Am J Physiol Regul Integr Comp Physiol 291: R1638 –R1643, 2006.
First published July 27, 2006; doi:10.1152/ajpregu.00241.2006.
Long-range negative correlation of glucose dynamics in humans and its
breakdown in diabetes mellitus
Hitomi Ogata,1 Kumpei Tokuyama,1 Shoichiro Nagasaka,2 Akihiko Ando,2 Ikuyo Kusaka,2 Naoko Sato,2
Akiko Goto,2 Shun Ishibashi,2 Ken Kiyono,3 Zbigniew R. Struzik,3 and Yoshiharu Yamamoto3
1
Graduate School of Comprehensive Human Sciences, University of Tsukuba, Tsukuba, Japan; 2Division of
Endocrinology and Metabolism, Department of Medicine, Jichi Medical School, Tochigi, Japan; and 3Educational
Physiology Laboratory, Graduate School of Education, The University of Tokyo, Tokyo Japan
Submitted 7 April 2006; accepted in final form 25 July 2006
continuous glucose monitoring system; detrended fluctuation analysis;
positive/negative correlation
glucose dynamics are also characterized by a long-range negative autocorrelation. Specifically, by continuously estimating
the blood glucose level from the interstitial fluid in humans and
using an extended random walk analysis, namely, detrended
fluctuation analysis (DFA) (19), we first demonstrate longrange negative autocorrelation to hold in the “long-range” (⬎2
h) regime in healthy individuals with normal glucose homeostasis. This indicates that in the long-range regime, the net
effects causing temporal changes in glucose concentration are
negatively correlated, meaning that the flux is duly compensated for by the reflux, and vice versa.
This glucose homeostasis is known to be impaired in diabetes (15, 34), and an elevated glucose level in the circulation and
a subsequent urinary loss—a hallmark of diabetes mellitus—is
critical in the development of diabetic complications. In the
present study, we further show that patients with diabetes
mellitus are associated with positively correlated glucose dynamics in the same regime, indicating that the net effects of the
flux and reflux persist for many hours. We suggest that the
presence of such an anomalous, long-range positive correlation
likely reflects the underlying pathogenic mechanisms of diabetes; that is, the lack of long-term stability of blood glucose
and the long-range negatively correlated glucose dynamics are
indeed functional in maintaining normal glucose homeostasis.
METHODS
that physiological signals often
display a power-law type negative autocorrelation (2, 6, 19, 20,
30, 37, 38) rather than an exponential one, which would be
expected for standard linear negative feedback systems. Although the origins of such long-range negatively correlated
dynamics remain largely unknown, it is often associated with a
dual antagonistic control mechanism such as the balance of the
sympathetic and parasympathetic nervous systems in heart rate
control (2, 30, 37, 38) and the antagonistic muscular control of
human postural balance (6).
The concentration of glucose in the circulation is more
closely controlled than any other fuels (11) and is also regulated based on a balance between the flux (ingestion and/or
breakdown) and the reflux (uptake and/or storage), respectively, by hyperglycemic hormones (cortisol, epinephrine,
growth hormone, and glucagon) and hypoglycemic insulin (5,
21). Thus, in the present study, we hypothesize that normal
Subjects. Twelve nondiabetic subjects and fifteen diabetes mellitus
patients participated in the study. The nondiabetic subjects and diabetes mellitus patients were age-matched with each other; the characteristics of both groups are shown in Table 1. In particular, the body
mass index showed no significant difference between the patient and
nondiabetic subject groups.
All the nondiabetic subjects were in good health, had no known
medical disorders, and took no medication. Except for one, all of the
diabetes mellitus patients were being treated with insulin (intensive
insulin therapy using mainly ultra-short-acting and ultra-long-acting
insulin for nine patients, and conventional insulin therapy using
biphasic insulin twice daily for five). Three patients received additional oral hypoglycemic agents (alpha-glucosidase inhibitor for one
and biguanide for the other two). The study was approved by the local
ethics committee of the Jichi Medical School and the University of
Tsukuba, and all of the subjects gave their informed consent to
participation.
In this study, we did not distinguish between Type 1 and Type 2
diabetes. The patient group contained more Type 2 diabetes mellitus
patients than Type 1 patients.
Address for reprint requests and other correspondence: Y. Yamamoto,
Graduate School of Education, Univ. of Tokyo, 7–3–1 Hongo, Bunkyo-ku,
Tokyo 113– 0033, Japan (e-mail: [email protected]).
The costs of publication of this article were defrayed in part by the payment
of page charges. The article must therefore be hereby marked “advertisement”
in accordance with 18 U.S.C. Section 1734 solely to indicate this fact.
THERE IS GROWING EVIDENCE
R1638
0363-6119/06 $8.00 Copyright © 2006 the American Physiological Society
http://www.ajpregu.org
Downloaded from http://ajpregu.physiology.org/ at Karger Libri AG on July 27, 2012
Ogata, Hitomi, Kumpei Tokuyama, Shoichiro Nagasaka, Akihiko Ando, Ikuyo Kusaka, Naoko Sato, Akiko Goto, Shun Ishibashi, Ken Kiyono, Zbigniew R. Struzik, and Yoshiharu
Yamamoto. Long-range negative correlation of glucose dynamics in
humans and its breakdown in diabetes mellitus. Am J Physiol Regul
Integr Comp Physiol 291: R1638 –R1643, 2006. First published July
27, 2006; doi:10.1152/ajpregu.00241.2006.—Diurnal fluctuations in
glucose levels continuously monitored during normal daily life are
investigated using an extended random walk analysis, referred to as
detrended fluctuation analysis (DFA), in 12 nondiabetic subjects and
15 diabetic patients. The DFA exponent ␣ ⫽ 1.25 ⫾ 0.29 for healthy
individuals in the “long-range” (⬎2 h) regime is shown to be significantly (P ⬍ 0.01) smaller than the reference “uncorrelated” value of
␣ ⫽ 1.5, suggesting that the instantaneous net effects of the dynamical
balance of glucose flux and reflux, causing temporal changes in
glucose concentration, are long-range negatively correlated. By contrast, in diabetic patients, the DFA exponent ␣ ⫽ 1.65 ⫾ 0.30 is
significantly (P ⬍ 0.05) higher than that in nondiabetic subjects,
evidencing a breakdown of the long-range negative correlation. It is
suggested that the emergence of such positive long-range glucose
correlations in diabetic patients—indicating that the net effects of the
flux and reflux persist for many hours—likely reflects pathogenic
mechanisms of diabetes, i.e., the lack of long-term stability of blood
glucose and that the long-range negatively correlated glucose dynamics are functional in maintaining normal glucose homeostasis.
LONG-RANGE CORRELATED GLUCOSE DYNAMICS
Table 1. Characteristics of nondiabetic subjects and
diabetes mellitus patients
n
Sex (M/F)
Type 1/Type 2
HbA1c, %
Diabetes duration, yr
Age, yr
Height, cm
Weight, kg
BMI, kg/m2
Nondiabetic
Diabetes Mellitus
12
5/7
na
na
na
47.3⫾1.4
163.5⫾2.6
58.2⫾3.6
21.7⫾1.1
15
5/10
5/10
8.5⫾0.4
11.8⫾1.2
53.4⫾2.8
156.1⫾2.2
56.2⫾2.5
23.2⫾1.1
Data are expressed as means ⫾ SD. BMI, body mass index; HbA1c: glycated
hemoglobin concentration (12); na, not applicable.
AJP-Regul Integr Comp Physiol • VOL
glucose instability—that is, the variability of glucose under ambulatory, fed conditions within 24 h. The MAGE parameter gives an
indication of “dispersion” of the glycemic patterns of differing configurations. To emphasize the major glucose swings and eliminate
minor ones, only glucose excursions that exceed one standard deviation are used for the calculation of MAGE.
We define “absolute” MAGE⫹/⫺ as an arithmetic mean of either
successive increases (MAGE⫹) or decreases (MAGE⫺) in glucose
levels exceeding one SD of the whole record; the smaller value
implies tighter regulation. In this study, all the MAGE values (i.e.,
MAGE⫹/⫺, MAGE⫹, and MAGE⫺) were calculated and compared
with the scaling exponent-␣ as defined above.
Statistical analyses. Data were expressed as means ⫾ SD. Differences in the scaling exponents among nondiabetic subjects, diabetes
mellitus patients, and the reference ␣ ⫽ 1.5, were evaluated by the
Student’s t-test. The DFA plots, that is, plots of log10 F(n) vs. log10 n,
for the nondiabetic subjects exhibit the so-called “crossover” phenomenon: there exists a difference in the glucose correlation properties
between the short- (␣1) and the long-range (␣2) regimes. These ␣1,2,
as well as the crossover point, were determined for each subject or
patient and for each group by the best two-line fit based on the ␹2-test
and on the Akaike Information Criterion (AIC) (1). The Pearson’s
product moment correlation coefficients were calculated for comparisons of the ␣1,2, the averaged F(n), and the MAGE values.
RESULTS
During a one-day measurement period, the glucose dynamics were measured by the CGMS for each subject. Fig. 1 shows
two representative records of glucose dynamics for an age- and
gender-matched nondiabetic subject and a Type 1 diabetes
mellitus patient. The record for the patient shows a day-long
excursion in addition to distinct bursts due to meals. In contrast, for the nondiabetic subject the glucose level is generally
stable throughout the day, with the exception of small mealinduced bursts. Compared with nondiabetic subjects, the mean
glucose level is higher (90.6 ⫾ 7.7 vs. 157.2 ⫾ 49.5 mg/dl,
P ⬍ 0.01), and the standard deviation is greater (11.6 ⫾ 3.2 vs.
46.5 ⫾ 17.3, P ⬍ 0.01) in diabetes mellitus patients.
The DFA plots for all of the nondiabetic subjects (Fig. 2A)
and for the group average F(n) (Fig. 2C) exhibit the crossover
phenomenon. The individual crossover points fall in a range
from 70 to 195 min, while the crossover point for the grouped
data occurs at about 2 h. The locations of the crossovers
coincide, irrespective of whether they were obtained using the
␹2-test or AIC for the crossover criterion. In the short-range
Fig. 1. Representative records of day-long glucose dynamics for a nondiabetic
subject and a diabetes mellitus patient. These data are representative of
responses to a typical daily schedule of meals and sleep. The Type 1 diabetes
mellitus patient received intensive insulin therapy using mainly ultra-shortacting and ultra-long-acting insulin application (indicated on the time axis by
triangles).
291 • DECEMBER 2006 •
www.ajpregu.org
Downloaded from http://ajpregu.physiology.org/ at Karger Libri AG on July 27, 2012
Measurements. A continuous glucose monitoring system (CGMS;
Medtronic MiniMed, Northridge, CA) was used to monitor subcutaneous cell interstitial fluid glucose levels (13, 24). The batteryoperated monitor collects average glucose level measurements every
5 min, providing 288 glucose readings over a 24-h period. In the
CGMS, the subjects are required to self-calibrate the interstitial
glucose measurements against finger-stick blood glucose measurements (self-monitoring of blood glucose) at least four times a day,
avoiding times immediately following meals, insulin delivery, or
exercise, when blood glucose is relatively unstable.
The subjects did not participate in rigorous physical activity and
did not eat between meals during a one-day measurement period. In
addition, they refrained from all forms of caffeine, because caffeine
has a nonnegligible effect on glucose levels (10, 22, 26). They were
also instructed not to have big meals. The subjects followed a
consistent daily schedule in their meal times (0800, 1200, and 1800),
paying attention to the time they woke up (0600 – 0700), and the time
they went to sleep (2100 –2300) during the measurement period.
DFA. To estimate the correlation property of the glucose fluctuations, we used an extended random walk analysis, namely, DFA (19),
defined as follows. First, we integrated the glucose time series, and
then we prepared equally sized sliding (one point at a time) windows
of length n. For each window, the polynomial trend (linear, quadratic,
or higher order; the third order in this study), representing nonstationarity in that window, is fit to the data. The detrended fluctuations F(n)
are then calculated as the root-mean-square deviation from the trend
in each window, which is summed up for all of the windows covering
the entire analyzed time series. The procedure is repeated for different
window sizes (i.e., scales) n.
For long-range, power law-correlated signals, the magnitude of the
fluctuations F(n) and the window size n have a relationship: F(n) ⬃
n␣. The scaling exponent ␣, quantifying the degree of the long-range
correlations, can be obtained from the slope of a straight line fit to
F(n) ⬃ n␣ on a log-log plot. Uncorrelated white noise yields ␣ ⫽ 0.5
and the Brownian motion—an integrated white noise with uncorrelated increments or changes—yields ␣ ⫽ 1.5. Therefore, the longrange negatively correlated fluctuations or changes in glucose level
are reflected in ␣ ⬍ 1.5 for the absolute glucose measures, while the
positively correlated fluctuations are reflected in ␣ ⬎ 1.5.
This method yields ␣ estimates (␣ˆ ’s) from 288 data points, which
are highly correlated with true, predetermined ␣ values— 420 samples
uniformly distributed within the range of 1.0 ⱕ ␣ ⱕ 2.0: ␣ˆ ’s ⫽
1.02␣ ⫺ 0.02 (correlation coefficient ⫽ 0.95, residual standard
error ⫽ 0.10) for shorter scales n ⬍ 22 below the crossover point at
n ⫽ 22 (Fig. 2), and ␣ˆ ’s ⫽ 1.00␣ ⫺ 0.03 (correlation coefficient ⫽
0.86, residual standard error ⫽ 0.18) for longer scales n ⬎ 22.
Mean amplitude of glycemic excursions. Clinically, the tightness of
glucose regulation has previously been evaluated based on continuous
glucose measurements, by using the mean amplitude of glycemic
excursions (MAGE) (28). The MAGE is a measure of within-day
R1639
R1640
LONG-RANGE CORRELATED GLUCOSE DYNAMICS
Fig. 2. DFA plots for nondiabetic subjects
(A), patients with diabetes mellitus (B), and
the group mean fluctuation functions F(n)
(C); plotted are group means with SD as
error bars. Note that there is a crossover at
log10 n ⬃ 1.35 or n ⬃ 22 in healthy controls.
As the glucose sample was taken every 5
min, this corresponds to about 2 h for nondiabetic subjects and about 3 h for patients
with diabetes mellitus.
patients, while they are significantly related to the glucose
variability only in nondiabetic individuals.
DISCUSSION
The present study investigates fluctuations of glucose using
an extended random walk analysis, referred to as DFA (19).
The advantage of the DFA method is that it can more accurately quantify the correlation property of original signals, even
if masked by nonstationarity (in the form of the quadratic trend
in this case—resulting from third-order polynomial detrending
applied to integrated time series data) (31), compared with
traditional methods such as autocorrelation and/or power spectrum analysis (8). Consequently, it is found that, in nondiabetic
subjects, continuous glucose dynamics are positively correlated within timescales shorter than about 2 h, but negatively
correlated in longer timescales. On the other hand, patients
with diabetes mellitus, although they possess a similar positive
correlation to that of nondiabetic subjects in short scales, are
associated with a lack of negative correlation at longer scales.
This suggests a loss of long-term glucose regulation by a
negative feedback mechanism normally resulting from the
actions of hyperglycemic and hypoglycemic hormones.
Long-range correlation. The scaling exponent ␣ derived by
the DFA method reflects the probability of a simultaneous
increase or decrease in the variability at two distant points in
time in the time series, revealed for all distances up to longrange timescales, hence probing the nature of “switching”
patterns between high and low values in a statistical sense.
Scaling exponents larger than 1.5 indicate positive temporal
autocorrelation or “persistency” in the subsequent increases or
decreases, and values lower than 1.5 correspond with negative
autocorrelation or “antipersistency”—thus, with a higher likelihood of the switching of direction. In the present study, we
find such negatively correlated glucose dynamics only in nondiabetic subjects at a longer timescale than ⬃2 h.
It is well known that normal glucose homeostasis is
achieved, through a negative feedback regulation, by a tightly
Fig. 3. Relationships between mean amplitude
of glycemic excursions (MAGE) and the scaling
exponent (␣1,2 for nondiabetic subjects and the
overall ␣ for diabetes mellitus patients). The
MAGE indices for glucose excursion were separately calculated from increasing (MAGE⫹;
left), decreasing (MAGE⫺; middle), and the
absolute (MAGE⫹/⫺; right) values. Solid
squares show short-range (⬍2 h) exponents for
nondiabetic subjects; open squares show longrange (⬎2 h) exponents for nondiabetic subjects,
and solid circles show the overall exponents for
patients with diabetes mellitus.
AJP-Regul Integr Comp Physiol • VOL
291 • DECEMBER 2006 •
www.ajpregu.org
Downloaded from http://ajpregu.physiology.org/ at Karger Libri AG on July 27, 2012
regime (⬍2 h), the scaling exponent estimated from the group
average of slope values is ␣1 ⫽ 1.77 ⫾ 0.32, meaning that the
glucose fluctuations are positively correlated (P ⬍ 0.01 from
␣ ⫽ 1.5) for short timescales in healthy subjects. This implies
the net effects of glucose flux/reflux persist within these shorter
timescales. On the contrary, in the long-range regime (⬎2 h),
the scaling exponent is ␣2 ⫽ 1.25 ⫾ 0.29, suggesting that the
glucose dynamics is negatively correlated (P ⬍ 0.01 from ␣ ⫽
1.5). This means that, in these longer timescales, the glucose
level is tightly regulated by a negative feedback mechanism,
giving rise to the long-term stability.
The DFA plots for most of the diabetic patients do not show
the presence of any crossover phenomenon (Fig. 2B). Although
the two-line regression identified a crossover point for the
grouped data approximating 3 h, the scaling exponents estimated from the slope values in the short-range regime (⬍3 h;
␣1 ⫽ 1.72 ⫾ 0.21) were not statistically (P ⬎ 0.05) different
from those in the long-range regime (⬎3 h; ␣2 ⫽ 1.65 ⫾ 0.30).
The scale dependency of the group mean F(n) (Fig. 2C) reveals
that these exponents are compatible with those of nondiabetic
subjects in the short-range regime (P ⬎ 0.05 for the difference), but significantly (P ⬍ 0.05) larger in the long-range
regime. This suggests that the patients with diabetes mellitus
are associated with positively correlated glucose dynamics,
even in the long-range regime, indicating that the net effects of
the flux and reflux persist for many hours.
The scaling exponent for the patients is significantly (P ⬍
0.05) correlated with the MAGE values. However, neither the
short- (␣1) nor the long-range (␣2) exponent for nondiabetic
subjects is significantly correlated with the MAGE values (Fig.
3). On the contrary, in nondiabetic subjects, the average log10
F(n) for ⬎2 h, that is, the long-term variability in glucose
levels, is significantly (P ⬍ 0.05) correlated with the MAGE
values (Fig. 4), while the correlations between the long-term
glucose variability and the MAGE values is rather weak in
patients with diabetes mellitus. Thus it can be said that the
MAGE indices are closely related to a mechanism responsible
for the long-range glucose correlation only in diabetes mellitus
R1641
LONG-RANGE CORRELATED GLUCOSE DYNAMICS
Fig. 4. Relationships between MAGE and the
average log10 F(n) for ⬎2 h. The MAGE indices
for glucose excursion were separately calculated
from increasing (MAGE⫹; left), decreasing
(MAGE⫺; middle), and the absolute
(MAGE⫹/⫺; right) values. Solid squares show
the log10 F(n) for nondiabetic subjects, and solid
circles show those for patients with diabetes
mellitus.
AJP-Regul Integr Comp Physiol • VOL
jects (Fig. 2C). Studies revealing disturbed high-frequency
pulsatile secretory patterns in first-degree relatives of Type 2
diabetic individuals suggest that an impairment of insulin
pulsatility may be an early marker of ␤-cell dysfunction (9, 14,
23). In vivo, the ␤-cell responds to glucose challenges with
characteristic first (within 10 min) and second (⬎10 min) phase
insulin responses (29), and a prominent feature of Type 2
diabetes is a dramatic reduction in the first-phase insulin
secretion (7, 35). Thus it is speculated here that the greater
magnitude of short-range glucose fluctuations in diabetes mellitus patients might reflect diminished compensatory effects of
pulsatile secretory patterns and/or the first-phase insulin secretion on ongoing glucose excursions. This hypothesis should,
however, be tested by the direct observation of plasma insulin
in further research.
Relationship with MAGE. Various attempts have been made
by means of analysis of blood glucose and urinary glucose data
to assess the degree of glucose homeostasis and to facilitate
monitoring and diagnosis of glucose regulatory system dysfunction (17, 18, 27, 28). The MAGE is one of the standard
clinical indices to measure the tightness of glucose regulation,
defined as an arithmetic mean of either successive increases
(MAGE⫹ in our case) or decreases (MAGE⫺) in glucose
levels exceeding one SD of the whole record. Consistent with
a previous report (28), the MAGE of diurnal blood glucose
changes in our study is indeed greater in diabetes mellitus
patients compared with nondiabetic subjects. Some degree of
correlation between the MAGE values and ␣2 in diabetes
mellitus patients should be expected, because a greater level of
long-range positive autocorrelation, reflecting the probability
of a simultaneous increase or decrease in the variability, should
lead to larger glucose excursions, that is, the successive increases or decreases.
However, in nondiabetic subjects, neither the short- (␣1) nor
the long-range (␣2) exponent is significantly correlated with
the MAGE values. The likely cause of this lies with the very
definition of MAGE: it does not consider excursions below one
SD of the record. In other words, in nondiabetic subjects with
negatively correlated glucose dynamics in the long-range regime, which counterbalances the excursion gain of the shortrange positive correlation, the glucose excursions remain naturally embedded in variability around the mean level. This is
why the MAGE indices are in this case more closely related to
the magnitude, rather than to the correlation property, of
long-term glucose variability. Thus the scaling exponents derived by the DFA method, particularly that for the long-range
regime (␣2) uniquely identified in nondiabetic subjects, potentially provide deeper insights into the dynamics of glucose
291 • DECEMBER 2006 •
www.ajpregu.org
Downloaded from http://ajpregu.physiology.org/ at Karger Libri AG on July 27, 2012
controlled balance between glucose delivery or flux (from the
liver in the postabsorptive state and from the gut in the
postprandial state) and glucose utilization or reflux (5). The
present study further reveals that this tight control is not very
effective in the short-range regime but is effective in timescales
longer than ⬃2 h. Indeed, this coincides with the standard
recommendation for the evaluation of the blood glucose level
2 h after oral glucose loading (4, 32, 33, 36). After 2 h,
compensation for the glucose load takes place in nondiabetic
individuals, and the resultant glucose level returns to normal by
an effective negative feedback mechanism. For diabetes mellitus patients, however, the glucose level remains high because
of an impaired negative feedback regulation reflected in the
resultant lack of the negative correlation in the longer timescales. Thus we believe that the current results provide deeper
insights into the dynamics of long-term glucose homeostasis in
health and disease.
It is also of note that the negative correlation observed and
characterized by the scaling exponent ␣2 is of a “power-law”
type rather than an exponential one, which would be expected
for standard linear negative feedback systems. Thus it can be
argued that the normal long-term glucose homeostasis is a
nontrivial and far more complex system than that which is
explained by the linear feedback regulation. Although diurnal
or even circadian variations in the neuroendocrine system,
which exert a strong regulatory effect on the carbohydrate
metabolism (16), may be related to this phenomenon, an exact
mechanism responsible for the power-law antipersistency in
healthy glucose dynamics remains unknown. There are further
possibilities to explore fundamental aspects of glucose control
in terms of models of systems displaying such long-range
correlated behavior and in terms of their dynamical properties,
as has been done in the case of heart rate variability (2, 19, 20,
30, 37, 38), a thoroughly investigated example of biological
complexity showing long-range anticorrelation.
Short-range correlation. Both in nondiabetic subjects and in
patients with diabetes mellitus, glucose dynamics in the shortrange regime is characterized by the positive correlation or
persistency. This persistent pattern implies that the net effects
of glucose flux/reflux persist within these shorter scales, presumably as a result of the short-term response to meals or other
behavioral effects. In addition, a reported pulsatile pattern of
insulin within these scales (9, 21) and the associated persistent
glucose excursions might contribute to this phenomenon.
Although the scaling exponents in the short-range regime are
not different between patients and controls, the root-meansquare of fluctuation (F(n)) of glucose in diabetes mellitus
patients is considerably greater than that in nondiabetic sub-
R1642
LONG-RANGE CORRELATED GLUCOSE DYNAMICS
ACKNOWLEDGMENTS
The authors would like to thank Fumiharu Togo for his help in data analyses
and discussion.
GRANTS
This study was supported in part by a Research Grant from the Medical and
Welfare Equipment Laboratory, Japan (to Y. Yamamoto) and a grant from the
Japanese Ministry of Education, Science, Sports and Culture (No. 17300205 to
K. Tokuyama).
REFERENCES
1. Akaike H. Information Theory and an Extension of the Maximum Likelihood Principle. Budapest: Akademiai Kiado, 1973, p. 267–281.
2. Amaral LAN, Ivanov PC, Aoyagi N, Hidaka I, Tomono S, Goldberger
AL, Stanley HE, and Yamamoto Y. Behavioral-independent features of
complex heartbeat dynamics. Phys Rev Lett 86: 6026 – 6029, 2001.
AJP-Regul Integr Comp Physiol • VOL
3. Bantle JP and Thomas W. Glucose measurement in patients with
diabetes mellitus with dermal interstitial fluid. J Lab Clin Med 130:
436 – 441, 1997.
4. Bennett PH. Impact of the new WHO classification and diagnostic
criteria. Diabetes Obes Metab 1 Suppl 2: S1–S6, 1999.
5. Cauter EV, Polonsky KS, and Scheen AJ. Roles of circadian rhythmicity and sleep in human glucose regulation. Endocr Rev 18: 716 –738, 1997.
6. Collins JJ and De Luca CJ. Open-loop and closed-loop control of
posture: a random-walk analysis of center-of-pressure trajectories. Exp
Brain Res 95: 308 –318, 1993.
7. Fehse F, Trautmann M, Holst JJ, Halseth AE, Nanayakkara N,
Nielsen LL, Fineman MS, Kim DD, and Nauck MA. Exenatide augments first- and second-phase insulin secretion in response to intravenous
glucose in subjects with type 2 diabetes. J Clin Endocrinol Metab 90:
5991–5997, 2005.
8. Gough DA, Kreutz-Delgado K, and Bremer TM. Frequency characterization of blood glucose dynamics. Ann Biomed Eng 31: 91–97, 2003.
9. Hollingdal M, Juhl CB, Pincus SM, Sturis J, Veldhuis JD, Polonsky
KS, Porksen N, and Schmitz O. Failure of physiological plasma glucose
excursions to entrain high-frequency pulsatile insulin secretion in type 2
diabetes. Diabetes 49: 1334 –1340, 2000.
10. Johnston KL, Clifford MN, and Morgan LM. Coffee acutely modifies
gastrointestinal hormone secretion and glucose tolerance in humans:
glycemic effects of chlorogenic acid and caffeine. Am J Clin Nutr 78:
728 –733, 2003.
11. Kahn CR and Weir GC. Hormone-fuel interrelationships: fed state,
starvation, and diabetes mellitus. In: Joslin’s Diabetes Mellitus. Philadelphia: Lea & Febiger, 1994.
12. Koenig RJ, Peterson CM, Jones RL, Saudek C, Lehrman M, and
Cerami A. Correlation of glucose regulation and hemoglobin A1c in
diabetes mellitus. N Engl J Med 295: 417– 420, 1976.
13. Koschinsky T and Heinemann L. Sensors for glucose monitoring:
technical and clinical aspects. Diabetes Metab Res Rev 17: 113–123, 2001.
14. Lang DA, Matthews DR, Burnett M, and Turner RC. Brief, irregular
oscillations of basal plasma insulin and glucose concentrations in diabetic
man. Diabetes 30: 435– 439, 1981.
15. Leibiger IB, Leibiger B, and Berggren PO. Insulin feedback action on
pancreatic ␤-cell function. FEBS Lett 532: 1– 6, 2002.
16. Merl V, Peters A, Oltmanns KM, Kern W, Hubold C, Hallschmid M,
Born J, Fehm HL, and Schultes B. Preserved circadian rhythm of serum
insulin concentration at low plasma glucose during fasting in lean and
overweight humans. Metabolism 53: 1449 –1453, 2004.
17. Molnar GD, Gastineau CF, Rosevear JW, and Moxness KE. Quantitative aspects of labile diabetes. Diabetes 14: 279 –288, 1965.
18. Molnar GD, Taylor WF, and Ho MM. Day-to-day variation of continuously monitored glycaemia: A further measure of diabetic instability.
Diabetologia 8: 342–348, 1972.
19. Peng CK, Havlin S, Stanley HE, and Goldberger AL. Quantification of
scaling exponents and crossover phenomena in nonstationary heartbeat
time series. Chaos 5: 82– 87, 1995.
20. Peng CK, Mietus J, Hausdorff JM, Havlin S, Stanley HE, and
Goldberger AL. Long-range anticorrelations and non-Gaussian behavior
of the heartbeat. Phys Rev Lett 70: 1343–1346, 1993.
21. Picardi A and Pozzilli P. Dynamic tests in the clinical management of
diabetes. J Endocrinol Invest 7: 99 –106, 2003.
22. Pizziol A, Tikhonoff V, Paleari CD, Russo E, Mazza A, Ginocchio G,
Onesto C, Pavan L, Casiglia E, and Pessina AC. Effects of caffeine on
glucose tolerance: A placebo-controlled study. Eur J Clin Nutr 52:
846 – 849, 1998.
23. Rahilly SO, Turner RC, and Matthews DR. Impaired pulsatile secretion
of insulin in relatives of patients with non-insulin-dependent diabetes.
N Engl J Med 318: 1225–1230, 1988.
24. Rebrin K and Steil GM. Can interstitial glucose assessment replace
blood glucose measurements? Diabetes Technol Ther 2: 461– 472, 2000.
25. Rebrin K, Steil GM, Van Antwerp WP, and Mastrototaro JJ. Subcutaneous glucose predicts plasma glucose independent of insulin: implications for continuous monitoring. Am J Physiol Endocrinol Metab 277:
E561–E571, 1999.
26. Robinson LE, Savani S, Battram DS, MaLaren DH, Sathasivam P,
and Graham TE. Caffeine ingestion before an oral glucose tolerance test
impairs blood glucose management in men with type 2 diabetes. J Nutr
134: 2528 –2533, 2004.
27. Schlichtkrull J, Munck O, and Jersild J. The m-value, an index of
blood-sugar control in diabetics. Acta Med Scand 177: 95–102, 1965.
291 • DECEMBER 2006 •
www.ajpregu.org
Downloaded from http://ajpregu.physiology.org/ at Karger Libri AG on July 27, 2012
homeostasis that cannot be captured by a simple thresholdbased index like the MAGE.
Limitations. There are some limitations to the present study.
First, we did not select nondiabetic subjects by using an oral
glucose tolerance test to exclude any underlying (slight) abnormality in their glucose control; however, judging from the
CGMS records over a period of 1 day, we do not see grounds
to diagnose the nondiabetic subjects as suffering from impaired
glucose tolerance on the basis of fasting glucose levels, postprandial glucose levels, and average glucose levels. Second, in
the present study, we opted for diabetes mellitus patients who
were severe cases. Thus the estimated difference in glucose
correlation properties between the patients and the controls is
at its limits. In the future, a comparison will be needed of
patients with impaired glucose tolerance and patients with mild
diabetes mellitus. Third, we estimated the glucose level from
the interstitial fluid. The CGMS is considered capable of
accurately tracking acute changes in plasma glucose, but the
absolute values for the interstitial glucose concentration are
known to be lower than the plasma values (25). The interstitial
glucose level with some time lag is highly correlated with the
plasma glucose level (r ⬎ 0.9) (3, 25), which makes alterations
in the scaling exponents unlikely. However, the effects of
factors, such as glucose diffusion from the capillary, its clearance from the interstitial space, and insulin dependency, should
be carefully evaluated in the future.
Implications. We have found that the normal long-term
glucose homeostasis is maintained by long-range negatively
correlated dynamics of the regulatory system. This negatively
correlated dynamics is weakened, entirely absent, or even
reversed to positive correlation in patients with diabetes mellitus. Owing to recent advances in sensing technologies, continuous and least-invasive data collection for monitoring glucose dynamics, like the CGMS, should be feasible. The present
study has emphasized that a state-of-the-art technique of time
series analyses such as DFA could be used to monitor and
evaluate, in theory in a continuous fashion, the performance of
glucose homeostasis in health and disease. It would be intriguing to investigate in a larger study whether the evaluation of the
momentary breakdown of long-range negative correlation or
the appearance of persistent positive correlation in glucose
dynamics could be helpful for better diagnosis and prognosis of
patients with diabetes mellitus in a clinical setting.
LONG-RANGE CORRELATED GLUCOSE DYNAMICS
28. Service FJ, Molnar GD, Rosevear JW, Ackerman E, Gatewood LC,
and Taylor WF. Mean amplitude of glycemic excursions, a measure of
diabetic instability. Diabetes 19: 644 – 655, 1970.
29. Steil GM, Panteleon AE, and Rebrin K. Closed-loop insulin delivery—the path
to physiological glucose control. Adv Drug Delivery Res 56: 125–144, 2004.
30. Struzik ZR, Hayano J, Sakata S, Kwak S, and Yamamoto Y. 1/f
scaling in heart rate requires antagonistic autonomic control [Online].
Phys Rev E Stat Nonlin Soft Matter Phys. 70: 050901 [4 November 2004].
31. Taqqu MS, Teverovsky V, and Willinger W. Estimators for long-range
dependence: an empirical study. Fractals 3: 785–798, 1995.
32. The Expert Committee on the Diagnosis, and Classification of Diabetes Mellitus. Expert Committee on the Diagnosis, and Classification of
Diabetes Mellitus. Report of the expert committee on the diagnosis and
classification of diabetes mellitus. Diabetes Care 20: 1183–1197, 1997.
33. The Expert Committee on the Diagnosis, and Classification of Diabetes Mellitus. Follow-up report on the diagnosis of diabetes mellitus.
Diabetes Care 26: 3160 –3167, 2003.
R1643
34. Tony KT, Gutierrez-Juarez LR, Pocai A, and Rossetti L. Regulation of
blood glucose by hypothalamic pyruvate metabolism. Science 309: 943–
947, 2005.
35. Turner RC, McCarthy ST, Holman RR, and Harris E. Beta-cell
function improved by supplementing basal insulin secretion in mild
diabetes. Br Med J 1: 1252–1254, 1976.
36. World Health Organization. Diagnosis and classification of diabetes
mellitus and its complications. In: Report of a WHO Consultation. Part 1,
Diagnosis and Classification of Diabetes Mellitus. Geneva, Switzerland:
World Health Organization, 1999.
37. Yamamoto Y and Hughson RL. On the fractal nature of heart rate
variability in humans: effects of data length and ␤-adrenergic blockade.
Am J Physiol Regul Integr Comp Physiol 266: R40 –R49, 1994.
38. Yamamoto Y, Nakamura Y, Sato H, Yamamoto M, Kato K, and
Hughson RL. On the fractal nature of heart rate variability in humans:
effects of vagal blockade. Am J Physiol Regul Integr Comp Physiol 269:
R830 –R837, 1995.
Downloaded from http://ajpregu.physiology.org/ at Karger Libri AG on July 27, 2012
AJP-Regul Integr Comp Physiol • VOL
291 • DECEMBER 2006 •
www.ajpregu.org