Page 125 - Advances in Biomechanics and Tissue Regeneration

7.2 EQUATIONS THAT GOVERN THE ELECTRICAL ACTIVITY OF THE HEART 121

inner and outer membrane surfaces, establishing, therefore, an electric field (a potential difference) within the mem-
brane. Because the ions are charged particles, this electric field exerts forces on the ions crossing the membrane that
oppose the diffusional forces established by the difference in ionic concentration. Therefore, to describe membrane ion
movements, electric-field forces and diffusional forces should be considered. In this regard, equilibrium is attained
when the diffusional force balances the electric field force for all permeable ions.
For a membrane that is permeable to only one type of ion, the electrochemical balance between forces due to the
concentration gradient and the potential gradient for a particular ion can be described by the Nernst-Planck-Einstein
equation:

RT c k,e
ln , (7.23)
E k ¼
z k F c k,i
where R, F, and T are the gas constant, the Faraday constant, and the absolute temperature constant, respectively; E k is
the equilibrium voltage across the membrane (Nernst potential) for the kth ion; z k is the valence of the kth ion; and c k, e
and c k, i are the extracellular and intracellular concentrations of the kth ion, respectively.

7.2.3.2.2 GOLDMAN-HODGKIN-KATZ EQUATION
Assuming that the cell membrane is permeable to a single ion only is not valid. However, it is assumed that when
several permeable ions are present, the flux of each is independent of the others (known as the independence principle).
According to this principle, and assuming: (i) the membrane is homogeneous and neutral, and (ii) the intracellular and
extracellular ion concentrations are uniform and unchanging, the membrane potential is governed by the well-known
Goldman-Hodgkin-Katz equation. For N monovalent positive ion species and for M monovalent negative ion species,
the potential difference across the membrane is as follows
N M
0 1
X + X
P j ½c Š +
j e
B j i P j ½c Š C
RT B j¼1 j¼1 C
C, (7.24)
B C
E ¼ logB N M
F B X X C
+
@ P j ½c Š + A
k i
j e P j ½c Š
j¼1 j¼1
where [c j ] i and [c j ] e are the intracellular and extracellular concentrations for the jth ion, P j is the permeability of the
intracellular and extracellular concentrations for the jth ion, and E is the membrane potential. The permeability for
the jth ion is defined as
D j β j
,
P j ¼
h
where h is the thickness of the membrane, D j is the diffusion coefficient, and β j is the water partition coefficient of the
membrane. Both D j and β j depend on the type of ion and the type of membrane.
7.2.3.3 Gates
Ion channels are specific units located in the cell membrane through which the ions can flow. These channels
open and close in response to potential differences or a change in ion concentration. The mechanisms by which
thesechannels openand close arestochasticinnatureand mayinvolve complexprocesses. Inthesimplestcase, the
channel, or the channel gate, is considered to be in only two possible states, either open or close. Assigned to these
sates, there is a possibility of opening O and a possibility of closure C, being stochastic the transition between
states. The density of opened channels is [O] and the density of closed channels [C]. Furthermore, assume that
the density of channels, [O]+[C], is constant. The change between the open state and closed state can be described
by a first-order reaction as
α
C>O, (7.25)
β

where α is the opening rate and β is the closing rate of the channels. These rates depend on the membrane potential V in
general, even though they could also be modulated by the ion concentration. By the law of mass action, the rate of
change from the open state to the closed state is proportional to the concentration of channels in the open state; equally,

I. BIOMECHANICS