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124 7. MULTISCALE NUMERICAL SIMULATION OF HEART ELECTROPHYSIOLOGY
regional ischemia (details will be discussed in the following sections). From a computational point of view, the TP06
model has 19 state variables, 14 ionic currents, and requires of a minimum time integration step of 0.02 ms.
7.2.4 Numerical Solution of the Electric Activity of the Heart
As discussed before, the monodomain model represents a major simplification of the bidomain model with impor-
tant advantages for mathematical and computational analysis, which is suitable for studying the electrical behavior of
the heart. This section focuses on the numerical solution of the monodomain model described by Eqs. (7.18), (7.19).
From a mathematical and computational view, the problem defined by Eqs. (7.16), (7.17) corresponds to the solution
of a linear partial differential equation, which describes the electrical conduction, coupled with a rigid nonlinear sys-
tem of ODEs describing the transmembrane ion currents, resulting in a problem of nonlinear reaction-diffusion. An
efficient way to solve Eqs. (7.16), (7.17) is by application of the splitting technique operators [49]. The decomposition
technique of operators has been applied to the monodomain equations [7, 50]. The basic steps are summarized below:
• Step 1: Use V (t) as the initial condition for integrating the equation
∂V
r DrVÞ ¼ C m + J ion ðV,uÞ, for t 2 t,t + Δt=2½ : (7.30)
∂t
ð
• Step 2: Use the result obtained in Step 1 as the initial condition to integrate
∂V
C m ¼ J ion ðV,uÞ, (7.31)
∂t
∂u
¼ fðu,V,tÞ, for t 2 t,t + Δt: (7.32)
½
∂t
• Step 3: Use the result obtained in Step 2 as the initial condition for integrating
∂V
r DrVÞ ¼ C m + J ion ðV,uÞ, for t 2 t + Δt=2,t + Δt½ : (7.33)
∂t
ð
In practice, Steps 1 and 3 can be combined into one, except for the first increment. Therefore, after the initial increase,
the algorithm has only two steps, Step I corresponding to the integration of ODEs (Step 2), and Step II corresponding to
the integration of the homogeneous parabolic equation (Steps 1 and 3).
k
• Step I: Use V (t ) as the initial condition to integrate the equation
∂V
C m ¼ J ion ðV,uÞ J stm ðtÞ,
∂t for t 2½t ,t + Δt: (7.34)
k
k
∂u
¼ fðu,V,tÞ,
∂t
• Step II: Use the result obtained in Step I as the initial condition to integrate
∂V k k
C m ¼r ðDrVÞ, for t 2½t ,t + Δt: (7.35)
∂t
7.2.4.1 Spatial-Temporal Discretization
When performing Step II, the computational domain must be discretized in space by a mesh of either finite elements
or finite differences to approximate the dependent variables of the problem, V and u, which allows writing Eq. (7.35)as
MV + KV ¼ 0, (7.36)
where M and K are the mass and stiffness matrices, respectively, obtained by assembling individual element matrices
over the entire computational domain.
The most well-known algorithms for integrating in time the semidiscrete system (7.36) are members of the gener-
k
k
alized trapezoidal family methods [51]. Let V and V denote vectors of the transmembrane potential and its time
k
derivative at each nodal point of the mesh at time t , respectively, where k is the index of the time step. Then at time
t k+1 we can write
k +1 k +1
MV + KV ¼ 0, (7.37)
I. BIOMECHANICS
