Page 65 - Control Theory in Biomedical Engineering

52 Control theory in biomedical engineering

analyzed for the condition m diet ¼0 (otherwise, m diet >0 corresponds to a
continuous meal containing cholesterol), whereas parameters m in and m out
describe the mean transport of cholesterol with bile, neglecting the role
of the gallbladder. Analysis of the sensitivity of the model to changes in
*
the parameters was performed based on Eq. (4). The calculated mass (m 2 )
divided by blood plasma volume (V 2 ) defines the concentration of total cho-
lesterol in the peripheral blood, designated as c 2 . The analysis was conducted
1
2
1
with the following set of values: k¼732mg min ; k 12 ¼1min ;
1 1 1
k 21 ¼3.58min ; m tis ¼0.243mgmin ; m in ¼0.8mgmin ; and
1
m out ¼1.2mgmin . In each of the case shown in Fig. 3A–F, only one
parameter was changed, which is shown in the description of the coordinate
system.
The analysis shows that changes in the values of parameters k 12 and k 21
indicate qualitatively different responses: c 2 grows linearly with k 12 exceeding
1
190mg/dL for k 12 >3.75min , whereas c 2 changes hyperbolically exceed-
1
ing 190mg/dL for k 21 <1min . It is worth noting that the parameters k 12
and k 21 representing the rate of cholesterol exchange between compartments
I and II appear in Eqs. (1), (2) in the same mathematical form.
The change in total cholesterol c 2 shown in Fig. 3C is in line with expec-
tation. With age, the tissue demand for cholesterol decreases due to the
decrease in the number of dividing cells as well as the decrease in the con-
centration of steroid hormones.
The parameter k responsible for the rate of de novo synthesis of cholesterol
causes a linear increase in c 2 concentration along with an increase in its value.
In our earlier work (Hrydziuszko et al., 2014, 2015; Wrona et al., 2015), we
considered larger values of parameter k because the c 2 concentration referred
to the whole blood. In this study, we reduced the range of changes in this
2
parameter to 1000mg min 1 because the concentration of the cholesterol
in analytical tests is referred against the volume of blood plasma.
Analysis of the model’s sensitivity to the changes in the parameter
(Fig. 3E and F) shows that the parameters m in and m out also have a significant
influence on total cholesterol in the peripheral blood. Because these param-
eters determine the amount of cholesterol circulating along with the bile,
this process of cholesterol circulation should be considered more carefully
taking into account the periodic accumulation of bile in the gallbladder.
If we assume that at the initial time t 0 , when a meal containing cholesterol
and fats is taken, the gallbladder is completely filled with bile and is expected
to release its contents into the duodenum during t 0 to t 1 . This period of time
varies between 30 and 100min depending on the food composition, the