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Far East Journal of Mathematic11I Sciences (FJMS) © 2013 Push pa Publishing House , Allahabad. l!1dia Available online al hnp://pphmj .com/joumalsifJms .htmS ecial Volume 2013, Pan VI. Pages 587-5% ..
.p d I . l "o1111111 S··1 Info S<i F111an<ll1I Mana" & 111111. s,1.)
(Devote 10 an1c ..:son · · .. ·· · · "
MA TH EMA TICAL MODEL OF THE INTRA VENOUS
GLUCOSE TOLERANCE TEST USING
MODIFIED MINIMAL MODEL
Nurullaeli, Agus Kartono and Kiagus Dahlan
Laboratory for Theoretical and Computational Physics
Physics Department
Faculty of Mathematical and Natural Sciences
Bogor Agricultural University (!PB) Jalan Meranti, Building Wing S, 2nd Floor Kampus IPB Darmaga, Bogor 16680, Indonesia
e-mail: aguskartono70@yahoo.com
Abstract
The modified minimal model of glucose and insulin plasma levels .is
I ｾ､＠ to analyze the results of glucose tolerance tests 111
common y ｵｳｾ＠ · . .
humans . In this paper. a mathematical model for dcscnh111g the
lucose infusion rate is introduced . The modified minimal model was
ｾｳ･､＠
to study a few sets of published data including healthy humansand type 2 diabetes . The gluc0se-i nsulin system is a dynamic
integrated physiological system and gcneratcs the rcal ｯｰｴＱＱｭｾ･､＠
model parameters from the experimental data using the 111od1hed
· · 1 d ·1 ·1·hc avcra,,ed ｒｾ＠ va lue between measured and
m11111na mo c . o
calculated plasma concentrations is 0.977 . which indicates excellent
agreement.
Received : October 26, 20 13 ; Accepted : Nov em her 22. 2013
2010 Mathematics Subject Classification: 37-XX. 37M05 .
Keywords and phrases: modified minimal model, intravenous glucose tolerance test.
588 Nurullaeli, Agus Kartono and Kiagus Dahlan
Introduction
Human bodies need to maintain a glucose concentration level in a narrow range (70-110 mg/di or 3.9-6.04 mmol/l after overnight fast) . lfone's glucose concentration level is significantly out of the normal range, then this person is considered to have a plasma glucose problem: hyperglycemia or hypoglycemia. Diabetes mellitus, or simply, diabetes, is characterized by high blood glucose levels resulted from defects in the glucose-insulin endocrine metabolic regulatory system, in which either the pancreas does not
ｲ･ｬ･ｾｳ･＠ insulin or the cells do not properly use insulin to uptake glucose.
. Diabetes mellitus has become an epidemic disease in the sense of life st)rle. To diagnose whether or not an individual subject is already a diabetic or has the potential to develop such disease, the so-called metabolic portrait, including the insulin sensitivity index and glucose effectiveness, of the subject needs to be sketched. To this end, several glucose tolerance tests have been developed and applied in clinics and experiments. The fundamental idea of such tests is to examine the response of insulin, called insulin sensitivity,
after a large amount of glucose is infused into one's body . A commonly used protocol is the intravenous glucose tolerance test (IVGTT).
Type 2 diabetes (formerly called non-insulin-dependent diabetes mellitus, NIDDM, or adult-onset diabetes), a more widespread metabolic disorder, is primarily characterized by insulin resistance, relative insulin deficiency and hyperglycemia. Some cases of type 2 diabetes also appear to be an autoimmune disease where the immune system attacks the P-cells, decreasing the function of producing insulin, while other type 2 diabetes cases may simply result from excessive body weight that strains the ability of
the P-cells to produce sufficient insulin.
In the procedure of IYGTT, overnight fast is required for the subject, and then the subject is given a bolus of glucose infusion intravenously, for example, 0.33g/kg body weight or 0.5g/kg body weight of a 50% solution, and is administered into an antecubital vein in approximately 2 min. To observe the metabolic regulation between the glucose and insulin, within the next 180 min, the plasma glucose and serum insulin of the subject
Mathematical Model of the Intravenous Glucose Tolerance Test . . . 589
arc sampled frequently at the time marks 2', 4', 6', 8', JO', 12', 14', 18', 21', 24', 30', 35', 40', 45', 50', 60', 70', 80', 90', 100', 120', 140', 160' and 180'. According to the rich information constituted in the sampled data, appropriate analysis can reveal the metabolic portrait.
One popular approach of the analysis is as follows : (I) fonnulate or choose a well-formulated kinetic model based on physiology ; (2) estimate the parameters of the IVGTT model with experimental data, and then (3) the parameter values are used to obtain physiological infonnation , for example, the insulin sensitivity and glucose effectiveness.
The intravenous glucose tolerance test (IVGTT), as an important experimental procedure for the estimation of glucose effectiveness and immlin sensitivity, has been widely used, because it is relatively easy to measure, and its analysis generates a lot of information. The test includes injecting a glucose bolus and then collecting a set of plasma glucose and insulin samples over a period of 3-5 h. For diabetic patients, it is sometimes necessary to have an exogenous insulin infusion because of insufficient endogenous insulin secretion despite stimulation by the glucose injection.
The glucose-insulin system as an in\egrated dynamic system m physiology, however, should be described mathematically as a whole. When an integrated dynamic system is split into two interacting sub-systems, and then, the system parameters are optimized by fitting the measured data separately, the generated parameters cannot be thought as optimal ones for the whole system. In the glucose-insulin physiological system both glucose and insulin have feedback effects on each other via pancreatic response and stimulation. It is worth to obtain optimized model parameters from measured plasma glucose and insulin concentrations simultaneously in the time course by using the regression method. This single-step parameter-fitting process could generate a real optimal approximation to the dynamical integrated glucose-insulin system without losing the interaction infom1ation implicitly contained in the measured concentration profiles.
a measure of the dependence of fractional glucose disappearance. The latter term represents a measure of the fractional ability of glucose to lower its own concentration in plasma independent of increased insulin at basal insulin, to normalize its own concentration within the extracellular glucose pool. This normalization occurs after glucose administration and may involve glucose-mediatcd inhibition of hepatic glucose output and/or accelerated glucose disposition.
The Modified Minimal Model from the IVGTT
The dynamic insulin and glucose responses to glucose injection were analyzed as previously described lo yield the individual parameters of the minimal model. The modified minimal model, which we proposed as the simplest representation that can account for the dynamics of glucose during the IVGTT, is described by the following differential equations :
dG(t)
-;ff= P1(Gh
- G(t))- X(t)G(t), G(to)=Go.
(1)dX(t)
-;ff=
-p2X(t) + p3(/(t)-Jb), X(to)=
0, (2)dl(t)
(ff= y(G(t) - a,,)1 - k(/(1)- 1,,) if G(t) > Gb, 1(10)
=
10, (3)､ｾｾｴＩ＠
=
-k(J(t)-1,,) if G(t) < G,,, /(10 )=
10 ,(4)
n
L(Y;
-
5';)2R2
=
l - ..:.i'--= l:...._ _ _ _n (5)
L(Y;
-
y;)2
i =I
where G(t) and J(t) are glucose and insulin concentrations measured in
plasma. X(t), insulin action, is proportional to insulin in a compartment
remote from plasma, and Pl, P2 and p3 are model parameters. Parameter
Pi represents the fractional ability of glucose to lower its own concentration
I
,.
I
JI
I!
!;
I
I
I.
II
I
I i
I I
I
I:
i:
·I
1.
in plasma; p2 and p3 are the fractional transfer coefficients of insulin into
and out of the remote compartment where it acts to accelerate glucose uptake
or decrease glucose production. G0 is the glucose concentration one would
obtain immediately afler glucose injection if there were instantaneous mixing
of glucose in the extracellular fluid compartment. Gh and 1,, arc the
pre-injcction values of glucose and insulin. Insulin sensitivity, represented as the
insulin sensitivity index, S / is calculated as p3 / P2 . Glucose effectiveness is
the ability of glucose, at basal insulin, to normalize its own concentration.
This latter parameter is represented as SG and is equal to p1 as defined in
equation ( 1 ), where
.v
is the prediction of the non-linear least-squares fit.The residuals between the best-fit curve and the data, Y; -
.v.
Glucose and insulin data [G(t} and /(1)) arc submitted to the program modified minimal model, which estimates the model parameters from the real data. This program is based on the non-linear least-squares estimation
method of Marquardt. /(1) is submitted to the program modified minimal
model, which predicts a glucose time course, G(t), which fits data G(t) as
closely as possible in the least-squares sense. In the course of the titting,
the model yields X(t), an estimate of X(t), as well as estimates of
parameters Pl· P2· p3 and Go. S1(P3/p2), S<;(pi). and the coefficient of
determination R2 are calculated from model parameter estimates.
Fitting the modified minimal model of glucose disappearance to standard
IVGTT data yields parameters representing glucose disappearance: S 1 and
SG. The fonncr tcnn is a measure of the effect of the dynamic insulin
response to augment glucose decline; the latter tcnn represents the ability of glucose per se (at basal insulin) to effect its own normalization . Parameter
s
1 , varied from 5.0 x ＱＰ Ｍ ｾ＠ to 2.2 x 10-.l (µU/Lr1min 1 and Sc; averaged 1.2x 10-3 to4 .5 x 10 2 min·1•
To examine the veracity of the Sc; value obtained from analysis of the
592 Nurullaeli, Agus Kartono and Kiagus Dahlan
Sa from experiments. In accordance with the modified minimal model, the
rate constant for the decline in glucose when the dynamic insulin response is
suppressed is equal to Sa (Sa = half-time/0.693) . Numerical Results and Discussion
In order to exemplify the computation of the proposed stability criteria, we considered sets of parameter values consistent with adaptation to data
from actual IVGTT experiments. Simulations are performed by using MATLAB ordinary differential equation solver ode45 . Experimental data published in [ l]. We obtained these data by manually measuring the data in related figures in [ l].
In the original data sets, the time mark 01 is the starting time of the bolus
glucose infusion. We further depart from published work in hypothesizing that glycemia increase between the start of infusion to the first observed glycemia point follows an ascending linear ramp. Moreover, in observing
that indeed the bolus glucose infusion belongs to the initial condition of the ordinary differential equation model ( 1 )-(4) .
Subject 6 Subject 6
"'°
m '
""
...
-""'
ｾ＠ 300 0 s
§ 1<1(1
ij 120
(>
100
IO ' -... cl U . o 0 0 ,
0
"
f ...
0 0ｾ＠ 200
\
..
r
- 100 100
..
0 D"
0-·
.. 0ｾ＠ 20 .. "' .. ｾ＠ ｾ＠ ｾ＠ ｾ＠ ｾ＠ "" 0 0
,.
.. .. ..
""'
120 140,
..
110 200 [image:5.843.471.776.62.208.2] [image:5.843.34.789.71.546.2]Tlmo(Mo) Tlmo(mo)
Figure 1. Profiles of subject 6 in [ l] produced by equations ( 1)-(4) with
parameters k = 0.12, y
=
0.015, Gb=
80 [mg.dr1] , lb = 40 [µU /ml],p2
=
0.06 [min-1], S1
=
6.5 x 10-4 [µu -1.ml.min-1], Sa=
0.025 [min- '],lo = 230 [µU/ml] and G0 = 500 [mg.dr'].
Mathematical Model of the Intravenous Glucose Tolerance Test . . . 593
1 '50 ﾷ ｾ＠
=
I ｾ＠ ' ..'
ｾ ＢＢ＠
::
ｾ＠
\ｾ＠ \
(; 150
100
\·
'°•!--ＭＮＮＬＬＬｾｾ＠ .. - .. ｾｾ＠ .. ｾ ＱＰＰ ｾＱｾ ＬＮＬ ＭＭＧＭ ＬＮＬ Ｍ .... ---;;o- -""'
Tlmo(mo) ,
..
,.,
"'
"
,.
;.:.
0 Subject 7.
\ II ,,
ｾ＠ j
..
100'"'
1 . . 160.
..
""'
Tlmo(mo)
Figure 2. Profiles of subject 7 in f I] produced by equations (I)-( 4) with
parameters k = 0.16, y = 0.0125, G"
=
90 fmg .dl 1]. I"
=
30 fµU /ml].P2
=
0.06 [min-1] , S1=
5.07 x 10-4 fµU 1.ml.min 1], Sc;=
0.035 fmin 1].lo
=
180 fµU /ml] and G0=
315 fmg.dl 1 ].""
Subject I'""
Subl•C1.""'
1000I"'° \
i ｾ＠ 000,,
g
I i""
I !
ｾ＠ 1"1
\
!,,,
0 I .s J
.,,
,. •\\' I 1\
J
""'
'<1ｾ ｉ＠
\,·--.
..
..
ｾ Ｎ＠
" r
I
..
0,.
-"
..
..
...
.
,.
140...
1!0""
0 ;.:.
..
..
...
,,.
.
..
...
11() _j"'°
Time(-)
,
...
,_,
Figure 3. Profiles of subject 8 in [ 1] produced by equations ( 1)-(4) with
parameters k
=
0.35, y = 0.18, G11=
80 !mg.di 11.
111
=
40 lµU /mll .P2 = 0.65 [min-1] . S1 = 2.5x10-4 lµU 1.ml.min
' I.
Sc;= 0.08 lmin'I.
100
..
(l v u 080
0 20 40 00 80 100 120 1'90 180 180 200
nN(nin)
Subject 13
ｾＮＭＭＭｾｾｾｾＮＭＭＭ ﾷｾｾｾｾ ｾｾｾ＠
()
ｴ
Ｑｾｾ
Ｑ
Ｂ＠
\- u
:§ HXI c.
,; Oo I')
"
ｾ＠
,, IJ • ﾷＭｾＭ _q ｾ＠ 0 iJ
..
ｾｾｾｾｾｾｾｾｾｾｾｾ｟Ｌ＠0 ｾ＠ ｾ＠ .. .. ｾ＠ ｾ＠ ｾ＠ ｾ＠ ｾ＠ ｾ＠
Tlnw(mln)
Figure 4. Profiles of subject 13 in [ 1] produced by equations ( 1)-(4) with
parameters k
=
0.45, y=
0.027, Gb=
75 [mg.dr1], lb=
20 [µU/ml],p2
=
0.45 [min-1], S1=
7.5 x 10-3 [µU-1.ml.min-1], S0=
0.026 [min-1],10
=
250 [µU/ml] and G0=
185 [mg.dr1].100
0
' -•. u
'-..0 fl .JJ ••• Q .u
0 0
80 10 100 120 140 180 180 200
Tlmo (ninl
'
1000
Figure S. Profiles of subject 27 in [ l] produced by equations ( l )-( 4) with
parameters k
=
0.345, y=
0.037, Gb=
75 [mg.dr1], lb=
30 [µU/ml], p2=
0.45 [min-1], S1=
3 .5 x 10-4 [µU-1.ml.min-1], Sa=
0.033 [min-1],10 = 1080 [µU/ml] and G0 = 345 [mg.dr1].
The optimized parameters obtained by titting the modified min
model to the experimental data of the live subjects. The averaged R2 v
including glucose and insulin concentrations for these four cases is 0.
this means that the modified minimal model was excellent. This coul.
explained by the increased flexibility of the modified minimal me
because of the assumption that the insulin decay rate is not always a ·
order process .
In the standard IVGTT, alier an injection of glucose bolus, the b
glucose reaches a higher concentration and shows an apparent d
immediately to the basal line in 1 hour. The corresponding in concentration stimulated by the injected glucose raises to form a peak,
an approximate exponential decay alierwards, and finally a secondary
appears.
Conclusions
The glucose-insulin time course in IV<iTT is a very complex pre
influenced by many factors. The interaction of glucose-insulin during IV
shows a great difference between subjects. The proposed modified mi1
model combined with the single-step filling process can be used to sim
the plasma glucose and insulin profiles subjects . The R2 va lue of ( between experimental and calculated time courses indicates the si mul
results are excellent. It also indicates that the modification to the c. minimal model improves the '!lodcl's llexibility. The modifications in·
an assumption that the decay rate of the plasma insulin is not alw;
first-order process. The single-step fitting process applied to the moc
model generates a set of real optimal model parameters to the glucosc-ir
system as a whole dynamic integrated physiological system .
Acknowlcdgmcnl
The authors would like to acknowledge fonding from the DANA ;
[image:6.843.45.802.15.554.2]596 Nurullaeli, Agus Kartono and Kiagus Dahlan
References
[I) J . Li, M. Wang, A. D. Gaetano, P. Palumbo and S. Panunzi, The range of time
delay and the global stability of the equilibrium for an IVGIT model, Math. Biosci. 235 (200 I}, 128-137.
(2) Y. Zheng and M. Zhao, Modified minimal model using a single-step fitting
process for the intravenous glucose tolerance test in type 2 diabetes and healthy humans, Comput. Meth. Prog. Biomed. 79 (2005), 73 -79.
(3
I
A. Kartono, Modified minimal model for effect of physical exercise on insulinsensitivity and glucose effectiveness in type 2 diabetes and healthy human, Theory Biosci. 132 (2013), 195-206.
[4J A. D. Gaetano and 0. Arino, Mathematical modeling of the intravenous glucose
tolerance test, J. Math. Biol. 40 (2000), 136-146.
FAR EAST JOURNAL OF MATHEMATICAL SCIENCES (FJ
Editorial Board
Gunjan Agrawal, India Natig M. Atakishiyev, Mexico Antonio Carbone, Italy Hasan Coskun, USA Zhenlu Cui, USA Manav Das, USA Shusheng Fu, China Wei Dong Gao, China Toshio Horiuchi, Japan Lisa M. James, USA Koji Kikuchi, Japan
Victor N. Krivtsov, Russian Federation Alison Marr, USA
Jong Seo Park, South Korea Alexandre J. Santana, Brazil
A. L. Smimov, Russian Federation Chun-Lei Tang, China
Carl A. Toews, USA Vladimir Tulovsky, USA Qing-Wen Wang, China Peter Wong, USA Pu Zhang, China
George S. Androulakis, Greec Carlo Bardaro, Italy
Yong Gao Chen, China Claudio Cuevas, Brazil Masiina Daros, Malaysia Massimiliano Ferrara, Italy Salvatore Ganci, Italy
Demetris P. K. Ghikas, Greec• Jay M. Jahangiri, USA Young Bae Jun, South Korea Hideo Kojima, Japan Yangming Li, China
Manouchehr Misaghian, USA
Cheon Seoung Ryoo, South K
K. P. Shum, China
Ashish K. Srivastava. USA E. Thandapani, India B. C. Tripathy, India Mitsuru Uchiyama, Japan G. Brock Williams, USA Chaohui Zhang, USA Kewen Zhao, China
![Figure 3. Profiles of subject 8 in [ 1] produced by equations ( 1)-(4) with](https://thumb-ap.123doks.com/thumbv2/123dok/278556.407926/5.843.34.789.71.546/figure-profiles-subject-produced-equations.webp)
![Figure 4. Profiles of subject 13 in [ 1] produced by equations ( 1)-(4) with](https://thumb-ap.123doks.com/thumbv2/123dok/278556.407926/6.843.45.802.15.554/figure-profiles-subject-produced-equations.webp)
