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1.AbstractBrain metastases (BM) remain a significant cause of morbidity and mortality in breast cancer (BC) patients. Specific factors promoting the process of BM and predilection for selected neuro-anatomical regions remain unknown, yet may have major implications for prevention or treatment. Anatomical spatial distributions of BM from BC suggest a predominance of metastases in the hindbrain and cerebellum. Systematic approaches to quantifying BM location or location-based analyses based on molecular subtypes, however, remain largely unavailable. We analyzed stereotactic Cartesian coordinates derived from 134 patients undergoing gamma-knife radiosurgery (GKRS) for treatment of 407 breast cancer BMs to quantitatively study BM spatial distribution along principal component axes and by intrinsic molecular subtype (ER,PR,Herceptin). We corroborated that BC BMs show a consistent propensity to arise posteriorly and caudally, and that Her2+ tumors are relatively more likely to arise medially rather than laterally. To compare the distributions among varying BC molecular subtypes, we used the notion of mutual information, which revealed that the ER-PR-Her2+ and ER-PR-Her2-subtypes showed the smallest amount of mutual information and were most molecularly distinct. Using kernel density estimators, we found a propensity for triple negative BC to arise in more superiorly or cranially situated BMs. BM location maps according to vascular and anatomical distributions using cartesian coordinates to aid in systematic classification of tumor locations were additionally developed. Further characterization of these patterns may have major impacts on treatment or management of cancer patients.SignificanceThe quantitative spatial distribution of breast cancer metastases to the brain, and the effects of breast cancer molecular subtype on distribution frequencies remain poorly understood. We present a novel and shareable workflow for characterizing and comparing spatial distributions of BM which may aid in identifying therapeutic or diagnostic targets and interactions with the tumor microenvironment.

We develop Bayesian methodologies for constructing and estimating a stochastic quasi-chemical model (QCM) for bacterial growth. The deterministic QCM, described as a nonlinear system of ODEs, is treated as a dynamical system with random parameters, and a variational approach is used to approximate their probability distributions and explore the propagation of uncertainty through the model. The approach consists of approximating the parameters’ posterior distribution by a probability measure chosen from a parametric family, through minimization of their Kullback–Leibler divergence.

We study the motion of point vortices on a sphere and, using the methods of linear algebra, find the symmetric configurations of relative equilibrium. Furthermore, we give a catalog of symmetric configurations based on regular polyhedrons. Finally, we investigate the stability of the equilibrium configurations found.

The interactions between two fish swimming side by side are examined. An inviscid fluid model is employed to argue that flapping out of phase can lead to locomotory advantages for both fish. Two effects are examined: (a) the effect of the cooperation between the wakes of the two fish in increasing their forward swimming velocities and (b) the effect of the coupling between the lateral oscillations of the fish and the circu- lations around their bodies in enhancing their forward swimming motion. The models are stripped to their simplest level in order to highlight these effects.

We describe the dynamical system governing the evolution of a system of point vortices on a rotating spherical shell, highlighting features which break what would otherwise be an integrable problem. The importance of the misalignment of the center-of-vorticity vector associated with a cluster of point vortices with the axis of rotation is emphasized as a crucial factor in the interpretation of dynamical features for many flow configurations. We then describe two important physical mechanisms which break what would otherwise be an integrable problem—the interactions between the local center-of-vorticity vectors of more than one region of concentrated vorticity, and the coupling between the center-of-vorticity vector and the background vorticity field which supports Rossby waves. Focusing on the Polar vortex splitting event of September 2002, we describe simple (i.e., low dimensional) mechanisms that can trigger instabilities whose subsequent development cause the onset of chaotic advection and global particle transport. At the linear level, eigenvalues that oscillate between elliptic and hyperbolic configurations initiate the pinch-off process of a passive patch representing the Polar vortex. At the nonlinear level, the evolution and topological bifurcations of the streamline patterns are responsible for its further splitting, stretching, and subsequent transport over the sphere. We finish by briefly describing how to incorporate conservation of potential vorticity and the development of a model governing the probability density function associated with the point vortex system.

We derive formulas for the long time evolution of passive interfaces in three “canonical” incompressible, inviscid, two-dimensional flow models. The point vortex models, introduced in Part 1 [1] are (i) a “restricted” three-vortex problem, (ii) a vortex and a particle in a closed circular domain, and (iii) a particle in the flowfield of a mixing layer model undergoing a vortex pairing instability. In each configuration, it was shown in Part 1 that the passive particle exhibits a geometric or Hannay-Berry phase over long time periods induced by the slowly varying periodic background field. In this paper we show how the formula for the evolution of a passive interface driven by the dynamics of the vortices inherits this geometric phase effect. The interface wraps into a spiral formation around the “parent” vortex, with a slowly varying component induced by the farfield vorticity. The length formula for the long time growth of the slowly rotating spiral decomposes into a “dynamic” part and a “geometric” part. The dynamic part is the length in the “unperturbed” system—i.e., in the absence of the background field—and the geometric part is the contribution of the geometric phase θg for a passive particle in the flow. We derive the following simple formula for an interface along a smooth curve joining two arbitrary particles labelled A and B. Define ΔL(T) as the difference in interface lengths between the “unperturbed” system and the “perturbed” slowly varying system at the end of the long time period T. In each case, \(\Delta L = \lim _{T \to \infty } \Delta L\left( T \right) = - \int_{\xi _A }^{\xi _B } {d\left( {\xi \theta _g } \right)}\), where ξ is the radial coordinate parametrizing the interface at t = 0.

The geometric, or Hannay-Berry, phase is calculated for three canonical point vortex configurations in the plane. The simplest configuration is the three-vortex problem with arbitrary (like signed) circulations, where two of the vortices are near each other compared to the distance between them and a third vortex. We show that the third (distant) vortex induces a geometric phase in the relative angle variable between the two nearby vortices. The second configuration is a particle and vortex in a circular domain. In this problem, the geometric phase is induced on the particle by the presence of the boundary. The third configuration is an infinite row of point vortices undergoing subharmonic pairing. In this case, a geometric phase is induced on a particle orbiting one of the vortices as the vortex pairs orbit each other. In each case we derive the formula for the geometric phase using an asymptotic procedure, then we give it a geometric interpretation. For the asymptotic derivation, we show how the geometric phase can be interpreted as the product of two terms, one of which goes to zero, the other to infinity. Because they go at rates that balance each other, there is a residual O(1) term in the limit e → 0. In this way, we can see that the e → 0 problem is fundamentally different from the e = 0 problem. For the geometric interpretation, we introduce a 1-form γ defined on the plane and show that the phase θg in the appropriate angle variable can be constructed by taking the contour integral of this 1-form over the closed vortex path. In each case this gives the simple formula θg = ∮ γ.