Adaptive radiation therapy re-optimizes treatment plans based on updated tumor geometries from magnetic resonance imaging scans. However, the imaging process is costly in labor and equipment. In this study, we develop a mathematical model that describes tumor evolution based on a Markov assumption. We then extend the model to predict tumor evolution with any level of information from a new patient: weekly MRI scans are used to estimate transition probabilities when available, otherwise historical MRI scans are used. In the latter case, patients in the historical data are clustered into two groups, and the model relates the new patient's behavior to the existing two groups. The models are evaluated with 33 cervical cancer patients from Princess Margaret Cancer Centre. The result indicates that our models outperform the constant volume model, which replicates the current clinical practice.

Adaptive radiation therapy re-optimizes treatment plans based on updated tumor geometries from magnetic resonance imaging scans. However, the imaging process is costly in labor and equipment. In this study, we develop a mathematical model that describes tumor evolution based on a Markov assumption. We then extend the model to predict tumor evolution with any level of information from a new patient: weekly MRI scans are used to estimate transition probabilities when available, otherwise historical MRI scans are used. In the latter case, patients in the historical data are clustered into two groups, and the model relates the new patient's behavior to the existing two groups. The models are evaluated with 33 cervical cancer patients from Princess Margaret Cancer Centre. The result indicates that our models outperform the constant volume model, which replicates the current clinical practice.

O câncer é caracterizado pelo crescimento anormal de células em consequência ao acúmulo de alterações no DNA. Diferentes tecidos apresentam variadas incidências de tumores. Em 2015, Tomasetti & Vogelstein demonstraram uma forte correlação positiva (r = 0.804) entre o número de divisões de células tronco e o risco de câncer em diferentes tecidos, em que tecidos com maior número de divisões de células tronco estão mais susceptíveis aos efeitos estocásticos da replicação do DNA, e, assim, mais propícios a desenvolverem tumores. Assim, esta correlação é justificada pelo número de mutações. Neste trabalho, propomos e testamos in silico um novo conceito para justificar parte desta correlação positiva entre o número de divisões de células tronco e o risco de câncer entre diferentes tecidos, o qual denominamos concentração de ancestrais (AC). Em nossa hipótese, tecidos com alta taxa de proliferação concentram mais seus ancestrais, amplificando as chances de perpetuar células ancestrais mutadas, e, por isso, estão relacionados a maiores riscos de câncer. Assim, tecidos com altos valores de divisão de células tronco apresentam um perfil de alto AC e tecidos com baixo número de divisão de células tronco apresentam um perfil de baixo AC Para comprovar nossa hipótese, simulamos a evolução tumoral através do software esi- Cancer e aplicamos diferentes valores de proliferação e morte (CTOR). Os resultados demonstraram uma correlação positiva de 0.995 entre o valor de CTOR e o perfil de AC (P = 0:002). Como esperado, demonstramos que maiores CTORs estão relacionados a maiores médias de gerações com esiTumors para valores totais de mutações por divisão iguais. Entretanto, esta relação se mantém quando aplicadas valores corrigidos conforme o número de divisões para os diferentes CTORs, a fim de o número de mutações totais ser igual. Logo, apenas variações não são suficientes para explicar a incidência observada em diferentes tecidos. Nossos resultados demonstram que tecidos com maior número de divisões de células tronco apresentam um perfil de alto AC, o qual amplifica as chances de concentrar ancestrais mutados, aumentando as chances de desenvolver tumores. Assim, justificando parte da correlação encontrada por Tomasetti & Vogelstein (2015). / Cancer is characterized by an abnormal replication of somatic cells as a result of DNA alterations. Different types of tissues present differences in cancer incidence. Tomasetti & Vogelstein (2015) have shown that lifetime cancer risk of different tissues presents a strong correlation of 0.804 with the number of stem cells divisions, in which tissues with higher number of stem cells divisions are more susceptible to stochastic effects of DNA replication and thus more likely to develop cancer. Thus, the number of mutations was used to explain this correlation. In our work, we propose and test in silico a new concept to explain this positive correlation, which we denominated ancestral concentration (AC). In our hypothesis, a tissue with high proliferation rates concentrates more their ancestral cells and increases the chance of a mutated ancestral to persist; which result in a higher risk of cancer. Tissues with a high number of stem cells divisions presents a high AC profile whereas tissues with a low number of stem cells divisions presents a low AC profile To prove our hypothesis, we simulated tumor evolution using esiCancer software and applied different initial rates of proliferation and death (CTOR). We observed a positive correlation of 0.995 between CTOR values and the AC profile (P = 0:002). Besides, higher CTOR values are associated to higher mean generations with esiTumors when equal mutation rates are applied. Nevertheless, this association still exist in simulations with mutation rates corrected by total number of divisions, whereas the total mutation rate is similar for different CTORs. This way, modifications of mutations solely are not sufficient to explain the observed cancer risks in different tissues. Our results showed that tissues with higher number of stem cells divisions present a high AC profile, which rises the probabilities of concentrate mutated ancestral cells, increasing the tumor risk. In this way, justifying partly the correlation that was founded by Tomasetti & Vogelstein (2015).

O câncer é caracterizado pelo crescimento anormal de células em consequência ao acúmulo de alterações no DNA. Diferentes tecidos apresentam variadas incidências de tumores. Em 2015, Tomasetti & Vogelstein demonstraram uma forte correlação positiva (r = 0.804) entre o número de divisões de células tronco e o risco de câncer em diferentes tecidos, em que tecidos com maior número de divisões de células tronco estão mais susceptíveis aos efeitos estocásticos da replicação do DNA, e, assim, mais propícios a desenvolverem tumores. Assim, esta correlação é justificada pelo número de mutações. Neste trabalho, propomos e testamos in silico um novo conceito para justificar parte desta correlação positiva entre o número de divisões de células tronco e o risco de câncer entre diferentes tecidos, o qual denominamos concentração de ancestrais (AC). Em nossa hipótese, tecidos com alta taxa de proliferação concentram mais seus ancestrais, amplificando as chances de perpetuar células ancestrais mutadas, e, por isso, estão relacionados a maiores riscos de câncer. Assim, tecidos com altos valores de divisão de células tronco apresentam um perfil de alto AC e tecidos com baixo número de divisão de células tronco apresentam um perfil de baixo AC Para comprovar nossa hipótese, simulamos a evolução tumoral através do software esi- Cancer e aplicamos diferentes valores de proliferação e morte (CTOR). Os resultados demonstraram uma correlação positiva de 0.995 entre o valor de CTOR e o perfil de AC (P = 0:002). Como esperado, demonstramos que maiores CTORs estão relacionados a maiores médias de gerações com esiTumors para valores totais de mutações por divisão iguais. Entretanto, esta relação se mantém quando aplicadas valores corrigidos conforme o número de divisões para os diferentes CTORs, a fim de o número de mutações totais ser igual. Logo, apenas variações não são suficientes para explicar a incidência observada em diferentes tecidos. Nossos resultados demonstram que tecidos com maior número de divisões de células tronco apresentam um perfil de alto AC, o qual amplifica as chances de concentrar ancestrais mutados, aumentando as chances de desenvolver tumores. Assim, justificando parte da correlação encontrada por Tomasetti & Vogelstein (2015). / Cancer is characterized by an abnormal replication of somatic cells as a result of DNA alterations. Different types of tissues present differences in cancer incidence. Tomasetti & Vogelstein (2015) have shown that lifetime cancer risk of different tissues presents a strong correlation of 0.804 with the number of stem cells divisions, in which tissues with higher number of stem cells divisions are more susceptible to stochastic effects of DNA replication and thus more likely to develop cancer. Thus, the number of mutations was used to explain this correlation. In our work, we propose and test in silico a new concept to explain this positive correlation, which we denominated ancestral concentration (AC). In our hypothesis, a tissue with high proliferation rates concentrates more their ancestral cells and increases the chance of a mutated ancestral to persist; which result in a higher risk of cancer. Tissues with a high number of stem cells divisions presents a high AC profile whereas tissues with a low number of stem cells divisions presents a low AC profile To prove our hypothesis, we simulated tumor evolution using esiCancer software and applied different initial rates of proliferation and death (CTOR). We observed a positive correlation of 0.995 between CTOR values and the AC profile (P = 0:002). Besides, higher CTOR values are associated to higher mean generations with esiTumors when equal mutation rates are applied. Nevertheless, this association still exist in simulations with mutation rates corrected by total number of divisions, whereas the total mutation rate is similar for different CTORs. This way, modifications of mutations solely are not sufficient to explain the observed cancer risks in different tissues. Our results showed that tissues with higher number of stem cells divisions present a high AC profile, which rises the probabilities of concentrate mutated ancestral cells, increasing the tumor risk. In this way, justifying partly the correlation that was founded by Tomasetti & Vogelstein (2015).

O câncer é caracterizado pelo crescimento anormal de células em consequência ao acúmulo de alterações no DNA. Diferentes tecidos apresentam variadas incidências de tumores. Em 2015, Tomasetti & Vogelstein demonstraram uma forte correlação positiva (r = 0.804) entre o número de divisões de células tronco e o risco de câncer em diferentes tecidos, em que tecidos com maior número de divisões de células tronco estão mais susceptíveis aos efeitos estocásticos da replicação do DNA, e, assim, mais propícios a desenvolverem tumores. Assim, esta correlação é justificada pelo número de mutações. Neste trabalho, propomos e testamos in silico um novo conceito para justificar parte desta correlação positiva entre o número de divisões de células tronco e o risco de câncer entre diferentes tecidos, o qual denominamos concentração de ancestrais (AC). Em nossa hipótese, tecidos com alta taxa de proliferação concentram mais seus ancestrais, amplificando as chances de perpetuar células ancestrais mutadas, e, por isso, estão relacionados a maiores riscos de câncer. Assim, tecidos com altos valores de divisão de células tronco apresentam um perfil de alto AC e tecidos com baixo número de divisão de células tronco apresentam um perfil de baixo AC Para comprovar nossa hipótese, simulamos a evolução tumoral através do software esi- Cancer e aplicamos diferentes valores de proliferação e morte (CTOR). Os resultados demonstraram uma correlação positiva de 0.995 entre o valor de CTOR e o perfil de AC (P = 0:002). Como esperado, demonstramos que maiores CTORs estão relacionados a maiores médias de gerações com esiTumors para valores totais de mutações por divisão iguais. Entretanto, esta relação se mantém quando aplicadas valores corrigidos conforme o número de divisões para os diferentes CTORs, a fim de o número de mutações totais ser igual. Logo, apenas variações não são suficientes para explicar a incidência observada em diferentes tecidos. Nossos resultados demonstram que tecidos com maior número de divisões de células tronco apresentam um perfil de alto AC, o qual amplifica as chances de concentrar ancestrais mutados, aumentando as chances de desenvolver tumores. Assim, justificando parte da correlação encontrada por Tomasetti & Vogelstein (2015). / Cancer is characterized by an abnormal replication of somatic cells as a result of DNA alterations. Different types of tissues present differences in cancer incidence. Tomasetti & Vogelstein (2015) have shown that lifetime cancer risk of different tissues presents a strong correlation of 0.804 with the number of stem cells divisions, in which tissues with higher number of stem cells divisions are more susceptible to stochastic effects of DNA replication and thus more likely to develop cancer. Thus, the number of mutations was used to explain this correlation. In our work, we propose and test in silico a new concept to explain this positive correlation, which we denominated ancestral concentration (AC). In our hypothesis, a tissue with high proliferation rates concentrates more their ancestral cells and increases the chance of a mutated ancestral to persist; which result in a higher risk of cancer. Tissues with a high number of stem cells divisions presents a high AC profile whereas tissues with a low number of stem cells divisions presents a low AC profile To prove our hypothesis, we simulated tumor evolution using esiCancer software and applied different initial rates of proliferation and death (CTOR). We observed a positive correlation of 0.995 between CTOR values and the AC profile (P = 0:002). Besides, higher CTOR values are associated to higher mean generations with esiTumors when equal mutation rates are applied. Nevertheless, this association still exist in simulations with mutation rates corrected by total number of divisions, whereas the total mutation rate is similar for different CTORs. This way, modifications of mutations solely are not sufficient to explain the observed cancer risks in different tissues. Our results showed that tissues with higher number of stem cells divisions present a high AC profile, which rises the probabilities of concentrate mutated ancestral cells, increasing the tumor risk. In this way, justifying partly the correlation that was founded by Tomasetti & Vogelstein (2015).

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]

Molecular mechanisms of lower-grade (II–III) diffuse gliomas (LGG) are still poorly understood, mainly because of their heterogeneity. They split into astrocytoma- (IDH-A) and oligodendroglioma-like (IDH-O) tumors both carrying mutations(s) at the isocitrate dehydrogenase (IDH) gene and into IDH wild type (IDH-wt) gliomas of glioblastoma resemblance. We generated detailed maps of the transcriptomes and DNA methylomes, revealing that cell functions divided into three major archetypic hallmarks: (i) increased proliferation in IDH-wt and, to a lesser degree, IDH-O; (ii) increased inflammation in IDH-A and IDH-wt; and (iii) the loss of synaptic transmission in all subtypes. Immunogenic properties of IDH-A are diverse, partly resembling signatures observed in grade IV mesenchymal glioblastomas or in grade I pilocytic astrocytomas. We analyzed details of coregulation between gene expression and DNA methylation and of the immunogenic micro-environment presumably driving tumor development and treatment resistance. Our transcriptome and methylome maps support personalized, case-by-case views to decipher the heterogeneity of glioma states in terms of data portraits. Thereby, molecular cartography provides a graphical coordinate system that links gene-level information with glioma subtypes, their phenotypes, and clinical context.

Die Intra-Tumor-Heterogenität spiegelt eine kontinuierliche Entwicklung zwischen den Zellen eines einzelnen Tumors wider. Sie ist eine der Hauptursachen für Arzneimittelresistenz bei der Krebsbehandlung. Um dieses Problem anzugehen, ist es daher wichtig, die Tumorevolution innerhalb eines einzelnen Patienten zu verstehen und erfolgreich zu modellieren. Bisherige Arbeiten haben sich nicht erfolgreich mit der Evolution von Tumoren befasst, deren Treiber strukturelle Veränderungen im Genom sind, wie z. B. somatische Kopienzahlveränderungen (SCNAs). Diese Arbeit befasst sich mit der Herausforderung, die Tumorevolution als Folge solcher Veränderungen zu charakterisieren. Wir verwenden einen phylogenetischen Ansatz zur Analyse von multiregionalen Datensätzen in einer großen Pan-Krebs-Kohorte. Wir untersuchen häufige SCNAs in verschiedenen Stadien der Tumorentwicklung und führen eine neue Methode, MEDICC2, ein, die die Tumorevolution innerhalb eines einzelnen Patienten rekonstruiert. In dieser Arbeit haben wir häufige SCNAs charakterisiert, die früh in der Tumorentwicklung auftreten. Aufgrund der Struktur der Kohorte ist die Charakterisierung der subklonalen SCNAs nicht eindeutig. Unsere neue Methode, MEDICC2, akzeptiert höhere Kopienzahlzustände und berücksichtigt die Verdopplung des gesamten Genoms, ein häufiges Ereignis in Tumoren, was eine genauere Modellierung der Tumorevolution ermöglicht. / Intra-tumor heterogeneity reflects an ongoing evolution among cells of a single tumor. It is one of the leading causes of drug resistance in cancer treatments. Therefore, to address this issue, it is important to understand and successfully model tumor evolution within a single patient. Previous work has failed to successfully address the evolution of tumors whose drivers are structural changes in the genome, such as somatic copy number alterations (SCNAs). This work addresses the challenge of characterizing tumor evolution as a result of such changes. We use a phylogenetic approach to analyze multi-region datasets in a large pan-cancer cohort. We investigate frequent SCNAs at different stages of tumor development, and introduce a new method, MEDICC2, which reconstructs tumor evolution within a single patient. In this work, we characterized frequent SCNAs that occur early in tumor development. Due to the structure of the cohort, the characterization of subclonal SCNAs remains inconclusive. Our new method, MEDICC2, accepts higher copy number states and takes into account whole-genome doubling, a frequent event in tumors, which allows for a more precise modeling of tumor evolution.