The Worst Drug Rule Revisited: Mathematical Modeling of Cyclic Cancer Treatments

In drug treatments of cancer, cyclic treatment strategies are characterized by alternating applications of two (or more) different drugs, given one at a time. One of the main problems of drug treatment in cancer is associated with the generation of drug resistance by mutations of cancerous cells. We use mathematical methods to develop general guidelines on optimal cyclic treatment scheduling, with the aim of minimizing the resistance generation. We define a condition on the drugs’ potencies which allows for a relatively successful application of cyclic therapies. We find that the best strategy is to start with the stronger drug, but use longer cycle durations for the weaker drug. We further investigate the situation where a degree of cross-resistance is present, such that certain mutations cause cells to become resistant to both drugs simultaneously. We show that the general rule (best-drug-first, worst-drug-longer) is unchanged by the presence of cross-resistance. We design a systematic method to test all strategies and come up with the optimal timing and drug order. The role of various constraints in the optimal therapy design, and in particular, suboptimal treatment durations and drug toxicity, is considered. The connection with the “worst drug rule” of Day (Cancer Res. 46:3876, 1986b) is discussed.     

Interaction of Tumor with Its Micro-environment: A Mathematical Model

This paper is concerned with early development of transformed epithelial cells (TECs) in the presence of fibroblasts in the tumor micro-environment. These two types of cells interact by means of cytokines such as transforming growth factor (TGF-β) and epidermal growth factor (EGF) secreted, respectively, by the TECs and the fibroblasts. As this interaction proceeds, TGF-β induces fibroblasts to differentiate into myofibroblasts which secrete EGF at a larger rate than fibroblasts. We monitor the entire process in silico, in a setup which mimics experiments in a Tumor Chamber Invasion Assay, where a semi-permeable membrane coated by extracellular matrix (ECM) is placed between two chambers, one containing TECs and another containing fibroblasts. We develop a mathematical model, based on a system of PDEs, that includes the interaction between TECs, fibroblasts, myofibroblasts, TGF-β, and EGF, and we show how model parameters affect tumor progression. The model is used to generate several hypotheses on how to slow tumor growth and invasion. In an Appendix, it is proved that the mathematical model has a unique global in-time solution.     

The Role of Viral Infection in Pest Control: A Mathematical Study

In this paper, we propose a mathematical model of viral infection in pest control. As the viral infection induces host lysis which releases more virus into the environment, on the average ‘κ’ viruses per host, κ∈(1,∞), so the ‘virus replication parameter’ is chosen as the main parameter on which the dynamics of the infection depends. There exists a threshold value κ ₀ beyond which the infection persists in the system. Still for increasing the value of κ, the endemic equilibrium bifurcates towards a periodic solution, which essentially indicates that the viral pesticide has a density-dependent ‘numerical response’ component to its action. Investigation also includes the dependence of the process on predation of natural enemy into the system. A concluding discussion with numerical simulation of the model is also presented.     

Interactions Between the Immune System and Cancer: A Brief Review of Non-spatial Mathematical Models

We briefly review spatially homogeneous mechanistic mathematical models describing the interactions between a malignant tumor and the immune system. We begin with the simplest (single equation) models for tumor growth and proceed to consider greater immunological detail (and correspondingly more equations) in steps. This approach allows us to clarify the necessity for expanding the complexity of models in order to capture the biological mechanisms we wish to understand. We conclude by discussing some unsolved problems in the mathematical modeling of cancer-immune system interactions.     

Modeling and Estimation of Kinetic Parameters and Replicative Fitness of HIV-1 from Flow-Cytometry-Based Growth Competition Experiments

Growth competition assays have been developed to quantify the relative fitness of HIV-1 mutants. In this article, we develop mathematical models to describe viral/cellular dynamic interactions in the assay system from which the competitive fitness indices or parameters are defined. In our previous HIV-viral fitness experiments, the concentration of uninfected target cells was assumed to be constant (Wu et al. 2006). But this may not be true in some experiments. In addition, dual infection may frequently occur in viral fitness experiments and may not be ignorable. Here, we relax these two assumptions and extend our earlier viral fitness model (Wu et al. 2006). The resulting models then become nonlinear ODE systems for which closed-form solutions are not achievable. In the new model, the viral relative fitness is a function of time since it depends on the target cell concentration. First, we studied the structure identifiability of the nonlinear ODE models. The identifiability analysis showed that all parameters in the proposed models are identifiable from the flow-cytometry-based experimental data that we collected. We then employed a global optimization approach (the differential evolution algorithm) to directly estimate the kinetic parameters as well as the relative fitness index in the nonlinear ODE models using nonlinear least square regression based on the experimental data. Practical identifiability was investigated via Monte Carlo simulations.     

Mathematical Modeling of Ischemia–Reperfusion Injury and Postconditioning Therapy

Reperfusion (restoration of blood flow) after a period of ischemia (interruption of blood flow) can paradoxically place tissues at risk of further injury: so-called ischemia–reperfusion injury or IR injury. Recent studies have shown that postconditioning (intermittent periods of further ischemia applied during reperfusion) can reduce IR injury. We develop a mathematical model to describe the reperfusion and postconditioning process following an ischemic insult, treating the blood vessel as a two-dimensional channel, lined with a monolayer of endothelial cells that interact (respiration and mechanotransduction) with the blood flow. We investigate how postconditioning affects the total cell density within the endothelial layer, by varying the frequency of the pulsatile flow and the oxygen concentration at the inflow boundary. We find that, in the scenarios we consider, the pulsatile flow should be of high frequency to minimize cellular damage, while oxygen concentration at the inflow boundary should be held constant, or subject to only low-frequency variations, to maximize cell proliferation.     

Mathematical Modeling of Heterogeneous Electrophysiological Responses in Human β-Cells

Electrical activity plays a pivotal role in glucose-stimulated insulin secretion from pancreatic -cells. Recent findings have shown that the electrophysiological characteristics of human -cells differ from their rodent counterparts. We show that the electrophysiological responses in human -cells to a range of ion channels antagonists are heterogeneous. In some cells, inhibition of small-conductance potassium currents has no effect on action potential firing, while it increases the firing frequency dramatically in other cells. Sodium channel block can sometimes reduce action potential amplitude, sometimes abolish electrical activity, and in some cells even change spiking electrical activity to rapid bursting. We show that, in contrast to L-type -channels, P/Q-type -currents are not necessary for action potential generation, and, surprisingly, a P/Q-type -channel antagonist even accelerates action potential firing. By including SK-channels and dynamics in a previous mathematical model of electrical activity in human -cells, we investigate the heterogeneous and nonintuitive electrophysiological responses to ion channel antagonists, and use our findings to obtain insight in previously published insulin secretion measurements. Using our model we also study paracrine signals, and simulate slow oscillations by adding a glycolytic oscillatory component to the electrophysiological model. The heterogenous electrophysiological responses in human -cells must be taken into account for a deeper understanding of the mechanisms underlying insulin secretion in health and disease, and as shown here, the interdisciplinary combination of experiments and modeling increases our understanding of human -cell physiology.     

A Mathematical Model of Schistosoma mansoni in Biomphalaria glabrata with Control Strategies

We describe and analyze a mathematical model for schistosomiasis in which infected snails are distinguished from susceptible through increased mortality and no reproduction. We based the model on the same derivation as Anderson and May (J. Anim. Ecol. 47:219–247, 1978), Feng and Milner (A New Mathematical Model of Schistosomiasis, Mathematical Models in Medical and Health Science, Nashville, TN, 1997. Innov. Appl. Math., Vanderbilt Univ. Press, Nashville, pp. 117–128, 1998), and May and Anderson (J. Anim. Ecol. 47:249–267, 1978), but used logistic growth both in human and snail hosts. We introduce a parameter r, the effective coverage of medical treatment/prevention to control the infection. We determine a reproductive number for the disease directly related to its persistence and extinction. Finally, we obtain a critical value for r that indicates the minimum treatment effort needed in order to clear out the disease from the population.     

Modeling and Analysis of the Effect of Training on \dot{V}O_{2} Kinetics and Anaerobic Capacity

In this paper, we present an application of a number of tools and concepts for modeling and analyzing raw, unaveraged, and unedited breath-by-breath oxygen uptake data. A method for calculating anaerobic capacity is used together with a model, in the form of a set of coupled nonlinear ordinary differential equations to make predictions of the kinetics, the time to achieve a percentage of a certain constant oxygen demand, and the time limit to exhaustion at intensities other than those in which we have data. Speeded oxygen kinetics and increased time limit to exhaustion are also investigated using the eigenvalues of the fixed points of our model. We also use a way of analyzing the oxygen uptake kinetics using a plot of vs which allows one to observe both the fixed point solutions and also the presence of speeded oxygen kinetics following training. A method of plotting the eigenvalue versus oxygen demand is also used which allows one to observe where the maximum amplitude of the so-called slow component will be and also how training has changed the oxygen uptake kinetics by changing the strength of the attracting fixed point for a particular demand.     

Maximum Urine Concentrating Capability in a Mathematical Model of the Inner Medulla of the Rat Kidney

In a mathematical model of the urine concentrating mechanism of the inner medulla of the rat kidney, a nonlinear optimization technique was used to estimate parameter sets that maximize the urine-to-plasma osmolality ratio (U/P) while maintaining the urine flow rate within a plausible physiologic range. The model, which used a central core formulation, represented loops of Henle turning at all levels of the inner medulla and a composite collecting duct (CD). The parameters varied were: water flow and urea concentration in tubular fluid entering the descending thin limbs and the composite CD at the outer-inner medullary boundary; scaling factors for the number of loops of Henle and CDs as a function of medullary depth; location and increase rate of the urea permeability profile along the CD; and a scaling factor for the maximum rate of NaCl transport from the CD. The optimization algorithm sought to maximize a quantity E that equaled U/P minus a penalty function for insufficient urine flow. Maxima of E were sought by changing parameter values in the direction in parameter space in which E increased. The algorithm attained a maximum E that increased urine osmolality and inner medullary concentrating capability by 37.5% and 80.2%, respectively, above base-case values; the corresponding urine flow rate and the concentrations of NaCl and urea were all within or near reported experimental ranges. Our results predict that urine osmolality is particularly sensitive to three parameters: the urea concentration in tubular fluid entering the CD at the outer-inner medullary boundary, the location and increase rate of the urea permeability profile along the CD, and the rate of decrease of the CD population (and thus of CD surface area) along the cortico-medullary axis.     

Response of Tumor Spheroids to Radiation: Modeling and Parameter Estimation

We propose a spatially distributed continuous model for the spheroid response to radiation, in which the oxygen distribution is represented by means of a diffusion-consumption equation and the radiosensitivity parameters depend on the oxygen concentration. The induction of lethally damaged cells by a pulse of radiation, their death, and the degradation of dead cells are included. The compartments of lethally damaged cells and of dead cells are subdivided into different subcompartments to simulate the delays that occur in cell death and cell degradation, with a gain in model flexibility. It is shown that, for a single irradiation and under the hypothesis of a sufficiently small spheroid radius, the model can be reformulated as a linear stationary ordinary differential equation system. For this system, the parameter identifiability has been investigated, showing that the set of unknown parameters can be univocally identified by exploiting the response of the model to at least two different radiation doses. Experimental data from spheroids originated from different cell lines are used to identify the unknown parameters and to test the predictive capability of the model with satisfactory results.     

Modeling Proximal Tubule Cell Homeostasis: Tracking Changes in Luminal Flow

During normal kidney function, there are routinely wide swings in proximal tubule fluid flow and proportional changes in Na⁺ reabsorption across tubule epithelial cells. This “glomerulotubular balance” occurs in the absence of any substantial change in cell volume, and is thus a challenge to coordinate luminal membrane solute entry with peritubular membrane solute exit. In this work, linear optimal control theory is applied to generate a configuration of regulated transporters that could achieve this result. A previously developed model of rat proximal tubule epithelium is linearized about a physiologic reference condition; the approximate linear system is recast as a dynamical system; and a Riccati equation is solved to yield the optimal linear feedback that stabilizes Na⁺ flux, cell volume, and cell pH. The first observation is that optimal feedback control is largely consigned to three physiologic variables, cell volume, cell electrical potential, and lateral intercellular hydrostatic pressure. Parameter modulation by cell volume stabilizes cell volume; parameter modulation by electrical potential or interspace pressure act to stabilize Na⁺ flux and cell pH. This feedback control is utilized in a tracking problem, in which reabsorptive Na⁺ flux varies over a factor of two, in order to represent a substantial excursion of glomerulotubular balance. The resulting control parameters consist of two terms, an autonomous term and a feedback term, and both terms include transporters on both luminal and peritubular cell membranes. Overall, the increase in Na⁺ flux is achieved with upregulation of luminal Na⁺/H⁺ exchange and Na⁺–glucose cotransport, with increased peritubular Na⁺–3HCO₃⁻ and K⁺–Cl⁻ cotransport, and with increased Na⁺, K⁺–ATPase activity. The configuration of activated transporters emerges as a testable hypothesis of the molecular basis for glomerulotubular balance. It is suggested that the autonomous control component at each cell membrane could represent the cytoskeletal effects of luminal flow.     

Optimization of Culture Conditions for the Production of Extracellular Agarases from Newly Isolated Pseudomonas Aeruginosa AG LSL-11

Summary An agar-degrading bacterium capable of utilizing agar as sole source of carbon and energy was isolated from sea water by enrichment culture technique. The bacterium was identified as Pseudomonas aeruginosa and the culture conditions were standardized for the maximal production of extracellular agarases. The bacterium grew in the pH range 5.0–11.0, optimal between pH 7.0 and 8.0; temperature between 25 °C and 37 °C, optimal at 30 °C and sodium chloride concentration 0–8% and optimal at 2% respectively. The agarases secreted by Pseudomonas aeruginosa AG LSL-11 were inducible by agar and not by any other simple sugars tested. Maximal agarase production was observed at pH 8.0, and temperature 30 °C. The bacterium had no requirement for NaCl for both growth and production of agarases. The bacterium did not utilize other polysaccharides like ĸ-carrageenan, alginate, cellulose and CMC. The activity staining of partially purified agarase preparation after native-PAGE revealed the presence of three different agarases, agarase LSL-11a, LSL-11b and LSL-11c, whose molecular weights were estimated to be 76, 64 and 46 kDa respectively.     

An Algorithm of Mathematical Modeling of Laser-Induced Indirect Thermotherapy of a Baker’s Cyst

Biophysics - Abstract—A thermophysical model of heat transfer from a liquid heated by radiation into biological tissues with a cyst of an individual picture of temperature fields, which takes...     

Extracellular α-amylase from Thermus filiformis Ork A2: purification and biochemical characterization

An extracellular α-amylase produced by the thermophilic bacterium Thermus filiformis Ork A2 was purified from cell-free culture supernatant by ion exchange chromatography. The molecular mass was estimated to be 60 000 Da by sodium dodecyl sulfate polyacrylamide gel electrophoresis. The enzyme was rich in both basic and hydrophobic amino acids, presenting the following NH₂-terminal amino acid sequence: Thr-Ala-Asp-Leu-Ile-Val-Lys-Ile-Asn-Phe. Amylolytic activity on soluble starch was optimal at pH 5.5–6.0 and 95°C, and the enzyme was stable in the pH range of 4.0–8.0. Calcium enhanced thermostability at temperatures above 80°C, increasing the half-life of activity to more than 8 h at 85°C, 80 min at 90°C, and 19 min at 95°C. Ethylenediaminetetraacetic acid (EDTA) inhibited amylase activity, the inhibition being reversed by the addition of calcium or strontium ions. The α-amylase was also inhibited by copper and mercuric ions, and p-chloromercuribenzoic acid, the latter being reversed in the presence of dithiothreitol. Dithiothreitol and β-mercaptoethanol activated the enzyme. The α-amylase exhibited Michaelis-Menten kinetics for starch, with a K _m of 5.0 mg·ml⁻¹ and k _cat/K _m of 5.2 × 10⁵ ml·mg⁻¹ s⁻¹. Similar values were obtained for amylose, amylopectin, and glycogen. The hydrolysis pattern was similar for maltooligosaccharides and polysaccharides, with maltose being the major hydrolysis product. Glucose and maltotriose were generated as secondary products, although glucose was produced in high levels after a 6-h digestion. To our knowledge this is the first report of the characterization of an α-amylase from a strain of the genus Thermus. Received: June 2, 1997 / Accepted: September 16, 1997