Biomedical Engineering Reference
In-Depth Information
Na
þ
,and
J
p
is the pump flow rate. Permeability
is a function of the membrane that describes the ease with which a particle moves through the
cell membrane and is discussed more fully in Chapter 12. Potassium has a high permeability
and sodium has a low permeability. To determine the cell volume, we replace [
K
þ
and
where
P
K
and
P
Na
are permeabilities for
A
]in
Eq. (7.19) with
a
V
,where
is the number of anions and
is the volume of the cell, giving
a
V
o
þ
a
V
RT K
þ
i
K
þ
o
þ
Na
þ
i
Na
þ
½
½
½
½
¼
R
m
Q
ð
7
:
21
Þ
At steady state, the net flow of ions,
Q
, and water is zero, so Eqs. (7.20) and (7.21) reduce to
J
p
¼
P
K
K
þ
i
K
þ
0
½
½
o
ð
7
:
22
Þ
þ
J
p
¼
P
Na
Na
þ
i
Na
þ
0
½
½
o
and
o
þ
a
K
þ
i
K
þ
o
þ
Na
þ
i
Na
þ
½
½
½
½
V
¼
0
ð
7
:
23
Þ
J
p
P
K
K
þ
i
K
þ
Na
þ
i
Na
þ
Equation (7.22) is solved for the concentrations,
½
½
o
¼
and
½
½
o
¼
J
p
P
Na
, and then substituted into Eq. (7.23), giving
J
p
P
K
J
p
P
Na
þ
a
P
Na
P
K
P
Na
P
K
þ
a
V
¼
J
p
V
¼
0
ð
7
:
24
Þ
Equation (7.24) is easily solved for
V
, yielding
aP
Na
P
K
J
p
P
K
P
Na
V
¼
ð
7
:
25
Þ
ð
Þ
, which is the case for the
mammalian cell membrane. Notice that the cell volume is inversely related to the pump rate,
so as the pump rate increases, cell volume decreases. The cell carefully controls the pump
rate so cell volume is maintained. Also, note that as the cell grows, evidenced by increasing
the number of impermeable proteins and molecules (
Positive cell volumes are possible in Eq. (7.25) when
P
K
>
P
Na
in Eq. (7.25)), the volume of the cell
increases according to Eq. (7.25). Finally, note that when the pump rate goes to zero (death),
the cell volume heads to infinity, but before getting there, the cell membrane bursts.
a
7.3.5 Capillary Diffusion
The movement of water, nutrients, electrolytes, and other particles through the capillary
wall is driven by osmotic and hydrostatic pressure. As we will see, these pressures cause fluid
to flow out of the capillary at the arterial end and flow into the capillaries at the venous end.
Figure 7.7 illustrates a capillary network. To reach the capillaries, blood first flows from
the heart to the aorta under high pressure. The blood leaves the aorta and flows into other
arteries until it reaches the arterioles, the smallest branch of the arterial system. From the
arterioles, blood flows through the capillaries, where diffusion into the interstitial volume
occurs in the lower portion of Figure 7.7. In the upper portion of Figure 7.7, diffusion from
the interstitial volume into the capillaries occurs, with the plasma then flowing into the
venules, the smallest branch of the venous system. From the venules, blood flows through
