Number of Surface Partitions Imposed by the Inner Scale <i>ℓ</i>.

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.</p

<i>δK</i> Superior to <i>δκ</i>_<i>g</i> 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.</p

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.</p

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.</p

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.</p

Contains all the data necessary to reproduce the main figures: Scan ID, mean curvedness, <i>δK</i>, total aortic area, aortic volume, mean aortic radius, max aortic radius, tortuosity index, cross sectional eccentricity, mean centerline curvature, question mark angle, radius ratio, and scan sequence.

The data correspond to the raw mask segmentations which can be obtained from the public GitHub: https://github.com/SurgBioMech/khabaz_2024/tree/main. (XLSX)</p

<i>δK</i> and <i>δκ</i>_<i>g</i> 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.</p

Supplementary information supporting the main text.

Fig A includes the demographic information for the non-pathologic aortic cohort. Fig B is the demographic information for the dissection cohort. Section titled “Aortic Segmentation and Post-Processing from CTA Imaging” includes details on the methods and procedures involved in Segmentation, Noise Reduction, Smoothing, Isolation of the Outer Surface of the aortic mask, and Meshing. The section on “Calculation of the Shape Operator” details our implementation of the Rusinkiewicz algorithm of calculating surface curvatures on a meshed surface which are the primary inputs into our shape and size calculations. The section “Artifact Removal” details the criteria used to remove the flat edges and and rims which are generated during the segmentations and constitute artifacts. The section “Jensen-Shannon Divergence of Partition Gaussian Curvature” details our implementation of the JSD as a measure of κg spatial gradients within partitions. The section titled “Sensitivity to Partition Size” details our exploration of how patch size impacts the distribution of data projected into the shape-size feature space. The section “Ideal Shapes” provides the analytical functions used to generate the idea shapes used for cross-validation of our methods in the manuscript. The section “Other Shape Metrics” shows our detailed exploration of other published functions quantifying shape and the projection of our data into each one of the individual shape-size feature spaces. The section “Finite Element Simulations” provides details of material model selection and element selection for the FEA simulations in the paper. And the final supplementary section “Analysis on Pre-Operative Data” projects only the last pre-operative scan into the shape-size feature space, this is a reduced dataset of the full data set provided in Fig 7 of the paper. (PDF)</p

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.</p

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.</p