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A mathematical model of the flow in the posterior communicating arteries.

This paper reports on a mathematical model designed to study the hemodynamics of one posterior communicating artery and its afferent and efferent vessels. The variables in the model are the diameter of the posterior communicating artery, the resistance in the vertebral artery and the ratio of the two peripheral resistances. In the model, the "posterior communicating artery" exhibits a compensatory capacity, as defined in the introduction, which appears to be independent of its diameter. The fluxes in the efferent vessels are dominated by the peripheral resistances.

Carotid Arteries↗

Mathematical modeling of humoral immune response suppression by passively administered antibodies in mice.

Although passively administered antibodies are known to suppress the humoral immune response, the mechanism is not fully understood. Here, we developed a mathematical model to better understand the suppression phenomena in mice. Using this model, we tested the generally accepted but difficult to prove "epitope masking hypothesis." To simulate the hypothesis and clearly observe masking of epitopes, we modeled epitope-antibody and epitope-B-cell receptor interactions at the epitope level. To validate this model, we simulated the effect of the antibody affinity and quantity as well as the timing of administration on the suppression, and we compared the results with experimental observations reported in the literature. We then developed a simulation to determine whether the epitope-masking hypothesis alone can explain known immune suppression phenomena, especially the conflicting results on F(ab')2 fragment-induced suppression, which has been shown to be no suppression, or similar to or up to 1000-fold weaker than the suppression by intact antibody. We found that suppression was caused by a synergistic effect of both epitope masking and rapid antigen clearance. Although the latter hypothesis has lost support because FcgammaRI/III mutant mice show antibody-mediated suppression, our simulations predict that, even in FcgammaRI/III mutant mice, the immune response can be suppressed according to the antibody affinity. Our model also effectively reproduced the conflicting results obtained using F(ab')2 fragments. Thus, in contrast to the idea that the F(ab')2 results prove the FcgammaRIIb involvement in suppression, our mathematical model suggests that the epitope-masking hypothesis together with rapid antigen clearance explains the conflicting results.

Animals↗

Mathematical models of action potentials in the periphery and center of the rabbit sinoatrial node.

Mathematical models of the action potential in the periphery and center of the rabbit sinoatrial (SA) node have been developed on the basis of published experimental data. Simulated action potentials are consistent with those recorded experimentally: the model-generated peripheral action potential has a more negative takeoff potential, faster upstroke, more positive peak value, prominent phase 1 repolarization, greater amplitude, shorter duration, and more negative maximum diastolic potential than the model-generated central action potential. In addition, the model peripheral cell shows faster pacemaking. The models behave qualitatively the same as tissue from the periphery and center of the SA node in response to block of tetrodotoxin-sensitive Na(+) current, L- and T-type Ca(2+) currents, 4-aminopyridine-sensitive transient outward current, rapid and slow delayed rectifying K(+) currents, and hyperpolarization-activated current. A one-dimensional model of a string of SA node tissue, incorporating regional heterogeneity, coupled to a string of atrial tissue has been constructed to simulate the behavior of the intact SA node. In the one-dimensional model, the spontaneous action potential initiated in the center propagates to the periphery at approximately 0.06 m/s and then into the atrial muscle at 0.62 m/s.

Action Potentials↗

[A semiempirical mathematical model for assessment of PH3 residues as exemplified by phosphine-fumigated hazelnuts].

A mathematical model for the calculation of residues of PH3 in fumigated hazelnuts in dependence on dosage, fumigation- and storage time is discussed. The model is based on the physical process of the diffusion of a gas out of a sphere. An empirical dependence of the diffusion coefficient on the residue levels can be observed. Adequate agreement is obtained between calculated and experimental values.

Food Contamination↗

[Mathematical model of oblique three-dimensional intertrochanteric detorsion varus-forming osteotomy of the femur by the Bernbeck method in surgical treatment of congenital hip dysplasia in children].

Whenever the conservative procedure fails to bring about congruence of the dysplastic hip joint, an operative procedure becomes indispensable. In Orthopaedic Clinic of the Pomeranian Medical Academy in Szczecin we implement the oblique three-dimensional intertrochanteric detorsion and varus forming osteotomy after Bernbeck in order to correct the proximal end of the femoral bone. Precise determination of the plane to be cut, prior to the operative procedure, simplifies and shortens the operation itself and facilitates the achieving of the planned angular values in all three planes. Mathematical model of osteotomy according to Bernbeck considering required angles of correction as well as angles determining the plane of osteotomy was worked out. In collaboration of the Szczecin Technical University, a simple computer program was elaborated which allowed the presentation of the results in the form of tables. With the help of tables the optimal cutting plane was chosen and created correct biomechanical and anatomical conditions as well as optimal conditions for stable osteosynthesis of dissected fragments of the femoral bone. That type of osteotomy is useful in most operative correcrions of the dysplastic hip joint (not great varus formation connected with relatively extensive detorsion). The achieved congruence in the 22 dysplastic hip joints operated on was the most important condition for their later physiological development. Short post-operative observations confirm the value of described mathematic model.

Acetabulum↗

A neurophysiologically-based mathematical model of flash visual evoked potentials.

Evidence is presented that a neurophysiologically-inspired mathematical model, originally developed for the generation of spontaneous EEG (electroencephalogram) activity, can produce VEP (visual evoked potential)-like waveforms when pulse-like signals serve as input. It was found that the simulated VEP activity was mainly due to intracortical excitatory connections rather than direct thalamic input. Also, the model-generated VEPs exhibited similar relationships between prestimulus EEG characteristics and subsequent VEP morphology, as seen in human data. Specifically, the large correlation between the N1 amplitude and the prestimulus alpha phase angle, and the insensitivity of P2 to the latter feature, as observed in actual VEPs to low intensity flashes, was also found in the model-generated data. These findings provide support for the hypothesis that the spontaneous EEG and the VEP are generated by some of the same neural structures and that the VEP is due to distributed activity, rather than dipolar sources.

Cerebral Cortex↗

Mathematical modeling of electrochemical remediation for soils under galvanostatic conditions.

This work proposes a mathematical model for the electrochemical remediation of clayey soils based on the total volume concept for a two-phase system. The mathematical formulation was done including contributions from theories for: groundwater, membranes, porous electrodes and environmental soil chemistry. The resulting model accounts for: free and complexed species in the soil matrix and the pore solution; chemical reactions taking place on either phase and/or between phases; a dynamic soil surface charge affected by the ion content of the pore solution; and electroneutrality of the total volume. Soil surface charge was included in a modified Ohm's law (voltage gradient) and in a modified Schlög's law (convective movement). Numerical implementation was done using orthogonal collocation on finite elements for spatial derivatives, and forward finite differences for time derivatives. Visual Fortran supported by IMSL subroutines was used for computer simulation. Model predictions were successfully compared with reported experimental data. Also, an analysis of pH profiles through the soil is provided for conditions when parameters including hydrostatic head, applied current density and initial pH are modified.

Electrochemistry↗

Mathematical model of man's tolerance to cold using morphological factors.

A mathematical model has been developed to anticipate the physiological responses and the thermal state of a naked human under exposure to cold, taking into account his morphological characteristics (skinfold, size, weight) and the environmental conditions (air or water temperature and velocity, barometric pressure and hygrometry). The skinfold conditions the body's thermal conductance and the metabolism depends both on rectal (Tre) and mean skin (Tsk) temperatures. After being tested, this model was used to study the evolution of Tre. It shows the influence of the skinfold which accounts for most of the inter-individual differences. It also permits discussion of survival possibilities during immersion and completes data provided by previously established curves.

Body Surface Area↗

[Woronoff rings in a mathematical model].

The width of a Woronoff ring is thought to be constant and independent of the diameter of the lesion. Clinical observations have allowed us to demonstrate that this is true only for lesions of a certain minimum size or greater. It is suggested that the paleness within the ring is caused by diffusion of a mediator from the psoriatic plaque into its environment. A mathematical model is formulated, and it is shown that clinical observations are consistent with the pathogenetic hypothesis. In addition, the mathematical model is used to derive some properties of the hypothetical mediator.

Diffusion↗

Mathematical modeling of human cardiovascular system for simulation of orthostatic response.

This paper deals with the short-term response of the human cardiovascular system to orthostatic stresses in the context of developing a mathematical model of the overall system. It discusses the physiological issues involved and how these issues have been handled in published cardiovascular models for simulation of orthostatic response. Most of the models are stimulus specific with no demonstrated capability for simulating the responses to orthostatic stimuli of different types. A comprehensive model incorporating all known phenomena related to cardiovascular regulation would greatly help to interpret the various orthostatic responses of the system in a consistent manner and to understand the interactions among its elements. This paper provides a framework for future efforts in mathematical modeling of the entire cardiovascular system.

Blood Pressure↗

Booked inpatient admissions and hospital capacity: mathematical modelling study.

OBJECTIVES: To investigate the variability of patients' length of stay in intensive care after cardiac surgery. To investigate potential interactions between such variability, booked admissions, and capacity requirements. DESIGN: Mathematical modelling study using routinely collected data. SETTING: A cardiac surgery department. SOURCE OF DATA: Hospital records of 7014 people entering intensive care after cardiac surgery. MAIN OUTCOME MEASURES: Length of stay in intensive care; capacity requirements of an intensive care unit for a hypothetical booked admission system. RESULTS: Although the vast majority of patients (89.5%) had a length of stay in intensive care of < or = 48 hours, there was considerable overall variability and the distribution of stays has a lengthy tail. A mathematical model of the operation of a hypothetical booking system indicates that such variability has a considerable impact on intensive care capacity requirements, indicating that a high degree of reserve capacity is required to avoid high rates of operation cancellation because of unavailability of suitable postoperative care. CONCLUSION: Despite the considerable enthusiasm for booked admissions systems, queuing theory suggests that caution is required when considering such systems for inpatient admissions. Such systems may well result in frequent operational difficulties if there is a high degree of variability in length of stay and where reserve capacity is limited. Both of these are common in the NHS.

Admitting Department, Hospital↗

The use of mathematical models to evaluate pelvic masses; can they beat an expert operator?

The pre-operative characterization of ovarian cysts remains a major challenge. Functional cysts and some other benign cysts should be managed conservatively, whereas persistent tumours may need removal. It is crucial to distinguish between malignant tumours, which are better treated by a gynaecological oncologist, and benign tumours, which may be suitable for minimal-access surgery. Over the past decade several ultrasound-based morphological scoring systems, colour Doppler parameters, logistic regression models and artificial neural networks have been proposed and tested in order to try to predict the histology of ovarian tumours. On prospective testing none of the current models can beat an expert sonologist. Signs of malignancy include the presence of papillary structures, irregular solid areas, septa and a strong vascularization at colour Doppler imaging. Further refinement of mathematical models and the results of multicentre trials need to be reviewed before the clinical use of mathematical models can be advocated.

Biomarkers, Tumor↗

A mathematical model of hepatitis a transmission in the United States indicates value of universal childhood immunization.

BACKGROUND: US recommendations issued in 1999 for hepatitis A (HA) childhood immunization varied according to regional HA incidences prior to vaccination. Mathematical models of HA transmission, especially those accounting for herd protection, can be useful in formulating new, highly effective recommendations that could lead to disease elimination. METHODS: A mathematical model of HA transmission was designed to assess the impact of different vaccination strategies on the evolution of HA infection over time in the United States. The model represents HA transmission dynamics and is stratified by age and regions defined in the Advisory Committee for Immunization Practices 1999 recommendations. The model accounts for herd protection and HA importation, using an age-dependent "force of infection" varying over time as a function of the prevalence of subjects with infectious HA. RESULTS: The model predicts a clear benefit of vaccinating all US children at as young an age as possible. Nationwide routine immunization at 1 year of age with 70% coverage would prevent 57% of additional cases during the period 1995-2029, compared with the continuation of the regional strategy of vaccinating children at 2 years of age, as recommended by the Advisory Committee for Immunization Practices in 1999. In contrast, the model also predicts that nationwide routine immunization for children 12 years of age only would result in a 14% increase of HA cases during the period 1995-2029, compared with the number of cases predicted with the regional strategy of the immunization of 2-year-olds. CONCLUSIONS: These findings highlight the importance of accounting for herd protection induced by early childhood HA vaccination. They also support the very recent Advisory Committee for Immunization Practices recommendations for universal HA immunization of 1-year-olds.

Adolescent↗

[A mathematical model of apolipoprotein B metabolism in human blood plasma].

Mathematical model was developed for transport of apo B in blood plasma, which allowed to estimate constants of metabolism of lipoproteins containing apo B, considering the alterations of apo B concentration in circulation followed by immuno- or hemosorption. Some cases were detected where stimulating effect of these procedures on the rate of apo B entering into circulation was found as well as definite ratios were shown between the constants using the parameters of alterations in apo B concentration after elimination of lipoproteins containing apo B from blood plasma. Existence of cases was predicted where fluctuations in apo B concentration attenuated after immuno- and hemosorption which were difficult to find in experiments.

Apolipoproteins B↗

A mathematical model of CD8+ lymphocyte dynamics in HIV infection.

The previously developed mathematical model of CD4+ lymphocyte dynamics in HIV infection incorporated a homeostatic mechanism regulating production of both CD4+ and CD8+ lymphocytes. The model simulated the CD4+ lymphocyte dynamics well, but simulation of CD8+ lymphocyte values was not satisfactory, because simulated numbers of these cells increased even at later stages of infection, when no further increase was observed in infected individuals. Modifications of the model were attempted to obtain better simulation results assuming the influence of HIV infection on CD8+ lymphocyte maturation. Satisfactory results were obtained, if the influx of immature CD4+ and CD8+ lymphocytes was constrained by HIV infection. This modified version was then used for simulation of the anti-CD8 antibody administration effect in HIV-infected persons.

CD8-Positive T-Lymphocytes↗

A mathematic model for relating the drug sensitivity of tumors to their spontaneous mutation rate.

A mathematic model has been developed relating the drug sensitivity of a tumor to its own spontaneous mutation rate towards phenotypic drug resistance. The proportion as well as the absolute numbers of resistant cells will increase with time and the fraction of resistant cells within tumor colonies of the same size with vary depending on whether mutation occurs as an early or late event. Analysis of the model indicates that the probability of the appearance of a resistant phenotype increases with the mutation rate. Furthermore, for any population of tumors with a non-zero mutation rate the likelihood of there being at least one resistant cell will go from a condition of low to high probability over a very short interval in the tumor's biologic history.

Antineoplastic Agents↗

[Principles in construction of mathematical model for neonatal corpse cooling studies].

The process of cooling of the body of a dead human was studied with consideration for the anatomical characteristics of adult and newborn human corpses. The body was simulated as a homogeneous elongated rotation ellipsoid and mathematical model was developed as function of relationship between body temperature and time. After appropriate characteristics in the mathematical model were established, cooling processes and effects of various factors on this process were analyzed for the final ellipsoid. The model suggests the possibility of amendments for environmental temperature and body parameters.

Body Temperature↗

[Problem of sphere formation in mathematical models of morphogenesis].

The model of a process of morphogenesis is described, being a new link in the series of mathematical models of morphogenesis based on the local interactions of cells (see [6]) and the first space model of the series. A corresponding mathematical problem is formulated. The results of the proper computer calculations are published, showing that the chosen rule of motion solves the problem of sphere formation for a very large class of initial states of all investigated types of nets. The processes taking place in solving the problem of sphere formation are considered.

Computers↗