A homological approach to a mathematical definition of pulmonary fibrosis and emphysema on computed tomography.

Three-dimensional imaging is essential to evaluate local abnormalities and understand structure-function relationships in an organ. However, quantifiable and interpretable methods to localize abnormalities remain unestablished. Visual assessments are prone to bias, machine learning methods depend on training images, and the underlying decision principle is usually difficult to interpret. Here, we developed a homological approach to mathematically define emphysema and fibrosis in the lungs on computed tomography (CT). With the use of persistent homology, the density of homological features, including connected components, tunnels, and voids, was extracted from the volumetric CT scans of lung diseases. A pair of CT values at which each homological feature appeared (birth) and disappeared (death) was computed by sweeping the threshold levels from higher to lower CT values. Consequently, fibrosis and emphysema were defined as voxels with dense voids having a longer lifetime (birth-death difference) and voxels with dense connected components having a lower birth, respectively. In an independent dataset including subjects with idiopathic pulmonary fibrosis (IPF), chronic obstructive pulmonary disease (COPD), and combined pulmonary fibrosis and emphysema (CPFE), the proposed definition enabled accurate segmentation with comparable quality to deep learning in terms of Dice coefficients. Persistent homology-defined fibrosis was closely associated with physiological abnormalities such as impaired diffusion capacity and long-term mortality in subjects with IPF and CPFE, and persistent homology-defined emphysema was associated with impaired diffusion capacity in subjects with COPD. The present persistent homology-based evaluation of structural abnormalities could help explore the clinical and physiological impacts of structural changes and morphological mechanisms of disease progression.NEW & NOTEWORTHY This study proposes a homological approach to mathematically define a three-dimensional texture feature of emphysema and fibrosis on chest computed tomography using persistent homology. The proposed definition enabled accurate segmentation with comparable quality to deep learning while offering higher interpretability than deep learning-based methods.

Exact solution to a Liouville equation with Stuart vortex distribution on the surface of a torus.

A steady solution of the incompressible Euler equation on a toroidal surface T R , r of major radius R and minor radius r is provided. Its streamfunction is represented by an exact solution to the modified Liouville equation, ∇ T R , r 2 ψ = c e d ψ + ( 8 / d ) κ , where ∇ T R , r 2 and κ denote the Laplace-Beltrami operator and the Gauss curvature of the toroidal surface respectively, and c, d are real parameters with cd < 0. This is a generalization of the flows with smooth vorticity distributions owing to Stuart (Stuart 1967 J. Fluid Mech. 29, 417-440. (doi:10.1017/S0022112067000941)) in the plane and Crowdy (Crowdy 2004 J. Fluid Mech. 498, 381-402. (doi:10.1017/S0022112003007043)) on the spherical surface. The flow consists of two point vortices at the innermost and the outermost points of the toroidal surface on the same line of a longitude, and a smooth vorticity distribution centred at their antipodal position. Since the surface of a torus has non-constant curvature and a handle structure that are different geometric features from the plane and the spherical surface, we focus on how these geometric properties of the torus affect the topological flow structures along with the change of the aspect ratio α = R/r. A comparison with the Stuart vortex on the flat torus is also made.

Stability of barotropic vortex strip on a rotating sphere.

We study the stability of a barotropic vortex strip on a rotating sphere, as a simple model of jet streams. The flow is approximated by a piecewise-continuous vorticity distribution by zonal bands of uniform vorticity. The linear stability analysis shows that the vortex strip becomes stable as the strip widens or the rotation speed increases. When the vorticity constants in the upper and the lower regions of the vortex strip have the same positive value, the inner flow region of the vortex strip becomes the most unstable. However, when the upper and the lower vorticity constants in the polar regions have different signs, a complex pattern of instability is found, depending on the wavenumber of perturbations, and interestingly, a boundary far away from the vortex strip can be unstable. We also compute the nonlinear evolution of the vortex strip on the rotating sphere and compare with the linear stability analysis. When the width of the vortex strip is small, we observe a good agreement in the growth rate of perturbation at an early time, and the eigenvector corresponding to the unstable eigenvalue coincides with the most unstable part of the flow. We demonstrate that a large structure of rolling-up vortex cores appears in the vortex strip after a long-time evolution. Furthermore, the geophysical relevance of the model to jet streams of Jupiter, Saturn and Earth is examined.

Point vortex interactions on a toroidal surface.

Owing to non-constant curvature and a handle structure, it is not easy to imagine intuitively how flows with vortex structures evolve on a toroidal surface compared with those in a plane, on a sphere and a flat torus. In order to cultivate an insight into vortex interactions on this manifold, we derive the evolution equation for N-point vortices from Green's function associated with the Laplace-Beltrami operator there, and we then formulate it as a Hamiltonian dynamical system with the help of the symplectic geometry and the uniformization theorem. Based on this Hamiltonian formulation, we show that the 2-vortex problem is integrable. We also investigate the point vortex equilibria and the motion of two-point vortices with the strengths of the same magnitude as one of the fundamental vortex interactions. As a result, we find some characteristic interactions between point vortices on the torus. In particular, two identical point vortices can be locally repulsive under a certain circumstance.

One-dimensional hydrodynamic model generating a turbulent cascade.

As a minimal mathematical model generating cascade analogous to that of the Navier-Stokes turbulence in the inertial range, we propose a one-dimensional partial-differential-equation model that conserves the integral of the squared vorticity analog (enstrophy) in the inviscid case. With a large-scale random forcing and small viscosity, we find numerically that the model exhibits the enstrophy cascade, the broad energy spectrum with a sizable correction to the dimensional-analysis prediction, peculiar intermittency, and self-similarity in the dynamical system structure.

Non-self-similar, partial, and robust collapse of four point vortices on a sphere.

This paper gives numerical examples showing that non-self-similar collapse can occur in the motion of four point vortices on a sphere. It is found when the four-vortex problem is integrable, in which the moment of vorticity vector is zero. The non-self-similar collapse has significant properties. It is partial in the sense that three of the four point vortices collapse to one point in finite time and the other one moves to the antipodal position to the collapse point. Moreover, it is robust with respect to perturbation of the initial configuration as long as the system remains integrable.

Efficient topological chaos embedded in the blinking vortex system.

We consider the particle mixing in the plane by two vortex points appearing one after the other, called the blinking vortex system. Mathematical and numerical studies of the system reveal that the chaotic particle mixing, i.e., the chaotic advection, is observed due to the homoclinic chaos, but the mixing region is restricted locally in the neighborhood of the vortex points. The present article shows that it is possible to realize a global and efficient chaotic advection in the blinking vortex system with the help of the Thurston-Nielsen theory, which classifies periodic orbits for homeomorphisms in the plane into three types: periodic, reducible, and pseudo-Anosov (pA). It is mathematically shown that periodic orbits of pA type generate a complicated dynamics, which is called topological chaos. We show that the combination of the local chaotic mixing due to the topological chaos and the dipole-like return orbits realize an efficient and global particle mixing in the blinking vortex system.