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.

Experimentally determined time dependent changes in the major components of this pathway were used to build a mathematical model describing pathway dynamics in the form of ordinary differential equations. The experimental results suggested existence of unknown negative control mechanisms that were tested numerically using the model. Together, experimental and numerical data show that the epithelial JAK-STAT pathway might be subjected to previously unknown dynamic negative control mechanisms: (1) activation of dormant phosphatases and (2) inhibition of nuclear import of IRF1.

Elevated temperature induces the heat shock (HS) response, which modulates cell proliferation, apoptosis, the immune and inflammatory responses. However, specific mechanisms linking the HS response pathways to major cellular signaling systems are not fully understood. Here we used integrated computational and experimental approaches to quantitatively analyze the crosstalk mechanisms between the HS-response and a master regulator of inflammation, cell proliferation, and apoptosis the Nuclear Factor κB (NF-κB) system. We found that populations of human osteosarcoma cells, exposed to a clinically relevant 43°C HS had an attenuated NF-κB p65 response to Tumor Necrosis Factor α (TNFα) treatment. The degree of inhibition of the NF-κB response depended on the HS exposure time. Mathematical modeling of single cells indicated that individual crosstalk mechanisms differentially encode HS-mediated NF-κB responses while being consistent with the observed population-level responses. In particular “all-or-nothing” encoding mechanisms were involved in the HS-dependent regulation of the IKK activity and IκBα phosphorylation, while others involving transport were “analogue”. In order to discriminate between these mechanisms, we used live-cell imaging of nuclear translocations of the NF-κB p65 subunit. The single cell responses exhibited “all-or-nothing” encoding. While most cells did not respond to TNFα stimulation after a 60 min HS, 27% showed responses similar to those not receiving HS. We further demonstrated experimentally and theoretically that the predicted inhibition of IKK activity was consistent with the observed HS-dependent depletion of the IKKα and IKKβ subunits in whole cell lysates. However, a combination of “all-or-nothing” crosstalk mechanisms was required to completely recapitulate the single cell data. We postulate therefore that the heterogeneity of the single cell responses might be explained by the cell-intrinsic variability of HS-modulated IKK signaling. In summary, we show that high temperature modulates NF-κB responses in single cells in a complex and unintuitive manner, which needs to be considered in hyperthermia-based treatment strategies.

BackgroundEpigenetic regulation contributes to many important processes in biological cells. Examples include developmental processes, differentiation and maturation of stem cells, evolution of malignancy and other. Cell cycle regulation has been subject of mathematical modeling by a number of authors that resulted in many interesting models and application of analytic techniques ranging from stochastic processes to partial differential equations and to integral, functional and operator equations. In this paper we address the question of how the regulation of protein contents influences the long-term dynamics of the population. To accomplish this, we follow the philosophy of a 1984 model by Kimmel et al., but adjust the details to fit the experimental data on protein PRC1 from a more recent paper.ResultsWe built a model of cell cycle dynamics of the PRC1 and fitted it to the data made available by Cohen and his co-authors. We have run the model for a large number of cell generations, recording the PRC1 contents in all cells of the resulting pedigree, at constant time intervals. During cell division the PRC1 is unequally divided between daughter cells. The picture emerging from simulations of Data set 1 is that of a very well-tuned regulatory circuit that provides a stable distribution of PRC1 contents and interdivision times. Data set 2 seems qualitatively different, with more variation in cell cycle duration.ConclusionsThe main question we address is whether the regulatory feedbacks deduced from single cell cycle data provide epigenetic regulation of cell characteristics in long run. PRC1 is a good candidate because of its role in setting timing of division. Findings of the current paper include tight regulation of the cell cycle (particularly the timing of the cell cycle) even that PRC1 is only one of the players in cell dynamics. Understanding that association, even close, does not necessarily imply causation, we consider this an interesting and important result.ReviewersThis article was reviewed by Ollivier Hyrien, Anna Marciniak-Czochra and Alberto d’Onofrio.

Studies proved that among all α1-adrenoceptors, cardiac myocytes functionally express only α1A- and α1B-subtype. Scientists indicated that α1A-subtype blockade might be beneficial in restoring normal heart rhythm. Therefore, we aimed to determine the role of α1-adrenoceptors subtypes (i.e., α1A and α1B) in antiarrhythmic effect of six structurally similar derivatives of 2-methoxyphenylpiperazine. We compared the activity of studied compounds with carvedilol, which is β1- and α1-adrenoceptors blocker with antioxidant properties. To evaluate the affinity for adrenergic receptors, we used radioligand methods. We investigated selectivity at α1-adrenoceptors subtypes using functional bioassays. We tested antiarrhythmic activity in adrenaline-induced (20 μg/kg i.v.), calcium chloride-induced (140 and 25 mg/kg i.v.) and barium chloride-induced (32 and 10 mg/kg i.v.) arrhythmia models in rats. We also evaluated the influence of studied compounds on blood pressure in rats, as well as lipid peroxidation. All studied compounds showed high affinity toward α1-adrenoceptors but no affinity for β1 receptors. Biofunctional studies revealed that the tested compounds blocked α1A-stronger than α1B-adrenoceptors, but except for HBK-19 they antagonized α1A-adrenoceptor weaker than α1D-subtype. HBK-19 showed the greatest difference in pA2 values—it blocked α1A-adrenoceptors around seven-fold stronger than α1B subtype. All compounds showed prophylactic antiarrhythmic properties in adrenaline-induced arrhythmia, but only the activity of HBK-16, HBK-17, HBK-18, and HBK-19 (ED50 = 0.18–0.21) was comparable to that of carvedilol (ED50 = 0.36). All compounds reduced mortality in adrenaline-induced arrhythmia. HBK-16, HBK-17, HBK-18, and HBK-19 showed therapeutic antiarrhythmic properties in adrenaline-induced arrhythmia. None of the compounds showed activity in calcium chloride- or barium chloride-induced arrhythmias. HBK-16, HBK-17, HBK-18, and HBK-19 decreased heart rhythm at ED84. All compounds significantly lowered blood pressure in normotensive rats. HBK-18 showed the strongest hypotensive properties (the lowest active dose: 0.01 mg/kg). HBK-19 was the only compound in the group, which did not show hypotensive effect at antiarrhythmic doses. HBK-16, HBK-17, HBK-18, HBK-19 showed weak antioxidant properties. Our results indicate that the studied 2-methoxyphenylpiperazine derivatives that possessed stronger α1A-adrenolytic properties (i.e., HBK-16, HBK-17, HBK-18, and HBK-19) were the most active compounds in adrenaline-induced arrhythmia. Thus, we suggest that the potent blockade of α1A-receptor subtype is essential to attenuate adrenaline-induced arrhythmia.

No abstract

No abstract

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 Pontryagin's 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).