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PubMed2001

Role of tissue hypoxia in cerebrovascular regulation: a mathematical modeling study.

This paper presents a mathematical model of cerebrovascular regulation, in which emphasis is given to the role of tissue hypoxia on cerebral blood flow (CBF). In the model. three different mechanisms are assumed to work on smooth muscle tension at the level of large and small pial arteries: CO2 reactivity, tissue hypoxia, and a third mechanism necessary to provide good reproduction of autoregulation to cerebral perfusion pressure (CPP) changes. Using a single set of parameters for the mechanism gains, assigned via a best fitting procedure, the model is able to reproduce the pattern of pial artery caliber and CBF under a large variety of physiological stimuli, either acting separately (hypoxia, CPP changes, CO2 pressure changes) or combination (hypercapnia+hypoxia; hypercapnia+hypotension). Furthermore, the model can explain the increase in CBF and the vasoconstriction of small pial arteries observed experimentally during hemodilution, ascribing it to the decrease in blood viscosity and to the antagonistic action of the flow-dependent mechanism (responsible for vasoconstriction) and of hypoxia (responsible for vasodilation). Finally, the interaction between hypoxia and intracranial pressure (ICP) has been analyzed. This interaction turns out quite complex, leading to different ICP time patterns depending on the status of the cerebrospinal fluid outflow pathways and of intracranial compliance.

Animals↗
PubMed1988

A minimal mathematical model of human periodic breathing.

Numerous mathematical models of periodic breathing (PB) currently exist. These models suggest mechanisms that may underlie many known causes of PB. However, each model that has been shown to simulate PB under reasonable conditions contains greater than 15 physiological parameters. Because some parameters exhibit a wide range of values in a population, such simulations cannot test a model's ability to account for the breathing patterns of individuals. Furthermore it is impractical to perform a direct experimental validation study that would require the estimation of each of 15 or more parameters for each subject. A minimal model of PB is presented that is suitable for direct validation. Analytic expressions are given that define the conditions for PB in terms of the following: 1) CO2 sensitivity, 2) Cardiac output, 3) Mixed venous CO2, 4) Circulation time, and 5) Mean lung volume for CO2. This model is shown to be consistent with previous models and experimental data regarding the degree of hypoxia or congestive heart failure required to produce PB. A quantitative measure of relative stability is defined as a metric of comparison to the human studies described in the accompanying paper (J. Appl. Physiol. 65: 1389-1399, 1988).

Blood Circulation Time↗
PubMed1992

Evaluation of a mathematical model of lactating sow metabolism.

A mathematical model of lactating sow metabolism was evaluated using three types of tests. First, 16 experimental treatments from four experiments reported in the literature were simulated with the model, and the simulated values for change in BW and protein and fat content were compared to reported or calculated values. Second, the model's response to level of feed intake, level of milk production, BW and composition at farrowing, and dietary lysine concentration was compared to expected responses. Third, the model's sensitivity to changes in several of its kinetic parameters was measured. There was good agreement between simulated and measured values for BW and fat loss and reasonable agreement for body protein loss. All responses to changes in external conditions were in expected directions and biologically reasonable. The model seemed rather robust with respect to changes in the kinetic parameters considered, although important changes in simulated values were found in some cases. Overall, the model seems sound. It can be useful in evaluation of feeding programs and in understanding biological relationships.

Adipose Tissue↗
PubMed1999

Mathematical models for peritoneal transport characteristics.

Four mathematical models and for the description of peritoneal transport of fluid solutes are reviewed. The membrane model is usually applied for (1) separation of transport components, (2) formulation of the relationship between flow components and their driving forces, and (3) estimation of transport parameters. The three-pore model provides correct relationships between various transport parameters and demonstrates that the peritoneal membrane should be considered heteroporous. The extended three-pore model discriminates between heteroporous capillary wall and tissue layer, which are assumed to be arranged in series; the model improves and modifies the results of the three-pore model. The distributed model includes all parameters involved in peritoneal transport and takes into account the real structure of the tissue with capillaries distributed at various distances from the surface of the tissue. How the distributed model may be applied for the evaluation of the possible impact of perfusion rate on peritoneal transport, as recently discussed for clinical and experimental studies, is demonstrated. The distributed model should provide theoretical bases for the application of other models as approximate and simplified descriptions of peritoneal transport. However, an unsolved problem is the theoretical description of bi-directional fluid transport, which includes ultrafiltration to the peritoneal cavity owing to the osmotic pressure of dialysis fluid and absorption out of the peritoneal cavity owing to hydrostatic pressure.

Biological Transport↗
PubMed2000

Review and recent progress: the mathematical modeling of mixed species Plasmodium infections.

Mathematical modeling serves numerous uses in biology, notably in the exploration of phenomena that are difficult to observe empirically. Here, I review recent progress in modeling the blood-stage dynamics of mixed-species Plasmodium infections, namely P. malariae-P. falciparum and P. vivax- P. falciparum mixed infections. Modeling reproduces features of such infections found in nature including the asymmetry of parasite blood-stage densities, inter-specific suppression, and parasite asexual-form recrudescence following long-standing sub-patency. Several findings which merit clinical attention are presented: the ability of P. malariae and P. vivax to reduce the peak parasitemia of co-infecting P. falciparum, and the potential recrudescence of a low-level P. falciparum infection following a P. malariae infection or P. vivax infection or relapse. The action of antimalarial drugs is discussed, highlighting some potential complications in treating mixed-species malaria infections. Most notably, if a mixed-species infection is misdiagnosed as a single-species P. vivax infection, treatment can lead to the dangerous appearance of "hidden" P. falciparum.

Animals↗
PubMedDate not supplied

[Evolution of rotifer population dynamics in a heterogeneous habitat: mathematical modeling].

We present the results of mathematical modeling of a rotifer species inhabiting two coupled habitats with different environmental conditions. We use the modified Consensus model and show that the exchange between the habitats can lead to chaotization of originally regular plankton dynamics and synchronization of plankton biomass oscillations. As a result, the invasion of a chaotic regime takes place.

Animals↗
PubMedDate not supplied

Mathematical modeling of erythropoietin therapy.

A simple mathematical model to describe hemoglobin (Hb) concentration response to recombinant human erythropoietin (EPO) therapy is proposed. The model is based on the assumption that Hb production increases linearly with EPO dose level. The resulting equation contains two patient parameters: 1) S, the proportionality constant between g Hb generated/L blood/wk and IU EPO administered/kg body weight/wk; and 2) tau, the patient erythrocyte lifetime in weeks. The model was applied retrospectively to 67 patients from the Canadian Erythropoietin Study, yielding an average error of 5.5 g/L between 27 measured and predicted Hb value pairs over the 27 week study. The model parameters, S (mean +/- SD = 0.015 +/- 0.005) and tau (14.0 +/- 4.1), varied over an order of magnitude. The model was also used to predict the EPO dose required to reach a target Hb of 110 g/L; the EPO requirements varied from 55 to 742 IU EPO/kg/wk (mean +/- SD = 225 +/- 124). It is recommended, based upon the model results, that EPO therapy be initiated at 3 IU EPO/kg/wk for each g/L difference between target and baseline Hb, with subsequent EPO dose adjustment guided by patient modeling.

Dose-Response Relationship, Drug↗
PubMedDate not supplied

[Long-term potentiation and depression of inhibitory transmission studied by using mathematical modelling of the postsynaptic processes].

The mathematical model of calcium-dependent posttetanic processes in a dendritic spine of a CA3 hippocampal pyramidal neuron which received excitatory and inhibitory afferents was used for studying the LTP and LTD of inhibitory transmission. It has been demonstrated that the inhibitory synaptic efficacy is defined by GABAa and GABAb dephosphorylation which, in turn, is determined by the Ca(2+)-dependent ratio between the active protein kinases and protein phosphatases. Posttetanic decrease/increase in intracellular Ca2+ concentration (Ca2+p) in respect of pretetanic Ca2+ level results in an increase/decrease in number of dephosphorylated GABA receptors and in the LTP/LTD of the efficacy of inhibitory transmission. The extent of modification depends on the ratio between the concentrations of excitatory and inhibitory transmitters in a synaptic cleft. The extent of inhibitory transmission modification is negligible if GABA concentration is very low or high.

Animals↗
PubMed2006

A mathematical model for adaptive transport network in path finding by true slime mold.

We describe here a mathematical model of the adaptive dynamics of a transport network of the true slime mold Physarum polycephalum, an amoeboid organism that exhibits path-finding behavior in a maze. This organism possesses a network of tubular elements, by means of which nutrients and signals circulate through the plasmodium. When the organism is put in a maze, the network changes its shape to connect two exits by the shortest path. This process of path-finding is attributed to an underlying physiological mechanism: a tube thickens as the flux through it increases. The experimental evidence for this is, however, only qualitative. We constructed a mathematical model of the general form of the tube dynamics. Our model contains a key parameter corresponding to the extent of the feedback regulation between the thickness of a tube and the flux through it. We demonstrate the dependence of the behavior of the model on this parameter.

Adaptation, Physiological↗
PubMed1980

Mathematical model of synaptic plasticity: III. Heterosynaptic changes.

A mathematical model, using differential equations, of heterosynaptic plasticity is proposed. The model is based on physiological studies of invertebrates in which nonspecific conditioning, such as sensitization and heterosynaptic inhibition, starts to be elucidated and behavioral studies of classical and instrumental conditioning, which we postulate to have the same mechanisms as those found in nonspecific conditioning. The model permits us to simulate the following heterosynaptic changes: sensitization, heterosynaptic inhibition, classical and instrumental conditioning--including short- and long-term memory--extinction and recuperation--spontaneously and by stimulation.

Animals↗
PubMed1998

Population dynamics of tuberculosis treatment: mathematical models of the roles of non-compliance and bacterial heterogeneity in the evolution of drug resistance.

SETTING: Patient non-compliance and/or spatial heterogeneity in drug concentration or effectiveness may contribute to the emergence of drug resistance during multiple-drug chemotherapy of tuberculosis. OBJECTIVE: Using mathematical models of mycobacterial population dynamics under antimicrobial treatment, to assess the effects of non-compliance, heterogeneity and other factors on the success of treatment. DESIGN: A mathematical model is used to generate predictions about the ascent of drug resistance in treated hosts with non-compliance and/or a 'protected compartment' of bacteria where only one drug is active; simulations of a more realistic version of this model take into account random mutation, and different assumptions about the size of, and growth rate of bacteria in, the protected compartment. RESULTS: The existence of a protected compartment can increase the likelihood of resistance to the single drug active in that compartment, but only if bacteria resistant to that drug can grow in the protected compartment or if the host is non-adherent to the treatment regimen. However, the protected compartment may also slow the ascent of bacteria resistant to drugs not active in it (e.g. isoniazid) by providing a reservoir of non-selected mycobacteria. The model predicts that relative rates of killing are more important than mutation rates in determining the order in which resistant mutants ascend. Model predictions, in combination with data about drug resistance patterns, suggest that non-compliance, but not heterogeneity, is an important cause of treatment failure. CONCLUSION: Patterns of acquired drug resistance may be used to infer processes of selection during treatment; mathematical models can aid in generating predictions about the relative impacts of treatment parameters in the evolution of resistance, and eventually in suggesting improved treatment protocols.

Antitubercular Agents↗
PubMed2005

Mathematical modeling of cancer radiovirotherapy.

Cancer virotherapy represents a dynamical system that requires mathematical modeling for complete understanding of the outcomes. The combination of virotherapy with radiation (radiovirotherapy) has been recently shown to successfully eliminate tumors when virotherapy alone failed. However, it introduces a new level of complexity. We have developed a mathematical model, based on population dynamics, that captures the essential elements of radiovirotherapy. The existence of corresponding equilibrium points related to complete cure, partial cure, and therapy failure is proved and discussed. The parameters of the model were estimated by fitting to experimental data. By using simulations we analyzed the influence of parameters that describe the interaction between virus and tumor cell on the outcome of the therapy. Furthermore, we evaluated relevant therapeutic scenarios for radiovirotherapy, and offered elements for optimization.

Algorithms↗
PubMed1978

A mathematical model of air and water caloric nystagmus.

A mathematical model is proposed to explain the induction of nystagmic eye movements in response to thermal stimulation of the ear by air and water. Laplace-transformed equations are set up to describe heat flow in the meatus lumen to the ear-drum and heat transmission into meatus wall. Heat transport to the lateral semicircular canal, resulting in convective endolymph flow, and the induction of reflectory eye movements are included in the mathematical description. Input of the model is the time-course of temperature at the irrigating tip, output is the time-course of eye position (in correspondence to experimental nystagmogramms). The predicted nystagmus is in good agreement with experimental results, thus supporting our assumptions on the thermal effects of air and water irrigations.

Air↗
PubMed1971

A mathematical model of the human menstrual cycle.

A mathematical model for the hormonal interactions of the human menstrual cycle is presented. The feedback effects of estrogen on the release of follicle-stimulating hormone (FSH) and luteinizing hormone (LH) are considered, including a mechanism describing the midcycle LH peak. Computer simulation with this model yields results which are periodic and in good agreement with physiological data.

Computers↗
PubMedDate not supplied

[Mathematical model of light diffraction on red blood cells].

Mathematical model diffraction of laser radiation on red blood smear for population erythrocytes with their normal distribution according to sizes is presented. It was shown during normal distribution of erythrocytes according to sizes that the first maximal pattern diffraction case does not correspond to the medium diameters, as it is given in literature. It was established that the depth of the first maximum is changed depending on the dispersion value according to sizes.

Erythrocytes↗
PubMed2006

Diffusivity and distribution of vinblastine in three-dimensional tumour tissue: experimental and mathematical modelling.

The distribution of chemotherapeutics in solid tumours is poorly understood and the contribution it makes to treatment failure is unknown. Novel approaches are required to understand how the three-dimensional organisation of cancer cells in solid tumours affects drug availability. Since convective drug transport is limited by increased interstitial pressure in poorly vascularised cancers, the aim of this study was to measure the diffusive hindrance exerted by solid tumour tissue. Multicell layer tumour models comprising DLD1 colon cancer cells were characterised and fluxes were determined for [3H]-vinblastine and [14C]-sucrose. The mathematical models provided the diffusion coefficients for both compounds and predicted higher exposure of cells in the vicinity of vessels. The diffusion of vinblastine was three times slower than that of sucrose. Although slow diffusion delays vinblastine penetration into the avascular regions of tumours, the proliferating cells are generally in the marginal area of tumours. The mathematical model that we have developed enabled accurate quantification of drug pharmacokinetic behaviour, in particular, the diffusivity of vinblastine within solid tissue. This mathematical model may be adapted readily to incorporate the influence of factors mediating pharmacokinetic drug resistance.

ATP Binding Cassette Transporter, Subfamily B, Mem↗
PubMed1996

Nurses' health study: log-incidence mathematical model of breast cancer incidence.

BACKGROUND: In 1983, Pike et al. developed a mathematical model to quantify the effects of reproductive risk factors on the incidence of breast cancer. In 1994, we modified that model to correct some deficiencies in the original model, including a lack of terms for spacing of births and an inability to easily accommodate births after age 40 years. Our extended Pike model, while improving on the original, still has serious disadvantages, such as difficulty in translating model parameters into relative risks (RRs) and an incomplete fit to data that slightly overestimated incidence for premenopausal women with an early age at first birth and that underestimated incidence for post-menopausal women with a late age at first birth. PURPOSE: We undertook both the development of a new mathematical model to quantify the effects of reproductive risk factors on breast cancer incidence and validation of the model. METHODS: A new log-incidence model of breast cancer incidence was developed using nonlinear regression methods, and a study population consisting of 89,132 women in the Nurses' Health Study from which a total of 2249 incident cases of breast cancer were identified. Subjects were followed from the return of the 1976 Nurses' Health Study questionnaire until June 1, 1990, or until the last questionnaire was returned, until the development of any cancer, or until death, yielding 1,148,593 person-years of follow-up. The log-incidence models were fitted using iteratively reweighted least squares analysis. RESULTS: The log-incidence model provided a better fit to that data than the extended Pike model, with parameter estimates interpretable in terms of RRs. This new model can be fitted using standard commercially available statistical software. In the model, younger parous women are generally at slightly higher risk than nulliparous women, which is true for both the observed and expected RRs, and older parous women, aged 55-64 years with an early age at first birth, are at lower risk than nulliparous women,while older women with a late age at first birth are at substantially higher risk than nulliparous women. CONCLUSION: Log-incidence models, such as this one, provide an efficient framework for modeling the effect of lifestyle risk factors on breast cancer incidence that may be specifically targeted to certain time periods of a woman's reproductive life.

Adult↗