Coronary Arteries Simplified with 3D Cylinders to Assess True Bifurcation Angles in Atherosclerotic Patients

Abstract

The geometry of coronary arteries affects regional atherogenic processes. Accurate images can be assessed using multislice computer tomography (MSCT) to estimate bifurcations angles. We propose a three-dimensional (3D) method to measure true bifurcation angles of coronary arteries and to determine possible correlations between plaque presence and angulations. The left main (LM) coronary artery, left anterior descendent (LAD) and left circumflex artery (LCX) were imaged in 40 atherosclerotic and 35 healthy patients, using 64-rows MSCT. This Y-junction was simplified fitting a 3D cylinder to each vessel to estimate true bifurcation angles and diameters. The method was tested in phantoms and interobserver variability was assessed. Geometrical results were compared between groups using an unpaired t-test. The cylinders fitted reasonably well with mean distances to measured points below 0.4 mm. LAD–LCX bifurcation angles were wider in the atherosclerotic group (p < 0.01). LAD (p < 0.01) and LCX (p < 0.05) diameters were also larger. In phantoms mean absolute difference between true and estimated angles (N = 27) was 0.44 ± 0.54°. Interobserver mean difference (N = 135) was 1.8 ± 5.8°. Simplifying coronary bifurcation with cylinders results in a reliable technique to assess coronary artery geometry in 3D, avoiding planar projections and decreasing interobserver variability. Geometrical risk factors should be incorporated to properly predict atherosclerosis processes.

Appendix

The problem of fitting a cylinder involves fitting a centerline (CL) and the radius (R) to the set of candidate points for each segment. A line in 3D can be defined as

$$ CL(\lambda ) = \lambda .\vec{v} + P = \lambda .\left( {v_{x} ,v_{y} ,v_{z} } \right) + \left( {P_{x} ,P_{y} ,P_{z} } \right) $$

(1)

where $\vec{v}$ is the direction vector, P is some point on the line and $\lambda$ is an arbitrary real scalar. The orthogonal projection (OP) of each candidate point (CP) to the unknown centerline can be calculated with a scalar product as (See Fig. 5):

$$ \left( {CP - OP} \right) \cdot \left( {v_{x} ,v_{y} ,v_{z} } \right) = 0 $$

(2)

Using (1), OP belongs to the centerline, thus replacing in (2)

$$ \left( {CP - \left( {\lambda .\left( {v_{x} ,v_{y} ,v_{z} } \right) + \left( {P_{x} ,P_{y} ,P_{z} } \right)} \right)} \right) \cdot \left( {v_{x} ,v_{y} ,v_{z} } \right) = 0 $$

and λ value for can be isolated:

$$ \lambda = {\frac{{\left( {CP - P} \right) \cdot \vec{v}}}{{\left\| {\vec{v}} \right\|^{2} }}} $$

The distance d from CP to OP can be calculated as

$$ \begin{gathered} d^{2} = \left\| {CP - OP} \right\|^{2} = \left( {\left( {CP_{x} - P_{x} } \right) - \lambda .v_{x} } \right)^{2} + \hfill \\ \, \left( {\left( {CP_{y} - P_{y} } \right) - \lambda .v_{y} } \right)^{2} + \left( {\left( {CP_{z} - P_{z} } \right) - \lambda .v_{z} } \right)^{2} \hfill \\ \end{gathered} $$

Fig. 5

Schematic representation of a centerline where $\vec{v}$ is the direction vector, P is some point on the line, CP is a candidate point and OP is its orthogonal projection on the line

To reduce the number of free variables, we imposed the arbitrary point P on centerlines to reside in the $z = 0$ plane: $P = \left( {P_{x} ,P_{y} ,0} \right)$ and the z component of the direction vector to be 1: $\vec{v} = \left( {v_{x} ,v_{y} ,1} \right).$ Therefore, two redundant parameters were eliminated. Finally, a sum function of distances to the hypothetical cylinder of radius R was constructed and minimized using the simplex method developed by Nead and Melder (Lagarias et al. 1998) that can be found in MATLAB^® fminsearch function. A root-mean square error (RMS) of approximating a cylinder to a vessel was defined as

$$ RMSerror = \left( {{\frac{{\sum\limits_{1}^{N} {\left( {CP - CYLp} \right)^{2} } }}{N}}} \right)^{{{\raise0.7ex\hbox{$1$} \!\mathord{\left/ {\vphantom {1 2}}\right.\kern-\nulldelimiterspace} \!\lower0.7ex\hbox{$2$}}}} $$

where N is the number of candidate points and CYL _p is the closest point in the cylinder from CP, orthogonal to its centerline. After cylinders determination (with direction vectors $\vec{v}_{1} ,\vec{v}_{2} ,$) the true angle in 3D between a pair of centerlines ($\theta$) was analytically calculated using the following formula:

$$ \cos \left( \theta \right) = {\frac{{\vec{v}_{1} \cdot \vec{v}_{2} }}{{\left\| {\vec{v}_{1} } \right\| \cdot \left\| {\vec{v}_{2} } \right\|}}} $$