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A mathematical overview of a computer simulation model of maternity histories with illustrative examples.

A mathematical overview of a stochastic computer simulation model of maternity histories is provided. Various components of human reproduction are accommodated in the model through distributions of waiting times among live births. Included in these components are distributions of age at first marriage in a cohort of women, waiting times to pregnancy for fecundable women, and the lengths of infecundable periods following live births. Probabilities that pregnancies end in either a live birth, induced abortion, or some other type of outcome are also included. Elements of renewal theory and semi-Markov processes in discrete time were the basic mathematical concepts used in the construction of the model. A brief description of an interactive software package called MATHIST, which may be used to implement the model on a computer, is also included. Four illustrative computer runs with MATHIST, pertinent to the operation of family planning programmes in Africa, are also described and discussed.

Adult↗

Population dynamics of HTLV-I infection: a discrete-time mathematical epidemic model approach.

Human T-cell Leukaemia Virus Type I (HTLV-I) is a retrovirus that causes Adult T-cell Leukaemia (ATL). The transmission routes of HTLV-I are (i) from infected mothers to their newborn babies, (ii) from infected males (husbands) to females (their wives) by long-term sexual intercourse, and (iii) from infected females (wives) to males (their husbands). Eshima et al. (2001) analysed a continuous-time HTLV-I model with no age structure in the population. In this paper, we consider the population dynamics of HTLV-I infection in a discrete-time mathematical model incorporating an age structure. The necessary and sufficient condition for the extinction of HTLV-I is derived from the mathematical model. A simulation of the HTLV-I infection based on the model demonstrates a rapid reduction of the HTLV-I infection proportion in Japan.

Adolescent↗

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↗

A mathematical model of zinc absorption in humans as a function of dietary zinc and phytate.

The quantities of zinc and phytate in the diet are the primary factors determining zinc absorption. A mathematical model of zinc absorption as a function of dietary zinc and phytate can be used to predict dietary zinc requirements and, potentially, enhance our understanding of zinc absorption. Our goal was to develop a model of practical and informative value based on fundamental knowledge of the zinc absorption process and then fit the model to selected published data to assess its validity and estimate parameter values. A model of moderate mathematical complexity relating total zinc absorption to total dietary zinc and total dietary phytate was derived and fit to 21 mean data from whole day absorption studies using nonlinear regression analysis. Model validity, goodness of fit, satisfaction of regression assumptions, and quality of the parameter estimates were evaluated using standard statistical criteria. The fit had an R(2) of 0.82. The residuals were found to exhibit a normal distribution, constant variance, and independence. The parameters of the model, A(MAX), K(R), and K(P), were estimated to have values of 0.13, 0.10, and 1.2 mmol/d, respectively. Several of these estimates had wide CI attributable in part to the small number and the scatter of the data. The model was judged to be valid and of immediate value for studying and predicting absorption. A version of the model incorporating a passive absorption mechanism was not supported by the available data.

Humans↗

Mathematical model to predict regions of chromatin attachment to the nuclear matrix.

The potentiation and subsequent initiation of transcription are complex biological phenomena. The region of attachment of the chromatin fiber to the nuclear matrix, known as the matrix attachment region or scaffold attachment region (MAR or SAR), are thought to be requisite for the transcriptional regulation of the eukaryotic genome. As expressed sequences should be contained in these regions, it becomes significant to answer the following question: can these regions be identified from the primary sequence data alone and subsequently used as markers for expressed sequences? This paper represents an effort toward achieving this goal and describes a mathematical model for the detection of MARs. The location of matrix associated regions has been linked to a variety of sequence patterns. Consequently, a list of these patterns is compiled and represented as a set of decision rules using an AND-OR formulation. The DNA sequence was then searched for the presence of these patterns and a statistical significance was associated with the frequency of occurrence of the various patterns. Subsequently, a mathematical potential value,MAR-Potential, was assigned to a sequence region as the inverse proportion to the probability that the observed pattern population occurred at random. Such a MAR detection process was applied to the analysis of a variety of known MAR containing sequences. Regions of matrix association predicted by the software essentially correspond to those determined experimentally. The human T-cell receptor and the DNA sequence from the Drosophila bithorax region were also analyzed. This demonstrates the usefulness of the approach described as a means to direct experimental resources.

Base Sequence↗

A method for quantification of absolute amounts of nucleic acids by (RT)-PCR and a new mathematical model for data analysis.

Accurate quantification of nucleic acids by competitive (RT)-PCR requires a valid internal standard, a reference for data normalization and an adequate mathematical model for data analysis. We report here an effective procedure for the generation of homologous RNA internal standards and a strategy for synthesizing and using a reference target RNA in quantification of absolute amounts of nucleic acids. Further, a new mathematical model describing the general kinetic features of competitive PCR was developed. The model extends the validity of quantitative competitive (RT)-PCR beyond the exponential phase. The new method eliminates the errors arising from different amplification efficiencies of the co-amplified sequences and from heteroduplex formation in the system. The high accuracy (relative error <2%) is comparable to the recently developed real time detection 5'-nuclease PCR. Also, corresponding computer software has been devised for practical data analysis.

Cell Line↗

Mathematical estimation of the potential effect of vascular remodelling/dilatation on B-mode ultrasound intima-medial thickness.

BACKGROUND: Arterial diameter changes are known to impact wall thickness, but the clinical relevance of the changes is unclear. AIM: To use known mathematical relationships to estimate anticipated changes in arterial wall thicknesses occurring with enlargement of atherosclerotic regions. DESIGN: Mathematical relationships between a cylinder's diameter and its wall thickness were used to calculate the theoretical effect of diameter enlargement on the thickness of an atherosclerotic wall. METHODS: Equating the wall areas of two cylinders, one of smaller diameter than the other, allowed estimation of the degree of thickening that would be needed to maintain intima-medial thickness (IMT) after arterial remodelling. The difference in cylinder diameters was based on arterial diameter enlargement reported with atherosclerosis progression. Thus, the calculated wall changes estimate arterial changes which could go undetected if only IMT is measured by ultrasound. RESULTS: The expected IMT change for diameter enlargement is not a linear function of the diameter change, but varies depending upon initial size (diameter and IMT). Thus a 0.6 mm arterial diameter enlargement would be expected to cause a 0.039-0.235 mm change in IMT, depending on artery size. The estimated IMT change is similar to that associated with major atherosclerotic risk factors. DISCUSSION: The level of vascular remodelling reported with atherosclerosis could have a measurable impact on IMT, suggesting that indicators incorporating both diameter and IMT may be better disease indicators than IMT alone. Arterial diameters, as well as IMT, should be obtained in ultrasound studies of atherosclerosis.

Arteriosclerosis↗

A mathematical model of aircraft for evaluating the effects of shielding structure on aircrew exposure.

To investigate the influence of the aircraft structures and contents on the exposure of aircrew to the galactic component of cosmic rays, a mathematical model of an aeroplane has been developed. The irradiation of the mathematical model in the cosmic ray environment has been simulated using the Monte Carlo transport code FLUKA. Effective dose andambient dose-equivalent rates have been determined inside the aircraft at several locations along the fuselage at a typicaI civil aviation altitude. A significant effect of the shielding of aircraft structures has been observed on the ambient dose-equivalent rates, while the impact on the effective dose rates seems to be minor. Care should be taken in positioning the detectors onboard when the measurements are aimed at validating the codes.

Aerospace Medicine↗

Prediction of inspiratory flow shapes during sleep with a mathematic model of upper airway forces.

STUDY OBJECTIVES: To predict the airflow dynamics during sleep using a mathematic model that incorporates a number of static and dynamic upper airway forces, and to compare the numerical results to clinical flow data recorded from patients with sleep-disordered breathing on and off various treatment options. DESIGN: Upper airway performance was modeled in virtual subjects characterized by parameter settings that describe common combinations of risk factors predisposing to upper airway collapse during sleep. The treatments effect were induced by relevant changes of the initial parameter values. SETTING: Computer simulations at our website (http://www.utu.fi/ml/sovmat/bio/). PARTICIPANTS: Risk factors considered in the simulation settings were sex, obesity, pharyngeal collapsibility, and decreased phasic activity of pharyngeal muscles. INTERVENTIONS: The effects of weight loss, pharyngeal surgery, nasal continuous positive airway pressure, and respiratory stimulation on the inspiratory flow characteristics were tested with the model. MEASUREMENTS AND RESULTS: Numerical predictions were investigated by means of 3 measurable inspiratory airflow characteristics: initial slope, total volume, and flow shape. The model was able to reproduce the inspiratory flow shape characteristics that have previously been described in the literature. Simulation results also supported the observations that a multitude of factors underlie the pharyngeal collapse and, therefore, certain medical therapies that are effective in some conditions may prove ineffective in others. CONCLUSIONS: A mathematic model integrating the current knowledge of upper airway physiology is able to predict individual treatment responses. The model provides a framework for designing novel and potentially feasible treatment alternatives for sleep-disordered breathing.

Arousal↗

Role of mathematical models in assessment of risk and in attempts to define management strategy.

Risk assessment of food-borne carcinogens is becoming a common practice at FDA. Actual risk is not being estimated, only the upper limit of risk. The risk assessment process involves a large number of steps and assumptions, many of which affect the numerical value estimated. The mathematical model which is to be applied is only one of the factors which affect these numerical values. To fulfill the policy objective of using the "worst plausible case" in estimating the upper limit of risk, recognition needs to be given to a proper balancing of assumptions and decisions. Interaction between risk assessors and risk managers should avoid making or giving the appearance of making specific technical decisions such as the choice of the mathematical model. The importance of this emerging field is too great to jeopardize it by inappropriately mixing scientific judgments with policy judgments. The risk manager should understand fully the points and range of uncertainty involved in arriving at the estimates of risk which must necessarily affect the choice of the policy or regulatory options available.

Animals↗

Maxillary distraction osteogenesis: a two-dimensional mathematical model.

Patients with cleft lip and palate with severe maxillary retrusion usually have a mandible with anterior-superior autorotation and subsequent overclosure and loss of the vertical facial dimension. Maxillary distraction osteogenesis can correct the sagittal maxillomandibular relationship and should simultaneously reestablish vertical dimension through maxillary vertical height increase and clockwise rotation of the mandible to restore facial balance. We present a two-dimensional mathematical model in the sagittal plane, which reestablishes sagittal and vertical skeletal deficiencies and proper occlusal alignment for planning maxillary advancement with distraction osteogenesis in patients with cleft lip and palate. The model is illustrated in a case of a 13-year-old boy with a complete bilateral cleft lip and palate and severe maxillary retrusion. The two-dimensional mathematical model described in this article allows the surgeon and orthodontist to calculate in a simple and accurate way the ideal distraction vector to advance the maxilla to its desired position.

Adolescent↗

Mathematic model to estimate change in burn scar length required for joint range of motion.

Burn scar contracture results from an insufficient amount of extensible tissue to permit complete range of motion. The purpose of this study was to develop a mathematic model to estimate additional tissue length required for full range of motion in the presence of a scar contracture. Seven areas with a known predilection for burn scar contracture were assessed. Twenty-five volunteers with normal range of motion had the length of their limbs measured at predetermined angles. Changes in limb length through range of motion were documented. On the basis of these changes, a mathematic model was developed to estimate the additional amount of tissue length required to complete range of motion for each area. This information may be useful to determine burn patient rehabilitation potential or need for reconstructive surgery.

Adult↗

A simple mathematical modification of TRISS markedly improves calibration.

BACKGROUND: TRISS has reigned as the preeminent trauma outcome prediction model for 20 years. Despite this endorsement, the calibration of TRISS has been poor in most data sets where it has been examined. We hypothesized that the lack of calibration of TRISS was because of the inappropriate mathematical specification of the model that TRISS is based on, rather than the predictors in the model. In particular, we hypothesized that the nonlinearity of the Injury Severity Score (ISS) in the log odds of death was responsible for the poor calibration of TRISS, and further, that this nonlinearity could be corrected by the simple addition of an ISS squared term to the TRISS model. METHODS: We examined ISS in the log odds of mortality for linearity in one large trauma data set, the National Pediatric Trauma Registry (NPTR) (n = 53,113 from 1985-1996; mortality, 1.3%); and two small data sets, the University of New Mexico (UNM) (n = 3,142 from 1991-1995; mortality, 8.6%) and Portland, Oregon (PORT) (n = 2,916 from 1990-1994; mortality, 1.75%). In addition, in the NPTR we compared the calibration of TRISS models with and without linearity in the log odds of death. RESULTS: In the NPTR, ISS was profoundly nonlinear in the log odds of death for both blunt and penetrating trauma (p < 0.001). Moreover, the overall calibration of the TRISS model for the NPTR data was significantly improved when the nonlinearity of ISS was corrected by the addition of a quadratic ISS term as demonstrated by a 70% reduction (improvement) in the Hosmer-Lemeshow statistic. Interestingly, the addition of the ISS squared term did not affect the discrimination of the model. The log odds of survival in the UNM and PORT data sets were also better modeled when an ISS squared term was added (UNM, p = 0 0.052; PORT, p = 0.014), but improvements in the Hosmer-Lemeshow statistic were smaller, possibly because of the small size of these data sets. CONCLUSION: The TRISS model for outcome prediction currently uses ISS in a mathematically inappropriate way that impairs the calibration, but not the discrimination, of its predictions. If TRISS is to continue as the prediction standard for trauma, a quadratic ISS term must be added to the model. In the future, outcome prediction models should undergo thorough statistical modeling and evaluation before being released. Injury severity descriptors other than ISS (such as ASCOT, ICISS, or NISS) may require other modeling techniques to optimize the calibration of survival models that use these injury scores.

Adolescent↗

Effect of mathematical modeling on the estimation of critical power.

PURPOSE: The purposes of this study were to re-examine the findings of previous studies by comparing the critical power (CP) estimates from five mathematical models and to determine the time to exhaustion during cycle ergometry at the lowest CP estimate from the five models. METHODS: Nine adult males performed a maximal incremental test to determine peak power and five or six randomly ordered trials on a cycle ergometer for the estimation of CP. Two linear, two nonlinear, and one exponential mathematical model were used to estimate CP. The subjects then completed two trials to exhaustion, or 60 min, at their lowest estimate of CP from the five models. RESULTS: The nonlinear three-parameter model (Nonlinear-3) produced a mean CP that was significantly (P < 0.05) less than the mean CP values derived from the other four models and was the lowest CP estimate for each subject. Two and three subjects, however, did not complete 60 min of cycling during the first and second trials at CP, respectively. At the end of the trials the subjects who completed 60 min of cycling had a mean heart rate of 92% of their maximum and a mean rating of perceived exertion of 17. CONCLUSION: These findings support previous studies that have indicated that in many cases CP overestimates the power output that can be maintained for at least 60 min.

Adult↗

Description of cutaneous excision and suture using a mathematical model.

Mathematical models are being developed at a fast rate in industry thanks to computer technology; they are used to simulate motion and deformation over time to test materials and objects in a virtual manner. These modeling techniques are being developed in medicine, but they remain, at this time, in the domain of biomechanical research. We report a mathematical model for cutaneous excision and suture, which we have used to predict the deformations and tensions that result when using four different forms of excision. The results are expressed in numerical and graphic form. The results obtained corresponded with our experience in dermatologic surgery. Uses and limits of this model are discussed.

Biomechanical Phenomena↗

A mathematical model of the pancreatic duct cell generating high bicarbonate concentrations in pancreatic juice.

OBJECTIVE: To develop a simple, physiologically based mathematical model of pancreatic duct cell secretion using experimentally derived parameters that generates pancreatic fluid bicarbonate concentrations of >140 mM after CFTR activation. METHODS: A new mathematical model was developed simulating a duct cell within a proximal pancreatic duct and included a sodium-2-bicarbonate cotransporter (NBC) and sodium-potassium pump (NaK pump) on a chloride-impermeable basolateral membrane, CFTR on the luminal membrane with 0.2 to 1 bicarbonate to chloride permeability ratio. Chloride-bicarbonate antiporters (Cl/HCO3 AP) were added or subtracted from the basolateral (APb) and luminal (APl) membranes. The model was integrated over time using XPPAUT. RESULTS: This model predicts robust, NaK pump-dependent bicarbonate secretion with opening of the CFTR, generates and maintains pancreatic fluid secretion with bicarbonate concentrations >140 mM, and returns to basal levels with CFTR closure. Limiting CFTR permeability to bicarbonate, as seen in some CFTR mutations, markedly inhibited pancreatic bicarbonate and fluid secretion. CONCLUSIONS: A simple CFTR-dependent duct cell model can explain active, high-volume, high-concentration bicarbonate secretion in pancreatic juice that reproduces the experimental findings. This model may also provide insight into why CFTR mutations that predominantly affect bicarbonate permeability predispose to pancreatic dysfunction in humans.

Animals↗

Quantification of lumbar intradiscal deformation during flexion and extension, by mathematical analysis of magnetic resonance imaging pixel intensity profiles.

STUDY DESIGN: A magnetic resonance imaging study of the internal kinematic response of normal lumbar intervertebral discs to non-weight-bearing flexion and extension. OBJECTIVES: To quantify the pattern of magnetic resonance imaging pixel intensity variation across discs, and noninvasively monitor displacement of the nucleus pulposus during sagittal-plane movements. SUMMARY OF BACKGROUND DATA: Invasive techniques used to study intradiscal movements of the nucleus pulposus have suggested that it moves posteriorly during flexion and anteriorly during extension. A noninvasive study based on magnetic resonance images gave similar results for normal young women. Quantification has been problematic, and the invasive procedures may have altered disc dynamics. METHODS: Ten male subjects (age, 21-38 years) with healthy backs were positioned in a magnetic resonance imaging portal with their lumbar spine stabilized in flexion and extension by supporting pads. For each disc, a T2-weighted image was obtained, as was a computer-generated profile of pixel intensities along a horizontal mid-discal transect. Mathematical curve-fitting regression analysis was used to characterize the shape of the intensity profile and to compute the point of maximum pixel intensity. RESULTS: A single equation fitted the profile for all normal discs. The intensity peak shifted posteriorly during flexion, anteriorly during extension. CONCLUSIONS: Automated mathematical modeling of magnetic resonance imaging pixel data can be used to describe the fundamental shape of the pixel intensity profile across a normal lumbar disc, to determine the precise location of the site of maximum pixel intensity, and to measure the movement of this peak with flexion and extension. This technique may be of value in recognizing incipient degenerative changes in lumbar discs.

Adult↗

A mathematical theory for identifying and measuring severity of episodes of care.

OBJECTIVES: We propose and test a method for constructing episodes of care from data within administrative databases and electronic health records. SUBJECTS: We created a measure for severity of episodes of illness for 565 randomly chosen developmentally delayed children who were enrolled in the Medicaid program. DESIGN: Regression analysis was conducted to test the percentage of variance explained by our proposed mathematical model in cost of care. DATA COLLECTION: Data included both hospitalizations and clinic visits obtained from Medicaid programs from one southeastern state. METHODS: For each patient, the likelihood that two diagnoses are part of the same episode is proportional to the similarity of the two diagnoses and to the short time interval between them. When this likelihood exceeds a preset cutoff, then the two diagnoses are part of the same episode. The cutoff is estimated by selecting number of days before two very similar diagnoses are considered to be part of separate episodes. The similarity between two diagnoses is assumed to be proportional to co-occurrence of the two diagnoses within a fixed period (usually 30 days). The severity of an episode was calculated using a Muliplicative Multiattribute Utility model, where severity of each diagnosis is aggregated to estimate the overall severity of the episode. Severity of each diagnosis was assumed to be proportional to average cost of a diagnosis-if patients do not die before care is delivered. The article includes an algorithm that can classify a patient's diagnosis into episodes of care and measure severity of the episodes from date of diagnoses, code for the diagnoses, and charges for the visit. To facilitate integration with existing database, the article includes a Standard Query Language computer program. To evaluate the method of constructing episodes of care, we regressed cost of care on the patient's number of episodes of care within the year, average severity of the episodes within the year, and the interaction between number and average severity of the episodes. RESULTS: The number of episodes (alpha = .001), the average severity of the episodes (alpha = .001), and the product of the two (alpha = .001) had statistically significant relationships to the average cost of the case. The 3 variables together explained 53% of variation in yearly cost of care. CONCLUSIONS: These data suggest that our proposed mathematical approach is reasonable and produces severity scores that are predictive of objective criteria such as cost of care.

Child, Preschool↗