Quantitation of organ symmetry by diagnostic ultrasonic imaging: a group theoretical approach

A method based on group theory is proposed to quantify the shape of an organ. It is proposed that a complex shape is characterized by a basic set of symmetry operations involving rotation about the center axis. Based on this concept, partial symmetry elements (PSE) were determined by stepwise rotation of the images and measurement of overlap area between the original and rotated images. These measurements were used to define a symmetry index (SI) and to "decompose" the complex shape into simpler shapes with regular geometry such as a circle, an ellipse, an equilateral triangle, a square, etc. Computer simulations and two-dimensional (2-D) ultrasonic images of the prostate were used to demonstrate the feasibility of the method. The method was also applied to a series of clinical ultrasonic cardiac images to determine the dynamic shape changes in left ventricle (LV) of a normal heart and an heart with infarction. The results of this study show that SI increased with an increase in the symmetry of the simulation models and the prostate. Cross-sectional images of prostate with triangular and ellipsoidal shapes exhibited larger proportion of three-fold and two-fold symmetries, respectively. Symmetry index of the LV changed during the cardiac cycle and exhibited a loop-like structure when plotted as a function of size. Quantitative measurements of SI indicated that the left ventricular chamber is rounder during the diastolic phase of the cardiac cycle. The loop-like structure was considerably reduced in the heart with infarction. Also, larger distortions in cardiac shape were observed over cardiac cycle in the heart with infarction, as compared to the normal heart. In conclusion, it is feasible to quantitatively characterize 2-D shapes of the organs by symmetry operations based on group theoretical analysis.