Fig 1.
Morphologic evolution of aortic shapes.
Eight representative aortas along the normal-to-diseased axis (left to right): a 3-year-old child, healthy adults, and type B aortic dissection (TBAD) patients at varying degrees of aneurysmal degeneration. Two clinical regimes exist: shape preserving growth and growth with shape changes.
Fig 2.
Aortas are segmented from CTA imaging scans of the chest, followed by smoothing of the segmentation, isolation of the segmentation outer surface, and triangular surface meshing. The noise reduction procedure encompasses the smoothing and meshing steps, in which multiple smoothing parameters and mesh density variations generate multiple plausible surface meshes representing the segmentation.
Fig 3.
Multi-Scale Surface Curvature Calculations.
By mapping the aortic surface to the unit sphere (Gauss map) [45], we have an independent measure of shape. The per-vertex shape operator is calculated using the Rusinkiewicz algorithm [46]. To minimize noise, the aorta is divided into multiple partitions with area Aj. The local integrated Gaussian curvature Kj is calculated as the product of each partition area and mean Gaussian curvature,
. Kj is equivalent to the signed partition area
mapped out by the normals projected onto the unit sphere. We define aortic shape by studying the statistics of the distributions of Kj. 〈K〉 and δK are the first and second distribution moments that define aortic shape geometry, respectively.
Fig 4.
Number of Surface Partitions Imposed by the Inner Scale ℓ.
Data for 302 aortas, including non-pathologic (black circles), pathologic with failed TEVAR (light gray circles), and pathologic with successful TEVAR (dark gray circles) aortas are plotted. The linear scaling can be used to define Aj ∼ ℓ2, which sets the number of partitions k used in the Gauss map calculations. The various linear fits are taken for different definitions of size: maximum aortic diameter (2Rm, red dashed line), mean radius (〈R〉, black solid line), median radius (, black dotted line), and mean inverse linearized aortic Casorati curvature (〈C1/2〉−1, black dashed line) are equivalent. Dimensionally scaled, aortic area (
, red dotted line) and volume (V1/3, red solid line) are also linear when plotted against ℓ = 2Rm. In this case, the fits are normalized by the pre-factors obtained from their fitting to the maximum dimeter (Fig 5). The normalized data is shown to demonstrate that k is independent of the specific size measure used to set the inner scale ℓ.
Fig 5.
Universal Scaling of Aortic Size.
Data for 302 aortas, including non-pathologic (black circles), pathologic with failed TEVAR (light gray circles), and pathologic with successful TEVAR (dark gray circles) aortas are plotted. A. shows that parameterizations of aortic size (mm) including maximum aortic diameter (2Rm, red dashed line), mean radius (〈R〉, black solid line), median radius (, black dotted line), and mean inverse linearized aortic Casorati curvature (〈C1/2〉−1, black dashed line) are equivalent. Dimensionally scaled, aortic area (
, red dotted line) and volume (V1/3, red solid line) are also linear when plotted against ℓ = 2Rm. All size measures can be collapsed onto a single master curve (B.), proving that all aortas scale as generalized bent cylinders parameterizable by a single length scale ℓ.
Fig 6.
Length-to-Size Ratio as Function of Size.
Ratio of centerline length to radial size ℓ. For the relationship
indicating a linear scaling between axial length and cross-sectional circular radius, we obtain c = 16.6±2.4. The yellow symbols indicate selected aortas shown in Fig 7A.
Fig 7.
Aortic Topological Invariance and Aortic Clustering in -space.
A. The eight canonical representative aortas along the normal-to-diseased axis (left to right): a 3-year-old child, healthy adults, and type B aortic dissection (TBAD) patients at varying degrees of aneurysmal degeneration. Two clinical regimes exist: shape preserving growth and growth with shape changes. B. shows the topologic equivalence of all aortic shapes to tori (red stars) and cylinders (red diamonds); the yellow symbols correspond to specific aortic shapes along the normal-to-diseased axis (A.). Red circles correspond to perfect spheres of varying size; pseudospheres and catenoids are depicted as red rightward-pointing triangles and upward-pointing triangles, respectively. C. shows the optimal two-dimensional aortic geometric feature space with independent axes for size and shape. The solid red curve is a best fit to the data. The power-like behavior is further supported by the probability distribution of δK (Fig 8). The aortas separate into shape invariant (normal) and shape fluctuating (diseased) populations. Furthermore, this feature space defines decision boundaries that correctly classify diseased patients based on success of TEVAR.
Fig 8.
Fits Power Law Distribution while Size is Gaussian.
Probability distributions of δK and Rm are plotted. A. The δK distribution is fitted to a power law in the form P = axb + c. C. A linear fit logP = blogδK + c achieves a high R2. B. The Rm distribution is well-fit to a two-term Gaussian in the form . D. When a linear fit is applied to the log-transformed data, logP = blogRm + c, a low R2 value results.
Fig 9.
Clustering Analysis in Geometric Feature Space Shows Superior Accuracy and Stability Compared to Size Alone.
The geometric feature space improves upon current sized-based methods. The clinical paradigm relies on size metrics alone to classify aortic disease states (green for normal aortas, blue for successful TEVAR, and red for failed TEVAR). However, broad within-group size distributions indicate considerable variability in aortic sizes within the general population. Clinicians routinely utilize statistical means of these distributions as thresholds for classifying disease states, but linear decision boundaries are highly sensitive to small changes in model setup. A. A 73.0% accuracy for classifying the 3 states is obtained when each threshold is defined as the mean 〈C1/2〉 of the two neighboring distributions. B. An 83.9% accuracy is achieved when the threshold is defined as the midpoint of the means of individual class distributions. C. An 87.0% accuracy is obtained when a logistic regression classifier is used. Thus, small changes in how a threshold is defined dramatically alter the perceived utility of size. D. The
shape and size-based geometric feature space allows for the utilization of two independent parameters to characterize aortic disease state. A 90.3% classification accuracy is obtained when defining thresholds according to the mean δK and 〈C1/2〉 of each patient group. E. A 92.8% mean accuracy with a standard deviation of only 1.7% is obtained using a logistic regression classifier with varying regularization parameters. The shaded region indicates the interquartile range of decision boundaries and demonstrates the robustness of the two-parameter space. Unlike the single parameter space, the presence of two physically interpretable and orthogonal asymptotic limits ensures more effective classification.
Fig 10.
Aortic Population Classification Based on Various Size and Shape Features.
Comparison of size and shape metrics in classifying aortic disease state from medical imaging. A. Measures of aortic size achieve similar classification accuracies and thus are functionally equivalent (corroborating Fig 5). The GLN and GAA are other size metrics. B. δK significantly outperforms measures of aortic shape from the clinical literature in classifying aortic disease state (normal non-diseased aortas, diseased aortas with desired outcomes following TEVAR, and aortas with failed outcomes following TEVAR). C. δK outperforms general shape metrics from the broad engineering literature. Error bars indicate ±1 standard deviation of the classification accuracies for the different classification methods.
Fig 11.
δK and δκg Equivalent for Small Changes in Overall Size.
Simulation of a sphere with pressurization followed by growth. A. ∑K is a topologic invariant and thus remains relatively constant throughout the simulation. B. δK captures the increasing degeneration of the spherical surface due to growth. C. δAj fails to capture this degeneration. D. δκg seems to capture the degeneration of the spherical surface as the value diverges for increasing size. However, the narrow scale of the x-axis indicates that there is little increase in overall size for this simulation. E. Surface geometries for selected frames in the simulation, with the undeformed geometry on the right side and the final geometry on the left side. The vectors indicate the direction of surface deformation.
Fig 12.
δK Superior to δκg with Significant Size Changes.
Simulation of an ideal aorta with pressurization followed by growth. A. ∑K is a topologic invariant and thus remains relatively constant as 〈C1/2〉 increases throughout the simulation. B. δK captures the increasing surface degeneration due to growth. C. δAj does not capture this degeneration, as evidenced by the increasing error with simulation progression. D. When size significantly changes, δκg no longer captures the geometric deformation. E. Surface geometries for selected frames in the simulation, with the undeformed geometry on the right side and the final geometry on the left side. The heatmap coloring indicates Kj the total curvature at the per-partition level.