Fiber generation following Bayer's method does not follow original convention.

[javijv4]

Description

In the fiber-generation codes, the Bayer convention for the beta angles does not match the original paper.
In Bayer's original paper, the beta angle induces a rotation about the fiber angle as the axis, whereas the code now performs a rotation about the rotated longitudinal axis.

Reproduction

Run utilities/fiber_generation/main_bayer.py

Expected behavior

The original Bayer paper and Piersanti's paper do

1. Rotate by $\alpha$ degrees using the transmural vector as the axis.
2. Rotate by $\beta$ degrees using the rotated fiber vector as an axis.

The code should be updated to do that, so it follows the original paper.

Additional context

When I was refactoring this code months ago, I thought there was a defined convention for the $\alpha$ and $beta$ angles, but I was wrong. Bayer and Piersanti perform the rotation as described previously. Doste performs the rotations as,

1. Rotate by $\alpha$ degrees using the transmural vector as the axis.
2. Rotate by $\beta$ degrees using the rotated apex-to-base vector as an axis.

Things to consider (I am just writing this here because the literature is very confusing, and I want to have records of this)

Vector notation

• Bayer and Piersanti call the vector pointing in the circumferential direction "longitudinal, $e_l$", and the vector pointing from the apex to the base "normal, $e_n$".
• Doste calls the vector pointing in the circumferential direction "circumferential, $e_c$", and the vector pointing from the apex to the base "longitudinal, $e_l$".

$\alpha, \beta$ angles versus helix, transverse angles.

Another thing I checked back then was experimental / imaging literature. I think there exists a consensus that the helix angle and the transverse angle are defined as in the image (from [this paper])(https://doi.org/10.1152/ajpheart.00059.2022):

If using the Doste convention, we can say that $\alpha$ is the helix angle and $\beta$ is the transverse angle.
If using the Bayer / Piersanti convention, then $\alpha$ = helix angle but $\beta$ != transverse angle.
Interestingly, if using Bayer / Piersanti convention, the transverse angle will always be zero, which would go against the transverse angles measured experimentally.

Code of Conduct

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[javijv4]

I'll be working in this branch.

I need to fix the issue, update the documentation and ensure everything is working correctly.

So far, I haven't been able to validate any of the methods against other software, but I found a paper Doste (the author) published last year, and his fiber-generation code is available on GitHub now (https://github.com/rdoste/InSilicoHeartGen/tree/main).

I will also try to run either https://github.com/KIT-IBT/LDRB_Fibers code or this code https://github.com/finsberg/fenicsx-ldrb to compare the Bayer implementation.

[michelebucelli]

Not sure if this is useful (if not, feel free to disregard) but both the Bayer and Doste methods are also available through lifex-fiber (paper and binary download with examples).

If I remember correctly that release only worked for single-ventricle models though - not sure if this would be a limitation for your verification purposes.

[javijv4]

@michelebucelli yeah, the first time I worked on this, I looked into using lifex-fiber for validation, but it is only implemented for single ventricles. Unfortunately, I think the most cumbersome part of generating BV fibers is combining the LV and RV sides.
Now that I think about it, the current implementation will probably only work for BV geometries, so this might be a good opportunity to add support for single ventricles and maybe add validation for single ventricles against lifex-fiber. I will give it a try. Thanks!

[dseyler]

Thanks for addressing this @javijv4 . I investigated a bit and noticed that fiber directions that lie entirely in the transmural plane (for both Bayer and Doste) give "better" cardiac mechanics results, i.e. more realistic apex to base shortening. I know the original Doste paper reports a 20-degree beta angle out of the transmural plane, but this seems to give much less realistic results. @michelebucelli, I was curious if you have any insights on this. It looks like the original Doste paper only compares electrophysiology but not mechanics

[michelebucelli]

@dseyler I remember always using the 20° beta angle, and I don't think I ever tried anything else, so I don't have a lot of insight. I remember that we observed some significant variation in wall motion depending on how the total active tension was distributed in the three fiber directions, so perhaps this is a different way of getting to a similar result.

I think you might find more on this in this paper.