DDMoRe Model Repository

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Model Definition
Estimation Steps

Short description:

Overgaard’s beta cell model for insulin secretion following IVGTT and OGTT

Format:	PharmML (0.6.1)
Related Publication:	Mathematical beta cell model for insulin secretion following IVGTT and OGTT. Overgaard RV, Jelic K, Karlsson M, Henriksen JE, Madsen H Annals of biomedical engineering, 8/2006, Volume 34, Issue 8, pages: 1343-1354 Affiliation: Informatics and Mathematical Modelling, Technical University of Denmark, Kongens Lyngby, Denmark. rvo@imm.dtu.dk Abstract: Evaluation of beta cell function is conducted by a variety of glucose tolerance tests and evaluated by a number of different models with less than perfect consistency among results obtained from different tests. We formulated a new approximation of the distributed threshold model for insulin secretion in order to approach a model for quantifying beta cell function, not only for one, but for several different experiments. Data was obtained from 40 subjects that had both an oral glucose tolerance test (OGTT) and an intravenous tolerance test (IVGTT) performed. Parameter estimates from the two experimental protocols demonstrate similarity, reproducibility, and indications of prognostic relevance. Useful first phase indexes comprise the steady state amount of ready releasable insulin A0 and the rate of redistribution krd, where both yield a considerable correlation (both r=0.67) between IVGTT and OGTT estimates. For the IVGTT, A0 correlates well (r=0.96) with the 10 min area under the curve of insulin above baseline, whereas krd represents a new and possibly more fundamental first phase index. For the useful second phase index gamma, a correlation of 0.75 was found between IVGTT and OGTT estimates.
Contributors:	Roberto Bizzotto

Context of model development:	Mechanistic Understanding; Clinical end-point;
Discrepancy between implemented model and original publication:	None;
Long technical model description:	Technical_description.pdf;
Model compliance with original publication:	Yes;
Model implementation requiring submitter’s additional knowledge:	No;
Modelling context description:	Before publication of the model by Overgaard et al., several descriptive mathematical models and model based methods had been proposed to calculate indexes for characterization of beta cell function from different tests, such as the oral glucose tolerance test, the meal tolerance test, the intravenous glucose tolerance test and the hyperglycemic clamp. Although useful for the analysis of a specific experiment type, those models can rarely be used across different tests, and similar indexes obtained from different experiments are not necessarily in agreement, leading to the conclusion that further work was needed for these indexes to be routinely used in clinical and epidemiological studies. Besides descriptive models for characterization, more comprehensive mathematical models had been used to communicate and increase the understanding of the physiological mechanisms behind insulin secretion. Of these, the distributed threshold hypothesis had been used to argue and derive many of the descriptive models. However, to the knowledge of Overgaard et al., descriptive insulin secretion models all failed to incorporate the fundamental mechanisms that have enabled the distributed threshold hypothesis to describe insulin secretion in response to a long series of glucose challenges, and hereby increase understanding of the beta cell function. The model by Overgaard et al. includes threshold distribution, redistribution, and incretin effects, and was applied to data from the IVGTT and the OGTT. At a longer perspective, this model was thought as a step towards a model for characterizing beta cell function, not only for one, but for many of the glucose tolerance tests; with parameters that are closer related to the physiology than those of more empirical models.;
Modelling task in scope:	estimation;
Nature of research:	Clinical research & Therapeutic use;
Therapeutic/disease area:	Endocrinology;

Validation Status:

Annotations are correct.

Certification Comment:

Model code and additional information has been scrutinized and code executed in the DDMoRe framework and demonstrated: i) successful execution in Beta Release of the Interoperability Framework; ii) agreement in parameters and uncertainty (following execution with this model using real original data) with the corresponding results in the related publication except for PPV_FRAC and POP_TRANS which could not be verified ; iii) agreement between simulations from the uploaded model and those presented in the related publication. The review concluded that this version of model code is executable, and described and coded in agreement with the related publication.

Mandatory file
- Executable_overgaard.xml

Additional Files

Model owner: Roberto Bizzotto
Submitted: Dec 11, 2015 11:53:37 AM
Last Modified: Sep 20, 2016 10:21:19 AM

Revisions

Version: 11
- Submitted on: Sep 20, 2016 10:21:19 AM
- Submitted by: Roberto Bizzotto
- With comment: Updated lst file and output plot, produced with Prod4_Beta installation
Version: 10
- Submitted on: Sep 6, 2016 11:24:41 AM
- Submitted by: Roberto Bizzotto
- With comment: Updated Comparison table
Version: 9
- Submitted on: Sep 6, 2016 10:11:46 AM
- Submitted by: Roberto Bizzotto
- With comment: Updated validationOvergaard.R, Comparison_table.pdf and Description_of_simulated_data.pdf
Version: 8
- Submitted on: Aug 11, 2016 8:54:06 PM
- Submitted by: Roberto Bizzotto
- With comment: Updated model annotations.
Version: 5
- Submitted on: May 18, 2016 3:56:38 PM
- Submitted by: Roberto Bizzotto
- With comment: Updated model annotations.
Version: 2
- Submitted on: Dec 11, 2015 11:53:37 AM
- Submitted by: Roberto Bizzotto
- With comment: Edited model metadata online.

Independent variable T

Function Definitions

combinedError2 (additive, proportional, f) = \sqrt{({proportional}^{2} + ({additive}^{2} \times f^{2}))}

Structural Model sm

Variable definitions

TMP = (G1 + \frac{((T - T1) \times (GLUC - G1))}{(TOBS - T1)})

GL = {\begin{matrix} 0.1 & if & (TMP < 0) \\ TMP & otherwise \end{matrix}

GL_1 = (G2 + \frac{((T - T1) \times (G3 - G2))}{(TOBS - T1)})

GLAB = (GL_1 - GB)

FG = \frac{{GL}^{HILL2}}{({G50}^{HILL2} + {GL}^{HILL2})}

FP = \frac{((HILL2 \times {GL}^{(HILL2 - 1)}) \times {G50}^{HILL2})}{{({G50}^{HILL2} + {GL}^{HILL2})}^{2}}

FONE = \frac{(((RI + A0) \times FP) \times GP1)}{(1 - FG)}

PROV = \frac{{GL}^{HILL}}{({GL}^{HILL} + {EC50}^{HILL})}

\frac{dPRO}{dT} = (- ALPHA \times (((PRO + P0) - (ES \times PROV)) - \frac{(EF \times GLAB)}{AUC}))

\frac{dRI}{dT} = ((((1 - FG) \times (PRO + P0)) - (KRD \times (RI + A0))) - FONE)

\frac{dRDI}{dT} = ((((FG \times (PRO + P0)) + (KRD \times (RI + A0))) + FONE) - (M \times (RDI + RI0)))

\frac{dPI}{dT} = ((M \times (RDI + RI0)) - (KI \times (PI + I0)))

IPLASMA = \frac{(PI + I0)}{V1}

logIPLASMA = {\begin{matrix} log (IPLASMA) & if & (IPLASMA > 0) \\ - 10000 & otherwise \end{matrix}

Initial conditions

PRO = 0

RI = 0

RDI = 0

PI = 0

Variability Model

Level	Type
DV	residualError
ID	parameterVariability

Covariate Model

Continuous covariate ID1

Continuous covariate TOBS

Continuous covariate BI

Continuous covariate GB

Continuous covariate GLUC

Continuous covariate G1

Continuous covariate T1

Continuous covariate AUC

Parameter Model

Parameters

POP_ALPHA

;

POP_KI

;

POP_K

;

POP_HILL

;

POP_ES

;

POP_TRANSF

;

POP_EOGF

;

POP_HILL2_1

;

POP_HILL2_2

;

POP_M

;

aa

;

b

;

PPV_KRD

;

PPV_FRAC

;

PPV_ES

;

PPV_EF

;

SIGMA

;

eta_PPV_KRD \sim N (0.0, PPV_KRD)

— ID

eta_PPV_FRAC \sim N (0.0, PPV_FRAC)

— ID

eta_PPV_ES \sim N (0.0, PPV_ES)

— ID

eta_PPV_EF \sim N (0.0, PPV_EF)

— ID

epsilon \sim N (0.0, SIGMA)

— DV

V1 = 3

ALPHA = POP_ALPHA

KI = POP_KI

HILL = POP_HILL

M = POP_M

HILL2_1 = POP_HILL2_1

HILL2_2 = POP_HILL2_2

HILL2 = {\begin{matrix} HILL2_1 & if & (ID1 > 100) \\ HILL2_2 & otherwise \end{matrix}

I0 = (BI \times V1)

RI0 = \frac{(I0 \times KI)}{M}

P0 = (I0 \times KI)

GP = \frac{(GLUC - G1)}{(TOBS - T1)}

GP1 = {\begin{matrix} GP & if & (GP > 0) \\ 0 & otherwise \end{matrix}

G2 = {\begin{matrix} GB & if & (G1 < GB) \\ G1 & otherwise \end{matrix}

G3 = {\begin{matrix} GB & if & (GLUC < GB) \\ GLUC & otherwise \end{matrix}

KRD = (POP_K \times exp (eta_PPV_KRD))

ES = {\begin{matrix} (POP_ES \times exp (eta_PPV_ES)) & if & (ID1 \geq 99) \\ (POP_ES \times exp (eta_PPV_ES)) & otherwise \end{matrix}

EF = {\begin{matrix} ((1000 \times POP_EOGF) \times exp (eta_PPV_EF)) & if & (ID1 \geq 99) \\ 0 & otherwise \end{matrix}

EMAX = ES

LF = exp ((POP_TRANSF + eta_PPV_FRAC))

FRAC = \frac{LF}{(1 + LF)}

G50 = (GB \times {\frac{(1 - FRAC)}{FRAC}}^{\frac{1}{HILL2}})

EC50 = {\begin{matrix} (GB \times {(\frac{EMAX}{P0} - 1)}^{\frac{1}{HILL}}) & if & (EMAX > P0) \\ 7 & otherwise \end{matrix}

A0 = \frac{(P0 \times (1 - FRAC))}{KRD}

Observation Model

Observation Y
Continuous / Residual Data

Parameters

Y = (logIPLASMA + (combinedError2 (logIPLASMA, aa, b) \times epsilon))

Estimation Steps

Estimation Step estimStep_1

Estimation parameters

Fixed parameters

$b = 0$
$SIGMA = 1$

Initial estimates for non-fixed parameters

$POP_ALPHA = 0.049$
$POP_KI = 0.128$
$POP_K = 0.0103$
$POP_HILL = 12.1$
$POP_ES = 116$
$POP_TRANSF = - 0.8$
$POP_EOGF = 8.2$
$POP_HILL2_1 = 6$
$POP_HILL2_2 = 5.28$
$POP_M = 1.35$
$aa = 0.0352$
$PPV_KRD = 0.284$
$PPV_FRAC = 0.034$
$PPV_ES = 0.3$
$PPV_EF = 0.3$

Estimation operations

1) Estimate the population parameters

Algorithm FOCEI

Step Dependencies

estimStep_1

DDMODEL00000101: Overgaard’s beta cell model for insulin secretion

Revisions

Function Definitions

Structural Model sm

Variability Model

Covariate Model

Parameter Model

Observation Model

Observation YContinuous / Residual Data

Estimation Steps

Estimation Step estimStep_1

Estimation parameters

Estimation operations

Step Dependencies

Observation Y
Continuous / Residual Data