Mathematical methods and models2
ISIB-CNR
Institute of Biomedical Engineering
Chairperson: Ferdinando GRANDORI
Mathematical Methods and Models for
Clinical Research on Metabolism, Diabetes
and Its Complications
Metodi e Modelli Matematici per la Ricerca Clinica
sul Metabolismo, il Diabete e sue Complicanze
Giovanni Bortolan, Andrea Mari, Giovanni Pacini, Karl Thomaseth, Andrea Tura
Collaboratori
Alessandra Brazzale, Valentina Nofrate, Alessandra Pavan, Stefano Sbrignadello
Diabetes: serious, widespread disease
1995
2000
2010
Type 1
3.5 million
4.4 million
5.5 million
Type 2
114.8 million
146.8 million
215.3 million
TOTAL
118.4 million
151.2 million
220.7 million
D ia be t e s
defect in t he regulat ion of blood glucose
glucose production
(liver)
8
exogenous
glucose
blood glucose
pancreatic
β cell
impaired
secretion
8
insulin
secretion
insulin
resistance
8
glucose utilization
(muscle, fat)
insulin
• Insulin Sensitivity
quantification of insulin action to promote glucose disappearance from blood (lowering
glycemia)
• Beta-cell Function
quantification of the ability of the beta-cell to respond to secretagogues (mainly glucose)
stimuli, by enhancing insulin production and release
SECREZIONE
CINETICA
Systemic circulation
potentiation
dose-response
function
C-Pep
β-cell model
secretion
Glucose
conc’n
glucose
measurements
clearance
secretion
LIVER
early secretion
Insulin
(function of
glucose derivative)
hepatic extraction
AZIONE
clearance
Design, development and use of clinical tests
• simple, accurate and of widespread clinical use
Oral glucose tolerance test (OGTT) with model analysis
-OGIS model for insulin sensitivity : ~100+ citations from other groups
-Insulin, C-peptide and proinsulin kinetic models : widely used for
physiological studies
-Insulin secretion model : ~30 articles on major journals on pharma agents
• high information content for specific analyses
on particular aspects
- Pharmacokinetic studies
- Urea kinetics in hemodialysis (artificial kidney)
- Evaluation of cardiologic parameters in diabetic neuropathies
Int’l
Italian
RISC
COOPERATIONS (Europe)
INTERNATIONAL COOPERATIONS (The World)
Japan
+
New Zealand
Australia
Consulting and Service Agreements
years 2004-2009 mostly with drug companies
Amylin-Eli Lilly (USA)
Bellco (Italia)
Glaxo-Smith & Kline (USA)
Fresenius (Germania)
Novartis, Basel (CH)
Bayer (Italia)
Takeda (UK)
Merck (USA)
Novartis, E.Hanover (USA)
Novo-Nordisk (DK)
Mannkind (USA)
La Roche (CH)
Int’l Projects
European Project RISC (5th FP)
IP Eur. Project NeuroFAST (7th FP) The Integrated Neurobiology of Food Intake,
Addiction and Stress (2009-14)
EFSD
Projects
● B-cell Function (2001-04)
● Insulin Secr. and Insulin Sens. Following Bariatric Surgery (2007-09)
● Prophylactic use of DPP-4 inhibition in glucocorticoid-induced beta-cell
dysfunction (2008-11)
Innovative Medicines Initiative (IMI) Project:
Surrogate markers for Micro- and Macro-vascular hard endpoints for Innovative
diabetes Tools” (SUMMIT) (2009-14)
Austrian Science Fund (FWF) Project:
Analysis of the metabolic state and vascular function in post GDM (2003-07)
Italian Projects
• Biologia e fisiologia clinica del tessuto adiposo. (PRIN – 2007-09)
• Sviluppo di un metodo accessibile via web per l'analisi con modelli matematici della
cinetica del glucosio. (Ricerca a tema libero CNR – 2007)
• Tessuto adiposo e farmaci: biologia e clinica. (PRIN – 2005-07)
• Biologia cellulare e fenotipo clinico nella sindrome metabolica. (PRIN – 2001-03;
rinnovato per 2003-05)
• Metodi e modelli matematici nello studio dei fenomeni biologici. (Progetto strategico
CNR – 1998-99)
• Development and validation of a mathematical model for the study of glucose
metabolism. (Progetto bilaterale CNR Italia-Australia – 1998-2000)
Other Activities
• Members of Editorial Board of journals in the field of Diabetes, Modeling and
Simulation.
• Invited Reviewers for prestigious int’l journals in the field of Diabetes, Modeling and
Simulation.
• Invited speakers and chairpersons at Congresses, Workshops and Schools.
PAPERS ON PEER-REVIEWED INTERNATIONAL JOURNALS
(source ISI Thompson)
35
30
25
20
15
10
5
0
2004
2005
2006
2007
2008
2009 *
( ) until July 2009; including 11 articles in press
*
Expertise
• Design, implementation and use of mathematical models for studies on
metabolism, pharmacokinetics and pharmacodynamics
• Analysis of experimental data for the estimation of physiological and
clinical parameters and their dependence on specific covariates (BW,
age, gender, BP,…)
• Design of experimental tests (based mostly on mathematical modelling)
for estimating insulin sensitivity, beta cell function, renal function,
assessament of ECG parameters and of those of the autonomic nervous
system
• Application to PK/PD studies for design and monitoring of clinical trials
with analysis of the results
case studies:
insulin sensitivity
beta cell function
Glucose Tolerance
Insulin
Resistance
Insulin
Secretion
normoglycemia
Why focussing on
insulin resistance ?
Insulin sensitivity measures insulin
resistance, which is strictly linked to several
vital diseases
Risk fa ct or s
Hyperglycaemia
Hyperinsulinaemia
Hypertension
Dyslipidaemia
Insulin
resistance
Decreased fibrinolytic
activity (↑PAI-1)
Endothelial dysfunction
Inflammatory markers
of atherosclerosis
Microalbuminuria
CVD risk
Insulin Resistance
Insulin resistance is measured by
Insulin sensitivity
that quantifies the insulin action of inhibiting
endogenous glucose production from the liver
and promoting peripheral glucose utilization by
muscle and fat.
The glucose clamp
insulin concentration
250
100
200
80
µU/ml
mU/min
insulin infusion
150
100
50
0
60
40
20
0
20
40
60
80
100
0
120
0
glucose infusion
10
100
8
80
6
4
2
0
0
20
40
60
80
100
120
glucose concentration
mg/dl
mg/min/kg
mean insulin
concentration: I
60
40
20
20
40
mean
infusion rate: M
60
80
100
120
0
0
20
40
60
80
100
120
Insulin sensitivity (clamp model) =
GOLD STANDARD
M
I
20
mM
15
10
5
nM
insulinemia
glycemia
IVGTT and the Minimal Model
0
2.5
2.0
1.5
1.0
0.5
0
-30
0
30
60
90
120
150 180
time
min
mathematical model
insulin
glucose
parameter estimation
2.0
20
10
1.0
0
0
60
120 180
Insulin sensitivity index (SI)
0
0
60
120 180
Derivation of the Insulin Sensitivity Index (SI)
decreasing complexity
clamp
minmod
TEST
MODEL
direct
measurement
estimated
parameter
validation
National
Research
Council
Metabolic Unit
Padova, Italy
Simple(r) method for the assessment of
insulin sensitivity from the IVGTT
Andrea Tura, Giovanni Pacini
with cooperation of
Stefano Sbrignadello
Is there a way of simplifying the estimation
of insulin sensitivity from an IVGTT ?
Glucose Conc.
Until the first
hour, glucose
keeps
decreasing
from the initial
peak
60
90
60
90
The slope of the line fitting the glucose decrease
yields a measurement of glucose disappearance
INSULIN SENSITIVITY FROM IVGTT
simplified formula
t2
slope (log G(t))
t1
CSI =
t4
1
t4 – t3
(I(t) – Ib) dt
t3
Units: min-1/(µU/ml)
A. Tura, S. Sbrignadello, G. Pacini. Diabetologia 2009
Relationship between computed (CSI) and minimal
model estimated (SIMM) insulin sensitivity in control
subjects of various age and weight
104min-1(µU/ml)-1
4.5
CSI
0
0
SI MM
N=144, r=0.934, p 0.2 vs. slope=1; r = 0.907, p < 0.0001
A. Tura, S. Sbrignadello, G. Pacini. Diabetologia 2009
Comparison between CSI and Clamp M in normo
glucose tolerant (circles), impaired tolerant (squares)
and diabetic (triangles) subjects
regression lines are virtually equivalent to the identity line
A. Tura, S. Sbrignadello, G. Pacini. Diabetologia 2009
further simplification
clamp
minmod
CSI
TEST
MODEL
FORMULA
direct
measurement
estimated
parameter
calculated
parameter
validation
CSI
•
•
•
•
•
•
Catanzaro
Copenhagen
Malmö
Lund
Melbourne (?)
Napoli-Finlandia
cooperation for the realization
use in minipigs
cooperation for the realization
use in mice
use in rats
use in humans
Methods for Measuring Insulin Sensitivity
calculation -- model
difficult
easy
IVGTT
(MINMOD)
lower
info
higher
info
OGTT
(OGIS, ISIcomp)
IVGTT
(KG)
basal
(HOMA)
experiment
eu- and hyper-glycemic
glucose clamp
difficult
Insulin sensitivity
IVGTT
(with simple formula and short protocol)
provides an index similar to the minimal model
and to the euglycemic glucose clamp
•
•
•
•
•
•
does not require a com put er program and expert ise t o solve
m at hem at ical m odels, j ust a spread sheet
requires a few sam ples
does not require addit ional inj ect ions of insulin
can be used also in larger populat ion size
t he exact BEST t im ing m ay be funct ion of t he t ype of
populat ion under st udy
t he possibilit y of including “ glucose effect iveness” int o t he
form ula is st ill t o be explored
Mathematical models for β-cell
function assessment in vivo
Andrea Mari
Andrea Tura, Valentina Nofrate
http://www.isib.cnr.it/~mari/view.php?page=5
Padova-Pisa:
more than 20 years friendship
Ele Ferrannini
Andrea Mari
1986-2000: tracer kinetics, insulin sensitivity
2001-2009: β-cell function
In vivo β-cell modeling project:
aims
● To understand how the β cell responds to
glucose stimulation in normal living
conditions quantitatively, using modeling
● To develop a widely applicable modelbased test for β-cell function based on an
oral glucose load or meal
● To use the model-based test in a large
variety of experimental situations
β-cell model for oral glucose tests:
reconsidering potentiation
potentiation
dose-response
function
secretion
glucose
concentration
glucose
insulin
secretion
P(t) f(G) + kd dG
dt +
early secretion
(function of
glucose derivative)
Mari … Ferrannini 2002
Roadmap
model ready
2001
IGIS meeting I
2002
2003
1st review
1st large population study
1st prospective study
2004
nateglinide 1st vildagliptin
1st bariatric surgery
2005
2006
1st genetic study (RISC)
effects of GLP-1
exenatide
liraglutide
2007
IGIS meeting II EASD meeting
2008
β-cell function & insulin sensitivity (RISC)
thiazolidinediones
incretin effects
2009
type 1 diabetes
Dissemination
~104 tests analyzed
Perspectives
● Still a long way to go with the current
approach
♦ The RISC study – Genetics
♦ Bariatric surgery
♦ Pharmas
● Modeling of in vitro insulin secretion to
integrate in vitro and in vivo information
Institute of Biomedical Engineering
Chairperson: Ferdinando GRANDORI
Mathematical Methods and Models for
Clinical Research on Metabolism, Diabetes
and Its Complications
Metodi e Modelli Matematici per la Ricerca Clinica
sul Metabolismo, il Diabete e sue Complicanze
Giovanni Bortolan, Andrea Mari, Giovanni Pacini, Karl Thomaseth, Andrea Tura
Collaboratori
Alessandra Brazzale, Valentina Nofrate, Alessandra Pavan, Stefano Sbrignadello
Diabetes: serious, widespread disease
1995
2000
2010
Type 1
3.5 million
4.4 million
5.5 million
Type 2
114.8 million
146.8 million
215.3 million
TOTAL
118.4 million
151.2 million
220.7 million
D ia be t e s
defect in t he regulat ion of blood glucose
glucose production
(liver)
8
exogenous
glucose
blood glucose
pancreatic
β cell
impaired
secretion
8
insulin
secretion
insulin
resistance
8
glucose utilization
(muscle, fat)
insulin
• Insulin Sensitivity
quantification of insulin action to promote glucose disappearance from blood (lowering
glycemia)
• Beta-cell Function
quantification of the ability of the beta-cell to respond to secretagogues (mainly glucose)
stimuli, by enhancing insulin production and release
SECREZIONE
CINETICA
Systemic circulation
potentiation
dose-response
function
C-Pep
β-cell model
secretion
Glucose
conc’n
glucose
measurements
clearance
secretion
LIVER
early secretion
Insulin
(function of
glucose derivative)
hepatic extraction
AZIONE
clearance
Design, development and use of clinical tests
• simple, accurate and of widespread clinical use
Oral glucose tolerance test (OGTT) with model analysis
-OGIS model for insulin sensitivity : ~100+ citations from other groups
-Insulin, C-peptide and proinsulin kinetic models : widely used for
physiological studies
-Insulin secretion model : ~30 articles on major journals on pharma agents
• high information content for specific analyses
on particular aspects
- Pharmacokinetic studies
- Urea kinetics in hemodialysis (artificial kidney)
- Evaluation of cardiologic parameters in diabetic neuropathies
Int’l
Italian
RISC
COOPERATIONS (Europe)
INTERNATIONAL COOPERATIONS (The World)
Japan
+
New Zealand
Australia
Consulting and Service Agreements
years 2004-2009 mostly with drug companies
Amylin-Eli Lilly (USA)
Bellco (Italia)
Glaxo-Smith & Kline (USA)
Fresenius (Germania)
Novartis, Basel (CH)
Bayer (Italia)
Takeda (UK)
Merck (USA)
Novartis, E.Hanover (USA)
Novo-Nordisk (DK)
Mannkind (USA)
La Roche (CH)
Int’l Projects
European Project RISC (5th FP)
IP Eur. Project NeuroFAST (7th FP) The Integrated Neurobiology of Food Intake,
Addiction and Stress (2009-14)
EFSD
Projects
● B-cell Function (2001-04)
● Insulin Secr. and Insulin Sens. Following Bariatric Surgery (2007-09)
● Prophylactic use of DPP-4 inhibition in glucocorticoid-induced beta-cell
dysfunction (2008-11)
Innovative Medicines Initiative (IMI) Project:
Surrogate markers for Micro- and Macro-vascular hard endpoints for Innovative
diabetes Tools” (SUMMIT) (2009-14)
Austrian Science Fund (FWF) Project:
Analysis of the metabolic state and vascular function in post GDM (2003-07)
Italian Projects
• Biologia e fisiologia clinica del tessuto adiposo. (PRIN – 2007-09)
• Sviluppo di un metodo accessibile via web per l'analisi con modelli matematici della
cinetica del glucosio. (Ricerca a tema libero CNR – 2007)
• Tessuto adiposo e farmaci: biologia e clinica. (PRIN – 2005-07)
• Biologia cellulare e fenotipo clinico nella sindrome metabolica. (PRIN – 2001-03;
rinnovato per 2003-05)
• Metodi e modelli matematici nello studio dei fenomeni biologici. (Progetto strategico
CNR – 1998-99)
• Development and validation of a mathematical model for the study of glucose
metabolism. (Progetto bilaterale CNR Italia-Australia – 1998-2000)
Other Activities
• Members of Editorial Board of journals in the field of Diabetes, Modeling and
Simulation.
• Invited Reviewers for prestigious int’l journals in the field of Diabetes, Modeling and
Simulation.
• Invited speakers and chairpersons at Congresses, Workshops and Schools.
PAPERS ON PEER-REVIEWED INTERNATIONAL JOURNALS
(source ISI Thompson)
35
30
25
20
15
10
5
0
2004
2005
2006
2007
2008
2009 *
( ) until July 2009; including 11 articles in press
*
Expertise
• Design, implementation and use of mathematical models for studies on
metabolism, pharmacokinetics and pharmacodynamics
• Analysis of experimental data for the estimation of physiological and
clinical parameters and their dependence on specific covariates (BW,
age, gender, BP,…)
• Design of experimental tests (based mostly on mathematical modelling)
for estimating insulin sensitivity, beta cell function, renal function,
assessament of ECG parameters and of those of the autonomic nervous
system
• Application to PK/PD studies for design and monitoring of clinical trials
with analysis of the results
case studies:
insulin sensitivity
beta cell function
Glucose Tolerance
Insulin
Resistance
Insulin
Secretion
normoglycemia
Why focussing on
insulin resistance ?
Insulin sensitivity measures insulin
resistance, which is strictly linked to several
vital diseases
Risk fa ct or s
Hyperglycaemia
Hyperinsulinaemia
Hypertension
Dyslipidaemia
Insulin
resistance
Decreased fibrinolytic
activity (↑PAI-1)
Endothelial dysfunction
Inflammatory markers
of atherosclerosis
Microalbuminuria
CVD risk
Insulin Resistance
Insulin resistance is measured by
Insulin sensitivity
that quantifies the insulin action of inhibiting
endogenous glucose production from the liver
and promoting peripheral glucose utilization by
muscle and fat.
The glucose clamp
insulin concentration
250
100
200
80
µU/ml
mU/min
insulin infusion
150
100
50
0
60
40
20
0
20
40
60
80
100
0
120
0
glucose infusion
10
100
8
80
6
4
2
0
0
20
40
60
80
100
120
glucose concentration
mg/dl
mg/min/kg
mean insulin
concentration: I
60
40
20
20
40
mean
infusion rate: M
60
80
100
120
0
0
20
40
60
80
100
120
Insulin sensitivity (clamp model) =
GOLD STANDARD
M
I
20
mM
15
10
5
nM
insulinemia
glycemia
IVGTT and the Minimal Model
0
2.5
2.0
1.5
1.0
0.5
0
-30
0
30
60
90
120
150 180
time
min
mathematical model
insulin
glucose
parameter estimation
2.0
20
10
1.0
0
0
60
120 180
Insulin sensitivity index (SI)
0
0
60
120 180
Derivation of the Insulin Sensitivity Index (SI)
decreasing complexity
clamp
minmod
TEST
MODEL
direct
measurement
estimated
parameter
validation
National
Research
Council
Metabolic Unit
Padova, Italy
Simple(r) method for the assessment of
insulin sensitivity from the IVGTT
Andrea Tura, Giovanni Pacini
with cooperation of
Stefano Sbrignadello
Is there a way of simplifying the estimation
of insulin sensitivity from an IVGTT ?
Glucose Conc.
Until the first
hour, glucose
keeps
decreasing
from the initial
peak
60
90
60
90
The slope of the line fitting the glucose decrease
yields a measurement of glucose disappearance
INSULIN SENSITIVITY FROM IVGTT
simplified formula
t2
slope (log G(t))
t1
CSI =
t4
1
t4 – t3
(I(t) – Ib) dt
t3
Units: min-1/(µU/ml)
A. Tura, S. Sbrignadello, G. Pacini. Diabetologia 2009
Relationship between computed (CSI) and minimal
model estimated (SIMM) insulin sensitivity in control
subjects of various age and weight
104min-1(µU/ml)-1
4.5
CSI
0
0
SI MM
N=144, r=0.934, p 0.2 vs. slope=1; r = 0.907, p < 0.0001
A. Tura, S. Sbrignadello, G. Pacini. Diabetologia 2009
Comparison between CSI and Clamp M in normo
glucose tolerant (circles), impaired tolerant (squares)
and diabetic (triangles) subjects
regression lines are virtually equivalent to the identity line
A. Tura, S. Sbrignadello, G. Pacini. Diabetologia 2009
further simplification
clamp
minmod
CSI
TEST
MODEL
FORMULA
direct
measurement
estimated
parameter
calculated
parameter
validation
CSI
•
•
•
•
•
•
Catanzaro
Copenhagen
Malmö
Lund
Melbourne (?)
Napoli-Finlandia
cooperation for the realization
use in minipigs
cooperation for the realization
use in mice
use in rats
use in humans
Methods for Measuring Insulin Sensitivity
calculation -- model
difficult
easy
IVGTT
(MINMOD)
lower
info
higher
info
OGTT
(OGIS, ISIcomp)
IVGTT
(KG)
basal
(HOMA)
experiment
eu- and hyper-glycemic
glucose clamp
difficult
Insulin sensitivity
IVGTT
(with simple formula and short protocol)
provides an index similar to the minimal model
and to the euglycemic glucose clamp
•
•
•
•
•
•
does not require a com put er program and expert ise t o solve
m at hem at ical m odels, j ust a spread sheet
requires a few sam ples
does not require addit ional inj ect ions of insulin
can be used also in larger populat ion size
t he exact BEST t im ing m ay be funct ion of t he t ype of
populat ion under st udy
t he possibilit y of including “ glucose effect iveness” int o t he
form ula is st ill t o be explored
Mathematical models for β-cell
function assessment in vivo
Andrea Mari
Andrea Tura, Valentina Nofrate
http://www.isib.cnr.it/~mari/view.php?page=5
Padova-Pisa:
more than 20 years friendship
Ele Ferrannini
Andrea Mari
1986-2000: tracer kinetics, insulin sensitivity
2001-2009: β-cell function
In vivo β-cell modeling project:
aims
● To understand how the β cell responds to
glucose stimulation in normal living
conditions quantitatively, using modeling
● To develop a widely applicable modelbased test for β-cell function based on an
oral glucose load or meal
● To use the model-based test in a large
variety of experimental situations
β-cell model for oral glucose tests:
reconsidering potentiation
potentiation
dose-response
function
secretion
glucose
concentration
glucose
insulin
secretion
P(t) f(G) + kd dG
dt +
early secretion
(function of
glucose derivative)
Mari … Ferrannini 2002
Roadmap
model ready
2001
IGIS meeting I
2002
2003
1st review
1st large population study
1st prospective study
2004
nateglinide 1st vildagliptin
1st bariatric surgery
2005
2006
1st genetic study (RISC)
effects of GLP-1
exenatide
liraglutide
2007
IGIS meeting II EASD meeting
2008
β-cell function & insulin sensitivity (RISC)
thiazolidinediones
incretin effects
2009
type 1 diabetes
Dissemination
~104 tests analyzed
Perspectives
● Still a long way to go with the current
approach
♦ The RISC study – Genetics
♦ Bariatric surgery
♦ Pharmas
● Modeling of in vitro insulin secretion to
integrate in vitro and in vivo information