Modeling Myocardial Fiber Orientation

To assign fiber orientation throughout the myocardium according to the rules R1–R6 above, the LDRB algorithm takes four functions as inputs, representing the desired α and β angles within the septum (s) and the ventricular walls (w). The angles α and β are in degrees and d is the transmural depth normalized from 0 to 1.
$${\alpha }_{\mathrm{s}}(d)={\alpha }_{\text{endo}}(1-d)-{\alpha }_{\text{endo}}\ast d$$
$${\alpha }_{\mathrm{w}}(d)={\alpha }_{\text{endo}}(1-d)+{\alpha }_{\text{epi}}\ast d$$
$${\beta }_{\mathrm{s}}(d)={\beta }_{\text{endo}}(1-d)+{\beta }_{\text{endo}}\ast d$$
$${\beta }_{\mathrm{w}}(d)={\beta }_{\text{endo}}(1-d)+{\beta }_{\text{epi}}\ast d$$
If non-linear interpolation functions are desired, these functions need to satisfy α_w(0) = α_s(0) = −α_s(1) modulo 180° and β_w(0) = β_s(0) = −β_s(1) modulo 180°. These conditions ensure that the coordinate systems used for assigning fiber orientation on the endocardium of Ω will be appropriately oriented, regardless of whether orientations are parametrized in the LV or RV wall.
The LDRB algorithm also requires the definition of the following surfaces in order to assign Dirichlet boundary conditions:

∂Ω_apex: the apex of the ventricles

∂Ω_base: the base of the ventricles

∂Ω_epi: the epicardial surface of the ventricles

∂Ω_lv: the endocardial surface of the LV

∂Ω_rv: the endocardial surface of the RV

For most studies, the surface ∂Ω_base can be extracted by taking a cutting plane at the apicobasal junction, and the surface ∂Ω_apex can be extracted by finding the point lying closest to the ventricular apex of Ω. The surfaces ∂Ω_epi, ∂Ω_lv, and ∂Ω_rv can then be extracted by choosing an arbitrary point on the epicardium, LV endocardium, and RV endocardium, then iteratively expanding from these three points along the surface of Ω until intersection with ∂Ω_base.