Cancer has recently overtaken heart disease as the world’s biggest killer. Cancer isinitiated by gene mutations that result in local proliferation of abnormal cells and theirmigration to other parts of the human body, a process called metastasis. The metastasizedcancer cells then interfere with the normal functions of the body, eventually leading todeath. There are two hundred types of cancer, classified by their point of origin. Most ofthem share some common features, but they also have their specific character. In thisarticle we review mathematical models of such common features and then proceed to describemodels of specific cancer diseases.

When invading the tissue, malignant tumour cells (i.e. cancer cells) need to detach fromneighbouring cells, degrade the basement membrane, and migrate through the extracellularmatrix. These processes require loss of cell-cell adhesion and enhancement of cell-matrixadhesion. In this paper we present a mathematical model of an intracellular pathway forthe interactions between a cancer cell and the extracellular matrix. Cancer cells usesimilar mechanisms as with normal cells for their interactions with the extracellularmatrix. We develop a model of cell-matrix adhesion that accounts for reactions between thecell surface receptor integrins, the matrix glycoprotein fibronectin, and the actinfilaments in the cytoskeleton. Each represents components for an intermediate compartment,the extracellular compartment, and the intracellular compartment, respectively. Binding offibronectin with integrins triggers a clustering of protein complexes, which thenactivates and phosphorylates regulatory proteins that are involved in actin reorganisationcausing actin polymerization and stress fibre assembly. Rearrangement of actin filamentswith integrin/fibronectin complexes near adhesion sites and interaction with fibrillarfibronectin produces the force necessary for cell migration, accounting for cell-matrixadhesion.

A computational framework for testing the effects of cytotoxic molecules, specific to agiven phase of the cell cycle, and vascular disrupting agents (VDAs) is presented. Themodel is based on a cellular automaton to describe tumour cell states transitions fromproliferation to death. It is coupled with a model describing the tumour vasculature andits adaptation to the blood rheological constraints when alterations are induced by VDAstreatment. Several therapeutic protocols in two structurally different vascular networkswere tested by varying the duration of cytotoxic drug perfusion and the periodicity oftreatment cycles. The impact of VDAs were also tested both experimentally from intravitalmicroscopy through a dorsal skinfold chamber on a mouse and numerically. Simulationresults show that combining cytotoxic treatment with a post treatment of VDA through ajudicious timing could favour the rapid eradication of the tumour. The computationalframework thus gives some insights into the outcome of cytotoxic and VDAs treatments on aqualitative basis. Future validation from our experimental setup could open up newperspectives towards Computer-Assisted Therapeutic Strategies.

The invasive capability is fundamental in determining the malignancy of a solid tumor.Revealing biomedical strategies that are able to partially decrease cancer invasiveness istherefore an important approach in the treatment of the disease and has given rise tomultiple in vitro and in silico models. We here developa hybrid computational framework, whose aim is to characterize the effects of thedifferent cellular and subcellular mechanisms involved in the invasion of a malignantmass. In particular, a discrete Cellular Potts Model is used to represent the populationof cancer cells at the mesoscopic scale, while a continuous approach of reaction-diffusionequations is employed to describe the evolution of microscopic variables, as the nutrientsand the proteins present in the microenvironment and the matrix degrading enzymes secretedby the tumor. The behavior of each cell is then determined by a balance of forces, such ashomotypic (cell-cell) and heterotypic (cell-matrix) adhesions and haptotaxis, and ismediated by the internal state of the individual, i.e. its motility. The resultingcomposite model quantifies the influence of changes in the mechanisms involved in tumorinvasion and, more interestingly, puts in evidence possible therapeutic approaches, thatare potentially effective in decreasing the malignancy of the disease, such as thealteration in the adhesive properties of the cells, the inhibition in their ability toremodel and the disruption of the haptotactic movement. We also extend the simulationframework by including cell proliferation which, following experimental evidence, isregulated by the intracellular level of growth factors. Interestingly, in spite of theincrement in cellular density, the depth of invasion is not significantly increased, asone could have expected.

Gliomas are highly invasive brain tumors that exhibit high and spatially heterogeneouscell proliferation and motility rates. The interplay of proliferation and migrationdynamics plays an important role in the invasion of these malignant tumors. We analyze theregulation of proliferation and migration processes with a lattice-gas cellular automaton(LGCA). We study and characterize the influence of the migration/proliferation dichotomy(also known as the “GO-or-Grow" mechanism) on avascular glioma invasion, in terms ofinvasion speed and width of the infiltration zone. We show that the invasive behavior ofthe (macroscopic) tumor colony is a highly complex phenomenon that cannot be extrapolatedby the sole knowledge of the (microscopic) individual cell phenotype.

There is evidence that cancer develops when cells acquire a sequence of mutations thatalter normal cell characteristics. This sequence determines a hierarchy among the cells,based on how many more mutations they need to accumulate in order to become cancerous.When cells divide, they exhibit telomere loss and differentiate, which defines anothercell hierarchy, on top of which is the stem cell. We propose a mutation-generation model,which combines the mutation-accumulation hierarchy with the differentiation hierarchy ofthe cells, allowing us to take a step further in examining cancer development and growth.The results of the model support the hypothesis of the cancer stem cell’s role in cancerpathogenesis: a very small fraction of the cancer cell population is responsible for thecancer growth and development. Also, according to the model, the nature of mutationaccumulation is sufficient to explain the faster growth of the cancer cell population.However, numerical results show that in order for a cancer to develop within a reasonabletime frame, cancer cells need to exhibit a higher proliferation rate than normalcells.

The cancer stem cell hypothesis has evolved to one of the most important paradigms inbiomedical research. During recent years evidence has been accumulating for the existenceof stem cell-like populations in different cancers, especially in leukemias. In thecurrent work we propose a mathematical model of cancer stem cell dynamics in leukemias. Weapply the model to compare cellular properties of leukemic stem cells to those of theirbenign counterparts. Using linear stability analysis we derive conditions necessary andsufficient for expansion of malignant cell clones, based on fundamental cellularproperties. This approach reveals different scenarios of cancer initiation and providesqualitative hints to possible treatment strategies.

In order to better understand the dynamics of acute leukemia, and in particular to findtheoretical conditions for the efficient delivery of drugs in acute myeloblastic leukemia,we investigate stability of a system modeling its cell dynamics.

The overall system is a cascade connection of sub-systems consisting of distributeddelays and static nonlinear feedbacks. Earlier results on local asymptotic stability areimproved by the analysis of the linearized system around the positive equilibrium. For thenonlinear system, we derive stability conditions by using Popov, circle and nonlinearsmall gain criteria. The results are illustrated with numerical examples andsimulations.

Existence and stability of periodic solutions are studied for a system of delaydifferential equations with two delays, with periodic coefficients. It models theevolution of hematopoietic stem cells and mature neutrophil cells in chronic myelogenousleukemia under a periodic treatment that acts only on mature cells. Existence of a guidingfunction leads to the proof of the existence of a strictly positive periodic solution by atheorem of Krasnoselskii. The stability of this solution is analysed.

We consider the early carcinogenesis model originally proposed as a deterministicreaction-diffusion system. The model has been conceived to explore the spatial effectsstemming from growth regulation of pre-cancerous cells by diffusing growth factormolecules. The model exhibited Turing instability producing transient spatial spikes incell density, which might be considered a model counterpart of emerging foci of malignantcells. However, the process of diffusion of growth factor molecules is by its nature astochastic random walk. An interesting question emerges to what extent the dynamics of thedeterministic diffusion model approximates the stochastic process generated by the model.We address this question using simulations with a new software tool called sbioPN (spatialbiological Petri Nets). The conclusion is that whereas single-realization dynamics of thestochastic process is very different from the behavior of the reaction diffusion system,it is becoming more similar when averaged over a large number of realizations. The degreeof similarity depends on model parameters. Interestingly, despite the differences, typicalrealizations of the stochastic process include spikes of cell density, which however arespread more uniformly and are less dependent of initial conditions than those produced bythe reaction-diffusion system.

We investigate optimal control of a cancer-immune cell interactive model with delay inthe interphase compartment. By applying the optimal control theory, we seek to minimizethe cost associated with the chemotherapy drug, minimize the accumulation of cancer cells,and increase the immune cell presence. Optimality conditions and characterization of thecontrol are provided. Numerical analyses are given to enhance the understanding of thedifficulties that occur in the control of cancer.

Recent discovery of cancer stem cells in tumorigenic tissues has raised many questionsabout their nature, origin, function and their behavior in cell culture. Most of currentexperiments reporting a dynamics of cancer stem cell populations in culture show theeventual stability of the percentages of these cell populations in the whole population ofcancer cells, independently of the starting conditions. In this paper we propose amathematical model of cancer stem cell population behavior, based on specific features ofcancer stem cell divisions and including, as a mathematical formalization of cell-cellcommunications, an underlying field concept. We compare the qualitative behavior ofmathematical models of stem cells evolution, without and with an underlying signal. Inabsence of an underlying field, we propose a mathematical model described by a system ofordinary differential equations, while in presence of an underlying field it is describedby a system of delay differential equations, by admitting a delayed signal originated byexisting cells. Under realistic assumptions on the parameters, in both cases (ODE withoutunderlying field, and DDE with underlying field) we show in particular the stability ofpercentages, provided that the delay is sufficiently small. Further, for the DDE case (inpresence of an underlying field) we show the possible existence of, either damped orstanding, oscillations in the cell populations, in agreement with some existingmathematical literature. The outcomes of the analysis may offer to experimentalists a toolfor addressing the issue regarding the possible non-stem to stem cells transition, bydetermining conditions under which the stability of cancer stem cells population can beobtained only in the case in which such transition can occur. Further, the provideddescription of the variable corresponding to an underlying field may stimulate furtherexperiments for elucidating the nature of “instructive" signals for cell divisions,underlying a proper pattern of the biological system.

Treating cancer patients with metastatic disease remains an ultimate challenge inclinical oncology. Because invasive cancer precludes or limits the use of surgery,metastatic setting is often associated with (poor) survival, rather than sustainedremission, in patients with common cancers like lung, digestive or breast carcinomas.Mathematical modeling may help us better identify non detectable metastatic status to inturn optimize treatment for patients with metastatic disease. In this paper we present afamily of models for the metastatic growth. They are based on four principles : to be assimple as possible, involving the least possible number of parameters, the maininformations are obtained from the primary tumor and being able to recover the variety ofphenomena observed by the clinicians. Several simulations of therapeutic strategies arepresented illustrating possible applications of modeling to the clinic.

Many cancer-associated genes and pathways remain to be identified in order to clarify themolecular mechanisms underlying cancer progression. In this area, genome-wideloss-of-function screens appear to be powerful biological tools, allowing the accumulationof large amounts of data. However, this approach currently lacks analytical tools toexploit the data with maximum efficiency, for which systems biology methods analyzingcomplex cellular networks may be extremely helpful. In this article we report such asystems biology strategy based on the construction of a Network for a biological processand specific for a given cell system (cell type). The networks are created fromgenome-wide loss-of-function screen datasets. We also propose tools to analyze networkproperties. As one of the tools, we suggest a mathematical model for discriminationbetween two distinct cell processes that may be affected by knocking down the activity ofa gene, i. e., a decreased cell number may be caused by arrested cell proliferation orenhanced cell death. Next we show how this discrimination between the two cell processeshelps to construct two corresponding subnetworks. Finally, we demonstrate an applicationof the proposed strategy to the identification and characterization of putative novelgenes and pathways significant for the control of lung cancer cell growth, based on theresults of a genome-wide proliferation/viability loss-of-function screen of human lungadenocarcinoma cells.