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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.

A new ecologically inspired paradigm in cancer treatment known as “adaptive therapy” capitalizes on competitive interactions between drug-sensitive and drug-resistant subclones. The goal of adaptive therapy is to maintain a controllable stable tumor burden by allowing a significant population of treatment-sensitive cells to survive. These, in turn, suppress proliferation of the less-fit resistant populations. However, there remain several open challenges in designing adaptive therapies, particularly in extending these therapeutic concepts to multiple treatments. We present a cancer treatment case study (metastatic castrate-resistant prostate cancer) as a point of departure to illustrate three novel concepts to aid the design of multidrug adaptive therapies. First, frequency-dependent “cycles” of tumor evolution can trap tumor evolution in a periodic, controllable loop. Second, the availability and selection of treatments may limit the evolutionary “absorbing region” reachable by the tumor. Third, the velocity of evolution significantly influences the optimal timing of drug sequences. These three conceptual advances provide a path forward for multidrug adaptive therapy.Significance:Driving tumor evolution into periodic, repeatable treatment cycles provides a path forward for multidrug adaptive therapy.

Figure S1, Tables S1, S2. Table S1: Competition parameters for Lupron and Abiraterone treatment; Table S2: Competition parameters for no treatment; Figure S1: After-action analysis of patient-specific subpopulations Normalized subpopulations over time (equation 4) for testosterone-dependent (T+, yellow), testosterone-producing (TP, blue), and testosterone-independent cells (T-, green). Timing of treatment received under clinical protocol is indicated (top) for Lupron & Abiraterone (red) and Lupron only (blue). Parameterization of the model is performed by least-squares fit to PSA data from clinical trial NCT02415621 (see figure 3).

PDF file - 232K, The supporting material describes in more detail the Markov chain model, and includes a full Table and two additional figures. Figure S1. Convergence plot for the lung cancer matrix. Figure S2. Metastatic tumor distributions after two-steps, compared with steady-state. Table S1. Comparative table of top two-step metastatic pathways of all types from Lung.

We use the Bernstein polynomials of degree d as the basis for constructing a uniform approximation to the rate of evolution (related to the fixation probability) of a species in a two-component finite-population frequency-dependent evolutionary game setting. The approximation is valid over the full range 0 ≤ w ≤ 1, where w is the selection pressure parameter, and converges uniformly to the exact solution as d → ∞. We compare it to a widely used non-uniform approximation formula in the weak-selection limit (w ∼ 0) as well as numerically computed values of the exact solution. Because of a boundary layer that occurs in the weak-selection limit, the Bernstein polynomial method is more efficient at approximating the rate of evolution in the strong selection region (w ∼ 1) (requiring the use of fewer modes to obtain the same level of accuracy) than in the weak selection regime.

Anyone who has visited a science museum has seen the demonstration where a beach (or ping-pong) ball is suspended in mid-air at a fixed position by constant blowing from below. After a while, the ball inevitably tumbles to the ground but can easily be rebalanced, by hand, again at the suspension point. Here, we ask a different more delicate question. Can we blow the ball from rest, starting at the nozzle opening (x = 0), moving it up to the suspension point x = x* above the nozzle? We show that it is not possible to do this using constant blowing because the point at which the downward gravitational force balances the upward blowing force is an elliptic fixed point of the governing equations, so there is no transfer trajectory that connects the origin to x*. To overcome this problem, we design time-dependent blowing schedules that achieve the transfer, making use of orbit transfer ideas developed in the orbital mechanics literature. Then, we ask which of these time-dependent schedules are optimal? We show that, generally, it is bang–bang (on–off) blowing schedules that achieve the transfer in minimal time, using minimal energy (action) and minimal air volume. For certain parameter values, however, there are more complicated blowing schedules that are optimal (with respect to energy), which can be designed using the Pontryagin Maximum Principle (PMP) and singular control. We use elementary concepts from mechanics, nonlinear dynamics, and control theory and challenge the inclined experimentalist to try to implement some of these nonconstant blowing schedules in the lab.

Background While it has been suspected that different primary cancers have varying predilections for metastasis in certain brain regions, recent advances in neuroimaging and spatial modeling analytics have facilitated further exploration into this field. Methods A systematic electronic database search for studies analyzing the distribution of brain metastases (BMs) from any primary systematic cancer published between January 1990 and July 2020 was conducted using PRISMA guidelines. Results Two authors independently reviewed 1957 abstracts, 46 of which underwent full-text analysis. A third author arbitrated both lists; 13 studies met inclusion/exclusion criteria. All were retrospective single- or multi-institution database reviews analyzing over 8227 BMs from 2599 patients with breast (8 studies), lung (7 studies), melanoma (5 studies), gastrointestinal (4 studies), renal (3 studies), and prostate (1 study) cancers. Breast, lung, and colorectal cancers tended to metastasize to more posterior/caudal topographic and vascular neuroanatomical regions, particularly the cerebellum, with notable differences based on subtype and receptor expression. HER-2-positive breast cancers were less likely to arise in the frontal lobes or subcortical region, while ER-positive and PR-positive breast metastases were less likely to arise in the occipital lobe or cerebellum. BM from lung adenocarcinoma tended to arise in the frontal lobes and squamous cell carcinoma in the cerebellum. Melanoma metastasized more to the frontal and temporal lobes. Conclusion The observed topographical distribution of BM likely develops based on primary cancer type, molecular subtype, and genetic profile. Further studies analyzing this association and relationships to vascular distribution are merited to potentially improve patient treatment and outcomes.

The Hawk-Dove mathematical game offers a paradigm of the trade-offs associated with aggressive and passive behaviors. When two (or more) populations of players (animals, insect populations, countries in military conflict, economic competitors, microbial communities, populations of co-evolving tumor cells, or reinforcement learners adopting different strategies) compete, their success or failure can be measured by their frequency in the population (successful behavior is reinforced, unsuccessful behavior is not), and the system is governed by the replicator dynamical system. We develop a time-dependent optimal-adaptive control theory for this nonlinear dynamical system in which the payoffs of the Hawk-Dove payoff matrix are dynamically altered (dynamic incentives) to produce (bang-bang) control schedules that (i) maximize the aggressive population at the end of timeT, and (ii) minimize the aggressive population at the end of timeT. These two distinct time-dependent strategies produce upper and lower bounds on the outcomes from all strategies since they represent two extremizers of the cost function using the Pontryagin maximum (minimum) principle. We extend the results forward to timesnT(n= 1, …, 5) in an adaptive way that uses the optimal value at the end of timenTto produce the new schedule for time (n+ 1)T. Two special schedules and initial conditions are identified that produce absolute maximizers and minimizers over an arbitrary number of cycles for 0 ≤T≤ 3. ForT> 3, our optimum schedules can drive either population to extinction or fixation. The method described can be used to produce optimal dynamic incentive schedules for many different applications in which the 2 × 2 replicator dynamics is used as a governing model.

Summary Organoid model systems recapitulate key features of mammalian tissues and enable high throughput experiments. However, the impact of these experiments may be limited by manual, non-standardized, static or qualitative phenotypic analysis. OrgDyn is an open-source and modular pipeline to quantify organoid shape dynamics using a combination of feature- and model-based approaches on time series of 2D organoid contour images. Our pipeline consists of (i) geometrical and signal processing feature extraction, (ii) dimensionality reduction to differentiate dynamical paths, (iii) time series clustering to identify coherent groups of organoids and (iv) dynamical modeling using point distribution models to explain temporal shape variation. OrgDyn can characterize, cluster and model differences among unique dynamical paths that define diverse final shapes, thus enabling quantitative analysis of the molecular basis of tissue development and disease. Availability and Implementation https://github.com/zakih/organoidDynamics (BSD 3-Clause License). Supplementary information Supplementary data are available at Bioinformatics online.

The prisoner's dilemma (PD) game offers a simple paradigm of competition between two players who can either cooperate or defect. Since defection is a strict Nash equilibrium, it is an asymptotically stable state of the replicator dynamical system that uses the PD payoff matrix to define the fitness landscape of two interacting evolving populations. The dilemma arises from the fact that the average payoff of this asymptotically stable state is suboptimal. Coaxing the players to cooperate would result in a higher payoff for both. Here we develop an optimal control theory for the prisoner's dilemma evolutionary game in order to maximize cooperation (minimize the defector population) over a given cycle time $T$, subject to constraints. Our two time-dependent controllers are applied to the off-diagonal elements of the payoff matrix in a bang-bang sequence that dynamically changes the game being played by dynamically adjusting the payoffs, with optimal timing that depends on the initial population distributions. Over multiple cycles $nT$ $(ng1)$, the method is adaptive as it uses the defector population at the end of the $n\mathrm{th}$ cycle to calculate the optimal schedule over the $n+1\mathrm{st}$ cycle. The control method, based on Pontryagin's maximum principle, can be viewed as determining the optimal way to dynamically alter incentives and penalties in order to maximize the probability of cooperation in settings that track dynamic changes in the frequency of strategists, with potential applications in evolutionary biology, economics, theoretical ecology, social sciences, reinforcement learning, and other fields where the replicator system is used.

The prisoner’s dilemma (PD) game offers a simple paradigm of competition between two players who can either cooperate or defect. Since defection is a strict Nash equilibrium, it is an asymptotically stable state of the replicator dynamical system that uses the PD payoff matrix to define the fitness landscape of two interacting evolving populations. The dilemma arises from the fact that the average payoff of this asymptotically stable state is sub-optimal. Coaxing the players to cooperate would result in a higher payoff for both. Here we develop an optimal control theory for the prisoner’s dilemma evolutionary game in order to maximize cooperation (minimize the defector population) over a given cycle-time T, subject to constraints. Our two time-dependent controllers are applied to the off-diagonal elements of the payoff matrix in a bang-bang sequence that dynamically changes the game being played by dynamically adjusting the payoffs, with optimal timing that depends on the initial population distributions. Over multiple cycles nT (n > 1), the method is adaptive as it uses the defector population at the end of the nth cycle to calculate the optimal schedule over the n + 1st cycle. The control method, based on Pontryagin’s maximum principle, can be viewed as determining the optimal way to dynamically alter incentives and penalties in order to maximize the probability of cooperation in settings that track dynamic changes in the frequency of strategists, with potential applications in evolutionary biology, economics, theoretical ecology, and other fields where the replicator system is used.PACS numbers02.50.Le; 02.30.Yy; 05.45.-a; 87.23.Kg; 87.23.Cc

Chemotherapeutic resistance via the mechanism of competitive release of resistant tumor cell subpopulations is a major problem associated with cancer treatments and one of the main causes of tumor recurrence. Often, chemoresistance is mitigated by using multidrug schedules (two or more combination therapies) that can act synergistically, additively, or antagonistically on the heterogeneous population of cells as they evolve. In this paper, we develop a three-component evolutionary game theory model to design two-drug adaptive schedules that mitigate chemoresistance and delay tumor recurrence in an evolving collection of tumor cells with two resistant subpopulations and one chemosensitive population that has a higher baseline fitness but is not resistant to either drug. Using the nonlinear replicator dynamical system with a payoff matrix of Prisoner's Dilemma (PD) type (enforcing a cost to resistance), we investigate the nonlinear dynamics of this three-component system along with an additional tumor growth model whose growth rate is a function of the fitness landscape of the tumor cell populations. A key parameter determines whether the two drugs interact synergistically, additively, or antagonistically. We show that antagonistic drug interactions generally result in slower rates of adaptation of the resistant cells than synergistic ones, making them more effective in combating the evolution of resistance. We then design evolutionary cycles (closed loops) in the three-component phase space by shaping the fitness landscape of the cell populations (i.e., altering the evolutionary stable states of the game) using appropriately designed time-dependent schedules (adaptive therapy), altering the dosages and timing of the two drugs. We describe two key bifurcations associated with our drug interaction parameter which help explain why antagonistic interactions are more effective at controlling competitive release of the resistant population than synergistic interactions in the context of an evolving tumor.

Chemotherapeutic resistance via the mechanism of competitive release of resistant tumor cell subpopulations is a major problem associated with cancer treatments and one of the main causes of tumor recurrence. Often, chemoresistance is mitigated by using multidrug schedules (two or more combination therapies) that can act synergistically, additively, or antagonistically on the heterogeneous population of cells as they evolve. In this paper, we develop a three-component evolutionary game theory model to design two-drug adaptive schedules (timing and dose levels associated with C1(t) and C2(t)) that mitigate chemoresistance and delay tumor recurrence in an evolving collection of tumor cells with two resistant subpopulations: R1 (sensitive to drug 1, resistant to drug 2), and R2 (sensitive to drug 2, resistant to drug 1). A key parameter, e, takes us from synergistic (e > 0), to additive (e = 0), to antagonistic (e < 0) drug interactions. In addition to the two resistant populations, the model includes a population of chemosensitive cells, S that have higher baseline fitness but are not resistant to either drug. Using the nonlinear replicator dynamical system with a payoff matrix of Prisoner’s Dilemma (PD) type (enforcing a cost to resistance), we investigate the nonlinear dynamics of the three-component system (S, R1, R2), along with an additional tumor growth model whose growth rate is a function of the fitness landscape of the tumor cell populations. We show that antagonistic drug interactions generally result in slower rates of adaptation of the resistant cells than synergistic ones, making them more effective in combating the evolution of resistance. We then design closed loops in the three-component phase space by shaping the fitness landscape of the cell populations (i.e. altering the evolutionary stable states of the game) using appropriately designed time-dependent schedules (adaptive therapy), altering the dosages and timing of the two drugs using information gleaned from constant dosing schedules. We show that the bifurcations associated with the evolutionary stable states are transcritical, and we detail a typical antagonistic bifurcation that takes place between the sensitive cell population S and the R1 population, and a synergistic bifurcation that takes place between the sensitive cell population S and the R2 population for fixed values of C1 and C2. These bifurcations help us further understand why antagonistic interactions are more effective at controlling competitive release of the resistant population than synergistic interactions in the context of an evolving tumor.

Background Biomechanical characterization of human performance with respect to fatigue and fitness is relevant in many settings, however is usually limited to either fully qualitative assessments or invasive methods which require a significant experimental setup consisting of numerous sensors, force plates, and motion detectors. Qualitative assessments are difficult to standardize due to their intrinsic subjective nature, on the other hand, invasive methods provide reliable metrics but are not feasible for large scale applications. Methods Presented here is a dynamical toolset for detecting performance groups using a non-invasive system based on the Microsoft Kinect motion capture sensor, and a case study of 37 cancer patients performing two clinically monitored tasks before and after therapy regimens. Dynamical features are extracted from the motion time series data and evaluated based on their ability to i) cluster patients into coherent fitness groups using unsupervised learning algorithms and to ii) predict Eastern Cooperative Oncology Group performance status via supervised learning. Findings The unsupervised patient clustering is comparable to clustering based on physician assigned Eastern Cooperative Oncology Group status in that they both have similar concordance with change in weight before and after therapy as well as unexpected hospitalizations throughout the study. The extracted dynamical features can predict physician, coordinator, and patient Eastern Cooperative Oncology Group status with an accuracy of approximately 80%. Interpretation The non-invasive Microsoft Kinect sensor and the proposed dynamical toolset comprised of data preprocessing, feature extraction, dimensionality reduction, and machine learning offers a low-cost and general method for performance segregation and can complement existing qualitative clinical assessments.

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.

The subject of fluid mechanics is a rich, vibrant, and rapidly developing branch of applied mathematics. Historically, it has developed hand-in-hand with the elegant subject of complex variable the...

A new ecologically inspired paradigm in cancer treatment known as “adaptive therapy” capitalizes on competitive interactions between drug-sensitive and drug-resistant subclones. The goal of adaptive therapy is to maintain a controllable stable tumor burden by allowing a significant population of treatment-sensitive cells to survive. These, in turn, suppress proliferation of the less-fit resistant populations. However, there remain several open challenges in designing adaptive therapies, particularly in extending these therapeutic concepts to multiple treatments. We present a cancer treatment case study (metastatic castrate-resistant prostate cancer) as a point of departure to illustrate three novel concepts to aid the design of multidrug adaptive therapies. First, frequency-dependent “cycles” of tumor evolution can trap tumor evolution in a periodic, controllable loop. Second, the availability and selection of treatments may limit the evolutionary “absorbing region” reachable by the tumor. Third, the velocity of evolution significantly influences the optimal timing of drug sequences. These three conceptual advances provide a path forward for multidrug adaptive therapy. Significance: Driving tumor evolution into periodic, repeatable treatment cycles provides a path forward for multidrug adaptive therapy.

We use a three-component replicator system with healthy cells, sensitive cells, and resistant cells, with a prisoner’s dilemma payoff matrix from evolutionary game theory, to model and control the nonlinear dynamical system governing the ecological mechanism of competitive release by which tumors develop chemotherapeutic resistance. The control method we describe is based on nonlinear trajectory design and energy transfer methods first introduced in the orbital mechanics literature for Hamiltonian systems. For continuous therapy, the basin boundaries of attraction associated with the chemo-sensitive population and the chemo-resistant population for increasing values of chemo-concentrations have an intertwined spiral structure with extreme sensitivity to changes in chemo-concentration level as well as sensitivity with respect to resistant mutations. For time-dependent therapies, we introduce an orbit transfer method to construct continuous families of periodic (closed) orbits by switching the chemo-dose at carefully chosen times and appropriate levels to design schedules that are superior to both maximum tolerated dose (MTD) schedules and low-dose metronomic (LDM) schedules, both of which ultimately lead to fixation of sensitive cells or resistant cells. By keeping the three subpopulations of cells in competition with each other indefinitely, we avoid fixation of the cancer cell population and regrowth of a resistant tumor. The method can be viewed as a way to dynamically shape the average population fitness landscape of a tumor to steer the chemotherapeutic response curve. We show that the method is remarkably insensitive to initial conditions and small changes in chemo-dosages, an important criterion for turning the method into an actionable strategy.

A tumor is made up of a heterogeneous collection of cell types, all competing on a fitness landscape mediated by microenvironmental conditions that dictate their interactions. Despite the fact that much is known about cell signaling, cellular cooperation, and the functional constraints that affect cellular behavior, the specifics of how these constraints (and the range over which they act) affect the macroscopic tumor growth laws that govern total volume, mass, and carrying capacity remain poorly understood. We develop a statistical mechanics approach that focuses on the total number of possible states each cell can occupy and show how different assumptions on correlations of these states give rise to the many different macroscopic tumor growth laws used in the literature. Although it is widely understood that molecular and cellular heterogeneity within a tumor is a driver of growth, here we emphasize that focusing on the functional coupling of states at the cellular level is what determines macroscopic growth characteristics.

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 (EGFRm) NSCLC patients, with exon 19 deletions or exon 21 L858R mutations, as compared to EGFR wild type (EGFRwt). The brain was classified as an anatomic “sponge”, with a higher ratio of incoming to outgoing spread, for EGFRm NSCLC. Bone metastases were more commonly identified in EGFRwt 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.

We describe a cell-molecular based evolutionary mathematical model of tumor development driven by a stochastic Moran birth-death process. The cells in the tumor carry molecular information in the form of a numerical genome which we represent as a four-digit binary string used to differentiate cells into 16 molecular types. The binary string is able to undergo stochastic point mutations that are passed to a daughter cell after each birth event. The value of the binary string determines the cell fitness, with lower fit cells (e.g. 0000) defined as healthy phenotypes, and higher fit cells (e.g. 1111) defined as malignant phenotypes. At each step of the birth-death process, the two phenotypic sub-populations compete in a prisoner's dilemma evolutionary game with the healthy cells playing the role of cooperators, and the cancer cells playing the role of defectors. Fitness, birth-death rates of the cell populations, and overall tumor fitness are defined via the prisoner's dilemma payoff matrix. Mutation parameters include passenger mutations (mutations conferring no fitness advantage) and driver mutations (mutations which increase cell fitness). The model is used to explore key emergent features associated with tumor development, including tumor growth rates as it relates to intratumor molecular heterogeneity. The tumor growth equation states that the growth rate is proportional to the logarithm of cellular diversity/heterogeneity. The Shannon entropy from information theory is used as a quantitative measure of heterogeneity and tumor complexity based on the distribution of the 4-digit binary sequences produced by the cell population. To track the development of heterogeneity from an initial population of healthy cells (0000), we use dynamic phylogenetic trees which show clonal and sub-clonal expansions of cancer cell sub-populations from an initial malignant cell. We show tumor growth rates are not constant throughout tumor development, and are generally much higher in the subclinical range than in later stages of development, which leads to a Gompertzian growth curve. We explain the early exponential growth of the tumor and the later saturation associated with the Gompertzian curve which results from our evolutionary simulations using simple statistical mechanics principles related to the degree of functional coupling of the cell states. We then compare dosing strategies at early stage development, mid-stage (clinical stage), and late stage development of the tumor. If used early during tumor development in the subclinical stage, well before the cancer cell population is selected for growth, therapy is most effective at disrupting key emergent features of tumor development.

We use a three-component replicator dynamical system with healthy cells, sensitive cells, and resistant cells, with a prisoner’s dilemma payoff matrix from evolutionary game theory to understand the phenomenon of competitive release, which is the main mechanism by which tumors develop chemotherapeutic resistance. By comparing the phase portraits of the system without therapy compared to continuous therapy above a certain threshold, we show that chemotherapeutic resistance develops if there are pre-exisiting resistance cells in the population. We examine the basin boundaries of attraction associated with the chemo-sensitive population and the chemo-resistant population for increasing values of chemo-concentrations and show their spiral intertwined structure. We also examine the fitness landscapes both with and without continuous therapy and show that with therapy, the average fitness as well as the fitness functions of each of the subpopulations initially increases, but eventually decreases monotonically as the resistant subpopulation saturates the tumor.

A tumor is made up of a heterogeneous collection of cell types all competing on a fitness landscape mediated by micro-environmental conditions that dictate their interactions. Despite the fact that much is known about cell signaling and cellular cooperation, the specifics of how the cell-to-cell coupling and the range over which this coupling acts affect the macroscopic tumor growth laws that govern total volume, mass, and carrying capacity remain poorly understood. We develop a statistical mechanics approach that focuses on the total number of possible states each cell can occupy, and show how different assumptions on correlations of these states gives rise to the many different macroscopic tumor growth laws used in the literature. Although it is widely understood that molecular and cellular heterogeneity within a tumor is a driver of growth, here we emphasize that focusing on the functional coupling of these states at the cellular level is what determines macroscopic growth characteristics.Significance statementA mathematical model relating tumor heterogeneity at the cellular level to tumor growth at the macroscopic level is described based on a statistical mechanics framework. The model takes into account the number of accessible states available to each cell as well as their long-range coupling (population cooperation) to other cells. We show that the degree to which cell populations cooperate determine the number of independent cell states, which in turn dictates the macroscopic (volumetric) growth law. It follows that targeting cell-to-cell interactions could be a way of mitigating and controlling tumor growth.

The development of chemotherapeutic resistance resulting in tumor relapse is thought largely to be a consequence of the mechanism of competitive release of pre-existing resistant cells in the tumor selected for growth after chemotherapeutic agents attack the previously dom- inant population of chemo-sensitive cells. To study this process, we use a mathematical model based on the replicator equations with a prisoner's dilemma payoff matrix defining fitness levels of three competing cell populations: healthy cells (cooperators), sensitive cells (defectors), and resistant cells (defectors). The model is shown to recapitulate prostate- specific antigen (PSA) measurement data of patients (a surrogate for tumor growth) from three randomized clinical trials with metastatic castration-resistant prostate cancer for pa- tients treated with 1) prednisone only, 2) mitoxantrone and prednisone and 3) docetaxel and prednisone. In each trial, continuous maximum tolerated dose (MTD) schedules reduce the sensitive cell population, initially shrinking tumor volume, but subsequently release the resistant cells to re-populate and eventually re-grow the tumor in a resistant form. The evolutionary model allows us to quantify responses to conventional therapeutic strategies as well as to design novel adaptive strategies which are able to maintain the tumor volume at reduced levels by keeping a sufficient number of sensitive cells to prevent tumor re-growth from the resistant population.

We describe the use of Markov chain models for the purpose of quantitative forecasting of metastatic cancer progression. Each site (node) in the Markov network (directed graph) is an organ site where a secondary tumor could develop with some probability. The Markov matrix is an N x N matrix where each entry represents a transition probability of the disease progressing from one site to another during the course of the disease. The initial state-vector has a 1 at the position corresponding to the primary tumor, and 0s elsewhere (no initial metastases). The spread of the disease to other sites (metastases) is modeled as a directed random walk on the Markov network, moving from site to site with the estimated transition probabilities obtained from longitudinal data. The stochastic model produces probabilistic predictions of the likelihood of each metastatic pathway and corresponding time sequences obtained from computer Monte Carlo simulations. The main challenge is to empirically estimate the N^2 transition probabilities in the Markov matrix using appropriate longitudinal data.

Competitive release is a bedrock principle of coevolutionary ecology and population dynamics. It is also the main mechanism by which heterogeneous tumors develop chemotherapeutic resistance. Understanding, controlling, and exploiting this important mechanism represents one of the key challenges and potential opportunities of current medical oncology. The development of sophisticated mathematical and computational models of coevolution among clonal and sub-clonal cell populations in the tumor ecosystem can guide us in predicting and shaping various responses to perturbations in the fitness landscape which is altered by chemo-toxic agents. This in turn can help us design adaptive chemotherapeutic strategies to combat the release resistant cells.

The paper by Dritschel et al. (J. Fluid Mech., vol. 783, 2015, pp. 1–22) describes the long-time behaviour of inviscid two-dimensional fluid dynamics on the surface of a sphere. At issue is whether the flow settles down to an equilibrium or whether, for generic (random) initial conditions, the long-time solution is periodic, quasi-periodic or chaotic. While it might be surprising that this issue is not settled in the literature, it is important to keep in mind that the Euler equations form a dissipationless Hamiltonian system, hence the set of equations only redistributes the initial vorticity, generating smaller and smaller scales, while keeping kinetic energy, angular impulse and an infinite family of vorticity moments (Casimirs) intact. While special solutions that never settle down to an equilibrium state can be constructed using point vortices, vortex patches and other distributions, the fate of random initial conditions is a trickier problem. Previous statistical theories indicate that the long-time state should be a stationary large-scale distribution of vorticity. By carrying out careful numerical simulations using two different methods, the authors make a compelling case that the generic long-time state resembles a large-scale oscillating quadrupolar vorticity field, surrounded by persistent small-scale vortices. While numerical simulations can never conclusively settle this issue, the results might help guide future theories that seek to prove the existence of such an interesting dynamical long-time state.

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.

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 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.

Recent experiments on the formation of vortex lattices in Bose-Einstein condensates has produced the need for a mathematical theory that is capable of predicting a broader class of lattice patterns, ones that are free of discrete symmetries and can form in a random environment. We give an overview of an $N$-particle based Hamiltonian theory which, if formulated in terms of the $O(N^2)$ interparticle distances, leads to the analysis of a nonnormal “configuration” matrix whose nullspace structure determines the existence or nonexistence of a lattice. The singular value decomposition of this matrix leads to a method in which all lattice patterns and the associated particle strengths, in principle, can be classified and calculated by a random-walk scheme which systematically uses the $m$ smallest singular values as a ratchet mechanism to home in on lattices with $m$-dimensional nullspaces, where $0 < m \le N$. The resulting distribution of singular values encodes detailed geometric properties of the lattice and allows us to identify and calculate important quantitative measures associated with each lattice, including its size (as measured by the Frobenius or 2-norms), distance between the lattices (hence lattice density), robustness, and Shannon entropy as a quantitative measure of its level of disorder. This article gives an overview of vortex lattice theory from 1957 to the present, highlighting recent experiments in Bose-Einstein condensate systems and formulating questions that can be addressed by understanding the singular value decomposition of the configuration matrix. We discuss some of the computational challenges associated with producing large $N$ lattices, the subtleties associated with understanding and exploiting complicated Hamiltonian energy surfaces in high dimensions, and we highlight ten important directions for future research in this area.

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.

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.