Elastic Shape Analysis of Surfaces with Second-Order Sobolev Metrics: A Comprehensive Numerical Framework.

This paper introduces a set of numerical methods for Riemannian shape analysis of 3D surfaces within the setting of invariant (elastic) second-order Sobolev metrics. More specifically, we address the computation of geodesics and geodesic distances between parametrized or unparametrized immersed surfaces represented as 3D meshes. Building on this, we develop tools for the statistical shape analysis of sets of surfaces, including methods for estimating Karcher means and performing tangent PCA on shape populations, and for computing parallel transport along paths of surfaces. Our proposed approach fundamentally relies on a relaxed variational formulation for the geodesic matching problem via the use of varifold fidelity terms, which enable us to enforce reparametrization independence when computing geodesics between unparametrized surfaces, while also yielding versatile algorithms that allow us to compare surfaces with varying sampling or mesh structures. Importantly, we demonstrate how our relaxed variational framework can be extended to tackle partially observed data. The different benefits of our numerical pipeline are illustrated over various examples, synthetic and real. Supplementary Information: The online version contains supplementary material available at 10.1007/s11263-022-01743-0.

Mechanistic Modeling of Longitudinal Shape Changes: Equations of Motion and Inverse Problems.

This paper examines a longitudinal shape evolution model in which a three-dimensional volume progresses through a family of elastic equilibria in response to the time-derivative of an internal force, or yank, with an additional regularization to ensure diffeomorphic transformations. We consider two different models of yank and address the long time existence and uniqueness of solutions for the equations of motion in both models. In addition, we derive sufficient conditions for the existence of an optimal yank that best describes the change from an observed initial volume to an observed volume at a later time. The main motivation for this work is the understanding of processes such as growth and atrophy in anatomical structures, where the yank could be roughly interpreted as a metabolic event triggering morphological changes. We provide preliminary results on simple examples to illustrate, under this model, the retrievability of some attributes of such events.

Landmark-free morphometric analysis of knee osteoarthritis using joint statistical models of bone shape and articular space variability.

Purpose: Osteoarthritis (OA) is a common degenerative disease involving a variety of structural changes in the affected joint. In addition to narrowing of the articular space, recent studies involving statistical shape analysis methods have suggested that specific bone shapes might be associated with the disease. We aim to investigate the feasibility of using the recently introduced framework of functional shapes (Fshape) to extract morphological features of OA that combine shape variability of articular surfaces of the tibia (or femur) together with the changes of the joint space. Approach: Our study uses a dataset of three-dimensional cone-beam CT volumes of 17 knees without OA and 17 knees with OA. Each knee is then represented as an object (Fshape) consisting of a triangulated tibial (or femoral) articular surface and a map of joint space widths (JSWs) measured at the points of this surface (joint space map, JSM). We introduce a generative atlas model to estimate a template (mean) Fshape of the sample population together with template-centered variables that model the transformations from the template to each subject. This approach has two potential advantages compared with other statistical shape modeling methods that have been investigated in knee OA: (i) Fshapes simultaneously consider the variability in bone shape and JSW, and (ii) Fshape atlas estimation is based on a diffeomorphic transformation model of surfaces that does not require a priori landmark correspondences between the subjects. The estimated atlas-to-subject Fshape transformations were used as input to principal component analysis dimensionality reduction combined with a linear support vector machine (SVM) classifier to identify the morphological features of OA. Results: Using tibial articular surface as the shape component of the Fshape, we found leave-one-out cross validation scores of ≈ 91.18 % for the classification based on the bone surface transformations alone, ≈ 91.18 % for the classification based on the residual JSM, and ≈ 85.29 % for the classification using both Fshape components. Similar results were obtained using femoral articular surfaces. The discriminant directions identified in the statistical analysis were associated with medial narrowing of the joint space, steeper intercondylar eminence, and relative deepening of the medial tibial plateau. Conclusions: The proposed approach provides an integrated framework for combined statistical analysis of shape and JSPs. It can successfully extract features correlated to OA that appear consistent with previous studies in the field. Although future large-scale study is necessary to confirm the significance of these findings, our results suggest that the functional shape methodology is a promising new tool for morphological analysis of OA and orthopedics data in general.

Joint spatial-angular sparse coding for dMRI with separable dictionaries.

Diffusion MRI (dMRI) provides the ability to reconstruct neuronal fibers in the brain, in vivo, by measuring water diffusion along angular gradient directions in q-space. High angular resolution diffusion imaging (HARDI) can produce better estimates of fiber orientation than the popularly used diffusion tensor imaging, but the high number of samples needed to estimate diffusivity requires longer patient scan times. To accelerate dMRI, compressed sensing (CS) has been utilized by exploiting a sparse dictionary representation of the data, discovered through sparse coding. The sparser the representation, the fewer samples are needed to reconstruct a high resolution signal with limited information loss, and so an important area of research has focused on finding the sparsest possible representation of dMRI. Current reconstruction methods however, rely on an angular representation per voxel with added spatial regularization, and so, for non-zero signals, one is required to have at least one non-zero coefficient per voxel. This means that the global level of sparsity must be greater than the number of voxels. In contrast, we propose a joint spatial-angular representation of dMRI that will allow us to achieve levels of global sparsity that are below the number of voxels. A major challenge, however, is the computational complexity of solving a global sparse coding problem over large-scale dMRI. In this work, we present novel adaptations of popular sparse coding algorithms that become better suited for solving large-scale problems by exploiting spatial-angular separability. Our experiments show that our method achieves significantly sparser representations of HARDI than is possible by the state of the art.

Age and Glaucoma-Related Characteristics in Retinal Nerve Fiber Layer and Choroid: Localized Morphometrics and Visualization Using Functional Shapes Registration.

Optical coherence tomography provides high-resolution 3D imaging of the posterior segment of the eye. However, quantitative morphological analysis, particularly relevant in retinal degenerative diseases such as glaucoma, has been confined to simple sectorization and averaging with limited spatial sensitivity for detection of clinical markers. In this paper, we present point-wise analysis and visualization of the retinal nerve fiber layer and choroid from cross-sectional data using functional shapes (fshape) registration. The fshape framework matches two retinas, or generates a mean of multiple retinas, by jointly optimizing the surface geometry and functional signals mapped on the surface. We generated group-wise mean retinal nerve fiber layer and choroidal surfaces with the respective layer thickness mapping and showed the difference by age (normal, younger vs. older) and by disease (age-matched older, normal vs. glaucomatous) in the two layers, along with a more conventional sector-based analysis for comparison. The fshape results visualized the detailed spatial patterns of the differences between the age-matched normal and glaucomatous retinal nerve fiber layers, with the glaucomatous layers most significantly thinner in the inferior region close to Bruch's membrane opening. Between the young and older normal cases, choroid was shown to be significantly thinner in the older subjects across all regions, but particularly in the nasal and inferior regions. The results demonstrate a comprehensive and detailed analysis with visualization of morphometric patterns by multiple factors.

Amotl1 mediates sequestration of the Hippo effector Yap1 downstream of Fat4 to restrict heart growth.

Although in flies the atypical cadherin Fat is an upstream regulator of Hippo signalling, the closest mammalian homologue, Fat4, has been shown to regulate tissue polarity rather than growth. Here we show in the mouse heart that Fat4 modulates Hippo signalling to restrict growth. Fat4 mutant myocardium is thicker, with increased cardiomyocyte size and proliferation, and this is mediated by an upregulation of the transcriptional activity of Yap1, an effector of the Hippo pathway. Fat4 is not required for the canonical activation of Hippo kinases but it sequesters a partner of Yap1, Amotl1, out of the nucleus. The nuclear translocation of Amotl1 is accompanied by Yap1 to promote cardiomyocyte proliferation. We, therefore, identify Amotl1, which is not present in flies, as a mammalian intermediate for non-canonical Hippo signalling, downstream of Fat4. This work uncovers a mechanism for the restriction of heart growth at birth, a process which impedes the regenerative potential of the mammalian heart.

Atlas-based shape analysis and classification of retinal optical coherence tomography images using the functional shape (fshape) framework.

We propose a novel approach for quantitative shape variability analysis in retinal optical coherence tomography images using the functional shape (fshape) framework. The fshape framework uses surface geometry together with functional measures, such as retinal layer thickness defined on the layer surface, for registration across anatomical shapes. This is used to generate a population mean template of the geometry-function measures from each individual. Shape variability across multiple retinas can be measured by the geometrical deformation and functional residual between the template and each of the observations. To demonstrate the clinical relevance and application of the framework, we generated atlases of the inner layer surface and layer thickness of the Retinal Nerve Fiber Layer (RNFL) of glaucomatous and normal subjects, visualizing detailed spatial pattern of RNFL loss in glaucoma. Additionally, a regularized linear discriminant analysis classifier was used to automatically classify glaucoma, glaucoma-suspect, and control cases based on RNFL fshape metrics.

Morphometry of anatomical shape complexes with dense deformations and sparse parameters.

We propose a generic method for the statistical analysis of collections of anatomical shape complexes, namely sets of surfaces that were previously segmented and labeled in a group of subjects. The method estimates an anatomical model, the template complex, that is representative of the population under study. Its shape reflects anatomical invariants within the dataset. In addition, the method automatically places control points near the most variable parts of the template complex. Vectors attached to these points are parameters of deformations of the ambient 3D space. These deformations warp the template to each subject's complex in a way that preserves the organization of the anatomical structures. Multivariate statistical analysis is applied to these deformation parameters to test for group differences. Results of the statistical analysis are then expressed in terms of deformation patterns of the template complex, and can be visualized and interpreted. The user needs only to specify the topology of the template complex and the number of control points. The method then automatically estimates the shape of the template complex, the optimal position of control points and deformation parameters. The proposed approach is completely generic with respect to any type of application and well adapted to efficient use in clinical studies, in that it does not require point correspondence across surfaces and is robust to mesh imperfections such as holes, spikes, inconsistent orientation or irregular meshing. The approach is illustrated with a neuroimaging study of Down syndrome (DS). The results demonstrate that the complex of deep brain structures shows a statistically significant shape difference between control and DS subjects. The deformation-based modelingis able to classify subjects with very high specificity and sensitivity, thus showing important generalization capability even given a low sample size. We show that the results remain significant even if the number of control points, and hence the dimension of variables in the statistical model, are drastically reduced. The analysis may even suggest that parsimonious models have an increased statistical performance. The method has been implemented in the software Deformetrica, which is publicly available at www.deformetrica.org.