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PubMed2003

The conceptual basis of mathematics in cardiology: (II). Calculus and differential equations.

This is the second in a series of four articles developed for the readers of Coronary Artery Disease. Without language ideas cannot be articulated. What may not be so immediately obvious is that they cannot be formulated either. One of the essential languages of cardiology is mathematics. Unfortunately, medical education does not emphasize, and in fact, often neglects empowering physicians to think mathematically. Reference to statistics, conditional probability, multicompartmental modeling, algebra, calculus and transforms is common but often without provision of genuine conceptual understanding. At the University of Vermont College of Medicine, Professor Bates developed a course designed to address these deficiencies. The course covered mathematical principles pertinent to clinical cardiovascular and pulmonary medicine and research. It focused on fundamental concepts to facilitate formulation and grasp of ideas. This series of four articles was developed to make the material available for a wider audience. The articles will be published sequentially in Coronary Artery Disease. Beginning with fundamental axioms and basic algebraic manipulations they address algebra, function and graph theory, real and complex numbers, calculus and differential equations, mathematical modeling, linear system theory and integral transforms and statistical theory. The principles and concepts they address provide the foundation needed for in-depth study of any of these topics. Perhaps of even more importance, they should empower cardiologists and cardiovascular researchers to utilize the language of mathematics in assessing the phenomena of immediate pertinence to diagnosis, pathophysiology and therapeutics. The presentations are interposed with queries (by Coronary Artery Disease abbreviated as CAD) simulating the nature of interactions that occurred during the course itself. Each article concludes with one or more examples illustrating application of the concepts covered to cardiovascular medicine and biology.

Cardiology↗
PubMedDate not supplied

[Rotation movements of the knee joint. A mathematical-kinematic model].

The kinematics of rotational motion of the knee joint is described by a three-dimensional mathematical model. It is on principle supposed, that cruciate ligaments interact in concert to guide motion. This concept is based on the idea to explain movement in the sagittal and horizontal plane by a four-bar kinematic chain. The path of the instantaneous centers of rotation is shown with the help of this concept.

Biomechanical Phenomena↗
PubMed2003

Decision analysis as a tool to support an analytical pattern-of-reasoning.

Decision analysis is offered as a tool to aid nurses' decision-making in complex and troublesome situations where there are mutually exclusive actions and time is available for deliberation. Decision analysis can be formal or informal. Formal decision analysis provides a structure for representing the decision situation and a mathematical procedure for prescribing the alternative action that is most consistent with what is known and what one values. Informal decision analysis uses the concepts and sometimes the structure of decision analysis, but usually does not include the mathematical calculations. In the present paper, the authors illustrate how formal and informal decision analysis might be used by nurses to: (i) enhance their own decision-making; (ii) assist patients or family caregivers with decision-making and; (iii) promote informed health care policy development. Finally, the advantages and limitations of decision analysis are discussed.

Caregivers↗
PubMed1995

A model for reproductive efficiency of dairy bulls.

Reproductive efficiency of bulls is usually measured by nonreturn rate, which is commonly defined as the proportion of cows that were inseminated and did not return for another service within a specified number of days. The AI organizations use nonreturn rate to evaluate fertility of a bull or performance of a technician. Measures derived from nonreturn rate, such as conception rate and calving rate, might be more reliable for evaluation than nonreturn rate itself. Estimated conception rate is a better early measure of efficiency than nonreturn rate, because conception rate depends on the population of spermatozoa at insemination and not on developmental potential of the conceptus after insemination. A mathematical function is presented to model reproductive efficiency of bulls by estimation of the probability of conception at time of insemination (conception rate) and the probability of completing gestation after insemination (calving rate) through the relationship of nonreturn rate to the concentration of spermatozoa at insemination and the time after insemination. The model is illustrated with three bulls, using nonreturn rates by 28, 56, and 84 d after insemination.

Animals↗
PubMed1978

Self-concept correlates in academic placement of secondary school students.

In a large school system in the Southeastern United States 26% of the eighth grade students in the lower distribution of scores on a standardized test in reading and mathematics were assigned to vocational education courses and a career education program. One of the school counselors administered a self-concept inventory, the Me Scale (Yeatts & Morrison, 1974), to a randomized group of 43 students who were in the lower 26% of scores. The same inventory was administered to 91 randomly selected students in the upper distribution of scores. Significant differences were noted for one of five factors in the Me Scale. Lower achieving students perceived themselves as more capable in manual dexterity than higher achieving students.

Achievement↗
PubMed1998

Biological variation and quantification of health: the emergence of the concept of normality.

Historical research on the concept of normality, its roots and its development show that this concept has its sources in very different areas of scientific and medical thinking. Of great importance were: (i) a new theory of disease arising early in the 19th century supposing a continuous change from the healthy to the diseased state; (ii) the examination of variation within and between species of plants and animals; (iii) the clinical theory of constitution developed to describe the wholeness of the individual determined both by genetic factors and the influence of the environment; (iv) the development of mathematical and statistical tools starting with the adaptation of Bernoulli's "law of the great numbers" and Gauss' and Legendre's "law of errors" to biological measurements by the Belgian astronomer Quetelet. At the end of the First World War the concept of normality was first discussed. An idealistic "value norm" was set against a statistical "frequency norm". Between 1920 and 1930 the principles of our present concept were accepted and the mathematical tools developed. It took several decades to introduce this concept into practical medicine finally being recommended by the IFCC.

Clinical Laboratory Techniques↗
PubMed2004

In silico design of clinical trials: a method coming of age.

OBJECTIVE: To determine the feasibility and potential usefulness of mathematical models in evaluating immunomodulatory strategies in clinical trials of severe sepsis. DESIGN: Mathematical modeling of immunomodulation in simulated patients. SETTING: Computer laboratory. MEASUREMENTS AND MAIN RESULTS: We introduce and evaluate the concept of conducting a randomized clinical trial in silico based on simulated patients generated from a mechanistic mathematical model of bacterial infection, the acute inflammatory response, global tissue dysfunction, and a therapeutic intervention. Trial populations are constructed to reflect heterogeneity in bacterial load and virulence as well as propensity to mount and modulate an inflammatory response. We constructed a cohort of 1,000 trial patients submitted to therapy with one of three different doses of a neutralizing antibody directed against tumor necrosis factor (anti-TNF) for 6, 24, or 48 hrs. We present cytokine profiles over time and expected outcome for each cohort. We identify subgroups with high propensity for being helped or harmed by the proposed intervention and identify early serum markers for each of those subgroups. The mathematical simulation confirms the inability of simple markers to predict outcome of sepsis. The simulation clearly separates cases with favorable and unfavorable outcome on the basis of global tissue dysfunction. Control survival was 62.9% at 1 wk. Depending on dose and duration of treatment, survival ranged from 57.1% to 80.8%. Higher doses of anti-TNF, although effective, also result in considerable harm to patients. A statistical analysis based on a simulated cohort identified markers of favorable or adverse response to anti-TNF treatment. CONCLUSIONS: A mathematical simulation of anti-TNF therapy identified clear windows of opportunity for this intervention as well as populations that can be harmed by anti-TNF therapy. The construction of an in silico clinical trial could provide profound insight into the design of clinical trials of immunomodulatory therapies, ranging from optimal patient selection to individualized dosage and duration of proposed therapeutic interventions.

Antibodies, Monoclonal↗
PubMed1990

Optimizing REM latency as a diagnostic test for depression using receiver operating characteristic analysis and information theory.

New diagnostic techniques must be evaluated for their intrinsic accuracy and for their applicability to particular patient groups in specific clinical settings. Using receiver operating characteristic (ROC) analysis and concepts from information theory, we have developed a new mathematical and graphical method that can evaluate, compare, and optimize the performance of diagnostic tests for any value of disorder prevalence. Our analytic method is appropriate to any test that sorts disordered from nondisordered subjects using a continuous or nonbinary diagnostic variable; its characterization of the fundamental properties of such tests thus has important implications for the evaluation and optimization of diagnostic modalities used by clinicians in all medical specialties. We demonstrate our method using published data from five studies that used sleep architecture as a "biological marker" for depression. Our analysis confirms that REM latency is comparable to the dexamethasone suppression test in its ability to discriminate depressed from control subjects. For each of the five studies, we show how optimal REM latency cut-off times may be selected so that diagnostic information yield is maximized, and we compare the ability of each study to detect depressed subjects in populations where the prevalence of affective disorder can be specified.

Adult↗
PubMed1995

Influence of dimensional stability of impression materials on the probability of acceptance of a prosthetic restoration.

The accuracy of rubber impression materials was evaluated by taking impressions of a steel model, pouring it with stone and comparing certain measurements between steel and stone models. It is assumed that acceptance is achieved when the difference in measurements is within permissible limits, defined by clinical criteria. A mathematical formulation of these criteria will lead to the concept of probability of acceptance. The probability is affected when the limits are changed. The particular case of satisfying two acceptance criteria that contain variables which are not statistically independent is considered, leading to the application of the bivariate normal distribution in the evaluation of acceptance probability.

Dental Implants↗
PubMed2001

A model for determination of motor vehicle emission factors from on-road measurements with a focus on submicrometer particles.

The multiplicity of parameters that influence traffic-related emissions and that are often very difficult to measure or predict, makes the assessment of traffic emissions a very complicated process, strongly dependent on local conditions and usually associated with a high degree of error. The aim of this paper was to develop, calibrate and test a simple model for 'on-road' measurements of traffic emission factors as part of a major program focussed on the assessment of traffic contribution to fine and ultrafine emissions to the whole air shed and to local areas in south-east Queensland, Australia. A mathematical model developed was based on the mass balance concept for on-road assessment of traffic-related emission rates. The model requires fewer experimental data points as input and is more applicable to the common on-road testing situation, when no more than two monitors of a specific pollutant can be used. The model was tested and calibrated using experimental data on particle number concentration collected at a road-monitoring site using the scanning mobility particle sizer, and was applied to assess the emission factors of submicrometer particles emitted by traffic. The average emission factor obtained using the box model and the experimental data from road measurements was 1.75 x 10(14) particles km(-1) vehicle(-1), with a standard error of 67.6%. While the emission factor obtained was comparable with some results obtained from dynamometer studies and applied to the vehicle mix at the sampling site, they were significantly higher than the factors reported by other studies.

Environmental Monitoring↗
PubMed1999

A model of molecular circadian clocks: multiple mechanisms for phase shifting and a requirement for strong nonlinear interactions.

A fundamental question in the field of circadian rhythms concerns the biochemical and molecular nature of the oscillator. There is strong evidence that circadian oscillators are cell autonomous and rely on periodic gene expression. In Drosophila, Neurospora, Aplysia, and vertebrates, circadian oscillators are thought to be based on molecular autoregulatory loops composed of transcription, translation, and negative feedback by proteins on nuclear transcription. By studying a mathematical model of molecular clocks based on this general concept, the authors sought to determine which features such clocks must have to generate robust and stable oscillations and to allow entrainment by external stimuli such as light. The model produced circadian oscillations as an emergent property even though a time delay in protein synthesis and rate constants of the feedback loop were much shorter than 24 h. Along with the delay in protein production, strong nonlinear interactions in macromolecular synthesis and nuclear feedback appeared to be required for the model to show well-behaved oscillatory behavior. Realistic phase-shifting patterns induced by external stimuli could be achieved by multiple mechanisms-namely, up- and downward perturbations of protein or mRNA synthesis or degradation rates. The model makes testable predictions about interactions between clock elements and mechanisms of entrainment and may help to understand the functions of the intricate molecular interactions governing circadian rhythmogenesis.

Animals↗
PubMed2001

Entropy generation method to quantify thermal comfort.

The present paper presents a thermodynamic approach to assess the quality of human-thermal environment interaction and quantify thermal comfort. The approach involves development of entropy generation term by applying second law of thermodynamics to the combined human-environment system. The entropy generation term combines both human thermal physiological responses and thermal environmental variables to provide an objective measure of thermal comfort. The original concepts and definitions form the basis for establishing the mathematical relationship between thermal comfort and entropy generation term. As a result of logic and deterministic approach, an Objective Thermal Comfort Index (OTCI) is defined and established as a function of entropy generation. In order to verify the entropy-based thermal comfort model, human thermal physiological responses due to changes in ambient conditions are simulated using a well established and validated human thermal model developed at the Institute of Environmental Research of Kansas State University (KSU). The finite element based KSU human thermal computer model is being utilized as a "Computational Environmental Chamber" to conduct series of simulations to examine the human thermal responses to different environmental conditions. The output from the simulation, which include human thermal responses and input data consisting of environmental conditions are fed into the thermal comfort model. Continuous monitoring of thermal comfort in comfortable and extreme environmental conditions is demonstrated. The Objective Thermal Comfort values obtained from the entropy-based model are validated against regression based Predicted Mean Vote (PMV) values. Using the corresponding air temperatures and vapor pressures that were used in the computer simulation in the regression equation generates the PMV values. The preliminary results indicate that the OTCI and PMV values correlate well under ideal conditions. However, an experimental study is needed in the future to fully establish the validity of the OTCI formula and the model. One of the practical applications of this index is that could it be integrated in thermal control systems to develop human-centered environmental control systems for potential use in aircraft, mass transit vehicles, intelligent building systems, and space vehicles.

Algorithms↗
PubMed1992

Magnesium, adenosine triphosphate, and catecholamine complexes: application of new means of identifying solute-solute interactions of biological significance using NMR chemical shift data.

The interaction of magnesium ions with biomolecular species is fundamental to the regulation of many biological processes. A mathematical technique to investigate such interactions is described. The concept of a pseudo-formation curve is discussed and the relevance and applicability of such constructions to the study of solute-solute interactions is outlined. Analysis of curves based on the published variation of chemical shift data of various nuclei of adrenaline and dichloroisoprenaline in aqueous solution, in the presence of increasing concentration of the complex Mg-ATP, is described. Unambiguous indentification of specific interactions and the stability of the species produced is shown to be feasible. Some variation of the values of formation constants deriving from different nuclei reflects the significance of molecular environmental factors on the parameters used.

Adenosine Triphosphate↗
PubMed1989

Modeling analysis of the global and microscopic distribution of immunoglobulin G, F(ab')2, and Fab in tumors.

In order to understand the pharmacology of monoclonal antibodies and their conjugates, one must consider both global and microscopic aspects of antibody distribution. Here we present an analysis of antibody distribution in tumors based on the following factors: (a) molecular weight and valence of the antibody; (b) global pharmacokinetic profile following i.v. bolus injection; (c) penetration through the vascular wall; (d) diffusive and convective transport through interstitial space in the tumor; (e) antigen-antibody interaction; (f) antibody metabolism. Partial differential equations were developed to incorporate these factors and then solved numerically using parameter values from animal experiments, from clinical protocols at our institution, from studies of antibody binding characteristics in vitro, and from the literature. Salient findings from this model are that (a) antigen-antibody interaction in the tumor can retard antibody percolation away from blood capillaries, thus constituting a "binding site barrier"; (b) high antibody affinity tends to decrease antibody penetration and result in a more heterogeneous distribution; (c) high molecular weight [IgG greater than F(ab')2 greater than Fab] slows percolation and results in less uniform spatial distribution; (d) the average antibody concentration in the tumor does not increase linearly with antibody dose; (e) raising the rate of antibody metabolism results in low concentration and poor percolation; (f) perhaps most interesting, there is predicted to be a range of antibody dose and affinity within which the specificity ratio and average concentration could be kept high while limiting the heterogeneity of distribution. PERC, the computer program package developed for these analyses, provides a convenient and flexible way to assess the impact of global and microscopic parameters on the distribution of immunoglobulin in tumors. For calculations presented here, the input data were obtained from experimental sources, and qualitative features of the output proved consistent with the few interpretable observations available. However, detailed validation would require much more data than are currently at hand. The mathematical findings should therefore be considered as aids to concept development and as a set of null hypotheses with which to guide experimentation. Experiments and simulations will continue in tandem. It should be noted that the PERC package (and also the general principles delineated here) can be applied as well to biological ligands other than antibodies.

Antibody Affinity↗
PubMed1987

[The effectiveness of dynamic CT as a tool for the analysis of cerebral hemodynamics].

It is generally accepted that dynamic CT is one of the best way to evaluate cerebral hemodynamics. Employing our original computer simulation method, the present study was undergone to examine the theoretical background of dynamic CT. When a single bolus injection of non-diffusable indicator is intravenously performed, the time density curve of a cerebral tissue consists of the following two components: the time density curve of a carotid artery (g(x)) and the dispersion curve through the cerebral tissue (f(x)). A mean transit time (MTT) of the latter curve can be mathematically calculated by height-over area method. The concept gave us an interesting idea that a computer simulation could produce the time density curve implying an MTT. Convolution technique was applied to synthesize it from f(x) and g(x). The latter curve (g(x)) is influenced by several conditions such as a dispersion through a lung, a cardiac output and systemic arterial pressure. Therefore, it is not essential to measure the definite parameter from the time density curve obtained by dynamic CT. Dynamic CT should be considered as a qualitative method rather than a quantitative one. Several parameters were given by the analysis of a simulated curve. A time to peak (PT), above all, was most linearly correlated to MTT. A prolongation of PT in the insulted area, which means a prolongation of MTT, leads to a decrease of cerebral blood flow. In addition, we examined if Gamma variate fitting curve could apply a substitute for an original time density curve.(ABSTRACT TRUNCATED AT 250 WORDS)

Cerebrovascular Circulation↗
PubMed1980

Incorporation of background in dose-response models.

Dose-response models are generally classified as either threshold-tolerance or mechanism models. These models are briefly reviewed and their impact upon low-dose estimation is discussed. The concept of independent versus additive backgrounds is explored. The mathematical implications are considered with the conclusion that additive background implies linearity and that the assumption of total independence of background is required to justify nonlinearity at low doses. This finding has important implications to low-dose extrapolations for the purpose of risk estimation.

Animals↗