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.

Modification of behaviour in response to changes in the environment or ambientconditions, based on memory, is typical of the human and, possibly, many animalspecies.One obvious example of such adaptivity is, for instance, switching to a saferbehaviour when in danger, from either a predator or an infectious disease. In humansociety such switching to safe behaviour is particularly apparent during epidemics.Mathematically, such changes of behaviour in response to changes in the ambient conditionscan be described by models involving switching. In most cases, this switching is assumedto depend on the system state, and thus it disregards the history and, therefore, memory.Memory can be introduced into a mathematical model using a phenomenon known as hysteresis.We illustrate this idea using a simple SIR compartmental model that is applicable inepidemiology. Our goal is to show why and how hysteresis can arise in such a model, andhow it may be applied to describe a variety of memory effects. Our other objective is tointroduce a unified paradigm for mathematical modelling with memory effects inepidemiology and ecology. Our approach treats changing behaviour as an irreversible flowrelated to large ensembles of elementary exchange operations that recently has beensuccessfully applied in a number of other areas, such as terrestrial hydrology, andmacroeconomics. For the purposes of illustrating these ideas in an application to biology,we consider a rather simple case study and develop a model from first principles. Weaccompany the model with extensive numerical simulations which exhibit interestingqualitative effects.

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.

Background

Independence models among variables is one of the most relevant topics in epidemiology,particularly in molecular epidemiology for the study of gene-gene and gene-environmentinteractions. They have been studied using three main kinds of analysis: regressionanalysis, data mining approaches and Bayesian model selection. Recently, methods ofalgebraic statistics have been extensively used for applications to biology. In this paperwe present a synthetic, but complete description of independence models in algebraicstatistics and a new method of analyzing interactions, that is equivalent to thecorrection by Markov bases of the Fisher’s exact test.

Methods

We identified the suitable algebraic independence model for describing the dependence oftwo genetic variables from the occurrence of cancer and exploited the theory of toricvarieties and Gröbner basis for developing an exact independence test based on theDiaconis-Sturmfels algorithm. We implemented it in a Maple routine and we applied it tothe study of gene-gene interaction in Gen-Air, an European case-control study. We computedthe p-value for each pair of genetic variables interacting with disease status and wecompared our results with the standard asymptotic chi-square test.

Results

We found an association among COMT Val158Met, APE1Asp148Glu and bladder cancer (p-value: 0.009). We also found the interactionamong TP53 Arg72Pro, GSTP1 Ile105Val and lung cancer(p-value: 0.00035). Leukaemia was observed to significantly interact with the pairsERCC2 Lys751Gln and RAD51 172 G > T (p-value0.0072), ERCC2 Lys751Gln and LIG4Thr9Ile (p-value:0.0095) and APE1 Asp148Glu and GSTP1 Ala114Val (p-value:0.0036).

Conclusion

Taking advantage of results from theoretical and computational algebra, the method wepropose was more selective than other methods in detecting new interactions, andnevertheless its results were consistent with previous epidemiological and functionalfindings. It also helped us in controlling the multiple comparison problem. In the lightof our results, we believe that the epidemiologic study of interactions can benefit ofalgebraic methods based on properties of toric varieties and Gröbner bases.

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.

Contact behavior plays an important role in influenza transmission. In the progression ofinfluenza spread, human population reduces mobility to decrease infection risks. In thispaper, a mathematical model is proposed to include adaptive mobility. It is shown that themobility response does not affect the basic reproduction number that characterizes theinvasion threshold, but reduces dramatically infection peaks, or removes the peaks.Numerical calculations indicate that the mobility response can provide a very goodprotection to susceptible individuals, and a combination of mobility response andtreatment is an effective way to control influenza outbreak.