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PubMedDate not supplied

[Mathematics and the heart].

Mathematical modeling in biomedical sciences is particularly difficult because of the great amount of variables that the biological phenomena imply, the difficulty in observation and experimentation, and the scarceness of mathematical methods suitable for incorporating all of these variables. Even so, many mathematical models can be found in biology which include reduccionist, structural or integrative approaches. Theoretical models in biology are designed upon clearly defined purposes and may extent from the molecular level to the intact organism. For a model to be useful, its variables, levels and links must be very explicit. Observable and measurable aspects of the heart allow a modeling approach of this organ, in normal or in pathological conditions. A few examples are mentioned. Mathematical modeling of the heart may help orient experimental research, join existing data and theories and it may also be of assistance as therapeutic guidance.

Humans↗

PubMed1998

Ocular fixation index and mathematical models.

Ocular fixation test and ocular fixation index (OFI) never have been interpreted in terms of mathematical models, despite their widespread diffusion. However, ocular fixation is a typical case of visual-vestibular interaction, and mathematical models have proven very helpful in interpreting some mechanisms of this interaction, e.g. those of the optokinetic-vestibular interaction. In the present paper, a first attempt is proposed toward a model interpretation of OFI. By using very simple mathematical models, the hypothesis is tested that visual suppression of vestibular nystagmus results from direct action of smooth pursuit system (SPS). The aim is to draw consequences and recognize possible limits of this hypothesis. Dependence of OFI on SPS performance is examined. Although the available experimental data are insufficient for comprehensive validation of the model, the results agree with the current interpretations. In particular, quantitative support is given to the sensitivity of OFI to central vestibular diseases. Although the interpretation of visual suppression and OFI in terms of mathematical models is still at a very preliminary stage, models may provide a theoretical reference framework for the interpretation of new experimental results and/or suggest new test protocols.

Fixation, Ocular↗

PubMed1998

[Mathematical model for the predictive value of a test in critically ill patients studies according to APACHE II score and pathology at admission].

OBJECTIVE: To find a predictive model for mortality at four different days from the admission for critically ill patients. DESIGN: Retrospective study on two consecutive series of critically ill patients admitted in ICU. SUBJECTS: 1254 critically ill patients, subdivided into two series of 813 (561 survivors and 252 non survivors) and 441 patients (291 survivors and 150 non survivors), respectively. INTERVENTIONS: None. MEASUREMENTS: All patients had APACHE II calculated within the first 24 hours from the admission in ICU and, if the patient was still in ICU, also at the 5th, 10th and 15th day from the admission. Casistics was subdivided into two unequal series, ratio 2:1, with a random selection made on each of the 6 considered years. On the 1st series, in 1st, 5th, 10th and 15th day, for mathematical predictive models were made, using stepwise logistic regression (BMDP, Los Angeles). In the 1st day the following independent variables were utilized: APACHE II score, the specific diagnosis at admission, fitted following Knaus' diagnostic criteria, united in 6 principal categories, while for the other 3 days the variation % of APACE II score as regards the previous day. RESULTS: For each of the considered day four mathematical models have been made. These models have been validated in both series in calibration from the Hosmer-Lemeshow Goodness-of-fit test and in discrimination from the ROC curves. For each day Y (Prob.% to die) = eLogit/1 + eLogit, where Logit = beta 0 (constant) + beta 1*APACHE II + beta 2*Variat.%APACE II (difference between actual APACHE II - APACHE II of the previous day/actual APACHE II) + beta k, (coefficient pertinent to pathology). CONCLUSIONS: The mathematical model, as other models do, stratifies enough the casistics according to the risk of death. Waiting for further studies to make more precise prognostic mathematical models, this one and others can help the clinical assessment in single patient evaluation.

APACHE↗

PubMed1978

Mathematical modeling of lag phases in microbial growth.

This paper describes a mathematical method of the lap phases of Saccharomyces cerevisiae that incorporates the basic concepts previously presented in a two-stage deterministic model for the growth of this organism under conditions of oxygen excess with a sugar as the growth-limiting substrate. The model structure was suggested by an extensive investigation of the causes of the lap phases of S. cerevisiae which found that, in contrast to the traditionally accepted trends, the length of the lap phase was not inoculum-size dependent. This was consistent with other previously published work which suggested that a major factor in the length of the lag phases in S. cerevisiae was the need to synthesize adequate levels of glycolytic and respiratory enzymes. These suggestions were confirmed experimentally with lag-age data. Based on this conclusion a mathematical model was developed incorporating a description of the levels of glycolytic and respiratory enzymes and their effect on the growth rate and metabolism. This model was tested experimentally and the initial results indicate that many aspects of the lag phase of this organism may be described mathematically. The experimental findings further support the concept of primary regulatory control proposed by Bijkerk and Hall.

Glycolysis↗

PubMed2005

Experimental and mathematical study of the influence of growth factors on the growth kinetics of adult human articular chondrocytes.

This study aimed at determining how kinetic parameters of adult human articular chondrocytes (AHAC) growth are modulated by the growth factor combination TGFbeta1, FGF-2, and PDGF BB (TFP), recently shown to stimulate AHAC proliferation. AHAC, isolated from cartilage biopsies of three individuals, were cultured in medium without (CTR) or with TFP. For growth curves, AHAC were seeded at 1,000 cells/cm(2) and cultured for 12 days, with cell numbers measured fluorimetrically in the same wells every 12 h. For microcolony tests, AHAC were seeded at 2.5 cells/cm(2) and cultured for 6 days, with cell numbers determined for each microcolony by phase contrast microscopy every 8 h. A mathematical model combining delay and logistic equations was developed to capture the growth kinetic parameters and to enable the description of the complete growth process of the cell culture. As compared to CTR medium, the presence of TFP increased the number of cells/well starting from the fifth day of culture, and a four-fold larger cell number was reached at confluency. For single microcolonies, TFP reduced the time for the first cell division by 26.6%, the time for subsequent cell divisions (generation time) by 16.8%, and the percentage of quiescent cells (Q(c)) by 42.5%. The mathematical model fitted well the experimental data of the growth kinetic. Finally, using both microcolony tests and the mathematical model, we determined that prolonged cell expansion induces an enrichment of AHAC with shorter first division time, but not of those with shorter generation time.

Aging↗

PubMed1986

A mathematical analysis of creatine kinase activity in the course of Duchenne muscular dystrophy.

Recently, clinical follow-up studies have shown that the age of first symptoms and the speed of progress may vary considerably in Duchenne muscular dystrophy (DMD). Prognostic factors would be of interest not only for the patients themselves, but also for the selection of DMD patients for therapeutic studies. Creatine kinase (CK) activity reflects the dystrophic degeneration of the muscle cells. Therefore, the coherence of the course of CK activity and its relationship to the clinical state was investigated with the aid of a heuristic mathematical formula. Two constants were calculated, b and c. The data of 88 DMD patients proved the validity of the mathematical formula. Constant b seemed to be of no clinical relevance. The height of constant c was related to the age at which the ability to walk was lost (r = -0.76). This constant provides an objective individual assessment of the prognosis in DMD. In addition, the mathematical analysis of the CK course opens the way for a prospective CK estimation, which is relevant for the evaluation of therapeutic trials.

Adolescent↗

PubMed2000

Gender differences in advanced mathematical problem solving.

Strategy flexibility in mathematical problem solving was investigated. In Studies 1 and 2, high school juniors and seniors solved Scholastic Assessment Test-Mathematics (SAT-M) problems classified as conventional or unconventional. Algorithmic solution strategies were students' default choice for both types of problems across conditions that manipulated item format and solution time. Use of intuitive strategies on unconventional problems was evident only for high-ability students. Male students were more likely than female students to successfully match strategies to problem characteristics. In Study 3, a revised taxonomy of problems based on cognitive solution demands was predictive of gender differences on Graduate Record Examination-Quantitative (GRE-Q) items. Men outperformed women overall, but the difference was greater on items requiring spatial skills, shortcuts, or multiple solution paths than on problems requiring verbal skills or mastery of classroom-based content. Results suggest that strategy flexibility is a source of gender differences in mathematical ability assessed by SAT-M and GRE-Q problem solving.

Adolescent↗

PubMed1994

Mathematics of microbial plasmid instability and subsequent differential growth of plasmid-free and plasmid-containing cells, relevant to the analysis of experimental colony number data.

Differential growth of plasmid-containing and plasmid-free microbial cells occurs in many and probably most plasmid systems. Misinterpretation of differential growth as replicational, recombinational, or segregational stability or instability can unfortunately result in grossly erroneous conclusions about replication, recombination, or segregation in plasmid model systems for studies of such phenomena. The differential growth rate should ideally be measured every time that the rate of loss due to instability per se is measured. Unfortunately, the possibility of differential growth has been ignored in most plasmid model systems, since the mathematics of instability and differential growth has not usually been dealt with in ways that are intuitively understandable to experimental microbiologists. Nevertheless, rapid diagnosis of differential growth, and accurate estimation of differential growth rate and rate of loss due to instability per se, can be done by analysis of colony number data using only the relatively simple mathematics described in this review. This review is intended for experimental microbiologists rather than for theoretical population geneticists or for pure mathematicians. However, the same mathematics described in this review is also applicable to certain simple model systems for plasmid ecology or evolution in natural or clinical environments.

Bacteria↗

PubMed1992

Comparison of mathematically determined blood lactate and heart rate "threshold" points and relationship with performance.

The purpose of this study was to investigate the relationship between threshold points for heart rate (Thfc) and blood lactate (Thla) as determined by two objective mathematical models. The models used were the mono-segmental exponential (EXP) model of Hughson et al. and the log-log (LOG) model of Beaver et al. Inter-correlations of these threshold points and correlations with performance were also studied. Seventeen elite runners (mean, SD = 27.5, 6.5 years; 1.73, 0.05 m; 63.8, 7.3 kg; and maximum oxygen consumption of 67.8, 3.7 ml.kg-1.min-1) performed two maximal multistage running field tests on a 183.9-m indoor track with inclined turns. The initial speed of 9 km.h-1 (2.5 m.s-1) was increased by 0.5 km.h-1 (0.14 m.s-1) every lap for the fc test and by 1 km.h-1 (0.28 m.s-1) every 4 min for the la test. After fitting the la or the fc data to the two mathematical models, the threshold speed was assessed in the LOG model from the intersection of the two linear segments (LOG-la; LOG-fc) and in the EXP model from a tangent point (TI-la; TI-fc). Thla and Thfc speeds computed with the two models were significantly different (P less than 0.001) and poorly correlated (LOG-la vs LOG-fc: r = 0.36, TI-la vs TI-fc: r = 0.13). In general, Thfc were less well correlated with performance than Thla. With two different objective mathematical models, this study has shown significant differences and poor correlations between Thla and Thfc.(ABSTRACT TRUNCATED AT 250 WORDS)

Adult↗

PubMedDate not supplied

Reduction of infarct size in vivo with ischemic preconditioning: mathematical evidence for protection via non-ischemic tissue.

We constructed a mathematical model of ischemic preconditioning based on experimental data obtained from rat hearts. In this animal model of low collateral blood flow, we found that infarct size in preconditioned hearts, expressed as a percentage of area at risk, increased as the size of the area at risk increased (r = 0.76, p = 0.0007). In contrast, infarct size in control hearts appeared independent of changes in area at risk. Similarly, the lateral distance between the edge of the area at risk and the edge of the area of necrosis did not vary with risk region in control hearts, but in preconditioned hearts, lateral distance decreased as the size of the area at risk increased (r = -0.67, p = 0.0046). We used these findings to develop a simple model which provided mathematical relationships between lateral distance and area at risk and between infarct size and area at risk for both control and preconditioned hearts that were consistent with the experimental data. These relationships led us to propose that in preconditioned hearts 1) a protective substance may be produced or activated throughout the heart, and 2) that the protective substance may be transported by diffusion. If we assumed uniform production of protective substance in an amount proportional to the size of the ischemic and non-ischemic areas, we were able to derive, using a simple diffusion model, relationships between the above variables that were consistent with our mathematical model and with the experimental data. Although our model does not identify the protective substance, its implications provide ideas for additional crucial experiments that may enhance our understanding of ischemic preconditioning.

Animals↗

PubMedDate not supplied

Mathematical three-dimensional solid modeling of biventricular geometry.

The characterization of regional myocardial stress distribution has been limited by the use of idealized mathematical representations of biventricular geometry. State-of-the-art computer-aided design and engineering (CAD/CAE) techniques can be used to create complete, unambiguous mathematical representations (solid models) of complex object geometry that are suitable for a variety of applications, including stress-strain analyses. We have used advanced CAD/CAE software to create a 3-D solid model of the biventricular unit using planar geometric data extracted from an ex vivo canine heart. Volumetric analysis revealed global volume errors of 4.7%, -1.3%, -1.6%, and -1.1% for the left ventricular cavity, right ventricular cavity, myocardial wall, and total enclosed volumes, respectively. Model errors for 34 in-plane area and circumference determinations (mean +/- SD) were 5.3 +/- 6.7% and 3.8 +/- 2.7%. Error analysis suggested that model volume errors may be due to operator variability. These results demonstrate that solid modeling of the ex vivo biventricular unit yields an accurate mathematical representation of myocardial geometry which is suitable for meshing and subsequent finite element analysis. The use of CAD/CAE solid modeling in the representation of biventricular geometry may thereby facilitate the characterization of regional myocardial stress distribution.

Animals↗

PubMed1994

Tumor growth in vivo and as multicellular spheroids compared by mathematical models.

In vivo volume growth of two murine tumor cell lines was compared by mathematical modeling to their volume growth as multicellular spheroids. Fourteen deterministic mathematical models were studied. For one cell line, spheroid growth could be described by a model simpler than needed for description of growth in vivo. A model that explicitly included the stimulatory role for cell-cell interactions in regulation of growth was always superior to a model that did not include such a role. The von Bertalanffy model and the logistic model could not fit the data; this result contradicted some previous literature and was found to depend on the applied least squares fitting method. By the use of a particularly designed mathematical method, qualitative differences were discriminated from quantitative differences in growth dynamics of the same cells cultivated in two different three-dimensional systems.

Animals↗

PubMed1991

Analysis of chemotactic bacterial distributions in population migration assays using a mathematical model applicable to steep or shallow attractant gradients.

The mathematical model developed by Rivero et al. (1989, Chem. Engng Sci. 44, 2881-2897) is applied to literature data measuring chemotactic bacterial population distributions in response to steep as well as shallow attractant gradients. This model is based on a fundamental picture of the sensing and response mechanisms of individual bacterial cells, and thus related individual cell properties such as swimming speed and tumbling frequency to population parameters such as the random motility coefficient and the chemotactic sensitivity coefficient. Numerical solution of the model equations generates predicted bacterial density and attractant concentration profiles for any given experimental assay. We have previously validated the mathematical model from experimental work involving a step change in the attractant gradient (Ford et al., 1991 Biotechnol. Bioengng, 37, 647-660; Ford and Lauffenburger, 1991, Biotechnol. Bioengng, 37, 661-672). Within the context of this experimental assay, effects of attractant diffusion and consumption, random motility, and chemotactic sensitivity on the shape of the profiles are explored to enhance our understanding of this complex phenomenon. We have applied this model to various other types of gradients with successful interpretation of data reported by Dalquist et al. (1972, Nature New Biol. 236, 120-123) for Salmonella typhimurium validating the mathematical model and supporting the involvement of high and low affinity receptors for serine chemotaxis by these cells.

Bacterial Physiological Phenomena↗

PubMed2002

Assessment of bone feature parameters from lumbar trabecular skeletal patterns using mathematical morphology image processing.

In this study, bone feature parameters were obtained from digital lumbar vertebral images to determine their potential usefulness in assessing binary trabecular skeletal patterns. The system consisted of a magnification radiographic technique, computed radiography (CR), microfocused X-ray computed tomography (mu-CT), and mathematical morphology image processing. A digital CR image was produced, using a ten-times magnified lumbar vertebral cancellous bone block (1.5 x 1.5 x 1.5 cm(3)). The information was then subjected to mathematical morphology image processing, to extract eight binary trabecular skeletal patterns having different continuities as the operation number ( n) increased. These skeletal patterns were used as test patterns and analyzed quantitatively by calculation of the skeletal pixel percentage (SKP) and star volume analysis (Vsk, volume of skeletal trabecular elements; Vsp, volume of nonskeletal elements). Then the binary skeletal pattern data were converted into the mu-CT format, using a general personal computer, and the images were quantitatively analyzed with mu-CT built-in image analysis software for microstructural indices, fractal dimension analysis, and node-strut analysis. The SKP and Vsk showed an expected decrease as n increased. Concurrently, the Vsp showed an expected increase as n increased. The skeletal thickness (Sk.Th) showed a constant value at n = 1 to 4 and n = 5 to 7, but decreased in a stepwise fashion at n = 1 and n = 5. The skeletal number (Sk.N), bone area fraction (B.Ar/T.Ar) and bone perimeter fraction (B.Pm/T.Ar), decreased as n increased. Skeletal space (Sk.Sp) and perimeter-to-area ratio (B.Pm/B.Ar) increased with increasing n. In the fractal dimension analysis, the values decreased with increasing n and showed changes similar to the image observations. In fact, the SKP, star volume analysis, Sk.N, B.Ar/T.Ar, B.Pm/T.Ar, Sk.Sp, and B.Pm/B.Ar all closely mirrored the observational analysis of the images. Linkage of the skeletal structure as functions of the following parameters could be quantitatively demonstrated. The parameters used were node (Nd) and terminus (Tm). Variations in the total strut length (TSL) and Tm could demonstrate quantitative changes in skeletal structure. These results indicate that the system consisting of a combination of a magnification radiographic technique, CR, mu-CT, and mathematical morphology image processing may be a useful tool for quantitative skeletal structure analysis and the structural assessment of lumbar vertebrae, for the assessment of skeletal structural changes. It is important to choose suitable parameters for the desired structural changes.

Bone and Bones↗

PubMed1999

Accuracy of stereotactic coordinate transformation using a localisation frame and computed tomographic imaging. Part I. Influence of the mathematical and physical properties of the CT on the image of the rods of the localisation frame and the determination of their centres.

The accuracy of coordinate transformation from the computed tomographic (CT) space to the stereotactic frame space was analysed for frame-based stereotactic systems which use a localisation frame and coordinate transformation based on matrix calculation. The coordinate transformation was divided into three consecutive steps: (1) transforming the localisation frame into the CT image built up from pixels with distinct attenuation values, (2) determining the rod centres of the localisation frame in the CT image, and (3) coordinate transformation from the image to the frame space using the centres of the rods in the image space and algebraic, matrix-based calculation. The error contribution at each step was evaluated separately and its effect on the subsequent mathematical operations was analysed. The first step dealt with the influences of the mathematical and physical properties of the CT on the image of the localisation frame. Noise, slice thickness, convolution filter, dimension of the pixel matrix, and image processing had an influence on the attenuation values in each pixel. Above all, the slice thickness had an effect on the shape of the oblique rods in the CT image. At the second step, the main error contribution was due to the method by which the centre of the rods was calculated. The most accurate method was to determine the centre of gravity using the attenuation values as single mass points (with accuracy in the range of +/-1/10 pixel, or +/-0.125 mm), followed by rounding off the centre of gravity and the highest pixel value in the square matrix R2(N) within 1 pixel. Pointing with a cursor under visual control was accurate to 1 pixel and the pixel with the highest attenuation value showed deviations of up to 2 pixels in the x and y axes. Thus, the methods differed by a factor of 20. The influence of the CT mathematics and physics on the determination of the centre of the fiducials was negligible in comparison to the method of calculation used. There was no systemic error due to the filtred back projection algorithm. Data input errors due to noise were in the range of 1/10 pixel. The effects of the remaining physical influences were all in the range of the error due to noise. In particular these results speak in favour of no influence of slice thickness on coordinate transformation.

Artifacts↗

PubMed2005

Prediction of smallpox outbreak and evaluation of control-measure policy in Japan, using a mathematical model.

Since the September 1 terrorist attacks and moreover, since the anthrax exposure events in 2001 in the United States, bioterrorism attacks seem to be a real threat. Of course, the public health authorities in Japan have started to prepare control measures for such events. We report here our attempts, using a mathematical model, to estimate outbreak size and to examine the most effective measures; comparing ring vaccination (contact tracing, isolation, and vaccination among contacts) and mass vaccination of the susceptible population in the area. The basic framework of the mathematical model follows a model used in previous research. The initial susceptible population is assumed to be 30 million persons. Concerning the important parameters, such as the number of initial-exposure cases, R0 (infectious power, or natural history) and, the starting day of intervention after the initial exposure, we checked the robustness of our conclusions by sensitivity analysis. We found that mass vaccination is preferable to ring vaccination when the values for the initial-exposure cases and R0 are high and when the start of intervention by public health authorities is delayed. In the base-case situation, the mass vaccination strategy needs almost 30 million vaccine doses. On the other hand, though ring vaccination needs fewer doses, it needs fewer than 50,000 doses in the worst-case scenario, that with larger first exposure, higher R0, or later start of public health authority intervention. This mathematical model can measure the prevalence of an infectious disease and can evaluate control measures for it before an outbreak. Especially, it is useful for the planning of the outbreaks of emerging diseases such as severe acute respiratory syndrome (SARS) or for bioterrorism attacks involving such diseases as smallpox. In further research, we will have to take into account the population people vaccinated of for smallpox, who account for about 70% of the total population in Japan.

Disease Outbreaks↗

PubMed1983

A mathematical approach to problems of cephalopelvic disproportion at the pelvic inlet.

So many problems have been left unsolved by x-ray pelvimetry for the prognostic diagnosis of cephalopelvic disproportion that the clinical usefulness of x-ray pelvimetry has been questioned and a more scientific approach to evaluation of cephalopelvic disproportion has been sought. By unifying various factors that impose problems in cephalopelvic relationships into one index via new mathematical models, we attempted to estimate quantitatively the potential dystocia caused by inlet disproportion. The index is calculated from ten measurements of data obtainable on roentgenograms, and the practical calculations can be readily done with a programmable calculator. The validity of the assumptions used in the mathematical models and the reliability of the developed index were retrospectively analyzed in 300 primiparous women with cephalic presentations. This study suggests that the mathematical index could serve as a useful prognostic guide to the proper management of labor in association with disproportion.

Cephalometry↗

PubMed1991

Modeling of immunosensors under nonequilibrium conditions. I. Mathematic modeling of performance characteristics.

Immunosensors for the detection of small analytes that use analyte-enzyme conjugates as signal generators require special attention if operated under nonequilibrium conditions. If the size of the analyte and the analyte-enzyme conjugate differ substantially, the two antigens do not diffuse at the same rate. This can cause time-dependent shifts in the sensitivity of competitive immunoassays. Therefore, immunosensors operating at short incubation times require precise timing that meets closely the specifications for which the sensors were calibrated. As an example, we have analyzed kinetic binding curves for the quantitative determination of progesterone with an immobilized monoclonal antibody and a conjugate between horseradish peroxidase and progesterone as signal generator. Mathematical paradigms have been developed to simulate the diffusion, antigen-antibody complex formation, and competitive binding processes in this analytical system. Dose-response curves obtained under nonequilibrium conditions can vary substantially from those obtained at equilibrium of antigen-antibody interaction. The degree of this variation depends on the performance characteristics of the major components of the immunosensor. The developed mathematical solutions reflect experimental results and can be used to model optimal conditions for immunosensors operating under nonequilibrium conditions. In this paper (Part I), we report on the mathematical modeling of the interaction between analyte, analyte-enzyme conjugate, and an immobilized antibody. In Part II (W. Schramm and S.-H. Paek (1991) Anal. Biochem. 196), we present experimental results and compare them with the theoretical models.

Antibodies↗