The geometric evolution of aortic dissections: Predicting surgical success using fluctuations in integrated Gaussian curvature

Clinical imaging modalities are a mainstay of modern disease management, but the full utilization of imaging-based data remains elusive. Aortic disease is defined by anatomic scalars quantifying aortic size, even though aortic disease progression initiates complex shape changes. We present an imaging-based geometric descriptor, inspired by fundamental ideas from topology and soft-matter physics that captures dynamic shape evolution. The aorta is reduced to a two-dimensional mathematical surface in space whose geometry is fully characterized by the local principal curvatures. Disease causes deviation from the smooth bent cylindrical shape of normal aortas, leading to a family of highly heterogeneous surfaces of varying shapes and sizes. To deconvolute changes in shape from size, the shape is characterized using integrated Gaussian curvature or total curvature. The fluctuation in total curvature (δK) across aortic surfaces captures heterogeneous morphologic evolution by characterizing local shape changes. We discover that aortic morphology evolves with a power-law defined behavior with rapidly increasing δK forming the hallmark of aortic disease. Divergent δK is seen for highly diseased aortas indicative of impending topologic catastrophe or aortic rupture. We also show that aortic size (surface area or enclosed aortic volume) scales as a generalized cylinder for all shapes. Classification accuracy for predicting aortic disease state (normal, diseased with successful surgery, and diseased with failed surgical outcomes) is 92.8±1.7%. The analysis of δK can be applied on any three-dimensional geometric structure and thus may be extended to other clinical problems of characterizing disease through captured anatomic changes

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

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

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.</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

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

Image Processing Workflow.

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.</p

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.</p

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.</p