Beyond Simpson's Rule: Accounting for Orientation and Ellipticity Assumptions

Abstract

Simpson's biplane rule (SBR) is considered the gold standard method for left ventricle (LV) volume quantification from echocardiography but relies on a summation-of-disks approach that makes assumptions about LV orientation and cross-sectional shape. We aim to identify key limiting factors in SBR and to develop a new robust standard for volume quantification. Three methods for computing LV volume were studied: (i) SBR, (ii) addition of a truncated basal cone (TBC) to SBR and (iii) a novel method of basal-oriented disks (BODs). Three retrospective cohorts representative of the young, adult healthy and heart failure populations were used to study the impact of anatomical variations in volume computations. Results reveal how basal slanting can cause over- and underestimation of volume, with errors by SBR and TBC >10 mL for slanting angles >6°. Only the BOD method correctly accounted for basal slanting, reducing relative volume errors by SBR from -2.23 ± 2.21% to -0.70 ± 1.91% in the adult population and similar qualitative performance in the other two cohorts. In conclusion, the summation of basal oriented disks, a novel interpretation of SBR, is a more accurate and precise method for estimating LV volume.

Keywords: Apical chamber views; Left ventricle volumes; Modified Simpson's biplane rule; Two-dimensional echocardiography.

Fig. 1

Definition of Simpson's biplane rule (SBR) and parametrization of the acquisition and anatomical assumptions that lead to errors in volume computations. (a) Schematic of SBR, the summation-of-disks method from two longitudinal planes. (b) Foreshortening that leads to volume underestimation (red plane = foreshortened view, black dot = apex). (c) Basal slanting that is parametrized with angle $b_{\theta }$ and leads to volume underestimation. (d) Planes not capturing the axes of the elliptical cross-sections, parametrized with the orientation angle A_m, leading to a varying range of volume over- and underestimations (green line = A4C, blue line = A2C, pink line = direction of major axis). (e) Varying viewing angles $v_{\theta }$ gradually from A3D (${90}^{\circ }$) to A2C (${60}^{\circ }$). (f) Varying eccentricities $\mu$ where a value of 1 corresponds to a perfect circle. LV = left ventricle; RV = right ventricle.

Fig. 2

(a) Schematic of the Simpson disk stacking for truncated basal cylinder (TBC), Simpson's bare rule (SBR) and basal-oriented-disks (BOD) methods. (b) Overestimation with TBC caused by choosing the largest $L$ (blue dotted line) rather than the correct length $L$ is represented by the red dotted line. (c) Variable errors with TBC depending on the curvature of the LV wall, where the truncated basal cylinder to approximate the basal region leads to either an overestimation (left) or underestimation (right). (d) SBR underestimation caused by poor fitting of cylindrical disks around the slanted base. The dotted lines represent the top of the basal plane of each view—see Figure S3B (online only) for an illustration in a real left ventricle anatomy.

Fig. 3

Performance of the three formulations (simple bare rule [SBR], truncated basal cylinder [TBC] and BOD: basal oriented disks [BODs]) on the synthetic data set. (a) Illustrative example of the disks formed by each formulation on a left ventricle represented by an ellipsoid with a large basal slanting angle $(b_{\theta }={20}^{\circ })$. (b) Error in volume estimation by the three formulations with varying basal slanting angles $(b_{\theta }=\left\{0^{\circ },{20}^{\circ }\right\})$ in two orientation angles ($A_m={90}^{\circ }\text{and}{0}^{\circ }$, top and bottom rows) and in three viewing angles ($v_{\theta }={90}^{\circ },{75}^{\circ },{60}^{\circ }$, in columns) and eccentricity $\mu =1.1$. (c) Relative error in volume computation with varying viewing ($v_{\theta })$ and orientation ($A_m)$ angles, with fixed eccentricity µ = 1.1 and basal slanting $b_{\theta }=5^{\circ }$ for both views. Viewing angle of $v_{\theta }={60}^{\circ }$ is representative of the angle between 4ch and 2ch views (highlighted by blue rectangles in panel C), and $v_{\theta }={90}^{\circ }$ is representative of the angle between 4ch and A3C views (highlighted by green rectangles in panel C).

Fig. 4

Percentage error in the estimation of left ventricle volume by the three methods (BOD, TBC, SBR) and in the viewing angles corresponding to the secondary view chosen as the A3C (V90) and A2c (V60), across the three cohorts (UKBB, YHC and HFC). BOD = basal oriented disk; HFC = heart failure cohort; SBR = Simpson's biplane rule; TBC = truncated basal cone; UKBB = UKBiobank (Ruijsink et al. 2020) (UKBB) database; YHC = young healthy cohort.