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PubMed1994

A mathematical model for appraisal of the impact of GH binding protein on GH receptor binding.

Discrepancies between GH measurements and growth rate of children have complicated diagnosis in a variety of clinical conditions. The competition of GH-BP with the GH-receptor towards GH-receptor binding can have a role in these discrepancies. A mathematical model was developed for appraising the availability of GH for receptor binding from measurements of serum GH by RIA and serum GH binding protein (BP) by a binding assay. Eighteen patients with high GH-BP (obesity), normal GH-BP (normal control) or low GH-BP (children, anorexia nervosa or cirrhosis of the liver) were the subjects of this study. Sera of patients with high, normal or low GH-BP levels were analyzed for their competition with [125I]hGH binding to rabbit liver membranes. Serum GH was measured by a commercial polyclonal RIA. Serum GH-BP was measured by a binding assay with dextran-coated charcoal separation. Receptor availability for GH was assessed by displacing of [125I]hGH from rabbit liver membranes. The decline in receptor availability for each hGH value, caused by GH-BP competition with the receptor, was calculated by subtraction of the percent displacement in the absence of GH-BP from the percent displacement in the presence of a given GH-BP value. The results were analyzed statistically to give a series of polynomes. These enabled the calculation of an activity factor for serum RIA GH levels, that should predict the receptor availability of each GH level, according to the concomitant GH-BP level.(ABSTRACT TRUNCATED AT 250 WORDS)

Animals↗

PubMed1995

Mathematical model and dimensional analysis of glycocholate binding to cholestyramine resin: implications for in vivo resin performance.

In large doses, cholestyramine resin lowers blood serum cholesterol by binding bile salts in the intestinal lumen and thus increases the fecal excretion of bile salts. In order to gain a better understanding of the low in vivo potency of cholestyramine, mathematical models estimating the amount of glycocholate bound per gram of cholestyramine and the free glycocholate concentration were derived and employ the capacity-corrected molar selectivity coefficient. Predictions of the quantity of glycocholate bound per gram of cholestyramine and of the free glycocholate concentration matched observed values (r2 = 0.993 and r2 = 0.998, respectively) over a wide range of conditions. Simulated binding studies indicated the relative importance of several biopharmaceutical parameters for improved resin in vivo performance. Increasing resin selectivity of glycocholate over chloride has greatest therapeutic impact if bile salt sequestering is most important in the upper portion of the intestines. Furthermore, ion exchange phenomena was subjected to dimensional analysis and revealed the controlling factors as components of two dimensionless numbers, GC* and Cl*. Placing physiologic limits on values of GC* and Cl* suggests requisite selectivity properties of more potent bile acid sequestrants and dosing strategies to optimize current resin therapy.

Binding Sites↗

PubMedDate not supplied

[The mathematical modelling of orthodontic designs supported on endosseous implants].

The priority problems in orthodontic treatment of denture total defects, making use of intraosseous implants, are discussed. The study was aimed at estimation of the optimal number of implants and sites of implantation, choice of a rational orthodontic design permitting the distribution of the masticatory load in proportion with the supporting tissue potentialities. The desired effect is achieved via estimation of the mathematical model of the denture design supported by intraosseous implants.

Biomechanical Phenomena↗

PubMed1981

[Use of regression analysis to construct mathematical models of the diameters of arteries].

By means of regressive analysis, using the cervicocerebral arteries as an example, the method to prognosticate the diameter in large arteries by measuring easily accessible external body parts was demonstrated. In 10 dogs body weight, diameter and length of the neck, height of the forelegs and diameter of the cervical arteries (in the roentgenograms) were measured. Statistically significant mathematical models on the diameter of the arteries studied were represented. The degree of influence of each of the four factors was stated for various cervical arteries. The data obtained could be explained from biological point of view.

Anatomy↗

PubMed2001

Mathematical modeling of capillary formation and development in tumor angiogenesis: penetration into the stroma.

The purpose of this paper is to present a mathematical model for the tumor vascularization theory of tumor growth proposed by Judah Folkman in the early 1970s and subsequently established experimentally by him and his coworkers [Ausprunk, D. H. and J. Folkman (1977) Migration and proliferation of endothelial cells in performed and newly formed blood vessels during tumor angiogenesis, Microvasc Res., 14, 53-65; Brem, S., B. A. Preis, ScD. Langer, B. A. Brem and J. Folkman (1997) Inhibition of neovascularization by an extract derived from vitreous Am. J. Opthalmol., 84, 323-328; Folkman, J. (1976) The vascularization of tumors, Sci. Am., 234, 58-64; Gimbrone, M. A. Jr, R. S. Cotran, S. B. Leapman and J. Folkman (1974) Tumor growth and neovascularization: an experimental model using the rabbit cornea, J. Nat. Cancer Inst., 52, 413-419]. In the simplest version of this model, an avascular tumor secretes a tumor growth factor (TGF) which is transported across an extracellular matrix (ECM) to a neighboring vasculature where it stimulates endothelial cells to produce a protease that acts as a catalyst to degrade the fibronectin of the capillary wall and the ECM. The endothelial cells then move up the TGF gradient back to the tumor, proliferating and forming a new capillary network. In the model presented here, we include two mechanisms for the action of angiostatin. In the first mechanism, substantiated experimentally, the angiostatin acts as a protease inhibitor. A second mechanism for the production of protease inhibitor from angiostatin by endothelial cells is proposed to be of Michaelis-Menten type. Mathematically, this mechanism includes the former as a subcase. Our model is different from other attempts to model the process of tumor angiogenesis in that it focuses (1) on the biochemistry of the process at the level of the cell; (2) the movement of the cells is based on the theory of reinforced random walks; (3) standard transport equations for the diffusion of molecular species in porous media. One consequence of our numerical simulations is that we obtain very good computational agreement with the time of the onset of vascularization and the rate of capillary tip growth observed in rabbit cornea experiments [Ausprunk, D. H. and J. Folkman (1977) Migration and proliferation of endothelial cells in performed and newly formed blood vessels during tumor angiogenesis, Microvasc Res., 14, 73-65; Brem, S., B. A. Preis, ScD. Langer, B. A. Brem and J. Folkman (1997) Inhibition of neovascularization by an extract derived from vitreous Am. J. Opthalmol., 84, 323-328; Folkman, J. (1976) The vascularization of tumors, Sci. Am., 234, 58-64; Gimbrone, M. A. Jr, R. S. Cotran, S. B. Leapman and J. Folkman (1974) Tumor growth and neovascularization: An experimental model using the rabbit cornea. J. Nat. Cancer Inst., 52, 413-419]. Furthermore, our numerical experiments agree with the observation that the tip of a growing capillary accelerates as it approaches the tumor [Folkman, J. (1976) The vascularization of tumors, Sci. Am., 234, 58-64].

Angiogenesis Inducing Agents↗

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↗

PubMed1999

A mathematical model of cell growth and alkane degradation in Wadden Sea sediment suspensions.

In a series of 16 experiments, hexadecane in different concentrations was mixed with natural Wadden Sea sediments, and the degradation of the alkane was observed under controlled conditions (bioreactor). The aim of the experiments was to quantify the impact of the dissolved oxygen tension and the initial alkane concentration on degradation time and degradation completeness. A mathematical model was constructed which is able to reproduce reasonably well the observed-time series under all conditions: aerobic and nearly anaerobic, diluted and alkane-saturated. The model contains hypotheses on the chain of reactions; it attempts to bridge existing gaps in the understanding of the degradation process.

Aerobiosis↗

PubMed2001

Probenecid-associated alterations in valproate glucuronide hepatobiliary disposition: mechanistic assessment using mathematical modeling.

The complexity of processes associated with the hepatobiliary disposition of xenobiotics may require a multiexperimental approach, including pharmacokinetic modeling, to assess mechanisms of drug interactions. The objective of this study was to examine the disposition of valproate glucuronide (VG) in the rat isolated perfused liver (IPL), and to determine the mechanisms of interaction with probenecid (PRB). Livers were isolated and perfused with standard techniques, and valproate (VPA) (20 mg) was administered in the absence and presence of PRB (approximately 75 microg/ml). Concentrations of VPA and VG in perfusate and bile were determined at timed intervals. In the absence of PRB, total recovery of VPA and VG in perfusate and bile was approximately 80%; PRB significantly increased this recovery to approximately 100%, suggesting a decrease in oxidative VPA metabolism. Similarly, pharmacokinetic modeling of the IPL data indicated that PRB competitively inhibited formation of oxidative VPA metabolites. PRB also significantly inhibited formation, biliary excretion, and sinusoidal egress of VG. These observations suggest a competitive interaction between PRB and VG for transport across the canalicular and sinusoidal membranes. Despite PRB-associated impairment of VG formation, mathematical modeling of the data revealed that hepatocyte VG concentrations were increased by PRB, presumably due to simultaneous inhibition of VG biliary excretion and sinusoidal egress by PRB. These results demonstrate the utility of pharmacokinetic modeling in elucidating the mechanisms of alteration in the hepatobiliary disposition of xenobiotics.

Animals↗

PubMed1998

Mathematical modeling of immunological reactions.

The immune system is a highly regulated, complex and integrated system which has evolved to provide the organism with substantial defenses against pathogenic organisms. Over the last several decades there has been an explosion of experimental data in this area, and new techniques in molecular and cellular biology have been crucial in deepening our understanding of immune processes. Most of these new techniques have allowed the isolation of the process or cell under study so that the results can be readily interpretable. At the present time, however, there is an emerging need to understand the system as it functions as a whole and the language of mathematics is the one best suited for this purpose. This review, written from the perspective of an experimental immunologist, describes some of the recent advances in the development of mathematical models of the immune system. Particular emphasis is placed on the rapidly growing field of modeling in HIV infection and T cell activation. Immunology as a whole will benefit from the introduction of the language of mathematics in much the same way as neuroscience has done in the last decade.

Animals↗

PubMed1994

Evolution of gene networks by gene duplications: a mathematical model and its implications on genome organization.

Networks of transcriptional regulators have key roles in metazoan development. Important forces in the evolution of these networks are gene duplications and gene deletions, events that may change the spatiotemporal expression pattern of network genes. A measure for the probability of such changes after gene-duplication events is proposed. This measure is based on a simple mathematical model that describes such networks as dynamical systems and on properties of ensembles of these dynamical systems. It is predicted that this probability depends only on the fraction of genes duplicated in a single event and that it is largest if approximately 40% of the genes in a network are duplicated. This property is robust with respect to variations in model parameters. On these grounds, it is argued that (i) evolution of gene networks should preferentially occur either by duplication of single genes or by duplication of all genes involved in a network, and that (ii) tight linkage ("clustering") or strong dispersal are the two evolutionarily most favorable forms of genomic organization of genes forming such networks.

Animals↗

PubMed2005

A mathematical model of fatty acid metabolism and VLDL assembly in human liver.

The lipid composition of very-low-density lipoprotein (VLDL) in plasma is crucial for human health. A pre-requisite for the alteration of VLDL composition is a co-ordinated understanding of the complex interactions in VLDL assembly. In order to determine the potential effects of changes in substrate availability on VLDL lipid composition, we constructed, parameterized and evaluated a mechanistic mathematical model of the biosynthesis of triglycerides, phospholipids, and cholesterol esters and the assembly of VLDL in human hepatocytes. Using published data on human liver metabolism, the model was also used to provide insight into the complex process of lipid metabolism and to estimate the affinities of different liver enzymes for different fatty acids (FA). For example, we found that Delta6-desaturase is 19 times more selective for C18:3n-3 than C18:2n-6, stearoyl-CoA-desaturase is 2.7 times more selective for C18:0 than C16:0, Delta5-desaturase desaturates C20:4n-3 preferentially over C20:3n-6 and FA elongase preferentially elongates C18:3n-6. The model was also used to predict the plasma free fatty acid (FFA) composition required to generate a prescribed change in plasma lipoprotein FA composition. Furthermore, the model was tested against a published human feeding trial that investigated the effect of changes in dietary FA composition on human plasma lipid FA composition. The model is a useful tool for predicting the effect of changes in plasma FFA composition on plasma lipoprotein lipid FA composition.

Dietary Fats↗

PubMed2006

Mathematical models in physiology.

Computational modelling of biological processes and systems has witnessed a remarkable development in recent years. The search-term (modelling OR modeling) yields over 58000 entries in PubMed, with more than 34000 since the year 2000: thus, almost two-thirds of papers appeared in the last 5-6 years, compared to only about one-third in the preceding 5-6 decades. The development is fuelled both by the continuously improving tools and techniques available for bio-mathematical modelling and by the increasing demand in quantitative assessment of element inter-relations in complex biological systems. This has given rise to a worldwide public domain effort to build a computational framework that provides a comprehensive theoretical representation of integrated biological function-the Physiome. The current and next issues of this journal are devoted to a small sub-set of this initiative and address biocomputation and modelling in physiology, illustrating the breadth and depth of experimental data-based model development in biological research from sub-cellular events to whole organ simulations.

Computer Simulation↗

PubMed1998

Mathematical modeling of biofilm structure with a hybrid differential-discrete cellular automaton approach.

A hybrid differential-discrete mathematical model has been used to simulate biofilm structures (surface shape, roughness, porosity) as a result of microbial growth in different environmental conditions. In this study, quantitative two- and three-dimensional models were evaluated by introducing statistical measures to characterize the complete biofilm structure, both the surface structure and volume structure. The surface enlargement, coefficient of roughness, fractal dimension of surface, biofilm compactness, and solids hold-up were found to be good measures of biofilm structure complexity. Among many possible factors affecting the biofilm structure, the influence of biomass growth in relation to the diffusive substrate transport was investigated. Porous biofilms, with many channels and voids between the "finger-like" or "mushroom" outgrowth, were obtained in a substrate-transport-limited regime. Conversely, compact and dense biofilms occurred in systems limited by the biomass growth rate and not by the substrate transfer rate. The surface complexity measures (enlargement, roughness, fractal dimension) all increased with increased transport limitation, whereas the volume measures (compactness, solid hold-up) decreased, showing the change from a compact and dense to a highly porous and open biofilm.

Automation↗

PubMed1996

Vagal control of sinoatrial rhythm: a mathematical model.

The ionic mechanisms underlying vagal control of the cardiac pacemaker were investigated using a new single cell mathematical model of sinoatrial node electrical activity. The model was formulated from a wide range of electrophysiological data available in the literature, with particular reference to whole cell recordings from enzymatically isolated sinoatrial node cells. Development of the model was prompted by the lack of an existing physiologically accurate formulation of sinoatrial node activity that could reproduce the known complex chronotropic response of the pacemaker to brief-burst vagal stimulation, as observed in whole animal and isolated sinus node preparations. Features of the model include the dynamic modulation of the hyperpolarisation-activated current (i(f)) and the L-type calcium current (iCa,L) by acetylcholine, the improved characterisation of the muscarinic potassium current (iK,ACh), assigning the entire background potassium current (ib,K) to spontaneous openings of its channels, and the utilisation of second order kinetics for acetylcholine within the neuroeffector junction. Simulations performed using brief vagal stimuli elicited a strong hyperpolarisation of the membrane which prolonged the cycle in which it was delivered in a phase-dependent manner. This phase-dependency was presented in the form of a standard phase response curve which was characterised by a positive linear slope region, a breakpoint characteristic and a "no effect" zone in which the vagal pulse could no longer prolong the cycle. The breakpoint was manifested as a discontinuity in the curve which was examined by bracketing this point at the limit of the double precision arithmetic employed. At these boundary points on either side of the breakpoint, the vagal stimulus was able to activate outward iK,ACh in such a manner as to finely balance the increasing inward iCa,L trying to generate phase 0 upstroke. On decay of iK,ACh, the membrane either subsequently repolarised or fired to produce an action potential depending on the precise phase of the stimulus. The positive linear slope portion of the PRC was characterised by a strong resetting type behaviour in which the membrane hyperpolarised to approximately the same value, irrespective of the phase of stimulus delivery. For vagal stimulus bursts applied throughout the "no effect" zone, outward iK,ACh was not sufficiently activated in order to overcome the strong inward drive of iCa,L and could not prevent upstroke occurring. For these vagal stimuli, the subsequent cycle was hyperpolarised and prolonged. The size of the "no effect" zone was directly related to the inherent latency incorporated in the activation characteristic of iK,ACh. In contrast to previous models of vagal pacemaker control, our new model was able to reproduce the classical triphasic chronotropic response to brief vagal stimulation characterised by a primary inhibition response, a postinhibitory rebound and a secondary inhibition response. In particular, the postinhibitory rebound was due to activation of the inward hyperpolarisation-activated current by the vagally-induced membrane hyperpolarisation, whilst the secondary inhibition phase resulted from the inhibition of the hyperpolarisation-activated current by acetylcholine. The model suggests that the complex chronotropic responses of the cardiac pacemaker to brief vagal stimulation arises from inherent ionic mechanisms operating within the sinoatrial node.

Acetylcholine↗

PubMed2000

A mathematical model for assessing changes in neurofilament protein levels in neurites and cell bodies of differentiating neuroblastoma cells.

A mathematical model which allows the calculation of the level of neurofilament protein in the cell body (x) and in the neurites (y) of differentiating SK-N-SH cells is presented. The model considers the changes in cell number (proliferating cells) and the number of cells with neurites (differentiating cells). It takes into account the fact that (i) when cells are cultured in differentiating conditions, an increase in cell number is initially observed and (ii) in a non-synchronized population of differentiating cells, the length of neurite extended by individual cells varies within the population. Total neurofilament protein levels in a population of cells were measured by enzyme-linked immunoabsorbant assay and application of the model to the data allowed values for x and y to be calculated. The validity of the model is supported by the fact that the predicted total neurofilament protein levels are highly correlated with the experimentally derived neurofilament protein levels. The model should be of use in temporal studies of cytoskeletal proteins involved in neuronal growth/differentiation and also in studies in which the system is a target of toxic insult.

Cell Differentiation↗

PubMed1983

Mathematical model for hormonal therapy (vasopressin, corticoids) in cerebral collapse and malignant tumors of the brain (36 cases).

A mathematical model of the adrenal postpituitary system has been used in cases of brain disease in which endocrine disturbances play an aggravating role. A computer simulation has suggested a type of therapy adding vasopressin to corticoids; this association seems to elicit better results than corticoids alone. Posttraumatic disorders of the cerebral hydration, on one hand, and inoperable or recurrent tumors, on the other hand, take advantage of such a systemic, formalized approach.

Adrenal Cortex Hormones↗

PubMed2001

A comparison of mammography spectral measurements with spectra produced using several different mathematical models.

Due to the relatively complex nature of spectral measurements from x-ray machines, many researchers use mathematical models to simulate the spectra they need. However, there is concern over their accuracy, and hence the impact that their accuracy may have, on subsequent calculations that rely upon the spectra modelled. With this in mind spectral measurements have been performed on a mammography machine and a comparison with spectra calculated using several different models is presented. Several different techniques have been investigated in the spectral measurements to allow for pulse pileup and other effects of high count rate. Comparison with half value layer (HVL) measurements shows that the use of a gating signal in conjunction with the air-free path provides accurate results without the need for a pinhole collimator. Comparison of the measured spectra with those calculated using different models proposed in the literature suggests that accurate results can be produced by all models, but only if the user attempts to match the calculated HVL of the modelled spectrum with the physically measured HVL. If this is not done the modelled spectra may be in error. The impact of such an error is demonstrated in calculations of mean glandular dose, which indicate a possible underestimate of the dose by up to 20%.

Female↗

PubMed1989

Chaotic activity in a mathematical model of the vagally driven sinoatrial node.

Phase-locking behavior and irregular dynamics were studied in a mathematical model of the sinus node driven with repetitive vagal input. The central region of the sinus node was simulated as a 15 x 15 array of resistively coupled pacemakers with each cell randomly assigned one of 10 intrinsic cycle lengths (range 290-390 msec). Coupling of the pacemakers resulted in their mutual entrainment to a common frequency and the emergence of a dominant pacemaker region. Repetitive acetylcholine (ACh; vagal) pulses were applied to a randomly selected 60% of the cells. Over a wide range of stimulus intensities and basic cycle lengths, such perturbations resulted in a large variety of stimulus/response patterns, including phase locking (1:1, 3:2, 2:1, etc.) and irregular (i.e., chaotic) dynamics. At a low ACh concentration (1 microM), the patterns followed the typical Farey sequence of phase-locked behavior. At a higher concentration (5 microM), period doubling and aperiodic patterns were found. When a single pacemaker cell was perturbed with repetitive ACh pulses, qualitatively similar results were obtained. In both types of simulation, chaotic behavior was investigated using phase-plane (orbital) plots, Poincaré mapping, and return mapping. Period-doubling bifurcations (2:2, 4:4, and 8:8) were found temporally and spatially within the array. Under certain conditions of stimulation, the attractor in the return map during chaotic activity of the single cell resembled the Lorenz tent map. However, when electrical coupling between cells was allowed, the interactions with neighboring cells exhibiting chaotic dynamics resulted in characteristic alterations of the attractor geometry. Our results suggest that irregular dynamics obeying the rules derived from other chaotic systems are present during vagal stimulation of the sinus node. In addition, application of the same analytical tools to the analysis of simulation of reflex vagal control of sinus rate suggests that chaotic dynamics can be obtained in the physiologically relevant case of the baroreceptor reflex loop. These results may provide insight into the mechanisms of dynamic vagal control of heart rate and may help to provide insights into clinically relevant disturbances of cardiac rate and rhythm.

Acetylcholine↗