Supplementary Data - Diabetes

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SUPPLEMENTARY DATA 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) ©2012 American <strong>Diabetes</strong> Association. Published online at http://diabetes.diabetesjournals.org/lookup/suppl/doi:10.2337/db11-1478/-/DC1 � G b (A7) (A9)
SUPPLEMENTARY DATA 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) ©2012 American <strong>Diabetes</strong> Association. Published online at http://diabetes.diabetesjournals.org/lookup/suppl/doi:10.2337/db11-1478/-/DC1 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.
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Supplementary Data - Diabetes

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