The Modified Oral Minimal Model to Know Incretin Hormone

Effect in Type 2 Diabetes Mellitus

Agus Kartono, Rojali Fadila, Heryanto Syafutra, Setyanto Tri Wahyudi, and Tony Sumaryada

Department of Physics, Faculty of Mathematical and Natural Sciences, Bogor Agricultural University (IPB)

Keywords: Incretin hormone, Insulin, Oral minimal model, Simulation, Type 2 diabetes,

Abstract: Simulation of glucose, insulin and incretin hormones concentrations in the blood is described using the

modified oral minimal model that can be learned easily. In this study, the modified oral minimal model can

also describe the kinetics of incretin hormone concentration to determine the level of insulin secretion in the

body. Incretin hormone has been recognized as a major stimulus for insulin secretion after an oral glucose

load. Based on the results of this study, subjects with type 2 diabetes have the smallest of incretin hormone

concentration (115 ng/dL). The normal subjects have the highest of incretin hormone concentration (300

ng/dL). The simulations which have been done show that the deterministic coefficient (R

) value of each

simulation has reached above 90%, meaning that all the simulations results are good.

1 INTRODUCTION

When glucose is infused intravenously, insulin

secretion is stimulated much less than it is when

glucose is taken orally so as to result in similar

glucose concentrations. The incretin hormones

function has the key physiological impact on glucose

homeostasis after oral glucose. The secretion of

insulin after ingestion of glucose compared to the

isoglycemic intravenous glucose challenges is called

the incretin effect. This effect is estimated to be

responsible for 50 to 70% of the insulin response to

glucose. Glucagon-like peptide-1 (GLP-1) and

glucose-dependent insulinotropic polypeptide (GIP)

are the two intestinal insulin-stimulating hormones

caused mainly the incretin effect. Incretins are

peptide hormones that originate in the gut and

increase the effectiveness of insulin secretion after

glucose ingestion in a glucose-dependent manner.

GLP-1 and GIP are the two dominant incretin

hormones. They play an essential role in maintaining

normal glucose homeostasis and in particular

postprandial glucose levels (Tura et al. 2014; Kaur

& Gautam 2006; Vilsbøll & Holst 2004; Ahren

2011).

In subjects with type 2 diabetes mellitus

(T2DM), the incretin effect is either greatly impaired

or absent, and it is assumed that this could contribute

to the inability of these subjects to adjust their

insulin secretion to their needs. In studies of the

mechanism of the impaired incretin effect in T2DM

subjects, it has been found that the secretion of GIP

is generally normal, whereas the secretion of GLP-1

is reduced, presumably as a consequence of the

diabetic state. It might be of even greater importance

that the effect of GLP-1 is preserved whereas the

effect of GIP is severely impaired (Michaliszyn et al.

2014).

Two types of minimal model for the glucose and

insulin dynamics and incretin effect have been

developed according to the different routes of

glucose administration: intravenous glucose

tolerance test (IVGTT) and oral glucose tolerance

test (OGTT). The IVGTT-based minimal model

consists of glucose and insulin subsystems, where

insulin in the plasma compartment passes the en-

dothelium and enters a remote interstitial

compartment to exert insulin action. However, the

intravenous administration of glucose is far from the

physiologic way of glucose intaking, which

necessitates further improvement of the model by

incorporating the physiology of oral glucose intake.

The most common oral administration test, the

standard method to determine glucose tolerance

status, is the oral glucose tolerance test (OGTT).

This test is vital for the characterization of the

metabolic syndrome and natural progression from

normal glucose tolerance (NGT) to prediabetes

(impaired glucose tolerance (NGT)) to T2DM. The

280

Kartono, A., Fadila, R., Syafutra, H., Wahyudi, S. and Sumaryada, T.

The Modiﬁed Oral Minimal Model to Know Incretin Hormone Effect in Type 2 Diabetes Mellitus.

DOI: 10.5220/0009009802800286

In Proceedings of the 7th Engineering International Conference on Education, Concept and Application on Green Technology (EIC 2018), pages 280-286

ISBN: 978-989-758-411-4

OGTT-based minimal model, is commonly called

the oral minimal model, can be easily applied to

examine the dynamic physiology of glucose

homeostasis. The oral minimal model can also be

readily adapted to various situations by modifying

its structure (Lim et al. 2016; De Gaetano et al.

2013).

The OGTT test is characterized by uncertainty

both in the amount of glucose absorbed and in its

absorption rate, which results in the time course of

exogenous glucose delivery to plasma. In the

OGTT-based minimal model, the time course of the

rate of appearance (R

) of exogenous glucose in

plasma cannot be specified a priori. Given the

ingested dose, indeed, several factors contribute

toward affecting the R

: the rate of gastric emptying

of ingested glucose, the extent of intestinal

absorption during the intestinal transit, and the

hepatic uptake of portal glucose. Because of these

uncertainties, the R

time course in the OGTT test

has been represented by a piecewise linear function,

with the time of breakpoints assigned and values that

are estimated from the data. The rate of gastric

emptying and the small intestine transit time appear

to be the main factors in determining the glucose R

profile (Salinari et al. 2011; Wilbaux et al. 2017).

Mathematical models of the glucose and insulin

dynamics have been used to represent the rela-

tionship mainly between glucose and insulin. The

glucose profiles during OGTTs are difficult to

predict due to highly variable factors such as the rate

of glucose absorption from the intestine and the

temporal delay of insulin action. Furthermore, the

incretin hormones have been recognized as a major

insulin secretory stimulus after an oral glucose load.

In the present model, the modified oral minimal

model that includes the incretin effect has been

developed to describe the glucose, insulin and

incretin hormone dynamics during the OGTT [Kim

et al. 2014; Brubaker et al. 2007; Seike et al. 2011).

This research introduced a modified oral minimal

model based on the incretin mechanisms for

investigating the incretin hormones. This study was

performed in healthy and subjects with T2DM that

the glucose excursion is very similar after ingestion

of 75 g due to an increase in the incretin effect

matching the increased glucose load.

2 MATERIALS AND METHODS

2.1 The Oral Minimal Model

The classic single-compartment IVGTT-based

minimal model was first introduced by Bergmann

and co-author. Minimal model is a one compartment

model, which means that the body is described as a

compartment with a basal concentration of glucose

and insulin. This model contains two minimal

models. The first model describes glucose kinetics,

how blood glucose concentration reacts to blood

insulin concentration and the second model

describes the insulin kinetics, how blood insulin

concentration reacts to blood glucose concentration.

However, the IVGTT used to obtain parameters in

this model is invasive and requires considerable

cooperation on the part of the patient [Bergman et al.

1981; Bergman et al. 1979; Kartono 2013).

The OGTT is a much simpler procedure to

perform, with both decreased invasiveness and

reduced burden on the patient. The OGTT tests are

thus routinely performed in clinical laboratories to

diagnose prediabetes (IGT) to T2DM. The OGTT-

based minimal model is commonly called oral

glucose minimal model resembles the IVGTT-based

minimal model but has a new element, the

gastrointestinal tract, which has as input the oral

dose.

The oral glucose minimal model has two

ordinary differential equations that represent the

changes in plasma glucose and insulin concentra-

tions. Variables and Parameters Of Oral Minimal

Model shows in Table 1. The general model

equations are described as follows (Cobelli et al.

2014):

 

 

 

GptGtXp

tdG





, (1)

 

GtG 

 

   

 

ItIptXp

tdX



, (2)

 

tX

 



















1 i





(3)

others

ittt

8,...,1,





The Modiﬁed Oral Minimal Model to Know Incretin Hormone Effect in Type 2 Diabetes Mellitus

281

Table 1: Variables and Parameters Of Oral Minimal Model.

Symbol

Unit

Notes

G(t)

mg/dL

glucose concentration when t after given input glucose by orally

I(t)

μU/mL

insulin concentration when t after input glucose is given orally

X(t)

min

-1

insulin action gives back glucose concentration to basal level when t

after given input glucose orally

mg/dL

basal glucose concentration before given input glucose orally

μU/mL

basal insulin concentration before input glucose given orally

mg/dL

theoretic glucose concentration in plasma when t equals zero which is

soon after input glucose is given orally

μU/mL

theoretic insulin concentration in plasma when t equals zero, more than

, is soon after input glucose is given orally.

min

-1

is effectivity of glucose, which is glucose absorption without insulin

in tissue.

min

-1

constant of increment of glucose absorption ability, in other words, the

fraction of the velocity of insulin which shows in interstitial plasma

min

-2

(μU/mL)

-1

improving the ability of glucose-dependent insulin absorption in tissue

per unit insulin concentration above basal insulin, in other words,

cleaning fraction insulin from interstitial compartment

(t)

mg. kg

-1

min

-1

the rate of input endogen glucose to the systemic circulation

dL/kg

the volume of distribution of glucose

mg. kg

-1

min

-1

the amplitude of glucose absorption

min

time of glucose absorption

The rate of plasma insulin secretion

concentration (dI/dt) is represented by the pancreatic

insulin secretion R

can be described as the sum of

two components: dynamic insulin secretion (R

) and

static insulin secretion (R

). The insulin circulation

rate of a single-compartment model with a rate

parameter p

(min

-1

) for insulin disappearance is

defined by the following equation:

21 III

RRR 

, (4)

, (5)

The secretion of rapidly releasable insulin stored

in β-cells in response to elevations in the glucose

level is represented by the parameter R

(µU mL

-1

min

-1

) can be written as follow:

if, (6)

where the sensitivity of dynamic insulin

secretion by the β-cells is described by the parameter

(µU mL

-1

dL). The secretion of newly

recruited insulin in response to an elevated glucose

level is represented by the parameter R

can be

written as follow:

if (7)

The sensitivity of static insulin secretion by β-

cells to an elevated glucose level with a time

constant parameter p

(min) is described by the

parameter p

(µU mL

-1

dL min

-1

2.2 The Modified Oral Minimal Model

2.2.1 The Incretin Hormones Model

The GLP-1 hormone is produced by L cell which is

available in ileum and colon, while GIP hormone is

produced by K cell in duodenum tract and jejunum.

Secretion of these hormones can increase quickly

after subjects consume food which contains

carbohydrate and lipid. Carbohydrate and lipid can

stimulate the secretion of incretin hormones. The

GLP-1 hormone can stimulate β-cells in the pancreas

to produce insulin and directly inhibit glucagon

secretion, so that blood glucose level decreased.

To design the incretin hormones model, it is

needed to know how glucose enters the body.

Glucose entry into the body during OGTT involves

two main compartments. These compartments are

the gastrointestinal (GI) tract and mesenteric

circulation. In the GI tract, liquid glucose is very

 

  

IbI

RItIp

tdI



 













tdG

 





tdG

  

 

















bII

GtGpR

 

GtG





EIC 2018 - The 7th Engineering International Conference (EIC), Engineering International Conference on Education, Concept and

Application on Green Technology

282

rapidly emptied from the stomach into the

duodenum. The equation to describe the rate of

delivery of glucose to the duodenum the duodenum

(Duod

) with a 75 g load OGTT is described by:













,0307.01014.5

Duod

75max







. (8)

where t

max100g

= 166.1 min, t is time since the

start of the simulation, and t

max

is the t-intercept of

the straight lines. The OGTT simulations began after

5 min of basal metabolism; that t

max

is used only to

refer to the intercepts of these curves and nowhere

else in the model; and that these straight lines are

simple mathematical descriptions of experimentally

measured glucose delivery rates (mmol min

-1

Duodenal delivery rates cannot be negative.

Furthermore, when the units are converted from

mmol back to grams:

27.414

1.166





dtDuod

mmol = 74.58 g, (9)

thus retrieving the ingested doses of glucose.

Duod

is used subsequently to determine the rate of

release of incretins

These equations determine the rate of secretion

of incretin hormones and the rate of absorption of

glucose into mesenteric circulation. There is an

additional ordinary differential equation to describe

more about incretin concentration which is written

below:

 

tInckDuodk

tdInc

Inc



, (10)

where

bInc

VInckRa



and basal plasma levels of

incretins (Inc

) were 200 ng/dL. The equations

above describe the concentration of incretin

hormones is based on GIP concentration during the

OGTT. Ra

Inc

is the basal rate of appearance of the

incretin, Inc

is the basal concentration determined

by the average fasting total GIP of all subjects and k

is the degradation which gives effect to the rate of

incretin hormones concentration. Parameter k

is the

rate of appearance of incretin due to Duod

2.2.2 Modified Insulin Secretion Model

The equation of insulin concentration kinetics in this

present model is from Brubaker and co-authors

which has some absorption parameters (Brubaker et

al. 2007). These absorption parameters give effect to

insulin kinetics in the body. The equation from

Brubaker and co-authors is modified by adding

constant R

from Seike et al. (2011). These are the

modified equations which describe insulin secretion

in pancreas and rate of insulin concentration:

 

     

RtIktIncktGk

tdI





987

, (11)

  

 















bIIb

GtGpRInc

 





(12)

Parameter k

is the rate of appearance of insulin

due to G(t), k

is the rate of appearance of I(t) due to

Inc(t), k

is the measure of degradation/clearance of I

and β is effects of additional regulators of I(t) on

insulin appearance.

3 RESULTS AND DISCUSSION

The simulation of the modified oral minimal

model is made using the Matlab language

programming. It is needed to simplify the numerical

calculation and simplify to make the graphic of the

glucose, insulin, and incretin hormones

concentration. Numerical analysis is also needed to

be done because it is very difficult to solve these

analytical equations. In this research, the

mathematical model used is the ordinary differential

equation (ODE). Since this research using ODE, the

most accurate numerical method is Runge-Kutta 45

or ode45. Analysis deterministic coefficient (R

) is

needed to figure out the validation value between

data from the model to data from the experiment

which is formulated by:

 





















tyy



, (13)





















SST



, (14)

SST

1

, (15)

where y

is experiment data which contains deviation

standard σ, y(t

) is modeling data, N is the data and

is the average value of the sum from experimental

data and model data.

The Modiﬁed Oral Minimal Model to Know Incretin Hormone Effect in Type 2 Diabetes Mellitus

283

These data were taken from 120 Japanese

subjects who were listed at Tokyo University

Hospital, Japan.

The potential subjects (male and female) were at

age 40 to 65 years. All subjects were given 75 g

sampled OGTT. After all, subjects fasted for

approximately 10 hours, their venous sample blood

was taken at 0 min. These samples were taken to

measure glucose, insulin, and incretin hormones

concentration during 10, 20, 30, 60, 90, 120, 150,

180, 240 and 300 min (Møller 2012).

Normal Japanese subjects.

T2DM Japanese subjects.

Normal Japanese subjects.

T2DM Japanese subjects.

Normal Japanese subjects.

T2DM Japanese subjects.

Figure 1: Simulation results of the glucose, insulin and incretin hormone concentration of the normal and

T2DM Japanese subjects and the deterministic coefficient (R

) is = 91%.

EIC 2018 - The 7th Engineering International Conference (EIC), Engineering International Conference on Education, Concept and

Application on Green Technology

284

The simulation results of the modified oral minimal

model for glucose, insulin and incretin hormones

concentration of normal Japanese subject data show

good results. This can be proved by the deterministic

coefficient R

from the simulation. The results of

fitting data which is represented by R

value are

more than 90%. The parameter G

value is 85 mg/dl.

This value shows that right value for the normal

subject because according to statistics data, the G

value for the normal subject is 80-100 mg/dL. In the

rate of glucose concentration, as seen in Figure 1,

the glucose level slowly increased, then it decreased

after 60 min. In the normal subject, the ability of

insulin to accelerate the losing glucose from plasma

is good. As seen in Fig. 1, the rate of insulin

concentration increased until 60 min, after an oral

glucose load. This research also determined the

parameter basal incretin hormones Inc

. The

parameter Inc

value is 300 ng/dL, according to

statistical data for the normal subject, the Inc

value

is more than 200 ng/dl. This parameter value can be

concluded as the normal subject because it is more

than 200 ng/dL.

The simulation results of the modified oral

minimal model for glucose, insulin and incretin

hormones concentration of T2DM Japanese

subjects’ data show also good results. The

deterministic coefficient results of fitting data which

is represented by R

value is more than 90%. The

parameter G

value is 140 mg/dL. This value shows

that right value for the T2DM subject because,

according to statistics data, the G

value of the

T2DM subject is 130-150 mg/dL. In this simulation,

the G

value is even higher than the previous

simulation results of the normal subject. This results

can show that the absorption of glucose in the body

for normal subjects is the highest than the T2DM

subjects.

The parameter Inc

value of T2DM subjects is

115 ng/dL, according to statistical data for the

T2DM subject, the Inc

value is less than 200 ng/dL.

So, this parameter value can be determined as

T2DM subjects. This parameter Inc

can show that

the incretin hormones response is even slower than

the normal subject. The summary of values and units

for constants used in the present model shows in

Table 2. As seen from Figure 1, the rate of incretin

concentration slowly increased from 0-120 min,

after input glucose orally, that it increased by 150

min and kept decreasing until 300 min.

Table 2: Values and units for constants used in the present

model.

Constant

Value

Units

85-140

mg/dL

10-11

µU/dL

Inc

115-300

ng/dL

0.01-0.05

min

-1

0.002-0.005

min

-1

0.0015-0.0025

min

-1

2.5-3.0

ng dL

-1

mmol

-1

0.018-0.024

min

-1

0.009-0.03

mU min

-1

mmol

1.3

-0.3

0.003-0.004

mU min

-1

1.3-1.6

min

-1

4 CONCLUSIONS

Kinematics of glucose, insulin and incretin

hormones concentration in the blood can be

described using a mathematical model. In this study,

the present model used is the modified oral minimal

model. Oral minimal model is modified to be able to

describe the rate of incretin hormones concentration

so that the level of secretion of insulin in the body

can be seen. Incretin hormones serve as a primary

stimulus of insulin secretion if there are incoming

glucose orally. The OGTT test data can show two

parameter values of the Incb and Gb from fit

between the present model and the OGTT

experimental results. Subjects with T2DM have the

highest basal glucose value parameter (G

), while

normal subjects have G

values lower than the

T2DM subject. Based on this study, subjects with

T2DM have an incretin hormone basal (Inc

)

parameter is the smallest (Inc

= 115 ng/dL). Normal

subjects have Inc

value exceeding 200 ng/dL.

ACKNOWLEDGEMENTS

This study was supported by grants from the

Direktorat Riset dan Pengabdian Masyarakat,

Direktorat Jenderal Penguatan Riset dan

Pengembangan, Kementerian Riset, Teknologi dan

Pendidikan Tinggi, Indonesia sesuai dengan Surat

Perjanjian Penugasan Pelaksanaan Program

Penelitian Nomor: 1624/IT3.11/PN/2018 tanggal 21

Februari 2018.

The Modiﬁed Oral Minimal Model to Know Incretin Hormone Effect in Type 2 Diabetes Mellitus

285

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Application on Green Technology

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