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We address the problem of subclassification of rare circulating cells using data driven feature selection from images of candidate circulating tumor cells from patients diagnosed with breast, prostate, or lung cancer. We determine a set of low level features which can differentiate among candidate cell types. We have implemented an image representation based on concentric Fourier rings (FRDs) which allow us to exploit size variations and morphological differences among cells while being rotationally invariant. We discuss potential clinical use in the context of treatment monitoring for cancer patients with metastatic disease.

Thrombotic events can herald the diagnosis of cancer, preceding any cancer-related clinical symptoms. Patients with cancer are at a 4- to 7-fold increased risk of suffering from venous thromboembolism (VTE), with ∼7,000 patients with lung cancer presenting from VTEs. However, the physical biology underlying cancer-associated VTE remains poorly understood. Several lines of evidence suggest that the shedding of tissue factor (TF)-positive circulating tumor cells (CTCs) and microparticles from primary tumors may serve as a trigger for cancer-associated thrombosis. To investigate the potential direct and indirect roles of CTCs in VTE, we characterized thrombin generation by CTCs in an interactive numerical model coupling blood flow with advection-diffusion kinetics. Geometric measurements of CTCs isolated from the peripheral blood of a lung cancer patient prior to undergoing lobectomy formed the basis of the simulations. Singlet, doublet, and aggregate circulating tumor microemboli (CTM) were investigated in the model. Our numerical model demonstrated that CTM could potentiate occlusive events that drastically reduce blood flow and serve as a platform for the promotion of thrombin generation in flowing blood. These results provide a characterization of CTM dynamics in the vasculature and demonstrate an integrative framework combining clinical, biophysical, and mathematical approaches to enhance our understanding of CTCs and their potential direct and indirect roles in VTE formation.

Tumor development is an evolutionary process in which a heterogeneous population of cells with different growth capabilities compete for resources in order to gain a proliferative advantage. What are the minimal ingredients needed to recreate some of the emergent features of such a developing complex ecosystem? What is a tumor doing before we can detect it? We outline a mathematical model, driven by a stochastic Moran process, in which cancer cells and healthy cells compete for dominance in the population. Each are assigned payoffs according to a Prisoner’s Dilemma evolutionary game where the healthy cells are the cooperators and the cancer cells are the defectors. With point mutational dynamics, heredity, and a fitness landscape controlling birth and death rates, natural selection acts on the cell population and simulated ‘cancer-like’ features emerge, such as Gompertzian tumor growth driven by heterogeneity, the log-kill law which (linearly) relates therapeutic dose density to the (log) probability of cancer cell survival, and the Norton-Simon hypothesis which (linearly) relates tumor regression rates to tumor growth rates. We highlight the utility, clarity, and power that such models provide, despite (and because of) their simplicity and built-in assumptions.

Human mobility has been studied extensively in various biomedical contexts with applications in clinical rehabilitation, disease diagnosis, health risk prognosis, and general performance assessments. In this paper, we present ATOMHP (Analytical Technologies to Objectively Measure Human Performance) Kinect: a system to objectively quantify human performance using the Microsoft Kinect as a single camera sensor to capture human mobility. We explore the viability of this noninvasive performance assessment system by studying a cohort of cancer patients undergoing various therapy regimens who are assigned a performance score based on a qualitative clinical test. The ATOM-HP Kinect is a clinically usable system which consists of tools for Kinect, clinical data collection, data quality validation, and mobility feature extraction, which can be used for downstream analysis of performance. Preliminary results based on the clinical case study indicate that ATOM-HP Kinect can quantify changes in kinematic parameters, and that these features are correlated with clinically measured risk factors which could be used for early prediction of diseases, or making decision on treatment modification.

We extended the classical tumor regression models such as Skipper's laws and the Norton–Simon hypothesis from instantaneous regression rates to the cumulative effect over repeated cycles of chemotherapy. To achieve this end, we used a stochastic Moran process model of tumor cell kinetics coupled with a prisoner's dilemma game-theoretic cell–cell interaction model to design chemotherapeutic strategies tailored to different tumor growth characteristics. Using the Shannon entropy as a novel tool to quantify the success of dosing strategies, we contrasted MTD strategies as compared with low-dose, high-density metronomic strategies (LDM) for tumors with different growth rates. Our results show that LDM strategies outperformed MTD strategies in total tumor cell reduction. This advantage was magnified for fast-growing tumors that thrive on long periods of unhindered growth without chemotherapy drugs present and was not evident after a single cycle of chemotherapy but grew after each subsequent cycle of repeated chemotherapy. The evolutionary growth/regression model introduced in this article agrees well with murine models. Overall, this model supports the concept of designing different chemotherapeutic schedules for tumors with different growth rates and develops quantitative tools to optimize these schedules for maintaining low-volume tumors. Cancer Res; 77(23); 6717–28. ©2017 AACR.

We extend classical tumor regression models, such as the Norton-Simon hypothesis, from instantaneous regression rates (i.e. the derivative) to the cumulative effect (i.e. the integral) over one (or many) cycles of chemotherapy. To achieve this end, we use a stochastic Moran process model of tumor cell kinetics, coupled with a prisoner’s dilemma game-theoretic cell-cell interaction model to design chemotherapeutic strategies tailored to different tumor growth characteristics. Using the Shannon entropy as a novel tool to quantify the success of dosing strategies, we contrast maximum tolerated dose (MTD) strategies as compared with low dose, high density metronomic strategies (LDM) for tumors with different growth rates. Our results show that LDM strategies can outperform MTD strategies in total tumor cell reduction (TCR). The advantage is magnified for fast growing tumors that thrive on long periods of unhindered growth without chemotherapy drugs present and is not evident after a single cycle of chemotherapy, but grows after each subsequent cycle of repeated chemotherapy. The model supports the concept of designing different chemotherapeutic schedules for tumors with different growth rates and develops quantitative tools to optimize these schedules for maintaining low volume tumors. The evolutionary model we introduce in this paper is compared with regression data from murine models and shown to be in good agreement.Major FindingsModel simulations show that metronomic (low dose, high density) therapies can outperform maximum tolerated dose (high dose, low density) therapies. This is due to the fact that tumor cell reduction is more sensitive to changes in dose density than changes in dose concentration, especially for faster growing tumors. This effect is negligible after a single cycle of chemotherapy, but magnified after many cycles. The model also allows for novel chemotherapeutic schedules and quantifies their performance according to tumor growth rate.

When a coin falls in water, its trajectory is one of four types determined by its dimensionless moment of inertia $I^\ast$ and Reynolds number Re: (A) steady; (B) fluttering; (C) chaotic; or (D) tumbling. The dynamics induced by the interaction of the water with the surface of the coin, however, makes the exact landing site difficult to predict a priori. Here, we describe a carefully designed experiment in which a coin is dropped repeatedly in water, so that we can determine the probability density functions (pdf) associated with the landing positions for each of the four trajectory types, all of which are radially symmetric about the center-drop line. In the case of the steady mode, the pdf is approximately Gaussian distributed, with variances that are small, indicating that the coin is most likely to land at the center, right below the point it is dropped from. For the other falling modes, the center is one of the least likely landing sites. Indeed, the pdf's of the fluttering, chaotic and tumbling modes are characterized by a "dip" around the center. For the tumbling mode, the pdf is a ring configuration about the center-line, with a ring width that depends on the dimensionless parameters $I^\ast$ and Re and height from which the coin is dropped. For the chaotic mode, the pdf is generally a broadband distribution spread out radially symmetrically about the center-line. For the steady and fluttering modes, the coin never flips, so the coin lands with the same side up as was dropped. For the chaotic mode, the probability of heads or tails is close to 0.5. In the case of the tumbling mode, the probability of heads or tails based on the height of the drop which determines whether the coin flips an even or odd number of times during descent.

Currently in patients with bladder cancer, various clinical evaluations (imaging, operative findings at transurethral resection and radical cystectomy, pathology) are collectively used to determine disease status and prognosis, and recommend neoadjuvant, definitive and adjuvant treatments. We analyze the predictive power of these measurements in forecasting two key long-term outcomes following radical cystectomy, i.e., cancer recurrence and survival. Information theory and machine learning algorithms are employed to create predictive models using a large prospective, continuously collected, temporally resolved, primary bladder cancer dataset comprised of 3503 patients (1971-2016). Patient recurrence and survival one, three, and five years after cystectomy can be predicted with greater than 70% sensitivity and specificity. Such predictions may inform patient monitoring schedules and post-cystectomy treatments. The machine learning models provide a benchmark for predicting oncologic outcomes in patients undergoing radical cystectomy and highlight opportunities for improving care using optimal preoperative and operative data collection.

The notion of a finite dipole is introduced as a pair of equal and opposite strength point vortices (i.e. a vortex dipole) separated by a finite distance. Equations of motion for N finite dipoles interacting in an unbounded inviscid fluid are derived from the modified interaction of 2 N independent vortices subject to the constraint that the inter-vortex spacing of each constrained dipole, ℓ, remains constant. In the absence of all other dipoles and background flow, a single dipole moves in a straight line along the perpendicular bisector of the line segment joining the two point vortices comprising the dipole, with a self-induced velocity inversely proportional to ℓ. When more than one dipole is present, the velocity of the dipole centre is the sum of the self-induced velocity and the average of the induced velocities on each vortex comprising the pair due to all the other dipoles. Each dipole orients in the direction of shear gradient based on the difference in velocities on each of the two vortices in the pair. Several numerical experiments are shown to illustrate the interactions between two and three dipoles in abreast and tandem configurations. We also show that equilibria (multi-poles) can form as a result of the interactions, and we study the stability of polygonal equilibria, showing that the N =3 case is linearly stable, whereas the N >3 case is linearly unstable.

We characterize all stationary equilibrium point singularity distributions in the plane of logarithmic type, allowing for real, imaginary or complex singularity strengths. The dynamical system follows from the assumption that each of the N singularities moves according to the flow field generated by all the others at that point. For strength vector , the dynamical system is the classical point vortex system obtained from a singular discrete representation of the vorticity field from ideal, incompressible fluid flow. When , it corresponds to a system of sources and sinks, whereas when the system consists of spiral sources and sinks discussed in Kochin et al (1964 Theoretical Hydromechanics 1 (London: Interscience)). We formulate the equilibrium problem as one in linear algebra, , , , where A is a N × N complex skew-symmetric configuration matrix which encodes the geometry of the system of interacting singularities. For an equilibrium to exist, A must have a kernel and must be an element of the nullspace of A. We prove that when N is odd, A always has a kernel, hence there is a choice of for which the system is a stationary equilibrium. When N is even, there may or may not be a non-trivial nullspace of A, depending on the relative position of the points in the plane. We provide examples of evenly and randomly distributed points on curves such as circles, figure eights, flower-petal configurations and spirals. We then show how to classify the stationary equilibria in terms of the singular spectrum of A.

We examine the vortical wake structure shed from a deformable Joukowski airfoil in an unbounded volume of inviscid and incompressible fluid. The deformable airfoil is considered to model a flapping fish. The vortex shedding is accounted for using an unsteady point vortex model commonly referred to as the Brown–Michael model. The airfoil’s deformations and rotations are prescribed in terms of a Jacobi elliptic function which exhibits, depending on a dimensionless parameter m , a range of periodic behaviors from sinusoidal to a more impulsive type flapping. Depending on the parameter m and the Strouhal number, one can identify five distinct wake structures, ranging from arrays of isolated point vortices to vortex dipoles and tripoles shed into the wake with every half-cycle of the airfoil flapping motion. We describe these regimes in the context of other published works which categorize wake topologies, and speculate on the importance of these wake structures in terms of periodic swimming and transient maneuvers of fish.

We investigate the interaction between multiple arrays of (reverse) von Karman streets as a model for the mid-wake regions produced by schooling fish. There exist configurations where an infinite array of vortex streets is in relative equilibrium, that is, the streets move together with the same translational velocity. We examine the topology of the streamline patterns in a frame moving with the same translational velocity as the streets. Fluid is advected along different paths depending on the distance separating two adjacent streets. When the distance between the streets is large enough, each street behaves as a single von Karman street and fluid moves globally between two adjacent streets. When the streets get closer to each other, the number of streets that enter into partnership in transporting fluid among themselves increases. This observation motivates a bifurcation analysis which links the distance between streets to the maximum number of streets transporting fluid among themselves. We describe a scaling law relating the number of streets that enter into partnership as a function of the three main parameters associated with the system, two associated with each individual street (determining the aspect ratio of the street), and a third associated with the distance between neighboring streets. In the final section we speculate on the timescale associated with the lifetime of the coherence of this mid-wake scaling regime.

A submerged spring–mass ring is analysed as a simple model for the way in which an underwater swimmer couples its body deformations to the surrounding fluid in order to accomplish locomotion. We adopt an inviscid, incompressible, irrotational assumption for the surrounding fluid and analyse the coupling response to various modes of excitation of the ring configuration. Due to the added mass effect, the surrounding fluid provides an environment which effectively couples the ‘normal modes’ of oscillation of the ring, leading to nonlinear trajectories if the ring is free to accelerate based on the effective forces the oscillations induce. Through a series of examples, we demonstrate various features that the model supports, including the locomotion on curved paths as a result of energy and angular momentum exchange with the surrounding fluid.

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.

Bladder cancer (BCa), the 6th commonest cancer in the U.S., is highly lethal when metastatic. Sites and patterns of patient-specific metastatic spread are deemed random and unpredictable. Whether BCa metastatic patterns can be quantified and predicted more accurately is unknown. We used a prospective, longitudinal dataset of 3,505 BCa patients who underwent definitive treatment following diagnosis and were continuously enrolled (1971-2016; mean follow-up 6.13 years). Metastases developed in 30% (n=1,040) of patients, with 5-year survival rate of 18%, compared to 72% in those without metastases (n=2,465). The three commonest metastatic sites at time-of-first progression were pelvis (n=283; 27.2%), bone (n=274; 26.3%), and lung (n=247; 23.8%). We illustrate metastatic pathway progression as color-coded, circular, tree-ring diagrams (primary in the center; metastatic sites outwards). Markov chain modeling, denoting nodes as potential anatomic metastatic sites, indicated higher probability of spread from the bladder to pelvis (15.6%), bone (15.6%), and lung (13.9%), categorizing 4 of 10 anatomical sites as ‘spreaders' and 2 as ‘sponges'. We created a dynamical, data-visualization, web-platform that displays temporal, spatial and Markov modeling figures with predictive capability. For contrasting subgroups, this platform indicated differences in transition probabilities, rank-ordering of metastatic sites, and spreader/sponge node classification. Spatiotemporal patterns of BCa metastasis and sites of spread indicate underlying organotropic mechanisms in the prediction of response. This recognition opens the possibility of organ site-specific therapeutic targeting in the oligo-metastatic disease setting. In the precision medicine era, visualization of complex, time-resolved clinical data will enhance management of post-operative, metastatic BCa patients. Citation Format: Jeremy M. Mason, Zaki Hasnain, Gus Miranda, Karanvir Gill, Paul K. Newton, Inderbir S. Gill, Peter Kuhn. Pathways of metastatic bladder cancer from a longitudinal patient data set [abstract]. In: Proceedings of the American Association for Cancer Research Annual Meeting 2018; 2018 Apr 14-18; Chicago, IL. Philadelphia (PA): AACR; Cancer Res 2018;78(13 Suppl):Abstract nr 4261.

6519Background: Functional status and predictors that identify those cancer patients at risk for unplanned hospitalization can have broad implications for the health care system and clinical trials...

We study properties of an ensemble of Sudoku matrices (a special type of doubly stochastic matrix when normalized) using their statistically averaged singular values. The determinants are very nearly Cauchy distributed about the origin. The largest singular value is , while the others decrease approximately linearly. The normalized singular values (obtained by dividing each singular value by the sum of all nine singular values) are then used to calculate the average Shannon entropy of the ensemble, a measure of the distribution of ‘energy’ among the singular modes and interpreted as a measure of the disorder of a typical matrix. We show the Shannon entropy of the ensemble to be 1.7331±0.0002, which is slightly lower than an ensemble of 9×9 Latin squares, but higher than a certain collection of 9×9 random matrices used for comparison. Using the notion of relative entropy or Kullback–Leibler divergence , which gives a measure of how one distribution differs from another, we show that the relative entropy between the ensemble of Sudoku matrices and Latin squares is of the order of 10 −5 . By contrast, the relative entropy between Sudoku matrices and the collection of random matrices has the much higher value, being of the order of 10 −3 , with the Shannon entropy of the Sudoku matrices having better distribution among the modes. We finish by ‘reconstituting’ the ‘average’ Sudoku matrix from its averaged singular components.

We answer the question of whether optimal packings of circles on a sphere are equilibrium solutions to the logarithmic particle interaction problem for values of N =3–12 and 24, the only values of N for which the optimal packing problem (also known as the Tammes problem) has rigorously known solutions. We also address the cases N =13–23 where optimal packing solutions have been computed, but not proven analytically. As in Jamaloodeen & Newton (Jamaloodeen & Newton 2006 Proc. R. Soc. Lond. Ser. A 462 , 3277–3299. ( doi:10.1098/rspa.2006.1731 )), a logarithmic, or point vortex equilibrium is determined by formulating the problem as the one in linear algebra, , where A is a N ( N −1)/2× N non-normal configuration matrix obtained by requiring that all interparticle distances remain constant. If A has a kernel, the strength vector is then determined as a right-singular vector associated with the zero singular value, or a vector that lies in the nullspace of A where the kernel is multi-dimensional. First we determine if the known optimal packing solution for a given value of N has a configuration matrix A with a non-empty nullspace. The answer is yes for N =3–9, 11–14, 16 and no for N =10, 15, 17–24. We then determine a basis set for the nullspace of A associated with the optimally packed state, answer the question of whether N -equal strength particles, , form an equilibrium for this configuration, and describe what is special about the icosahedral configuration from this point of view. We also find new equilibria by implementing two versions of a random walk algorithm. First, we cluster sub-groups of particles into patterns during the packing process, and ‘grow’ a packed state using a version of the ‘yin-yang’ algorithm of Longuet-Higgins (Longuet-Higgins 2008 Proc. R. Soc. A (doi:10.1098/rspa.2008.0219)). We also implement a version of our ‘Brownian ratchet’ algorithm to find new equilibria near the optimally packed state for N =10, 15, 17–24.

We formulate the problem of finding equilibrium configurations of N -point vortices in the plane in terms of a gradient flow on the smallest singular value of a skew-symmetric matrix M whose nullspace structure determines the (real) strengths, rotational frequency and translational velocity of the configuration. A generic configuration gives rise to a matrix with empty nullspace, and hence is not a relative equilibrium for any choice of vortex strengths. We formulate the problem as a gradient flow in the space of square covariance matrices M T M . The evolution equation for drives the configuration to one with a real nullspace, establishing the existence of an equilibrium for vortex strengths that are elements of the nullspace of the matrix. We formulate both the unconstrained gradient flow problem where the point vortex strengths are determined a posteriori by the nullspace of M and the constrained problem where the point vortex strengths are chosen a priori and one seeks configurations for which those strengths are elements of the nullspace.

The oscillations of a class of submerged mass–spring systems are examined. An inviscid fluid model is employed to show that the hydrodynamic effects couple the normal modes of these systems. This coupling of normal modes can excite the displacement mode – yielding passive locomotion of the system – even when starting with zero displacement velocity. This is in contrast with the fact that under similar initial conditions but without the hydrodynamic coupling, such systems cannot achieve a net displacement. These ideas are illustrated via two examples of a two-mass and a three-mass system.

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 describe a Brownian ratchet scheme that we use to calculate relative equilibrium configurations of N point vortices of mixed strength on the surface of a unit sphere. We formulate it as a problem in linear algebra, A Γ =0, where A is a N ( N −1)/2× N non-normal configuration matrix obtained by requiring that all inter-vortical distances on the sphere remain constant and Γ ∈ N is the (unit) vector of vortex strengths that must lie in the null space of A . Existence of an equilibrium is expressed by the condition det( A T A )=0, while uniqueness follows if Rank( A )= N −1. The singular value decomposition of A is used to calculate an optimal basis set for the null space, yielding all values of the vortex strengths for which the configuration is an equilibrium and allowing us to decompose the equilibrium configuration into basis components. To home in on an equilibrium, we allow the point vortices to undergo a random walk on the sphere and, after each step, we compute the smallest singular value of the configuration matrix, keeping the new arrangement only if it decreases. When the smallest singular value drops below a predetermined convergence threshold, the existence criterion is satisfied and an equilibrium configuration is achieved. We then find a basis set for the null space of A , and hence the vortex strengths, by calculating the right singular vectors corresponding to the singular values that are zero. We show a gallery of examples of equilibria with one-dimensional null spaces obtained by this method. Then, using an unbiased ensemble of 1000 relative equilibria for each value N =4→10, we discuss some general features of the statistically averaged quantities, such as the Shannon entropy (using all of the normalized singular values) and Frobenius norm, centre-of-vorticity vector and Hamiltonian energy.

The evolution, interaction and scattering of 2Npoint vortices grouped into equal and opposite pairs (N-dipoles) on a rotating unit sphere are studied. A new coordinate system made up of centres of vorticity and centroids associated with each dipole is introduced. With these coordinates, the nonlinear equations for an isolated dipole diagonalize and one directly obtains the equation for geodesic motion on the sphere for the dipole centroid. When two or more dipoles interact, the equations are viewed as an interacting billiard system on the sphere—charged billiards—with long-range interactions causing the centroid trajectories to deviate from their geodesic paths. Canonical interactions are studied both with and without rotation. For two dipoles, the four basic interactions are described asexchange-scattering,non-exchange-scattering,loop-scattering(head on) and loop-scattering (chasing) interactions. For three or more dipoles, one obtains a richer variety of interactions, although the interactions identified in the two-dipole case remain fundamental.

A theory capable of producing equilibrium configurations of point vortices in the plane, along with a numerical scheme to compute them, is described. The theory is formulated as a problem in linear algebra where one must find solutions to the matrix equation , where A is the (1/2) N ( N −1)× N non-normal configuration matrix obtained by requiring that all intervortical distances remain fixed, and are the N -vortex strengths. For existence of an equilibrium, A must have a non-trivial nullspace. We consider the singular values of A ; when this has one or more zero singular values, the nullspace of A is non-empty and an equilibrium exists for some choice of Γ . New equilibrium configurations are found numerically by randomly depositing N points in the plane, which generically gives rise to a configuration matrix A with empty nullspace. Using the sum of squares of the k smallest singular values of A as a ‘ratchet’, we ‘thermally fluctuate’ the configuration, allowing each point to execute a random walk in the plane, retaining only those configurations which reduce this quantity at the next step. The configuration is thus driven to one with nullspace ( A )= k >0. These converged states are not necessarily nearby their initial configurations, typically they are asymmetric, and often we can drive the same initial state to several different equilibria. A reverse-ratchet method is also described, which can produce initial conditions that would evolve to a specified equilibrium state. Once a converged final state is achieved, the full singular value decomposition of A is used to calculate an optimal basis set for the nullspace of A and thus all allowable Γ . The distribution of the singular values gives important information on the size of each equilibrium state (as measured by Frobenius norm), their distance from each other (spacing and density) and how far a randomly chosen system of N points in the plane is from the nearest equilibrium configuration with a specified rank, as well as its Shannon entropy.

We study the evolution of N -point vortices in ring formation embedded in a background flowfield that initially corresponds to solid-body rotation on a sphere. The evolution of the point vortices is tracked numerically as an embedded dynamical system along with the M contours which separate strips of constant vorticity. The full system is a discretization of the Euler equations for incompressible flow on a rotating spherical shell, hence a ‘barotropic’ model of the one-layer atmosphere. We describe how the coupling creates a mechanism by which energy is exchanged between the ring and the background, which ultimately serves to break up the structure. When the centre-of-vorticity vector associated with the ring is initially misaligned with the axis of rotation of the background field, it sets up the propagation of Rossby waves around the sphere which move retrograde to the solid-body rotation. These waves pass energy to the ring (in the case when the solid-body field and the ring initially co-rotate) or extract energy from the ring (when the solid-body field and the ring initially counter-rotate), hence the Hamiltonian and the centre-of-vorticity vector associated with the N -point vortices are no longer conserved as they are for the one-way coupled model described by Newton & Shokraneh. In the first case, energy is transferred to the ring, the length of the centre-of-vorticity vector increases, while its tip spirals in a clockwise manner towards the North Pole. The ring stays relatively intact for short times, but ultimately breaks-up on a longer time-scale. In the latter case, energy is extracted from the ring, the length of the centre-of-vorticity vector decreases while its tip spirals towards the North Pole and the ring loses its coherence more quickly than in the co-rotating case. The special case where the ring is initially oriented so that its centre-of-vorticity vector is perpendicular to the axis of rotation is also examined as it shows how the coupling to the background field breaks this symmetry. In this case, both the length of the centre-of-vorticity vector and the Hamiltonian energy of the ring achieve a local maximum at roughly the same time.

We consider the collision dynamics produced by three beads with masses (m 1, m 2, m 3) sliding without friction on a ring, where the masses are scaled so that m 1 = 1/ɛ, m 2 = 1, m 3 = 1 − ɛ, for 0 ⩽ ɛ ⩾ 1. The singular limits ɛ = 0 and ɛ = 1 correspond to two equal mass beads colliding on the ring with a wall, and without a wall respectively. In both these cases, all solutions are periodic and the eigenvalue distributions (around the unit circle) associated with the products of collision matrices are discrete. We then numerically examine the regime which parametrically connects these two states, i.e. 0 < ɛ < 1, and show that the eigenvalue distribution is generically uniform around the unit circle, which implies that the dynamics are no longer periodic. By a sequence of careful numerical experiments, we characterize how the uniform spectrum collapses from continuous to discrete in the two singular limits ɛ → 0 and ɛ → 1 for an ensemble of initial velocities sampled uniformly on a fixed energy surface. For the limit ɛ → 0, the distribution forms Gaussian peaks around the discrete limiting values ± 1, ± i, with variances that scale in power law form as σ 2 ∼ αɛ β. By contrast, the convergence in the limit ɛ → 1 to the discrete values ±1 is shown to follow a logarithmic power-law σ 2 ∼ log(ɛ β).

The problem of finding and classifying all relative equilibrium configurations of $N$-point vortices in the plane is first described as a classical variational principle and then formulated as a problem in linear algebra. Given a configuration of $N$ points in the plane, one must understand the structure of the $N(N-1)/2 \times N$ configuration matrix $A$ obtained by requiring that all interparticle distances remain fixed in time. If the determinant of the square, symmetric $N \times N$ covariance matrix $A^T A$ is zero, there is a non-trivial nullspace of $A$ and a basis set for this nullspace can be used to determine all vortex strengths $\vec{\Gamma} \in R^N$ for which the configuration remains rigid. Optimal basis sets are obtained by using the singular value decomposition of $A$ which allows one to categorize exact equilibria, approximate equilibria, and the distance between different equilibria in the appropriate vector space, as characterized by the Frobenius norm.

Physical theories of active matter continue to provide a quantitative understanding of dynamic cellular phenomena, including cell locomotion. Although various investigations of the rheology of cells have identified important viscoelastic and traction force parameters for use in these theoretical approaches, a key variable has remained elusive both in theoretical and experimental approaches: the spatiotemporal behavior of the subcellular density. The evolution of the subcellular density has been qualitatively observed for decades as it provides the source of image contrast in label-free imaging modalities (e.g., differential interference contrast, phase contrast) used to investigate cellular specimens. While these modalities directly visualize cell structure, they do not provide quantitative access to the structures being visualized. We present an established quantitative imaging approach, non-interferometric quantitative phase microscopy, to elucidate the subcellular density dynamics in neutrophils undergoing chemokinesis following uniform bacterial peptide stimulation. Through this approach, we identify a power law dependence of the neutrophil mean density on time with a critical point, suggesting a critical density is required for motility on 2D substrates. Next we elucidate a continuum law relating mean cell density, area, and total mass that is conserved during neutrophil polarization and migration. Together, our approach and quantitative findings will enable investigators to define the physics coupling cytoskeletal dynamics with subcellular density dynamics during cell migration.

The chaotic advection of tracer particles in the field of a perturbed latitudinal ring of point vortices on a sphere is considered. We consider a restricted four-vortex problem where three vortices have equal strength, while the fourth has strength zero. The equal-strength vortices are initially spaced evenly on a ring of fixed latitude in the northern hemisphere. The equilateral triangle formed by the vortices is known to be a nonlinearly stable relative equilibrium configuration. When perturbed, the vortex motion induces chaotic particle advection analyzed by means of stroboscopic Poincare maps as a function of the dimensionless energy of the system, which can be related to the size of the perturbation from equilibrium. A critical energy is identified which separates the vortex motion into two distinct dynamical regimes. For energies below critical, the vortices undergo periodic partner exchange while retaining their relative orientation. For values above critical, the relative orientation of the vortices changes throughout the periodic cycle. We consider how the streamline topologies bifurcate both as a function of the energy and during the course of their evolution, as well as the role that the evolution of instantaneous streamline structures plays in the mixing and transport of particles. The geometric extent of the mixing region on the full sphere is considered (measured as a percentage of the surface area of the sphere) and dynamical properties in the region, such as mixing and stretching rates as well as computational evidence of ergodicity, are obtained. Global mixing on the sphere does not seem to increase monotonically with energy, but appears to be maximized for values near critical.

We describe a new method of constructing point vortex equilibria on a sphere made-up of N vortices with different strengths. Such equilibria, called heterogeneous equilibria , are obtained for the five Platonic solid configurations, hence for . The method is based on calculating a basis set for the nullspace of a matrix obtained by enforcing the necessary and sufficient condition that the mutual distances between each pair of vortices remain constant. By symmetries inherent in the Platonic solid configurations, this matrix is reduced for each case and we call the dimension of the nullspace the degree of heterogeneity of the structure. For the tetrahedron ( N =4) and octahedron ( N =6), the degree of heterogeneity is 4 and 6, respectively, hence we are free to choose each of the vortex strengths independently. For the cube ( N =8), the degree of heterogeneity is 5, for the icosahedron ( N =12) it is 7, while for the dodecahedron ( N =20) it is 4. Thus, the entire set of equilibria based on the Platonic solid configurations is obtained, including substructures associated with each configuration constructed by taking different linear combinations of the basis elements.

The problem ofN-point vortices moving on a rotating unit sphere is considered. Through a sequence of linear coordinate transformations, which takes into account the orientation of the centre of vorticity vector with respect to the axis of rotation, we show how to reduce the problem to that on a non-rotating sphere, where the centre of vorticity vector is aligned with thez-axis. As a consequence, we prove that integrability on the rotating sphere is the same as on the non-rotating sphere, namely, the three-vortex problem on the rotating sphere is integrable for all vortex strengths, while the four-vortex problem is integrable in the special case where the centre of vorticity is zero. Rigid multi-frequency configurations that retain their shape while rotating about two independent axes with two independent frequencies are obtained, and necessary conditions for one- and two-frequency motions are derived. Examples including dipoles which exhibit global ‘wobbling’ and ‘tumbling’ dynamics, rings, and Platonic solid configurations are shown to undergo either periodic or quasi-periodic evolution on the rotating sphere depending on the ratio of the solid-body rotational frequencyΩto the rotational frequencyωassociated with the rigid structure.

The probability of winning a game, a set, and a match in tennis are computed, based on each player's probability of winning a point on serve, which we assume are independent identically distributed (iid) random variables. Both two out of three and three out of five set matches are considered, allowing a 13-point tiebreaker in each set, if necessary. As a by-product of these formulas, we give an explicit proof that the probability of winning a set, and hence a match, is independent of which player serves first. Then, the probability of each player winning a 128-player tournament is calculated. Data from the 2002 U.S. Open and Wimbledon tournaments are used both to validate the theory as well as to show how predictions can be made regarding the ultimate tournament champion. We finish with a brief discussion of evidence for non-iid effects in tennis, and indicate how one could extend the current theory to incorporate such features.

Motivated by problems in electrostatics and vortex dynamics, we develop two general methods for constructing Green's function for simply connected domains on the surface of the unit sphere. We prove a Riemann mapping theorem showing that such domains can be conformally mapped to the upper hemisphere. We then categorize all domains on the sphere for which Green's function can be constructed by an extension of the classical method of images. We illustrate our methods by several examples, such as the upper hemisphere, geodesic triangles, and latitudinal rectangles. We describe the point vortex motion in these domains, which is governed by a Hamiltonian determined by the Dirichlet Green's function.

Lung cancer is often classified by the presence of oncogenic drivers, such as epidermal growth factor receptor (EGFR), rather than patterns of anatomical distribution. While metastatic spread may seem a random and unpredictable process, we explored the possibility of using its quantifiable nature as a measure of describing and comparing different subsets of disease. We constructed a database of 664 non-small cell lung cancer (NSCLC) patients treated at the University of Southern California Norris Comprehensive Cancer Center and the Los Angeles County Medical Center. Markov mathematical modeling was employed to assess metastatic sites in a spatiotemporal manner through every time point in progression of disease. Our findings identified a preferential pattern of primary lung disease progressing through lung metastases to the brain amongst EGFR mutated (EGFR m) NSCLC patients, with exon 19 deletions or exon 21 L858R mutations, as compared to EGFR wild type (EGFR wt). The brain was classified as an anatomic "sponge", with a higher ratio of incoming to outgoing spread, for EGFR m NSCLC. Bone metastases were more commonly identified in EGFR wt patients. Our study supports a link between the anatomical and molecular characterization of lung metastatic cancer. Improved understanding of the differential biology that drives discordant patterns of anatomic spread, based on genotype specific profiling, has the potential to improve personalized oncologic care.

Adrenal gland metastases are common in lung cancer. It is well recognized that aggressive treatment of solitary adrenal metastases leads to improved outcomes but the exact nature of adrenal deposits is not well understood. Controversy exists as to the routing of cancer cells to the adrenal gland with some believing that this transmission is lymphatic, in contrast to the more generally accepted theory of hematogenous spread. Using an autopsy dataset of 3827 untreated cancer patients, we use the metastatic distribution of common primary cancer types to create Markov models of progression. The anatomical sites of spread in the body represent states in the model, while the transition probability between states represents the probability of metastatic spread from one site to another. We then use the Markov models to run Monte Carlo simulations of random walkers representing circulating tumor cells traveling within the body to simulate metastatic spread. We calculate mean first passage times (MFPT) to each site as a representative of time to metastasis formation. Analysis of 6 common cancer types (bladder [n=120 autopsies; 289 metastases], breast [n=432; 2235], colorectal [n=161; 420], lung [n=560; 859], ovarian [n=418; 806], and prostate [n=62; 212]) from the dataset showed distinct metastatic distributions across the populations. MFPT calculations to anatomical sites in the 6 analyzed primary cancer types indicated model progression times to the adrenal gland similar to the regional and distal lymph nodes in only lung cancer. The times associated with adrenal gland in the other primary cancers were similar to other metastatic sites of hematogenous spread. The Markov models created strongly support the lymphatic theory for metastatic spread to the adrenal glands. After performing a literature review to look for the biological plausibility of the simulated results, we believe evidence supports this theory and validates the models. This could explain improved survival for patients in whom solitary adrenal metastases are managed aggressively with surgical or radiation modalities. In order to further validate this theory, we are calling for clinical trials prospectively testing this hypothesis. Citation Format: Jeremy M. Mason, AnneMarie Ciccarella, Lori Marx-Rubiner, Lyudmila Bazhenova, Paul K. Newton, Kelly Bethel, Jorge J. Nieva, Peter Kuhn. Adrenal metastases in lung cancer: Clinical implications of a mathematical model [abstract]. In: Proceedings of the American Association for Cancer Research Annual Meeting 2017; 2017 Apr 1-5; Washington, DC. Philadelphia (PA): AACR; Cancer Res 2017;77(13 Suppl):Abstract nr 4532. doi:10.1158/1538-7445.AM2017-4532

Adrenal gland metastases are common in lung cancer. It is well recognized that aggressive treatment of solitary adrenal metastases leads to improved outcomes but the exact nature of adrenal deposits is not well understood. Controversy exists as to the routing of cancer cells to the adrenal gland with some believing that this transmission is lymphatic, in contrast to the more generally accepted theory of hematogenous spread. Recently published mathematical modeling of cancer progression strongly supports the lymphatic theory. With that in mind, we performed a literature review to look for biological plausibility of simulation results and believe that evidence supports the contention that metastases to the adrenal gland can be routed by means of lymphatic channels. This could explain improved survival for patients in whom solitary adrenal metastases are managed aggressively with surgical or radiation modalities. We are calling for clinical trials prospectively testing this hypothesis.

We describe a geometric phase that arises when two elliptical vortex patches corotate. Using the Hamiltonian moment model of Melander, Zabusky, and Styczek [J. Fluid Mech. 167, 95–115 (1986)] we consider two corotating uniform elliptical patches evolving according to the second order truncated equations of the model. The phase is computed in the adiabatic setting of a slowly varying Hamiltonian as in the work of Hannay [J. Phys. A 18, 221–230 (1985)] and Berry [Proc. R. Soc. London, Ser. A 392, 45–57 (1984)]. We also discuss the geometry of the symplectic phase space of the model in the context of nonadiabatic phases. The adiabatic phase appears in the orientation angle of each patch—it is similiar in form and is calculated using a multiscale perturbation procedure as in the point vortex configuration of Newton [Physica D 79, 416–423 (1994)] and Shashikanth and Newton [J. Nonlinear Sci. 8, 183–214 (1998)], however, an extra factor due to the internal stucture of the patch is present. The final result depends on the initial orientation of the patches unlike the phases in the works of Hannay and Berry [J. Phys. A 18, 221–230 (1985)]; [Proc. R. Soc. London, Ser. A 392, 45–57 (1984)]. We then show that the adiabatic phase can be interpreted as the holonomy of a connection on the trivial principal fiber bundle pi:T2×S1-->S1, where T2 is identified with the product of the momentum level sets of two Kirchhoff vortex patches and S1 is diffeomorphic to the momentum level set of two point vortex motion. This two point vortex motion is the motion that the patch centroids approach in the adiabatic limit.

We describe the instantaneous streamline patterns that occur on the surface of a two-dimensional sphere in the presence of point vortices of general strength for cases in which the system is completely integrable. After stating some general results based on the spherical topology, we categorize all possible instantaneous streamline patterns and describe their stagnation point structure for the cases of two and three vortices. It is found that for the case of two vortices, the only non-degenerate topologies that can arise are a figure eight (lemniscate) or a limacon, which are homotopically equivalent. For the case of three vortices, there are 12 topologically distinct primitives, from which an additional 23 patterns can be produced via continuous deformations on the sphere (homotopies). All possible streamline patterns that arise from three vortex motion can be obtained via linear superposition of the primitive topologies and their homotopic equivalents. In this sense, the primitives can be viewed as the ‘building blocks’ for the general flow patterns. The equations of motion and streamline patterns in the stereographic plane are obtained using the projected equations of motion in Hamiltonian form. We describe streamline patterns for three vortex fixed equilibria and relative equilibria as seen in both a fixed and a rotating frame of reference. Non-equilibrium streamline patterns for a collapsing and special periodic solution are studied, although to generally understand and classify the many bifurcations associated with non-equilibrium patterns would require more extensive computation. We conclude with a discussion of the relevance of the three vortex topological classification scheme and the bifurcation of streamline topologies for understanding global atmospheric weather patterns (spherical isobars) and large scale mixing phenomena.

We consider the motion of a point vortex on the surface of a sphere with solid boundaries. This problem is of interest in oceanography, where coherent vortex structures can persist for long times, and move over distances large enough so that the curvature of the Earth becomes important (see Gill [1982], Chaos [1994]). In this context, the boundary is a first step in modeling the presence of coastlines and shores using inviscid theory. Using the equations of motion for the vortex projected onto the stereographic plane, we construct the appropriate Green’s function using classical image method ideas, as long as the domain has certain symmetry properties. After the solution is obtained in the stereographic plane, it is projected back down to the sphere, yielding the sought after solution to the problem. We demonstrate the utility of the method by solving for the vortex trajectories and streamlines for several canonical examples, including a spherical cap, longitudinal wedge, half longitudinal wedge, channel, and rectangle. The results are compared with the corresponding ones in the physical plane in order to highlight the effect of the spherical geometry.

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.

This is the second in a two-part series of articles in which we analyze a system similar in structure to the well-known Zakharov equations from weak plasma turbulence theory, but with a nonlinear conservation equation allowing finite time shock formation. In this article we analyze the incompressible limit in which the shock speed is large compared to the underlying group velocity of the dispersive wave (a situation typically encountered in applications). After presenting some exact solutions of the full system, a multiscale perturbation method is used to resolve several basic wave interactions. The analysis breaks down into two categories: the nonlinear limit and the linear limit, corresponding to the form of the equations when the group velocity to shock speed ratio, denoted by e, is zero. The former case is an integrable limit in which the model reduces to the cubic nonlinear Schrodinger equation governing the dispersive wave envelope. We focus on the interaction of a “fast” shock wave and a single hump soliton. In the latter case, the e=0 problem reduces to the linear Schrodinger equation, and the focus is on a fast shock interacting with a dispersive wave whose amplitude is cusped and exponentially decaying. To motivate the time scales and structure of the shock-dispersive wave interactions at lowest orders, we first analyze a simpler system of ordinary differential equations structurally similar to the original system. Then we return to the fully coupled partial differential equations and develop a multiscale asymptotic method to derive the effective leading-order shock equations and the leading-order modulation equations governing the phase and amplitude of the dispersive wave envelope. The leading-order interaction equations admit a fairly complete analysis based on characteristic methods. Conditions are derived in which: (a) the shock passes through the soliton, (b) the shock is completely blocked by the soliton, or (c) the shock reverses direction. In the linear limit, a phenomenon is described in which the dispersive wave induces the formation of a second, transient shock front in the rapidly moving hyperbolic wave. In all cases, we can characterize the long-time dynamics of the shock. The influence of the shock on the dispersive wave is manifested, to leading order, in the generalized frequency of the dispersive wave: the fast-time part of the frequency is the shock wave itself. Hence, the frequency undergoes a sudden jump across the shock layer.In the last section, a sequence of numerical experiments depicting some of the interesting interactions predicted by the analysis is performed on the leading-order shock equations.

We solve the equations governing the relative motion of three point vortices of arbitrary strength moving on the surface of a sphere of radius R. The system is more general than the corresponding one in the plane [1,3,5,19,28,33,34,37,38] and reduces to it in the limit R → ∞, as long as the three vortices remain sufficiently close to each other during the course of their motion. Instead of using spherical coordinates, we use cartesian coordinates in which the vector χi points from the center of the sphere to the vortex with strength Γi. An important conserved quantity is the center of vorticity vector, c = (∑Γ i χ i ) (∑Γ i ) which, with no loss of generality, we align with the z-axis. Based on the size of this vector relative to the radius of the sphere, we classify the motion into one of five types: super-radial, radial, sub-radial, degenerate, or a limiting super-radial case. This categorization allows us to draw several conclusions about the qualitative motion of the vortices. We then fully characterize all fixed and relative equilibria on the sphere. For fixed equilibria, the vortices must lie on great circles (geodesics). If the strengths are equal, they form an equilateral triangle. Otherwise, the triangle shape is specified once the strength of the vortices is given. The relative, equilibria are classified as either degenerate (c = 0) or non-degenerate (c ≠ 0). For each type, the shape of the vortex triangle is described and the frequency of rotation is computed. As in the planar problem, it is possible to introduce trilinear coordinates and study the motion in a phase plane, which allows us to locate all the equilibria, as well as to characterize more complex relative dynamics. We then describe self-similar vortex collapse on the sphere, stating necessary and sufficient conditions for collapse to occur, computing the collapse times and vortex trajectories on the route towards collapse. Comparisons are made with the collapse formulas in the plane derived in [1,30].

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.

Neutrophils serve as a first line of defense in innate immunity owing in part to their ability to rapidly migrate towards chemotactic factors derived from invading pathogens. As a migratory function, neutrophil chemotaxis is regulated by the Rho family of small GTPases. However, the mechanisms by which Rho GTPases orchestrate cytoskeletal dynamics in migrating neutrophils remain ill-defined. In this study, we characterized the role of p21-activated kinase (PAK) downstream of Rho GTPases in cytoskeletal remodeling and chemotactic processes of human neutrophils. We found that PAK activation occurred upon stimulation of neutrophils with f-Met-Leu-Phe (fMLP), and PAK accumulated at the actin-rich leading edge of stimulated neutrophils, suggesting a role for PAK in Rac-dependent actin remodeling. Treatment with the pharmacological PAK inhibitor, PF3758309, abrogated the integrity of RhoA-mediated actomyosin contractility and surface adhesion. Moreover, inhibition of PAK activity impaired neutrophil morphological polarization and directional migration under a gradient of fMLP, and was associated with dysregulated Ca(2+) signaling. These results suggest that PAK serves as an important effector of Rho-family GTPases in neutrophil cytoskeletal reorganization, and plays a key role in driving efficient directional migration of human neutrophils.

This paper describes the occurence of phase anholonomies in the context of point vortex problems for two-dimensional incompressible flows. After giving a brief description of anholonomic effects in other contexts, we focus attention on the restricted three-vortex problem and a simpler modified problem where the "Hannay-Berry" phase can be computed using multi-scale asymptotic methods. Our main emphasis in this paper is to show how the Hannay-Berry phase arises as the leading term in an asymptotic expansion as the result of a non-uniform limit process. We show how it arises when computing the long time growth rate of passive scalar interfaces as they wrap around vortex cores in the presence of a slowly varying background field due to other vortices, and discuss the results in the context of "spiral-vortex" models for 2D turbulence.

We consider the initial value problem for the forced one dimensional nonlinear Schrodinger equation (NLS), where the forcing is assumed to be fast compared to the evolution of the unforced equation. This suggests the introduction of two time scales. Solutions to the forced NLS are sought by expressing the dependent variable in modulus‐phase form and expanding in powers of a small parameter, which is inversely related to the forcing time scale. This system is similar to a forced eikonal‐transport system arising in nonlinear geometrical optics. We focus on the effect that the forcing has on the NLS standing solitary wave. Solutions to second order in the expansion are computed analytically for some specific choices of the forcing function. A general conclusion of this work is that the effect of the forcing on the phase variable is quite important in determining the overall structure of the forced solitary wave.

We investigate several unsteady two-dimensional models of increasing complexity for the nonlinear evolution of the temporally growing mixing layer. In the simplest model, we use a subharmonically perturbed point vortex row which serves initially to identify certain key dynamical and kinematic aspects of the transition process, in which small scales in the flow are rapidly generated as a result of the vortex pairing process. A slightly more realistic vortex blob model is used at the next level. Finally, Navier-Stokes calculations are performed in order to identify which features are artifacts of the earlier models, and which ones are likely to be inherent to the physics. An important kinematic aspect of the transition, present in all of the models for small subharmonic perturbations, is a change in the topology of the streamline pattern when viewed in a rotating reference frame. This topological change contributes to the formation of spiral arm structures seen in other work and consequently to the intense production of small scales. For larger subharmonic perturbation amplitudes, this effect does not occur. We attempt to establish a connection between the aspects of our two-dimensional models and the findings of other authors in three-dimensional computations and experiments.

We describe a Markov chain mathematical model of lung cancer progression based on a longitudinal dataset of 722 lung cancer patients. Specifically, we investigate the patterns of metastatic spread in lung cancer and how the VEGF inhibitor, bevacizumab, alters these metastatic patterns. Stochastic models were used to simulate metastatic spread by means of random walk processes on directed graphs. We created spatiotemporal diagrams of cancer progression to analyze the differences between patients treated and not treated with bevacizumab. Patients with squamous lung cancer or brain metastases at baseline were ineligible for bevacizumab and excluded from analysis. Our results demonstrated that not only does bevacizumab extend survival, but it also alters patterns of metastatic spread. Patients treated with bevacizumab were characterized by greater heterogeneity in their pathways of metastatic progression, compared to those patients not treated with bevacizumab, which showed more homogeneity in their pathways. Furthermore, we quantify this spread and characterize metastatic sites as ‘spreaders’ and ‘sponges’ based on their probability of spreading the disease. Citation Format: Jeremy M. Mason, Paul K. Newton, Gino K. In, Sonia Lin, Peter Kuhn, Jorge J. Nieva. Spatiotemporal progression patterns in metastatic lung cancer treated with bevacizumab. [abstract]. In: Proceedings of the 107th Annual Meeting of the American Association for Cancer Research; 2016 Apr 16-20; New Orleans, LA. Philadelphia (PA): AACR; Cancer Res 2016;76(14 Suppl):Abstract nr 2711.