Supplementary Data - Diabetes

SUPPLEMENTARY DATA 

Oral Glucose Minimal Model 

The model consists of the following differential equations: 

where G is plasma glucose concentration, I is plasma insulin concentration, suffix “b” denotes basal 

values, X is insulin action, V is the distribution volume, SG, p2 and p3 are model parameters. Specifically, 

SG is the fractional (i.e., per unit distribution volume) glucose effectiveness measuring glucose ability 

per se to promote glucose disposal and inhibit glucose production; p2 is the rate constant describing the 

dynamics of insulin action; p3 is the parameter governing the magnitude of insulin action. 

The parametric description of Rameal is a piecewise-linear function with known break-point ti and 

unknown amplitude �i: 

Ra 

meal 

� 

�� 

i 

( t) 

� � 

� 

�0 

�. 

1 

� i � � i 

� 

t � t 

i 

�. 

1 

i�1 

� 

�t�t� i�1 

per t 

i- 1 

otherwise 

� t � t 

i � 1... 

7 

with the breakpoints allocated at 0,10,30,60,90,120,180 and 360 min. 

©2012 American Diabetes Association. Published online at http://diabetes.diabetesjournals.org/lookup/suppl/doi:10.2337/db11-1478/-/DC1 

i 

(A2) 

SI is estimated as: 

p3 

SI� � V (dl/kg/min per µU/ml) (A3) 

p 

2 

Oral C-peptide and Insulin Minimal Model 

The model used to describe C-peptide secretion assumes that the pancreatic insulin secretion rate (ISR) 

is linked to plasma C-peptide concentration by the two-compartment model of C-peptide kinetics 

originally proposed (1):


where CP1 and CP2 (pmol) are C-peptide masses in the accessible and peripheral compartments 

respectively, C is the C-peptide plasma concentration, and k01, k12, and k21 (min -1 ) are C-peptide kinetic 

parameters and Vc the C-peptide volume of distribution, fixed to standard values (2) to assure numerical 

identification of the overall model. The model also assumes that ISR is made up of a basal (ISRb), a 

static (ISRs) and a dynamic (ISRd) component: 

ISRb is equal to k01� CP1b�V to guarantee the steady state conditions. 

ISRs is assumed equal to the provision of releasable insulin to β-cells, controlled by glucose 

concentration above a threshold level h: 

� 

ISRs 

�� 

� 

Y ( t) 

� � 

�� 

� �Y 

( t) 

� � 

�Y( t) 

� ��G( t) 

� h�� 

if G � h 

if G � h 

ISRd represents the secretion of insulin from the promptly releasable pool and is proportional to the rate 

of increase of glucose: 

� dG(t) 

�K 

� 

ISRd(t) � � dt 

� 

�0 

dG(t) 

if � 0 

dt 

otherwise 

Static β-cell responsivity (Φs) is defined as: 

� 

� 

� �G( t) 

� h� 

and G(t) 

0 

� s � � 

� � 

(A8) 

dt 

0 

ISRs( 

t) 

dt 

Dynamic β-cell responsivity is defined as: 

� 

d 

� 

� 

� 

0 

� 

ISRd( 

t) 

dt 

� 

0 

dG( 

t) 

dt 

dt 

Gmax 

� 

G 

KdG 

b � � K 

G � G 

max 

b 

assuming that G(t) decreases monotonously after reaching its maximum (Gmax), that G(0)=Gb and the 

integral at the denominator is calculated only when the derivative of G is positive. 

The model used to describe insulin kinetics is a single compartment model: 

IDR t 

I� ( ) 

( t) 

� �n 

� I( 

t) 

� 

V 

I( 

0) 

� I b 

(A10) 

I 

(A6) 


� G 

b 

(A7) 

(A9)


where I is plasma insulin concetration, n is the fractional insulin clearance rate, VI is insulin volume of 

distribution and IDR is insulin delivery rate, i.e. the rate of appearance of insulin in plasma after hepatic 

extraction (HE). IDR, ISR and HE are linked by the following relationship: 

ISR( 

t) 

� IDR( 

t) 

HE( 

t) 

� 

ISR( 

t) 

thus 

ISR t � HE t � 

I� ( ) � 1 � ( ) 

( t) 

� �n 

� I( 

t) 

� 

I( 

0) 

� I 

V 

I 

According to Campioni et al 2009, HE is described with a piecewise linear function: 

� HE 

�HEi�1 

� 

HE( 

t) 

� � t 

� 

�HE0 

� HEb 

i 

i 

� HE 

� t 

i�1 

i�1 

� 

�t�t� i�1 

t 

i�1 

� t � t 

(A11) 


i 

b 

i �1, 

2,... 

6 

wehere HEb is the basal hepatic extraction, that can be derived from basal conncetrations: 

HE 

b 

k 

� 

01 

� Cb 

�Vc 

� k 01 � I 

k � C �V 

01 

b 

b 

�V 

I 

(A12) 

(A13) 

(A14) 

A table containing the definition of all the minimal model indices can be found in [Cobelli et al, Am. J. 

Physiol 2007], Table 2(3). 

Parameter Estimation 

Oral Glucose Minimal Model was numerically identified by nonlinear least squares, implemented in 

Matlab ® R2010b. Measurement error on glucose data was assumed to be independent, gaussian, with 

zero mean and known standard deviation (CV = 2%). Insulin concentration is the model forcing function 

and was assumed to be known without error. Model identification requires a number of assumptions 

which were discussed in detail in [Dalla Man et al, IEEE Trans Biomed Eng 2002] (4). Briefly, to ensure 

its a priori identifiability one has to assume values for V and SG. Here we fixed them to the median 

obtained with a triple tracer method in [Dalla Man et al Am J. Physiol, 2004] (5), i.e. V=1.45 dl/kg, 

SG=0.025 min -1 (median was preferred to mean values since parameters are not normally distributed). 

To improve numerical identifiability of the remaining parameters p2, p3, �i (i=1..7) a gaussian bayesian 

prior was considered on the square root of p2 ref (SQRp2 ref ), which is normally distributed: SQRp2 =0.11 

min -1/2 , SD=10%. Finally, a constraint was imposed to guarantee that the area under the estimated Rameal 

equals the total amount of ingested glucose, D, multiplied by the fraction of the ingested dose that is 

actually absorbed, f (fixed to the median of reference values f=0.9): this constraint provides an 

additional relationship among the unknown parameters �i, thus reducing the number of unknowns by 

one.


Oral C-peptide and Insulin Minimal Model was numerically identified by nonlinear least squares, 

implemented in Matlab ® R2010b. Measurement error on glucose data was assumed to be independent, 

Gaussian, with zero mean and known standard deviation, as reported in Toffolo et al 2006 (6). Glucose 

concentration is the model forcing function and was assumed to be known without error. 

The models fit glucose, C-peptide and insulin data well. Weighted residuals are shown in Figure 

attached online. 

References 

1. Eaton RP, Allen RC, Schade DS, Erickson KM, Standefer J: Prehepatic insulin production in man: 

Kinetic analysis using peripheral connecting peptide behavior. J.Clin.Endocrinol.Metab. 1989;51:520- 

528 

2. Van Cauter E, Mestrez F, Sturis J, Polonsky KS: Estimation of insulin secretion rates from C-peptide 

levels. Comparison of individual and standard kinetic parameters for C-peptide clearance. Diabetes 

1992;41:368-377 

3. Cobelli C, Toffolo GM, Dalla Man C, Campioni M, Denti P, Caumo A, Butler P, Rizza R: 

Assessment of beta-cell function in humans, simultaneously with insulin sensitivity and hepatic 

extraction, from intravenous and oral glucose tests. American journal of physiology. Endocrinology and 

metabolism 2007;293:E1-E15 

4. Dalla Man C, Caumo A, Cobelli C: The oral glucose minimal model: estimation of insulin sensitivity 

from a meal test. IEEE Trans Biomed Eng 2002;49:419-429 

5. Dalla Man C, Caumo A, Basu R, Rizza R, Toffolo G, Cobelli C: Minimal model estimation of 

glucose absorption and insulin sensitivity from oral test: validation with a tracer method. Am J Physiol 

Endocrinol Metab 2004;287:E637-643 

6. Toffolo G, Basu R, Dalla Man C, Rizza R, Cobelli C: Assessment of postprandial glucose 

metabolism: conventional dual- vs. triple-tracer method. Am J Physiol Endocrinol Metab 

2006;291:E800-806 

©2012 American Diabetes Association. Published online at http://diabetes.diabetesjournals.org/lookup/suppl/doi:10.2337/db11-1478/-/DC1

Supplementary Data - Diabetes

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