Processes 2018, 6, 134
2.3. Non-Dimensionalisation
The model was non-dimensionalised according to the following rescalings
¯
t =
t
¯
δ mh
, ¯
m h = ¯
s 0 m h , ¯
m r = ¯
s 0 m r , ¯
h = ¯
s 0 h, ¯
l E = ¯
l E0 l E ,
¯
l RB = ¯
l E0 l RB , ¯
l I = ¯
l E0 l I , ¯
v E = ¯
v E0 v E , ¯
v RB = ¯
v E0 v RB , ¯
v I = ¯
v E0 v I ,
(35)
¯
r f = ¯
r f 0 r f , ¯
r I = ¯
r f 0 r I , ¯
c = ¯
c T c,
where c T = 1.89 × 10 19 molecules/mL and s 0 = 8.21 × 10 16 molecules/mL are the total concentrations
of cholesterol and SREBP-2 in a hepatocyte [12,18], respectively. Substituting these rescalings into
Equations (22)–(34) leads to
J
dm h
dt
=
μ mh
1 +
κ mh (1 +(
c
κc ) xc )
x h − δ mh m h ,
(36)
J
dm r
dt
=
μ mr
1 +
κ mr (1 +(
c
κc ) xc )
xr − δ mr m r ,
(37)
dh
dt
= μ h m h − δ h h,
(38)
W
dl E
dt
= −α L r f l E + α −L l RB + Wχ v ρ v v E ,
(39)
dl RB
dt
= α L r f l E − α −L l RB − β L l RB ,
(40)
dl I
dt
= β L l RB − γ L l I ,
(41)
W
dv E
dt
= −α v r f v E + α −v v RB − Wχ v v E + Wω,
(42)
dv RB
dt
= α v r f v E − α −v v RB − β v v RB ,
(43)
dv I
dt
= β v v RB − γ v v I ,
(44)
dr f
dt
= γ r r I +
m l
ϑ l
−β L
r f l RB
1 − r f
− α L l E r f + α −L l RB
−β 0 r
P
f +
m v
ϑ v
−β v
r f v RB
1 − r f
− α v v E r f + α −v v RB
,
(45)
dr I
dt
= −γ r r I + f β 0 r
P
f + f
m l
ϑ l
(1 +
r f
1 − r f
)β L l RB
+ f
m v
ϑ v
(1 +
r f
1 − r f
)β v v RB + μ r m r ,
(46)
dc
dt
= R
chol
L σ l γ L l I + R
chol
v σ v γ v v I + μ c h − δ c c,
(47)
with the initial conditions
m h (0)=m h0 , m r (0)=m r0 , h(0)=h 0 , l E (0)=1,
l RB (0)=0, l I (0)=0, v E (0)=v E0 , v RB (0)=0,
(48)
v I (0)=0, r f (0)=r f 0 , r I (0)=0 and c(0)=0.
All non-dimensional parameters in terms of dimensional ones and their values are shown in
Table 2.
88