a circulatory model for calculating non-steady-state glucose fluxes. validation and comparison with...

A circulatory model for calculating non-steady-state glucosefluxes. Validation and comparison with compartmental models

Andrea Mari a,*, L. Stojanovska b, J. Proietto c, A.W. Thorburn c

a Institute of Systems Science and Biomedical Engineering, National Research Council, LADSEB-CNR, Corso Stati Uniti 4, 35127

Padova, Italyb Faculty of Engineering and Science, Victoria University of Technology, PO Box 14428, Melbourne City 8001, Australia

c Department of Medicine, Royal Melbourne Hospital, Parkville, Vic. 3040, Australia

Accepted 5 July 2002

Abstract

This study presents a circulatory model of glucose kinetics for application to non-steady-state conditions, examines

its ability to predict glucose appearance rates from a simulated oral glucose load, and compares its performance with

compartmental models. A glucose tracer bolus was injected intravenously in rats to determine parameters of the

circulatory and two-compartment models. A simulated oral glucose tolerance test was performed in another group of

rats by infusing intravenously labeled glucose at variable rates. A primed continuous intravenous infusion of a second

tracer was given to determine glucose clearance. The circulatory model gave the best estimate of glucose appearance,

closely followed by the two-compartment model and a modified Steele one-compartment model with a larger total

glucose volume. The standard one-compartment model provided the worst estimate. The average relative errors on the

rate of glucose appearance were: circulatory, 10%; two-compartment, 13%; modified one-compartment, 11%; standard

one-compartment, 16%. Recovery of the infused glucose dose was 939/2, 949/2, 929/2 and 859/2%, respectively. These

results show that the circulatory model is an appropriate model for assessing glucose turnover during an oral glucose

load.

# 2002 Elsevier Science Ireland Ltd. All rights reserved.

Keywords: Glucose kinetics; Tracer method; Oral glucose test; Mathematical models

1. Introduction

The study of glucose metabolism often requires

the calculation of glucose fluxes in non-steady-

state conditions, for instance during an oral

glucose load or a glucose clamp. This calculation

is based on tracer methodology, and requires a

mathematical model. Unless the specific activity of

the tracer is kept constant, which is often difficult

or impossible, the accuracy of the calculated

glucose fluxes depends on the model used.The most commonly used models are compart-

mental models with one or two compartments

[1,2]. These models have proved useful in many

* Corresponding author. Tel.: �/39-049-829-5753; fax: �/39-

049-829-5763

E-mail address: [email protected] (A. Mari).

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0169-2607/02/$ - see front matter # 2002 Elsevier Science Ireland Ltd. All rights reserved.

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situations. However, discrepancies between theglucose fluxes calculated with these models and

with independent methods have been found.

Steele’s one-compartment model [1] has been

shown to underestimate glucose production during

a standard euglycemic hyperinsulinemic glucose

clamp [3]. In experiments which simulate an oral

load with an exogenous glucose infusion, both

Steele’s model and the two-compartment model byRadziuk and colleagues [2] predicted the rate of

appearance with some distortion [2,4]. During an

oral glucose load, the rate of appearance calcu-

lated using the two-compartment model and the

arteriovenous difference method were also some-

what different [5].

The problems encountered with the compart-

mental models may not originate entirely from theinadequacy of the models (cf. [5]). Nevertheless,

compartmental models do not represent the phy-

siological system as it is [6,7], and for this reason

may introduce errors [8]. To avoid this drawback,

circulatory models have been developed [8,9],

which more appropriately represent the physiolo-

gical system and are based on the solid principles

of the theory developed for organ kinetics byZierler [10,11]. Circulatory models have been used

in various artificial non-steady-state conditions

(e.g. [12,13]), but a model for general use has not

been proposed and tested. In this work, we present

a circulatory model for the calculation of non-

steady-state glucose fluxes in the general case. We

evaluate the model performance in experiments in

anesthetized rats that simulate an oral glucose loadthrough an exogenous glucose infusion, by com-

paring the model-calculated rate of appearance

with the known glucose infusion. We also compare

the circulatory model performance with that of the

more traditional compartmental approaches.

2. Methods

2.1. Experimental procedures

2.1.1. Basal glucose kinetics

To determine the parameters for basal glucose

kinetics, six adult male Sprague�/Dawley rats were

anesthetized with pentobarbitone sodium (Nem-

butal, Boehringer Ingelheim, NSW, Australia)administered by intraperitoneal injection (60 mg

kg�1 body weight). A catheter was placed in the

right jugular vein (for tracer infusion) and left

carotid artery (for blood sampling) [14]. The rats

were given a single injection of [6-3H]glucose (10

mCi) and blood samples were taken at 0.5, 1, 1.5, 2,

3, 4, 5, 6, 8, 10, 12, 15, 20, 25, 30, 40, 50 and 60 min

for analysis of plasma tracer.

2.1.2. Simulated oral load

Another group of five male Sprague�/Dawley

rats were anesthetized and implanted with cathe-

ters as for the basal experiments. At time 0 min, a

primed (185 mmol kg�1) continuous infusion (3.8

mmol min�1 kg�1) of [6,6-D2]glucose (Tracer

Technologies, Somerville, MA, USA) in 0.9%NaCl was commenced. Three blood samples were

taken at 60, 65 and 70 min for the estimation of the

tracer-to-tracee ratio of [6,6-D2]glucose and

plasma glucose. At 70 min, an infusion of 15 mCi

ml�1 [6-3H]glucose in 25% glucose commenced in

an algorithm designed to mimic the appearance of

gut-derived glucose following the administration

of an oral glucose load. From time 70 to 90 minthe glucose infusion rate was increased every

minute to a maximum infusion rate of �/48

mmol min�1. This infusion rate was maintained

for 15 min (time 90�/105 min) after which the

infusion rate was decreased every minute until it

was stopped at 125 min. Blood samples (300 ml)

were taken at 5 min intervals between 70 and 125

min for measurement of plasma glucose, thetracer-to-tracee ratio of [6,6-D2]glucose, and

[6-3H]glucose specific activity. At the end of the

study, timed collections of the [6,6-D2]glucose

infusate were taken for accurate measurement of

the constant [6,6-D2]glucose infusion rate. Timed

collections of the variable glucose infusate were

also taken to determine accurate time courses of

the cold glucose delivery rate in each experiment.The [6-3H]glucose specific activity of the variable

glucose infusate was also determined.

2.2. Analytical methods

Plasma and infusate glucose levels, and plasma

[6-3H]glucose tracer radioactivity and [6,6-D2]glu-

A. Mari et al. / Computer Methods and Programs in Biomedicine 71 (2003) 269�/281270

cose enrichment were measured as described pre-viously [14,15]. [6,6-D2]glucose enrichment was

analyzed using a gas chromatograph mass spectro-

meter (Shimadzu model GCSM-QP2000, Shi-

madzu Corporation, Kyoto, Japan) on ion

monitoring mode to determine the relative inten-

sity of the 98 and 100 molecular weight fragments

i.e. (peak area at 100)/(peak area at 100�/peak

area at 98). A standard curve was run in parallel toconvert relative intensity into relative tracer abun-

dance (A ). The tracer-to-tracee ratio was then

calculated as A /(1�/A ). The study protocol was

approved by the Royal Melbourne Hospital’s

Animal Ethics Committee.

2.3. Circulatory model

2.3.1. Model structure

Glucose kinetics are described using the circu-

latory model of Fig. 1, the mathematical theory of

which has been presented previously [8,9,16]. In

the model, the heart chambers and the lungs are

represented in the heart�/lungs block (upperblock), while the remaining tissues are lumped

into the periphery block (lower block). Blood flow

for both blocks is cardiac output. In the experi-

mental configuration adopted here, glucose is

injected at the venous side, while blood is sampled

at the arterial side (as indicated in Fig. 1).

The two tissue blocks shown in Fig. 1 can be

regarded as single inlet�/single outlet organs,coupled in a feedback arrangement. The basis of

the mathematical description of a block (as for an

organ) is its impulse response [17]. The impulse

response is the tracer efflux (concentration times

blood flow) observed at the organ outlet after a

bolus injection of a unit tracer dose at the organ

inlet (this definition assumes tracer does not

recirculate). Cardiac output and the impulseresponses of the blocks fully determine the circu-

latory model, and are specified in this section.

Cardiac output was expressed as flow of blood

per kilogram of body weight. Cardiac output was

fixed to the value of 236 ml min�1 kg�1, which is

appropriate for anesthetized rats [18]. Since glu-

cose concentration was measured in plasma and

not in blood, this cardiac output value wascorrected using the ratio between glucose concen-

tration in whole blood and in plasma (0.53 in our

rats) to ensure that the product of concentration

and flow gives the actual glucose mass flux. A

constant cardiac output value during the test was

assumed (see Section 4).

The impulse response of the heart�/lungs block

[rlung(t)] was modeled as a two-exponential func-tion, which is the minimal representation ensuring

a time course of the response in agreement with the

experimental data. The impulse response starts

from 0, rapidly increases to a peak value, and

returns to zero as a single-exponential function. To

simplify the notation, it is convenient to represent

rlung(t ) as the convolution of two single-exponen-

tial functions:

rlung(t)�be�bt�ve�vt (1)

where the symbol �/ denotes the convolution

operator. As the coefficients of the single-expo-

nential functions equal their exponents, the inte-gral from 0 to � of each exponential is 1. This also

implies that the integral from 0 to � of rlung(t) is 1,

i.e. the glucose fractional extraction in the heart�/

lungs system is 0 [12]. The exponent that char-

acterizes the rising phase of rlung(t) was fixed (b�/

15 min�1, cf. [12]). The exponent of the decaying

phase (v , min�1) was calculated using the prop-

erty that the mean transit time of rlung(t) is theratio of volume to blood flow [17]. From this

property, the following equation was obtained (cf.

[12]):

v�bCO

bVlung � CO(2)

where cardiac output (CO) and the heart�/lungs

glucose volume (Vlung) were assumed knownFig. 1. The circulatory model.

A. Mari et al. / Computer Methods and Programs in Biomedicine 71 (2003) 269�/281 271

(CO�/0.53�/236 ml min�1 kg�1, see above;Vlung�/17 ml kg�1, see [12]).

The impulse response of the periphery block

[rper(t)] in the basal steady-state period can be

expressed as [17]:

rper(t)� (1�E)p(t) (3)

where E (dimensionless) is the glucose fractional

extraction and p (t) is the glucose transit time

density function.The transit time density function p (t ) was

modeled as a four-exponential function, starting

from zero, rapidly increasing to a peak value, and

returning to 0 as a three-exponential function.

Similarly to rlung(t), we have represented p(t) as a

convolution of a single-exponential function with

a three-exponential function:

p(t)�ge�gt

� [w1a1e�a1t�w2a2e�a2t

�(1�w1�w2)a3e�a3t] (4)

where g , a1, a2, a3 (min�1) and w1, w2 (dimension-

less) are parameters.

In Eq. (4), the integral from 0 to � of the three-exponential function in square brackets is w1�/

w2�/(1�/w1�/w2), i.e. 1. This ensures that the

integral from 0 to � of p (t) is also 1, as required

for a transit time density function. The parameters

w1 and w2 determine the contribution in the total

response of the three exponential terms. As b in

rlung(t), g determines the initial rising phase of

p (t ). The parameter g was fixed (g�/10 min�1,[12]), while a1, a2, a3, w1, w2 and E were estimated

from the basal kinetic experiments.

Eq. (3) can be extended to the non-steady-state

[9]. In non-steady-state, the glucose fractional

extraction E varies with time under the action of

insulin, and the transit time density function p(t)

may also change. However, in the present model

we have assumed that the transit time densityfunction is not affected by insulin. This is sup-

ported by the finding that the main effect of insulin

is exerted on the fractional extraction of the

periphery, while the transit time density function

does not change substantially [8,12]. Thus, Eq. (3)

was assumed to be valid also in the non-steady-

state period, with the same p (t) obtained from thebasal period, and E (t) time-varying.

To simulate the model of Fig. 1, the multi-

exponential equations for the impulse responses of

the heart�/lungs and periphery blocks are first

represented as ordinary differential equations.

These differential equations are then combined

according to the feedback arrangement of Fig. 1,

and then solved numerically using MATLAB (seeAppendix A for details).

2.3.2. Analysis of basal steady-state experiments

For the present circulatory model global iden-tifiability of the model parameters is ensured in

steady-state [16]. Mean parameters of the periph-

ery transit time density function p (t) and the basal

glucose fractional extraction E (six parameters)

were obtained by least-squares fit of the mean

[3-3H]glucose concentration in the basal kinetic

experiments. The estimated parameters were sub-

sequently used in the analysis of the non-steady-state data (a single parameter set for all non-

steady-state experiments). Parameter estimation

was performed using a Levenberg�/Marquardt

least-squares algorithm (MATLAB function

leastsq).

2.3.3. Analysis of non-steady-state experiments

In non-steady-state, as p(t) was assumed to be

time-invariant, the only time-varying parameter to

be determined was E (t). E (t ) was assumed con-

stant in the basal period preceding the start of

glucose infusion, and represented as a piecewiseconstant function of time over 2 min time intervals

thereafter (28 elements, E1�/E28, were necessary to

cover the 55 min non-steady-state period). The Ek

values were estimated from [6,6-2H2]glucose con-

centration using the model and a method tradi-

tionally used for deconvolution, as previously

described [12]. In brief, the Ek values were

estimated by least-squares fit of [6,6-2H2]glucoseconcentration, including in the least-squares a

penalty term dependent on the second derivative

of E (t ), calculated by forward differences of the

Ek values. This additional term is necessary to

obtain a smooth E (t), which would otherwise

exhibit spurious oscillations.


2.3.4. Exogenous glucose appearance

By dividing the [3-3H]glucose concentration by

the specific activity of the glucose infusate, the

glucose concentration component due to the

exogenous glucose infusion was calculated (‘exo-

genous’ glucose concentration). From the exogen-

ous glucose concentration and the model (using

the tracer-determined E (t)), the rate of appearance

of exogenous glucose was calculated using adeconvolution method [12]. For this purpose, we

have approximated the exogenous glucose appear-

ance rate as a piecewise constant function of time

over 2 min time intervals, as E (t).

2.3.5. Glucose production

The glucose concentration component due to

glucose production (‘endogenous’ glucose concen-

tration) was calculated by subtracting the exogen-

ous glucose concentration from the total

(measured) glucose concentration. From the en-

dogenous glucose concentration and the model,

glucose production was calculated using a decon-

volution method, as for exogenous glucose. Glu-cose production was approximated as a constant

value before t�/70 min, and a piecewise constant

function of time over 2 min time intervals there-

after. The calculated values were corrected for the

non-negligible mass flux of the stable-isotope

tracer (�/3.8 mmol min�1 kg�1), as the endogen-

ous glucose concentration component includes the

contribution of the tracer.

2.4. Compartmental models

2.4.1. One-compartment model

The glucose rate of appearance [Ra(t )] was

calculated every 2 min (time instants t0�/0, t1�/

2,. . . min) using Steele’s single-compartment

model (Fig. 2 top, [1]), according to the equation

(see [19]):

Ra(tk)�Ra�(tk)

a(tk)�VS

G(tk)a(tk)

a(tk)(5)

where Ra�(tk ) is the tracer infusion rate, G (tk ) is

arterial glucose concentration, a (tk ) is the glucose

specific activity (the dot represents the time

derivative), and VS is the compartment volume.

We have used two values for the glucose

distribution volume VS. The first value representsa standard volume choice (pool fraction of 0.65

with a total glucose volume of 250 ml kg�1, i.e.

VS�/0.65�/250 ml kg�1, [19]). The second value

is the total glucose volume as calculated with the

circulatory model in the basal period (VS�/191 ml

kg�1).

The time course of tracer and glucose concen-

tration (exogenous or endogenous), and the deri-vative of specific activity were obtained from the

data fit of the circulatory model (see Appendix A).

This provides the necessary data smoothing and

ensures that the differences between Steele’s model

and the circulatory model are not due to differ-

ences in data smoothing and interpolation.

2.4.2. Two-compartment model

The rate of glucose appearance was calculated

using the two-compartment model of Fig. 2,

according to the equations developed in [19]:

Ra(tk)�Ra�(tk)

a(tx)�V1

G(tk)a(tk)

a(tk)�V1

k12k21

k12

��

g�(tk)

a(tk)�g(tk)

�

g(tk�1)�b1g(tk)�b2G(tk)�b3G(tk�1);

g(t0)�G(t0)

Fig. 2. Compartmental models. Top: single-compartment mod-

el; bottom: two-compartment model. See Eqs. (5) and (6) for

the symbols.


g�(tk�1)�b1g�(tk)�b2G�(tk)�b3G�(tk�1);

g�(t0)�G�(t0)(6)

where the star denotes the tracer, V1 is the volume

of the first compartment, k21 and k12 are the

intercompartmental rate constants (cf. Fig. 2), g(t)

and g*(t ) are auxiliary variables (delayed glucose

concentration), and the bk ’s (cf. [19]) are constants

calculated from k12 and the calculation interval

tk�1�/tk , which was 2 min as for Steele’s and the

circulatory model.The parameters of the two-compartment model

were obtained from the basal steady-state experi-

ments, as for the circulatory model. The two-

compartment model did not fit the initial part of

the tracer disappearance curves accurately, as

expected [20]. We have thus discarded the first 2

min in the analysis of the basal tracer curves.

As for Steele’s model, concentrations of tracer

and glucose, and the derivative of specific activity

were obtained from the data fit of the circulatory

model.

2.5. Evaluation of the model accuracy

The accuracy of the models was evaluated by

comparing the model-calculated rate of appear-

ance with the actual glucose infusion rate, the

error being the difference of the two. Besides the

direct comparison of the time course of the mean

rate of appearance, we have calculated the follow-

ing indices of model performance: (1) percent

recovery of the infused glucose dose, i.e. the ratio

between the integral of the model-calculated rate

of appearance and the total quantity of infused

glucose, expressed in percent; (2) integral absolute

error, i.e. the integral of the absolute value of the

error on the rate of appearance; (3) cumulative

distribution of the percent error (absolute value of

the error divided by the actual glucose infusion

rate). The cumulative distribution function was

calculated by pooling the error values in all rats

and at all time points. The frequency of occurrence

was divided into ten percentiles.

3. Results

3.1. Basal glucose kinetics

3.1.1. Circulatory model

The parameters of the circulatory model esti-

mated from the tracer bolus injection in the basal

state are reported in Table 1. The coefficients of

variation of the parameters provided by the least-squares algorithm were below 23%. The circula-

tory model predicted the [3-3H]glucose concentra-

tion curve accurately.

3.1.2. Two-compartment model

The two-compartment model parameters were:

V1�/119 ml kg�1; k01�/0.033; k21�/0.073; k12�/

0.11 min�1; clearance�/3.9 ml min�1 kg�1; totalvolume�/197 ml kg�1. The coefficients of varia-

tion were below 20%. In the first 2 min, which were

excluded from the fit, the two-compartment model

underestimated [3-3H]glucose concentration, while

in the remaining period the fit was accurate.

3.2. Non-steady-state glucose kinetics

3.2.1. Glucose and tracer concentrations

Fig. 3 shows the mean [6,6-2H2]glucose concen-

tration, total glucose concentration, and its exo-

genous and endogenous components. The solid

lines represent the circulatory model fit.

3.2.2. Rate of appearance of exogenous glucose

The mean rate of appearance of exogenous

glucose calculated with the circulatory and thetwo-compartment model are shown in Fig. 4. The

results for Steele’s model, obtained using the two

glucose distribution volumes, are shown in Fig. 5.

The standard Steele calculation (Fig. 5 top) was

the least accurate, while the performance of the

other models was not markedly different on

average. All the models underestimated the down-

slope of the glucose rate of infusion. The two-compartment model most accurately predicted the

downslope, while the circulatory model most

accurately predicted the upslope. The performance

of the modified Steele’s model with the volume set

to the total glucose volume was similar to that of

the circulatory model.


Table 2 reports the recovery of the infused

glucose and the average absolute integral error

for all models. Depending on the model, the

correlation coefficient between the integral of the

calculated rate of appearance and the total quan-

tity of infused glucose was �/0.85�/0.88, with a

borderline significance (P :/0.05�/0.07). These in-

tegral accuracy indexes basically confirmed the

results of Figs. 4 and 5.

The distribution of the pooled relative error is

shown in Fig. 6. The curves show the percentage of

the rate of appearance values that are affected by

an error which does not exceed a specified value.

For instance, for the circulatory model 60% of the

calculated rates of appearance (read off the

ordinate) are affected by an error which is not

greater than 12% (read off the abscissa). In this

representation, the curves of the more accurate

models lie above those of the less accurate models.

According to this analysis, which shows the error

Table 1

Model parameters in the basal state for the anesthetized rats

CO ml min�1 kg�1 E Heart�/lungs Periphery

b (min�1) v (min�1) g (min�1) a1 (min�1) a2 (min�1) a3 (min�1) w1 w2

125 0.0293 15.0 14.4 10.0 3.69 0.534 0.0788 0.684 0.265

CO, cardiac output; E , fractional extraction. For the remaining symbols see Eqs. (1)�/(4). In the basal state, the clearance was 3.7 ml

min�1 kg�1 and the total volume 191 ml kg�1. The parameter v was calculated from cardiac output and the volume of the heart�/

lungs block (Vlung�/17 ml kg�1) using Eq. (2). The cardiac output value reported in the table is corrected for the distribution of

glucose between plasma and red cells (CO�/0.53�/236�/12 ml min�1 kg�1, see Section 2).

Fig. 3. Mean (9/S.E.M.) tracer concentration, total glucose

concentration, exogenous glucose concentration and endogen-

ous glucose concentration. The solid line represents the

circulatory model fit.

Fig. 4. Mean (9/S.E.M.) exogenous glucose rate of appearance

calculated with the circulatory (top) and the two-compartment

model (bottom). The solid lines represent the true mean glucose

infusion rate (standard errors are omitted for clarity).


in 90% of the calculated points for clarity, the

overall performance of the circulatory model is the

best and that of the standard Steele’s model the

worst. However, with the exception of the stan-dard Steele’s model, the differences are small.

3.2.3. Glucose production

Fig. 7 shows mean glucose production calcu-

lated using the circulatory model. Glucose produc-

tion decreased to 0 at 25 min after the start of

glucose infusion, and slowly returned towards the

Fig. 5. Mean (9/S.E.M.) exogenous glucose rate of appearance

calculated with Steele’s model and the two glucose volume

values indicated in the figure. The solid lines represent the true

mean glucose infusion rate (standard errors are omitted for

clarity).

Table 2

Recovery and integral absolute error

Circulatory

model

Two-compartment

model

Steele’s model V�/162.5 ml

kg�1

Steele’s model V�/191 ml

kg�1

Recovery (%) 939/2 949/2 859/2 929/2

Integral absolute error (mmol

kg�1)

7099/89 7179/72 8629/106a 7719/93

a P B/0.05 or less vs. circulatory model (paired t -test).

The recovery was significantly less than 100% for all models (P B/0.05 or less, one-sample t -test).

Fig. 6. Cumulative distribution of the pooled relative error.

Results for all models are reported, as indicated in the figure.

The ordinate shows the percentage of the calculated rates of

appearance that do not have an error which exceeds the level

shown on the abscissa. For clarity, only 90% of the calculated

rates are included. The error value on the abscissa which

corresponds to 100% of the calculated rates (maximal error) is

in fact much higher: circulatory model, 145%; two-compart-

ment model, 121%; Steele’s model, 107% (V�/162.5 ml kg�1)

and 134% (V�/191 ml kg�1).

Fig. 7. Mean (9/S.E.M.) glucose production calculated with

the circulatory model.


basal value thereafter. Glucose production ascalculated using the compartmental models was

similar to that reported in Fig. 7. The largest

difference was observed with the standard Steele’s

model, which yielded a higher glucose production

on the average, but the maximum difference was

less that 4 mmol min�1 kg�1.

4. Discussion

In our experiments, the circulatory model pre-

dicted the actual glucose infusion rate with an

average error of about 10%. The rising phase of

the glucose infusion was precisely matched, while

the falling phase was underestimated. The biggest

error was observed in this period, but on average

the error was about 15%, and in 90% of all thecalculated rates of appearance it did not exceed

30%. According to the distribution of the error on

the rate of appearance (Fig. 6, Table 2), the

performance of the circulatory model was the

best. However, none of the models matched the

glucose infusion rate accurately, and the evalua-

tion of the relative performance is thus somewhat

subjective. Indeed, the accuracy of the two-com-partment model and Steele’s model with the larger

volume was only slightly inferior to that of the

circulatory model.

As concerns the circulatory model, the error on

the glucose rate of appearance may originate from

two major sources. The first is the use of mean

instead of individual parameters. Non-steady-state

calculations are made using a fixed parameter setfor all rats, while it is expected that the individual

model parameters are somewhat different from rat

to rat. This may introduce an error in the

individual rates of appearance. We could not

evaluate the magnitude of this error because

experimental limitations prevented us from asses-

sing the basal parameters in each rat used for the

non-steady-state experiments. However, it is un-likely that this error is also significant for the mean

rate of appearance, as the interindividual para-

meter differences are random and tend to cancel

out in the mean. The second error concerns limited

information on the system. The circulatory model

is a physical representation of the system, but

assumes a fixed cardiac output and a time-invariant distribution of the transit times p(t) of

the periphery because the little is known about the

changes that the cardiac output and p (t ) undergo

in this and other experimental conditions. During

glucose challenges or hyperinsulinemia in con-

scious human subjects, changes in cardiac output

do not exceed 20% [21�/23]. Changes from basal in

p(t) have been determined during hyperinsuline-mic euglycemic glucose clamps, directly in dog [12]

and indirectly in man [8], and in both cases are not

substantial. In hyperinsulinemic hyperglycemic

glucose clamps in anesthetized rats, we have also

found small changes in p (t ) (�/20% difference in

mean transit time, results not shown). These

findings justify the assumption of this study of

constant cardiac output and time-invariant p (t).Additional experimental and theoretical work are,

however, needed to resolve this issue, and to

clarify whether more precise assumptions result

in a better model performance, or if factors other

than the model are the cause of the observed

discrepancies, as pointed out previously (e.g. [5]).

Unfortunately, the determination of the time

course of cardiac output and the changes of p (t )is a difficult problem to solve, and this limits the

possibilities of model improvement.

In regard to cardiac output, we have assumed

the most appropriate value for anesthetized rats

[18]. In principle, cardiac output cannot be

ignored, because it does have a role in glucose

kinetics and thus in the calculation of glucose

fluxes. However, the calculations are not verysensitive to the cardiac output value. Only with

rapid changes of fluxes, concentrations and spe-

cific activity, which are not observed in the present

experiments, is it expected that the calculated

glucose fluxes are influenced by the cardiac output

value. It is in fact known that in steady-state, or

with negligible changes in specific activity, cardiac

output is not relevant for calculating glucoseturnover.

The compartmental models considered here

performed differently. The two-compartment

model performance was only slightly inferior to

that of the circulatory model. The standard Steele

equation markedly distorted the shape of the

glucose infusion rate, but the poor performance


of this model could be corrected by using a larger

glucose volume. This stratagem has theoretical

support when the specific activity changes slowly

[24]. However, the use of larger glucose volume

does not ensure sufficient accuracy in general. For

instance, the calculation of glucose production

during a standard glucose clamp is not accurate

with Steele’s model, regardless of the glucose

volume used [24]. Therefore, in this experimental

condition compartmental models are sufficiently

accurate, but, as demonstrated for the Steele’s

model, it is not necessarily true that the error due

to the incorrect representation of the physiological

system is small in all situations.

The circulatory model presented here is suitable

for application to general non-steady-state condi-

tions, as is the Steele’s one-compartment model

and the two-compartment model. To use the

circulatory model, the cardiac output value and

the parameters of p (t) are required. It is clearly

important that the parameters used are appropri-

ate for the specific experimental condition. If

possible, the model parameters should be esti-

mated in each individual experiment, using for

instance the tracer equilibration period that nor-

mally precedes the actual non-steady-state phase.

In this work, however, a single set of parameters

(Table 1) was used for all rats because individual

estimates could not be obtained. These values are

appropriate for anesthetized rats. As it is useful to

provide analogous values for humans, we report in

Table 3 a set of parameters derived from a study in

lean and obese subjects [13]. We also present in

Appendix B some results useful to extend the

parameters of Tables 1 and 3 to experimentalconditions in which the cardiac output value is

different from that reported here.

In conclusion, we have presented a circulatory

model for the calculation of the rate of appearance

in non-steady-state conditions and we have as-

sessed its accuracy experimentally. The circulatory

model gives a physiological representation for

glucose kinetics, and its accuracy is to some extentsuperior to that of the more standard Steele and

two-compartment models.

Acknowledgements

This study was supported in part by a grant

from the Italian National Research Council.

Appendix A

In this appendix, the differential equations of

the model are derived. Additional details on the

equations can be found in [12], which presents a

similar model using the same notation.

Basic differential equation

The key for the transformation of the impulse

responses (Eqs. (1), (3) and (4)) into differential

equations is the representation of the convolution

by means of a differential equation. If the impulse

response of the heart�/lungs block were the single-

Table 3

Model parameters in the basal state for humans (from [13])

CO (ml min�1 kg�1) E Heart�/lungs Periphery

b (min�1) v (min�1) g (min�1) a1 (min�1) a2 (min�1) a3 (min�1) w1 w2

66 0.0223 15.0 5.16 10.0 3.05 0.382 0.0388 0.430 0.505

CO, cardiac output; E , fractional extraction. For the remaining symbols see Eqs. (1)�/(4). The total volume was 223 ml kg�1. The

parameter v was calculated from cardiac output and the volume of the heart�/lungs block (Vlung�/17 ml kg�1) using Eq. (2). The

cardiac output value reported in the table is corrected for the distribution of glucose between plasma and red cells (see Section 2). In

[13], cardiac output (flow of blood) was assumed to be 78 ml min�1 kg�1, and the correction factor 0.84 (CO�/0.84�/78�/66 ml

min�1 kg�1). In [13], cardiac output was expressed in ml min�1 m�2 (cardiac index), and is converted here into ml min�1 kg�1

assuming a body surface area to body weight ratio of 1.7/70.


exponential function ke�lt , the relationship be-tween the input (glucose concentration in the right

atrium, Gra(t)) and the output (arterial glucose

concentration, Ga(t)) would be the convolution

Ga(t)�ke�lt�Gra(t) (A1)

This convolution can be represented by means of

the differential equation

x(t)��lx(t)�Gra(t)

Ga(t)�kx(t) (A2)

where x (t) is an additional variable (state vari-able), and the dot represents the time derivative.

Eq. (A2) is a state space representation of a linear

system, in a canonical form. Its state variables do

not represent physical quantities. The initial con-

dition for Eq. (A2) (and the successive equations)

depends on whether endogenous glucose, oral

glucose or the tracer is considered. For oral

glucose and the tracer, which are not present inthe system before time 0, x (0)�/0. For endogen-

ous glucose, x (0) is the steady-state value, i.e. the

solution of the algebraic equations obtained from

Eq. (A2) by letting x(t)�0:/

Heart�/lung differential equations

Because the two-exponential impulse response

of the heart�/lungs block (Eq. (1)) is represented as

a convolution of two single-exponential functions,

the relationship between Gra(t ) and Ga(t) can be

expressed as a cascade of two differential equa-

tions like Eq. (A2), i.e.

x1��vx1(t)�Gra(t)

x2��bx2(t)�vx1(t)

Ga(t)�bx2(t) (A3)

where x1 and x2 are state variables.

Periphery differential equations

Similarly, the impulse response of the periphery

block (Eqs. (3) and (4)) is represented by the four

differential equations:

x3(t)��a1x3(t)�w1(1�E)Ga(t)

x4(t)��a2x4(t)�w2(1�E)Ga(t)

x5(t)��a3x5(t)�(1�w1�w2)(1�E)Ga(t)

x6(t)��gx6(t)�a1x3(t)�a2x4(t)�a3x5(t)

Gmv(t)�gx6(t) (A4)

where Gmv(t) is mixed-venous glucose concentra-

tion (output of the periphery block), and x3�/x6 are

state variables. In Eq. (A4), the three exponentialterms in the square brackets in Eq. (4) are

represented by a sum of three differential equa-

tions, while the last two equations express the

convolution with the exponential function gegt

.

Whole-body differential equations

In the feedback arrangement of Fig. 1, the inputof the heart�/lungs block is the output of the

periphery block [Gmv(t)] plus a term representing

the appearance of glucose, which is Ra(t)/CO.

Therefore, the whole set of differential equations

describing the circulatory model is:

x1(t)��vx1(t)�gx6(t)�Ra(t)

CO

x2(t)��bx2(t)�vx1(t)

x3(t)��a1x3(t)�w1(1�E)bx2(t)

x4(t)��a2x4(t)�w2(1�E)bx2(t)

x5(t)��a3x5(t)�(1�w2�w2)(1�E)bx2(t)

x6(t)��gx6(t)�a1x3(t)�a2x4(t)�a3x5(t)

Ga(t)�bx2(t) (A5)

Numerical solution

Eq. (A5) is a standard representation in state

space form of a linear system of differential

equations. The solution of the system yields the

glucose concentration (tracer, exogenous or en-

dogenous) that corresponds to a given rate of


appearance. Eq. (A5) is valid both in steady andnon-steady-state. In steady-state E is constant,

and Eq. (A5) represents a linear time-invariant

system. In non-steady-state E is time-varying, and

the system is time-varying. However, as E(t) is

assumed to be piecewise constant, the system is in

reality piecewise time-invariant, which facilitates

the calculation of the solution.

To solve the differential equations numerically,Eq. (A5) was expressed in matrix form as:

x(t)�Ax(t)�BRa(t)

Ga(t)�Cx(t) (A6)

where A, B and C are the standard system

matrices, x (t) is the state vector in which the statevariables xk (t) are stacked, and the initial value

x (0) is as discussed for Eq. (A2). Because E (t) is

assumed to be piecewise constant, the matrix A is

also piecewise constant.

Eq. (A6) was integrated by updating the state

vector step by step over the time grid resulting

from the union of the data time points and the

time grid for E (t) and Ra(t). In this way, in eachstep A, B and C are constant matrices, and x (tk�1)

was updated from x (tk) after calculation of the

matrix exponential eA(tk�1

�/t

k) (MATLAB function

expm), using the standard equations for state

update with piecewise constant inputs [25].

Except for Ra(t ) and E (t), all the parameters of

Eq. (A5) are reported in Table 1 (Table 3 for

humans).

Specific activity derivative

The numerical solution of Eq. (A5) provides a

smooth time course of Ga(t) (glucose and tracer

concentration) and also of its derivative, Ga(t):Ga(t) is in fact proportional to x2(t) (Eq. (A5), last

equation), and is thus a linear combination of x1(t)and x2(t) (Eq. (A5), second equation), which are

smooth. From the glucose and tracer smooth Ga(t)

and Ga(t); a smooth time course of the specific

activity derivative can be obtained. The smoothed

variables were used with the compartmental mod-

els (Eqs. (5) and (6)).

Appendix B

This appendix briefly addresses the problem of

adapting the parameters of Tables 1 and 3, which

are appropriate for anesthetized rats or lean and

obese humans, to experimental conditions in

which the cardiac output value is different from

that reported here.

The glucose distribution volume in the peripheryblock is the product of cardiac output, 1-fractional

extraction and the mean transit time of the

periphery [8]. The glucose distribution volume is

likely to be substantially independent from the

cardiac output value. If the circulatory model is

used in conditions in which cardiac output is

different (e.g. conscious vs. anesthetized rats), it

is necessary to change the periphery mean transittime in order that the glucose volume remains

constant. This is achieved by adjusting the ex-

ponents ak of the periphery impulse response as

follows.

The glucose volume is calculated from the

periphery impulse response (Eq. (4)) as the sum

of volume terms Vk , one for each exponential (the

volume contribution due to g is negligible):

Vk�CO(1�E)wk

ak

(B1)

To maintain the total glucose volume constant, it

is logical to increase in each volume term the

exponents ak in proportion to the cardiac output

value. Thus, in an experimental condition in which

cardiac output is CO?, the new exponents, ak? ?, willbe:

ak? �CO?

COak (B2)

where CO and ak are the values reported in Table

1 or Table 3.

Eq. (B2) is an approximate correction, whichassumes that cardiac output does not affect

substantially the distribution of glucose. If there

are reasons to think that cardiac output affects

glucose distribution, the model parameters should

be determined experimentally.

The cardiac output problem is not apparent

using compartmental models. However, this does


not imply that the compartmental model para-meters are unaffected by cardiac output. It is in

fact expected that in conditions in which cardiac

output is different, also the compartmental model

parameters (e.g. k12 and k21) differ [8].

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