Abstract
The main aim of this research was to test if fractional-order differential equation models could give better fits than integer-order models to continuous glucose monitoring (CGM) data from subjects with type 1 diabetes. In this research, real continuous glucose monitoring (CGM) data was analyzed by three mathematical models, namely, a deterministic first-order differential equation model, a stochastic first-order differential equation model with Brownian motion, and a deterministic fractional-order model. CGM data was analyzed to find optimal values of parameters by using ordinary least squares fitting or maximum likelihood estimation using a kernel-density approximation. Matlab and R programs have been developed for each model to find optimal values of the parameters to fit observed data and to test the usefulness of each model. The fractional-order model giving the best fit has been estimated for each subject. Although our results show that fractional-order models can give better fits to the data than integer-order models in some cases, it is clear that the models need further improvement before they can give satisfactory fits.
Highlights
1 Introduction Insulin and glucagon are hormones that are produced in the pancreas and which control the level of glucose in the blood
If blood glucose is high, the pancreas secretes insulin into the bloodstream to decrease glucose level
3 Fractional differential equation models Because the first-order models do not fit the data, we look at higher-order models
Summary
Insulin and glucagon are hormones that are produced in the pancreas and which control the level of glucose in the blood (see, e.g., [ – ]). Type is sometimes called non-insulin-dependent or adult-onset diabetes In this type, the pancreas either produces insufficient insulin with respect to the heightened demands of relatively insulin-resistant peripheral tissues or the cells of the body do not react to Sakulrang et al Advances in Difference Equations (2017) 2017:150. . The fit using ordinary least-squares gives a constant value for G(t) and an estimate for the ratio of parameter values kGX kXG and not separate values for kGX and kXG, i.e., it gives the steady-state solution of equation ( ). Numerical solution using the Euler-Maruyama method requires a step size that is much smaller than the time ( minutes) between measurements in the CGM data. The first-order models do not give a good fit to the CGM data They have been useful for developing R-programs and testing some of the algorithms to be used in the stochastic fractional differential equation models. For < α ≤ , the initial value of the second derivative G( )( ) is required to uniquely specify the solution
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