introduction: i n d i a bangalore 2008 – insulin/glucose modelling
TRANSCRIPT
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