EPILOGUE
Minimal model SG
overestimation and SI
underestimation: improved accuracy by a Bayesian two-compartment
model
Claudio
Cobelli1,
Andrea
Caumo2, and
Matteo
Omenetto1
1 Department of Electronics and
Informatics, University of Padova, 35131 Padova; and
2 Scientific Institute San
Raffaele, 20132 Milano, Italy
 |
ABSTRACT |
The intravenous glucose tolerance test (IVGTT)
single-compartment minimal model (1CMM) method has recently been shown
to overestimate glucose effectiveness and underestimate insulin
sensitivity. Undermodeling, i.e., use of single- instead of
two-compartment description of glucose kinetics, has been advocated to
explain these limitations. We describe a new two-compartment minimal
model (2CMM) into which we incorporate certain available knowledge on
glucose kinetics. 2CMM is numerically identified using a Bayesian
approach. Twenty-two standard IVGTT (0.30 g/kg) in normal humans were
analyzed. In six subjects, the clamp-based index of insulin sensitivity
(ScI) was also measured. 2CMM glucose
effectiveness (S2G) and insulin
sensitivity (S2I) were, respectively,
60% lower (P < 0.0001) and 35%
higher (P < 0.0001) than the
corresponding 1CMM S1G and
S1I indexes: 2.81 ± 0.29 (SE) vs.
S1G = 4.27 ± 0.33 ml · min
1 · kg
1
and S2I = 11.67 ± 1.71 vs.
S1I = 8.68 ± 1.62 102
ml · min
1 · kg
1
per µU/ml. S2I was not different from
ScI = 12.61 ± 2.13 102
ml · min
1 · kg
1
per µU/ml (nonsignificant), whereas S1I
was 60% lower (P < 0.02). In
conclusion, a new 2CMM has been presented that improves the accuracy of
glucose effectiveness and insulin sensitivity estimates of the classic
1CMM from a standard IVGTT in normal humans.
glucose effectiveness; insulin sensitivity; glucose kinetics; glucose clamp technique
 |
INTRODUCTION |
THE SINGLE-COMPARTMENT minimal model (1CMM) method (4)
is widely used in clinical and epidemiological studies to estimate indexes of glucose effectiveness
(SG) and insulin sensitivity (SI) from an intravenous glucose
tolerance test (IVGTT). However, recent reports indicate that
SG is overestimated (11, 18, 20)
and SI underestimated (22).
Undermodeling of glucose kinetics by 1CMM during the highly dynamic
IVGTT perturbation, i.e., the use of a single- instead of a
two-compartment description, has been advocated to explain
SG overestimation and
SI underestimation (8-10).
Unfortunately, a two-compartment model is only resolvable if a tracer
is added to the IVGTT bolus (7, 14, 23). However, the labeled IVGTT is
not going to reach the widespread application of the standard IVGTT
because of the additional technicalities and costs involved. It is
therefore of interest to determine whether use of certain available a
priori knowledge on glucose kinetics allows us to resolve a
two-compartment model.
This was exactly the aim of this paper. We formulated a two-compartment
minimal model (2CMM) by appending a second, nonaccessible compartment
to the classic 1CMM. Theory shows that resolution of this 2CMM from an
IVGTT requires a priori knowledge on glucose exchange kinetics. We
incorporated such knowledge (12, 13, 17, 19) into the 2CMM in a
probabilistic context by using the Bayesian approach (24). Indexes of
glucose effectiveness and insulin sensitivity were then derived from
the 2CMM and compared with those provided by the 1CMM. In addition, in
a subset of six subjects (6 of 22), the indexes of insulin sensitivity
provided by the two minimal models were also compared with the insulin sensitivity index provided by the glucose clamp technique. Our results
in normal humans show that this approach is able to improve the
accuracy of SG and
SI estimation of the 1CMM from a
standard IVGTT.
 |
MATERIALS AND METHODS |
The IVGTT Data Base
Twenty-two standard IVGTT [dose 302 ± 7 (SE) mg/kg]
performed in normal humans (age 28 ± 1 yr; body weight 72 ± 2 kg) were considered. Sixteen IVGTT have already been
published (2, 3, 23), whereas six are new. In these last six subjects,
insulin sensitivity was also measured by the euglycemic
hyperinsulinemic clamp technique, with insulin infused at 1 mU · min
1 · kg
1
(16; and unpublished data of Dr. R. C. Bonadonna). The protocol was
approved by the ethical committee of the University of Verona, School
of Medicine, Verona, Italy).
The Single-Compartment Minimal Model
The classic 1CMM (Fig.
1A)
(4, 13) can be conveniently written with its uniquely identifiable
parameters as
|
(1)
|
|
(2)
|
|
(3)
|
where
Q is glucose mass (mg/kg), with
Qb denoting its
basal (end-test) steady-state value; D is the glucose dose (mg/kg); X is a variable related to insulin
concentration (deviation from basal) in a compartment remote from
plasma,
X(t) = (k4+k6)I'(t), where k4 and
k6 are rate
parameters (min
1);
I(t) is plasma insulin concentration
(µU/ml), with Ib denoting its
basal value; G is plasma glucose concentration, with
Gb denoting its basal value; V is
the distribution volume per unit body weight (ml/kg); and
p1=
k1+k5,
p2 = k3, and
p3 = k2(k4+k6)
are rate parameters expressed in min
1,
min
1, and
min
2 · µU
1 · ml,
respectively. Clearly one has
Qb = GbV. From the 1CMM one can derive
indexes of glucose effectiveness (S1G)
and insulin sensitivity (S1I)
|
(4)
|
|
(5)
|
where the subscript ss denotes steady state.

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Fig. 1.
The classic single-compartment minimal model (1CMM,
A) and the two-compartment minimal
model (2CMM, B). For meaning of
abbreviated terms, see MATERIALS AND
METHODS.
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|
It is important to note that S1G and
S1I, at variance with the fractional
(i.e., per unit volume) indexes SG
and SI commonly expressed
elsewhere, have the same units of the analogous glucose clamp indexes,
thus allowing a direct comparison (10). Also, because the 1CMM and 2CMM
have different accessible pool volumes (see Model
structure), the choice of these indexes is
the most appropriate for comparing the two models.
The 1CMM parameters were estimated (4, 6, 14) by weighted nonlinear
least squares [using the ADAPT software (15), see
Bayesian identification] with
weights optimally chosen, i.e., equal to the inverse of the variance of
the glucose measurement error assumed to be additive, independent,
Gaussian zero mean, and with a constant coefficient of variation (CV)
of 2%. Precision of parameter estimates was obtained from the inverse
of the Fisher information matrix (6). As normally done with the 1CMM,
the first 8- to 10-min glucose samples were ignored in model
identification. The glucose dose administration was described as a
1-min rectangular infusion.
The Two-Compartment Minimal Model
Model structure.
The 2CMM is the natural evolution of the classic 1CMM: a second,
nonaccessible compartment is appended to it (Fig. 1,
B), and the only difference is the
exchange between the accessible and nonaccessible pools. It is
described by
|
(6)
|
|
(7)
|
|
(8)
|
|
(9)
|
where
Q1 and
Q2 (mg/kg) denote
the glucose masses in the accessible and nonaccessible compartments,
respectively, with subscript b denoting their basal (end-test)
steady-state values; V1 is the volume of the accessible compartment (ml/kg);
k12 and
k21
(min
1) are rate
parameters describing glucose exchange kinetics; D, G, I,
X,
p1,
p2, and
p3 are variables
and parameters already defined for the 1CMM. One has
Q1b = GbV1,
and thus Q1(0)
has a similar expression to Q(0) of
Eq. 1, with
V1 in place of V. Q2(0) is
k21Q1b/k12 from the steady-state constraint.
From the 2CMM one can calculate, as for the 1CMM, indexes of glucose
effectiveness (at basal insulin) and insulin sensitivity. The 2CMM
glucose effectiveness (S2G; superscript
"2" denotes the second compartment) is
|
(10)
|
and
insulin sensitivity (S2I)
is
|
(11)
|
The 2CMM differs from the 1CMM only in allowing an exchange of glucose
between the accessible and the nonaccessible compartment, i.e., the
terms
k21Q1
and
k12Q2
in Eqs. 1 and 2 (cf. Fig. 1). Unfortunately, this
small added complexity brings a priori identifiability problems. By
employing the a priori identifiability analysis method for nonlinear
models described in Ref. 6, it can be shown (see APPENDIX A) that only
V1,
p2, and
p3 are uniquely
identifiable, whereas
p1,
k21, and
k12 are not. In
particular, one can only estimate their aggregates
p1+k21
and
k21k12.
This means that unique identifiability of the 2CMM can only be reached
by resorting to additional independent knowledge of glucose exchange
kinetic parameters k21 and
k12.
Glucose tracer kinetic studies do in principle contain this
information. We have reanalyzed published tracer bolus injection data
obtained in the basal state in normal humans (12, 13, 17, 19) with a
two-compartment model corresponding to that of Fig.
1B, i.e., with no irreversible loss
from the nonaccessible pool. The model has been numerically identified
in fourteen subjects, and population values of
k21 and
k12 were obtained
(in addition to values of V1 and
k01). This
kinetic knowledge has been used to resolve the 2CMM with the strategy
described in Bayesian
identification.
Bayesian identification.
Bayesian estimation allows a flexible, theoretically sound
incorporation of a priori knowledge into model identification. In
particular, the so-called Maximum a Posteriori (MAP) Bayesian estimator
(24) was chosen. Briefly, the unknown model parameters are partitioned
into two uncorrelated components. The first is formed by
V1,
p1,
p2, and
p3, of which we
assume to have no a priori knowledge, i.e., their estimates will be
data driven. The second component is formed by the glucose exchange
kinetic parameters k21 and
k12, of which we
assume to have some prior knowledge available, i.e., their estimates
will be both data and a priori knowledge driven. In particular, from
the above reanalysis of basal state tracer data,
k21 and
k12 are assumed
to be normally distributed, with means and standard deviations of 0.050 ± 0.013 and 0.070 ± 0.018 min
1, respectively, and
with a correlation coefficient of 0.90.
The cost function for a MAP Bayesian estimator is similar to that of
weighted nonlinear least squares, with an additional term pertaining to
k21 and
k12 a priori
knowledge (APPENDIX B, Eq. B3). As for the 1CMM, weights
were chosen optimally, and precision of parameter estimates was
obtained from the inverse of the Fisher information matrix (6). All
glucose data (usually starting from 2 min) were used in model
identification. A 1-min rectangular infusion was used to describe the
glucose administration format. Parameter estimation was performed with
the ADAPT software (15), which contains a MAP Bayesian estimation feature.
Glucose Clamp Insulin Sensitivity
Insulin sensitivity measured with the euglycemic hyperinsulinemic
glucose clamp technique (ScI) was
calculated as in Ref. 3a
|
(12)
|
where
GIR and
I are increments of the exogenous glucose infusion rate
and plasma insulin concentration, respectively, and Gb is basal plasma glucose concentration.
Statistical Analysis
Results are given as means ± SE. Student's t-test for paired
data was used to evaluate differences between indexes estimated with
1CMM, 2CMM, and the glucose clamp. In addition, the value of
ScI has been compared with the 2CMM and
the 1CMM by the use of a statistical approach that assesses the
agreement between two methods for measuring a clinical variable by
displaying on the y-axis the
difference between methods and on the
x-axis the mean of the two methods
(1).
 |
RESULTS |
The individual results of
Bayesian
identification of the 2CMM are shown in Tables 1 and 2; Fig.
2 shows the mean weighted residuals. The
residuals (Fig. 2) have a satisfactory behavior, in terms of both
pattern and amplitude: in particular, the 2CMM (A) is able to describe the initial
portion of the IVGTT (8-10 min), which is not possible with the
1CMM (B). Parameters were generally
estimated with an acceptable precision (Tables 1 and 2). In a few
circumstances, the exchange kinetic parameters, and particularly so
k12, were
difficult to resolve with precision. As expected, the 2CMM accessible
pool volume was lower than the 1CMM volume (128.9 ± 7.4 vs. 152.9 ± 5.2 ml/kg).
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Table 2.
Glucose effectiveness and insulin sensitivity estimated from the 2CMM
and the 1CMM minimal models, and insulin sensitivity measured in 6 subjects by the glucose clamp technique
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Fig. 2.
Mean weighted residuals, i.e., difference between data and model
predictions divided by the standard deviation averaged over all
subjects, of the 2CMM (A) and 1CMM
(B).
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|
The individual estimates of 2CMM (S2G,
S2I) and 1CMM
(S1G, S1I)
and glucose effectiveness and insulin sensitivity shown in Table 2 are
summarized in Fig. 3, together with the
insulin sensitivity clamp index
(ScI). S2G was almost one-half of
(P < 0.0001)
S1G [2.81 ± 0.29 (SE)
vs. 4.27 ± 0.33 ml · min
1
· kg
1]
and S2I was 35% higher
(P < 0.0001) than
S1I (11.67 ± 1.71 vs. 8.68 ± 1.62 102
ml · min
1 · kg
1
per µU/ml). Precision of the 2CMM indexes was comparable to that of
the 1CMM indexes: standard deviations are virtually the same for
S2G, S1G
and S2I,
S1I, whereas the CV is greater for
S2G (less for
S2I), because the parameter estimate is
less (greater for S2I).

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Fig. 3.
Indexes of glucose effectiveness
(SG) and insulin sensitivity
(SI) for the 1CMM
(S1G, S1I)
and the 2CMM (S2G,
S2I). Insulin sensitivity measured with
the glucose clamp technique (ScI) is also
shown and compared with S1I and
S2I in the same subset of subjects. Data
are expressed as means ± SE.
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|
Of interest is the comparison of S2I and
S1I with the glucose clamp measure
ScI in the subsample of six subjects.
S2I was not statistically different from
ScI [9.52 ± 1.98 (SE)
vs. 12.61 ± 2.13 102
ml · min
1 · kg
1
per µU/ml], whereas S1I (6.91 ± 1.07) was significantly lower (P < 0.02). To better assess the agreement of
S2I and S1I
with ScI, the "difference against average of methods" comparison plots were examined (Fig.
4). The small amount of data prevents any
definite conclusion. However, one can safely say that 2CMM
S2I underestimates
ScI (B) much less than 1CMM
S1I
(A).

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Fig. 4.
Insulin sensitivity comparison of the glucose clamp method, with 1CMM
(A) and the 2CMM
(B) minimal model methods.
Difference between methods (y-axis) is shown
against the average of the methods
(x-axis) with regression line.
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 |
DISCUSSION |
One of the assumptions of the 1CMM method is that glucose exhibits
single-compartment kinetics. To favor the domain of validity of this
assumption, the initial portion of the glucose concentration data
(usually the first 8-10 min) is not used in model identification. In fact, albeit the necessity of a two-compartment description of
glucose kinetics in a highly dynamic nonsteady state like the IVGTT is
a well-established notion, it is virtually impossible to resolve from
an IVGTT a two-compartment model without the addition of a glucose
tracer. However, evidence has become available that undermodeling of
glucose kinetics, i.e., the use of a one- instead of a two-compartment
description, is the major factor responsible for 1CMM overestimation of
SG and underestimation of
SI (8-11). The goal of this
paper was to improve on these 1CMM limitations by exploiting available
knowledge on glucose kinetics and by using a Bayesian approach to
identify a 2CMM.
The new 2CMM provides estimates of glucose effectiveness,
S2G, and insulin sensitivity,
S2I, which improve the accuracy of the
1CMM S1G and
S1I (Fig. 3):
S2G = 2.81 ± 0.29 ml · min
1 · kg
1
is almost one-half of the value of S1G,
and S2I = 11.67 ± 1.71 102
ml · min
1 · kg
1
per µU/ml is 35% higher than S1I.
These values are in agreement with recently published values measured
with the glucose clamp technique in normal humans. Best et al. (5)
report for glucose effectiveness measured with the glucose clamp,
ScG, a value of 2.4 ml · min
1 · kg
1
and a value for the 1CMM S1G (using the
1CMM volume of 150 ml/kg) 62% higher than
ScG. Saad et al. (22) report for
ScI a value of 10.1 102
ml · min
1 · kg
1
per µU/ml, and for the 1CMM S1I (using
the 1CMM volume of 150 ml/kg) a value 53% lower than
ScI. This trend is also confirmed by the
glucose clamp insulin sensitivity measurements we perfomed in a subset
of subjects: S2I, but not
S1I, was not different from
ScI (Fig. 3), and association of
S2I with
ScI is stronger than that with S1I (Fig. 4).
Theory indicates that resolving a 2CMM from a standard (nonlabeled)
IVGTT requires independent knowledge of glucose kinetics. A
theoretically sound approach to incorporate such knowledge in probabilistic terms is the so-called MAP Bayesian approach (24), which,
although frequently used in pharmacokinetic/pharmacodynamic studies
(see references in Ref. 15), has not yet been fully exploited in the
endocrine-metabolic area. The glucose kinetic parameters
k21 and
k12 are described
as Gaussian variables, with their mean, variance, and covariance
determined from independent studies. The theory of this approach is
well established (24), and software for Bayesian model identification
is available (15). The results were very satisfactory both in terms of
capability of the model to describe the data (Fig. 2) and in terms of
parameter estimation (Tables 1 and 2).
The structure chosen for the 2CMM follows in some sense a minimum
assumption strategy, i.e., the added complexity to the 1CMM (Fig.
1A) is simply a nonaccessible
compartment attached to it (Fig.
1B). However, one should note that
this description of glucose kinetics, i.e., with a time-varying
irreversible loss in the accessible pool and no loss in the
nonaccessible pool, is equivalent to that proposed by Radziuk et al.
(21), which has become the most widely used model to analyze
non-steady-state glucose kinetics. Although the description of glucose
kinetics incorporated into the 2CMM is reasonable, the question arises
of its physiological plausibility compared with other descriptions that
have been proposed in the literature. Other commonly used structures
also have a constrained irreversible loss in the nonaccessible pool
(13, 17, 21). The use of an irreversible loss in the nonaccessible pool
(even without the one in the accessible pool) means additional
complexity: in fact, a new parameter is required in the 2CMM to
separate the effect of insulin on glucose production from that on
glucose utilization (this is not required with an irreversible loss in
the accessible pool only). A priori knowledge of this additional
parameter is scarce and would make even the Bayesian approach a
difficult route to follow. Therefore, the proposed description of
glucose kinetics is a reasonable necessity. An important plus of the
chosen structure with a single irreversible loss in the accessible pool
is that it is the one that makes the glucose exchange parameters
k21 and k12 less
dependent on insulin levels with respect to structures with an
irreversible loss also, or exclusively, in the nonaccessible pool;
(when the basal and elevated insulin data of Refs. 13 and 17 are
reanalyzed with this model,
k21 and
k12 in the
elevated insulin state are not statistically different from the basal
ones). Thus with respect to other models, the chosen structure makes the assumption that
k21 and
k12 do not vary
appreciably during the IVGTT more tenable.
The glucose clamp technique is considered in the literature the gold
standard for measuring insulin sensitivity. The availability of this
measure in a subset of subjects (n = 6) thus allows us to address the validity of the 1CMM and the 2CMM
measurements. Usually this comparison is made in the literature by
resorting to correlation plots and correlation coefficients (see, e.g., Ref. 22) as indicators of agreement. However, this strategy is not the
most appropriate one (1). A plot of the difference against the mean of
the methods is more informative (1) (Fig. 4). Albeit the amount of data
is small, one can state that 2CMM S2I is
providing much closer values to ScI (B) than 1CMM
S1I
(A). However, an underestimation is
still present and, in addition, one can note an increase of the
difference between the two methods with the increase of the insulin
sensitivity value. Another issue of relevance here is more general: do
we really have to expect a "one to one" concordance between the
minimal model and the glucose clamp methods? In the literature there is
almost a unanimous consensus on the glucose clamp technique being
considered the gold standard. The answer is yes in theory, because both
methods rely on the same insulin sensitivity definition. In practice,
however, for S2I (and
S1I) to be equivalent to
ScI, a number of conditions must be met
(also described in Ref. 10), the most important of which are that
1) insulin dose independence of the glucose clamp technique across the insulin range experienced during an
IVGTT, i.e., insulin effect of the aggregation of glucose production and utilization, increases linearly with insulin concentration, and
that 2) the 2CMM description of
glucose kinetics and their control by insulin is "correct." There
is good evidence that neither of these requirements is fully met. For
instance, the nonlinear effect of insulin on glucose production is a
well accepted notion, and this renders the glucose clamp measurement of
"local" validity, i.e., dose dependent. On the other hand, the
way in which the 2CMM depicts glucose production, distribution, and
utilization bears some approximation, e.g., glucose utilization may not
be accurately described by the single accessible pool irreversible loss, and the description of glucose and insulin control on glucose production embodied in the 2CMM (and 1CMM) may be too
rude. Given this scenario, one should interpret with caution the plots
of Fig. 4: the reassuring "take home message" is the closer
association with ScI of
S2I than of
S1I.
In conclusion, a new 2CMM approach for the estimation of glucose
effectiveness and insulin sensitivity from an IVGTT has been presented
that improves on the 1CMM limitation. The present studies in normal
humans are an obvious prerequisite, but further work is needed to
assess the reliability of this approach. For instance, investigations
are required to better define the role of the description of glucose
production currently embodied in the 2CMM and to assess the reliability
of the Bayesian approach in other situations, like the insulin-modified
IVGTT, and pathophysiological states.
 |
APPENDIX A |
We analyze here the a priori identifiability of the 2CMM described by
Eqs. 6-9. The model
is nonlinear, and one has to resort to the Taylor series expansion of
the measured variable, i.e., glucose concentration, around
time 0 (immediately after the bolus) to check a priori identifiability (6). The unknown
parameters are
p1,
p2,
p3,
k21,
k12, and
V1. The exhaustive summary of the model is given by
|
(A1)
|
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(A2)
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(A3)
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(A4)
|
|
(A5)
|
where G(0), GI(0),
GII(0),
GIII(0), and
GIV(0) are the known glucose
concentration and its first, second, third, and fourth derivatives at
time 0; (the fifth derivative does not
add independent knowledge).
By solving the system of Eqs.
A1-A5, one sees immediately that whereas
Eqs. A1, A2, and A3 give
V1,
p1 + k21, and
k12k21,
respectively, Eqs. A4 and A5 only provide a relationship between
k12 and
p3 and among
k12,
p3, and
p2, respectively.
Therefore, the model is a priori nonidentifiable, and unique
identifiability can only be reached by using an independent additional
relationship between k21 and
k12.
 |
APPENDIX B |
We briefly review here some fundamentals of Bayesian estimation.
Consider the problem of estimating a parameter vector
p = (p1, ..., pP)T
from a set of N noisy measurements
|
(B1)
|
where
y(ti,
p) is the model prediction at
time
ti, and
vi denotes the
(additive) error that affects the i-th
measurement zi.
To solve the problem, a commonly used approach is nonlinear least
squares (LS) (6). A more sophisticated but less used approach is
maximum a posteriori (MAP) estimation. The major difference between
these two approaches is that MAP estimation exploits not only the data
but also certain a priori available statistical information on the
unknown parameters. In other words, while LS is a Fisherian approach to
parameter estimation, i.e., data are the only information supplied to
the estimator, MAP is a Bayesian approach, i.e., a priori information,
e.g., obtained from population studies, is used in addition to the data
(termed a posteriori information) in the numerical estimation of the
model parameters. Bayesian estimation can be of relevant interest
because it can significantly improve the precision of parameter
estimates with respect to Fisher estimation or allow (as in this paper)
the adoption of more complex, and thus more physiologically plausible,
models than those resolvable by a Fisherian approach. Clearly, one has to pay a price for this, i.e., the supply of a priori information.
Let's now turn to a more formal framework. As previously stated,
Bayesian estimation is based on the concept of a priori information on
the unknown parameter vector p,
mathematically specified by its a priori probability density function
fp. For
instance, one can expect a priori, i.e., before having "seen" the
data vector z = (z1, ..., zN)T,
that the unknown vector p is sampled
from a normal distribution with mean
µ and covariance matrix
. After having "observed" the
data vector z, the probability density
function from which we expect that p
is sampled obviously changes. This function, conditional on the data
vector z, goes under the name of an a
posteriori probability density function and is denoted by
fp|z
(p|z);
(p|z reads as "p given
z" and stays for
"p given the data
z"). The MAP estimate of
p is the vector
,
which maximizes the a posteriori probability density function
fp|z(p|z)
|
(B2)
|
Equation B2 gives the general
definition of the MAP estimator. In practical applications, the
functional in the right side of Eq. B2
depends on the specific form of both
fp(p)
and noise statistics in Eq. B1. For
example, if vector p is Gaussian, with
mean µ and covariance matrix
, and the measurement errors
vi are also
Gaussian, with zero mean and variance
2i, by applying the Bayes
theorem it is easily shown (see Ref. 24 for details) that
Eq. B2 turns into
|
(B3)
|
It
is worth noting that in the cost function of Eq. B3 there are two contributions. The first term, which
coincides with the cost function of LS estimation (6), measures the
goodness of fit, i.e., the adherence to the a posteriori information.
The second term weights the adherence of the candidate estimate to the
available a priori knowledge of the parameter vector. This is why Bayes
estimators are said to establish a trade-off between a priori and a
posteriori information, linked to expectations and data, respectively.
It is also worth noting that if the a priori knowledge becomes weaker
and weaker (i.e., the covariance matrix
tends to infinity), the last term
of Eq. B3 can be neglected, and the
MAP estimator collapses into the LS estimator (only a posteriori
information, i.e., the data, are exploited).
In the 2CMM, the vector p is made up
of two components, i.e., p = [k|q]T,
with k = [V1,p1,p2,p3]T,
and q = [k1,k12]T;
µ is the vector of the a priori mean
of p, i.e., µ = [µk,µq]T,
and
is the a priori covariance
matrix of p.
is made up of two components,
k and
q, related to
k and
q, respectively, which are
uncorrelated, and this brings the zero value of the off-diagonal
components
|
(B4)
|
Because no a priori knowledge is imposed on
k (data-driven parameters), the 4 × 4
k
matrix has its diagonal elements, i.e., the variances equal to infinity
and its off-diagonal elements equal to zero. In contrast, we impose a
priori knowledge on q, i.e.,
q is a 2 × 2 matrix with its diagonal and off-diagonal elements given,
respectively, by the population variances and covariances of
k21 and
k12.
 |
ACKNOWLEDGEMENTS |
We thank Dr. R. C. Bonadonna for sharing some unpublished data,
and Glaxo-Wellcome Research and Development (Middlesex, UK) for having
made available the data of Ref. 19. We also acknowledge the expert
advice of Dr. Giovanni Sparacino in writing APPENDIX B in a mathematical as well as descriptive language,
which will hopefully make it less impenetrable to the physiological readership. Finally, we thank the reviewer for an outstanding report,
which allowed us to improve the quality of the manuscript.
 |
FOOTNOTES |
This work was partially supported by National Center for Research
Resources Grants RR-02176, RR-11095, and RR-12609.
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. §1734 solely to indicate this fact.
Address for correspondence and reprint requests: C. Cobelli,
Dipartimento di Elettronica e Informatica, Università degli Studi
di Padova, Via Gradenigo 6a 35131 Padova, Italy (E-mail:
cobelli{at}dei.unipd.it).
Received 7 August 1998; accepted in final form 6 May 1999.
 |
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