Mathematical Oncology 2013

This paper is a review including some original elements of a family of reaction-diffusion models of carcinogenesis exhibiting diffusion-driven (Turing) instability (DDI), but consisting of a single reaction-diffusion equation coupled with a system of ordinary differential equations (ODE). Such models are very different from the classical Turing-type models in that they exhibit qualitatively new patterns of behavior of solutions, including, in some cases, a strong dependence of the emerging pattern on initial conditions and quasi-stability followed by rapid growth of solutions, which may take the form of isolated spikes, corresponding to discrete foci of proliferation. However, the process of diffusion of growth factor molecules is by its nature a stochastic random walk. An interesting question emerges to what extent the dynamics of the deterministic diffusion model approximates the stochastic process generated by the model. We address this question using simulations with a software tool called sbioPN (spatial biological Petri Nets). The picture emerging suggests that some of the generic features of the deterministic system, such as spike formation, and dependence of the number of spikes on diffusivity, are preserved. However, new elements, such as spike competition and appearance of spikes at isolated random locations, are also present. We also discuss the relevance of the model and particularly the cell cooperativity hypothesis, underlying transition to the DDI.

We review mathematical models of tumor growth based on conservation laws in the full system of cells and interstitial liquid. First we deal with tumor cords evolving in axisymmetric geometry, where cells motion is simply passive and compatible with the saturation condition. The model is characterized by the presence of free boundaries with constraints driving the free boundary conditions, which in our opinion are particularly important, especially in the presence of treatments. Then a tumor spheroid is considered in the framework of the so-called two-fluid scheme. In a multicellular spheroid, on the appearance of a fully degraded necrotic core, the analysis of mechanical stresses becomes necessary to determine the motion via momentum balance, requiring the specification of the constitutive law for the “cell fluid.” We have chosen a Bingham-type law that presents considerable difficulties because of the presence of a yield stress, particularly with reference to the determination of an asymptotic configuration. Finally, we report some recent PDE-based models addressing complex processes in multicomponent tumors, more oriented to clinical practice.

In this chapter I present the state-of-the-art theoretical models for avascular tumor growth which are well established nowadays. I focus on models able to treat morphologic instabilities and phase segregation, two typical features of skin cancer for example melanoma. Contrary to experiments made in vitro on growing colonies, I show that the geometry of melanoma confined in the epidermis in the early stages of tumor growth suppresses the necrotic core and is responsible of inhomogeneities due to aggregation of cancerous cells. A relatively simple model consisting in the adaptation of the two-phase mixture model is enough to explain the morphologies of the tumor not only qualitatively but also quantitatively. Despite the complexity of the nonlinear partial differential equations that results from this model, I also present analytical treatments based on the techniques of nonlinear physics and W.K.B approximation to explain the observed structures in dermatology.

An analysis is given of a continuum model of a proliferating cell population, which incorporates cell movement in space and cell progression through the cell cycle. The model consists of a nonlinear partial differential equation for the cell density in the spatial position and the cell age coordinates. The equation contains a diffusion term corresponding to random cell movement, a nonlocal dispersion term corresponding to cell–cell adhesion, a cell age-dependent boundary condition corresponding to cell division, and a nonlinear logistic term corresponding to constrained population growth. Basic properties of the solutions are proved, including existence, uniqueness, positivity, and long-term behavior dependent on parametric input. The model is illustrated by simulations applicable to in vitro wound closure experiments, which are widely used for experimental testing of cancer therapies.

In this paper we review methods that allow the construction of a consistent set of models that may describe the interactions between a tumor and the immune system on microscopic, mesoscopic, and macroscopic scales. The presented structures may be a basis for a description on the sub–cellular, cellular, and macroscopic levels. Important open problems are indicated.

Cancer is increasingly recognized as not solely a disease of the genes and chromosomes but as a systemic disease that affects numerous components of the host including blood vessel formation, immune cell function, and nutrient recycling. This review summarizes a variety of time-dependent mathematical models that focus on the consequences of tumor growth within an evolving microenvironment, represented by a dynamic carrying capacity. Transcending the specifics of each model, their overview reveals that the key to tumor control really lies in controlling the support furnished the tumor by its microenvironment.

The immune response is an important factor in the progression of cancer, and this response has been harnessed in a variety of treatments for a range of cancers. In this chapter we develop mathematical models that describe the immune response to the presence of a tumor. We then use these models to explore a variety of immunotherapy treatments, both alone and in combination with other therapies.

Cancer is a complex multiscale disease involving inter-related processes across a wide range of temporal and spatial scales. Multiscale mathematical models can help in studying cancer progression and serve as an in silico test base for comparing and optimizing various multi-modality anticancer treatment protocols. Here, we discuss one such hybrid multiscale approach, interlinking individual cell behavior with the macroscopic tissue scale. Using this technique, we study the spatio-temporal dynamics of individual cells and their interactions with the tumor microenvironment. At the intracellular level, the internal cell-cycle mechanism is modelled using a system of coupled ordinary differential equations, which determine cellular growth dynamics for each individual cell. The evolution of these individual cancer cells are modelled using a cellular automaton approach. Moreover, we have also incorporated the effects of oxygen distribution into this multiscale model as it has been shown to affect the internal cell-cycle dynamics of the cancer cells. The hybrid multiscale model is then used to study the effects of cell-cycle-specific chemotherapeutic drugs, alone and in combination with radiotherapy, with a long-term goal of predicting an optimal multimodality treatment plan for individual patients.

In this short review paper, I will present the mathematical models that have been designed in the frame of continuous deterministic cell population dynamics that aim at optimization of cancer treatments using chronotherapeutics. Many authors have dealt with chronobiology of cancer, less with continuous mathematical models and even less with the declared aim to optimize chronotherapeutics. The biological and theoretical bases for these models are sketched, started from a historical viewpoint, and the main theoretical results are presented, with biological suggestions to account for them. Chronotherapeutics that leads to therapeutic optimization with the constraint of limiting unwanted toxicity of anticancer drugs towards healthy cell populations is put in a medical perspective together with the other main pitfall of cancer therapeutics, for which optimization procedures should have little to do with circadian biology, i.e., emergence of drug resistance in cancer cell populations, which is amenable to the use of other sorts of models, that are briefly mentioned.

In this paper, results about the structure of cancer treatment protocols that can be inferred from an analysis of mathematical models with the methods and tools of optimal control are reviewed. For homogeneous tumor populations of chemotherapeutically sensitive cells, optimal controls are bang-bang corresponding to the medical paradigm of maximum tolerated doses (MTD). But as more aspects of the tumor microenvironment are taken into account, such as heterogeneity of the tumor cell population, tumor angiogenesis and tumor-immune system interactions, singular controls which administer agents at specific time-varying reduced dose rates become optimal and give an indication of what might be the biologically optimal dose (BOD).