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At least 1,405 records · Page 78Linked to original sources
PubMed1996

A mathematical model of the first steps of tumour-related angiogenesis: capillary sprout formation and secondary branching.

The growth of a solid tumour is dependent on an adequate supply of nutrients. A tumour can establish a blood supply by inducing neighbouring blood vessels to sprout and grow towards it, a process known as angiogenesis. The tumour cells may secrete a number of diffusible chemicals which stimulate endothelial cell to migrate, to rearrange themselves into capillary tubes or sprouts, and to proliferate. In this paper we focus firstly upon the early stage of angiogenesis wherein the endothelial cells group together in the parent vessel to form the initial capillary-sprout buds. A mathematical model for the formation of the capillary buds is presented which focuses on the potential role that haptotaxis may play. In Section 2 we turn attention to the endothelial cells within the growing and developing capillary sprouts as they migrate towards the tumour cells. Once again the potential role of haptotaxis is focused upon.

Animals↗
PubMed2001

Mathematical models and simulations of bacterial growth and chemotaxis in a diffusion gradient chamber.

The diffusion gradient chamber (DGC) is a novel device developed to study the response of chemotactic bacteria to combinations of nutrients and attractants [7]. Its purpose is to characterize genetic variants that occur in many biological experiments. In this paper, a mathematical model which describes the spatial distribution of a bacterial population within the DGC is developed. Mathematical analysis of the model concerning positivity and boundedness of the solutions are given. An ADI (Alternating Direction Implicit) method is constructed for finding numerical solutions of the model and carrying out computer simulations. The numerical results of the model successfully reproduced the patterns that were observed in the experiments using the DGC.

Bacteria↗
PubMed2003

Mathematical modeling of drug release from bioerodible microparticles: effect of gamma-irradiation.

Bioerodible polymers used in controlled drug delivery systems, such as poly(lactic-co-glycolic acid) (PLGA) undergo radiolytic degradation during gamma-irradiation. In spite of the considerable practical importance, yet only little knowledge is available on the consequences of this sterilization method on the resulting drug release patterns in a quantitative way. The major objectives of the present study were: (i) to monitor the effects of different gamma-irradiation doses on the physicochemical properties of drug-free and drug-loaded, PLGA-based microparticles; (ii) to analyze the obtained experimental results using adequate mathematical models; (iii) to get further insight into the occurring physical and chemical phenomena; and (iv) to relate the applied gamma-irradiation dose in a quantitative way to the resulting drug release rate. 5-Fluorouracil-loaded, PLGA-based microparticles were prepared with an oil-in-water solvent extraction method and exposed to gamma-irradiation doses ranging from 0 to 33 kGy. Size exclusion chromatography, differential scanning calorimetry, scanning electron microscopy, particle size analysis, determination of the actual drug loading and in vitro drug release kinetics were used to study the effects of the gamma-irradiation dose on the physicochemical properties of the microparticles. Two mathematical models-a simplified and a more comprehensive one-were used to analyze the experimental results. The simplified model considers drug diffusion based on Fick's second law for spherical geometry and a Higuchi-like pseudo-steady-state approach. The complex model combines Monte Carlo simulations (describing polymer erosion) with partial differential equations quantifying drug diffusion with time-, position- and direction-dependent diffusivities. Interestingly, exponential relationships between the gamma-irradiation dose and the initial drug diffusivity within the microparticles could be established. Based on this knowledge both models were used to predict the resulting drug release kinetics as a function of the gamma-irradiation dose. Importantly, the theoretical predictions were confirmed by experimental results.

Dose-Response Relationship, Radiation↗
PubMed2004

Reduction of mathematical models of signal transduction networks: simulation-based approach applied to EGF receptor signalling.

Biological systems and, in particular, cellular signal transduction pathways are characterised by their high complexity. Mathematical models describing these processes might be of great help to gain qualitative and, most importantly, quantitative knowledge about such complex systems. However, a detailed mathematical description of these systems leads to nearly unmanageably large models, especially when combining models of different signalling pathways to study cross-talk phenomena. Therefore, simplification of models becomes very important. Different methods are available for model reduction of biological models. Importantly, most of the common model reduction methods cannot be applied to cellular signal transduction pathways. Using as an example the epidermal growth factor (EGF) signalling pathway, we discuss how quantitative methods like system analysis and simulation studies can help to suitably reduce models and additionally give new insights into the signal transmission and processing of the cell.

Algorithms↗
PubMed1994

Mathematical modelling for the generation of L-[3-2H,3-13C]lactic acid isotopomers by erythrocytes exposed to either D-[1-13C]glucose or D-[6-13C]glucose in the presence of 2H2O.

The production of C3-trisdeuterated, bisdeuterated, monodeuterated or non-deuterated L-[3-13C]lactate by human erythrocytes exposed to either D-[1-13C]glucose or D-[6-13C]glucose in the presence of 2H2O can be assessed by 13C NMR spectroscopy. Such a deuteration may occur at the level of the reactions catalyzed by phosphoglucoisomerase, phosphomannoisomerase, pyruvate kinase and glutamate-pyruvate transaminase. In this report, a mathematical model is proposed for the analysis of experimental data. It allows to estimate the relative extent of deuteration at each step of D-glucose metabolism. This approach may thus provide novel information on the extent of back-and-forth interconversion of either hexose 6-phosphates in both the phosphoglucoisomerase and phosphomannoisomerase reactions or pyruvate and L-alanine in the reaction catalyzed by glutamate-pyruvate transaminase.

Alanine Transaminase↗
PubMed1979

[Mathematical models for calculating volume growth, represented by the yield tables for the principal wood-producing species of Czechoslovakia].

Methods for the construction of volume increment tables for stands on the basis of the volume mean stem are described. The volume increment of the stand (iv) is expressed by the equation (Formula: see text). The stem number of stands (N) is determined directly by 100% or representative enumeration. The mean annual increment (iv) is calculated with the aid of the regression function (Formula: see text) (v) being volume of mean stem and (t) age of stand. The relations (2) were derived for 5 tree species (spruce, fir, pine, beach, oak) from the data of the new Czechoslovak yields tables (1975). The mathematical model for the tree species spruce and pine was tested on the basis of empirical material collected by the Tharandt department of forest management an forest yield science (within the framework of joint research activities) and good results were obtained. The results should be regarded as provisional ones. Further tests are envisaged.

Czechoslovakia↗
PubMed1981

[Mathematical modelling of glycolysis and adenine nucleotide metabolism of human erythrocytes. I. Reaction-kinetic statements, analysis of in vivo state and determination of starting conditions for in vitro experiments].

A mathematical model of energy metabolism of human red cells is presented, which includes besides the glycolytic reactions the adenine nucleotide metabolism. The model is based on the network of chemical reactions, the thermodynamic equilibrium constants of fast reversible reactions and on the kinetic equations for irreversible enzyme reactions. The model consists of a system of 16 differential equations and allows the mathematical evaluation of metabolic levels in the steady state of energy metabolism corresponding to the in vivo state erythrocytes with the kinetic data for the enzymes derived from in vitro experiments. The dependence of the levels of metabolites in the steady state on the activity of some enzymes is analysed to characterize the regulatory properties of the system. The comparison of the steady state levels of the model with experimental data makes it possible to estimate values of some controversial enzyme parameters. Estimates of the kinetic parameters of the following intracellular processes are presented: 1) rate constant of AMP-phosphatase, 2) maximum rate of adenylate deaminase, 3) activity of adenine phosphoribosylpyrophosphate transferase and 4) adenosine transport through the cell membrane. The simulation of the preparatory phase before incubation of erythrocytes indicates, that the model also permits to compute the time course of changes of levels of metabolites. To solve the initial problem the stiff differential equation system is integrated numerically by an efficient program without the application of the quasi-steady-state approximation.

Adenine Nucleotides↗
PubMed2000

Effect of feeding strategy on Zymomonas mobilis CP4 fed-batch fermentations and mathematical modeling of the system.

In this work, the effect of the feeding strategy in Zymomonas mobilis CP4 fed-batch fermentations on the final biomass and ethanol concentrations was studied. Highest glucose yields to biomass (0.018 g/g) and to ethanol (0.188 g/g) were obtained in fed-batch fermentations carried out using different feeding rates with a glucose concentration in the feed equal to 100 g/l. Lower values (0.0102 g biomass/g glucose and 0.085 g ethanol/ g glucose) were obtained when glucose accumulated to levels higher than 60 g/l. On the other hand, the highest biomass (5 g/l) and ethanol (39 g/l) concentrations were obtained using a glucose concentration in the feed equal to 220 g/l and exponentially varied feeding rates. Experimental data were used to validate the mathematical model of the system. The prediction errors of the model are 0.39, 14.36 and 3.24 g/l for the biomass, glucose and ethanol concentrations, respectively. Due to the complex relationship for describing the specific growth rate, a fed-batch culture in which glucose concentration is constant would not optimize the process.

Ethanol↗
PubMed1991

A mathematical model for the diffusion of tumour angiogenesis factor into the surrounding host tissue.

Unless they are furnished with an adequate blood supply and a means of disposing of their waste products by a mechanism other than diffusion, solid tumours cannot grow beyond a few millimetres in diameter. It is now a well-established fact that, in order to accomplish this neovascularization, solid tumours secrete a diffusable chemical compound known as tumour angiogenesis factor (TAF) into the surrounding tissue. This stimulates nearby blood vessels to migrate towards and finally penetrate the tumour. Once provided with the new supply of nutrient, rapid growth takes place. In this paper, a mathematical model is presented for the diffusion of TAF into the surrounding tissue. The complete process of angiogenesis is made up of a sequence of several distinct events and the model is an attempt to take into account as many of these as possible. In the diffusion equation for the TAF, a decay term is included which models the loss of the chemical into the surrounding tissue itself. A threshold distance for the TAF is incorporated in an attempt to reflect the results from experiments on corneal implants in test animals. By formulating the problems in terms of a free boundary problem, the extent of the diffusion of TAF into the surrounding tissue can be monitored. Finally, by introducing a sink term representing the action of proliferating endothelial cells, the boundary of the TAF is seen to recede, and hence the position and movement of the capillaries can be indirectly followed. The changing concentration gradient observed as the boundary recedes may offer a possible explanation for the initiation of anastomosis. Several functions are considered as possible sink terms and numerical results are presented. The situation where the tumour (i.e. the source of TAF) is removed is also considered.

Angiogenesis Inducing Agents↗
PubMed2005

Predicting human chronically paralyzed muscle force: a comparison of three mathematical models.

Chronic spinal cord injury (SCI) induces detrimental musculoskeletal adaptations that adversely affect health status, ranging from muscle paralysis and skin ulcerations to osteoporosis. SCI rehabilitative efforts may increasingly focus on preserving the integrity of paralyzed extremities to maximize health quality using electrical stimulation for isometric training and/or functional activities. Subject-specific mathematical muscle models could prove valuable for predicting the forces necessary to achieve therapeutic loading conditions in individuals with paralyzed limbs. Although numerous muscle models are available, three modeling approaches were chosen that can accommodate a variety of stimulation input patterns. To our knowledge, no direct comparisons between models using paralyzed muscle have been reported. The three models include 1) a simple second-order linear model with three parameters and 2) two six-parameter nonlinear models (a second-order nonlinear model and a Hill-derived nonlinear model). Soleus muscle forces from four individuals with complete, chronic SCI were used to optimize each model's parameters (using an increasing and decreasing frequency ramp) and to assess the models' predictive accuracies for constant and variable (doublet) stimulation trains at 5, 10, and 20 Hz in each individual. Despite the large differences in modeling approaches, the mean predicted force errors differed only moderately (8-15% error; P=0.0042), suggesting physiological force can be adequately represented by multiple mathematical constructs. The two nonlinear models predicted specific force characteristics better than the linear model in nearly all stimulation conditions, with minimal differences between the two nonlinear models. Either nonlinear mathematical model can provide reasonable force estimates; individual application needs may dictate the preferred modeling strategy.

Adult↗
PubMed1996

Making sense of the combined effect of interleukin-2 and interleukin-4 on lymphocytes using a mathematical model.

The cytokines are the information superhighway of the immune system. They are an important component of the integrated behavior of the system. In order to be able to have a good understanding of the immune system, we must be able to model the effect of cytokines and their combined effect. This work is a step in that direction. We study the combined effect of two cytokines: interleukin-2 (IL-2) and interleukin-4 (IL-4) on some cells of the immune system. Interleukin-2 and interleukin-4 are important growth and differentiation factors for B and T cells. Interleukin-4 antagonizes the effect of interleukin-2 on B cells and some T cells while it synergizes with interleukin-2 on other T cells. We build a mathematical model of the interaction of both cytokines on T and B cells as a building block toward a model of the Th1/Th2 cross-regulation. The response of a given cell to the combination of interleukin-2 and interleukin-4 is shown to involve competing dynamical effects which can lead to either antagonistic or synergistic combined effect.

Animals↗
PubMed1998

A mathematical model for the generation and control of a pH gradient in an immobilized enzyme system involving acid generation.

An optimal pH control technique has been developed for multistep enzymatic synthesis reactions where the optimal pH differs by several units for each step. This technique separates an acidic environment from a basic environment by the hydrolysis of urea within a thin layer of immobilized urease. With this technique, a two-step enzymatic reaction can take place simultaneously, in proximity to each other, and at their respective optimal pH. Because a reaction system involving an acid generation represents a more challenging test of this pH control technique, a number of factors that affect the generation of such a pH gradient are considered in this study. The mathematical model proposed is based on several simplifying assumptions and represents a first attempt to provide an analysis of this complex problem. The results show that, by choosing appropriate parameters, the pH control technique still can generate the desired pH gradient even if there is an acid-generating reaction in the system.

Acids↗
PubMed1981

Comparison of four mathematical models for the calculation of radioimmunoassay data of LH, FSH, and GH.

Weighted linear logit-log regression, point-to-point logit-log interpolation, smoothing spline approximation and the four-parameter logistic function calculated by non-linear regression have been compared. The data for comparison have been obtained from two different pool-sera for each of the LH-, FSH- and GH-RIA and from the basal serum LH values of two populations of children. The Wilcoxon matched pairs signed rank test was used for comparison: For GH there is no significant difference between all methods, for FSH the weighted linear logit-log regression and spline approximation appeared to be equivalent, but for LH no unequivocal assertion can be made. There is no significant difference between the mathematical models for determination of hormone concentration within one assay-run of a population as exemplified for LH. In addition, pool-sera data were subjected to an analysis of variance and the comparison of the results revealed that the different models did not lead to different statements about assay performance. The point-to-point logit-log interpolation is proposed as most simple curvilinear approximation for assays which cannot be linearized by logit-log transformation.

Adolescent↗
PubMed1998

Mathematical models to correlate amniotic fluid index and amniotic fluid volume.

PURPOSE: To describe a predictable relationship that relates amniotic fluid index (AFI) to amniotic fluid volume (AFV) and improve the accuracy of AFI to detect true oligohydramnios. METHODS: Data from 42 parturients (group I) who underwent measurements of amniotic fluid sonographically (amniotic fluid index) as well as by dye-dilution technique was used to relate AFI to AFV. Subsequently, 22 consecutive women (group II) were used to test the accuracy of the equation to predict true oligohydramnios. RESULTS: In group II, 11 of 22 patients had true oligohydramnios and the sensitivity, specificity, positive and negative predictive values of AFI < or = 5.0 to detect a confirmed AFV < 500 mL were 0%, 91%, 0%, and 48%, respectively. These values of AFI, when used in conjunction with the equation, improved to 73%, 55%, 62%, 67%, respectively. With AFI and the equation, significantly more patients in group II with true oligohydramnios (8 of 11) could be detected than with using AFI alone (0 of 11; p = 0.002). CONCLUSION: AFI is poor predictor of true oligohydramnios. Using the mathematical model, the detection rate of oligohydramnios is significantly improved.

Amniotic Fluid↗
PubMed1997

A mathematical model for the capillary endothelial cell-extracellular matrix interactions in wound-healing angiogenesis.

Angiogenesis, the process by which new blood capillaries grow into a tissue from surrounding parent vessels, is a key event in dermal wound healing, malignant-tumour growth, and other pathologic conditions. In wound healing, new capillaries deliver vital metabolites such as amino acids and oxygen to the cells in the wound which are involved in a complex sequence of repair processes. The key cellular constituents of these new capillaries are endothelial cells: their interactions with soluble biochemical and insoluble extracellular matrix (ECM) proteins have been well documented recently, although the biological mechanisms underlying wound-healing angiogenesis are incompletely understood. Considerable recent research, including some continuum mathematical models, have focused on the interactions between endothelial cells and soluble regulators (such as growth factors). In this work, a similar modelling framework is used to investigate the roles of the insoluble ECM substrate, of which collagen is the predominant macromolecular protein. Our model consists of a partial differential equation for the endothelial-cell density (as a function of position and time) coupled to an ordinary differential equation for the ECM density. The ECM is assumed to regulate cell movement (both random and directed) and proliferation, whereas the cells synthesize and degrade the ECM. Analysis and numerical solutions of these equations highlights the roles of these processes in wound-healing angiogenesis. A nonstandard approximation analysis yields insight into the travelling-wave structure of the system. The model is extended to two spatial dimensions (parallel and perpendicular to the plane of the skin), for which numerical simulations are presented. The model predicts that ECM-mediated random motility and cell proliferation are key processes which drive angiogenesis and that the details of the functional dependence of these processes on the ECM density, together with the rate of ECM remodelling, determine the qualitative nature of the angiogenic response. These predictions are experimentally testable, and they may lead towards a greater understanding of the biological mechanisms involved in wound-healing angiogenesis.

Animals↗
PubMed1990

[The predictability of the radiotherapy response in epidermoid tumors of the head and neck. A review of the literature and mathematical models for choosing the fractionated doses].

Literature data show that the term "squamous-cell head and neck cancer" includes a wide range of epidermoid cell subgroups, each of them with its own intrinsic radiosensitivity (Do values ranging from 107 to 184 in primary tumors, and 146-263 in recurrences; n values ranging from 1 to 5; and, if we consider linear-quadratic model alpha values from 0.273 to 0.490 and beta values from 0.029-0.045). Different sublethal and potential lethal repair times are also observed (4-6 hours and 12-24 hours, respectively), and structural tissue heterogeneity (hypoxic fraction oscillating 5%-30% of the neoplasm). Most important, different kinetic parameters are demonstrated, with Labelling Index ranging from 4% to 30%, phase-S time from 6 to 19 hours, and potential doubling time from 2 to 20 days. On the basis of Fowler's and Barendsen's mathematical models and knowing the potential doubling time and Labelling Index values (derived from bioptic specimens), as well as alpha/beta ratio for both tumor and normal tissue, we tried to identify the optimal fractionation (standard, accelerated, hyperfractionated) for slow/fast-growth tumors, also evaluating the relative acute and late side effects. Our analysis shows that: 1) tumors with aggressive biological behavior (Labelling Index greater than 15%, aneuploidy, potential doubling time less than 5 days) seem to respond to accelerated fractionation/hyperfractionation without split better than to standard regimens: 2) tumors with slow growth (Labelling Index less than 15%, potential doubling time greater than 5 days, euploidy) seem to respond not only to standard regimens, but also--and mainly--to hyperfractionation.

Carcinoma, Squamous Cell↗