Link between the data-driven model and physiological models.

As introduced in Section3.1, physiologically based models are widely accepted in glucose

dynamics modelling. Among them, the compartmental maximal model (M M) suggested

by Dalla Man et al. [135] was designed to simulate the postprandial transient responses to food intake for subjects with and without DM. Twelve differential equations and thirty five parameters are included in the modelM M. Different time series representing different subjects can be simulated by changing the values of the 35 model parameters. Three postprandial glucose responses characterising three non-DM subjects were simu- lated using the Simulating the Glucose-Insulin Response script provided in SimBiology

toolbox [155]. The standard parameter values suggested [135] were used: (i) without any signs of DM, (ii) with low insulin sensitivity, and (iii) with impairedβ cell function. Cases (ii) and (iii) describe potential T2D patients with partially impaired pancreatic function, and are considered as pre-DM cases. The impaired function is compensated by either secreting more insulin or increasing tissue sensitivity, and thus keeping the blood glucose levels still within the healthy range. Note thatMLandM2 belong to a different

class of model compared to M M. As a physiologically based model, the glucose varia- tion is only one of the variables that is generated by theM M using a set of population based parameter values that describe a typical pre-diabetes subject.

Treating these three simulated peaks the same way as the measurement time series

in our cohort, the VB method was used to select between ML and M2 and infer the

corresponding parameters. The purpose of inferring simulated time series is threefold. 1) If eitherMLorM2 can be fitted well to the simulated time series, theMLandM2 are

proved to be consistent with the well-recognised compartmentalM M. 2) The differences in the inferred parameter values between three simulated subjects can reveal how the parameter values change when a healthy person starts to develop T2D in theory. 3) A simulated time series is free from the limitations brought about by using the CGM devices, including a 3–12 minute delay from measurements and actual blood glucose level, measurement noise, and a fixed sampling frequency of one data point every five minutes.

Figure 3.13: The dotted line is the simulated time series from the maximal model for

a non-DM case without any signs of DM and the solid line is the deterministic solution using the parameter values inferred from modelML.

All three simulated time series can be fitted by the linear model ML, and an example

result of fitting for case (i) is shown in Fig. 3.13. Values of the inferred parameter θM M_k are as follows respectively: (i) 0.027, (ii) 0.023, (iii) 0.021 [min−1], and the values of θ₁M M are (i) 0.4, (ii) 0.25, (iii) 0.16 [min−2]. They have been compared with θk

and θ1 obtained for all measured peaks fitted by ML for our cohort of subjects. The

values of θk for all the peaks in the control group that modelled by ML are presented

as the left box in the boxplot shown in Fig. 3.14 (a), and the values of θk for all the

peaks in the T2D group modelled by ML is presented on the right box in Fig. 3.14

(b). The value of θ_kM M for the simulated non-DM subject locates in the interquartile range of the box for the control group and in the top quartile of the box for the T2D group. The values of θ_kM M for both pre-DM cases locate in the lower part of the box for the control group, but within the top part of the interquartile of the box for T2D. A qualitatively similar result is observed for parameterθM M₁ (Fig. 3.14(b)): the simulated value for case (i) is within the interquartile of the control group distribution and beyond the interquartile of T2D. Similarly, θM M₁ for both pre-DM simulated subject locate within the bottom quartile of the control group and within the interquartile of the T2D range. This clearly shows that these two simulated pre-DM cases fall into the area between the non-DM and T2D distributions. It is worth noting that this observation is based on three simulated glucose variation time series, and it is too early to draw any significant conclusions without further validation from glucose dynamic data of pre- diabetes patients. This was an outcome that arose though the modelling performed. However, a close observation of the trend of the parameters θk and θ1 might provide

crucial information for early diagnosis of DM, particularly if such trends identify early abnormalities in glucose dynamics before the rise in the blood glucose concentration is considered significant.

This result provides same evidence of the robustness of the model (3.1a–3.1b) and its validation by comparison with an established phenomenological model [135]. The deter- ministic solution of our model (3.1a) in Fig.3.13indicated by the solid line contains only two parameters and matches the dynamics of the time series simulated using the M M [135] with thirty-five parameters indicated by the dotted line in Fig. 3.13. Being data- driven, our model takes full advantage of the CGM data, and, at the same time, reflects the intrinsic characteristics of the glucose-insulin system without detailed knowledge of the underlying physiological mechanisms.

Figure 3.14: Boxplots for (a)θk0 and (b)θ1 for all measured peaks fitted by ML in our cohort of participants. Horizontal lines mark θM M

k0 and θ

M M

1 for no signs of DM (upper dashed green line), low insulin sensitivity (middle solid line) and impairedβ-cell

function (lower dashed pink line) cases.