Glycemic control (polynomial version)

Classification

# of variables # of modes # of jumps
4 6,9 10,18
Type Continuous dynamics Guards & Invariants Resets
hybrid non-linear polynomial linear polynomial identity

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Flow* Strategy I diabetic_1.model
Flow* Strategy II diabetic_2.model

Model description

We consider a model of the glycemic control in diabetic patients such that all dynamics are defined by polynomials. The modeling ODE is given by

     \[ \left\{ \begin{array}{lcl} \dot{G} & = & -p_1 \cdot G - X\cdot(G + G_B) + g(t) \\ \dot{X} & = & -p_2\cdot X + p_3\cdot I \\ \dot{I} & = & -n\cdot (I + I_B) + \frac{1}{V_I}\cdot i(t) \end{array} \right. \]

such that G is plasma glucose concentration above the basal value G_B and I is the plasma insulin concentration above the basal value I_B. X is the insulin concentration in an interstitial chamber. The constant parameters are given by p_1 = 0.01, p_2 = 0.025, p_3 = 0.000013, V_I = 12, n = 0.093, G_B = 4.5, I_b = 15.

The influx of glucose g(t) after a meal is modeled as

     \[ g(t) = \left\{ \begin{array}{ll} \frac{t}{60} & t \leq 30 \\ \frac{120-t}{180} & t\in (30,120] \\ 0 & t > 120 \end{array}\right. \]

while the insulin control strategies i(t) due to [1] and [2] are given by

     \[ i_1(t) = \left\{ \begin{array}{ll} \frac{25}{3} & G(t) \leq 4 \\ \frac{25}{3} (G(t) -3) & G(t) \in (4,8] \\ \frac{125}{3} & G(t) > 8 \end{array} \right.\] \[ i_2(t) = \left \{ \begin{array}{ll} 1 + \frac{2G(t)}{9} & G(t) < 6 \\ \frac{50}{3} & G(t) > 6 \end{array}\right. \]

respectively.

Reachability settings

We consider the initial condition G(0) \in [-2,2], X(0) = 0, I(0) \in [-0.1,0.1], and the time horizon is given by [0,360].

Results

The following figures show an over-approximation computed by Flow* for the first and the second strategies:

diabetic_1

diabetic_2

References

[1] 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, HighWire Press, 1985.

[2] 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.