Biomedical Engineering Reference
In-Depth Information
2.4 First-Generation and Second-Generation Models
In the early ionic models of cardiac cells, all ion concentrations were constant, so that
no provision had to be made for pumps and exchangers to regulate these
concentrations. We refer to those models as 'first-generation' models, in contrast to
'second-generation' models with time-varying ion concentrations that are regulated
by pumps and exchangers. As set out in detail by Krogh-Madsen et al. [42], there are
two major problems with the more physiologically realistic second-generation
models, in which, besides membrane potential and gating variables, ion
concentrations vary in time. The first is drift, with very slow long-term trends in some
of the variables, primarily ion concentrations. Drift has been dealt with in several
ways, including stimulus current assignment to a specific ionic species (e.g. [41]). The
other major problem noted with second-generation models is 'degeneracy'. This
means that there is a continuum of equilibrium points, i.e. 'steady-state solutions' to
the set of differential equations defining the model, rather than isolated equilibrium
points, e.g. the resting potential of a quiescent system depends on the initial
conditions. This can be overcome through a 'chemical' approach using an explicit
formula for the membrane potential of cells in terms of the intracellular and
extracellular ion concentrations [20]. A practical solution to the problems of drift and
degeneracy is to set the intracellular sodium and potassium concentrations to constant
values, which reflects the buffering of these ion concentrations in patch-clamp
experiments through the pipette solution.
When using cardiac cell models, one should realize that first-generation models
tend to reach a steady state within only a few beats, whereas second-generation
should be run for a much longer time to reach steady state. For example, in the canine
atrial cell model by Kneller et al. [41] action potential duration reaches steady state
after approximately 40 min of pacing, which limits its suitability for performing large-
scale (whole-heart) simulations, where simulating a single beat may already take
several hours on a state-of-the-art computer [69]. Furthermore, it should be
emphasized that the equations representing the calcium subsystem of the second-
generation models are still evolving and even the latest comprehensive models fail to
adequately represent fundamental properties of calcium handling and inactivation of
L-type calcium current by intracellular Ca
2+
(see, e.g. [38, 82] and primary references
cited therein).
2.5 Deterministic, Stochastic and Markov Models
Until a few years ago, cardiac cell models employed the traditional Hodgkin-Huxley
formulation of ion channel gating, with one or more independent gates that can flip
between their open and closed state (Fig. 3a), although it had been noted that many
currents are more accurately described by Markov-type models. Markov models are
not restricted to the Hodgkin-Huxley concept of independent gates defining the
channel state diagram. Thus, they allow for more complex state diagrams with
specific transitions between open, closed, or inactivated channel states (Fig. 3b), with
the advantage of being more closely related to the underlying structure and
