an event-based point of view on the control of insulin...

An event-based point of view on the control ofinsulin-dependent diabetes

Brigitte Bidegaray-Fesquet

Laboratoire Jean KuntzmannUniv. Grenoble Alpes, France

Reunion e-BaCCuSSSeptember 4th 2018

B. Bidegaray-Fesquet (LJK) Event-based control of diabetesReunion e-BaCCuSS, 04/09/2018 1 /

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Introduction Treatment of insulin-dependent diabetes

Treatment of insulin-dependent diabetescontinuous vs. discrete control

The treatment of insulin-dependent diabetes involves the artificial controlof the patient’s plasma glucose rate via insulin infusion.

Today’s main research: artificial pancreas,which combines a glycaemia sensor and aninsulin pump → continuous control.

Most patients do not have this device.The sensor and the control are decoupled.Glycaemia measurements and insulin infusionsare not continuous in time but triggered atdiscrete times.→ discrete control, event- and self-triggeredcontrol.


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Introduction Approach

Signal processing vs. ODE approach

Engineering approach: signal processingGlycaemia is measured, producing a discrete signal, with a 1 minutesampling time on current devices.Its value should be within a target interval Itarget.When it departs from this target, glycaemia can be pushed back byinfusing insulin, or ingesting carbohydrates.

Mathematical approach: ODEsA patient is modeled by a big system of Ordinary Differential Eqs:

Y = f (Y ,P,U)

involving many variables Y , patient-dependent parameters P and acomplex control procedure U.Only one variable, glycaemia, is observed:

y = CY .


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Cobelli’s model Constructing a patient-dependent model

Model definition

ODE model based on Cobelli’s model ([DM07,DM14]).

Time evolution of main variables:

G: plasma glucose rate,I: plasma insulin rate,H: plasma glucagon rate.

and secondary variables.

It involves many patient-dependent parameters.

It is driven by external outputs (meals M, physical exercise) andcontrol procedures.

[DM07] Chiara Dalla Man, Robert A. Rizza, and Claudio Cobelli, Meal simulation model of the glucose-insulin system, IEEETransactions on Biomedical Engineering 54 (2007), no. 10, 1740–1749.[DM14] Chiara Dalla Man, Francesco Micheletto, Dayu Lv, Marc Breton, Boris Kovatchev, and Claudio Cobelli, TheUVA/PADOVA type 1 diabetes simulator: New features, Journal of Diabetes Science and Technology 8 (2014), no. 1, 26–34.


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Compartments

Carbohydrates

Insulin

Glucagon

Meals

Stomach

healthydiabetic diabetic

including delay

event-based

Carbohydrate Compartment

EGP = kp1 − kp2GP − kp3Id

E = max(ke1(GP − ke2), 0)

Uii = Fcns

Uid = (Vm0 + VmxX )Gt/(Km0 + Gt)

GP = EGP + Ra− Uii − E − k1GP + k2Gt

Gt = −Uid + k1GP − k2Gt

G = GP/VG

Insulin Compartment

Isc1 = −(kd + ka1)Isc1 + IIR(M,G )

Isc2 = kd Isc1 − ka2Isc2

Il = −(m1 + m3)Il + m2Ip

Ip = −(m2 + m4)Ip + m1Il + ka1Isc1 + ka2Isc2

I = Ip/VI

I1 = −ki (I1 − I )

Id = −ki (Id − I1)

X = −p2UX + p2U(I − Ib)


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Parameter definition

Parametrization of the model in many steps.

Direct model: Define a first model with mean patient parameters:

Y = f (Y ,P,U).

Three types of patients: healthy, insulin-dependent diabetic, noninsulin-dependent diabetic.

Parameter sensitivity: Study the model’s parameter sensitivity in thevicinity of mean parameters.Parameter estimation: Identify the parameters with real patientglycaemia measures, when knowing the inputs and controls (meals,insulin infusions):

Y = f (Y ,P,U).

Control parameters: Adapt the control procedures to this specificpatient :

Y = f (Y ,P,U).


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Y = f (Y ,P,U).

Three types of patients: healthy, insulin-dependent diabetic, noninsulin-dependent diabetic.Parameter sensitivity: Study the model’s parameter sensitivity in thevicinity of mean parameters.

Parameter estimation: Identify the parameters with real patientglycaemia measures, when knowing the inputs and controls (meals,insulin infusions):

Y = f (Y ,P,U).


Y = f (Y ,P,U).


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Y = f (Y ,P,U).

Three types of patients: healthy, insulin-dependent diabetic, noninsulin-dependent diabetic.Parameter sensitivity: Study the model’s parameter sensitivity in thevicinity of mean parameters.Parameter estimation: Identify the parameters with real patientglycaemia measures, when knowing the inputs and controls (meals,insulin infusions):

Y = f (Y ,P,U).


Y = f (Y ,P,U).


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Y = f (Y ,P,U).

Three types of patients: healthy, insulin-dependent diabetic, noninsulin-dependent diabetic.Parameter sensitivity: Study the model’s parameter sensitivity in thevicinity of mean parameters.Parameter estimation: Identify the parameters with real patientglycaemia measures, when knowing the inputs and controls (meals,insulin infusions):

Y = f (Y ,P,U).


Y = f (Y ,P,U).B. Bidegaray-Fesquet (LJK) Event-based control of diabetes

Reunion e-BaCCuSS, 04/09/2018 6 /18


Parameter estimation

The model’s parameters can be classified in many categories:

a few known, and constant, subject parameters

weight. . .

a whole bunch of not exactly known and/or variable subjectparameters

rate constants for intestinal glucose absorptionliver responsiveness to glucagon. . .

There are also time-dependent inputs

physical activityemotionsgrowth hormones for children. . .


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Cobelli’s model Numerical simulation

Numerical simulation

Numerical scheme:

simple forward Euler scheme

time step 1 min

works fine! (as good as more sophisticated schemes)

1 min is also the data sampling time obtained from usual glucosesubcutaneous sensors.

good feature for parameter estimation.

Displayed results: 30 hours of

Meals

Glucose rate of appearance

Plasma insulin concentration

Plasma glucose concentration

ItargetB. Bidegaray-Fesquet (LJK) Event-based control of diabetes


Cobelli’s model Three types of in silico patients

In silico healthy patient

10 20 30t [h]

0

2

4

[g]

MealsM

10 20 30t [h]

0

5

10

[mg/

kg/m

in]

Glucose rate of appearanceR

10 20 30t [h]

0

200

400

600

[pm

ol/l]

Plasma insulin concentrationI

10 20 30t [h]

50

100

150

200[m

g/dl

]

Plasma glucose concentrationG


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In silico insulin-dependent diabetic patient

10 20 30t [h]

0

2

4

[g]

MealsM

10 20 30t [h]

0

5

10

[mg/

kg/m

in]


10 20 30t [h]

0

100

200

300

[pm

ol/l]


10 20 30t [h]

50

100

150

200

250

[mg/

dl]



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In silico non-insulin-dependent diabetic patient

10 20 30t [h]

0

2

4

[g]

MealsM

10 20 30t [h]

0

2

4

6

[mg/

kg/m

in]


10 20 30t [h]

0

100

200

300

[pm

ol/l]


10 20 30t [h]

100

200

300[m

g/dl

]



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Event-based and self-triggered controls Control goal

Target and secure intervals

Control goal:

Keep the plasma glucose rate within a target range Itarget (typically[80, 150] mg/dl).

There is a larger secure range Isecure = [40, 350] mg/dl, out of whichthe subject is in immediate danger (hypo- or hyperglycemia).

Mathematical issues to address:

Under which conditions can Isecure be ensured while using Itarget togenerate events?

Can self-triggered control delays be designed to ensure that most timeis spent in Itarget?

Trade-off between event-based control and an ”expensive” continuouscontrol.


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Event-based and self-triggered controls Self-triggered control

Self-triggered control

When a control is applied a decision is made on the next time thesubcutaneous glucose rate will be measured: self-triggered control.Typical values:

3 hours after an insulin infusion,1 hours after glucose ingestion.

The presence of a control and its type (insulin infusion or glucoseingestion) triggers the time of the next measure and possibly the nextcontrol.

The value of the control depends on the new state of the system(glycaemia).

This prescribed time delay can of course can be interrupted by an otherevent (event-triggered control).


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Event-based and self-triggered controls Event-triggered control

Event-triggered control

Events:

Theoretically: crossing the Itarget boundaries.Usually not detected if alarms are disabled.

Symptoms of hypo-/hyper-glycaemia.

Meals, sporting activities or other events that can strongly impact theglucose rate.

Actions:

a measure of glycaemia is done,

the proper control is applied (insulin infusion or carbohydrateingestion).

The control algorithm is very simple and only depends on the measuredglycaemia, and the quantity of ingested carbohydrates if a meal is involved.


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Event-based and self-triggered controls Event-triggered control

An example: insulin infusion rate

The control is far from continuous in terms of its variables, M and G :

IIR = IIRc + α(t)M + β(t)bG − Gmax

Gstepc+ − β(t)bGmin − G

Gstepc+,

Gstep and b·c+: because computations are made with a tabulatedvalues, and infusion can only be made in integer multiples of a givenamount of insulin.

α(t) and β(t) are only nonzero close to meal times.

It is delayed, since it is infused not directly in blood:

I1 = −γ1I1 + IIR(t,M,G ),

I2 = −γ2I2 + δ1I1,

I = −γI + ε1I1 + ε2I2.

(All the coefficients are patient dependent.)B. Bidegaray-Fesquet (LJK) Event-based control of diabetes


Work in progress

Work in progress

Still many things to do:

Parameter sensitivity study.

Parameter estimation.

Response to mathematical questions.

Challenging issues:

The control is both event- and self-triggered.

The value that is measured by sensors (sub-cutaneous glucose rate) isnot the one that we want to keep in a predefined range (bloodglucose rate), but reflects its value a few minutes before.

What in the case of capillary blood measures which cannot be donemore than a few times a day (not enough to ensure a proper control)?


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Work in progress

A toy model: minimal glucose-insulin model

Only three variables and equations.

∂tG (t) = −Sg (G (t)− Gb)− X (t)G (t), G (t0) = G0,

∂tX (t) = k3(Si (I (t)− Ib)− X (t)), X (t0) = 0,

∂t I (t) = γ(G (t)− Gt)+(t − t0)− kI (t), I (t0) = I0.

Very few and easy to interpret parameters.

Test sensitivity

Control analysis


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Conclusion

Conclusion

The treatment of insulin-dependent diabetes can be modeled as acontrolled ordinary differential system.

This approach, contrarily to purely automation approaches, needs tocarefully estimate the numerous parameters of the model.

The analysis as an event-based control will help better understandand calibrate the treatment of diabetic subjects.


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