Computation of the alignment of myocardial contractile pathways using a magnetic tablet and an optical method

Abstract

The computation of the inclination angle of myocardial contractile pathways, based on the data from (1) optically and (2) manually digitized hearts is described.

The measured raw data comprised: (1) A list $\mathit{epi}$ of points on an ‘epicardial’ surface S. (2) For each selected contractile pathway f, a list of points along the contractile pathway.

For any point $\mathit{p}$ on a contractile pathway f, the angle of inclination ${\mathit{\alpha }}_{\mathit{p}}={\mathit{\alpha }}_{\mathit{p}}\mathrm{(}\mathit{p}\mathrm{,}\mathit{f}\mathrm{,}\mathit{S}\mathrm{)}$ is defined to be the angle (in degrees) between the tangent ${\mathrm{t}}_{\mathit{p}}={\mathrm{t}}_{\mathit{p}}\mathrm{(}\mathit{f}\mathrm{)}$ to the contractile pathway f at the point $\mathit{p}$ and the tangent plane ${\mathit{T}}_{{\mathit{v}}_{\mathit{p}}}$ to the surface S at the surface point ${\mathit{v}}_{\mathit{p}}=\mathit{v}\mathrm{(}\mathit{p}\mathrm{,}\mathit{S}\mathrm{)}$ which is nearest to $\mathit{p}$ . Thus ${\mathit{\alpha }}_{\mathit{p}}$ is a generalization of the imbrication angle of Streeter.

The angle of inclination was computed using two separate numerical methods: (1) A discrete method, applying finite differences to the raw data, to compute the tangents ${\mathrm{t}}_{\mathit{p}}$ and the tangent planes ${\mathit{T}}_{{\mathit{v}}_{\mathit{p}}}$ , after which the results were smoothed. (2) A smoothing method in which the data was first smoothed to obtain an approximation ${\mathit{S}}_{\mathit{epi}}$ to the epicardial surface and spline approximations to the contractual pathways f.

We describe the results for two typical hearts: a manually digitized dilated pig heart and an optically digitized constricted cow heart.

For each heart we first present the depths and angles of inclination of typical contractual pathways and then summarize the results in the form of histograms. The results are discussed in detail in the accompanying paper of Lunkenheimer, Redmann et al. [P.P. Lunkenheimer, K. Redmann, K.-H. Dietl, C. Cryer, K.-D. Richter, W.F. Whimster and P. Niederer, The heart’s fibre alignment assessed by comparing two digitizing systems. Methodological investigation into the inclination angle towards wall thickness, Technology and Health Care 5 (1997), 65–77], where the digitization methods are also described.