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PubMedDate not supplied

Mathematical modelling of stimulus-secretion coupling in the pancreatic B-cell. I. Dynamics of insulin release.

A mathematical model for cyclic AMP-Ca2+ interactions in pancreatic islets, defined in a previous study, was modified to include two new features. First, the cytosol and vacuolar Ca2+ pools were stratified in a cortical layer (representing 1 to 20% of the total cellular pool) and central core. Second, the changes in Ca2+ inflow and Ca2+ fractional outflow rate evoked by a rise in glucose concentration were discontinued during the 3rd and 4th min of stimulation. In this onion leaf sheet model, the biphasic pattern of glucose-induced insulin release could be simulated. This new model emphasizes the significance of both cytosol heterogeneity and signal discontinuity in the dynamics of insulin secretion.

Animals↗

PubMed1990

Using mathematical models to estimate drug resistance and treatment efficacy via CT scan measurements of tumour volume.

A previously described mathematical model designed to evaluate resistance and tumour-kill for individual patients, and to predict changing tumour sizes, has been applied to patients with small cell lung cancer. The model requires tumour volume measurements, and these were obtained via computed tomography scans of the chest. The model fitted the data well, and was able to predict later tumour volumes using earlier ones, as well as suggesting times at which to change or abandon treatment for individual patients. The model gave estimates for resistance and tumour-kill which may provide additional useful outcome measures for clinical trials, and help in the design of future studies.

Drug Resistance↗

PubMed1984

Role of mathematical models in assessment of risk and in attempts to define management strategy.

Risk assessment of food-borne carcinogens is becoming a common practice at FDA. Actual risk is not being estimated, only the upper limit of risk. The risk assessment process involves a large number of steps and assumptions, many of which affect the numerical value estimated. The mathematical model which is to be applied is only one of the factors which affect these numerical values. To fulfill the policy objective of using the "worst plausible case" in estimating the upper limit of risk, recognition needs to be given to a proper balancing of assumptions and decisions. Interaction between risk assessors and risk managers should avoid making or giving the appearance of making specific technical decisions such as the choice of the mathematical model. The importance of this emerging field is too great to jeopardize it by inappropriately mixing scientific judgments with policy judgments. The risk manager should understand fully the points and range of uncertainty involved in arriving at the estimates of risk which must necessarily affect the choice of the policy or regulatory options available.

Animals↗

PubMed1993

Dynamics of a mathematical model of Chrysomya megacephala (Diptera: Calliphoridae).

The laboratory population dynamics of Chrysomya megacephala (F.) was explored with a mathematical model of density-dependent growth. Fecundity and survival decreased significantly as a function of larval density. Parameters in the exponential regressions fitted to the fecundity and survival data were incorporated into a finite-difference equation that incorporates the delayed effect of larval density on fecundity and survival of adults. The theoretical population model of C. megacephala showed cyclic behavior with a stable limit cycle of two points for adults and immatures.

Analysis of Variance↗

PubMed1994

A mathematical model for the determination of total area under glucose tolerance and other metabolic curves.

OBJECTIVE: To develop a mathematical model for the determination of total areas under curves from various metabolic studies. RESEARCH DESIGN AND METHODS: In Tai's Model, the total area under a curve is computed by dividing the area under the curve between two designated values on the X-axis (abscissas) into small segments (rectangles and triangles) whose areas can be accurately calculated from their respective geometrical formulas. The total sum of these individual areas thus represents the total area under the curve. Validity of the model is established by comparing total areas obtained from this model to these same areas obtained from graphic method (less than +/- 0.4%). Other formulas widely applied by researchers under- or overestimated total area under a metabolic curve by a great margin. RESULTS: Tai's model proves to be able to 1) determine total area under a curve with precision; 2) calculate area with varied shapes that may or may not intercept on one or both X/Y axes; 3) estimate total area under a curve plotted against varied time intervals (abscissas), whereas other formulas only allow the same time interval; and 4) compare total areas of metabolic curves produced by different studies. CONCLUSIONS: The Tai model allows flexibility in experimental conditions, which means, in the case of the glucose-response curve, samples can be taken with differing time intervals and total area under the curve can still be determined with precision.

Blood Glucose↗

PubMed1991

The influence of transport parameters and enzyme kinetics of the fibrinolytic system on thrombolysis: mathematical modelling of two idealised cases.

Experimental data obtained by magnetic resonance imaging and photographing clot dissolution in vitro have shown that whole blood clots dissolve almost two orders of magnitude faster when urokinase is introduced into the clot by pressure induced permeation than when its access is limited to diffusion. In view of these findings, two mathematical models have been developed that quantitatively link the enzymatic and transport properties of the fibrinolytic system to the velocity of thrombolysis. Without a pressure gradient across the thrombus, the plasminogen activator molecules diffuse into the thrombus through the blood-thrombus boundary plane. The blood-thrombus boundary slowly moves inwards due to thrombolysis that is spatially restricted to a relatively narrow zone. The velocity of thrombolysis is primarily limited by the diffusion constants of the plasminogen activator and plasmin. In contrast, when plasminogen activator is rapidly distributed along the thrombus by pressure induced bulk flow, lysis occurs at each segment of the thrombus after a lag period that is due to plasmin activation and sufficient fibrin degradation. The lag time is determined primarily by the catalytical properties of the plasminogen activator and plasmin. The mathematical models with the observations of the clot boundaries during lysis permit the characterization of plasmin action on the fibrin network.

Biological Transport↗

PubMed2002

PLGA-based microparticles: elucidation of mechanisms and a new, simple mathematical model quantifying drug release.

The two major aims of this study were: (i) to elucidate the underlying release mechanisms from drug-loaded, erodible microparticles based on poly(lactic-co-glycolic acid) (PLGA) showing biphasic drug release behavior: an initial 'burst' effect, followed by a zero order release phase; and (ii) to develop a new, simple mathematical model that allows the quantitative description of the observed in vitro drug release patterns from this type of delivery system. PLGA-based microparticles offer various advantages, such as the possibility to control the resulting drug release rate accurately over prolonged periods of time, easiness of administration (e.g., by stereotaxic injection), good biocompatibility and complete erosion (avoiding the removal of empty remnants). Consequently, the practical importance of these advanced drug delivery systems is remarkably increasing. However, only little knowledge is yet available concerning the processes controlling the release rate of the drug out of these devices. Various chemical and physical phenomena are involved, rendering the identification of the crucial mechanisms and the mathematical description of the resulting drug release kinetics difficult. In the present study, different physicochemical characterization methods (e.g., DSC, SEM, SEC, particle size analysis) were used to monitor the changes occurring within anticancer drug-loaded PLGA microparticles upon exposure to phosphate buffer pH 7.4. Based on these experimental findings, the most important underlying drug release rate controlling mechanisms were identified and a new mathematical model was developed that allows the quantitative description of the resulting release patterns.

Buffers↗

PubMed1999

A mathematical model for simulating virus transport through synthetic barriers.

Synthetic barriers such as gloves, condoms and masks are widely used in efforts to prevent disease transmission. Due to manufacturing defects, tears arising during use, or material porosity, there is inevitably a risk associated with use of these barriers. An understanding of virus transport through the relevant passageways would be valuable in quantifying the risk. However, experimental investigations involving such passageways are difficult to perform, owing to the small dimensions involved. This paper presents a mathematical model for analyzing and predicting virus transport through barriers. The model incorporates a mathematical description of the mechanisms of virus transport, which include carrier-fluid flow, Brownian motion, and attraction or repulsion via virus-barrier interaction forces. The critical element of the model is the empirically determined rate constant characterizing the interaction force between the virus and the barrier. Once the model has been calibrated through specification of the rate constant, it can predict virus concentration under a wide variety of conditions. The experiments used to calibrate the model are described, and the rate constants are given for four bacterial viruses interacting with a latex membrane in saline. Rate constants were also determined for different carrier-fluid salinities, and the salt concentration was found to have a pronounced effect. Validation experiments employing laser-drilled pores in condoms were also performed to test the calibrated model. Model predictions of amount of transmitted virus through the drilled holes agreed well with measured values. Calculations using determined rate constants show that the model can help identify situations where barrier-integrity tests could significantly underestimate the risk associated with barrier use.

Condoms↗

PubMed1991

[A mathematical model of the growth of a population of Legionella pneumophila in the presence of Tetrahymena pyriformis protozoa].

The mathematical model describing the dynamics of the growth of L. pneumophila in aqueous environment in the presence of protozoa has been worked out. The model has demonstrated considerable heterogeneity of the initial population of virulent L. pneumophila strains. The number of bacteria capable of multiplication in Infusoria is no more than 0.1% of the initial population. The time of the generation of the infective agent inside Tetrahymena pyriformis is 2.8 hours.

Animals↗

PubMed1980

[Mathematical model for energy metabolism in erythrocytes. Independence of scaled glycolytic characteristics of individual features of the donors].

A mathematical model for energy metabolism of erythrocytes includes the first three enzymes of the Embden-Meyerhof pathway as well as total ATPase. It was assumed that the ATPase is strongly inhibited by AMP. The model was used to calculate the dependence of the rate of ATP production and glucose consumption on ATP concentration, i. e. characteristics of glycolysis. The scaled characteristics were plotted in relative units, the normal physiological values of the rates of glucose consumption and ATP concentration being taken for 100%. It was shown that the scaled characteristics show a coincidence, the precision being 20%, when the values of the enzyme activities and other parameters of the model vary within 100%. This agrees with the data, where the scaled characteristics of glycolysis in different donors (in all donors) coincide with experimental precision, while the values of the rates and concentrations with respect to the absolute value varied within 100%.

Adenosine Triphosphatases↗

PubMed1998

A simple mathematical model applied to selection of the sodium profile during profiled haemodialysis.

BACKGROUND: Among dialysis patients in the last 10 years the incidence of intradialytic dysequilibrium syndrome and symptomatic hypotension has increased significantly. Profiled haemodialysis (PHD), a new dialysis technique based on intradialytic modulation of the dialysate sodium concentration according to pre-elaborated individual profiles, has been set up to reduce intradialytic imbalances and the incidence of dysequilibrium syndrome and symptomatic hypotension. The present paper illustrates a new mathematical model for solute kinetics, single-compartment for sodium and two-compartment for urea, aimed at improving the use of PHD. The model allows the sodium profile to be elaborated a priori, before each dialysis session, according to the patient's clinical needs and respecting the individual sodium mass removal and weight gain. METHOD: The mathematical model was first derived and then applied to determining a rational dialysate sodium profile. A procedure which allows the method to be tuned to individual clinical needs on the basis of routine measurements performed before each session is also presented. The proposed method was validated in vivo during seven dialysis sessions, each performed on a different patient. RESULTS: The comparison between data predicted by the model and those obtained in vivo shows a good correspondence in particular concerning the time pattern of blood urea and sodium. The comparison between the model prediction and in vivo determined sodium and urea plasma curves showed standard deviations (2.25 mEq/l for sodium and 0.87 mmol/l for urea) only slightly higher than those attributable to laboratory measurement errors. Moreover, in vivo implementation of PHD by our model enables one to remove an amount of sodium mass comparable with the a priori quantity predicted by the model.

Aged↗

PubMed2004

A mathematical model for chronic myelogenous leukemia (CML) and T cell interaction.

In this paper, we propose and analyse a mathematical model for chronic myelogenous leukemia (CML), a cancer of the blood. We model the interaction between naive T cells, effector T cells, and CML cancer cells in the body, using a system of ordinary differential equations which gives rates of change of the three cell populations. One of the difficulties in modeling CML is the scarcity of experimental data which can be used to estimate parameters values. To compensate for the resulting uncertainties, we use Latin hypercube sampling (LHS) on large ranges of possible parameter values in our analysis. A major goal of this work is the determination of parameters which play a critical role in remission or clearance of the cancer in the model. Our analysis examines 12 parameters, and identifies two of these, the growth and death rates of CML, as critical to the outcome of the system. Our results indicate that the most promising research avenues for treatments of CML should be those that affect these two significant parameters (CML growth and death rates), while altering the other parameters should have little effect on the outcome.

Cell Death↗

PubMed2001

Study and application of a mathematical model for the provisional assessment of areas and nasal resistance, obtained using acoustic rhinometry and active anterior rhinomanometry.

Nasal resistance (NR) depends on the geometrical features and tortuosity of the nasal airway and on the air flow. Knowing the longitudinal distribution of cross-sectional areas (CSAs) in the nasal cavity (which can be obtained using acoustic rhinometry) and the laminar nasal resistance (obtainable by processing the rhinomanometric results), it is possible to calculate, utilizing a mathematical model elaborated on the basis of fluid dynamics, the differential nasal resistance (NRdiff) and the cumulative nasal resistance (NRcum), thus localizing the position at which the highest resistance is concentrated and the related longitudinal distribution. Using a mathematical model, we integrated the sigmoid curves DeltaP/Q of rhinomanometry with the cross-sectional areas obtained using acoustic rhinometry, thus obtaining the normal distribution of differential and cumulative nasal resistances. Afterwards, we empirically reduced the cross-sectional areas corresponding to the head, body, tail and the whole inferior turbinate, recalculating the differential and cumulative nasal resistance distribution curves. The results show that reduction of up to 50% of cross-sectional areas does not substantially affect the resistivity role of the nasal valve, while greater reductions move the highest resistivity point to an area at the junction of the body and the head of the inferior turbinate. The study of the differential nasal resistance trend curves as a function of the reduction of cross-sectional areas shows that the resistance variation of the body and the whole inferior turbinate prevail with reductions of up to 40%, while the variation of cross-sectional areas of the body bordering the inferior turbinate head is predominant with higher reductions. The cross-sectional areas of the nasal airway cavity with highest resistivity are mainly located in an anterior position, where the differential nasal resistances are higher, but there are substantial variations produced by reducing the cross-sectional area of the posterior nasal airway. A similar model can produce provisional values for the results obtainable with functional nasal surgery.

Acoustics↗

PubMed1997

The interaction of growth rates and diffusion coefficients in a three-dimensional mathematical model of gliomas.

This paper is a natural three-dimensional extension of a simple two-dimensional mathematical model of glioma growth and diffusion. The model was originally constructed to simulate a case of recurrent anaplastic astrocytoma treated with chemotherapy, and then modified to allow estimation of the effects of the extent of surgical resection and of variations in growth and diffusion to cover the whole range of glioma behavior. Growth is considered to be constant and exponential (analogous to continuously compounding interest) and is expressed as a decimal fraction per day; the diffusion coefficient is expressed as cm2 per day. Model predictions suggest that diffusion, practically ignored until the present, is a more important component of glioma growth than the growth rate. Even with very early diagnosis, only those tumors with a low diffusion coefficient and a rapid growth rate benefit from a wide resection. Surgical resections generally fail, just as dropping fire-fighters into the burned out center of a forest fire fails, the action being on the periphery as the tumor cells or fires spread out from the center.

Astrocytoma↗

PubMed1985

A mathematical model for gas exchange in the fish gill based on non-linear blood gas equilibrium curves.

A mathematical model for gas exchange in fish gills is presented, which makes allowance for the non-linear nature of the oxygen and carbon dioxide equilibrium curves of blood, for the uneven distribution of diffusion conductance along the secondary lamellae, and for coupling of oxygen and carbon dioxide exchange through the Bohr and Haldane effects. The model demonstrates that for oxygen loading in the gill the sigmoid equilibrium curve is superior to a linear one, whereas the non-linearity of the carbon dioxide equilibrium curve does not significantly affect carbon dioxide exchange and that the Bohr and Haldane effects have importance only for carbon dioxide exchange. It is also shown that the arterial and expired gas tensions and concentrations are unaffected by whether the bulk of the diffusion conductance is at either the afferent or the efferent end of the individual lamellae, provided that the total conductance is unchanged.

Animals↗

PubMed1989

Mathematical models of cell survival after ionizing radiation: application to radiotherapy planning.

The work described here represents an attempt to use mathematical models of single-cell survival for radiotherapy planning. The aim of the study is to develop a procedure in which the distribution of the effects achieved by optimizing physical radiation dose is implemented by taking into account radiobiological terms. An algorithmic model has been developed to evaluate the probability of tumor control and of excessive damage to normal tissue on the basis of the linear-quadratic model of cell survival. In its present preliminary form, the procedure can be used to predict differential isoeffect distributions obtained by varying the total dose and the fraction size of multifraction radiotherapy courses. This approach can also be used to extract from historical clinical results, by maximum-likelihood methods, parameters related to the cellular response to radiation (for instance, alpha and beta of the linear-quadratic model).

Cell Survival↗

PubMed1989

Mathematical model of the frog skeletal muscle--analysis of non-linear mechanical properties.

A mathematical model of the skeletal muscle, consisting of contractile and series elastic components of the contracting muscle as well as viscoelastic components of the resting muscle is presented, in which the contractile component obeying the force-load-velocity relation is expressed by two components, a force generator and a viscous-like component. Since it is one of the most significant properties of the muscle that the mechanical properties depend upon the contractile force (the intensity of the active state), detailed attention is given to the explanation of those non-linear properties. Parameters of the model are determined based on physiological observations obtained from fiber bundles prepared from the frog semitendinosus muscle. While viscoelasticity is constant at the resting state, the series elastic coefficient of the contracting muscle increases with an increase in tension and the viscous-like coefficient also increases linearly with increasing contractile force. This model is checked by digital simulation against responses to ramp stretch of the muscle of the steady and transient contractions. A close agreement is shown between the simulated results and the experimental ones. The model can explicitly account for the non-linear mechanical properties of the contracting muscle.

Animals↗

PubMed1986

A mathematical model of survival kinetics. II. Parameter estimation.

Procedures for the estimation of the four parameters of a new mathematical model of survival and mortality kinetics are given. A formulation of the model has been found which had the advantage of maintaining three of four parameters independent of the unit chosen for the age; in addition, two of these parameters have values falling in a narrow range, even when the model is applied to rather different curves. Since, in any problem of this type, the initial estimate of the parameters plays a major role in the achievement of good final estimates, some simple methods of estimation are indicated based upon the characteristics of the function. The initial estimates may enter three different types of procedures; the best one can be chosen according to the precision of the initial estimates. The method is capable of fitting both survivorship and dying functions directly to the empirical data. An interactive approach to the computer facilities has been used as at each step the operator has to decide whether or not to apply a corrective factor. Goodness of fit, usually high, is estimated by chi 2 test.

Aging↗