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PubMed2004

Using mathematical models to help understand biological pattern formation.

One of the characteristics of biological systems is their ability to produce and sustain spatial and spatio-temporal pattern. Elucidating the underlying mechanisms responsible for this phenomenon has been the goal of much experimental and theoretical research. This paper illustrates this area of research by presenting some of the mathematical models that have been proposed to account for pattern formation in biology and considering their implications.

Animals↗
PubMed2004

A mathematical study of a two-regional population growth model.

The paper provides a mathematical study of a model of urban dynamics, adjusting to an ecological model proposed by Lotka and Volterra. The model is a system of two first-order non-linear ordinary differential equations. The study proposed here completes the original proof by using the main tools such as a Lyapunov function.

Humans↗
PubMed2004

Statistical evaluation of mathematical models for microbial growth.

The aim of this study was to evaluate the suitability of several mathematical functions for describing microbial growth curves. The nonlinear functions used were: three-phase linear, logistic, Gompertz, Von Bertalanffy, Richards, Morgan, Weibull, France and Baranyi. Two data sets were used, one comprising 21 growth curves of different bacterial and fungal species in which growth was expressed as optical density units, and one comprising 34 curves of colony forming units counted on plates of Yersinia enterocolitica grown under different conditions of pH, temperature and CO(2) (time-constant conditions for each culture). For both sets, curves were selected to provide a wide variety of shapes with different growth rates and lag times. Statistical criteria used to evaluate model performance were analysis of residuals (residual distribution, bias factor and serial correlation) and goodness-of-fit (residual mean square, accuracy factor, extra residual variance F-test, and Akaike's information criterion). The models showing the best overall performance were the Baranyi, three-phase linear, Richards and Weibull models. The goodness-of-fit attained with other models can be considered acceptable, but not as good as that reached with the best four models. Overall, the Baranyi model showed the best behaviour for the growth curves studied according to a variety of criteria. The Richards model was the best-fitting optical density data, whereas the three-phase linear showed some limitations when fitting these curves, despite its consistent performance when fitting plate counts. Our results indicate that the common use of the Gompertz model to describe microbial growth should be reconsidered critically, as the Baranyi, three-phase linear, Richards and Weibull models showed a significantly superior ability to fit experimental data than the extensively used Gompertz.

Bacteria↗
PubMed2004

Three-dimensional mathematical reconstruction of the spinal shape, based on active contours.

To reduce the amount of radiographs needed for patients with a scoliosis, a radiation-free method based on topographic images of the back was developed. An active contour model simulating spinal stiffness has been applied to video rasterstereographic (VRS) data. The aim of the present study is (a) to evaluate the applicability of active contours to improve the accuracy and the reliability of the three-dimensional (3D) spinal midline reconstruction from back surface data and (b) to design a more robust method to detect the spinal midline. To evaluate the reliability and accuracy, the active contour-based method is compared to a conventional procedure, which has been specifically developed for scoliosis; both methods produce a 3D curve of the spinal midline. The frontal projections and surface rotations of these spinal midlines are compared; r.m.s. deviations of 0.9 mm between the frontal curves and 0.4 degrees between the surface rotations were obtained. Applying the active contour-based method does therefore not result in a substantial difference in accuracy to the conventional procedure. As a conclusion the active contour method is a valuable mathematical method that can accurately reconstruct the spinal midline based on back surface data. In addition, the method can be applied to various postures.

Algorithms↗
PubMed2005

Mathematical modeling and in vitro study of controlled drug release via a highly swellable and dissoluble polymer matrix: polyethylene oxide with high molecular weights.

A mathematical model is developed to describe the transport phenomena of a water-soluble small molecular drug (caffeine) from highly swellable and dissoluble polyethylene oxide (PEO) cylindrical tablets. Several important aspects in drug release kinetics were taken into account simultaneously in this theoretical model: swelling of the hydrophilic matrix and water penetration, three-dimensional and concentration-dependent diffusion of drug and water, and polymer dissolution. The moving boundary conditions are explicitly derived, and the resulting coupled partial differential equations are solved numerically. In vitro study of swelling, dissolution behavior of PEOs with different molecular weights and drug release are also carried out. When compared with experimental results, this theoretical model agrees with the water uptake, dimensional change and polymer dissolution profiles very well for pure PEO tablets with two different molecular weights. Drug release profiles using this model are predicted with a very good agreement with experimental data at different initial loadings. The overall drug release process is found to be highly dependent on the matrix swelling, drug and water diffusion, polymer dissolution and initial dimensions of the tablets. Their influences on drug release kinetics from PEO with two different molecular weights are also investigated.

Delayed-Action Preparations↗
PubMed2004

Strategy choices in simple and complex addition: Contributions of working memory and counting knowledge for children with mathematical disability.

Groups of first-grade (mean age = 82 months), third-grade (mean age = 107 months), and fifth-grade (mean age = 131 months) children with a learning disability in mathematics (MD, n = 58) and their normally achieving peers (n = 91) were administered tasks that assessed their knowledge of counting principles, working memory, and the strategies used to solve simple (4+3) and complex (16+8) addition problems. In all grades, the children with MD showed a working memory deficit, and in first grade, the children with MD used less sophisticated strategies and committed more errors while solving simple and complex addition problems. The group differences in strategy usage and accuracy were related, in part, to the group difference in working memory and to group and individual differences in counting knowledge. Across grade-level and group, the switch from simple to complex addition problems resulted in a shift in the mix of problem-solving strategies. Individual differences in the strategy mix and in the strategy shift were related, in part, to individual differences in working memory capacity and counting knowledge.

Adolescent↗
PubMed2006

Working memory, reading, and mathematical skills in children with developmental coordination disorder.

The aim of the present study was investigate the relationship between working memory and reading and mathematical skills in 55 children diagnosed with developmental coordination disorder (DCD). The findings indicate a pervasive memory deficit in all memory measures. In particular, deficits observed in visuospatial short-term and working memory tasks were significantly worse than in the verbal short-term memory ones. On the basis of these deficits, the sample was divided into high and low visuospatial memory ability groups. The low visuospatial memory group performed significantly worse on the attainment measures compared to the high visuospatial memory group, even when the contribution of IQ was taken into account. When the sample was divided into high and low verbal working memory ability groups, verbal working memory skills made a unique contribution to attainment only when verbal IQ was taken into account, but not when performance IQ was statistically controlled. It is possible that the processing demands of the working memory tasks together with the active motor component reflected in the visuospatial memory tasks and performance IQ subtest both play a crucial role in learning in children with DCD.

Child↗
PubMedDate not supplied

Mathematical neuroscience: from neurons to circuits to systems.

Applications of mathematics and computational techniques to our understanding of neuronal systems are provided. Reduction of membrane models to simplified canonical models demonstrates how neuronal spike-time statistics follow from simple properties of neurons. Averaging over space allows one to derive a simple model for the whisker barrel circuit and use this to explain and suggest several experiments. Spatio-temporal pattern formation methods are applied to explain the patterns seen in the early stages of drug-induced visual hallucinations.

Action Potentials↗
PubMed2006

Analysis of the orientation of primary cilia in growth plate cartilage: a mathematical method based on multiphoton microscopical images.

The chondrocytic primary cilium has been hypothesized to act as a mechano-sensor, analogously to primary cilium of cells in epithelial tissues. We hypothesize that mechanical inputs during growth, sensed through the primary cilium, result in directed secretion of the extracellular matrix, thereby establishing tissue anisotropy in growth plate cartilage. The cilium, through its orientation in three-dimensional space, is hypothesized to transmit to the chondrocyte the preferential direction for matrix secretion. This paper reports on the application of classical mathematical methods to develop an algorithm that addresses the particular challenges relative to the assessment of the orientation of the primary cilium in growth plate cartilage, based on image analysis of optical sections visualized by multiphoton microscopy. Specimens are prepared by rapid cold precipitation-based fixation to minimize possible artifactual post-mortem alterations of ciliary orientation. The ciliary axoneme is localized by immunocytochemistry with antibody acetylated-alpha-tubulin. The method is applicable to investigation of ciliary orientation in different zones of the growth plate, under either normal or altered biomechanical environments. The methodology is highly flexible and adaptable to other connective tissues where tissue anisotropy and directed secretion of extracellular matrix components are hypothesized to depend on the tissue's biomechanical environment during development and growth.

Algorithms↗
PubMed2004

A mathematical model for chronic myelogenous leukemia (CML) and T cell interaction.

In this paper, we propose and analyse a mathematical model for chronic myelogenous leukemia (CML), a cancer of the blood. We model the interaction between naive T cells, effector T cells, and CML cancer cells in the body, using a system of ordinary differential equations which gives rates of change of the three cell populations. One of the difficulties in modeling CML is the scarcity of experimental data which can be used to estimate parameters values. To compensate for the resulting uncertainties, we use Latin hypercube sampling (LHS) on large ranges of possible parameter values in our analysis. A major goal of this work is the determination of parameters which play a critical role in remission or clearance of the cancer in the model. Our analysis examines 12 parameters, and identifies two of these, the growth and death rates of CML, as critical to the outcome of the system. Our results indicate that the most promising research avenues for treatments of CML should be those that affect these two significant parameters (CML growth and death rates), while altering the other parameters should have little effect on the outcome.

Cell Death↗
PubMed2004

Autoimmunity as an immune defense against degenerative processes: a primary mathematical model illustrating the bright side of autoimmunity.

Self-tolerance, or the ability of the immune system to refrain from destroying the organism's own tissues, is a prerequisite for proper immune system operation. How such self-tolerance is achieved is still a subject of debate. The belief that autoimmunity poses a continuous threat to the organism was challenged by data demonstrating that autoimmunity has a protective function after traumatic injury to the central nervous system. This finding led us to suggest the 'comprehensive immunity' approach by which autoimmunity is viewed as a special case of immunity, namely as a defense mechanism that operates by fighting against the threat of potential destructive activity originated or mediated within the organism, similarly to the immune defense that operates against the threat from exogenous pathogens. We present a primary mathematical spatio-temporal model that supports this concept. The numerical solutions of this model illustrate the beneficial operation of a well-controlled immune response specific to self-antigens residing in the site of lesion. The model also explains how the response to self might be tolerated on a day-to-day basis. In addition, we demonstrate that the same autoimmune response, operating at different levels of regulation, can lead to either an autoimmune disease or a degenerative disorder. This preliminary qualitative model supports our contention that the way autoimmunity is perceived should be revised.

Animals↗
PubMed2004

A general mathematical framework to model generation structure in a population of asynchronously dividing cells.

In otherwise homogeneous cell populations, individual cells undergo asynchronous cell cycles. In recent years, interest in this fundamental observation has been boosted by the wide usage of CFSE, a fluorescent dye that allows the precise estimation by flow cytometry of the number of divisions performed by different cells in a population, and thus the generation structure. In this work, we propose two general mathematical frameworks to model the time evolution of generation structure in a cell population. The first modeling framework is more descriptive and assumes that cell division time is distributed in the cell population, due to intrinsic noise in the molecular machinery in individual cells; while the second framework assumes that asynchrony in cell division stems from randomness in the interactions individual cells make with environmental agents. We reduce these formalisms to recover two preexistent models, which build on each of the hypotheses. When confronted to kinetics data on CFSE labeled cells taken from literature, these models can fit precursor frequency distributions at each measured time point. However, they fail to fit the whole kinetics of precursor frequency distributions. In contrast, two extensions of those models, derived also from our general formalisms, fit equally well both the whole kinetics and individual profiles at each time point, providing a biologically reasonable estimation of parameters. We prove that the distribution of cell division times is not Gaussian, as previously proposed, but is better described by an asymmetric distribution such as the Gamma distribution. We show also that the observed cell asynchrony could be explained by the existence of a single transitional event during cell division. Based on these results, we suggest new ways of combining theoretical and experimental work to assess how much of noise in internal machinery of the cell and interactions with the environmental agents contribute to the asynchrony in cell division.

CD4-Positive T-Lymphocytes↗
PubMed2004

A mathematical model of tumour angiogenesis, regulated by vascular endothelial growth factor and the angiopoietins.

Angiogenesis--the growth of new blood vessels from existing ones--is a prerequisite for the growth of solid tumours beyond a diameter of approximately 2 mm. In recent years, the angiopoietins have emerged as important regulators of angiogenesis. They mediate a delicate balance between vascular quiescence, regression and new growth, but their mechanism of action is not fully understood. This work attempts to provide a mathematical description of the role of the angiopoietins in angiogenesis. The model is formulated within the framework of reinforced random walks, which allows easy transition between the continuum (macroscopic) and discrete (microscopic) forms. Model predictions are in qualitative agreement with experimental observations, and may have implications for anti-cancer therapies based on the prevention of angiogenesis.

Angiopoietins↗
PubMed2006

A mathematical model of plants as ecosystem engineers.

Understanding the structure and dynamics of plant communities in water-limited systems often calls for the identification of ecosystem engineers--key species that modify the landscape, redistribute resources and facilitate the growth of other species. Shrubs are excellent examples; they self-organize to form patterns of mesic patches which provide habitats for herbaceous species. In this paper we present a mathematical model for studying ecosystem engineering by woody plant species in drylands. The model captures various feedbacks between biomass and water including water uptake by plants' roots and increased water infiltration at vegetation patches. Both the uptake and the infiltration feedbacks act as mechanisms for vegetation pattern formation, but have opposite effects on the water resource; the former depletes the soil-water content under a vegetation patch, whereas the latter acts to increase it. Varying the relative strength of the two feedbacks we find a trade-off between the engineering capacity of a plant species and its resilience to disturbances. We further identify two basic soil-water distributions associated with engineering at the single patch level, hump-shaped and ring-shaped, and discuss the niches they form for herbaceous species. Finally, we study how pattern transitions at the landscape level feedback to the single patch level by affecting engineering strength.

Biomass↗
PubMed2006

A mathematical model of the treatment and survival of patients with high-grade brain tumours.

More years of life per patient are lost as the result of primary brain tumours than any other form of cancer. The most aggressive of these is known as glioblastoma (GBM). The median survival time of patients with GBM is under 10 months and the outlook has hardly improved over the past 20 years. Generally, these tumours are remarkably resistant to radiotherapy and yet about 2-3% of all GBMs appear to be cured. The objectives of this study were to formulate a mathematical and phenomenological model of tumour growth in a population of patients with GBM to predict survival, and to use the model to extract biological information from clinical data. The model describes the growth of the tumour and the resulting damage to the normal brain using simple concepts borrowed from chemical reaction engineering. Death is assumed to result when the amount of surviving normal brain falls to a critical level. Radiotherapy is assumed to destroy tumour but not healthy brain. Simple rules are included to represent approximately the clinician's decisions about what type of treatment to offer each patient. A population of patients is constructed by assuming that key parameters can be sampled from statistical distributions. Following Monte Carlo simulation, the model can be fitted to data from clinical trials. The model reproduces clinical data extremely accurately. This suggests that the long-term survivors are not a separate sub-population but are the 'lucky tail' of a unimodal distribution. The estimated values of radiation sensitivity (represented as SF2, the survival fraction after 2Gy) suggest the presence of severe hypoxia, which renders cells less sensitive to radiation. The model can predict the probable age distribution of tumours at presentation. The model shows the complicated effects of waiting times for treatment on the survival outcomes, and is used to predict the effects of escalation of radiotherapy dose. The model may aid the design of clinical trials using radiotherapy for patients with GBM, especially in helping to estimate the size of trial required. It is also designed in a generic form, and might be applicable to other tumour types.

Age Distribution↗
PubMed2006

Mathematical model comparisons of potential non-typeable Haemophilus influenzae vaccine effects.

Vaccines to prevent acute otitis media (AOM) caused by non-typeable Haemophilus influenzae (NTHi) are under development. Because NTHi is highly variable and colonization rates are high, special vaccine characteristics and trial designs might be needed. We examined in mathematical models the equilibrium NTHi-caused AOM rate given hypothetical vaccines that generated immunity identical to corresponding maximal naturally acquired immunity. Vaccines were examined with single effects and combinations of immunity affecting (1) AOM rates given colonization (pathogenicity), (2) susceptibility to colonization, and (3) contagiousness given colonization. Percent reductions in AOM across all preschool children were (1) 34%, (2) 31%, (3) 9%, (1 and 2) 57%, (2 and 3) 50%, and (1, 2, and 3) 75%. Effects on children in daycare vs. not in daycare were (1) 18 vs. 48%, (2) -1 vs. 57%, (3) 13 vs. 5%, (1 and 2) 30 vs. 79%, (2 and 3) 33 vs. 60%, and (1, 2, and 3) 64 vs. 85%. Pure pathogenicity effects (1 alone) will need to be supplemented by transmission effects. The effects of susceptibility (2 alone) are diminished or negative because children protected against colonization have lower levels of immunity to (1) and (3) than unvaccinated children. For trials to predict population effects, both colonization and AOM outcomes must be studied and all three effects must be evaluated. This need arises because, unlike H. influenzae type B, high NTHi exposure diminishes cumulative vaccine effects and high colonization rates generate rapid accumulation of natural immunity that alters the indirect effects of vaccine immunity on transmission differently by age and daycare status.

Acute Disease↗
PubMed2005

Mathematical model to assess the control of Aedes aegypti mosquitoes by the sterile insect technique.

We propose a mathematical model to assess the effects of irradiated (or transgenic) male insects introduction in a previously infested region. The release of sterile male insects aims to displace gradually the natural (wild) insect from the habitat. We discuss the suitability of this release technique when applied to peri-domestically adapted Aedes aegypti mosquitoes which are transmissors of Yellow Fever and Dengue disease.

Aedes↗
PubMed2006

Nursing students' mathematic calculation skills.

This mixed method study used a pre-test/post-test design to evaluate the efficacy of a teaching strategy in improving beginning nursing student learning outcomes. During a 4-week student teaching period, a convenience sample of 54 sophomore level nursing students were required to complete calculation assignments, taught one calculation method, and mandated to attend medication calculation classes. These students completed pre- and post-math tests and a major medication mathematic exam. Scores from the intervention student group were compared to those achieved by the previous sophomore class. Results demonstrated a statistically significant improvement from pre- to post-test and the students who received the intervention had statistically significantly higher scores on the major medication calculation exam than did the students in the control group. The evaluation completed by the intervention group showed that the students were satisfied with the method and outcome.

Adult↗