The methodology implemented to perform the desired objective is discussed in this section. To implement the heart movement on COMSOL the FitzhughNagumo equations and Complex LandauGinzburg Equations is used.
a. FitzhughNagumo equations
Fitzhugh – Nagumo equations are the primary equations that may be used to model the heart, together with COMSOL, to observe the behavior and check the influence of standard equations and their reaction to the given stimulus.
∂u1/δt = Δu +(αu1) (u11) u1+(u2) (1)
δu2/δt = ε(β.u1γu2δ) (2)
An action potential is represented by the variable of activator u1, and a variable of gate is represented by the gate variable u2 [23]. The insulating boundary conditions for u1 assume that no current flows into or out of the heart. The corner cases, from (1) & (2)
u1 (0,x,y,z) = v0 ((x + d) > 0) ((z + d) > 0),
u2 (0,x,y,z) = v2 ((x + d) > 0) ((z + d) > 0).
The condition initially defines an u1intial potential, where one of the heart quadrants is at constant, however remaining been at rest, the v0 is elevated potential. The values ‘x’ and ‘d’ for the changes should be less than 0 for dependent variable v0.
Similarly, v2 must keep the value of (x + d) <‘0'. To satisfy the equation, the value (z + d) must be greater than zero. The following logical expressions can be used to implement this initial distribution, where TRUE equals 1 and FALSE equals 0. Here, d equals 10− 5, which is used in the formulas for raising the raised slightly potential over the major axes.
b. The Complex LandauGinzburg Equations
δv1/δt  Δ(v1c1*v2) = v1(v1c3*v2) (v12v22) (3)
The activator and inhibitor (3) variables are v1 and v2, respectively. Constants c3 and c1 are parameters that reflect the material's properties. Existence and nature of stable solutions are also determined by these constants. The following are the initial conditions that result in a smooth transition step near z = 0:

v1(0,x,y,z) = tanh(z)

v2(0,x,y,z) = –tanh(z).
c. Parameters
The parameter used to define equations 1, 2, and 3 are expressed in Table 1. The Table 1 shown here accepts various values for the simulation of the heart. The values for alpha, beta, epsilon, gamma, delta, V0, nu0, d, c1, and c3 constants used are predefined using Table 1.
Table 1
Parameters Defined for the Equations
Name

Expression

Description

epsilon

0.01

excitability

gamma

1

system’s parameter

alpha

0.1

excitation threshold

beta

0.5

system’s parameter

nu0

0.3

inhibitor value elevated

delta

0

system’s parameter

V0

1

potential value elevated

d

1E − 5

shift distance offaxis

c3

−0.2

PDE parameter

c1

2

PDE parameter

The values for alpha, gamma, beta, and delta are used to implement in the equations (1) & (2) for modeling of heart structure on the simulation tool. The constants c1 and c3 have 2 and − 0.2 values for the PDE parameter as shown in Table 1. The values such as less element quality, average quality of element, triangle, tetrahedron, element edge, and element vertex used for implementation are shown in Table 2 for various models that are used to analyze heart on the COMSOL tool.
d. Models
The different models are designed to have better results to compare on the heart simulations modeled from the equation. The model 1 and model 4 has the same mesh value but different cavity as signifying the expansion of heart tubes due to ischemia. Similarly, model 2 and model 3 have the same mesh value but the different cavity in them. These models are important to analyze on different aspects which will be discussed in the section [24].
The minimum element quality is in the range of 0–1 that is because it considers the values in terms of percentage and their value for all the models is almost equal to ~ 2.5. Similarly, the average element quality for the 4 models is ~ 0.7. The edge value for model 1 to model 4 is arranged in decreasing order [25]. In Fig. 2, the mesh analysis is shown for various models i.e., models 1, 2, 3 & 4. This is modeled using Table 1 and Table 2.
Table 2
Parameters Defined for the Equations
Description

Value(Model 1)

Value(Model 2)

Value(Model 3)

Value(Model 4)

Minimum element quality

0.2592

0.2346

0.2518

0.2715

Average element quality

0.709

0.7065

0.6863

0.7186

Tetrahedron

28498

28621

17127

4572

Triangle

3802

4074

2762

1188

Edge element

306

336

276

164

Vertex element

16

23

23

16
