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Dr. Vittorio Dall'Aglio - Medico Chirurgo specialista in Cardiologia - Pordenone

Computerized two-dimensionalechocardiographic evaluation of left ventricular volumes by multiconic theory: theoretical approach and preliminary results


Divisione di Cardiologia - Ospedale Civile di Pordenone

Estratto dal Volume CURRENT CONCEPTS ON ULTRASOUND « II Giornate Italo-Jugoslave di Ultrasuoni»
Chieti, 1-3 May, 1980 Publ. « D. Guanella », Roma 1980


Ejection fraction has been shown to be a reliable index of left ventricular (LV) function in clinical setting. Usually ejection fraction (EF) is calculated from LV volumes obtained by cineangiography. More recently two dimensional echocardiography has been proposed as a non invasive method to evaluate LV volumes. Several mathematical approaches are reported in the literature. The results show large variability and great scattering in the individual data as compared with the cineangiography (1, 2, 3, 4, 5). The purpose of this paper is to present a new computerized two dimensional ecocardiographic approach based on multiconic surface theory for evaluating LV volumes.

Material and methods

Preliminary studies in evaluating LV volumes by cineangiography and two dimensional echocardiography (TDE) were obtained in 5 patients. LV angiograms were performed by standard techniques in RAO 30° projection. TDE were recorded within 48 hours from cardiac catheterization by using Toshiba SSH 10A wide angle phased array system with a 2.4 MHz transducer. The LV echocardiographic images were obtained in a modified apical view including mitral and aortic valvular planes. The echo data were stored in real time on a JVC-3800E video tape recorder with slow motion and stop frame capability, by using Philips video camera. Data were then transferred to a HP-21MX mini-computer with the usual peripherical equipments.

Computational Aspects

The boundary of LV on a two dimensional echocardiogram can be considered to be a closed two dimensional curve. The only information on the LV solid is the intersection of its surface with a plane. Intersection is assumed to be a simple closed curve. The curve is not completely specified; only a finite sequence of points can be given. A digital device is used to digitally measure the position of the stylus on the surface of the digitizing tablet. The sequence of points is used to construct a piecewise linear approximation (6) of the original silhouette. A finite sequence of weighted points is obtained (7). When the boundary is reconstructed the area of ventricular chamber for a continuous boundary which encloses the area A of the ventricular region R can be computed:

With the trapezoidal rule the LV area is calculated. Subsequently by a worst case project the computer builds-up a solid from the planar figure using minor semiaxis of each LV boundary point (obtained by polar coordinates), to trace multiple sequential circumference (Fig. 1 top). Each single LV point on the reconstructed silhouette is the extreme of one semiaxis or of one circumferential radius. Each circumference is then used as a base or as a top of a truncated cone. For each truncated cone (Fig. 1 top) two points of the LV boundary are identified. The curve joining these points can be regarded as a control curve (planar cubic splines) (8).

Convex cubic spline

Consider a cubic defined by P2 (x2, y2), m2 and P1 (x1, y1), mi where m2 and m1 are associated finite slopes (Fig. 2, top), the equation (8) of the cubic segment is:


the value of u for the inflection point are:

Since the cubic dealt with is completely defined (9) by to points and their associated slopes, a set of data P1, m1, P2, m2 may also be called a cubic or cubic segment. For infinite slope case the goal here is to find a spline segment on P1, P2 passing through P2, P0, P1 with their m2, m1 slopes and for Po with infinite slope. Bernstein-Bazier cubics (8) can be used to solve the problem (Fig. 2, bottom).

Multiconic Surface

For adiacent pair of the spline segments a surface called polyconic can be generated. The term "multiconic" is meant to imply that the control curves used are more general than the pseudo polynomials which define policonic surfaces. Multiconic surfaces
are finite connected surfaces for which a family of parallel planes intersects the surface in curves that are conic sections (9). The conic (Fig. 2 top) passes through the points P1, P2 and is respectively tangent to the lines P2, Px and P1, Px. Each point of the set corresponds to a value con X’ axis (Fig. 2 top). Such surfaces (9) are usually defined as space curves in terms of their Y’ and Z’ coordinates:

The functions Y’ (x) and Z’ (x) are called control curves. From this information a single conic can be specified by providing a number Q which is the shape factor (9).

Area and Volume

The volume contained in a region bounded by a multiconic surface defined by a set of triangular bases (Fig. 1 and Fig. 2) is obtained for the domain (x2, x1). The procedure requires the sum of infinitesimals A (x) dx where A (x) is the area of conic section at x contained in the bounded region contained in the triangle (P1, Pi, P2) (Fig. 2 top). In order to compute that area it is convenient to translate the origin to P2 and rotate the coordinates so that P1 lies on the new y-axis. The surface area of a multiconic can be written as:

derivatives yx, zx are, computed for s constant and ys and zs are computed for x costant. If variable s is changed:
s=(u)2/(u2+(1-u)2 it can be obtained the A (•):

The double integral can be evaluated numerically except for the case where control curves have an infinite slope. In this case further analysis are required. In this way the LV volumes, surface shape and the changes of the surface can be estimated.

Results and Discussion

Results are reported in table 1. Heart rate "during LV angiography and TDE "were similar. End diastolic volumes ranged from 67.81 to 93.70 mis, arid systolic volumes ranged from 26.02 to 52.33 mis, by angiography. Patient number 5, who presented the largest LV volumes, had coronary artery disease with LV saccular aneurism. The EF was normal in all but in patient N° 5. Volumes and EF obtained by TDE compared patient by patient very closely with the data obtained by angiography. The coefficient of correlation between all volumes obtained by angiography and correspondent volumes obtained by TDE was r = 0.9954. When comparing only diastolic volumes coefficient of correlation was even better (r = 0.9962). For systolic volumes the coefficient of correlation was sligthly less (r == 0.9599).

HRC= heart rate during cineangiography; HRB = heart rate during two-dimensional study; DVC == diastolic volume by cineangiography; DVB = diastolic volume by echocardiography; SVC = systolic volume by cine; SVB = systolic volume by echo; EFC = ejection fraction by cine; EFB = ejection fraction by echo.

Coefficient correlation for EF was also very good (r = 0.9697). The standard error of estimate was 1.23 mls, with a t = 3.19 (Pr < 0.01). This indicates that for each single volume from 26.02 to 93.70 mls the estimation by TDE was very close to cineangiography and very satisfactory. Validation of volumes calculated by multiconic theory from LV cineangiograms was previously reported (6). The validation was obtained by comparing LV volumes evaluated by angiography and by fluid displacement in human casts (r = 0.999, SEE = 0.22 mis) (6, 7). The excellent correlation between volumes obtained by the reported method (6, 7) and volumes calculated by using the same theory in TDE emphasizes the possibility of studying volumes and EF by ultrasound two dimensional system. In previous studies (1, 2, 3, 4, 5) comparison between LV volumes and EF obtained by angiography and TDE have been made. Coefficient of correlation ranged from r =0.55, (5) to r = 0.93, (1). Better results were abtained in isolated canine hearts studies (10) examining cross sectional images taken at 3 mm intervals along the vertical axis (r = 0.972). A difference between the clinical data and the results obtained by the present method can be noted. The difference is most probably related to the mathematical approach. Even though the small number of cases considered in this study suggests caution, the method seems to apply accurately to different volumes. In conclusion the multiconic surface theory seems to be very useful in evaluating LV volumes by TDE with excellent correlation with LV angiography. Further studies on larger series of patients are needed to confirm these preliminary results.


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7) ZANUTTINI D., TARGA S., NICOLOSI G.L.: Evaluation of left ventricular function: a new semiautomated technique. Abstracts-1 of the VIII World Congress of Cardiology, 17th-23rd September 1978, Tokyo, pag. 392 n. 1201.
8) WANG A.P-: On conic surface. Report n. 320-2667. IBM Scientific Center, Los Angeles, California, 1974.
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10) EATON L.W., MAUGHAN W.L., SHOUKAS A.A., WEISS J.L.: Accurate volume determination in the isolated ejecting canine left ventricle by two-dimensional echocardiography. Circulation 60: 320-326, 1979.


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