MODELING IN PHYSIOLOGY
A kinetic mass balance model for 1,5-anhydroglucitol: applications
to monitoring of glycemic control
Douglas
Stickle1 and
John
Turk1,2
1 Division of Laboratory
Medicine, Department of Pathology, and
2 Division of Endocrinology,
Diabetes and Metabolism, Department of Medicine, Washington University
School of Medicine, St. Louis, Missouri 63110
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ABSTRACT |
The polyol 1,5-anhydroglucitol (AG) present in
human plasma is derived largely from ingestion and is excreted
unmetabolized. Reduction of plasma [AG] has been noted in
diabetics and is due to accelerated excretion of AG during
hyperglycemia. Plasma [AG] has therefore been proposed as a
marker for glycemic control. A precise understanding of its utility
relies on a quantitative understanding of the mass balance for AG. In
this study, non-steady-state data from the literature were analyzed to
develop a dynamic mass balance model for AG that is based on the
two-compartment model proposed by Yamanouchi et al. [T.
Yamanouchi, Y. Tachibana, H. Akanuma, S. Minoda, T. Shinohara, H. Moromizato, H. Miyashita, and I. Akaoka. Am. J. Physiol. 263 (Endocrinol.
Metab. 26): E268
E273, 1992]. The data are
consistent with a model in which exchange between tissue and plasma
pools is rapid and in which the tissue compartment mass is two to three
times the mass of the plasma compartment. According to model estimates,
accelerated excretion of AG due to hyperglycemia can cause marked net
depletion of total AG over a time scale of days. Recovery from a
depleted state is slow because the total body capacity represents >5
wk of normal intake. Accordingly, AG monitoring should be able to
indicate the presence of past glucosuric hyperglycemic episodes during a period of days to weeks, as well as provide information on the extent
to which high deviations from the average plasma glucose concentration
are operative.
mathematical model; diabetes; hyperglycemia; glycated
hemoglobin
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INTRODUCTION |
1,5-ANHYDROGLUCITOL (AG,
1-deoxyglucose) is a polyol present in human plasma and cerebrospinal
fluid (11, 17, 23). It is derived largely from ingestion (~5 mg/day),
and its major route of elimination is urinary excretion as the
unmetabolized substance (38). The plasma concentration of AG (~20
µg/ml or ~130 µM) is essentially stable in the normal population
(17). A diminished plasma [AG] has been noted in diabetics
(2, 16, 23, 24, 29, 32, 40). The reduction in [AG] appears
to be due to accelerated urinary excretion of AG in parallel with
urinary excretion of glucose during hyperglycemia (1, 8, 17, 20, 36) and is likely attributable to competition between glucose and AG for
tubular reabsorption (16). For these reasons, plasma AG monitoring has
been suggested and/or advocated as a marker for glycemic
control (3, 17, 29, 40). At present, however, the interpretation of
serial AG measurements has been based more on a qualitative than a
quantitative understanding of the data regarding the relationship
between hyperglycemia and AG excretion (38). The purpose of this study
was to characterize quantitatively the mass balance for AG and the
kinetics of AG disposal by use of data existing in the literature to
define more precisely the information that might be derived from
monitoring of AG.
A steady-state two-compartment mass balance model for AG has previously
been proposed by Yamanouchi et al. (38). In this model (Fig.
1), the ingestion rate
(k1) and a
modest rate of endogenous production
(k5) are
balanced in the steady state by the excretion rate
(k4), with
steady-state exchange between plasma
[A, within volume of
distribution (plasma volume)
VA] and tissue pools
[B, within volume of
distribution (VB)]
occurring with rate constants
k2 and
k3. The effect of
high glucose to inhibit reabsorption of AG increases the excretion rate
and can lead to a net depletion of AG. Unknowns in this system are
1) the mass within the tissue pool,
B, and its volume of distribution,
VB; 2) the rate constants
k2 and
k3 for exchange
between the plasma and tissue pools; and
3) the functional dependence of the
AG excretion rate
k4 on plasma
glucose. As will be shown, the steady-state mass balance
model proposed by Yamanouchi et al. (38) can be expanded to include
the kinetic characteristics of AG disposal by an evaluation of these
unknowns with use of data from published studies. The implications of
the kinetic model for the use of AG monitoring in the evaluation of
glycemic control are discussed.
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KINETIC MODEL ANALYSIS OF AG MASS BALANCE |
Two-Compartment Mass Balance Model for AG
To begin a more detailed analysis of the steady-state two-compartment
mass balance model for AG of Yamanouchi et al. (38), the model was
expanded to include kidney function to account explicitly for
filtration, reabsorption, and excretion of plasma AG, as shown in Fig.
1. In this scheme, the fractional reabsorption,
r, represents the fraction of AG that
is reabsorbed after filtration (i.e., r is the fraction of the amount that
is filtered but is not excreted). The rate of AG excretion is given by
the product of the glomerular filtration rate (F), the plasma AG
concentration ([A]), and
the fraction of AG that is not reabsorbed (1
r)
|
(1)
|
where
r is a variable that is presumed to
depend on the plasma glucose concentration, [Glc]. In this
model the instantaneous rates of change of masses
A and
B are given by
|
(2)
|
|
(3)
|
For the case of constant
coefficients
(k1-5),
the mass balance Eqs. 2 and 3 for the model shown in Fig.
2 have simple analytic
solutions for A(t) and
B(t) of the form
A(t) =
+
+
B(t) =
+
+
3. Such an analytic solution
enables evaluation of the instantaneous rates of change for
A and
B and the overall kinetics of AG
disposal if all coefficients are known for all conditions. The rate
constants k2 and
k3, the volume of
distribution VB, and the rate k4 are all
unknowns. As will be shown below, existing data enable a
condition-dependent determination of the excretion rate
k4 by characterization of the dependence of
r on [Glc]. In addition, the data suggest a rapid exchange of AG between
A and
B, leading to a simpler form of the
mass balance model for which
k2,
k3, and
VB are not explicit variables.

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Fig. 1.
Two-compartment steady-state mass balance model for 1,5-anhydroglucitol
(AG) based on model proposed by Yamanouchi et al. (38). Ingestion rate
(k1 mass/time) and a modest rate of endogenous production
(k5 mass/time) are balanced in steady state by excretion rate
(k4 mass/time), with exchange between plasma
(A) and tissue
(B mass) pools, of volumes of
distribution VA and
VB, occurring with exchange rates
k2 and
k3 ( volume/time) as shown. Model assumes no metabolism of AG. Kidney
processes include filtration, reabsorption, and excretion. Glomerular
filtration rate (F volume/time) permits mass entry of AG according
to F[A] ( mass/time); a
fraction r of that mass is reabsorbed,
whereas a fraction (1 r) is
excreted. Reabsorption can be inhibited by high plasma
[Glc]. Unknowns in this model system are the mass of AG
within the tissue pool, B, the volume
of distribution for the tissue pool,
VB, the rates of exchange between
the plasma and tissue pools
(k2,
k3), and the
functional dependence of the excretion rate
k4 on
[Glc] or on
[A].
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Fig. 2.
Concentrations of blood glucose (A),
plasma AG (B), urinary glucose
(C), and urinary AG
(D) vs. time during an oral glucose
tolerance test (OGTT). Data were redrawn from original study by Akanuma
et al. (1). Patients were grouped according to World Health
Organization standards on the basis of blood glucose profiles: normal
(N, ), impaired (I, ), and diabetic (D, ) subjects. In
C, data points were connected by a
cubic spline procedure for use in model simulations. In
B and
D, plasma and urinary AG profiles
(dotted lines) were obtained by model simulation, as described in text,
with initial condition of time 0 data.
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Characterization of Fractional Reabsorption of AG as a Function
of Plasma Glucose
The relationship between r and
[Glc] can be characterized empirically by analysis of data
from a study by Akanuma et al. (1), in which plasma and urine AG were
measured during the oral glucose tolerance test (OGTT) (Fig. 2). On the
basis of the temporal measurements of blood glucose (Fig.
2A), subjects in this study were
classified within three groups according to World Health Organization
standards: normal (N), impaired glucose tolerance (I), and diabetic (D)
subjects. As shown in Fig. 2B,
zero-time plasma AG concentrations were distinct for each group,
conforming to the expectation that the degree of impairment in glucose
tolerance should correlate with the degree of AG depletion compared
with normal subjects; these values remained constant throughout the
test period. Urine glucose followed an expected pattern for the three
groups (Fig. 2C), in which glucose was excreted during the test period in both I and D groups but not for
the N group. The patterns for excretion of AG were distinct for each
group (Fig. 2D), and these patterns
were in qualitative accord with the schematic model expectation: for N
subjects, there was a constant and low excretion rate derived from a
fixed plasma AG in the absence of hyperglycemia; for I and D subjects,
excretion from lower plasma AG concentrations was increased during
periods of hyperglycemia and exceeded the excretion observed in N
subjects.
The AG data from the OGTTs do not directly constitute a mass balance,
because urine concentration of AG was measured rather than urine output
of AG. However, the data can be analyzed to obtain an estimate of
r as a function of [Glc],
with the assumptions of a normal value for urine output rate (
,
ml/min) and an average and equivalent normal glomerular filtration rate
(F, ml/min) among the three groups. First, AG clearance (
) can be
calculated according to
|
(4)
|
where
[U] is urine concentration
(mass/volume) and [A] is
the plasma concentration of AG. Second,
r can be calculated from
given F
according to
|
(5)
|
when
clearance is assumed to involve filtration and reabsorption with no
intervening secretion of AG.
The glomerular filtration rate F was assumed to be a normal value of
100 ml/min, and a normal value for urine output rate,
= 51 ml/h,
was assumed for use in the calculations. On this basis the urine
excretion rate for N subjects was equal to the designated net intake
rate of 5 mg/day. According to the calculations, when these values are
used, AG clearance in N subjects was small compared with F (
< 1 ml/min) and was increased in I and D groups in rough proportion to
glucose excretion (Fig. 3), where glucose excretion occurred for plasma glucose >8 mM (also shown in Fig. 3).
Acute urinary excretion of AG accompanying and in proportion to
glucosuria is in general accordance with data obtained in previous studies on longer time scales in humans (1, 14, 35) and in rats (8, 33,
36, 39). The relationship of r to
glucose excretion is not of signal importance with respect to the
kinetic mass balance for AG except as a rough confirmation of the
validity of the assumptions used in the calculation by the
correspondence of the results to those observed in previous studies. Of
greater importance is the relationship of
r to [Glc] that can be
obtained from the data in Fig. 2. As is shown in Fig.
4, r as a
function of [Glc] decreased in rough proportion to
[Glc] when [Glc] exceeded a threshold of
[Glc]
8 mM, from an apparently constant value of >0.99
below the threshold to <0.97 for [Glc] = 20 mM. The AG reabsorption data for the N, I, and D groups appear to be connected by
a continuous function of plasma glucose alone, despite the fact that
each condition involved widely different but essentially constant
values of [A]. The
reduction in r is significant in terms of clearance of AG and the kinetics of excretion, because it represents a fourfold increase in the clearance rate (i.e., a 4-fold increase in
the excretion fraction, 1
r)
compared with the N group over the range of values for
[Glc] observed for the D group.

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Fig. 3.
A: AG clearance vs. [Glc]
excretion. Clearance was calculated assuming a constant urine output of
51 ml/h and using data in Fig. 2, B
and D; value of 51 ml/h was chosen
such that excretion rate for N subjects was equal to net input rate of
5 mg/day. B: glucose excretion vs.
plasma [Glc]. Plasma glucose was calculated as the average
(AVG) during intervals connecting points in Fig. 2 by use of spline
data from Fig. 2A. , N; , I; and
, D glucose tolerance groups.
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Fig. 4.
Fractional reabsorption of AG vs. plasma [Glc]. Fractional
reabsorption (r) was calculated
assuming a constant glomerular filtration rate (F = 100 ml/min), given
AG clearance in Fig. 3, and with assumption of no tubular secretion of
AG. , N; , I; and , D glucose tolerance groups. Lines shown
connect constant r in normal range
(r = 0.9984, [Glc] <7.4
mM) and as a function of [Glc] for hyperglycemia
(r = 0.0026
mM 1 [Glc] + 1.018, [Glc] >7.4 mM) obtained from a linear regression
of data points for I and D groups.
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Simplification of the Model, Assuming Rapid Exchange Between Pools A
and B
The data in Fig. 2 also provide some information about the kinetics of
the exchange of AG between the tissue and plasma pools in the
two-compartment model. Although the total excretion for the D group
over this time represented ~10% of the total mass within the plasma
pool for these subjects, the plasma level of AG did not change on the
time scale of the glucose tolerance test for this group (or for the I
or N groups). Despite the variability in the data in Fig.
2B, they suggest that loss of mass
from the plasma pool via excretion can be rapidly replaced, as would
occur with relatively rapid exchange between tissue and plasma AG
pools, such that the mass available for excretion is drawn from both pools even within a short time frame. The interpretation of relatively rapid redistribution and availability to plasma of the majority of the
total body AG mass is broadly consistent with the high flux of AG into
urine during the major, rapid phase of plasma AG depletion and
accompanying tissue AG depletion that occur in glucose-overloaded or
streptozotocin-treated rats (8, 33) and with the
observation of rapid plasma-erythrocyte exchange of AG (17). In
glucose-overloaded rats, an estimate of the average tissue content
(mass/volume) decreased to 66% of control within a 2-h period, during
which plasma [AG] decreased to 49% of control (33),
indicating coupling of the masses within the two pools on a relatively
short time scale. The supposition that the total AG mass partitions
rapidly between plasma and tissue pools leads to a simplification of
the mass balance model, as shown in Fig. 5.
In the revised model, the mass of the tissue pool is proportional at
all times to the mass in the plasma pool, according to the relationship
|
(6)
|
such
that B = KA, where
K = k2/k3
VB/VA.
According to this model, the instantaneous rate of change of the total
body mass of AG (C, where
C = A + B) is given by
|
(7)
|
where
ki = k1 + k5. Because the
plasma pool mass A is proportional to
the total mass C by the relation
A = C/(1+K),
Eq. 7 can be written in terms of
C only
|
(8)
|
This
equation has a simple analytic solution for constant coefficients
|
(9)
|
where
= VA(1+K)/[F(1
r)] is the time
constant for changes in C, and where
is dependent on r as a function of
[Glc]. The steady-state value for
C is given by the product ki. From the
standpoint of data analysis, because r
as a function of [Glc] is not constant in hyperglycemia
unless [Glc] is constant, the utility of
Eq. 9 is the provision of an explicit
function for the rate of change of C
that is calculable given the known dependence of
r on [Glc]
|
(10)
|
A(t)
and
B(t)
are calculable from
C(t)
by the relations A = C/(1+K)
and B = CK/(1+K)
if K is known. At this stage, then, a
complete specification of the kinetic mass balance model lacks only a
value for K.

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Fig. 5.
Simplified kinetic mass balance model assuming rapid partitioning
between pools A and
B. Production and ingestion rates have
been combined to form a single constant input rate,
ki = 5 mg/day.
K is proportionality constant for
partitioning of mass between pools,
K = B/A.
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Estimation of the Mass of AG in Pools A and B and of the
Proportionality Constant K
An estimation for the proportionality constant
K can be obtained in two ways by using
Eq. 9 when any value for
is given that was obtained under conditions in which
r is constant and known. First, a
value for
specifies the mass of AG in the total body pool
(C), given a known value for the
overall input rate, ki, because
steady-state C is equal to the product
ki
. Because the steady-state amount of AG in the plasma pool
A is known directly by measurement,
the size of the B pool is known by the
difference between C and
A, and from this distribution a value
for the proportionality constant, K,
is obtained by K = B/A.
Second, because
is an explicit function of
VA,
K, F,
and r {
= VA(1+K)/[F(1
r)]}, then by
rearrangement, K =
F(1
r)/VA
1. Thus an estimation for the proportionality constant
K can be obtained from a value for
, obtained under conditions of a known and constant value for
r, given that
VA and F are known constants.
The recovery of plasma AG from a depleted state to a normal state after
initiation of "glycemic control" in type II diabetic subjects
[Fig. 6; data from the study of
Akanuma et al. (1)] is an example of data demonstrating changes
in plasma [AG] under conditions of a relatively constant
r associated with normoglycemia. Such
data are suitable for determination of
for the estimation of
K. In these data, plasma
[AG] demonstrated an increase to a steady state of ~20
µg/ml after 10 wk, and the time course of the increase is well
characterized by a single time constant,
5.6 wk. The recovery
data thus are compatible with the analytic solution to the simplified
mass balance model for
A(t)
with a constant r. As described above,
K can now be computed in two ways.

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Fig. 6.
Recovery of depleted plasma AG after initiation of "glycemic
control" in type II diabetics. Data were redrawn from original study
by Yamanouchi et al. (35). Line, fit of data to Eq. 9, with assumption of a constant fractional
reabsorption, r, for normoglycemic
state, namely, r equal to that for N
group obtained from Fig. 4 (r = 0.998). For [A] = [A]s
[1
e (t t0)/ ],
steady-state value for plasma [AG]
([A]s) = 20.4 µg/ml, = 5.6 wk. Initial condition was assumed to be
[A] = 0, and effective
time 0 point
(t0 = 1.1 wk) was
allowed to shift in fit to accommodate a period of establishment of
glycemic control; first 2 data points were excluded from fit.
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1) K
computed from
given
ki.
According to Eq. 9, the steady-state value for the
total body content C is the product
ki
. Given
5.6 wk and
ki = 5 mg/day, then C is calculated to be 196 mg. The
mass of AG within the tissue pool can be calculated as the difference
between the total mass, C, and the
mass within the plasma pool: B = C
A, where
A is known (calculated as
[A]VA,
where VA is assumed to be a round
value for normal subjects of 3,000 ml). Given the steady-state value
for [A], ~21 µg/ml,
then A, the mass of AG present in the
plasma pool, is ~63 mg. The tissue pool
B is thus (196
64) = 132 mg,
approximately two times the mass of AG within the plasma pool. Given
that K = B/A,
then K = 2.1. This value is sensitive
to the precision used for the calculated values for A and
C; with use of round numbers,
C = 200 mg and
A = 60 mg, and then
K = 2.3.
2) K
computed from
given
VA, F, and
r.
By use of K = F(1
r)/(VA
1), then K can be obtained,
given values for
, VA, and F
and r as constants. For
= 5.6 wk,
VA = 3,000 ml, F = 100 ml/min,
and r = 0.9984, then
K = 2.0. This value is obviously most
sensitive to the precision of the value used for
r. Decreasing
r to three significant figures (to
r = 0.998) increases
K to
K = 2.8.
Thus two approaches to the calculation of the value of
K result in the range of values
K = 2.0-2.8. These numbers can be
compared with an estimated upper bound for
K that can be obtained from the same
data. The maximum possible total body content,
C, from the recovery data would be the
integral of the input rate,
ki, during the
recovery period of ~10 wk, by assuming zero excretion; this would
lead to a value for C of ~300 mg,
corresponding to an upper bound of K = 4 [by K = B/A = (300
60)/60 = 4]. Another way to estimate an upper bound
for K with the assumption of zero excretion is to take the increase per unit time in
A and calculate the partitioning into
B of the mass input rate
ki that would
have to occur to account for the net increase in
C due to
ki. The data in
Fig. 6 show a maximum rate of increase in
[A] of 1.94 µg · ml
1 · wk
1;
with the assumption VA = 3,000 ml, then this is equal to an increase of 5.8 mg/wk in
A; for
ki = 5 mg/day
(35 mg/wk), then the increase in B
would be equal to
ki
5.8 mg/wk = 29.2 mg/wk with the assumption of zero excretion; this
partitioning would correspond to K = (29.2/5.8) = 5. Thus an upper limit for K is in the range of
K = 4-5, which is a factor of 2 greater than that obtained with the kinetic model equations analysis.
This calculation is simply for comparison with the values obtained above, given that there are no data for
r at low
[A] in normoglycemia; to
say that excretion was insignificant except at near-normal values for
[A], then the value of
K would be at most in the range of
4-5. If r is even
greater than 0.998 at low
[A], then the value for
K obtained by the analysis using
Eq. 9 may be regarded as a lower
limit.
From this, the parameters necessary to characterize completely the
kinetic mass balance for AG have been defined and evaluated (Table
1): the proportionality constant for the
exchange plasma and tissue pools
(K), which gives their relative
amounts, and the dependence of the excretion rate on glucose
(r = f [Glc]), which
determines the time constant (
) for changes in total AG and the
associated steady-state values for A
and B, given a constant input rate,
ki. As a check
on the model parameters, the results of model simulations are compared
with the OGTT data in Fig. 2. Simulations were performed from the
initial conditions
{[A](0) for each
group} by successive calculations of Eq. 9 at 1-min simulation intervals for which
[Glc](t) was given from
the spline interpolation in Fig. 2A,
and from which r was calculated using
the correlation given in Fig. 4. The correspondence of simulation
results to the urine excretion data in Fig.
2D was inexact only because of the difference between the correlation of
r to [Glc] and the exact values of the small number of measurements for
r. For the AG recovery data in Fig. 6,
model values of
[A](t)
can be calculated directly from Eq. 9
by use of the model parameter values in Table 1. Because this is the
same equation as that used in fitting the data, the only difference
between the model predictions and the data set is a slightly different
value for the steady-state
[A] between the recovery
data fit (20.4 µg/ml) and the value used in Table 1 based on the OGTT
data for the N group (21 µg/ml).
Time Constants for Changes in
[A] Derived from Changes in
[Glc]
Previous studies of long-term profiles for [Glc] and
[A] in individuals have shown that episodes of
hyperglycemia are tracked by decreases in
[A], such that
[A] can be used as a
monitor of hyperglycemia on a time scale of days or weeks (31, 33). To examine this potential use from the standpoint of the model
calculations, the time constant (
) for changes in total AG and
steady-state values for
[A] for constant
coefficient conditions of the model were calculated using the parameter
values of Table 1 and the function
r([Glc]) from Fig. 4. As
shown in Fig. 7, values for
ranged from
months to days for glucose concentration, ranging from 5 to 20 mM, with
the corresponding steady-state values for [A] decreasing with
glucose concentration from normal to 5% of normal. According to the
calculation, marked net depletion of AG via the plasma pool due to
hyperglycemia can thus occur relatively quickly depending on the
magnitude of hyperglycemia. For instance, a step change to 20 mM
glucose produces a steady state in which plasma AG is reduced to ~5%
of its initial value, with a time constant for approach to the steady
state of 44 h, or ~1.8 day. Note that the analytic function given by
Eq. 9 only gives a time solution for
AG mass under the condition of constant coefficients (e.g., for a step
change to a different and fixed [Glc], with an associated
step change in r). Although step
changes in fixed glucose are not part of normal or disease physiology,
the model value for
indicates the instantaneous rate of change that
would be operative under specified conditions, and it depicts
boundaries on the mass within the system under different steady-state
conditions.

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Fig. 7.
Time constants ( ) for changes in plasma [AG] and
steady-state values for plasma [AG]
([A]s)
as a function of [Glc] according to Eq. 9, with use of model parameters given in Table 1.
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Because of the dependence of excretion of AG on both the magnitude and
duration of glucosuric hyperglycemic excursions on short time scales,
serial measurements of plasma AG probably cannot be used as a
retrospective indicator of the sequence of conditions that were
operative during the interval between any two measurements. That is to
say, AG monitoring on the usual time scale for evaluation of diabetic
care will not identify a unique set for the characteristics (frequency,
duration, magnitude) of possible intervening hyperglycemic episodes,
although serial measurement may place boundaries on estimates of the
characteristics of such episodes when they occur. Moreover, because the
repletion rate for AG is slow, AG measurement should be able to
indicate accurately the occurrence of glycemic episodes during
monitoring periods on the scale of days or weeks, as has previously
been suggested (38).
Monitoring of AG Compared With Monitoring of Glycated Hemoglobin
Measurement of glycated hemoglobin (gHb) assesses the average of blood
glucose concentrations on the basis of weeks (12, 26), but it provides
essentially no information about the magnitude of short-term glycemic
excursions about the average that characterize unstable diabetic
control (22). In contrast, because AG excretion is accelerated under
conditions of glucosuric hyperglycemia, monitoring of AG can
theoretically provide retrospective information on glycemic excursions
that would not be apparent from monitoring of gHb alone, as has
previously been suggested (9, 31, 35, 37). Specifically, according to
the model, AG monitoring should be able to distinguish between
conditions in which a set average plasma glucose is accompanied by
greater or lesser glucosuric hyperglycemic excursions. A hypothetical example is shown in Fig. 8. Three different
glucose profiles, repeating with the same period and with identical
average glucose values, were used as inputs to determine
r and to calculate plasma [AG] as a function of time according to a 90-day simulation
of the AG mass balance model, beginning in each case with the same initial condition for [AG]. By design, each of these
glucose profiles would in principle produce no significant change in
gHb concentration during the simulation period, and, in addition, there
should be no difference in gHb concentrations among the profiles. In
contrast, AG measurements for the two profiles that were designed with
excursions of [Glc] above the glucosuric threshold show
continuing changes in AG as well as differing extents of the changes in
AG as a function of time. AG depletion is more rapid for greater
excursions from the mean and from the glucosuric threshold, and the
depletion indicates circumstances that would not be discernible via
analysis of gHb measurements alone (37). The results accord with human studies tracking the relationship of
[A] to [Glc]
in individuals (31, 33), although the time resolution of the data in
those studies is too low to permit a direct comparison of those results with the predictions of the model.

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|
Fig. 8.
A: 3 theoretical time profiles for
plasma [Glc] for which mean [Glc] values are
equal (6.5 mM) but with varying degrees of excursion from the mean. A
variable amplitude sine waveform was chosen for profile of greatest
excursion amplitude (profile 1); a
simple sine waveform (profile 2) and
a constant waveform (profile 3) were
chosen to match the same mean as for profile
1. B: time courses of
plasma [AG] according to simulations of mass balance model
for plasma glucose profiles 1-3
shown in A. For simulation,
Eq. 9 with constant
ki was used to
obtain successive values for C, by
calculating Ci+1 = Ci+dC/dt
t, with t = 5 min and with updated for each 5-min interval, using [Glc] from glucose
profile to obtain r from correlation
of Fig. 4 to calculate . Plasma [AG]
([A]) was calculated from
C by
[A] = C/(1+K)/VA.
Initial condition for C was set equal
to steady state for constant plasma glucose profile (corresponding to
[A] = 130 µmol/l).
C: simulated time courses of plasma
[AG] continued from (B)
with 3 plasma glucose profiles [profiles
1-3 in
(A)] repeated over a period of
90 days.
|
|
The simulation data illustrate the point that monitoring data will
probably be unable to specify uniquely the conditions that generate
them. On a long time scale, the simulation data are well characterized
by a single exponential with a single time constant (Fig. 8), but this
composite time constant greatly underestimates the actual magnitude of
the glucose excursions above the mean compared with the relationship of
the instantaneous time constants
to glucose concentrations (Fig.
7). Thus weekly or monthly monitoring data that result from fluctuating
blood glucose concentrations (the usual case) would be consistent with
numerous patterns of blood glucose profiles in addition to the one
profile that generated the result.
 |
DISCUSSION |
Because of the coupling of AG excretion to blood glucose
concentrations, the potential use of measuring plasma AG to screen for
diabetes or to monitor glycemic control in diabetes has been widely
investigated. A two-compartment mass balance model proposed by
Yamanouchi et al. (38) has thus far provided the basic framework whereby AG measurement for monitoring has been interpreted.
Schematically, the model states that a slow rate of AG ingestion is
balanced by a slow rate of excretion in N subjects, whereas in D
subjects a decrease in AG results from increased excretion in the
presence of acute glucosuric hyperglycemia that is not replaced in
balance by ingestion. Thus decreases in plasma AG reflect past
glucosuric hyperglycemia on a time scale that is derived from the
relative excretion and ingestion rates of AG. The kinetic aspects of
the model for the nonsteady state have not heretofore been
quantitatively characterized. In this study, the model of Yamanouchi et
al. was expanded to attempt to characterize quantitatively the
parameters on which dynamic changes in AG depend to better define the
basis for the use of AG measurement in monitoring glycemic control. This analysis relied on relevant data from two previous studies (2,
35). The data are consistent with a two-compartment kinetic mass
balance model in which 1) the
distribution between tissue and plasma pools of AG is in the ratio of
approximately 2:1; 2) the exchange
between tissue and plasma pools is sufficiently rapid to be treated as
an equilibrium partitioning between the two pools; and
3) reabsorption of AG is between 99 and 100% in normoglycemia but decreases significantly in hyperglycemia
in approximate proportion to the extent of hyperglycemia above the
glucosuric threshold.
The conclusions of the model analysis are in accordance with previous
interpretations of the relationship of AG to hyperglycemia (see review
in Ref. 31). The total body pool of AG can be depleted relatively
rapidly by glucosuric hyperglycemia, on the time scale of days in the
presence of overt hyperglycemia. Because AG is derived from dietary
intake and because the normal daily intake of AG represents a small
fraction of the total body pool of AG in the replete normoglycemic
steady state, recovery from a significantly depleted state is slow even
under continuously normoglycemic conditions. Thus decreased plasma AG
reflects glucosuric hyperglycemia within a time scale of days or
weeks.
According to the model, depletion of plasma AG depends on both the
magnitude and duration of glucosuric hyperglycemia, and measurement
of plasma AG will not be attributable to a unique set or sequence of
prior episodes of hyperglycemia. AG monitoring may nonetheless be
useful to identify episodes of glucosuric hyperglycemia. As in the
example given and in accordance with the conclusion from previous
studies (33), AG monitoring may provide adjunct information about the
characteristics of glycemic episodes that is distinct from that derived
from monitoring of gHb. Whether identification of such circumstances
would be useful from a clinical standpoint remains to be established.
AG monitoring might be useful in diabetes management both to assess
fluctuation of blood glucose concentrations and to document the
continuous absence of glucosuric hyperglycemia. It is emphasized that
the AG mass balance is only significantly affected by excursions of
glucose when the renal threshold for glucose reabsorption is exceeded;
AG monitoring would not be useful for monitoring of hyperglycemia of
patients who are not well controlled but whose glucose levels
nonetheless stay below the renal threshold for glucose reabsorption.
Individual variations in glomerular filtration rate would need to be
considered in interpretation of AG monitoring results. The fractional
reabsorption data shown in Fig. 4 for high glucose concentrations
probably represent an upper boundary for
r values, because whereas an average
and "normal" glomerular filtration rate was assumed for the
calculation of r, it is not improbable that F for the D group would be less than average and that due to
osmotic diuresis the urine output rate would be greater than average.
In this circumstance the calculated fractional reabsorption r would be less than that shown in
Fig. 4. A reduction in r would have
the net effect of further decreasing the calculated time constants for
changes in AG in the D cases.
Kidney function can also affect AG clearance in other ways. Decreases
in AG occur in uremia unrelated to diabetes (15, 18). Chronic renal
failure alone can lead to decreased plasma [AG], presumably
due to tubular damage and ineffective reabsorption (4, 30), and a
recent paper discusses the potential use of AG measurement to
distinguish chronic from acute renal failure (30). Interindividual
variations in F may be part of the reason that AG measurement would
appear to be better used in serial measurements for monitoring of
diabetic control than as a single measurement to screen for diabetes.
Diabetic screening is a complex issue (5, 7, 10), and AG screening
studies have yielded differing results (13, 21, 28, 34). Complexities
for AG use in screening for diabetes include the observation of a
difference in average plasma [AG] among normal subjects (as
defined by an OGTT) depending on whether there exists a family history
of non-insulin-dependent diabetes mellitus (27). Age is also a factor
in defining normal reference values for plasma AG (6, 28), and possible
variations in the ingestion rate of AG or factors attributable to
ethnic differences may influence plasma [AG] (5).
As with any modeling study, the analysis of the data in terms of the
mass balance model is only a demonstration that the model is consistent
with the data rather than evidence that the model is correct. There are
other, more complicated models that would also be consistent with the
data. A more detailed model that includes a physiological rather than
an empirical characterization of the dependence of
r on [Glc] could
accommodate the data by use of a greater number of parameters
characterizing binding and transport functions in the way that glucose
reabsorption has been characterized (25). The dependence of AG
excretion on [Glc] as a function of plasma [AG]
also requires better characterization, such as with data obtained from
AG clearance during short-term hyperglycemic clamp of N subjects, or
measurement of urine output during "recovery" of depleted AG.
Another more detailed model might specify a dependence of AG excretion
on [AG] itself. For instance, the model does not specify
the excretion that would occur on a bolus addition of AG, wherein the
fractional reabsorption would be expected to decrease in the presence
of overtly high concentrations of AG (19). In normal physiology,
however, this would appear to be unimportant (19), because the
steady-state excretion in the setting of constant dietary intake
results in a constant level of plasma [AG].
In summary, the mass balance for AG has been characterized for both the
steady state and the nonsteady state on the basis of a two-compartment
model. The study determined the empirical relationship between glucose
and the fractional reabsorption of AG, characterized the distribution
between plasma and tissue compartments for AG, and determined the
functional form of the time constant that characterizes changes in AG
in the nonsteady state according to the model formulation. The results
of this study may be useful in guiding interpretation of AG monitoring
or in suggesting areas for further study of the physiology of AG
disposal and of the clinical use of AG measurement.
 |
ACKNOWLEDGEMENTS |
Present address of D. Stickle: Brooke Army Medical Center, Dept. of
Pathology, Area Laboratory, Bldg. 2630, 2472 Schofield Rd., Fort Sam
Houston, TX 78234.
 |
FOOTNOTES |
Address for reprint requests: J. Turk, Washington University School of
Medicine, Box 8127, 660 S. Euclid Av., St. Louis, MO 63110.
Received 4 March 1997; accepted in final form 19 June 1997.
 |
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