`Fibre Angle Stimulation I - Rotation`

*Module:* `tutorials.02_EP_tissue.14_bidm_rotation.run`

*Section author: Laura Marx <laura.marx@medunigraz.at>*

The aim of this tutorial is to study the effect of fiber orientation on the electrical behavior of cardiac tissue. Generally, electrical properties of the myocardial tissue can be described by the bidomain model accounting for the intracellular and extracellular domain. The domains are of anisotropic nature, the intracellular domain even more so than the extracellular domain resulting in ‘’unequal anisotropy ratios’’ affecting the heart’s electrical behavior causing two effects ‘’rotation’’ and ‘’distribution’‘. This tutorial focuses on the ‘’rotation’’ effect, which occurs when two spaces have different degrees of anisotropy. If considering an isotropic tissue the electrical field and current density would point in the same direction. It follows that with g being the scalar conductivity. In anisotropic tissue this is not the case as the conductiviy tensor rotates towards the fiber direction relative to . As anisotropy ratios are unequal in the intra- and extracellular space current densities and are rotated by different amounts. [3]

**Analytical vs. Numerical Setup:**
Given is a 2D strip of infinite length of width W=10mm in which fibers are oriented with (shown as grey dotted lines in the figure below).
Conductivities are given as S/m, S/m, S/m and S/m
where L and T indicate directions along and transverse to the fiber orientation, respectively (Anisotropy ratios are unequal!). The
tissues membrane surface-to-volume ratio and the resistance are given as and
. For numerical analysis we assume that the length L of the strip is not infinite, but
. For the data given choosing L=60mm should suffice. The spatial resolution should be used as height of the strip (only one layer of 3D elements).
The analytical solutions should be used to verify the numerical results in 3D.

**Model Settings:** A hexahedral FE model with dimensions of 60mm x 10mm x 1mm shall be generated functioning as cardiac tissue strip.
An orthotropic fiber setup is used with a sheet angle of .

**Electrical Settings:**
As passive ionic model the modified Beeler-Router Druhard-Roberge model as implemented in LIMPET is used with set to and .
Current is injected by Electrode A at x=-L/2 and withdrawn by electrode B at x=+L/2. Electrode C is located at x=0 and can be gounded or not.

**Conductivity Settings:** Unequal anisotropy is assigned using the following values:

```
gregion[0].g_il = 0.2
gregion[0].g_it = 0.02
gregion[0].g_in = 0.02
gregion[0].g_el = 0.2
gregion[0].g_et = 0.08
gregion[0].g_en = 0.08
```

**Solution Method:** The simulation is solved using the parabolic formulation of the bidomain equations with the Crank-Nicolson method.

Go to the following directory to run underlying experiment:

```
cd ${TUTORIAL}/02_EP_tissue/14_bidm_rotation/
```

The following experiment specific options are available (default values are indicated):

```
./run.py --angle #pick fiber angle (0 is along the long edge of the preparation) (default:45)
--resolution #choose mesh resolution in um (default: 250)
--grounded {on, off} #turn on/off ground (default:off)
```

Note

If a high mesh resolution is chosen it may take longer to run. You may use the `--np`

option to increase number of cores.
To visualize results with meshalyzer use `--visualize`

.

`--grounded`

- A ground can be used or not. In the latter case, no grounding electrode
is used, that is, a pure Neumann-problem is solved. A rank-one stabilization is used
in this case which essentially enforces the integral over the extracellular potential
to be zero at each instant in time, but no specific point is clamped.

At electrical current is injected into the tissue strip and the same current is withdrawn at . Initially, an electrical field is setup at the center of the strip at y=0, which is oriented along x, but current density is oriented towards the fiber orientation (running against a sealed boundary under angle , see Fig. A) with

Since current will be directed towards the sealed ends at . Due to charge will accumulate at the sealed ends which sets up an electrical field (see Fig. B). From

and the fact that at must be zero it can be concluded that makes and angle with x given by

Due to and neither nor can depend on y.

Now we considering a bidomain with unequal anisotropy ratios. and must be zero individually at sealed boundaries. Thus,

As the tissue strip is infinitely long cannot be a function of x thus

and it follows that

Thus,

It can be concluded that there must be a y-directed gradient of causing membrane polarisation near the boundary (see Fig. C).

Near the center of the strip at y=0 the net current must be in direction of x with , but due to rotation intra- and extracellular densities and make an angle with the x-axis. Intra- and extracellular field are equal (see Fig. D):

which implies that the vanishes near the center as

**Analytical Solution:**

First conductivity ratios are computed as

followed by conductivity tensors in the fiber coordinate system

and in the cartesian coordinate system assuming a fiber angle of :

and

At the sealed boundaries and must be zero individually at sealed boundaries thus,

As the tissue strip is infinitely long cannot be a function of x thus

and it follows that

Thus,

holds with as well as

with .

It can be concluded that the y-directed gradient of causes membran polarisation near the boundaries (see results of numerical solution).

Near the center of the strip at y=0 the net current must be in direction of x with , but due to rotation intra- and extracellular densities and make an angle with the x-axis. Intra- and extracellular field are equal:

which implies that the vanishes near the center (see results of numerical solution) as

**Numerical Solution:** Front and rear view of simulated tissue strip for :

Simulation results confirm theoretical expectations. Gradient hyperpolarizes and depolarizes edges of the tissue strip at and and vanishes at the center of the strip.

[1] | Sepulveda, Nestor G., Bradley J. Roth, and John P. Wikswo Jr., |

[2] | Bradley J. Roth, and Marcella C. Woods, |

[3] | Bradley J. Roth, |