Organizers
Several dynamic processes play an important role in the response of a cell to therapy. This workshop will focus on the dynamic interplay of biological factors that determine the outcome of chemotherapy of cancer. The key factors are: (a) delivery of therapy to target tumor cells, (b) mechanisms of drug action, (c) growth and differentiation of cell populations, (d) initiation and development of resistance, and (e) optimization of chemotherapy protocols.
 Delivery of therapy to target tumor cells. Over 80% of human cancers are solid tumors. Presentation of a drug to cells in a solid tumor and the accumulation and retention of a drug in tumor cells depend on the drug delivery from the site of administration, the ability of the drug to diffuse through the interstitial space, and the binding of the drug to intracellular macromolecules. Some of these factors are also time and drug concentrationdependent. For example, the interstitial space, which determines the porosity and therefore the diffusion coefficient, may be expanded due to druginduced apoptosis. Mathematical models to depict how these processes affect the drug delivery to tumor cells are useful to identify the treatment regimens that will result in the most effective drug concentration and residence time in the target sites.
 Mechanisms of drug action. Most anticancer drugs act on specific molecular targets that are often involved in the regulation of cell growth, cell differentiation, and cell death. Mathematical models to link the effective drug concentration in the tumor cells with the molecular targets, in a time and concentrationdependent manner, are needed to improve the understanding of drugtarget interaction.
 Mathematical modeling of growth and differentiation of cell populations. This is one of the oldest and best developed topics in biomathematics. It involves modeling of growth and differentiation of laboratory cell populations, of populations of normal cells, and of cell in tumors. Precise mathematical models exist for the processes of haemopoiesis (blood cell production) and selfrenewal of colon epithelium. Mathematical tools used vary from stochastic processes (useful when describing small colonies or early stages of cancer) particularly branching processes, to nonlinear ordinary differential equations (useful for modeling feedbacks of cellproduction systems), to integral equations and partial differential equations (useful for modeling heterogeneous populations). The challenges involve integrating newly described genetic and molecular mechanisms in the models of proliferation, as well as mathematically modelled geometric growth of tumors in various phases (prevascular, vascular, anoxic), and heterogeneity of tumor populations. Mathematical tools needed involve partial differential equations with free boundary, bifurcation in systems of many nonlinear ordinary differential equations, and branching processes with infinite type space.
 Genetic basis, initiation, and development of resistance. Cancer cells are genetically unstable and can acquire genetic and phenotypic changes that permit them to escape cytotoxic insults. Development of drug resistance is a major problem in cancer chemotherapy, and is usually acquired after exposure to a drug. Development of drug resistance is often a function of the frequency, intensity and duration of drug exposure, as well as the chronological age of the cells. These biological parameters can be described in mathematical terms.
 odeling and optimization of chemotherapy protocols. This is an area of potentially great practical importance. Classical models involve populations of normal and cancer cells described as systems of ordinary differential equations with control terms representing treatment intervention. The most common approach involves defining a performance index, which summarizes efficiency of the therapy and damage done to normal (noncancer) cells, and using methods of control theory to find the best value of the index. These models had a lot of appeal in the early days of chemotherapy, when the complexity of tumor cell populations was not entirely appreciated. There exist models taking into account emerging resistance (like the ColdmanGoldie clonal resistance model), and heterogeneity (e.g. geneamplification), but they are based on unrealistic biological hypotheses. Challenges for the field involve more realistic models of drug action and cell proliferation and heterogeneity, as well as new methods for parameter estimation. Mathematical tools needed involve robust optimal control in systems of ordinary differential equations, resonance results for periodic dynamical systems, and control of infinitelydimensional and distributed systems.
The use of mathematical models to describe these biological processes will improve the understanding of the dynamic interplay between these processes and the ability to translate the basic science findings to clinical application. The challenges involved will undoubtedly lead to new mathematical problems and give rise to the development of new mathematical and computational methods.
Accepted Speakers
Monday, November 10, 2003  

Time  Session 
09:15 AM 10:15 AM   Computational Modeling & Cancer Therapy Development: How we used it to shift Paradigms Computational Modeling & Cancer Therapy Development: How we used it to shift Paradigms 
10:30 AM 11:30 AM  Marek Kimmel  Modeling Progression of Lung Cancer: From Genetic Susceptibility to Tumor Growth and Metastasis The talk is an overview of results by the presenter and his colleagues, concerning probabilistic and statistical modeling of lung cancer. The underlying processes studied are (1) carcinogenesis as a random process being function of genetic susceptibility and behavioral factors, (2) tumor growth, with emphasis on stochasticity and ascertainment phenomena, and (3) cancer spread through metastasis. Methodology includes stochastic processes, estimation theory and Monte Carlo simulations. The interplay between underlying biology and medical observations (detection) is discussed. The models presented, beside mathematical and scientific interest, have health policy implications. 
01:30 PM 02:30 PM  John Weinstein  Genomics and Bioinformatics in Cancer Drug Discovery: A Tale of Two Scientific Cultures The first challenge after a microarray or other 'omic' (1,2) experiment is to analyze the data statistically. The second is to interpret the resulting lists of genes biologically. The third is to integrate the data with other types of molecular and pharmacological information ('IntegromicsTM'). We have developed a number of practical software tools for meeting those three challenges: MedMiner (3), which speeds up 510 fold the organization of biomedical literature on genes and drugs; CIMminer (4,5), which flexibly produces Clustered Image Maps ('heat maps'); MatchMiner (6), which translates fluently among the many types of gene and protein identifiers; GoMiner (7), which leverages the Gene Ontology for discovery of functional order in lists of genes; MethMiner, which organizes patterns of sequence information from DNA methylation studies; LeadScope/ LeadMiner (8), which links genomic and proteomic information to the molecular substructures of potential drugs; and AbMiner, a relational database of information on antibodies available for proteomic studies. Development of these computer resources has been motivated in part by our studies of 60 human cancer cell lines (the NCI60) used by the NCI to screen >100,000 chemical compounds since 1990 to find new drugs for cancer therapy. These cells provide detailed information about mechanisms of drug action and resistance (9,10). We and our collaborators have generated multifaceted molecular target profiles of the NCI60 using 2D gel electrophoresis (6), 'reversephase' protein microarrays (11), cDNA microarrays (12,13), Affymetrix oligo chips (14), realtime RTPCR, arrayCGH, SKY, SNP chips, and DNA methylationsequencing. Clinical molecular markers identified are validated by tissue microarray (11). Such integrated databases will have a great impact on cancer drug discovery and individualization (15). In this talk, I will try to provide the necessary elements of background in biology and will emphasize the roles of bioinformatics, biostatistics, and other areas of computational biology in current, cutting edge biomedical research. See http://discover.nci.nih.gov.

03:00 PM 04:00 PM  Edison Liu  Expression Genomics and the Cellular Pharmacology of Cancer Therapeutics Expression Genomics and the Cellular Pharmacology of Cancer Therapeutics 
Tuesday, November 11, 2003  

Time  Session 
09:40 AM 10:20 AM  Steven Kern  Modeling MultipleDrug Interactions with Response Surfaces Drug delivery strategies that maximize positive effects and minimize side effects often employ drug combinations. For cancer chemotherapy, this approach represents the standard of care. There are two primary methods for characterizing pharmacodynamic interactions: isoboles and response surfaces. Isoboles model interactions at a specific level of drug effect. Response surfaces characterize the interaction over a range of effects and are therefore more generally applicable for understanding interactions. Past approaches for modeling response surfaces have presented many problems that limit their generalizability. These include: inability to converge to simple models under constrained conditions, the creation of illogical surfaces, particularly for antagonistic reaction, lack of meaningful parameters that can be compared between different combinations, and the inability to model assymetric interactions surfaces. We have developed a new method for modeling response surfaces of drug interactions that overcome limitations of previous models. Our proposed model is based on a Hill concentrationresponse profile that considers each drug combination as a virtual drug acting in a sigmoid manner. The model use polar coordinates to fit synergistic, additive, and antagonistic interactions as defined by Loewe.[1] We have used simulated data sets to assess the ability of the model to fit a number of different types of drug interactions and have compared these results to other response surface models that have been proposed in the literature. The model was also applied to clinical data for the interaction of the opioid alfentanil with the induction agent propofol that was previously reported by Short et al and also modeled by Minto et al. using response surfaces.[2,3] Aikike Information Criteria (AIC) was used to compare the models. The proposed model has greater flexibility in terms of adequately fitting a number of different interaction conditions from the simulated data. This included asymmetric interactions, competitive antagonistic interactions, and inverse agonist interactions. The model interaction parameter can be statistically assessed to evaluate the significance of the interaction. The proposed model also fit the clinical data well with a comparable AIC to that reported by Minto et al. Further application to antiproliferative agents and leukemia treatments are under way. The flexibility and adequacy of this new model will enhance its application to characterizing the nature and extent of interaction of coadministered drugs. References:

10:40 AM 11:20 AM  Andrzej Swierniak  Phase Specificity and Drug Resistance in Optimal Protocols Design for Cancer Chemotherapy Mathematical modeling of cancer chemotherapy has hadmore than four decades of history. It has contributed to the development of ideas of chemotherapy scheduling, multidrug protocols, and recruitment. It has also helped in the refinement of mathematical tools of control theory applied to the dynamics of cell populations[10]. However, regarding practical results it has been, with minor exceptions, a failure. The reasons for that failure are not always clearly perceived. They stem from the direction of both biomedicine and mathematics: important biological processes are ignored and crucial parameters are not known, but also the mathematical intricacy of the models is not appreciated. In this talk, we would like to outline several directions of research which may play a role in improving the situation and realizing the obvious potential existing in the mathematical approach. We are concerned with three issues:
The main purpose of the talk is to outline our own views on the issues involved. The talk will be in large part a critical survey of published work by us and others. It also includes material not published before. Wherever appropriate, we give credit to others, without attempts at an exhaustive review. The philosophy of this talk is related to our professional experience. The first author has been involved for a decade in attempts to develop a satisfactory theory of optimal control of bilinear systems resulting from a description of chemotherapy action using ordinary differential equations. The second author has spent a similar period in a cancer research institute trying to develop models of the cell cycle for the purpose of estimation of cellcyclephase specificaction of anticancer drugs. More recently, he has investigated gene amplification as the mechanism of resistance of cancer cells.The last two authors have been engaged in mathematical projects on higher order conditions of optimality and recently have used their results to clarify the status of the candidates for optimal protocols worked out by the first two authors. The cellcyclephase specificity is essential for the initial period of chemotherapy, when at issue is the most efficient reduction of the cancer burden. This seems to be of practical importance in nonsurgical cancers such as for example leukemias. Emergence of clones of cancer cells resistant to chemotherapy is important in treatment and prevention of systemic spread of disease. This comprises potential treatment of metastasis and all variants of adjuvant chemotherapy. Cellcyclephase specificity of some cytotoxic drugs is important since itmakes sense to apply anticancer drugs when cells gather in the sensitive phases of the cell cycle. It can be approached by considering dissection of the cell cycle into an increasing number of disjoint compartments, with drug action limited to only some of them. We provide a classification of several simplest models of this kind. Mathematical problems encountered include singularity and nonuniqueness of solutions of the optimization problems. There exist also conceptual problems. One of them is that of the "resonances", postulated by many authors (eg.Dibrov[2], Agur[1]) as the way to either maximize the efficacy of treatment or to spare the organism's normal cells. The emergence of resistance to chemotherapy has been first considered in a point mutation model of Coldman and Goldie[4] and then in the framework of gene amplification by Agur and Harnevo[5]. The main idea is that there exist spontaneous or induced mutations of cancer cells towards drug resistance and that the scheduling of treatment should anticipate these mutations. The point mutation model can be translated into simple recommendations, which have even been recently tested in clinical trials. The gene amplification model[6] was extensively simulated and also resulted in recommendations for optimized therapy. We present a model of chemotherapy based on a stochastic approach to evolution of cancer cells[7]. Asymptotic analysis of this model results in some understanding of its dynamics[11]. This, in our opinion, is the first step towards a more rigorous mathematical treatment of the dynamics of drug resistance and/or metastasis[12]. The simplest cellcyclephase dependent models of chemotherapy can be classified based on the number of compartments and types of drug action modeled[14]. 2 In all these models the attempts at finding optimal controls are confounded by the presence of singular and periodic trajectories, and multiple solutions[13],[15] . However, efficient numerical methods have been developed[3]. Moreover recently singularity of optimal arcs was excluded for a broad class of the models and sufficient conditions for optimal bangbang strategies were found[8],[9]. In simpler cases, it is possible to provide exhaustive classification of solutions. We have reviewed analytic and computational methods which are available. The traditional area of application of ideas of cell synchronization, recruitment and rational scheduling of chemotherapy including multidrug protocols, is in treatment of leukemias[14]. It is there where the cellcyclephase dependent optimization is potentially useful. Concerning the emergence of drug resistance, we have presented the problem in the framework of gene amplification, although much of what is written may apply to different mechanisms which are reversible and occur at high frequency. We have defined a mathematical model which can be used to pose and solve an optimal chemotherapy problem under evolving resistance. We have shown preliminary results regarding dynamics of this model. Analysis of variants of this model should give insight into possible scheduling strategies of chemotherapy in the situations when drug resistance is a significant factor. All possible applications of the mathematical models of chemotherapy are contingent on our ability to estimate their parameters. There has been a progress in that direction, particularly concerning precise estimation of drug action in culture and estimation of cell cycle parameters of tumor cells in vivo. Also, more is known about the mutation rates of evolving resistant cell clones. The emergence of resistant clones is a universal problem of chemotherapy. However, it seems that its most acute manifestation is the failure to treat metastasis. A part of this problem is the imperfect effectiveness of adjuvant chemotherapy as the tool to eradicate undetectable micrometastases. In view of toxicity of anticancer drugs, optimal scheduling is potentially useful in improving these treatments. This research was supported by NSF and Polish Academy of Science in the form of addendum to NSF grant DMS 0205093 for three authors(AS, UL, HS) and by the internal grant BK275/RAu1/03 of SUT for two authors(AS, MK). References

01:30 PM 02:10 PM  Ollivier Hyrien  Analysis of the Effect of an AntiCancer Drug on Cell Proliferation In this talk, we propose a method to analyze the effect of an anticancer drug on the proliferation of oligodendrocytes and O2A progenitor cells in culture conditions. The dynamic of the cell population is represented by a multitype BellmanHarris branching process, which describes the division and differentiation processes as well as the potential action of the drug. A statistical method is also described for quantitative inference from clonal data and the proposed methodologies are illustrated on a real data set. 
02:20 PM 03:00 PM  Jaroslaw Smieja  Models of Cancer Population Evolution Combining MultiDrug Chemotherapy and Drug Resistance A factor that can have a strong influence on the evolution of drug resistance of cancer cells is gene amplification. This process includes an increase in the number of copies of a gene coding for a protein that supports either removal or metabolization of the drug. The more copies of the gene present, the more resistant the cell, in the sense that it can survive under higher concentrations of the drug. Increase of drug resistance by gene amplification has been observed in numerous experiments with in vivo and cultured cell populations. In addition it has been established that, at least in some experimental systems, tumor cells may increase the number of copies of an oncogene in response to unfavorable environment. Mathematical modeling of gene amplification has provided good fits to experimental data. These results suggest that drug resistance and other processes altering the behavior of cancer cells may be better described by multistage mechanisms, including a gradual increase in number of discrete units. The multistage stepwise model of gene amplification or, more generally, of transformations of cancer cells, leads to new mathematical problems and results in novel dynamic properties of the systems involved. The mathematical modeling results suggest that under gene amplification dynamics with high amplification probability, protocols involving frequent lowconcentration dosing may result in the rapid evolution of large fully resistant residual tumors; the same total doses divided into highconcentration doses applied at larger intervals may result in partial or complete remission. Most of existing forms of therapy consist in using several drugs, instead of a single one, since such chemotherapy might reduce drug resistance effects. Then, modelling should take into account increasing drug resistance to each of the used chemotherapeutic agents. Moreover, each drug affects cell being in particular cell phase and it makes sense to combine these drugs so that their cumulative effect on the cancer population would be the greatest. So far, phasespecific chemotherapy has been considered only in the finitedimensional case, without any regard to problems stemming from increasing drug resistance The talk will deal with models that take into account both the phenomenon of gene amplification and multidrug chemotherapy, in their different aspects, so far been studied separately. Combining infinite dimensional model of drug resistance with the multidrug and/or phasespecific model of chemotherapy should move mathematical modelling much closer to its clinical application. Different examples will be discussed, each of them addressing different aspects of cancer cell modelling. As the first one, a model taking into account partial sensitivity of the resistant subpopulation will be introduced. In this case, it is assumed that the resistant subpopulation consists of two parts  one, which is sensitive to the drug (but, contrary to previous works, may contain cells of different drug sensitivity), and another one, completely drugresistant. Subsequently, an attempt to model multidrug (but not phasespecific) protocols will be presented that take into account increasing drug resistance to each used chemotherapeutic agent used. Finally, different cases of phasespecific control of the drugsensitive cancer population will be addressed. 
03:20 PM 04:00 PM  Urszula Ledzewicz , Heinz Schaettler  Mathematical Methods for the Analysis of Optimal Controls in Compartmental Models for Cancer Chemotherapy Mathematical Methods for the Analysis of Optimal Controls in Compartmental Models for Cancer Chemotherapy 
Wednesday, November 12, 2003  

Time  Session 
09:00 AM 09:40 AM  Cynthia Sung  Interspecies Allometric Modeling of the Pharmacokinetics, Biodistribution and Dosimetry of LymphoRad131, a Radiolabeled Cytokine Targeted to B Cells LymphoRad131 (LR131) is iodine131 labeled BLyS protein, a cytokine that binds to B lineage cells, but not T cells, monocytes, natural killer cells or granulocytes. This unique binding profile suggests that LR131 may be a useful treatment for B cell neoplasias such as B cell lymphomas and multiple myeloma. The pharmacokinetics and biodistribution of iodine125 BLyS after intravenous injection into normal and tumorbearing mice will be described. These data were used to predict radiation dosimetry in human subjects by means of interspecies allometric modeling and MIRDose, a program for internal dose assessment in nuclear medicine. Clinical trials of LR131 are currently being conducted in patients with multiple myeloma and nonHodgkin's lymphoma. Whole body gamma scintigraphy is performed on each paitent in order to obtain radiation dosimetry estimates for major organs and tumors. Results from the first cohort of patients will be compared to those predict ed from allometric modeling. 
09:40 AM 10:20 AM  Jan Lankelma  Transport of SmallMolecule Drugs, from Injection Site to the Target After injection into the blood drugs will be cleared from the body, and transported into tissues and sometimes metabolized. As a result after an intravenous bolus injection the blood concentration will decrease and this decrease can be described by pharmacokinetic models, which can be linear or nonlinear models. At the time scales mostly met in these models the blood concentration can be regarded as homogeneous (stirred tank model). Presently, in cancer chemotherapy mostly smallmolecule drugs (mol. wt < 1000 Da) are being used. When compared to large molecules, such as proteins, smallmolecule drugs diffuse relatively fast. However, in some cases, e.g. when tissue components have a high binding capacity, the effective diffusion from the capillary blood vessels can be slow for small molecules, as well. In the tissue the observed drug concentrations can then be different from cell to cell. This was demonstrated for the fluorescent drug doxorubicin in islets of human breast cancer, where concentration gradients were found at 224 h after i.v. injection, with the highest concentrations at the rim of the islet (Lankelma et al. 1999). A mathematical model was developed describing doxorubicin transport by diffusion from the smallest blood capillaries into the tumor tissue (Lankelma et al. 2000). Using transport parameters measured in vitro for doxorubicin, the model could explain the observed gradients. The model showed that the radius of the islet and the width of the interstitium between the cells could have a significant influence on the steepness of the gradient. We could also calculate the drug tissue concentrationversustime profiles at different distances from the rim of the islets, using the blood concentrationversustime profile as a boundary condition. The profiles after an i.v. injection were mimicked in vitro using MCF7 breast cancer cells. Extrapolating to the in vivo situation, the model predicted less druginduced cell damage at the rim when compared to the center of the islets (Lankelma, 2003). Other drugs may also show concentration gradients in tumor tissue (Lankelma, 2002). In the absence of autofluorescence, the presence of gradients can be detected by autoradiography (ex vivo) or potentially by immunohistochemistry of proteins that will be induced by the drug in a concentration dependent way. References:

10:40 AM 11:40 AM  Guill Wientjes  Drug Delivery to Tumors  Determinants and Barriers Drug Delivery to Tumors  Determinants and Barriers 
01:30 PM 02:10 PM  Paolo Ubezio  Kinetics of Cell Cycle Response of Cancer Cells to Drug Treatment Cells respond to a drug challenge by activating programs of cell cycle arrest or suicide (apoptosis). The knowledge of the kinetics of such events in applied research can support the design of rationales of drug scheduling or drug combinations. In basic research it can contribute to the knowledge of the mechanisms of druginduced cell death and of the drug interactions with cell cycle checkpoints. However, no substantial progress has been made on how to describe these effects in quantitative terms. The problem is complicated by the fact that the response to treatment is heterogeneous even in populations of genetically identical cells, like a cell line growing in vitro. Only a fraction of cells (not all) is blocked, some cells repair DNA damage and recycle, some others are killed. Then, the values of these fractions depend on the treatment dose. In order to tackle the complexity of such situation we explored a mixed experimentaltheoretical approach. We used an ovarian carcinoma cell line (IGROV1) growing in vitro and we made measures at different drug concentrations and times with different techniques (particularly by flow cytometry), with a particular experimental design. Then a mathematical model of cellular proliferation kinetics was used to reconstruct the cell flows into the different phases of the cell cycle (G1, S and G2M) after a treatment. The inputs are parameters ("effect descriptors") directly describing the biological effects induced by the treatment, i.e. cell cycle arrest, DNA repair and cell death in G1, S and G2M, in probabilistic terms. The output is a set of values that are equivalent to the measured data, like absolute number of cells or flow cytometric phase percentages, and can be directly compared with them. The aim of the analysis is to find a set (or the sets) of descriptors coherent with the data, i.e. producing simulated measures in the range of precision of the real measures. In case of the coexistence of morethanone scenarios consistent with the data, the discrimination between them is performed experimentally (not mathematically, e.g. with best fit procedures), by additional experiments suggested by the simulation itself. At the end of the procedure, only a single set of parameter values will give the scenario coherent with all experimental measures. This methodology has been successfully applied in studies on classical and new anticancer drugs. 
02:00 PM 03:00 PM  Sandy Anderson  Modelling Solid Tumour Invasion: The Importance of Adhesion The development of a primary solid tumour (e.g., a carcinoma) begins with a single normal cell becoming transformed as a result of mutations in certain key genes (e.g., P53), this leads to uncontrolled proliferation. An individual tumour cell has the potential, over successive divisions, to develop into a cluster (or nodule) of tumour cells consisting of approximately 106 cells. This avascular tumour cannot grow any further, owing to its dependence on diffusion as the only means of receiving nutrients and removing waste products. For any further development to occur the tumour must initiate angiogenesis  the recruitment of blood vessels. After the tumour has become vascularised via the angiogenic network of vessels, it now has the potential to grow further and invade the surrounding tissue. There is now also the possibility of tumour cells finding their way into the circulation and being deposited in distant sites in the body, resulting in metastasis. In this talk we present a hybrid discrete/continuum mathematical model, which describes the invasion of host tissue by tumour cells and examines how changes in key cell attributes (e.g. P53 mutation, cellcell adhesion, invasiveness) affect the tumour's growth. In the model, we focus on four key variables implicated in the invasion process, namely, tumour cells, host tissue (extracellular matrix, ECM), and matrixdegrative enzymes (MDE) associated with the tumour cells and oxygen supplied by the angiogenic network. The continuous mathematical model consists of a system of partial differential equations describing the production and/or activation of degradative enzymes by the tumour cells, the degradation of the matrix, oxygen consumption, and the migratory response of the tumour cells. The hybrid model focuses on the microscale (individual cell) level and uses a discrete technique developed in previous models of angiogenesis. This technique enables one to model migration and invasion at the level of discrete cells whilst still allowing the chemicals (e.g., MDE, ECM, oxygen) to remain continuous. Hence it is possible to include microscale processes both at the cellular level (such as, proliferation, cellcell adhesion) and at the subcellular level (such as, cell mutation properties). This in turn allows us to examine the effects of such microscale changes upon the overall tumour geometry and subsequently the potential for metastatic spread. 
Thursday, November 13, 2003  

Time  Session 
09:00 AM 10:00 AM  Zvia Agur  InterDosing Interval Can Determine Efficacy/toxicity Tradeoff in Cytotoxic and Supportive Cancer Therapy: Prospective Validation of a Mathematical Theory The phenomenon of resonance in population dynamics  enhancement of population growth when the period of the imposed loss process coincides with the inherent periodicity of the population has been applied in many areas of disease control, including African Trypanosomiasis, measles and cancer. The latter application for improving efficacy of chemotherapy, denoted the "ZMethod," has been validated experimentally in mice, suggesting that it is feasible to control cancer load as well as host toxicity by rational drug scheduling. The above concept was further investigated in a comprehensive effort to put forward clinically validated improved cancer treatments. Thus, sets of detailed computerized mathematical models of the full process of tumor progression and of haematopoiesis have been constructed and thoroughly investigated. One of the conclusions is that reducing the dosing interval of standard chemotherapy will increase the efficacy of nonHodgkin's lymphomas (NHL) treatment. Thrombocytopenia was shown to be significantly associated with NHL chemotherapy. Thrombopoietin (TPO), has been developed as a therapeutic agent to attenuate thrombocytopenia, but its immunogenicity is a serious impediment to further pharmaceutical development. To overcome this problem a computerimplemented mathematical model for thrombopoiesis has been employed, predicting that platelet counts, similar to those obtained with accepted TPO dose scheduling, can also be achieved by new schedules, having significantly reduced immunogenicity and improved efficacy. These predictions have been prospectively validated in preclinical trials, thus substantiating the benefit of further TPO development. 
10:30 AM 11:30 AM  Mark Chaplain  Mathematical Modelling of the SpatioTemporal Response of Cytotoxic Tlymphocytes to a Solid Tumour In this talk we will present a mathematical model describing the growth of a solid tumour in the presence of an immune system response. In particular, attention is focussed upon the interaction of tumour cells with socalled tumourinfiltrating cytotoxic lymphocytes (TICLs), in a small, multicellular tumour, without central necrosis and at some stage prior to (tumourinduced) angiogenesis. At this stage the immune cells and the tumour cells are considered to be in a state of dynamic equilibrium (cancer dormancy). The lymphocytes are assumed to migrate into the growing solid tumour and interact with the tumour cells in such a way that lymphocytetumour cell complexes are formed. These complexes result in either the death of the tumour cells (the normal situation) or the inactivation (sometimes even the death) of the lymphocytes. The migration of the TICLs is determined by a combination of random motility and chemotaxis in response to the presence of specialized interleukins (chemokines). The resulting system of four nonlinear partial differential equations (TICLs, tumour cells, complexes and chemokines) is analysed and numerical simulations are presented. The numerical simulations demonstrate the existence of cell distributions that are quasistationary in time but unstable and heterogeneous in space. A linear stability analysis of the underlying (spatially homogeneous) ODE kinetics coupled with a numerical investigation of the ODE system reveals the existence of a stable limit cycle. This is verified further when a subsequent bifurcation analysis is undertaken using a numerical continuation package. These results then explain the complex heterogeneous spatiotemporal dynamics observed in the PDE system. 
01:30 PM 02:10 PM  Gary Schwartz  Development of Cell Cycle Inhibitors in Combination with Chemotherapy for the Treatment of Human Malignancies Despite recent advances in the treatment of some types of metastatic solid tumors, patients still do poorly and cures are quite rare. The ultimate cure of cancer will depend on finding novel ways to kill cancer cells. Cell death proceeds through a process called apoptosis. Apoptosis is tightly regulated by a series of parallel signal transduction pathways: one leading to cellular survival and the other to cell death. The failure of current chemotherapy, in fact, represents the inability to activate those signaling events that direct the tumor cell to its own demise, and/or the inability to interrupt the signaling events that promote tumor cell survival. Therefore, the future of cancer therapy depends on tipping the balance of these tightly regulated reciprocal pathways away from tumor cell survival to cell death. One approach that appears especially promising is to combine chemotherapy with small targeted molecules that enhance chemotherapyinduced apoptosis and result in an increased antitumour effect. Two promising candidate drugs include flavopiridol, a synthetic flavone, and UCN01, 7OHstaurosporine. They have been identified in the NCI drug screen as potent inhibitors of the cyclin dependent kinases (CDK's) and induce cell cycle arrest. Clinically, though, there has been little evidence of single agent activity. However, both drugs potently enhance the induction of apoptosis by a wide range of chemotherapeutic agent. These include irinotecan (CPT11), gemcitabine, cisplatin and docetaxel, as well as radiation. The effects of these combinations are best achieved with sequential therapy, such that the chemotherapy (or radiation) must come before the flavopiridol or the UCN01. For example, Hct116 colon cancer cells can be sensitized to undergo apoptosis in vitro by adding nanomolar concentrations of flavopiridol AFTER treatment with SN38 (the active metabolite of CPT11). Similarly, in vivo, single agent CPT11 induced some tumor regressions but no complete responses (CR) in the Hct116 xenografts. However, CPT11 followed by flavopiridol resulted in over a doubling of tumor regressions and a 30% CR rate. These preclinical studies have been translated into phase I clinical trials of sequential combination therapy. These combinations have proven generally well tolerated and micromolar concentrations of these agents can be achieved. We have seen promising antitumor activity. Thus, this class of drug may provide a completely new therapeutic strategy in the treatment of patients with advanced cancers. (Supported by NCI R01CA67819) 
02:40 PM 03:20 PM  Li Deng  Modeling the Cell Proliferation, Carcinogenesis in Lung Cancer: Taking the Interaction Between Genetic Factors and Smoking into Account A stochastic twostage carcinogenesis model has been widely used to model the mechanism of tumor development for varieties of cancers and some interesting results have been revealed by this approach. In our research, we are focusing on studies of several risk factors' influence on initiation and promotion of lung cancer by applying such a model. We modify a traditional twostage (MVK) model and integrate the environmental exposure, namely cigarette smoking and genetic information into both mutation stages and the cell proliferation rate of intermediate cells. Some experiment data, which measure the cigarette metabolism capacity and DNA repair capacity, enable us to explore the risk of individual's genetic susceptibility in the development of lung cancer. Through the estimates of some important biological parameters, we can make inference on the impact of the several risk factors and their interaction in the carcinogenesis of lung cancer. 
03:20 PM 04:20 PM  Leonid Hanin  Distibution of the Number of Clonogenic Tumor Cells Surviving Fractionated Radiation We solve, under realistic biological assumptions, the following longstanding problem: To find the distribution of the number, N, of clonogenic tumor cells surviving a given arbitrary schedule of fractionated radiation. We show that the distribution of the number N at any time t after treatment belongs to the class of generalized negative binomial distributions, find an explicit computationally feasible formula for the distribution in question, and identify its limiting forms. In particular, for t = 0 the limiting distribution turns out to be Poisson, and an estimate of the rate of convergence in the total variation metric similar to the classical Law of Rare Events is obtained. 
Friday, November 14, 2003  

Time  Session 
09:00 AM 10:00 AM  Alberto Gandolfi  Modelling the Regression and Regrowth of Tumour Cords Following Cell Killing In some human and experimental tumours, cylindrical arrangements of tumour cells growing around central blood vessels and generally surrounded by necrosis have been observed [1]. These structures were called tumour cords. Oxygen and nutrient deprivation are considered to be the main factors in determining the occurrence of necrosis at the cord periphery. In [2], a mathematical model has been developed that describes in cylindrical symmetry and according to the continuum approach the behaviour of a cord under the influence of a cell killing treatment. The diffusion of a chemical critical for cell viability, assumed to be the oxygen, is taken into account. Cells proliferate at a rate depending on the oxygen concentration and become quiescent below a threshold value of this concentration. Massive cell death occurs when the concentration reaches another threshold at a lower value, marking the cord boundary. The model also accounts for both spontaneous and treatment induced cell death within the cord. The necrotic material produced by cell death is removed according to a first order kinetics. Under the assumption that the volume fraction occupied by cells and necrotic material is constant within the cord, the velocity field that describes cell motion is obtained. To describe the effect of chemotherapy, the model has been coupled to a single equation describing drug diffusion from the vessels. The response to different singledose treatments (radiation or drugs), starting from the stationary state of the cord, has been simulated [3]. The model evidences the existence of a transient phase of reoxygenation after treatment, due to cell death and cord shrinkage. Thus, a time window exists in which the surviving cells should exhibit an increased sensitivity to a successive dose of the therapeutic agent.

10:30 AM 11:30 AM  Nicola Bellomo  Multiscale Modelling of Cellular Systems in the Competition between Tumor and Immune System Multiscale Modelling of Cellular Systems in the Competition between Tumor and Immune System 
Name  Affiliation  

Agur, Zvia  agur@imbm.org  Institute for Medical Biomathematics 
Anderson, Alexander  Division of Mathematics, University of Dundee  
Bellomo, Nicola  nicola.bellomo@polito.it  Dipartimento di Matematica, Politecnico di Torino 
Bertuzzi, Alessandro  bertuzzi@iasi.rm.cnr.it  InformaticaCNR, Instituto de Analisi dei Sistemi ed 
Best, Janet  Mathematics, The Ohio State University  
Borisyuk, Alla  borisyuk@mbi.osu.edu  Mathematical Biosciences Institute, The Ohio State University 
Boushaba, Khalid  boushaba@iastate.edu  PDE/Math Biology Group, Iowa State University 
Chaplain, Mark  chaplain@maths.dundee.ac.uk  Mathematics, University of Dundee 
Cracium, Gheorghe  craciun@math.wisc.edu  Dept. of Mathematics, University of WisconsinMadison 
Danthi, Sanjay  danthi.1@osu.edu  Staff Scientist II, Genzyme Corporation 
Deng, Li  deng@microsoft.com  Department of Statistics, Rice University 
Dougherty, Daniel  dpdoughe@mbi.osu.edu  Mathematical Biosciences Institute, The Ohio State University 
Eladdadi, Amina  eladdadi@yahoo.com  Mathematical Sciences, Rensselaer Polytechnic Institute 
Frank, Steve  safrank@uci.edu  Ecology and Evolutionary Biology, University of California, Irvine 
French, Donald  french@math.uc.edu  Department of Mathematical Sciences, University of Cincinnati 
Gandolfi, Alberto  gandolfi@iasi.rm.cnr.it  "A. Ruberti" CNR, Instituto di Analisi dei Sistemi ed Informatica 
Globus, Stephanie  stephglo@mac.com  Department of Mathematics, The Ohio State University 
Goel, Pranay  goelpra@helix.nih.gov  NIDDK, Indian Institute of Science Education and Research 
Gray, Joe  gray@cc.ucsf.edu  University of California, San Diego 
Greco, William  William.Greco@RoswellPark.org  Department of Biostatistics, Roswell Park Cancer Institute 
Grever, Michael  grever1@medctr.osu.edu  Department of Internal Medicine, The Ohio State University 
Guo, JongShenq  jsguo@cc.ntnu.edu.tw  Department of Mathematics, National Taiwan Normal University 
Guo, Yixin  yixin@math.drexel.edu  Department of Mathematics, The Ohio State University 
Hanin, Leonid  hanin@isu.edu  Department of Mathematics, Idaho State University 
Hinow, Peter  peter.hinow@vanderbilt.edu  Department of Mathematics, Vanderbilt University 
Hyrien, Ollivier  Ollivier_Hyrien@URMC.Rochester.edu  Biostatistics & Computational Biology, University of Rochester 
Isaacson, David  isaacd@rpi.edu  Mathematical Sciences, Rensselaer Polytechnic Institute 
Kao, LieJane  ljkao@mbi.osu.edu  Department of Industrial Engineering, DaYeh University 
Karunanayaka, Prasanna  kar4rp@cchmc.org  Dept. of Radiology, Cincinnati Children's Hospital Medical Center 
Kern, Steven  skern@remi.med.utah.edu  Pharmaceutics, Anesthesiology, & Bioeng, University of Utah 
Kimmel, Marek  Department of Statistics, Rice University  
King, John  Theoretical Mechanics Division, University of Nottingham  
Lankelma, Jan  j.lankelma@vumc.nl  Prof. of Tumor Cell Biology, VU Medical Center, #BR230 
Ledzewicz , Urszula  uledzew@siue.edu  Mathematics and Statistics, Southern Illinois University 
Levine, Howard  halevine@iastate.edu  Department of Mathematics, Iowa State University 
Lim, SookKyung  Mathematical Biosciences Institute, The Ohio State University  
Liu, Edison  gisanga@nus.edu.sg  Genome Institute of Singapore 
MarciniakCzochra, Anna  Anna.Marciniak@iwr.uniheidelberg.de  Institute of Applied Mathematics, University of Heidelberg 
Nie, Qing  qnie@math.uci.edu  Mathematical Biosciences Institute, The Ohio State University 
Rejniak, Katarzyna  rejniak@mbi.osu.edu  Mathematical Biosciences Institute, The Ohio State University 
Roe, Rachel  roer@rpi.edu  Mathematical Sciences, Rensselaer Polytechnic Institute 
Sadee, Wolfgang  wolfgang.sadee@osumc.edu  Department of Pharmacology, The Ohio State University 
Sandstede, Bjorn  sandsted@math.ohiostate.edu  Division of Applied Mathematics, Brown University 
Schaettler, Heinz  hms@wustl.edu  Electrical & Systems Engineering, Washington University 
Schwartz, Gary  schwartg@mskcc.org  Gastrointestinal Oncology Service, Memorial SloanKettering Cancer Ctr 
Smieja, Jaroslaw  jsmieja@zeus.polsl.gliwice.pl  Department of Automatic Control, Silesian University of Technology 
Sneyd, James  sneyd@mbi.osu.edu  Mathematics, The University of Auckland 
Sung, Cynthia  cynthia_sung@hgsi.com  Clinical & Preclinical Pharmacology, Human Genome Sciences 
Swierniak, Andrzej  Department of Automatic Control, Silesian University of Technology  
Tan, WaiYuan  waitan@memphis.edu  Mathematical Sciences, University of Memphis 
Terman, David  terman@math.ohiostate.edu  Mathemathics Department, The Ohio State University 
Thomson, Mitchell  Mathematical Biosciences Institute, The Ohio State University  
Tsai, ChihChiang  tsaijc@mbi.osu.edu  Department of Mathematics, National Taiwan Normal University 
Tzafriri, Rami  ramitz@mit.edu  Health Science and Technology, Massachusetts Institute of Technology 
Ubezio, Paolo  ubezio@pop.marionegri.it  Instituto di Ricerche Farmacologiche "Mario 
Wechselberger, Martin  wm@mbi.osu.edu  Mathematical Biosciences Insitute, The Ohio State University 
Weinstein, John  weinstein@dtpax2.ncifcrf.gov  Lab of Molecular Pharmacology, CCR, National Cancer Institute 
Wientjes, Guill  wientjes.1@osu.edu  College of Pharmacy, The Ohio State University 
Wright, Geraldine  School of Biology, Newcastle University  
Yakovlev, Andrei  andrei_yakovlev@urmc.rochester.edu  Biostatistics & Computational Biology, University of Rochester 
You, Yuncheng  you@math.usf.edu  Department of Mathematics, University of South Florida 
The phenomenon of resonance in population dynamics  enhancement of population growth when the period of the imposed loss process coincides with the inherent periodicity of the population has been applied in many areas of disease control, including African Trypanosomiasis, measles and cancer. The latter application for improving efficacy of chemotherapy, denoted the "ZMethod," has been validated experimentally in mice, suggesting that it is feasible to control cancer load as well as host toxicity by rational drug scheduling.
The above concept was further investigated in a comprehensive effort to put forward clinically validated improved cancer treatments. Thus, sets of detailed computerized mathematical models of the full process of tumor progression and of haematopoiesis have been constructed and thoroughly investigated. One of the conclusions is that reducing the dosing interval of standard chemotherapy will increase the efficacy of nonHodgkin's lymphomas (NHL) treatment.
Thrombocytopenia was shown to be significantly associated with NHL chemotherapy. Thrombopoietin (TPO), has been developed as a therapeutic agent to attenuate thrombocytopenia, but its immunogenicity is a serious impediment to further pharmaceutical development. To overcome this problem a computerimplemented mathematical model for thrombopoiesis has been employed, predicting that platelet counts, similar to those obtained with accepted TPO dose scheduling, can also be achieved by new schedules, having significantly reduced immunogenicity and improved efficacy. These predictions have been prospectively validated in preclinical trials, thus substantiating the benefit of further TPO development.
The development of a primary solid tumour (e.g., a carcinoma) begins with a single normal cell becoming transformed as a result of mutations in certain key genes (e.g., P53), this leads to uncontrolled proliferation. An individual tumour cell has the potential, over successive divisions, to develop into a cluster (or nodule) of tumour cells consisting of approximately 106 cells. This avascular tumour cannot grow any further, owing to its dependence on diffusion as the only means of receiving nutrients and removing waste products. For any further development to occur the tumour must initiate angiogenesis  the recruitment of blood vessels. After the tumour has become vascularised via the angiogenic network of vessels, it now has the potential to grow further and invade the surrounding tissue. There is now also the possibility of tumour cells finding their way into the circulation and being deposited in distant sites in the body, resulting in metastasis.
In this talk we present a hybrid discrete/continuum mathematical model, which describes the invasion of host tissue by tumour cells and examines how changes in key cell attributes (e.g. P53 mutation, cellcell adhesion, invasiveness) affect the tumour's growth. In the model, we focus on four key variables implicated in the invasion process, namely, tumour cells, host tissue (extracellular matrix, ECM), and matrixdegrative enzymes (MDE) associated with the tumour cells and oxygen supplied by the angiogenic network. The continuous mathematical model consists of a system of partial differential equations describing the production and/or activation of degradative enzymes by the tumour cells, the degradation of the matrix, oxygen consumption, and the migratory response of the tumour cells. The hybrid model focuses on the microscale (individual cell) level and uses a discrete technique developed in previous models of angiogenesis. This technique enables one to model migration and invasion at the level of discrete cells whilst still allowing the chemicals (e.g., MDE, ECM, oxygen) to remain continuous. Hence it is possible to include microscale processes both at the cellular level (such as, proliferation, cellcell adhesion) and at the subcellular level (such as, cell mutation properties). This in turn allows us to examine the effects of such microscale changes upon the overall tumour geometry and subsequently the potential for metastatic spread.
Multiscale Modelling of Cellular Systems in the Competition between Tumor and Immune System
In this talk we will present a mathematical model describing the growth of a solid tumour in the presence of an immune system response. In particular, attention is focussed upon the interaction of tumour cells with socalled tumourinfiltrating cytotoxic lymphocytes (TICLs), in a small, multicellular tumour, without central necrosis and at some stage prior to (tumourinduced) angiogenesis. At this stage the immune cells and the tumour cells are considered to be in a state of dynamic equilibrium (cancer dormancy). The lymphocytes are assumed to migrate into the growing solid tumour and interact with the tumour cells in such a way that lymphocytetumour cell complexes are formed. These complexes result in either the death of the tumour cells (the normal situation) or the inactivation (sometimes even the death) of the lymphocytes. The migration of the TICLs is determined by a combination of random motility and chemotaxis in response to the presence of specialized interleukins (chemokines). The resulting system of four nonlinear partial differential equations (TICLs, tumour cells, complexes and chemokines) is analysed and numerical simulations are presented. The numerical simulations demonstrate the existence of cell distributions that are quasistationary in time but unstable and heterogeneous in space. A linear stability analysis of the underlying (spatially homogeneous) ODE kinetics coupled with a numerical investigation of the ODE system reveals the existence of a stable limit cycle. This is verified further when a subsequent bifurcation analysis is undertaken using a numerical continuation package. These results then explain the complex heterogeneous spatiotemporal dynamics observed in the PDE system.
A stochastic twostage carcinogenesis model has been widely used to model the mechanism of tumor development for varieties of cancers and some interesting results have been revealed by this approach. In our research, we are focusing on studies of several risk factors' influence on initiation and promotion of lung cancer by applying such a model. We modify a traditional twostage (MVK) model and integrate the environmental exposure, namely cigarette smoking and genetic information into both mutation stages and the cell proliferation rate of intermediate cells. Some experiment data, which measure the cigarette metabolism capacity and DNA repair capacity, enable us to explore the risk of individual's genetic susceptibility in the development of lung cancer. Through the estimates of some important biological parameters, we can make inference on the impact of the several risk factors and their interaction in the carcinogenesis of lung cancer.
In some human and experimental tumours, cylindrical arrangements of tumour cells growing around central blood vessels and generally surrounded by necrosis have been observed [1]. These structures were called tumour cords. Oxygen and nutrient deprivation are considered to be the main factors in determining the occurrence of necrosis at the cord periphery. In [2], a mathematical model has been developed that describes in cylindrical symmetry and according to the continuum approach the behaviour of a cord under the influence of a cell killing treatment. The diffusion of a chemical critical for cell viability, assumed to be the oxygen, is taken into account. Cells proliferate at a rate depending on the oxygen concentration and become quiescent below a threshold value of this concentration. Massive cell death occurs when the concentration reaches another threshold at a lower value, marking the cord boundary. The model also accounts for both spontaneous and treatment induced cell death within the cord. The necrotic material produced by cell death is removed according to a first order kinetics. Under the assumption that the volume fraction occupied by cells and necrotic material is constant within the cord, the velocity field that describes cell motion is obtained. To describe the effect of chemotherapy, the model has been coupled to a single equation describing drug diffusion from the vessels. The response to different singledose treatments (radiation or drugs), starting from the stationary state of the cord, has been simulated [3]. The model evidences the existence of a transient phase of reoxygenation after treatment, due to cell death and cord shrinkage. Thus, a time window exists in which the surviving cells should exhibit an increased sensitivity to a successive dose of the therapeutic agent.
 Tannock, I.F. (1968). The relation between cell proliferation and the vascular system in a transplanted mouse mammary tumour. Br. J. Cancer, 22, 258273.
 Bertuzzi, A., Fasano, A., & Gandolfi, A. A free boundary problem with unilateral constraints describing the evolution of a tumour cord under the influence of cell killing agents. Manuscript submitted for publication.
 Bertuzzi, A., d'Onofrio, A., Fasano, A., & Gandolfi, A. (2003). Regression and regrowth of tumour cords following singledose anticancer treatment. Bull. Math. Biol., 65, 903931.
We solve, under realistic biological assumptions, the following longstanding problem: To find the distribution of the number, N, of clonogenic tumor cells surviving a given arbitrary schedule of fractionated radiation. We show that the distribution of the number N at any time t after treatment belongs to the class of generalized negative binomial distributions, find an explicit computationally feasible formula for the distribution in question, and identify its limiting forms. In particular, for t = 0 the limiting distribution turns out to be Poisson, and an estimate of the rate of convergence in the total variation metric similar to the classical Law of Rare Events is obtained.
In this talk, we propose a method to analyze the effect of an anticancer drug on the proliferation of oligodendrocytes and O2A progenitor cells in culture conditions. The dynamic of the cell population is represented by a multitype BellmanHarris branching process, which describes the division and differentiation processes as well as the potential action of the drug. A statistical method is also described for quantitative inference from clonal data and the proposed methodologies are illustrated on a real data set.
Drug delivery strategies that maximize positive effects and minimize side effects often employ drug combinations. For cancer chemotherapy, this approach represents the standard of care. There are two primary methods for characterizing pharmacodynamic interactions: isoboles and response surfaces. Isoboles model interactions at a specific level of drug effect. Response surfaces characterize the interaction over a range of effects and are therefore more generally applicable for understanding interactions. Past approaches for modeling response surfaces have presented many problems that limit their generalizability. These include: inability to converge to simple models under constrained conditions, the creation of illogical surfaces, particularly for antagonistic reaction, lack of meaningful parameters that can be compared between different combinations, and the inability to model assymetric interactions surfaces. We have developed a new method for modeling response surfaces of drug interactions that overcome limitations of previous models.
Our proposed model is based on a Hill concentrationresponse profile that considers each drug combination as a virtual drug acting in a sigmoid manner. The model use polar coordinates to fit synergistic, additive, and antagonistic interactions as defined by Loewe.[1] We have used simulated data sets to assess the ability of the model to fit a number of different types of drug interactions and have compared these results to other response surface models that have been proposed in the literature. The model was also applied to clinical data for the interaction of the opioid alfentanil with the induction agent propofol that was previously reported by Short et al and also modeled by Minto et al. using response surfaces.[2,3] Aikike Information Criteria (AIC) was used to compare the models.
The proposed model has greater flexibility in terms of adequately fitting a number of different interaction conditions from the simulated data. This included asymmetric interactions, competitive antagonistic interactions, and inverse agonist interactions. The model interaction parameter can be statistically assessed to evaluate the significance of the interaction. The proposed model also fit the clinical data well with a comparable AIC to that reported by Minto et al. Further application to antiproliferative agents and leukemia treatments are under way. The flexibility and adequacy of this new model will enhance its application to characterizing the nature and extent of interaction of coadministered drugs.
References:
 Loewe, S. (1953). The problem of synergism and antagonism of combined drugs. Arzneim. Forsch, 3, 2.
 Short, T.G., Plummer, J.L., Chui, P.T. (1992). Hypnotic and anaesthetic interactions between midazolam, propofol and alfentanil. Br J Anaesth, 69, 1627.
 Minto, C. F., Schnider, T. W., Short, T. G., Gregg, K. M., Gentilini, A., & Shafer, S. L. (2000). Response surfaces for anesthetic drug interactions. Anesthesiology, 92,16031616.
The talk is an overview of results by the presenter and his colleagues, concerning probabilistic and statistical modeling of lung cancer. The underlying processes studied are (1) carcinogenesis as a random process being function of genetic susceptibility and behavioral factors, (2) tumor growth, with emphasis on stochasticity and ascertainment phenomena, and (3) cancer spread through metastasis. Methodology includes stochastic processes, estimation theory and Monte Carlo simulations. The interplay between underlying biology and medical observations (detection) is discussed. The models presented, beside mathematical and scientific interest, have health policy implications.
After injection into the blood drugs will be cleared from the body, and transported into tissues and sometimes metabolized. As a result after an intravenous bolus injection the blood concentration will decrease and this decrease can be described by pharmacokinetic models, which can be linear or nonlinear models. At the time scales mostly met in these models the blood concentration can be regarded as homogeneous (stirred tank model). Presently, in cancer chemotherapy mostly smallmolecule drugs (mol. wt < 1000 Da) are being used. When compared to large molecules, such as proteins, smallmolecule drugs diffuse relatively fast. However, in some cases, e.g. when tissue components have a high binding capacity, the effective diffusion from the capillary blood vessels can be slow for small molecules, as well. In the tissue the observed drug concentrations can then be different from cell to cell. This was demonstrated for the fluorescent drug doxorubicin in islets of human breast cancer, where concentration gradients were found at 224 h after i.v. injection, with the highest concentrations at the rim of the islet (Lankelma et al. 1999).
A mathematical model was developed describing doxorubicin transport by diffusion from the smallest blood capillaries into the tumor tissue (Lankelma et al. 2000). Using transport parameters measured in vitro for doxorubicin, the model could explain the observed gradients. The model showed that the radius of the islet and the width of the interstitium between the cells could have a significant influence on the steepness of the gradient. We could also calculate the drug tissue concentrationversustime profiles at different distances from the rim of the islets, using the blood concentrationversustime profile as a boundary condition.
The profiles after an i.v. injection were mimicked in vitro using MCF7 breast cancer cells. Extrapolating to the in vivo situation, the model predicted less druginduced cell damage at the rim when compared to the center of the islets (Lankelma, 2003).
Other drugs may also show concentration gradients in tumor tissue (Lankelma, 2002). In the absence of autofluorescence, the presence of gradients can be detected by autoradiography (ex vivo) or potentially by immunohistochemistry of proteins that will be induced by the drug in a concentration dependent way.
References:
 Lankelma, J., Dekker, H., Luque, F. R., Luykx, S., Hoekman, K., van der Valk, P., et al. (1999). Doxorubicin gradients in human breast cancer. Clin Cancer Res, 5, 17037.
 Lankelma, J., Fernandez Luque, R., Dekker, H., & Pinedo, H. M. (2003). Simulation model of doxorubicin activity in islets of human breast cancer cells. Biochim Biophys Acta, 1622, 16978.
 Lankelma, J., Fernandez Luque, R., Dekker, H., Schinkel, W., & Pinedo, H. M.. (2000). A mathematical model of drug transport in human breast cancer. Microvasc Res, 59, 14961.
 Lankelma, J. (2002). Tissue transport of anticancer drugs. Curr Pharm Des, 8, 19871993.
Mathematical Methods for the Analysis of Optimal Controls in Compartmental Models for Cancer Chemotherapy
Expression Genomics and the Cellular Pharmacology of Cancer Therapeutics
Mathematical Methods for the Analysis of Optimal Controls in Compartmental Models for Cancer Chemotherapy
Despite recent advances in the treatment of some types of metastatic solid tumors, patients still do poorly and cures are quite rare. The ultimate cure of cancer will depend on finding novel ways to kill cancer cells. Cell death proceeds through a process called apoptosis. Apoptosis is tightly regulated by a series of parallel signal transduction pathways: one leading to cellular survival and the other to cell death. The failure of current chemotherapy, in fact, represents the inability to activate those signaling events that direct the tumor cell to its own demise, and/or the inability to interrupt the signaling events that promote tumor cell survival. Therefore, the future of cancer therapy depends on tipping the balance of these tightly regulated reciprocal pathways away from tumor cell survival to cell death.
One approach that appears especially promising is to combine chemotherapy with small targeted molecules that enhance chemotherapyinduced apoptosis and result in an increased antitumour effect. Two promising candidate drugs include flavopiridol, a synthetic flavone, and UCN01, 7OHstaurosporine. They have been identified in the NCI drug screen as potent inhibitors of the cyclin dependent kinases (CDK's) and induce cell cycle arrest. Clinically, though, there has been little evidence of single agent activity. However, both drugs potently enhance the induction of apoptosis by a wide range of chemotherapeutic agent. These include irinotecan (CPT11), gemcitabine, cisplatin and docetaxel, as well as radiation. The effects of these combinations are best achieved with sequential therapy, such that the chemotherapy (or radiation) must come before the flavopiridol or the UCN01. For example, Hct116 colon cancer cells can be sensitized to undergo apoptosis in vitro by adding nanomolar concentrations of flavopiridol AFTER treatment with SN38 (the active metabolite of CPT11). Similarly, in vivo, single agent CPT11 induced some tumor regressions but no complete responses (CR) in the Hct116 xenografts. However, CPT11 followed by flavopiridol resulted in over a doubling of tumor regressions and a 30% CR rate.
These preclinical studies have been translated into phase I clinical trials of sequential combination therapy. These combinations have proven generally well tolerated and micromolar concentrations of these agents can be achieved. We have seen promising antitumor activity. Thus, this class of drug may provide a completely new therapeutic strategy in the treatment of patients with advanced cancers. (Supported by NCI R01CA67819)
A factor that can have a strong influence on the evolution of drug resistance of cancer cells is gene amplification. This process includes an increase in the number of copies of a gene coding for a protein that supports either removal or metabolization of the drug. The more copies of the gene present, the more resistant the cell, in the sense that it can survive under higher concentrations of the drug. Increase of drug resistance by gene amplification has been observed in numerous experiments with in vivo and cultured cell populations. In addition it has been established that, at least in some experimental systems, tumor cells may increase the number of copies of an oncogene in response to unfavorable environment. Mathematical modeling of gene amplification has provided good fits to experimental data. These results suggest that drug resistance and other processes altering the behavior of cancer cells may be better described by multistage mechanisms, including a gradual increase in number of discrete units. The multistage stepwise model of gene amplification or, more generally, of transformations of cancer cells, leads to new mathematical problems and results in novel dynamic properties of the systems involved. The mathematical modeling results suggest that under gene amplification dynamics with high amplification probability, protocols involving frequent lowconcentration dosing may result in the rapid evolution of large fully resistant residual tumors; the same total doses divided into highconcentration doses applied at larger intervals may result in partial or complete remission. Most of existing forms of therapy consist in using several drugs, instead of a single one, since such chemotherapy might reduce drug resistance effects. Then, modelling should take into account increasing drug resistance to each of the used chemotherapeutic agents. Moreover, each drug affects cell being in particular cell phase and it makes sense to combine these drugs so that their cumulative effect on the cancer population would be the greatest. So far, phasespecific chemotherapy has been considered only in the finitedimensional case, without any regard to problems stemming from increasing drug resistance The talk will deal with models that take into account both the phenomenon of gene amplification and multidrug chemotherapy, in their different aspects, so far been studied separately. Combining infinite dimensional model of drug resistance with the multidrug and/or phasespecific model of chemotherapy should move mathematical modelling much closer to its clinical application. Different examples will be discussed, each of them addressing different aspects of cancer cell modelling. As the first one, a model taking into account partial sensitivity of the resistant subpopulation will be introduced. In this case, it is assumed that the resistant subpopulation consists of two parts  one, which is sensitive to the drug (but, contrary to previous works, may contain cells of different drug sensitivity), and another one, completely drugresistant. Subsequently, an attempt to model multidrug (but not phasespecific) protocols will be presented that take into account increasing drug resistance to each used chemotherapeutic agent used. Finally, different cases of phasespecific control of the drugsensitive cancer population will be addressed.
LymphoRad131 (LR131) is iodine131 labeled BLyS protein, a cytokine that binds to B lineage cells, but not T cells, monocytes, natural killer cells or granulocytes. This unique binding profile suggests that LR131 may be a useful treatment for B cell neoplasias such as B cell lymphomas and multiple myeloma. The pharmacokinetics and biodistribution of iodine125 BLyS after intravenous injection into normal and tumorbearing mice will be described. These data were used to predict radiation dosimetry in human subjects by means of interspecies allometric modeling and MIRDose, a program for internal dose assessment in nuclear medicine. Clinical trials of LR131 are currently being conducted in patients with multiple myeloma and nonHodgkin's lymphoma. Whole body gamma scintigraphy is performed on each paitent in order to obtain radiation dosimetry estimates for major organs and tumors. Results from the first cohort of patients will be compared to those predict ed from allometric modeling.
Mathematical modeling of cancer chemotherapy has hadmore than four decades of history. It has contributed to the development of ideas of chemotherapy scheduling, multidrug protocols, and recruitment. It has also helped in the refinement of mathematical tools of control theory applied to the dynamics of cell populations[10]. However, regarding practical results it has been, with minor exceptions, a failure. The reasons for that failure are not always clearly perceived. They stem from the direction of both biomedicine and mathematics: important biological processes are ignored and crucial parameters are not known, but also the mathematical intricacy of the models is not appreciated. In this talk, we would like to outline several directions of research which may play a role in improving the situation and realizing the obvious potential existing in the mathematical approach. We are concerned with three issues:
 The inner structure of the cell cycle and the cellcyclephase specificity of some chemotherapy agents.
 The dynamics of emergence of resistance of cancer cells to chemotherapy, as understood based on recent progress in molecular biology.
 Estimation of quantitative parameters of the cell cycle, drug action and cell mutation to resistance.
The main purpose of the talk is to outline our own views on the issues involved. The talk will be in large part a critical survey of published work by us and others. It also includes material not published before. Wherever appropriate, we give credit to others, without attempts at an exhaustive review.
The philosophy of this talk is related to our professional experience. The first author has been involved for a decade in attempts to develop a satisfactory theory of optimal control of bilinear systems resulting from a description of chemotherapy action using ordinary differential equations. The second author has spent a similar period in a cancer research institute trying to develop models of the cell cycle for the purpose of estimation of cellcyclephase specificaction of anticancer drugs. More recently, he has investigated gene amplification as the mechanism of resistance of cancer cells.The last two authors have been engaged in mathematical projects on higher order conditions of optimality and recently have used their results to clarify the status of the candidates for optimal protocols worked out by the first two authors.
The cellcyclephase specificity is essential for the initial period of chemotherapy, when at issue is the most efficient reduction of the cancer burden. This seems to be of practical importance in nonsurgical cancers such as for example leukemias. Emergence of clones of cancer cells resistant to chemotherapy is important in treatment and prevention of systemic spread of disease. This comprises potential treatment of metastasis and all variants of adjuvant chemotherapy.
Cellcyclephase specificity of some cytotoxic drugs is important since itmakes sense to apply anticancer drugs when cells gather in the sensitive phases of the cell cycle. It can be approached by considering dissection of the cell cycle into an increasing number of disjoint compartments, with drug action limited to only some of them. We provide a classification of several simplest models of this kind. Mathematical problems encountered include singularity and nonuniqueness of solutions of the optimization problems. There exist also conceptual problems. One of them is that of the "resonances", postulated by many authors (eg.Dibrov[2], Agur[1]) as the way to either maximize the efficacy of treatment or to spare the organism's normal cells.
The emergence of resistance to chemotherapy has been first considered in a point mutation model of Coldman and Goldie[4] and then in the framework of gene amplification by Agur and Harnevo[5]. The main idea is that there exist spontaneous or induced mutations of cancer cells towards drug resistance and that the scheduling of treatment should anticipate these mutations. The point mutation model can be translated into simple recommendations, which have even been recently tested in clinical trials. The gene amplification model[6] was extensively simulated and also resulted in recommendations for optimized therapy. We present a model of chemotherapy based on a stochastic approach to evolution of cancer cells[7]. Asymptotic analysis of this model results in some understanding of its dynamics[11]. This, in our opinion, is the first step towards a more rigorous mathematical treatment of the dynamics of drug resistance and/or metastasis[12].
The simplest cellcyclephase dependent models of chemotherapy can be classified based on the number of compartments and types of drug action modeled[14]. 2 In all these models the attempts at finding optimal controls are confounded by the presence of singular and periodic trajectories, and multiple solutions[13],[15] . However, efficient numerical methods have been developed[3]. Moreover recently singularity of optimal arcs was excluded for a broad class of the models and sufficient conditions for optimal bangbang strategies were found[8],[9]. In simpler cases, it is possible to provide exhaustive classification of solutions. We have reviewed analytic and computational methods which are available. The traditional area of application of ideas of cell synchronization, recruitment and rational scheduling of chemotherapy including multidrug protocols, is in treatment of leukemias[14]. It is there where the cellcyclephase dependent optimization is potentially useful.
Concerning the emergence of drug resistance, we have presented the problem in the framework of gene amplification, although much of what is written may apply to different mechanisms which are reversible and occur at high frequency. We have defined a mathematical model which can be used to pose and solve an optimal chemotherapy problem under evolving resistance. We have shown preliminary results regarding dynamics of this model. Analysis of variants of this model should give insight into possible scheduling strategies of chemotherapy in the situations when drug resistance is a significant factor. All possible applications of the mathematical models of chemotherapy are contingent on our ability to estimate their parameters. There has been a progress in that direction, particularly concerning precise estimation of drug action in culture and estimation of cell cycle parameters of tumor cells in vivo. Also, more is known about the mutation rates of evolving resistant cell clones. The emergence of resistant clones is a universal problem of chemotherapy. However, it seems that its most acute manifestation is the failure to treat metastasis. A part of this problem is the imperfect effectiveness of adjuvant chemotherapy as the tool to eradicate undetectable micrometastases. In view of toxicity of anticancer drugs, optimal scheduling is potentially useful in improving these treatments.
This research was supported by NSF and Polish Academy of Science in the form of addendum to NSF grant DMS 0205093 for three authors(AS, UL, HS) and by the internal grant BK275/RAu1/03 of SUT for two authors(AS, MK).
References
 Agur, Z. (1988). The effect of drug schedule on responsiveness to chemotherapy. Annals N.Y.Acad.Sci., 504, 274277.
 Dibrov, B.F., Zhabotinsky, A.M., Neyfakh, L.A., Orlova, H.P., & Churikova, L.I. (1985). Mathematical model of cancer chemotherapy. Periodic schedules of phasespecific cytotoxicagent administration increasing the selectivity of therapy. Math.Biosci., 73, 131.
 Duda, Z. (1994). Evaluation of some optimal chemotherapy protocols by using gradient method. Appl.Math.and Comp.Sci.,special issue: Control and 3 Modelling of Cancer Cell Population, 4, 257263.
 Goldie, J.H. & Coldman, A.J. (1978). A mathematical model for relating the drug sensitivity of tumors to their spontaneous mutation rate. Cancer Treat. Rep., 63, 17271733.
 Harnevo, L.E. & Agur, Z. (1991). The dynamics of gene amplification described as a multitype compartmental model and as a branching process. Math. Biosci., 103, 115138.
 Kimmel, M. & Axelrod, D.E. (1990) Mathematical models of gene amplification with applications to cellular drug resistance and tumorigenicity. Genetics, 125, 633644.
 Kimmel, M., Swierniak, A., & Polanski, A. (1998). Infinite dimensional model of evolution of drug resistance of cancer cells. J.Mathematical Systems, Estimation and Control, 8, 116.
 Ledzewicz, U. & Schattler, H. (2002). Optimal bangbang controls for a 2 compartment model in cancer chemotherapy. J. of Optimization Theory and Applications, 114, 609637.
 Ledzewicz, U. & Schattler, H. (2002). Analysis of a cellcycle specific model for cancer chemotherapy. J. of Biological Systems, 10,183206.
 Swan, G.W. (1990). Role of optimal control theory in cancer chemotherapy. Math. Biosci., 101, 237284.
 Swierniak, A., Polanski, A., Kimmel, M., Bobrowski, A., & Smieja, J. (1999). Qualitative analysis of controlled drug resistance model  inverse Laplace and semigroup approach. Control and Cybernetics, 28, 6175.
 Swierniak, A. & Smieja, J. (2001). Cancer chemotherapy optimization under evolving drug resistance. Nonlinear Analysis, 47, 375386.
 Swierniak, A. & Duda, Z. (1994). Singularity of optimal control problems arising in cancer chemotherapy. Math.and Comp.Modeling, 19, 255262.
 Swierniak, A. & Kimmel, M. (1984). Optimal control application to leukemia chemotherapy protocols design. ZN Pol.Sl., s.Autom., 74, 261277 (in Polish).
 Swierniak, A. & Polanski, A. (1994). Irregularity of optimal control problem in scheduling of cancer chemotherapy. Appl.Math.and Comp.Sci., 4, 263271.
 Swierniak, A., Polanski, A., & Kimmel, M. (1996). Optimal control problems arising in cellcyclespecific cancer chemotherapy. Cell Proliferation, 29, 1 23.
Cells respond to a drug challenge by activating programs of cell cycle arrest or suicide (apoptosis). The knowledge of the kinetics of such events in applied research can support the design of rationales of drug scheduling or drug combinations. In basic research it can contribute to the knowledge of the mechanisms of druginduced cell death and of the drug interactions with cell cycle checkpoints.
However, no substantial progress has been made on how to describe these effects in quantitative terms. The problem is complicated by the fact that the response to treatment is heterogeneous even in populations of genetically identical cells, like a cell line growing in vitro. Only a fraction of cells (not all) is blocked, some cells repair DNA damage and recycle, some others are killed. Then, the values of these fractions depend on the treatment dose. In order to tackle the complexity of such situation we explored a mixed experimentaltheoretical approach. We used an ovarian carcinoma cell line (IGROV1) growing in vitro and we made measures at different drug concentrations and times with different techniques (particularly by flow cytometry), with a particular experimental design. Then a mathematical model of cellular proliferation kinetics was used to reconstruct the cell flows into the different phases of the cell cycle (G1, S and G2M) after a treatment. The inputs are parameters ("effect descriptors") directly describing the biological effects induced by the treatment, i.e. cell cycle arrest, DNA repair and cell death in G1, S and G2M, in probabilistic terms. The output is a set of values that are equivalent to the measured data, like absolute number of cells or flow cytometric phase percentages, and can be directly compared with them. The aim of the analysis is to find a set (or the sets) of descriptors coherent with the data, i.e. producing simulated measures in the range of precision of the real measures. In case of the coexistence of morethanone scenarios consistent with the data, the discrimination between them is performed experimentally (not mathematically, e.g. with best fit procedures), by additional experiments suggested by the simulation itself. At the end of the procedure, only a single set of parameter values will give the scenario coherent with all experimental measures.
This methodology has been successfully applied in studies on classical and new anticancer drugs.
The first challenge after a microarray or other 'omic' (1,2) experiment is to analyze the data statistically. The second is to interpret the resulting lists of genes biologically. The third is to integrate the data with other types of molecular and pharmacological information ('IntegromicsTM'). We have developed a number of practical software tools for meeting those three challenges: MedMiner (3), which speeds up 510 fold the organization of biomedical literature on genes and drugs; CIMminer (4,5), which flexibly produces Clustered Image Maps ('heat maps'); MatchMiner (6), which translates fluently among the many types of gene and protein identifiers; GoMiner (7), which leverages the Gene Ontology for discovery of functional order in lists of genes; MethMiner, which organizes patterns of sequence information from DNA methylation studies; LeadScope/ LeadMiner (8), which links genomic and proteomic information to the molecular substructures of potential drugs; and AbMiner, a relational database of information on antibodies available for proteomic studies.
Development of these computer resources has been motivated in part by our studies of 60 human cancer cell lines (the NCI60) used by the NCI to screen >100,000 chemical compounds since 1990 to find new drugs for cancer therapy. These cells provide detailed information about mechanisms of drug action and resistance (9,10). We and our collaborators have generated multifaceted molecular target profiles of the NCI60 using 2D gel electrophoresis (6), 'reversephase' protein microarrays (11), cDNA microarrays (12,13), Affymetrix oligo chips (14), realtime RTPCR, arrayCGH, SKY, SNP chips, and DNA methylationsequencing. Clinical molecular markers identified are validated by tissue microarray (11). Such integrated databases will have a great impact on cancer drug discovery and individualization (15). In this talk, I will try to provide the necessary elements of background in biology and will emphasize the roles of bioinformatics, biostatistics, and other areas of computational biology in current, cutting edge biomedical research. See http://discover.nci.nih.gov.
 Weinstein. (1998). Science, 282, 628.
 Weinstein, & Curr. (2002). Opinion in Pharmacol., 2, 361.
 Tanabe, et al. (1999). BioTechniques, 27, 1210.
 Weinstein, et al. (1997). Science, 275, 343.
 Myers, et al. (1997). Electrophoresis, 18, 647.
 Bussey. (2003). Genome Biology, 4, R27.
 Zeeberg. (2003). Genome Biol., 4, R28.
 Blower, Jr. (2002). The Pharmacogenomics Journal (Nature), 2, 259.
 Paull, et al. (1989). J. Natl. Cancer Inst., 81, 1088.
 Weinstein, et al. (1992). Science, 258, 343.
 Nishizuka, et al. (2003). Cancer Res., 65, 5243.
 Ross, et al. (2000). Nature Genetics, 24, 227.
 Scherf, et al. (2000). Nature Genetics, 24, 236.
 Staunton, et al. (2001). Proc. Natl. Acad. Sci. U.S.A., 98, 10787.
 Reinhold. (2003). Cancer Res., 63, 1000.
Drug Delivery to Tumors  Determinants and Barriers