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PubMedDate not supplied

Evaluation of a predictive mathematical model of di-(2-ethylhexyl) adipate plasticizer migration from PVC film into foods.

The diffusion coefficient of the plasticizer di-(2-ethylhexyl)adipate (DEHA) in Cheddar cheese (Df) was determined by measuring the extent to which DEHA penetrated cheese that was placed in intimate contact with artificially DEHA-contaminated cheese. Slices (20 microns) of cheese from the boundary layer, into which DEHA had migrated, were microtomed at -40 degrees C, and analyzed for DEHA by gas chromatography (GC). Mean values of Df determined by graph fitting experimental and calculated data were 1.5 x 10(-9) cm2 s-1 at 5 degrees C and 3.0 x 10(-8) cm2 s-1 at 25 degrees C. The partition coefficient (K) of DEHA between cheese and PVC film was derived from the partition coefficients of DEHA between acetonitrile (ACN) and cheese lipid, ACN and cheese solid, and ACN and PVC film. The mean values of K between cheese and PVC film were estimated to be 0.70 at 5 degrees C and 0.58 at 25 degrees C. The estimated values of Df and K were then used in a mathematical model (Till et al. 1982) to predict migration levels of DEHA into cheese. Good agreement with previously published experimental data was obtained. Extrapolation of the prediction of DEHA migration into fatty foods, such as salami and avocado, was also successful (ratio of experimental to predicted results within a factor of two). The values of Df and K for cheese are, however, inappropriate for modelling non-fatty foods such as meat, cakes, fruit and vegetables. Predicted values for these foods were typically 3-10-fold too high. More accurate predictions would probably be obtained if values of Df more relevant to these foods are used. The predictions were relatively insensitive to the value of K.

Adipates↗

PubMed2001

Mathematical models of cognitive recovery.

Longitudinal psychological test results are used as dependent variables to explore the complex relationship between length of coma, time of testing on the recovery curve, and corresponding cognitive status after traumatic brain injury (TBI). A database containing 319 TBI patients with a broad spectrum of coma duration was used. Statistical analysis of mixed effects modelling was applied to longitudinal WAIS-R (Wechsler Adult Intelligence Scale-Revised) scores to construct two mathematical models (verbal IQ and performance IQ). The models predict the course of recovery (initial cognitive level post-coma, eventual recovery level, and level of cognitive functioning at any point on the recovery curve) when the duration of coma is known. Performance IQ was found to recover at a rate that is almost four times slower than verbal IQ. The results have important clinical rehabilitation implications. This statistical modelling technique also enables the medical researcher to investigate disease progression or recovery using structured assessments, which would normally be part of the routine medical monitoring.

Adult↗

PubMed1978

A new family of mathematical models describing the human growth curve.

A new family of mathematical functions to fit longitudinal growth data is described. All members derive from the differential equation dh/dt = s(t). (h1-h) where h1 is adult size and s(t) is a function of time. The form of s(t) is given by one of many functions, all solutions of differential equations, thus generating a family of different models. Three versions were compared. All were superior to previously described models. Model 1, in which s(t) was defined by ds/dt = (s1 - s)(s - s0) was especially accurate and robust, containing only five parameters to describe growth in stature from age two to maturity. Derived "biological" parameters such as Peak Height Velocity were very consistent between these three members of the family but, in some cases, differed signficantly from previous estimates.

Body Height↗

PubMed1980

Loss of chlorpyrifos in pond water: examination of results using three simple mathematical models.

Four outdoor artificial pools were fortified with two formulations of chlorpyrifos (the active ingredient in Dursban insecticide), as emulsifiable concentrate, and a slow-release granular formulation. From concentration-time data collected during summer months, the half life of chlorpyrifos was found to be 5 +/- 3 h and 14 +/- 4 h for the emulsifiable concentrate and slow release formulations respectively. The concentration-time data was analyzed by three mathematical models; the power rate model, the hyperbolic rate model, and an empirical power function. Their average correlation coefficients, determined by linear regression, were respectively, 0.78 +/- 0.08, 84 +/- 0.07, 0.76 +/- 0.16.

Chlorpyrifos↗

PubMed1993

A mathematical model for cell killing by heat applied to a C3H mammary carcinoma in vivo.

Jung (1986) has proposed a mathematical model for cell killing by hyperthermia which assumes that heat killing involves two steps: the production (p) of non-lethal lesions at random and a subsequent conversion (c) into lethal lesions. The p & c model has been shown to predict the survival of CHO cells heated in vitro even when complicated biological phenomena such as thermotolerance and step-down heating (SDH) are involved (Jung 1986, 1991). In the present study the objective was to test the p & c model's ability to describe the effect of single heating and SDH in an experimental tumour in vivo. The endpoint was tumour growth delay (GD). The doubling times (DT) for untreated and heated tumours were similar, and the surviving fraction (SF) could therefore be estimated using: SF = -in(2).GD/DT. SF was fitted to the model by non-linear regression. The p & c model adequately described the GD obtained by SDH (39-44.5 degrees C) and single heating above 42.5 degrees C. Multiple linear regression showed that the residuals for single heating and SDH were independent of both heating time and temperature. However, the residuals for single heating (41-44.5 degrees C) were significantly correlated to heating time when analysed separately. The GD obtained by the use of extended single heating times at or below 42.5 degrees C was therefore overestimated by the model. Development of chronic thermotolerance during heating may account for the observed divergence. The Arrhenius plots for both p and c were log-linear with activation energies of 678 and 311 kJ/mol, respectively. Jung (1986) has previously reported similar p and c activation energies above 42.5 degrees C for CHO cells in vitro.

Animals↗

PubMed1997

Progression rate of radiation damage to the mouse kidney: a quantitative analysis of experimental data using a simple mathematical model of the nephron.

Mouse kidneys were irradiated bilaterally with a range of single or fractionated X-ray doses. After an interval of 2 weeks or 26 weeks, the animals were reirradiated with a range of single X-ray doses. The rate of development of functional kidney damage was assessed repeatedly by the 51Cr-EDTA clearance assay. The rate at which the damage is expressed was found to depend on the primary dose, on the interval between primary treatment and retreatment, and on the retreatment dose. A subset of the data was analysed using a mathematical model of nephron function. In the model, the residual activity of 51Cr-EDTA depends on the glomerular filtration rate (GFR). The GFR is related to the cellularities of three target cell populations. The filtration capacity of the glomerulus is assumed to depend on the numbers of glomerular endothelial cells and mesangial cells. The reabsorption capacity of the tubule is related to the number of tubular epithelial cells. The impact of tubulo-glomerular feedback and the reserve capacity of the kidney on residual activity is considered. The target cell populations are assumed to be of a flexible type, i.e. to consist of cells which are all both functional and self-renewing. Free parameters of the model were optimized by minimizing the residual sum of squares. With the optimized parameter values, the measured and the model-predicted rates of progression of the functional damage correspond well for a wide range of irradiation schedules. The model analysis suggests a pronounced role of tubulo-glomerular feedback in the development of functional injury in the kidney. It is concluded that the model represents a good starting point for quantitative studies of the cellular basis of radiation nephropathy.

Animals↗

PubMedDate not supplied

Deposition of inhaled particulate matter in the upper respiratory tract, larynx, and bronchial airways: a mathematical description.

A mathematical description of inhaled particle behavior suitable for analysis of factors affecting deposition in the human upper respiratory tract (nasopharyngeal and oropharyngeal compartments), larynx, and ciliated airways is presented. When upper respiratory tract and larynx filtering efficiencies are incorporated in the model, theoretical calculations of aerosol deposition agree with tracheobronchial (TB) data reported in the open literature. The model predicts the sigmoidal shape of TB deposition patterns measured in human subjects, the slopes of the quasilinear segments of these curves, and the positions of their two asymptotes. The model could therefore be used to evaluate the health effects of toxic airborne particles.

Aerosols↗

PubMed1986

A simple mathematical model for diffusional sampler operation.

A simple mathematical model of the molecular basis for the function of a diffusional sampler for dilute mixtures of gaseous contaminants in supporting gases is presented. The model is based on the movement of single molecules of the contaminant between sections of a tubular diffusion path on a step-by-step basis; the length of the step and of each section of the tube are equal to the mean free path, lambda, under the specified conditions. When the model is used, the coefficient of diffusion, D, can be calculated from lambda and the average velocity, v, of the contaminant molecule. Both lambda and v were calculated independently using equations which involved the minimum number of assumptions. The value of D so estimated was of the same order as that in the literature, differing by a factor of less than 2. It should be emphasized that the model represents a statistical, thermodynamic approach to understanding diffusional samplers, and its utility is independent of the means of estimating lambda and v for specific gas pairs.

Diffusion↗

PubMed1986

Stomach emptying of milk diets in pigs. A mathematical model allowing description and comparison of the emptying pattern.

The effect of doubling the nutrient intake in pigs fed a liquid cow's milk diet was investigated. This was done, either by doubling the volume given, or by doubling the dry matter content of the milk. There were two experiments in which the cumulative outflow of fresh contents, dry matter, crude protein and total carbohydrates, relative to the intake, was studied as affected by the treatments. This was done using pigs equipped with a simple duodenal cannula and an occluding balloon. The cumulative outflow curves indicated that the emptying process was biphasic both for the fresh contents and the nutrients emptied. A rapid emptying, followed by a period during which the emptying was inhibited was seen during the first emptying phase, the so called initial emptying. Thereafter the emptying was resumed again during the second phase: the final emptying. The individual emptying patterns were described very accurately by an adapted mathematical function with four parameters: y = c (1-2-(at)b)+dt3. The parameters c, b and a describe the initial emptying which is intrinsically exponential while d is descriptive for the final emptying. The appearance of the outflowing material, changing with time from unchanged milk, whey plus clot fragments, pure whey to whey with granular clot fragments indicated that the clotting of the milk was complete after about 15 min. The clotting certainly was responsible for the biphasic appearance of the emptying pattern. The emptying pattern of fresh contents was not related to the emptying pattern of dry matter and nutrients. Due to the rather high variability, there were no significant differences between the emptying patterns of dry matter and nutrients studied, although there were clear indications that crude protein was emptied at a slower rate than the other nutrients. Doubling the nutrient intake either by doubling the volume given, or by doubling the dry matter content of the milk did not change the initial emptying pattern to a great extent, except in the first experiment, where doubling the volume increased the relative amount of dry matter and nutrients emptied. A difference of gastric capacity related to a difference in age between the pigs used in the two experiments is thought to be the reason for this discrepancy. The relative amounts of dry matter and nutrients emptied during the final emptying phase on the other hand were inversily related to the amount of nutrients given.(ABSTRACT TRUNCATED AT 400 WORDS)

Animals↗

PubMedDate not supplied

Mathematical model for energy and protein balance simulation in broilers.

The authors used the empirical equations of mathematical modelling for the energy and protein balance simulation, with the view of calculating the body weight gain in broilers fed various diets. The results found by using the model were compared with the experimental data obtained by several authors. A standard deviation of +/- 1.20% and a mean error of +/- 0.28% and a mean error of +/- 0.28% which proved that the model average was a sufficient estimate of the experimental average were found. Furthermore, using the regression method, a significant correlation of variables was evidentiated.

Animal Feed↗

PubMed1976

Flow cytofluorometric analysis of cell cycle distributions using propidium iodide. Properties of the method and mathematical analysis of the data.

In order to better characterize the new rapid staining method for flow cytofluorometry proposed by Krishan, we have tested its stability and several other properties, and have carried out a quantitative comparison of the fluorescence histograms obtained using propidium iodide or the acriflavine-Feulgen staining procedure. Using a human hematopoietic cell line in the logarithmic phase of growth, and analyzing the data by means of a mathematical method we have devised, we found that the fluorescence intentsity of cells stained with propidium iodide remains stable for at least 48 h; it is insensitive to dye concentration between 0.025 and 0.10 mg/ml (37-150 muM); it is not affected by incubation with ribonuclease before staining; propidium iodide in 0.1% sodium citrate remains stable for at least 20 days; and quantitative estimates of the fractions of cells in the different phases of the cell cycle are in good agreement with those obtained from acriflavine-Feulgen staining and from autoradiography after pulse labeling with tritiated thymidine. We conclude that this method is useful for the measurement of relative DNA content by flow cytofluorometry, although modifications in the technique are necessary for some cell types which grow in monolayers.

Acriflavine↗

PubMed1975

A mathematical model for the determination of cardiac regurgitant and ejection fractions from radioisotope angiocardiograms.

A mathematical model taking into account the pulsatile pumping action of the heart has been developed which describes, in terms of a set of four recursive equations, the activity in the atrium and in the ventricle of the left or right heart at successive end-systoles and end-diastoles after bolus injection. The resulting closed form solutions permit calculations of chamber forward ejection fractions and valvular regurgitant fractions from computerized radioisotope angiocardiogram data. The method has been applied to a series of over thirty clinical studies with encouraging results.

Angiocardiography↗

PubMed1982

A mathematical model to calculate temperature distributions in human and rabbit eyes during hyperthermic treatment.

A mathematical model based on the finite difference method has been developed to calculate transient and steady state temperature distributions in normal unexposed human and rabbit eyes, and human and rabbit eyes heated by various heating techniques. The normal steady state temperature distributions in human and rabbit eyes are given. The model has been experimentally fitted to data obtained from measurements on rabbit eyes. The heat transfer from the choroid to the body core temperature of the rabbit is described by the heat transfer coefficient hs = 65 W m-2 degrees C-1, and from the cornea to the surrounding air temperature by hc = 20 W m-2 degrees C-1. The thermal conductivity and the specific heat of the lens of the rabbit eye were determined empirically to be 0.40 W m-1 degree C-1 and 3.0 J g-1 degree C-1 respectively. The thermal properties of the vitreous humour were taken to be equal to the thermal properties of water.

Animals↗

PubMed1985

A mathematical model for the quantitative study of left to right cardiac shunt.

Non-invasive radioisotope cardiographic techniques have become a useful tool for studying the anatomy and function of the heart. The mathematical model described justifies the phenomenological analysis of pulmonary time-activity histograms and was invented for quantitative study of left to right cardiac shunt. The model also gives a theoretical insight into such an empirically proposed diagnostic method and represents an adequate framework for the understanding of other possible approaches to the problem of left to right shunt.

Blood Circulation↗

PubMed1987

Basic mathematical and electromagnetic concepts of the biomagnetic inverse problem.

In this paper basic mathematical and physical concepts of the biomagnetic inverse problem are reviewed with some new approaches. The forward problem is discussed for both homogeneous and inhomogeneous media. Geselowitz' formulae and a surface integral equation are presented to handle a piecewise homogeneous conductor. The special cases of a spherically symmetric conductor and a horizontally layered medium are discussed in detail. The non-uniqueness of the solution of the magnetic inverse problem is discussed and the difficulty caused by the contribution of the electric potential to the magnetic field outside the conductor is studied. As practical methods of solving the inverse problem, a weighted least-squares search with confidence limits and the method of minimum norm estimate are discussed.

Animals↗

PubMed1987

Mathematical circulation model for the blood-flow-heat-loss relationship in the rat tail.

A mathematical model for the heat-loss-blood-flow relationship is developed for the rat tail. When supplied with experimental values of heat loss and blood flow, the model allows one to compute the distribution of flow in deep and cutaneous vessels as a function of body core and tail temperature and to determine the savings in heat loss that result from alterations in the pattern of circulation and from counter-current heat transfer. Blood flow in the cutaneous and deep lying veins of the tail is controlled by both central and local temperatures and increases fairly linearly with deep body temperature. However, the distribution of blood flow in the tail is controlled only by local tail temperature and is independent of deep body temperature. The change in venous distribution of flow has a great impact on the conservation of heat and can reduce the heat loss from the circulating blood by more than 50% when venous return is directed to deep lying veins. On the other hand, counter-current heat transfer is of only minor importance in the control of heat loss from the tail, resulting at most in a 10% saving of heat loss, and that only at the smallest rate of blood flow.

Animals↗

PubMed1990

A single-tube mathematical model of reactive hyperaemia.

A mathematical model of reactive hyperaemia is developed using quasi-steady flow in a single tube to represent blood flow in the vascular bed. The role of the myogenic response during reactive hyperaemia is examined by suggesting a linear relationship between tube cross-sectional area S and pressure p, in which S decreases as p increases, thereby modelling the response of the smooth muscle in the blood vessel walls to increases in p which the myogenic mechanism proposes. However, this simple relationship, together with the equations of continuity and Poiseuille flow, lead to an unstable equation for p which is inconsistent with the known boundary conditions. It is necessary to make S a function of p and delta p/delta t in order to achieve a stable response which implies that the myogenic response must be rate sensitive to pressure changes. The resulting equations are then solved for p, S, and flow Q by numerical integration and give results for Q which are in broad agreement with experiment. The model also suggests that the changing pressure gradient governs the flow in reactive hyperaemia rather than changes in the resistance of the blood vessels.

Humans↗

PubMed1992

Mathematical modelling of objectives in radiation therapy treatment planning.

We view radiation therapy treatment planning (RTTP) as a problem of constrained optimization. In this context, the 'optimal' treatment is one that maximizes some objective function subject to the inherent constraints of the problem. We discuss two objective functions which are motivated by biological considerations rather than mathematical expedience. One of these measures the 'goodness' of a treatment in terms of its probability of success. The other measures the degree to which a treatment succeeds in satisfying a system of constraints that are mutually contradictory. We describe how to actually perform the optimization of these objective functions in practice and show some numerical results which demonstrate advantages of these objective functions over their more conventional counterparts.

Humans↗