Page 69 - Control Theory in Biomedical Engineering

Mathematical modeling of cholesterol homeostasis 55

8% of the dry component of bile) (Guyton and Hall, 2016). During contrac-
tion, the gallbladder releases 75% of the accumulated bile—m bile . Finally, the
mass of bile ejected from the gallbladder during contraction from t 0 to t 1 can
be expressed as (Eq. 11):

ð
t 1
2 t 1 t 0
ð
ð
M b sin ω b t t 0 ÞÞdt ¼ M b ¼ 0:75m bile (11)
2
t 0
Thus, the amount of cholesterol transported by bile into the duodenum
*
represented by M out is 0.08 0.75m bile .
Cholesterol carried by bile returns to the liver (first compartment) after
its absorption in the small intestine, and the gallbladder is refilled with liver
bile approximately three times slower compared to its emptying (Di Ciaula
et al., 2012). The process of absorption of cholesterol along with bile can also
be described by Eq. (8). In the case of a vegetarian diet, the amount of cho-
*
lesterol returning with the bile to the liver (M in ) should be equal to the
amount of cholesterol ejected along with the bile from the gallbladder
reduced by losses with feces (w), that is, about 6% (Guyton and Hall,
2016) (Eq. 12):
∗ ∗
M ¼ M ð 1 wÞ (12)
in out
If we define the time of absorption of cholesterol with bile as t 3 t 2 , then
*
the amount of cholesterol returning to the liver (M in ) can be expressed as:
t ð 3
∗ 2 t 3 t 2
M ¼ M in sin ω in t t 2 Þð ð Þdt ¼ M in (13)
in
2
t 2
Based on relation (12), we can determine the amplitude (M in ) using
Eq. (14):

t 1 t 0
M in ¼ M out ð 1 wÞ (14)
t 3 t 2
where
0:12
M out ¼ m bile (15)
t 1 t 0
In the case of a cholesterol-containing diet, most of the cholesterol is pri-
marily absorbed by the lymphatic system and the expected increase in its
concentration in the circulating blood is delayed compared to the time of
the meal. It is worth noting here that in the intestine, the bile is also enriched