# Non Linear Deconvolution

Background: Deconvolution methods using stochastic differential equations have been investigated in publications [1,2,3]. Using this method the most likely outcome of an unobservable internal process can be quantified. The (extended) Kalman Filter is used for the quantification of the internal process which is based on the observations and the model specification.

Examples are often simple linear models and furthermore do the deconvolved profiles often not satisfy normal physiological properties. For instance can measurement error easily result in deconvolved profiles with negative values. The problem originates from the modelling approach where the unobservable process is represented as a random walk or an integrated random walk.

Aim: Demonstrate non linear deconvolution on a simulated example further extended with a non negativity constraint on the unobservable process.

Methods: The glucose insulin oral minimal model [4] is used for simulation and deconvolution. The model is a non linear model that describes glucose dynamics as explained by insulin. The model simulation and deconvolution were performed using the R-package PSM [5]. The model was first used to simulate different sets of observations with different sampling schedules and different noise levels. The simulated observations were then used to demonstrate non linear deconvolution which enabled a quantification of the meal input. The model structure from the simulation model was reused and the non linear deconvolution covered only the estimation of the variance component for the random walk.

The non negativity constraint was implemented as the exponential to the random walk. This parameterization ensured positive contributions from the meal input throughout the profile. The drawback was a relatively numerically unstable model which was evident in the parameter estimation. The solution was a recursive parameter estimation where boundaries were updated in every iteration.

Results and Discussion: Non linear deconvolution was demonstrated on a non linear model where classical deconvolution only works on linear systems. The method was able to determine rate of appearance into a non linear model given the model and the observations. Furthermore, it was demonstrated how different samplings schedules and noise levels affect the deconvolution.

Publications:

1. “A matlab framework for estimation of NLME models using stochastic differential equations: applications for estimation of insulin secretion rates”, Mortensen et al, 2007, Journal of PK/PD
2. “A Deconvolution Method for Linear and Nonlinear Systems based on Stochastic Differential Equations”, Kristensen, 2004, Poster at PAGE in Uppsala
3. “Deconvolution of Insulin Secretion Rates”, Klim, 2008, Oral presentation at PAGE in Marseille
4. “Insulin sensitivity by oral glucose minimal models: validation against clamp”, Dalla Man et al, 2005, Am J Physiol Endocrinol Metab 289: E954–E959, 2005.
5. Population Stochastic Modelling: www.imm.dtu.dk/psm