Local basis functions

In this demo we will show how to generate local coordinate basis functions in the longitudinal, circumferential and radial direction. These vectors are very important in several applications. For example in ultrasound strain estimation strain in along these directions are important metrics to quantify cardiac function.

Let us first import the relevant libraries

import ldrb
import dolfin

We will show how to to this on an LV and a BiV geometry. The BiV geometry is a bit tricky compared to the LV.

case = "lv"
# case = "biv"

And depending on the case we create a geometry

if case == "lv":
    geometry = ldrb.create_lv_mesh()
else:
    geometry = ldrb.create_biv_mesh()

And let us also extract the relevant variables

# The mesh
mesh = geometry.mesh
# The facet function (function with marking for the boundaries)
ffun = geometry.ffun
# A dictionary with keys and values for the markers
markers = geometry.markers
# Space for the vector functions
space = "P_2"

In the case of a BiV we do a little trick where me mark the RV as LV in order to find the longitudinal vector field. We run the algorithm again with some different angles to get the circumferential vector field and finally take the cross product to get the radial vector field. (if you find a better way of doing this, please submit a PR).

if case == "biv":
    lv_ffun = dolfin.MeshFunction("size_t", mesh, 2)
    lv_ffun.array()[:] = ffun.array().copy()
    lv_ffun.array()[ffun.array() == markers["rv"]] = markers["lv"]
    lv_markers = markers.copy()
    lv_markers.pop("rv")

    long, _, _ = ldrb.dolfin_ldrb(
        mesh=mesh,
        fiber_space=space,
        ffun=lv_ffun,
        markers=lv_markers,
        alpha_endo_lv=-90,
        alpha_epi_lv=-90,
        beta_endo_lv=0,
        beta_epi_lv=0,
    )

    circ, _, _ = ldrb.dolfin_ldrb(
        mesh=mesh,
        fiber_space=space,
        ffun=ffun,
        markers=markers,
        alpha_endo_lv=0,
        alpha_epi_lv=0,
        beta_endo_lv=0,
        beta_epi_lv=0,
    )

    rad = dolfin.project(dolfin.cross(circ, long), circ.function_space())

For the single ventricle it is much simpler

else:
    long, circ, rad = ldrb.dolfin_ldrb(
        mesh=mesh,
        fiber_space=space,
        ffun=ffun,
        markers=markers,
        alpha_endo_lv=-90,
        alpha_epi_lv=-90,
        beta_endo_lv=0,
        beta_epi_lv=0,
        save_markers=True,
    )

Finally we save the vector fields to a file

ldrb.fiber_to_xdmf(long, "long")
ldrb.fiber_to_xdmf(circ, "circ")
ldrb.fiber_to_xdmf(rad, "ran")

This resulting BiV and LV vector field are shown in Figure 1 and Figure 2 respectively.

Fig. 1 LV longitudinal, circumferential and radial vector fields

Fig. 2 BiV longitudinal, circumferential and radial vector fields