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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↗
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↗
PubMed2002

Effect of rotary blood pump failure on left ventricular energetics assessed by mathematical modeling.

In this study, we used a mathematical model to study the influence of backflow through a failing rotary blood pump. We performed simulations based on animal experiments that were published earlier by Nishida et al., who used the Medos Microdiagonal pump to assess the acute effect of sudden pump failure. The mathematical model consists of validated cardiac and arterial modules and a pump module. We could evaluate the influence of pump failure with mechanoenergetic parameters and wall stress obtained from model output. Simulations were performed at baseline and after 15 min of backflow in a control group and a heart failure group. Simulation results agreed well with the experiment. Stroke volume, aortic flow, and stress time integral increased significantly because of pump failure. However, total systemic flow and arterial pressure were not altered by backflow, and a life-threatening situation did not appear.

Animals↗
PubMed1982

[Use of mathematical models in the analysis of gastric digestion].

Mathematical models of gastric hydrolysis of different proteins were elaborated in dogs. These models consider the spatialtemporal coordinates of distribution of pepsin, hydrogen ions and hydrolysates in the stomach, carry out the analysis of factors determining the gastric digestion. The main part belongs to topography and hydrodynamics of gastric content as well as to pepsin concentration. The concentration of "total hydrolysates" is proposed as the final criterion of the gastric digestion. An algorithm for mathematical modelling of gastric digestion is suggested.

Animals↗
PubMedDate not supplied

Arteriovenous extracorporeal carbon dioxide removal: a mathematical model and experimental evaluation.

To explore the feasibility and operating limits of arteriovenous extracorporeal CO2 removal (AVCO2R) for support of acute respiratory failure, the authors developed a mathematical model to simulate (AVCO2R), evaluate the effects of several parameters used in its application, and predict the feasibility and necessary conditions for total CO2 removal. The mathematical model incorporated compartments representing blood, pulmonary alveoli, pulmonary capillaries, peripheral tissues and capillaries, and an extracorporeal gas exchange device. The model was validated against an animal model of extracorporeal CO2 removal. This model consisted of anesthetized and mechanically ventilated piglets. An extracorporeal CO2 removal device was placed by cannulation of a femoral artery and vein. Dynamic and steady state measurements of CO2 transfer were made and compared with simulations using the mathematical model. There was good agreement between experimental and simulated data, validating the mathematical model under a variety of conditions. The mathematical model was used to determine operating parameters for total CO2 removal. Relationships between extracorporeal blood flow, device diffusing capacity, and device gas sweep flow were established for CO2 removal at various levels of CO2 production. These simulations indicate that it is possible to achieve total CO2 removal using an extracorporeal shunt fraction of 10%-15% of cardiac output, a device diffusing capacity of 0.5 ml x min(-1) x torr(-1) (kg body weight)(-1), and a gas:blood flow of 5 or greater.

Animals↗
PubMed1996

Sexually transmitted diseases and sexual behavior: insights from mathematical models.

The major role of mathematical models of transmission dynamics and population biology of sexually transmitted diseases is helping understand the influence of the many biologic, social, and behavioral factors that influence the incidence or prevalence of infection. Various models can examine heterogeneity in sexual behavior and determine how individual variation influences epidemiologic pattern within a population. In the cases of heterogeneity in sex acts and in sex partner numbers, heterogeneity acts to enhance the likelihood of the persistence of infection. Also important is the pattern of mixing or sexual contact within a community. Assortative mixing promotes rapid spread in high-sexual-activity classes but results in a lower endemic equilibrium state compared with random mixing. In these models, each facet of behavior is treated separately. The obvious next goal of modeling is to meld processes together into a single mathematical framework; however, quantitative epidemiologic information on each factor is still needed.

Humans↗
PubMed2002

Mathematical model to predict individual survival for patients with renal cell carcinoma.

PURPOSE: To develop a multivariate model and mathematical formula capable of calculating personalized survival for renal cell carcinoma (RCC) patients with clinically available variables. PATIENTS AND METHODS: A total of 477 patients out of 661 undergoing nephrectomy at the University of California Los Angeles between 1989 and 1999 were eligible for evaluation and formed the analyzed cohort for this retrospective study. Time to death was the primary end point assessed. Univariate analysis for 14 to 20 variables was conducted, followed by a multivariate Cox analysis. The variables that provided independent information as to the time of death for metastatic and nonmetastatic patients were coded and incorporated into a function based on the Nadas equation principle. RESULTS: For nonmetastatic patients, the significant variables in the multivariate analysis were Fuhrman's grade and Eastern Cooperative Oncology Group performance status. For the metastatic patients, Fuhrman's grade, 1997 classification T stage, number of symptoms, nodal involvement, and immunotherapy were independent predictors for survival. These variables, based on the Cox multivariate regression model, were implanted into an exponential Nadas equation. The expected survival predicted by use of the Nadas equations faithfully describes the actual survival based on Kaplan-Meier curves. CONCLUSION: We have developed mathematical equations for estimating survival after radical nephrectomy for RCC. The resulting formulas are capable of better tailoring survival estimates for a specific patient and are based on widely accepted clinical prognostic variables. On validation with external data, this type of representation can be used as a tool for the determination of personalized prognosis and may be useful for patient education and counseling.

Carcinoma, Renal Cell↗
PubMed1990

Mathematical model of chest wall mechanics: a phenomenological approach.

A mathematical model of chest wall mechanics, based on a phenomenological approach to force balances, provides a quantitative framework for analyzing many types of chest wall movements by using orthogonal displacement coordinates. The moveable components of the ventilatory system include the rib cage, diaphragm, and abdomen. A distinction is made between the lung-apposed and diaphragm-apposed actions on the rib cage. The model equations are derived from "pressure" balances and geometrical relations of the compartments; the stress-displacement relations are hyperbolic. With this model we simulated stiff and flaccid chest wall behavior under normal and constrained conditions associated with abdominal compression, a Mueller maneuver, and a diaphragmatic isometric inspiration. We also examined situations that produce paradoxical as well as orthodox inspiratory movements. The results of these simulations were quantitatively consistent with available data from the literature. A phenomenon predicted by the stiff-wall model during quasi-static inspiration is that the rib cage displacement is negligible near residual volume, but then increases dramatically with lung volume. Since this mathematical model has a sound physical basis and is more comprehensive than previous models, it can be used to predict and analyze the behavior of the chest wall under a wide variety of circumstances.

Abdominal Muscles↗
PubMed2000

Mathematical modeling of diffusion-mediated release from bulk degrading matrices.

The release of active agent from a bulk degrading matrix is formulated as a linear reaction diffusion problem. Two pools of active agent are assumed to contribute to the release: a pool of mobile active agent which readily diffuses out of the matrix upon immersion in an aqueous medium and a pool of immobilized active agent which can diffuse only after matrix degradation. Due to the linearity of our model, the dynamics of the two pools of active agent can be considered separately, for any mode of bulk degradation kinetics. For definiteness, we consider the case of first order degradation kinetics and a rectangular parallelepiped shaped matrix. A closed form solution is obtained for the release under perfect sink conditions which is then used to describe the in vitro release of the PerioChip¿trade mark omitted¿. This solution can explain the bi-phasic release profile characteristic of many hydrolytically degradable matrices. The case of mass transfer boundary conditions is solved numerically using the finite element method (FEM). This analysis indicates that under ordinary mixing conditions the diffusion layer is not rate limiting and the release is very well approximated by the analytical result for perfect sink conditions.

Aerosols↗
PubMed1998

Human amniotic fluid mathematical model: determination and effect of intramembranous sodium flux.

OBJECTIVE: A recently described mathematical model of human amniotic fluid dynamics used known and estimated rates of fetal fluid production (lung liquid and urine) and composition (osmolality) to enable calculation of previously unmeasured routes of amniotic fluid resorption, including fetal swallowing and intramembranous (across the amnion) water flow. This "osmolar" model assumed that only free water resorption occurred across the intramembranous route. We hypothesized that intramembranous flow also may include solutes and electrolytes because significant concentration gradients exist between amniotic fluid and fetal plasma. We used mass balance analysis to determine the direction and magnitude of intramembranous sodium flux and to assess the ability of a newly described "sodium" model to predict changes in amniotic fluid volume in response to changes in intramembranous electrolyte flow. Mathematical modeling was used to predict changes in amniotic fluid volume in response to changes in intramembranous electrolyte flow. STUDY DESIGN: Model predictions were calculated using published values for human amniotic fluid and fetal urine composition and volume. Ovine studies were used to derive lung fluid volumes and composition. Fetal swallowing and intramembranous flow were independently determined using net amniotic fluid osmolar (osmolality model) and sodium (sodium model) balance. Differences between osmolality and sodium model predictions were normalized to calculate the net intramembranous sodium flux, assuming a net balance of intramembranous osmotic solute flow. RESULTS: Both sodium and osmolality models predicted swallowed volume to be greater than intramembranous flow until 28 to 32 weeks' gestation, after which the relationship reversed. However, the sodium model predicted greater intramembranous flow and lower swallowing rates compared with the osmolality model at all gestational ages. Osmolar mass balance required daily intramembranous sodium flux into the amniotic fluid, which increased with gestational age. Furthermore, assuming stable swallowing and intramembranous water flow, the model predicts that 5% increases or decreases in amniotic fluid solute concentrations caused by intramembranous flux result in polyhydramnios or oligohydramnios, respectively. CONCLUSION: Sodium and osmolality models demonstrate similarities in determinations of amniotic fluid dynamics. However, mass balance equations demonstrate a net intramembranous flow of sodium into the amniotic fluid under normal conditions. Mathematical modeling suggests that small alterations in daily intramembranous sodium flux may evoke large changes in amniotic fluid volume.

Amniotic Fluid↗
PubMed2002

Which approach to anticoagulation management is best? Illustration of an interactive mathematical model to support informed decision making.

BACKGROUND: Among patients with atrial fibrillation or mechanical heart valves, determining the best approach to oral anticoagulation largely depends on comparing the costs of anticoagulation management with the costs of events (thromboembolism and bleeding) averted. The Anticoagulation Management Event/Cost Model (ACME) is an interactive mathematical model intended to help clarify these trade-offs. METHODS: The ACME is a series of linked, nested spreadsheets. At the least detailed level, the user specifies the percentage of patients falling into various management strategies (no anticoagulation, usual physician care, anticoagulation service, patient self-testing/self-management), and the ACME estimates event rates and costs. At more detailed levels the ACME performs a series of weighted average calculations combining, for example, utilization times unit price. Cost categories are divided into event-related and management-related costs (costs of management, testing, and medication). RESULTS: Regardless of how anticoagulation is subsequently managed, perhaps the greatest benefit is obtained by moving patients who are not currently receiving anticoagulation onto warfarin. Additional benefits can be obtained by eliminating outliers (extremely high or extremely low anticoagulation levels). If changing to a more intensive approach also serves to reduce the tendency for physicians to prescribe anticoagulate below the optimal range, additional savings can be anticipated. The cost calculation typically involves a trade-off between increased up-front costs of anticoagulation management versus greater down-line savings associated with a decreased number of events. To assess the quality of anticoagulation within a given organization, it is critical to know the distribution of clotting levels for the population under anticoagulation. CONCLUSIONS: Interactive mathematical models, if sufficiently well documented, can be helpful in clarifying decisions regarding costs and benefits of various methods of anticoagulation.

Anticoagulants↗
PubMed1985

Studies on the release of solubilized drugs from ointment bases. Part 9: Modelling of drug release process of active substance from emulsive ointments (W/O)--the mathematical model.

A new mathematical model of the release process of solubilized active substance from emulsive (W/O) ointments is proposed. This model is based on Fick's diffusion law and additional assumptions. The most important assumption is recognition of the heterogeneous character of emulsive ointments. An equation has been derived, which expresses the functional relationship between the quantity of substance liberated from the ointment and the period of release.

Emulsions↗
PubMed2002

Mathematical models of human CD4+ T-cell population kinetics.

We review how mathematical models help the interpretation of data measuring CD4+ T-cell kinetics by two recently-developed techniques. Mathematical models are developed for the average content of T-cell receptor excision circles (TRECs) and the average telomeric restriction fragment (TRF) in T-cells in the peripheral blood. Changes in the TRECs were supposed to indicate changes in thymic production. The rate at which naive and memory CD4+ T-cells erode their telomeres was supposed to reflect their respective division rates. Analysing the mathematical models, we show that rapid changes in the TRECs per naive T-cell are most likely due to changes in the division rates, and that the rates of telomere erosion fail to reflect naive and memory division rates. The model is applied to explain data showing that rheumatoid arthritis (RA) patients have abnormal TRECs and telomeres.

Arthritis, Rheumatoid↗
PubMed1991

Mathematical modelling of flow through an irregular arterial stenosis.

A mathematical model of flow through an irregular arterial stenosis is developed. The model is two-dimensional and axi-symmetric with the stenosis outline obtained from a three-dimensional casting of a mildly stenosed artery. Agreement between modelled and experimental pressure drops (obtained from an axi-symmetric machined stenosis with the same profile) is excellent. Results are also obtained for a smooth stenosis model, similar to that used for most mathematical modelling studies. This model overestimates the pressure drop across the stenosis, as well as the wall shear stress and separation Reynolds number. Also, the smooth model predicts one instead of three recirculation zones present in the irregular model. The original stenosis is modified to increase the severity from 48 and 87% areal occlusion, while maintaining the same general shape. This has the effect of increasing the pressure drop by an order of magnitude and decreasing the number of recirculation zones to one, with a lower separation Reynolds number.

Arterial Occlusive Diseases↗
PubMed1994

A study of the singularities in a mathematical model for circadian rhythms.

One of the models that has been suggested for describing circadian rhythms mathematically is an extension of the van der Pol equation given by ÿ + 0.5(y2 + y-2 - 3)y + (1 + 0.6 y) y = z + z + z, where y is the oscillating variable, and z is the light intensity assumed to excite the oscillator. In order for the equation to exhibit self-sustained oscillations, z has to be within the oscillatory range (0.847 < z < 3.189). This equation has been shown to simulate several of the features possessed by circadian systems (Wever, R., 1984, Toward a mathematical model of circadian rhythmicity, in: Mathematical Models of the Circadian Sleep-Wake Cycle, M.C. Moore-Ede and C.A. Czeisler (eds.) (Raven Press, New York) pp. 17-79). Physiological experiments have been performed which show that circadian rhythms can have stable singularities. Therefore, it was of interest to investigate whether or not the equation given above also has this property. We have studied the stability of the two singularities of the model system above. One of the singularities is unstable and corresponds to non-physiological conditions. The other one is an unstable spiral point if the light conditions are such that oscillations can occur in the system. We conclude that the model mentioned above is unsuitable to describe circadian systems which have stable singularities. The model has been simulated, and pulses have been applied to the system by temporarily changing the value of z to find appropriate conditions forcing the system into its singularity. The strategy to find such pulses is discussed.

Animals↗
PubMedDate not supplied

[Mathematical modeling and optimization of plasmadiafiltration].

A mathematical model of mass transport of toxic substances with small, middle and large molecules weight in the body compartments and in the extracorporal system was worked out and used in the clinic for individual optimization and prediction of final results when treating patients with acute hepatic and renal failure in plasmadiafiltration. Permeability and the sieving coefficients were found "in vivo" in the plasma for 3 types of dialysers with different membranes. For practical use of this model a program was written by an interactive dialogue for the personal computer.

Acute Kidney Injury↗
PubMed2002

A mathematical model of metabolic insulin signaling pathways.

We develop a mathematical model that explicitly represents many of the known signaling components mediating translocation of the insulin-responsive glucose transporter GLUT4 to gain insight into the complexities of metabolic insulin signaling pathways. A novel mechanistic model of postreceptor events including phosphorylation of insulin receptor substrate-1, activation of phosphatidylinositol 3-kinase, and subsequent activation of downstream kinases Akt and protein kinase C-zeta is coupled with previously validated subsystem models of insulin receptor binding, receptor recycling, and GLUT4 translocation. A system of differential equations is defined by the structure of the model. Rate constants and model parameters are constrained by published experimental data. Model simulations of insulin dose-response experiments agree with published experimental data and also generate expected qualitative behaviors such as sequential signal amplification and increased sensitivity of downstream components. We examined the consequences of incorporating feedback pathways as well as representing pathological conditions, such as increased levels of protein tyrosine phosphatases, to illustrate the utility of our model for exploring molecular mechanisms. We conclude that mathematical modeling of signal transduction pathways is a useful approach for gaining insight into the complexities of metabolic insulin signaling.

Animals↗