The mathematics of cancer: integrating quantitative models

Philipp M. Altrock; Liu, Lin; Franziska Michor

Altrock, P.M., et al., 2018. A computational model of MGUS progression to Multiple Myeloma identifies optimum screening strategies and their effects on mortality. JCO Clinical Cancer Informatics , 2018 (2018(2) , pp. 1-12. Publisher's Version Abstract

altrock2018cci.pdf

PURPOSE: Recent advances uncovered therapeutic interventions that might reduce the risk of progression of premalignant diagnoses, such as Monoclonal Gammopathy of Undetermined Significance (MGUS) to multiple myeloma (MM). It remains unclear how to best screen populations at risk and how to evaluate the ability of these interventions to reduce disease prevalence and mortality at the population level. To address these questions, we developed a computational modeling framework. METHODS: We used individual-based computational modeling of MGUS incidence and progression across a population of diverse individuals, to determine best screening strategies in terms of screening start, intervals, and risk-group specificity. Inputs were life tables, MGUS incidence and baseline MM survival. We measured MM-specific mortality and MM prevalence following MGUS detection from simulations and mathematical precition modeling. RESULTS: We showed that our framework is applicable to a wide spectrum of screening and intervention scenarios, including variation of the baseline MGUS to MM progression rate and evolving MGUS, in which progression increases over time. Given the currently available progression risk-point estimate of 61% risk, starting screening at age 55 and follow-up screening every 6yrs reduced total MM prevalence by 19%. The same reduction could be achieved with starting age 65 and follow-up every 2yrs. A 40% progression risk reduction per MGUS patient per year would reduce MM-specific mortality by 40%. Generally, age of screening onset and frequency impact disease prevalence, progression risk reduction impacts both prevalence and disease-specific mortality, and screeenign would generally be favorable in high-risk individuals. CONCLUSION: Screening efforts should focus on specifically identified groups of high lifetime risk of MGUS, for which screening benefits can be significant. Screening low-risk MGUS individuals would require improved preventions.

Gerlee, P. & Altrock, P.M., 2017. Extinction rates in tumour public goods games. Royal Society Interface , 14 (134). Publisher's Version Abstract

20170342.full_.pdf

Cancer evolution and progression are shaped by cellular interactions and Darwinian selection. Evolutionary game theory incorporates both of these principles, and has been proposed as a framework to understand tumour cell population dynamics. A cornerstone of evolutionary dynamics is the replicator equation, which describes changes in the relative abundance of different cell types, and is able to predict evolutionary equilibria. Typically, the replicator equation focuses on differences in relative fitness. We here show that this framework might not be sufficient under all circumstances, as it neglects important aspects of population growth. Standard replicator dynamics might miss critical differences in the time it takes to reach an equilibrium, as this time also depends on cellular turnover in growing but bounded populations. As the system reaches a stable manifold, the time to reach equilibrium depends on cellular death and birth rates. These rates shape the time scales, in particular, in coevolutionary dynamics of growth factor producers and free-riders. Replicator dynamics might be an appropriate framework only when birth and death rates are of similar magnitude. Otherwise, population growth effects cannot be neglected when predicting the time to reach an equilibrium, and cell-type-specific rates have to be accounted for explicitly.

Altrock, P.M., Traulsen, A. & Nowak, M.A., 2017. Evolutionary games on cycles with strong selection. Physical Review E , 95 , pp. 022407. Publisher's Version Abstract

altrock2017pre.pdf

Evolutionary games on graphs describe how strategic interactions and population structure determine evolutionary success, quantified by the probability that a single mutant takes over a population. Graph structures, compared to the well-mixed case, can act as amplifiers or suppressors of selection by increasing or decreasing the fixation probability of a beneficial mutant. Properties of the associated mean fixation times can be more intricate, especially when selection is strong. The intuition is that fixation of a beneficial mutant happens fast in a dominance game, that fixation takes very long in a coexistence game, and that strong selection eliminates demographic noise. Here we show that these intuitions can be misleading in structured populations. We analyze mean fixation times on the cycle graph under strong frequency-dependent selection for two different microscopic evolutionary update rules (death-birth and birth-death). We establish exact analytical results for fixation times under strong selection and show that there are coexistence games in which fixation occurs in time polynomial in population size. Depending on the underlying game, we observe inherence of demographic noise even under strong selection if the process is driven by random death before selection for birth of an offspring (death-birth update). In contrast, if selection for an offspring occurs before random removal (birth-death update), then strong selection can remove demographic noise almost entirely.

Altrock, P.M., et al., 2016. Mathematical modeling of erythrocyte chimerism informs genetic intervention strategies for sickle cell disease. American Journal of Hematology , 91 (9) , pp. 931-937. Publisher's Version Abstract

Altrock2016ajh.pdf

Recent advances in gene therapy and genome-engineering technologies offer the opportunity to correct sickle cell disease (SCD), a heritable disorder caused by a point mutation in the β-globin gene. The developmental switch from fetal γ-globin to adult β-globin is governed in part by the transcription factor (TF) BCL11A. This TF has been proposed as a therapeutic target for reactivation of γ-globin and concomitant reduction of β-sickle globin. In this and other approaches, genetic alteration of a portion of the hematopoietic stem cell (HSC) compartment leads to a mixture of sickling and corrected red blood cells (RBCs) in periphery. To reverse the sickling phenotype, a certain proportion of corrected RBCs is necessary; the degree of HSC alteration required to achieve a desired fraction of corrected RBCs remains unknown. To address this issue, we developed a mathematical model describing aging and survival of sickle-susceptible and normal RBCs; the former can have a selective survival advantage leading to their overrepresentation. We identified the level of bone marrow chimerism required for successful stem cell-based gene therapies in SCD. Our findings were further informed using an experimental mouse model, where we transplanted mixtures of Berkeley SCD and normal murine bone marrow cells to establish chimeric grafts in murine hosts. Our integrative theoretical and experimental approach identifies the target frequency of HSC alterations required for effective treatment of sickling syndromes in humans. Our work replaces episodic observations of such target frequencies with a mathematical modeling framework that covers a large and continuous spectrum of chimerism conditions.

$The cancer stem cell fraction in hierarchically organized tumors can be estimated using mathematical modeling and patient-specific treatment trajectories$

Werner, B., et al., 2016. The cancer stem cell fraction in hierarchically organized tumors can be estimated using mathematical modeling and patient-specific treatment trajectories. Cancer Research , 76 , pp. 1705-1713. Publisher's Version Abstract

werner2016cr.pdf

Many tumors are hierarchically organized and driven by a sub-population of tumor initiating cells (TICs), or cancer stem cells. TICs are uniquely capable of recapitulating the tumor and are implied to be highly resistant to radio- and chemotherapy. Macroscopic patterns of tumor expansion before treatment and tumor regression during treatment are tied to the dynamics of TICs. Until now, quantitative information about the fraction of TICs from macroscopic tumor burden trajectories could not be inferred. In this study, we generated a quantitative method based on a mathematical model that describes hierarchically organized tumor dynamics and patient-derived tumor burden information. The method identifies two characteristic equilibrium TIC regimes during expansion and regression. We show that tumor expansion and regression curves can be leveraged to infer estimates of the TIC fraction in individual patients at detection and after continued therapy. Furthermore, our method is parameter-free; it solely requires knowledge of a patient's tumor burden over multiple time points to reveal microscopic properties of the malignancy. We demonstrate proof of concept in the case of chronic myeloid leukemia (CML), wherein our model recapitulated the clinical history of the disease in two independent patient cohorts. Based on patient-specific treatment responses in CML, we predict that after one year of targeted treatment, the fraction of TICs increases 100-fold and continues to increase up to 1000-fold after five years of treatment. Our novel framework may significantly influence the implementation of personalized treatment strategies and has the potential for rapid translation into the clinic.

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Dr. Philipp M. Altrock

The mathematics of cancer: integrating quantitative models

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