Introduction:

I N D I A

Bangalore 2008 – Insulin/Glucose modelling

India, diabetes capital of the world(before China and US as No. of cases; data: WHO)

Zimmet, Nature 2001

India:2000:32 mill2020: 81 mill

Prevalence depends on: Age Residence(urban/rural) Obesity Physical activity Ethnicity

Type 2 DM: global epidemic

Rising Prevalence of Obesity in Urban India

BMI >27 kg/m2

11.2

22.3

13.2

29.7

0

5

10

15

20

25

30

Male Female

19942001

Gupta et al, IHJ 2002

Obese people develop Diabetes

RR risk of DM in females (ref. BMI < 22)• 22-23: 3.0

• 24-25: 5.0

• > 31: 40(Colditz & al, Ann Int Med, 1995, 122; 481-6)

Rising prevalence of diabetes in Southern India

0

2

4

6

8

10

12

14

16

18

IGT DM

1989

1995

2000

Ramchandran et al: Diab Care 92,Diabetol 97, Diabetol 2001

so what?

Diabetes and CAD risk7 year incidence of CV events (%)

0

5

10

15

20

25

30

35

40

45

MyocardialInfarction

Stroke CardiovascularDeaths

No DM, No prior MI

No DM, Yes prior MI

Yes DM, No prior MI

Yes DM, Yes prior MI

Haffner SM et al. N Engl J Med 1998;339:229-234.

Pathophysiology of the glucose/insulin system

Andrea De Gaetano

CNR IASI BioMatLab – Rome Italy

Bangalore 2008

CNR

Consiglio Nazionale delle Ricerche (Italian National Research Council): the research organization of the Italian Government, 6000+ researchers distributed over 100+ Institutes in the Country.

Research ranging from humanities to genomics, linguistics, aerospace engineering, pure mathematics, …

CNR IASI

IASI, Istituto di Analisi dei Sistemi ed Informatica “A. Ruberti” (Institute for Systems Analysis and Informatics) in

Rome: 30+ researchers, 20 administrative/technical personnel. seven research areas

Systems and Control Theory Mathematical Programming in Operations Research Mathematical Modeling in Biology and Medicine Algorithms, data structures and networks Language and Programming theory Information Systems and Knowledge Bases Pathophysiology of Metabolism and Immunology

CNR IASI BioMatLab BioMathematics Lab, within the Catholic University

School of Medicine (2000 bed hospital), Rome 5 full-time lab researchers (1 biomathematician, 1

statistician, 3 engineers), clerical personnel, part-time associates.

ODE, DDE, SDE models: analytical study of behavior of solutions, numerical integration, statistical parameter estimation

www.biomatematica.it

Hypoglicemia Brain works on sugar Little sugar: hunger, irritability, confusion,

hyperactivity, cold sweat, tremor (adrenergic response) No sugar: brain death.

COUNTERREGULATION: Adrenalin (fight-or-flight), glucagon, cortisol, Growth Hormone all INCREASE blood glucose levels.

Food......

…but, Hyperglycemia Acute above renal threshold: sweet, abundant urine

(Diabetes Mellitus), dehydration. Chronic: microvascular damage in retina (blindness),

kidneys (renal insufficiency), extremities; peripheral neuropathy.

Endogenous Glucose production

(liver, kidney)

Diabetes type 1 and 2

glycemiainsulinemia

pancreatic-cellInsulin

secretion

Insulin independent Glucose utilization

(brain)

Exhogenous glucose administration

lack of secretion

Insulin dependent Glucose utilization

(muscle)

Modified from A.Mari 2001

Insulin resistance

Insulin Proinsulin (86 AA) = C-Peptide (35 AA) +

Insulin(51=A+B chain)

Secreted from pancreatic beta-cells (Langerhans islets) in response to: GLUCOSE, AA, neurotransmitters (AC, like after a meal), hormones (glucagon); FFA?

Increases Glycogen synthesis, inhibits Gluconeogenesis, inhibits lipases and increases FFA deposition in Adipose tissue

Portal circulation

Insulin resistance: operational definition

Insulin resistance may be defined as

inappropriately high glycemia for the insulinemia,

or again as

inappropriately high insulinemia for the glycemia

Insulin sensitivity

Insu

lin

secr

etio

n

Increasing Glycemia

Disposition Index

An overview of energy metabolism

following diagrams ...

GlycolysisGLUCOSE

GLUCOSE-6-P

ATP

ADP

FRUCTOSE-6-P

GLUCOSE-1-P

ATP

ADP

FRUCTOSE-1,6-dP

GLYCERALDEHYDE-3-P + DIHYDROXYACETONE-P

GLYCOGEN + P

NAD+

NADH

1,3-dP-GLYCERATE

3-P-GLYCERATE

ADP

ATP

2-P-GLYCERATE

P-ENOLPIRUVATE

ADP

ATP

PYRUVATE

Krebs’ Cycle

OXALOACETATE

ISOCITRATE

SUCCINYL-CoA

SUCCINATE

FUMARATE

MALATE

CITRATE

NAD+

NADH

FADH2

FAD

H2O

GTP

GDP + P

alpha-KETOGLUTARATE

CoA

CO2

NAD+

NADH

NADP+

NADPH, CO2

ACETYL CoA

ATP, CO2

ADP

NAD+ NADH, CO2

PIRUVATE

ALANINE

NH2

LACTATE

H2

lipid -oxidation

glycolysisprotein breakdown

DA oxidation

Randle’s Cycle

1963, Sir Philip Randle: cardiac and skeletal muscle shifts back and forth between CHO and fat oxidation depending on the availability of FFA.

In vivo infusion of lipid increases fat oxidation and decreases glucose oxidation

Hyperinsulinemia

Insulin secretion

Fat storageInhibitionof Lipases

Glucose Uptake

TG

FFA

Hyperglycemia(Randle)

How McDonald & KFC make you diabetic!

Insulinresistance

Bariatric Surgery

BPD and insulin resistance

Insulin resistance after BPD drops dramatically, well before body weight does: Using EHC, whole body glucose uptake increased from 18.18.6 to 35.5 9.9 moles/min/kgbw after an average weight loss of only 11 kg reached 3 months after BPD. A marked reduction of both plasma FFA and TG was observed together with the therapeutic lipid malabsorption (Mingrone, Castagneto et al. Diabetologia 1997).

Also in normal weight subjects with a genetic defect of LPL activity, insulin resistance and frank diabetes mellitus were reversed by lowering plasma TG through lipid malabsorption induced by BPD (Mingrone, Castagneto et al. Diabetes 1999).

Models of the glucose-insulin system

Why modelling the G/I system?

To identify the components of insulin resistance and measure its level:

Diabetologist approach (lots of data, make a diagnosis)

Standard modeling approach (less data, try to figure out the whole system )

Models Tracer “hot” vs. “cold” models Why cold? Our perspective is the clinical application.

TRACERS: Steele 1956 traced glucose constant infusion with approx computation of SteadyState cold inflow.

eqs/hr

Bolie 1961 First attempt to understand actual time-concentration

points in plasma.

Introduces plasma insulin and LGE Problems?

01 2 3

dGG I(t) , G 0 Gp p p

dt

Glycemia

Insu

line

mia

G1

I1

dG

dt - p1 G - p2 I + p3

I2

G2

qualitative analysis reveals ... the actual model functional form, which allows negative

solutions to appear, must have something in it which goes against the physiology as we think we know it

Bolie: no matter how little glucose there is in blood, by increasing insulin we would be able to make the tissues extract as much more as we wanted, linearly with insulin levels.

Mechanism seems wrong. Better to change model.

IVGTT three days of standard composition diet (55%

carbohydrate, 30% fat, 15% protein) ad libitum with at least 250g carbohydrates per day

Overnight fast, at 8:00 AM 0.33 g/kgBW IV Glucose Contralateral IV samples at -30, -15, 0, 2, 4, 6, 8, 10, 12,

15, 20, 25, 30, 35, 40, 50, 60, 80, 100, 120, 140, 160, 180 minutes (23 pts.)

On each blood sample determine Glucose, Insulin (C-peptide).

GlycogenolysisGluconeogenesis

cell

Glucoseincreases the ATP/ADP ratio

Ca2+

The K+ channel opens causing depolarization

Depolarization cause Ca2+

influx

-

0 10 20 30 40 50 minutes

Plasma Insulin

0 10 20 30 40 50 minutes

Plasma Glucose

IVGTT

Bergman, Cobelli 1979/1981

1 1 b 0

d G tb X t G t b G , G 0 b

d t

2 3 b

d X tb X t b I t I , X 0 0

d t

4 5 6 b 7 b

d I tb G t b t b I t I , I 0 b I

d t

Sample run (IVGTT+MM)

min

0

50

100

150

200

250

300

350

400

450

0 20 40 60 80 100 120 140 160 180

SI : derivation

I

d GE , S E

G dt I

1 1 b 1

d Gb X t G t b G b X t

G dt G

EX t

I I

SI : derivation

Solving Eq.2 MM for X

22t b t sb t

3 b0X t X 0 e e b I s I ds

2 2 2 2t b t s b t b t b t3 3

3 302 2 2

E 1 b be b ds b e e 1 e

I b b b

SI

For infinite time, SI = b3/b2

in one third to one half of studies on obese subjects SI

cannot be estimated, due to insufficient variation of glucose decrement with insulin.

An IVGTT obvious for insulin resistance (high constant insulin levels) yields no estimable SI.

Applications of MM Physicians want a single test returning a single measure

of insulin resistance, like M/I or SI

MM applied to diabetes, aging, hyperthyroidism, hyperparathyroidism, myotonic dystrophy, pregnancy and gynecological conditions, obesity, hypertension, cirrhosis, ethnical subpopulations, in siblings of diabetic patients, during pharmacological tests

Minimal model Whole body, cold

Can compute SI …

…de-facto standard

Minimal problems Models only IVGTT (nonautonomous)

Fitting: piecewise?

SI strictly valid at infinite time, MM “valid” for 3 hrs.

SI not estimable in many interesting cases.

Structural problems

5t

limsupG t b

tXsuplimt

Suppose Gb > b5,

Then

In other words, for any value b5 < Gb the system does not admit an equilibrium.

Estimation problems

Two-step procedure advocated by Authors Each step fits one arm of feedback cycle Interpolated observed concentrations used as forcing

function 1 1 b 0

d G tb X t G t b G , G 0 b

d t

2 3 b

d X tb X t b I t I , X 0 0

d t

4 5 6 b 7 b

d I tb G t b t b I t I , I 0 b I

d t

GI

We would like: single model, single fit of both feedback arms positiveness, boundedness of solutions stability WRT parameters & initial conditions good fit, identifiability direct physiological meaning

The SDM

ghxgI

g

TdG tK I t G t

dt V

gb b

g

DG t G t ,0 , G 0 G G , where G

V

g

*

ig maxxi

ig

*

G t

GTdI tK I t

dt V G t1

G

b GI 0 I I G

,

The SDM insulin sensitivity index

ghxgI xgI

g

TdG K G(t)I(t) K

I G dt I G V

SDM characteristics

Single locally attractive equilibrium at baseline Positive, limited solutions Global stability guaranteed under conditions on

parameters* Physiologically limited pancreatic secretion ability Single pass GLS estimation

*Giang, Lenbury, Palumbo, Panunzi, De Gaetano, 2006-2007

0 100 200

5

10

15

0 100 2000

100

200

300

400

500

SDM: Subject with BMI >= 40

0 100 2000

200

400

600

Plasma Insulin (pM)

SDM: Subject with BMI >=24

0 100 2004

6

8

10

12

14

16

Plasma Glucose (mM)

0 100 2004

6

8

10

12

0 100 2000

100

200

300

SDM: Subject with BMI >24 and <=30

0 100 200

5

10

15

20

0 100 2000

100

200

300

400

SDM: Subject with BMI > 30 and <= 40

SDM vs. MM Over 74 subjects with widely varying BMI (20 – 60)

KxgI from the SDM identifiable (CV < 52%) in 73 out of 74 subjects (one 68%) All estimates within physiological limits (1.25 × 10-5 to 4.36 ×

10-4 )

SI from the MM not identifiable in 36 subjects out of 74, with coefficients of

variation ranging from 52.76 % to 2.3610+9 % in 11 subjects estimates doubtfully large (from 3.99 to 890) in 8 subjects estimates very small (≤ 1.5 × 10-6, “zero-SI”)

EHC, the Euglycemic Hyperinsulinemic Clamp

Administer a large I.V. infusion of insulin Prevent hypoglycemia by external glucose I.V. infusion,

with rate adjusted q5’ on the basis of glycemia determination and algorithm (Defronzo, 1979).

EHC: interpretation Large insulin infusion: suppression of Liver Glucose

Excretion, then… …at SteadyState exogenous administration (measured)

and Tissue Uptake must be equal. Hence: Smaller than normal M (average G administration rate)

implies insulin resistance.

EHC: problems Clearly, M = M(mass, age, sex,…) and normalizations

necessary. Still, M = M(I), hopefully monotonic increasing (in fact,

nonlinear saturating) First correction: M/I. But this implies linearity, which is

false:

I

M

M/I

The industrious diabetologist Second correction: two-step clamp, and compute ΔM/ΔI. This also assumes linearity (and a 3-5hr session):

I

M

M/I

EHC: more problems Doubtful physiological meaning of index derived from

several hours maximum insulinization. Obese: typically depressed M at 2 hrs, normal at 5 hrs…

EHC: huge success! MOST research diabetologists use EHC over modelling

methods: “No need to perform complicated CALCULATIONS,

this is something we understand” Doubtful attitude towards validity of models “You can

show anything and its opposite…” (e.g. compartmental assumptions)

EHC: a gold data mine?

Decades of experimentation have produced a huge amount of EHC data.

A deterministic Clamp model

gx g gh

xg xgI bg 0

T t T tdG t G(t)T K s I t s ds G t , G(0) = G

dt V 0.1 G(t)

iG ix

xi bi

T G t T (t)dI tK I t , I(t) = I t 0

dt V

gh gh max gh ghb ghmax b b0

T (t) T exp G(t) s I t s ds , T (0) = T = T exp(-λG I )

2 -αsgx g ix ixbω(s) = α se , T (s) = 0 s [-τ ,0] and T (0) = T .

2005 Picchini et al. TBMM

a good subject

What’s wrong?

NO model we could think of fits the peaks/troughs ACCIDENTAL factors generate/shift oscillations, hence

… … a deterministic model will do its best to AVERAGE

disturbances OR … … be overparametrized and fit perfectly only one

individual realization.

Need something else!

Stochastic model

gx g gh

xg xgIg

iG ixxi

i

The model is represented byaStochastic (Ito) Differential Equation with delay

T t T t G td G t T K G t I t dt G t I t dW(t)

V 0.1 G t

T G t T td I t K I t dt

V

xgI xgI xgI t

t t

random oscillationsin K : K K W

where W dt d W and where isa constant.

Deterministic: 1

CV: (0.05, 0.15), subj 1

CV: (0.03, 0.15), subj 1

Deterministic: 9

CV: (0.05, 0.15), subj 9

CV: (0.03, 0.15), subj 9

Deterministic: 10

CV: (0.05, 0.15), subj 10

CV: (0.03, 0.15), subj 10

Please do not forget …

Denmark 2008 (more about this from Susanne…) and …

Sicily (Italy) 13-26 Sept. 2009

Parameter Estimation in Dynamical Models Glucose/Insulin Modelling

www.biomatematica.it

Thank you