Assessment of type II diabetes mellitus using irregularly sampled measurements with missing data

Abstract

Diabetes mellitus is one of the leading diseases in the developed world. In order to better regulate blood glucose in a diabetic patient, improved modelling of insulin-glucose dynamics is a key factor in the treatment of diabetes mellitus. In the current work, the insulin-glucose dynamics in type II diabetes mellitus can be modelled by using a stochastic nonlinear state-space model. Estimating the parameters of such a model is difficult as only a few blood glucose and insulin measurements per day are available in a non-clinical setting. Therefore, developing a predictive model of the blood glucose of a person with type II diabetes mellitus is important when the glucose and insulin concentrations are only available at irregular intervals. To overcome these difficulties, we resort to online sequential Monte Carlo (SMC) estimation of states and parameters of the state-space model for type II diabetic patients under various levels of randomly missing clinical data. Our results show that this method is efficient in monitoring and estimating the dynamics of the peripheral glucose, insulin and incretins concentration when 10, 25 and 50 % of the simulated clinical data were randomly removed.

Appendix

Glucose sub-model

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

$$\begin{aligned} V{^G_{\rm BC}} \frac{\mathrm{d}G_{\rm BC}}{\mathrm{d}t}= Q{^G_{B}}(G _{H}-G_{\rm BC})-\frac{V{^G_{\rm BF}}}{T{^G_{B}}}(G _{\rm BC}-G_{\rm BF}), \end{aligned}$$

(7.1)

$$\begin{aligned} V{^G_{\rm BF}} \frac{\mathrm{d}G_{\rm BF}}{\mathrm{d}t}= \frac{V{^G_{\rm BF}}}{T{^G_{B}}}(G _{\rm BC}-G_{\rm BF})-r_{\rm BGU}, \end{aligned}$$

(7.2)

$$\begin{aligned} V{^G_{H}} \frac{\mathrm{d}G_{H}}{\mathrm{d}t}= Q{^G_{B}}G_{\rm BC}+Q{^G_{L}}G_{L}+Q{^G_{K}}G_{K}+Q{^G_{P}}G_{\rm PC}+Q{^G_{H}}G_{H}-r_{\rm BCU}, \end{aligned}$$

(7.3)

$$\begin{aligned} V{^G_{G}} \frac{\mathrm{d}G_{G}}{\mathrm{d}t}= Q{^G_{G}}(G _{H}-G_{G})-r_{\rm GGU}+Ra, \end{aligned}$$

(7.4)

$$\begin{aligned} V{^G_{L}} \frac{\mathrm{d}G_{L}}{\mathrm{d}t}= Q{^G_{A}}G_{H}+Q{^G_{G}}G_{G}-Q{^G_{L}}G_{L}+r_{\rm HGP}-r_{\rm HGU}, \end{aligned}$$

(7.5)

$$\begin{aligned} V{^G_{K}} \frac{\mathrm{d}G_{K}}{\mathrm{d}t}= Q{^G_{K}}(G _{H}-G_{K})-r_{\rm KGE}, \end{aligned}$$

(7.6)

$$\begin{aligned} V{^G_{\rm PC}} \frac{\mathrm{d}G_{\rm PC}}{\mathrm{d}t}= Q{^G_{P}}(G _{H}-G_{\rm PC})-\frac{V{^G_{\rm PF}}}{T{^G_{P}}}(G _{\rm PC}-G_{\rm PF}), \end{aligned}$$

(7.7)

$$\begin{aligned} V{^G_{\rm PF}} \frac{\mathrm{d}G_{\rm PF}}{\mathrm{d}t}= \frac{V{^G_{\rm PF}}}{T{^G_{P}}}(G _{\rm PC}-G_{\rm PF})-r_{\rm PGU}, \end{aligned}$$

(7.8)

A detailed description about the metabolic rates is available in [15]. The general form of the metabolic production and consumption rates in each organ is as follows:

$$\begin{aligned} r= M^IM^GM^{\Gamma }M^B, \end{aligned}$$

(7.9)

and multipliers have the following general form:

$$\begin{aligned} M^C= a+b\tanh \left( \frac{C}{C^B}-d\right) , \end{aligned}$$

(7.10)

The glucose absorption model that calculates the glucose appearance rate into the blood stream following an oral glucose intake is considered in the gut compartment of the glucose sub-model as follows:

$$\begin{aligned} \frac{\mathrm{d}q_{\rm Ss}}{\mathrm{d}t}= -k_{12}q_{\rm Ss}+\delta (t), \end{aligned}$$

(7.11)

$$\begin{aligned} \frac{\mathrm{d}q_{\rm SI}}{\mathrm{d}t}= -k_{\rm empt}q_{\rm Ss}+k_{12}q_{\rm SI}, \end{aligned}$$

(7.12)

$$\begin{aligned} \frac{\mathrm{d}q_{\rm int}}{\mathrm{d}t}= -k_{\rm abs}q_{\rm int}+k_{\rm empt}q_{\rm SI}, \end{aligned}$$

(7.13)

$$\begin{aligned} k_{\rm empt}=k_{\rm min}+\frac{k_{\rm max}-k_{\rm min}}{2}\{\tanh [\varphi _1(q_{\rm Ss}+q_{\rm SI}-x_{1}D)]-\tanh [\varphi _2(q_{\rm Ss}+q_{\rm SI}-x_{2}D)]\}+2, \end{aligned}$$

(7.14)

$$\begin{aligned} \varphi _1= \frac{5}{2D(1-x_1)}, \end{aligned}$$

(7.15)

$$\begin{aligned} \varphi _2= \frac{5}{2Dx_2}, \end{aligned}$$

(7.16)

$$\begin{aligned} Ra= fk_{\rm abs}q_{\rm int}, \end{aligned}$$

(7.17)

Incretins sub-model

The incretins production is calculated from the following differential equation:

$$\begin{aligned} \frac{\mathrm{d}\psi }{\mathrm{d}t}= \varsigma k_{\rm empt}q_{S2}-r_{I\Psi P}, \end{aligned}$$

(7.18)

$$\begin{aligned} r_{I\Psi P}= f\frac{\psi }{\tau \psi }, \end{aligned}$$

(7.19)

The mass balance equation over the incretins compartment results in:

$$\begin{aligned} \frac{\mathrm{d}\psi }{\mathrm{d}t}= \varsigma k_{empt}q_{S2}-r_{I\Psi P}, \end{aligned}$$

(7.20)

$$\begin{aligned} r_{P\Psi C}= r_{M\Psi C}\psi , \end{aligned}$$

(7.21)

Insulin sub-model

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

$$\begin{aligned} V{^I_{B}} \frac{\mathrm{d}I_{B}}{\mathrm{d}t}= Q{^I_{B}}(I _{H}-I_{B}), \end{aligned}$$

(7.22)

$$\begin{aligned} V{^I_{H}} \frac{\mathrm{d}I_{H}}{\mathrm{d}t}= Q{^I_{B}}I_{B}+Q{^I_{L}}I_{L}+Q{^I_{K}}I_{K}+Q{^I_{P}}I_{PV}-Q{^I_{H}}I_{H}, \end{aligned}$$

(7.23)

$$\begin{aligned} V{^I_{G}} \frac{\mathrm{d}I_{G}}{\mathrm{d}t}= Q{^I_{G}}(I_{H}-I_{G}), \end{aligned}$$

(7.24)

$$\begin{aligned} V{^I_{L}} \frac{\mathrm{d}I_{L}}{\mathrm{d}t}= Q{^I_{A}}I_{H}+Q{^I_{G}}I_{G}-Q{^I_{L}}I_{L}+r_{\rm PIR}-r_{\rm LIC}, \end{aligned}$$

(7.25)

$$\begin{aligned} V{^I_{K}} \frac{\mathrm{d}I_{K}}{\mathrm{d}t}= Q{^I_{K}}(I _{H}-I_{K})-r_{\rm KIC}, \end{aligned}$$

(7.26)

$$\begin{aligned} V{^I_{\rm PC}} \frac{\mathrm{d}I_{\rm PC}}{\mathrm{d}t}= Q{^I_{P}}(I _{H}-I_{\rm PC})-\frac{V{^I_{\rm PF}}}{T{^I_{P}}}(I _{\rm PC}-I_{\rm PF}), \end{aligned}$$

(7.27)

$$\begin{aligned} V{^I_{\rm PF}} \frac{\mathrm{d}I_{\rm PF}}{\mathrm{d}t}= \frac{V{^I_{\rm PF}}}{T{^I_{P}}}(I _{\rm PC}-I_{\rm PF})-r_{\rm PIC}, \end{aligned}$$

(7.28)

The pancreas model contains two compartments. The pancreas model equations include mass balance equations over compartments and correlations between variables. The mass balance equation over each compartment results in:

$$\begin{aligned} \frac{\mathrm{d}m}{\mathrm{d}t}= K'm _S Km+\gamma P-S, \end{aligned}$$

(7.29)

$$\begin{aligned} \frac{\mathrm{d}m_S}{\mathrm{d}t}= Km-K'm _S-\gamma P, \end{aligned}$$

(7.30)

The steady state mass balance equation around the storage compartment is:

$$\begin{aligned} K'm _S= Km_0, \end{aligned}$$

(7.31)

where $m_0$ is the labile insulin quantity at a glucose concentration of zero. The rest of the equations for the pancreas model are:

$$\begin{aligned} \frac{\mathrm{d}P}{\mathrm{d}t}= \alpha (P_\infty -P), \end{aligned}$$

(7.32)

$$\begin{aligned} \frac{\mathrm{d}R}{\mathrm{d}t}= \beta (X-R), \end{aligned}$$

(7.33)

$$\begin{aligned} \begin{aligned} S= [N_1Y+N_2(X-R)+\xi _1\psi ]m\,\quad x>R,\\ S= [N_1Y+\xi _1\psi ]m\,\quad\,x\le R, \end{aligned} \end{aligned}$$

(7.34)

$$\begin{aligned} P_\infty =Y= X^{1.11}+\xi _2\psi , \end{aligned}$$

(7.35)

$$\begin{aligned} X=\frac{G{^{3.27}_{H}}}{{132}^{3.27}+5.93 G{^{3.02}_{H}}} \end{aligned}$$

(7.36)

Glucagon sub-model

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

$$\begin{aligned} V^{\Gamma } \frac{\mathrm{d}\Gamma }{\mathrm{d}t}= r_{P\Gamma R}-r_{P\Gamma C}, \end{aligned}$$

(7.37)