Dimensional analysis of MINMOD leads to definition of the disposition index of glucose regulation and improved simulation algorithm

Abstract

Background: Frequently Sampled Intravenous Glucose Tolerance Test (FSIVGTT) together with its mathematical model, the minimal model (MINMOD), have become important clinical tools to evaluate the metabolic control of glucose in humans. Dimensional analysis of the model is up to now not available.

Methods: A formal dimensional analysis of MINMOD was carried out and the degree of freedom of MINMOD was examined. Through re-expressing all state variable and parameters in terms of their reference scales, MINMOD was transformed into a dimensionless format. Previously defined physiological indices including insulin sensitivity, glucose effectiveness, and first and second phase insulin responses were re-examined in this new formulation. Further, the parameter estimation from FSIVGTT was implemented using both the dimensional and the dimensionless formulations of MINMOD, and the performances were compared utilizing Monte Carlo simulation as well as real human FSIVGTT data.

Results: The degree of freedom (DOF) of MINMOD was found to be 7. The model was maximally simplified in the dimensionless formulation that normalizes the variation in glucose and insulin during FSIVGTT. In the new formulation, the disposition index (Dl), a composite parameter known to be important in diabetes pathology, was naturally defined as one of the dimensionless parameters in the system. The numerical simulation using the dimensionless formulation led to a 1.5-5 fold gain in speed, and significantly improved accuracy and robustness in parameter estimation compared to the dimensional implementation.

Conclusion: Dimensional analysis of MINMOD led to simplification of the model, direct identification of the important composite factors in the dynamics of glucose metabolic control, and better simulations algorithms.

Figure 1

Flowchart of the Monte Carlo simulation.

Figure 2

The comparison of parameter estimations for S_G (A) and S_I (B) between the simulations using the dimensional (dim'nl) and the dimensionless (dim'less) formulation of MINMOD. The results are from the 100 simulated data sets. Good agreements are observed in both cases.

Figure 3

An example of model fitting to the Monte Carlo simulation data. The fitting algorithm using the dimensionless formulation of MINMOD is able to achieve better fit to data, and is less sensitive to noise.

Figure 4

Comparison of the robustness against noise. Data shown are the results from 100 Monte Carlo simulations, with 0–10% noise added. (A) The average goodness-of-fit R² is plotted against the noise level. Overall the dimensionless implementation is able to achieve better model fitting at all noise levels. (B) The number of failed fitting (defined to be those with R² < 0) is more in the dimensional implementation than the dimensionless counterpart at all noise levels.

Figure 5

The ratio between estimated and true values of S_G (top panels) and S_I (bottom panels) is compared between the dimensional (left panels) and dimensionless implementation (right panels), for the 100 data sets with 1% noise.

Figure 6

Comparison of the robustness in parameter estimation. The right panels show the number of failed estimations (defined by > 5 fold difference with the true values) for S_G and S_I as functions of noise level. The dimensionless implementation is better at recovering the true values of the parameters. The right panels plot the ratio of the estimated to true parameter values for those after removing failed fittings. The data points at 0% noise was normalized to one.

Figure 7

Contour plot of convergence in error minimization versus the number of iterations, for subject 9 in the FUSION data set. The levels of the cost-function in the contour plot are represented through their color, and is labeled on each contour. Top: Simulation using the dimensional formulation. Bottom: Simulation using the dimensionless formulation.

Figure 8

An example (subject 2) of the glucose measurements of the FUSION FSIVGTT data, and the model fittings using both dimensional and dimensionless formulation of MINMOD.

Figure 9

Comparison of the model fitting between the two methods for the FUSION data sets. (A) R². (B) Number of iterations it takes to reach a fit. (C) Estimated S_G values. (D) Estimated S_I values. (D-E) FSD in parameter estimation of S_G and S_I.