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Assessment of minimal model applicability to longitudinal studies

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Assessment of minimal model applicability to longitudinal studies

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Content ASSESSMENT OF MINIMAL MODEL APPLICABILITY TO
LONGITUDINAL STUDIES
by
ANTONIOS E. PANTELEON
A Thesis Presented to the
FACULTY OF THE GRADUATE SCHOOL
UNIVERSITY OF SOUTHERN CALIFORNIA
In Partial Fulfillment of the
Requirements for the Degree
MASTER OF SCIENCE
(Biomedical Engineering)
August 2000
Copyright 2000 Antonios E. Panteleon
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UMI Number: 1407920
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This thesis, written by
AtiToM Ios £ . ?ANT£L60fJ
under the guidance of his/her Faculty Committee and
approved by all its members, has been presented to and
accepted by the School of Engineering in partial
fulfillment of the requirements for the degree of
M A S T 6 ^ O f
aw BlOMf-DlCAL 6
Date: - - M U G + M S l -----------------------------------------
Faculty Committee
Dft.
Chairma,
_ _ P . .P . ^ yJP—- D .’ . £ L 9.
D B . Y f \ S S t L ) $ _ ^ L A P M A R a l5
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TABLE OF CONTENTS
LIST OF FIGURES................................................................................. iii
LIST OF TABLES................................................................................... vi
A.BACKGROUN D ................................................................................. 1
I. INTRODUCTION............................................................................... 1
II. ROLE OF GLUCOSE EFFECTIVENESS IN THE
DETERMINATION OF GLUCOSE TOLERANCE.......................... 6
III. CONCERNS ON MINIMAL MODEL......................................... 10
IV: APPLICATION OF THE MINIMAL MODEL TO
LONGITUDINAL STUDIES................................................................ 12
B. CASE STUDY 1.................................................................................. 15
I. METHODS........................................................................................... 15
II. RESULTS............................................................................................ 18
III. DISCUSSION.................................................................................... 27
C. CASE STUDY II................................................................................ 32
I. METHODS........................................................................................... 32
II. RESULTS............................................................................................. 33
III. DISCUSSION.................................................................................... 41
D. CASE STUDY III.............................................................................. 44
I.METHOD S............................................................................................. 44
II. RESULTS............................................................................................. 45
III. DISCUSSION.................................................................................... 52
E. CONCLUSIONS AND DISCUSSION............................................ 55
REFERENCES......................................................................................... 59
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LIST OF FIGURES
Figure 1: Typical glucose and insulin profiles after a FSIGT Page 2
Figure 2: Schematic representation of the minimal Model Page 3
Figure 3: Typical glucose and insulin profiles after a modified
FSIGT Page 5
Figure 4: Organizational chart for the simulated longitudinal
study. For every one of the 100 subjects 2 FSIGTs are generated
and both minimal model and LongMod are tested on these 200
datasets Page 15
Figure 5: Insulin profiles used for the generation of the data
sets Page 16
Figure 6: Regression analysis between theoretical Si and
Minimal Model calculated insulin sensitivity index, for the no
noise c a se Page 19
Figure 7: Regression analysis between theoretical Si and
LongMod Si for the no noise case Page 20
Figure 8: Regression analysis between theoretical ASi and
Minimal Model or LongMod ASi in the case of absence of
measurement error Page 21
Figure 9: Regression analysis between theoretical Si and Minimal
Model calculated insulin sensitivity index, for the 2%
measurement error case..................................................................... Page 23
Figure 10: Regression analysis between theoretical Si and
LongMod Si (2% measurement error case) Page 24
Figure 11: Regression analysis between theoretical ASi and
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Minimal Model and LongMod ASi ? for the 2% noise case Page 25
Figure 12: Regression analysis between theoretical Si and
Minimal Model calculated insulin sensitivity index, for the 4%
measurement error case Page 26
Figure 13: Regression analysis between theoretical Sf and
LongMod Si (4% measurement error case) Page 27
Figure 14: Regression analysis between theoretical ASi and
Minimal Model or LongMod ASi, for the 4% noise case................ Page 29
Figure 15: Average pre- and post-treatment insulin profiles used
for the generation of data sets Page 33
Figure 16: Regression analysis between theoretical Si and
Minimal Model Si with no measurement error superposed to the
data........................................................................................................ Page 34
Figure 17: Regression analysis between theoretical Si and
LongMod Si with no measurement error superposed to the data... Page 35
Figure 18: Regression analysis for ASiin the no noise scenario.... Page 36
Figure 19: Regression analysis between theoretical Si and
Minimal Model Si with 2% measurement error superposed to the
data........................................................................................................ Page 37
Figure 20: Regression analysis between theoretical Si and Page 38
LongMod Si with 2% measurement error superposed to the
data........................................................................................................
Figure 21: Regression analysis for ASi in the 2% noise
scenario................................................................................................. Page 39
iv
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Figure 22: Regression analysis between theoretical Si and
Minimal Model Si with 4% measurement error superposed to the
data Page 40
Figure 23: Regression analysis between theoretical Si and
LongMod Si with 4% measurement error superposed to the
data........................................................................................................ Page 41
Figure 24: Regression analysis for ASi in the 4% noise
scenario Page 42
Figure 25: Regression between theoretical Si and minimal model
Si for the no noise case Page 46
Figure 26: Regression between theoretical Si and LongMod Si for
the no noise case................................................................................. Page 47
Figure 27: Regression analysis for ASi in the 0% noise
scenario Page 48
Figure 28: Regression between theoretical Si and minimal model
Si for the 2% noise case Page 49
Figure 29: Regression between theoretical Si and LongMod Si for
the 2% noise case................................................................................ Page 50
Figure 30: Regression analysis for ASi in the 2% noise
scenario................................................................................................. Page 51
Figure 31: Regression between theoretical Si and minimal model
Si for the 4% noise case...................................................................... Page 52
Figure 32: Regression between theoretical Si and LongMod Si for
the 4% noise case.................................................................................. Page 53
Figure 33: Regression analysis for ASi in the 4% noise
scenario................................................................................................. Page 54
v
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LIST OF TABLES
Table 1: Mean and standard deviation of the parameters used for
the generation of the synthetic data sets............................................ Page 17
vi
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A. BACKGROUND
/ . INTRODUCTION
Historically, the study of the pathogenesis of diabetes was in the
traditional pattern of endocrinology. During this intense effort to discover
the etiology of the disease and decode the labyrinth of underlying
physiological phenomena, a big number of methods have been proposed in
order to quantify the severity of the disease and monitor the condition of
patients.
The discovery of insulin has been a hallmark for the study of diabetes.
The vital role of this hormone has been studied extensively and one of the
many findings was the identification of an unexpected variability among
diabetics in the ability of insulin to normalize the blood sugar level. This
finding triggered researchers to move towards an objective method for the
quantification of insulin sensitivity.
Among the most established and widely used experimental techniques
for the assessment of insulin sensitivity one can refer to the glucose clamp,
the Oral Glucose Tolerance Test (OGTT) and the Frequently-Sampled
Intravenous Glucose Tolerance Test (FSIGT) (1).
1
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The FSIGT (Figure 1) has been specifically designed to extract
insulin sensitivity from a dynamic response to a glucose injection, which
acts as a stimulus. Glucose achieves an elevated value and then
G lu c o se Insulin
40
E
3
a
c
144 180 36 72 108
T im e (m in )
300
240
"I 180
E,
a )
( / >
o
o
| 120
60
180 72 108
T im e (m in )
144 36
Figure 1: Typical glucose and insulin profiles after a FSIGT
begins to decline. Rapidly after the glucose injection, an abrupt insulin
response takes place. The insulin acts to accelerate the glucose decline at a
rate faster than that expected in insulin’s absence. Therefore, the extent to
which a given plasma insulin response accelerates the decline of glucose
after injection must, in some sense, be reflective o f insulin sensitivity. In
order to analyze the relationship between the pattern of insulin response and
2
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the rate of glucose decline the Minimal Model (2) has been proposed and
widely accepted. The Minimal Model (Figure 2) assumes a single-
compartmental distribution of glucose kinetics and also represents the
insulin action by the use of a “remote” compartment X. The model is
differential, in that it accounts for the
HGO(t)
Plasma Insulin
Insulin
Effect
►
Figure 2: Schematic representation of the minimal Model
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effect of an increment of plasma insulin to enhance glucose utilization. The
mathematical expression of the model is implemented by use of a simplified
bilinear set of 2 equations with 4 unknown parameters, pi through p4:
~ = -lP,+XlG + P l Gt (1 )
at
^ - = -p2 X + p ,[ H » - h \ P)
at
Where Gb is the basal value of glucose and lb is the basal value of insulin.
The mere act of modeling the FSIGT focuses on the factors that conspire to
enhance insulin-dependent glucose utilization: transendothelial insulin
transport (p3 ), the effect o f glucose on regulating itself (pi) and the effect of
insulin on glucose disappearance (embodied in X). As the effect of plasma
insulin to enhance glucose disappearance is determined both by
transendothelial transport and by the effect of interstitial insulin on insulin
sensitive cells it is of interest to derive a single parameter to encompass both
processes. This is achieved by calculating the dependence of the glucose-
independent glucose disappearance upon insulin. This calculation leads to
the definition of the minimal model insulin sensitivity index, Sp
S I= ^~ (3)
P i
4
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In its initial stage, minimal model had been applied towards the
extraction of the glucose effectiveness and insulin sensitivity indices by
measurements of glucose and insulin taken during a FSIGT. However, the
accuracy of the estimated indices was poor and this finding was mainly
attributed to the lack of adequate stimuli during the experimental procedure.
Therefore, the modified FSIGT protocol (6) was proposed. In this modified
protocol, a bolus of synthetic insulin is injected to the subject shortly after
the decay of the subject’s insulin response. The modified FSIGT (Figure 3)
G lucose Insulin
9 0 0 3 0 0
8 0 0
2 4 0
7 0 0
T 3
180
5 0 0
4 0 0
« 1 2 0
3 0 0
200
60
100
108 144 180 7 2 108 144 180
T i m e (min)
36 72 36
T i m e (min)
Figure 3: Typical glucose and insulin profiles after a modified FSIGT
5
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protocol offers better deconvolution of the glucose-insulin feedback loop
and, thus, increased accuracy of the minimal model indices for glucose
effectiveness and insulin sensitivity.
Minimal model has been extensively used for the last 15 years. It has
been proven to be a simple, cost effective and reliable method to evaluate
insulin sensitivity. Moreover, there has been proof that the minimal model
extracted insulin sensitivity index is well correlated with analogous indices
acquired by use of the clamp technique both in dogs and humans. (3, 7)
II. ROLE OF GLUCOSE EFFECTIVENESS IN THE
DETERMINATION OF GLUCOSE TOLERANCE
Insulin secretion and action are key factors in the determination of
glucose tolerance. However, glucose itself has been shown to enhance
glucose disposal and simultaneously suppress hepatic glucose production,
independent of changes in insulin. This finding led to the consideration and
thorough investigation of an additional factor that determines glucose
tolerance coined as glucose effectiveness.
6
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Glucose effectiveness has been measured in many studies in which
glucose disposal and production have been quantified at basal levels of
insulin and varying levels of glycemia. It has been shown that the effect of
glucose on glucose disposal is such that for an increase of 100 mg/dl in
plasma glucose, at basal insulin, an increase of 1.63 mg/[min kg] is noticed
in glucose disposal. Moreover, the same increase of 100 mg/dl in plasma
glucose has been found to suppress endogenous glucose output by 0.79
mg/[min kg]. Thus, two-thirds of glucose effectiveness in humans can be
accounted for the disposal effect and one-third is due to regulation of
endogenous glucose production (10).
The contribution of insulin independent glucose uptake depends on
insulinemia. During clamp experiments, with insulin maintained at basal
levels, at least 60% of the uptake is insulin-independent (11), which
decreases to -30% at insulin levels >100 pU/ml. However, the contribution
of insulin-independent glucose utilization to the overall glucose disposal is
of major importance at the existence of insulin resistance, because of the
combined effect of decreased insulin actions and prevailing hyperglycemia
(12).
7
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The clamp experimental technique and improved tracer methods have
provided an excellent insight of the two components of glucose
effectiveness, (rate of disposal of glucose) and HGO (hepatic glucose
output). However, concerns have arisen on whether glucose effectiveness is
the same on steady and non-steady state conditions. Specifically, during the
IVGTT, peak glucose and insulin levels are generated simultaneously and
this complicates the task of dissecting out the relative effects of glucose and
insulin on glucose normalization during the test.
Intuitively one might expect that, in healthy subjects, the insulin
dependent processes dominate the glucose disposal and that glucose auto
regulation only plays a minor role. In fact, while insulin plays the key role
in glucose tolerance, there has been evidence that its contribution to glucose
dynamics during an IVGTT might have been overestimated.
Although during an IVGTT glucose and insulin attain peak
concentrations rapidly, their action on glucose dynamics follows different
patterns (13). Insulin action on glucose uptake is delayed because of the
sluggish insulin transport through the capillary endothelial barrier. In
addition, the insulin effect on regulating hepatic glucose output is also slow.
8
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In contrast, hyperglycemia per se induces a rapid increase in and
suppression of HGO.
Due to these dynamic differences between glucose and insulin action
on glucose regulation, the importance of glucose effectiveness during
rapidly changing patterns has been underestimated. Researchers have used
simulations and experimental techniques to point out that glucose
effectiveness dominates the early phase of such tests like the OGTT and
IVGTT.
The minimal model has been extensively used to quantify glucose
effectiveness under not steady state conditions. Due to the complicated
nature of simultaneous action o f insulin and glucose on glucose regulation,
the scientific community has expressed concerns on whether the minimal
model induced glucose effectiveness can be used as a reliable index of
glucose auto-regulation. These concerns are being discussed on the next
section.
9
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III. CONCERNS ON MINIMAL MODEL
Minimal model has been proven to be an exquisite way for the
estimation of insulin sensitivity. However, over the years, there has been
increased concern (4), (5) about the accuracy of the minimal model indices.
The main objection that has arisen concerned the use of single-compartment
glucose distribution kinetics when strong evidence exists for two- or three-
compartments of glucose distribution kinetics. Moreover, minimal model
fails to distinguish between the two discrete ways of glucose auto-regulation
i.e. suppression of hepatic glucose output (HGO) and increase of the
glucose-dependent glucose uptake. These two facts led to the conclusion
that minimal model’s estimate for glucose effectiveness is oversimplified
and consequently induces some deviation to the calculation of insulin
sensitivity.
One of the most common and extensive applications of the minimal
model has been the analysis of data from FSIGTs, which have been
conducted during longitudinal studies. Minimal model’s simplicity, speed
and validity combined with the cost effectiveness of the FSIGT experimental
protocol constitute a charming combination for the design of such studies.
However, the problem of undermodeling of glucose distribution kinetics
10
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remains a fundamental obstacle to the validity of the results of longitudinal
studies.
Two of the most commonly known pre-diabetic stages, impaired
glucose tolerance and insulin resistance are attributed mostly to changes in
insulin sensitivity. In the case of longitudinal studies, changes that occur in
the glucose patterns of the same subjects are primarily attributed to changes
in insulin sensitivity and not in glucose effectiveness. This fact, combined
with the oversimplified representation of the glucose auto-regulation
mechanism in minimal model brings up the question of a possible
modification of the minimal model. Through this modification glucose
effectiveness might be estimated in a more accurate way than by the
Minimal Model or even eliminated in order to acquire more accurate
estimates o f insulin sensitivity.
11
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IV: APPLICATION OF THE MINIMAL MODEL TO
LONGITUDINAL STUDIES.
During a longitudinal study, two or more modified FSIGTs are being
performed on the same subject within a certain period o f time, during which
the subject is either undergoing drug treatment or exposed to physical
activity or exposed to a certain dietary pattern. If Gi, Ii, P21, P31, Gbi, G01,
Ibl) are the minimal model parameters for the first FSIGT, G2 , I2, P22, P32, Gb 2 ,
G0 2 , Ib2 are the minimal model parameters for the second FSIGT and pi is the
minimal model glucose effectiveness, common for both subjects, we can
write the minimal model equations, by use of equations (1) and (2):
[Pi +Xi]'Gi +Pi 'Gu> G ,(0) = G01, (4a)
P ^ + P m T W m]. X j(0) = 0, (4b)
[.Pi + x 2 1 • G 2 + Pi- G b2 » G 2 (0) = G02, (4c)
■p2-X2+ p 32-[I2- I b2], X 2(0) = 0, (4d)
Taking into account that glucose effectiveness pi changes minimally
during the time in between tests, we can eliminate it from equations (4a) and
(4c):
12
dGx
dt
dX,
dt
dG2
dt
dX2
dt
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Equation (5) does not include glucose effectiveness pi but is an
estimate of it at each time point. If our hypothesis o f non-changing glucose
effectiveness is true, we expect the average value of glucose effectiveness
from both tests to be the same. In mathematical terms, we expect:
~ final
dGl{t)+xi( 0 • G,(0 dG p + x 2 (t)• g2(0
y [— dl---------------------------- dt ] = 0, (6)
u Eq. (6) sets glucose effectiveness from both data sets to be equal at
any time point but does not hinder any of the two algebraic expressions on
both sides of eq. (5) to change with time. Therefore, it is necessary to add an
extra constraint which will set glucose effectiveness for both data sets to be
time invariant. In order to achieve this, we set the derivative of both sides of
eq. (5) equal to zero (see eq. (7)) and then use least squares fitting to find the
optimal values for the unknown parameters.
d- [ G m + X x{t)-Gx(t)
± [ A ] = 0, (7a)
dt Gj (/) - Gb
, ~ [ G 2(t)] + X 2(t)-G2(t)
i f [A ] = o (7b)
dt G2 (t) - Gb
13
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In order to examine the validity of the proposed method, Monte Carlo
simulation techniques were used. From this point on, we shall refer to the
new model as LongMod (i.e. Longitudinal Model).
14
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B. CASE STUDY I
I. METHODS
In order to examine the validity of the proposed method, the Monte
Carlo simulation technique was used. The simulation was designed as a
theoretical representation of a longitudinal study with 100 participants
(Figure 4). For each of the virtual subjects two simulated FSIGT profiles
Simulation Protocol
G eneration o f synthetic FSIG T datasets by use o f random
distribution o f p i, p2, p3, GO,Gb
2nd week FSJGT 2nd week F3GT 1st week FSGT
1st Subject 100th subject
N= 100
Run MfrriinaL Model Run Mmirnal Model
Run Minimal Model 1 1 Run Minimal Model
results comparison
Figure 4: Organizational chart for the simulated longitudinal study. For every
one of the 100 subjects 2 FSIGTs are generated and both minimal model and
LongMod are tested on these 200 data sets
15
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have been created using the minimal model equations. The first data set for
each subject (pre-treatment case) was set to have greater insulin sensitivity
than the second one (post-treatment case). During the creation of the
synthetic data sets each subject was assumed to have the same glucose
effectiveness p i, the same plasma volume of distribution, which is
represented as the value of G0 in minimal model and also the same basal
value of glucose between the pre-treatment and post-treatment cases. The
same two plasma insulin profiles were used for the generation of the
synthetic data sets in the pre- and post-treatment case (Figure 5).
M o d ifie d F S I G T profile
S e c o n d insulin profile--*”' Firstinsulin profile
4 0 0
Z )
3 2 0
2 4 0
C
3
t /)
C
2 4 0 3 2 0 4 0 0 0 8 0 1 6 0
Time (min)
Figure 5: Insulin profiles used for the generation of the data sets
16
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The minimal model parameters (Table 1) that have been used for the
generation of the simulated data sets have been produced by use of
MATLAB’s built in random number generator. The mean values and
standard deviations of the simulation parameters have been set equal to the
corresponding values of already published data. (6)
SIMULATION PARAMETERS
Mean Standard Deviation
P i
0.024 (min1 ) 0.012 (min'1 )
P2
0.038 (min_ 1 ) 0.006 (min_ 1 )
P3
13.971x10^ [min2 / (pU/ml)] 6.4x10"° [min'2 / (jiU/ml)]
Go 225 (mg/dl) 52 (mg/dl)
Sj 3.74 [ min'7(p,U/ml)] 1.824 [ min 7(pU/ml)]
Table 1: Mean and standard deviation of the parameters used for the generation of
the synthetic data sets
In order to calculate the derivative of the glucose profile, a cubic
spline approximation of the glucose profile was used. The derivative was
17
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then smoothed for purposes of eliminating the error in the estimation of the
derivative due to discontinuities of the original profile.
In order to study both the theoretical and practical aspects of the
proposed method, noise has been superposed to all 200 data sets. In order to
cover the cases of “average” and “extreme” measurement error, white noise
of 2% and 4% variance was added to the initial data sets. Minimal Model
and LongMod were run in all 200 data sets and three cases (no noise, 2%
noise and 4% noise) and indices of insulin sensitivity derived from both
methods were compared.
Parameters for LongMod and minimal model were identified by
weighted nonlinear least squares fitting using a Levenberg-Marquardt
algorithm. Smoothing was performed by MLAB's built-in smoothing
routine. Data are presented as means±SE unless otherwise stated.
II. RESULTS
In the absence of measurement error, insulin sensitivity indices from both
minimal model and LongMod seem to be well correlated with the theoretical
Si values. The correlation coefficient between Minimal Model Sr (3.76±2.0
18
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10'4 min'VfjLxU/ml]) and the theoretical Si (3.74±1.82 10'4 min'1 / [pU/ml]) is
r=0.96 and the corresponding coefficient for LongMod Si (4.05±2.27-10‘4
min'VfpU/ml]) and theoretical Si values is r=0.91. After applying linear
regression between Minimal Model Si and the theoretical Si values we got a
linear relationship Si M in m o d = l -09-Si th-0.33 (Figure 6). The corresponding
10
I
1 .09*x - 0 .3 3 , r = 0 .96
r—I
c
‘ E
d >
d,
CO
2 5 7 8 10 0 1 3 4 6 9
Theoretical S , (10-4 rrin-l fa&J/fulD
Figure 6: Regression analysis between theoretical SI and Minimal Model calculated
insulin sensitivity index, for the no noise case
relationship for LongMod Si and the theoretical Si values Figure 1 Figure 7:
Regression analysis between theoretical SI and LongMod SI, for the no
19
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noise caseis Si L ongMod=l-13-Si *-0.15 (Figure 7). There was no statistically
significant difference between the slope or the intercept in both cases.
'E
6
- o ~
10
y = 1 .1 3 * x - 0 .1 5 ,r = 0 .9 1
8 -
6 -
A *
T h e o r e t i c a l , §L 0 4 m i n 1 /[ a e U / m I])
Figure 7: Regression analysis between theoretical SI and LongMod S I, for the no
noise case
Statistical analysis between the theoretical SI values and minimal
model or LongMod insulin sensitivity indices prove that minimal model
accurately represents the theoretical SF data. On the contrary, LongMod
tends to overestimate the theoretical Si values (p<0.01).
In order to examine whether LongMod overestimates both the pre-
and post-treatment Sr or demonstrates biased behavior in favor of only one
20
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of the two indices, AS( (defined as the algebraic difference between pre- and
post-treatment Si) was calculated. In the case of no noise, both LongMod
and Minimal Model seem to predict the theoretical ASi accurately. The
correlation coefficient between ASith and ASi L o n g M o d is 0.95 and the one
between ASI th and ASi minm od is 0.96. After linear regression, A S IMiNm od=
ASjlongmod^I-04-ASIth (Figure 8).
3 SI (No noise)
o M n i m a l IVbdel ▼ L o n g M o d
m m w d = l M % r = 0 3 6
=1j 0 4 % i-=0.95
C D
LongMod
•I#
1 o
0 1 2 3 4 5 6 7
T h e o r e t i c a l * S
Figure 8: Regression analysis between theoretical ASI and Minimal Model or
LongMod ASI in the case of absence of measurement error
21
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In the case of 2% measurement error the strong linearity between the
theoretical St values and Minimal Model values (3.74±1.8 10'4 min"1 /
[pU/ml]) is preserved (r=0.93). The comparison between the theoretical
values of Si and LongMod Si indices (4.08±2.2 10"4 min"1 / [pU/ml]) gives a
much lower coefficient of correlation (r=0.77). Linear regression performed
between theoretical Si and Minimal Model Si showed that SiM in m o d = l-l 1'Sith-
0.33 (Figure 9). The results from linear regression between theoretical Si
and LongMod showed that SiL o n g M o d =0.92-Sith+0.64 (Figure 10).
Statistical analysis performed between minimal model Si and theoretical Si as
well as between LongMod Si and theoretical Si shows that the minimal
model method retains the statistical characteristics of the theoretical Si
population. In the 2% measurement error case, LongMod tends to
overestimate the theoretical Si (p<0.01).
22
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1 0
y = l.ll* x - 0 .3 3 , r= 0 .9 3
8
6
'" - 0 3 o O
4
2
0
10 9 7 8 3 5
Theoretical S , (10-4 min-1 /[aeU/ml])
4 2 0 1
Theoretical
Figure 9 : Regression analysis between theoretical SI and Minimal Model calculated
insulin sensitivity index, for the 2% measurement error case
ASi fr°m minimal model seems to be well correlated with theoretical
(r=0.89, A S IMIn m o d =1.09-ASi* +0.02). LongMod gives a less correlated
representation of Si* in this case of 2% noise (r=0.71, ASi L o n g M o d -0.81-ASi*
+0.37) (Figure 11).
In the case of poor measurements (4% measurement error) a strong
correlation of SIth and Si m inm od (3.98+2.3 10'4 min V[pU/ml]) is noticed
(r=0.89). The correlation coefficient between SI* and SiL o n g M o d
(4.63+2.6 10‘4 min‘V[pU/ml]) is lower (r=0.72). The statistical difference,
which has been noted at both noise levels mentioned above, is also present
23
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here (p<0.01). Linear regression between Si* and Sim inm od and Si* and Si
L o n g M o d show that while the minimal model is giving a robust representation
of the actual Si values (Si m in m o d = 1 -1 2 -S i* -0 .2 10'4 min"V[pU/ml]) (Figure
12) LongMod overestimates SI* by almost 0.84-1 O '4 min"V[pU/ml]. (Si
LongMod=1.02-Si*+0.83 10"4 min'V[pU/ml]) (Figure 13).
1
C
E
c o
■ a
i
o o
c
o
10
y = Q 9 2 f h (- iQ 6 f l, r=Q77
8
6
4 M
• • •
/ V
2
0
Initial |S (10^ min4 /{a£Mnl]) 10 8 0 2
Figure 10: Regression analysis between theoretical SI and LongMod SI (2%
measurement error case)
24
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
3 SI (2% noise)
o Minimal Model,3 S * LongMod 3 ,S
10.00
O
7.58 y M , N M o n l - ^ ^ r t ) . 8 9
yL ongM ^x-40.37,P0.71
o
5.16 -
2.75 -
o
o
o
o
o
” 0 o.sT^
° ° f
o
o
0.33
f 'f
o
o
(}°
-2.09 * -
0.00
_L J _
1.40 2.80 4.20
Theoretical3 S
5.60 7.00
Figure 11: Regression analysis between theoretical ASI and Minimal Model and
LongMod ASI, for the 2% noise case
25
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ASj from minimal model seems to be well correlated with
theoretical Sj (r=0.79, A S M inm od =1.05-ASIth -0.05). LongMod gives a less
10
c
£
o
r— H
N w /
0 0 -
a 3
■ o
o
( D
E
O
y = 1 .1 2 * x -0 .2 , r = 0 .8 9
o O
9) O o
o
Q >
o
o
o
0 C O f)
Q 0. 0
q O o
o
o
o
o
£o O qQ -
o
o
o
o
0
o
o o 0o ° ° ^°o
° ° " ° o
)(^>o ° 0
O O
o
o
to
9 )
o
o
,0*
0
- e -
_ L
o
o
3 4 5 6 7
Theoretical ^ lO 4 min1 - /[s e ll/m l])
10
Figure 12: Regression analysis between theoretical SI and Minimal Model
calculated insulin sensitivity index, for the 4% measurement error case
correlated representation of SIth in this case of 4% noise (r=0.62, AS[
L o n g M o d = l-l-ASIth +0.09) (Figure 14).
26
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III. DISCUSSION
In this first part of this study, we systematically explored the effect of
the single compartment assumption on the estimation of insulin sensitivity
and checked the validity of an alternative method for the estimation of Si.
The comparison of the indices acquired after using both the minimal model
C
E
T 3
O
bo
c
o
9 . 0 0
7 . 2 0
5 . 4 0
3 . 6 0
1 . 8 0
0.00
• * • / • •
• • * • • •
* ** • * *
j v . :
' • • • $ •
. * • . *
— •••— 1 —»
• •
y=1..02*x+0.83, r=0.7 2
0,00 1 . 8 0 3 . 6 0 5 . 4 0
Initial ^ ( 1 0 - 4 m in -1 / [ a e U / m l ] )
7 . 2 0 9 . 0 0
Figure 13: Regression analysis between theoretical SI and LongMod SI (4%
measurement error case)
27
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
and LongMod to account for the insulin sensitivity o f the synthetic data sets
showed a reliable and robust performance o f the minimal model independent
of the level of measurement error to the data. The simulation results also
revealed the sensitivity of LongMod to measurement error.
LongMod seems to give reliable and accurate estimates of Sj in the
case of no measurement error superposed to the data and also demonstrates a
slightly higher Sj than the Minimal Model. The latter has also been noticed
by various researchers who have noted an underestimation in the minimal
model induced insulin sensitivity index due to the single compartmental
assumption of the glucose kinetics (10) and the interaction between glucose
effectiveness and the first phase insulin secretion.
LongMod demonstrates increased sensitivity to measurement error
compared to the minimal model. The latter demonstrates an astonishing
robustness even at levels of measurement error that are considered extremely
high for today's standards. SIL o n g M o d correlation with the theoretical values o f
Si declines proportionally to the amplitude o f measurement error.
28
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In order to explore the causes of the decreased performance of
LongMod on noisy data sets, LongMod was rerun for all "noisy" data sets
3 SI (4% noise)
O Minimal Mode( 3 S ▼ LongMod 3 S
. 0.00
/m ,n m o ^ -0 5 * x -0 .0 5 , r= 0.79
MINMOD
6.91
3.82
0.72
-2.37 ,L
-5.46
5.60 7.00 0.00 1.40 2.80 4.20
Theoretical , 3 S
Figure 14: Regression analysis between theoretical ASI and Minimal Model or
LongMod ASI, for the 4% noise case
using the derivative estimate acquired before the superposition o f synthetic
measurement error to the data. The performance of LongMod was
significantly increased but still remained worse than that of the Minimal
29
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Model. The inherent numerical instability of the equations of LongMod,
especially during the undershoot which occurs in the glucose profile, seems
to deprive the model of its potential to extract accurate results.
However, if low levels of measurement noise are present in the
glucose measurements, LongMod proved to give a very good alternative for
the calculation of insulin sensitivity indices in longitudinal studies, provided
that the glucose effectiveness of the subject changes minimally during the
treatment period. It is widely established that insulin sensitivity can be easily
manipulated by pharmacological or physical interventions. However, there
has been very little proof in the literature about similar effects on glucose
effectiveness. Therefore, attention must be paid to the nature of the
treatment and its impact on the glucose auto-regulation mechanism. In
cases where there are hints o f significant changes in glucose effectiveness,
other experimental techniques should be considered.
LongMod offers an interesting insight into numerical methods. The
use of a cubic spline for the approximation of the derivative of glucose and
the subsequent smoothing of the derivative can be a major source of error
and instability during the optimization process. Non-convergence of the
30
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method implies a significant amount of measurement error in the glucose
profile.
31
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C CASE ST U D Y II
I. METHODS
The simulation results that have been extensively discussed on the
second part of this thesis used the one single pair of insulin profiles for every
pair of synthetic data sets produced. In this part, the goal is to investigate
the effect the increased variability of the insulin profile in Minimal Model
and LongMod performance.
In this case study, synthetic data sets were generated by use of the
minimal model equations using randomly generated values for the
parameters. Instead of using just one standard pair of insulin profiles, like
we did on case study I, we created a pool of 25 pairs of typical modified
FSIGT insulin profiles (Figure 15). After randomly assigning one of the 25
pairs of insulin profiles in each subject a simulated longitudinal study was
conducted in exactly the same way like in the 2n d part of this thesis.
32
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II. RESULTS
In the absence of measurement error, LongMod (3.27±1.6 10"4 min"1 /
[jjU/ml]) and Minimal Model (3.14±1.6 10"4 min'1 / [|j,U/ml]) seem to give a
well-correlated estimate of the theoretical Si (3.74±1.82 10"4 min"1 /
[jj.U/ml]). The correlation coefficient between LongMod Si and theoretical
Si is r=0.80 and the correlation coefficient between Minimal Model Si and
theoretical Si is r=0.89.
• Pre-treatment insulin Post treatm ent insulin
200
100
0
300 180 240 0 60 120
Time (min)
Figure 15: Average pre- and post-treatment insulin profiles used for the generation
of data sets
33
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Linear regression between the theoretical Si and Minimal Model Sj
shows that both models slightly underestimate the theoretical value for the
insulin sensitivity index. Specifically, St M in m o d = :0-8‘SI th+0-19 (Figure 16)
and Sj L o „ g M o d =0.69-Si *+0.72 (Figure 17). Statistical analysis also reveals
that both LongMod and Minimal Model tend to underestimate the theoretical
Sr value (p<0.01).
Minimal Model SI vs. Theoretical SI
10
y=0.8*x+0.19, r=0.89
8
6
4
2
0
10 4 8 2 6 0
I n itia l £ ( 1 0 # m in-1 / [ a e U / m l ] )
Figure 16: Regression analysis between theoretical SI and Minimal Model SI with
no measurement error superposed to the data
34
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ASi seems to be underestimated by both models. Linear regression between
theoretical Si and Minimal Model or LongMod ASi (Figure 18) shows that
ASMiNMOD=0.79-ASith+0.19, r=0.80 and also that ASiL o N G M O D =0.79-ASith-
0.48, r=0.77. Statistical analysis also verified that both models
underestimate the theoretical value of ASi (p<0.01).
LongMod SI vs. Theoretical SI
10
y=0.69*x+0.72, r=0.80
8
6
M •
4
2
0
0 4 6 8 10 2
Initial $ ( 1 0 - 4 m in-1 / [ a e l l / m l ] )
Figure 17: Regression analysis between theoretical SI and LongMod SI with no
measurement error superposed to the data
The trend that both models exhibit towards underestimation of the theoretical
Si can be noticed in the 2% error superposition. In this case, Minimal Model
35
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Si (3.18±1.7 10'4 min1 / QiU/ml]) and LongMod Si (3.16±1.5 10'4 min 1 /
[pU/ml]) exhibit a statistically significant lower average value than
theoretical Si (3.74±1.82 10'4 min1 / [|iiU/ml]) (p<0.01). The correlation
coefficient between the theoretical Si values and the minimal model induced
Si is r=0.88, significantly higher that the correlation coefficient between the
theoretical Si and the LongMod Si(r=0.73).
3 SI (0% noise)
o Minimal Model3 S ▼ LongMod3 ^
10.00
=0.79*x+0.19, r=0.80
=0.79*x-0.48, r=0.77
MINMOD
7.52
LongMod
to
T 3
< U 5.04
a
o.
2.57 o
O o
0.09
-2.39
7.00 4.20 5.60 1.40 2.80 0.00
Theoretical
Figure 18: Regression analysis for ASI in the no noise scenario
36
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Linear regression between the theoretical Si and Minimal Model Si verifies
the no noise case finding of underestimation of the theoretical insulin
sensitivity indexFigure 1 Figure 19: Regression analysis between theoretical
SI and Minimal Model SI with 2% measurement error superposed to the data
by both models. Specifically,Si M i n m o d = 0 - 8 3 - S Ith+ 0 . 0 9 (Figure 19) and Si
L on gM od = 0 .59-S ith+ 0.97 (Figure 20).
Minimal Model SI vs. Theoretical SI
10
y=0.83*x+0.09, r=0.88
8
6
• • •
4
«•
• ! « J
2
0
10 8 6 0 2 4
I n i t i a l ty(10-4 m i n - 1 / [ a e U / m l ] )
Figure 19: Regression analysis between theoretical SI and Minimal Model SI with
2% measurement error superposed to the data
37
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ASi in this case is expressed more accurately by the Minimal Model than by
LongMod. Linear regression (r=0.74, A Sim inm od -0.85-ASIth +0.14) shows
that even if both minimal model and LongMod (r=0.65, ASi L o n g M o d
=0.59-ASith +0.06) (Figure 21) underestimate (p<0.01) the theoretical ASi
values, minimal model gives a closer estimate of the expected ASi value.
LongMod SI vs. Theoretical SI
10
y=0.59*x+0.97, r=0.73
8
6
4
2
0
10 2 6 8 0 4
Initial j3(10# min* /[asll/m l])
Figure 20: Regression analysis between theoretical SI and LongMod SI with 2%
measurement error superposed to the data
38
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In the case of poor measurements (4% measurement error) a high
correlation of SIth and Si m i n m o d (3.22±1.8 10' 4 min'‘/[pU/ml]) is noticed
3 SI (2% noise)
O M in im a l M o d e l sjS ^ L o n g M o d 3S|
10.00
= 0 .8 4 * x + 0 .1 4 , r = 0 .7 4
= 0 .5 9 * x + 0 .0 6 , r = 0 .6 5
MINMOD
7.05
LongMod
O Q
4.11
1.16 cr
-1.78
-4.73
7.00 5.60 0.00 1.40
Figure 21: Regression analysis for ASI in the 2% noise scenario
(r=0.84). The correlation coefficient between SI* and Si L o n g M o d (3.12±1.5
1 0 " 4 min'V[pU/ml]) is lower (r=0.73). The statistical difference, which has
been noted at both noise levels mentioned above, is also present here
(p<0 .0 1 ). Linear regression between Si* and S i m i n m o d and Si* and Si L o n g M o d
show that while the minimal model represents robustly the actual Si values
39
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(Si M iN M O D =0-84-Sith+0.07 10' 4 min‘1/[p,U/ml]) (Figure 22) LongMod is not
as accurate (SxL o n g M o d =0.59-SIth +0.94 10' 4 min''/[pU/ml]) (Figure 23).
ASi from minimal model seems to be correlated with theoretical Si
(r=0.70, ASiminmod =0.94-AS[lh -0.03). LongMod gives a slightly less
correlated representation of S[th in this case of 4% noise (r=0.64, ASi L o n g M o d
=0.58-ASith+0.07) (Figure 24).
Minimal Model SI vs. Theoretical SI
10
y=0.84*x+0.07, r=0.J4
i — — 1
E
3
E
o
a)
T3
0
1
to
E
c
5
10 6 8 4 0 2
I n i t i a l £ ( 1 0 ^ m in -1 / [ a e L I /m l] )
Figure 22: Regression analysis between theoretical SI and Minimal Model SI with
4% measurement error superposed to the data
40
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I ll DISCUSSION
The use of changing insulin profiles during the generation of the
synthetic data sets and the subsequent effort to fit the synthetic data with
LongMod and the minimal model after superposing noise at various levels
lead to useful conclusions about the behavior of the minimal model. To
LongMod SI vs. Theoretical SI
10
y=0.59*x+0.94, r=0.73
8
6
4
2
0
10 8 2 4 6 0
Initial S, (1 0 -4 m in-1 /[aaU /m l])
Figure 23: Regression anafysis between theoretical SI and LongMod SI with 4%
measurement error superposed to the data
41
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begin with, the theoretical Si value tends to be underestimated by both
LongMod and the minimal Model. This finding contradicts with the
conclusion drawn from case study I, where there is no significant statistical
J SI (4% noise)
O M in im a l M o d e l L o n g M o d 3S .
10.00
7 . 4 7 -
'MINMOD
= 0 .9 4 * x -0 .0 3 , r = 0 .7 0 o
y, M = 0 .5 8 * x + 0 .0 7 , r = 0 .6 4
LongMod ’
O
▼ o
- 2 .6 5
0.00 1 . 4 0 2 . 8 0 4 . 2 0
T h e o r e t i c a l 3S,
5 . 6 0 7 . 0 0
Figure 24: Regression analysis for ASI in the 4% noise scenario
difference of the mean values between S fth and S i m i n m o d b u t only between
the mean values of SIth and SiL ongM od- It is obvious that the increased
variability of the insulin profiles causes the minimal model to slightly
underestimate the insulin sensitivity index for which the data were
42
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generated. Further studies have to be made in order to understand what the
impact of different insulin profiles is in the minimal model.
The correlation coefficient between S Ith and S m inm od is higher in all
three levels of noise than that between S Ith and SiLongMod. Moreover, linear
regression analysis performed in all three cases shows that by using minimal
model the underestimation of S^h from the minimal model lies around the
area of 80%-85% while the underestimation that occurs from LongMod is
somewhere between 60%-70% of S^h. This shows that the minimal model
gives a more accurate index of insulin sensitivity than LongMod. This
finding has to be further investigated and verified since the synthetic data
have been acquired by use of the minimal Model. In order to check whether
the performance of the minimal model is due to the single compartmental
nature of the synthetic data or is due to the more straightforward approach of
the model, real or two-compartment synthetic data have to be used.
43
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D. CASE STU DY III
LMETHODS
The protocol that has been used for the generation of the synthetic
data in case study I and case study II was the modified FSIGT protocol (6 ).
In this chapter our goal is to investigate the performance o f LongMod and
the minimal model when the simple FSIGT protocol is used, i.e. no injection
of synthetic insulin takes place after the endogenous insulin response to the
glucose bolus. There has been proof in the literature (6 ) that the indices
acquired by use of this protocol are characterized by decreased accuracy.
This problem is obvious especially in diabetic subjects, where the lack of an
adequate endogenous insulin response does not offer enough information to
help minimal model fit the glucose measurements.
The same method of Monte Carlo simulation was used in this case
study in order to evaluate the performance of both the minimal model and
LongMod under a regular FSIGT protocol. 200 synthetic data sets were
generated by use of previously published mean and standard deviation
values for the minimal model parameters (Table I). The insulin profiles
were randomly selected from a pool of 25 pairs of real insulin profiles taken
44
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from experimental data of longitudinal studies. Minimal model and
LongMod were run on the synthetic data sets and insulin sensitivity indices
from both models were compared to the theoretical values that were used to
generate the data. Following the pattern that has been established on the
previous case studies, noise of 2% and 4% variance was superposed to the
synthetic data sets in order to evaluate the theoretical as well as the practical
aspects of our methods.
II. RESULTS
In the absence of noise, the theoretical values of Sj (3.74±1.8 10' 4
min' 1 / [pU/ml]) seem to be correlated well with minimal model (3.24±1.9
10' 4 min' 1 / [pU/ml], r=0.86) and LongMod (3.44±1.9 10"4 min"1 / [pU/ml],
r=0.77) insulin sensitivity indices. However, both minimal model and
LongMod seem to demonstrate a statistically significant (p<0.01) lower
mean value than the theoretical one.
This finding is further supported by the results of linear regression
analysis. For minimal model Si M in m o d =0.87-SIth (Figure 25) while for
LongMod SILo n g M o d = 0.81-SIth+0.42 (Figure 26).
45
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A S i seems to be underestimated by both methods. More specifically,
ASiMiNMOD^-^ASith+O.l, r=0.72 and A Snx>N G M O D = 0 - 8 1 - A S i th - 0 . 5 6 , r=0.68.
(Figure 27). However, ASi from the minimal model is not significantly
Minimal Model SI vs. Theoretical SI
No noise
o
t— i
w
< u
T 3
< D
E
c
10
y=0.87*x-0.007, r=0.86
8
6
4
» •
2
0
2 0 4 6 8 10
Initial $(10-4 min-1 /[aeU/ml])
Figure 25: Regression between theoretical SI and minimal model SI for the no noise
case
different from the theoretical value of the difference between pre- and post
treatment Si (p=0.39) while LongMod ASi demonstrates a significant
difference (p<0 .0 1 ).
46
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In the case o f 2% measurement error, minimal model (3.22±L8 1G "4
min"1 / [pU/ml], r=0.85) exhibits a very robust performance compared to
LongMod (3.46±1.8 1G "4 min' 1 / jjiU/ml], r=0.63). Statistical analysis
LongMod SI vs. Theoretical SI
No noise
10
y=0.81*x+0.42, r=0.77
/— S
r — n
E
c
'e
o
I — I
# •
«-
73
O
6 8 10 0 2 4
Initial 5 j> ( 1 0 - 4 min- 1 /[aeU/ml])
Figure 26: Regression between theoretical SI and LongMod SI for the no noise case
between theoretical Sf and minimal model or LongMod induced S{ show
that both methods give significantly lower insulin sensitivity values
(p<0.05). Linear regression between LongMod Sj and theoretical Sj (Figure
28) give an especially high zero-intercept value
47
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(S, i.o n g M o d =0-64-Si th +l .08) while same style analysis between minimal model
Si and theoretical S] exhibits a fairly robust performance of the minimal
model (SIMnm «r0^ 4-S fth+0.09) (Figure 29).
3 SI (0% noise)
o Minimal Model 3 S ▼ LongMod
10.00
6.93
to
3.86
o
~ o
< D
Q_
5 0.78
o
x+0.1, r— 0.72
yL o n g 0 o?:O- 8 ^ X -O-5 6 ’ O^O . 6 8
o-
o
O o ▼
0 8
CL
o
o
f J H T
v o
o
- 6 ° '
gy ¥
-2.29
-5.36
0.00 1.40 2.80 4.20
Theoretical 3 S
5.60 7.00
Figure 27: Regression analysis fur ASI in the Q% noise scenario
Statistical analysis for ASf shows that the minimal model does not
alter the difference significantly. On the contrary, AS[f,0 n g M o d is significantly
4S
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
lower (p<0.01). Linear regression shows that AS{ M iN M O D =0-82-AStfh +0-25,
i=0.67 and ASnxwoMCD^O.SO-ASith-O^, r=0.53 (Figure 30).
Minimal Model SI vs. Theoretical SI
2% noise
10
y=0.84*x+0.09, r=0.85
8
6
4
2
0
10 0 2 4 6 8
Initial ^(1Q - 4 min- 1 /[seU/ml])
Figure 28: Regression between theoretical SI and minimal model SI for the 2%
noise case
In the ease o f 4% measurement error, minimal model (3.26±2
1G ' 4 min"1 / [pU/ml], r=0.76) still performs rohustly even at such high level
o f measurement eiror. LongMod (3.44±2 10"4 min' 1 / [pli/ml], r=0.56)
seems to be more noise sensitive than die minimal model. Statistical
49
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analysis between theoretical Sj and minimal model or LongMod induced Sj
shows that both the minimal model Si and LongMod Si are significantly
lower than S{ th (p<0.05). Linear recession between LongMod Si and
LongMod SI vs. Theoretical SI
2% noise
10
y=0.64*x+1.08f c r=0.63
8
6
4
• •
2
0
10 0 2 4 6 8
Initial ^(1 Q -4 m in -1 /[aeU/ml])
Figure 29: Regression between theoretical SI and LongMod SI for the 2% noise ease
theoretical S{ (Figure 31) give an especially high zero-intercept value (Sj
L o n g M o d =0-62-Sf th + 1.13) while same style analysis between minimal model Si
and theoretical Si exhibits similar performance o f the minimal model to the
2 % noise case (Si Mmmn^O.&LSi t h + 0 - 1 1 ) (Figure 32).
5 0
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Statistical analysis for ASj shows that the minimal model does not
alter the difference significantly (p-0. 38). On the contrary, ASiL o n g M o d is
significantly lower (p<0.01). Linear regression shows that ASmnmqd=
0.97-ASflh -0.09, r=0.63 and A S ILOn g m o d = 0 - 4 9 - A S I{1i- 0 . 0 3 , r=0.48 (Figure 33).
3 SI (2% noise)
0 Minimal Model, 3 S ▼ LongMod 3 jS
C O
-a
a >
- a
a >
10.00
6.73
3.46
a >
g 0.20
-3.07
-6.34
V»-,nM o d= 0 - 8 2 * x + 0 . 2 5 (5 r=0.67
y ^ S # 0 -5 6 * * - 0 -0 3 . r = 0 - 5 3
< t > ’ o
0
f ▼ « o
o
o
o
0 .
--GT3
o
o
7
T O
' J A ' -
0
JO.-"'
o
o
o
0.00 1.40 2.80 4.20 5.60 7.00
Theoretical 3 S
Figure 30: Regression analysis lor ASI in the 2% noise scenario
51
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
ILL DISCUSSION
In all three levels of noise, the theoretical value o f Sj seems to be
underestimated by either minimal model or LongMod. Minimal model
underestimates St* by a level o f 80%-85% while LongMod's
underestimation ranges from 65%-80%. Moreover, the intercept o f
LongMod results is systematically higher than the corresponding index
acquired from the minimal model.
Minimal Model SI vs. Theoretical SI
4% noise
10
y=0.83*x+0.11, r=0.76
8
6
v f j . ..
4
2
0
10 8 2 4 6 0
I n it ia l S , ( 1 0 - 4 min-1 > £ a e U /m lI >
Figure 31: Regression between theoretical SI and minimal model SI for the 4%
noise case
52
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
LongMod exhibits increased sensitivity to noise. In the 2% and 4%
noise case the slope abruptly declines from 0.81 {no noise) to 0.64 and 0.62
respectively. On the contrary* the slope o f the linear recession between
minimal Model S{ and theoretical Sf exhibits a remarkable stability regardless
of the level of noise on the data (0.87 for 0% noise* 0.84 for 2% noise and
0.84 for 4% noise). In the case of the minimal model the intercept lies close
to 0 {0.007 for the 0% noise case, 0.09 for the 2% noise case and 0.11 for the
4% noise case).
LongMod SI vs. Theoretical SI
4% noise
10
y=0.62*x+1.13, r=0.56
8
• • • 6
4
2
0
8 10 4 6 0 2
Imtlal S (10-4 min-1 /£aeU/m+}>
Figure 32: Regression between theoretical SI and LongMod SI for the 4% noise ease
53
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M o d el P re d ic te d 3SI
Minimal model outperforms LongMod when the regular FSIGT
experimental protocol is used. This is also evident when ASj data are
examined. The results o f linear regression between ASIth and A S !Minm od and
ASjfe and AS!L o n g M o c j show that minimal model, once again, performs better
and represents AS{ th in a more robust way.
3 S 1 (4% noise)
O M in im a l M o d e l 3S } ? L o n g M o d 3S ,
10.00
= 0 .9 7 * x - 0 .0 9 , r = 0 .6 3 1
= 0 .4 9 * x - 0 .0 3 , r ^ 0 .4 8
7.25
MINMOD
LongMod
4.50
'O
1.75
-0.99
-3.74
7.00 0.00 1.40 2.80 4.20 5.60
Theoretical 3 Sr
Figure 33: Regression analysis for ASI in the 4% noise scenario
54
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E. CONCLUSIONS AND DISCUSSION
The goal of this thesis has been to implement a novel approach for the
estimation of insulin sensitivity indices from results acquired from
longitudinal studies. This novel approach relied on the elimination of
glucose effectiveness Sg from the minimal model equations and scoped on
attaining increased accuracy of Si by eliminating the interaction between
insulin and glucose action on glucose auto-regulation. In mathematical
terms, the proposed model was of less degrees of freedom than the minimal
model, which has to be used separately for each one of the two data sets.
Monte Carlo simulations were conducted under different experimental
protocols (regular and modified insulin injection) and conditions (increased
variability on the insulin profiles and different levels of noise). Although the
method constitutes an excellent theoretical way of calculating Sf, it seems to
demonstrate vulnerability to noise and decreased sensitivity to insulin
changes compared to the original minimal model.
One of the most significant barriers to the performance o f LongMod
was its inherent numerical instability. The use o f cubic splines or other
55
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derivative estimating algorithms (OOPSEG) was successful when low levels
of noise were present on the data but failed proportionally to the levels of
measurement error of the glucose profile.
Another potential pitfall of LongMod is the optimization method used.
Non linear least squares fitting has been validated as an excellent way to
extract the parameters of the minimal model. However, its performance was
not tested against other optimization strategies (global optimization,
different line search techniques).
The performance of LongMod was evaluated based on results
acquired by the IVGTT. This test has been designed to extract the highest
amount of information related to the glucose insulin feedback system for the
minimal model. Due to the aforementioned numerical instability of
LongMod, it will be useful to use different experimental protocols that will
provide less rapidly changing glucose patterns. Thus, a great portion of the
numerical instability caused by lack of precise computation of the derivative
of the glucose profile would be eliminated.
Another possible way to improve the performance of LongMod will
be to try different cost functions and weighting schemes. Given the fact that
the action of glucose effectiveness is dominant during the first minutes of an
56
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IVGTT, the cost function and weighting scheme can be modified in order to
accommodate this fact. More specifically, since the glucose action per se
does not change in longitudinal studies, one can assume that difference of
glucose action on itself on the data sets of the longitudinal studies will differ
only because of physiological variation. Therefore, one can extract the
residuals of glucose action per se for both cases and come to useful
conclusions about the model parameters following the assumption that they
have to follow a random distribution.
In order to get a better initial estimate for pi, statistical pools for
normal, insulin resistant and diabetic subjects can be used. Another possible
solution would be to use adaptive least-squares algorithms, in order to
gradually detect the optimal values for insulin sensitivity indices.
Another concern is whether LongMod can be used to model more
than two data sets. In such a case, the power for estimating SG is much
bigger than that of the minimal model and can lead to a significantly smaller
confidence interval for the estimation of Sg .
Intuitively, one would think that in longitudinal studies P3 or p2 do not
change considerably on the time interval between tests. Simulation results
from our lab show that even when Si or even AS[ follow certain distributions
57
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it is not sure that p3 or p2 fulfill the assumption o f changing according to a
specific pattern. This finding makes any effort to use Sf as a pivotal quantity
to increase accuracy on indices from longitudinal studies extremely hard.
Overall, LongMod is based on the plausible idea of minimally
changing glucose effectiveness in the time interval between tests. The
results of the simulation showed that LongMod performs successfully on the
absence of high levels of noise and increases the accuracy of the predicted
indices. Understanding the numerical properties of the method and the
impact of the experimental protocol to the effectiveness of LongMod is of
primary concern and should be further investigated in order to allow the
method to be widely accepted.
58
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REFERENCES
1. Bergman R.N, Finegood D, Ader M.: Assessment of Insulin Sensitivity
In Vivo, Endocrine Reviews, Vol. 6 , No. 1
2. Bergman R.N, Lovejoy J: The Minimal Model Approach and
Determinants of Glucose Tolerance, Pennington Center Nutrition Series,
Louisiana State University Press, Baton Rouge And London, 1995
3. Finegood D., Pacini G., Bergman R.N, The Insulin Sensitivity Index:
Correlation in Dogs Between Values Determined from the Intravenous
Glucose Tolerance Test and the Euglycemic Glucose Clamp, Diabetes,
Vol. 33, No. 4, 1984.
4. Vicini P., Caumo A., Cobelli C.: Glucose Effectiveness and Insulin
Sensitivity from the Minimal Models: Consequences o f Undermodeling
Assessed by Monte Carlo Simulation, IEEE Transaction on Biomedical
Engineering, Vol. 46, No. 2 ., 1999
5. Ta-Chen Ni, Marilyn Ader, Bergman R.N: Reassessment of Glucose
Effectiveness and Insulin Sensitivity From Minimal Model Analysis,
Diabetes, Vol. 46, 1997
6 . Richard N. Bergman, James C. Beard, Mei Chen: The Minimal
Modeling Method, Assessment of Insulin Sensitivity and p-cell function
in Vivo, Methods in Diabetes Research, Volume II: Clinical methods,
1986
7. Bergman R.N, Prager R., Volund Aage and Jerold M. Olefsky:
Equivalence of the Insulin Sensitivity Index in Man Derived by the
Minimal Model Method and the Euglycemic Glucose Clamp, JCI, Vol.
79:790-800, 1987.
8 . IRAS team: The Insulin Resistance Atherosclerosis Study (IRAS)
Objectives, Design and Recruitment Results, AEP Vol. 5, No. 6:464-472,
1995
59
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
9. Marilyn Ader, Ta-Chen Ni and Richard Bergman: Glucose Effectiveness
Assessed under Dynamic and Steady State Conditions, JCI, Vol. 99-
6:1187-1199, 1997
10 Jam es D. Best, Steven E. Kahn, Marilyn Ader, Richard Watanabe, Ta-
Chen Ni, Richard N. Bergman: Role of Glucose Effectiveness in the
Determination of Glucose Tolerance, Diabetes Care, Vol. 19, 9, 1996.
11. Baron AD, Brechtel G, Wallace P, Edelman SV: Rates and tissue sites
o f non-insulin- and insulin-mediated glucose uptake in humans, Am J
Physiol., 255:E-769-E774, 1988
12. Alzaid AA, Dinneen SF, Turk DJ, Cauomo A, Cobelli C, Rizza RA.:
Assessment of insulin action and glucose effectiveness in diabetic and
non-diabetic humans. J Clin Invest, 94:2341-2348, 1994
13. Marilyn Ader: Physiological Principles Underlying Glucose
Effectiveness, The Minimal Model Approach and Determination of Glucose
Tolerance, 159-186, Pennington Center Nutrition Series, Louisiana State
University Press, 1997
60
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Asset Metadata

Creator Panteleon, Antonios E. (author)

Core Title Assessment of minimal model applicability to longitudinal studies

School Graduate School

Degree Master of Science

Degree Program Biomedical Engineering

Publisher University of Southern California (original), University of Southern California. Libraries (digital)

Tag biology, animal physiology,engineering, biomedical,mathematics,OAI-PMH Harvest

Language English

Contributor Digitized by ProQuest (provenance)

Advisor Bergman, Richard (committee chair), D'Argenio, David (committee member), Marmarelis, Vasilis (committee member)

Permanent Link (DOI) https://doi.org/10.25549/usctheses-c16-41120

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Legacy Identifier 1407920.pdf

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Document Type Thesis

Rights Panteleon, Antonios E.

Type texts

Source University of Southern California (contributing entity), University of Southern California Dissertations and Theses (collection)

Access Conditions The author retains rights to his/her dissertation, thesis or other graduate work according to U.S. copyright law. Electronic access is being provided by the USC Libraries in agreement with the au...

Repository Name University of Southern California Digital Library

Repository Location USC Digital Library, University of Southern California, University Park Campus, Los Angeles, California 90089, USA