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PubMedDate not supplied

Mathematical models to predict long bone lengths by transvaginal scan at 11-16 weeks' gestation.

OBJECTIVE: To propose a new mathematical model to estimate the length of fetal long bones in early pregnancy that can be used in the routine clinical setting. METHODS: In a group of 400 singleton normal fetuses, referred for transvaginal ultrasound examination between 11 and 16 weeks' gestation prior to genetic amniocentesis, a regression analysis was performed to evaluate the relationship between femur length (FLl) (mm) and biparietal diameter (BPD) (mm) and gestational age (GA) (days), as well as between humerus length (HL) (mm) and BPD (mm) and GA (days). The confidence intervals (CIs) of the predicted values for different values of BPD and for different gestational periods and CIs for the regression coefficients are stated as the mean +/- SD of standardized residuals. The accuracy of our best models obtained were evaluated at each gestational week between 11 and 16 with a 10% error cut-off limit. RESULTS: The best relationships between FL and HL versus BPD and GA are: expected FL = -16.92108 + 0.4569402 BPD +0.171617 GA (R(2) = 0.86) and expected HL = -16.28531 + 0.4283019 BPD + 0.1696017 GA (R(2) = 0.88), respectively. When a cut-off limit of 10% in estimating fetal long bones was utilized, the mathematical models revealed a good accuracy particularly at 13-14 weeks' gestation, a period when transvaginal biometric and morphologic examination is advisable and the highest percentage of scans are performed. DISCUSSION: Our proposed two linear models had a very good ability to estimate FL and HL and, due to the simplicity of the calculations, would be particularly useful in the routine clinical setting.

Bone Diseases↗
PubMed2003

Validation of a mathematical model predicting the response to growth hormone treatment in prepubertal children with idiopathic growth hormone deficiency.

OBJECTIVE: To validate a mathematical model developed by Ranke et al. (J Clin Endocrinol Metab 1999;84:1174-7783) to predict the GH response during the first years of GH replacement therapy. PATIENTS AND METHODS: 38 children with idiopathic GH deficiency (GHD) met all inclusion criteria for the prediction model, but the group differed in some characteristics from the cohort from which the model was derived. RESULTS: Using the model for the 1st year including maximum GH after stimulation and the equation for the 6th year, the predicted value corresponded well with actual height gain. Differences were found when the growth response of the 1st year excluding maximum GH and that of the 2nd-5th year were calculated, resulting in a significant underestimation of actual height gain (-0.63 to -1.07 cm/year). CONCLUSION: The mathematical prediction model tended to underpredict the growth response to GH treatment in our patients with pronounced GHD. The severity of GHD seems to be an important parameter for the 1st year prediction.

Body Height↗
PubMed2005

Mathematical model to characterize internal filtration.

Convective-diffusive dialysis techniques have recently gained considerable favor. Indeed, convective fluxes through dialyzer membranes have been demonstrated to play a role in enhancing the clearance of middle-molecular-weight solutes. An interesting opportunity is given by exploiting the internal filtration (IF)/back filtration mechanism that occurs spontaneously in high-flux dialyzers, but is difficult to quantify. In view of overcoming this drawback, a semi-empirical, lumped-parameter mathematical model for characterization of IF phenomena was developed. The model considers a dialyzer as composed by N adjacent axial blocks. For each block, hydrodynamics in the blood and dialysate compartments are determined considering hydraulic resistance and calculating local filtration. Blood viscosity and oncotic pressure are calculated locally based on hematocrit and protein concentration. Resistance parameters were determined experimentally for the BS-UL (Toray Industries Inc., Tokyo, Japan) dialyzers. The set of equations describing the model, implemented into a software program, is solved using a numerical method. Simulations allow highlighting the role of device-, treatment- and patient-dependent parameters in affecting IF. Provided an extensive validation is carried out, the use of a mathematical model could be the key to make IF more understandable and its use reliable in clinical practice.

Computer Simulation↗
PubMed2005

Mathematical modeling and computer simulation in blood coagulation.

Over the last two decades, mathematical modeling has become a popular tool in study of blood coagulation. The in silico methods were able to yield interesting and significant results in the understanding of both individual reaction mechanisms and regulation of large sections of the coagulation cascade. The objective of this paper is to review the development of theoretical research in hemostasis and thrombosis, to summarize the main findings, and outline problems and possible prospects in the use of mathematical modeling and computer simulation approaches. This review is primarily focused on the studies dealing with: (1) the membrane-dependent reactions of coagulation; (2) regulation of the coagulation cascade, including effects of positive and negative feedback loops, diffusion of coagulation factors, and blood flow.

Blood Coagulation↗
PubMed1988

Ventricular volume regulation: a mathematical model and computer simulation.

A mathematical model of ventricular volume regulation based on fluid mechanical principles has been constructed using a systems engineering approach. The parameters used in the model are based on clinical observation, laboratory investigation, and presumptions that will be tested later. The model was constructed to be the basis of a computer simulation. Using the computer simulation, information obtained from the literature and laboratory hypotheses regarding pathophysiology, several enigmatic conditions were tested. The model predicted that over-production of cerebrospinal fluid, as in the case of choroid plexus papilloma, could by itself lead to distention of the ventricular system. In simulating pseudotumor cerebri, if cerebrospinal fluid absorption at the arachnoid villi is impaired and the brain itself is rendered incompressible by swelling, intracranial pressure rises and ventricular volume diminishes. Conversely, in normal-pressure hydrocephalus, if cerebrospinal fluid flow is restricted between the spinal and cortical subarachnoid spaces and the brain is made more compressible, the ventricular volume increases with minimal increases in intracranial pressure. This mathematical model and its associated computer simulation is useful in predicting the behavior of the volume of the cerebral ventricles to a variety of pathological phenomena.

Animals↗
PubMed1988

Pollen count, symptom and medicine score in birch pollinosis. A mathematical approach.

This study investigates the correlation between the daily birch pollen counts, hay fever symptoms and medicine scores. Fifteen birch pollinosis patients were studied during two consecutive birch pollen seasons. All had a positive history for birch hay fever and a positive skin prick test, nasal provocation test and/or conjunctival provocation test to birch pollen. The patients recorded daily symptom and medicine scores during February through May for two seasons. According to nasal/conjunctival sensitivity and medicine consumption the group was divided into three groups: very sensitive, sensitive, and fairly sensitive. The mathematical calculations were based only on the results from the two most sensitive groups. The relationship between symptom scores and medicine scores as a function of the pollen load was nonlinear. A mathematical model was calculated. It was found that simply adding symptom scores and medicine scores to a total symptom/medication score was not meaningful as a basis for a quantitative analysis. It was further shown that the response caused by a given pollen load decays exponentially with time and that this decay had a characteristic half-life period of about 1-2 days indicating a long-lasting effect, i.e. contribution of the late allergic reaction to symptoms. Both groups showed the development of increased medicine intake during the season for a constant pollen load. This indicates the development of a higher sensitivity to birch pollen during the season. The overall response was divided into characteristic levels based on dose-response relationships, and pollen concentration intervals for forecasting purposes are suggested.

Adolescent↗
PubMedDate not supplied

Automated rescreening in cervical cytology. Mathematical models for evaluating overall process sensitivity, specificity and cost.

OBJECTIVE: To develop mathematical models to assist decision makers with the difficult task of evaluating the use of automated rescreening in the process of screening cervical smears. STUDY DESIGN: Using assumptions about incidence, per smear screening costs, and the sensitivity and specificity of cytotechnologists, pathologists and the rescreening device, basic probability models were developed to describe the overall sensitivity, specificity and cost of the screening process. RESULTS: The optimal screening policy is highly dependent on assumptions, and an automated system can significantly affect the overall system cost and accuracy. CONCLUSION: Mathematical planning models are valuable tools to assist decision makers in the design of a screening process for cervical smears.

Automation↗
PubMed1982

Electrotonic metabolism of pacemaker activity. Further biological and mathematical observations on the behavior of modulated parasystole.

An in vitro biologic model of parasystole and a mathematical model of parasystole based on the phase-response relationships derived from the biologic model were used in tandem to further develop our understanding of the patterns of ectopic activity that might arise as a consequence of the interaction of two pacemakers across a zone of block. Superfusion of the central segment of a dog Purkinje fiber with an ion-free isotonic sucrose solution provided a narrow region of block. The modulation of pacemaker activity by electronic potentials transmitted across the area of block was shown to be importantly influenced by the position of the "ectopic" pacemaker relative to the site of block. Effects of repetitive electrotonic influences on a single pacemaker cycle, the degree of entrance block and capture of the pacemaker during a phase of supernormal excitability were also studied in both the biologic and mathematical models. Our results indicate that marked shifts in the incidence and pattern of manifest ectopic activity can occur as a result of slight changes in heart rate, ectopic pacemaker rate, level of block and the position of the parasystolic pacemaker relative to the block border.

Action Potentials↗
PubMed2005

Mechanisms of atrial fibrillation termination by pure sodium channel blockade in an ionically-realistic mathematical model.

The mechanisms by which Na+-channel blocking antiarrhythmic drugs terminate atrial fibrillation (AF) remain unclear. Classical "leading-circle" theory suggests that Na+-channel blockade should, if anything, promote re-entry. We used an ionically-based mathematical model of vagotonic AF to evaluate the effects of applying pure Na+-current (I(Na)) inhibition during sustained arrhythmia. Under control conditions, AF was maintained by 1 or 2 dominant spiral waves, with fibrillatory propagation at critical levels of action potential duration (APD) dispersion. I(Na) inhibition terminated AF increasingly with increasing block, terminating all AF at 65% block. During 1:1 conduction, I(Na) inhibition reduced APD (by 13% at 4 Hz and 60% block), conduction velocity (by 37%), and re-entry wavelength (by 24%). During AF, I(Na) inhibition increased the size of primary rotors and reduced re-entry rate (eg, dominant frequency decreased by 33% at 60% I(Na) inhibition) while decreasing generation of secondary wavelets by wavebreak. Three mechanisms contributed to I(Na) block-induced AF termination in the model: (1) enlargement of the center of rotation beyond the capacity of the computational substrate; (2) decreased anchoring to functional obstacles, increasing meander and extinction at boundaries; and (3) reduction in the number of secondary wavelets that could provide new primary rotors. Optical mapping in isolated sheep hearts confirmed that tetrodotoxin dose-dependently terminates AF while producing effects qualitatively like those of I(Na) inhibition in the mathematical model. We conclude that pure INa inhibition terminates AF, producing activation changes consistent with previous clinical and experimental observations. These results provide insights into previously enigmatic mechanisms of class I antiarrhythmic drug-induced AF termination. The full text of this article is available online at http://circres.ahajournals.org

Algorithms↗
PubMed1989

Circulatory assistance by intrathoracic pressure variations: optimization and mechanisms studied by a mathematical model in relation to experimental data.

The hemodynamic effects of phasic variations in intrathoracic pressure (ITP) timed to the cardiac cycle were predicted by a mathematical model and were compared with data from canine experimental studies. The model was used to predict the hemodynamic effects of changing the onset of the ITP rise relative to the start of cardiac systole, as well as the hemodynamic effects of changes in the duration and amplitude of the ITP rise. The predictions of the model were compared with hemodynamic data from seven anesthetized dogs. Cardiac function was depressed with large doses of verapamil and propranolol, and the hearts were atrioventricular sequentially paced at a rate of 72 beats/min. Phasic ITP variations were generated by a perithoracic vest and were electronically timed to the cardiac cycle. The model predicted, and the experimental data confirmed, that phasic intrathoracic pressure variations generated by vest inflation, timed to the cardiac cycle, can augment both peak and mean aortic flow. The following predictions of the model were also confirmed by the experimental data: 1) Maximum flow augmentation occurs when the onset of the ITP rise is simultaneous with the onset of left ventricular isovolumic contraction, and the ITP rise has a duration of 400 msec. 2) The magnitude of the flow augmentation is a function of the amplitude of the ITP rise. The experimental data showed that there was little further flow augmentation when the ITP rise was greater than 30-40 mm Hg. 3) The magnitude of flow augmentation was inversely proportional to the peak left ventricular elastance (Emax). The best fit between the measured and predicted flow augmentations was obtained for an assumed Emax of 0.5 mm Hg/ml, while Emax measurements in three dogs, using a volume conductance catheter and transient vena caval occlusion, yielded values of 0.4-1.6 mm Hg/ml. Thus, both the mathematical model and canine experiments showed that relatively low-amplitude ITP variations, rising synchronously with the onset of cardiac systole and having an optimal duration, assist the failing heart by augmentation of aortic flow. The degree of cardiac assistance decreases if the ITP variations do not rise synchronously with the onset of systole, or if their duration is not optimal. Thus, properly applied ITP variations may be used as an efficient, noninvasive method to temporarily assist the failing heart.

Animals↗
PubMed2001

Development of a noninvasive ultrasound color M-mode means of estimating pulmonary vascular resistance in pediatric pulmonary hypertension: mathematical analysis, in vitro validation, and preliminary clinical studies.

BACKGROUND: Accurate determination of pulmonary vascular resistance (PVR) is an important component in the evaluation and treatment of pediatric patients with pulmonary hypertension. We developed a novel technique, based on the concept of flow propagation, to estimate PVR noninvasively. The hypothesis is that changes in PVR cause changes in the velocity propagation (Vel(prop)) within the main pulmonary artery and that Vel(prop) can be quantified using color M-mode imaging. METHODS AND RESULTS: We tested the hypothesis using mathematical modeling, in vitro experiments, and preliminary clinical studies. The mathematical model showed that pressure and velocity tracings are closely correlated in time and that 6 to 18 ms time resolution was needed to resolve propagation times within typical main pulmonary artery lengths (2 to 5 cm). The in vitro experiments demonstrated that it was feasible to use color M-mode to measure Vel(prop) and that Vel(prop) correlated well with downstream resistance [y=(-1.01x)+22.77; R=0.96]. The method was then evaluated on patients undergoing acute pulmonary reactivity testing (n=22 measurements). Good correlation between Vel(prop) and PVR was found [y=(-1.71x)+26.0; R=0.90; SEE=2.41]. CONCLUSION: This newly developed method promises to be useful in the noninvasive evaluation of adults and children with pulmonary hypertension.

Adolescent↗
PubMed1972

A threshold distribution hypothesis for packet storage of insulin and its mathematical modeling.

Phases of insulin release were studied in the perfused pancreas during a variety of glucose stimulation patterns. Patterns included staircase stimulations, constant prolonged single steps, restimulations, and ramp functions. Except at low concentrations, prolonged single steps of glucose elicited early spikes of insulin and a slowly rising second phase. Total insulin in the initial spikes increased with higher glucose concentrations. However, the time-related pattern of these spikes was similar in all cases; ratios of initial secretion rate to total insulin released were constant. Total insulin released in this early phase approximated a sigmoidal function of glucose concentration; mathematical differentiation of this function gave a skewed bell-shaped distribution curve. Staircase stimulations caused insulin to be released as a series of transient spikes which did not correlate with the increment of glucose but rather to the available insulin for a given glucose concentration minus that released in previous steps. The sum of total insulin released as spikes in a staircase series leading to a given glucose concentration was the same as when that concentration was used as a single step. Interrupted prolonged glucose infusions indicated the second phase of insulin release could prime the pancreas and that the first and second phases were interrelated. When glucose was perfused as ramp functions of slow, increasing, concentration, phasic response disappeared.A previous two-compartmental model was expanded to include a threshold or sensitivity distribution hypothesis. This hypothesis proposes that labile insulin is not stored in a homogeneous form but as packets with a bell-shaped distribution of thresholds to glucose. These packets respond quickly when their threshold levels to glucose are reached or exceeded. Data from single step stimulations were utilized for constructing a mathematical model which simulated satisfactorily the various stimulation patterns.

Animals↗
PubMed1995

Kinetic modeling and mathematical analysis indicate that acute phase gene expression in Hep 3B cells is regulated by both transcriptional and posttranscriptional mechanisms.

To evaluate the possible role of posttranscriptional mechanisms in the acute phase response, we determined the kinetics of transcription (by nuclear run-on assay) and mRNA accumulation of five human acute phase genes in Hep 3B cells incubated with conditioned medium from LPS-stimulated monocytes. Increase in mRNA accumulation was comparable to increase in transcription rate for fibrinogen-alpha and alpha-1 protease inhibitor, suggesting largely transcriptional regulation. In contrast, mRNA accumulation was about 10-20-fold greater than transcriptional increase for serum amyloid A, C3, and factor B, suggesting participation of posttranscriptional mechanisms. Since finding a disparity between the magnitudes of increase in mRNA and transcription does not definitively establish involvement of posttranscriptional mechanisms, we subjected our data to modeling studies and dynamic mathematical analysis to evaluate this possibility more rigorously. In modeling studies, accumulation curves resembling those observed for these three mRNAs could be generated from the nuclear run-on results only if posttranscriptional regulation was assumed. Dynamic mathematical analysis of relative transcription rates and relative mRNA abundance also strongly supported participation of posttranscriptional mechanisms. These observations suggest that posttranscriptional regulation plays a substantial role in induction of some, but not all acute phase proteins.

Acute-Phase Reaction↗
PubMed2003

Application of a mathematical model to prevent in vivo amplification of antibiotic-resistant bacterial populations during therapy.

The worldwide increase in the prevalence of multi-antibiotic-resistant bacteria has threatened the physician's ability to provide appropriate therapy for infections. The relationship between antimicrobial drug concentration and infecting pathogen population reduction is of primary interest. Using data derived from mice infected with the bacterium Pseudomonas aeruginosa and treated with a fluoroquinolone antibiotic, a mathematical model was developed that described relationships between antimicrobial drug exposures and changes in drug-susceptible and -resistant bacterial subpopulations at an infection site. Dosing regimens and consequent drug exposures that amplify or suppress the emergence of resistant bacterial subpopulations were identified and prospectively validated. Resistant clones selected in vivo by suboptimal regimens were characterized. No mutations were identified in the quinolone resistance-determining regions of gyrA/B or parC/E. However, all resistant clones demonstrated efflux pump overexpression. At base line, MexAB-OprM, MexCD-OprJ, and MexEF-OprN were represented in the drug-resistant population. After 28 hours of therapy, MexCD-OprJ became the predominant pump expressed in the resistant clones. The likelihood of achieving resistance-suppression exposure in humans with a clinically prescribed antibiotic dose was determined. The methods developed in this study provide insight regarding how mathematical models can be used to identify rational dosing regimens that suppress the amplification of the resistant mutant population.

Animals↗
PubMed1992

The effects of cognitive and metacognitive strategy instruction on the mathematical problem solving of middle school students with learning disabilities.

This study investigated the effects of cognitive and metacognitive strategy instruction on the mathematical problem solving of six middle school students with learning disabilities. Conditions of the multiple baseline, across-subjects design included baseline, two levels of treatment, setting and temporal generalization, and retraining. For Treatment 1, subjects received either cognitive or metacognitive strategy instruction. Treatment 2 consisted of instruction in the complementary component of the instructional program so that all subjects eventually received both cognitive and metacognitive strategy instruction. This design allowed a componential analysis of the content as well as sequence of instruction. Generally, subjects improved their mathematical problem solving as measured by performance on one-, two-, and three-step word problems. Discussion focused on effectiveness of treatment, acquisition and application of strategic knowledge, error pattern analysis, and the need to tailor instruction to the learner's individual characteristics.

Adolescent↗
PubMedDate not supplied

Roach's mathematical equations in predicting pathological stage in men with clinically localized prostate cancer.

AIMS AND BACKGROUND: The therapeutic choice in patients with clinically localized prostate cancer depends on preoperative clinical stage. Diagnostic instruments currently available for such an evaluation--considered separately--have not shown enough efficacy. Roach has recently introduced three simple mathematical equations that--on the basis of prostate-specific antigen and the biopsy Gleason score--are aimed at calculating the definitive pathological stage. We retrospectively analyzed our radical prostatectomy data base to assess the accuracy of the equations in predicting the final stage in patients with clinically localized prostate cancer. METHODS STUDY DESIGN: The study included 173 patients who had undergone radical retropubic prostatectomy at our Institution. Patients were divided into 25 groups, depending on preoperative PSA and the biopsy Gleason score. The risk of extracapsular neoplastic growth, seminal vesicle involvement and lymph node involvement was calculated for each group by means of Roach's equations. On the basis of definitive histological examinations, we compared the expected risk to the observed risk. RESULTS: The observed risk fell within the interval of expected risk in 16 of 17 groups (94%) regarding the evaluation of extracapsular growth, in 15 of 17 (88%) regarding the analysis of seminal vesicle involvement, and in 14 of 17 (82%) regarding the evaluation of lymph node involvement. Therefore, the observed event was in agreement with the expected event in 45 of 51 groups (88%). CONCLUSIONS: The equations represent a practical and effective instrument for preoperative clinical staging in patients with localized prostate cancer. By means of these mathematical formulas, one can assess the correct prognosis and--above all--plan the best therapeutic approach.

Aged↗
PubMed1999

Interactive mathematical models of subjective alertness and cognitive throughput in humans.

The authors present here mathematical models in which levels of subjective alertness and cognitive throughput are predicted by three components that interact with one another in a nonlinear manner. These components are (1) a homeostatic component (H) that falls in a sigmoidal manner during wake and rises in a saturating exponential manner at a rate that is determined by circadian phase during sleep; (2) a circadian component (C) that is a function of the output of our mathematical model of the effect of light on the circadian pacemaker, with the amplitude further regulated by the level of H; and (3) a sleep inertia component (W) that rises in a saturating exponential manner after waketime. The authors first construct initial models of subjective alertness and cognitive throughput based on the results of sleep inertia studies, sleep deprivation studies initiated across all circadian phases, 28-h forced desynchrony studies, and alertness and performance dose response curves to sleep. These initial models are then refined using data from nearly one hundred fifty 30- to 50-h sleep deprivation studies in which subjects woke at their habitual times. The interactive three-component models presented here are able to predict even the fine details of neurobehavioral data from sleep deprivation studies and, after further validation, may provide a powerful tool for the design of safe shift work and travel schedules, including those in which people are exposed to unusual patterns of light.

Algorithms↗
PubMedDate not supplied

A mathematical model for the estimation of human embryonic and fetal age.

Precise determination of donor age in human embryonic and fetal tissue is crucial for cell transplantation due to the existence of distinct time windows within which successful grafting is possible. This study demonstrates that between 4-12 wk postconception embryonic and fetal age can be estimated based on various morphometric parameters measured on a routine basis in suction abortion material. The greatest length, the neck-rump length, the foot length, and the proximal and distal arm and leg length were correlated with the anamnestic and ultrasonographically estimated age. Multivariate regression analyses showed a linear correlation between age and the logarithmic value of the various morphometric parameters. The best correlation was found for a mathematical model combining the limb parameters (r = 0.904; p < 0.001; n = 37). A prospective follow-up study (n = 40) was carried out to test the validity of the mathematical model. A high correlation was found between the calculated age and the estimated age based on anamnestic data (r = 0.749, p < 0.001). Outliers due to errors in the anamnestic data were readily identified by comparing anamnestic with calculated age. This method allows determination of embryonic and fetal age within and beyond the age group of the Carnegie classification and may, therefore, be useful for the needs of experimental and clinical cell transplantation.

Animals↗