## Workshop 2: Mathematical Models of Cell Proliferation and Cancer Chemotherapy

### Organizers

Marek Kimmel
Department of Statistics, Rice University

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.

1. 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- concentration-dependent. For example, the interstitial space, which determines the porosity and therefore the diffusion coefficient, may be expanded due to drug-induced 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.
2. 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 concentration-dependent manner, are needed to improve the understanding of drug-target interaction.
3. 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 self-renewal 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 cell-production 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.
4. 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.
5. 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 (non-cancer) 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 Coldman-Goldie clonal resistance model), and heterogeneity (e.g. gene-amplification), 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 infinitely-dimensional 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

Zvia Agur
Institute for Medical Biomathematics
Sandy Anderson
Division of Mathematics, University of Dundee
Nicola Bellomo
Dipartimento di Matematica, Politecnico di Torino
Mark Chaplain
Mathematics, University of Dundee
Li Deng
Department of Statistics, Rice University
Alberto Gandolfi
"A. Ruberti" CNR, Instituto di Analisi dei Sistemi ed Informatica
Joe Gray
University of California, San Diego
Leonid Hanin
Department of Mathematics, Idaho State University
Ollivier Hyrien
Biostatistics & Computational Biology, University of Rochester
Steven Kern
Pharmaceutics, Anesthesiology, & Bioeng, University of Utah
Jan Lankelma
Prof. of Tumor Cell Biology, VU Medical Center, #BR230
Gary Schwartz
Gastrointestinal Oncology Service, Memorial Sloan-Kettering Cancer Ctr
Jaroslaw Smieja
Department of Automatic Control, Silesian University of Technology
Cynthia Sung
Clinical & Pre-clinical Pharmacology, Human Genome Sciences
Andrzej Swierniak
Department of Automatic Control, Silesian University of Technology
Paolo Ubezio
Instituto di Ricerche Farmacologiche "Mario
John Weinstein
Lab of Molecular Pharmacology, CCR, National Cancer Institute
Guill Wientjes
College of Pharmacy, The Ohio State University
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 5-10 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 NCI-60) 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 multi-faceted molecular target profiles of the NCI-60 using 2-D gel electrophoresis (6), 'reverse-phase' protein microarrays (11), cDNA microarrays (12,13), Affymetrix oligo chips (14), real-time RT-PCR, array-CGH, SKY, SNP chips, and DNA methylation-sequencing. 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.

1. Weinstein. (1998). Science, 282, 628.

2. Weinstein, & Curr. (2002). Opinion in Pharmacol., 2, 361.

3. Tanabe, et al. (1999). BioTechniques, 27, 1210.

4. Weinstein, et al. (1997). Science, 275, 343.

5. Myers, et al. (1997). Electrophoresis, 18, 647.

6. Bussey. (2003). Genome Biology, 4, R27.

7. Zeeberg. (2003). Genome Biol., 4, R28.

8. Blower, Jr. (2002). The Pharmacogenomics Journal (Nature), 2, 259.

9. Paull, et al. (1989). J. Natl. Cancer Inst., 81, 1088.

10. Weinstein, et al. (1992). Science, 258, 343.

11. Nishizuka, et al. (2003). Cancer Res., 65, 5243.

12. Ross, et al. (2000). Nature Genetics, 24, 227.

13. Scherf, et al. (2000). Nature Genetics, 24, 236.

14. Staunton, et al. (2001). Proc. Natl. Acad. Sci. U.S.A., 98, 10787.

15. Reinhold. (2003). Cancer Res., 63, 1000.

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 Multiple-Drug 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 concentration-response 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 co-administered drugs.

References:

1. Loewe, S. (1953). The problem of synergism and antagonism of combined drugs. Arzneim. Forsch, 3, 2.

2. Short, T.G., Plummer, J.L., Chui, P.T. (1992). Hypnotic and anaesthetic interactions between midazolam, propofol and alfentanil. Br J Anaesth, 69, 162-7.

3. 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, 92,1603-1616.

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:

1. The inner structure of the cell cycle and the cell-cycle-phase specificity of some chemotherapy agents.

2. The dynamics of emergence of resistance of cancer cells to chemotherapy, as understood based on recent progress in molecular biology.

3. 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 cell-cycle-phase 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 cell-cycle-phase 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.

Cell-cycle-phase 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 non-uniqueness 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 cell-cycle-phase 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 bang-bang 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 cell-cycle-phase 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

1. Agur, Z. (1988). The effect of drug schedule on responsiveness to chemotherapy. Annals N.Y.Acad.Sci., 504, 274-277.

2. Dibrov, B.F., Zhabotinsky, A.M., Neyfakh, L.A., Orlova, H.P., & Churikova, L.I. (1985). Mathematical model of cancer chemotherapy. Periodic schedules of phase-specific cytotoxic-agent administration increasing the selectivity of therapy. Math.Biosci., 73, 1-31.

3. 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, 257-263.

4. 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, 1727-1733.

5. 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, 115-138.

6. Kimmel, M. & Axelrod, D.E. (1990) Mathematical models of gene amplification with applications to cellular drug resistance and tumorigenicity. Genetics, 125, 633-644.

7. 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, 1-16.

8. Ledzewicz, U. & Schattler, H. (2002). Optimal bang-bang controls for a 2- compart-ment model in cancer chemotherapy. J. of Optimization Theory and Applications, 114, 609-637.

9. Ledzewicz, U. & Schattler, H. (2002). Analysis of a cell-cycle specific model for cancer chemotherapy. J. of Biological Systems, 10,183-206.

10. Swan, G.W. (1990). Role of optimal control theory in cancer chemotherapy. Math. Biosci., 101, 237-284.

11. 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, 61-75.

12. Swierniak, A. & Smieja, J. (2001). Cancer chemotherapy optimization under evolving drug resistance. Nonlinear Analysis, 47, 375-386.

13. Swierniak, A. & Duda, Z. (1994). Singularity of optimal control problems arising in cancer chemotherapy. Math.and Comp.Modeling, 19, 255-262.

14. Swierniak, A. & Kimmel, M. (1984). Optimal control application to leukemia chemotherapy protocols design. ZN Pol.Sl., s.Autom., 74, 261-277 (in Polish).

15. Swierniak, A. & Polanski, A. (1994). Irregularity of optimal control problem in scheduling of cancer chemotherapy. Appl.Math.and Comp.Sci., 4, 263-271.

16. Swierniak, A., Polanski, A., & Kimmel, M. (1996). Optimal control problems arising in cell-cycle-specific cancer chemotherapy. Cell Proliferation, 29, 1- 23.

01:30 PM
02:10 PM
Ollivier Hyrien - Analysis of the Effect of an Anti-Cancer Drug on Cell Proliferation

In this talk, we propose a method to analyze the effect of an anti-cancer drug on the proliferation of oligodendrocytes and O-2A progenitor cells in culture conditions. The dynamic of the cell population is represented by a multitype Bellman-Harris 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 Multi-Drug 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 low-concentration dosing may result in the rapid evolution of large fully resistant residual tumors; the same total doses divided into high-concentration 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, phase-specific chemotherapy has been considered only in the finite-dimensional 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 phase-specific 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 drug-resistant. Subsequently, an attempt to model multidrug (but not phase-specific) protocols will be presented that take into account increasing drug resistance to each used chemotherapeutic agent used. Finally, different cases of phase-specific control of the drug-sensitive 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 LymphoRad-131, a Radiolabeled Cytokine Targeted to B Cells

LymphoRad-131 (LR131) is iodine-131 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 iodine-125 BLyS after intravenous injection into normal and tumor-bearing 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 LR-131 are currently being conducted in patients with multiple myeloma and non-Hodgkin'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 Small-Molecule 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 small-molecule drugs (mol. wt < 1000 Da) are being used. When compared to large molecules, such as proteins, small-molecule 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 2-24 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 concentration-versus-time profiles at different distances from the rim of the islets, using the blood concentration-versus-time profile as a boundary condition.

The profiles after an i.v. injection were mimicked in vitro using MCF-7 breast cancer cells. Extrapolating to the in vivo situation, the model predicted less drug-induced 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:

1. 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, 1703-7.

2. 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, 169-78.

3. 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, 149-61.

4. Lankelma, J. (2002). Tissue transport of anti-cancer drugs. Curr Pharm Des, 8, 1987-1993.

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 drug-induced 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 experimental-theoretical approach. We used an ovarian carcinoma cell line (IGROV-1) 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 more-than-one 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, cell-cell 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 matrix-degrative 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 micro-scale (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 micro-scale processes both at the cellular level (such as, proliferation, cell-cell adhesion) and at the sub-cellular level (such as, cell mutation properties). This in turn allows us to examine the effects of such micro-scale 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 - Inter-Dosing 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 "Z-Method," 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 non-Hodgkin'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 computer-implemented 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 pre-clinical trials, thus substantiating the benefit of further TPO development.

10:30 AM
11:30 AM
Mark Chaplain - Mathematical Modelling of the Spatio-Temporal Response of Cytotoxic T-lymphocytes 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 so-called tumour-infiltrating cytotoxic lymphocytes (TICLs), in a small, multicellular tumour, without central necrosis and at some stage prior to (tumour-induced) 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 lymphocyte-tumour 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 quasi-stationary 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 spatio-temporal 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 chemotherapy-induced apoptosis and result in an increased anti-tumour effect. Two promising candidate drugs include flavopiridol, a synthetic flavone, and UCN-01, 7-OH-staurosporine. 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 (CPT-11), 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 UCN-01. For example, Hct116 colon cancer cells can be sensitized to undergo apoptosis in vitro by adding nanomolar concentrations of flavopiridol AFTER treatment with SN-38 (the active metabolite of CPT-11). Similarly, in vivo, single agent CPT-11 induced some tumor regressions but no complete responses (CR) in the Hct116 xenografts. However, CPT-11 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 R01-CA67819)

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 two-stage 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 two-stage (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 long-standing 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 single-dose 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.

1. Tannock, I.F. (1968). The relation between cell proliferation and the vascular system in a transplanted mouse mammary tumour. Br. J. Cancer, 22, 258-273.

2. 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.

3. Bertuzzi, A., d'Onofrio, A., Fasano, A., & Gandolfi, A. (2003). Regression and regrowth of tumour cords following single-dose anticancer treatment. Bull. Math. Biol., 65, 903-931.

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 Email Affiliation
Agur, Zvia agur@imbm.org Institute for Medical Biomathematics
Anderson, Alexander alexander.ra.anderson@gmail.com 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 Informatica-CNR, Instituto de Analisi dei Sistemi ed
Best, Janet jbest@mbi.osu.edu 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 Wisconsin-Madison
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
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 grever-1@medctr.osu.edu Department of Internal Medicine, The Ohio State University
Guo, Jong-Shenq 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, Lie-Jane ljkao@mbi.osu.edu Department of Industrial Engineering, Da-Yeh 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 kimmel@stat.rice.edu Department of Statistics, Rice University
King, John john.king@nottingham.ac.uk; 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, Sook-Kyung Mathematical Biosciences Institute, The Ohio State University
Liu, Edison gisanga@nus.edu.sg Genome Institute of Singapore
Marciniak-Czochra, Anna Anna.Marciniak@iwr.uni-heidelberg.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
Sandstede, Bjorn sandsted@math.ohio-state.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 Sloan-Kettering 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 & Pre-clinical Pharmacology, Human Genome Sciences
Swierniak, Andrzej Department of Automatic Control, Silesian University of Technology
Tan, Wai-Yuan waitan@memphis.edu Mathematical Sciences, University of Memphis
Terman, David terman@math.ohio-state.edu Mathemathics Department, The Ohio State University
Thomson, Mitchell Mathematical Biosciences Institute, The Ohio State University
Tsai, Chih-Chiang 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 wright.572@osu.edu 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
Inter-Dosing 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 "Z-Method," 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 non-Hodgkin'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 computer-implemented 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 pre-clinical trials, thus substantiating the benefit of further TPO development.

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, cell-cell 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 matrix-degrative 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 micro-scale (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 micro-scale processes both at the cellular level (such as, proliferation, cell-cell adhesion) and at the sub-cellular level (such as, cell mutation properties). This in turn allows us to examine the effects of such micro-scale 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

Multiscale Modelling of Cellular Systems in the Competition between Tumor and Immune System

Mathematical Modelling of the Spatio-Temporal Response of Cytotoxic T-lymphocytes 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 so-called tumour-infiltrating cytotoxic lymphocytes (TICLs), in a small, multicellular tumour, without central necrosis and at some stage prior to (tumour-induced) 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 lymphocyte-tumour 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 quasi-stationary 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 spatio-temporal dynamics observed in the PDE system.

Modeling the Cell Proliferation, Carcinogenesis in Lung Cancer: Taking the Interaction Between Genetic Factors and Smoking into Account

A stochastic two-stage 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 two-stage (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.

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 single-dose 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.

1. Tannock, I.F. (1968). The relation between cell proliferation and the vascular system in a transplanted mouse mammary tumour. Br. J. Cancer, 22, 258-273.

2. 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.

3. Bertuzzi, A., d'Onofrio, A., Fasano, A., & Gandolfi, A. (2003). Regression and regrowth of tumour cords following single-dose anticancer treatment. Bull. Math. Biol., 65, 903-931.

Distibution of the Number of Clonogenic Tumor Cells Surviving Fractionated Radiation

We solve, under realistic biological assumptions, the following long-standing 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.

Analysis of the Effect of an Anti-Cancer Drug on Cell Proliferation

In this talk, we propose a method to analyze the effect of an anti-cancer drug on the proliferation of oligodendrocytes and O-2A progenitor cells in culture conditions. The dynamic of the cell population is represented by a multitype Bellman-Harris 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.

Modeling Multiple-Drug 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 concentration-response 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 co-administered drugs.

References:

1. Loewe, S. (1953). The problem of synergism and antagonism of combined drugs. Arzneim. Forsch, 3, 2.

2. Short, T.G., Plummer, J.L., Chui, P.T. (1992). Hypnotic and anaesthetic interactions between midazolam, propofol and alfentanil. Br J Anaesth, 69, 162-7.

3. 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, 92,1603-1616.

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.

Transport of Small-Molecule 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 small-molecule drugs (mol. wt < 1000 Da) are being used. When compared to large molecules, such as proteins, small-molecule 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 2-24 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 concentration-versus-time profiles at different distances from the rim of the islets, using the blood concentration-versus-time profile as a boundary condition.

The profiles after an i.v. injection were mimicked in vitro using MCF-7 breast cancer cells. Extrapolating to the in vivo situation, the model predicted less drug-induced 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:

1. 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, 1703-7.

2. 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, 169-78.

3. 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, 149-61.

4. Lankelma, J. (2002). Tissue transport of anti-cancer drugs. Curr Pharm Des, 8, 1987-1993.

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

Expression Genomics and the Cellular Pharmacology of Cancer Therapeutics

Expression Genomics and the Cellular Pharmacology of Cancer Therapeutics

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

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 chemotherapy-induced apoptosis and result in an increased anti-tumour effect. Two promising candidate drugs include flavopiridol, a synthetic flavone, and UCN-01, 7-OH-staurosporine. 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 (CPT-11), 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 UCN-01. For example, Hct116 colon cancer cells can be sensitized to undergo apoptosis in vitro by adding nanomolar concentrations of flavopiridol AFTER treatment with SN-38 (the active metabolite of CPT-11). Similarly, in vivo, single agent CPT-11 induced some tumor regressions but no complete responses (CR) in the Hct116 xenografts. However, CPT-11 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 R01-CA67819)

Models of Cancer Population Evolution Combining Multi-Drug 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 low-concentration dosing may result in the rapid evolution of large fully resistant residual tumors; the same total doses divided into high-concentration 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, phase-specific chemotherapy has been considered only in the finite-dimensional 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 phase-specific 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 drug-resistant. Subsequently, an attempt to model multidrug (but not phase-specific) protocols will be presented that take into account increasing drug resistance to each used chemotherapeutic agent used. Finally, different cases of phase-specific control of the drug-sensitive cancer population will be addressed.

Interspecies Allometric Modeling of the Pharmacokinetics, Biodistribution and Dosimetry of LymphoRad-131, a Radiolabeled Cytokine Targeted to B Cells

LymphoRad-131 (LR131) is iodine-131 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 iodine-125 BLyS after intravenous injection into normal and tumor-bearing 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 LR-131 are currently being conducted in patients with multiple myeloma and non-Hodgkin'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.

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:

1. The inner structure of the cell cycle and the cell-cycle-phase specificity of some chemotherapy agents.

2. The dynamics of emergence of resistance of cancer cells to chemotherapy, as understood based on recent progress in molecular biology.

3. 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 cell-cycle-phase 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 cell-cycle-phase 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.

Cell-cycle-phase 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 non-uniqueness 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 cell-cycle-phase 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 bang-bang 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 cell-cycle-phase 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

1. Agur, Z. (1988). The effect of drug schedule on responsiveness to chemotherapy. Annals N.Y.Acad.Sci., 504, 274-277.

2. Dibrov, B.F., Zhabotinsky, A.M., Neyfakh, L.A., Orlova, H.P., & Churikova, L.I. (1985). Mathematical model of cancer chemotherapy. Periodic schedules of phase-specific cytotoxic-agent administration increasing the selectivity of therapy. Math.Biosci., 73, 1-31.

3. 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, 257-263.

4. 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, 1727-1733.

5. 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, 115-138.

6. Kimmel, M. & Axelrod, D.E. (1990) Mathematical models of gene amplification with applications to cellular drug resistance and tumorigenicity. Genetics, 125, 633-644.

7. 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, 1-16.

8. Ledzewicz, U. & Schattler, H. (2002). Optimal bang-bang controls for a 2- compart-ment model in cancer chemotherapy. J. of Optimization Theory and Applications, 114, 609-637.

9. Ledzewicz, U. & Schattler, H. (2002). Analysis of a cell-cycle specific model for cancer chemotherapy. J. of Biological Systems, 10,183-206.

10. Swan, G.W. (1990). Role of optimal control theory in cancer chemotherapy. Math. Biosci., 101, 237-284.

11. 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, 61-75.

12. Swierniak, A. & Smieja, J. (2001). Cancer chemotherapy optimization under evolving drug resistance. Nonlinear Analysis, 47, 375-386.

13. Swierniak, A. & Duda, Z. (1994). Singularity of optimal control problems arising in cancer chemotherapy. Math.and Comp.Modeling, 19, 255-262.

14. Swierniak, A. & Kimmel, M. (1984). Optimal control application to leukemia chemotherapy protocols design. ZN Pol.Sl., s.Autom., 74, 261-277 (in Polish).

15. Swierniak, A. & Polanski, A. (1994). Irregularity of optimal control problem in scheduling of cancer chemotherapy. Appl.Math.and Comp.Sci., 4, 263-271.

16. Swierniak, A., Polanski, A., & Kimmel, M. (1996). Optimal control problems arising in cell-cycle-specific cancer chemotherapy. Cell Proliferation, 29, 1- 23.

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 drug-induced 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 experimental-theoretical approach. We used an ovarian carcinoma cell line (IGROV-1) 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 more-than-one 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.

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 5-10 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 NCI-60) 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 multi-faceted molecular target profiles of the NCI-60 using 2-D gel electrophoresis (6), 'reverse-phase' protein microarrays (11), cDNA microarrays (12,13), Affymetrix oligo chips (14), real-time RT-PCR, array-CGH, SKY, SNP chips, and DNA methylation-sequencing. 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.

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3. Tanabe, et al. (1999). BioTechniques, 27, 1210.

4. Weinstein, et al. (1997). Science, 275, 343.

5. Myers, et al. (1997). Electrophoresis, 18, 647.

6. Bussey. (2003). Genome Biology, 4, R27.

7. Zeeberg. (2003). Genome Biol., 4, R28.

8. Blower, Jr. (2002). The Pharmacogenomics Journal (Nature), 2, 259.

9. Paull, et al. (1989). J. Natl. Cancer Inst., 81, 1088.

10. Weinstein, et al. (1992). Science, 258, 343.

11. Nishizuka, et al. (2003). Cancer Res., 65, 5243.

12. Ross, et al. (2000). Nature Genetics, 24, 227.

13. Scherf, et al. (2000). Nature Genetics, 24, 236.

14. Staunton, et al. (2001). Proc. Natl. Acad. Sci. U.S.A., 98, 10787.

15. Reinhold. (2003). Cancer Res., 63, 1000.

Drug Delivery to Tumors - Determinants and Barriers

Drug Delivery to Tumors - Determinants and Barriers

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