A note on a method for determining advantageous properties of an anti-arrhythmic drug based on a mathematical model of cardiac cells

Regional hyperkalemia during acute ischemia may provoke cardiac arrhythmias such as ventricular fibrillation. Despite intense research efforts over the last decades, the problem of finding an efficient anti-arrhythmic drug without dangerous side effects is still open. One approach to analyze the effect of anti-arrhythmic drugs is to do simulations based on mathematical models of collections of cardiomyocytes. Such simulations have recently illuminated the pro-arrhythmic capability of well-established anti-arrhythmic drugs.The purpose of the present note is to introduce a method intended for computing advantageous properties of an anti-arrhythmic drug. For a given model of a normal and an ischemic cell, we introduce a drug as a vector of non-negative real numbers whose components are multiplied by individual terms representing specific ionic currents. The drug vector is computed such that the action potentials of the resulting drugged cells are as close as possible to the action potential of a normal (not drugged) cell. Numerical simulations based on the Luo-Rudy I model and the Hund-Rudy model show that the classical shortened action potential obtained due to hyperkalemia is prolonged by using the drug computed by this method. Furthermore, for both models a 2D collection of spatially coupled ischemic cells give arrhythmogenic solutions before the drug is applied, and stable solutions after the drug is applied. It is emphasized that we do not address the possibility of realizing a drug with the properties computed in this note.     

Kinetic Analysis of Drug-Induced G2 Block In Vitro

The data on cell-cycle effects of two prospective antitumour agents, (+)-1,2,-bis(3,5-dioxopiperazine-l-yl)propane (soluble ICRF; NSC 169780) and 1,4-bis(2′chloroethyl)-1,4-diazabicyclo [2.2.1] heptane diperchlorate (CBH; NSC 57198) were used to determine whether a modified stathmokinetic experiment could predict the effects of continuous, long-term (0–48 hr) drug exposure in an in vitro L1210 murine leukaemia cell system. Generally, continuous drug exposure of exponentially growing cells does not provide sufficient quantitative information concerning cell-cycle-phase-specific mechanisms of drug action. Alternatively, stathmokinetic experiments, which are usually limited to some fraction of one cell doubling time, provide little information about long-term drug effects. By using mathematical models constructed for this purpose, however, stathmokinetic data can predict the overall proportion of cells affected by a drug though failing to discern between various kinds of drug action (e.g. reversible v. irreversible block, blocking v. killing action, etc.), especially when it occurs in G₂ phase. In addition, it can be shown that for at least one of the drugs (soluble ICRF) the stathmokinetic experiment fails to predict ‘after-effects’ of drug treatment which extend into the following cell cycle(s). It also becomes clear that the degradation of exponential growth characteristics of quickly dividing cells during long-term, continuous drug exposure makes prediction of cell-cycle kinetic perturbations uncertain when derived from short-duration stathmokinetic experiments. However, with care, the joint application of ‘short term’ (e.g. stathmokinesis) and ‘long term’ (e.g. continuous exposure) techniques allow adequate quantitative insight into drug-perturbed cell-cycle kinetics. the applicability of modelling techniques is discussed: in the present instance it is limited to lower drug concentrations. For higher drug concentrations, effects like increased ploidy, ineffective division, etc., make it impossible in the present study to obtain a clear picture of the kinetics.     

Competing Risks and Time‐Dependent Covariates

Time‐dependent covariates are frequently encountered in regression analysis for event history data and competing risks. They are often essential predictors, which cannot be substituted by time‐fixed covariates. This study briefly recalls the different types of time‐dependent covariates, as classified by Kalbfleisch and Prentice [The Statistical Analysis of Failure Time Data, Wiley, New York, 2002] with the intent of clarifying their role and emphasizing the limitations in standard survival models and in the competing risks setting. If random (internal) time‐dependent covariates are to be included in the modeling process, then it is still possible to estimate cause‐specific hazards but prediction of the cumulative incidences and survival probabilities based on these is no longer feasible. This article aims at providing some possible strategies for dealing with these prediction problems. In a multi‐state framework, a first approach uses internal covariates to define additional (intermediate) transient states in the competing risks model. Another approach is to apply the landmark analysis as described by van Houwelingen [Scandinavian Journal of Statistics 2007, 34 , 70–85] in order to study cumulative incidences at different subintervals of the entire study period. The final strategy is to extend the competing risks model by considering all the possible combinations between internal covariate levels and cause‐specific events as final states. In all of those proposals, it is possible to estimate the changes/differences of the cumulative risks associated with simple internal covariates. An illustrative example based on bone marrow transplant data is presented in order to compare the different methods.     

Kinetic analysis of drug-induced G2 block in vitro

The data on cell-cycle effects of two prospective antitumour agents, (+)-1,2,-bis(3,5-dioxopiperazine-1-yl)propane (soluble ICRF; NSC 169780) and 1,4-bis(2'chloroethyl)-1,4-diazabicyclo [2.2.1] heptane diperchlorate (CBH; NSC 57198) were used to determine whether a modified stathmokinetic experiment could predict the effects of continuous, long-term (0-48 hr) drug exposure in an in vitro L1210 murine leukaemia cell system. Generally, continuous drug exposure of exponentially growing cells does not provide sufficient quantitative information concerning cell-cycle-phase-specific mechanisms of drug action. Alternatively, stathmokinetic experiments, which are usually limited to some fraction of one cell doubling time, provide little information about long-term drug effects. By using mathematical models constructed for this purpose, however, stathmokinetic data can predict the overall proportion of cells affected by a drug though failing to discern between various kinds of drug action (e.g. reversible v. irreversible block, blocking v. killing action, etc.), especially when it occurs in G2 phase. In addition, it can be shown that for at least one of the drugs (soluble ICRF) the stathmokinetic experiment fails to predict 'after-effects' of drug treatment which extend into the following cell cycle(s). It also becomes clear that the degradation of exponential growth characteristics of quickly dividing cells during long-term, continuous drug exposure makes prediction of cell-cycle kinetic perturbations uncertain when derived from short-duration stathmokinetic experiments. However, with care, the joint application of 'short term' (e.g. stathmokinesis) and 'long term' (e.g. continuous exposure) techniques allow adequate quantitative insight into drug-perturbed cell-cycle kinetics. The applicability of modelling techniques is discussed: in the present instance it is limited to lower drug concentrations. For higher drug concentrations, effects like increased ploidy, ineffective division, etc., make it impossible in the present study to obtain a clear picture of the kinetics.     

Macrophage-Based Anti-Cancer Therapy: Modelling Different Modes of Tumour Targeting

Tumour hypoxia is associated with poor drug delivery and low rates of cell proliferation, factors that limit the efficacy of therapies that target proliferating cells. Since macrophages localise within hypoxic regions, a promising way to target hypoxic tumour cells involves engineering macrophages to express therapeutic genes under hypoxia. In this paper we develop mathematical models to compare the responses of avascular tumour spheroids to two modes of action: either the macrophages deliver an enzyme that activates an externally applied prodrug (bystander model), or they deliver cytotoxic factors directly (local model). The models we develop comprise partial differential equations for a multiphase mixture of tumour cells, macrophages and extracellular fluid, coupled to a moving boundary representing the spheroid surface. Chemical constituents, such as oxygen and drugs, diffuse within the multiphase mixture. Simulations of both models show the spheroid evolving to an equilibrium or to a travelling wave (multiple stable solutions are also possible). We uncover the parameter dependence of the wave speed and steady-state tumour size, and bifurcations between these solution forms. For some parameter sets, adding extra macrophages has a counterintuitive deleterious effect, triggering a bifurcation from bounded to unbounded tumour growth. While these features are common to the bystander and local models, the crucial difference is where cell death occurs. The bystander model is comparable to traditional chemotherapy, with poor targeting of hypoxic tumour cells; however, the local mode of action is more selective for hypoxic regions. We conclude that effective targeting of hypoxic tumour cells may require the use of drugs with limited mobility or whose action does not depend on cell proliferation.     

Modeling positive regulatory feedbacks in cell-cell interactions

Our current understanding of molecular mechanisms of cellular regulation still does not support quantitative predictions of the overall growth kinetics of normal or malignant tissues. However, discernment of the role of growth-factor mediated cell-cell communication in tissue kinetics is possible by the use of simple mathematical models. Here we discuss the design and use of mathematical models in quantifying the contribution of autocrine and paracrine (i.e., humoral) interactions to the kinetics of tissue growth. We present models that include a humorally mediated regulatory feedback among cells built into phenomenological mathematical models of growth. Application of these models to data exemplifies the finite contributions of positive feedback in cell-cell interactions to the overall tissue growth. In addition, we propose a perturbation approach to allow separation of cell-cell interactions dependent on the perturbing agent (such as hormone antagonists in hormone-dependent tissues) from cell-cell interactions independent of it.     

Calculation of the cumulative distribution function of the time to a small observable tumor

Multistage mathematical models of carcinogenesis (when applied to tumor incidence data) have historically assumed that the growth kinetics of cells in the malignant state are disregarded and the formation of a single malignant cell is equated with the emergence of a detectable tumor. The justification of this simplification is, from a mathematical point of view, to make the estimation of tumor incidence rates tractable. However, analytical forms are not mandatory in the estimation of tumor incidence rates. Portier et al. (1996b, Math. Biosci. 135, 129–146) have demonstrated the utility of the Kolmogorov backward equations in numerically calculating tumor incidence. By extending their results, the cumulative distribution function of the time to a small observable tumor may be numerically obtained.     

Ethanol-based proliposome delivery systems of paclitaxel for in vitro application against brain cancer cells

In this study the anticancer activity of paclitaxel-loaded nano-liposomes on glioma cell lines was investigated. Soya phosphatidylcholine:cholesterol (SPC:Chol), hydrogenated soya phosphatidylcholine:cholesterol (HSPC:Chol) or dipalmitoylphosphatidylcholine:cholesterol (DPPC:Chol) in 1:1?mole ratio were used to prepare ethanol-based proliposomes. Following hydration of proliposomes, the size of resulting vesicles was subsequently reduced to nanometer scale via probe-sonication. The resulting formulations were characterized in terms of size, zeta potential and morphology of the vesicles, and entrapment efficiency of paclitaxel (PX) as well as the final pH of the preparations. DPPC-liposomes entrapped 35–92% of PX compared to 27–74% and 25–60% entrapped by liposomes made from SPC and HSPC formulations respectively, depending on drug concentration. The entrapment efficiency of liposomes was dependent on the lipid bilayer properties and ability of PX to modify surface charge of the vesicles. In vitro cytotoxicity studies revealed that PX-liposome formulations were more selective at inhibiting the malignant cells. The cytotoxicity of PX-liposomes was dependent on their drug-entrapment efficiency. This study has shown PX-liposomes generated from proliposomes have selective activity against glioma cell lines, and the synthetic DPPC phospholipid was most suitable for maximized drug entrapment and highest activity against the malignant cells in vitro.     

X-ray-induced G2 arrest in ataxia telangiectasia lymphoblastoid cells

Sensitivity to X-ray-induced G2 arrest was compared between ataxia telangiectasia (AT) lymphoblastoid cells and normal human cells. Flow cytometrical analysis of cells following X-ray irradiation revealed that the fraction of cells with 4n DNA content was greater in AT cells than in normal cells as previously reported by other investigators. However, the other parameters for cell-cycle progression kinetics including mitotic indices, cumulative mitotic indices and cumulative labelled mitotic indices indicated that X-ray-induced G2 arrest as a function of dose in AT cells was indistinguishable from that in normal cells. Moreover, no significant difference in cell viability was noted between AT and normal cells until 48 h following X-irradiation up to 2.6 Gy, although X-irradiated AT cells, compared to normal cells, showed a significantly decreased survival in terms of cell multiplication in growth medium and colony formation in soft agar. These data collectively suggest that the greater accumulation of AT cells with 4n DNA content in flow cytometry cannot be attributed to more stringent irreversible blockage of cell-cycle progression at the G2 phase and eventual cell death there. The possible reasons for this greater accumulation are discussed.     

Selenite induces apoptosis in sarcomatoid malignant mesothelioma cells through oxidative stress

Malignant mesothelioma cells differentiate into sarcomatoid or epithelioid phenotypes. The sarcomatoid cell type is more resistant to chemotherapy and gives a worse prognosis. We have investigated whether selenite alone and in combination with doxorubicin induced apoptosis in variously differentiated mesothelioma cells. Selenite in concentrations that could potentially be administered to patients strongly inhibited the growth of the sarcomatoid mesothelioma cells (IC50 = 7.5 microM), whereas epithelioid cells were more sensitive to doxorubicin. Benign mesothelial cells remained largely unaffected. Selenite potentiated doxorubicin treatment. Apoptosis was the dominating mode of cell death. The toxicity of selenite was mediated by oxidative stress. Furthermore the activity of the thioredoxin system was directly dependent on the concentration of selenite. This offers a possible mechanism of action of selenite treatment. Our findings suggest that selenite is a promising new drug for the treatment of malignant mesothelioma.     

A structured-population model of Proteus mirabilis swarm-colony development

In this paper we present continuous age- and space-structured models and numerical computations of Proteus mirabilis swarm-colony development. We base the mathematical representation of the cell-cycle dynamics of Proteus mirabilis on those developed by Esipov and Shapiro, which are the best understood aspects of the system, and we make minimum assumptions about less-understood mechanisms, such as precise forms of the spatial diffusion. The models in this paper have explicit age-structure and, when solved numerically, display both the temporal and spatial regularity seen in experiments, whereas the Esipov and Shapiro model, when solved accurately, shows only the temporal regularity. The composite hyperbolic-parabolic partial differential equations used to model Proteus mirabilis swarm-colony development are relevant to other biological systems where the spatial dynamics depend on local physiological structure. We use computational methods designed for such systems, with known convergence properties, to obtain the numerical results presented in this paper.     

Timed,sequential administration of paclitaxel improves its cytotoxic effectiveness in a cell culture model

Paclitaxel (taxol) is a chemotherapeutic agent frequently used in combination with other anti-neoplastic drugs. It is most effective during the M phase of the cell-cycle and tends to cause synchronization in malignant cells lines. In this study, we investigated whether timed, sequential treatment based on the cell-cycle characteristics could be exploited to enhance the cytotoxic effect of paclitaxel. We characterized the cell-cycle properties of a rapidly multiplying cell line (Sp2, mouse myeloma cells) by propidium-iodide DNA staining such as the lengths of various cell cycle phases and population duplication time. Based on this we designed a paclitaxel treatment protocol that comprised a primary and a secondary, timed treatment. We found that the first paclitaxel treatment synchronized the cells at the G2/M phase but releasing the block by stopping the treatment allowed a large number of cells to enter the next cell-cycle by a synchronized manner. The second treatment was most effective during the time when these cells approached the next G2/M phase and was least effective when it occurred after the peak time of this next G2/M phase. Moreover, we found that after mixing Sp2 cells with another, significantly slower multiplying cell type (Jurkat human T-cell leukemia) at an initial ratio of 1:1, the ratio of the two different cell types could be influenced by timed sequential paclitaxel treatment at will. Our results demonstrate that knowledge of the cell-cycle parameters of a specific malignant cell type could improve the effectivity of the chemotherapy. Implementing timed chemotherapeutic treatments could increase the cytotoxicity on the malignant cells but also decrease the side-effects since other, non-malignant cell types will have different cell-cycle characteristic and be out of synch during the treatment.     

Drug scheduling of cancer chemotherapy based on natural actor-critic approach

Recently, reinforcement learning methods have drawn significant interests in the area of artificial intelligence, and have been successfully applied to various decision-making problems. In this paper, we study the applicability of the NAC (natural actor-critic) approach, a state-of-the-art reinforcement learning method, to the drug scheduling of cancer chemotherapy for an ODE (ordinary differential equation)-based tumor growth model. ODE-based cancer dynamics modeling is an active research area, and many different mathematical models have been proposed. Among these, we use the model proposed by de Pillis and Radunskaya (2003), which considers the growth of tumor cells and their interaction with normal cells and immune cells. The NAC approach is applied to this ODE model with the goal of minimizing the tumor cell population and the drug amount while maintaining the adequate population levels of normal cells and immune cells. In the framework of the NAC approach, the drug dose is regarded as the control input, and the reward signal is defined as a function of the control input and the cell populations of tumor cells, normal cells, and immune cells. According to the control policy found by the NAC approach, effective drug scheduling in cancer chemotherapy for the considered scenarios has turned out to be close to the strategy of continuing drug injection from the beginning until an appropriate time. Also, simulation results showed that the NAC approach can yield better performance than conventional pulsed chemotherapy.     

Experimental analysis and modelling of in vitro HUVECs proliferation in the presence of various types of drugs

Objectives: This study focuses on experimental analysis and corresponding mathematical simulation of in vitro HUVECs (human umbilical vein endothelial cells) proliferation in the presence of various types of drugs. Materials and methods: HUVECs, once seeded in Petri dishes, were expanded to confluence. Temporal profiles of total count obtained by classic haemocytometry and cell size distribution measured using an electronic Coulter counter, are quantitatively simulated by a suitable model based on the population balance approach. Influence of drugs on cell proliferation is also properly simulated by accounting for suitable kinetic equations. Results and discussion: The models’ parameters have been determined by comparison with experimental data related to cell population expansion and cell size distribution in the absence of drugs. Inhibition constant for each type of drug has been estimated by comparing the experimental data with model results concerning temporal profiles of total cell count. The reliability of the model and its predictive capability have been tested by simulating cell size distribution for experiments performed in the presence of drugs. The proposed model will be useful in interpreting effects of selected drugs on expansion of readily available human cells.     

Antiproliferative activity of long chain acylated esters of quercetin-3-O-glucoside in hepatocellular carcinoma HepG2 cells

Despite their strong role in human health, poor bioavailability of flavonoids limits their biological effects in vivo. Enzymatically catalyzed acylation of fatty acids to flavonoids is one of the approaches of increasing cellular permeability and hence, biological activities. In this study, six long chain fatty acid esters of quercetin-3-O-glucoside (Q3G) acylated enzymatically and were used for determining their antiproliferative action in hepatocellular carcinoma cells (HepG2) in comparison to precursor compounds and two chemotherapy drugs (Sorafenib and Cisplatin). Fatty acid esters of Q3G showed significant inhibition of HepG2 cell proliferation by 85 to 90% after 6 h and 24 h of treatment, respectively. The cell death due to these novel compounds was associated with cell-cycle arrest in S-phase and apoptosis observed by DNA fragmentation, fluorescent microscopy and elevated caspase-3 activity and strong DNA topoisomerase II inhibition. Interestingly, Q3G esters showed significantly low toxicity to normal liver cells than Sorafenib (P < 0.05), a chemotherapy drug for hepatocellular carcinoma. Among all, oleic acid ester of Q3G displayed the greatest antiproliferation action and a high potential as an anti-cancer therapeutic. Overall, the results of the study suggest strong antiproliferative action of these novel food-derived compounds in treatment of cancer.     

The proliferation rate paradox in antimitotic chemotherapy

Cytotoxic cancer chemotherapy drugs are believed to gain selectivity by targeting cells that proliferate rapidly. However, the proliferation rate is low in many chemosensitive human cancers, and it is not clear how a drug that only kills dividing cells could promote tumor regression. Four potential solutions to this "proliferation rate paradox" are discussed for the microtubule-stabilizing drug paclitaxel: drug retention in tumors, killing of quiescent cells, targeting of noncancer cells in the tumor, and bystander effects. Testing these potential mechanisms of drug action will facilitate rational improvement of antimitotic chemotherapy and perhaps cytotoxic chemotherapy more generally.     

A mathematical model of the effects of hypoxia on the cell-cycle of normal and cancer cells

The evolution of the cell-cycle is known to be influenced by environmental conditions, including lack of extracellular oxygen (hypoxia). Notably, hypoxia appears to have different effects on normal and cancer cells. Whereas both experience hypoxia-induced arrest of the G1 phase of the cell-cycle (i.e. delay in the transition through the restriction point), experimental evidence suggests that only cancer cells undergo hypoxia-induced quiescence (i.e. the transition of the cell to a latent state in which most of the cell functions, including proliferation, are suspended). Here, we extend a model for the cell-cycle due to Tyson and Novak (J. Theor. Biol. 210 (2001) 249) to account for the action of the protein p27. This protein, whose expression is upregulated under hypoxia, inhibits the activation of the cyclin dependent kinases (CDKs), thus preventing DNA synthesis and delaying the normal progression through the cell-cycle. We use a combination of numerical and analytic techniques to study our model. We show that it reproduces many features of the response to hypoxia of normal and cancer cells, as well as generating experimentally testable predictions. For example our model predicts that cancer cells can undergo quiescence by increasing their levels of p27, whereas for normal cells p27 expression decreases when the cellular growth rate increases.