Biomedical Engineering Reference
In-Depth Information
P K d þ qV ½ K þ o
J K
e q KT ¼
d
KT
ð
12
:
26
Þ
P K d þ qV ½ K þ i
J K
KT
d
Solving for
J K
in Eq. (12.26) gives
0
@
1
A
qV
KT
½ K þ o ½ K þ i e
J K ¼ qVP K
KT
ð
12
:
27
Þ
qV
KT
e
1
Chlorine Ions
The same derivation carried out for
K þ can be repeated for
Cl , which yields
0
@
1
A
qV
KT ½ Cl i
½ Cl o e
J Cl ¼ qVP Cl
KT
ð
12
:
28
Þ
qV
KT
1
e
Cl .
where
P Cl
is the permeability for
Summarizing for Potassium and Chlorine Ions
Using space charge neutrality,
J K ¼ J Cl
, and Eqs. (12.27) and (12.28) gives
qV
KT
qV
KT ½ Cl i
P K ½ K þ o ½ K þ i e
¼ P Cl ½ Cl o e
ð
12
:
29
Þ
Solving for the exponential terms yields
KT ¼ P K ½ K þ o þ P Cl ½ Cl i
P K ½ K þ i þ P Cl ½ Cl o
qV
e
ð
12
:
30
Þ
Solving for
gives
V
P K ½ K þ o þ P Cl ½ Cl i
P K ½ K þ i þ P Cl ½ Cl o
V ¼ v o v i ¼ KT
q
ln
ð
12
:
31
Þ
or in terms of
V m
P K ½ K þ o þ P Cl ½ Cl i
P K ½ K þ i þ P Cl ½ Cl o
V m ¼ KT
q
ln
ð
12
:
32
Þ
This equation is called the
Goldman equation
. Since sodium is also important in membrane
K þ ,
Cl , and
Na þ can be derived as
potential, the Goldman equation for
P K ½ K þ o þ P Na ½ Na þ o þ P Cl ½ Cl i
P K ½ K þ i þ P Na ½ Na þ i þ P Cl ½ Cl o
V m ¼ KT
q
ln
ð
12
:
33
Þ
Na þ . To derive Eq. (12.33), first find
where
is the permeability for
J Na and then use space
P Na
charge neutrality
J K þ J Na ¼ J Cl . Equation (12.33)
then follows.
In general, when the
Search WWH ::




Custom Search