Schedule for: 18w5115 - Mathematical Challenges in the Analysis of Continuum Models for Cancer Growth, Evolution and Therapy

Diffuse low-grade gliomas are slowly growing tumors. After tens of years, they transform inexorably into more aggressive forms, jeopardizing the patient’s life. Mathematical modeling could help clinicians to have a better understanding of the natural history of these tumors and their response to treatments. We present here different models of these tumors: the first one is discrete and describes the appearance of the first glioma cells and the genesis of a tumor. The second model is continuous and consists in a PDE that describes the evolution of the cell density. This model can describe the natural evolution of gliomas, their response to treatments such as radiotherapy and the changes in their dynamics in pregnant women. The discrete and the continuous models are designed to be close to the biological reality. The results are quantitatively compared with either biological data or clinical data, at the cellular level (histological samples) and at the tissue level (MRI scans).

Glioblastoma (GBM) is the most common malignant brain tumour and has an extremely poor prognosis. The invasion of tumour cells into normal brain tissue makes complete surgical removal impossible; GBM is also resistant to treatment with chemotherapy and radiotherapy. However it is not clear how cells move within the tumour, nor how they move in the surrounding brain tissue, nor how their behaviour is affected by drugs. Experimental studies of cell movement have traditionally been done on 2D plates; however this does not well represent the 3D tissue environment. Recently VS lab have established an experimental system which allows them to image live cells in 3D spheroid systems. For spheroid systems built on glioblastoma cell lines, they have been able to obtain a large data set of cell trajectories and investigate the effect of different chemotherapy treatments on cell motility. The trajectories can be analysed to calculate not only cell speed, but also the type of movement, e.g. how tortuous is the trajectory. Furthermore because of the 3D imaging facility, they can examine how these properties change with position in the spheroid, and also can examine how motility patterns are dependent on the local environment, for example, whether cells moving through regions of the spheroid that are densely populated behave differently to those moving through the external matrix gel. By tracking individual cells they also can examine the full range of population behaviours, and are able to identify, for example, fast outliers. Mathematical models of cell motility have been developed which link individual-level behaviour (e.g. speed, turning frequency) to population-level descriptors (drift, diffusion). This can be done using classical probability theory (e.g. deriving diffusion approximations from simple random walks can be a simple application of the central limit theorem), or using tools from classical continuum mechanics which have been developed to describe solid & fluid mechanics, e.g. partial differential equation conservation equations. There are active theoretical research questions being explored in this area, for example, when cell-cell interactions become important, or individual behaviour is density dependent. Furthermore, much mathematical theory is developed with very limited experimental data; the imaging data obtained in VS lab is a fantastic data set from which to build and then test models. EH has a long-term interest in Ada Lovelace’s idea to create a Calculus of the Nervous System: a mathematical model for how the brain would give rise to thought, and nerves to feelings.
After exploring this by initially composing an orchestral work “Calculus of the Nervous System” (Wien Modern Commission 2011, subsequently performed at BBC Proms 2012), treating the structure of the work as a ‘neural network with its roots in strictly engineered time, pitch and rhythmic calculations using values derived from one source (an exponential equation and its derivative), subsequently muddled and reordered by chance processes’, Howard then metaphorically put “Calculus of the Nervous System” under the microscope to create orchestral work “Axon” (a BBC Commission for BBC Philharmonic, 2013) where the compositional process is one of “passing musical material through an imagined axon”. Most recently, Howard composed “Afference” (2014-15) for string quartet (short- listed for a British Composer Award 2016) where throughout she “uses the idea of the axon transferring an impulse to a synapse to inform the music’s trajectory, and the interplay of parts”. The new proposed work will explore the brain in a different angle. It will focus on how cancer cells in the brain have a resilient ability to migrate and infiltrate in different brain regions, making brain tumours very hard to cure.

Glioblastoma (GBM) is the most frequent and lethal malignant brain tumor in adults due to its invasive capability and resistance to conventional therapies. GBM typically shows an heterogenous microenviron- ment including necrotic and hypoxic areas, abnormal vasculature and different tumor cell phenotypes. We will focus on mathematical models based on PDE that try to reproduce this complex system in order to understand it and better predict the tumor behaviour [1, 2]. In addition, we will discuss in-silico simu- lations based on a mathematical model for brain tumor response to the combination of antithrombotic, antioxidants and radiation therapies [3]. Our mathematical results predict a synergistic decrease in tumor volume when both, cytotoxic therapies and antioxidants were applied. In vitro and in vivo results have confirmed this benefit not only in terms of tumor reduction but also in terms of toxicity reduction. Combined mathematical simulations and on-chip validation of malignant cellular structures formation in GBM have confirmed their usability to better understand the tumor behaviour [4, 5]. Considering the promising results, a clinical trial is being designed.

Glioblastoma is an aggressive form of glioma often having diffuse boundaries, making the definition of treatment area challenging. We aim to more precisely determine the location of tumor cells by using modern diffusion tensor imaging (DTI) data to advance current mathematical models. The brain is composed of white matter and grey matter, where glioma cells travel faster along white matter fibers faster than within grey matter. DTI is a magnetic resonance imaging (MRI)-based neuroimaging technique that locates white matter fibers and determines the rate of diffusion along these fibers. Previous uses of DTI data have increased the accuracy of glioma models in predicting tumor volume. Anisotropic diffusion tensors are a way of specifying that rate of glioma diffusion at each point in the brain and it has been shown that anisotropic models more accurately capture tumor volume. Our goal is to provide a more accurate model of tumor growth using a continuum mechanics approach to model the tumor and the forces acting on the tumor by the brain and skull. The tumor will be modeled as a viscoelastic body with anisotropic diffusion (i.e. white vs. grey matter) using DTI data to define diffusion tensors. Using a continuum mechanics approach treats the glioma as a continuous mass, allowing for the incorporation of more physical and mechanical aspects than previous models.

The quest for personalized anticancer treatment has fostered the development of new technologies enabling the measurement of multi-type data in individual patients. A critical current challenge lays in translating such individual datasets into personalized therapies. We here develop a multi-scale systems pharmacology approach to personalize multi-agent chemotherapies based on multi-type tumor data. The model disease is here glioblastoma (GBM), the most common and devastating primary brain tumor currently associated with a median overall survival below 18 months. The project focuses on temozolomide (TMZ), the cornerstone of GBM treatments, and its combination with targeted therapies. We have developed a framework of mechanistic ODE-based models representing TMZ pharmacokinetics-pharmacodynamics (PK-PD) in a heterogeneous cell population, and at the whole-body level in mice and in patients. The effect of pH on TMZ activation was studied as a potential therapeutic target as healthy and cancer cells regulate their intracellular pH differently. The cellular PK-PD model was fitted to multi-type data obtained in GBM patient-derived cell lines and aims to design cell line-specific TMZ-based combination chemotherapies. Next, the cellular PK model was included into a hybrid tumour model to further investigate spatial effects and cell-to-cell variability (see Angélique Stéphanou's presentation). Such modeling framework would ultimately allow for the integration of both multi-omics and imaging tumor data to personalize GBM treatment.

Pharmacokinetics-pharmacodynamics (PK-PD) models are standardly used to assess the availability and effects of a drug. However those models expressed with ordinary differential equations (ODEs) only describe the evolution of the drug concentration with time assuming that all cells receive the same amount and are targetted homogeneously in the same way. In a tumour case however, the cells states and the local cell environment – in terms of oxygenation and acidity – vary depending on the cells location in the tumour (periphery versus core). As a consequence it might prove useful to integrate spatiality in the models in order to get a more accurate evaluation of the drug uptake by the cells. In this presentation, we show how the effects of temozolomide – a pH-dependent drug directed against brain tumours – can be over-evaluated by the standard PK approach. The integration of the spatial component also shows how the healthy tissue might also be affected by the drug and gives a mean to evaluate collateral effects. The model is thus very helpful to highlight the weaknesses of this therapy and to suggest some new means to significantly improve it.

A new way to tackle cancer modeling has been introduced during recent years by means of models describing heteroge- neous cell populations. Populations are tipically structured in spatial and/or phenotypical variables. New mathematical mod- els of continuous cell population dynamics have been considered in the form of structured partial differential equations of reaction- diffusion type. Discrete counterparts have been also considered as a way to describe stochastic fluctuations linked with the struc- tural variables; those fluctuations may be central to an accurate description of invasive phenomena such as tumor growth. We present a set of multiscale population models along the previous lines (R. de la Cruz, P. Guerrero, J. Calvo, T. Alarco ́n, Journal of Computational Physics 2017). Discrete frameworks provide the most detailed description, however they are computationally ex- pensive. We use coarse-graining procedures to derive mean-field and hybrid deterministic-stochastic representations, together with computational simulation methods. Hybrid computational models provide a suitable balance between an accurate description and a reasonable simulation time. These representations enable us to assess the role of stochastic fluctuations at the leading edge of invasion fronts.

Understanding the unique microenvironment and the architecture of the stem cell niche and its surrounding neighbors is crucial to determine the origins of cancer. Knowledge of the division patterns of cells, both healthy and malignant, can suggest ways of altering the microenvironment of the tissue, with the aim of controlling the growth rate of the cells, and possibly minimizing the size of the tumor. Two widely studied model system for adult stem cell proliferation and differentiation are the hematopoetic system and intestinal crypts. The standard models of evolutionary dynamics in tissues consider well-mixed populations. I will talk about the effect of cell division pattern along the intestinal crypt on mutant production, assuming that the division rate of each cell depends on its location. I will also discuss the effects of multiple feedback loops governing tissue division.

Chronic myeloid leukemia is a blood cancer for which there exists a very efficient tar-
geted therapy (Tyrosine Kinase Inhibitors). While this has revolutionnized the long
term prognosis of treated patients, the next question is the possibility of stopping
the treatment and thereby entering so called Treatment Free Remission (TFR). We
present recent results on the mathematical modelling of chronic myeloid leukemia.
Describing the interaction between chronic myeloid leukemia and autologous im-
mune response, we propose an interpretation of treatment free remission as a stability property. The interpretation is then a control of the disease by the immune
system rather than an eradication.


We investigate a mathematical model of hematopoietic stem cell dynamics. This model takes into account a finite number of stages in blood production, characterised by cell maturity levels, which enhance the difference, in the hematopoiesis process, between dividing cells that differentiate (by going to the next stage) and dividing cells that keep the same maturity level (by staying in the same stage). For each maturity level, we take two cell populations into account, quiescent and proliferating one, and we note the difference between dividing cells that enter directly to the quiescent phase and dividing cells that return to the proliferating phase to divide again. The resulting mathematical model is a system of nonlinear differential-difference equations. We start by studying the existence and positivity of the solutions. We then investigate the boundedness and unboundedness of the solutions of the system. We also discus the existence of steady states. Sufficient conditions for the global asymptotic stability of the trivial steady state as well as conditions for its instability are obtained. Numerical simulation is carried out to show that a blocking of the differentiation at some stage may lead to an over-expression of immature cells. Such situation corresponds to the observation in the case of Acute Myelogenous Leukemia.

The emergence of drug-resistance is a major challenge in chemotherapy. In this talk we will present our recent mathematical models for describing the dynamics of drug-resistance in solid tumors. Our models follow the dynamics of the tumor, assuming that the cancer cell population depends on a phenotype variable that corresponds to the resistance level to a cytotoxic drug. We incorporate the dynamics of nutrients and two different types of drugs: a cytotoxic drug, which directly impacts the death rate of the cancer cells, and a cytostatic drug that reduces the proliferation rate. Through analysis and simulations, we study the impact of spatial and phenotypic heterogeneity on the tumor growth under chemotherapy. We demonstrate that heterogeneous cancer cells may emerge due to the selection dynamics of the environment. Our models predict that under certain conditions, multiple resistant traits emerge at different locations within the tumor. We show that a higher dosage of the cytotoxic drug may delay a relapse, yet, when this happens, a more resistant trait emerges. Moreover, we estimate the expansion rate of the tumor boundary as well as the time of relapse, in terms of the resistance trait, the level of the nutrient, and the drug concentration. Finally, we propose an efficient drug schedule aiming at minimizing the growth rate of the most resistant trait. By combining the cytotoxic and cytostatic drugs, we demonstrate that the resistant cells can be eliminated.

In the majority of cancers, secondary tumors (metastases) and associated complications are the main cause of death. To design the best therapy for a given patient, one of the major current challenge is to estimate, at diagnosis, the burden of invisible metastases and the future time of emergence of these, as well as their growth speed. In this talk, I will present the current state of our research efforts towards the establishment of a predictive computational tool for this aim. I will first shortly present the model used, which is based on a physiologically-structured partial differential equation for the time dynamics of the population of metastases, combined to a nonlinear mixed-effects model for statistical representation of the parameters’ distribution in the population. Then, I will show results about the descriptive power of the model on data from clinically relevant ortho-surgical animal models of metastasis (breast and kidney tumors). The main part of my talk will further be devoted to the translation of this modeling approach toward the clinical reality. Using clinical imaging data of brain metastasis from non-small cell lung cancer, several biological processes will be investigated to establish a minimal and biologically realistic model able to describe the data. Integration of this model into a biostatistical approach for individualized prediction of the model’s parameters from data only available at diagnosis will also be discussed. Together, these results represent a step forward towards the integration of mathematical modeling as a predictive tool for personalized medicine in oncology.

Metastasis — the spread of cancer from a primary to a distant secondary location — is implicated in over 90% of all cancer related deaths. Despite its importance in patient outcome, a full understanding of the metastatic process remains elusive, largely because of the difficulty in studying the phenomenon experimentally. Recent experimental evidence — including the discovery of the so-called ‘pre-metastatic niche’ — has suggested that metastasis may be a more precisely controlled process than previously thought. In particular, it has been suggested that a developing tumor may be able to corrupt, or ‘educate’, infiltrating immune cells and have them switch from cytotoxic to tumor-promoting roles. Such ‘tumor educated’ immune cells can then travel to distant sites and establish favorable conditions for the settlement and growth of circulating tumor cells. In order to investigate the consequences of tumor-mediated immune education on metastatic spread and growth, we have developed an ordinary differential equation model for tumor-immune dynamics in the metastatic context. The model is studied analytically and numerically, with an examination of the effects of tumor education of immune cells on metastatic dormancy and metastatic blow-up.

Several treatments are used to deal with bone metastases formation, but they are palliative since the disease is considered incurable. Computational and mathematical models are used to understand the underlying mechanisms of how bone metastasis evolves. In this way, new therapies aiming to reduce or eliminate the metastatic burden in the bone tissue may be proposed. We present an optimal control approach to analyze some common treatments for bone metastasis. In particular, we focus on denosumab treatment, an anti-resorptive therapy, and radiotherapy treatment which has a cell killing action.

Glioblastoma Multiforme (GBM) is a malignant heterogenous cancer in the brain. We propose modeling GBM with heterogeneity in cell phenotypes using a random differential equation version of the reaction-diffusion equation, where the parameters describing diffusion (D) and proliferation (ρ) are random variables. We investigate the ability to perform the inverse problem to recover the probability distributions of D and ρ solely from spatiotemporal data, for a variety of probability distribution functions. We test the ability to perform the inverse problem for noisy synthetic data. We then examine the predicted effect of treatment, specifically, chemotherapy, when assuming such a heterogeneous population and compare with predictions from a homogeneous cell population model.

For decades, elucidate the dynamics of the microenvironment of solid tumors has been considered a major research challenge. The fact is that different therapies have not succeeded to eliminate entirely cancer cells in tumors. This resistance feature to anticancer drugs is often attributed to genetic or even epigenetic causes. Another important, but less appreciated cause of this resistance -a possible manifestation of the former, is the geometrical and physical heterogeneity within the tumor microenvironment that leads to marked gradients in the rate of cell proliferation and to regions of hypoxia and acidity, all of which can influence the sensitivity of the tumor cells to drug treatment. There have been different approaches to overcome this resistance, mainly altering solid tumors' microenvironment, for instance, promoting angiogenesis to help the entrance of drugs. Radiation some times is not an option due to the hypoxia in the tumor deep tissue. On the other hand, these physical and geometrical factors have been identified to be responsible of the unsuccessful drug disperse in tumors in some specific time and spatial scales. In this direction, we present an update of our model of drug transport in solid tumors, that quantifies these factors in terms of space-dependent coefficients of the Fokker-Planck equation. The model follows experiments conducted in the Laboratory of Medical Physics and Molecular Imaging of the National Institute for Cancer (INCan) and the Institute of Physics (IFUNAM).

In both normal tissue and disease states, cells interact with one another, and other tissue components using cellular adhesion proteins. These interactions are fundamental in determining tissue fates, and the outcomes of normal development, wound healing and cancer metastasis. Traditionally continuum models (PDEs) of tissues are based on purely local interactions. However, these models ignore important nonlocal effects in tissues, such as long-ranged adhesion forces between cells. For this reason, a mathematical description of cell adhesion had remained a challenge until 2006, when Armstrong et. al. proposed the use of an integro-partial differential equation (iPDE) model. The initial success of the model was the replication of the cell-sorting experiments of Steinberg (1963). Since then this approach has proven popular in applications to embryogenesis (Armstrong et. al. 2009), zebrafish development (Painter et. al. 2015), and cancer modelling (e.g. Painter et. al. 2010, Domschke et. al. 2014, Bitsouni et. al. 2018). While popular, the mathematical properties of this non-local term are not yet well understood. I will begin this talk by outlining, the first systematic derivation of non-local (iPDE) models for adhesive cell motion. The derivation relies on a framework that allows the inclusion of cell motility and the cell polarization vector in s stochastic space-jump process. The derivation's significance is that, it allows the inclusion of cell-level properties such as cell-size, cell protrusion length or adhesion molecule densities into account. In the second part, I will present the results of our study of the steady-states of a non-local adhesion model on an interval with periodic boundary conditions. The significance of the steady-states is that these are observed in experiments (e.g. cell-sorting). Combining global bifurcation results pioneered by Rabinowitz, equivariant bifurcation theory, and the mathematical properties of the non-local term, we obtain a global bifurcation result for the branches of non-trivial solutions. Using the equation’s symmetries the solutions of a branch are classified by the derivative’s number of zeros. We further show that the non-local operator’s properties determine whether a sub or super-critical pitchfork bifurcation occurs. Finally, I want to demonstrate how the equation's derivation from a stochastic random walk can be extended to derive different non-local adhesion operators describing cell-boundary adhesion interactions. The significance is that in the past, boundary conditions for non-local equations were avoided, because their construction is subtle. I will describe the three challenges we encountered, and their solutions.

The lamellipodium is a thin, sheet-like structure that is found in the propagating front of fast moving cells like fibroblasts, keratocytes, cancer cells, and more. It is a dense network of linear biopolymers of the protein actin, termed actin-filaments. These actin-filaments are highly dynamic structures that participate in a plethora of processes such as polymerization, nucleation, capping, fragmentation, and more. These processes are important for the structure and functionality of the lamellipodium and the motility of the cell. They are, to a large extent, affected by the extracellular environment; for example, the chemical landscape in which the cell of resides and the local composition and architecture of the Extracellular Matrix (ECM), lead to biased motility responses of the cell. When in proximity to each other, they develop cell-cell adhesion via specialized transmembrane proteins of the \textit{cadherin} family. Collectively, they coagulate to clusters of cells that eventually merge to form cell monolayers. We model these phenomena using the Filament Based Lamellipodium Model (FBLM); an anisotropic, two-phase, two-dimensional, continuum model that describes the dynamics the lamellipodium at the level of actin-filaments and their interactions. The model distinguishes between two families (phases) of filaments and includes the interactions between them, as well as between the network of the filaments and the extracellular environment. The FBLM was first proposed in [1] and later extended in [2,4,5]. The FBLM is endowed with a problem specific Finite Element Method (FEM) that we have previously developed in [3]. In this talk we present the basic components of the FBLM and the FEM and focus on a series of simulations reproducing fundamental components of the motility of the cells, such us chemotaxis, haptotaxis, interaction with the environment [3, 4]. We also present our new findings with respect to cell-cell collision and adhesion, as well as the formation of clusters of cells and cell monolayers [5]. To confront the increased computational needs of the monolayer, we have developed a parallel version of our numerical method which we also address in this talk. Literature: [1] D. Oelz, C. Schmeiser. How do cells move? in Cell mechanics: from single scale-based models to multiscale modeling, Chapman and Hall, (2010). [2] A. Manhart, D. Oelz, C. Schmeiser, N. Sfakianakis, An extended Filament Based Lamellipodium: Model produces various moving cell shapes in the presence of chemotactic signals. J. Theor. Biol. (2015). [3] A. Manhart, D. Oelz, C. Schmeiser, N. Sfakianakis. Numerical treatment of the filament based lamellipodium model (FBLM) in Modelling Cellular Systems. (2016) [4] N. Sfakianakis, A. Brunk. Stability, convergence, and sensitivity analysis of the FBLM and the corresponding FEM, Bull. Math. Biol. (2018) [5] N. Sfakianakis, D. Peurichard, C. Schmeiser, and A. Brunk. The FBLM-FEM: from cell-cell adhesion to cluster formation, (in review).

I will first present current philosophical discussions in the context of modelling and systems biology, but then also map out a future agenda for philosophers of science. Starting less than a decade ago, philosophical interest in systems biology was among other things a reaction to the hitherto dominant notion of mechanistic explanation. Mechanists maintained that only mechanistic (part-whole) explanations count as genuine explanations, sometimes deeming mathematical models to be non-explanatory. In response, other philosophers have pointed to systems biology to argue that mathematical models can be mechanistic (‘dynamic mechanistic explanation’) or that there are explanations in terms of mathematical features that are fully explanatory yet non-mechanistic (topological and structural explanation). My position is that philosophers’ attempts to articulate an authoritative notion of ‘(non-)mechanistic explanation’ is fruitless, and the agenda should instead be to investigate the diversity of explanations in systems biology. Indeed, going beyond the traditional topic of explanation, the future agenda for philosophy of science ought to be to examine modelling and other research strategies, including strategies for analyzing and representing large datasets and strategies for developing and analyzing mathematical models of complex systems. Some work by philosophers of biology have addressed strategies and investigative practice, but limited attention has been devoted to the modelling of cancer. In this latter context, what I view as in need of future philosophical attention are scientific strategies to deal with variability across systems (as a challenge to model validation), to deal with heterogeneity and robustness within systems (as a feature to be modeled), to deal with phenomena across different levels and scales, as well as strategies for coordinating across experimental, modelling, and clinical work.

There is great interest in the molecular characterisation of intestinal metaplasia, such as Barrett’s esophagus (BE), to understand the basic biology of metaplastic development from a tissue of origin. BE is asymptomatic, so it is not generally known how long a patient has lived with this precursor of esophageal adenocarcinoma (EAC) when initially diagnosed in the clinic. Since curative interventions carry patient risks and the annual risk of cancer progression is < 1%, BE is usually left in the body but knowledge of BE tissue age may be advantageous in predicting a patient’s future risk of developing cancer. A recently published BE clock model uses patient-specific methylation data to estimate BE onset times using Bayesian inference techniques, and thus obtain the biological age of BE tissue (Curtius et al. 2016). We find such epigenetic drift to be widely evident in BE tissue (Luebeck et al. 2017) and the corresponding tissue ages show large inter-individual heterogeneity in two patient populations (52 patients). From a basic biological standpoint, we also do not fully understand mechanistically how the Barrett’s tissue forms in the human esophagus, and such information is critical to inform such biomarkers of risk based on biological tissue age. We analysed multi-region samples from 17 BE patients (including multiple phenotypes) to 1) measure the spatial heterogeneity in biological tissue age and 2) use these ages to calibrate mathematical models of the mechanisms for formation of the segment itself. Mathematical challenges arise when attempting to combine such rich molecular data into mechanistic models of evolutionary processes at the cell level, which for BE can currently only be inferred from tissue biopsies or resections sampled in vivo with spatial and practical constraints. For example, there may be more than one plausible biological assumption in a model that can explain the patterns observed from cross-sectional data. Using the epigenetic clock to estimate tissue ages, the main questions we explored with simulations using an agent-based computational model were 1) What are the important biophysical assumptions needed to be capture evolution from a tissue of origin? 2) What are the timescales involved in both the growth of BE to a certain size? 3) Which mechanistic parameters (e.g., cellular birth and death rates) are found to be patient-specific and which evolutionary “rules” seem to be universal in the population? Most importantly, we found that tissue must be regenerated nearer to the stomach, perhaps driven by wound healing caused by exposure to reflux, implying a gastric tissue of origin for the lesions observed in BE. Combining bioinformatics and mechanistic modelling allowed us to infer evolutionary processes that cannot be clinically observed and we believe there is great promise for the community to develop such novel hybrid methods to better understand multiscale cancer data. References: Curtius K, Wong C, Hazelton WD, Kaz AM, Chak A, et al. (2016) A Molecular Clock Infers Heterogeneous Tissue Age Among Patients with Barrett's Esophagus. PLoS Comput Biol 12(5): e1004919 Luebeck EG, Curtius K, Hazelton WD, Made S, Yu M, et al. (2017) Identification of a key role of epigenetic drift in Barrett’s esophagus and esophageal adenocarcinoma. J Clin Epigenet 9:113

We analyzed a model of an avascular tumour growth that considers the basic biological principles of cell proliferation, motility, dead and genes mutations. We identified two sets of genes –a set of sixteen and six genes– that are believed to play an important role in tumour growth. Gene mutations were modelled as a markovian process and mutation rate was assumed to depend on nutrient concentration. Thus, mutations dynamics was coupled to a set of reaction-diffussion equations that describe the transport of nutrients. Tumour malignancy was characterized by its fractal dimension and we measured genetic heterogeneity with the Shannon index. The results suggest that tumour malignancy and heterogeneity arise from a relatively small number of events that are driven by stochasticity and nutrients concentration. J.R.R.A. would like to acknowledge financial support from CONACyT-FORDECyT under Grant No. 265667. G.R.S. would like to acknowledge financial support from DGAPA- UNAM Grant No. IN108916. J.X.V.H. would like to ac- knowledge financial support from DGAPA-UNAM Grant No. IN110917.

The immunotherapy using dendriticcells(DCs) against different varieties of cancer is an approach that has been previously explored which induces a specific immune response. This work presents a mathematical model of DCs immunotherapy for melanoma in mice based on work by Experimental Immunotherapy Laboratory of the Medicine Faculty in the Universidad Autonoma de Mexico (UNAM). The model is a five delay differential equation(DDEs) which represents a simplified view of the immunotherapy mechanisms. The mathematical model takes into account the interactions between tumor cells, dendritic cells, naive cytotoxic T lymphocytes cells (inactivated cytotoxic cells), effector cells (cytotoxic T activated cytotoxic cells) and transforming growth factor β cytokine (TGF − β). The model is validated comparing the computer simulation results with biological trial results of the immunotherapy developed by the research group of UNAM.

Acquisition by cancer cells of a plethora of resistance-conferring genetic alterations greatly limits the clinical utility of most anti- cancer drugs. Therefore, there is a need to improve the effective- ness of treatment before mutational-acquired resistance prevails. Relapse is driven by a small subpopulation of residual or ‘‘drug-tolerant’’ cells, which are traditionally called ‘‘minimal residual disease’’ (MRD), that remain viable upon drug exposure. Recent in vitro findings have indicated that the emergence of these per- sisters is unlikely due to mutational mechanisms. A non-mutually exclusive scenario proposes that the drug-tolerant phenotype is transiently acquired by a small pro- portion of cancer cells through non-mutational mechanisms. To gain insights into the biology of MRD, we applied single-cell RNA sequencing to malignant melanoma BRAF mutated cells, and we identified a subpopulation of melanoma cells is tolerant to targeted therapy via metabolic reprogramming. Cancer cells were known to reprogram their metabolic profiles geared toward glycolysis, despite sufficient oxygen available to support oxidative phosphorylation (OXPHOS), a phenomenon known as the Warburg effect. We found that melanoma MRD can switch their metabolic program from glycolysis towards mitochondrial OXPHOS alimented by fatty acid oxidation (FAO), thereby renders the melanoma MRD highly sensitive to FAO inhibition in vitro and in mouse tumor models. This MRD-directed metabolic reprogramming suggests a more clever treatment combination regimen to fight against cancer resistance.

The inherent capacity of somatic cells to switch their phenotypic status in response to damage stimuli \textit{in vivo} might have a pivotal role in ageing and cancer. However, how the entry-exit mechanisms of phenotype reprogramming are established remains poorly understood. In an attempt to elucidate such mechanisms, we herein introduce a stochastic model of combined epigenetic regulation (ER)-gene regulatory network (GRN) to study the plastic phenotypic behaviours driven by ER heterogeneity. Furthermore, based on the existence of multiple scales, we formulate a method for stochastic model reduction, from which we derive an efficient hybrid simulation scheme that allows us to deal with such complex systems. Our analysis of the coupled system reveals a regime of tristability in which pluripotent stem-like and differentiated steady-states coexist with a third indecisive state. Crucially, ER heterogeneity of differentiation genes is for the most part responsible for conferring abnormal robustness to pluripotent stem-like states. We then formulate epigenetic heterogeneity-based strategies capable of unlocking and facilitating the transit from differentiation-refractory (pluripotent stem-like) to differentiation-primed epistates. The application of the hybrid numerical method validated the likelihood of such switching involving solely kinetic changes in epigenetic factors. Our results suggest that epigenetic heterogeneity regulates the mechanisms and kinetics of phenotypic robustness of cell fate reprogramming. The occurrence of tunable switches capable of modifying the nature of cell fate reprogramming from pathological to physiological might pave the way for new therapeutic strategies to regulate reparative reprogramming in ageing and cancer.

I will present an evolutionary viewpoint on cancer, seen as the 2 time scales of (large-time) evolution in the genomes and of (short-time) evolution in the epigenetic landscape of a constituted genome. These views, based on works by Lineweaver, Davies and Vincent (cancer as anatomically located backward evolution in multicellular organisms, aka atavistic theory of cancer [3, 8]) and by Sui Huang and collaborators (the Waddington epigenetic landscape revisited [6]), respectively, may serve as guidelines to propose a global conception of cancer, including towards possible innovating therapeutic strategies. Drug-induced drug resistance, the biological and medical question I am tackling from a theoretical point of view, may be due to biological mechanisms of different natures, mere local regulation, epigenetic modifications (reversible, nevertheless heritable [7]) or genetic mutations (irreversible), according to the extent to which the genome of the cells in the population is affected. In this respect, the modelling framework of adaptive dynamics I will present is more likely to correspond biologically to epigenetic modifications, although eventual induction of emergent resistant cell clones due to mutations under drug pressure is never to be excluded. From the biologist’s point of view, I study phenotypically heterogeneous, but genetically homogeneous, cancer cell populations under stress by drugs [2]. The built-in targets for theoretical therapeutic control present in the phenotype-structured PDE models I advocate are not supposed to represent well-defined molecular effects of the drugs in use, but rather functional effects, i.e., related to cell death (cytotoxic drugs), or to cell proliferation [1] in the sense of slowing down the cell division cycle without killing cells (cytostatic drugs). I propose that cell life-threatening drugs (cytotoxics) induce by farmore resistance in the highly plastic cancer cell populations than drugs that only limit their growth (cytostatics), and that a rational combination of the two classes of drugs - and possibly others, adding relevant targets to the model - may be optimised to propose therapeutic control strategies to avoid the emergence of drug resistance in tumours. I address this optimal control problem in the context of two populations, healthy and cancer, both endowed with phenotypes evolving with drug pressure, and competing for space and nutrients in a non-local Lotka-Volterra-like way [4, 5], taking into account a double constraint of limiting unwanted adverse effects and avoiding the emergence of drug resistance. I conclude by proposing a list of open challenging questions to modellers and mathematicians about the emergence and evolution of cancer. References [1] Chisholm, R.H., Lorenzi, T., Lorz, A., Larsen, A.K., Almeida, L., Escargueil, A., Clairambault, J. Emergence of reversible drug tolerance in cancer cell populations: an evolutionary outcome of selection, non-genetic instability and stress-induced adaptation. Cancer Research, 75(6):930-939, 2015 [2] Chisholm, R.H., Lorenzi, T., Clairambault, J. Cell population heterogeneity and evolution towards drug resistance in cancer: biological and mathematical assessment, theoretical treatment optimisation. BBA General Subjects (special issue on system genetics), 1860:2627-2645, 2016 [3] Davies P.C., Lineweaver C.H. Cancer tumors as Metazoa 1.0: tapping genes of ancient ancestors. Phys Biol, 8:015001, 2011 [4] Lorz, A., Lorenzi, T., Clairambault, J., Escargueil, A., Perthame, B. Effects of space structure and combination therapies on phenotypic heterogeneity and drug resistance in solid tumors. Bull Math Biol, 77(1):1-22, 2015 [5] Pouchol, C., Clairambault, J., Lorz, A., Trélat, E. Asymptotic study and optimal control of integrodifferential systems modelling healthy and cancer cells exposed to chemotherapy. Journal de Mathématiques Pures et Appliquées, published online, October 2017 [6] Huang S. Genetic and non-genetic instability in tumor progression: Link between the fitness landscape and the epigenetic landscape of cancer cells, Canc Metastasis Rev, 32:423-448, 2013 [7] Sharma, S.V., Lee, D.Y. , Li, B., Quinlan, M.P., Takahashi, F., Maheswaran, S., McDermott, U., Azizian, N., Zou, L., Fischbach, M.A., et al. A chromatin-mediated reversible drug-tolerant state in cancer cell subpopulations, Cell 141(1):69-80, 2010

The mathematical modelling of metastasis is a challenge. The occurrence of metastasis is basically random, hence the use of stochastic modelling seems appropriate. We introduce a stochastic process called branched random walk with settlement to derive equations for the expected number of particles, the variance, the furthest particle and the extinction probability. We are able to identify a parameter R_0, such that metastasis spread for R_0>1 and they die out for R_0<1. Hence we call R_0 the metastatic reproduction number. We compare this index to experimental outcomes in animal studies and we discuss its relevance for the treatment of metastasis. (Joint work with A. Rhodes and C. Frei).