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PubMed2004

Estimation of parameters for a mathematical model of growth hormone secretion.

Here, we describe partial calibration of a parsimonious mathematical model of growth hormone (GH) secretion. From first principles, we derived a model of the effects on GH secretion from pituitary somatotrophs of stimulation by GH-releasing factor (GRF) or GH secretagogue, and of inhibition by somatostatin. We obtained a concise model by collapsing the many processes of the signal transduction cascade into a single step broadly reflecting the initial binding of GRF to its receptors. In the model, GH secretion is proportional to the rate of binding of GRF to activatable receptors. Desensitization occurs because of reduction of free receptors/available effector units, and resensitization occurs as those lost are replaced. This replacement is speeded up in the presence of somatostatin, which also inhibits GH secretion by reducing the constant of proportionality between the rate of GH secretion and the rate of GRF binding. We derived simple mathematical equations for the rate of GH secretion and cumulative secretion. Using these, we tested the model against data obtained from experiments performed in vitro, and made it quantitative using rigorous statistical approaches to optimize parameter estimates. The behaviour of the calibrated model matches experimental observations closely.

Algorithms↗
PubMed2004

A mathematical model of factors that influence the performance of accommodative intraocular lenses.

In this work a mathematical model of capsule movement during pseudophakic accommodation is described to allow identification and evaluation of factors that may explain the variation in effect of accommodative intraocular lenses (IOLs) between patients. The model assumes that increasing vitreous pressure pushes the lens capsule forward as a circular diaphragm and that this movement is from a fixed fulcrum. With an IOL in situ, the capsule is taken to have a non-uniform thickness due to the presence of the anterior capsulorhexis. The model assumes a uniform capsular elasticity and ignores contributions from cellular elements such as posterior capsule opacification. Using our model and a regression formula to calculate capsular bag size, taking into account axial length and keratometry values, we are able to predict accommodative effect in individual patients. By simple geometry we have developed a mathematical model to identify variables that are important in pseudophakic accommodation. It provides the basis for the development of a more complex model that would address the movement of a lens taking into account the influence of the zonular system during accommodation.

Accommodation, Ocular↗
PubMed1997

Nonlinear mathematical analysis of the hemodynamic parameters during left ventricular assistance with oscillated blood flow.

For the development of a totally implantable ventricular assist system (VAS), we have been developing the vibrating flow pump (VFP), which can generate oscillated blood flow with a relatively high frequency (10-50 Hz) for a totally implantable system. In this study, effects of left ventricular assistance with this unique oscillated blood flow were analyzed by nonlinear mathematics for evaluation as the entire circulatory regulatory system, not as a separate part of the system. Left heart bypasses using VFPs from the left atriums to the descending aortas were performed in chronic animal experiments using healthy adult goats. Electrocardiogram (ECG), arterial blood pressure, VFP pump flow, and flow of the descending aorta data taken while the goats were awake were recorded in the data recorder and analyzed in the personal computer system through the AD convertor. Using nonlinear mathematics, time series data were embedded into the phase space, and the Lyapunov numerical method, fractal dimension analysis, and power spectrum analysis were performed to evaluate the nonlinear dynamics. During left ventricular assistance with the VFP, Mayer wave fluctuations were decreased in the power spectrum, the fractal dimension of the hemodynamics was significantly decreased, and peripheral vascular resistance was significantly decreased. These results suggest that nonlinear dynamics, which mediate the cardiovascular dynamics, may be affected during LV bypass with oscillated flow. Decreased power of the Mayer wave in the spectrum caused the limit cycle attractor of the hemodynamics and decreased the peripheral resistance. Decreased sympathetic discharges may be the origin of the decreased Mayer wave and fractal dimension. These nonlinear dynamical analyses may be useful to design the optimal VAS control.

Animals↗
PubMed2004

The power-law mathematical model for blood damage prediction: analytical developments and physical inconsistencies.

Blood trauma caused by medical devices is a major concern. Complications following the implantation/application of devices such as prosthetic heart valves, cannulae, blood pumps, tubing, and throttles lead to sublethal and lethal damage to platelets and erythrocytes. This damage is provided by the alterations in fluid dynamics, providing a mechanical load on the blood corpuscle's membrane by means of the shear stress. An appropriate quantification of the shear-induced hemolysis of artificial organs is thought to be useful in the design and development of such devices in order to minimize device-induced blood trauma. To date, a power-law mathematical relationship using the time of exposure of a blood corpuscle to a certain mechanical load and the shear stress itself (derived under the peculiar condition of uniform shear stress) has served as a basic model for the estimation of the damage to blood, investigated by means of numerical and/or experimental fluid dynamical techniques. The aim of the present article is to highlight the effect of a time-varying mechanical loading acting on blood cells based on the usual power-law model; furthermore, the effect of the loading history of a blood particle is discussed, showing how the past history of the shear acting on a blood corpuscle is not taken into account, as researchers have done until now. The need for a reassessment of the power-law model for potential blood trauma assessment is discussed by using a mathematical formulation based on the hypotheses of the existence of damage accumulation for blood with respect to time and with respect to shear stress, to be applied in complex flow fields such as the ones established in the presence of artificial organs.

Algorithms↗
PubMed2004

A mathematical model of cell salvage compared and combined with normovolemic hemodilution.

BACKGROUND: Mathematical models have been used to describe the factors that affect cell salvage (CS) and normovolemic hemodilution (ANH). Here, the CS and ANH models were used to compare these two techniques alone or in combination with each other. STUDY DESIGN AND METHODS: Variables used for a hypothetical patient included an estimated blood volume of 5000 mL, a presurgery hematocrit (Hct) of 45 percent, and a transfusion trigger of 21 percent. The model accounts for both the effect of decreasing the Hct due to blood loss and the effect of increasing Hct due to the readministration of blood in an isovolemic patient. The efficacy of CS and ANH is defined to be the maximum allowable blood loss for a fixed blood volume and a fixed transfusion trigger. RESULTS: Comparison of CS with ANH showed that 3 units of ANH was comparable to CS when CS recovery rates ranged from 19 to 24 percent. For a patient with a blood volume of 5000 mL and a starting Hct of 40 percent, 3 units of ANH would allow for 3972 mL of blood to be lost before crossing a 21-percent transfusion trigger, whereas CS with a 125-mL bowl would allow for 7611 mL. CONCLUSION: When comparing ANH to CS, this mathematical model would suggest that CS has the potential to offer significantly greater red blood cell avoidance than does ANH; however, the combination of ANH with CS may offer allogeneic avoidance superior to either technique alone.

Blood Loss, Surgical↗
PubMed1991

Cardiac electrophysiological experiments in numero, Part I: Concepts and strategies of mathematical and computer models.

This article is the first of three articles that review mathematical and computer models of the heart and describe their construction, development, research potential, and clinical utility. This article explains the methodological principles of mathematical and computer simulation of biomedical systems. The strategies of model construction, testing, and application are presented; the advantages and limitations of computer simulation studies are explained, and the basic value of computer simulation for cardiological research and practice is discussed.

Cardiac Pacing, Artificial↗
PubMed1996

Interaction between opioid and muscarinic receptors in the guinea-pig ileum preparation: a mathematical model.

Fentanyl and pethidine are opioid agonists and muscarinic antagonists in the guinea-pig ileum preparation. In this preparation an opioid agonist reduces the release of acetylcholine. Therefore an opiate may influence the potency of an anticholinergic drug. A mathematical model was developed to characterize this putative interaction between opioid and muscarinic receptors. The model is based on the assumption that the drugs interact with the receptors in a competitive manner according to the law-of-mass action. Concentration-response experiments were performed in the guinea-pig ileum preparation to test the model. The mathematical model describes the concentration-response curves very well and estimates the IC50 values for the two components with good precision. The study shows that an opioid agonist can potentiate the effect of an anticholinergic drug substantially. This is interesting with regard to the central anticholinergic syndrome. The conclusion is that the model describes the interaction adequately.

Animals↗
PubMed2005

New mathematical methods in pharmacokinetic modeling.

In recent years, several new methods for the mathematical modeling have gradually emerged in pharmacokinetics, and the development of pharmacokinetic models based on these methods has become one of the most rapidly growing and exciting application-oriented sub-disciplines of the mathematical modeling. The goals of our MiniReview are twofold: i) to briefly outline fundamental ideas of some new modeling methods that have not been widely utilized in pharmacokinetics as yet, i.e. the methods based on the following concepts: linear time-invariant dynamic system, artificial-neural-network, fuzzy-logic, and fractal; ii) to arouse the interest of pharmacological, toxicological, and pharmaceutical scientists in the given methods, by sketching some application examples which indicate the good performance and perspective of these methods in solving pharmacokinetic problems.

Animals↗
PubMed2005

Mathematical models of periodic breathing and their usefulness in understanding cardiovascular and respiratory disorders.

Periodic breathing is an unusual form of breathing with oscillations in minute ventilations and with repetitive apnoeas or near apnoeas. Reported initially in patients with heart failure or stroke, it was later recognized to occur especially during sleep. The recurrent hypoxia and surges of sympathetic activity that often occur during the apnoeas have serious health consequences. Mathematical models have helped greatly in the understanding of the causes of recurrent apnoeas. It is unlikely that every instance of periodic breathing has the same cause, but many result from instability in the feedback control involved in the chemical regulation of breathing caused by increased controller and plant gains and delays in information transfer. Even when it is not the main cause of the periodic breathing, unstable control modifies the ventilatory pattern and sometimes intensifies the recurrent apnoeas. The characteristics of disturbances to breathing and their interaction with the control system can be critical in determining ventilation responses and the occurrence of periodic breathing. Large abrupt changes in ventilation produced, for example, in the transition from waking to sleep and vice versa, or in the transition from breathing to apnoea, are potent factors causing periodic breathing. Mathematical models show that periodic breathing is a 'systems disorder' produced by the interplay of multiple factors. Multiple factors contribute to the occurrence of periodic breathing in congestive heart failure and cerebrovascular disease, increasing treatment options.

Animals↗
PubMed2005

Mathematical model for pressure losses in the hemodialysis graft vascular circuit.

Stenosis-induced thrombosis and abandonment of the hemodialysis synthetic graft is an important cause of morbidity and mortality. The graft vascular circuit is a unique low-resistance shunt that has not yet been systematically evaluated. In this study, we developed a mathematical model of this circuit. Pressure losses (deltaPs) were measured in an in vitro experimental apparatus and compared with losses predicted by equations from the engineering literature. We considered the inflow artery, arterial and venous anastomoses, graft, stenosis, and outflow vein. We found significant differences between equations and experimental results, and attributed these differences to the transitional nature of the flow. Adjustment of the equations led to good agreement with experimental data. The resulting mathematical model predicts relations between stenosis, blood flow, intragraft pressure, and important clinical variables such as mean arterial blood pressure and hematocrit. Application of the model should improve understanding of the hemodynamics of the stenotic graft vascular circuit.

Animals↗
PubMed1985

The hearing aid feedback path: mathematical simulations and experimental verification.

Acoustic feedback in hearing aids has received little attention in the literature. Feedback occurs when stability conditions of the open-loop transfer function of an in situ hearing aid are violated. Solving the feedback problem will first require knowledge of the open-loop transfer function. Included in the open-loop transfer function is the acoustical path by which sound emanating from the earmold vent returns to the microphone (i.e., the feedback path). Reported herein are two different mathematical procedures for simulating transfer functions of the feedback path of an eyeglass-type hearing aid. In one procedure the vent exit was modeled as a point source of sound located on a flat plane, while it was treated as a point source on a sphere in the other. Results of laboratory experiments indicate that the mathematical models accurately predict those acoustic phenomena for which they were intended: point sources on plane and spherical baffles. Results of manikin experiments showed both models to be less accurate for simulating the feedback path around the human head. The maximum difference between experiment and theory was 6 dB at one frequency. Surprisingly, the flat-baffle model produced better agreement with experimental results than did the sphere model.

Acoustics↗
PubMed1990

An alternative mathematical description of the relationship between noise exposure and hearing loss.

Retrospective investigation of large populations has provided means for determining quantitative relationships between the hearing levels of non-noise-exposed populations and age, and between the hearing levels of noise-exposed populations and age and noise exposure. In the latter case, noise exposures have been predominantly steady state over many years and no way of measuring the effects of noise alone, excluding the effects of age, has ever been demonstrated. In the following paper, attention is confined to the problem of developing a mathematical description of an existing set of empirically determined hearing level data; questions of audiology are not of concern here. It is shown that the mathematical analysis traditionally used to determine the contribution of noise exposure alone to hearing level is not unique; an alternative formulation is possible and indeed is demonstrated. Whereas the traditional formulation leads to the conclusion that noise-induced hearing loss scales on the integral of sound pressure squared with time, and thus, to the equal energy hypothesis, the alternative formulation leads to the conclusion that noise-induced hearing loss scales on the integral of pressure with time. Since either formulation adequately describes the data, and the equal energy hypothesis has never been adequately substantiated, use of the latter hypothesis to extend the findings of steady-state exposures to application for unsteady exposures is not justified. The alternative formulation presented here is recommended for consideration.

Adolescent↗
PubMed1990

Mathematical modeling of antimicrobial susceptibility data of Staphylococcus haemolyticus for 11 antimicrobial agents, including three experimental glycopeptides and an experimental lipoglycopeptide.

Antimicrobial MIC data were obtained for 96 strains of Staphylococcus haemolyticus and the following 11 antimicrobial agents: methicillin, gentamicin, rifampin, fusidic acid, ciprofloxacin, vancomycin, teicoplanin; three experimental glycopeptides, MDL 62,873, MDL 62,208, and MDL 62,224; and an experimental lipoglycopeptide, ramoplanin. Resistance to methicillin and gentamicin was present in over 50% of the strains, although resistance to the other agents was present in less than 10%. It is shown how application of mathematical modeling techniques can add to the understanding of such MIC data. MICs of methicillin and gentamicin were highly correlated, suggesting that evolutionary pressures for development of resistance to these agents were similar. The structural relationships among the glycopeptides were accurately reflected in their spatial relationships within the model. MICs of ramoplanin were negatively correlated with MICs of some other antimicrobial agents, particularly gentamicin, suggesting that this agent is more active against gentamicin-resistant strains. Methicillin-resistant strains were more tightly clustered than were methicillin-susceptible strains, suggesting that methicillin-resistant strains were more closely related to each other than were methicillin-susceptible strains. Mathematical modeling techniques enable more detailed analysis of MIC data.

Anti-Bacterial Agents↗
PubMed1991

Application of mathematical model to multiple-dose experimental chemotherapy for fatal murine pneumonia.

Two beta-lactam antibiotics, cefazolin and cefmenoxime, were administered for 7 days to mice with pneumonia caused by Klebsiella pneumoniae by using dosage regimens that would simulate multiple dosing in usual clinical treatments at dosing intervals of 8 or 12 h. Viable numbers of the bacteria in the lungs were measured at 12- or 24-h intervals. The mathematical model established in a previous single-dose study was applied in this study to explain the time courses of the changes in bacterial count over 7 days. However, because the error in viable count measurements was larger than that in the previous study, the time course of the changes in mean viable count was not regular and the viable count reduction rate changed during multiple dosing, and therefore it was difficult to explain the time course by repeated application of the mathematical model described previously. This study suggests that the changes in pharmacokinetic and pharmacodynamic parameters during multiple dosing need to be considered.

Animals↗
PubMed2006

Bdellovibrio predation in the presence of decoys: Three-way bacterial interactions revealed by mathematical and experimental analyses.

Bdellovibrio bacteriovorus is a small, gram-negative, motile bacterium that preys upon other gram-negative bacteria, including several known human pathogens. Its predation efficiency is usually studied in pure cultures containing solely B. bacteriovorus and a suitable prey. However, in natural environments, as well as in any possible biomedical uses as an antimicrobial, Bdellovibrio is predatory in the presence of diverse decoys, including live nonsusceptible bacteria, eukaryotic cells, and cell debris. Here we gathered and mathematically modeled data from three-member cultures containing predator, prey, and nonsusceptible bacterial decoys. Specifically, we studied the rate of predation of planktonic late-log-phase Escherichia coli S17-1 prey by B. bacteriovorus HD100, both in the presence and in the absence of Bacillus subtilis nonsporulating strain 671, which acted as a live bacterial decoy. Interestingly, we found that although addition of the live Bacillus decoy did decrease the rate of Bdellovibrio predation in liquid cultures, this addition also resulted in a partially compensatory enhancement of the availability of prey for predation. This effect resulted in a higher final yield of Bdellovibrio than would be predicted for a simple inert decoy. Our mathematical model accounts for both negative and positive effects of predator-prey-decoy interactions in the closed batch environment. In addition, it informs considerations for predator dosing in any future therapeutic applications and sheds some light on considerations for modeling the massively complex interactions of real mixed bacterial populations in nature.

Bacillus subtilis↗
PubMed2004

Relationship between inflammation and joint destruction in early rheumatoid arthritis: a mathematical description.

BACKGROUND: The relationship between inflammation and joint destruction in rheumatoid arthritis (RA) has not been unequivocally characterised. Joint destruction may result from the cumulative inflammatory burden over time, modified by an individual constant factor. OBJECTIVE: To test the hypothesis that the relationship between radiological progression and inflammation can mathematically be expressed as: [equation: see text] where Re is a factor that varies from person to person. METHODS: Clinical data and radiographs of 76 patients with early RA receiving different disease modifying antirheumatic drugs were analysed. Radiographs were quantified using the modified Larsen score and the "X-Ray RheumaCoach" software. The cumulative inflammatory burden was estimated by the time integrated 28 joint Disease Activity Score (DAS28), calculated as the area under the curve. RESULTS: 76 patients with early RA who started treatment with methotrexate (n = 20), sulfasalazine (n = 37), or oral gold (n = 19) monotherapy were evaluated. The mean (SEM) DAS28 decreased from 4.6 (0.1) at baseline to 2.3 (0.1) after 2 years. The mean (SEM) DeltaLarsen score from baseline to year 2 was 10.3 (1.5). Correlation between cumulative inflammation and radiographic change was poor. In contrast, when calculating a person's factor Re in year 1 ( Re 1) and year 2 ( Re 2), a strong and significant correlation (r = 0.58, p<0.000001) was seen between Re 1 and Re 2. CONCLUSIONS: Joint destruction is the result of the cumulative burden of inflammation over time, modified by an individual factor Re that remains relatively constant over the first 2 years of observation. The data support a mathematical model that expresses the interrelationship between inflammation and joint destruction.

Aged↗
PubMed2002

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↗
PubMed2003

A simple approximate mathematical model to predict the number of severe acute respiratory syndrome cases and deaths.

BACKGROUND: Severe acute respiratory syndrome (SARS) is currently spreading in many countries. This paper proposes a simple approximate mathematical model for public health practitioners to predict the number of SARS cases and deaths. METHODS: The model is based on four parameters: R(o) (basic reproductive number), F (case-fatality rate), i (incubation period), and d (duration of disease). The calculations can be done by hand or by using a computer spreadsheet. RESULTS: The best parameters to fit Canadian data as of 6 April 2003 (before infection controls took effect) are R(o) = 1.5, F = 30%, i = 5 days, d = 14 days. On 6 April (day 40) there were 74 cases and 7 deaths. If this trend continues, SARS numbers in Canada are predicted to be as follows: 387 cases and 34 deaths by 26 April (day 60), 4432 cases and 394 deaths by 26 May (day 90), and 50 500 cases and 4489 deaths by 25 June (day 120). By comparison, the best parameters to fit Hong Kong data as of 10 April 2003 are R(o) = 2.0, F = 20%, i = 5 days, d = 14 days. CONCLUSIONS: Using the proposed mathematical model, it was estimated that about 1.5 to 2 new infectious cases were produced per infectious case every five days. Also, about 20% to 30% of the cases die within 14 days. The case-fatality may therefore be considerably higher than initially thought. The model indicates that SARS can spread very fast when there are no interventions.

Canada↗