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Abstract

In this research two kinds of stochastic model were studied to describe the decay of a glucose tracer in the blood plasma of a group of female subjects. The group was divided into two subgroups, normal (non-obese) and obese subjects, to study the effect of obesity on the stochastic models. The data were acquired as part of a study at St. Thomas' Hospital, London (Bowes et al., 1996). A known mass of glucose and glucose tracer was injected intravenously in each subject and blood samples were taken at various times. To build a stochastic model of the decay of plasma glucose tracer concentration, a two-compartmental system was considered. Two different stochastic models were discussed.

In model A Soong's approach (Soong, 1971) was applied. The model consists of two differential equations for the concentration of tracer in each compartment. The four parameters of the model were estimated using blood sample data for each subject. The variability in the set of glucose tracer curves was assumed to be due to variability of the parameters and each parameter was assumed to have a quadrivariate lognormal frequency distribution. The mean plasma glucose tracer concentration was calculated using a four dimensional integration at times between 0 and 180 minutes.

A comparison was made between the stochastic and deterministic models for the mean concentration of glucose tracer in total group of subjects and in both subgroups. A small difference was found between the two models in the total group of subjects and the two subgroups. Also, the differences between deterministic and stochastic curves in the non- obese group was larger than in the total group of subjects and the obese subgroup. In all groups choosing the deterministic values as the mean concentration of glucose tracer yields a small overestimate error in the mean concentration.

The S. E. of the stochastic and deterministic models for the concentration of the glucose tracer was also calculated together with the S. E. for the original data and for all groups.

There were some large differences between the deterministic and stochastic values particularly for times between 20 minutes to 100 minutes. The differences are quite large in the total group of subjects but small in the non-obese and obese subgroups. The S. E. for the original data are larger than the stochastic models in total and obese subjects and also at large times in non-obese subject. Therefore, using the simple S. E. of the original data at each time point overestimates of the S. E. as predicted by the stochastic model.

For the stochastic model B Limic's approach (Limic,1989) was used. In this approach the uncertainty in the parameters of the model is incorporated into a compartmental matrix where all elements of the matrix fluctuate randomly under a normal distribution. The mathematical calculation for the mean concentration of glucose tracer is complex. To simplify the model, it was assumed that the fluctuations are the same (ie. non- independent) for all of the elements of the compartmental matrix.

A comparison was made between the deterministic and stochastic models for the mean concentration. It was found that the differences between the deterministic and the stochastic curve for the first compartment were small as in the case of model A. This similarity may be due to the small sample size and/or the dependent random processes for every element of the matrix. Therefore, model B is not recommended since all the elements of the compartmental matrix fluctuate together and this simplification does not represent actual physiological processes which are very likely to have independent fluctuations.

In conclusion, although both methods have a similar mean for the concentration of glucose tracer, the large difference between the S. E. of stochastic and deterministic models is probably due to the small number of measurement or the small sample size. The error should decrease with a greater number of measurements and increasing the sample size. In building the stochastic models this problem does not arise since the data are considered as a distribution, but taking a larger sample should produce a more accurate result.

The above results mean that in building a stochastic model in future studies, we need to consider a larger sample size, a larger number of measurements and restrict the number of the compartments to less than three (model A) to overcome the complexity of the calculation. The compartmental matrix should have elements which fluctuate independently.

Publication Type:	Thesis (Doctoral)
Subjects:	R Medicine > RZ Other systems of medicine
Departments:	School of Health & Psychological Sciences School of Health & Psychological Sciences > School of Health & Psychological Sciences Doctoral Theses Doctoral Theses

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