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PubMedDate not supplied

Mathematical models and their applications in medicine and health.

Mathematical models have great potentialities as regards their utility in different disciplines of medicine and health. This paper attempts to elucidate their uses in the field. A brief mention of some models has also been made. Mathematical models are useful in epidemiologic research, planning and evaluation of preventive and control programmes, clinical trials, measurement of health, cost-benefit analysis, diagnosis of patients and in maximizing effectiveness of operations aimed at attaining specified goals within existing resources.

Health Services Research↗

PubMed1998

Distributing working versions of published mathematical models for biological systems via the Internet.

Mathematical models are useful tools for investigating complex systems. By representing physiological systems as models, theories can be tested quantitatively against data from the system. Models can be used to explore new theories prior to experimentation and to design studies to optimize experimental resources. They can also be used as teaching tools to illustrate physiochemical principles. In spite of their usefulness and the time invested in developing models, published models are often underused due to the difficulty in obtaining working versions of the model. To address this problem we have designed a library for mathematical models of biological systems on the Internet. The library contains published models of biological systems in formats compatible with several modeling packages, from the fields of physiology, metabolism, endocrinology, biochemistry, and chemistry. The models can be viewed graphically, model solutions can be viewed as plots against data, and models can be downloaded to be run with software on the user's own system. The address of the library is: http://biomodel.georgetown.edu/model/ Investigators are invited to submit working versions of published models to the library. Models can be submitted electronically at the time a manuscript is accepted for publication. As journals go online, articles containing models can be linked to working versions of the models in the library. By increasing access to working versions of models, more of the investment in kinetic studies and model development can be realized.

Computer Simulation↗

PubMed1995

Psychovegetative syndrome diagnosis: an automated psychophysiological investigation and mathematical modeling approach.

1. INTRODUCTION. The main purpose of our work was to create the informational expert system of psychovegetative syndrome diagnosis by applying clinical data and estimating the functioning of the central and peripheral part of regulatory apparatus of the human organism, taking into consideration parallel and consecutive sensory, motor, associative, emotional drive systems, and internal body state. We used automatized psychophysiological investigation and mathematical models. For this purpose the following principal tasks have been prepared: the creation of database of quantifiable estimation patient state; the definition and automation of psychophysiological investigation; mathematical modeling of vegetative functions using a non-invasive sample and its connection with real psychophysiological experiment; mathematical modeling of organisms inner medium homeostasis; and the creation of an informational-expert system of psychovegetative syndrome diagnosis. 2. DATABASE OF ESTIMATION OF PATIENTS STATE. The medical records of the DB "PATIENT" contain data on patient psychic and somatoneurological status. 3. AUTOMATED PSYCHOPHYSIOLOGICAL INVESTIGATION. Psychophysiological investigation enables estimation of the functioning of several subsystems of the human organism and establishes an interrelationship between them by means of electrophysiological data and performance parameters. The study of psychophysiological provision of behavior by psychophysiological investigation enables us to get information about adaptational mechanisms of the patient under certain environmental loads. By means of special mathematical provision, the mathematical elaboration of biosignals as performance parameters has been realized; also realized were the formation of received parameters in the database, the estimation of separated parameters in the view of informativity, and the establishment of diagnostic patterns. 4. MATHEMATICAL MODELS FOR ESTIMATION INTERNAL BODY STATE. The proposed mathematical models allow investigation of homeostatic regulation in various intensities of the metabolic processes and external load. This approach in mathematical models allows us to characterize the relations between the central and peripheral parts of the regulatory mechanisms, using non-invasive samples under psychophysiological investigation and the simulation of different surroundings for brain cells functioning. 5. INFORMATION-EXPERT SYSTEM. Proceeding from the principle of psychoneural unity, we characterized the functioning mechanisms of CNS by means of automatized EEG analysis, visually evoked potential analysis, estimated psychic status, and the characteristics of neural system biochemical processes received by mathematical modeling. By using the indices of the viscero-vegetative and somatolocomotor system (as well as parameters received by automatized analysis of ECG), the EEG respiratory signal--from mathematical models of vegetative functions decision support system applied in estimation of a peripheral block of the regulatory system--is reflected in diagnosis of certain syndromes. The informational expert system, proceeding from the functioning of the human organism's regulatory apparatus, diagnosed psychovegetative syndrome and described the mechanisms of its development. 6. CONCLUSION. The informational expert system enables estimation of the functioning of a human organism as a whole and can be introduced in the sphere of practical medicine and professional orientation as well as in laboratories of experimental psychology and neurosciences.

Brain↗

PubMed1997

Compartmentalized energy transfer in cardiomyocytes: use of mathematical modeling for analysis of in vivo regulation of respiration.

The mathematical model of the compartmentalized energy transfer system in cardiac myocytes presented includes mitochondrial synthesis of ATP by ATP synthase, phosphocreatine production in the coupled mitochondrial creatine kinase reaction, the myofibrillar and cytoplasmic creatine kinase reactions, ATP utilization by actomyosin ATPase during the contraction cycle, and diffusional exchange of metabolites between different compartments. The model was used to calculate the changes in metabolite profiles during the cardiac cycle, metabolite and energy fluxes in different cellular compartments at high workload (corresponding to the rate of oxygen consumption of 46 mu atoms of O.(g wet mass)-1.min-1) under varying conditions of restricted ADP diffusion across mitochondrial outer membrane and creatine kinase isoenzyme "switchoff." In the complete system, restricted diffusion of ADP across the outer mitochondrial membrane stabilizes phosphocreatine production in cardiac mitochondria and increases the role of the phosphocreatine shuttle in energy transport and respiration regulation. Selective inhibition of myoplasmic or mitochondrial creatine kinase (modeling the experiments with transgenic animals) results in "takeover" of their function by another, active creatine kinase isoenzyme. This mathematical modeling also shows that assumption of the creatine kinase equilibrium in the cell may only be a very rough approximation to the reality at increased workload. The mathematical model developed can be used as a basis for further quantitative analyses of energy fluxes in the cell and their regulation, particularly by adding modules for adenylate kinase, the glycolytic system, and other reactions of energy metabolism of the cell.

Adenosine Diphosphate↗

PubMed2002

A minimal mathematical model of calcium homeostasis.

A mathematical model of calcium homeostasis is presented in which the controlling factors are the plasma concentrations of calcium, PTH, and calcitriol, and the effector organs are the parathyroids, bone, kidney, and intestine. Other factors can be added as the need arises. The model is aimed at simulating what happens in a single individual, but its parameters and variables were adjusted to the corresponding published average values. Simulations of published observations in humans undergoing the infusion of calcium or its chelators are presented. With a single exception, these simulations provided a good fit to the data. The response of the system to extrinsic perturbations was characterized by simulating chronic infusions of calcium, PTH, and calcitriol. Finally, the steady state response to perturbations in some of its parameters (the secretory mass of the parathyroids and the affinity and/or sensitivity of the calcium, PTH, and calcitriol receptors) and to renal failure were also investigated in an attempt to analyze the pathogenesis of clinical hypo- or hypercalcemias. In its present form the model cannot be used to base clinical decisions in individual cases. However, it requires modest computational resources, and clinicians with a modest mathematical background can manipulate it. It is a useful tool for the analysis of general mechanisms of the diseases of calcium metabolism and for the design of clinical experiments aimed at characterizing these diseases. The model can also be the core of future autoadaptive extensions to be used in individual patients.

Calcitriol↗

PubMed2001

Mathematical modelling of physicochemical reactions and transport processes occurring around a platinum cathode during the electrochemical treatment of tumours.

The electrochemical treatment (EChT) of tumours is an anti-tumour therapy in which a continuous direct current is applied to electrodes, placed in or near a tumour. Promising results have been reported from clinical trials in China, where more than 10,000 patients have been treated with EChT during the past 10 years. Before clinical trials can be conducted outside of China, a reliable dose-planning strategy has to be developed. One approach in achieving this is the use of physicochemical simulation models. A simplified mathematical model of the physicochemical processes, occurring around a spherical platinum cathode during EChT, is developed and visualized in three steps in this paper. In the final step, tissue is treated as an aqueous solution of sodium chloride, containing a bicarbonate buffer system and organic constituents susceptible to reactions with hydroxyl ions. This model is shown to give a good description of the pH profile obtained around the cathode after EChT. The simulation results reveal a strong correlation between the pH profiles and size of experimentally measured lesions, thus indicating that it is the spreading of hydroxyl ions that determines the extent of tissue destruction around the cathode. In addition, the simulations indicate that the model could be of use in predicting the size of a lesion produced by EChT.

Animals↗

PubMed2001

Mathematical modelling of patients flow through an accident and emergency department.

OBJECTIVES: The objectives of this project; (1) to evaluate the method, (2) to assess the information required for a more detailed model, and (3) to determine if it was worthwhile to undertake the data collection needed for a more detailed model. METHODS: A mathematical model was constructed using the operational research method of discreet event simulation. The effect of different SHO shift patterns on waiting time was assessed with the model. RESULTS: The model constructed was not an accurate representation of patient flow because of the large number of assumptions that had to be made in this preliminary model. However, the model predicted that an SHO shift pattern that more closely matched the patient arrival pattern would produce shorter waiting times. CONCLUSIONS: This method can be applied to an accident and emergency department. Extension of this approach with the collection of additional data and the development of more sophisticated models seems worthwhile.

Data Collection↗

PubMed2005

Mathematical models for pressure controlled ventilation of oleic acid-injured pigs.

One-compartment, mathematical models for pressure controlled ventilation, incorporating volume dependent compliances, linear and nonlinear resistances, are constructed and compared with data obtained from healthy and (oleic acid) lung-injured pigs. Experimental data are used to find parameters in the mathematical models and were collected in two forms. Firstly, the P(e)-V curves for healthy and lung injured pigs were constructed; these data are used to compute compliance functions for each animal. Secondly, dynamic data from pressure controlled ventilation for a variety of applied pressures are used to estimate resistance parameters in the models. The models were then compared against the collected dynamic data. The best mathematical models are ones with compliance functions of the form C(V) = a + bV where a and b are constants obtained from the P(e)-V curves and the resistive pressures during inspiration change from a linear relation P(r) = RQ to a nonlinear relation P(r) = RQ(epsilon) where Q is the flow into the one-compartment lung and epsilon is a positive number. The form of the resistance terms in the mathematical models indicate the possible presence of gas-liquid foams in the experimental data.

Animals↗

PubMed1998

Ocular fixation index and mathematical models.

Ocular fixation test and ocular fixation index (OFI) never have been interpreted in terms of mathematical models, despite their widespread diffusion. However, ocular fixation is a typical case of visual-vestibular interaction, and mathematical models have proven very helpful in interpreting some mechanisms of this interaction, e.g. those of the optokinetic-vestibular interaction. In the present paper, a first attempt is proposed toward a model interpretation of OFI. By using very simple mathematical models, the hypothesis is tested that visual suppression of vestibular nystagmus results from direct action of smooth pursuit system (SPS). The aim is to draw consequences and recognize possible limits of this hypothesis. Dependence of OFI on SPS performance is examined. Although the available experimental data are insufficient for comprehensive validation of the model, the results agree with the current interpretations. In particular, quantitative support is given to the sensitivity of OFI to central vestibular diseases. Although the interpretation of visual suppression and OFI in terms of mathematical models is still at a very preliminary stage, models may provide a theoretical reference framework for the interpretation of new experimental results and/or suggest new test protocols.

Fixation, Ocular↗

PubMed1989

A mathematical model of the kinetics of blood coagulation.

Linear mathematical models of the kinetics of blood coagulation have previously been presented (Levine, 1966, Science, N.Y. 152, 651; Martorana & Moro, 1974, Math. Biosci. 21, 77). In this paper a non-linear mathematical model of the extrinsic pathway of blood coagulation is presented to take into account a positive feedback. The feedback is due to factor Va as a co-factor involved in thrombin formation. The extrinsic pathway is shown to function as an amplifier cascade if a vessel wall injury exceeds a threshold value. For sub-threshold stimulation, the extrinsic pathway does not function.

Blood Coagulation↗

PubMed2006

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.

Algorithms↗

PubMed1980

Mathematical models of synaptic plasticity: I. Posttetanic potentiation.

A mathematical model of post-tetanic potentiation is proposed. The model uses differential equations and is based upon physiological postulates of the electrical, metabolic, and neuroendocrine activities that are related to synaptic connectivity. These activities may modify some important parameters in synaptic function. In the proposed model these parameters are restricted to the presynapse in view of the physiological evidence indicating that posttetanic potentiation is probably due to presynaptic mechanisms. The model takes into consideration the size of the transmitter pool available for release, the mobilization of transmitter from and to this pool, and the fraction of transmitter released. Based upon the above postulates, we have simulated different phases of the phenomenon of posttetanic potentiation, and we have presented the results of several preparations in which this event has been studied. This work represents a successful attempt to reproduce the dynamics of posttetanic potentiation based upon physiological results with a mathematical model.

Calcium↗

PubMed2001

A three-dimensional mathematical model of temporomandibular joint loading.

OBJECTIVE: A mathematical model of the temporomandibular joint was developed to study the magnitude and direction of the compressive loading experienced at the temporomandibular joint during clenching. DESIGN: The model was based on the principles of static equilibrium in three dimensions. BACKGROUND: Direct measurement of temporomandibular joint loading in humans is extremely difficult. Animal models have provided an alternative in the past. However, evidence suggests that primates are not the most accurate human analogues for temporomandibular joint studies. A mathematical model was used as an alternative to direct measurement. METHODS: The EMG activity of two masticatory muscles was combined with their cross-sectional areas to calculate the force exerted by each muscle. Experimentally determined forces were implemented into a quadratic programming model to solve for the compressive forces on the joint. Two objective functions were chosen and their ability to predict muscle and joint forces was evaluated. RESULTS: The maximum bite forces for normal men, normal women, and women with temporomandibular joint disorders were 300 N (SD 102 N), 210 N (SD 57.7 N), and 120 N (SD 77.1 N), respectively. The calculated joint force for normal males was 260 N (SD 84.1 N). Normal females and female temporomandibular joint disorder patients produced temporomandibular joint forces of 172 N (SD 37.5 N) and 152 N (SD 44.2 N), respectively.

Electromyography↗

PubMedDate not supplied

Mathematical modelling and quantitative methods.

The present review reports on the mathematical methods and statistical techniques presently available for hazard characterisation. The state of the art of mathematical modelling and quantitative methods used currently for regulatory decision-making in Europe and additional potential methods for risk assessment of chemicals in food and diet are described. Existing practices of JECFA, FDA, EPA, etc., are examined for their similarities and differences. A framework is established for the development of new and improved quantitative methodologies. Areas for refinement, improvement and increase of efficiency of each method are identified in a gap analysis. Based on this critical evaluation, needs for future research are defined. It is concluded from our work that mathematical modelling of the dose-response relationship would improve the risk assessment process. An adequate characterisation of the dose-response relationship by mathematical modelling clearly requires the use of a sufficient number of dose groups to achieve a range of different response levels. This need not necessarily lead to an increase in the total number of animals in the study if an appropriate design is used. Chemical-specific data relating to the mode or mechanism of action and/or the toxicokinetics of the chemical should be used for dose-response characterisation whenever possible. It is concluded that a single method of hazard characterisation would not be suitable for all kinds of risk assessments, and that a range of different approaches is necessary so that the method used is the most appropriate for the data available and for the risk characterisation issue. Future refinements to dose-response characterisation should incorporate more clearly the extent of uncertainty and variability in the resulting output.

Animals↗

PubMed2005

RF tumour ablation: computer simulation and mathematical modelling of the effects of electrical and thermal conductivity.

This study determined the effects of thermal conductivity on RF ablation tissue heating using mathematical modelling and computer simulations of RF heating coupled to thermal transport. Computer simulation of the Bio-Heat equation coupled with temperature-dependent solutions for RF electric fields (ETherm) was used to generate temperature profiles 2 cm away from a 3 cm internally-cooled electrode. Multiple conditions of clinically relevant electrical conductivities (0.07-12 S m-1) and 'tumour' radius (5-30 mm) at a given background electrical conductivity (0.12 S m-1) were studied. Temperature response surfaces were plotted for six thermal conductivities, ranging from 0.3-2 W m-1 degrees C (the range of anticipated clinical and experimental systems). A temperature response surface was obtained for each thermal conductivity at 25 electrical conductivities and 17 radii (n=425 temperature data points). The simulated temperature response was fit to a mathematical model derived from prior phantom data. This mathematical model is of the form (T=a+bRc exp(dR) s(f) exp(g)(s)) for RF generator-energy dependent situations and (T=h+k exp(mR)+n?exp(p)(s)) for RF generator-current limited situations, where T is the temperature (degrees C) 2 cm from the electrode and a, b, c, d, f, g, h, k, m, n and p are fitting parameters. For each of the thermal conductivity temperature profiles generated, the mathematical model fit the response surface to an r2 of 0.97-0.99. Parameters a, b, c, d, f, k and m were highly correlated to thermal conductivity (r2=0.96-0.99). The monotonic progression of fitting parameters permitted their mathematical expression using simple functions. Additionally, the effect of thermal conductivity simplified the above equation to the extent that g, h, n and p were found to be invariant. Thus, representation of the temperature response surface could be accurately expressed as a function of electrical conductivity, radius and thermal conductivity. As a result, the non-linear temperature response of RF induced heating can be adequately expressed mathematically as a function of electrical conductivity, radius and thermal conductivity. Hence, thermal conductivity accounts for some of the previously unexplained variance. Furthermore, the addition of this variable into the mathematical model substantially simplifies the equations and, as such, it is expected that this will permit improved prediction of RF ablation induced temperatures in clinical practice.

Catheter Ablation↗

PubMed2004

Mathematical models of the embryo and fetus for use in radiological protection.

This development of new mathematical models arose from our current work in external neutron dosimetry for the embryo and fetus when pregnant women travel at commercial aircraft altitudes. A problem of concern in radiation protection is exposure of pregnant women to ionizing radiation because of the high radiosensitivity of the embryo and fetus. Special regulations and dosimetric considerations are necessary for pregnant women at the work place and in the public. To perform dosimetry, mathematical models for the embryo and the fetus, together with the modified adult female model for pregnant woman, are required. There are no models available for embryo. Models developed for the fetus need to be updated with the new reference values such as those in ICRP Publication 89. This article presents mathematical models for the embryo and fetus at different stages: the embryo at 8 wk and the fetus at the end of each trimester. In addition to fetal skeleton, the fetal brain is explicitly modeled because of its high radiosensitivity. All model parameters are determined from the most recent reference values available. The models are designed so that an interpolation can be easily performed to generate a model of embryo/fetus at any given stage of development. This feature also allows convenient adaptation of the models to different reference values representing various ethnic populations. The new mathematical models presented here were developed for external dosimetry. They can also be used for internal dosimetry purposes, if other organs inside the female phantom are adjusted accordingly.

Adult↗

PubMed1996

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.

Analgesia↗

PubMed2005

Evaluation of nonlinear chromatographic performance by frontal analysis using a simple multi-plate mathematical model.

A multi-plate (MP) mathematical model was proposed by frontal analysis to evaluate nonlinear chromatographic performance. One of its advantages is that the parameters may be easily calculated from experimental data. Moreover, there is a good correlation between it and the equilibrium-dispersive (E-D) or Thomas models. This shows that it can well accommodate both types of band broadening that is comprised of either diffusion-dominated processes or kinetic sorption processes. The MP model can well describe experimental breakthrough curves that were obtained from membrane affinity chromatography and column reversed-phase liquid chromatography. Furthermore, the coefficients of mass transfer may be calculated according to the relationship between the MP model and the E-D or Thomas models.

Chromatography↗