A comprehensive compartmental model of blood glucose regulation for healthy and type 2 diabetic subjects

Abstract

We have expanded a former compartmental model of blood glucose regulation for healthy and type 2 diabetic subjects. The former model was a detailed physiological model which considered the interactions of three substances, glucose, insulin and glucagon on regulating the blood sugar. The main drawback of the former model was its restriction on the route of glucose entrance to the body which was limited to the intravenous glucose injection. To handle the oral glucose intake, we have added a model of glucose absorption in the gastrointestinal tract to the former model to address the resultant variations of blood glucose concentrations following an oral glucose intake. Another model representing the incretins production in the gastrointestinal tract along with their hormonal effects on boosting pancreatic insulin production is also added to the former model. We have used two sets of clinical data obtained during oral glucose tolerance test and isoglycemic intravenous glucose infusion test from both type 2 diabetic and healthy subjects to estimate the model parameters and to validate the model results. The estimation of model parameters is accomplished through solving a nonlinear optimization problem. The results show acceptable precision of the estimated model parameters and demonstrate the capability of the model in accurate prediction of the body response during the clinical studies.

Appendix

The following nomenclature is adopted throughout the Sorensen model description.

Model variables in the glucose sub-model

D :

Oral glucose amount (mg)

G :

Glucose concentration (mg/dl)

M :

Multiplier of metabolic rates (dimensionless)

q :

Glucose amount in GI tract (mg)

Q :

Vascular blood flow rate (dl/min)

r :

Metabolic production or consumption rate (mg/min)

Ra :

Rate of glucose appearance in the blood stream (mg/min)

T :

Transcapillary diffusion time constant (min)

t :

Time (min)

V :

Volume (dl)

Model variables in the insulin sub-model

I :

Insulin concentration (mU/l)

M :

Multiplier of metabolic rates (dimensionless)

m :

Labile insulin mass (U)

P :

Potentiator (dimensionless)

Q :

Vascular blood flow rate (l/min)

R :

Inhibitor (dimensionless)

r :

Metabolic production or consumption rate (mU/min)

S :

Insulin secretion rate (U/min)

T :

Transcapillary diffusion time constant (min)

t :

Time (min)

V :

Volume (l)

X :

Glucose-enhanced excitation factor (dimensionless)

Y :

Intermediate variable (dimensionless)

Model variables in the glucagon sub-model

$\varGamma$ :

Normalized glucagon concentration (dimensionless)

M :

Multiplier of metabolic rates (dimensionless)

r :

Metabolic production or consumption rate (dl/min)

V :

Volume (dl)

t :

Time (min)

Model variables in the incretin sub-model

$\varPsi$ :

Incretin concentration (pmol/l)

r :

Metabolic production or consumption rate (pmol/min)

V :

Volume (l)

t :

Time (min)

First superscript

Γ:

Glucagon

Ψ:

Incretins

B :

Basal condition

G :

Glucose

I :

Insulin

Second superscript

$\infty$ :

Final steady-state value

Metabolic rate subscripts

BGU:

Brain glucose uptake

GGU:

Gut glucose uptake

HGP:

Hepatic glucose production

HGU:

Hepatic glucose uptake

$I\varPsi R$ :

Intestinal incretins release

IVG:

Intravenous glucose infusion

IVI:

Intravenous insulin infusion

KGE:

Kidney glucose excretion

KIC:

Kidney insulin clearance

LIC:

Liver insulin clearance

$M\varGamma C$ :

Metabolic glucagon clearance

$P\varGamma C$ :

Plasma glucagon clearance

$P\varPsi C$ :

Plasma incretins clearance

$P\varGamma R$ :

Pancreatic glucagon release

PGU:

Peripheral glucose uptake

PIC:

Peripheral insulin clearance

PIR:

Pancreatic insulin release

RBCU:

Red blood cell glucose uptake

First subscripts

A :

Hepatic artery

B :

Brain

G :

Gut

H :

Heart and lungs

L :

Liver

P :

Periphery

S :

Stomach

$\infty$ :

Final steady-state value

Second subscripts (if required)

C :

Capillary space

F :

Interstitial fluid space

l :

Liquid

s :

Solid

1.1 Glucose sub-model

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

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

(5)

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

(6)

$$V_{H}^{G} \frac{{{\text{d}}G_{H} }}{{{\text{d}}t}} = Q_{B}^{G} G_{BC} + Q_{L}^{G} G_{L} + Q_{K}^{G} G_{K} + Q_{P}^{G} G_{PC} - Q_{H}^{G} G_{H} - r_{\text{RBCU}} + r_{\text{IVG}}$$

(7)

$$V_{G}^{G} \frac{{{\text{d}}G_{G} }}{{{\text{d}}t}} = Q_{G}^{G} \left( {G_{H} - G_{G} } \right) - r_{\text{GGU}} + Ra$$

(8)

$$V_{L}^{G} \frac{{{\text{d}}G_{L} }}{{{\text{d}}t}} = Q_{A}^{G} G_{H} + Q_{G}^{G} G_{G} - Q_{L}^{G} G_{L} + r_{\text{HGP}} - r_{\text{HGU}}$$

(9)

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

(10)

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

(11)

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

(12)

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

$$r_{\text{BGU}} = 70$$

(13)

$$r_{\text{RBCU}} = 10$$

(14)

$$r_{\text{GGU}} = 20$$

(15)

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

(16)

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

(17)

$$M_{\text{PGU}}^{I} = \frac{{7.03 + 6.52\tanh \left[ {c\left( {I_{\text{PF}} /I_{\text{PF}}^{B} - d} \right)} \right]}}{{7.03 + 6.52\tanh \left[ {c\left( {1 - d} \right)} \right]}}$$

(18)

$$M_{\text{PGU}}^{G} = G_{\text{PF}} /G_{\text{PF}}^{B}$$

(19)

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

(20)

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

(21)

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

(22)

$$M_{\text{HGP}}^{I\infty } = \frac{{1.21 - 1.14\tanh \left[ {c\left( {I_{L} /I_{L}^{B} - d} \right)} \right]}}{{1.21 - 1.14\tanh \left[ {c\left( {1 - d} \right)} \right]}}$$

(23)

$$M_{\text{HGP}}^{G} = \frac{{1.42 - 1.41\tanh \left[ {c\left( {G_{L} /G_{L}^{B} - d} \right)} \right]}}{{1.42 - 1.41\tanh \left[ {c\left( {1 - d} \right)} \right]}}$$

(24)

$$M_{\text{HGP}}^{\varGamma } = 2.7\tanh \left[ {0.39\varGamma /\varGamma^{B} } \right] - f$$

(25)

$$\frac{d}{{{\text{d}}t}}f = 0.0154\left[ {\left( {\frac{{2.7\tanh \left[ {0.39\varGamma /\varGamma^{B} } \right] - 1}}{2}} \right) - f} \right]$$

(26)

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

(27)

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

(28)

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

(29)

$$M_{\text{HGU}}^{I\infty } = \frac{{2.0\tanh \left[ {c\left( {I_{L} /I_{L}^{B} - d} \right)} \right]}}{{2.0\tanh \left[ {c\left( {1 - d} \right)} \right]}}$$

(30)

$$M_{\text{HGU}}^{G} = \frac{{5.66 + 5.66\tanh \left[ {c\left( {G_{L} /G_{L}^{B} - d} \right)} \right]}}{{5.66 + 5.66\tanh \left[ {c\left( {1 - d} \right)} \right]}}$$

(31)

$$r_{\text{KGE}} = 71 + 71\tanh \left[ {0.11\left( {G_{K} - 460} \right)} \right]\quad 0 \le G_{K} < 460$$

(32)

$$r_{\text{KGE}} = - 330 + 0.872G_{K} \quad G_{K} \ge 460$$

The model of glucose absorption in the GI tract proposed by Dalla Man et al. [16] is added to the glucose sub-model. The model equations are:

$$\frac{{{\text{d}}q_{Ss} }}{{{\text{d}}t}} = - k_{12} q_{Ss} + {\text{D}}\delta \left( t \right)$$

(33)

$$\frac{{{\text{d}}q_{Sl} }}{{{\text{d}}t}} = - k_{\text{empt}} q_{Sl} + k_{12} q_{Ss}$$

(34)

$$\frac{{{\text{d}}q_{\text{int}} }}{{{\text{d}}t}} = - k_{\text{abs}} q_{\text{int}} + k_{\text{empt}} q_{Sl}$$

(35)

$$k_{\text{empt}} = k_{\hbox{min} } + \frac{{k_{\hbox{max} } - k_{\hbox{min} } }}{2}\left\{ {\tanh \left[ {\varphi_{1} \left( {q_{Ss} + q_{Sl} - x_{1} D} \right)} \right] - \tanh \left[ {\varphi_{2} \left( {q_{Ss} + q_{Sl} - x_{2} D} \right)} \right] + 2} \right\}$$

(36)

$$\varphi_{1} = \frac{5}{{2D\left( {1 - x_{1} } \right)}}$$

(37)

$$\varphi_{2} = \frac{5}{{2{\text{D}}x_{2} }}$$

(38)

$$Ra = fk_{\text{abs}} q_{\text{int}}$$

(39)

where $\delta \left( t \right)$ is the impulse function.

1.2 Insulin sub-model

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

$$V_{B}^{I} \frac{{{\text{d}}I_{B} }}{{{\text{d}}t}} = Q_{B}^{I} \left( {I_{H} - I_{B} } \right)$$

(40)

$$V_{H}^{I} \frac{{{\text{d}}I_{H} }}{{{\text{d}}t}} = Q_{B}^{I} I_{B} + Q_{L}^{I} I_{L} + Q_{K}^{I} I_{K} + Q_{P}^{I} I_{PV} - Q_{H}^{I} I_{H}$$

(41)

$$V_{G}^{I} \frac{{{\text{d}}I_{G} }}{{{\text{d}}t}} = Q_{G}^{I} \left( {I_{H} - I_{G} } \right)$$

(42)

$$V_{L}^{I} \frac{{{\text{d}}I_{L} }}{{{\text{d}}t}} = Q_{A}^{I} I_{H} + Q_{G}^{I} I_{G} - Q_{L}^{I} I_{L} + r_{\text{PIR}} - r_{\text{LIC}} + r_{\text{IVI}}$$

(43)

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

(44)

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

(45)

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

(46)

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

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

(47)

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

(48)

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

(49)

As mentioned, the pancreatic insulin release model used in the Sorensen model has been proposed by Landahl and Grodsky [35]. The aim of Landahl and Grodsky’s model is to mimic the biphasic behavior of pancreatic insulin secretion in response to a glucose stimulus. 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 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).

The mass balance equation over each compartment results in:

$$\frac{{{\text{d}}m}}{{{\text{d}}t}} = K^{\prime}m^{\prime} - Km + \gamma P - S$$

(50)

$$\frac{{{\text{d}}m^{\prime}}}{{{\text{d}}t}} = Km - K^{'} m^{\prime} - \gamma P$$

(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:

$$K^{\prime}m^{\prime} = Km_{0}$$

(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:

$$\frac{{{\text{d}}P}}{{{\text{d}}t}} = \alpha \left( {P_{\infty } - P} \right)$$

(53)

$$\frac{{{\text{d}}R}}{{{\text{d}}t}} = \beta \left( {X - R} \right)$$

(54)

$$S = \left[ {N_{1} Y + N_{2} \left( {X - R} \right) + \xi_{2} \varPsi } \right]m X > R$$

(55)

$$S = \left( {N_{1} Y + \xi_{2} \varPsi } \right)m \quad X \le R$$

$$P_{\infty } = Y = X^{1.11} + \xi_{1} \varPsi$$

(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 Glucagon sub-model

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

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

(58)

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

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

(59)

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

(60)

$$M_{{{\text{P}}\varGamma {\text{R}}}}^{G} = 1.31 - 0.61\tanh \left[ {1.06\left( {G_{H} /G_{H}^{B} - 0.47} \right)} \right]$$

(61)

$$M_{{{\text{P}}\varGamma {\text{R}}}}^{I} = 2.93 - 2.09\tanh \left[ {4.18\left( {I_{H} /I_{H}^{B} - 0.62} \right)} \right]$$

(62)

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

(63)

1.4 Incretin sub-model

Similar to the glucagon sub-model, the incretin sub-model has one compartment. The incretin model equations comprise two ordinary differential equations, one represents the production of the incretins following the presence of the glucose in the small intestine and the other one represents the mass balance over the compartment. The incretin production is calculated from the following differential equation:

$$\frac{{{\text{d}}\psi }}{{{\text{d}}t}} = \varsigma k_{\text{empt}} q_{Sl} - r_{I\varPsi P}$$

(64)

where $\psi$ is the amount of produced incretins, $k_{\text{empt}} q_{Sl}$ is the rate of glucose entrance to the small intestine, $r_{I\varPsi P}$ is the rate of incretin absorption into the blood stream, and $\varsigma$ is a constant.

$r_{I\varPsi P}$ is calculated from the following equation:

$$r_{I\varPsi P} = \frac{\psi }{{\tau_{\varPsi } }}$$

(65)

where $\tau_{\varPsi }$ is the time constant of the incretin absorption process into the blood stream. The mass balance equation over the incretin compartment results in:

$$V^{\varPsi } \frac{{{\text{d}}\varPsi }}{{{\text{d}}t}} = r_{I\varPsi P} - r_{P\varPsi C}$$

(66)

where $V^{\varPsi }$ is the incretin distribution volume, $\varPsi$ is the blood incretin concentration, and $r_{P\varPsi C}$ is the rate of plasma incretin clearance which depends on the incretin concentration. The clearance rate is calculated from the following equation:

$$r_{P\varPsi C} = r_{M\varPsi C} \varPsi$$

(67)

where $r_{M\varPsi C}$ is the mean incretin clearance rate and is a constant.

The model constant parameters are available in [49].