Paul K. Newton

Recent viral coinfection experiments show that recombinant genomes are generated readily and depend strongly on infection timing and order, yet only a small fraction give rise to persistent lineages. We develop a hybrid deterministic–stochastic framework that resolves this discrepancy by coupling density-dependent recombinant generation with stochastic establishment of rare lineages. The resulting hazard of successful lineage formation is generically non-monotonic, increasing with parental abundance through enhanced generation while decreasing under competitive suppression, and exhibits a unique interior maximum in parental-density space. As parental populations evolve, their trajectory across this hazard landscape defines a sharply localized temporal window of evolutionary opportunity. These results reveal a general principle: evolutionary success is determined not only by intrinsic fitness, but by when variants arise within a dynamically changing ecological context.

We introduce a mathematical model of the cancer-immunity cycle and use it to test several hypotheses regarding the combination, timing, and optimization associated with chemotherapy and immunotherapy dosing schedules in the context of competition and time-dependent selection pressure. A key idea is the value of synchronizing the dosing schedules with the fundamental period of the cancer-immunity cycle. The competitors in the population dynamics evolutionary game are the cancer cells, healthy (normal) cells, and T cells, which conceptually form a nontransitive rock-paper-scissor chain. The chemotherapy and immunotherapy dosing schedules each act as control functions whose timing and magnitudes we synchronize with the fundamental period of the underlying nonlinear dynamical system. With the model, we show among other more detailed results, that chemotherapy and immunotherapy pulse-dosing schedules do not commute; the best duration of the chemotherapy is one-quarter of the cancer-immunity cycle, whereas for immunotherapy it is one-half cycle; immunotherapy dosing should precede chemotherapy dosing and last twice as long. A general conclusion is that optimized timing of the dosing schedules can make up for lower total dose, opening up new possibilities for designing less toxic and more efficacious dosing regimens with drugs currently in use. Obtaining and calibrating more accurate measurements of the cycle-period across patient populations would be an important step in making some of these ideas clinically actionable.

We describe a Q-learning approach to optimized chemotherapy dose scheduling in a stochastic finite-cell Markov process that models tumor cell natural selection dynamics. The three competing subpopulations comprising our virtual tumor are a chemo-sensitive population (S), and two chemo-resistant populations, R 1 and R 2 , each resistant to one of two drugs, C 1 and C 2 . The two drugs are toggled off or on which constitute the actions (selection pressure) imposed on our state-variables ( S, R 1 , R 2 ), measured as proportions in our finite state-space of N cancer cells ( S + R 1 + R 2 = N ). After the converged chemo-dosing policies are obtained, corresponding to a given reward structure, we focus on three important aspects of chemotherapy dose scheduling. First, we identify the most likely evolutionary paths of the tumor cell populations in response to the optimized (converged) policies. Second, we quantify the robustness in the ability to reach our target of balanced co-existence in light of incomplete information in both the initial cell-populations as well as the state-variables at each step. Third, we evaluate the efficacy of simplified policies which exploit the symmetries uncovered from an examination of the full policy. Our reward structure is designed to delay the onset of chemo-resistance in the tumor by rewarding a well-balanced mix of co-existing states, while punishing unbalanced subpopulations to avoid extinction.

We introduce a mathematical model of the cancer-immunity cycle and use it to test several hypotheses regarding the combination, timing, and optimization associated with chemotherapy and immunotherapy dosing schedules in the context of competition and selection pressure. A key conceptual idea is the value of synchronizing the dosing schedules with the fundamental period of the cancer-immunity cycle. The competitors in the population dynamics evolutionary game are the cancer cells, healthy cells, and T-cells, which form a non-transitive rock-paper-scissor chain, mediated by the tumor microenvironment. The chemotherapy and immunotherapy dosing schedules each act as control functions whose timing we synchronize with the fundamental period of the underlying nonlinear dynamical system. With the model, we show among other more detailed results, that chemotherapy and immunotherapy schedules are non-transitive; the best duration of the chemotherapy is around one-quarter of the cancer-immunity cycle, whereas for immunotherapy it is one-half cycle; immunotherapy dosing should preceed chemotherapy dosing. A general conclusion is that optimized timing of the dosing schedules can make up for lower total dose, opening up new possibilities for designing less toxic and more efficacious dosing regimens with drugs currently in use. Obtaining and calibrating more accurate measurements of the cycle-period across patient populations would be an important step in making some of these ideas clinically actionable.

We use a replicator evolutionary game in conjunction with control theory to design a two-stage multidrug chemotherapy schedule where each stage has a specific design objective. In the first stage, we use optimal control theory that minimizes a cost function to design a transfer orbit which takes any initial tumor-cell frequency composition and steers it to a state-space region of three competing clonal subpopulations in which the three populations co-exist with a relatively equal abundance (high-entropy co-existence region). In the second stage, we use adaptive control with continuous monitoring of the subpopulation balance to design a maintenance orbit which keeps the subpopulations trapped in the favorable co-existence region to suppress the competitive release of a resistant cell population in order to avoid the onset of chemoresistance. Our controlled replicator dynamics model consists of a chemo-sensitive cell phenotype S , which is sensitive to both drugs, and two resistant cell phenotypes, R 1 and R 2 , which are sensitive to drugs 1 and 2 respectively, but resistant to drug 2 and 1. The 3 × 3 payoff matrix used to define the fitness function associated with the interactions of the competing populations is a prisoner’s dilemma matrix which ensures that in the absence of chemotherapy, the S population (defectors) has higher fitness (reproductive prowess) than the two resistant cell populations, reflecting an inherent cost of resistance which our chemotherapy design methodology seeks to exploit. In our model, the two drugs C 1 and C 2 can act synergistically, additively, or antagonistically on the populations of cells as they compete and evolve under natural and artifical selection dynamics. Our model brings to light the inherent trade-offs between navigating to the maintenance orbit in minimal time vs. arriving there using the least total drug dose and also that the optimal balance of synergystic or antagonistic drug combinations depends the frequency balance of the populations of cells.

Brain metastases (BM) are associated with poor prognosis and increased mortality rates, making them a significant clinical challenge. Studying BMs can aid in improving early detection and monitoring. Systematic comparisons of anatomical distributions of BM from different primary cancers, however, remain largely unavailable. To test the hypothesis that anatomical BM distributions differ based on primary cancer type, we analyze the spatial coordinates of BMs for five different primary cancer types along principal component (PC) axes. The dataset includes 3949 intracranial metastases, labeled by primary cancer types and with six features. We employ PC coordinates to highlight the distinctions between various cancer types. We utilized different Machine Learning (ML) algorithms (RF, SVM, TabNet DL) models to establish the relationship between primary cancer diagnosis, spatial coordinates of BMs, age, and target volume. Our findings revealed that PC1 aligns most with the Y axis, followed by the Z axis, and has minimal correlation with the X axis. Based on PC1 versus PC2 plots, we identified notable differences in anatomical spreading patterns between Breast and Lung cancer, as well as Breast and Renal cancer. In contrast, Renal and Lung cancer, as well as Lung and Melanoma, showed similar patterns. Our ML and DL results demonstrated high accuracy in distinguishing BM distribution for different primary cancers, with the SVM algorithm achieving 97% accuracy using a polynomial kernel and TabNet achieving 96%. The RF algorithm ranked PC1 as the most important discriminating feature. In summary, our results support accurate multiclass ML classification regarding brain metastases distribution.

Objective Brain metastases (BM) are associated with poor prognosis and increased mortality rates, making them a significant clinical challenge. Therefore, studying BMs can aid in developing better diagnostic tools for their early detection and monitoring. Systematic comparisons of anatomical distributions of BM from different primary cancers, however, remain largely unavailable. Methods To test the hypothesis that anatomical BM distributions differ based on primary cancer type, we analyze the spatial coordinates of BMs for five different primary cancer types along principal component (PC) axes which optimizes their largest spread along each of the three PC axes. Data used in this analysis is taken from the International Radiosurgery Research Foundation (IRRF) and all patients underwent gamma-knife radiosurgery (GKRS) for the treatment of BMs which are labeled based on the primary cancer types Breast, Lung, Melanoma, Renal, and Colon. The dataset consists of six features including sex, age, target volume, and stereotactic Cartesian coordinates X, Y, and Z of a total of 3949 intracranial metastases. We employ PC coordinates to reduce the dimensionality of our dataset and highlight the distinctions in the anatomical spread of BMs between various cancer types. We utilized different Machine Learning (ML) algorithms: Random Forest (RF), Support Vector Machine (SVM), and TabNet Deep Learning (DL) model to establish the relationship between primary cancer diagnosis, spatial coordinates of BMs, age, and target volume. Results Our findings demonstrate that the first principal component (PC1) exhibits a greater alignment with the Y axis, followed by the Z axis, with a minimal correlation observed with the X axis. Based on our analysis of the PC1 versus PC2 plots, we have determined that the pairs of Breast and Lung cancer, as well as Breast and Renal cancer, exhibit the most notable distinctions in their anatomical spreading patterns. In contrast, we find that the pairs of Renal and Lung cancer, as well as Lung and Melanoma, were most similar in their patterns. Our ML and DL results indicate high accuracy in distinguishing the distribution of BM for different primary cancers, with the SVM algorithm achieving a 97% accuracy rate when using a polynomial kernel and TabNet a 96% accuracy. The RF algorithm ranks PC1 as the most important discriminating feature. Conclusions Taken together, the results demonstrate an accurate multiclass machine learning classification with respect to the distribution of brain metastases.

We develop a finite-cell model of tumor natural selection dynamics to investigate the stochastic fluctuations associated with multiple rounds of adaptive chemotherapy. The adaptive cycles are designed to avoid chemoresistance in the tumor by managing the ecological mechanism of competitive release of a resistant subpopulation. Our model is based on a three-component evolutionary game played among healthy (H), sensitive (S), and resistant (R) populations of N cells, with a chemotherapy control parameter, C(t), which we use to dynamically impose selection pressure on the sensitive subpopulation to slow tumor growth and manage competitive release of the resistant population. The adaptive chemoschedule is designed based on the deterministic (N→∞) adjusted replicator dynamical system, then implemented using the finite-cell stochastic frequency dependent Moran process model (N=10K-50K) to ascertain the cumulative effect of the stochastic fluctuations on the efficacy of the adaptive schedules over multiple rounds. We quantify the stochastic fixation probability regions of the R and S populations in the HSR trilinear phase plane as a function of the control parameter C∈[0,1], showing that the size of the R region increases with increasing C. We then implement an adaptive time-dependent schedule C(t) for the stochastic model and quantify the variances (using principal component coordinates) associated with the evolutionary cycles over multiple rounds of adaptive therapy. The variances increase subquadratically through several rounds before the evolutionary cycle begins to break down. Despite this, we show the stochastic adaptive schedules are more effective at delaying resistance than standard maximum tolerated dose and low-dose metronomic schedules. The simplified low-dimensional model provides some insights on how well multiple rounds of adaptive therapies are likely to perform over a range of tumor sizes (i.e., different values of N) if the goal is to maintain a sustained balance among competing subpopulations of cells to avoid chemoresistance via competitive release in a stochastic environment.

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.

<div>Abstract<p>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.</p>Significance:<p>Driving tumor evolution into periodic, repeatable treatment cycles provides a path forward for multidrug adaptive therapy.</p></div>

<p>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).</p>

<p>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).</p>

<div>Abstract<p>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.</p>Significance:<p>Driving tumor evolution into periodic, repeatable treatment cycles provides a path forward for multidrug adaptive therapy.</p></div>

<p>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.</p>

<p>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.</p>

<div>Abstract<p>The classic view of metastatic cancer progression is that it is a unidirectional process initiated at the primary tumor site, progressing to variably distant metastatic sites in a fairly predictable, although not perfectly understood, fashion. A Markov chain Monte Carlo mathematical approach can determine a pathway diagram that classifies metastatic tumors as “spreaders” or “sponges” and orders the timescales of progression from site to site. In light of recent experimental evidence highlighting the potential significance of self-seeding of primary tumors, we use a Markov chain Monte Carlo (MCMC) approach, based on large autopsy data sets, to quantify the stochastic, systemic, and often multidirectional aspects of cancer progression. We quantify three types of multidirectional mechanisms of progression: (i) self-seeding of the primary tumor, (ii) reseeding of the primary tumor from a metastatic site (primary reseeding), and (iii) reseeding of metastatic tumors (metastasis reseeding). The model shows that the combined characteristics of the primary and the first metastatic site to which it spreads largely determine the future pathways and timescales of systemic disease. <i>Cancer Res; 73(9); 2760–9. ©2013 AACR</i>.</p></div>

<div>Abstract<p>The classic view of metastatic cancer progression is that it is a unidirectional process initiated at the primary tumor site, progressing to variably distant metastatic sites in a fairly predictable, although not perfectly understood, fashion. A Markov chain Monte Carlo mathematical approach can determine a pathway diagram that classifies metastatic tumors as “spreaders” or “sponges” and orders the timescales of progression from site to site. In light of recent experimental evidence highlighting the potential significance of self-seeding of primary tumors, we use a Markov chain Monte Carlo (MCMC) approach, based on large autopsy data sets, to quantify the stochastic, systemic, and often multidirectional aspects of cancer progression. We quantify three types of multidirectional mechanisms of progression: (i) self-seeding of the primary tumor, (ii) reseeding of the primary tumor from a metastatic site (primary reseeding), and (iii) reseeding of metastatic tumors (metastasis reseeding). The model shows that the combined characteristics of the primary and the first metastatic site to which it spreads largely determine the future pathways and timescales of systemic disease. <i>Cancer Res; 73(9); 2760–9. ©2013 AACR</i>.</p></div>