Dynamics of Glucose-Insulin Regulation: Insulin Injection
Regime for Patients with Diabetes Type 1'
Canan Herdem and Hakan Yasarcan
Industrial Engineering Department
Bogazici University
Bebek — Istanbul 34342 — Turkey
canan.herdem@boun.edu.tr; hakan.yasarcan@boun.edu.tr
Abstract
A healthy human body regulates blood glucose concentration via regulating the insulin
concentration. Diabetes type 1 patients’ bodies cannot produce insulin. Therefore, blood
glucose needs to be regulated by insulin injections. This is not an easy task because there
are dynamic complexities such as accumulation processes, delays, nonlinearities, and
feedback loops in the system. Moreover, the task is a critical one because both low and
high levels of glucose are harmful for the body. In this work, we first developed and
calibrated a system dynamics model for “the two time delay model” as described by Li et
al., 2006. Later, we introduced a penalty formulation to be able to evaluate different
cases. We also deleted the insulin production flow and added insulin injections to the
base model in order to obtain the model for a diabetes type 1 patient. According to the
initial results of the study, the suggested decision making heuristic would yield
satisfactory results. However, further tests under different glucose infusion rate patterns
and improvement to the heuristic are necessary.
Keywords: glucose-insulin regulation; control problem; diabetes type 1; insulin
injections; medical modeling; decision making heuristic.
Introduction
Diabetes Mellitus is a disease associated with body insulin deficiency or inefficient
use of it. A patient with diabetes either cannot produce insulin to absorb glucose and turn
it into energy (diabetes type 1) or cannot properly respond to insulin (diabetes type 2)
(Alberti and Zimmet, 1998). Regulation of glucose is very critical because both
hyperglycemia (high blood glucose) and hypoglycemia (low blood glucose) are harmful
for the organs (Cryer, 2001; Ruderman et al., 1992, Sanlioglu et al., 2008). In a healthy
human body, glucose is regulated via regulating the blood insulin concentration.
However, diabetes type 1 patients’ bodies cannot produce insulin, which leaves glucose
' Supported by Bogazici University Research Fund; grant no: 5025
unregulated. Insulin injection is the main treatment for such patients (Sanlioglu et al.,
2008). However, one should be very careful with insulin injections because if more
insulin than necessary is injected, this may lead to hypoglycemia and, if less insulin than
necessary is injected, this may lead to hyperglycemia (Cryer, 2001; Sanlioglu et al.,
2008). Regulating the blood glucose and insulin concentrations by insulin injections is
not a simple task because of the existing dynamic complexities in the regulatory system.
There are many studies and different models on the glucose-insulin regulatory
system (Makroglou et al., 2006). According to experiments, insulin secretion rate has
three different oscillatory patterns superimposed on each other. The first oscillatory
pattern is the fastest and it has a period of tens of seconds; The second oscillatory pattern
is called rapid oscillation and has a period of 5-15 minutes; The third oscillatory pattern
is called ultradian oscillations and has a period of 50-120 minutes (Makroglou et al.,
2006). Sturis et al. (1991) developed a six dimensional differential equation model to
analyze the ultradian oscillations. The model introduced by Sturis et al. (1991) separates
insulin stock (compartment) into two distinct stocks and contains a third order delay of
insulin effectiveness and a glucose stock. Tolic et al. (2000) improved the model
developed by Sturis et al. (1991) by simplifying it?. Li et al. (2006) proposed a
delay-differential-equation model that uses the same functions and parameter values as
in the models of Sturis et al. (1991) and Tolic et al. (2000). This model is named as the
two time delay model and includes two distinct time delays for both insulin effectiveness
and glucose effectiveness (Li et al., 2006).
In this work, we first constructed a system dynamics model of the two time delay
model introduced by Li et al. (2006). We run our model for the different cases discussed
in the Li et al. (2006) paper and confirmed that our model produces the same dynamics
as the Li et al. (2006) model in all cases. In order to save some space, we did not provide
those runs in the paper. After the calibration of the system dynamics model, we set the
delay parameters equal to the values used in section 3 of the Li et al. (2006) paper and
obtained base run dynamics for a healthy person as a benchmark. We introduced a
penalty formulation to be able to evaluate different cases. Later, we adapted the base
model for diabetes type 1 patients. In order to obtain the model for a diabetes type 1
patient, we deleted the insulin production flow from and added insulin injections to the
base model. We also suggested a decision heuristic for insulin injections.
* If correctly applied model simplification increases the usefulness of models (Saysel and Barlas, 2006;
Yasarcan, 2010).
Base Model for a Healthy Person
We constructed a system dynamics model of the two time delay model introduced by
Li et al. (2006). The stock-flow diagram of this model is given in Figure 1. Note that the
equations (1-37) of this model are all taken from Li et al. (2006) and we also tried to
provide the related information presented in the Li et al. (2006) paper by adding
footnotes to the equations. Note that this model represents the glucose-insulin regulatory
system for a healthy person.
ja Glucose Concentration
Glucose
oxe——
Glucose Infusion Rate
&
pe
ae “6: es 69
Hepatic GlucoseProduction Insulin Independent/Utilization
|
| |
ur b. b_.%
t \ @{ |) Delayed Value of/Glucose
8
4 \
{6
f li
Insulin Concentration 65
¥
|
| | ‘ ~
by L
Insulin
Xd \ p
Ex a rege 3 £yRm
_ ei MG
Insulin Degradation Insulin Production Stimulated
and Clearance by Glucose Concentration
Figure 1. Stock-flow diagram of the two time delay model
e
Initial values and approximate integral equations for the stock variables:
Glucose, = 10,500 [milligram ] qd)
Glucose Infusion Rate
+ Hepatic Glucose Production
pr = Glucose, + . ae *DT [milligram] (2)
— Insulin Dependent Utiliation
Glucose,
— Insulin Independent Utilization
Insulin, = 90 [milliunit | (3)
Insulin Production Stimulated by
Insulin, p7 = Insulin, + Glucose Concentration *DT [milliunit] (4)
— Insulin Degradation and Clearance
Flow variables:
Glucose Infusion Rate = 108 [milligram [minute] (5)
. . Rg milligram
Hepatic Glucose Production = [+ @ Hi Delaved Value of lnsulin]¥p-C5) [ miniie | (6)
Insulin Dependent Utilization =
Glucose «| U0+ (Um - U0) milligram (7)
C3 «Vg in liters) en ee minute
Insulin Independent Utilization = Ub* f - el e 5
minute
“= eal ®)
(Ci-Delayed Value of Glucose [Vg in liters)
by Glucose Concentration
Insulin Production Stimulated \ _ Rm [ae] ()
1 minute
+e al
(10)
Insulin Degrdation and Clearance = Insulin + di [ee
minute
e Other variables:
Delayed Value of Glucose = DELAY (Glucose, Taul, Glucose) [milligram ] ais
Delayed Value of Insulin = DELAY (Unsulin, Tau2, Insulin ) [milliunit | (12)
Glucose Concentration = Glucose/Vg in deciliters [milligram [deciliter | (13)
Insulin Concentration = Insulin]Vp |milliunit [liter | (14)
e Parameters:
al = 300 [milligram [liter | (15)
Alfa = 300 [milligram /liter | (16)*
Beta =1.77 [dimensionless] (7p
C1 = 2000 [milligram /liter] (18)
C2 = 144 [milligram [liter | (19)
C3 = 1000 [milligram /liter | (20)
C4 =80 [milliunit /liter | (21)
C5 = 26 [milliunit [liter ] (22)
C5 = 26 [milliunit /liter] (23)
di = 0.06 [1/minute] (24)°
E = 0.2 [liter /minute] (25)’
* DELAY(input, delay time, initial value) is a function that creates a delayed version of the input as its
output such that if Y= DELAY(X, tl, Yo), this means that Y, +.) =X,
* The software that we used to develop the system dynamics model does not allow symbols. In the Li et al.
(2006) paper A/a is represented with the symbol a.
* In the Liet al. (2006) paper Bera is represented with the symbol B.
° diis the clearance fraction.
7 Bis the diffusion transfer rate.
Rg =180 [milligram /minute| (26)
Rm = 210 [milliunit [minute] (27)
Taul =7 [minute] (28)°
Tau2 =12 [minute] 29)
ti =100 [minute] (30)"°
U0 = 40 [milligram /minute] G1)
Ub =72 [milligram [minute] (32)
Um = 940 [milligram [minute] (33)
Vg in deciliters = 100 [deciliter ] (34)""
Vg in liters = 10 [liter] (35)
Vi =11 [liter] (36)
Vp =3 [liter] 37)?
We simulated the model for 1440 minutes (1 day) in order to obtain the benchmark
dynamics. The glucose and insulin concentration dynamics for this run is given in Figure
2. Glucose concentration level varies approximately between 83 and 106. Insulin
concentration level varies approximately between 25-43. According to Li et al. (2006),
the oscillatory behavior observed in Figure 2 is in agreement with physiological data.
* In the Li et al. (2006) paper Zaw/ is represented with the symbol 1; and it is the insulin transportation
delay time.
° In the Li et al. (2006) paper Tau2 is represented with the symbol 1 and it is the time lag for insulin effect
on liver.
'° tis the insulin degradation time constant.
u Vg in deciliters and Vg in liters are actually the same parameter. We separated Vg into two parameters
because the software that we used cannot handle unit transformation.
'° Wiis the effective volume of the intercellular space.
8 Vp is the plasma insulin distribution volume.
® +: ctucose Concentration 2: Insulin Concentration
1 1064
a
|
1 60
2 25.0:
5
Be
o8
ny
0.00 360.00 720.00 1080.00 1440.00
Page 1 min
?
Figure 2. Base run: glucose and insulin concentration dynamics for a healthy person
Suggested Penalty Formulation
We introduced the following penalty formulation (equations 38-40):
Penalty, =0 [milligram - minute [deciliter | (38)
Penalty,,y7 = Penalty, + Penalty Generation * DT [ eee mine (39)
deciliter
Penalty Generation = | 94.25 — Glucose Concentration [milligram [deciliter] (40)
Approximately, 94.25 is the average glucose concentration. Penalty is the
accumulated absolute difference between 94.25 and Glucose Concentration (equations
38-40). Penalty is 10,179 for the base run in Figure 2.
Changes in the Model for a Diabetes Type 1 Patient
We changed Equation 4 to the following (Equation 41) by replacing the inflow of
Insulin stock, which is Insulin Production Stimulated by Glucose Concentration, with
Insulin Injections:
Insulin Injections
Insulin, , pr = Insulin, + : . *DT [milliunit] (41)
— Insulin Degradation and Clearance
We assumed that the actual value of Glucose Concentration is not available to the
decision maker. Hence, the following equations (42-45) are added to the model:
illi
Measured Glucose Concentration, = Glucose Concentration [ae | (42)
deciliter
Measured Measured Ui
feasurement milligram
Glucose =| Glucose + . °DT ASTM: (43)
. . Formation deciliter
Concentration Concentration
14+DT t
Measured
Glucose
__|—| Glucose
Concentration .
Measurement Concentration milligram 44)
Formation 7 Measurement Delay Time deciliter * minute
Measurement Delay Time = 2 [minute] (45)
We assume that a dynamic decision making heuristic control the automatic insulin
injection unit attached to the patient. Patient should use the unit 24 hours a day. The
equations for the suggested dynamic decision making heuristic, which controls the
injections, are given below (equations 46-52):
IF (Measured Glucose Concentration > 94.25)
rn AND NOT(Remaining Time > 0) milliunit
Insulin Injections = ——— | (46)
THEN Amount of Injection/DT minute
ELSE 0
Amount of Injection = 200 [milliunit] (47)
Remaining Time, = 10 [minute] (48)
Time — Count Down
Remaining Remaining Restart Remaining Time
= +
Time “DT t
| * DT [minute] (49)
Restart IF Insulin Injections > 0
Remaining |= {THEN Minimum Time Between Injections
DT
[dimensionless] (50)
fine ELSE 0
Minimum Time Between Injections =15 [minute] (51)
Count Down = {IF Remaining Time > 0 THEN 1 ELSE o} [dimensionless] (52)
The resulting behavior can be seen in Figure 3. The associated penalty is
approximately 8,801, which is even less than the penalty (10,179) obtained for a healthy
person. Thus, we can conclude that the proposed heuristic is successful under the
conditions presented in this paper.
B® +: cucose Concentration 2: Insulin Concentration
1 108% 1
2: 500.0
A
1 1
a4]
2 250.0 |
he
1 60 4 >
2 0.0+4 T T
0.00 360.00 720.00 1080.00 1440.00
Page 1 min
Figure 3. Run for a diabetes type | patient
Conclusions
In this work, we first constructed a system dynamics model of the two time delay
model introduced by Li et al. (2006). This model represents the glucose-insulin
regulatory system in a healthy person. We simulated the model for 1440 minutes (1 day)
and obtained the benchmark dynamics given in Figure 2. Later, we introduced a penalty
formulation and calculated the penalty as 10,179 for the benchmark.
We adapted the model for a diabetes type 1 patient by replacing the insulin
production with /nsulin Injections. We assumed that an automatic insulin injection unit is
attached to the patient 24 hours a day. We also introduced a dynamic decision making
heuristic that can be utilized in the control of the unit. The suggested decision making
heuristic generated a penalty value (8,801) less than the benchmark penalty (10,179).
Hence, we conclude that the proposed heuristic is successful under the conditions
presented in this paper. However, further tests under different glucose infusion rate
patterns would be required before utilizing the heuristic. Moreover, the performance of
the heuristic should be improved by optimizing its parameters.
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