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PubMed2002

Is reduction of pocket probing depth correlated with the baseline value or is it "mathematical coupling"?

Previous studies using correlation or regression analysis have showed that treatment effects measured by the change in clinical parameters are often associated with baseline values of the same parameters. These studies, however, have a methodological weakness. Correlation/regression between baseline measures and the derived change variable invalidates the statistical procedures of testing the null hypothesis: that the coefficient of correlation/regression is zero. This is due to the phenomenon of mathematical coupling. To investigate the impact that this has on the observed correlation/regression coefficient when in reality this is zero, we used random simulations of hypothetical data to model the treatment of periodontal pockets. Results showed a strong probability of obtaining statistically significant correlation/regression coefficients. To separate this artificial effect of mathematical coupling from the true underlying biological relationship, one must apply appropriate analytical strategies to re-evaluate previous evidence within the periodontal literature.

Algorithms↗
PubMed2005

Mathematical model of the rupture mechanism of intracranial saccular aneurysms through daughter aneurysm formation and growth.

OBJECTIVES: Daughter aneurysms have been strongly associated with saccular aneurysm rupture. We constructed a mathematical model to help explain this association as a possible hemodynamic mechanism for intracranial saccular aneurysm rupture. METHODS: Our model is based on the assumption that when an aneurysm reaches a state of imminent rupture, the weakest area of the aneurysm wall responds passively to a surge of intra-aneurysmal pressure by forming a daughter aneurysm that will be the site of the eventual rupture. The daughter and parent aneurysms were assumed to be spherical. Using mathematical modeling, the growth of the daughter aneurysm was observed. To obtain the change in tensile stress in the daughter aneurysm wall under constant pressure and changing geometry, the Law of Laplace was applied to the parent and the daughter aneurysms. RESULTS: The model reveals that the stress factor, i.e. tensile stress in the daughter aneurysm wall relative to the wall strength (rupture point), is dependent on two geometric parameters: the orifice factor (mu), which represents the relative size of the daughter aneurysm orifice radius to the parent aneurysm radius; and the aspect ratio (lambda), which represents the height-to-orifice ratio of the daughter aneurysm. As the daughter aneurysm develops, the stress factor first decreases to protect against rupture. Minimal stress is attained at an aspect ratio (lambda) of 0.577 regardless of the orifice factor. This is a relatively stable state. Further growth of the daughter aneurysm results in an increase of stress above the minimum, eventually leading to rupture at a stress factor of 1. A smaller orifice factor mu allows this aneurysm to grow to a higher aspect ratio lambda before rupture. DISCUSSION: Daughter aneurysm formation is a likely path to aneurysm rupture. The formation of a daughter aneurysm temporarily decreases the tensile stress within a parent aneurysm in which rupture is imminent, indicating a temporary protective role of daughter aneurysm development. Aneurysms harboring daughter aneurysms are at a more advanced stage of development, hence at a greater risk for rupture. The severity of the rupture risk can be estimated on the basis of daughter aneurysm geometry; aspect ratio lambda > 0.577 indicates a greater risk of rupture. Furthermore, daughter aneurysms with larger orifices are associated with a greater risk of rupture.

Aneurysm, Ruptured↗
PubMed1998

Parameterization of inoculum effect via mathematical modeling: aminoglycosides against Staphylococcus aureus and Escherichia coli.

Inoculum effect describes the inoculum size dependent changes in minimum inhibitory concentrations (MIC) exhibited by antibiotic-bacterium combinations demonstrating such effect. Traditionally, inoculum effect has been loosely defined based on the extent of increase in the MIC with respect to the increase in inoculum size. In most studies, assessment of MIC data has relied on the arbitrary selection of a point of reference for both baseline MIC and inoculum size. More importantly, this conventional method of assessment does not permit information conveyed in a complete MIC versus inoculum size profile to be fully explored. To undertake these issues, a mathematical model was developed for the description of the entire inoculum effect profile. With the employment of three key parameter estimates, i.e., the baseline MIC, the threshold inoculum size at which the increase in MIC commences, and the rate of increase in MIC with respect to inoculum size, both the shape and location of the profile could be adequately defined. To verify the application of this model, a series of four aminoglycosides were tested against standard strains of E. coli and S. aureus. Results showed a good degree of organism specificity and antibiotic-class dependency of the inoculum effect profiles. Analysis of the parameter estimates obtained provided further support for these observations. In conclusion, the mathematical model developed in the present study adequately described the inoculum effect exhibited by the various aminoglycoside-bacterium combinations tested. The parameter estimates generated by the modeling approach allowed comparison and quantitative analysis of the inoculum effect profiles with minimal difficulties.

Amikacin↗
PubMed2004

Improved methods for the mathematically controlled comparison of biochemical systems.

The method of mathematically controlled comparison provides a structured approach for the comparison of alternative biochemical pathways with respect to selected functional effectiveness measures. Under this approach, alternative implementations of a biochemical pathway are modeled mathematically, forced to be equivalent through the application of selected constraints, and compared with respect to selected functional effectiveness measures. While the method has been applied successfully in a variety of studies, we offer recommendations for improvements to the method that (1) relax requirements for definition of constraints sufficient to remove all degrees of freedom in forming the equivalent alternative, (2) facilitate generalization of the results thus avoiding the need to condition those findings on the selected constraints, and (3) provide additional insights into the effect of selected constraints on the functional effectiveness measures. We present improvements to the method and related statistical models, apply the method to a previously conducted comparison of network regulation in the immune system, and compare our results to those previously reported.

Animals↗
PubMed2005

A mathematical model of the euglycemic hyperinsulinemic clamp.

BACKGROUND: The Euglycemic Hyperinsulinemic Clamp (EHC) is the most widely used experimental procedure for the determination of insulin sensitivity, and in its usual form the patient is followed under insulinization for two hours. In the present study, sixteen subjects with BMI between 18.5 and 63.6 kg/m(2) were studied by long-duration (five hours) EHC. RESULTS: From the results of this series and from similar reports in the literature it is clear that, in obese subjects, glucose uptake rates continue to increase if the clamp procedure is prolonged beyond the customary 2 hours. A mathematical model of the EHC, incorporating delays, was fitted to the recorded data, and the insulin resistance behaviour of obese subjects was assessed analytically. Obese subjects had significantly less effective suppression of hepatic glucose output and higher pancreatic insulin secretion than lean subjects. Tissue insulin resistance appeared to be higher in the obese group, but this difference did not reach statistical significance. CONCLUSION: The use of a mathematical model allows a greater amount of information to be recovered from clamp data, making it easier to understand the components of insulin resistance in obese vs. normal subjects.

Blood Glucose↗
PubMed2006

Mathematical modeling of tumor therapy with oncolytic viruses: effects of parametric heterogeneity on cell dynamics.

BACKGROUND: One of the mechanisms that ensure cancer robustness is tumor heterogeneity, and its effects on tumor cells dynamics have to be taken into account when studying cancer progression. There is no unifying theoretical framework in mathematical modeling of carcinogenesis that would account for parametric heterogeneity. RESULTS: Here we formulate a modeling approach that naturally takes stock of inherent cancer cell heterogeneity and illustrate it with a model of interaction between a tumor and an oncolytic virus. We show that several phenomena that are absent in homogeneous models, such as cancer recurrence, tumor dormancy, and others, appear in heterogeneous setting. We also demonstrate that, within the applied modeling framework, to overcome the adverse effect of tumor cell heterogeneity on the outcome of cancer treatment, a heterogeneous population of an oncolytic virus must be used. Heterogeneity in parameters of the model, such as tumor cell susceptibility to virus infection and the ability of an oncolytic virus to infect tumor cells, can lead to complex, irregular evolution of the tumor. Thus, quasi-chaotic behavior of the tumor-virus system can be caused not only by random perturbations but also by the heterogeneity of the tumor and the virus. CONCLUSION: The modeling approach described here reveals the importance of tumor cell and virus heterogeneity for the outcome of cancer therapy. It should be straightforward to apply these techniques to mathematical modeling of other types of anticancer therapy. REVIEWERS: Leonid Hanin (nominated by Arcady Mushegian), Natalia Komarova (nominated by Orly Alter), and David Krakauer.

Journal Article↗
PubMed2006

Mathematical modeling of tumor therapy with oncolytic viruses: regimes with complete tumor elimination within the framework of deterministic models.

BACKGROUND: Oncolytic viruses that specifically target tumor cells are promising anti-cancer therapeutic agents. The interaction between an oncolytic virus and tumor cells is amenable to mathematical modeling using adaptations of techniques employed previously for modeling other types of virus-cell interaction. RESULTS: A complete parametric analysis of dynamic regimes of a conceptual model of anti-tumor virus therapy is presented. The role and limitations of mass-action kinetics are discussed. A functional response, which is a function of the ratio of uninfected to infected tumor cells, is proposed to describe the spread of the virus infection in the tumor. One of the main mathematical features of ratio-dependent models is that the origin is a complicated equilibrium point whose characteristics determine the main properties of the model. It is shown that, in a certain area of parameter values, the trajectories of the model form a family of homoclinics to the origin (so-called elliptic sector). Biologically, this means that both infected and uninfected tumor cells can be eliminated with time, and complete recovery is possible as a result of the virus therapy within the framework of deterministic models. CONCLUSION: Our model, in contrast to the previously published models of oncolytic virus-tumor interaction, exhibits all possible outcomes of oncolytic virus infection, i.e., no effect on the tumor, stabilization or reduction of the tumor load, and complete elimination of the tumor. The parameter values that result in tumor elimination, which is, obviously, the desired outcome, are compatible with some of the available experimental data. REVIEWERS: This article was reviewed by Mikhail Blagosklonny, David Krakauer, Erik Van Nimwegen, and Ned Wingreen. OPEN PEER REVIEW: Reviewed by Mikhail Blagosklonny, David Krakauer, Erik Van Nimwegen, and Ned Wingreen. For the full reviews, please go to the Reviewers' comments section.

Journal Article↗
PubMed2003

Mathematical model predicts clinical ocular motor syndromes.

Clinical ocular motor syndromes were compared with ocular motor syndromes simulated by a mathematical model of the vestibuloocular reflex. The mathematical sensorimotor feedforward model of otolith control of three-dimensional binocular eye position is based on relevant anatomical connections of the vestibuloocular reflex from the utricles to extraocular eye muscles. This is the first attempt to simulate static ocular motor syndromes for unilateral utricular or vestibular nerve failure, lesions of the vestibular nucleus, and lesions of the ascending vestibuloocular reflex pathways. Comparison of the predicted syndromes with those found in patients with unilateral disorders of the vestibular nerve (herpes zoster neuritis), the vestibular nucleus (medullary infarction), and the medial longitudinal fasciculus (pontine infarction) showed good agreement as regards the direction of horizontal, vertical, and torsional eye deviations. The ability of the model to simulate complete or incomplete failures of single elements or entire pathways allows us to pose direct clinical questions about as yet unknown ocular motor syndromes or about the localization of the damage as well as the mechanism involved in syndromes already known.

Aged↗
PubMed2003

Hemostasis and hemodilution: a quantitative mathematical guide for clinical practice.

UNLABELLED: Quantitative changes of hemostasis during hemodilution remain unclear. With the increasing popularity of artificial blood substitutes (ABS), which solely provide oxygen-transport capacity, this issue becomes even more complex. We developed a mathematical model to quantitatively analyze hemostasis during hemodilution and validated it by recalculating patient data. We calculated and compared maximal allowable blood losses (MABL) related to minimal acceptable hematocrit, platelet concentration, and plasma fibrinogen concentration. MABL is the maximal blood loss that can be tolerated without any additional blood products. The variable with the smallest MABL thus limits hemodilution foremost. Hemodilution included isovolemic replacement of blood loss with colloid or acute normovolemic hemodilution (ANH) followed by isovolemic replacement of blood loss with colloid and ABS. We also related our findings to preoperative patient data (n = 204). The decline in platelet concentrations rarely (<2% of all patients) limits hemodilution. By contrast, critical plasma fibrinogen (< or =100 mg/dL) concentrations can often (< or =20% of all patients) limit hemodilution if their initial concentrations are within the lower normal range (<300 mg/dL). These findings become more frequent if ANH is combined with ABS. Under those circumstances ANH blood products are solely required for stabilization of hemostasis, thereby defeating the original purpose of combining ANH with ABS. IMPLICATIONS: The causes of quantitative changes of hemostasis during hemodilution, as well as their clinical effect and relevance, remain unclear. Using a validated, realistic mathematical model, we demonstrate that hemostasis, especially plasma fibrinogen, can limit the extent of hemodilution. This phenomenon is particularly prominent when acute normovolemic hemodilution is combined with artificial blood substitutes.

Blood Loss, Surgical↗
PubMed2003

Validation of an original mathematical model of CO(2) elimination and dead space ventilation.

UNLABELLED: We present an original, mathematical model of ventilation and gas-exchange. Our aim was to validate it using data from previous clinical investigations, allowing our use of it in future investigations. The first previous investigation used a low-dead space, double-lumen, tracheal tube (DLT). We matched the model's PaCO(2) and airway pressures (P(AW)) to the patient mean during use of the DLT and a single-lumen tube (SLT). The model's resulting PaCO(2), PECO(2) and P(AW) were compared with the patients' as tidal volume (VT) changed with constant minute volume. The second investigation examined dead space during anesthesia. The model's VT, respiratory rate, CO(2) production, temperature, and alveolar and anatomical dead spaces were matched to each mechanically ventilated subject. Bias and precision in predictions of PaCO(2) and PECO(2) were calculated. The model's bias in prediction of dead space reduction by the DLT was 6.9%. Bias in prediction of P(AW) was 0.1% (peak) and -5.13% (mean), of PaCO(2) was 1.2% (DLT) and 1.5% (SLT) and of PECO(2) was 1.7% (DLT) and 1.3% (SLT). Prediction of PaCO(2) and PECO(2) in the second investigation (as 95% confidence interval of bias): PaCO(2) -2.6% to 0.8% and PECO(2) -4.9% to 1.2%. This validation allows future application of our model in appropriate theoretical investigations. IMPLICATIONS: We present an original, mathematical model of ventilation and gas exchange. We validate it against previously published clinical data to allow its use in future theoretical investigations where data may be unavailable from patients.

Airway Resistance↗
PubMed2004

Coordination of cell growth and cell division: a mathematical modeling study.

Although there is general agreement that cell growth and division are functionally coordinated, the mechanisms that link the two processes are poorly understood. In this study, we have developed a mathematical model based on current biological concepts of the signaling transduction pathways involved in cell growth, which predicts that cell growth rate is proportional to cell surface area at birth. To investigate the relationship between growth control and cell division, we then applied our mathematical model to three classic experiments measuring cycle time versus cell birth size in fission yeast and Xenopus laevis, and the cell cycle delay in mammalian cells after serum withdrawal. When coupled to a cell cycle exhibiting 'sizer' and 'timer' phases, we show that a simple model in which growth rate is proportional to the cell surface area immediately after division reproduces the experimental observations including the relationship between cycle time and birth size in fission yeast and Xenopus laevis. The model also accounts for the cell cycle delay seen in restriction point experiments performed in HeLa cells.

Animals↗
PubMed1997

Segmentation and numerical analysis of microcalcifications on mammograms using mathematical morphology.

The top-hat and watershed algorithms of mathematical morphology have been applied to detect automatically and segment microcalcifications on mammograms digitized to a pixel resolution of 40 microns using a CCD camera. The database comprised 38 cases from the breast assessment clinic in Liverpool. For all cases, both craniocaudal (CC) and lateral oblique (LO) views were available. 19 cases were proven to be benign and 19 malignant based on cytology and histology. Malignant clusters contained more microcalcifications (14 malignant, 10 benign), occupied a larger area (37 mm2, 9 mm2) and had longer cluster perimeters than benign clusters (33.2 mm, 15.5 mm). Malignant microcalcifications exhibited a wider variety of shapes and were more heterogeneous in terms of image signal intensity than benign microcalcifications. Further mathematical morphology algorithms were applied to describe microcalcification shape in terms of the presence or absence of infoldings, elongation, narrow irregularities and wide irregularities. The three largest microcalcifications were selected for each case and, using a "leave-one-out" approach, each microcalcification was classified in respect of its five nearest neighbours as either malignant or benign. The area under the curve of a receiver operating characteristic (ROC) analysis of the proportion of the three microcalcifications which agreed with the true diagnosis increased from 0.73 (CC) and 0.63 (LO) to 0.79 when both views were considered. Next, each cluster in turn was ranked according to its agreement with the database as a whole over 21 features. An ROC analysis was performed to investigate the effect on sensitivity and specificity of the proportion of the nine nearest neighbours that agreed with the true classification. The largest area under the ROC curve was 0.84 produced by the four features of proportion of irregular microcalcifications, proportion of round microcalcifications, number of microcalcifications in the cluster and the interquartile range of microcalcification area. The shape of microcalcifications is confirmed as being of overriding importance in classifying cases as either malignant or benign. This observation motivates a further study enhanced by using magnified views digitized to a higher resolution by a laser scanner. This will enable the reliable assessment of the shape of a greater number of microcalcifications in each cluster, which is likely to increase further the discriminating power of the image analysis routines and lead to the development of an expert system for automatic mammographic screening.

Aged↗
PubMed1999

Mathematical models for predicting indoor air quality from smoking activity.

Much progress has been made over four decades in developing, testing, and evaluating the performance of mathematical models for predicting pollutant concentrations from smoking in indoor settings. Although largely overlooked by the regulatory community, these models provide regulators and risk assessors with practical tools for quantitatively estimating the exposure level that people receive indoors for a given level of smoking activity. This article reviews the development of the mass balance model and its application to predicting indoor pollutant concentrations from cigarette smoke and derives the time-averaged version of the model from the basic laws of conservation of mass. A simple table is provided of computed respirable particulate concentrations for any indoor location for which the active smoking count, volume, and concentration decay rate (deposition rate combined with air exchange rate) are known. Using the indoor ventilatory air exchange rate causes slightly higher indoor concentrations and therefore errs on the side of protecting health, since it excludes particle deposition effects, whereas using the observed particle decay rate gives a more accurate prediction of indoor concentrations. This table permits easy comparisons of indoor concentrations with air quality guidelines and indoor standards for different combinations of active smoking counts and air exchange rates. The published literature on mathematical models of environmental tobacco smoke also is reviewed and indicates that these models generally give good agreement between predicted concentrations and actual indoor measurements.

Air Pollution, Indoor↗
PubMed2025

Integrating multiscale mathematical modeling and multidimensional data reveals the effects of epigenetic instability on acquired drug resistance in cancer.

Biological and dynamic mechanisms by which Drug-tolerant persister (DTP) cells contribute to the development of acquired drug resistance have not been fully elucidated. Here, by integrating multidimensional data from drug-treated PC9 cells, we developed a novel multiscale mathematical model from an evolutionary perspective that encompasses epigenetic and cellular population dynamics. By coupling stochastic simulation with quantitative analysis, we identified epigenetic instability as the most prominent kinetic feature related to the emergence of DTP cell subpopulations and the effectiveness of intermittent treatment. Moreover, we revealed the optimal schedule for intermittent treatment, including the optimal area for therapeutic time and drug holidays. By leveraging single-cell RNA-seq data characterizing the drug tolerance of lung cancer, we validated the predictions made by our model and further revealed previously unrecognized biological features of DTP cells, such as cell autophagy and migration, as well as new biomarker genes of therapeutic tolerance. Our work not only provides a paradigm for the integration of multiscale mathematical models with newly emerging genomics data but also improves our understanding of the crucial roles of DTP cells and offers guidance for developing new intermittent treatment strategies against acquired drug resistance in cancer.

Drug Resistance, Neoplasm↗
PubMed2005

A mathematical approach for optimizing dendritic cell-based immunotherapy.

Adoptive dendritic cell (DC)-based immunotherapy represents a promising approach to overcome peripheral tolerance against autologous tumor antigens and to maintain protective antitumor immunity. The translation of successful preclinical studies, however, appears to be hampered by new complexities associated with the clinical situation. Mathematical modeling provides the means for qualitative and quantitative analysis, predictions for complex dynamic systems in immunology, and for the design and improvement of therapeutic approaches. We present here a workable computational methodology for developing meaningful data- and hypothesis-driven mathematical models for DC-based immunotherapy with a particular focus on numerical parameter estimation and sensitivity analysis.

CD8-Positive T-Lymphocytes↗
PubMed1999

Mathematical modeling of lipase and protease production by Penicillium restrictum in a batch fermenter.

This work presents a mathematical model that describes time course variations of extracellular lipase and protease activities for the batch fermentation of the fungus Penicillium restrictum, a new and promising strain isolated from soil and wastes of a Brazilian babassu coconut oil industry. The fermentation process was modeled by an unstructured model, which considered the following dependent variables: cells, fat acid, dissolved oxygen concentrations, lipase and protease activities, and cell lysate concentration. The last variable represents the amount of cells that has been lysed by the shear stress and natural cell death. Proteases released to the medium, as consequence of this process, enhance lipase inactivation. The model is able to predict the effects of some operation variables such as air flow rate and agitation speed. The mathematical model was validated against batch-fermentation data obtained under several operating conditions. Because substrate concentration has antagonistic effects on lipase activity, a typical optimization scheme should be developed in order to minimize these deleterious effects while maximizing lipase activity.

Journal Article↗
PubMed2000

Mathematical modeling of controlled-release systems of herbicides using lignins as matrices. A review.

The herbicides applied in soils can be easily lost, owing to leaching, volatilization, and bio- and photodegradation. Controlled-release systems using polymeric matrices claim to solve these problems. The movement of the herbicides in the soil is also an important phenomenon to be studied in order to evaluate the loss processes. The development of mathematical models is a relevant requirement for simulation and optimization of such systems. This study reviews mathematical models as an initial step for modeling data obtained for controlled-release systems of herbicides (diuron, 2,4-dichlorophenoxyacetic acid, and ametryn) using sugarcane bagasse lignin as a polymeric matrix. The release kinetic studies were carried out using several acceptor systems including a water bath, soil, and soil-packed columns. Generally, these models take into account phenomena such as unsteady-state mass transfer by diffusion (Fick's law) and convection, consumption by several processes, and partitioning processes, resulting in partial differential equations with respect to time and space variables.

Delayed-Action Preparations↗
PubMedDate not supplied

[Vector mathematical model for the analysis of complex movements in 3-dimensional space exemplified by the divided symphysis].

For the analysis of complex movements in the divided pubic symphysis of cadavers a vector-mathematical model was developed. Overall movements is analysed in components, as translations in the three dimensions of space and rotations around the three axes of space, as error-free and as simply as possible by measuring the distances to different defined points. Experimental conditions, the development of a vector-mathematical system of analysis, error analysis and interpretation of results are shown. With this method it also possible to analyse similar problems in the same way.

Biomechanical Phenomena↗