Jaroslaw Smieja

Mathematical modeling as a tool for planning anticancer therapy

We review a large volume of literature concerning mathematical models of cancer therapy, oriented towards optimization of treatment protocols. The review, although partly idiosyncratic, covers such major areas of therapy optimization as phase-specific chemotherapy, antiangiogenic therapy and therapy under drug resistance. We start from early cell cycle progression models, very simple but admitting explicit mathematical solutions, based on methods of control theory. We continue with more complex models involving evolution of drug resistance and pharmacokinetic and pharmacodynamic effects. Then, we consider two more recent areas: angiogenesis of tumors and molecular signaling within and among cells. We discuss biological background and mathematical techniques of this field, which has a large although only partly realized potential for contributing to cancer treatment.

CELL CYCLE AS AN OBJECT OF CONTROL

This chapter is devoted to models of cancer growth and anticancer therapies that put special emphasis on the dependence of therapy efficiency on cell cycle. First, biological background is introduced and detailed description of a cell cycle is given, based on the review of biological literature. It is supplemented with information about chosen chemotherapeutic drugs and their efficacy with respect to the cell cycle. Next, pharmacokinetic and pharmacodynamics aspects of chemotherapeutics are briefly described. They along with cell cycle specificity of drugs may lead to various phenomena of resonances and aftereffects that need to be taken into account in therapy and synchronization of treatment protocols. These issues are mentioned in a separate section of this chapter. Finally, models that incorporate evolution of drug resistance are presented. For all models, the problem of finding a suitable treatment protocol is formulated as a problem of control optimization and some results of application of optimization theory to solve these problems are presented.

Gradient Method for Finding Optimal Scheduling in Infinite Dimensional Models of Chemotherapy

One of the major obstacles against succesful chemotherapy of cancer is the emergence of resistance of cancer cells to cytotoxic agents. Applying optimal control theory to mathematical models of cell cycle dynamics can be a very efficient method to understand and, eventually, overcome this problem. Results that have been hitherto obtained have already helped to explain some observed phenomena, concerning dynamical properties of cancer populations. Because of recent progress in understanding the way in which chemotherapy affects cancer cells, new insights and more precise mathematical formulation of control problem, in the meaning of finding optimal chemotherapy, became possible. This, together with a progress in mathematical tools, has renewed hopes for improving chemotherapy protocols. In this paper we consider a population of neoplastic cells stratified into subpopulations of cells of different types. Due to the mutational event a sensitive cell can acquire a copy of the gene that makes it resistant to the agent. Likewise, the division of resistant cells can result in the change of the number of gene copies. We convert the model in the form of an infinite dimensional system of ordinary differential state equations discussed in our previous publications (see e.g. Swierniak etal., 1996b; Polariski etal., 1997; Swierniak etaL, 1998c), into the integro-differential form. It enables application of the necessary conditions of optimality given by the appropriate version of Pontryagins maximum principle, e.g. (Gabasov and Kirilowa, 1971). The performance index which should be minimized combines the negative cumulated cytotoxic effect of the drug and the terminal population of both sensitive and resistant neoplastic cells. The linear form of the cost function and the bilinear form of the state equation result in a bang-bang optimal control law. To find the switching times we propose to use a special gradient algorithm developed similarly to the one applied in our previous papers to finite dimensional problems (Duda 1994; 1997).

Integrated System Supporting Research on Environment Related Cancers

There are many impediments to progress in cancer research. Insufficient or low quality data and computational tools that are dispersed among various sites are one of them. In this paper we present an integrated system that combines all stages of cancer studies, from gathering of clinical data, through elaborate patient questionnaires and bioinformatics tools, to data warehousing and preparation of analysis reports.

Optimal control for the model of drug resistance resulting from gene amplification

Abstract In this paper we are concerned with an infinite dimensional model of emergence of drug resistance of cancer cells, as understood based on recent progress in molecular biology. Optimization of treatment protocols for resistant cancer population is formulated as an optimal control problem. The necessary conditions for optimal control of drug resistant population are found. To achieve this, the primary infinite dimensional differential model is transformed into one described by single integro-differential equation. A gradient method based approach for finding the solution of the stated problem is presented.

Modelling growth of drug resistant cancer populations as the system with positive feedback

We are concerned with dynamical properties of models of emergence of resistance of cancer cells to chemotherapy, based on recent progress in molecular biology. The model of the process has a form of the infinite system of linear and bilinear state equations. We show that, it can be decomposed into two parts: the first one is finite dimensional bilinear, and the second one is infinite dimensional linear, coupled by the positive feedback. Then, we use some results of the theory of feedback systems to study the asymptotic properties of the process.

Control theoretic approach to random branching walk models arising in molecular biology

Techniques commonly used in control theory based on Laplace transforms, spectral analysis, semigroup for state equations, methods of analysis and design of feedback systems enable to characterize the asymptotic behavior of telomeres shortening of which is supposed to be the mechanism of aging and death and cancer cells with increasing number of copies genes responsible for coding proteins causing drug removal or metabolisation. We model these two phenomena by branching random walk processes representing dynamical evolution of particles in them that leads to the systems of infinitely many linear or bilinear state equations. Their transformation into equivalent integro-differential or integral equations allow to design control strategies which are optimal in the sense of some performance indices.

Nucleotide Composition Based Measurement Bias in High Throughput Gene Expression Studies

High throughput gene expression profiling methods suffer from various sources of measurement bias inherent to the experimental procedures used. Most of the commonly used data standardization methods, designed to reduce the sample-to-sample variability of technical origin, do not account for probe- or transcript-specific effects. However, the efficiency of RNA isolation, cDNA synthesis and amplification does depend on the percentage of GC nucleotides in the transcript sequences and therefore constitutes a strong bias for the analysis of gene expression data. This work is focused on analysis of how and to what extent GC-content bias of oligonucleotide microarray probes affects the measurement data. We propose a mechanism explaining this phenomenon, the implications of GC-content bias for differentially expressed genes (DEGs) detection, and propose a new data standardization method, which by using sample-specific background intensity estimation and LOESS regression, allows to counteract the described effects.

Deterministic modeling of interferon-beta signaling pathway

Abstract The paper is concerned with dynamical modeling of early and late gene expression in one of Interferon- β induced signaling pathways. First, the biological background is given. Subsequently, the importance of stochastic nature of biological processes is addressed. Then, the assumptions underlying their deterministic modeling are presented and the mathematical model is introduced. Finally, the methodology for identification of model parameters is briefly described, followed by the presentation of experimental and simulation results. In addition to analysis of the particular regulatory network, this work also builds a framework for modeling of other signalling pathways whose not all components are known.

Sensitivity of signaling pathway dynamics to plasmid transfection and its consequences.

This paper deals with development of signaling pathways models and using plasmid-based experiments to support parameter estimation. We show that if cells transfected with plasmids are used in experiments, the models should include additional components that describe explicitly effects induced by plasmids. Otherwise, when the model is used to analyze responses of wild type, i.e. non-transfected cells, it may not capture their dynamics properly or even lead to false conclusions. In order to illustrate this, an original mathematical model of miRNA-mediated control of gene expression in the NFκB pathway is presented. The paper shows what artifacts might appear due to experimental procedures and how to develop the models in order to avoid pursuing these artifacts instead of real kinetics.