Evolution of a structured cell population endowed with plasticity of traits under constraints on and between the traits

Confronted with the biological problem of managing plasticity in cell populations, which is in particular responsible for transient and reversible drug resistance in cancer, we propose a rationale consisting of an integro-differential and a reaction-advection-diffusion equation, the properties of which are studied theoretically and numerically. By using a constructive finite volume method, we show the existence and uniqueness of a weak solution and illustrate by numerical approximations and their simulations the capacity of the model to exhibit divergence of traits. This feature may be theoretically interpreted as describing a physiological step towards multicellularity in animal evolution and, closer to present-day clinical challenges in oncology, as a possible representation of bet hedging in cancer cell populations.

Cell plasticity in cancer cell populations

In this review, we propose a recension of biological observations on plasticity in cancer cell populations and discuss theoretical considerations about their mechanisms.

Identification of a transient state during the acquisition of temozolomide resistance in glioblastoma

Drug resistance limits the therapeutic efficacy in cancers and leads to tumor recurrence through ill-defined mechanisms. Glioblastoma (GBM) are the deadliest brain tumors in adults. GBM, at diagnosis or after treatment, are resistant to temozolomide (TMZ), the standard chemotherapy. To better understand the acquisition of this resistance, we performed a longitudinal study, using a combination of mathematical models, RNA sequencing, single cell analyses, functional and drug assays in a human glioma cell line (U251). After an initial response characterized by cell death induction, cells entered a transient state defined by slow growth, a distinct morphology and a shift of metabolism. Specific genes expression associated to this population revealed chromatin remodeling. Indeed, the histone deacetylase inhibitor trichostatin (TSA), specifically eliminated this population and thus prevented the appearance of fast growing TMZ-resistant cells. In conclusion, we have identified in glioblastoma a population with tolerant-like features, which could constitute a therapeutic target.

Adaptive dynamics of hematopoietic stem cells and their supporting stroma: a model and mathematical analysis

We propose a mathematical model to describe the evolution of hematopoietic stem cells (HSCs) and stromal cells in considering the bi-directional interaction between them. Cancerous cells are also taken into account in our model. HSCs are structured by a continuous phenotype characterising the population heterogeneity in a way relevant to the question at stake while stromal cells are structured by another continuous phenotype representing their capacity of support to HSCs. We then analyse the model in the framework of adaptive dynamics. More precisely, we study single Dirac mass steady states, their linear stability and we investigate the role of parameters in the model on the nature of the evolutionary stable distributions (ESDs) such as monomorphism, dimorphism and the uniqueness properties. We also study the dominant phenotypes by an asymptotic approach and we obtain the equation for dominant phenotypes. Numerical simulations are employed to illustrate our analytical results. In particular, we represent the case of the invasion of malignant cells as well as the case of co-existence of cancerous cells and healthy HSCs.

Control in dormancy or eradication of cancer stem cells: Mathematical modeling and stability issues

OBJECTIVE

Modeling and analysis of cell population dynamics enhance our understanding of cancer. Here we introduce and explore a new model that may apply to many tissues.

ANALYSES

An age-structured model describing coexistence between mutated and ordinary stem cells is developed and explored. The model is transformed into a nonlinear time-delay system governing the dynamics of healthy cells, coupled to a nonlinear differential-difference system describing dynamics of unhealthy cells. Its main features are highlighted and an advanced stability analysis of several steady states is performed, through specific Lyapunov-like functionals for descriptor-type systems.

RESULTS

We propose a biologically based model endowed with rich dynamics. It incorporates a new parameter representing immunoediting processes, including the case where proliferation of cancer cells is locally kept under check by the immune cells. It also considers the overproliferation of cancer stem cells, modeled as a subpopulation of mutated cells that is constantly active in cell division. The analysis that we perform here reveals the conditions of existence of several steady states, including the case of cancer dormancy, in the coupled model of interest. Our study suggests that cancer dormancy may result from a plastic sensitivity of mutated cells to their shared environment, different from that - fixed - of healthy cells, and this is related to an action (or lack of action) of the immune system. Next, the stability analysis that we perform is essentially oriented towards the determination of sufficient conditions, depending on all the model parameters, that ensure either a regionally (i.e., locally) stable dormancy steady state or eradication of unhealthy cells. Finally, we discuss some biological interpretations, with regards to our findings, in light of current and emerging therapeutics. These final insights are particularly formulated in the paradigmatic case of hematopoiesis and acute leukemia, which is one of the best known malignancies for which it is always hard, in presence of a clinical and histological remission, to decide between cure and dormancy of a tumoral clone.

Genotype- or Phenotype-Targeting Anticancer Therapies? Lessons from Tumor Evolutionary Biology

Despite the efficacy of most cancer therapies, drug resistance remains a major problem in the clinic. The eradication of the entire tumor and the cure of the patient by chemotherapy alone are rare, in particular for advanced disease. From an evolutionary perspective, the selective pressure exerted by chemotherapy leads to the emergence of resistant clones where resistance can be associated with many different functional mechanisms at the single cell level or can involve changes in the tumor micro-environment. In the last decade, tumor genomics has contributed to the improvement of our understanding of tumorigenesis and has led to the identification of numerous cellular targets for the development of novel therapies. However, since tumors are by nature extremely heterogeneous, the drug efficacy and economical sustainability of this approach is now debatable. Importantly, tumor cell heterogeneity depends not only on genetic modifications but also on non-genetic processes involving either stochastic events or epigenetic modifications making genetic biomarkers of uncertain utility. In this review, we wish to highlight how evolutionary biology can impact our understanding of carcinogenesis and resistance to therapies. We will discuss new approaches based on applied ecology and evolution dynamics that can be used to convert the cancer into a chronic disease where the drugs would control tumor growth. Finally, we will discuss the way metabolic dysfunction or phenotypic changes can help developing new delivery systems or phenotypetargeted drugs and how exploring new sources of active compounds can conduct to the development of drugs with original mechanisms of action.

Abstract 92: Acquisition of temozolomide resistance: Identification of a new drug tolerant stage in glioblastoma cells

Glioblastoma multiforme (GBM) are the most aggressive and common brain tumors in adults. Despite surgery and combined radio-chemotherapy with temozolomide (TMZ), tumor reccurence always occurs. The median survival time for patients diagnosed with GBM is about 14 months with less than 5% survival at 5 years. Today the main marker of TMZ resistance is the methylation status of MGMT promoter. Patients with a methylated promoter usually have a better response to treatment than patients with an unmethylated promoter. Indeed MGMT is an enzyme involved in DNA repair mechanisms that abrogates TMZ effects. However, in clinical trials targeting the MGMT enzyme, median survival of patients was not improved. It is thus essential to decipher the mechanisms involved in the acquisition of TMZ resistance to identify new therapeutic targets. To achieve this goal TMZ resistant cells were generated by continuous treatment of the U251 human glioblastoma cell line. These cells are sensitive to TMZ and do not express MGMT. We performed transcriptomic analysis by RNA-Seq on U251 treated with TMZ (50µM) for different time and selected differentially expressed genes. We also evaluated target genes expression in single cells by RT-qPCR using C1-HD-Biomark technology (Fluidigm). Transcriptome profiling allowed to identify a transient phase with TMZ tolerant cells before acquisition of complete resistance. These cells are characterized by a modified morphology and a no proliferative state. In this population we identified a subset of genes with the same transient overexpression. Similar results were observed in other glioblastoma cell lines and under other stress conditions. In contrast, stable expression of MGMT appeared later with the emergence of the TMZ resistant population. Interestingly single cell qPCR showed that MGMT expression in the resistant cells could not be explained by clonal selection of MGMT positive cells. Drug screening on the TMZ tolerant cells revealed a potent killing activity of histone deacetylating agents (HDAC inhibitors). In conclusion, we have shown that glioblastoma cells become resistant after a transient state of TMZ tolerance. Identification of this singular population highlights new molecular targets and a new therapeutic window. Targeting these tolerant cells could avoid emergence of resistance and tumor recurrence, thereby patients survival could be improved.

Citation Format: Marion Rabé, Hicham Janati, Solenne Dumont, Christelle Thibault-Carpentier, Jean Clairambault, François M. Vallette, Catherine Gratas. Acquisition of temozolomide resistance: Identification of a new drug tolerant stage in glioblastoma cells [abstract]. In: Proceedings of the American Association for Cancer Research Annual Meeting 2017; 2017 Apr 1-5; Washington, DC. Philadelphia (PA): AACR; Cancer Res 2017;77(13 Suppl):Abstract nr 92. doi:10.1158/1538-7445.AM2017-92

Tracking the evolution of cancer cell populations through the mathematical lens of phenotype-structured equations

Background

A thorough understanding of the ecological and evolutionary mechanisms that drive the phenotypic evolution of neoplastic cells is a timely and key challenge for the cancer research community. In this respect, mathematical modelling can complement experimental cancer research by offering alternative means of understanding the results of in vitro and in vivo experiments, and by allowing for a quick and easy exploration of a variety of biological scenarios through in silico studies.

Results

To elucidate the roles of phenotypic plasticity and selection pressures in tumour relapse, we present here a phenotype-structured model of evolutionary dynamics in a cancer cell population which is exposed to the action of a cytotoxic drug. The analytical tractability of our model allows us to investigate how the phenotype distribution, the level of phenotypic heterogeneity, and the size of the cell population are shaped by the strength of natural selection, the rate of random epimutations, the intensity of the competition for limited resources between cells, and the drug dose in use.

Conclusions

Our analytical results clarify the conditions for the successful adaptation of cancer cells faced with environmental changes. Furthermore, the results of our analyses demonstrate that the same cell population exposed to different concentrations of the same cytotoxic drug can take different evolutionary trajectories, which culminate in the selection of phenotypic variants characterised by different levels of drug tolerance. This suggests that the response of cancer cells to cytotoxic agents is more complex than a simple binary outcome, i.e., extinction of sensitive cells and selection of highly resistant cells. Also, our mathematical results formalise the idea that the use of cytotoxic agents at high doses can act as a double-edged sword by promoting the outgrowth of drug resistant cellular clones. Overall, our theoretical work offers a formal basis for the development of anti-cancer therapeutic protocols that go beyond the ‘maximum-tolerated-dose paradigm’, as they may be more effective than traditional protocols at keeping the size of cancer cell populations under control while avoiding the expansion of drug tolerant clones.

Reviewers

This article was reviewed by Angela Pisco, Sébastien Benzekry and Heiko Enderling.

Electronic supplementary material

The online version of this article (doi:10.1186/s13062-016-0143-4) contains supplementary material, which is available to authorized users.

Cell population heterogeneity and evolution towards drug resistance in cancer: Biological and mathematical assessment, theoretical treatment optimisation

BACKGROUND

Drug-induced drug resistance in cancer has been attributed to diverse biological mechanisms at the individual cell or cell population scale, relying on stochastically or epigenetically varying expression of phenotypes at the single cell level, and on the adaptability of tumours at the cell population level.

SCOPE OF REVIEW

We focus on intra-tumour heterogeneity, namely between-cell variability within cancer cell populations, to account for drug resistance. To shed light on such heterogeneity, we review evolutionary mechanisms that encompass the great evolution that has designed multicellular organisms, as well as smaller windows of evolution on the time scale of human disease. We also present mathematical models used to predict drug resistance in cancer and optimal control methods that can circumvent it in combined therapeutic strategies.

MAJOR CONCLUSIONS

Plasticity in cancer cells, i.e., partial reversal to a stem-like status in individual cells and resulting adaptability of cancer cell populations, may be viewed as backward evolution making cancer cell populations resistant to drug insult. This reversible plasticity is captured by mathematical models that incorporate between-cell heterogeneity through continuous phenotypic variables. Such models have the benefit of being compatible with optimal control methods for the design of optimised therapeutic protocols involving combinations of cytotoxic and cytostatic treatments with epigenetic drugs and immunotherapies.

GENERAL SIGNIFICANCE

Gathering knowledge from cancer and evolutionary biology with physiologically based mathematical models of cell population dynamics should provide oncologists with a rationale to design optimised therapeutic strategies to circumvent drug resistance, that still remains a major pitfall of cancer therapeutics. This article is part of a Special Issue entitled "System Genetics" Guest Editor: Dr. Yudong Cai and Dr. Tao Huang.

Emergence of drug tolerance in cancer cell populations: an evolutionary outcome of selection, nongenetic instability, and stress-induced adaptation

In recent experiments on isogenetic cancer cell lines, it was observed that exposure to high doses of anticancer drugs can induce the emergence of a subpopulation of weakly proliferative and drug-tolerant cells, which display markers associated with stem cell-like cancer cells. After a period of time, some of the surviving cells were observed to change their phenotype to resume normal proliferation and eventually repopulate the sample. Furthermore, the drug-tolerant cells could be drug resensitized following drug washout. Here, we propose a theoretical mechanism for the transient emergence of such drug tolerance. In this framework, we formulate an individual-based model and an integro-differential equation model of reversible phenotypic evolution in a cell population exposed to cytotoxic drugs. The outcomes of both models suggest that nongenetic instability, stress-induced adaptation, selection, and the interplay between these mechanisms can push an actively proliferating cell population to transition into a weakly proliferative and drug-tolerant state. Hence, the cell population experiences much less stress in the presence of the drugs and, in the long run, reacquires a proliferative phenotype, due to phenotypic fluctuations and selection pressure. These mechanisms can also reverse epigenetic drug tolerance following drug washout. Our study highlights how the transient appearance of the weakly proliferative and drug-tolerant cells is related to the use of high-dose therapy. Furthermore, we show how stem-like characteristics can act to stabilize the transient, weakly proliferative, and drug-tolerant subpopulation for a longer time window. Finally, using our models as in silico laboratories, we propose new testable hypotheses that could help uncover general principles underlying the emergence of cancer drug tolerance.

The dynamics of p53 in single cells: physiologically based ODE and reaction-diffusion PDE models

The intracellular signalling network of the p53 protein plays important roles in genome protection and the control of cell cycle phase transitions. Recently observed oscillatory behaviour in single cells under stress conditions has inspired several research groups to simulate and study the dynamics of the protein with the aim of gaining a proper understanding of the physiological meanings of the oscillations. We propose compartmental ODE and PDE models of p53 activation and regulation in single cells following DNA damage and we show that the p53 oscillations can be retrieved by plainly involving p53-Mdm2 and ATM-p53-Wip1 negative feedbacks, which are sufficient for oscillations experimentally, with no further need to introduce any delays into the protein responses and without considering additional positive feedback.

Reaction-diffusion systems for spatio-temporal intracellular protein networks: A beginner's guide with two examples

Spatio-temporal dynamics of a variety of proteins is, among other things, regulated by post-translational modifications of these proteins. Such modifications can thus influence stability and biochemical activities of the proteins, activity and stability of their upstream targets within specific signalling pathways. Commonly used mathematical tools for such protein–protein (and/or protein-mRNA) interactions in single cells, namely, Michaelis–Menten and Hill kinetics, yielding a system of ordinary differential equations, are extended here into (non-linear) partial differential equations by taking into account a more realistic spatial representation of the environment where these reactions occur. In the modelling framework under consideration, all interactions occur in a cell divided into two compartments, the nucleus and the cytoplasm, connected by the semipermeable nuclear membrane and bounded by the impermeable cell membrane. Passive transport mechanism, modelled by the so-called Kedem–Katchalsky boundary conditions, is used here to represent migration of species throughout the nuclear membrane. Nonlinear systems of partial differential equations are solved by the semi-implicit Rothe method. Examples of two spatial oscillators are shown. Namely, these are the circadian rhythm for concentration of the FRQ protein in Neurospora crassa and oscillatory dynamics observed in the activation and regulation of the p53 protein following DNA damage in mammalian cells.

The p53 protein and its molecular network: modelling a missing link between DNA damage and cell fate

Various molecular pharmacokinetic-pharmacodynamic (PK-PD) models have been proposed in the last decades to represent and predict drug effects in anticancer chemotherapies. Most of these models are cell population based since clearly measurable effects of drugs can be seen much more easily on populations of cells, healthy and tumour, than in individual cells. The actual targets of drugs are, however, cells themselves. The drugs in use either disrupt genome integrity by causing DNA strand breaks, and consequently initiate programmed cell death, or block cell proliferation mainly by inhibiting factors that enable cells to proceed from one cell cycle phase to the next through checkpoints in the cell division cycle. DNA damage caused by cytotoxic drugs (and also cytostatic drugs at high concentrations) activates, among others, the p53 protein-modulated signalling pathways that directly or indirectly force the cell to make a decision between survival and death. The paper aims to become the first-step in a larger scale enterprise that should bridge the gap between intracellular and population PK-PD models, providing oncologists with a rationale to predict and optimise the effects of anticancer drugs in the clinic. So far, it only sticks at describing p53 activation and regulation in single cells following their exposure to DNA damaging stress agents. We show that p53 oscillations that have been observed in individual cells can be reconstructed and predicted by compartmentalising cellular events occurring after DNA damage, either in the nucleus or in the cytoplasm, and by describing network interactions, using ordinary differential equations (ODEs), between the ATM, p53, Mdm2 and Wip1 proteins, in each compartment, nucleus or cytoplasm, and between the two compartments. This article is part of a Special Issue entitled: Computational Proteomics, Systems Biology & Clinical Implications.

Age-structured cell population model to study the influence of growth factors on cell cycle dynamics

Cell proliferation is controlled by many complex regulatory networks. Our purpose is to analyse, through mathematical modeling, the effects of growth factors on the dynamics of the division cycle in cell populations. Our work is based on an age-structured PDE model of the cell division cycle within a population of cells in a common tissue. Cell proliferation is at its first stages exponential and is thus characterised by its growth exponent, the first eigenvalue of the linear system we consider here, a growth exponent that we will explicitly evaluate from biological data. Moreover, this study relies on recent and innovative imaging data (fluorescence microscopy) that make us able to experimentally determine the parameters of the model and to validate numerical results. This model has allowed us to study the degree of simultaneity of phase transitions within a proliferating cell population and to analyse the role of an increased growth factor concentration in this process. This study thus aims at helping biologists to elicit the impact of growth factor concentration on cell cycle regulation, at making more precise the dynamics of key mechanisms controlling the division cycle in proliferating cell populations, and eventually at establishing theoretical bases for optimised combined anticancer treatments.

Applying ecological and evolutionary theory to cancer: a long and winding road

Since the mid 1970s, cancer has been described as a process of Darwinian evolution, with somatic cellular selection and evolution being the fundamental processes leading to malignancy and its many manifestations (neoangiogenesis, evasion of the immune system, metastasis, and resistance to therapies). Historically, little attention has been placed on applications of evolutionary biology to understanding and controlling neoplastic progression and to prevent therapeutic failures. This is now beginning to change, and there is a growing international interest in the interface between cancer and evolutionary biology. The objective of this introduction is first to describe the basic ideas and concepts linking evolutionary biology to cancer. We then present four major fronts where the evolutionary perspective is most developed, namely laboratory and clinical models, mathematical models, databases, and techniques and assays. Finally, we discuss several of the most promising challenges and future prospects in this interdisciplinary research direction in the war against cancer.

A spatial physiological model for p53 intracellular dynamics

In this paper we design and analyse a physiologically based model representing the accumulation of protein p53 in the nucleus after triggering of ATM by DNA damage. The p53 protein is known to have a central role in the response of the cell to cytotoxic or radiotoxic insults resulting in DNA damage. A reasonable requirement for a model describing intracellular signalling pathways is taking into account the basic feature of eukaryotic cells: the distinction between nucleus and cytoplasm. Our aim is to show, on a simple reaction network describing p53 dynamics, how this basic distinction provides a framework which is able to yield expected oscillatory dynamics without introducing either positive feedbacks or delays in the reactions. Furthermore we prove that oscillations appear only if some spatial constraints are respected, e.g. if the diffusion coefficients correspond to known biological values. Finally we analyse how the spatial features of a cell influence the dynamic response of the p53 network to DNA damage, pointing out that the protein oscillatory dynamics is indeed a response that is robust towards changes with respect to cellular environments. Even if we change the cell shape or its volume or better its ribosomal distribution, we observe that DNA damage yields sustained oscillations of p53.

A combined experimental and mathematical approach for molecular-based optimization of irinotecan circadian delivery

Circadian timing largely modifies efficacy and toxicity of many anticancer drugs. Recent findings suggest that optimal circadian delivery patterns depend on the patient genetic background. We present here a combined experimental and mathematical approach for the design of chronomodulated administration schedules tailored to the patient molecular profile. As a proof of concept we optimized exposure of Caco-2 colon cancer cells to irinotecan (CPT11), a cytotoxic drug approved for the treatment of colorectal cancer. CPT11 was bioactivated into SN38 and its efflux was mediated by ATP-Binding-Cassette (ABC) transporters in Caco-2 cells. After cell synchronization with a serum shock defining Circadian Time (CT) 0, circadian rhythms with a period of 26 h 50 (SD 63 min) were observed in the mRNA expression of clock genes REV-ERBα, PER2, BMAL1, the drug target topoisomerase 1 (TOP1), the activation enzyme carboxylesterase 2 (CES2), the deactivation enzyme UDP-glucuronosyltransferase 1, polypeptide A1 (UGT1A1), and efflux transporters ABCB1, ABCC1, ABCC2 and ABCG2. DNA-bound TOP1 protein amount in presence of CPT11, a marker of the drug PD, also displayed circadian variations. A mathematical model of CPT11 molecular pharmacokinetics-pharmacodynamics (PK-PD) was designed and fitted to experimental data. It predicted that CPT11 bioactivation was the main determinant of CPT11 PD circadian rhythm. We then adopted the therapeutics strategy of maximizing efficacy in non-synchronized cells, considered as cancer cells, under a constraint of maximum toxicity in synchronized cells, representing healthy ones. We considered exposure schemes in the form of an initial concentration of CPT11 given at a particular CT, over a duration ranging from 1 to 27 h. For any dose of CPT11, optimal exposure durations varied from 3h40 to 7h10. Optimal schemes started between CT2h10 and CT2h30, a time interval corresponding to 1h30 to 1h50 before the nadir of CPT11 bioactivation rhythm in healthy cells.

Treatment timing within the 24-h timescale, that is, circadian (circa, about; dies, day) timing, can change by several fold the tolerability and antitumor efficacy of anticancer agents both in experimental models and in cancer patients. Chronotherapeutics aims at improving the tolerability and/or the efficacy of medications through the administration of treatments according to biological rhythms. Recent findings highlight the need of individualizing circadian delivery schedules according to the patient genetic background. In order to address this issue, we propose a combined experimental and mathematical approach in which molecular mathematical models are fitted to experimental measurements of critical biological variables in the studied experimental model or patient. Optimization procedures are then applied to the calibrated mathematical model for the design of theoretically optimal circadian delivery patterns. As a first proof of concept we focused on the anticancer drug irinotecan. A mathematical model of the drug molecular PK-PD was built and fitted to experimental data in Caco-2 colon cancer cells. Numerical algorithms were then applied to theoretically optimize the chronomodulated exposure of Caco-2 cells to irinotecan.

Optimizing cancer pharmacotherapeutics using mathematical modeling and a systems biology approach

Research in mathematics and in mathematical biology on cancer and its treatments has been soaring in the past 10 years at an unprecedented speed. Such thriving is likely due as much to new findings in fundamental biology as to an emerging general interest from mathematicians and engineers towards applications in biology and medicine and to their subsequently designed representations and predictions of tumor processes that are now allowed by modern means of computation and simulation. This article, which does not claim the status of an extended review paper on mathematical models of cancer and its treatment, is focused on modeling in a systems biology perspective. I will list here the most necessary mathematical methods, in my opinion, which, while enforcing already existing methods, should be further developed towards designing and applying optimized individualized treatments of cancer in the clinic.

Modelling the genesis and treatment of cancer: the potential role of physiologically based pharmacodynamics

Physiologically based modelling of pharmacodynamics/toxicodynamics requires an a priori knowledge on the underlying mechanisms causing toxicity or causing the disease. In the context of cancer, the objective of the expert meeting was to discuss the molecular understanding of the disease, modelling approaches used so far to describe the process, preclinical models of cancer treatment and to evaluate modelling approaches developed based on improved knowledge. Molecular events in cancerogenesis can be detected using 'omics' technology, a tool applied in experimental carcinogenesis, but also for diagnostics and prognosis. The molecular understanding forms the basis for new drugs, for example targeting protein kinases specifically expressed in cancer. At present, empirical preclinical models of tumour growth are in great use as the development of physiological models is cost and resource intensive. Although a major challenge in PKPD modelling in oncology patients is the complexity of the system, based in part on preclinical models, successful models have been constructed describing the mechanism of action and providing a tool to establish levels of biomarker associated with efficacy and assisting in defining biologically effective dose range selection for first dose in man. To follow the concentration in the tumour compartment enables to link kinetics and dynamics. In order to obtain a reliable model of tumour growth dynamics and drug effects, specific aspects of the modelling of the concentration-effect relationship in cancer treatment that need to be accounted for include: the physiological/circadian rhythms of the cell cycle; the treatment with combinations and the need to optimally choose appropriate combinations of the multiple agents to study; and the schedule dependence of the response in the clinical situation.

Circadian Timing in Cancer Treatments

The circadian timing system is composed of molecular clocks, which drive 24-h changes in xenobiotic metabolism and detoxification, cell cycle events, DNA repair, apoptosis, and angiogenesis. The cellular circadian clocks are coordinated by endogenous physiological rhythms, so that they tick in synchrony in the host tissues that can be damaged by anticancer agents. As a result, circadian timing can modify 2- to 10-fold the tolerability of anticancer medications in experimental models and in cancer patients. Improved efficacy is also seen when drugs are given near their respective times of best tolerability, due to (a) inherently poor circadian entrainment of tumors and (b) persistent circadian entrainment of healthy tissues. Conversely, host clocks are disrupted whenever anticancer drugs are administered at their most toxic time. On the other hand, circadian disruption accelerates experimental and clinical cancer processes. Gender, circadian physiology, clock genes, and cell cycle critically affect outcome on cancer chronotherapeutics. Mathematical and systems biology approaches currently develop and integrate theoretical, experimental, and technological tools in order to further optimize and personalize the circadian administration of cancer treatments.

Implications of circadian clocks for the rhythmic delivery of cancer therapeutics

The circadian timing system (CTS) controls drug metabolism and cellular proliferation over the 24 hour day through molecular clocks in each cell. These cellular clocks are coordinated by a hypothalamic pacemaker, the suprachiasmatic nuclei, that generates or controls circadian physiology. The CTS plays a role in cancer processes and their treatments through the downregulation of malignant growth and the generation of large and predictable 24 hour changes in toxicity and efficacy of anti-cancer drugs. The tight interactions between circadian clocks, cell division cycle and pharmacology pathways have supported sinusoidal circadian-based delivery of cancer treatments. Such chronotherapeutics have been mostly implemented in patients with metastatic colorectal cancer, the second most common cause of death from cancer. Stochastic and deterministic models of the interactions between circadian clock, cell cycle and pharmacology confirmed the poor therapeutic value of both constant-rate and wrongly timed chronomodulated infusions. An automaton model for the cell cycle revealed the critical roles of variability in circadian entrainment and cell cycle phase durations in healthy tissues and tumours for the success of properly timed circadian delivery schedules. The models showed that additional therapeutic strategy further sets the constraints for the identification of the most effective chronomodulated schedules.

Physiologically based modelling of circadian control on cell proliferation

The molecular circadian clock which is present in almost all cells of animal organisms exerts a control on the cell division cycle in proliferating tissues by modulating the activity of cyclins and cyclin dependent kinases (CDKs), the proteins which determine transitions from one phase of the cell cycle to the following one, until effective division. Each peripheral cell circadian clock is under the synchronising control of a central hypothalamic pacemaker which itself receives inputs, synchronising or disruptive, from external light and from circulating molecules such as cytokines. Principles for modelling these interacting systems are exposed. They rely on age-structured partial differential equations for cell proliferation in a population of cells and ordinary differential equations for the control of cell cycle phase transitions and for the circadian system presented as a network of oscillators with synchronisation and desynchronisation. These physiological cellular systems are coupled together and subject to pharmacological inputs, e.g. from anticancer therapies, which may be synchronised with cell cycle timing by the knowledge of the body circadian clock status, investigated by noninvasive measurements. The output of the controlled cell proliferation is a population growth exponent identifiable by in vivo tissue measurements; it allows to assess the proliferative status of the tissues under investigation, as a function of the circadian clock status, well fit or disrupted, and of pharmacological inputs such as used in anticancer treatments.

Modeling oxaliplatin drug delivery to circadian rhythms in drug metabolism and host tolerance

To make possible the design of optimal (circadian and other period) time-scheduled regimens for cytotoxic drug delivery by intravenous infusion, a pharmacokinetic-pharmacodynamic (PK-PD, with circadian periodic drug dynamics) model of chemotherapy on a population of tumor cells and its tolerance by a population of fast renewing healthy cells is presented. The application chosen for identification of the model parameters is the treatment by oxaliplatin of Glasgow osteosarcoma, a murine tumor, and the healthy cell population is the jejunal mucosa, which is the main target of oxaliplatin toxicity in mice. The model shows the advantage of a periodic time-scheduled regimen, compared to the conventional continuous constant infusion of the same daily dose, when the biological time of peak infusion is correctly chosen. Furthermore, it is well adapted to using mathematical optimization methods of drug infusion flow, choosing tumor population minimization as the objective function and healthy tissue preservation as a constraint. Such a constraint is in clinical settings tunable by physicians by taking into account the patient's state of health.

[Toxicity and anti-tumour efficacy of oxaliplatin on Glasgow osteosarcoma induced in mice: a mathematical model]

To study time-scheduled regimens in the treatment of tumours by cytotoxic drugs delivered by IV injection, we propose a mathematical model of the action of a chemotherapy on the population of tumoral cells on the one hand, on a population of fast renewing healthy cells on the other hand. We chose for model parameter identification the treatment by oxaliplatin of Glasgow Osteosarcoma in mice.

Period shift induction by intermittent stimulation in a Drosophila model of PER protein oscillations

PER protein circadian oscillations in Drosophila have been described by Goldbeter according to a five-dimensional model that includes the possibility of genetic mutation described by changing one parameter, the maximum degradation rate of the PER protein. Assuming that, in a mutant Drosophila this parameter is unreachable, we modify another parameter, the translation rate between the mRNA and the nonphosphorylated form of PER protein, by periodic intermittent activation or inhibition. We show how such a modification, simulated in the model by a periodic, on/off, piecewise constant stimulation (which increases or decreases this parameter) allows the entrainment of oscillations exactly at, or close to, a desired period. In a different context, this suggests that some diseases may be corrected using pharmacological agents according to specific periodic delivery schedules.

Myocardial determinants in regulation of the heart rate

Heart rate is a function of at least three factors located in the sinus node, including the pacemaker and the activity of the sympathetic and vagal pathways. Heart rate varies during breathing and exercising. The is far from being a purely academic question because, after myocardial infarction or in cardiac insufficiency, reduced heart rate variability (HRV) represents the most valuable prognostic factor. HRV is usually considered index of the sympathovagal balance and is explored using time domain analysis, such as spectral analysis. Nevertheless, methods such as the Fast Fourier Transformation are not applicable to small rodents which have an unstable heart rate with asymmetric oscillations. Nonlinear methods show chaotic behavior under some conditions. A time and frequency domain method of analysis, the Wigner-Villé Transform, has been proposed for the study of HRV in both humans and small rodents, as a compromise between linear and nonlinear methods. We developed a method to quantify both arrhythmias and HRV in unanesthetized rodents. Such a method allows study of the relationship between the physiological parameters and the myocardial phenotype. Ventricular premature beats are more frequent in 16-month-old spontaneously hypertensive rats than in age-matched controls. In addition, HRV is attenuated in spontaneously hypertensive rats, as in compensatory cardiac hypertrophy in humans, and such attenuation is considered a prognostic index. Converting enzyme inhibition reduces in parallel arterial hypertension, cardiac hypertrophy, and ventricular fibrosis; it prevents ventricular premature beats and normalizes heart rate variability. It can be demonstrated that the incidence of ventricular premature beats is linked to the myocardial phenotype in terms of both cardiac hypertrophy and fibrosis. The two factors act as independent variables. HRV is correlated with the incidence of arrhythmias, suggesting that the beneficial effects of converting enzyme inhibition are related to prevention of arrhythmias.

Heart rate and heart rate variability, a pharmacological target

Heart rate varies with respiration, blood pressure, emotion, etc., and heart rate variability (HRV) is presently one of the best indices to predict fatal issues in cardiac failure and after myocardial infarction. HRV depends on various reflexes. In addition, parallel studies of HRV and the myocardial adrenergic and muscarinic transduction system in experimental models of cardiac hypertrophy (CH) have suggested that the myocardial phenotype at the sinus-node level may also play a role. A transgenic strain of mice with atrial overexpression of the beta 1-adrenergic receptors was generated with attenuated HRV, which demonstrates that the phenotype itself is a determinant of HRV. HRV is explored by noninvasive techniques, including simple determination of the standard error of the mean, time-domain analysis, and Fourier transformation. We recently developed a time and frequency domain method of analysis, the smoothed pseudo-Wigner-Ville transformation, which allows better exploration of nonstationarity. Nonlinear methods have also been applied due to the extreme complexity of the biological determinants, and have provided evidence of a chaotic attractor in certain conditions. It is proposed that in steady state a very simple process, which is not completely deterministic, could better explain intermit interval regulations than chaotic behavior. In contrast, under extreme circumstances the regulation proceeds using chaotic behavior. Arrhythmias and HRV can be quantitated in 16-month-old unanesthetized spontaneously hypertensive rats (SHR). Ventricular premature beats are more frequent in SHR than in age-matched controls; they disappear after converting enzyme inhibition (CEI) relative to the reduction of both cardiac hypertrophy and ventricular fibrosis. HRV is attenuated in SHR, as it is in compensatory CH in humans. When CH is prevented, HRV returns to normal. CEI is therefore antiarrhythmic. Another pharmacological application of this concept concerns the bradycardic agents that may improve HRV.

Decreased heart rate variability in transgenic mice overexpressing atrial beta 1-adrenoceptors

Heart rate variability (HRV) depends on various reflexes, including the baroreflex or respiratory reflex. Experimental studies have suggested that the sinoatrial node density in G protein-linked receptors may be involved. Transgenic mice, with a specific eightfold atrial overexpression of human beta 1-adrenoceptor (beta 1-AR), have been generated to evaluate the role of the atrial beta 1-AR density on HRV. The heart rate was monitored using telemetry, and the signal was analyzed using a quantitative time-frequency domain analysis, the smoothed pseudo-Wigner-Ville method, and phase portrait maps. 1) Heart rate was unchanged, but the two normal components of HRV were decreased in transgenic mice. Transgenic mice have an unshortened life span and no arrhythmias. 2) Challenge of the animals by propranolol showed no modulation of the HRV in transgenic mice compared with controls. 3) In isolated atrial strips from transgenic mice, basal contractility was increased and there was no isoproterenol-induced inotropic effect. 4) The basal level of adenosine 3',5'-cyclic monophosphate production was lowered in transgenic mice, suggesting a shift in adenylate cyclase isoforms.

Linear and non-linear analyses of heart rate variability: a minireview

To complete traditional time- and frequency-domain analyses, new methods derived from non-linear systems analysis have recently been developed for time series studies. A panel of the most widely used methods of heart rate analysis is given with computations on mouse data, before and after a single atropine injection.

Heart rate and heart rate variability during sleep in small-for-gestational age newborns

To assess the influence of intrauterine growth retardation on heart rate (HR) and HR variability during sleep, we performed polygraphic recordings in 10 small-for-gestational age (SGA) and 16 appropriate-for-gestational age (AGA) newborns. Both groups were clinically and neurologically normal and were at 37 to 41 wk conceptional age. RR intervals were analyzed using the short-time Fourier transform in three frequency bands: 1) high frequency, with a period 3-8 heartbeat; 2) mid frequency, with a period 10-25 heartbeat; and 3) low frequency, with a period 30-100 heartbeat. In both active and quiet sleep, SGA newborns significantly differed from AGA newborns by having a shorter RR interval (p < 0.01) and lower amplitude of HR variability in all bands (p < 0.05) except low frequency in quiet sleep. Quiet sleep differed from active sleep by having a longer RR interval (p < 0.05), higher high-frequency variability (p < 0.02) in both SGA and AGA newborns, and lower low-frequency variability (p < 0.005 for AGA newborns). Our data give evidence of clear modifications of both sympathetic and parasympathetic HR control in the at-risk SGA population. Similarity of between-state characteristics suggests maintained CNS control of HR in SGA as well as in AGA newborns. We speculate that between-group HR and HR variability differences may be related to augmented metabolic rate in SGA compared with AGA newborns.

Heart-rate variability in low-risk prematurely born infants reaching normal term: a comparison with full-term newborns

To investigate the influence of prematurity and postnatal age on the maturation of the autonomic nervous system function, we analysed heart-rate and heart-rate variability in twelve prematurely born infants (< 37 weeks gestational age) reaching the conceptional age of 37-41 weeks. These neonates were compared with sixteen 37-41 week conceptional age newborns (< 10 days postnatal age). Heart-rate variability was analysed by spectral analysis of interbeat intervals using Short-Time Fourier Transform. We found that during both active and quiet sleep, the durations of RR-intervals were shorter and the amplitude of heart-rate variability in different frequency bands was lower in prematures reaching term than in newborns of the same conceptional age (P < 0.001). Between-state comparison showed differences in both groups. In both groups, low-frequency heart-rate variability was higher in active sleep than in quiet sleep. Between-state differences of RR-intervals and high-frequency heart-rate variability were present only in newborns (P < 0.01). Discrimination between newborns and prematures reaching term, based on RR-intervals and heart-rate variability, was correct in both sleep states with errors between 7 to 16%. However, in both newborns and prematures reaching term, between-state discrimination showed less reliable results, especially for quiet sleep discrimination with 24% (in PRT) and 20% (in NB) of errors. Our results, especially information given by factor analysis, suggest that the differences between newborns and prematures reaching term, concerning RR-interval and heart-rate variability, may be related to a changed balance between the sympathetic and parasympathetic nervous systems with a diminished parasympathetic component of heart rate control in prematures reaching term, as compared to newborns.

Heart rate variability in normal sleeping full-term and preterm neonates

To assess maturation of the Autonomic Nervous System (ANS) and sleep states, Heart Rate Variability (HRV) was studied in 24 healthy sleeping newborns, aged from 31 to 41 weeks, conceptional age (CA). Spectral analysis of the interbeat interval (RR) signal, was performed by Short-Time Fourier Transform, in three frequency bands: high (HF), of purely vagal origin, mid (MF), and low (LF), vagal and sympathetic, thus allowing evaluation of both branches of the ANS, observed in Active Sleep (AS = REM Sleep) and in Quiet Sleep (QS = nREM Sleep). Principal Component Analysis, Discriminant Analysis, and hypothesis tests were used to investigate the evolution of spectral variables and their relation with sleep states. HF, MF, LF, and mean RR all increased with age; the differences from the premature to the full-term group, were more marked, as a whole, in AS than in QS. HF showed the highest increase from the premature (31-36 weeks CA) to the intermediate (37-38) group, whereas LF showed equal differences from the premature to the intermediate, and from the intermediate to the full-term (39-41) groups. These results suggest a steep increase in vagal tone at 37-38 weeks CA, with stability afterwards, and a more regular increase in sympathetic tone from 31 to 41 weeks CA.