As is well known in embryology and pathology, the normal development of various anatomical structures and their response to pathological processes are associated with changes in shape. Evaluation of shapes is a routine part of ultrasound examinations and is used to determine whether a given structure appears 'normal' or 'abnormal'. In serial examinations, changes in shape over time are followed to assess the normalcy of development or the progression of a detected abnormality. Because these shape evaluations are usually subjective, impressions are recorded in statements such as 'The fetal heart appears normal' or 'The obstruction of the urinary tract appears more severe'. It's desired that a quantitative method to assess the shape of anatomical structures be developed and evaluated. This study presents a novel procedure for deriving a single number, the 'shape score,' from the surface point coordinates of standardized gestational sacs. This number, which characterizes the shape of the gestational sac quantitatively, is called the gestational sac shape score (GSSS).

Shape sample acquisition
The study population consisted of 30 pregnant women of mean maternal age 29.0±5.3 years. Ultrasound volume acquisitions were performed using 3D transabdominal sonography [1]. Multiplanar views of each gestational sac were obtained using a customized research version of four-dimensional (4D) View software, Version 5.0 (GE Healthcare). Electronic markers were first placed on the anterior and posterior sac walls. Gestational sac profiles were then generated and their contours manually traced using a graphics pen and tablet at 6-degree intervals until completion of a 180-degree rotational sweep. For each sac, a total of 30 sampling slices were generated with the 6-degree step. Each slice contains 122 surface points, which leads to 3660 surface points per sac. The surface point coordinates of the sac were exported to a text file for additional analysis. To determine the appropriate value for the sampling step, we compared the sampling efficiency of different sampling steps. The smaller sac samples were obtained by selecting every other slice (12-degree interval) and every fifth slice (30-degree interval). Figure 1 shows a gestational sac under different sampling resolution. All the sac samples of different sampling interval were subject to the same procedure of analysis as follows.

Figure 1 Gestational sac slice sampling. From left to right, slices of the gestational sac are generated at 6-degree, 12-degree and 30-degree intervals.

Surface interpolation
A cubic-spline interpolation procedure was used in this experiment to estimate the radial values at standard locations with 6-degree intervals for azimuth angle θ and 3-degree interval for polar angle φ [2]. Splines are piece-wise, defined smooth and continuous polynomials whose individual curves and their first and second derivatives meet at the endpoints of intervals. Cubic splines require the degree of polynomials to be three or less. This procedure was implemented using MATLAB function 'interp1.' The cubic spline interpolation procedure was used with the coordinate data obtained from the 12-degree and 30-degree sampling steps to evaluate how well these two samples could retain surface details. The results show that the accuracy of the spline interpolation method was reasonably good. So the procedure was used with all 12-degree and 30-degree samples to generate the coordinates of a standard set of surface points, which has a sampling density equivalent to the 6-degree sample.

Shape descriptor: from surface to volume, and to shape vector
To make valid shape comparisons, the effect of differences in location, orientation and size must be removed from the surface-point coordinate data. We first generated a triangular surface mesh from the standard surface point coordinate data, then applied a voxelization algorithm to the mesh to generate volumetric data, and computed the location, orientation and size information from the volumetric data [3]. The voxelization algorithm uses a uniform three-dimensional grid to sample the space defined by the contours. The resolution of the 3D grid was 0.4×0.4×0.4 mm (6.4×10^-5 mL) and the volume enclosed by this elementary component of the grid is referred to as a voxel. Voxels were assigned a value of 1 or 0 depending on whether they were inside the space defined by the contours (1) or outside this space (0). This is done by determining the vector between the center of the voxel and the center of the nearest surface triangle of the voxel. This vector was projected on to the triangle's normal unit vector that originates from the triangle's center and projects outward, by calculating the vector dot product. Vector dot products greater than zero indicate locations outside the sac while those less than zero indicate locations inside the sac. The volume of each sac was computed from this volumetric descriptor by counting the total number of 1's in the 3D grid and multiplying it by the voxel volume. We assumed the density of a sac is homogeneous so the center of gravity of a sac has coordinates that are the means of the three coordinates of all the voxels inside the sac. Using these data, the surface point coordinates of each sac were transformed such that the center of gravity of the sac is at the origin of the external coordinate system. Similarly, the inertial tensor of the sac, a matrix containing the moments and products of inertia associated with the three Cartesian axes, can be computed from the coordinates of all the voxels inside of the gestational sac. The angles between the principal axes of inertia and the axes of the external coordinate system were extracted. These data were used to transform the surface-point coordinates of each sac such that its principal axes of inertia are aligned with the axes of the external coordinate system. The volumes of these sacs were standardized to 1 mL by dividing all the surface coordinates by the cube root of the sac volume.

As indicated previously, standard voxels inside the sac are assigned a value of 1 and those outside the sac a value of 0. However, the number of standard voxels is so high that a lower resolution 3D grid must be used to generate a more reasonable volumetric shape descriptor. This was achieved by placing a 3×3×3cm (27 mL) box, centered at the origin, around each sac. The volume of this box is divided into 21×21×21 standard cubes (total number = 9261) having equal volumes of 0.0029 mL. The value assigned to each cube is determined by summing the number of voxels of value 1 inside the cube and dividing this sum by the total number of voxels within the cube.

The shape vector used in the principal components analysis (PCA) was obtained by reshaping the 21×21×21 3D matrix of cube scores into a one-dimensional (1D) vector of size 9261; i.e. the 3D cube scores were sequentially placed in a single row, the shape information being retained by the score's position in the row. The shape vectors of the 20 sacs were stored in a two-dimensional (2D) matrix of size 20×9261. If the 20 entries of a column vector in the 2D matrix were all zeros or ones, it indicated that the corresponding cubes were either outside or inside of all the sacs. Such columns can be removed from the 2D matrix because they do not provide shape variation information. Only the column vectors that are a mix of different values in the range of 0 to 1 need be retained for PCA. In these studies there were 810 such column vectors. To explore how the PCA results might be affected by different sampling resolution, we have generated the shape vectors of all 20 sacs from the data obtained in 30-slice, 15-slice and six-slice sampling procedures.

Derivation of shape score
Initially, PCA was carried out on the set of 20 shape vectors derived from the coordinate data obtained from the 30 slices made on each gestational sac [4]. Since there were 810 column vectors containing shape information, 810 principal components (PCs) were defined. The shape variance accounted for and the set of weighting factors were calculated for each PC from the data for each sac. As approximately 90% of the shape variance was accounted for by the first 10 PCs, subsequent analysis was limited to these PCs. Using the set of 810 weighting factors for each PC and the corresponding cube scores, PCi scores (PCiSs) were calculated for the first 10 PCs for each of the 20 sacs. A series of seven 'shape scores' (SSs) were calculated by adding weighted PCiSs serially (e.g. SS-1 = PC1S; SS-2 = w1PC1S + w2PC2S; ). In these linear combinations, each PCiS was weighted by the fraction of shape variance accounted for by the PC. Evaluation of the seven SSs was carried out with the following tests and the results obtained used to select the GSSS: 1) Assessment of changes in individual SSs as the number of included PCiSs increased; 2) Evaluation of the normalcy of the SS distributions; 3) Calculation of mean and standard deviation of SSs; 4) Comparison of individual SSs using the paired t-test; 5) Examination of SS histograms.

Results
As the 30-slice sets provide measurements of gestational sac surface-point coordinates at 6-degree intervals, they represent the most detailed characterization of sac shapes and utilize no interpolated surface points in the standard surface point sample. Therefore these data sets were taken as the best representatives of the gestational sac shapes. For the 30-slice sets, the first seven PCs accounted for 28.1%, 18.4%, 11.0%, 7.4%, 6.3%, 4.7% and 4.0% of the cube score variance/covariance (total for first seven PCs = 80.5%; total for first 10 PCs = 89.3%). As seen in Table 1, the mean percent change between SS-4 and SS-5 (6.3%) was considerably less than that between SS-3 and SS-4 (35.3%) and the SDs associated with these mean percent changes (40.0% vs. 101.8%) were quite different. The difference in percent change was not significant by t-test (P = 0.25), probably owing to the small sample size of 20 and high random variability. However, the mean SS-4 to SS-5 percent change was similar to those seen between SSs containing more PCiSs. These results suggest relative stability in individual SS values beginning with SS-4. SS-4 accounted for 67.6% of the shape variance/covariance. Based on these findings, subsequent analysis was limited to the first four SSs. The distributions of the first four SSs did not show significant deviations from a normal distribution and their means and standard deviations were essentially the same. The means did not differ from zero and the standard deviations were around one. Paired t-tests showed no significant differences in comparisons of individual SS values. Based on these findings and those presented in Table 1, SS-4 was chosen as the GSSS (GSSS-30).

GSSS-15 and GSSS-6 were derived using 15-slice and 6-slice data sets by similar procedure. To be consistent with the decisions made for the 30-slice data set, SS-4 was chosen as the GSSS for both the 15-slice and 6-slice samples (GSSS-15, GSSS-6). Linear regression analysis using GSSS-15 as the dependent variable and GSSS-30 as the independent variable gave an R² of 95.1% with an intercept of 0.00001 and a slope of 0.982. A similar analysis using GSSS-6 as the dependent variable gave an R² of 97.8% with an intercept of 0.0000 and a slope of 1.005. These results indicate no effect of slice sample size on the GSSS. Therefore the more practical six-slice sample can be used without significant loss of shape information in quantitative studies of gestational sac shape.

References

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