# DDMODEL00000111: Magni_2004_diabetes_IVGTT

Short description:
Insulin minimal model (MM) for the Bayesian estimation of insulin secretion rate (ISR) and other physiological indexes (e.g,. beta-cell sensitivity) in presence of a uncertain C-peptide kinetics.
 Format: PharmML 0.8.x (0.8.1) Related Publication: .hiddenContent {display:none;} Insulin minimal model indexes and secretion: proper handling of uncertainty by a Bayesian approach. Sparacino G, Bellazzi R, Toffolo GM, Cobelli C, Magni Paolo Annals of biomedical engineering, 7/2004, Volume 32, Issue 7, pages: 1027-1037 Affiliation: Dipartimento di Informatica e Sistemistica, Università degli Studi di Pavia via Ferrata, Pavia, Italy. paolo.magni@unipv.it Abstract: The identification of the insulin minimal model (MM) for the estimation of insulin secretion rate (ISR) and physiological indexes (e.g. beta-cell sensitivity) requires the knowledge of C-peptide (CP) kinetics. The four parameters of the two-compartment model of CP kinetics in a given individual can be derived either from an additional bolus experiment or, more frequently, from a population model. However, in both situations, the CP kinetics is uncertain and, in MM identification, it should be treated as such. This paper shows how to handle CP kinetics uncertainty by using a Bayesian methodology. In seven subjects, MM indexes and ISR were estimated together with their confidence intervals, using either the bolus data or the population model to assess CP kinetics. The two main results that arise from the application of the new methodology are: (i) the use of the population model in place of the bolus data to determine CP kinetics does not affect, on average, the point estimates of ISR profile and MM parameters but only the confidence intervals which becomes wider (less than 50%); (ii) in both the bolus and population situation neglecting the uncertainty of CP kinetics, as done in MM literature so far, introduces no bias, on average, on point estimates of MM indexes but only an underestimation of confidence intervals. Contributors: Paolo Magni
 Context of model development: Clinical end-point; Mechanistic Understanding; Variability sources in PK and PD (CYP, Renal, Biomarkers); Discrepancy between implemented model and original publication: Compared to the model described in the paper, the expression of ISR has been herein simplified as ISR=m*X, instead of using a piecewise function. This assumption has been done for computational problems, exactly as it was carried out in the Matlab target code of the original publication (available from the authors). The described simplification was also adopted in other previous works on the glucose-insulin minimal model (e.g., Toffolo G. et al., 2006, doi:10.1152/ajpendo.00473.2004).; Model compliance with original publication: Yes; Model implementation requiring submitter’s additional knowledge: No; Modelling context description: The identification of the insulin minimal model (MM) for the estimation of insulin secretion rate (ISR) and physiological indexes (e.g. beta-cell sensitivity) requires the knowledge of C-peptide (CP) kinetics. The four parameters of the two-compartment model of CP kinetics in a given individual can be derived either from an additional bolus experiment or, more frequently, from a population model. However, in both situations, the CP kinetics is uncertain and, in MM identification, it should be treated as such. This paper shows how to handle CP kinetics uncertainty by using a Bayesian methodology. In seven subjects, MM indexes and ISR were estimated together with their confidence intervals, using either the bolus data or the population model to assess CP kinetics. The two main results that arise from the application of the new methodology are: (i) the use of the population model in place of the bolus data to determine CP kinetics does not affect, on average, the point estimates of ISR profile and MM parameters but only the confidence intervals which becomes wider (less than 50%); (ii) in both the bolus and population situation neglecting the uncertainty of CP kinetics, as done in MM literature so far, introduces no bias, on average, on point estimates of MM indexes but only an underestimation of confidence intervals.; Modelling task in scope: estimation; simulation; Nature of research: Clinical research & Therapeutic use; Therapeutic/disease area: Endocrinology;
 Validation Status: Annotations are correct. Certification Comment: This model is not certified.
• Model owner: Paolo Magni
• Submitted: Dec 11, 2015 11:43:47 PM
• Last Modified: Nov 8, 2017 4:42:03 PM
##### Revisions
• Version: 14
• Submitted on: Nov 8, 2017 4:42:03 PM
• Submitted by: Paolo Magni
• With comment: Updated model annotations.
• Version: 11
• Submitted on: Oct 11, 2016 5:41:52 PM
• Submitted by: Paolo Magni
• With comment: Update MDL syntax to the version 1.0 and R script to SEE version 2.0.0. Added prior distributions Code automatically generated/manually modified for WinBUGS
• Version: 8
• Submitted on: Jul 16, 2016 3:22:58 PM
• Submitted by: Paolo Magni
• With comment: Model revised without commit message
• Version: 4
• Submitted on: Dec 11, 2015 11:43:47 PM
• Submitted by: Paolo Magni
• With comment: Edited model metadata online.

### Name

Magni_2004_diabetes_IVGTT

 T

### Function Definitions

 $\mathrm{combinedError1}:\mathrm{real}\left(\mathrm{additive}:\mathrm{real},\mathrm{proportional}:\mathrm{real},f:\mathrm{real}\right)=\mathrm{additive}+\mathrm{proportional}\cdot f$

### Covariate Model: $\mathrm{cm}$

#### Continuous Covariates

$\mathrm{GLUC}$

### Parameter Model: $\mathrm{pm}$

#### Random Variables

${\mathrm{eps_RES_CP1}}_{\mathrm{vm_err.DV}}~\mathrm{Normal2}\left(\mathrm{mean}=0,\mathrm{var}=1\right)$

#### Population Parameters

$\mathrm{data1}$
$\mathrm{data_k_joint}$
${\mathrm{POP_joint}}_{\mathrm{vm_mdl.MDL__prior}}~\mathrm{RandomSample}\left(\right)$
$\mathrm{POP_K01}=\mathrm{pm.POP_joint}\left(1\right)$
$\mathrm{POP_K12}=\mathrm{pm.POP_joint}\left(2\right)$
$\mathrm{POP_K21}=\mathrm{pm.POP_joint}\left(3\right)$
${\mathrm{POP_m}}_{\mathrm{vm_mdl.MDL__prior}}~\mathrm{Normal1}\left(\mathrm{mean}=\mathrm{pm.MU_POP_m},\mathrm{stdev}=\mathrm{pm.SIGMA_POP_m}\right)$
${\mathrm{POP_alpha}}_{\mathrm{vm_mdl.MDL__prior}}~\mathrm{Normal1}\left(\mathrm{mean}=\mathrm{pm.MU_POP_alpha},\mathrm{stdev}=\mathrm{pm.SIGMA_POP_alpha}\right)$
${\mathrm{POP_beta}}_{\mathrm{vm_mdl.MDL__prior}}~\mathrm{Normal1}\left(\mathrm{mean}=\mathrm{pm.MU_POP_beta},\mathrm{stdev}=\mathrm{pm.SIGMA_POP_beta}\right)$
${\mathrm{POP_x0}}_{\mathrm{vm_mdl.MDL__prior}}~\mathrm{Normal1}\left(\mathrm{mean}=\mathrm{pm.MU_POP_x0},\mathrm{stdev}=\mathrm{pm.SIGMA_POP_x0}\right)$
${\mathrm{POP_h}}_{\mathrm{vm_mdl.MDL__prior}}~\mathrm{Normal1}\left(\mathrm{mean}=\mathrm{pm.MU_POP_h},\mathrm{stdev}=\mathrm{pm.SIGMA_POP_h}\right)$
$\mathrm{CV}=0.06$
$\mathrm{GB}=87$
$\mathrm{MU_POP_m}=0.5$
$\mathrm{MU_POP_alpha}=0.06$
$\mathrm{MU_POP_beta}=11$
$\mathrm{MU_POP_x0}=1.8$
$\mathrm{MU_POP_h}=\mathrm{pm.GB}$
$\mathrm{SIGMA_POP_m}=0.5$
$\mathrm{SIGMA_POP_alpha}=0.06$
$\mathrm{SIGMA_POP_beta}=11$
$\mathrm{SIGMA_POP_x0}=1.8$
$\mathrm{SIGMA_POP_h}=\mathrm{pm.GB}\cdot 0.03$

#### Individual Parameters

$\mathrm{K01}=\mathrm{pm.POP_K01}$
$\mathrm{K12}=\mathrm{pm.POP_K12}$
$\mathrm{K21}=\mathrm{pm.POP_K21}$
$m=\mathrm{pm.POP_m}$
$\mathrm{alpha}=\mathrm{pm.POP_alpha}$
$\mathrm{beta}=\mathrm{pm.POP_beta}$
$\mathrm{x0}=\mathrm{pm.POP_x0}$
$h=\mathrm{pm.POP_h}$
$\mathrm{x0new}=\mathrm{pm.x0}\cdot 1000$

### Structural Model: $\mathrm{sm}$

#### Variables

$\mathrm{ISR}=\mathrm{pm.m}\cdot \mathrm{sm.X}$
$\mathrm{dYi}=\left\{\begin{array}{lll}-\mathrm{pm.alpha}\cdot \left(\mathrm{sm.Yi}-\mathrm{pm.beta}\cdot \left(\mathrm{cm.GLUC}-\mathrm{pm.h}\right)\cdot 0.05551\right)& \text{if}& \mathrm{cm.GLUC}>\mathrm{pm.h}\\ -\mathrm{pm.alpha}\cdot \mathrm{sm.Yi}& \text{otherwise}& \end{array}$
$\begin{array}{c}\frac{d}{dT}\mathrm{CP1}=-\left(\mathrm{pm.K01}+\mathrm{pm.K21}\right)\cdot \mathrm{sm.CP1}+\mathrm{pm.K12}\cdot \mathrm{sm.CP2}+\mathrm{sm.ISR}\\ \mathrm{CP1}\left(T=0\right)=0\end{array}$
$\begin{array}{c}\frac{d}{dT}\mathrm{CP2}=-\mathrm{pm.K12}\cdot \mathrm{sm.CP2}+\mathrm{pm.K21}\cdot \mathrm{sm.CP1}\\ \mathrm{CP2}\left(T=0\right)=0\end{array}$
$\begin{array}{c}\frac{d}{dT}X=-\mathrm{sm.ISR}+\mathrm{sm.Yi}\\ X\left(T=0\right)=\mathrm{pm.x0new}\end{array}$
$\begin{array}{c}\frac{d}{dT}\mathrm{Yi}=\mathrm{sm.dYi}\\ \mathrm{Yi}\left(T=0\right)=0\end{array}$

### Observation Model: $\mathrm{om1}$

#### Continuous Observation

$Y=\mathrm{sm.CP1}+\mathrm{combinedError1}\left(\mathrm{additive}=1.0E-4,f=\mathrm{sm.CP1},\mathrm{proportional}=\mathrm{pm.CV}\right)+\mathrm{pm.eps_RES_CP1}$

## External Dataset

 OID $\mathrm{data1}$

### File Specification

 Format $\mathrm{csv}$ Delimiter comma File Location prior_magni2004.csv

### Column Definitions

Column ID Position Column Type Value Type
$\mathrm{data_k01}$
$1$
$\mathrm{undefined}$
$\mathrm{real}$
$\mathrm{data_k12}$
$2$
$\mathrm{undefined}$
$\mathrm{real}$
$\mathrm{data_k21}$
$3$
$\mathrm{undefined}$
$\mathrm{real}$

### Column Mappings

Column Ref Modelling Mapping
$data_k01$
$\mathrm{pm.data_k_joint}\left(1\right)$
$data_k12$
$\mathrm{pm.data_k_joint}\left(2\right)$
$data_k21$
$\mathrm{pm.data_k_joint}\left(3\right)$

## External Dataset

 OID $\mathrm{nm_ds}$ Tool Format NONMEM

### File Specification

 Format $\mathrm{csv}$ Delimiter comma File Location Simulated_magni2004_data.csv

### Column Definitions

Column ID Position Column Type Value Type
$\mathrm{ID}$
$1$
$\mathrm{id}$
$\mathrm{int}$
$\mathrm{TIME}$
$2$
$\mathrm{idv}$
$\mathrm{real}$
$\mathrm{DV}$
$3$
$\mathrm{dv}$
$\mathrm{real}$
$\mathrm{GLUC}$
$4$
$\mathrm{covariate}$
$\mathrm{real}$
$\mathrm{EVID}$
$5$
$\mathrm{evid}$
$\mathrm{real}$

### Column Mappings

Column Ref Modelling Mapping
$TIME$
$T$
$DV$
$\mathrm{om1.Y}$
$GLUC$
$\mathrm{cm.GLUC}$

## Estimation Step

 OID $\mathrm{estimStep_1}$ Dataset Reference $\mathrm{nm_ds}$

### Parameters To Estimate

Parameter Initial Value Fixed? Limits
pm.POP_joint
false
$\left(,\right)$
pm.POP_m
false
$\left(,\right)$
pm.POP_alpha
false
$\left(,\right)$
pm.POP_beta
false
$\left(,\right)$
pm.POP_x0
false
$\left(,\right)$
pm.POP_h
false
$\left(,\right)$

### Operations

#### Operation: $1$

 Op Type generic
##### Operation Properties
Name Value
algo
$\text{mcmc}$

#### Operation: $2$

 Op Type BUGS
##### Operation Properties
Name Value
burnin
$1000$
inits
$\text{POP_m=0.73, POP_beta=10.07, POP_x0=1.5, POP_alpha=0.044, POP_h=88.83}$
nchains
$1$
niter
$20000$
odesolver
$\text{LSODA}$
parameters
$\text{POP_K01, POP_K12, POP_K21}$
winbugsgui
$\text{false}$

## Step Dependencies

Step OID Preceding Steps
$\mathrm{estimStep_1}$