Detection of Abnormalities in Type II Diabetic Patients Using Particle Filters

Abstract

This study proposes a strategy for detecting possible dysfunction of organs such as the liver, pancreas, muscles and adipose tissues in a group of type II diabetic patients. Several in silico clinical trials are performed on a previously developed type II diabetes model. Since the pancreatic insulin secretion rate and glucose metabolic rates of different organs represent the functional behavior of the corresponding organs, calculated values of these rates are analyzed and compared with the corresponding rates calculated from a healthy subject’s model to detect possible abnormalities. These rates are calculated from estimated values of glucose and insulin concentrations inside the corresponding organs/tissues from the physiological model. Estimation of the concentrations in body organs/tissues is carried out using a sequential Monte Carlo filtering method called particle filters. The results show that the proposed strategy is capable of detecting deficiencies in hepatic and peripheral glucose disposal, hepatic glucose production and pancreatic insulin secretion. The information provided by this strategy can potentially be used to tailor patient dietary requirements and/or select appropriate medications for the patients.

Appendix

1.1 Appendix 1: Glucose Sub-model

The mass balance equation over each compartment in the glucose sub-model results in following equations:

$$V_{BC}^{G} \frac{{dG_{BC} }}{dt} = Q_{B}^{G} \left( {G_{H} - G_{BC} } \right) - \frac{{V_{BF}^{G} }}{{T_{B}^{G} }}\left( {G_{BC} - G_{BF} } \right)$$

(12)

$$V_{BF}^{G} \frac{{dG_{BF} }}{dt} = \frac{{V_{BF}^{G} }}{{T_{B}^{G} }}\left( {G_{BC} - G_{BF} } \right) - r_{BGU}$$

(13)

$$V_{H}^{G} \frac{{dG_{H} }}{dt} = Q_{B}^{G} G_{BC} + Q_{L}^{G} G_{L} + Q_{K}^{G} G_{K} + Q_{P}^{G} G_{PC} - Q_{H}^{G} G_{H} - r_{RBCU}$$

(14)

$$V_{G}^{G} \frac{{dG_{G} }}{dt} = Q_{G}^{G} \left( {G_{H} - G_{G} } \right) - r_{GGU}$$

(15)

$$V_{L}^{G} \frac{{dG_{L} }}{dt} = Q_{A}^{G} G_{H} + Q_{G}^{G} G_{G} - Q_{L}^{G} G_{L} + r_{HGP} - r_{HGU}$$

(16)

$$V_{K}^{G} \frac{{dG_{K} }}{dt} = Q_{K}^{G} \left( {G_{H} - G_{K} } \right) - r_{KGE}$$

(17)

$$V_{PC}^{G} \frac{{dG_{PC} }}{dt} = Q_{P}^{G} \left( {G_{H} - G_{PC} } \right) - \frac{{V_{PF}^{G} }}{{T_{P}^{G} }}\left( {G_{PC} - G_{PF} } \right)$$

(18)

$$V_{PF}^{G} \frac{{dG_{PF} }}{dt} = \frac{{V_{PF}^{G} }}{{T_{P}^{G} }}\left( {G_{PC} - G_{PF} } \right) - r_{PGU}$$

(19)

The metabolic rates for the glucose sub-model are summarized below:

$$r_{BGU} = 70$$

(20)

$$r_{RBCU} = 10$$

(21)

$$r_{GGU} = 20$$

(22)

$$r_{PGU} = M_{PGU}^{I} M_{PGU}^{G} r_{PGU}^{B}$$

(23)

$$r_{PGU}^{B} = 35$$

(24)

$$M_{PGU}^{I} = 2.552 + 1.66\tanh \left[ {0.69\left( {{\raise0.7ex\hbox{${I_{PF} }$} \!\mathord{\left/ {\vphantom {{I_{PF} } {I_{PF}^{B} }}}\right.\kern-0pt} \!\lower0.7ex\hbox{${I_{PF}^{B} }$}} - 3.454} \right)} \right]$$

(25)

$$M_{PGU}^{G} = {\raise0.7ex\hbox{${G_{PF} }$} \!\mathord{\left/ {\vphantom {{G_{PF} } {G_{PF}^{B} }}}\right.\kern-0pt} \!\lower0.7ex\hbox{${G_{PF}^{B} }$}}$$

(26)

$$r_{HGP} = M_{HGP}^{I} M_{HGP}^{G} M_{HGP}^{\varGamma } r_{HGP}^{B}$$

(27)

$$r_{HGP}^{B} = 35$$

(28)

$$\frac{d}{dt}M_{HGP}^{I} = 0.04\left( {M_{HGP}^{I\infty } - M_{HGP}^{I} } \right)$$

(29)

$$M_{HGP}^{I\infty } = 0.872 - 0.797\tanh \left[ {0.993\left( {{\raise0.7ex\hbox{${I_{L} }$} \!\mathord{\left/ {\vphantom {{I_{L} } {I_{L}^{B} }}}\right.\kern-0pt} \!\lower0.7ex\hbox{${I_{L}^{B} }$}} - 1.164} \right)} \right]$$

(30)

$$M_{HGP}^{G} = 1.42 - 1.41\tanh \left[ {0.62\left( {{\raise0.7ex\hbox{${G_{L} }$} \!\mathord{\left/ {\vphantom {{G_{L} } {G_{L}^{B} }}}\right.\kern-0pt} \!\lower0.7ex\hbox{${G_{L}^{B} }$}} - 0.497} \right)} \right]$$

(31)

$$M_{HGP}^{\varGamma } = 2.7\tanh \left[ {0.39{\raise0.7ex\hbox{$\varGamma $} \!\mathord{\left/ {\vphantom {\varGamma {\varGamma^{B} }}}\right.\kern-0pt} \!\lower0.7ex\hbox{${\varGamma^{B} }$}}} \right] - f$$

(32)

$$\frac{d}{dt}f = 0.0154\left[ {\left( {\frac{{2.7\tanh \left[ {0.39{\raise0.7ex\hbox{$\varGamma $} \!\mathord{\left/ {\vphantom {\varGamma {\varGamma^{B} }}}\right.\kern-0pt} \!\lower0.7ex\hbox{${\varGamma^{B} }$}}} \right] - 1}}{2}} \right) - f} \right]$$

(33)

$$r_{HGU} = M_{HGU}^{I} M_{HGU}^{G} r_{HGU}^{B}$$

(34)

$$r_{HGU}^{B} = 20$$

(35)

$$\frac{d}{dt}M_{HGU}^{I} = 0.04\left( {M_{HGU}^{I\infty } - M_{HGU}^{I} } \right)$$

(36)

$$M_{HGU}^{I\infty } = 0.6623 + 0.731\tanh \left[ {0.985\left( {{\raise0.7ex\hbox{${I_{L} }$} \!\mathord{\left/ {\vphantom {{I_{L} } {I_{L}^{B} }}}\right.\kern-0pt} \!\lower0.7ex\hbox{${I_{L}^{B} }$}} - 0.493} \right)} \right]$$

(37)

$$M_{HGU}^{G } = 1.855 + 1.855\tanh \left[ {2.047\left( {{\raise0.7ex\hbox{${G_{L} }$} \!\mathord{\left/ {\vphantom {{G_{L} } {IG_{L}^{B} }}}\right.\kern-0pt} \!\lower0.7ex\hbox{${G_{L}^{B} }$}} - 1.244} \right)} \right]$$

(38)

$$\begin{aligned} r_{KGE} &= 71 + 71\tanh \left[ {0.11\left( {G_{K} - 460} \right)} \right]0 \le G_{K} < 460 \hfill \\ r_{KGE} &=71 + 71\tanh \left[ {0.11\left( {G_{K} - 460} \right)} \right]G_{K} \ge 460 \hfill \\ \end{aligned}$$

(39)

1.2 Appendix 2: Insulin Sub-model

The mass balance equation over the compartments in the insulin sub-model results in following equations:

$$V_{B}^{I} \frac{{dI_{B} }}{dt} = Q_{B}^{I} \left( {I_{H} - I_{B} } \right)$$

(40)

$$V_{H}^{I} \frac{{dI_{H} }}{dt} = Q_{B}^{I} I_{B} + Q_{L}^{I} I_{L} + Q_{K}^{I} I_{K} + Q_{P}^{I} I_{PV} - Q_{H}^{I} I_{H}$$

(41)

$$V_{G}^{I} \frac{{dI_{G} }}{dt} = Q_{G}^{I} \left( {I_{H} - I_{G} } \right)$$

(42)

$$V_{L}^{I} \frac{{dI_{L} }}{dt} = Q_{A}^{I} I_{H} + Q_{G}^{I} I_{G} - Q_{L}^{I} I_{L} + r_{PIR} - r_{LIC}$$

(43)

$$V_{K}^{I} \frac{{dI_{K} }}{dt} = Q_{K}^{I} \left( {I_{H} - I_{K} } \right) - r_{KIC}$$

(44)

$$V_{PC}^{I} \frac{{dI_{PC} }}{dt} = Q_{P}^{I} \left( {I_{H} - I_{PC} } \right) - \frac{{V_{PF}^{I} }}{{T_{P}^{I} }}\left( {I_{PC} - I_{PF} } \right)$$

(45)

$$V_{PF}^{I} \frac{{dI_{PF} }}{dt} = \frac{{V_{PF}^{I} }}{{T_{P}^{I} }}\left( {I_{PC} - I_{PF} } \right) - r_{PIC}$$

(46)

The metabolic rates for the insulin sub-model are summarized below:

$$r_{LIC} = 0.4\left[ {Q_{A}^{I} I_{H} + Q_{G}^{I} I_{G} + r_{PIR} } \right]$$

(47)

$$r_{KIC} = 0.3Q_{K}^{I} I_{K}$$

(48)

$$r_{PIC} = \frac{{I_{PF} }}{{\left[ {\left( {\frac{1 - 0.15}{{0.15Q_{P}^{I} }}} \right) - \frac{20}{{V_{PF}^{I} }}} \right]}}$$

(49)

The pancreatic insulin release model used in the type II diabetes model has been proposed by Landahl and Grodsky [46]. The graphical representation of Landahl and Grodsky’s model is depicted in Fig. 6 in Appendix. The aim of Landahl and Grodsky’s model is to mimic the biphasic behavior of pancreatic insulin secretion in response to a glucose stimulus.

Fig. 6

Landahl and Grodsky’s model for pancreatic insulin secretion

In this model, a small labile insulin compartment is assumed to exchange insulin with a large storage compartment. The rate at which insulin flows into the labile compartment is regulated by a glucose-stimulated factor, P. The rate of insulin secretion, S, is dependent on the glucose concentration, the amount of labile insulin, m, and the difference between the instantaneous level of glucose-enhanced excitation factor, X, and its inhibitor, R. This functionality provides a mathematical description of the pancreas biphasic response to a glucose stimulus. The first phase insulin release is caused by an instantaneous increase in the glucose-enhanced excitation factor (X) followed by a rapid increase in its inhibitor (R). The second phase release results from the direct dependence of the insulin secretion rate (S) on the glucose stimulus and the gradual increase in the level of the labile compartment filling factor (P).

$$\frac{dm}{dt} = K^{\prime}m_{S} - Km + \gamma P - S$$

(50)

$$\frac{{dm_{S} }}{dt} = Km - K^{'} m_{S} - \gamma P$$

(51)

It is assumed that the capacity of the storage compartment is large enough and remains at steady state. For a glucose concentration of zero, P is set to zero. Therefore, the steady-state mass balance equation around the storage compartment is:

$$K^{\prime}m_{S} = Km_{0}$$

(52)

where $m_{0}$ is the labile insulin quantity at a glucose concentration of zero. The rest of the equations for the pancreas model are:

$$\frac{dP}{dt} = \alpha \left( {P_{\infty } - P} \right)$$

(53)

$$\frac{dR}{dt} = \beta \left( {X - R} \right)$$

(54)

$$\begin{aligned} S & = \left[ {N_{1} Y + N_{2} \left( {X - R} \right)} \right] \quad m \, X > R \hfill \\ S & = N_{1} Ym \quad X \le R \hfill \\ \end{aligned}$$

(55)

$$P_{\infty } = Y = X^{1.11}$$

(56)

$$X = \frac{{G_{H}^{3.27} }}{{132^{3.27} + 5.93G_{H}^{3.02} }}$$

(57)

$P_{\infty }$ and Y reflect the glucose-induced stimulation effects on the liable compartment filling factor and the insulin secretion rate, respectively.

1.3 Appendix 3: Glucagon Sub-model

The glucagon sub-model has one mass balance equation over the whole body as follows:

$$V^{\varGamma } \frac{d\varGamma }{dt} = r_{P\varGamma R} - r_{P\varGamma C}$$

(58)

The metabolic rates for the glucagon sub-model are summarized below:

$$r_{P\varGamma C} = 9.1\varGamma$$

(59)

$$r_{P\varGamma R} = M_{P\varGamma R}^{G} M_{P\varGamma R}^{I} r_{P\varGamma R}^{B}$$

(60)

$$M_{P\varGamma R}^{G} = 1.31 - 0.61\tanh \left[ {1.06\left( {{\raise0.7ex\hbox{${G_{H} }$} \!\mathord{\left/ {\vphantom {{G_{H} } {G_{H}^{B} }}}\right.\kern-0pt} \!\lower0.7ex\hbox{${G_{H}^{B} }$}} - 0.47} \right)} \right]$$

(61)

$$M_{P\varGamma R}^{I} = 2.93 - 2.09\tanh \left[ {4.18\left( {{\raise0.7ex\hbox{${I_{H} }$} \!\mathord{\left/ {\vphantom {{I_{H} } {I_{H}^{B} }}}\right.\kern-0pt} \!\lower0.7ex\hbox{${I_{H}^{B} }$}} - 0.62} \right)} \right]$$

(62)

$$r_{P\varGamma R}^{B} = 9.1$$

(63)

The model parameters are summarized in Table 3.

Table 3 Model parameters