Jeffrey West

Chemotherapeutic Dose Scheduling Based on Tumor Growth Rates Provides a Case for Low-Dose Metronomic High-Entropy Therapies

We extended the classical tumor regression models such as Skippers 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 prisoners 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.

Nonlinear dynamical shaping of the fitness landscape of an evolving tumor to combat competitive release

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 dominant 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 patients 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.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 that are selected for growth after chemotherapeutic agents attack the population of chemo-sensitive cells which had previously dominated the collection of competing subclones. To study this process, we use an evolutionary game theory model, with a prisoner9s dilemma payoff matrix, based on a system of coupled replicator equations quantifying the clonal competition among three groups of cells: healthy cells (H), sensitive cells (S), and resistant cells (R). Maximum tolerated dose (MTD) schedules are effective at reducing the sensitive cell population which initially shrinks the tumor volume, but releases the resistant cells to re-populate and eventually re-grow the tumor in a more dangerous resistant form. By monitoring the state space associated with the three populations of cells as a coupled nonlinear dynamical system and using the nullcline structure of the system, we show how one can steer the tumor away from the resistant state with an adaptive chemotherapeutic schedule. The control parameters in our model adjust the selection pressure on the various subclones, which effectively allows us to tailor the fitness landscape to suppress the growth of the resistant population while keeping the sensitive population at low enough levels so the tumor volume remains small.

An Evolutionary Model of Tumor Cell Kinetics and the Emergence of Molecular Heterogeneity Driving Gompertzian Growth

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 prisoners 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 prisoners 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.

Optimizing chemo-scheduling based on tumor growth rates

We review the classic tumor growth and regression laws of Skipper and Schable based on fixed exponential growth assumptions, and Norton and Simon’s law based on a Gompertzian growth assumption. We then discuss ways to optimize chemotherapeutic scheduling using a Moran process evolutionary game-theory model of tumor growth that incorporates more general dynamical and evolutionary features of tumor cell kinetics. Using this model, and employing the quantitative notion of Shannon entropy which assigns high values to low-dose metronomic (LDM) therapies, and low values to maximum tolerated dose (MTD) therapies, we show that low-dose metronomic strategies can outperform maximum tolerated dose strategies, particularly for faster growing tumors. The general concept of designing different chemotherapeutic strategies for tumors with different growth characteristics is discussed.

Competitive release in tumors

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.

Evolutionary exploitation of PD-L1 expression in hormone receptor positive breast cancer

Based on clinical data from hormone positive breast cancer patients, we determined that there is a potential tradeoff between reducing tumor burden and altering metastatic potential when administering combination therapy of aromatase inhibitors and immune checkpoint inhibitors. While hormone-deprivation therapies serve to reduce tumor size in the neoadjuvant setting pre-surgery, they may induce tumors to change expression patterns towards a metastatic phenotype. We used mathematical modeling to explore how the timing of the therapies affects tumor burden and metastatic potential with an eye toward developing a dynamic prognostic score and reducing both tumor size and risk of metastasis.

The immune checkpoint kick start: Optimization of neoadjuvant combination therapy using game theory

An upcoming clinical trial at the Moffitt Cancer Center for women with stage 2/3 ER+breast cancer combines an aromatase inhibitor and a PD-L1 checkpoint inhibitor, and aims to lower a preoperative endocrine prognostic index (PEPI) that correlates with relapse-free survival. PEPI is fundamentally a static index, measured at the end of neoadjuvant therapy before surgery. We develop a mathematical model of the essential components of the PEPI score in order to identify successful combination therapy regimens that minimize both tumor burden and metastatic potential, based on time-dependent trade-offs in the system. We consider two molecular traits, CCR7 and PD-L1 which correlate with treatment response and increased metastatic risk. We use a matrix game model with the four phenotypic strategies to examine the frequency-dependent interactions of cancer cells. This game was embedded into an ecological model of tumor population growth dynamics. The resulting model predicts both evolutionary and ecological dynamics that track with changes in the PEPI score. We consider various treatment regimens based on combinations of the two therapies with drug holidays. By considering the trade off between tumor burden and metastatic potential, the optimal therapy plan was found to be a 1 month kick start of the immune checkpoint inhibitor followed by five months of continuous combination therapy. Relative to a protocol with both therapeutics given together from the start, this delayed regimen results in transient sub-optimal tumor regression while maintaining a phenotypic constitution that is more amenable to fast tumor regression for the final five months of therapy. The mathematical model provides a useful abstraction of clinical intuition, enabling hypothesis generation and testing of clinical assumptions.

Cellular cooperation shapes tumor growth: a statistical mechanics mathematical model

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 statement A 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.