Spatial Heterogeneity in Drug Concentrations Can Facilitate the Emergence of Resistance to Cancer Therapy

Acquired resistance is one of the major barriers to successful cancer therapy. The development of resistance is commonly attributed to genetic heterogeneity. However, heterogeneity of drug penetration of the tumor microenvironment both on the microscopic level within solid tumors as well as on the macroscopic level across metastases may also contribute to acquired drug resistance. Here we use mathematical models to investigate the effect of drug heterogeneity on the probability of escape from treatment and the time to resistance. Specifically we address scenarios with sufficiently potent therapies that suppress growth of all preexisting genetic variants in the compartment with the highest possible drug concentration. To study the joint effect of drug heterogeneity, growth rate, and evolution of resistance, we analyze a multi-type stochastic branching process describing growth of cancer cells in multiple compartments with different drug concentrations and limited migration between compartments. We show that resistance is likely to arise first in the sanctuary compartment with poor drug penetrations and from there populate non-sanctuary compartments with high drug concentrations. Moreover, we show that only below a threshold rate of cell migration does spatial heterogeneity accelerate resistance evolution, otherwise deterring drug resistance with excessively high migration rates. Our results provide new insights into understanding why cancers tend to quickly become resistant, and that cell migration and the presence of sanctuary sites with little drug exposure are essential to this end.     

Intracellular potassium compartments in Nitella axillaris

Three intracellular compartments for potassium exchange have been observed in intact cells of the giant-celled alga, Nitella axillaris. These compartments have been compared with the exchange properties of isolated subcellular structures. The smallest and fastest compartment (apparent half-time, 23 seconds) appears to involve passive absorption on the cell wall. The next largest (apparent half-time, 5 hours) may represent exchange with the cytoplasmic layer through the plasma membrane, the chloroplasts being in rapid equilibrium with the surrounding cytoplasm. The largest and slowest compartment (apparent half-time, 40 days) has been identified with the central vacuole. The vacuolar membrane and the plasma membrane have similar properties with respect to K permeability. Thus, the experimental data from the whole cell can be accounted for by a structural model of the compartments. Cyanide in concentrations up to 10(-3)M causes no net loss of K. The fastest compartment in Nitella and in higher plants is compared, and the ecological significance of the slow rate of potassium transport in Nitella is discussed.     

A mathematical model to study the effects of drugs administration on tumor growth dynamics

A mathematical model for describing the cancer growth dynamics in response to anticancer agents administration in xenograft models is discussed. The model consists of a system of ordinary differential equations involving five parameters (three for describing the untreated growth and two for describing the drug action). Tumor growth in untreated animals is modelled by an exponential growth followed by a linear growth. In treated animals, tumor growth rate is decreased by an additional factor proportional to both drug concentration and proliferating cells. The mathematical analysis conducted in this paper highlights several interesting properties of this tumor growth model. It suggests also effective strategies to design in vivo experiments in animals with potential saving of time and resources. For example, the drug concentration threshold for the tumor eradication, the delay between drug administration and tumor regression, and a time index that measures the efficacy of a treatment are derived and discussed. The model has already been employed in several drug discovery projects. Its application on a data set coming from one of these projects is discussed in this paper.     

The development of thermotolerance in hyperthermal injury to the villus compartment of mouse small intestine

The development and decay of thermotolerance in the villus compartment of the intestinal mucosa of mouse was investigated by giving a primary treatment of 41.5 degrees C for 1 hour (subthreshold for thermal injury) at various intervals before a second, test treatment of 43.0 degrees C for 30 min. The test treatment was given 65 hours after an intraperitoneal injection of 3H-thymidine (i.e. at a time when the heavily labelled cells could be seen to have moved from the crypts on to the upper halves of the villi) and thermal damage assessed by loss of radioactive label. A transient tolerance to the second treatment was induced by the primary treatment. This 'thermotolerance' was maximal 3-13 hours after the first treatment and had decayed by 24 hours. Both the extent and time course of expression and decay of thermotolerance in this post-mitotic functional compartment were very similar to those previously reported for damage to the proliferative epithelium as assayed by crypt loss. This suggests either that the kinetics of thermotolerance are not dependent on the proliferative status of the tissue or that there is a common limiting factor in thermotolerance development, despite the apparent difference between the two mucosal compartments in their susceptibilities to thermal injury.     

Modeling multi-drug chemotherapy: tailoring treatment to individuals

BACKGROUND: Predicting and tailoring optimal cancer treatments presents a major challenge. METHODS: A computational model (kinetically tailored treatment, or KITT model) is developed to predict drug combinations, doses, and schedules likely to be effective in reducing tumor size and prolonging patient life. Treatment strategies may be tailored to individuals based on tumor cell kinetics. The model incorporates intra-tumor heterogeneity and evolution of drug resistance, apoptotic rates, and cell division rates. Tumor growth may follow an exponential or a Gompertzian trajectory. Drug pharmacodynamic and pharmacokinetic models are used. Toxicity is modeled in several ways. RESULTS: A key prediction of KITT is that including cytostatic drugs like tamoxifen and herceptin during treatment with cytotoxic drugs substantially increases the probability of cure and prolongs patient life. Results also suggest that altering drug scheduling may be more effective but not more toxic than dose escalation. CAF chemotherapy (cyclophosphamide, adriamycin, and 5-fluorouracil) is predicted to be more effective than CMF (cyclophosphamide, methotrexate, and 5-fluorouracil). KITT also suggests that tumors with a high proliferative index (PI) may respond better to drug combinations incorporating two cell-cycle phase-specific drugs than do tumors with a low PI. Tumors with a low PI, in contrast, are predicted to respond better to regimens involving two cell-cycle phase-non-specific drugs than do tumors with a high PI. These predictions are borne out by clinical trial results published in the literature, which are discussed.Simulated predictions of the model match well with results from a clinical trial by Silvestrini et al. (2000. Int. J. Cancer 87, 405). The results of simulating the growth of 26896 tumors are used to construct a decision tree for prognosis to identify the key tumor and treatment variables. CONCLUSION: Additional tests of the model are needed in which physicians collect information on apoptotic and proliferative indices, cell-cycle times, and drug resistance from biopsies of each individual's tumor. Computational models may become important tools to help optimize and tailor cancer treatments.     

Modeling of lipid and protein depletion during total starvation

This study presents a model describing lipid and protein depletion of an individual facing total starvation. The model distinguishes two compartments of body mass: a metabolic compartment and a structural compartment. It is considered that the lipids and the proteins of the metabolic compartment ensure the totality of physiological functions. The main assumptions of the model lie in the definitions of lipid mass and protein mass of the metabolic compartment, which are related to total lipid mass and total body mass, respectively. Under these assumptions, for a given individual, the ratio of lipid and protein utilization rates is proportional to the adiposity. The model accounts for the protein sparing observed at high adiposity levels and enables us to discuss the individual's survival in relation to the levels of lipid and protein depletion. The time course of changes in lipid and protein depletion rates can be calculated by introducing the energy expenditure of the individual. In simulations, it was assumed that specific energy expenditure was constant during starvation and that mortality occurred at a critical level of protein depletion. The most characteristic results derived from these simulations concern the kinetics of protein depletion, which depend markedly on initial adiposity. Accordingly, in obese subjects, the rate of protein losses remains fairly constant during fasting, whereas it increases from the onset of the fast in lean subjects, in agreement with experimental observations. In the model, protein and lipid depletion rates are both proportional to energy expenditure, which needs to be confirmed from complementary data.     

A stochastic approach for the interpretation of single pulse experiments in morphological multicompartments of renewing and exponentially growing cell populations

A general approach for the interpretation of single pulse experiments in multicompartment systems is presented. It allows an extension of the classical single compartment Barret's methodology, and has been detailed in the pipeline model case to show its capabilities. FLM, FLC, grain count curves, labelling indices and compartment size ratios of each compartment, are fitted into a coherent scheme that fully describes the statistical aspects of the phase durations including possible losses. It is shown that if the compartments are identical, irrespective of the feedback coupling between them, the system may be treated as a single compartment. It is also shown that the FLC information is necessary to identify the existence of losses in the system, and how to correct the "apparent" transit times if losses are present. The pipeline model is treated and a suggestion is made to reconcile the "British" and "American" interpretations of the erythroid system. As a corollary, simple formulae are derived in the deterministic case through a coupling matrix describing the interaction between compartments. Computer codes are described and have been implemented in the J.E.N. Thermoecology Laboratory.     

A hypothesis and theoretical model speculating the possible role of therapy mediated neoplastic cell loss in promoting the process of glioblastoma relapse

Tumor recurrence is considered to be one of the biggest culprits, behind the poor prognosis of glioblastomas. Using published facts on primary glioblastomas, with special reference to cancer stem cells and their recently described heterogeneity, a hypothesis is being proposed that speculates the possible role of therapy mediated neoplastic cell loss in promoting the process of relapse in these tumors. The mechanisms by which such a phenomenon could be functional has been integrated into a double version theoretical model, which envisages glioblastomas as neoplasms comprising of multiple, differentially regulated and dynamically distinct neoplastic compartments (named as active and back-up compartments in this article) supported by their own complement of cancer stem cells, wherein therapy mediated cell loss, which mainly affects the size of the active compartment, results in abrogating the inhibitory effect of the active compartment on the back-up compartment, thereby leading to the activation of the back-up compartment. This activation contributes towards tumor recurrence. The possibility of the existence of such a phenomenon could have strong implications on management and prognosis of these tumors. This article aims to provoke discussion and generate new ideas for further research.     

Dopamine Uptake by Rat Striatal Synaptosomes: A Compartmental Analysis

Abstract: Dopamine (DA) uptake into synaptosomes from rat corpus striatum was studied in the presence of a monoamine oxidase (MAO) inhibitor and dithiothreitol, by means of a filtration technique. Under these conditions a steady state develops rapidly in which the synaptosomal DA content remains constant while the continuing DA uptake is counterbalanced by DA efflux from the synaptosome. Exchange of synaptosomal [³H]DA and [¹⁴C]DA was measured under these conditions. In timecourse experiments it was found that exchange could be described significantly better by a three-compartment model than by a two-compartment model. However, if synaptosomes from reserpine-pretreated animals were used, analysis according to a three-compartment model did not result in a significantly better fit compared with a two-compartment model. Subsequently, kinetic transfer parameters describing DA fluxes between compartments at different DA concentrations were calculated from the fitted exchange curves. A Michaelis-Menten kinetic analysis indicated that only the in-series three-compartment configuration, in which DA is taken up from the medium into one synaptosomal compartment, from which it can subsequently be transferred to a second compartment without direct access to the medium, gave kinetically acceptable results. Transfer parameters in synaptosomes from reserpine-treated rats were comparable to those parameters describing DA transport between the medium and the first intrasynaptosomal compartment as measured under control conditions. Morover, it was found that potassium depolarization of synaptosomes resulted in a release of DA in a quantity similar to that found in the second intrasynaptosomal compartment. It is suggested that the two intrasynaptosomal compartments found correspond to a cytoplasmatic and vesicular DA pool, respectively. The functional significance of these findings is discussed in terms of the regulation of DA levels within the nerve terminal.     

Computer Simulations of Protein Diffusion in Compartmentalized Cell Membranes

The diffusion of proteins in the cell membrane is investigated using computer simulations of a two-dimensional model. The membrane is assumed to be divided into compartments, with adjacent compartments separated by a barrier of stationary obstacles. Each compartment contains traps represented by stationary attractive disks. Depending on their size, these traps are intended to model either smaller compartments or binding sites. The simulations are intended to model the double-compartment model, which has been used to interpret single molecule experiments in normal rat kidney cells, where five regimes of transport are observed. The simulations show, however, that five regimes are observed only when there is a large separation between the sizes of the traps and large compartments, casting doubt on the double compartment model for the membrane. The diffusive behavior is sensitive to the concentration and size of traps and the strength of the barrier between compartments suggesting that the diffusion of proteins can be effectively used to characterize the structure of the membrane.     

Optimal control for a cancer chemotherapy problem with general growth and loss functions

We investigate a model of a cancer chemotherapy problem where the aim is to minimize the tumor burden at the end of the treatment period while maintaining a normal cell population above a lower level as a limit of toxicity. The analysis is performed for general classes of growth and loss functions. The optimal drug dose is maximum initially so that the normal cell population is driven down to its lower level, and then the drug level is chosen to maintain the normal cell population there until the end of treatment. During treatment the number of tumor cells is always decreasing.     

Theoretical study of flow,pressure and solute concentration in double membrane systems

A model system consisting of two rigidly held membranes in series was investigated through the application of the Kedem and Katchalsky thermodynamic single membrane flow equations. This analysis results in predictions of the steady state flow properties as well as values for the solute concentration and pressure of the internal compartment when the system is under the influence of a constant solute concentration or hydrostatic pressure gradient. It is demonstrated that although the flow properties and internal compartment pressure are complicated functions of the membrane permeability coefficients and driving gradient across the system, the relationships are greatly simplified by the explicit appearance of the internal compartment steady state solute concentration in the equations. It is shown that the steady state volume flow rate depends on the absolute value of the solute concentration in the external compartments, as well as the solute concentration gradient across the system. The properties of non-linear dependence of volume flow on concentration gradient, and rectification of volume flow are discussed and shown to be independent properties of the system. For the system under the influence of a solute concentration gradient, the internal compartment pressure can be greater or less than the ambient pressure, and depends mainly on the order in which the membranes are encountered by the volume flow. These properties are qualitatively correlated with certain available experimental observations in biological systems.     

NSAIDs induce apoptosis in nonproliferating ovarian cancer cells and inhibit tumor growth in vivo

Ovarian cancer (OC) is one of the most lethal gynaecological cancers, which usually has a poor prognosis due to late diagnosis. A large percentage of the OC cell population is in a nonproliferating and quiescent stage, which poses a barrier to success when using most chemotherapeutic agents. Recent studies have shown that several nonsteroidal anti-inflammatory drugs (NSAIDs) are effective in the treatment of OC. Furthermore, we have previously described the molecular mechanisms of NSAIDs' induction of cancer apoptosis. In this report, we evaluated various structurally distinct NSAIDs for their efficacies in inducing apoptosis in nonproliferating OC cells. Although several NSAIDs-induced apoptosis, Flufenamic Acid, Flurbiprofen, Finasteride, Celocoxib, and Ibuprofen were the most potent NSAIDs inducing apoptosis. A combination of these agents resulted in an enhanced effect. Furthermore, we demonstrate that the combination of Flurbiprofen, which targets nonproliferative cells, and Sulindac Sulfide, that affects proliferative cells, strongly reduced tumor growth when compared with a single agent treatment. Our data strongly support the hypothesis that drug treatment regimens that target nonproliferating and proliferating cells may have significant efficacy against OC. These results also provide a rationale for employing compounds or even chemically modified NSAIDs, which selectively and efficiently induce apoptosis in cells during different stages of the cell cycle, to design more potent anticancer drugs.     

Effect of proinflammatory cytokines, tumor necrosis factor-alpha and interferon-gamma on epithelial barrier function and matrix metalloproteinase-9 in Madin Darby canine kidney cells.

BACKGROUND: Elevated matrix metalloproteinase-9 production during inflammation may be deleterious to epithelial barrier function. Therefore we examined the effect of proinflammatory cytokines on the expression and regulation of matrix metalloproteinase-9 in a model renal epithelial cell system. Tight junctions limit diffusion between compartments and permit directional transport of solutes. Impairment of these junctional complexes by proteolysis may contribute to renal failure through loss of barrier function. METHODS: The renal epithelial cell model, MDCK cells were employed to examine metalloproteinase activity and mRNA expression. Epithelial barrier function was determined using paracellular flux studies. RESULTS: We found that matrix metalloproteinase-9 expression (MMP-9) and activity is markedly elevated in response to tumor necrosis factor-alpha exposure through a mitogen-activated protein kinase dependent pathway. The MMP-9 is predominately secreted into the apical compartment and elevated MMP-9 expression correlates with impaired cell barrier function that was restored using a specific inhibitor of MMP activity. Addition of recombinant MMP-9 to the apical compartment of MDCK cultures significantly elevated paracellular flux rate. CONCLUSIONS: We provide direct evidence for a MMP-9-mediated mechanism that produces junctional disruption. Collectively, these findings support the hypothesis that impaired epithelial barrier function due to activation of tissue/matrix degrading mechanisms occurs in response to specific inflammatory cues.     

Cytochalasin B-induced pseudo-cleavage of mouse oocytes in vitro: asymmetric localization of mitochondria and microvilli associated with a stage-specific response.

Mouse oocytes are induced by cytochalasin B to undergo 'pseudo-cleavage' in vitro into 2 equally sized and separable compartments. This response to the drug is dependent upon the meiotic state of the oocytes, as well as upon the presence of an intact zona pellucida. The resulting 2 cellular compartments can be completely separated from another and cultured in vitro. Each of the compartments possesses characteristic structural features. The most pronounced structural differences include: (i) the presence of a nucleus (germinal vesicle) and nucleolus in one compartment; (ii) the presence of microvilli on the surface of the anucleate, but not the nucleate, compartment; and (iii) the localization (segregation) of mitochondria at the periphery of the anucleate, but not the nucleate, compartment. The results presented suggest that pseudo-cleavage induced by cytochalasin B arises as a consequence of a limited interaction of the drug with the oocyte surface and/or cortex and that it may represent a topographical dissociation of transporting and non-transporting regions of the membrane. These and other features of mouse oocytes treated with cytochalasin B are of interest in view of the involvement of the oocyte zona pellucida and plasma membrane during meiotic maturation, fertilization, and early embryogenesis.     

The discrete-time kinetic model analysis of DNA content distributions in experimental tumour cells.

A method was developed to analyse and characterize FMF measurements of DNA content distribution, utilizing the discrete time kinetic (DTK) model for cell kinetics analysis. The DTK model determines the time sequence of the cell age distribution during the proliferation of a tumor cell population and simulates the distribution pattern of the DNA content of cells in each age compartment of the cell cycle. The cells in one age compartment are distributed and spread into several compartments of the DNA content distribution to allow for different rates of DNA synthesis and instrument dispersion effects. It is assumed that the DNA content of cells in each age compartment has a Gaussian distribution. Thus, for a given cell age distribution the DNA content distribution depends on two parameters of the cells in each age compartment: the average DNA content and its coefficient of variation. As the DTK model generates the best fit DNA content distribution to the FMF measurement data, it enables one to estimate specific values of these two parameters in each stage of the cell cycle and to determine the fraction of cells in each cycle phase. The method was utilized to fit FMf measurements of DNA content distributions and to analyse their relationship tothe cell kinetic parameters, namely cell loss rate, cell cycle times and grwoth graction of exponentially growing Chinese hamster ovary cells in vitro and, also, with a wide range of coeffficients of variation, of the L1210 ascites tumour during the growth period.     

THE DISCRETE–TIME KINETIC MODEL ANALYSIS OF DNA CONTENT DISTRIBUTIONS IN EXPERIMENTAL TUMOUR CELLS

A method was developed to analyse and characterize FMF measurements of DNA content distribution, utilizing the discrete time kinetic (DTK) model for cell kinetics analysis. The DTK model determines the time sequence of the cell age distribution during the proliferation of a tumor cell population and simulates the distribution pattern of the DNA content of cells in each age compartment of the cell cycle. The cells in one age compartment are distributed and spread into several compartments of the DNA content distribution to allow for different rates of DNA synthesis and instrument dispersion effects. It is assumed that the DNA content of cells in each age compartment has a Gaussian distribution. Thus, for a given cell age distribution the DNA content distribution depends on two parameters of the cells in each age compartment: the average DNA content and its coefficient of variation. As the DTK model generates the best fit DNA content distribution to the FMF measurement data, it enables one to estimate specific values of these two parameters in each stage of the cell cycle and to determine the fraction of cells in each cycle phase. The method was utilized to fit FMF measurements of DNA content distributions and to analyse their relationship to the cell kinetic parameters, namely cell loss rate, cell cycle times and growth fraction of exponentially growing Chinese hamster ovary cells in vitro and, also, with a wide range of coefficients of variation, of the L1210 ascites tumour during the growth period.     

The relationship of intercompartmental excitation transfer rate constants to those of an underlying physical model

In studies on photosynthetic systems it is common practice to interpret the results of time-resolved fluorescence experiments on the basis of compartmental, or target, models. Each compartment represents a group of molecules with similar fluorescence characteristics. In cases of practical interest, the members of each compartment are spatially contiguous and make up part of an overall energy-transferring system. Since a rate constant describing the overall transfer between compartments is not that of any pair of molecules in the system, this question naturally rises: what do we learn about the microscopic structure from these data? In this note we introduce ‘compartment melting’, a smooth mathematical connection between the compartmental and microscopic levels. We then show, on the basis of model calculations on finite lattices in one, two, and three dimensions, that average microscopic rates at the interfaces between compartments may be estimated from observed intercompartmental rates. The estimate involves a modest number of structural assumptions about the system. As examples of the method, which is applicable mainly to systems containing homogeneous pigment pools, some recent chlorophyll-protein antenna studies are analyzed.     

矮嵩草草甸生长季节生产者亚系统分室模型

本文报道了高寒矮嵩草草甸生产者亚系统六分室模型(GCS 01模型)。六分室包括地上部分的禾草类(V1)、莎草类(V2)、杂类草(V3)、枯枝落叶(V4)四个分室以及地下部分的活根(V5)、死根(V6)两个分室。并建造了相应的微分方程组数学模型,用含有龙格库塔法的BASIC程序可求其数值解。本模型可供高寒草甸生态系统研究人员研究系统行为、探索系统内部规律使用。     

Stochastic modeling of controlled-drug release

A drug release process by the oral route is random in nature and thus is subject to constant fluctuations. Moreover, individuals have varied tolerances to such fluctuations. The objective of this work is to characterize these fluctuations by a stochastic formalism. The system under consideration, i.e., the gastrointestinal tract consists of four consecutive compartments, i.e., stomach, duodenum, jejunum, and ileum. The master equation of the system as well as the governing equations for the means, variances, and covariances of the random variables, each representing the number of microspheres in an individual compartment, have been derived through the probabilistic population balance. These equations have been numerically solved to predict the total release fraction of drug and its internal fluctuations, and the dynamic statistics (means, variances, and covariances) of the amount of drug in each compartment at any time after administration. The dissolution-intensity functions in the model have been recovered from the available in vitro dissolution data from controlled-release pellets of isosorbide-5-nitrate (IS-5-N) by assuming that the rate of release is of the first order. The residence times and transition-intensity functions of drug in the individual compartments have been estimated from the available data generated by the gamma scintigraphies of IS-5-N pellets labeled by ¹¹¹In. Based on these parameters, the total numbers of dissolved drug microspheres and their fluctuations at any instance have been calculated. The model is in accord with the existing in vivo dissolution data of the same drug independently obtained through plasma analysis. More important, the model predicts that fluctuations in terms of the standard deviations of the numbers of particles in the duodenum, jejunum, and ileum can be of the same orders of magnitude as the corresponding mean numbers when 100 microspheres are simultaneously administered orally; in practice, such fluctuations characterized by these deviations could result in an undesirable release profile. Discussion is given of the potential direct clinical application of the results obtained as well as the plausible indirect application of these results and the model derived to the analyses of chemical and biochemical reactors.