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PubMed1997

Mathematical modeling of biofilms.

A set of mathematical equations constitutes a mathematical model if it aims to represent a real system and is based on some theory of that system's operation. On this definition, mathematical models, some very simple, are everywhere in science. A complex system like a biofilm requires modeling by numerical methods and, because of inevitable uncertainties in its theoretical basis, may not be able to make precise predictions. Nevertheless, such models almost always give new insight into the mechanisms involved, and stimulate further investigation. The way in which diffusion coefficients are measured for use in a model, particularly whether they include effects of reversible reaction, is a key element in the modeling. Reasons are given for separating diffusion from reversible reaction effects and dealing with them in a separate subroutine of the model.

Bacterial Adhesion↗
PubMed2002

Mathematical methods relevant to the study of wound healing.

Over recent years there has been a growing interest in the application of mathematical techniques and methods to improve understanding of clinical problems. Mathematical methods permit accurate description and quantification of observations and processes. Wound healing is a clinical problem that needs support in order to improve management and understanding of the mechanisms that may disarrange healing processes. This article attempts to provide an overview of currently available methods that have a direct relevance to the management of lower extremity wounds. Mathematics can provide valuable assistance and insight to clinical understanding, but only if applied correctly. It is hoped that examples of good practice will promote the utilization of this versatile tool.

Journal Article↗
PubMed2006

Mathematical modeling of endovenous laser treatment (ELT).

BACKGROUND AND OBJECTIVES: Endovenous laser treatment (ELT) has been recently proposed as an alternative in the treatment of reflux of the Great Saphenous Vein (GSV) and Small Saphenous Vein (SSV). Successful ELT depends on the selection of optimal parameters required to achieve an optimal vein damage while avoiding side effects. Mathematical modeling of ELT could provide a better understanding of the ELT process and could determine the optimal dosage as a function of vein diameter. STUDY DESIGN/MATERIALS AND METHODS: The model is based on calculations describing the light distribution using the diffusion approximation of the transport theory, the temperature rise using the bioheat equation and the laser-induced injury using the Arrhenius damage model. The geometry to simulate ELT was based on a 2D model consisting of a cylindrically symmetric blood vessel including a vessel wall and surrounded by an infinite homogenous tissue. The mathematical model was implemented using the Macsyma-Pdease2D software (Macsyma Inc., Arlington, MA, USA). Damage to the vein wall for CW and single shot energy was calculated for 3 and 5 mm vein diameters. In pulsed mode, the pullback distance (3, 5 and 7 mm) was considered. For CW mode simulation, the pullback speed (1, 2, 3 mm/s) was the variable. The total dose was expressed as joules per centimeter in order to perform comparison to results already reported in clinical studies. RESULTS: In pulsed mode, for a 3 mm vein diameter, irrespective of the pullback distance (2, 5 or 7 mm), a minimum fluence of 15 J/cm is required to obtain a permanent damage of the intima. For a 5 mm vein diameter, 50 J/cm (15W-2s) is required. In continuous mode, for a 3 mm and 5 mm vein diameter, respectively 65 J/cm and 100 J/cm are required to obtain a permanent damage of the vessel wall. Finally, the use of different wavelengths (810 nm or 980 nm) played only a minor influence on these results. DISCUSSION AND CONCLUSION: The parameters determined by mathematical modeling are in agreement with those used in clinical practice. They confirm that thermal damage of the inner vein wall (tunica intima) is required to achieve the tissue alterations necessary in order to lead the vein to permanent occlusion. However, in order to obtain a high rate of success without adverse events, the knowledge of the vein diameter after tumescent anesthesia is recommended in order to use the optimal energy. As clearly demonstrated by our calculations, both pulsed and continuous mode operations of the laser can be efficient. An interesting observation in our model is that less amount of energy is required in pulsed mode than in continuous mode. Damaging the vein sequentially along its entire length may lead to permanent occlusion. However, the pulsed mode requires a very precise positioning of the fiber after each pullback and the duration of the treatment is much longer. For these reasons, continuous irradiation seems to be preferred by most clinicians. This model should serve as a useful tool to simulate and better understand the mechanism of action of the ELT.

Animals↗
PubMed2006

A critical review of mathematical models and data used in diabetology.

The literature dealing with mathematical modelling for diabetes is abundant. During the last decades, a variety of models have been devoted to different aspects of diabetes, including glucose and insulin dynamics, management and complications prevention, cost and cost-effectiveness of strategies and epidemiology of diabetes in general. Several reviews are published regularly on mathematical models used for specific aspects of diabetes. In the present paper we propose a global overview of mathematical models dealing with many aspects of diabetes and using various tools. The review includes, side by side, models which are simple and/or comprehensive; deterministic and/or stochastic; continuous and/or discrete; using ordinary differential equations, partial differential equations, optimal control theory, integral equations, matrix analysis and computer algorithms.

Algorithms↗
PubMed2005

A multilevel modelling solution to mathematical coupling.

Owing to mathematical coupling, statistical analyses relating change to baseline values using correlation or regression are erroneous, where the statistical procedure of testing the null hypothesis becomes invalid. Alternatives, such as Oldham's method and the variance ratio test, have been advocated, although these are limited in the presence of measurement errors with non-constant variance. Furthermore, such methods prohibit the consideration of additional covariates (e.g., treatment group within trials) or confounders (e.g., age and gender). This study illustrates the more sophisticated approach of multilevel modelling (MLM) which overcomes these limitations and provides a comprehensive solution to the analysis of change with respect to baseline values. Although mathematical coupling is widespread throughout applied research, one particular area where several studies have suggested a strong relationship between baseline disease severity and treatment effect is guided tissue regeneration (GTR) within dental research. For illustration, we use GTR studies where the original data were found to be available in the literature for reanalysis. We contrast the results from an MLM approach and Oldham's method with the standard (incorrect) approach that suffers from mathematical coupling. MLM provides a robust solution when relating change to baseline and is capable of simultaneously dealing with complex error structures and additional covariates and/or potential confounders.

Data Interpretation, Statistical↗
PubMed2003

Mathematical extrapolation of image spectrum for constraint-set design and set-theoretic superresolution.

Several powerful iterative algorithms are being developed for the restoration and superresolution of diffraction-limited imagery data by use of diverse mathematical techniques. Notwithstanding the mathematical sophistication of the approaches used in their development and the potential for resolution enhancement possible with their implementation, the spectrum extrapolation that is central to superresolution comes in these algorithms only as a by-product and needs to be checked only after the completion of the processing steps to ensure that an expansion of the image bandwidth has indeed occurred. To overcome this limitation, a new approach of mathematically extrapolating the image spectrum and employing it to design constraint sets for implementing set-theoretic estimation procedures is described. Performance evaluation of a specific projection-onto-convex-sets algorithm by using this approach for the restoration and superresolution of degraded images is outlined. The primary goal of the method presented is to expand the power spectrum of the input image beyond the range of the sensor that captured the image.

Journal Article↗
PubMed2005

Mathematical definition and analysis of the retinex algorithm.

We present a detailed mathematical analysis of the original Retinex algorithm due to Land and McCann [J. Opt. Soc. Am. 61, 1 (1071)]. To this end, we propose an analytic formula that describes the algorithm behavior. More than one Retinex version (e.g., with and without threshold) is examined. The behavior of Retinex varying the number of paths is predicted, and its recursive iterations are mathematically analyzed using the formula. The mathematical setting presented serves as a common ground for the various Retinex implementations. Its validity is confirmed by the tests on images that we have performed.

Color Perception↗
PubMed2025

Mathematical modeling suggests 14-3-3 proteins modulate RAF paradoxical activation.

RAF inhibitor "paradoxical activation" (PA) is a phenomenon where RAF kinase inhibitors increase RAF kinase signaling. Through mathematical modeling and experimental data analysis, we recently demonstrated that the combination of conformational autoinhibition (CA) with the disruption of CA by RAF inhibitors plays an important role in PA. 14-3-3 proteins are known to modulate RAF CA and RAF dimerization. We here extend our mathematical model to include both roles of 14-3-3 proteins, and we derive rigorous analytical expressions of RAF signal regulation as modulated by 14-3-3 proteins. We then use the model to investigate how 14-3-3 proteins may modulate PA. We mathematically show 14-3-3 protein stabilization of the autoinhibited form of RAF should potentiate PA, while 14-3-3 protein stabilization of the active RAF dimer should reduce PA. Our analysis suggests that the net-effect will often be a potentiation of PA, and that 14-3-3 proteins may be capable of inducing PA for RAF inhibitors that normally show little to no PA. We test model-based insights experimentally with two different approaches: forced increases in 14-3-3 expression (which we find amplifies PA) and evolved resistance assays (which suggest increased 14-3-3 expression may contribute to resistance to RAF inhibitors). Overall, this work supports a role for 14-3-3 in modulating RAF-inhibitor mediated paradoxical activation.

14-3-3 Proteins↗
PubMed2006

Program design based on a mathematical model using rating of perceived exertion for an elite Japanese sprinter: a case study.

We investigated the effects of program design on 400-m sprint time by applying a Rating of Perceived Exertion (RPE) mathematical model to training performance. The subject was 24 years old and had been training for 9 years. His best performance in 400-m sprint competitions was 45.50 seconds. Body weight, resting heart rate, training time and RPE were monitored daily after training sessions. Similarly, performance in 400-m races was recorded 9 times during 2003. At the World Championships in Athletics in France, the subject's team placed eighth in the 1,600-m relay. The RPE mathematical model was able to predict changes in performance. Rate of matching was statistically significant (r(2) = 0.83, F ratio = 34.27, p < 0.001). Application of the RPE mathematical model to the design of a training program specific to the needs of a 400-m sprinter indicates a potentially powerful tool that can be applied to accurately assess the effects of training on athletic performance.

Adult↗
PubMed2004

Improving methods for epidemiological control of canine visceral leishmaniasis based on a mathematical model. Impact on the incidence of the canine and human disease.

The mathematical model described by Dye (1996) condemned the epidemiological canine visceral leishmaniasis control campaign, considering it non-efficient. Using this model, we mathematically demonstrate that the control is not efficient, only at low kappa values (rate at which latent and infectious dogs are lost by the destruction program) which match the canine seropositivity observed in the field by the immunofluorescency (IF) blood eluates analysis. With higher k values, corresponding to IF (kappa = 0.07) or ELISA (kappa = 0.25) results in sera samples, the number of infectious dogs declines to a Ro =1 or Ro =0, respectively, interrupting the transmission and the advancement of epidemics. We also experimentally demonstrate that the dog removal, following the results of IF of sera, instead of eluates lead to a 57% (p < 0.005) decrease in canine cases and 87.5% (p < 0.005) in human cases. Our mathematical and experimental results indicate that the control campaign become more efficient by enhancing the sensitivity of the diagnostic assay.

Animals↗
PubMedDate not supplied

A mathematical investigation of the multiple pathways to recurrent prostate cancer: comparison with experimental data.

Considerable research is aimed at determining the mechanisms by which hormone-refractory prostate cancer develops. In an effort to assist in the understanding of recurrent prostate cancer and the cellular processes that mediate this disease, a mathematical model is presented that describes both the pretreatment growth and the posttherapy relapse of human prostate cancer xenografts. Our goal is to evaluate the interplay between the multiple mechanisms that have been postulated as causes of androgen-independent relapse. Simulations of the model show that molecular events that render the androgen receptor irrelevant to disease progression, such as upregulation of BCL2, can result in relapse after androgen deprivation therapy. However, decreased apoptosis of androgen-independent cells alone overestimates the effects of hormone therapy when compared to experimental data. When decreased apoptosis is combined with continual androgen receptor activation, the posttherapy growth dynamics are in excellent correlation with experimental observations of the growth of LuCaP xenografts. Furthermore, the mathematical model predicts that upregulation of the androgen receptor, together with its increased activation, is alone sufficient to result in the androgen-independent growth of LNCaP xenografts. Recent experimental studies that suggest that the posttherapy increase in and continual activation of the androgen receptor are common and crucial features of recurrent prostate cancer provide validation of the model predictions. This approach provides a framework for using mathematical techniques to study novel therapeutic strategies aimed at controlling this disease.

Androgens↗
PubMed1998

Mathematical relations between the lengths of the metacarpal bones and phalanges: surgical significance.

The mathematical relationship between the lengths of the metacarpals and phalanges is important in hand surgery because of its role in determining the lengthening of the tubular bone size or the length of a prosthesis. The purpose of the present study is to perform the measurements on the hand radiographs, and to determine possible mathematical relations between the lengths of the metacarpals and phalanges. The lengths and widths of metacarpals and phalanges were measured in millimeters on hand roentgenograms of 100 normal voluntary subjects (50 men, 50 women) and the results were evaluated with a descriptive analysis test. The relations between the lengths of the metacarpals and phalanges were mathematically investigated on several sets of data. Unchangeable proportions between the metacarpals and phalanges lengths were found and formulated. Because any defects or faults of the hand, especially shortness, can corrected with current improved surgical techniques, a knowledge of the morphometric relations between the lengths of metacarpals and phalanges may be useful for such surgical interventions.

Adult↗
PubMed1968

Effects of manipulating an antecedent event on mathematics response rate.

The purpose of this investigation was to assess the effects on performance rate of simply writing the answers to mathematics problems versus verbalizing the problems before making a written response. The subject was an 11-yr-old boy whose response accuracy on mathematics problems was very erratic. Three experiments were conducted, each consisting of three phases. In each first phase, the subject was requested to write the answers to sets of mathematics problems. In the second, he was required to verbalize the problem before writing the answer. In the third phase, the subject was told to write the answer again without prior verbalization. The results indicated that the subject's correct answer rate increased and his error rate decreased as a result of his verbalizing the problems before making a written response. Results further revealed that in the final phase of each experiment, the return to the original conditions, his correct answer rate continued to increase.

Journal Article↗
PubMed1992

The nonequivalence of behavioral and mathematical equivalence.

Sidman and his colleagues derived behavioral tests for stimulus equivalence from the axiom in logic and mathematics that defines a relation of equivalence. The analogy has generated abundant research in which match-to-sample methods have been used almost exclusively to study interesting and complex stimulus control phenomena. It has also stimulated considerable discussion regarding interpretation of the analogy and speculation as to its validity and generality. This article reexamines the Sidman stimulus equivalence analogy in the context of a broader consideration of the mathematical axiom than was included in the original presentation of the analogy and some of the data that have accumulated in the interim. We propose that (a) mathematical and behavioral examples of equivalence relations differ substantially, (b) terminology is being used in ways that can lead to erroneous conclusions about the nature of the stimulus control that develops in stimulus equivalence experiments, and (c) complete analyses of equivalence and other types of stimulus-stimulus relations require more than a simple invocation of the analogy. Implications of our analysis for resolving current issues and prompting new research are discussed.

Animals↗
PubMed2005

Mathematical modelling and controlled drug delivery: matrix systems.

This paper deals with the physical and mathematical modelling description of drug release from matrix systems. In the introduction, matrix systems are considered in the wide frame of the controlled release systems and the concept of mathematical model is briefly discussed. Then, matrix structure and topology are matched, analysing the characteristics of the three-dimensional network constituting them. In this context, drug release mechanisms are considered with particular emphasis on the key factors ruling the release kinetics, such as matrix swelling, erosion, drug dissolution (re-crystallisation), drug diffusion, drug - polymer interaction, initial drug distribution and particle size distribution (for powdered matrix systems). The mathematical modelling section firstly considers the empirical and semi-empirical models that have the great advantage of showing analytical solutions. Then, the attention is focused on theoretical approaches regarding matrix swelling equilibrium and kinetics, drug dissolution, drug diffusion, drug - polymer interaction, initial drug distribution and matrix erosion. Finally, release kinetics from polydispersed spherical particles is studied. This review points out the fact that the comprehension of the phenomena ruling drug release from matrix systems is appropriate from both the physical and modelling point of view, although further improvements are always possible and desirable.

Algorithms↗
PubMed1992

Mathematical performance before, during, and following cycling at workloads of low and moderate intensity.

Mental arithmetic performance before, during, and following low (40% maximal heart-rate reserve; approximately 90 watts exercise for 15 min.) and moderate (60% maximal heart-rate reserve; approximately 150 watts exercise for 10 min.) intensity cycling by 20 male students (M age = 28.1 yr.) was studied. Subjects were grouped, by using a median-split on their total mathematical performance scores, into a group of 10 low in arithmetic skill and a group of 10 high in arithmetic skill. The numbers and percentages of right answers to 1-min. mathematical problem-sets of either group were not different in the various conditions, suggesting that 25 min. of progressive cycling exercise did not influence mathematical problem-solving efficacy.

Adult↗
PubMed1987

Predicting intake and digestibility using mathematical models of ruminal function.

Intake and digestibility of feeds by ruminants are influenced by characteristics of the feed, animal and feeding situation. Integration of these characteristics in mathematical models is critical to future progress in forage evaluation and optimal formulation of diets for ruminants. The physiological and physical theories of intake regulation can be described by simple mathematical equations. These equations indicate that intake is a linear function of animal characteristics, such as body weight and production level, and a reciprocal function of feed characteristics, such as fill effect and energy content. Theoretical equations were developed to predict intake when the neutral detergent fiber and energy content of the diet and the energy requirements of the animal are known. The theoretical model also can be used to predict the maximum intake that will maintain a given level of animal production by solving the physiological and physical intake equations at their intersection. Psychogenic intake regulation, which is related to the animal's behavioral response to factors not related to physiological or physical characteristics, can be described mathematically as a multiplier. Digestibility can be predicted by summing the contents of ideal nutritive entities in feeds, which have true digestibilities near 100%, subtracting their associated endogenous losses and adding the variable digestible fiber content. Steady-state models indicate fractional rates of digestion and passage can be used to define ideal nutritive entities and predict digestibility over a range of kinetic characteristics. The steady-state solutions are particularly useful in understanding and predicting the depression in digestibility associated with changes in rates of passage at high levels of feed intake.

Animals↗
PubMedDate not supplied

A mathematical model for laser in situ keratomileusis and photorefractive keratectomy.

PURPOSE: The purpose of ablation refractive surgery is to remove the refractive error from the inherently asymmetric aspheric cornea. Although the technique is gaining wide acceptance and popularity, some patients are left with irregular corneas. Our objective was to develop a methodology to analyze corneal shape, reduce the shape to arcs, modify the local arc value to the desired new arc value, and render a new continuous Euclidean surface without discontinuity. METHODS: The method to reconstruct the corneal surface consists of importing scanner output elevation data points into a computer-aided design (CAD) application to form the surface model. The corneal arc measurements are derived at 5 degrees increments and centered about the Gauss point of symmetry. Each arc is manipulated to adjust the corresponding arc on the proposed corneal surface to reflect the new arc value, correcting for the refractive error. The method determines the amount of corneal tilt and ablation depth at a given diameter required for the refractive error with a smooth transition zone to the base cornea. RESULTS: The case example is a patient who began with a spherical refraction of -8.75 D and after LASIK was emmetropic, but had irregular astigmatism and 20/30 best spectacle-corrected Snellen visual acuity. The proposed mathematical model compares the achieved surface shape to the mathematically planned surface contour. An enhancement procedure to remove the LASIK-induced corneal irregularity was designed. CONCLUSION: A mathematical technique to plan myopic ablative surgery to make the corneal surface regular and symmetric is proposed.

Adult↗