Multiple Instance Learning for Heterogeneous Images: Training a CNN for Histopathology

Multiple instance (MI) learning with a convolutional neural network enables end-to-end training in the presence of weak image-level labels. We propose a new method for aggregating predictions from smaller regions of the image into an image-level classification by using the quantile function. The quantile function provides a more complete description of the heterogeneity within each image, improving image-level classification. We also adapt image augmentation to the MI framework by randomly selecting cropped regions on which to apply MI aggregation during each epoch of training. This provides a mechanism to study the importance of MI learning. We validate our method on five different classification tasks for breast tumor histology and provide a visualization method for interpreting local image classifications that could lead to future insights into tumor heterogeneity

Joint and individual analysis of breast cancer histologic images and genomic covariates

A key challenge in modern data analysis is understanding connections between complex and differing modalities of data. For example, two of the main approaches to the study of breast cancer are histopathology (analyzing visual characteristics of tumors) and genetics. While histopathology is the gold standard for diagnostics and there have been many recent breakthroughs in genetics, there is little overlap between these two fields. We aim to bridge this gap by developing methods based on Angle-based Joint and Individual Variation Explained (AJIVE) to directly explore similarities and differences between these two modalities. Our approach exploits Convolutional Neural Networks (CNNs) as a powerful, automatic method for image feature extraction to address some of the challenges presented by statistical analysis of histopathology image data. CNNs raise issues of interpretability that we address by developing novel methods to explore visual modes of variation captured by statistical algorithms (e.g. PCA or AJIVE) applied to CNN features. Our results provide many interpretable connections and contrasts between histopathology and genetics

Passing to the Limit in a Wasserstein Gradient Flow: From Diffusion to Reaction

We study a singular-limit problem arising in the modelling of chemical reactions. At finite {\epsilon} > 0, the system is described by a Fokker-Planck convection-diffusion equation with a double-well convection potential. This potential is scaled by 1/{\epsilon}, and in the limit {\epsilon} -> 0, the solution concentrates onto the two wells, resulting into a limiting system that is a pair of ordinary differential equations for the density at the two wells. This convergence has been proved in Peletier, Savar\'e, and Veneroni, SIAM Journal on Mathematical Analysis, 42(4):1805-1825, 2010, using the linear structure of the equation. In this paper we re-prove the result by using solely the Wasserstein gradient-flow structure of the system. In particular we make no use of the linearity, nor of the fact that it is a second-order system. The first key step in this approach is a reformulation of the equation as the minimization of an action functional that captures the property of being a curve of maximal slope in an integrated form. The second important step is a rescaling of space. Using only the Wasserstein gradient-flow structure, we prove that the sequence of rescaled solutions is pre-compact in an appropriate topology. We then prove a Gamma-convergence result for the functional in this topology, and we identify the limiting functional and the differential equation that it represents. A consequence of these results is that solutions of the {\epsilon}-problem converge to a solution of the limiting problem.Comment: Added two sections, corrected minor typos, updated reference

On thermodynamically consistent Stefan problems with variable surface energy

A thermodynamically consistent two-phase Stefan problem with temperature-dependent surface tension and with or without kinetic undercooling is studied. It is shown that these problems generate local semiflows in well-defined state manifolds. If a solution does not exhibit singularities, it is proved that it exists globally in time and converges towards an equilibrium of the problem. In addition, stability and instability of equilibria is studied. In particular, it is shown that multiple spheres of the same radius are unstable if surface heat capacity is small; however, if kinetic undercooling is absent, they are stable if surface heat capacity is sufficiently large.Comment: To appear in Arch. Ration. Mech. Anal. The final publication is available at Springer via http://dx.doi.org/10.1007/s00205-015-0938-y. arXiv admin note: substantial text overlap with arXiv:1101.376

Geometric Observers for Dynamically Evolving Curves

This paper proposes a deterministic observer design for visual tracking based on nonparametric implicit (level-set) curve descriptions. The observer is continuous discrete with continuous-time system dynamics and discrete-time measurements. Its state-space consists of an estimated curve position augmented by additional states (e.g., velocities) associated with every point on the estimated curve. Multiple simulation models are proposed for state prediction. Measurements are performed through standard static segmentation algorithms and optical-flow computations. Special emphasis is given to the geometric formulation of the overall dynamical system. The discrete-time measurements lead to the problem of geometric curve interpolation and the discrete-time filtering of quantities propagated along with the estimated curve. Interpolation and filtering are intimately linked to the correspondence problem between curves. Correspondences are established by a Laplace-equation approach. The proposed scheme is implemented completely implicitly (by Eulerian numerical solutions of transport equations) and thus naturally allows for topological changes and subpixel accuracy on the computational grid

Thalamocortical Connectivity Correlates with Phenotypic Variability in Dystonia

Dystonia is a brain disorder characterized by abnormal involuntary movements without defining neuropathological changes. The disease is often inherited as an autosomal-dominant trait with incomplete penetrance. Individuals with dystonia, whether inherited or sporadic, exhibit striking phenotypic variability, with marked differences in the somatic distribution and severity of clinical manifestations. In the current study, we used magnetic resonance diffusion tensor imaging to identify microstructural changes associated with specific limb manifestations. Functional MRI was used to localize specific limb regions within the somatosensory cortex. Microstructural integrity was preserved when assessed in subrolandic white matter regions somatotopically related to the clinically involved limbs, but was reduced in regions linked to clinically uninvolved (asymptomatic) body areas. Clinical manifestations were greatest in subjects with relatively intact microstructure in somatotopically relevant white matter regions. Tractography revealed significant phenotype-related differences in the visualized thalamocortical tracts while corticostriatal and corticospinal pathways did not differ between groups. Cerebellothalamic microstructural abnormalities were also seen in the dystonia subjects, but these changes were associated with genotype, rather than with phenotypic variation. The findings suggest that the thalamocortical motor system is a major determinant of dystonia phenotype. This pathway may represent a novel therapeutic target for individuals with refractory limb dystonia

Qualitative behavior of solutions for thermodynamically consistent Stefan problems with surface tension

The qualitative behavior of a thermodynamically consistent two-phase Stefan problem with surface tension and with or without kinetic undercooling is studied. It is shown that these problems generate local semiflows in well-defined state manifolds. If a solution does not exhibit singularities in a sense made precise below, it is proved that it exists globally in time and its orbit is relatively compact. In addition, stability and instability of equilibria is studied. In particular, it is shown that multiple spheres of the same radius are unstable, reminiscent of the onset of Ostwald ripening.Comment: 56 pages. Expanded introduction, added references. This revised version is published in Arch. Ration. Mech. Anal. (207) (2013), 611-66

The visual perception of natural motion: abnormal task-related neural activity in DYT1 dystonia

Although primary dystonia is defined by its characteristic motor manifestations, non-motor signs and symptoms have increasingly been recognized in this disorder. Recent neuroimaging studies have related the motor features of primary dystonia to connectivity changes in cerebello-thalamo-cortical pathways. It is not known, however, whether the non-motor manifestations of the disorder are associated with similar circuit abnormalities. To explore this possibility, we used functional magnetic resonance imaging to study primary dystonia and healthy volunteer subjects while they performed a motion perception task in which elliptical target trajectories were visually tracked on a computer screen. Prior functional magnetic resonance imaging studies of healthy subjects performing this task have revealed selective activation of motor regions during the perception of \u27natural\u27 versus \u27unnatural\u27 motion (defined respectively as trajectories with kinematic properties that either comply with or violate the two-thirds power law of motion). Several regions with significant connectivity changes in primary dystonia were situated in proximity to normal motion perception pathways, suggesting that abnormalities of these circuits may also be present in this disorder. To determine whether activation responses to natural versus unnatural motion in primary dystonia differ from normal, we used functional magnetic resonance imaging to study 10 DYT1 dystonia and 10 healthy control subjects at rest and during the perception of \u27natural\u27 and \u27unnatural\u27 motion. Both groups exhibited significant activation changes across perceptual conditions in the cerebellum, pons, and subthalamic nucleus. The two groups differed, however, in their responses to \u27natural\u27 versus \u27unnatural\u27 motion in these regions. In healthy subjects, regional activation was greater during the perception of natural (versus unnatural) motion (P \u3c 0.05). By contrast, in DYT1 dystonia subjects, activation was relatively greater during the perception of unnatural (versus natural) motion (P \u3c 0.01). To explore the microstructural basis for these functional changes, the regions with significant interaction effects (i.e. those with group differences in activation across perceptual conditions) were used as seeds for tractographic analysis of diffusion tensor imaging scans acquired in the same subjects. Fibre pathways specifically connecting each of the significant functional magnetic resonance imaging clusters to the cerebellum were reconstructed. Of the various reconstructed pathways that were analysed, the ponto-cerebellar projection alone differed between groups, with reduced fibre integrity in dystonia (P \u3c 0.001). In aggregate, the findings suggest that the normal pattern of brain activation in response to motion perception is disrupted in DYT1 dystonia. Thus, it is unlikely that the circuit changes that underlie this disorder are limited to primary sensorimotor pathways