While drawing a link between the papers contained in this issue and those present in a previous one (Vol. 2, Issue 3), this introductory article aims at putting in evidence some trends and challenges on cancer modelling, especially related to the development of multiphase and multiscale models.

This review aims at presenting asynoptic, if not exhaustive, point of view on some of the problemsencountered by biologists and physicians who deal with naturalcell proliferation and disruptions of its physiological control incancer disease. It also aims at suggesting how mathematicians arenaturally challenged by these questions and how they might help,not only biologists to deal theoretically with biologicalcomplexity, but also physicians to optimise therapeutics, on whichlast point the focus will be set here. To this purpose,mathematical modelling should represent proliferating cellpopulation dynamics with natural built-in control targets (whichimplies modelling the cell division cycle), together with thedistribution of drugs in the organism and their molecular actionson different targets at the cell level on proliferation, i.e.,molecular pharmacokinetics-pharmacodynamics of antiproliferativedrugs. This should make possible optimal control of drug deliverywith constraints to be determined according to the mainpharmacological issues encountered in the clinic: unwanted toxicside-effects, occurrence of drug resistance. Mathematicalmodelling should also take into account physiological determinantsof cell and tissue proliferation, such as intervention of theimmune system, circadian control on cell cycle checkpointproteins, and activity of intracellular drug processing enzymestogether with individual variations in the activities of theseproteins (genetic polymorphism). Taking these points into accountwill add to the rich scenery of normal or disrupted cell andtissue regulations, and their corrections by drugs, a naturalenvironmental, whole body physiological, frame. It is necessaryindeed to consider such a framework if one wants to eventually beactually helpful to clinicians who routinely treat by combinationsof drugs living Humans with their complex whole body regulations,often dependent on genotypic variations, and not isolated cells ortissues.

Many tumours undergo periods in whichthey apparently do not grow but remain at a roughly constant sizefor extended periods. This is termed tumour dormancy. Themechanisms responsible for dormancy include failure to develop aninternal blood supply, individual tumour cells exiting the cellcycle and a balance between the tumour and the immune response toit. Tumour dormancy is of considerable importance in the naturalhistory of cancer. In many cancers, and in particular in breastcancer, recurrence can occur many years after surgery to removethe primary tumour, following a long period of occult disease.Mathematical modelling suggested that continuous growth of tumourswas incompatible with data of the times of recurrence in breastcancer, suggesting that tumour dormancy was a common phenomenon.Modelling has also been applied to understanding the mechanismsresponsible for dormancy, how they can be manipulated and theimplications for cancer therapy. Here, the literature onmathematical modelling of tumour dormancy is reviewed. Inconclusion, promising future directions for research arediscussed.

In this paper we investigate the role of spatial effects in determining thedynamics of a subclass of signalling pathways characterised by their ability todemonstrate oscillatory behaviour. To this end, we formulate a simple spatial model of thep53 network that accounts for both a negative feedback and a transcriptional delay. We show that the formation of protein density patterns can depend on the shape of the cell, position of the nucleus, and the protein diffusion rates. Thetemporal changes in the total amounts of protein are also subject to spatial influences. The level of DNA damage required to induce sustained oscillations, forinstance, depends on the morphology of the cell. The model also provides a newinterpretation of experimentally observed undamped oscillations in p53 levels in single cells.Our simulations reveal that alternate sequences of high- and low-amplitudeoscillations can occur. We propose that the digital pulses may correspond to snap-shots ofour high-amplitude sequences. Shorter waiting-times between subsequent time-lapse fluorescence microscopy images in combination with lower detection thresholds may reveal the irregular high-frequency oscillations suggested by our spatialmodel.

Tumor growth and progression is a complex phenomenon dependent on theinteraction of multiple intrinsic and extrinsic factors. Necessary for tumordevelopment is a small subpopulation of potent cells, so-called cancer stemcells, that can undergo an unlimited number of cell divisions and which areproposed to divide symmetrically with a small probability to produce more cancerstem cells. We show that the majority of cells in a tumor must indeed benon-stem cancer cells with limited life span and limited replicative potential. Tumor development is dependent as well on the proliferative potential and deathof these cells, and on the migratory ability of all cancer cells. Withincreasing number of cells in the tumor, competition for space limits tumorprogression, and in agreement with in vitro observation, the majority of cancercells become quiescent, with proliferation primarily occurring on the outer rimwhere space is available. We present an agent-based model of early tumordevelopment that captures the spatial heterogeneity of stemness andproliferation status. We apply the model to simulations of radiotherapy topredict treatment outcomes for tumors with different stem cell pool sizes anddifferent quiescence radiosensitivities. We show by first presuming homogeneousradiosensitivity throughout the tumor, and then considering the greaterresistance of quiescent cells, that stem cell pool size and stem cellrepopulation during treatment determine treatment success. The results fortumor cure probabilities comprise upper bounds, as there is evidence that cancerstem cells are also more radioresistant than other tumor cells. Beyond justdemonstrating the influence of mass effects of stem to non-stem cell ratios andproliferating to quiescent cell ratios, we show that the spatiotemporalevolution of the developing heterogeneous population plays a pivotal role indetermining radioresponse and treatment optimization.

The present paper introduces a tumormodel with two time scales, the time t during which the tumorgrows and the cycle time of individual cells. The model alsoincludes the effects of gene mutations on the population densityof the tumor cells. The model is formulated as a free boundaryproblem for a coupled system of elliptic, parabolic and hyperbolicequations within the tumor region, with nonlinear and nonlocalterms. Existence and uniqueness theorems are proved, andproperties of the free boundary are established.

Most mammalian tissues are organized into a hierarchical structure of stem,progenitor, and differentiated cells. Tumors exhibit similar hierarchy, even ifit is abnormal in comparison with healthy tissue. In particular, it is believedthat a small population of cancer stem cells drives tumorigenesis in certainmalignancies. These cancer stem cells are derived from transformed stem cells ormutated progenitors that have acquired stem-cell qualities, specifically theability to self-renew. Similar to their normal counterparts, cancer stem cellsare long-lived, can self-renew and differentiate, albeit aberrantly, and arecapable of generating tissue, resulting in tumor formation. Although identifiedand characterized in several forms of malignancy, the specific multi-stepprocess that causes the formation of cancer stem cells is uncertain. Here, amaturity-structured mathematical model is developed to investigate thesequential order of mutations that causes the fastest emergence of cancer stemcells. Using model predictions, we discuss conditions for which geneticinstability significantly speeds cancer onset and suggest that unbalancedstem-cell self-renewal and inhibition of progenitor differentiation contributeto aggressive forms of cancer. To our knowledge, this is the first continuousmaturity-structured mathematical model used to investigate mutation acquisitionwithin hierarchical tissue in order to address implications of cancer stem cellsin tumorigenesis.

We study the growth rate of a cell population that follows anage-structured PDE with time-periodic coefficients. Our motivationcomes from the comparison between experimental tumor growth curvesin mice endowed with intact or disrupted circadian clocks,known to exert their influence on the cell division cycle.We compare the growth rate of the model controlled by a time-periodiccontrol on its coefficients with the growth rate of stationarymodels of the same nature, but with averaged coefficients. We firstly derive a delay differential equation which allows us to proveseveral inequalities and equalities on the growth rates. We alsodiscuss about the necessity to take into account the structure ofthe cell division cycle for chronotherapy modeling. Numerical simulationsillustrate the results.

The paper is devoted to mathematical modelling of erythropoiesis, production of red blood cells in the bone marrow. We discuss intra-cellular regulatory networks which determine self-renewal and differentiation of erythroid progenitors. In the case of excessive self-renewal, immature cells can fill the bone marrow resulting in the development of leukemia. We introduce a parameter characterizing the strength of mutation. Depending on its value, leukemia will or will not develop. The simplest model of treatment of acute myeloid leukemia with chemotherapy allows us to determine the conditions of successful treatment or of its failure. We show that insufficient treatment can worsen the situation. In some cases curing may not be possible even without resistance to treatment. Modelling presented in this work is based on ordinary differential equations, reaction-diffusion systems and individual based approach.