Introduction

Imaging modalities such as computed tomography (CT) provide sophisticated three-dimensional representations of the human body and are ubiquitously relied upon in clinical practice [1–4]. At its core, imaging captures anatomic changes that are linked to underlying pathologic processes. Anatomy is the spatial organization of tissues in space. As such, geometry is the natural mathematical framework to quantitate anatomy. Geometric approaches have widely been used to characterize and classify multiple diseases including pulmonary nodules [1], liver cirrhosis [2], and thyroid masses [3, 4]. In all of these examples, the physician’s human eye defines the hallmark of disease by the appearance of nodularity or spiculation. Even the process of aging or healing often leads to the appearance of undulations or wrinkles, which is a departure from the baseline smooth geometry of healthy skin [5]. These examples highlight the ubiquitous classification of anatomic objects based upon the surfaces that define their boundaries. These surfaces are easily appreciated when the anatomy is directly visualized with gross inspection (such as skin or with dissection in the operating room). They also appear as level sets in cross-sectional imaging, most often x-ray based tomography, because of the intrinsic difference in x-ray absorption and scattering of various tissues. The problem of disease progression mathematically becomes the transformation of a smooth surface into a rough surface, with the appearance of multiple new length scales.

Quantifying what the human eye so easily discerns has proven challenging. Recent work has sought to translate successes from deep learning (DL) to utilize imaging for disease diagnosis and clinical planning [6, 7]. Challenges in the availability of high-quality data [8, 9], the lack of reproducibility and generalizability [9, 10], the disconnect between technical accuracy and clinical efficacy [11, 12], and interpretability of inherently “black-box” DL models have made the application of DL in medicine particularly fraught [13, 14]. Sophisticated approaches harnessing the machinery of Riemannian geometry have been successful in certain neuroanatomy problems [15]; however, they are limited to tissues with well-defined internal landmarks allowing for global coordinate systems, e.g. Talairach space or MNI coordinate system in brain imaging [16], which do not exist for cardiovascular tissues.

We take a fundamentally mathematical approach to developing imaging-based geometric descriptors of disease evolution in clinically meaningful systems. We focus on the challenging problem of classification in aortic dissections. The aorta is the largest blood vessel in the human body. It functions as a conduit carrying blood from the left ventricle of the heart to provide blood flow to the head, arms, abdominal organs, and legs [17, 18]. Mechanically, the aorta is akin to a pressurized distensible cylindrical shell; aortic pathologies are mechanical in nature, with aortic rupture leading to near-certain death. Aortic dissections are partial tears in the aortic wall. Mechanically, dissections are cracks that do not penetrate through the entire aortic thickness; rather, they propagate between the different concentric layers making up the aortic wall generating aortic blisters (medically termed the false lumen). Dr. Michael DeBakey, the pioneer of aortic dissection surgery, famously described diseased thoracic aortas as ‘torturous’, ‘globular’, ‘U-shaped’, ‘dilated’, and ‘idiosyncratically varying’ [19]. These qualitative descriptors underpin the complex changes in shape that accompany aortic disease and its morphologic evolution. The current standard of care for type B aortic dissection (TBAD) is anchored on the concept of preventing “dangerous” shape changes, such as aneurysmal degeneration, which contribute to worse outcomes [17, 18]. Thus, the concept of shape and pathology are well-accepted clinically; however, accurate quantification of these changes is limited. Thoracic aortic stabilization by placement of a fabric-covered stent into the diseased thoracic aorta, clinically termed Thoracic Endovascular Aortic Repair (TEVAR), is the preferred modern surgical approach to TBAD [17, 18].

Since the 1950’s and DeBakey’s initial work, the ubiquitous anatomic classifier clinically used to trigger aortic surgery has been maximum aortic diameter (2R_m, where R_m is the maximum radius) [17, 18], yet clinicians have long appreciated that size by itself is an inadequate descriptor of aortic anatomy [17, 20–22] and have resorted to qualitative descriptors of aortic morphologic evolution. The urgent need for a richer description of aortic anatomy has been highlighted by several clinical trials using TEVAR, which showed that using size alone to trigger intervention did not improve mortality [23–26]. The cardiovascular surgical community realizes that better identification of patients who would benefit from TEVAR is needed to shift the post-intervention mortality curve in the positive direction [25–28]. Identifying anatomic parameters that impact the ability of an aorta to remodel [29] is instrumental in guiding therapy.

Geometry and mechanics have driven the use of maximum diameter as a singular scalar descriptor of aortic stability and, therefore, the dominant trigger of intervention. Geometrically, since aortas are generalized bent cylinders, the cylinder’s radius naturally appears as a principal length scale describing the geometry. Since diseased aortas expand non-uniformly in the radial direction as a function of axial distance (or distance along the aortic centerline), the gradient of aortic radius along the centerline is maximized near the largest diameter. Clinically, largely for the ease of application, the gradient has been replaced by simply measuring the maximum diameter and assuming a uniform ‘normal’ aortic radius. This assumption is fraught with problems because normal aortic diameters, while being uniform for any given individual, show a broad distribution in the general population [30]. The problem is further complicated by the fact that uniform aortic growth (∼0.5 mm/year) occurs with aging [31]. Lacking appropriate normalization, the maximum diameter is only a descriptor of geometry in comparison to population means.

Standard biomechanics literature asserts that stability of dilated aortas is dominated by the Law of Laplace, where the stress (σ) is a simple linear function of internal pressure (P), aortic thickness (h), and maximum diameter (2R_m): σ = P ⋅ R_m/h. By Laplace, larger aortas will have linearly proportional larger stresses. Since aortic mechanical stability is a fracture problem, once the wall stress supersedes a critical stress, the aorta will crack. Given the distribution of R_m values in the population, 〈R_m〉 + δR_m ∼ 1.25±0.5cm, and the fact that R_m is not constant with time, ∂_t〈R_m〉 = 0.5mm/year, the calculation of a broadly applicable critical stress becomes nearly impossible. Furthermore, the mathematical derivation of Laplace’s law assumes homogeneous isotropic shells of uniform thickness, which is not true even for non-diseased aortas [32]. More advanced mechanical failure analysis of aortic aneurysms and dissections utilizes finite element approaches (FEA); however, these ultimately lack the simple relationships between geometry (derived from clinical imaging data) and stress contained in Laplace’s law. As such, their wide clinical application has been limited [33, 34]. Nearly all clinical data gathered about aortas is imaging-based geometric information. The herculean efforts to translate intricate mechanical stability models using FEA to clinical practice largely failed due to lack of interpretability and understanding of these models by practicing physicians. This failure should be heeded when developing sophisticated AI-based deep-learning models, which take raw CT scan data as inputs; the challenge of interpreting such ‘black-box’ models in medical applications is well known [13, 14].

We derive and validate a mathematical descriptor of aortic shape; a single variable that scales appropriately with DeBakey’s famous qualitative descriptors of aortic shapes [19, 20]. As D’Arcy Thompson wrote a century ago: “In a very large part of morphology, our essential task lies in the comparison of related forms rather than in the precise definition of each” [35]. The problem of morphogenesis is deeply linked to the stability of self-reproducing structures; Rene Thom postulated that any morphological process is divided into islands of determinism, dominated by morphogenic fields whose precise mathematical form is defined by topologic homeomorphisms, separated by regions of instability and indeterminacy [36]. Topology is now a fundamental tool in the classification of rapidly changing shapes and structures in developmental biology, where anatomic descriptions are translated into topological language allowing the quantification of purely geometric transformations in growing tissues [37–39]. The generalization of a tissue or body into sets of smooth, closed, orientable surfaces (i.e. membrane systems) makes them topological objects [36, 39]. Local topological surgeries, cutting or gluing, lead to global topological changes in biological forms through regions of topological catastrophes [36, 38, 39]. We apply such methods to the study and generalization of aortic morphogenesis throughout the life-cycle of an aorta (see Fig 1).

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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.

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We work with aortic surfaces derived from cross-sectional imaging such as CT scans from both normal patients and patients with aortic dissections at various stages of disease. Fig 1 outlines the problem of defining a continuous geometric parameter space that can smoothly evolve from a normal pediatric aorta to the most diseased aortic dissection. Visual inspection of the objects in Fig 1 easily leads to a qualitative description of their anatomy: the aortas grow left to right with an initial rapid change followed by a more subtle increase in size (scale), a trend that reverses when surface shape is considered. The rapidly growing aortas are self-similar concerning their shape (bent cylinders with the toroidal aortic arch followed by the cylindrical descending thoracic aorta); a shape symmetry-breaking transition occurs as the aorta further moves towards the right. We follow this transition by projecting the CT-derived anatomy of 302 aortas, distributed from normal to diseased states, into a novel geometric space spanned by orthogonal axes for shape and size defined using the shape operator and total curvature of the aortic surface. The local geometry of the aortic surface is defined by its principal curvatures, k₁ and k₂. Specific measures of aortic shape in the literature have focused on functions of mean (κ_m = (k₁ + k₂)/2) or Gaussian (κ_g = k₁k₂) curvature [40, 41], cross-sectional deviations [42], and calculations on centerline reductions of the aortic surface [21, 43, 44]. A mathematically robust and reproducible definition of aortic shape does not exist despite these efforts because of spatial over-reduction and convolution of shape and size information. Centerline calculations lose information when the aortic surface deforms heterogeneously [45], and κ_g, the dominant shape function, proves itself an ill-defined shape measure because it convolutes shape and size changes [45]. We solve this problem by using the Gauss-Bonnet Theorem and the total curvature, K = ∬_Aκ_gdA, as the primary measure of shape. We prove that all aortic shapes are homeomorphic to T²; as such, all aortas, no matter how deformed, remain generalized bent cylinders. Using simulated shapes with evolving surface roughness, we show that κ_g and K hold the same shape information, provided overall changes in size are small.

Topology preservation is by far not the norm in biologically growing surfaces that are non-conservative and can locally add mass or remove mass without the constraint of elastic deformations or linkage to some global manifold. The existence of a homeomorphism implies that as aortas deform through normal growth or pathology, every increase in K in some region must be balanced by a proportional decrease somewhere else on the aortic surface. Therefore, the distribution of K across the aortic surface holds information about shape. This allows us to classify shape by studying the statistical properties of these distributions. Since 〈K〉 is constant, the variance of K captures the balance of positive and negative curvature regions across the surface: δK = 〈K²〉 − 〈K〉². δK captures the heterogeneous morphologic evolution by characterizing local shape changes. We hypothesize and demonstrate that this novel shape measure correlates with aortic disease evolution and can be used to characterize treatment response for aortic dissections when appropriately coupled with a size metric.

Methods

Meshing algorithm and geometric parameterization

Three-dimensional aortic models are created from CTA image data using a custom workflow in Simpleware ScanIP (S-2021.06-SP1, Synopsys, Mountain View, CA). Aortic geometry is extracted from scans using a five-step algorithm which includes 1. segmentation, 2. noise reduction, 3. smoothing, 4. isolation of the segmentation outer surface, and 5. surface meshing. A representative schematic of the process is shown in Fig 2, and more information on the process can be found in SI Aortic Segmentation and Post-Processing from CTA Imaging. A triangular mesh for the outer surface is generated for each smoothed segmentation in ScanIP for analysis in Matlab (2021b, Mathworks, Natick, MA). A total of 15 meshed surfaces are generated for each segmentation (sampling 5 mesh densities and 3 smoothing variations), allowing for control of process-derived variance in surface curvature calculations.

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Fig 2. 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.

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Once a meshed geometry is created for each aorta, the Rusinkiewicz algorithm is used to calculate the per-vertex shape operator , where k_1i and k_2i are the per-vertex principal curvatures. Briefly, the algorithm calculates the per-vertex shape operator as a weighted average of the shape operators of immediately adjacent faces. The per-face tensors are computed using a finite-difference approximation defined in terms of the directional derivative of the surface normal [46] (see SI Calculation of the Shape Operator for more details). Therefore, each vertex in the mesh is accompanied by a first principal curvature and second principal curvature that approximate the local shape at the intersection of neighboring faces. SI Artifact Removal outlines the artifact removal process that was implemented.

Size and shape characterization

Size.

Fig 3 shows our computational workflow for calculating aortic shape and size features. Aortic size is parameterized using the centerline length and aortic radius ℓ. While is singular for each aorta, ℓ is a distribution, especially in diseased aortas with heterogeneous shapes, and therefore a family of radii. Multiple measured values can be used to represent this distribution to characterize aortic size: mean aortic radius (〈R〉), median aortic radius (), and maximum aortic diameter 2R_m. ℓ can also be calculated directly from surface curvatures. The mean Frobenious norm of the per-vertex shape operator , , is the per-vertex Casorati curvature [47–49]. Averaging over the entire aorta gives the mean Casorati curvature, (1) where is the per-partition mean, k is the number of partitions, A_j is the area of a partition, and p the number of vertices within each partition. The need for dividing the surface into k smaller partitions is not necessary to obtain accurate mean Casorati curvatures; however as explained below, it becomes necessary when performing shape calculations. We keep it here from an implementation consistency standpoint because C_i is calculated from principal curvatures. The inverse mean Casorati curvature 〈C^1/2 〉⁻¹ is a measure of aortic size along the direction of greatest curvature, which is the aortic radius. Lastly, higher-order functions of size such as total aortic area, A_T, and aortic volume, V, are calculated directly from the surface segmentations. Total aortic area is also calculated from the sum of per-element areas: , where g is the total number of triangular mesh elements on a given aortic surface. It is important to note that only 〈C^1/2〉⁻¹ allows for a measure of aortic size using only information contained in ; all other size measures necessitate additional information.

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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 A_j. The local integrated Gaussian curvature K_j is calculated as the product of each partition area and mean Gaussian curvature, . K_j 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 K_j. 〈K〉 and δK are the first and second distribution moments that define aortic shape geometry, respectively.

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Shape.

The size-independent parameterization of shape is obtained through an effective mapping of normals on the external aortic surface to the unit sphere S². Operationally, we do not directly perform the Gauss mapping. Instead, we calculate the total curvature over local regions (partitions) of the aortic surface with area A_j, which by the local Gauss-Bonnet theorem is the holonomy angle of the circumnavigated area, a topologic quantity [47]. In our analysis, given geometrically highly heterogenous shapes, a meshed surface is employed and the per-vertex Gaussian curvature is extrinsically calculated from the shape operator: . The area elements are intrinsic to the surface and ; however, since a generic triangular mesh is used, no information about the surface metric exists. Knowledge of the metric would be needed to carry out the integration of per-vertex Gaussian curvature explicitly. Our computational approach therefore relies on splitting the integration into a product , where . To bring the mean Gaussian curvature within a partition outside the integral, κ_gi must vary weakly within A_j. Clearly, this variance is minimized when A_j = a_i. As shown in SI Sensitivity to Partition Size, calculating K at this scale holds little information about overall aortic shape. To understand the necessity of sub-dividing the aortic surfaces into partitions where a_j < A_j < A_T, the scale space nature of the problem needs to be explored and appreciated. Since the per-vertex Gaussian curvature is calculated on a mesh with an average element spacing proportional to the CT scan resolution ( mm); this value assumes the mesh-imposed inner scale. However, as is appreciated by examining the evolution in Fig 1, the change in aortic shape, the increasing ‘bumpiness’, is occurring on a much larger length scale. Visual inspection of the shapes shows this scale to be on the order of the aortic radius or ℓ. The mesh-imposed inner scale is ∼ℓ/1000. As discussed in SI Aortic Segmentation and Post-Processing from CTA Imaging, the mesh resolution was set by the inherent resolution of the CT data and by the need of a finely enough spaced computational mesh to calculate discrete derivatives of surface normals. However, from an aortic shape classification standpoint, using the mesh-imposed inner scale runs the risk of imposing a large amount of “spurious resolution” or “false detail”, commonly seen with scale space problems when an inappropriately small inner scale is selected [47]. We impose ℓ as the inner-scale by sub-dividing the aortic surface into k area partitions via a Voronoi decomposition using the k-means algorithm, in which k = A_T/ℓ² (see Fig 4 which shows that k is independent of how ℓ is measured and relatively constant with respect to ℓ). For subsequent analysis, (indicating the median R₂, where R₂ = 1/k_2i is calculated per-vertex from ). We use the k-means++ algorithm to find initial centroid seeds, and the k-means algorithm is performed with a maximum of 10000 iterations. SI Sensitivity to Partition Size shows how k can vary by a factor of 10 and does not impact the results discussed in the following sections. Lastly, SI Jensen-Shannon Divergence of Within-Partition Gaussian Curvature outlines our use of the Jensen-Shannon Divergence to check that at this inner-scale Gaussian curvatures still show acceptably small variance, allowing us to bring the mean Gaussian curvature within a partition out of the area integral and replace the area integral with the sum of mesh element areas.

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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 A_j ∼ 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 (2R_m, red dashed line), mean radius (〈R〉, black solid line), median radius (, black dotted line), and mean inverse linearized aortic Casorati curvature (〈C^1/2〉⁻¹, black dashed line) are equivalent. Dimensionally scaled, aortic area (, red dotted line) and volume (V^1/3, red solid line) are also linear when plotted against ℓ = 2R_m. 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.

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Having partitioned the aortic surface into k partitions of size ∼ℓ², the per partition total curvature (2) is calculated. To characterize the entire aorta, the sum of per partition total curvatures (a topologic invariant) (3) and the fluctuations in total curvature (a normalized measure of shape) across the entire manifold surface are calculated: (4)

The mean Gaussian curvature (5) and the fluctuation in Gaussian curvature of the manifold (6) are also calculated.

Statistics.

Each aorta contains k partitions. All calculations are performed for each of the 15 meshed models per aorta. Of note, the k-means partitioning is independently performed for each mesh. An average value for 〈C^1/2〉, ∑K, δK, 〈κ_g〉, and δκ_g is computed for the 15 meshes, such that each scan becomes represented by a single value that aggregates the variation from the smoothing and meshing algorithms. To quantify variability from the partitioning procedure itself, the entire process (partitioning, partition-level curvature calculations, manifold-level curvature calculation, and averaging between meshes) is repeated 10 times to obtain 10 replicates of each value per aorta. The mean of these replicates is reported and a sensitivity analysis is performed in SI Sensitivity to Partition Size. The error bars in all data plotted in the two-dimensional feature space represent ±1 standard deviation to quantify the variability in the results from the partitioning replicates.

Results

Three major and two minor results come from our analysis. The first major result is that aortic size scales as a generalized bent cylinder and depends only on a single length scale, ℓ. The second major result is that projecting aortas into a two-dimensional space defined by (δK, ℓ⁻¹) separates aortic geometries into shape-preserving self-similar growth and growth with shape changes; moreover, all aortic geometries are shown to be topologically homeomorphic to T². The third major result shows that the (δK, ℓ⁻¹)-feature space outperforms all other size and shape measures when applied to the predictive classification of TEVAR success. The first minor result shows that the data projected into the normalized () space are best fit to a simple power law (). The second minor result shows that δK and δκ_g hold similar information, however, δK provides a stronger signal of shape changes when shape and size changes occur simultaneously.

Universal scaling of aortic size

The analysis includes a variety of size measures previously described in the literature, including the traditional metric of maximum diameter (2R_m) and higher-dimensional values of area (A_T) or volume (V) (Fig 5). When each length scalar is plotted versus maximum diameter, the data collapses onto lines, with linear size measures having a slope of 1 and higher-dimensional measures ( and V^1/3) having slopes that correspond to modeling aortic size as a generalized bent cylinder. In generalized bent cylinders, area and volume scale as and . Hypothesizing a linear relationship between axial length and cross-sectional circular radius ℓ, , the reparameterized equations A_T ∼ 2πcℓ² and V ∼ πcℓ³ are obtained.

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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 (2R_m, red dashed line), mean radius (〈R〉, black solid line), median radius (, black dotted line), and mean inverse linearized aortic Casorati curvature (〈C^1/2〉⁻¹, black dashed line) are equivalent. Dimensionally scaled, aortic area (, red dotted line) and volume (V^1/3, red solid line) are also linear when plotted against ℓ = 2R_m. 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.

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Fig 5 shows that and V^1/3 ∼ 3.5ℓ, which implies that A_T ∼ 100ℓ² and V ∼ 42.88ℓ³. Solving for c, we obtain c ∼ 15.9 for area and c ∼ 13.6 for volume. The pre-factor c can alternatively be calculated using directly. is best approximated using the length of the aortic centerline measured from the segmentations. Thus, can be calculated on each aortic geometry (Fig 6), with c = 16.6±2.4 when ℓ ∼ 〈C^1/2〉⁻¹ is used. The values of c ∼ 15.0, c ∼ 13.6 and c ∼ 16.6 indicate similar constants that are produced using independent methods and that quantify a linear relationship between aortic axial length and cross-sectional size. This demonstrates that aortic geometry follows a universal linear scaling between different size measures and globally retains an invariant toroidal-cylindrical geometry. This indicates that additional information is unlikely to exist with higher-dimensional size measurements that have been the focus of recent literature and provides geometric reasoning behind the well-known utility of maximum diameter in aortic management.

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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.

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Role of shape deviation

Fig 7 plots the data from 302 aortic segmentations into two spaces: and . Size is represented in both by , where 〈ℓ〉_norm is the mean of the normal aortas. As was shown above, the different measures of ℓ differ only by a constant pre-factor. Therefore, the same normalized size axis is obtained irrespective of how ℓ is measured. ∑K is the total integrated Gaussian curvature of each aorta and mathematically defines its topology. Fig 7B plots the aortic data points (black and grey circles) along with the representative aortas in yellow. Red points are calculated on analytically generated surfaces (see SI Ideal Shapes) analyzed through the same analysis pipeline as the aortic data: spheres (red circles), tori (red stars), cylinders (red diamonds), pseudospheres (red rightward pointing triangle), and catenoids (red upward pointing triangle) of various sizes.

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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.

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These ideal shapes represent the three canonical geometries: spherical, parabolic, and hyperbolic. Their total curvature values of ∑K = 4π (spherical), ∑K = 0 (parabolic or Euclidian), and ∑K = −4π (hyperbolic) are in agreement with analytical results from differential geometry [57]. This agreement for the ideal surfaces is an important validation of the computational methodology used in this paper. The aortic data clearly align along the ∑K = 0 line and cluster with the tori and cylinders. The normal aortas (solid black dots) show a very tight distribution with very little variance along ∑K. The diseased aortas show more spread, which is likely a consequence of the complex shapes encountered in the group. Nevertheless, what is striking in this data is that throughout the morphologic life span of an aorta (during normal growth and diseased degeneration), any given aorta is topologically equivalent to a generalized bent cylinder or torus homeomorphic to T². Consequently, by the Gauss-Bonnet Theorem, aortic topologic invariance implies that as aortas deform, be it through normal growth or pathology, every increase in K somewhere must be balanced by a proportional decrease elsewhere on the aortic surface [45, 57, 58]. Therefore, the distribution of K across the aortic surface should hold information about shape. Since ∑K is constant, the variance δK = 〈K²〉−〈K〉² captures the balance of positive and negative curvature regions across the surface and is a measure of shape deformation. Normalized δK is defined as , where 〈δK〉_norm is the mean δK of the normal aortas. Examination of the data in Fig 7C shows that quantitatively captures the above-mentioned qualitative descriptions of aortic shape. Our three aortic populations naturally separate into three regions in -space. In the asymptotic limit , self-similar aortic growth occurs; all normal aortas exist along this limit. In the other limit, , fluctuations in shape become seemingly independent of size change. The most distorted and convoluted aortas, which also correspond to patients with failed endovascular surgeries, cluster in this limit. The two limits are joined through a transition region that appears as an elbow in the two-dimensional shape-size feature space. Interestingly, the dissection patients who had successful surgery predominantly cluster in this region. The utilization of this novel -space for aortic classification is further developed below.

Power law fit in the feature space

In the geometric feature space, the data can be fit to a power law . Here we follow an established statistical framework of discerning power-law behavior in empirical data to determine whether the data is truly consistent with a power-law [59]. Fig 8 shows separate probability distributions for δK and R_m and then plots these on doubly logarithmic axes. Logarithmic distributions, p(x) ∝ x^−a, are linear in log-log space: lnp(x) = αlnx + c, where c is a constant. The log-log transformed δK distribution suitably fits a straight line with R² ≈ 0.74, while the R_m data does not fit a line (R² ≈ 0.002). This indicates that a power law may be an appropriate fit for the shape metric δK and is not an appropriate fit for the size parameter R_m. The size data are well approximated by a Gaussian distribution, a fact demonstrated in the literature on aortic size distributions [30].

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Fig 8. Fits Power Law Distribution while Size is Gaussian.

Probability distributions of δK and R_m are plotted. A. The δK distribution is fitted to a power law in the form P = ax^b + c. C. A linear fit logP = blogδK + c achieves a high R². B. The R_m 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 = blogR_m + c, a low R² value results.

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Classification with predictive modeling

The strong correlation between δK divergence and the evolution of aortic pathology begs application to treatment planning. Fig 9 compares the effectiveness of aortic size, the clinical standard, with the shape-size feature space in determining aortic disease states (normal aortas, successful TEVAR, and failed TEVAR). Significant variability in aortic size within the broader patient population is captured by this single institutional dataset. Furthermore, there is overlap between the normal and diseased populations and as such, classification accuracy dramatically changes due to slight differences in model parameterizations (from 73% in Fig 9A, 83.9% in Fig 9B, and 87.0±2.3% in Fig 9C). The dual parameterization of both aortic size and shape—two orthogonal metrics—expands the geometric characterization into a two-dimensional space (Fig 9D and 9E). Classification accuracy increases to 90.3% in Fig 9D and 92.8±1.7% in Fig 9E.

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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 〈C^1/2〉 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 〈C^1/2〉 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.

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As shown in Fig 10A, there is no significant difference amongst size predictors such as aortic volume, surface area, median and maximum diameters shown in Fig 5, as well as Gaussian-curvature based size measures like the L2-norm of the Gaussian curvature (GLN) and the area-averaged Gaussian curvature (GAA) [40, 60, 61], see SI Other Shape Metrics. There is a major difference with shape measures: clinical shape measures, which are based on the aortic centerline and include tortuosity index [21, 43, 44, 62], question mark angle [44, 63], cross-sectional eccentricity [42, 64], and mean centerline curvature [43], significantly underperform δK in aortic disease state classification (Fig 10B). Similarly, δK outperforms other shape measures described in the biomedical engineering literature including the sphericity index (χ) [65], flatness index (γ) [65], Gaussian curvature (κ_g) [40, 41], area-averaged mean curvature (MAA), and L2-norm of the mean curvature (MLN) [40, 60, 61] (Fig 10C).

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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.

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Shape evolution modeling

In this paper, K is shown to be a topologic invariant across aortic anatomies and δK a strong function of aortic shape changes. The integrated total curvature ∑K is a topologic invariant and remains approximately constant across all aortic anatomies (Fig 7B). δK is approximately constant across the range of normal aortas, matching empirical knowledge that non-pathologic aortas experience shape-preserving growth. δK diverges with rapidly degenerating aortas, especially those where TEVAR ultimately failed. The role of δK in this divergence is elucidated with FE simulations on spheres (Fig 11) and an idealized aorta (Fig 12). More information on the simulation parameterizations can be found in SI Finite Element Simulations.

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Fig 11. δK and δκ_g 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. δA_j fails to capture this degeneration. D. δκ_g 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.

https://doi.org/10.1371/journal.pcbi.1011815.g011

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Fig 12. δK Superior to δκ_g 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 〈C^1/2〉 increases throughout the simulation. B. δK captures the increasing surface degeneration due to growth. C. δA_j does not capture this degeneration, as evidenced by the increasing error with simulation progression. D. When size significantly changes, δκ_g 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 K_j the total curvature at the per-partition level.

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Fig 11 shows how for the case of a small change in global size, as exhibited by the narrow range of the x-axis (9.5%), the fluctuation in Gaussian curvature (δκ_g) accurately captures surface deformation just as well as the fluctuation in total curvature (δK). This is to be expected as the main function of total curvature is to normalize size effects by mapping the aortic surface to S². Thus, with little change in overall size, Gaussian curvature remains an effective indicator of shape. This is to be contrasted with what happens when size does change, as in the real patient data (400%) or the idealized aorta shown in Fig 12 (50%). Gaussian curvature is no longer an effective indicator of shape when size also changes because it convolutes shape and size effects. On the other hand, total curvature normalizes size effects and effectively captures the degeneration in the shape of the ideal aorta. This experimental evidence in a controlled simulated system bolsters the empirical data and mathematical formulation, proving the significance of δK.

Discussion

Type B aortic dissection (TBAD) is a life-threatening disease with significant associated morbidity and mortality [17, 19, 20]. While the old paradigm of open surgical repair was fraught with peri-operative risk, new minimally invasive approaches like TEVAR often trade a decrease in initial operative risk for a higher risk of long-term repair failure. Proper identification of patients for TEVAR is therefore critical and necessitates the definition of an appropriate classification scheme [17, 22, 66, 67]. While previous work has focused on linking changes in aortic anatomy and suitability for repair, there remains a dire need to improve our understanding of how best to define geometric changes and to understand their impact on patient outcomes.

Projection of aortic anatomy into the (δK, ℓ⁻¹)-space provides an improved ability to differentiate aortas along the entire spectrum of growth and pathology, including both normal size-related development and pathologic shape-related changes that occur secondary to aortic dissection. We demonstrate that in normal conditions, the aorta undergoes shape-invariant growth (see Fig 7), while, in diseased states, the aorta experiences shape fluctuations defined by increasing δK. As shown in Fig 10, δK significantly outperforms all other available measures of shape in predicting clinical treatment outcomes, including tortuosity, which is prevalent in the clinical arena.

Furthermore, because of the invariant global cylindrical geometry of the aorta, the parameterization of size is dependent only on the single length scale ℓ⁻¹. Higher-dimensional characterizations of size, including area A_T and volume V, do not provide additional information [68, 69]. Thus, current efforts to replace 2R_m with area or volume [68, 69] are unlikely to yield substantially more information (Figs 5 and 10). This universal size scaling provides the quantitative basis behind the utility of maximum diameter throughout the decades of aortic management and further validates the study of shape [67].

Fig 9 compares the efficacy of a single-variable space defined by maximum diameter (2R_m), the clinical standard, with an enhanced shape-size feature space for predicting treatment outcomes. Using size as the sole metric of disease change, the cornerstone of imaging-based practice in other clinical contexts, is inherently problematic because a critical point between closely-spaced and overlapping populations is very sensitive to the available data. For instance, a well-known problem in R_m-based criteria is the bias against smaller-statured female patients because of the heavily male-weighted population-based statistics [70, 71]. This is best illustrated by Fig 9. While arbitrarily defined size classifiers did discriminate amongst normal aortas, successful TEVAR, and failed TEVAR, such scalars lack physical meaning and clinical generalizability outside of the specific cohort being analyzed. This is partly due to natural population-level variation in aortic size in addition to the inherently operator-dependent nature of aortic size measurements (such as diameter) [30].

The addition of the second axis (δK) alleviates this issue by providing a quantifiable and reproducible shape scalar. δK also captures the global geometry of the aorta and is operator-independent. The combined -feature space demonstrates greater than 90% classification accuracy for the same three cohorts. The addition of a tangible shape axis also offers enhanced interpretability of the underlying geometric trends driving aortic pathology. Such a classification space can be clinically applied to pre-operative treatment planning for aortic dissection patients. δK outperforms previously described shape metrics in both the clinical and engineering literatures. Clinically-derived shape measures, such as the tortuosity index, are predominately acquired from aortic centerlines [21, 72]. These measures underperform compared to δK and are no better than ℓ⁻¹ alone for characterizing aortic disease pathology from geometry. As such, future analysis of δK is liable to demonstrate substantial clinical application of δK as a clinical outcomes predictor.

No general theory exists on the meaning of δK divergence. In soft matter systems such as spherical vesicles, where size weakly changes, and in dynamic systems, rapid fluctuations of Gaussian curvature have been linked to so-called ‘topologic catastrophes’ indicative of physical instability [48, 49, 73]. While we do not inherently study aortic stability as it relates to clinical rupture, we show that aortas with high δK independently classify as high risk for clinical complications and poor outcomes post-TEVAR. It is therefore reasonable to conjecture that vertical divergence in the -space is a sign of aortic instability and an indicator of suboptimal suitability for endovascular repair.

The morphologic evolution of biological structures is tightly integrated with disease development in many other contexts. δK is a size-independent shape metric, and because it only requires extrinsic geometric information, this procedure can be applied to any surface mesh geometry. For instance, 3D imaging is extensively used to analyze lung nodules for malignancy [1], breast lesions for tumor growth [74], liver irregularities for cirrhosis [2], cerebral aneurysms [75], and the left ventricle for heart failure [76].

As with the aorta, size-based criteria form the mainstay of clinical approaches, while shape is qualitatively used but has been proven difficult to quantify until now. This methodology is based on a general geometric and topologic foundation, and future analysis will be needed to validate its extension to other clinically relevant problems of characterizing disease through analysis of shape change in medical imaging.

Supporting information

Acknowledgments

We are grateful to Enrique Cerda, Efi Efrati, Haim Diamant, and Thomas Witten for careful reading of the text and detailed comments.