is the minimal model too minimal?

4
Diabetologia (1996) 39:997-1000 Diabetologia Springer-Verlag 1996 For debate Is the minimal model too minimal? A. Caumo 1, P. Vicilli 2, C. Cohelli 2 1San Raffaele Scientific Institute, Milano, Italy 2Department of Electronics and Informatics, University of Padova, Padova, Italy The intravenous glucose tolerance test (IVGTT: stan- dard or modified with a tolbutamide or insulin injec- tion) interpreted with the classic minimal model of glucose disappearance [1, 2] is a powerful non-inva- sire tool with which to investigate glucose metabo- lism in physiopathology studies. The model provides two metabolic indices measuring glucose effective- ness, Sc~ and insulin sensitivity, SI, in a single individ- ual. S c and S I are composite parameters, i.e. they measure the net effect of glucose and insulin, respec- tively, to promote glucose disappearance and inhibit endogenous glucose production. The minimal model method has gained increasing popularity (approximately 250 papers appeared be- fore 1995) and is used by investigators around the world because it is simple and non-invasive. Unfortu- nately several reports in the last few years have indi- cated that some assumptions of the minimal model method may not be entirely correct [3-7]. In particu- lar, Quon et al. [6] have shown in a study on insulin- dependent diabetic patients that the decay of glucose during an IVGTT in which insulin is maintained at the basal level is not monoexponential as predicted by the minimal model and that S G is overestimated. More recently, Saad et al. [7] have shown that in nor- motolerant subjects S~ obtained from an insulin-mod- ified IVGTT is underestimated when compared to in- sulin sensitivity obtained with the glucose clamp tech- nique and that in non-insulin-dependent diabetic (NIDDM) patients SI is both inaccurate and impre- cise. However, no clear explanation has been offered as to why S c is overestimated and S I underestimated. Corresponding author: Professor C. Cobelli, Department of Electronics and Informatics, University of Padova, 1-35131 Padova, Italy Abbreviations: IVGTT, Intravenous glucose tolerance test; S o glucose effectiveness; SI, insulin sensitivity; NIDDM, non-in- sulin-dependent diabetes mellitus This is highly desirable in order to better define the domain of validity of the minimal model which, as for all models, is limited. In our previous work [3-5] we pointed out the in- adequacy of the single-compartment description on which the minimal model is founded. In particular, we have sl~own that a two-compartment description is necessary in order to obtain a reliable profile of en- dogenous glucose production during the IVGTT [5]. Our goal here is to explain the findings of Quon et al. [6] and Saad et al. [7] by using a two-compartment paradigm for glucose kinetics. This exercise will also allow us to better define the domain of validity of S G and S I and explain why the classic minimal model works with difficulty in NIDDM patients. Glucose effectiveness S c measures glucose effectiveness at basal insulin, i.e. the ability of glucose to favour its own disposal and inhibit its own production when insulin is basal. If Sa were a valid measure of glucose effectiveness at basal insulin its value should be the same whether estimated with or without an insulin response above basal..Quon et al. [6] measured S G in insulin-depen- dent diabetic subjects during an IVGTT with a nor- mal hyperinsulinaemic profile (reproduced by a com- puter controlled insulin infusion) and during an IV- GTT with insulin maintained at the basal level. They then compared the glucose profile measured at basal insulin with the one predicted by the minimal model using the value of S~ estimated during the IVGTT with the normal hyperinsulinaemic profile. They found that the minimal model prediction was in good agreement during the first 20 min, but signifi- cantly lower than the observed glucose concentration for the rest of the experiment. This finding led these authors to conclude that S~ is overestimated.

Upload: a-caumo

Post on 10-Jul-2016

218 views

Category:

Documents


2 download

TRANSCRIPT

Page 1: Is the minimal model too minimal?

Diabetologia (1996) 39:997-1000 Diabetologia �9 Springer-Verlag 1996

For debate

Is the minimal model too minimal? A. Caumo 1, P. Vicilli 2, C. Cohelli 2

1 San Raffaele Scientific Institute, Milano, Italy 2 Department of Electronics and Informatics, University of Padova, Padova, Italy

The intravenous glucose tolerance test (IVGTT: stan- dard or modified with a tolbutamide or insulin injec- tion) interpreted with the classic minimal model of glucose disappearance [1, 2] is a powerful non-inva- sire tool with which to investigate glucose metabo- lism in physiopathology studies. The model provides two metabolic indices measuring glucose effective- ness, Sc~ and insulin sensitivity, SI, in a single individ- ual. S c and S I are composite parameters, i.e. they measure the net effect of glucose and insulin, respec- tively, to p romote glucose disappearance and inhibit endogenous glucose production.

The minimal model me thod has gained increasing populari ty (approximately 250 papers appeared be- fore 1995) and is used by investigators around the world because it is simple and non-invasive. Unfortu- nately several reports in the last few years have indi- cated that some assumptions of the minimal model me thod may not be entirely correct [3-7]. In particu- lar, Quon et al. [6] have shown in a study on insulin- dependent diabetic patients that the decay of glucose during an IV G TT in which insulin is maintained at the basal level is not monoexponent ia l as predicted by the minimal model and that S G is overestimated. More recently, Saad et al. [7] have shown that in nor- motolerant subjects S~ obtained from an insulin-mod- ified IVGTT is underes t imated when compared to in- sulin sensitivity obtained with the glucose clamp tech- nique and that in non-insul in-dependent diabetic (NIDDM) patients SI is both inaccurate and impre- cise. However, no clear explanation has been offered as to why S c is overest imated and S I underest imated.

Corresponding author: Professor C. Cobelli, Department of Electronics and Informatics, University of Padova, 1-35131 Padova, Italy Abbreviations: IVGTT, Intravenous glucose tolerance test; S o glucose effectiveness; SI, insulin sensitivity; NIDDM, non-in- sulin-dependent diabetes mellitus

This is highly desirable in order to better define the domain of validity of the minimal model which, as for all models, is limited.

In our previous work [3-5] we pointed out the in- adequacy of the single-compartment description on which the minimal model is founded. In particular, we have sl~own that a two-compar tment description is necessary in order to obtain a reliable profile of en- dogenous glucose product ion during the IVGTT [5]. Our goal here is to explain the findings of Quon et al. [6] and Saad et al. [7] by using a two-compar tment paradigm for glucose kinetics. This exercise will also allow us to better define the domain of validity of S G and S I and explain why the classic minimal model works with difficulty in N I D D M patients.

Glucose effectiveness

S c measures glucose effectiveness at basal insulin, i.e. the ability of glucose to favour its own disposal and inhibit its own product ion when insulin is basal. If Sa were a valid measure of glucose effectiveness at basal insulin its value should be the same whether est imated with or without an insulin response above basal . .Quon et al. [6] measured S G in insulin-depen- dent diabetic subjects during an IVGTT with a nor- mal hyperinsulinaemic profile ( reproduced by a com- puter controlled insulin infusion) and during an IV- GTT with insulin maintained at the basal level. They then compared the glucose profile measured at basal insulin with the one predicted by the minimal model using the value of S~ est imated during the IVGTT with the normal hyperinsulinaemic profile. They found that the minimal model prediction was in good agreement during the first 20 min, but signifi- cantly lower than the observed glucose concentrat ion for the rest of the experiment. This finding led these authors to conclude that S~ is overestimated.

Page 2: Is the minimal model too minimal?

998 A. Caumo et al.: Minimal model indices

16-

--~- 12- O

E 8- E

4 -

0 0

16-

--=-12-

E

4-

0

A

~ oncentration

Minimal model (one compartment)

0.06- B

0.05 -

~. 0.04

0.03

0.02

0.01

Fractional decay rate

5'0 120 2;.0 300 3;0 ~ 5'0 120 1;0 240 3;0 ' 360

Two compartment model

C 0.05 D 0.O5

. 0.04 Fractional decay rate

0.03

0.02 -

0.01 -

0 i I i I

o 5'0 12o 1;o 2;o 300 3;0 o 6'0 240 3;0 Time (min) Time (rain)

Fig. 1 A-D. Time courses of glucose concentration (A, C) and of the fractional glucose decay rate (B, D) during an IVGTT at basal insulin as predicted by the minimal model (A, B) and a two-compartment model (C, D)

_.=-18-

~E 16- s

12-

10- "E

8- C o 6"

4 -

2-

0 0 6'0 120

~ rtment model

Minimal model

360 Time (min)

Fig.2. Comparison between the glucose concentration profiles predicted by the minimal model and a two-compartment mod- el during an IVGTT at basal insulin

To elucidate the reasons for the discrepancy ob- served by Quon et al. [6] it is useful to examine in de- tail what happens during an IVGTT in which insulin is mainta ined at the basal level. The minimal model, due to its monocompar tmen ta l structure, predicts that under these conditions glucose decay is monoex- ponential (Fig. 1, A) and the fractional decay rate (fraction of glucose concentrat ion above basal declin- ing per unit time) is constant and equal to S O (Fig. 1, B):

g(t) = gb -F [g0--gb]e-Sa t (1)

where gb is basal glucose concentrat ion and go is glu- cose concentrat ion immediately after the glucose bo- lus.

However, it is well-accepted that the true glucose system does not exhibit single but at least two-

Fig. 3. A two-compartment picture of the glucose system with the accessible and the nonaccessible pools. S~ measures not only events pertaining to glucose uptake and production, but also to the exchange kinetics between the accessible and the nonaccessible pool

compar tment kinetics [8-10]. A two-compar tment model predicts that glucose decays with a two-expo- nential profile (Fig. 1, C) and that the fractional glu- cose decay rate is no longer constant (Fig. 1, D): it is higher at the beginning of the IVGTT, when the fast componen t of glucose disappearance plays a major role, and lower at the end of the IVGTT, when only the slow component remains in play. The undermod- elling associated with the monocompar tmenta l de- scription of glucose kinetics is likely to be the major source of the discrepancy observed by Quon et al. [6]. In fact, plotting together the monoexponent ia l glucose decay curve predicted by the minimal model and the two-exponential profile generated using the two-compar tment model (Fig. 2) we have been able to reproduce the experimental results shown in the paper of Quon et al. [6]: the mono- and the two-expo- nential glucose profiles are almost superimposable during the first 20 min of the IVGTT, but diverge thereafter. This suggests that the validity of S o as des- criptor of glucose effectiveness is l imited to the initial port ion of the IVGTT. Afterwards the glucose profile predicted by the minimal model returns to the base- line too quickly (Fig. 2) because S o overestimates the true fractional glucose decay rate that becomes slow- er and slower as the contribution of the fast exponen- tial componen t fades away and only the slow compo- nent remains in play (Fig. 1, D).

Should we conclude f rom this, like Quon et al. [6], that S o overestimates glucose effectiveness "tout court"? Alternatively, having shown that S o well-de- scribes the initial port ion of glucose decay, why not take home this parameter with its l imited domain of validity? With this last attitude, one could say that S o is probably an accurate descriptor of glucose effec- tiveness during the initial part of the IVGTT, say be- tween 10 and 20 min. However, since in that part of the IVGTT both the fast and the slow components of glucose disappearance are active, So not only re- flects the ability of glucose to p romote its uptake and suppress its own product ion but also the ex- change kinetics between the accessible and the non- accessible compar tment (Fig. 3).

Page 3: Is the minimal model too minimal?

A. Caumo et al.: Minimal model indices 999

Insulin sensitivity

The above analysis of the domain of validity of S G helps to throw light on the findings of Saad et al. [7] concerning the minimal model estimate of insulin sensitivity S~. These authors found that S~ was 60 % lower in normal subjects than that estimated from the glucose clamp. In addition they found that the minimal model, when applied to N I D D M subjects, often failed to give precise estimates of S I and that the correlation with the analogous index estimated with the glucose clamp was weak.

We reason here that the marked underestimation of S I may be the result of undesired compensations between glucose effectiveness and insulin sensitivity. We have seen that S G approximates the fractional de- cay rate of glucose per se in the initial part of the IV- GTT, but overestimates it thereafter. Since glucose and insulin are the only factors that determine glu- cose disappearance in the minimal model, to fit the glucose data the model is forced to compensate for the S~ overestimation by underestimating insulin ac- tion and possibly insulin sensitivity. The analysis of the bias affecting insulin action of the minimal model would require the comparison with the insulin action predicted by a two-compartment model. However, the gross features of the bias can be appreciated by the following reasoning. The equations of the mini- mal model are:

dg(t) _ [Pl + x(t)]g(t) + Plgb g(O) = go (2) dt

dx(t) dt = -pzx(t) + p3[i(t) - ib] x(0) = 0 (3)

where pa, P2, P3 are rate parameters, x is insulin ac- tion, i is plasma insulin concentration, and i b its basal value. By dividing both members of Eq. 2 by g(t) and remembering that S G = Pl and (dg/dt)/g(t) equals d[lng(t)]/dt, Eq. 2 can be rewritten:

d ln[g(t)] = x(t) + S o g(t) gb

g(t) (4)

Integrating Eq. 3 from 0 to infinity and remembering that x(0) = x(~) = 0 one obtains:

SI = P_2 = Jo x(t)dt p2 Jo[i(t)--ib]dt (5)

Eq. 4 shows that the slope of the logarithm of glucose concentration is partitioned into two components, one depending on insulin action and the other on SG. Fitting the glucose data means accurately describing the slope of the logarithm of glucose concentration. To do so with the second term of the right member

of Eq. 4 overestimated generates a compensatory bias on x. In particularl since S G is weighted by the ra- tio [g(t) - gb]/g(t),x will be underestimated from ap- proximately 20 rain up to g = gb and overestimated thereafter. Since S I is proportional to the integral of insulin action (Eq. 5), S I will be underestimated if the integral of the initial underestimation of insulin action is larger than that of the subsequent overesti- mation. The results of Saad et al. [7] seem to support this hypothesis.

Of note is that the underestimation of insulin ac- tion and insulin sensitivity due to the monocompart- mental approximation may contribute to explaining the difficulty in precisely estimating S I in NIDDM pa- tients and the unsatisfactory agreement with the cor- responding clamp-based estimate. In fact, if the true insulin action is already low because the subject is re- sistant to insulin and a considerable portion is used to compensate for the inadequacy of the monocompart- mental description of glucose kinetics, the minimal model profile of insulin action will become so tow and slow so as to degrade the precision of S I or even prevent its estimation. Even in NIDDM subjects in which S~ can be precisely estimated, the underestima- tion of this parameter will considerably narrow the range of the values of insulin sensitivity of this group, thus worsening the correlation between the IVGTT and the clamp-based measurement of insulin sensitiv- ity. This may explain why the degree of correlation is worse in NIDDM than in normotolerant subjects.

Conclusions

We have shown that the recent results obtained by Q u o n et al. [6] and Saad et al. [7] can be attributed to the approximation inherent to the single pool de- scription of glucose kinetics of the minimal model. S G estimated with the classic single compartment minimal model measures the fractional rate of glu- cose disappearance per se in the initial portion of the IVGTT, approximately between 10 and 20 min and reflects not only the ability of glucose to promote its disappearance and suppress its own production, but also the exchange kinetics between the accessible and the nonaccessible glucose pool.

The limited domain of validity of S G as a descriptor of glucose effectiveness determines undesired com- pensations on the profile of insulin action which can lead to an underestimation of S I. This makes the as- sessment of insulin sensitivity with the minimal mod- el in N I D D M subjects particularly problematic since in this group insulin sensitivity is already very low.

Acknowledgements. This work was partially supported by a grant from the Italian Ministero della Universita' e della Ricerca Scientifica e Tecnologica (MURST 40%) on Bio- ingegneria dei Sistemi Metabolici e Cellulari and by National Institutes of Health Grant RR-02176.

Page 4: Is the minimal model too minimal?

1000 A. Caumo et al.: Minimal model indices

References

1. Bergman RN, Ider YZ, Bowden CR, Cobelli C (1979) Quantitative estimation of insulin sensitivity. Am J Physiol 236:E667-E677

2. Bergman RN, Phillips NLS, Cobelli C (1981) Physiologic evaluation of factors controlling glucose tolerance in man. Measurement of insulin sensitivity and beta-cell sensitivity from the response to intravenous glucose. J Clin Invest 68: 1456-1467

3. Cobelli C, Pacini G, Toffolo G, Sacca L (1986) Estimation of insulin sensitivity and glucose clearance from minimal model: new insights from labeled IVGTT. Am J Physiol 250:E591-E598

4. Caumo A, Giacca A, Morgese M, Pozza G, Micossi R Co- belli C (1991) Minimal model of glucose disappearance: lessons from the labelled IVGTT. Diabet Med 8:822-832

5. Caumo A, Cobelli C (1993) Hepatic glucose production during the labeled IVGTT: estimation by deconvolution with a new minimal model. Am J Physiol 264:E829-E841

6. Quon MJ, Cochran C, Taylor SI, Eastman RC (i994) Non- insulin mediated glucose disappearance in subjects with IDDM: discordance between experimental results and minimal model analysis. Diabetes 43:890-896

7. Saad MF, Anderson RL, Laws A e t al. for the IRAS (1994) A comparison between the minimal model and the glucose clamp in the assessment of insulin sensitivity across the spectrum of glucose tolerance. Diabetes 43:1114-1121

8. Radziuk J, Norwich KH, Vranic M (1978) Experimental validation of measurements of glucose turnover in non- steady state. Am J Physiol 234:E84-E93

9. Cobelli C, Toffolo G, Ferrannini E (1984) A model of glu- cose kinetics and their control by insulin, compartmental and noncompartmental approaches. Math Biosci 72: 291- 315

10. Cobelli C, Mari A, Ferrannini E (1987) Non-steady-state: error analysis of Steele's model and developments for glu- cose kinetics. Am J Physiol 252:E679-E689