## Deconvolution of Insulin Secretion Rates

Soren Klim, Stig Mortensen and Henrik Madsen

Technical University of Denmark

Abstract is based on Master Project [4] and article [3] by Soren Klim and Stig Mortensen

Objectives: Insulin and C-peptide are secreted in equi-molar amounts from the beta-cells then passing through the liver before entering systemic circulation. The variable first pass effect on insulin due to hepatic extraction makes the insulin observations unusable for determining the secretion rate. C-peptid passes completely through the liver and has well-established kinetics. These properties of C-peptide enable deconvolution of C-peptide secretion which is equivalent to insulin secretion rate (ISR).

Methods: Deconvolution of C-peptide measurements has been performed in many different ways. Classic deconvolution is very sensitive to observation noise. A modelling approach is to model insulin secretion rates as piecewise constant and then estimate the secretion levels. This method was presented at PAGE in Copenhagen by C. Dansirikul [1].
In this project the deconvolution was performed using a framework able to handle Non-Linear Mixed Effects models based on Stochastic Differential Equations (SDEs). The framework was first developed for Matlab but has recently been made available to R.
The structural model and parameters were based on the article by Van Cauter et al. [2]. The insulin secretion rate was modelled as a random walk. This assumption is useful for quantifying the insulin secretion as the estimated parameter then becomes the variance scaling factor. This is an unrealistic assumption but it enables the compromise between observation noise and natural variation in secretion. The final deconvolved secretion rate is the optimal according to both stochastic factors.
The assumption on insulin secretion rate being modelled as a random walk induces problems when the model is extended into simulation purposes. The insulin secretion rate can under this assumption assume unphysiological negative values. A model structure using 2 compartments to build a double exponential decay to model the insulin secretion was added to the assumed random walk. This model was named the intervention model from the inspiration in time series analysis. It uses the information on when the meals are served to trigger insulin secretion. The insulin secretion rates were estimated using this extended model for ISR.

Results: The insulin secretion rates and uncertainty can be estimated with the framework. The stochastic deconvolution technique is less sensitive to observation error and enables the incorporation of existing model structures.
The intervention model is furthermore able to fit insulin secretion rates and provides a model with simulation properties.

Conclusions: The stochastic deconvolution technique can be used to determine insulin secretion rates. The deconvolved rates are optimal according to measurement noise and structural model. The technique enables a possibility to determine rates and uncertainty at any time point based on all observations i.e. smoothed estimates.

References:
[1] C. Dansirikul et al. PAGE 16 (2007) Abstract 1142
[2] E. Van Cauter et al. Diabetes 41: 368-77 (1992)
[3] S. Mortensen et al. Journal of PK/PD. 2007 Oct; 34(5):623-42
[4] S. Klim and S. Mortensen. DTU-IMM, Master Project 2006

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