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PubMed1999

Mathematical modeling of the retention and clearance of low-toxicity particles in the lung.

A mathematical model has been formulated to describe the mechanisms that determine the retention or clearance of insoluble inhaled particles in the rat lung. The hypotheses underlying the model are described-for example, the phagocytosis of free particles by macrophages, the transport of particles in macrophages from the alveolar region, the effect of the life cycle of macrophages leading to the eventual release of phagocytosed particles, the effect of lung burden on the macrophage activity, the transport of particles into the interstitium, the role of interstitial macrophages, the formation of granulomata, and transport of interstitialized particles to the thoracic lymph nodes. With these hypotheses, the fate of particles is described mechanistically via the cellular response of the lung. The mathematical model expresses these particle transitions as differential equations quantifying the transport of particles from one compartment to another, where the compartments represent the alveolar surface, the alveolar macrophages, overloaded alveolar macrophages, the interstitium, interstitial macrophages, and the thoracic lymph nodes. A companion article describes the application of the model to a data set from rats exposed to a low-toxicity dust at several concentrations and for a range of exposure times.

Algorithms↗
PubMed2000

Mathematical methods in biological dosimetry: the 1996 Iranian accident.

PURPOSE: To report 18 months of cytogenetic follow-up for an Iranian worker accidentally overexposed to 192Ir, the mathematical extrapolation and comparison with clinical data. MATERIAL AND METHODS: Unstable chromosome aberrations were measured using conventional cytogenetic tests by French and Iranian biological dosimetry laboratories on five occasions after the exposure. The decrease in dicentrics over time was analysed mathematically. In addition, Dolphin and Qdr extrapolations were applied to the data to check the exposure estimates. FISH determination of translocation yields was performed twice by the French laboratory and the results compared with the Dolphin and Qdr corrected values. RESULTS: Dose estimates based on dicentrics decreased from 3.1 +/- 0.4 Gy at 5 days after the accident to 0.8 +/- 0.2 Gy at 529 days. This could be fitted by double-exponential regression with an inflexion point between rapid and slow decrease of dicentrics after about 40 days. Dose estimates of 3.4 +/- 0.4 Gy for the Qdr model and 3.6 +/- 0.5 Gy for the Dolphin model were calculated during the post-exposure period and were remarkably stable. FISH translocation data at 26 and 61 days appeared consistent with the Dolphin and Qdr estimates. CONCLUSION: Dose correction by the Qdr and Dolphin models and translocation scoring appeared consistent with the clinical data and provided better information about the radiation injury than did crude estimates from dicentric scoring alone. Estimation by the Dolphin model of the irradiated fraction of the body seemed unreliable: it correlated better with the fraction of originally irradiated lymphocytes.

Biometry↗
PubMed1999

Estimates of essential amino acid requirements from dose-response studies with rainbow trout and broiler chicken: effect of mathematical model.

A total of 37 dose-response experiments with essential amino acids performed with rainbow trout and broiler chicken were re-evaluated with different mathematical approaches: an exponential model, a four-parameter logistic function, the saturation kinetics model and the broken line approach. The different approaches were compared both with regard to the goodness of fit (r2 and sy.x) and with regard to the allowances which were derived regarding the optimal amino acid level in the diet. The experimental design, particularly the chosen range in dietary amino acid concentration was found to be important for the comparison of models. Amongst the non-linear models, the four-parameter logistic function and the saturation kinetics model appeared superior to the exponential approach, when the range in dietary amino acid concentration was very wide and included both a severely deficient basal level and a level that exceeded the needs of the animal by approximately the factor 2. In these cases, allowances derived from individual experiments were considerably different depending on the model. The allowances based on the exponential and the saturation kinetics approach were 27.7 and 20.7 g lysine/kg DM and 8.0 and 6.3 g methionine/kg DM, respectively, for rainbow trout. For other amino acids studied in rainbow trout the difference due to model was less. Consequently, the predicted 'ideal protein' for rainbow trout was considerably different depending on the model used. The maximum deviation found in different experiments with broiler chicken for the exponential vs. the saturation kinetics approach was 13.0 and 9.7 g lysine/kg and 11.4 and 8.2 g sulfur-containing amino acids/kg, respectively. However, the more restricted the range in dietary concentration was, the lesser became the differences between the different non-linear models. No definite recommendation can therefore be extracted regarding the most suitable, generally applicable mathematical model.

Amino Acids, Essential↗
PubMed1983

Quantitative studies on T cell diversity. IV. Mathematical analysis of multiple limiting populations of effector and suppressor T cells.

Limiting dilution (LD) analyses of polyclonally activated T cells yielded results suggesting the existence of multiple paired populations of effector and suppressor precursors for a number of different T cell functions and specificities analyzed. These populations occur at graded frequencies and suppression occurs within a pair but not between pairs. In this paper, we establish the mathematical basis for the interpretation of these multi-component limiting dilution results. First, we derive equations for a number of mathematical models and identify one model that both makes biological sense and can be used to reproduce experimental data. Second, within this model, we identify parameters such as the frequency of suppressive cells and the number of suppressive cells required for suppression. The results suggest that within each paired population, suppressor precursors are 20 times more frequent that effector precursors. Furthermore, a similar but variable excess of suppressor cells is required for suppression to become effective. Together with the high frequency (1/50-1/500) of most effector T cell precursors previously reported, the results suggest that up to 40% of the T cells can become involved in suppression of an antigen-specific effector T cell population. These studies may provide exact estimates for predictions to be tested in experiments on immune regulation.

Cell Count↗
PubMed2005

A mathematical model quantifying the impact of antibiotic exposure and other interventions on the endemic prevalence of vancomycin-resistant enterococci.

BACKGROUND: Mathematical modeling can be used to describe the interdependent and dynamic interactions that contribute to the transmission dynamics of vancomycin-resistant enterococci (VRE). A model was developed to quantify the contribution of antibiotic exposure and of other modifiable factors to the dissemination of VRE in the hospital setting. METHODS: The model consists of 4 compartments: patients colonized with VRE receiving and not receiving antibiotics and uncolonized patients receiving and not receiving antibiotics. A series of differential equations describe the movement between these compartments. Baseline parameter estimates were obtained from pharmacy, infection-control, and clinical databases. RESULTS: The main predictions of this model are that (1) preventing the initiation or enhancing the discontinuation of unnecessary antimicrobial therapy will have a greater impact if it is targeted to patients who are not colonized with VRE; (2) increasing the number of patients harboring VRE at the time of hospital admission substantially increases the endemic prevalence of VRE; and (3) eliminating the influx of VRE results in the eradication of this pathogen from the hospital. A decrease in the endemic prevalence of VRE also occurs with a decrease in the length of hospital stay of colonized patients, increased hand hygiene compliance, and a lower ratio of health-care workers : patients. CONCLUSION: This mathematical model provides a framework to assist in targeting necessary interventions aimed at limiting the spread of VRE.

Anti-Bacterial Agents↗
PubMed2006

A mathematical model of hepatitis a transmission in the United States indicates value of universal childhood immunization.

BACKGROUND: US recommendations issued in 1999 for hepatitis A (HA) childhood immunization varied according to regional HA incidences prior to vaccination. Mathematical models of HA transmission, especially those accounting for herd protection, can be useful in formulating new, highly effective recommendations that could lead to disease elimination. METHODS: A mathematical model of HA transmission was designed to assess the impact of different vaccination strategies on the evolution of HA infection over time in the United States. The model represents HA transmission dynamics and is stratified by age and regions defined in the Advisory Committee for Immunization Practices 1999 recommendations. The model accounts for herd protection and HA importation, using an age-dependent "force of infection" varying over time as a function of the prevalence of subjects with infectious HA. RESULTS: The model predicts a clear benefit of vaccinating all US children at as young an age as possible. Nationwide routine immunization at 1 year of age with 70% coverage would prevent 57% of additional cases during the period 1995-2029, compared with the continuation of the regional strategy of vaccinating children at 2 years of age, as recommended by the Advisory Committee for Immunization Practices in 1999. In contrast, the model also predicts that nationwide routine immunization for children 12 years of age only would result in a 14% increase of HA cases during the period 1995-2029, compared with the number of cases predicted with the regional strategy of the immunization of 2-year-olds. CONCLUSIONS: These findings highlight the importance of accounting for herd protection induced by early childhood HA vaccination. They also support the very recent Advisory Committee for Immunization Practices recommendations for universal HA immunization of 1-year-olds.

Adolescent↗
PubMed1990

On the biophysical interpretation of the mathematical product of dose and relative biological effectiveness.

The mathematical product of dose and relative biological effectiveness (DR) is commonly used empirically as the 'effective' dose of radiations. It is often interpreted as the equivalent dose of a reference radiation, such that the individual DRs of the radiation components are summed like physical doses in a mixture of radiations with different RBE values. It is shown that such a physical interpretation of DR would be both mathematically and logically inconsistent unless the action of each radiation has a constant RBE value for all end-effects. This is contrary to general experimental findings. Based on the isoeffect biological connotation in the definition of RBE, a biophysical interpretation is being introduced in this paper in which DR is always interpreted with respect to a particular end-effect on which the RBE value is evaluated, somewhat similar to having an extra biological dimension. Hence, only DRs evaluated for the same end-effect can be meaningfully computed together in a mixture. From the empirical results of radiobiological experiments using mixtures of radiations of different qualities, DRs of radiation components are shown to be additive in a mixture just like physical doses. A convenient linear computation framework is, therefore, available for the use of DRs in the empirical calculation of effect of mixtures of radiations of different qualities. The bearing of this biophysical interpretation of DR on radiation protection and treatment planning is discussed.

Biophysical Phenomena↗
PubMed2000

Improvement of internal dose calculations using mathematical models of different adult heights.

In internal dosimetry for both nuclear medicine and radiation protection, the adult morphology is represented by a limited number of anthropomorphic models that may not be suitable for all patients. To develop more patient-specific dosimetry, we derived six mathematical models for adults of different height. Three male models (160 cm, 170 cm and 180 cm) and three female models (150 cm, 160 cm and 170 cm), based on the MIRD model design, were developed from the statistical analysis of anthropometric data gathered from autopsies. Monte Carlo calculations were used to provide an example of estimations of S value for these new models for iodine 131 uniformly distributed successively in the stomach or in the urinary bladder. On average, for both male and female models, an increase in the model height of 10 cm leads to a mean reduction in the S value for iodine-131 by 20% and 29% when the stomach and the urinary bladder respectively are selected as source regions. Similarly, when the model height increases by 20 cm, the S values decrease on average by 35% and 48%. This study presents the use of anthropometric data to develop new mathematical models for adults of different height, and shows the significant influence of the morphology on dosimetric parameters.

Adult↗
PubMed2002

Edema-induced increase in tumour cell survival for 125I and 103Pd prostate permanent seed implants--a bio-mathematical model.

Edema caused by the surgical procedure of prostate seed implantation expands the source-to-point distances within the prostate and hence decreases the dose coverage. The decrease of dose coverage results in an increase in tumour cell survival. To investigate the effects of edema on tumour cell survival, a bio-mathematical model of edema and the corresponding cell killing by continuous low dose rate irradiation (CLDRI) was developed so that tumour cell surviving fractions can be estimated in an edematous prostate for both 125I and 103Pd seed implants. The dynamic nature of edema and its resolution were modelled with an exponential function V(T) = V(p)(1 + M exp(-0.693T/ T(e))) where V(p) is the prostate volume before implantation, M is the edema magnitude and T(e) is edema half-life (EHL). The dose rate of a radioactive seed was calculated according to AAPM TG43, i.e. D = SkAg(r)phi(an)/r2, where r is the distance between a seed and a given point. The distance r is now a function of time because of edema. The g(r) was approximated as 1/r(0,4) and 1/r(0.8) for 125I and 103Pd, respectively. By expanding the mathematical expression of the resultant dose rate in a Taylor series of exponential functions of time, the dose rate was made equivalent to that produced from multiple fictitious radionuclides of different decay constants and strengths. The biologically effective dose (BED) for an edematous prostate implant was then calculated using a generalized Dale equation. The cell surviving fraction was computed as exp(-alphaBED), where alpha is the linear coefficient of the survival curve. The tumour cell survival was calculated for both 125I and 103Pd seed implants and for different tumour potential doubling time (TPDT) (from 5 days to 30 days) and for edemas of different magnitudes (from 0% to 95%) and edema half-lives (from 4 days to 30 days). Tumour cell survival increased with the increase of edema magnitude and EHL. For a typical edema of a half-life of 10 days and a magnitude of 50%. the edema increased tumour cell survival by about 1 and 2 orders of magnitude for 125I and 103Pd seed implants respectively. At the extreme (95% edema magnitude and an edema half-life of 30 days), the increase was more than 3 and 5 orders of magnitude for 125I and I03Pd seed implants respectively. The absolute increases were almost independent of TPDT and the prostate edema did not significantly change the effective treatment time. Tumour cell survival for prostate undergoing CLDRI using 125I or 103Pd seeds may be increased substantially due to the presence of edema caused by surgical trauma. This effect appears to be more pronounced for 103Pd than 125I because of the shorter half-life of 103Pd. If significant edema is observed post implantation, then a boost to the prostate using external beam radiotherapy may be considered as a part of the treatment strategy.

Brachytherapy↗
PubMed2005

A predictive mathematical model for the calculation of the final mass of Graves' disease thyroids treated with 131I.

Substantial reductions in thyroid volume (up to 70-80%) after radioiodine therapy of Graves' hyperthyroidism are common and have been reported in the literature. A relationship between thyroid volume reduction and outcome of 131I therapy of Graves' disease has been reported by some authors. This important result could be used to decide individually the optimal radioiodine activity A0 (MBq) to administer to the patient, but a predictive model relating the change in gland volume to A0 is required. Recently, a mathematical model of thyroid mass reduction during the clearance phase (30-35 days) after 131I administration to patients with Graves' disease has been published and used as the basis for prescribing the therapeutic thyroid absorbed dose. It is well known that the thyroid volume reduction goes on until 1 year after therapy. In this paper, a mathematical model to predict the final mass of Graves' diseased thyroids submitted to 131I therapy is presented. This model represents a tentative explanation of what occurs macroscopically after the end of the clearance phase of radioiodine in the gland (the so-called second-order effects). It is shown that the final thyroid mass depends on its basal mass, on the radiation dose absorbed by the gland and on a constant value alpha typical of thyroid tissue. Alpha has been evaluated based on a set of measurements made in 15 reference patients affected by Graves' disease and submitted to 131I therapy. A predictive equation for the calculation of the final mass of thyroid is presented. It is based on macroscopic parameters measurable after a diagnostic 131I capsule administration (0.37-1.85 MBq), before giving the therapy. The final mass calculated using this equation is compared to the final mass of thyroid measured 1 year after therapy administration in 22 Graves' diseased patients. The final masses calculated and measured 1 year after therapy are in fairly good agreement (R = 0.81). The possibility, for the physician, to decide a therapeutic activity based on the desired decrease of thyroid mass instead of on a fixed thyroid absorbed dose could be a new opportunity to cure Graves' disease.

Computer Simulation↗
PubMed2006

Development of pregnant female, hybrid voxel-mathematical models and their application to the dosimetry of applied magnetic and electric fields at 50 Hz.

This paper describes the development of 2 mm resolution hybrid voxel-mathematical models of the pregnant female. Mathematical models of the developing foetus at 8-, 13-, 26- and 38-weeks of gestation were converted into voxels and combined with the adult female model, NAOMI. This set of models was used to calculate induced current densities and electric fields in the foetus from applied 50 Hz magnetic and electric fields. The influence of foetal tissue conductivities was investigated and implications for electromagnetic field guidelines discussed.

Body Burden↗
PubMed2001

Analysis by mathematical model of haemodynamic data in the failing Fontan circulation.

Several late complications jeopardize the clinical performance of recipients of the Fontan operation. The underlying causes have been referred to disturbed flow dynamics in the cavopulmonary connections. Presumably, the large pressure drops occurring in the inferior and superior connections play a pivotal role in the pressure level of the entire circulation, especially in the venous. To address this issue, we retrospectively reviewed catheterization data of six patients with failing Fontan circulation and compared them with those of six patients with functioning Fontan circulation. The impact on the systemic and pulmonary pressure of the increase in the cavopulmonary connection resistances was studied through a steady-state mathematical model of the univentricular closed-loop circulation. In the patients with failing Fontan, pressure in the venae cavae was found to be significantly higher, especially at the inferior cava (19.3 +/- 2.2 versus 12.5 +/- 2.3 mmHg) with the pressure drop at the inferior cavopulmonary connection significantly increased (4.7 +/- 3.1 versus 0.33 +/- 0.82 mmHg). The proposed mathematical model permits us to clearly relate the pressure increase in the venae cavae to an increased resistance in the cavopulmonary connections. Therefore, the present analysis confirms that, to avoid possible congestion of venous circulation, the definitive palliation of univentricular heart should not cause pressure drops at the cavopulmonary connections.

Adolescent↗
PubMed2001

A new mathematical model of dynamic cerebral autoregulation based on a flow dependent feedback mechanism.

A new mathematical model representing dynamic cerebral autoregulation as a flow dependent feedback mechanism is presented. Two modelling parameters are introduced, lambda, the rate of restoration, and tau, a time delay. Velocity profiles are found for a general arterial blood pressure, allowing the model to be applied to any experiment that uses changes in arterial blood pressure to assess dynamic cerebral autoregulation. Two such techniques, thigh cuffs and a lower body negative pressure box, which produce step changes and oscillatory variations in arterial blood pressure respectively, are investigated. Results derived using the mathematical model are compared with data from the two experiments. The comparisons yield similar estimates for lambda and tau, suggesting these parameters are independent of the pressure change stimulus and depend only on the main features of the dynamic cerebral autoregulation process. The modelling also indicates that for imposed oscillatory variations in arterial blood pressure a small phase difference between pressure and velocity waveforms does not necessarily imply impaired autoregulation. It is shown that the ratio between the variation in maximum velocity and pressure variation can be used, along with the phase difference, to indicate the nature of the autoregulatory response.

Blood Flow Velocity↗
PubMed1980

A mathematical model of creatine metabolism in normal males--comparison between theory and experiment.

Based upon knowledge that the body creatine pool in normal males undergoes a constant fractional conversion rate to creatinine and that 24-hr urinary creatinine excretion is dependent upon creatine and protein intakes, we developed a mathematical model with feedback which describes the creatine pool size and, consequently, the 24-hr urinary creatinine excretion as a function of time after change in diet. Validity of the model was tested by comparing calculated with experimental changes in 24-hr urinary creatinine excretion rates of male volunteers who were participating in studies of effects of "high" and "low" protein diets on mineral requirements. The model was further verified by comparing published changes in creatinine excretion rates with changes predicted by the model. It is concluded that the effects of changes in creatine and protein intakes upon the body creatine pool size in healthy males can be described mathematically by a model with a feedback component. It is also cautioned that 24-hr urinary creatinine excretion may not be a good reference for quantifying other urinary substances under circumstances where the creatine or protein intakes are not constant.

Adult↗
PubMed1977

A mathematical model of a blood-gas service.

A mathematical model depicting operation of a blood-gas workstation was developed by two systems analysts working closely with two clinical pathologists. This model was used to provide estimates of average as well as maximum turnaround times under various conditions of workload, specimen types (capillary vs. syringe), methodology (use of IL 513 vs. IL 313 for capillary samples), and reporting procedures (report each sample as analyzed vs. report after analysis of all samples in batch). These estimates have been validated against actual experience in our laboratory. Such an objective mathematical model can be used to plan optimal service.

Blood Gas Analysis↗
PubMed1999

Use of a mathematical model of rodent in vitro benzene metabolism to predict human in vitro metabolism data.

Benzene, a ubiquitous environmental pollutant, is known to cause leukemia and aplastic anemia in humans and hematotoxicity and myelotoxicity in rodents. Toxicity is thought to be exerted through oxidative metabolites formed in the liver, primarily via pathways mediated by cytochrome P450 2E1 (CYP2E1). Phenol, hydroquinone and trans-trans-muconaldehyde have all been hypothesized to be involved in benzene-induced toxicity. Recent reports indicate that benzene oxide is produced in vitro and in vivo and may be sufficiently stable to reach the bone marrow. Our goal was to improve existing mathematical models of microsomal benzene metabolism by including time course data for benzene oxide, by obtaining better parameter estimates and by determining if enzymes other than CYP2E1 are involved. Microsomes from male B6C3F1 mice and F344 rats were incubated with [(14)C]benzene (14 microM), [(14)C]phenol (303 microM) and [(14)C]hydroquinone (8 microM). Benzene and phenol were also incubated with mouse microsomes in the presence of trans-dichloroethylene, a CYP2E1 inhibitor, and benzene was incubated with trichloropropene oxide, an epoxide hydrolase inhibitor. These experiments did not indicate significant contributions of enzymes other than CYP2E1. Mathematical model parameters were fitted to rodent data and the model was validated by predicting human data. Model simulations predicted the qualitative behavior of three human time course data sets and explained up to 81% of the total variation in data from incubations of benzene for 16 min with microsomes from nine human individuals. While model predictions did deviate systematically from the data for benzene oxide and trihydroxybenzene, overall model performance in predicting the human data was good. The model should be useful in quantifying human risk due to benzene exposure and explicitly accounts for interindividual variation in CYP2E1 activity.

Animals↗
PubMed2006

Mathematical models for predicting the epidemiologic and economic impact of vaccination against human papillomavirus infection and disease.

Infection with human papillomavirus (HPV) is the primary cause of cervical cancer, other anogenital cancers, genital warts, and recurrent respiratory papillomatosis. Clinical studies have demonstrated that a prophylactic HPV vaccine can prevent infection, genital warts, and the precancerous lesions that lead to cervical cancer. Given the absence of data on the long-term effectiveness of HPV vaccination, a number of mathematical models have been developed to provide insight to policy makers by projecting the long-term epidemiologic and economic consequences of vaccination and evaluate alternative vaccination policies. This paper reviews the state of these models. Three types of HPV mathematical models have been reported in the literature: cohort, population dynamic, and hybrid. All have demonstrated that vaccination can significantly reduce the incidence of cervical cancer in the long term. However, only the cohort and hybrid models have evaluated the cost-effectiveness of vaccination strategies for preventing cervical cancer. These models have generally shown that vaccinating females can be cost-effective. None has accounted for the potential benefits of vaccinating the population to reduce the burden of recurrent respiratory papillomatosis and cancers of the vagina, vulva, anus, penis, and head/neck. Given that only the population dynamic model can account for both the direct and indirect (i.e., herd immunity effects) benefits of vaccination in the population, future research should focus on further development of dynamic models by expanding the range of epidemiologic outcomes tracked and including the ability to assess the cost-effectiveness of alternative vaccination policies.

Cost-Benefit Analysis↗
PubMed2003

Mathematical issues arising from the directed mutation controversy.

During the past 14 years or so a large body of new evidence that supposedly supports the directed mutation hypothesis has accumulated. Interpretation of some of the evidence depends on mathematical reasoning, which can be subtler than it appears at first sight. This article attempts to clarify some of the mathematical issues arising from the directed mutation controversy, thereby offering alternative interpretations of some of the evidence.

Data Interpretation, Statistical↗