in [5].
The rate u is 0.5 U h^{-1} when G < 4 mmol L^{-1},
and it is 2.5 U h^{-1} when G > 8 mmol L^{-1}.
If G is in the range from 4 to 8 mmol L^{-1}, we use u = (0.5\cdot G - 1.5) U h^{-1} which is a linear transition between the rates 4 and 8 mmol L^{-1}.

Strategy II is taken from [6], it is similar to Strategy I but considers the rates 2 and 12 mmol L^{-1}.
That is, we set u = 0.5 U h^{-1} when G < 2 mmol L^{-1},
and u = 2.5 U h^{-1} when G > 12 mmol L^{-1}.
For the case that G is between 2 and 12 mmol L^{-1}, u is defined by (0.2\cdot G + 0.1)$ U h$^{-1}.

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 G in the beginning of every 3 hours and do the following job.
If G \geq 6 mmol L^{-1}, then we use the input rate u = G\cdot (0.41 - 0.0094\cdot G) U min^{-1}.
Otherwise u is set to be the linear form 0.007533\cdot(1 + 0.22\cdot G) U min^{-1}.

Reachability settings

We consider the initial condition G \in [13,14], X = 0, and I = 0.5 under all control strategies.


An over-approximation computed by Flow* for Strategy I is shown below. The time horizon is [0,720]:


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




[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 \beta-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,

[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.