Cancer, a leading cause of death globally, is characterized by the uncontrolled growth of abnormal cells evolving due to natural selection. A cancerous tumor is a complex ecosystem of heterogeneous cell populations that, over time, acquire new traits like therapy resistance. Despite progress in experimental methods, measuring genetic and phenotypic processes on time scales relevant to tumor evolution is still challenging. As a result, the mechanisms that lead to tumor heterogeneity, evolution, progression, and response to treatment remain largely unclear. Mathematical models can help address this challenge, allowing us to test hypotheses, predict cellular behavior, and optimize cancer treatment. In this thesis, I investigate the role of genetic and phenotypic heterogeneity in tumor evolution using mathematical models and analysis.
Discrete stochastic models are well-suited to study tumor evolution due to the involvement of rare stochastic events and small populations. Here, I introduce evolutionary lattice-gas cellular automata (evo-LGCA), a generalization of classical lattice-gas cellular automata (LGCA). LGCA are discrete mathematical models describing the interactions of moving agents, such as cancer cells, on a regular lattice, with discretized velocities, and in discrete time steps. Agents are indistinguishable and obey an exclusion principle that prevents them from being simultaneously in the same state, causing unwanted behavior. In contrast, in evo-LGCA, agents are distinguishable, have unique properties, and can be in the same state, minimizing model artifacts. This makes evo-LGCA particularly suitable for studying the complexity of tumors.
Using this framework, I investigate the interplay of evolutionary dynamics and population growth. In particular, I am interested in the role of the distribution of fitness effects (DFE). The DFE determines the strength and frequency of the effect of mutations. I present an evo-LGCA model for tumor evolution, in which cells can divide, die, move, and mutate given an arbitrary but fixed DFE. From the dynamics of the evo-LGCA model, I derive an integro-partial differential equation, predicting the
distribution in fitness space over time. This equation is equivalent to the replicator-mutator equation, establishing a connection to population genetics and evolutionary game theory. Additionally, I derive a generalized version of Fisher’s fundamental theorem of natural selection, a classic theorem stating that a population’s change in mean fitness is proportional to the population’s variance in fitness. However, it neglects the effect of mutations and the dynamics of higher moments, such as the variance. My generalization is a hierarchy of equations for the time evolution of all moments of the fitness distribution, depending on the DFE. Through simulations of the evo-LGCA model, I show that continuum approximations are suitable in regimes of frequent mutations with weak effects on fitness and large, well-mixed populations. I further establish that the fastest-growing cells spearhead spreading populations, accelerating the expansion speed.
Next, I examine the evolutionary dynamics within small, clinically undetectable tumors. Cancer cells quickly accumulate weakly disadvantageous passenger mutations, whereas beneficial driver mutations are rare but have a significant effect. Previous studies have shown that this leads to competition between passenger and driver mutations, affecting population fitness. Populations below a critical population size accumulate deleterious mutations too quickly, leading to extinction. I highlight how small cancer cell populations can bypass potential extinction through swift invasion of their microenvironment. This invasion can be seen as an adaptation to counteract the accumulation of disadvantageous mutations.
Lastly, I examine the complex relationship between evolution and phenotypic plasticity, focusing on the phenotypic change between proliferative and migratory phenotypes relevant to tumors like glioblastoma, a deadly brain tumor. Contrary to previous studies, I propose that evolution acts on the cellular decision-making process in response to the environment rather than on phenotypic traits like cell motility. I study this hypothesis with an evo-LGCA model that tracks individual cells’ phenotypic and genetic states. I assume cells change between migratory and proliferative states controlled by inherited and mutation-driven genotypes and the cells’ microenvironment in the form of cell density. Cells at the tumor edge evolve to favor migration over proliferation and vice versa in the tumor bulk. Notably, this phenotypic heterogeneity can be realized by two distinct regulations of the phenotypic switch. I predict the outcome of the evolutionary process with a mathematical analysis, revealing a dependence on microenvironmental parameters. The emerging synthetic tumors display varying levels of heterogeneity, which I show are predictors of the cancer’s recurrence time after treatment. Interestingly, higher phenotypic heterogeneity predicts poor treatment outcomes, unlike genetic heterogeneity.
In conclusion, this thesis offers a mathematical framework for studying heterogeneous populations. Applying it to tumor evolution, I gained new insights into the relationship between discrete and continuous evolution models and the interplay of population growth and evolutionary dynamics. I also proposed a novel perspective on phenotypic plasticity accounting for cell decision-making, demonstrating the predictive value of phenotypic heterogeneity.:1. Introduction [13]
1.1 Background on Cancer [13]
1.1.1 Definition [13]
1.1.2 Hallmarks of Cancer [13]
1.1.3 Cancer as a Genetic Disease [14]
1.1.4 Tumor Evolution [15]
1.1.5 Tumor Heterogeneity [17]
1.2 Mathematical Models of Tumor Evolution and Heterogeneity [19]
1.2.1 Overview [19]
1.2.2 Deterministic Approaches [20]
1.2.3 Agent-Based Approaches [24]
1.2.4 Hybrid Models [26]
1.2.5 Evolutionary Game Theory [27]
1.3 Research Questions and Dissertation Outline [27]
2. Evolutionary Lattice-Gas Cellular Automata [31]
2.1 Cellular Automaton Basics [31]
2.2 Lattice-Gas Cellular Automata [33]
2.2.1 Origins [33]
2.2.2 Definition [34]
2.2.3 Extensions [39]
2.3 Evolutionary Lattice-Gas Cellular Automata [43]
2.3.1 Concept [43]
2.3.2 State Space [44]
2.3.3 Dynamics [45]
2.4 Discussion [49]
3. Bridging Micro- and Macroscale of Evolutionary Dynamics [51]
3.1 Connecting Discrete and Continuous Models of Evolution [51]
3.2 Model Definition [53]
3.3 Mathematical Analysis [55]
3.3.1 Mean-Field Approximation of Evolutionary Dynamics [55]
3.3.2 A Generalized Fundamental Theorem of Natural Selection [57]
3.3.3 Derivation of Local Replicator-Mutator Equation [61]
3.3.4 Finite-Size Correction [62]
3.3.5 Spatial Growth Dynamics [63]
3.4 Comparison with Agent-Based Simulations [64]
3.4.1 Well-Mixed Populations [64]
3.4.2 Expanding Populations [68]
3.5 Discussion [69]
4. The Interplay of Invasion and Mutational Meltdown [73]
4.1 Muller’s Ratchet in Tumors [73]
4.2 Influence of Invasion on Evolutionary Dynamics [74]
4.3 Model Parameterization [74]
4.4 Tug-of-War between Driver and Passenger Mutations [76]
4.5 Invasion as a Strategy against Mutational Meltdown [79]
4.6 Discussion [80]
5. Evolution under the Go-or-Grow Dichotomy [85]
5.1 Phenotypic Plasticity [85]
5.2 The Role of Cell Decision-Making in Evolutionary Dynamics [86]
5.3 Model Definition [87]
5.4 Emergence of Phenotypic and Genetic Heterogeneity [90]
5.4.1 Migratory Phenotype Favored by Minimal Apoptosis Rates [91]
5.4.2 Emerging Spatial Heterogeneity for Low Switching Threshold [91]
5.4.3 Repulsive Strategy Favored by High Switching Threshold [92]
5.4.4 Prediction of Optimal Go-or-Grow Strategy [92]
5.5 Heterogeneity as a Predictor of Treatment Outcomes [95]
5.6 Discussion [98]
6. Discussion & Outlook [103]
A. Mathematical Derivations [107]
B. Supplementary Simulations [113]
C. Software [119]
Bibliography [121]
| Identifer | oai:union.ndltd.org:DRESDEN/oai:qucosa:de:qucosa:89825 |
| Date | 21 February 2024 |
| Creators | Syga, Simon |
| Contributors | Deutsch, Andreas, Behme, Anita, Altrock, Philipp, Voigt, Axel, Technische Universität Dresden |
| Source Sets | Hochschulschriftenserver (HSSS) der SLUB Dresden |
| Language | English |
| Detected Language | English |
| Type | info:eu-repo/semantics/publishedVersion, doc-type:doctoralThesis, info:eu-repo/semantics/doctoralThesis, doc-type:Text |
| Rights | info:eu-repo/semantics/openAccess |
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