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PubMed2001

Understanding the spread of antibiotic resistant pathogens in hospitals: mathematical models as tools for control.

As microorganisms become more resistant to antimicrobial agents, effective infection control measures will become increasingly important. However, despite multiple studies on infection prevention, few data exist on the quantitative effects of the individual aspects of infection control strategies. The combination of epidemiologic surveillance, molecular genotyping, observational studies on compliance, and mathematical modeling may improve our ability to determine the quantitative effects of individual infection control measures. This may help to design more effective infection control programs. In this study, we review several of the models that have been published and speculate on the usefulness of mathematical modeling for improving the prevention of infection.

Anti-Bacterial Agents↗

PubMed1993

Using mathematical morphology to determine left ventricular contours.

The aim of this study is to investigate the use of mathematical morphology for the determination of left ventricular contours in scintigraphic images using multigated radionuclide angiography. We have developed a completely automatic method that first restores the image with a Wiener filter, then finds the region where the left ventricle is contained, and finally segments the left ventricle contour and a background zone. The contours depend on the values of the parameters that appear in the mathematical morphology method, which are related to the height and the slope of the count distribution. Results obtained with this method are compared with the contours and the background zones outlined by experts on the basis of the number of counts. We study the values of the parameters with which the optimum correlation is obtained.

Humans↗

PubMed1996

Quantification of dopamine metabolism in man: a mathematically justifiable approach.

Model derivation, the proposal of the best model to fit the measured data, is a crucial and often neglected step in compartmental analysis. Since the model which best fits the data in a statistical sense may be simpler than the known physiology of the system, the order (mathematical complexity) of the model that can be accepted must be determined independently of physiological considerations. Although [18F]6-fluoro-L-dopa (F-dopa) has been used as a tracer for the investigation of presynaptic dopamine metabolism for a number of years, a definitive method of quantitative analysis has not yet emerged. Simple graphical analyses have been used predominantly in clinical studies, while compartmental analyses have been used to gain a better understanding of F-dopa kinetics. Over the years quantitative approaches to F-dopa metabolism have grown increasingly complicated, and complex models that rely on the use of parameter constraints, or simplifying assumptions in order to collapse the model to solvable dimensions, have been developed. We propose a simple compartmental model to quantify the metabolism of F-dopa in the striatum. Both compartmental and graphical analyses were performed using this model, and the mathematical expression which relates the results of the two analyses was derived. The ability of our approach to reveal and quantify differences in the dopaminergic metabolism of individuals has been demonstrated in a small number of studies.

Adult↗

PubMed1999

Application of mathematical methods in dynamic nuclear medicine studies.

Dynamic nuclear medicine studies can generate large quantities of data, and their analysis consists essentially of a reduction of these data to a small number of relevant parameters which will assist in clinical decision making. This review examines some of the mathematical techniques that have been used in the process of data reduction and attempts to explain the principles behind their application. It particularly identifies the techniques that have stood the test of time and demonstrated their usefulness, many of which are now available as standard tools on nuclear medicine processing computers. These include curve processing tools such as smoothing, fitting and factor analysis, as well as tools based on empirical models, such as the Patlak/Rutland plot and deconvolution. Compartmental models and vascular models are also examined and the review finishes with a summary of some functional images and condensed images. It is concluded that an appreciation of the principles and limitations of these mathematical tools is valuable for their correct usage and interpretation of the results produced.

Cardiovascular System↗

PubMed1983

Mathematical modelling of aqueous humour outflow from the eye through the pores in the lining endothelium of Schlemm's canal.

Aqueous humour leaves the eye mainly by way of pore openings in vacuolar swellings in the endothelial cells which line the wall of Schlemm's canal. Calculation of the resistance to fluid flow offered by these small pores requires data on their incidence and dimensions linked to a suitable mathematical model. Data are derived from scanning electron microscopy (SEM) studies conducted on rhesus monkey eyes maintained at an intraocular pressure (IOP) of either 8 or 15 mmHg and on human eyes either untreated controls or treated with pilocarpine. Mathematical models of flow through the pore openings are developed. Application of a Venturi tube model to the SEM data shows that the pore resistance of rhesus monkey eyes at 8 mmHg was seventeen times greater than that at 15 mmHg while the pore resistance in untreated human eyes was five times greater than in eyes which received pilocarpine.

Animals↗

PubMed2002

MAT3D: a virtual reality modeling language environment for the teaching and learning of mathematics.

Virtual Reality Modeling Language (VRML) is an independent platform language that allows the creation of nonimmersive virtual environments (VEs) and their use through the Internet. In these VEs, the viewer may navigate and interact with virtual objects, moving around and visualizing them from different angles. Students can benefit from this technology, because it permits them access to objects, which describe the topics covered in their studies in addition to oral and written information. In this work, we investigate the aspects involved in the use of VEs in teaching and learning and propose a conceptual model, called MAT3D, as a learning environment that can be used for the teaching and learning of mathematics. A case study is also presented, in which students use a virtual environment modeled in VRML. Data resulting from this study is analyzed statistically to evaluate the impact of this prototype when applied to the actual teaching and learning of mathematics.

Educational Technology↗

PubMed2006

Listeria monocytogenes in multiple habitats and host populations: review of available data for mathematical modeling.

Listeria monocytogenes has the ability to survive and multiply in diverse habitats and to cause infection in a variety of animal species and humans. We evaluated the literature on survival and multiplication within and transmission among multiple host populations and habitats, including man, sewage, general environment (soil, water, and vegetation), silage (fermented plant material), animals (including wild and domestic animals), and food processing plants. The available knowledge on L. monocytogenes transmission dynamics was translated into the key process nodes of interrelated host- and habitat-specific mathematical models, providing a starting framework for future modeling work and the ultimate development of a system-wide model for evaluation of its transmission, and strategies to reduce human exposure. Because of the ability of L. monocytogenes to survive and multiply in many habitats and hosts, and the number of possible transmission routes, it is highly unlikely that it could be eradicated from any habitat or host, including man. However, L. monocytogenes load within and transmission among habitats and host populations could probably be reduced. Based on the published information, we hypothesize that three recent anthropogenic practices increase the load within and transmission among reviewed habitats and host populations: extended refrigerated storage of ready-to-eat foods allowing L. monocytogenes growth in foods that are contaminated during production or subsequent handling; feeding domestic ruminants with silage often contaminated with L. monocytogenes; and dispersal of contaminated products of sewage treatment to agricultural fields and waters. Future mathematical modeling work could test how much the reduction of L. monocytogenes load and transmission in hosts and habitats associated with these anthropogenic practices would reduce human exposure and consequently human listeriosis.

Animal Feed↗

PubMed2000

Extending the method of mathematically controlled comparison to include numerical comparisons.

MOTIVATION: The method of mathematically controlled comparison has been used for some time to determine which of two alternative regulatory designs is better according to specific quantitative criteria for functional effectiveness. In some cases, the results obtained using this technique are general and independent of parameter values and the answers are clear-cut. In others, the result might be general, but the demonstration is difficult and numerical results with specific parameter values can help to clarify the situation. In either case, numerical results with specific parameter values can also provide an answer to the question of how much larger the values might be. In contrast, a more ambiguous result is obtained when either of the alternatives can have the larger value for a given systemic property, depending on the specific values of the parameters. In any case, introduction of specific values for the parameters reduces the generality of the results. Therefore, we have been motivated to develop and apply statistical methods that would permit the use of numerical values for the parameters and yet retain some of the generality that makes mathematically controlled comparison so attractive. RESULTS: We illustrate this new numerical method in a step-by-step application using a very simple didactic example. We also validate the results by comparison with the corresponding results obtained using the previously developed analytical method. The analytical approach is briefly present for reference purposes, since some of the same key concepts are needed to understand the numerical method and the results are needed for comparison. The numerical method confirms the qualitative differences between the systemic behavior of alternative designs obtained from the analytical method. In addition, the numerical method allows for quantification of the differences and it provides results that are general in a statistical sense. For example, the older analytical method showed that overall feedback inhibition in an unbranched pathway makes the system more robust whereas it decreases the stability margin of the steady state. The numerical method shows that the magnitudes of these differences are not comparable. The differences in stability margins (1-2% on average) are small when compared to the differences in robustness (50-100% on average). Furthermore, the numerical method shows that the system with overall feedback responds more quickly to change than the otherwise equivalent system without overall feedback. These results suggest reasons why overall feedback inhibition is such a prevalent regulatory pattern in unbranched biosynthetic pathways.

Computational Biology↗

PubMed1989

Mathematical model of jet ventilation.

The design of jet ventilators has always been empirical because no theory linked jet geometry, driving pressure, gas density and lung mechanics. A mathematical means has been developed linking these parameters to model jet ventilation. The mathematical model predicts the flow into a physical lung model, but under-estimates flow into the lungs when used to predict the effects of jet ventilation on an animal model at high rates of ventilation. This is attributed to a reduction in compliance with increasing ventilatory frequency.

Animals↗

PubMed1995

Mathematical modelling in risk/exposure assessment of tobacco related lung cancer.

The existence of a dose-related increase of lung cancer risk in cigarette smokers has been indisputably established. This finding, however, is not confirmed at low doses (< 5 cigarettes/day), there still being a lack of epidemiological data. The use of mathematical models of carcinogenesis to extrapolate from higher doses allows estimation of the risk for very light smokers. The present study has been designed to compare a set of mathematical models, i.e. one-hit, two-stage, multi-stage, logit, probit, and Weibull, in extrapolating relative risks at low doses from the data of nine large cohort studies on cigarette smokers reported in the IARC Scientific Monograph on tobacco smoking. All models evaluated, apart from the one-hit, achieved a good fit, with the proportion of explained variance ranging between 61% and 67%. The relative risk estimates for passive smokers from the most updated epidemiological studies were taken into account to evaluate, on the basis of these models, the corresponding exposure in terms of 'cigarette equivalent' smoked. These values ranged from 0.21 to 0.43 cigarettes/day for the two-stage and multi-stage model, while probit, logit and Weibull models, yielded estimates one or even two orders of magnitude lower. The authors emphasize the substantial agreement between the estimates of 'cigarette equivalent' based on the application of two-stage and multi-stage models to the epidemiological evidence on the effect of passive smoking and to the data based on the comparison of tobacco metabolites in active and passive smokers.

Dose-Response Relationship, Drug↗

PubMed2005

A mathematical model of atherogenesis as an inflammatory response.

We construct a mathematical model of the early formation of an atherosclerotic lesion based on a simplification of Russell Ross' paradigm of atherosclerosis as a chronic inflammatory response. Atherosclerosis is a disease characterized by the accumulation of lipid-laden cells in the arterial wall. This disease results in lesions within the artery that may grow into the lumen restricting blood flow and, in critical cases, can rupture causing complete, sudden occlusion of the artery resulting in heart attack, stroke and possibly death. It is now understood that when chemically modified low-density lipoproteins (LDL cholesterol) enter into the wall of the human artery, they can trigger an immune response mediated by biochemical signals sent and received by immune and other cells indigenous to the vasculature. The presence of modified LDL can also corrupt the normal immune function triggering further immune response and ultimately chronic inflammation. In the construction of our mathematical model, we focus on the inflammatory component of the pathogenesis of cardiovascular disease (CVD). Because this study centres on the interplay between chemical and cellular species in the human artery and bloodstream, we employ a model of chemotaxis first given by E. F. Keller and Lee Segel in 1970 and present our model as a coupled system of non-linear reaction diffusion equations describing the state of the various species involved in the disease process. We perform numerical simulations demonstrating that our model captures certain observed features of CVD such as the localization of immune cells, the build-up of lipids and debris and the isolation of a lesion by smooth muscle cells.

Atherosclerosis↗

PubMed2006

A mathematical model gives insights into nutritional and genetic aspects of folate-mediated one-carbon metabolism.

Impaired folate-mediated 1-carbon metabolism has been linked to multiple disease outcomes. A better understanding of the nutritional and genetic influences on this complex biochemical pathway is needed to comprehend their impact on human health. To this end, we created a mathematical model of folate-mediated 1-carbon metabolism. The model uses published data on folate enzyme kinetics and regulatory mechanisms to simulate the impact of genetic and nutritional variation on critical aspects of the pathway. We found that the model predictions match experimental data, while providing novel insights into pathway kinetics. Our primary observations were as follows: 1) the inverse association between folate and homocysteine is strongest at very low folate concentrations, but there is no association at high folate concentrations; 2) the DNA methylation reaction rate is relatively insensitive to changes in folate pool size; and 3) as folate concentrations become very high, enzyme velocities decrease. With regard to polymorphisms in 5,10-methylenetetrahydrofolate reductase (MTHFR), the modeling predicts that decrease MTHFR activity reduces concentrations of S-adenosylmethionine and 5-methyltetrahydrofolate, as well as DNA methylation, while modestly increasing S-adenosylhomocysteine and homocysteine concentrations and thymidine or purine synthesis. Decreased folate together with a simulated vitamin B-12 deficiency results in decreases in DNA methylation and purine and thymidine synthesis. Decreased MTHFR activity superimposed on the B-12 deficiency appears to reverse the declines in purine and thymidine synthesis. These mathematical simulations of folate-mediated 1-carbon metabolism provide a cost-efficient approach to in silico experimentation that can complement and help guide laboratory studies.

Betaine↗

PubMed1996

Nurses' health study: log-incidence mathematical model of breast cancer incidence.

BACKGROUND: In 1983, Pike et al. developed a mathematical model to quantify the effects of reproductive risk factors on the incidence of breast cancer. In 1994, we modified that model to correct some deficiencies in the original model, including a lack of terms for spacing of births and an inability to easily accommodate births after age 40 years. Our extended Pike model, while improving on the original, still has serious disadvantages, such as difficulty in translating model parameters into relative risks (RRs) and an incomplete fit to data that slightly overestimated incidence for premenopausal women with an early age at first birth and that underestimated incidence for post-menopausal women with a late age at first birth. PURPOSE: We undertook both the development of a new mathematical model to quantify the effects of reproductive risk factors on breast cancer incidence and validation of the model. METHODS: A new log-incidence model of breast cancer incidence was developed using nonlinear regression methods, and a study population consisting of 89,132 women in the Nurses' Health Study from which a total of 2249 incident cases of breast cancer were identified. Subjects were followed from the return of the 1976 Nurses' Health Study questionnaire until June 1, 1990, or until the last questionnaire was returned, until the development of any cancer, or until death, yielding 1,148,593 person-years of follow-up. The log-incidence models were fitted using iteratively reweighted least squares analysis. RESULTS: The log-incidence model provided a better fit to that data than the extended Pike model, with parameter estimates interpretable in terms of RRs. This new model can be fitted using standard commercially available statistical software. In the model, younger parous women are generally at slightly higher risk than nulliparous women, which is true for both the observed and expected RRs, and older parous women, aged 55-64 years with an early age at first birth, are at lower risk than nulliparous women,while older women with a late age at first birth are at substantially higher risk than nulliparous women. CONCLUSION: Log-incidence models, such as this one, provide an efficient framework for modeling the effect of lifestyle risk factors on breast cancer incidence that may be specifically targeted to certain time periods of a woman's reproductive life.

Adult↗

PubMed1992

An evaluation of the mathematical skills of beginner Swedish nursing students.

During the autumn of 1989 a mathematical test was given to 997 students attending a 2-year program of nursing education and 188 students on a 3-year program of nursing education. The test consisted of 65 items. Apart from the four fundamental rules of arithmetics, the test included calculations concerning fractions, scales, transformation of units and percentages. The level of the test was comparable to the 9-year compulsory school curriculum. The main purpose of the present study was to investigate whether nursing students had adequate knowledge and skills in mathematics when they started their studies. The result showed that approximately 50% of the items were solved correctly. The nursing students attending a 3-year nursing education program had higher average scores than the group attending a 2-year education program. The result is discussed in connection with medical errors. The Swedish Medical Responsibility Board supervises nurses' professional work and penalizes those who give wrong doses of drugs to patients.

Adolescent↗

PubMed2004

TNF-alpha neutralization in cytokine-driven diseases: a mathematical model to account for therapeutic success in rheumatoid arthritis but therapeutic failure in systemic inflammatory response syndrome.

OBJECTIVES: Neutralization of TNF-alpha with either monoclonal antibodies or soluble receptors, although not curative, has significant clinical benefit in patients with rheumatoid arthritis (RA). In contrast, blockade of TNF-alpha has little clinical benefit in the majority of patients with systemic inflammatory response syndrome (SIRS) in spite of the identification of TNF-alpha as a key factor in its pathology. It is not clear why there is such a significant difference in the responses to TNF-alpha neutralization in these two conditions. Here we use mathematical modelling to investigate this discrepancy. METHODS: Using the known pharmacokinetic and pharmacodynamic properties of TNF-alpha-blocking biological agents, we constructed a mathematical model of the biological actions of soluble(s) TNFR2, Etanercept and Infliximab. RESULTS: Our model predicts that all three inhibitors, but especially Etanercept, are effective at controlling TNF-alpha levels in RA, which we propose is a condition in which TNF-alpha production and inhibition are in equilibrium. However, when free TNF-alpha drops to a low level, as can occur in SIRS, which we propose is a non-equilibrium condition, the sequestered TNF-alpha can act as a slow-release reservoir, thereby sabotaging its effectiveness. CONCLUSIONS: These results may explain the effectiveness of TNF-alpha blockade in the equilibrium condition RA and the ineffectiveness in the non-equilibrium condition SIRS.

Antibodies, Monoclonal↗

PubMed1985

The mathematical versus anatomical methods of stature estimate from long bones.

The mathematical and anatomical methods of estimating living stature from long limb bones are discussed. In forensic cases, when one has a nearly complete skeleton, the anatomical method is preferable to the mathematical method. The anatomical method may also be used to derive stature estimation equations in samples where living statures or cadaver lengths are unavailable, such as some dissection hall and museum collections.

Anatomy↗

PubMed2002

Assessing pediatrics residents' mathematical skills for prescribing medication: a need for improved training.

PURPOSE: To evaluate residents' skills in performing basic mathematical calculations used for prescribing medications to pediatric patients. METHOD: In 2001, a test of ten questions on basic calculations was given to first-, second-, and third-year residents at Miami Children's Hospital in Florida. Four additional questions were included to obtain the residents' levels of training, specific pediatrics intensive care unit (PICU) experience, and whether or not they routinely double-checked doses and adjusted them for each patient's weight. The test was anonymous and calculators were permitted. The overall score and the score for each resident class were calculated. RESULTS: Twenty-one residents participated. The overall average test score and the mean test score of each resident class was less than 70%. Second-year residents had the highest mean test scores, although there was no significant difference between the classes of residents (p =.745) or relationship between the residents' PICU experiences and their exam scores (p =.766). There was no significant difference between residents' levels of training and whether they double-checked their calculations (p =.633) or considered each patient's weight relative to the dose prescribed (p =.869). Seven residents committed tenfold dosing errors, and one resident committed a 1,000-fold dosing error. CONCLUSION: Pediatrics residents need to receive additional education in performing the calculations needed to prescribe medications. In addition, residents should be required to demonstrate these necessary mathematical skills before they are allowed to prescribe medications.

Drug Prescriptions↗

PubMed2001

Mathematical coupling does not explain the relationship between right ventricular end-diastolic volume and cardiac output.

OBJECTIVE: To evaluate the clinical significance of mathematical coupling on the correlation between cardiac output and right ventricular end-diastolic volume (RVEDV) through measurement of cardiac output by two independent techniques. DESIGN: Prospective, observational study. SETTING: Surgical intensive care unit in a level 1 trauma center. PATIENTS: Twenty-eight critically ill surgical patients who received mechanical ventilation and hemodynamic monitoring with a pulmonary artery catheter. INTERVENTIONS: A pulmonary artery catheter designed to measure right ventricular ejection fraction (RVEF) and cardiac output by the intermittent bolus thermodilution (TDCO) method and continuous cardiac output by the pulsed thermal energy technique was placed. A computerized data logger was used to collect data simultaneously from the RVEF/TDCO system and the continuous cardiac output system. MEASUREMENTS AND MAIN RESULTS: Two hundred forty-nine data sets from 28 patients were compared. There is statistical correlation between TDCO and continuous cardiac output measurements (r = 0.95, p < 0.0001) with an acceptable bias (-0.11 L/min) and precision (+/-0.74 L/min). The correlation was maintained over a wide range of cardiac outputs (2.3-17.8 L/min). There is a high degree of correlation between RVEDV and both TDCO (r = 0.72, p < 0.0001) and independently measured continuous cardiac output (r = 0.68, p < 0.0001). These correlation coefficients are not statistically different (p = 0.15). CONCLUSIONS: The continuous cardiac output technique accurately approximates cardiac output measured by the TDCO method. RVEDV calculated from TDCO correlates well with both TDCO and independently measured continuous cardiac output. Because random measurement errors of the two techniques differ, mathematical coupling alone does not explain the correlation between RVEDV estimates of preload and cardiac output.

Adult↗