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PubMed2004

Mathematical modeling of complex regulatory networks.

Cellular regulation comprises overwhelmingly complex interactions between genes and proteins that ultimately will only be rendered understandable by employing formal approaches. Developing large-scale mathematical models of such systems in an efficient and reliable way, however, requires careful evaluation of structuring principles for the models, of the description of the system dynamics, and of the experimental data basis for adjusting the models to reality. We discuss these three aspects of model development using the example of cell cycle regulation in yeast and suggest that capturing complex dynamic networks is feasible despite incomplete (quantitative) biological knowledge.

Cell Cycle↗

PubMed1999

Withdrawing low risk women from cervical screening programmes: mathematical modelling study.

OBJECTIVE: To evaluate the impact of policies for removing women before the recommended age of 64 from screening programmes for cervical cancer in the United Kingdom. DESIGN: A mathematical model of the clinical course of precancerous lesions which accounts for the influence of infection with the human papillomavirus, the effects of screening on the progression of disease, and the accuracy of the testing procedures. Two policies are compared: one in which women are withdrawn from the programme if their current smear is negative and they have a recent history of regular, negative results and one in which women are withdrawn if their current smear test is negative and a simultaneous test is negative for exposure to high risk types of human papillomavirus. SETTING: United Kingdom cervical screening programme. MAIN OUTCOME MEASURES: The incidence of invasive cervical cancer and the use of resources. RESULTS: Early withdrawal of selected women from the programme is predicted to give rise to resource savings of up to 25% for smear tests and 18% for colposcopies when withdrawal occurs from age 50, the youngest age considered in the study. An increase in the incidence of invasive cervical cancer, by up to 2 cases/100 000 women each year is predicted. Testing for human papillomavirus infection to determine which women should be withdrawn from the programme makes little difference to outcome. CONCLUSIONS: This model systematically analyses the consequences of screening options using available data and the clinical course of precancerous lesions. If further audit studies confirm the model's forecasts, a policy of early withdrawal might be considered. This would be likely to release substantial resources which could be channelled into other aspects of health care or may be more effectively used within the cervical screening programme to counteract the possible increase in cancer incidence that early withdrawal might bring.

Adult↗

PubMed1998

Interaction among autoregulation, CO2 reactivity, and intracranial pressure: a mathematical model.

The relationships among cerebral blood flow, cerebral blood volume, intracranial pressure (ICP), and the action of cerebrovascular regulatory mechanisms (autoregulation and CO2 reactivity) were investigated by means of a mathematical model. The model incorporates the cerebrospinal fluid (CSF) circulation, the intracranial pressure-volume relationship, and cerebral hemodynamics. The latter is based on the following main assumptions: the middle cerebral arteries behave passively following transmural pressure changes; the pial arterial circulation includes two segments (large and small pial arteries) subject to different autoregulation mechanisms; and the venous cerebrovascular bed behaves as a Starling resistor. A new aspect of the model exists in the description of CO2 reactivity in the pial arterial circulation and in the analysis of its nonlinear interaction with autoregulation. Simulation results, obtained at constant ICP using various combinations of mean arterial pressure and CO2 pressure, substantially support data on cerebral blood flow and velocity reported in the physiological literature concerning both the separate effects of CO2 and autoregulation and their nonlinear interaction. Simulations performed in dynamic conditions with varying ICP underline the existence of a significant correlation between ICP dynamics and cerebral hemodynamics in response to CO2 changes. This correlation may significantly increase in pathological subjects with poor intracranial compliance and reduced CSF outflow. In perspective, the model can be used to study ICP and blood velocity time patterns in neurosurgical patients in order to gain a deeper insight into the pathophysiological mechanisms leading to intracranial hypertension and secondary brain damage.

Animals↗

PubMed1975

Drug interactions: the effects of alcohol and meprobamate applied singly and jointly in human subjects. III. The concentrations of alcohol and meprobamate in the blood and their effects on performance; application of mathematical models.

The relations between the levels of alcohol and meprobamate in the blood and performance on a visual-motor coordination tracking task were analyzed by a general system of mathematical models, using data from Experiment V by Carpenter et al. [J. Stud. Alc., Suppl. No. 7, pp. 54-139, 1975]. The derivation of the models is described. In general, the relationship between blood alcohol concentration (BAC) and performance was nonmonotonic: best performance occurred at BACS of 10 to 20 mg per 100 ml. The relationship between meprobamate concentration (BMC) and performance was monotonic: performance deteriorated with increasing BMC. The results of the reaction latency measure, howevr, showed no consistent relationship with BAC or BMC. The action of alcohol can be represented by a model which involves 2 distinct sites of action; that of meprobamate, 1 site. It could not be determined whether the site of action of meprobamate is distinct from those of alcohol because the blood levels of the drugs were not high enough. The implications of the results are discussed, with particular reference to the quantitative description of the joint action of drugs and the design of future experiments.

Drug Combinations↗

PubMedDate not supplied

[Mathematical model for prognostication of work capacity recovery in patients with myocardial infarct].

The impact of clinical and instrumental findings on prognosis was evaluated in patients with myocardial infarction and a mathematical model was proposed to predict their recovered working ability. The developed predictive index for prediction of recovered working ability is of high informative value and accuracy. It may be practically used by a cardiologist to choose a protocol of medical and rehabilitative measures in myocardial infarction.

Data Interpretation, Statistical↗

PubMed2006

A mathematical model of cell reorientation in response to substrate stretching.

It is well documented that in response to substrate stretching adhering cells alter their orientation. Generally, the cells reorient away from the direction of the maximum substrate strain, depending upon the magnitude of the substrate strain and the state of cell contractility. Theoretical models from the literature can describe only some aspects of this phenomenon. In the present study, we developed a more comprehensive mathematical model of cell reorientation than the current models. Using the framework of theory of non-linear elasticity, we found that the problem of cell reorientation was a stability problem, with the global (Maxwell's) criterion for stability. For the case of uniaxial substrate stretching, we showed that cells would orient away from the direction of substrate strain such that the angle between the cell long axis and the direction of the substrate strain would increase with increasing magnitude of the strain. We also showed that at a given substrate strain this angle would be greater in cells having greater contractile strain. These results are consistent with experimental observations reported in the literature.

Animals↗

PubMed2002

A mathematical model to detect inspiratory flow limitation during sleep.

The physiological significance of inspiratory flow limitation (IFL) has recently been recognized, but methods of detecting IFL can be subjective. We sought to develop a mathematical model of the upper airway pressure-flow relationship that would objectively detect flow limitation. We present a theoretical discussion that predicts that a polynomial function [F(P) = AP(3) + BP(2) + CP + D, where F(P) is flow and P is supraglottic pressure] best characterizes the pressure-flow relationship and allows for the objective detection of IFL. In protocol 1, step 1, we performed curve-fitting of the pressure-flow relationship of 20 breaths to 5 mathematical functions and found that highest correlation coefficients (R(2)) for quadratic (0.88 +/- 0.10) and polynomial (0.91 +/- 0.05; P < 0.05 for both compared with the other functions) functions. In step 2, we performed error-fit calculations on 50 breaths by comparing the quadratic and polynomial functions and found that the error fit was lowest for the polynomial function (3.3 +/- 0.06 vs. 21.1 +/- 19.0%; P < 0.001). In protocol 2, we performed sensitivity/specificity analysis on two sets of breaths (50 and 544 breaths) by comparing the mathematical determination of IFL to manual determination. Mathematical determination of IFL had high sensitivity and specificity and a positive predictive value (>99% for each). We conclude that a polynomial function can be used to predict the relationship between pressure and flow in the upper airway and objectively determine the presence of IFL.

Humans↗

PubMed1989

Mathematical models of the flow in the basilar artery.

The flow in the basilar artery arises from the merging of the flows from the two vertebral arteries. This study deals with the question whether a parabolic (Poiseuille) profile will have been established before the basilar artery divides into both posterior cerebral arteries. The inlet length (that is, the downstream distance needed for the flow to become approximately equal to the limiting Poiseuille flow) and velocity profiles have been computed from two- and three-dimensional mathematical models in which flow pulsatility and vessel wall distensibility have been neglected and the complex geometry of the junction has been taken into account in a simplified form. The results show that the flow at the end of the basilar artery is far from being parabolic and that an asymmetry in the entrance flow will be carried along towards the end of the basilar artery, thus affecting flows in the circle of Willis.

Basilar Artery↗

PubMed1990

Quantification of neonatal cerebral ventricular volume by real-time ultrasonography. Derivation and in vitro confirmation of a mathematical model.

The derivation and in vitro confirmation of a method for the calculation of cerebral ventricular volume determined by cranial ultrasonography is described. The derivation depends on use of the cylindrical coordinate method. Confirmation was performed by construction of ultrasound phantoms from latex balloons embedded in a talc/gelatin matrix. Seventeen phantoms of volumes ranging from 10 to 60 mL were measured. The known volume was then compared to the calculated volume by linear regression analysis. The regression analysis indicated a close correlation between the actual volume of the phantoms and the calculated volume with r = 0.96. Various sources of error in the method are discussed. Our results indicate that the cylindrical coordinate mathematical model when applied to cranial ultrasonography images accurately measures cerebral ventricular volumes.

Analysis of Variance↗

PubMed2001

Parasitoids may determine plant fitness--a mathematical model based on experimental data.

The present paper deals with the problem of enhancement of plant fitness due to parasitization of herbivores. The experimental evidence for such situations is reviewed. Two mathematical models, plant-herbivore (two trophic) and plant-herbivore-parasitoid (three trophic) are considered to analyse the experimental observations. The effect of environmental fluctuation in the tritrophic system is also observed and optimum values of the inaccessible parameters involved in the system are estimated for purposes of biological control.

Animals↗

PubMed2004

Femoral loads during passive, active, and active-resistive stance after spinal cord injury: a mathematical model.

OBJECTIVE: The purpose of this study was to estimate the loading environment for the distal femur during a novel standing exercise paradigm for people with spinal cord injury. DESIGN: A mathematical model based on experimentally derived parameters. BACKGROUND: Musculoskeletal deterioration is common after spinal cord injury, often resulting in osteoporotic bone and increased risk of lower extremity fracture. Potential mechanical treatments have yet to be shown to be efficacious; however, no previous attempts have been made to quantify the lower extremity loading during passive, active, and active-resistive stance. METHODS: A static, 2-D model was developed to estimate the external forces; the activated quadriceps forces; and the overall bone compression and shear forces in the distal femur during passive (total support of frame), active (quadriceps activated minimally), and active-resistive (quadriceps activated against a resistance) stance. RESULTS: Passive, active, and active-resistive stance resulted in maximal distal femur compression estimates of approximately 45%, approximately 75%, and approximately 240% of body weight, respectively. Quadriceps force estimates peaked at 190% of body weight with active-resistive stance. The distal femur shear force estimates never exceeded 24% of body weight with any form of stance. CONCLUSIONS: These results support our hypothesis that active-resistive stance induces the highest lower extremity loads of the three stance paradigms, while keeping shear to a minimum. RELEVANCE: This model allows clinicians to better understand the lower extremity forces resulting from passive, active, and active-resistive stance in individuals with spinal cord injury.

Ankle Joint↗

PubMed1999

Studies of antibiotic resistance within the patient, hospitals and the community using simple mathematical models.

The emergence of antibiotic resistance in a wide variety of important pathogens of humans presents a worldwide threat to public health. This paper describes recent work on the use of mathematical models of the emergence and spread of resistance bacteria, on scales ranging from within the patient, in hospitals and within communities of people. Model development starts within the treated patient, and pharmacokinetic and pharmacodynamic principles are melded within a framework that mirrors the interaction between bacterial population growth, drug treatment and the immunological responses targeted at the pathogen. The model helps identify areas in which more precise information is needed, particularly in the context of how drugs influence pathogen birth and death rates (pharmacodynamics). The next area addressed is the spread of multiply drug-resistant bacteria in hospital settings. Models of the transmission dynamics of the pathogen provide a framework for assessing the relative merits of different forms of intervention, and provide criteria for control or eradication. The model is applied to the spread of vancomycin-resistant enterococci in an intensive care setting. This model framework is generalized to consider the spread of resistant organisms between hospitals. The model framework allows for heterogeneity in hospital size and highlights the importance of large hospitals in the maintenance of resistant organisms within a defined country. The spread of methicillin resistant Staphylococcus aureus (MRSA) in England and Wales provides a template for model construction and analysis. The final section addresses the emergence and spread of resistant organisms in communities of people and the dependence on the intensity of selection as measured by the volume or rate of drug use. Model output is fitted to data for Finland and Iceland and conclusions drawn concerning the key factors determining the rate of spread and decay once drug pressure is relaxed.

Anti-Bacterial Agents↗

PubMed2005

Substrate size as a determinant of fibrillatory activity maintenance in a mathematical model of canine atrium.

Tissue size has been considered an important determinant of atrial fibrillation (AF), but recent work has questioned the critical size hypothesis. Here, we use a previously developed mathematical model of the two-dimensional canine atrium with realistic action potential, ionic, and conduction properties to address substrate size effects on the maintenance of fibrillatory activity. Cholinergic AF was simulated at different acetylcholine (ACh) concentrations ([ACh]) and distributions, with substrate area varied 11.1-fold. Automated phase singularity detection was used to facilitate the analysis of arrhythmic activity. The duration of activity induced by a single extrastimulus increased with increasing substrate dimensions. Two general mechanisms underlying activity were observed and were differentially affected by substrate size. For large mean [ACh], single primary rotors anchored in low-[ACh] zones maintained activity and substrate dimensions were not critical. At lower mean [ACh], extensive spiral wave meander prevented the emergence of single stable rotors. Prolonged activity was favored when substrate size permitted a sufficiently large number of simultaneous longer-lasting rotors that extinction of all was unlikely. Thus either single dominant rotor or multiple reentrant spiral generator mechanisms could maintain fibrillatory activity in this model and were differentially dependent on substrate size. These results speak to recent debates about the role in AF of single driver rotors versus multiple reentrant circuit mechanisms by suggesting that either may maintain fibrillatory atrial activity depending on atrial size and electrophysiological properties.

Action Potentials↗

PubMed2004

[Significance of the mathematical model of cardiac electrical field for the interpretation of experimental data].

BACKGROUND: In concurrence of our recent findings of the elevation of QT dispersion (QTd) in the group of pregnant women, mathematical approaches were developed aimed to give possible geometrical explanation whether the observed changes result from the rotation or from the changed position of the heart. METHODS AND RESULTS: Mathematical model of the cardiac electrical field approximated as a time variable dipole in a homogenous spatial conductor was developed. From the experimental vectocardiographic records, representing time course of the cardiac dipole, body surface potential maps were calculated on the basis of the model. To validate the adequacy of the model, the reconstructed electrocardiograms were compared with the empiric data. To determine the effects of rotation, original empiric VCG data of the control group were transformed accordingly the hypothetic pregnancy related changes. Calculated surface electrocardiograms were then compared with empiric cardiograms of the pregnant women. CONCLUSIONS: Based on the results, several conclusions can be drawn: 1) QT dispersion is associated also with the geometrical relations between the direction of cardiac vector during the terminal phase of repolarization and the direction of axes in the given system of leads. The dispersion then has its typical occurrence at the thoracic surface--minimums of the QT duration are found in the plane perpendicular to the axis of the terminal vector lead. 2) When the duration of repolarization is estimated from the classic thoracic leads within the phisiological variations of terminal-depolarization vector orientations, can exist that in some cases the minimum of QT interval is and in others it is not recorded by the lead system. Value of QT dispersion between these two extremes will be significantly different. 3) In case of the horizontal declination of the heart, the ECG signal in most of the leads of the body surface mapping has a higher voltage than in case of vertical declination due to a smaller angle between axes of the terminal vector and most of the leads. Such fact will contribute to more accurate reading of the T wave end and to the estimation of QT interval, usually with smaller value of QTd. 4) The change of the cardiac electrical field corresponding to the changed position of the heart (rotation) does not result by itself in QTd changes, if it is evaluated from the records from the whole thorax. Obversely, horizontalization of the heart contributes more to the evaluation of lower QTd values, as it is given above. 5) More then the result of geometrical changes, QT dispersion found in the group women in high level of pregnancy is an effect of changes in the T loop morphology, which was observed in this group. Another possible explanation of the observed dispersion is the non-dipolar character of the electrical field changes during pregnancy.

Adult↗

PubMed1998

The humoral immune response to Haemophilus influenzae type b: a mathematical model based on T-zone and germinal center B-cell dynamics.

Through careful mapping of the physiology of the T-zone and GC B-blast dynamics to a mathematical representation of the cell processes including proliferation, migration, differentiation, and cell death, a mathematical model is constructed to capture the dominant nominal primary, late follicular, and secondary humoral response to Haemophilus influenzae Type b. This model explicitly incorporates the dynamics of memory B-cells, T-zone and GC B-dynamics, IgM and IgG antibodies, avidity maturation, and IC presentation by FDCs into a coherent framework. This paper describes the relevant immunology, the pertinent physiological assumptions, the developed model, and the parameter identification procedure. The model parameters were found using a parameter identification procedure that capitalizes on the timing and interactions of certain dominant physiological attributes. Simulation results and validation tests indicate that the model reflects not only a nominal primary and secondary humoral immune response but also the tertiary and T-independent responses. The model shows robustness to variations in infection dosage, bacterial growth rate (virulence of the strain), and onset-timing of the secondary response. The utility of this model in studying the humoral immune response is demonstrated through suggested physiological assumptions, mechanisms, and rates to be eventually clinically evaluated as well as insights into vaccination design. The model and parameter identification techniques are easily adapted to other diseases which primarily evoke a humoral immune response.

Antibodies, Bacterial↗

PubMed2005

The bisexual branching process with population-size dependent mating as a mathematical model to describe phenomena concerning to inhabit or re-inhabit environments with animal species.

We consider the bisexual Galton-Watson branching process with population-size dependent mating as a mathematical model adequate for the description of some natural phenomena. More specifically we are interested in studying some questions about the problem of populating an environmental with new animal species or re-populating it with species which have previously disappeared.

Algorithms↗

PubMed2002

Mathematical modeling of laser ablation in liquids with application to laser ultrasonics.

The use of laser ablation as a means of generating ultrasonic waves in liquid metals is studied in this paper. A mathematical model for predicting the onset of ablation is developed, as is a model of the ablation process based on steady state, one-dimensional gas dynamics in which the vapor phase is treated as an ideal gas. The results of this model are then used in a quasi-two-dimensional model of laser ablation that accounts for the spatial distribution of intensity in the laser beam. Model predictions are compared with experiments conducted on liquid mercury and excellent agreement is obtained. Based on these results, a simplified model is developed that shows excellent agreement with both the theory and the experiments.

Journal Article↗

PubMed2002

The Plasmodium falciparum var gene switching rate, switching mechanism and patterns of parasite recrudescence described by mathematical modelling.

Recrudescing Plasmodium falciparum parasitaemia is attributed to the switching of PfEMP1, a variant antigen family encoded by the var gene repertoire, and the host's immune response. We have developed a mathematical model which incorporates var gene switching, and variant specific, non-variant specific and non-specific immunity. By conducting a sensitivity analysis of the model we have defined the parameter limits which produce chronic and recrudescing infections. We explore 3 switching mechanisms: ordered, random and uncoupled switching. We show that if var genes switch on and off independently at variable rates through the repertoire a chronic clinical infection is predicted. The fastest switching-on rate that produces a chronic infection is 0.03% per generation. The model predicts that non-variant specific immunity plays an important role in reducing disease severity. This work illustrates the complex relationship between the malaria parasite and its host and shows that var gene switching at rates substantially slower than 2% are essential for parasite survival.

Animals↗