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Please cite thisarticle in press as: R. Burattini,M. Morettini, Identification of an integrated mathematicalmodel of standard oralglucosetolerancetest for characterization of insulin potentiation in health, Comput. Methods Programs Biomed. (2011), doi:10.1016/j.cmpb.2011.07.002

ARTICLE IN PRESSCOMM-3248; No.of Pages14

computer methods and programs in b i omed ic ine x x x ( 2 0 1 1 ) xxxxxx

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Identification ofan integrated mathematical model of

standard oral glucose tolerance test for characterization of

insulin potentiation in health

Roberto Burattini , Micaela Morettini

Department of Information Engineering, Polytechnic University of Marche, Via Brecce Bianche, 60131Ancona, Italy

a r t i c l e i n f o

Article history:

Received 10 December 2010

Received in revised form

14 May 2011

Accepted 4July 2011

Keywords:

Glucose-insulin system

Gastric emptying

Oral glucose absorption

GIP

GLP-1

Incretin effect

Parameter estimation

a b s t r a c t

Two new formulations, respectively denominated INT M1 and INT M2, of an integrated

mathematical model to describe the glycemic and insulinemic responses to a 75 g oral

glucose tolerance test (OGTT) are proposed and compared. The INT M1 assumes a sin-

gle compartment for the intestine and the derivative of a power exponential function

for the gastric emptying rate, while, in the INT M2, a nonlinear three-compartment sys-

tem model is adopted to produce a more realistic, multiphase gastric emptying rate. Both

models were implemented in a Matlab-based, two-step procedure for estimation of seven

adjustable coefficients characterizing the gastric emptying rate and the incretin, insulin and

glucose kinetics. Model behaviour was testedvs. mean plasma glucagon-like peptide 1 (GLP-

1), glucose-dependent insulinotropic polypeptide (GIP), glucose and insulin measurements

from two different laboratories, where glycemic profiles observed during a 75 g OGTT were

matched in healthy subjects (HC1- and HC2-group, respectively) by means of an isoglycemic

intravenous glucose (I-IVG) infusion. Under the hypothesis of an additive effect of GLP-1

and GIP on insulin potentiation, our results demonstrated a substantial equivalence of the

two models in matching the data. Model parameter estimates showed to be suitable mark-

ers of differences observed in the OGTT and matched I-IVG responses from the HC1-group

compared to the HC2-group. Model implementation in our two-step parameter estimation

procedure enhances the possibility of a prospective application for individualization of the

incretin effect in a single subject, when his/her data are plugged in.

2011 Elsevier Ireland Ltd. All rights reserved.

1. Introduction

Mathematical modelling in the assessment of insulinglucose

interactions has a longstanding tradition, and reported mod-

els show a wide degree of complexity depending on their

purpose. Besides models of minimal complexity, some of

whichare referredto as minimal models, afterBergman et al.

[1], increasing relevance is being assumed by integrated simu-

Corresponding author. Tel.: +39 071 2204458; fax: +39 071 2204224.E-mail address: [email protected] (R. Burattini).

lation models of the glucose-insulin control system during an

oral glucose tolerance test (OGTT) or meal glucose tolerancetest (MGTT) to improve knowledge of diabetes pathophys-

iology and assess the efficacy of hypoglycemic agents in

clinical drug development [26]. In this context, a key issue

concerns the modelling of glucose transit through the gastro-

intestinal tract, to describe glucose absorption [79] and the

related incretin effect [8,1019], mostly due to the contribu-

tion of glucagon-like peptide-1 (GLP-1) and glucose-dependent

0169-2607/$ see front matter 2011 Elsevier Ireland Ltd. All rights reserved.doi:10.1016/j.cmpb.2011.07.002
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ARTICLE IN PRESSCOMM-3248; No.of Pages142 computer methods and programs in b iomed ic ine x x x ( 2 0 1 1 ) xxxxxx

insulinotropic polypeptide (GIP) released in response to nutri-

ent ingestion, which stimulate oral glucose dependent insulin

secretion.

A promising simulation model of the oral glucose tolerance

test, primarily intended to illustrate the importance of incretin

within the normal ranges observed clinically in humans, has

recently been proposed by Brubaker et al. [4]. Unfortunately,

a simple empiric description of the rate of gastric emptyingof ingested glucose into the gut, Gempt , constrains the model

to the reproduction of glycemia and insulinemia responses

to 50 and 100 g oral glucose loads, and does not allow an

effective model testing versus experimental data measured

during clinical OGTT protocols, which generally consist of the

administration of a standard 75g glucose dose. In its orig-

inal formulation [4] the model was characterized by fifteen

parameters, ten of which were given numerical values known

apriori from previously reported measurements, thus leaving

five adjustable parameters. However, the lack of a validation

against standard OGTT data limits the reliability of the values

assumed for these adjustable parameters, thus affecting the

predictive capabilities of the model. Based on these consider-ations, the aim of the present study was to improve the model

formulation by incorporating a mathematical description of

oral glucose absorption that allows model application to stan-

dardOGTT data, such that adjustable model parameters can

be better assessed by fitting to clinical data.

Two competing integrated mathematical models of oral

glucose tolerance test, denominated INT M1 and INT M2,

respectively, were obtained by incorporating two alterna-

tive representations of glucose absorption. The former model

incorporates a single compartment for the intestine and

the derivative of a power exponential function for the

gastric emptying rate [7]. In the latter, a nonlinear three-

compartment system model is adopted to produce a morerealistic multiphase gastric emptying rate [9]. A comparative

analysis of INT M1 and INT M2 behaviour was performed in

terms of their ability to reproduce the augmented glucose-

dependent plasma insulin concentration (generally referred

to as incretin-induced insulin potentiation) observed after an

OGTT, as it compares with the insulin response to an intra-

venous glucose infusion given in amount sufficient to match

the profile of glucose concentration observed during OGTT

(isoglycemic intravenous glucose, I-IVG, infusion) [15,18,19].

Especially, mean data of OGTT and I-IVG responses from two

groups of metabolically healthy subjects (here denominated

HC1- and HC2-group), respectively reported by Muscelli et al.

[18] and Nauck et al. [15], were used to test our models capabil-

ity to reproduce and interpret differences in essential aspects

of insulin potentiation in health observed in the data sets from

the two different laboratories.

2. Methods

2.1. Model formulation

The model previously proposed by Brubaker et al. [4] to sim-

ulate the glycemic and insulinemic responses to a 50 and a

100 g oral glucose load, with explicit incorporation of incretin

action, was used as a basis to build-up an integrated model

upon an OGTT protocol consisting of oral administration of

a standard 75 g oral glucose load (i.e. 1 g of glucose per kg

body weight, BW, administered at time t= 0 min, and stan-

dardized to a 75 kg individual). To this aim the description

of the oral glucose absorption was modified by incorporat-

ing either a one-compartment model [7], denoted as M1, or

a three-compartment model [9], denoted as M2. The two ver-

sions of our integrated model, the flow diagram of which isdepicted in Fig.1, were denoted asINT M1 when incorporating

the M1, and INT M2 when incorporating the M2, respectively.

The overallmathematical description is given in the Appendix

A.

Description, basal values and units for both the INT M1

and the INT M2 variables are given in Table 1. In the

absence of measurements of basal hepatic glucose bal-

ance, HGBb, the value of 0.77 mmolmin1 was deduced

from the literature by multiplying the BW of 75 kg times

10.25 103 mmolmin1 kg1 determined as the mean of two

hepatic glucose balance values per unit BW reported by Bell

et al. [20] (and summarized at points 3 and 4 of Table A.1

in [21]) for humans with mean glycemia of 5.28 mmolL1

and mean insulinemia of 10mU L1, which are consistent

with the mean fasting values of glycemia and insulinemia

characterizing our HC1-group (5.4 mmolL1 and 10.2 mU L1,

respectively) and HC2-group (5.5 mmolL1 and 7.6 mU L1,

respectively).

Besides the basal values of variables as given in Table 1,

and considered that HGBb is treated as a fixed parameter,

the INT M1 formulation consists of fifteen further indepen-

dent parameters; namely, k3, k4, k5, k6, k7, k8, k9, p, V, M,

, kabs, f, ke and . Instead, the INT M2 is characterized by

seventeen parameters, because in this model the ke and

characteristic parameters of the monophase waveshape for

the rate of gastric emptying (Gempt; Eq. (A1)) are replaced bythe c, b, kmin, and kmax coefficients of the emptying function

kempt(qsto) of Eq. (A10). Quantification of all these parameters

was accomplished as follows. The k3, k4, k6, k9, p and V were

assumed as fixed and were given numerical values (Table 2)

known apriori from observations reported in the literature, as

discussed by Brubaker et al. [4]. The kabs, f, b and c parame-

ters pertaining to glucose absorption were also assumed as

fixed (Table 2). In accordance with the validation studies on

glucose-absorption models reported by Dalla Man et al. [9,22],

the value of 0.22 min1 was assumed for the rate constant,

kabs, of intestinal glucose absorption, while the value of 0.90

was assumed for the fraction, f, of the ingested glucose dose,

D, that is actually absorbed. Eventually, the b and c dimen-sionless coefficients were given the values of 0.85 and 0.25,

respectively, thus fixing the percentage of the glucose dose,

D, for which the emptying function kempt (Eq. (A10)) decreases

(b) and subsequently increases (c) at (kmax kmin)/2 [9]. The

remaining independent model parameters, denoted as k5, k7,

k8, M and , which were assumed as adjustable in the orig-

inal model of Brubaker et al. [4] and presumably manually

adjusted, were considered as free parameters in addition to

the ke and in the INT M1 model formulation, and in addition

to kmin and kmax in the INT M2 model formulation. One novel

aspect of the present work is that these free parameters were

automatically estimated by a two step procedure as described

in Section 2.3.
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ARTICLE IN PRESSCOMM-3248; No.of Pages14computer methods and programs in b iomed ic ine x x x ( 2 0 1 1 ) xxxxxx 3

Fig. 1 Flow diagram ofour integrated model incorporating either one oftwo alternative formulations ofglucose

absorption, denoted as M1 and M2. The M1 representation assumes the derivative ofa power exponential function for the

gastric emptying rate (Gempt) ofthe oral glucose dose (D), and a single compartment for the gut ( qgut). In the M2

configuration, the oral glucose dose enters a two compartment model (qsto1 and qsto2) ofthe stomach, while a gastric

emptying function (kempt), dependent on the total amount ofglucose in the stomach (qsto, equal to the sum ofqsto1 and

qsto2), plays a key role in determining the time course ofboth the rate ofemptying oforal glucose load into the gut (Gempt)

and the glucose quantity in the gut compartment (qgut). Inboth our integrated model formulations (denominated INT M1

and INT M2, respectively), the Gempt is transmuted by the k5 coefficient into a signal that enters the compartment ofplasma

incretin (INC). Similarly, qgut is multiplied by the kabs rate constant ofintestinal absorption and theffraction ofthe intestinal

absorption that actually appears into plasma, to predict the RaG rate ofglucose absorption from the gut into the mesenteric

circulation. This feeds the glucose compartment (G). Solid lines with arrows indicate directional flows. Dashed lines with

arrows indicate control actions ofincretin (INC) and glucose (G) on post-hepatic insulin secretion, as well as control actions

ofinsulin (I) on hepatic glucose balance (HGB) and glucose delivery to the peripheral tissues.


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Table 1 Description, basal values and units for model variables.

Constant Description Value Units

qsto1b Amount of glucose in the compartment 1 of stomach. 0 mmol

qsto2b Amount of glucose in the compartment 2 of stomach. 0 mmol

qgutb Glucose mass in the intestine. 0 mmol

Gb Plasma glucose concentration. Fasting measurement mmolL1

Ib Plasma insulin concentration. Fasting measurement mU L

1

INCb Plasma incretin concentration. Fasting measurement ng L1

HGBb Hepatic glucose balance. 0.77 mmolmin1

Gemptb Rate of emptying of glucose load into the gut. 0 mmolmin1

RaGb Rate of appearance of glucose in the peripheral circulation. 0 mmolmin1

In accordance with Dalla Man et al. [9], to favour

numerical identifiability of the INT M2 model, the con-

straint k21 = kmax was imposed. After INT M1 and INT M2

model parameter estimation was accomplished, the remain-

ing dependent parameters 1, 2, RaINCb, , k1, and k2 were

computed by Eqs. (A11), (A12), (A15), (A17), (A20) and (A21),

respectively.

2.2. Model testing vs. reported experimental data

Our INT M1 and INT M2 model outputs were tested against

data of plasma glucagon-like peptide-1 (GLP-1), glucose-

dependent insulinotropic polypeptide (GIP), glucose and

insulin concentration, averaged over two groups of metabol-

ically healthy subjects, here denominated HC1- and HC2-

group, respectively reported by Muscelli et al. [18] and Nauck

et al. [15].

Essentially, the subjects involved in these previous stud-

ies underwent two different glucose-challenge protocols. On

the first occasion (OGTT protocol) the subject ingested 75 g of

glucose, whereas on the second occasion (I-IVG protocol) an

intravenous glucose infusion was given in amount sufficientto match the profile of glucose concentration observed dur-

ing OGTT [15,18]. An augmented glucose-dependent insulin

secretion (insulin potentiation), observed in these circum-

stances after OGTT, compared to the matched I-IVG infusion,

is attributed to the influence of the so called incretin effect

mostly due to the gut-derived incretin hormones, GLP-1 and

GIP, which enhance glucose-dependent insulin secretion by

binding to specific receptors on the -cell [2325]. These

hormones are released during glucose or meal intake in pro-

portion to nutrient transport across the intestinal epithelium

[10], their effect seems to be additive, and they stimulate

insulin secretion both at fasting and postprandial plasma glu-

cose level [14]. On this basis, the mean levels of GLP-1 and GIP

(expressed in pmol L1) reported in [18] for the HC1-group, and

in [15] for the HC2-group, were combined as follows to definean INC(t) signal for each group. First, the reported mean lev-

els of GLP-1 and GIP, expressed in pmolL1, were converted

into ng L1 (conversion factors were: 1 pmolL1 = 3.30 ng L1

for the GLP-1, and 1 pmolL1 = 4.98 ng L1 for the GIP) and,

then, for each group the INC(t) signal was determined as the

sum of related GLP-1 and GIP. Eventually, this INC(t) signal was

used together with the glycemia, G(t), and insulinemia, I(t),

data, and the related Gb, Ib and INCb, fasting values, for model

parameter estimation by the procedure described in Section

2.3.

2.3. Parameter estimation procedure

As explained in Section 2.1, the INT M1 and the INT M2 are

characterized by seven free parameters, i.e., k5, k7, k8, M, ,

and ke, the former, and k5, k7, k8, M, , kmin and kmax, the

latter. To define an optimal value for each one of these param-

eters, a two-step parameter estimation procedure was set-up

as follows.

Table 2 Description, values and units for fixed INT M1 and INT M2 model parameters.

Coefficient Description Value Units

kabs Rate constant of intestinal absorption. 0.22 min1

f Fraction of the intestinal absorption which actuallyappears into plasma.

0.90

c Glucose dose percentage corresponding to the first flex

point of the kempt(qsto) function (Eq. (A10)).

0.25

b Glucose dose percentage corresponding to the second

flex point of the kempt(qsto) function (Eq. (A10)).

0.85

p Ratio of non-insulin mediated to insulin mediated

glucose uptake (NIMGU/IMGU; Eq. (A19)).

2

k3 Slope of renal glucose clearance. 0.0718 L min1

k4 Intercept of renal glucose clearance. 0.717 mmolmin1

k6 Measure of degradation/clearance of incretin. 0.1 min1

k9 Measure of degradation/clearance of insulin. 0.1 min1

V Volume of distribution. 15 L

Source: Sources of the values given top, k3, k4, k6, k9 and V are found in Brubaker et al. [4].

Values ofkabs, f, c and b are taken from Dalla Man et al. [9,22].


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The first step was to estimate the k7 parameter by run-

ning either the INT M1 and the INT M2 to fit mean plasma

insulin concentrations from the I-IVG protocol. To this aim,

Gempt (and, with it, the dynamic component of plasma incretin

inflow in Eq. (A14)) was set to zero, while the incretin level

was maintained at the basal value, INCb, and the glucose pro-

file matched to that measured during OGTT in the HC1- and

HC2-group, respectively, was used as model input instead ofthe description by Eq. (A18). Under these conditions, glucose

absorption is not involved, so that k7 (Eq. (A16)) is the only one

free parameter that remains to be estimated from fitting the

model predicted I(t) to I-IVG insulin data.

In a subsequent step, the k7 estimate was filled into the

INT M1 and the INT M2, which were run to generate incretin,

glycemia and insulinemia responses to a 75 g OGTT, to be

simultaneously fitted to the OGTT data(corresponding to each

I-IVG), in order to estimate the six remaining unknown param-

eters (k5, k8, M, , ke and ) of the INT M1, as well as the six

remaining unknown parameters (k5, k8, M, , kmin, and kmax)

of the INT M2.

All model formulations were implemented inMatlabSimulink environment. Data fit and parameter

estimation were accomplished by means of a weighted

least squares (WLS) procedure [26]. The following sum of

square error (SSE) expression was used to fit the model

predicted I-IVG insulin output at the i-th instant, Im(ti), to the

corresponding insulin measurement, I(ti):

SSEI-IVG =

N=1

Im(ti) I(ti)

Ii

2(1)

The following SSE expression was subsequently used for

simultaneous fit of model predicted OGTT incretin, INCm(ti),

glycemia, Gm(ti), and insulinemia, Im(ti), outputs at the i-th

instant, to the corresponding incretin, INC(ti), glycemia, G(ti),

and insulinemia, I(ti), measurements:

SSEOGTT =

N=1

INCm(ti) INC(ti)

INCi

2+

Gm(ti) G(ti)

Gi

2

+

Im(ti) I(ti)

Ii

2(2)

The weights, INCi, Gi, Ii, were assumed equal to 5.5%,

1.5% and 4% of INC(ti), G(ti) and I(ti), respectively, under the

assumption of normal distribution with zero mean for the

measurement errors [18,26]. The goodness of fit was evaluated

by calculation of the root-mean-square errors, RMSEI-IVG =SSEI-IVG/N and RMSEOGTT =

SSEOGTT/3N

Precision of all parameter estimates was expressed as per-

cent coefficient of variation: CV(pi)% = SDpi/pi 100, where piis the i-th component of the model parameters vector and

SDpi is the standard deviation ofpi, which is calculated as the

square root of the diagonal terms of the inverse of the Fisher

information matrix.

2.4. Insulin potentiation

In accordance with Nauck et al. [11], the quantity of-cell

secretory response evoked by factors other than glucose itself

(insulin potentiation) was deduced from the difference in

integrated incremental responses (over basal) of insulinemia

between OGTT and matched I-IVG protocols. This insulin

potentiation, IP, can be quantified as the percentage of theOGTT response by the equation:

IP% =AUCIOGTT AUCII-IVG

AUCIOGTT 100 (3)

where AUCIOGTT and AUCII-IVG represent the area under

the curve of incremental insulin concentration over the OGTT

and the matched I-IVG duration, respectively, according to the

trapezoidal rule.

3. Results

Mean fasting values ofGb, Ib, and INCb were 5.4 mmolL1,

10.2 mU L1, and 145 ng L1, respectively, for the HC1-group,

and 5.5mmolL1, 7.6 mU L1, and 90 ng L1, respectively, for

the HC2-group.

Fitting the INT M1 and INT M2 outputs (dash-dot line

in Fig. 2C and F) to mean insulinemia data from OGTT-

matched I-IVG protocols, reported for the HC1-group and

HC2-group (open circles in Fig. 2C and F, respectively), yielded

the RMSEI-IVG and the estimates ofk7 given in Table 3 for the

former model and in Table 4 for the latter. After feeding theINT M1 and the INT M2 with the k7 values obtained from the

previous step, the INC(t), G(t) and I(t) responses to a 75 g OGTT

were generated and fitted to incretin, glycemia and insuline-

mia data from OGTT protocols corresponding to each I-IVG,

thus estimating the k5, k8, M, , ke and values reported in

Table 3 for the INT M1, and the k5, k8, M, , kmin and kmaxvalues reported in Table 4 for the INT M2. Eventually, compu-

tation of1 (Eq. (A11)), 2 (Eq. (A12)), RaINCb (Eq. (A15)), (Eq.

(A17)), k1 (Eq. (A20)) and k2 (Eq. (A21)) yielded the values given

in Table 5. Fig. 2 displays the quality of data fit obtained from

applying the INT M1 (dashed line) and the INT M2 (solid line)

to mean OGTT data from the HC1-group (panels AC), and the

HC2-group (panels DF). Values of RMSEOGTT are reported inTable 3 for the INT M1 and in Table 4 for the INT M2.

Fig. 3 displays the time course of the Gempt and RaG pro-

files predicted by the INT M1 (dashed line) and the INT M2

(solid line) for the HC1-group (panels A and B, respectively)

and the HC2-group (panels C and D, respectively). The grey

area in panels B and D represents the range of variability of

RaG, measured with the multiple tracer,tracer-to-tracee clamp

technique during OGTT as reported by Dalla Man et al. [9].

Values of the insulin potentiation index, IP%, computed

by Eq. (3) from OGTT and matched I-IVG insulinemia outputs

produced by the INT M1 and INT M2 models are compared

in Table 6 with the corresponding IP% values computed from

mean insulinemia data of the HC1- and HC2-group.


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HC2HC1

2402101801501209060300

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5

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8

9

10

11

12B

G

(t)(mmolL-1)

Time (min)

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0

20

40

60

80

100

120C

I

(t)(mUL-1)

Time (min)

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100

200

300

400

500

D

INC

(t)(ngL-1)

Time (min)

2402101801501209060300

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8

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10

11

12E

Time (min)

G

(t)(mmolL-1)

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120F

I(

t)(mUL-1)

Time (min)

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100

200

300

400

500

A

INC

(t)(ngL-1)

Time (min)

Fig. 2 Mean data ofplasma incretin (closed triangles in panels A and D), glucose (closed squares in panels B and E) and

insulin (closed circles in panels C and F) concentrations in response to 75 g oral glucose challenge in the HC1-group (left

hand column) and the HC2-group (right hand column) are matched by the profiles ofINC(t), G(t), and I(t) outputs ofthe

INT M1 (dashed lines) and the INT M2 (solid lines) models after free parameter optimization. Open circles in panels C and F

are mean plasma insulin concentration data from the HC1- and HC2-group, respectively, measured after the glycemic

profiles observed following glucose ingestion (OGTT) were matched by means ofan isoglycemic intravenous glucose (I-IVG)

infusion (open squares in panels B and E). The dash-dot line in panels C and F describes the best fitting I-IVG insulin outputprovided by both the INT M1 and the INT M2 model.

4. Discussion

This work yields an improved formulation of a mathematical

model previously proposed by Brubaker et al. [4] to simu-

late the glycemia and insulinemia responses to 50 and 100 g

oral glucose administration, by explicitly incorporating the

incretin effect. The incretin response, INC(t), is thought to

be a direct effect of the rate of emptying of ingested glucose

into the gut, Gempt(t), through a zero-order transfer function,

with k5 gain, between the Gempt and the dynamic component

of plasma incretin inflow (Eq. (A14)). An improved descrip-

tion of intestinal glucose absorption in response to 75 g OGTT,

was accomplished here by replacing the previously proposed

empirical description, restricted to 50 and 100 g glucose chal-

lenge [4], with either a one-compartment model (M1 in Fig. 1)

and a three-compartment model (M2 in Fig. 1), thus giving rise

to two integrated models denominated INT-M1 and INT-M2,

respectively.

Besides the k3, k4, k6, k9, p, and V parameters (Table 2),

which were fixed at numerical values known a priori from


7/14



2402101801501209060300

0

1

2

3

4

5

6A

Time (min)

2402101801501209060300

-1

0

1

2

3

4

5

6B

Time (min)

2402101801501209060300

0

1

2

3

4

5

6C

Gempt

(t)(mmolmin-1)

Time (min)

2402101801501209060300

-1

0

1

2

3

4

5

6D

RaG(t)(mmolmin-1)

Gempt

(t)(mmolmin-1)

RaG(t)(mmolmin-1)

Time (min)

HC2HC1

Fig. 3 The profiles ofthe rate ofemptying oforal glucose load into the gut, Gempt(t), and the rate ofglucose absorption from

the gut into the mesenteric circulation, RaG(t), for the HC1-group (Panel A and B, respectively) and the HC2-group (Panel C

and D, respectively) as predicted by INT M1 (dashed line) and the INT M2 (solid line). The grey area in panels B and D

represents the range ofvariability ofRaG, measured with the multiple tracer, tracer-to-tracee clamp technique during OGTT

as reported by Dalla Man et al. [9].

previously reported measurements, as discussed by Brubaker

et al. [4], the basal hepatic glucose balance, HGBb, and the four

more glucose absorption parameters, kabs, c, b andf, were fixed

at numerical values taken from the literature as described

in Section 2.1. Consequently, seven free parameters (k5, k7,

k8, M, , ke and ) characterize the INT M1 (Table 3), as well

as seven are the free parameters (k5, k7, k8, M, , kmin, and

kmax) that characterize the INT M2 (Table 4). Among these,

the five parameters denominated k5, k7, k8, M and are the

same as those defined adjustable by Brubaker et al. [4], and

presumably manually adjusted in their study for qualitative

representation of OGTT responses. The remaining two param-eters, i.e. ke and for the INT M1 and kmin and kmax for the

INT M2, are free parameters that allow individualization of

the rate of ingested glucose absorption. Granted the suitabil-

ity of all fixed parameters, a novel aspect of the present study

was the set-up of a two-step procedure that allows estima-

tion of the free parameters of our INT M1 and INT M2 model

formulations by fitting to incretin, glycemia and insulinemia

data taken from previously published studies [15,18], where

glycemic profiles observed in two groups, (HC1 and HC2) of

metabolically healthy subjects, during an OGTT, were used

to quantify incretin-induced insulin potentiation by compar-

ing the insulin response to OGTT with that obtained from an

isoglycemic intravenous glucose (I-IVG) infusion.

According to our two-step estimation procedure, the k7parameter was first estimated from fitting to I-IVG insuline-

mia data. To this aim the Gempt (and, with it, the dynamic

component of plasma incretin inflow in Eq. (A14)) was set

to zero and the plasma glucose compartment was fed with

the glucose profile matched to that measured during OGTT

(isoglycemic profile). Under these conditions the alternative

formulations (M1 and M2) of glucose absorption do not come

into play. This explains why both the INT M1 and the INT M2

produce the same k7 estimates and related RMSEI-IVG val-

ues, as given in Tables 3 and 4, respectively. The higher k7

estimate of 0.648 mU min1

mmol1.3

L0.3

for the HC1-group,compared to the estimate of 0.158 mU min1 mmol1.3 L0.3

for the HC2-group provides quantitative information on an

enhanced insulin response to intravenous glucose infusion

in the former group. This is consistent with the experimen-

tal finding of a higher value of the ratio (AUCII-IVG/AUCGI-IVG)

between the areaunder the curve of I-IVG suprabasalinsulin to

the area under the curve of the corresponding suprabasal pro-

file of glucose infused intravenously in the HC1-group (ratio of

14.3 mU mmol1 from data ofFig. 2B and C) compared to that

of the HC2-group (ratio of 5.7 mU mmol1 from data ofFig. 2E

and F).

Once the estimated k7 values were filled into the INT M1

and the INT M2, these two models produced a comparable


8/14


9/14



T

able5ComputedvaluesofINTM1

andINTM2modelparameters.

1 mmol1

2mmol1

RaINCb

ngmin1

mUL1min1

k1min1

k2

m

molmU1min1

HC1

HC2

HC1

HC2

HC1

HC2

HC1

HC2

HC1

HC2

H

C1

HC2

INTM1

217

135

8.66

2.22

0.0064

0.0062

0.0017

0.0022

INTM2

0.040

0.024

0.040

0.024

217

135

8.41

2.21

0.0064

0.0062

0.0017

0.0022

H

C1,groupof11subjectswithnormalgluco

setolerance(NGT)from

Muscellietal.[18];HC2,groupof10metabolicallyhealthysubjectsfrom

Naucketal.[15];1and2,resp

ectively,quantifythe

rateofdecreaseandsubsequentincreaseof

thegastricemptyingfunction(Eqs.(A11)and(A12),respectively);RaINCb,basalrateofappearanceofincretinintheperipheralcirculation(Eq.(A15));,

e

ffectsofadditionalregulatorsofI(t)oninsu

linappearance(Eq.(A17));k1andk2arecoe

fficientsoflinearglucosemediatedandins

ulinmediatedglucoseuptake,respectively

(Eqs.(A20)and(A21),

r

espectively).

fit to incretin (closed triangles in Fig. 2A and D), glycemia

(closed squares in Fig. 2B and E) and insulinemia (closed

circles in Fig. 2C and F), as judged from the dashed-line

profiles for the INT M1 and the solid-line profiles for the

INT M2, and the related RMSEOGTT reported in Tables 3 and 4,

respectively.

Due to their ability to approximate both the OGTT

and the matched I-IVG insulin response, the two modelsyielded estimates of the IP% index of insulin poten-

tiation (Eq. (3)) consistent with the corresponding val-

ues computed from measured data in the HC1- and

HC2-group (Table 6).

The augmented insulin secretion observed after oral glu-

cose load, compared with intravenous glucose infusion, at

similar plasma glucose concentration, is attributed to the

influence of the incretin effect, most of which is due to

the GLP-1 and GIP secretion [8,14,16,18,2325]. As the con-

tribution of intestinal hormones to oral glucose-stimulated

insulin secretion is relevant, and being the major stimu-

lus to the release of these hormones clearly related to the

rate of ingested glucose delivery to the gut, Gempt(t), theINC(t) signal described in Eq. (A14) by the product k5 Gempt(t)

was derived from the GLP-1 and GIP responses to OGTT

reported in [18] for the HC1-group and in [15] for the HC2-

group, respectively, by hypothesizing an additive effect as

described in Section 2.2. With this assumption, the closed-

triangle data ofFig. 2A and D were obtained. The lower k5estimates provided by both the INT M1 (Table 3) and the

INT M2 (Table 4) for the HC1-group, compared to those pro-

vided for the HC2-group, reflect the experimental observation

of a lower area under the suprabasal curve of incretin in the

HC1-group(25.6103 ng L1 over 180 min, Fig.2A)comparedto

that of the HC2-group (40.0103 ng L1 over 180 min, Fig. 2D)

in response to the 75 g oral glucose challenge. On this basis,the k5 free parameter appears a suitable marker of the ampli-

tude of the incretin response to oral glucose administration.

In spite of lower incretin response, the suprabasal insuline-

mia response of the HC1-group, quantified by the AUCIOGTTvalue of 9.03103 mU L1 over 180 min, is higher than that,

7.70103 mU L1 over 180 min, observed in the HC2-group

(compare Fig. 2C and F). This implies the presence, in the HC1-

group, of a compensatory increase of the insulin response to

the incretin, which is reflected in the increased k8 values esti-

mated in the same group by the INT M1 (Table 3) and the

INT M2 (Table 4), compared to those of the HC2-group. In the

presence of this compensation, the enhanced insulin potenti-

ation (higher IP% index; Table 6) inthe HC2-group, compared to

the HC1-group, is mainly explained by the observed reduction

Table 6 Insulin potentiation.

IP%

INT M1 INT M2 EXP

HC1 63.4 62.9 63.0

HC2 81.0 81.1 78.1

HC1, group of11 subjects with normal glucose tolerance (NGT) from

Muscelli et al. [18]; HC2, group of 10 metabolically healthy subjects

fromNauck et al. [15]; IP%, insulin potentiationcomputedby Eq. (3).


10/14

Pleasecitethisarticleinpressas:RBurattiniM

MorettiniIdentificationofanintegratedmathematicalmodelofst

andardoralglucosetolerance

Table 7 Estimates ofINT M1 model parameters in the presence ofHGBb equal to 0.92mmol min1.

k7 (CV%)

mU min1 mmol1.3 L0.3

k5 (CV%)

ng L1 mmol1

k8 (CV%)

mU min1 ng1

(CV%)

mmolmU1

M (CV%)

L2 mU1 min1

ke (CV%)

min1

HC1 0.648

(2.2%)

6.86

(5.7%)

0.026

(4.8%)

0

()

0

()

0.0093

(7.8%)

1.0

(2

HC2 0.158

(2.3%)

8.85

(1.3%)

0.017

(1.2%)

0.0489

(5.0%)

0.0017

(14%)

0.0156

(1.0%)

1.4

(0

See legend ofTable 3 for meaning of symbols.

Table 8 Estimates ofINT M2 model parameters in the presence ofHGBb equal to 0.92mmol min1.

k7 (CV%)mU min1 mmol1.3 L0.3

k5 (CV%)ng L1 mmol1

k8 (CV%)mU min1 ng1

(CV%)mmolmU1

M (CV%)L2 mU1 min1

kmin (CV%)min1

kmaxmin

HC1 0.648

(2.2%)

5.95

(3.8%)

0.025

(4.8%)

0.0032

(65%)

0.0043

(9.1%)

0.016

(2.4%)

0.037

(2.5%

HC2 0.158

(2.3%)

8.93

(2.7%)

0.016

(3.3%)

0.0566

(4.9%)

0.0021

(8.0%)

0.026

(1.8%)

0.037

(1.4%

See legend ofTable 4 for meaning of symbols.


11/14



of suprabasal insulin response to the OGTT-matched profile of

glucose infused intravenously, as marked by a lower k7 in the

HC2-group.

In spite of a similar behaviour of the INT M1 and INT-M2

models in accounting for variations ofk5, k7 and k8 estimates

between the HC1- and the HC2-groupof healthy subjects,a rel-

evant diversity is seen in that, in the HC1-group, the INT M1

model estimated a zero value for the and a value ofM 66%lower than that produced by the INT M2. No model inde-

pendent information is available to judge the reliability of

estimated values for,since the small derivative dI/dt term

was empirically introduced by Brubaker et al. [4] to sharpen

the peak in glucose level following entry of glucose into the

circulation, and was set equal to 0.06 when RaG > 0. This value

is comparable with that estimated here for the HC2-group

by both the INT M1 and the INT M2 (Tables 3 and 4). The

zero-value estimated for by the INT M1 in the HC1 group

(Table 3), as well as the corresponding very low estimate

(with 84% estimation error) provided by the INT M2 (Table 4)

might be ascribed to a vanishing role of the dI/dt in the

presence of a limited glycemia dynamics as observed inthe HC1-group compared to the HC2-group (Fig. 2B with E).

Concerning the M parameter, which modulates the bilin-

ear glucose-insulin control on hepatic balance, it is likely

that its estimate is affected by the value of HGBb (Eq. (A13))

assumed as fixed to favour model identifiability. Because

the value of 10.25103 mmolmin1 kg1 assumed here for

basal hepatic glucose balance per unit BW (yielding HGBbequal to 0.77 mmolmin1 for a standardized 75kg individ-

ual), differs from the value of 12.22103 mmolmin1 kg1

(yielding HGBb equal to 0.92 mmolmin1) previously assumed

by Brubaker et al. [4], the effect of the latter HGBb value

on the estimates of our free model parameters was tested.

Results are given in Table 7 for the INT M1 and in Table 8for the INT M2 model. Comparison with Tables 3 and 4,

respectively, confirms that the HGBb has a major effect on

the M parameter estimation. In particular, the M values of

Tables 7 and 8 are significantly underestimated with respect

to the corresponding values of Tables 3 and 4. The zero-

value of M produced by the INT M1 for the HC1 group

(Table 7) implies a stationary unidirectional HGB flux (Eq.

(A13)), which is unlikely to occur under normal conditions

[2729]. For this reason the parameter estimates obtained

from our HGBb assumption (Tables 3 and 4) are more physi-

ologically consistent. The lack of quantitative information on

the HGB dynamics in humans does not allow assessment of

the reliability of absolute values ofM when different fromzero.

In describing the glucose kinetics by Eq. (A18), a lin-

ear non-insulin mediated glucose uptake, k1 G(t), was

assumed here, rather than the k1 G(t)1.3 power term origi-

nally assumed by Brubaker et al. [4]. Under our simplifying

assumption, which is consistent with that characterizing

minimal models of glucose kinetics [1], the estimated val-

ues for the k1 coefficient (Table 5) increased in both our

HC1 and HC2 groups by about 65%, compared to the k1values computed in the presence of the k1 G(t)1.3 power

term. Practically no effect was observed in the RMSEOGTTof data fit and in the estimates of free model parame-

ters.

4.1. Monophase vs. multiphase waveshape ofgastric

emptying ofglucose

In conceptual terms, our INT M1 model differs from the

INT M2 in that the former incorporates the hypothesis of a

monophasic waveshape for the rate of gastric emptying of

ingested glucose into the gut, (Gempt ; Eq. (A1) and dashed line

in Fig. 3A and C), characterized by the two identifiable param-eters, ke and (Table 3). Instead, the INT M2 configuration

assumes a three-compartment model of glucose absorption,

with a nonlinear gastric emptying function (kempt; Eq. (A10)),

that yields a multiphase Gempt (solid line in Fig. 3A and C).

Unfortunately, the kempt is characterized by four independent

parameters (b, c, kmin and kmax) a couple of which needs to be

fixed to favour numerical identifiability of the INT M2 model.

Our choice was, then, to fix the b and c values (Table 2), which

respectively locate the flex points ofkempt in correspondence

of the percentage of the glucose dose, D, for which the emp-

tying function kempt (Eq. (A10)) decreases and subsequently

increases at (kmax kmin)/2 [9]. The need of fixing two of the

four characteristic parameters of the kempt function yields alimitation in the INT M2 ability to match the data, such that

no relevant benefit is seen in the RMSEOGTT, compared to the

INT M1 output.

4.2. Summary and conclusions

An improvement over a mathematical model previously pro-

posed by Brubaker et al. [4] to simulate the glycemia and

insulinemia responses to an OGTT, was accomplished here

by replacing the previously proposed empirical description

of glucose absorption, restricted to 50 and 100 g glucose

challenge, with either the M1 and the M2 model configu-

rations depicted in Fig. 1 and mathematically described inthe Appendix A. This new arrangement, gave rise to our

INT M1 and INT M2 integrated models, respectively, that allow

a reliable description of glycemic and insulinemic responses

to a clinically consistent 75 g OGTT. Under the hypothesis

of an additive effect of GLP-1 and GIP on insulin potenti-

ation, our results showed a substantial equivalence of the

two alternative INT M1 and INT M2 models in reproducing

insulin potentiation as observed by comparing mean OGTT

responses to a 75 g oral glucose challenge with matched

I-IVG responses, as reported in [18] and in [15] for two dif-

ferent groups of healthy subjects, here denominated HC1

and HC2, respectively. Especially, the implementation of both

the INT M1 and INT M2 models in our two-step parameter

estimation procedure allowed assessment of k5, k7 and k8parameters, which appear as suitable markers of the differ-

ences in the incretin, glucose and insulin responses between

the two groups. Namely, k5 is a marker of the amplitude of

the incretin response to oral glucose administration; k8 is

a marker of the insulin response to the incretin, while k7is a marker of the insulin response irrespective of incretin

effect. A higher k8 value estimated by both the INT M1 and

the INT M2 in the HC1-group, compared to the HC2, indicates

the presence, in the former group of an increase of insulin

response to incretin, which compensates for a lower incretin

response to oral glucose load (lower k5 inthe HC1). In the pres-

ence of this compensation, the enhanced insulin potentiation


12/14



(higher IP% index; Table 6) in the HC2-group, compared to the

HC1-group, is mainly explained by the observed reduction of

suprabasal insulin response to the OGTT-matched profile of

glucose infused intravenously, as marked by a lower k7 in the

HC2-group.

Owing to the substantial equivalence of INT M1 and INT-

M2 behaviour, the former model appears the best compromise

between simplicity and predictive capability. Model imple-mentation in our two-step parameter estimation procedure

enhances the possibility of a prospective application for indi-

vidualization of the incretin effect in a single subject, when

his/her data are plugged in.

Appendix A. Mathematical formulation ofour75 g OGTT integrated models

A.1. M1 model oforal glucose absorption

This model describes glucose absorption by the gut as shown

in the M1 panel ofFig. 1. Under the assumption that, afteringestion of a glucose dose, D (75 g, equivalent to 417 mmol),

the fraction of glucose in the duodenum follows a power expo-

nential function increase, the rate of emptying of the oral

glucose load into the gut, Gempt (mmolmin1), is described by

the following equation [7,9]:

Gempt(t) = D ke t

1 e[(ket) ] (A1)

where ke (min1) is a fractional transfer coefficient and a

dimensionless shape factor. Denoting the amount of glucose

inthe gut asqgut (mmol),the rate constant of intestinal absorp-

tion as kabs (min1), the fraction of the intestinal absorption

which actually appears in plasma asf, and the rate of appear-

ance of ingested glucoseinto plasma as RaG (mmol min1), the

following equations hold [7,9]:

qgut(t) = kabs qgut(t) + Gempt(t) (A2)

RaG(t) = f kabs qgut(t) (A3)

A.2. M2 model oforal glucose absorption

The representation of glucose absorption displayed in the M2

panel ofFig. 1 assumes two compartments (qsto1 and qsto2,

respectively) for the stomach, and a single compartment, qgut,for the intestine. Mathematical description is as follows [9]:

qsto1(t) = k21 qsto1(t) + D (t) (A4)

qsto2(t) = kempt(qsto) qsto2(t) + k21 qsto1(t) (A5)

qgut(t) = kabs qgut(t) + kempt(qsto) qsto2(t) (A6)

qsto(t) = qsto1(t) + qsto2(t) (A7)

Gempt(t) = kempt(qsto) qsto2(t) (A8)

RaG(t) = f kabs qgut(t) (A9)

with the initial conditions being the basal values qsto1b,

qsto2b and qgutb reported in Table 1.

In these equations (t) is the impulse function, D (mmol)

is the amount of ingested glucose, k21 (min1) is the rate of

grinding, Gempt (mmol min1) is the rate of emptying of the

oral glucose load into the gut, RaG (mmol min1) is the rate of

glucose absorption from the gut into the mesenteric circula-

tion; kabs (min1) is the rate constant of intestinal absorption,f is the fraction of the intestinal absorption which actually

appears in the plasma, and kempt (min1) is a gastric empty-

ing function, which depends on the total amount of glucose

in the stomach, qsto (mmol), as follows [9]:

kempt(qsto) = kmin +kmax kmin

2 {tanh[1 (qsto b D)]

tanh[2 (qsto c D)] + 2} (A10)

As reported by Dalla Man et al. [9], this model is suit-

able to describe glucose absorption during an OGTT under

the assumption ofkempt = kmax at qsto =D (full stomach) and atqsto = 0 (empty stomach). This assumption yields the following

expressions for the 1 and 2 parameters:

1 =5

2 D (1 b)(A11)

2 =5

2 D c(A12)

A.3. Hepatic glucose balance

Hepatic glucose balance (HGB; mmolmin1) represents the

net flux of glucose G (mmolL1

) across the hepatic bed,thereby reflecting the sum of glucose production and glucose

uptake from the mesenteric circulation (e.g., following the oral

glucose load). In accordance with Brubaker et al. [4] the fol-

lowing equation is assumed to describe the HGB(t) dynamics

in healthy conditions:

HGB(t) = HGBb +M [Gb G(t)] I(t) (A13)

In this equation, HGBb is basal hepatic glucose balance, Gbis basal glucose concentration, while G(t) and I(t) represent

the time course of glucose and insulin concentrations. M is

a modulating factor in the bilinear control term.

A.4. Incretin response

The following differential equation was assumed to describe

the kinetics of the incretin [4]:

dINC(t)

dt=

RaINCbV

+ k5 Gempt(t) k6 INC(t) (A14)

According to this equation, three terms determine the rate

of variation with time of circulating incretin concentrations

(INC), based on the combined contribution of glucagon-

like peptide-1 (GLP-1) and glucose-dependent insulinotropic

polypeptide (GIP) [8,14,16,18,2325]. The first term is the

basal rate of appearance of the incretin (RaINCb; ng min1),


13/14



normalized to distribution volume (V); the second term is

proportional, through the k5 constant (ng L1 mmol1), to the

profile of the rate of gastric emptying into the gut (Gempt;

mmolmin1); and, eventually the third term accounts for a

negative contribution, proportional through the k6 (min1) to

the time course ofthe incretin (INC; ng L1) due to the removal

of bioactive GIP and GLP-1 from the circulation [30]. The quan-

tity

RaINCb = k6 V INCb (A15)

is obtained from Eq. (A14) by setting the derivative equal to

zero, which occurs when Gempt equals zero.

A.5. Insulin response


insulin kinetics [4]:

dI(t)

dt= k7 G(t)

1.3+ k8 INC(t) k9 I(t) + (A16)

Eq. (A16) is based on the assumption that four terms deter-

mine the rate of variation with time of circulating insulin.

The first nonlinear term, assumed equal to k7 G(t)1.3, accel-

erates the effect of plasma glucose on insulin release, with

respect to a linear term; the second and the third terms are

proportional, through the k8 (mU min1 ng1) and k9 (min1)

constants, respectively, to the circulating concentrations of

the incretin and insulin; and eventually, the constant term,

(mU L1 min1), which is expressed, in steady state, in terms

ofthe known basal valuesofG and I,after setting the derivativeterm, dI/dt, equal to zero and solving for :

= k9 Ib k8 INCb k7 G1.3b (A17)

A.6. Plasma glucose kinetics


the glucose kinetics [4]:

dG(t)

dt=

RaG(t)

V+

HGB(t)

V k1 G(t) k2 I(t) +

dI(t)

dt

(A18)

According to this equation, the rate of change of glucose

concentration during an OGTT depends on glucose absorp-

tion from the gut into the mesenteric circulation (RaG(t)/V);

net absorption/production of glucose by the liver (HGB(t)/V);

insulin mediated glucose uptake, k2 I(t); and a small cor-

rective term, dI(t)/dt, which has the effect of sharpening

the peak in glucose level following entry of glucose into the

circulation. A linear, k1 G(t), non-insulin mediated glucose

uptake was assumed here, rather than the k1 G(t)1.3 power

term originally assumed by Brubaker et al. [4]. Under these

assumptions, denoting as p the ratio of non-insulin mediated

to insulin mediated glucose uptake, and assuming a value of

2 for basal conditions [31,32], the following relation holds:

p =k1 Gbk2 Ib

= 2 (A19)

From Eq. (A18) written for steady state conditions (dG/dt= 0

and dI/dt= 0) and Eq. (A19), the values of the constants k1(min1) and k2 (mmol min1 mU1) are given by the following

equations [4]:

k1 =p

p+ 1

HGBbGb V

(A20)

k2 =1

p+ 1

HGBbIb V

(A21)

As reported by Brubaker et al. [4], the Eq. (A18) works

well under normoglycemic conditions. To account for urinary

glucose loss during hyperglycemia (G> 10 mmolL1), an addi-

tional term is added as shown in the following equation:

dG(t)

dt=

RaG(t)

V+

HGB(t)

V k1 G(t) k2 I(t) +

dI(t)

dt

k3 G(t) k4V

(A22)

r e f er enc es

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[4] P.L. Brubaker, E.L. Ohayon, L.M. DAlessandro, K.H. Norwich,A mathematical model of the oral glucose tolerance testillustrating the effects of the incretin , Ann. Biomed. Eng. 35(2007) 12861300.

[5] C. Dalla Man, R.A. Rizza, C. Cobelli, Meal simulation modelof the glucose-insulin system , IEEE Trans. Biomed. Eng. 54

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