Biophysical Models of the Cardiovascular System - Systems Biology: Networks, Models, and Applications - page 272

set of coupled nonlinear ordinary differential algebraic equations
describing the time rate of change of model state variables. These state
variables are: (a) probability of occupancy of ion channel states such
as those of eq. (2) and current flux through open channels as in eq. (4);
(b) concentrations of ion species in model compartments as in eq. (5);
and (c) time evolution of membrane potential. Currently, all biophysi-
cally detailed models of the myocyte assume that since these cells
are spatially compact, they are isopotential, with time rate of change
of membrane potential given by:

dv t
dt

()

& 5

pump

ion

=− ∑ I [ ( )] vt + ∑ I [ ( ), ( )] vt ct

*

(6)

i i

i i

where v ( t ) is membrane potential, I ion [ v ( t )] is current carried by the i th
membrane current, and I i

pump [ v ( t ), c ( t )] is current through the i th
membrane pump/exchanger, which can depend on both membrane
potential v ( t ) and the relevant ion concentration c ( t ).

Successes and Failures of Common Pool Models
Common pool models have been used to explore diverse properties
of the cardiac myocyte. In this section we focus on some major suc-
cesses of these models, but more importantly, on a recently discovered,
significant weakness of this modeling approach.
Figures 9.4a and 9.4b show examples of APs (figure 9.4a) and Ca 2+
transients (figure 9.4b) measured from ventricular myocytes iso-
lated from normal (solid line) and failing (dashed line) canine hearts.
Figures 9.4c-d show corresponding APs (figure 9.4c) and Ca 2+ tran-
sients (figure 9.4d) computed by numerical solution of the model initial
value problem. These results were obtained using the canine ventricu-
lar myocyte model of Winslow et al. [31] for normal (solid line) and
failing (dashed line) myocytes. These data demonstrate that common
pool models have been quite successful in reconstruction of the AP and
the time-varying waveform of the cytosolic Ca 2+ transient.
Results of a more challenging simulation, obtained using the
Jafri-Rice-Winslow model of the guinea pig ventricular myocyte [11],
are shown in figure 9.5. In this simulation, the model was used to
elucidate mechanisms of the interval-force relation, a classical charac-
terization of cardiac muscle in which developed force is strongly
determined by pacing history. In these simulations, the model cell
was paced at an interstimulus interval of 1.5 s until a steady state was
reached. Panels (a) and (b) of figure 9.5 show model Ca 2+ (figure 9.5a)
and isometric force (figure 9.5b) transients in response to the pacing
stimuli. Isometric force was computed using a model developed by
Rice et al. [11,32] that relates cytosolic Ca 2+

level to isometric force.

Ca 2+ and force transients labeled SS in figures 9.5a-b are responses