A WAY FOR PREDICTING AND MANAGING THE GLYCAEMIC

INSTABILITY OF THE DIABETIC PATIENT

Farida Benmakrouha, Christiane Hespel

IRISA-INSA,20 Avenue des Buttes de Coesmes, 35043 Rennes cedex, France

Mikhail V. Foursov

IRISA-Universite de Rennes-1, 35042 Rennes cedex, France

Jean-Pierre Hespel

Centre Hospitalier Universitaire de Rennes, 35000 Rennes, France

Keywords:

Dynamical system, Bilinear model, Regulation, Stability, Insulin perfusion.

Abstract:

We study the Bounded-Input-Bounded-Output (BIBO) stability of the system modeling the behavior “insulin

delivery/glycaemia” of the diabetic patient, under continuous insulin infusion, continuous glucose monitoring,

in order to point out that the patient is entering in a period of stable/unstable equilibrium.

The model is a bilinear dynamical system predicting for an interval of 15 minutes, with an average error of

15%. In case of stable equilibrium, the prediction will be valid for a longer time interval, when in case of

unstable equilibrium, it will leads one to reduce the time intervals.

The BIBO stability is studied by computing the generating series G of the model. This series, generalization

of the transfer fuction, is a tool for analyzing the stability of bilinear systems. It is a rational power series in

noncommutative variables and by evaluating it, a formal expression of the output in form of iterated integrals

is provided. Three cases arise: ﬁrstly, the output can be explicitly computed; secondly, the output can be

bounded/unbounded if the input is bounded; thirdly, no conclusion seems available about the BIBO stability

by using G. We propose a stabilizing constant input η by studying the univariate series G

η

.

1 INTRODUCTION

Among the different medical possibilities to adminis-

ter insulin, the sub-cutaneous route is most secure and

easy to implement, but it lacks reliability. The intra-

venous route is the most rapidly responding method,

but it may cause vascular complications. The in-

traperitoneal route seems to be the most physiological

one.

In order to carry out a glycaemic regulation by an in-

traperitonal infusion of insulin, it is necessary to pre-

dict the glycaemia as a function of the insulin infu-

sion rate, for a given patient and a given insulin. The

ﬁrst closed-loop regulation method was developed by

A. Albisser back in 1974 (A. M. Albisser, ). Among

other methods, we can mention (J. L. Selam, 1992).

But, in spite of many positive aspects of these meth-

ods, none of them was unanimously accepted by the

medical community. This is partly due to insufﬁcient

frequency of glycaemic sampling and the difﬁculty to

vary rapidly the insulin infusion rates.

Recent technical progress made it possible to over-

come these difﬁculties. In 2000 appeared the ﬁrst

holter glycaemic device: the CGMS (Continuous

Glucose Monitoring System) which allows one to

measure the glycaemia every 3 minutes. A ﬁrst reg-

ulation system based on the CGMS was developed in

2001 by E. Renard of CHU of Montpellier in collab-

oration with Medtronic Minimed (Renard, 2003). In

2006, E. Renard (E. Renard, 2006), analyzes the im-

plantation of three components: a pump for peritoneal

insulin delivery, a central intravenous glucose sensor

and a controller. Meals are preceded by a handheld

programmed bolus calculed for pre-meals. A prob-

lem remains in case of unstability of the glycaemia

close to a meal, a stress, a physical effort.

335

Benmakrouha F., Hespel C., Foursov M. and Hespel J. (2009).

A WAY FOR PREDICTING AND MANAGING THE GLYCAEMIC INSTABILITY OF THE DIABETIC PATIENT.

In Proceedings of the International Conference on Biomedical Electronics and Devices, pages 335-338

DOI: 10.5220/0001121003350338

Copyright

c

SciTePress

Then we proposed a bilinear modeling giving a

good approximation of the behavior “insulin deliv-

ery/glycaemia” on an interval of 15 minutes, in stan-

dard conditions (C. Hespel, 2000).

Once the model is known, the regulation consists

in inverting the input/output behavior of the system

(M. V. Foursov, 2003; M. V. Foursov, 2004). One has

to calculate the input in terms of the output function

one wishes to obtain (Fig.1). This regulation is par-

Figure 1: Regulation.

tially closed-loop because the glycaemic values are

only used every 15 minutes in order to compute the in-

sulin delivery. On constant intervals of time [t

i

,t

i+1

]

i

,

we compute some model M

i

and function insulin de-

livery u

i

(t) in order to follow an ideal trajectory y

i

(t)

for the glycaemia. On every time interval, the trajec-

tory is recalculed because of the variation between the

ideal trajectory and the true trajectory.

The crucial point consists in determining the size of

the time intervals, i.e. the frequency of the changes of

the insulin delivery. The study of the stability leads

us to reduce the size of the interval when the system

is unstable.

2 PRELIMINARIES: MODELING,

REGULATION

Two techniques are used to achieve this goal:

Phenomenological modeling requires a knowledge of

the equations governing the evolution of the process.

Such models of the glycaemic behavior of diabetics

were developed (Albisser, ).

In the behavioral modeling, the system is regarded as

a black box (J. Sjoberg, 1995). The goal is to con-

struct a model that approximates the system with a

desired precision (Chen and Chen, ). The parame-

ters involved in the obtained system of equations have

no practical signiﬁcance, but their number depends on

the required precision.

A commonly-used class of models is formed by linear

dynamical systems. Linear-model-based regulation is

simple to implement and it gives quite satisfactory re-

sults in many cases. It did not seem to be sufﬁcient

for regulating the glycaemia of diabetics.

Another class of models consists of bilinear systems.

A bilinear system is quite similar to a linear one: it is

additionally linear as a function of the input. One has

thus more leeway to approximate the real system with

a better precision. We choose to model the system by

a bilinear system whose dimension is not ﬁxed.

Since a diabetic does not respond in the same way to

equal doses of insulin at different times of the day, we

suppose that she is described by different systems at

different time instants. So we construct a collection

of models that describe the behavior of the glycaemia

under certain conditions. Each model is thus valid

only for a certain period of time (at least 15 minutes).

Mathematically, the problem is to identify locally

(near t

0

) up to a given order k, a single input sys-

tem with drift considered as a black box, when only a

sample of the input/output data is known. Our method

involves the identiﬁcation up to order k of the gener-

ating series G of the unknown system and the con-

struction of a bilinear system (B

k

) approximating the

unknown system up to k.

An advantage of this method is the possibility to pro-

vide, in terms of the order k, a system (B

k

) approxi-

mating the unknown system. (B

k

) is chosen so that its

output and the output of the unknown system coincide

up to k. The problem of identiﬁcation of dynamical

systems in a neighborhood of t

0

is the following: to

determine the generating series of the unknown sys-

tem, up to a given order k, given the Taylor series of

the input and corresponding output functions.

The identiﬁcation involves the following 3 steps. Dur-

ing the ﬁrst step, one obtains a system of linear equa-

tions that express the relationship between the deriva-

tives of the input and the output. The unknown pa-

rameters are certain linear combinations of the coefﬁ-

cients of the generating series. On the next step, these

linear combinations of the coefﬁcients are identiﬁed

from the available data, by choosing appropriate in-

put/output sets. Finally, the coefﬁcients are identiﬁed

by solving another system of linear equations.

A bilinear system corresponding to a rational series of

minimal rank is constructed providing a local model.

2.1 The Bilinear Model

A bilinear system (B) with a single input u

1

(t) and a

drift u

0

(t) ≡ 1 is given by its state equations

(B)

x

(1)

(t) = (M

0

+ u

1

(t)M

1

)x(t)

y(t) = λ.x(t)

(1)

x(t) ∈ an R−vector space Q, M

0

, M

1

, λ are R−linear.

We consider the alphabet Z = {z

0

, z

1

}, where z

0

codes

the drift and z

1

codes the input. The expansion of the

generating series G built on Z, by noting w a word

∈ Z

∗

, is G =

∑

w∈Z

∗

hG|wiw.

BIODEVICES 2009 - International Conference on Biomedical Electronics and Devices

336

G is a rational series deﬁned from (1) by:

G = λ.x(0) +

∑

ν≥0

1

∑

j

0

,···, j

ν

=0

λ.M

j

0

·· ·M

j

ν

x(0)z

j

0

·· ·z

j

ν

(2)

We compute the rational expression associated with

(2), by generalizing the Schutzenberger’s method

(Schutzenberger, 1961).

By “evaluating” the expression of G, we obtain a for-

mal expression of the output (Fliess, 1981)

y(t) =

∑

w∈Z

∗

hG|wi

Z

t

0

δ(w) =

Z

t

0

δ(G) = ε(G) (3)

where the iterated integrals are recursively deﬁned by:

Z

t

0

δ(w) =

1 i f w = 1

Z

R

t

0

(

R

τ

0

δv)u

i

(τ)dτ i f w = vz

i

(4)

And we compute the iterated integral

R

t

0

δ(G).

2.2 The Regulation Method

Once the model is known, the regulation consists in

inverting the input/output behavior. One has to cal-

culate the input in terms of the output function one

wishes to obtain. This regulation is partially closed-

loop, since the glycaemic values are used to recalcu-

late the insulin infusion rates every 15 minutes.

In a previous paper (M. V. Foursov, 2002) we have

shown that we are capable of ﬁnding the Taylor se-

ries expansion of the command from the Taylor series

expansion of the desired output trajectory, using gen-

erating series techniques similar to those used during

the identiﬁcation and the modeling. The algorithm

consists in sequential solving a system of polynomial

equations. If the model of a diabetic were an ex-

act one, this would be sufﬁcient to regulate the gly-

caemia. But since our bilinear model is an approxi-

mation of the actual one, the glycaemic behavior will

eventually deviate from the chosen trajectory. Then,

the trajectory has to be recalculated in order to com-

pensate for these deviations and the insulin infusion

device has to be reprogrammed accordingly.

An important point is to determine the frequency of

change of the insulin infusion rates. The ﬁrst tests of

our modeling method (C. Hespel, 2000) showed that

we can predict the glycaemia over 15-minute inter-

vals with an error of about 10% or 15%. We need to

provide a new data to the pump every 15 minutes.

3 STUDY OF THE BIBO

STABILITY

A dynamical system is Bounded-Input-Bounded-

Output (BIBO) stable if its output y(t) is deﬁned and

bounded for every bounded input u(t).

The output of a bilinear dynamical system can be

computed in evaluating its generating series. This

evaluation consists in integrating every term of this

series and in summing. We use the theorem of Hoang

(Hoang, 1990) :

Theorem 1. ∀k, let us suppose that G

k

is exchange-

able and let us denote ε(G

k

) by g

k

(ξ(t))

g

k

(ξ(t)) = g

k

(t, ξ

1

(t), ··· , ξ

m

(t)) (5)

where ξ

j

(t) is the primitive of the input u

j

(t) can-

celling for t = 0. Then, ∀k, the series

S

k

= G

0

z

i

1

G

1

·· · z

i

k

G

k

(6)

where z

i

1

, · · · , z

i

k

∈ Z, has the following evaluation:

ε(S

k

) = y(t) =

R

t

0

R

τ

k

0

·· ·

R

τ

2

0

g

0

(ξ(τ

1

))g

1

(ξ(τ

2

) − ξ(τ

1

))· · ·

g

k

(ξ(t)− ξ(τ

k

))dξ

z

i

1

(τ

1

)· · · dξ

z

i

k

(τ

k

)

(7)

Three cases occur: In some cases, this process is easy

and y(t) can be explicitly computed. In other cases, if

we assume that u(t) is bounded by 2 values Min, Max,

then we can know if so is y(t), without computing

explicitly y(t). Lastly, in some difﬁcult cases, we only

try to ﬁnd some stabilizant constant inputs u(t) = η

such that the output remains bounded, if it is possible.

We prove that the output of the bilinear system for the

input u(t) = η consists in evaluating some univariate

series G

η

. This series being rational, can be written

as a quotient of 2 polynomials. We can then use 2

propositions (F. Benmakrouha, 2007) dealing with the

poles of G

η

in order to decide that a stability exists for

u(t) = η

Proposition 1. A necessary condition for the BIBO

stability of (B), is that, for every η ∈ R, the real part

of the poles of G

η

is ≤ 0 and the imaginary poles of

G

η

are single.

Proposition 2. If there exists η such that every pole

of G

η

has a negative real part and if every imaginary

pole is single, then u(t) = η is a stabilizing input

3.1 Example 1

For an insulin delivery u(t), a glycaemia y(t), the state

equations of the system (B

2

) are

x

(1)

(t) = (

0 0

a b

+ u(t)

0 0

1 0

)x(t)

y(t) = (1.5 1) x(t)

(8)

The generating series is :

G

2

= (z

1

+ az

0

)(bz

0

)

∗

+ 1.5 (9)

G

2

= G

21

+ G

22

+ 1.5 and y(t) = ε(G

2

)

with G

21

= z

1

(bz

0

)

∗

, G

22

= az

0

(bz

0

)

∗

A WAY FOR PREDICTING AND MANAGING THE GLYCAEMIC INSTABILITY OF THE DIABETIC PATIENT

337

We obtain an explicit expression of y(t):

ε(G

21

) = e

bt

R

t

0

e

−bτ

1

dξ

1

(τ

1

)

ε(G

22

) = ae

bt

Z

t

0

e

−bτ

1

dτ

1

(10)

For u(t) = η then ξ

1

(τ

1

) = ητ

1

, we get :

y

2,η

(t) =

η+a

b

(e

bt

− 1) + x(0)

This system is not BIBO for b > 0.

This system is BIBO for b < 0 (if M

1

≤ u(t) ≤ M

2

then y(t) is bounded)

For instance, for a > 0, b < 0, 0 ≤ u(t) ≤ M, then

y(t) ≤ x(0) +

M+a

−b

3.2 Example 2

The state equations of the bilinear system (B

3

) are

x

(1)

(t) = (

0 0 0

a b c

0 a 0

+u(t)

0 0 0

1 0 0

0 1 0

)x(t)

y(t) = (1.5 1 0) x(t)

(11)

The generating series is

G

3

= (z

1

+ az

0

)(bz

0

+ (z

1

+ az

0

)cz

0

)

∗

+ 1.5 (12)

We compute G

3,η

by substituting ηz

0

to z

1

in G

3

:

G

3,η

= 1.5 +

(a+η)z

0

1−bz

0

−(a+η)cz

2

0

.

If η 6= −a then we decompose G

3,η

in partial fractions

for studying the constant stabilizing inputs

u(t) = η depending on the parameters a, b, c.

4 CONCLUSIONS AND FUTURE

WORKS

The BIBO stability of a bilinear system cannot be

generally studied by considering its state equation. In

this paper, we use the “evaluation” of its generating

series G. If the rational expression of G is simple or

obtained by concatenating some simple rational ex-

pressions, then the use of the generating series of the

system provides an answer about the stability and a

bound for the output. Otherwise, we can look for a

stabilizant constant input u(t) = η by using the uni-

variate series G

η

.

By applying this method to the bilinear model approx-

imating the behavior “insulin delivery/glycaemia”,

we expect an information about the stability of the

system describing really this behavior.

A speciﬁc surveillance depending on whether the sys-

tem is stable/unstable will be set. Rather than take

constant interval of 15 minutes for recalculate the

ideal trajectory of the glycaemia, we propose that the

time intervals depend on this information about the

stability. In case of unstability, the varying size of the

intervals of time would be deﬁned in order to keep the

glycaemia between some moderate bounds.

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