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Establishment of a discriminant mathematical model for diagnosis of deficiency-cold syndrome using gene expression profiling.

OBJECTIVE: To screen diagnostic markers of Deficiency-Cold syndrome by gene expression profile and to establish a discriminant mathematical milliliters model for the clinical diagnosis of this syndrome based on a support vector machine (SVM). METHODS: A family suffering from Deficiency-Cold syndrome is chosen for this study. This family has 5 patients with Deficiency-Cold syndrome and 10 normal members. The peripheral blood samples for these 5 patients and 5 normal members are tested by using cDNA microarray with 18,816 clones to get their differential expression genes. These genes are further explored to understand their biological functions and pathways through existing databases. A SVM model for clinical diagnosis is then developed based on these differential expression genes. RESULTS: A total of 83 differential expression genes were identified between patients and normal members, in which 21 genes were recorded in the FATIGO database and 16 genes were related to metabolism. Eight (8) pathways were sorted out in the KEGG database, and half pathways were associated with human metabolism. A discriminant mathematical model based on a support vector machine successfully predicted a normal person and a patient with heavy Deficiency-Cold syndrome based on their gene differential expression profiles. Thus, this model may classify the Deficiency-Cold syndrome. CONCLUSION: This work demonstrates that the differential expression genes can be used to identify normal persons and patients with Deficiency-Cold syndrome. Deficiency-Cold syndrome is mainly associated with the metabolism-related gene regulations. In addition, the discriminant mathematical model based on a support vector machine is applicable to the clinical diagnosis for Deficiency-Cold syndrome.

Adolescent↗

A mathematical programming approach for gene selection and tissue classification.

MOTIVATION: Extracting useful information from expression levels of thousands of genes generated with microarray technology needs a variety of analytical techniques. Mathematical programming approaches for classification analysis outperform parametric methods when the data depart from assumptions underlying these methods. Therefore, a mathematical programming approach is developed for gene selection and tissue classification using gene expression profiles. RESULTS: A new mixed integer programming model is formulated for this purpose. The mixed integer programming model simultaneously selects genes and constructs a classification model to classify two groups of tissue samples as accurately as possible. Very encouraging results were obtained with two data sets from the literature as examples. These results show that the mathematical programming approach can rival or outperform traditional classification methods.

Algorithms↗

Mathematical discretization of size-exclusion chromatograms applied to commercial corn maltodextrins.

Discretization of a size-exclusion chromatography (SEC) chromatogram is shown here to be an important calculation for characterizing the distribution of a polydisperse polymer, especially when the polydispersity is large. Commercial poly-glucose maltodextrins are known to have such a polydispersity. A mathematical discretization method with Gaussian peaks centered on each individual degree of polymerization is proposed and is performed on the entire SEC chromatogram for three different grades of corn maltodextrins. Because SEC and high-performance anion exchange chromatography with pulsed amperometric detection (HPAEC-PAD) are based on different separation mechanisms, they can be considered orthogonal techniques, and HPAEC-PAD was therefore used to validate the SEC discretization procedure. Because this validation proved satisfactory for all commercially available oligomers, the discretization is extended to all of their SEC chromatograms. Comparing the number-average molar weight and the weight-average molar weight before and after the mathematical discretization verifies that such a mathematical treatment does not denaturate the chromatogram. This approach tentatively leads to a more exhaustive characterization of a broadly polydisperse sample, such as maltodextrins, than was previously available, as it (i) gets rid of the apparent, chemically irrelevant, continuous molar weight distribution obtained by raw SEC and (ii) addresses the current detection and quantitation limits of the HPAEC-PAD technique without any sample treatment.

Calibration↗

The implantation of every embryo facilitates the chances of the remaining embryos to implant in an IVF programme: a mathematical model to predict pregnancy and multiple pregnancy rates.

BACKGROUND: We aimed to assess the validity of a theoretical mathematical model to predict the pregnancy rate and the multiple pregnancy rate in IVF/oocyte donation programmes on the basis of the implantation rate and the number of transferred embryos. METHODS: A total of 1835 embryo transfers corresponding to three different programmes in two centres with different implantation rates were analysed. Pregnancy and multiple pregnancy rates observed in the aforementioned programmes were compared with those obtained following different mathematical models. Four models were tested: binomial model, ground model, maternal variability model and collaborative model. The goodness of fit was performed by means of the maximum likelihood fit method. RESULTS: The binomial model could not predict the pregnancy rate, and especially the multiple pregnancy rate. The multiple pregnancy rate predicted following the binomial model was much lower than observed, up to 40-fold reduced. Ground model and maternal variability model adjusted to the data with more precision, but were still not accurate. Finally, the collaborative model reproduced with very great accuracy both pregnancy rate and the multiple pregnancy rate. A collaborative parameter of 22% was found, implying that the implantation probability of each embryo is increased by 22% for every embryo previously implanted. CONCLUSIONS: Embryonic implantation does not follow a binomial law, showing that the implantation is not independent from the number of embryos implanted. The best fit to the data is obtained following a collaborative model by which the implantation of one embryo is facilitated by the implantation of other embryo(s). The mathematical formula of the collaborative model predicts very accurately the pregnancy rate and the multiple pregnancy rate in IVF/oocyte donation programmes, based on the implantation rate of this specific programme and the number of embryos transferred up to five embryos. We recommend using the aforementioned formula to quantify the pregnancy rate and the risk of multiple pregnancy in the counselling of the infertile couple at embryo transfer. Such a formula is freely available at www.ifca.unican.es/matorras/mathpreg/.

Embryo Implantation↗

A new mathematical model for relative quantification in real-time RT-PCR.

Use of the real-time polymerase chain reaction (PCR) to amplify cDNA products reverse transcribed from mRNA is on the way to becoming a routine tool in molecular biology to study low abundance gene expression. Real-time PCR is easy to perform, provides the necessary accuracy and produces reliable as well as rapid quantification results. But accurate quantification of nucleic acids requires a reproducible methodology and an adequate mathematical model for data analysis. This study enters into the particular topics of the relative quantification in real-time RT-PCR of a target gene transcript in comparison to a reference gene transcript. Therefore, a new mathematical model is presented. The relative expression ratio is calculated only from the real-time PCR efficiencies and the crossing point deviation of an unknown sample versus a control. This model needs no calibration curve. Control levels were included in the model to standardise each reaction run with respect to RNA integrity, sample loading and inter-PCR variations. High accuracy and reproducibility (<2.5% variation) were reached in LightCycler PCR using the established mathematical model.

Animals↗

Applicability and limitations of the Adam mathematical phantom with respect to radiological protection.

Monte Carlo (MC) simulation of radiation transport is applied to an anthropomorphic mathematical (ADAM) or Zubal's voxel phantom, representing a male adult. The purpose is to compare absorbed energy in various organs (liver, kidneys, lungs, pancreas, spleen, adrenals and heart) in the simplified (mathematical) and more realistic (voxel) anatomy. A broad beam of monodirectional and monoenergetic photons (20 keV to 10 MeV), perpendicular to the longitudinal body axis, is incident on the front (AP) or the back (PA) of the phantom. Two MC codes, MCNP-4C and MCNPX-2.1.5, are used for the calculations. Specific absorbed fraction as a function of energy reflects the shielding of an organ by other organs. Comparison of the results for the two phantoms enables an evaluation of the applicability and the limitations of ADAM with respect to radiological protection. The cases studied indicate no urgent need to replace the (commonly used) mathematical phantom by a more sophisticated voxel phantom.

Computer Simulation↗

Volumetric assessment of preload in trauma patients: addressing the problem of mathematical coupling.

The availability of the volumetric thermodilution pulmonary artery catheter allows preload assessment based on ventricular volume rather than pressure. This technique has been shown clinically to be a better measure of preload than the pulmonary artery occlusion pressure (PAOP). Critics of the technique argue that the use of thermodilution to measure cardiac output (CO) accounts for the better correlation between right ventricular end-diastolic volume (RVEDV) and CO than PAOP and CO, since stroke volume derived from the CO is a common term to both RVEDV and CO. Previous studies have attempted mathematical corrections for this coupling effect, but direct comparisons using a nonthermodilution measure of CO have not been reported. Our objective was to evaluate the importance of mathematical coupling between RVEDV and CO by assessing the ability of RVEDV to predict CO measured by thermodilution (COTH) compared with CO simultaneously determined by the Fick principle (COFICK). We performed a prospective study of 53 consecutive trauma patients admitted to a Level I trauma center between 10/1/94 and 6/1/95 who received a volumetric pulmonary artery catheter. Using linear regression analysis, RVEDV and PAOP were correlated with simultaneous measurements of both COFICK determined via indirect calorimetry and COTH. Fisher's z-transformation was used to evaluate the correlation coefficients for significant differences (p < .05). The correlation coefficients for RVEDV vs. COTH and RVEDV vs. COFICK were similar (.48 vs. 0.45, p = .76). There was a significant correlation between COTH and COFICK (r = .74, p < .001). RVEDV was significantly better than PAOP at predicting both COTH (p < .001) and COFICK (p = .04). Multivariate regression analysis confirmed that RVEDV was the only estimate of preload which was significantly related to CO. We conclude that mathematical coupling does not have a significant clinical effect on the relationship between RVEDV and CO.

Blood Pressure↗

Optimal phasic tracheal gas insufflation timing: an experimental and mathematical analysis.

OBJECTIVE: To investigate the modulation of CO2 clearance by changes in the duration of tracheal gas flow application during tracheal gas insufflation (TGI). DESIGN: Combination of bench studies using a commercial test lung and a commercially available intensive care ventilator and mathematical analysis using a clearance model derived from first principles. SETTING: University pulmonary research laboratory. PATIENTS: None. INTERVENTIONS: Experiments using TGI were performed on a test lung at two combinations of tidal volume and frequency. TGI was limited to part of the expiratory phase (the terminal 10-100% of expiration), and two different TGI catheter flow rates were studied. Permutations over a range of compliances, dead-space volumes, catheter flows, and TGI durations were collected. A mathematical model incorporating key ventilatory and TGI-related variables was developed to provide a first-principles theoretical foundation for interpreting the experimental results. MEASUREMENTS AND MAIN RESULTS: In the physical model, alveolar Pco2 attained a minimum value with TGI flow applied during the terminal 40-60% of the expiratory phase, a finding that was consistent over an almost eight-fold range of expiratory time constants. The mathematical model shows the same qualitative pattern as the experimental model, indicating that the observed behaviors are not an experimental artifact. CONCLUSION: The optimal duration of expiratory TGI flow application is stable over a wide range of impedance characteristics. Such stability suggests that near maximal effect of expiratory TGI could be obtained by applying TGI flow solely within the final 50% of the expiratory phase. Such uniform restriction of the application profile might both simplify technique implementation and decrease adverse consequences.

Carbon Dioxide↗

The evolution of mathematical modeling of glioma proliferation and invasion.

Gliomas are well known for their potential for aggressive proliferation as well as their diffuse invasion of the normal-appearing parenchyma peripheral to the bulk lesion. This review presents a history of the use of mathematical modeling in the study of the proliferative-invasive growth of gliomas, illustrating the progress made in understanding the in vivo dynamics of invasion and proliferation of tumor cells. Mathematical modeling is based on a sequence of observation, speculation, development of hypotheses to be tested, and comparisons between theory and reality. These mathematical investigations, iteratively compared with experimental and clinical work, demonstrate the essential relationship between experimental and theoretical approaches. Together, these efforts have extended our knowledge and insight into in vivo brain tumor growth dynamics that should enhance current diagnoses and treatments.

Animals↗

Role of Concentration-dependent Unloading in Mathematical Models of Münch Transport.

It has been erroneously claimed (Goeschl et al. [1976] Plant Physiol. 58: 556-562) that concentration-dependent unloading is required for a correct mathematical solution for Münch phloem transport. Although it may prove to be physiologically correct, concentration-dependent unloading is not a mathematical necessity. Furthermore, its use in a mathematical model may not be desirable, because there is an infinite family of solutions for any given unloading distribution, irrespective of whether concentration-dependent unloading is assumed. Some illustrative numerical results are presented.

Journal Article↗

Modelling skin disease: lessons from the worlds of mathematics, physics and computer science.

Theoretical biology is a field that attempts to understand the complex phenomena of life in terms of mathematical and physical principles. Likewise, theoretical medicine employs mathematical arguments and models as a methodology in approaching the complexities of human disease. Naturally, these concepts can be applied to dermatology. There are many possible methods available in the theoretical investigation of skin disease. A number of examples are presented briefly. These include the mathematical modelling of pattern formation in congenital naevi and erythema gyratum repens, an information-theoretic approach to the analysis of genetic networks in autoimmunity, and computer simulations of early melanoma growth. To conclude, an analogy is drawn between the behaviour of well-known physical processes, such as earthquakes, and the spatio-temporal evolution of skin disease. Creating models in skin disease can lead to predictions that can be investigated experimentally or by observation and offer the prospect of unexpected or important insights into pathogenesis.

Humans↗

Estimation of fetal weight in twins: a new mathematical model.

OBJECTIVES: Evaluation of new mathematical formula (Femur 4) derived from a twin population to estimate fetal weight in twins using ultrasound. Comparison of Femur 4 is with conventional mathematical models. DESIGN: Retrospective analysis of ultrasonic measurements of 297 twin babies from 24 to 40 weeks of gestation who were born within 10 days of ultrasound examination. SETTING: Aberdeen Maternity Hospital. METHODS: With ultrasonic measurements obtained from twin babies, estimated fetal weight was calculated using the mathematical models of Campbell, Shepard and Hadlock. The calculations were repeated for the model of Femur 4. All models were compared against Femur 4. RESULTS: The coefficient of determination of the linear regression between the actual and predicted weight was highest for Femur 4 (0.852). Femur 4 had the highest proportion of babies with estimated weights within 10% of actual birthweight (71.4%). In babies who weighed between 2000 and 3000 g, Femur 4 had the least systematic and random error of -1.69 and 8.96, respectively. For babies below the 10th centile for weight, Femur 4 had comparable positive and negative predictive values of 76.0% and 92.3%, respectively. Femur 4 was equally poor at predicting growth discordancy with positive and negative predictive values of 70.0% and 86.5% only. CONCLUSION: Femur 4 requires measurements of femur length and abdominal circumference only, hence avoiding the need to obtain difficult head measurements which is a common problem in twins. It is a good model for estimation of fetal weight in twins. However, prediction of growth discordancy remains problematic.

Body Weight↗

Prediction of the distance from the skin to the lumbar epidural space in the Greek population, using mathematical models.

BACKGROUND AND OBJECTIVES: The skin to lumbar epidural space distance (SLED) is variable, and therefore the ability to clinically predict the SLED may help increase the success of epidural anesthesia/analgesia. The goal of this study was to determine the relationship between the SLED and demographic/anthropometric variables in the Greek population, and develop a mathematical model for its prediction. METHODS: This prospective randomized study enrolled 406 male and female Greek patients who required an epidural block as part of their anesthetic management. With patients placed in the left lateral and knee-chest position, the lumbar epidural space was located by the loss of resistance to normal saline technique. Statistical analysis was used to identify the relationship between SLED, and the following variables were evaluated: age, weight, height, body mass index, body surface area, intervertebral space used, pregnancy, and geographic origin within Greece. RESULTS: No adverse events or dural punctures occurred. Mean SLED in the general population was 4.98 +/- 0.95 cm, with values significantly higher in males (5.37 +/- 0.88 cm) compared with females (4.83 +/- 0.93 cm). SLED was best associated with weight, body surface area, and body mass index. Mathematical formulae for prediction of SLED in the general population and the female population were derived from linear regression analysis. These formulae were able to predict approximately half of the observed variability in SLED. CONCLUSIONS: While mathematical models of SLED can be a useful tool, they should not be exclusively relied on in the clinical setting, but rather should be used as an adjunct to standardized techniques to improve the safety and efficacy of epidural anesthesia/analgesia.

Journal Article↗

Biogeneric tooth: a new mathematical representation for tooth morphology in lower first molars.

A mathematical representation of tooth morphology may help to improve and automate restorative computer-aided design processes, virtual dental education, and parametric morphology. However, to date, no quantitative formulation has been identified for the description of dental features. The aim of this study was to establish and to validate a mathematical process for describing the morphology of first lower molars. Stone replicas of 170 caries-free first lower molars from young patients were measured three-dimensionally with a resolution of about 100,000 points. First, the average tooth was computed, which captures the common features of the molar's surface quantitatively. For this, the crucial step was to establish a dense point-to-point correspondence between all teeth. The algorithm did not involve any prior knowledge about teeth. In a second step, principal component analysis was carried out. Repeated for 3 different reference teeth, the procedure yielded average teeth that were nearly independent of the reference (less than +/- 40 microm). Additionally, the results indicate that only a few principal components determine a high percentage of the three-dimensional shape variability of first lower molars (e.g. the first five principal components describe 52% of the total variance, the first 10 principal components 72% and the first 20 principal components 83%). With the novel approach presented in this paper, surfaces of teeth can be described efficiently in terms of only a few parameters. This mathematical representation is called the 'biogeneric tooth'.

Algorithms↗

The mathematical modelling of human culture and its implications for psychology and the human sciences.

Recent years have seen the growth of a new and exciting field of theoretical research concerned with the mathematical modelling of human culture, and of its interaction with genetics. Drawing on analogies between genetic and cultural processes, mathematically sophisticated biologists have used population genetics models as the basis for the development of analogous models of cultural transmission, cultural evolution and gene-culture co-evolution. These models are designed to describe and analyse the diffusion of cultural traits through populations, under the influence of various cultural and evolutionary forces. They have already been applied to address many problems of interest to psychologists. Here I present an introduction to these models, explaining the mathematics in simple terms, and giving examples of the work of leading theorists. I go on to discuss the findings of most relevance to psychology, critically analysing the most important conclusions. Finally, I suggest some areas of psychology in which these models might usefully be applied. I argue that these models constitute a major theoretical innovation, and that there is considerable potential for their application in psychology.

Behavior↗

Short-term autonomic control of cardiovascular function: a mini-review with the help of mathematical models.

In this work the main aspects of the short-term regulation of the cardiovascular system are reviewed and critically discussed, laying special emphasis on the role of the autonomic neural mechanisms involved, on their mutual interrelationships and complex integration. All these aspects are summarized with the help of mathematical models developed by the authors in past years. The main characteristics of the uncontrolled system (i.e., the heart and vessels) and of the efferent neural branches (sympathetic and vagal) working on it are first described. Then, the afferent pathways which participate in feedback mechanisms (baroreceptors, chemoreceptors, lung-stretch receptors, direct CNS response), and the feedforward mechanisms anticipating cardiovascular requirements are introduced, and their role discussed with reference to various cardiovascular perturbations (hemorrhage or posture changes, hypoxia, asphyxia, dynamic exercise). Analysis of physiological data via mathematical equations, and results of computer simulations, emphasize the great complexity, richness and variability of the autonomic cardiovascular control, including redundant mechanisms and antagonistic requirements. The use of mathematical models is essential to capture this richness, and to summarize apparent contradictory data into a coherent and comprehensive theoretical setting.

Animals↗

A mathematical model for the quantification of mitral regurgitation. Experimental validation in the canine model using contrast echocardiography.

BACKGROUND: Because the clearance of contrast from the left atrium (LA) relative to the left ventricle (LV) depends on the degree of mitral regurgitation (MR), we hypothesized that a mathematical model can be developed that would provide a quantitative estimation of MR from the washout of contrast from these chambers. METHODS AND RESULTS: After mathematically developing the model, we performed experiments in two groups of dogs with the use of contrast echocardiography. Group 1 consisted of nine dogs in which different degrees of MR were produced by creating ischemic LV dysfunction. Contrast was injected into the LV, and MR was graded visually on a scale of from 0 to 4+. Videointensity plots generated from the LA and LV were provided to the model. There was excellent correlation between visual assessment of MR and model-derived regurgitant fraction in the 33 stages: y = 0.16x + 0.002 (r = 0.97, p less than 0.001, SEE = 0.06). To obtain a more quantitative validation, we placed electromagnetic flow probes on the aorta and just cephalad to the mitral annulus in six dogs (group 2) during cardiopulmonary bypass. Different degrees of MR were produced by chordal traction and/or myocardial ischemia. Regurgitant fraction was calculated at each stage from the flow probe and videointensity data. There was excellent correlation between flow probe and model-derived regurgitant fraction (y = 0.90x + 0.03; r = 0.96, p less than 0.001, SEE = 0.06), and close interobserver and intraobserver correlations were noted using flow probe and contrast echocardiographic data. CONCLUSIONS: A mathematical model that uses the clearance of contrast from the LA relative to the LV can be used to accurately measure the severity of MR. These findings may have important practical implications for the quantification of MR.

Animals↗

The effects of computer-based attribution retraining on the attributions, persistence, and mathematics computation of students with learning disabilities.

The purpose of the present study was to examine the impact of attribution retraining, embedded within a mathematics computer-assisted instructional (CAI) program, on students' attributions, persistence, and mathematics computation. Twenty-nine school-identified students with learning disabilities from five urban schools participated in the study. The sample's mean age was 13.3 years. After blocking on initial attributional patterns, students were randomly assigned to a mathematics CAI program that provided either attribution retraining or neutral feedback. Students used their assigned program for eight 30-minute sessions. Results did not support the contention that attribution retraining would have a significant impact on students' attributions. However, students who participated in the attribution retraining condition completed significantly more levels of the program than their counterparts who received neutral feedback. Attribution retraining students also obtained significantly higher scores on a test of problems practiced during the CAI program. These results suggest that attribution retraining may be a desirable addition to the type of feedback typically provided by CAI programs. However, they also highlight the need for further research that examines the conditions under which specific attributions are most advantageous.

Achievement↗