Agent-Based and Continuum Models for Spatial Dynamics of Infection by Oncolytic Viruses.

The use of oncolytic viruses as cancer treatment has received considerable attention in recent years, however the spatial dynamics of this viral infection is still poorly understood. We present here a stochastic agent-based model describing infected and uninfected cells for solid tumours, which interact with viruses in the absence of an immune response. Two kinds of movement, namely undirected random and pressure-driven movements, are considered: the continuum limit of the models is derived and a systematic comparison between the systems of partial differential equations and the individual-based model, in one and two dimensions, is carried out. In the case of undirected movement, a good agreement between agent-based simulations and the numerical and well-known analytical results for the continuum model is possible. For pressure-driven motion, instead, we observe a wide parameter range in which the infection of the agents remains confined to the center of the tumour, even though the continuum model shows traveling waves of infection; outcomes appear to be more sensitive to stochasticity and uninfected regions appear harder to invade, giving rise to irregular, unpredictable growth patterns. Our results show that the presence of spatial constraints in tumours' microenvironments limiting free expansion has a very significant impact on virotherapy. Outcomes for these tumours suggest a notable increase in variability. All these aspects can have important effects when designing individually tailored therapies where virotherapy is included.

A phenotype-structured model to reproduce the avascular growth of a tumor and its interaction with the surrounding environment.

We here propose a one-dimensional spatially explicit phenotype-structured model to analyze selected aspects of avascular tumor progression. In particular, our approach distinguishes viable and necrotic cell fractions. The metabolically active part of the disease is, in turn, differentiated according to a continuous trait, that identifies cell variants with different degrees of motility and proliferation potential. A parabolic partial differential equation (PDE) then governs the spatio-temporal evolution of the phenotypic distribution of active cells within the host tissue. In this respect, active tumor agents are allowed to duplicate, move upon haptotactic and pressure stimuli, and eventually undergo necrosis. The mutual influence between the emerging malignancy and its environment (in terms of molecular landscape) is implemented by coupling the evolution law of the viable tumor mass with a parabolic PDE for oxygen kinetics and a differential equation that accounts for local consumption of extracellular matrix (ECM) elements. The resulting numerical realizations reproduce tumor growth and invasion in a number scenarios that differ for cell properties (i.e., individual migratory behavior, duplication, and mutation potential) and environmental conditions (i.e., level of tissue oxygenation and homogeneity in the initial matrix profile). In particular, our simulations show that, in all cases, more mobile cell variants occupy the front edge of the tumor, whereas more proliferative clones are selected at more internal regions. A necrotic core constantly occupies the bulk of the mass due to nutrient deprivation. This work may eventually suggest some biomedical strategies to partially reduce tumor aggressiveness, i.e., to enhance necrosis of malignant tissue and to promote the presence of more proliferative cell phenotypes over more invasive ones.

A Mathematical Study of the Influence of Hypoxia and Acidity on the Evolutionary Dynamics of Cancer.

Hypoxia and acidity act as environmental stressors promoting selection for cancer cells with a more aggressive phenotype. As a result, a deeper theoretical understanding of the spatio-temporal processes that drive the adaptation of tumour cells to hypoxic and acidic microenvironments may open up new avenues of research in oncology and cancer treatment. We present a mathematical model to study the influence of hypoxia and acidity on the evolutionary dynamics of cancer cells in vascularised tumours. The model is formulated as a system of partial integro-differential equations that describe the phenotypic evolution of cancer cells in response to dynamic variations in the spatial distribution of three abiotic factors that are key players in tumour metabolism: oxygen, glucose and lactate. The results of numerical simulations of a calibrated version of the model based on real data recapitulate the eco-evolutionary spatial dynamics of tumour cells and their adaptation to hypoxic and acidic microenvironments. Moreover, such results demonstrate how nonlinear interactions between tumour cells and abiotic factors can lead to the formation of environmental gradients which select for cells with phenotypic characteristics that vary with distance from intra-tumour blood vessels, thus promoting the emergence of intra-tumour phenotypic heterogeneity. Finally, our theoretical findings reconcile the conclusions of earlier studies by showing that the order in which resistance to hypoxia and resistance to acidity arise in tumours depend on the ways in which oxygen and lactate act as environmental stressors in the evolutionary dynamics of cancer cells.

Effects of mutations and immunogenicity on outcomes of anti-cancer therapies for secondary lesions.

Cancer development is driven by mutations and selective forces, including the action of the immune system and interspecific competition. When administered to patients, anti-cancer therapies affect the development and dynamics of tumours, possibly with various degrees of resistance due to immunoediting and microenvironment. Tumours are able to express a variety of competing phenotypes with different attributes and thus respond differently to various anti-cancer therapies. In this paper, a mathematical framework incorporating a system of delay differential equations for the immune system activation cycle and an agent-based approach for tumour-immune interaction is presented. The focus is on those metastatic, secondary solid lesions that are still undetected and non-vascularised. By using available experimental data, we analyse the effects of combination therapies on these lesions and investigate the role of mutations on the rates of success of common treatments. Findings show that mutations, growth properties and immunoediting influence therapies' outcomes in nonlinear and complex ways, affecting cancer lesion morphologies, phenotypical compositions and overall proliferation patterns. Cascade effects on final outcomes for secondary lesions are also investigated, showing that actions on primary lesions could sometimes result in unexpected clearances of secondary tumours. This outcome is strongly dependent on the clonal composition of the primary and secondary masses and is shown to allow, in some cases, the control of the disease for years.

Combination therapies and intra-tumoral competition: Insights from mathematical modeling.

Drug resistance is one of the major obstacles to a successful treatment of cancer and, in turn, has been recognized to be linked to intratumoral heterogeneity, which increases the probability of the emergence of cancer clones refractory to treatment. Combination therapies have been introduced to overcome resistance, but the design of successful combined protocols is still an open problem. In order to provide some indications on the effectiveness of medical treatments, a mathematical model is proposed, comprising two cancer populations competing for resources and with different susceptibilities to the action of immune system cells and therapies: the focus is on the effects of chemotherapy and immunotherapy, used singularly or in combination. First, numerical predictions of the model have been tested with experimental data from the literature and next therapeutic protocols with different doses and temporal order have been simulated. Finally the role of competitive interactions has been also investigated, to provide some insights on the role of competitive interactions among cancer clones in determining treatment outcomes.

Emergence of spatial patterns in a mathematical model for the co-culture dynamics of epithelial-like and mesenchymal-like cells.

Accumulating evidence indicates that the interaction between epithelial and mesenchymal cells plays a pivotal role in cancer development and metastasis formation. Here we propose an integro-differential model for the co-culture dynamics of epithelial-like and mesenchymal-like cells. Our model takes into account the effects of chemotaxis, adhesive interactions between epithelial-like cells, proliferation and competition for nutrients. We present a sample of numerical results which display the emergence of spots, stripes and honeycomb patterns, depending on parameters and initial data. These simulations also suggest that epithelial-like and mesenchymal-like cells can segregate when there is little competition for nutrients. Furthermore, our computational results provide a possible explanation for how the concerted action between epithelial-cell adhesion and mesenchymal-cell spreading can precipitate the formation of ring-like structures, which resemble the fibrous capsules frequently observed in hepatic tumours.

Mathematical model reveals how regulating the three phases of T-cell response could counteract immune evasion.

T cells are key players in immune action against the invasion of target cells expressing non-self antigens. During an immune response, antigen-specific T cells dynamically sculpt the antigenic distribution of target cells, and target cells concurrently shape the host's repertoire of antigen-specific T cells. The succession of these reciprocal selective sweeps can result in 'chase-and-escape' dynamics and lead to immune evasion. It has been proposed that immune evasion can be countered by immunotherapy strategies aimed at regulating the three phases of the immune response orchestrated by antigen-specific T cells: expansion, contraction and memory. Here, we test this hypothesis with a mathematical model that considers the immune response as a selection contest between T cells and target cells. The outcomes of our model suggest that shortening the duration of the contraction phase and stabilizing as many T cells as possible inside the long-lived memory reservoir, using dual immunotherapies based on the cytokines interleukin-7 and/or interleukin-15 in combination with molecular factors that can keep the immunomodulatory action of these interleukins under control, should be an important focus of future immunotherapy research.

A mathematical model for the dynamics of cancer hepatocytes under therapeutic actions.

This paper deals with the development of a mathematical model for the in vitro dynamics of malignant hepatocytes exposed to anti-cancer therapies. The model consists of a set of integro-differential equations describing the dynamics of tumor cells under the effects of mutation and competition phenomena, interactions with cytokines regulating cell proliferation as well as the action of cytotoxic drugs and targeted therapeutic agents. Asymptotic analysis and simulations, developed with an exploratory aim, are addressed to enlighten the role played by the biological phenomena under consideration in the dynamics of hepatocellular carcinoma, with particular reference to the intra-tumor heterogeneity and the response to therapies. The obtained results suggest that cancer progression selects for highly proliferative clones. Moreover, it seems that intra-tumor heterogeneity makes targeted therapeutic agents to be less effective than cytotoxic drugs and a joint action of these two classes of agents may mutually increase their efficacy. Finally, it is highlighted how targeted therapeutic agents might act as an additional selective pressure leading to the selection for the most fitting, and then most resistant, cancer clones.

A mathematical model for progression and heterogeneity in colorectal cancer dynamics.

This paper deals with the development of a mathematical model that describes cancer dynamics at the cellular scale. The selected case study concerns colon and rectum cancer, which originates in colorectal crypts. Cells inside the crypts are assumed to be organized according to a compartmental-like arrangement and to be homogeneously mixing. A mathematical model for cancer progression is proposed here. This model describes the generation of multiple clonal sub-populations of cells at different progression stages in a single crypt. Asymptotic analysis and simulations are developed with an exploratory aim. The obtained results offer some insights into the role played by mutation, proliferation and differentiation phenomena on cancer dynamics. In particular, the acquisition of an additional growing power and a reduction for cellular differentiation seem more likely to be the driving force behind carcinogenesis rather than an increase in the mutation rate. The mutation rate instead seems to affect progression dynamics and intra-tumor heterogeneity. The role played by cells, at different differentiation stages, in the onset and progression of colorectal cancer is highlighted. The results support the fact that stem cells play a key role in cancer development and the idea that transit-amplifying cells could also take on an active role in carcinogenesis.