Angle Definitions

This page explains how helix and intrusion angles are calculated from the 3D eigenvector field derived by cardiotensor.

Coordinate System

A transformation to a cylindrical coordinate system is defined for each voxel based on an approximation of the left ventricle (LV) centerline.

• Radial (r): outward from the LV center
• Circumferential (θ): tangential around the ventricle
• Longitudinal (z): base to apex direction

To compute local fiber angles consistently, all eigenvectors are first rotated into this cylindrical coordinate frame. This alignment is performed using Rodrigues' rotation formula, which computes the minimal-angle rotation that maps the global reference axis (here the z-axis) onto the local longitudinal axis at each point. This allows a robust comparison of orientations across the myocardium.

Helix Angle (HA)

The helix angle is defined as the angle between the third eigenvector \( \vec{v}_3 \) (smallest eigenvalue direction) and the local circumferential plane.

It captures the transmural variation of fiber orientation from epicardium to endocardium.

Typical pattern: - ~−60° at epicardium - ~0° in mid-wall - ~+60° at endocardium

Intrusion Angle (IA)

The intrusion angle is the angle between \( \vec{v}_3 \) and the tangential plane (longitudinal + circumferential).

It captures radial deviation of fiber aggregates and can help identify wall thickening or microstructural disruptions.

Angle Ranges

Both angles are reported in degrees: - HA: −90° to +90° - IA: −90° to +90°

Angles are defined in a left-handed cylindrical coordinate system aligned to the LV.