A mathematical model for the leukocyte filtration process

Leukocyte filters are applied clinically to remove leukocytes from blood. In order to optimize leukocyte filters, a mathematical model to describe the leukocyte filtration process was developed by modification of a general theoretical model for depth filtration. The model presented here can be used to predict the time-dependent leukocyte filtration as a function of cell-cell interaction in the filter, filter efficiency, filter capacity, filter dimensions, and leukocyte concentration in the suspension applied to the filter. The results of different leukocyte filtration experiments previously reported in the literature could be well described by the present model. (c) 1995 John Wiley & Sons, Inc.     

Infanticide as an alternative male reproductive strategy in langurs: A mathematical model

This paper makes explicit some of the assumptions underlying the view that infanticide is one of two (or more) alternative reproductive strategies utilized by adult male langur monkeys (genus Presbytis). A mathematical model has been developed from these assumptions, and formulae derived which give the equilibrium proportion of infanticidal males expected in langur populations under any given set of reproductive and demographic conditions. Together with estimates of adult male reproductive success obtained from a previous analysis of langur infanticide, these formulae were then used to calculate the precise proportion of infanticidal males expected in natural populations of langurs characterized by specific average male tenures. Further, the number of generations required for such populations to reach their predicted equilibrium was estimated using a simple computer model of langur population dynamics. The present work has thus produced several quantitative predictions which are directly falsifiable with observational data obtainable from wild populations of langur monkeys. With only slight modification, the present model may also be applied to the numerous other primate and non-primate species whose mating systems include many of the same general features as that of langurs.     

On the mathematical modelling of pain

In this review a case is presented for the use of mathematical modelling in the study of pain. The philosophy of mathematical modelling is outlined and a recommendation is made for the use of modern nonlinear techniques and computational neuroscience in the modelling of pain. Classic and more recent examples of modelling in neurobiology in general and pain in particular, at three different levels—molecular, cellular and neural networks—are described and evaluated. Directions for further progress are indicated, particularly in plasticity and in modelling brain mechanisms. Major advantages of mathematical modelling are that it can handle extremely complex theories and it is non-invasive, and so is particularly valuable in the investigation of chronic pain. Special issue dedicated to Dr. Herman Bachelard     

A note on equivalent mathematical models

A mathematical model of a process contains parameters supposedly characterizing the system which manifests the process. If the parameters are statistically distributed in a population of such systems, the process manifested by the entire population will in general be described by a different mathematical model. Thus a choice is always at hand between two or more mathematical models, depending on which parameters (if any) are assumed to be distributed and, if so, how. Examples of such alternative interpretations are given for mathematical models of some behavioral processes.     

Continuous and discrete mathematical models of tumor-induced angiogenesis

Angiogenesis, the formation of blood vessels from a pre-existing vasculature, is a process whereby capillary sprouts are formed in response to externally supplied chemical stimuli. The sprouts then grow and develop, driven initially by endothelial-cell migration, and organize themselves into a dendritic structure. Subsequent cell proliferation near the sprout tip permits further extension of the capillary and ultimately completes the process. Angiogenesis occurs during embryogenesis, wound healing, arthritis and during the growth of solid tumors. In this paper we present both continuous and discrete mathematical models which describe the formation of the capillary sprout network in response to chemical stimuli (tumor angiogenic factors, TAF) supplied by a solid tumor. The models also take into account essential endothelial cell-extracellular matrix interactions via the inclusion of the matrix macromolecule fibronectin. The continuous model consists of a system of nonlinear partial differential equations describing the initial migratory response of endothelial cells to the TAF and the fibronectin. Numerical simulations of the system, using parameter values based on experimental data, are presented and compared qualitatively with in vivo experiments. We then use a discretized form of the partial differential equations to develop a biased random-walk model which enables us to track individual endothelial cells at the sprout tips and incorporate anastomosis, mitosis and branching explicitly into the model. The theoretical capillary networks generated by computer simulations of the discrete model are compared with the morphology of capillary networks observed in in vivo experiments.     

A method for quantification of absolute amounts of nucleic acids by (RT)-PCR and a new mathematical model for data analysis

Accurate quantification of nucleic acids by competitive (RT)-PCR requires a valid internal standard, a reference for data normalization and an adequate mathematical model for data analysis. We report here an effective procedure for the generation of homologous RNA internal standards and a strategy for synthesizing and using a reference target RNA in quantification of absolute amounts of nucleic acids. Further, a new mathematical model describing the general kinetic features of competitive PCR was developed. The model extends the validity of quantitative competitive (RT)-PCR beyond the exponential phase. The new method eliminates the errors arising from different amplification efficiencies of the co-amplified sequences and from heteroduplex formation in the system. The high accuracy (relative error <2%) is comparable to the recently developed real time detection 5'-nuclease PCR. Also, corresponding computer software has been devised for practical data analysis.     

A method for quantification of absolute amounts of nucleic acids by (RT)–PCR and a new mathematical model for data analysis

Accurate quantification of nucleic acids by competitive (RT)–PCR requires a valid internal standard, a reference for data normalization and an adequate mathematical model for data analysis. We report here an effective procedure for the generation of homologous RNA internal standards and a strategy for synthesizing and using a reference target RNA in quantification of absolute amounts of nucleic acids. Further, a new mathematical model describing the general kinetic features of competitive PCR was developed. The model extends the validity of quantitative competitive (RT)–PCR beyond the exponential phase. The new method eliminates the errors arising from different amplification efficiencies of the co-amplified sequences and from heteroduplex formation in the system. The high accuracy (relative error <2%) is comparable to the recently developed real time detection 5′-nuclease PCR. Also, corresponding computer software has been devised for practical data analysis.     

A mathematical model of iron chelation with desferrioxamine

Removal of body iron is a severe clinical problem in patients affected by hemochromatosis or iron-loading anemias. Desferrioxamine (DFO) is the most potent artificial iron-chelating agent. This work deals with the problem of describing DFO action in man by means of a mathematical model, formulated on the basis of the current knowledge about DFO and iron metabolism. Since most clinical data are concerned with DFO-induced urinary iron excretion, only this process, which involves iron stored in reticuloendothelial (RE) cells, was described in detail. Most of the parameters of the model were estimated from data obtained by studying the single processes involved in iron chelation with DFO. A computer simulation study then allowed us to assign meaningful values to the remaining parameters. Different DFO treatments were simulated, and the results obtained seem to show us that the model can reproduce the main experimental findings and the relationship between the amount of chelated iron and the iron status of the patient.     

Pressure-jump kinetics of bovine beta-casein micellization

Two samples of highly purified bovine beta-casein supplied to us by Dr. T.A.J. Payens of the National Institute for Dairy Research, Ede, The Netherlands, were studied over a range of concentration from just below the critical micelle concentration (CMC) to 0.46%, in 0.2 ionic strength phosphate buffer (pH 7.0), at 20 and 25 degrees C. The relaxation process studied by pressure jump using a 90 degree scattered light detector was also confirmed by the temperature-jump method. In the pressure-jump experiments, the process could be separated into two general time domains, with an approximate ratio of 10-25:1, a behavior reminiscent of that found for synthetic micellar systems. The faster relaxation process was still exhibited below the CMC, however. The concentration dependence of the faster relaxation time agreed very satisfactorily with predictions from the micelle model described in the companion paper.     

A mathematical model for dorsal closure

During embryogenesis, drosophila embryos undergo epithelial folding and unfolding, which leads to a hole in the dorsal epidermis, transiently covered by an extraembryonic tissue called the amnioserosa. Dorsal closure (DC) consists of the migration of lateral epidermis towards the midline, covering the amnioserosa. It has been extensively studied since numerous physical mechanisms and signaling pathways present in DC are conserved in other morphogenetic events and wound healing in many other species (including vertebrates).We present here a simple mathematical model for DC that involves a reduced number of parameters directly linked to the intensity of the forces in the presence and which is applicable to a wide range of geometries of the leading edge (LE). This model is a natural generalization of the very interesting model proposed in Hutson et al. (2003). Being based on an ordinary differential equation (ODE) approach, the previous model had the advantage of being even simpler, but this restricted significantly the variety of geometries that could be considered and thus the number of modified dorsal closures that could be studied.A partial differential equation (PDE) approach, as the one developed here, allows considering much more general situations that show up in genetically or physically perturbed embryos and whose study will be essential for a proper understanding of the different components of the DC process. Even for native embryos, our model has the advantage of being applicable since an early stages of DC when there is no antero-posterior symmetry (approximately verified only in the late phases of DC).We validate our model in a native setting and also test it further in embryos where the zipping force is perturbed through the expression of spastin (a microtubule severing protein). We obtain variations of the force coefficients that are consistent with what was previously described for this setting.     

Dynamic mathematical modeling of sulfate reducing gas-lift reactors

This paper presents a mathematical model able to simulate under dynamic conditions the physical, chemical and biological processes prevailing in a biological sulfate reducing gas-lift reactor. The proposed model is based on differential mass balance equations for substrates, products and bacterial groups involved in a sulfate reduction process. Heterotrophic sulfate reducing bacteria (HSRB), autotrophic sulfate reducing bacteria (ASRB), homoacetogenic bacteria (HB), methanogenic archaea (MA) and acetate degraders (AD) are the microbial groups taken into account in the model. The model is also used to validate a steady-state design model previously proposed by Esposito et al. [1].The proposed model is able to simulate the competition between the biological bacteria growing in the reactor, and predict the performance of a gas-lift reactor. The model includes two main parts: (1) a kinetic part including growth, metabolism and competition of SRB, HB, MA and AD in the system and (2) a mass-transfer part describing the thermodynamic concentration equilibria of gaseous components in the liquid and gas phase. The model has been validated using experimental data obtained by operating a laboratory-scale gas-lift reactor as described in Esposito et al. [2].The model can be applied to simulate the sulfate reduction process in a gas-lift reactor for several purposes, such as the evaluation of the optimal process conditions in terms of COD:SO₄^2? ratio, hydraulic retention time and gas input flow. In particular, model simulations reported in this paper show the model capability to predict the prevailing bacterial species and concentrations in the reactor as a function of the hydraulic retention time.     

On the solution of mathematical models of herd immunity in human helminth infections

The general solution of the mathematical model of herd immunity to human helminth infections recently proposed by Anderson and May [3] is obtained. The numerical solution of a more accurate biological model is indistinguishable from the corresponding exact solution of a more tractable mathematical model. Computer simulations of some particular cases of this model support the notion that both ecological and immunological factors determine the observed convex patterns of age-prevalence and age-intensity curves of human helminth infections.This work was made thanks to the advise and support of Dr. Robert M. May while the author was Postdoctoral Fellow at Princeton University     

Photoelectric properties of the “yellow strips” inVespa orientalis; a mathematical model

The “yellow strips” on the cuticle of the Oriental Hornet (Vespa orientalis, Hymenoptera, Vespinae), present photoelectric properties. A mathematical model for the relative changes in resistance as a photoconductive process conforms to the general model for a semiconductor with traps.     

Some mathematical aspects of the medical diagnostic process. I: a general mathematical model

Making a medical diagnosis consists of correlating knownpatterns of disease with the various classes of clinical data elicited from the history, physical examination, and batteries of tests relative to the diagnostic dynamics symbolized by atree branching into the various possible diagnostic decisions. In this paper a relational mathematical model of the reasoning aspects of the conventional medical diagnostic process is suggested as a way of extracting a general, formal concept of medical diagnosis. Computer implementation of the model is discussed briefly.     

Study of the combined action of an antibiotic and an immunostimulator using mathematical modeling

A mathematical model of antibiotic and immunostimulator (IMS) combined effect on various elements of the immune system and general state of patients with infectious diseases is described. The model was constructed as a system including 6 usual differential equations of the 1st order. With the use of this model and a computer many diverse variants of infection development under conditions of treatment with IMS at the background of antibiotic therapy were modeled. Ii was shown that IMS-antibiotic complexes markedly improved the indices of antibiotic therapy as compared to the use of the antibiotics alone. In combined use of IMS and antibiotics it was possible to lower the antibiotic doses without lowering the antimicrobial effect. The use of IMS at the optimal period led to balanced activation of the host specific and nonspecific resistance factors at the background of antibacterial therapy. The results of the mathematical modeling corresponded to the data on protective effect of salmozan (IMS) and doxycycline (antibiotic) combination in animals (albino mice). It was concluded that the described mathematical model was adequate for validation and optimization of schemes for combined use of IMS and antibacterial agents.     

A mathematical model of the sleep/wake cycle

We present a biologically-based mathematical model that accounts for several features of the human sleep/wake cycle. These features include the timing of sleep and wakefulness under normal and sleep-deprived conditions, ultradian rhythms, more frequent switching between sleep and wakefulness due to the loss of orexin and the circadian dependence of several sleep measures. The model demonstrates how these features depend on interactions between a circadian pacemaker and a sleep homeostat and provides a biological basis for the two-process model for sleep regulation. The model is based on previous “flip–flop” conceptual models for sleep/wake and REM/NREM and we explore whether the neuronal components in these flip–flop models, with the inclusion of a sleep-homeostatic process and the circadian pacemaker, are sufficient to account for the features of the sleep/wake cycle listed above. The model is minimal in the sense that, besides the sleep homeostat and constant cortical drives, the model includes only those nuclei described in the flip–flop models. Each of the cell groups is modeled by at most two differential equations for the evolution of the total population activity, and the synaptic connections are consistent with those described in the flip–flop models. A detailed analysis of the model leads to an understanding of the mathematical mechanisms, as well as insights into the biological mechanisms, underlying sleep/wake dynamics.     

The Hospital Bed and the General Practitioner: A Pilot Study of Two Practices with Respect to Hospital Bed Usage and Patient Management

Fourteen hundred randomized patient records of two urban practitioners covering a one-year period were studied. The two practitioners represented two very different types of general practice. Dr. A was an active staff member of a teaching hospital; Dr. B confined his work to office and home visits.The age, marital status, occupation, diagnoses, and patient management were coded on IBM cards, and the data analyzed with the aid of a computer program.The findings indicate that over 90% of patients seen by both practitioners were managed entirely from their offices. Dr. A admitted 41 (5.9%) of the 700 patients seen and referred 24 (3.4%); Dr. B admitted no patients and referred 44 (6.3%). These and other findings suggest that the role and function of the urban general practitioner is changing rapidly in this era of specialization.     

A critical review of mathematical models and data used in diabetology

The literature dealing with mathematical modelling for diabetes is abundant. During the last decades, a variety of models have been devoted to different aspects of diabetes, including glucose and insulin dynamics, management and complications prevention, cost and cost-effectiveness of strategies and epidemiology of diabetes in general. Several reviews are published regularly on mathematical models used for specific aspects of diabetes. In the present paper we propose a global overview of mathematical models dealing with many aspects of diabetes and using various tools. The review includes, side by side, models which are simple and/or comprehensive; deterministic and/or stochastic; continuous and/or discrete; using ordinary differential equations, partial differential equations, optimal control theory, integral equations, matrix analysis and computer algorithms.     

A mathematical analysis of small mammal populations

Populations of Microtus montanus, the montane vole, have been extensively studied. It is known that their reproductive activity is closely linked to the availability of the chemicals in growing plants. We use a mathematical model here to study how the length of the vegetative season and the natural reproduction rhythm of voles are involved in the long term dynamics of the population numbers. In particular, we use data obtained from Timpie Springs, Utah, and from Jackson Hole, Wyoming, to formulate a model. The novelty of this model is its use of littering curves that highlight the temporally discrete nature of vole reproduction. The model shows how the timing of the vegetative season can influence vole population sizes.This work was supported in part by NSF Grant No. MCS-82-08986 and DMS85-14000 (FCH) and by the University of Utah where Dr. Murphy was a visitor     

A survey of unequal crossover systems and their mathematical properties

We present a model of gene duplication by means of unequal crossover (UCO) where the probability of any given pairing between homologous sequences scales as a penalty factor p ^z ≤ 1, with z the number of mismatches due to asymmetric sequence alignment. From this general representation, we derive several limiting case models of UCO, some of which have been treated elsewhere in the literature. One limiting case is random unequal crossover (RUCO), obtained by setting p = 1 (corresponding to equiprobable pairings at each site). Another limiting case scenario (the ‘Krueger-Vogel’ model) proposes an optimal ‘endpoint’ alignment which strongly penalizes both overhang and deviations from endpoint matching positions. For both of these scenarios, we make use of the symmetry properties of the transition operator (together with the more general UCO properties of copy number conservation and equal parent-offspring mean copy number) to derive the stationary distribution of gene copy number generated by UCO. For RUCO, the stationary distribution of genotypes is shown to be a negative binomial, or alternatively, a convolution of geometric distributions on ‘haplotype’ frequencies. A different type of model derived from the general representation only allows recombination without overhang (internal UCO or IntUCO). This process has the special property of converging to a single copy length or a distribution on a pair of copy lengths in the absence of any other evolutionary forces. For UCO systems in general, we also show that selection can readily act on gene copy number in all of the UCO systems we investigate due to the perfect heritability (h ² = 1) imposed by conservation of copy number. Finally, some preliminary work is presented which suggests that the more general models based on misalignment probabilities seem to also converge to stationary distributions, which are most likely functions of parameter value p. An erratum to this article is available at .