Page 41 - Artificial Intelligence for Computational Modeling of the Heart

Chapter 1 Multi-scale models of the heart for patient-specific simulations 11

rent. J in is the inward current and is computed using

hC(v)
J in (v,h) = , (1.2)
τ in
2
where the cubic function C(v) = v (1−v) describes the voltage de-
pendence of the inward current, which mimics the behavior of the
fast acting ionic channel gates. τ in is a time constant representing
the time required for the ion channels to open. J out is the outwards
current, and is computed using

v
J out (v) =− , (1.3)
τ out
where τ out is a time constant representing the time required for the
ion channels to close. Being an outwards current, it has a negative
sign. Finally, J stim is the stimulation current, which is applied ex-
ternally by a pacemaker for instance.
The gating variable h(t) is a non-dimensional variable which
varies between 0 (gate is closed) and 1 (gate is open). The evolu-
tion of h(t) is governed by:

dh 1 − h h
= H(v − v gate ) − H(v gate − v), (1.4)
dt τ open τ close
where H is the Heaviside function, τ open and τ close are the time con-
stants governing the duration of the phases in which the gates are
open and closed, respectively, and v gate is the change-over voltage.
Fig. 1.6 illustrates how these parameters relate to the shape
of the action potential. Other important quantities are the action
potential duration (APD) and the diastolic interval (DI), which
represent the total time in which the cell is depolarized, and the
time interval from cell repolarization to the next depolarization.
The M-S model can produce different action potentials based on
when the stimulus is provided. If the system is not starting from
equilibrium because of a short diastolic interval, a shorter action
potential will be generated. The relationship between APD and DI
is called the restitution curve and plays a crucial role in complex
arrhythmias like ventricular tachycardia or premature ventricular
contractions.
The M-S model is characterized by several useful properties
that make it particularly suitable for patient-specific simulations:
• The state variables are directly related to the ionic currents that
cross the cellular membrane.
• The parameters are closely related to the shape of the action
potential.