Introduction

Radiomics is the process of extracting high-throughput quantitative data from medical images to contribute to current paradigms in disease diagnosis, staging, management and prognostication1,2,3. In recent years, there has been a rapid increase in publications on radiomics, but their routine utilization in the clinics is still to be achieved.

In complement to textural features, shape features are often extracted in radiomic analysis to describe tumor aggressiveness. Using CT (Computed Tomography) or MR (Magnetic Resonance) images, some tumors are described as spiculated or having ‘ill-defined borders’, which indicates potential to spread to contiguous structures and association with advanced stages4. On the contrary, less aggressive and benign tumors frequently have well-defined margins5. Shape-based features have been extracted in a number of studies, but not all have retained them in their final radiomic signature. Compactness index has been found to be helpful in differentiating benign from malignant lung nodules6,7, to aid in nodule segmentation8, and to be associated with distant metastases. In head and neck cancers9, the same index has been likewise shown to predict survival2 and HPV (Human Papillomavirus) status10. Spherical disproportion has been associated with prediction of malignant lung nodules6, distant metastases in lung cancers9, and HPV status in head and neck cancers10. Sphericity was also linked to increased micropapillary component, which portends poorer prognosis, in lung adenocarcinoma11. Table 1 summarizes the shape features found significant in publications.

Table 1 Shape features found significant in publications.
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Radiomic features often suffer from being highly correlated, either with tumor volume or with each other, making some of them redundant12,13. Too many features, compared to sample size, result in high false discovery rates, over-fitting, and decreased generalizability1,14,15. A recent study has shown that certain radiomic features extracted from CT scans, including a shape-based one (compactness), have a high dependency on tumor volume, with Spearman rank correlation coefficients ranging from 0.71 to 0.9816.

Moreover, with the field still developing, standards with regards to feature extraction or selection are few and not universally accepted. Because of this, feature nomenclature is not homogenous among publications. A specific example is compactness, which describes how much the shape of a tumor resembles that of a sphere, which has at least three different formulas in the literature8,17. Published studies likewise suffer from lack of standardization, making reproducibility of the results a challenge. To tackle these issues, a list of formulas for radiomic feature calculation was proposed by the Image Biomarker Standardization Initiative18.

Although different types of medical images such as MRI19 or Positron-Emission Tomography (PET) scan20 can be analyzed for shape characterization, the use of CT imaging has dramatically increased in the past years21,22 leading to numerous shape-based studies using this modality. However, datasets are usually retrospective with a wide range of imaging equipment, acquisition techniques and reconstruction parameters used23. Several publications have focused on the influence of these technical aspects on radiomic features (Table 2). A phantom study showed the significant impact of slice thickness on textural features24, although shape features generally exhibit more stability13. In lung nodules, feature extraction from seven different centers showed that 68% of shape features are robust to segmentation, with concordance correlation coefficients (CCC) > 75%25. Another multicentric study showed that shape features were repeatable on test-retest CTs, with standard deviations of 3 to 11%26. By adding uncorrelated noise to original images, shape descriptors were however shown to vary more importantly in CT than in PET imaging, with values of 13% and 4%, respectively27.

Table 2 Radiomic articles on methodology, detailing effects of different acquisition and reconstruction parameters on shape features.
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Thus, the aims of this study were (i) to create mathematical models of tumors with increasing degrees of spiculatedness/complexity, (ii) to extract radiomic shape features and determine their relationship with spiculatedness, and (iii) to evaluate the impact of slice thickness, resampling, algorithm used for surface and volume calculations, and change in volume on these features. The ultimate goal is to identify shape features that are least affected by technical parameters such as slice thickness, resampling, and volume, and thus may be prioritized in future radiomic studies.

Results

The validity of the use of spherical harmonics to model the tumor complexity was verified by asking five different operators to independently classify the forms in order of spiculatedness based on their visual assessment. They were able to classify the tumors correctly, with a maximum error of five forms for one participant, which were consecutive models (d = 47 and 50, 86, 89 and 92).

Comparison of features computed directly from stl models versus after 3d printing, acquisition, segmentation and meshing

Compared to the 2 mm, the 0.6 mm slice thickness results to increased differences for the feature Volume for all four meshing methods, as seen in Fig. 1. For the 2 mm slice thickness, the differences between the STL derived and the post-processing shape features are lower for the M3 meshing method with mean differences equal to 4.8% (1.0–14.8%, Surface-to-volume), 4.4% (0.7–14.0%, Compactness1), 10.8% (1.5–36.5%, Compactness2), 1.8% (0.3–6.2%, Compactness3), 3.6% (0.5–12.3%, Spherical Disproportion), 3.6% (0.5–12.3%, Sphericity). Standard deviations are slightly decreased for the M1 meshing method compared to M2 and M3 for all features, with values ranging from 1.2 (Compactness3) to 7.1 (Compactness2) and 1.6 (Compactness3) to 9.5 (Compactness2), for the M1 and M3 methods, respectively, when the slice thickness is equal to 2 mm. Standard deviations are lower for the 2 mm slice thickness compared to the 0.6 mm slice thickness, with values ranging from 2.8 (Compactness3) to 16.2 (Compactness2) and 1.6 (Compactness3) to 9.5 (Compactness2), for the 0.6 mm and 2 mm slice thicknesses of the M3 method, respectively.

Figure 1
figure 1

Relative differences between reference shape feature values computed from STL format compared with shape features evaluated after the whole radiomics process including 3D printing, acquisition, image segmentation, and meshing. M1, M2, M3 and M4 meshing methods as well as two slice thicknesses are illustrated here for comparison.

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Impact of slice thickness on feature values

The nine shape features were first extracted from native binary masks obtained from 2 mm and 0.6 mm slice thickness CT images without any pixel size modification. Change of the slice thickness during the acquisition process leads to mean differences equal to 12.3% (Surface-to-volume, 6.2–26.6%), 13.4% (Compactness1, 5.6–29.1%), 21.4% (Compactness2, 13.1–44.9%), 3.6% (Compactness3, 2.2–7.6%), 7.2% (Spherical Disproportion, 4.4–15.2%), 7.2% (Sphericity, 4.4–15.2%) and 2.6% (0.1–7.5%, Fractional Concavity) for the M1 meshing method. Values were equal to 9.3% (Surface-to-volume, 2.8–23.3%), 11.3% (Compactness1, 3.2–26.9%), 12.7% (Compactness2, 4.1–35.2%), 2.1% (Compactness3, 0.7–5.9%), 4.2% (Spherical Disproportion, 1.3–11.8%), 4.2% (Sphericity, 1.3–11.8%) and 3.0% (0.1–6.1%, Fractional Concavity) for the M3 meshing method.

Impact of the meshing algorithm on feature values

Pixel dimensions varied from 0.643 mm to 0.835 mm (X and Y directions) and from 0.565 to 0.733 mm (Z direction) after volume equalization for the masks extracted from the 0.6 mm slice thickness images. These values ranged between 0.639 mm and 0.840 mm (X and Y directions) and between 1.870 to 2.459 mm (Z direction) for the masks extracted from the 2 mm slice thickness images. Figure 2 shows the meshes obtained from the 0.6 mm slice thickness acquisitions for a representative model d = 47 for the four algorithms. Figure 3 illustrates the impact of the meshing algorithm and slice thickness on feature values for all the 28 tumor models after volume equalization. Outliers are observed using the Boundary M4 method. Differences ranged from 4.1% (Compactness3, d = 77) to 43.2% (Compactness2, d = 11) between M1 and M2 methods. Surface area, Surface-to-volume, and Spherical Disproportion had direct relationships with spiculatedness (increasing value with increasing tumor spiculatedness). The three formulas for Compactness, Sphericity and Fractional Concavity had inverse relationships. All features exhibit large variations between d = 11 to d = 41. Compactness2 is able to highlight shape differences even for the least spiculated models. For this feature, no slope breaking is observed until d = 65 irrespective of the slice thickness and meshing method. Subsequent analyses were performed on the 2 mm slice thickness acquisitions, as this is more coherent with what is used in the clinics.

Figure 2
figure 2

Representation of the meshes obtained for the d = 47 tumor model using the M1, M2, M3 and M4 meshing algorithms. CT-images acquired with a 0.6 mm slice thickness were used to extract the binary masks.

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Figure 3
figure 3

Variation of radiomic shape features as a function of slice thickness, tumor spiculatedness and meshing algorithm used for surface and volume calculation.

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Correlations between tumor complexity and features

Spearman’s rho correlation coefficients between each shape feature and tumor complexity were computed for the M1, M2 and M3 meshing algorithms (Table 3). Seven of the eight features had strong significant correlations with tumor model complexity irrespective of the meshing algorithm (r > 0.98, p < 10-4). Fractional Concavity showed the lowest correlation coefficient (r = 0.83, p < 10-4, M2).

Table 3 Spearman’s correlation coefficients evaluating the relationship of each shape feature with tumor complexity.
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Correlations between shape features

Correlations among the eight shape features, with the exception of volume that was fixed at a constant value, were calculated using Spearman’s rho coefficients for the M1, M2 and M3 meshing algorithms. Almost all of the features were highly correlated with each other with r = 1, as seen in the correlation matrix plots (Supplementary Figure S1). Only Fractional Concavity was slightly less correlated with the others, with r values from 0.85 to 0.99, with M2 having the lowest correlation.

Effect of grid resampling

All of the feature values changed when resampling on a 1 × 1 × 1 mm3 grid was performed on native masks deduced from the 2 mm thickness original CT images (M1 and M3 meshing methods, no volume equalization). Absolute mean percent changes from the 28 tumor models were equal to 7.8% (5.5–15.7%) for Volume, 7.8% for Surface Area (7.1–10.7%), 1.4% for Surface-to-volume (0.0–5.0%), 2.7% for Compactness1 (0.7–8.6%), 7.5% for Compactness2 (0.6–10.8%), 1.3% for Compactness3 (0.1–1.8%), 2.6% for Spherical Disproportion (0.2–3.6%), 2.6% for Sphericity (0.2–3.6%) and 14.7% for Fractional Concavity (13.8–17.4%) for M1 (Fig. 4) and 7.9% (5.7–16.0%) for Volume, 2.6% for Surface Area (2.0–4.9%), 5.3% for Surface-to-volume (3.2–11.2%), 6.2% for Compactness1 (4.1–12.8%), 8.0% for Compactness2 (4.1–17.4%), 1.3% for Compactness3 (0.7–2.9%), 2.6% for Spherical Disproportion (1.4–5.8%), 2.6% for Sphericity (1.3–5.8%) and 9.5% for Fractional Concavity (8.8–11.6%) for M3 (Figure 4). The ranking of the models was not influenced by the resampling.

Figure 4
figure 4

Comparison of feature values extracted from 2 mm thickness original images for each model without and with resampling. Resampling was performed on a 1 × 1 × 1 mm3 grid. M3 method was used for feature extraction.

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Impact of change in volume on shape features

The absolute percent changes of each feature with regards to the change in volume from 25% to 150% were compared to the percent change observed when modifying the spiculatedness of the model from d = 11 to d = 92 for M3 (Table 4). Compactness2, Compactness3, Spherical Disproportion, Sphericity and Fractional Concavity were less affected by volume changes. Surface-to-volume and Compactness1 were more affected by change in volume than tumor complexity, with the Surface-to-volume feature having a 69.9% change from the least to most spiculated models versus 54.2% change with volume; and Compactness1 having a 47.5% change with spiculatedness compared to 90.4% change with volume. Figure 5 illustrates the change in feature values with changes in volume for the three representative phantoms d = 11, 47, 92.

Table 4 Absolute percent changes in shape feature values between the most (d = 11) and least spiculated models (d = 92, first column), and with change in volume of 25% to 150% for the 3 representative models (d = 11, 47, 92) obtained for the M3 meshing algorithm.
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Figure 5
figure 5

Graphs depicting the change in feature value for changes in volume of 25, 50, 75, 100, 125, and 150% for the three representative features d = 11, 47, 92 (M3).

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Figure 6 summarizes the effect of the technical parameters slice thickness, resampling, and change in volume on the radiomic shape features, reiterating that Compactness1 has important variations with changes both in slice thickness and in volume, and Surface-to-volume with change in volume.

Figure 6
figure 6

Summary of the effect of technical parameters on the radiomic shape features (M3 meshing method). Effects of the different parameters were compared to the ability of each feature to distinguish change in spiculatedness. Green cases correspond to a ratio of less than 5% between the effect of the technical parameter to the percent change observed when modifying the spiculatedness of the model from d = 11 to d = 92. Orange cases correspond to ratios ranging from 10 to 20% and red cases to ratios superior to 20%.

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Comparison with actual tumors

To confirm the clinical relevance and applicability of our method, three representative patients from the RIDER database were contoured, and radiomic shape features were extracted thereafter. The range of values for each of the features among the three patients are within the range of the values from the shape phantoms, even if the volume after equalization was greater in the phantoms (range: 28257–28265 mm3). Patient volumes ranged from 5890 mm3 to 21045 mm3. The results for the three patients as well as the mean values for the phantoms are shown in the Supplementary Table S2.

Discussion

This is the first methodological study that directly demonstrates the relationship of tumor spiculatedness with radiomic shape features. Models with increasing degrees of spiculatedness were created to examine the behavior of quantitative shape features with known incremental degrees of tumor border complexity. It was seen that specific features increase monotonically with increasing tumor spiculatedness, in particular Surface Area, Surface-to-volume, and Spherical Disproportion. Conversely, certain features exhibit a monotone decreasing correlation with increasing spiculatedness (Compactness, Sphericity, Fractional Concavity). Quantitative extracted shape features have already been demonstrated to give insights on tumor behavior, underlying their importance in radiomic analysis. Based on CT scans, several publications have shown that shape features differentiate between benign and malignant nodules6,7,8 as well as correlate with patient outcomes2,9,28. In addition, a radiomic study of pre-treatment contrast-enhanced T1 MRI images in glioblastoma showed that tumor surface regularity was a powerful predictor of survival in the discovery (p = 0.005, hazard ratio [HR] = 1.61) and validation groups (p = 0.05, HR = 1.84)29.

It can be seen that many of the features exhibit strong correlations with each other, either positive or negative. If the behaviors of certain features are known to depend on specific parameters, calculating all may not necessarily give complementary information but instead redundant ones. In particular, Surface-to-volume, Compactness, Spherical Disproportion and Sphericity are all calculated from tumor volume and surface area17, which explains the strong relationships among them. In this regard, features might eventually be grouped into clusters instead of being analyzed individually12. For instance, different formulas for compactness have been previously published and used in radiomic studies8,17. In clinical studies, it needs to be determined whether correlations seen with radiomic shape features are inherent, or if tumor volume is a confounding factor. In our study, it is seen that Surface-to-volume and Compactness1 are affected with volume changes, and should thus be used with caution when comparing tumors with differing volumes. Indeed, even if it has been widely used in previous publications, the result obtained from the formula for Compactness1 is not dimensionless, and thus is not ideal in feature analysis. Compactness2, Compactness3, Spherical Disproportion, and Sphericity’s percent changes between d = 11 and d = 92 were noticeably higher than the percent change with volume variations, which may make these features more useful in analyses of patient tumors as they are not volume dependent. In addition, it was also seen that in general, the features highlight differences in complexity better in more spiculated tumors. For instance, in this study, the slope of the relationship between feature and spiculatedness was steep until d = 41, and thereafter relatively flattened out for the less spiculated models.

In this study, four meshing algorithms have been used and the differences between the features computed directly from the STL files and those computed after meshing have been compared (Fig. 1). Values extracted using M3 meshing method are the closest to the STL reference ones, with mean differences lower than 10.8% (Compactness2) for all features. The decrease of the standard deviation when comparing the values from STL files versus from M1 and M3 with 0.6 or 2 mm slice thickness (Fig. 1) validates the fact that a 2 mm slice thickness should be preferred for shape-based radiomic analysis. With a 2 mm slice thickness, shape features are impacted in a more homogeneous way by the entire radiomics process. The associations of the deduced shape features with change in tumor complexity have also been analyzed for each method (Fig. 3). Using the MATLAB Boundary function, a non-monotone behavior of the features with phantom spiculatedness was observed, with the presence of outliers. The use of this function is thus not recommended in in-house MATLAB-based softwares. Comparison of the M1, M2 and M3 methods shows that different meshing implementations can lead to different quantitative values. As a consequence, thresholds determined in the literature should be used with caution. Numerical phantoms such as the ones developed in this study can be also of major interest for the evaluation of the pertinence of meshing algorithms as well as for the development of new shape features. Notable is that in this study, we chose to use meshing for volume extraction, which is not performed in most of the radiomic software that typically multiplies the voxel size by the number of voxels in the volume of interest30. This choice is of importance for maintaining consistency between surface and volume quantities.

Comparisons of feature values between scans acquired with 0.6 mm versus 2 mm slice thickness reveal that this parameter affects all radiomic shape features, with changes of up to 12.7% (Compactness1, M3). Resampling the CT images on a 1 × 1 × 1 mm3 grid likewise resulted in small differences of between 1.3 to 9.6% changes in extracted features for the isosurface remesher M3 method. In a phantom study that computed the differences between original features and those resampled on a 1 × 1 × 2 mm3 grid with original pixel sizes ranging from 0.39 to 0.98 mm, shape features belonged to the group that were generally not significantly affected by resampling31. In this study, the Credence Cartridge Radiomics (CCR) phantoms used were rectangular in shape and created primarily for texture analysis, whereas ours had fine spiculations specifically created for shape analysis. Another study using the same phantoms showed that radiomic features were affected by slice thickness, but that this effect could be reduced by resampling the images before feature extraction. However, this study focused on 114 first order and textural features and did not include shape32. In yet another phantom study using spherical, elliptical, lobulated and spiculated forms, it was shown that shape features were significantly different between 1.25 and 5 mm slice thickness scans24, from which we can infer that voxel size affects results of feature extraction. At present, we therefore recommend not to constitute a cohort with images having too different slice thicknesses particularly if the Compactness1, Surface-To-Volume and Fractional Concavity indices are computed, given their dependence on slice thickness. However, the ideal is prospective studies with homogenous acquisition parameters, as resampling alone does not completely eliminate bias resulting from differences in acquisition such as slice thickness.

There are disadvantages to this study. First, only 3D features were calculated as the tumor models were contoured on axial CT slices and had discontinuous islets on some slices (usually at the top and edges of the tumor) because of the spiculations. In addition, the tumor phantoms were printed with a flat base, instead of a spherical-based shape with no flat edges due to technical considerations for 3D printing. However, all the phantoms were created in the same manner (with a flat base) such that all shape feature variations are expected to be comparable. Another limitation is that although the shape phantoms have increasing degrees of complexity, the variations of these are all based on the formula of spherical harmonics and thus have a consistent mathematical progression. Actual tumors are rarely symmetric and regularly shaped. However, theoretical knowledge of how radiomic shape features vary remains of value in deducing the complexity of actual tumors. Also, in studying variations with volume, the volumes were modified mathematically by recomputing the pixel sizes, which are inherently correct; but another way would have been to do a 3D reprint of each model with each corresponding volume change. Another limitation in the conduct of radiomics studies in general is that there is no generally accepted and universally utilized meshing method, and as illustrated in this study, different methods do not result to identical values.

In summary, majority of radiomic shape features have strong monotone direct or inverse correlations with tumor spiculatedness. However, we have shown that quantitative values of these features can vary with slice thickness, volume, and resampling; and depend on the meshing algorithm used for surface and volume extraction. The radiomic shape features Compactness2, Compactness3, Spherical Disproportion, and Sphericity have been shown to have minimal variations with the aforementioned parameters, and should thus be prioritized in future studies. It is clear that quantitative radiomic shape features provide important information on tumor characteristics, underlining the importance of their integration into future radiomic models and notably their combination with clinical, textural and genomic features. Refinements in the methodology of conducting radiomic studies as well as transparency in the exact nomenclature and formula used for each feature are indispensable to enable its eventual translatability to clinical utility.

Material and Methods

Shape phantoms

Spherical harmonics were used to create mathematical tumor models with increasing degrees of complexity28.

$${Y}_{l}^{m}(\theta ,\phi )=\sqrt{\frac{(2l+1)}{4\pi }\frac{(l-m)!}{(l+m)!}}{P}_{l}^{m}(cos(\theta )).{e}^{im\phi }$$

with ℓ the degree, m = {− ℓ, − (ℓ −1), …, 0, …, ℓ − 1, ℓ | ℓ ∈ ℕ} indicating the order, θ and φ the polar and azimuthal coordinates and \({P}_{l}^{m}\) an associated Legendre polynomial33. Using the real component of \({Y}_{l}^{m}(\theta ,\phi )\) cartesian coordinates x, y and z were deduced as follows:

$$x=\rho .\,\sin (\theta ).\,\cos (\phi )$$
$$y=\rho .\,\sin (\theta ).\,\sin (\phi )$$
$$z=\rho .\,\cos (\theta )$$

with

$$\rho =d+\frac{A.Re\{{Y}_{l}^{m}(\theta ,\phi )\}}{{\rm{\max }}(abs(Re\{{Y}_{l}^{m}(\theta ,\phi )\}))}$$

In the presented work, ℓ was arbitrarily set to 10, m to 5 and A to 10. The degree of the spherical harmonic, d, was increased in increments of 3 from 11 to 92 to create a total of 28 models. Model “11” corresponded to the most spiculated model and model “92” to the least one.

The 3D models were cut in the middle of the horizontal plane which permitted these to have a flat base for printing. Then, models were set with identical bases of 5 cm by adjusting the height ratio of the original models to the new base. Each of the tumor models was created using a 3D printer (Discoeasy200, dagoma.fr), using a polylactic acid filament (ρ = 1.25 g.cm−3) with standard printing speed (Supplementary Figure S3).

The models were then scanned using a Siemens Sensation Open CT scan (Siemens Healthineers, Erlangen, Germany) with 0.6 mm and 2 mm slice thicknesses, 100 kVp tube voltage, 300 mAs and a 350 mm reconstruction field of view. The phantoms were scanned on top of a cardboard box, with only the bases being in contact with a surface. Original pixel size was 0.68 mm × 0.68 mm in the transverse planes. Scans were contoured using the thresholding function of 3DSlicer (http://www.slicer.org), with lower limit at −700 Hounsfield Units (HU) and upper limit at 3000 HU, resulting in binary masks.

Feature extraction

Four different approaches based on MATLAB libraries (R2017b software (The Mathworks Inc., Natick, MA, USA) were used to generate a mesh for all tumor models, which involved creating a surface mask from the contours (Supplementary Table S4). The first method (M1) used the Isosurface function of MATLAB. This method connects points having the same value to generate the mask. The isovalue was set to 0.9. The second method (M2), Isosurface filter, consisted of smoothing the triangulated mesh generated with the first method by using the normalized curvature operators as weights. The mesh was mainly smoothed in the normal direction to preserve the original ratio in length between edges. One smoothing iteration was used and the smoothing quantity was set to 5. The third strategy (M3), Isosurface remesher, is an iterative triangle optimization for meshing. In this method, all the closed meshes obtained with the first method are cleaned according to a targeted edge length. The edge length was set to 2 and only one iteration was used. The last method (M4) used the Boundary function of MATLAB that returns a triangulation corresponding to a single conforming 3D boundary around the points. A shrinking factor of 1 was used to obtain the concave hull of the shape of interest.

Nine three-dimensional (3D) shape features were deduced from the surface and volume values extracted using the four meshing methods previously described. These included Volume, Surface Area, Surface-to-volume, three formulas for Compactness8, Spherical Disproportion, Sphericity17 and Fractional Concavity34. Table 5 shows the description and formulas of the computed features. The surface of the convex hull included in the fractional concavity formula was obtained using the Boundary function and a shrinking value of 0.

Table 5 Radiomic shape feature formulas.
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To quantify the impact of 3D printing, acquisition, segmentation and meshing on the radiomic shape features, features were computed directly from the stereolithography (STL) file format that was used for 3D printing, and compared to those computed using the aforementioned Matlab functions.

To study the effect of resampling, masks extracted from the original images were resampled on a 1 × 1 × 1 mm3 grid using a 3D linear interpolation. To remove the inherent variation on volume between objects, a homothetic transformation was then applied to bring back all the volumes to the value corresponding to the average of all volume values calculated for the 28 shapes for the M1 method. Finally, three representative phantoms (d = 11, 47, 92) were resampled to have 25, 50, 75, 125 and 150% of their original volume.

To validate the clinical relevance of the phantoms, CT-scans of lung tumors of three patients from the publicly available RIDER database35 were contoured. Radiomic features were extracted and compared to the range of the values extracted from the 28 printed phantoms.

Statistical analysis

All statistical analyses were performed with R version 3.3.2 (https://www.r-project.org/). Differences between features computed directly from the STL files and those computed after meshing were compared. Percent changes between scans acquired with 0.6 and 2 mm slice thicknesses were quantified. To determine the relationship of each shape feature with tumor complexity, Spearman’s rank-order correlation coefficients were computed for each of the four meshing methods. Complexity was considered as an ordinal variable with numeric values ranging from 11 (most spiculated) to 92 (least spiculated). Pairwise correlations among variables were also computed. To evaluate the effect of resampling on feature variation, percent changes were computed between features extracted from original and resampled (1 × 1 × 1 mm3 grid) 3D masks for the M1 and M3 meshing methods. To evaluate the effect of changes in volume, the percent change in shape features when the volume varied from 25% to 150% was computed for d = 11, 47 and 92 and compared to the percent change observed when modifying the spiculatedness of the model from d = 11 to d = 92 for M3. Figure 7 summarizes the general schema of the methodology.

Figure 7
figure 7

Schema of the steps undertaken in the study. Broken arrows represent comparison between the original extracted features and after resampling or after volume changes.

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