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PubMedDate not supplied

Mathematical modeling of human embryonic and fetal growth rates.

A mathematical model for human embryonic/fetal growth data from implantation to birth is developed. In previous work, it was shown that an unbiased estimate for human fetal growth data from about day 50 post-conception until term could be calculated from the Gompertz equation. This period represents a range of embryonic/fetal weights from one to 3500 g. When the Gompertz equation is extended, with no change of parameters, to the prenatal period before 50 days, the predicted weights have a consistent bias which might have a biological basis. Early embryonic growth immediately following fertilization is exponential; i.e., one cell goes to 2, then 4, then 8... etc., with essentially no decrease in relative growth rate. Except for possible small changes in cell size and cell mitosis cycle time, such exponential growth can be considered as a special case of the Gompertz equation with a, the relative rate of decrease of the relative growth rate, equal to zero. The relative growth rate begins to decrease about 20 days post-conception, at the time of cell differentiation into organ precursors. Although the "Hayflick Limit" of the maximum of 50 to 60 cell divisions for human cells would tend to cause a decrease in growth rate, it can be shown that the effect is insignificant during embryonic/fetal growth. The observed decrease in the growth rate might be a result of a decreasing fraction of cells in the pool of dividing cells. For the Gompertz equation model, a at this time changes from zero to a positive number. Analysis of fetal growth data shows that a rapidly becomes large and then decreases over a period of several days to become a constant positive value for the remainder of the prenatal term. Good fits of empirical embryonic/fetal growth data were obtained by nonlinear regression with calculation of the embryonic/fetal weights through numerical integration of the differential Gompertz equations and the functionality of alpha.

Body Weight↗
PubMed1983

Active anterior rhinomanometry in pre- and postoperative evaluation, use of Broms' mathematical model.

The authors studied the value of the Broms' mathematical model for active anterior rhinomanometry in a pathological population. They compared all different variables of a pathological group with a normal group and found significant differences for v0, v2 and R. There existed no difference between the means of v0 before and after surgery. There existed, however, a strongly positive correlation between the expiration and inspiration value. The influence of surgery was always significant for the variables v1, v2, v3 and R. Furthermore, the absolute deviation between the computed values and the recorded values tended to be very small. So, the authors concluded that rhinomanometry is a valuable aid in judging pre- and postoperative results.

Adolescent↗
PubMed1979

A mathematical model of hiking positions in a sailing dinghy.

A mathematical model of the human body designed to calculate the resultant muscle torques required at the hip and knee joints for specific hiking techniques is presented. Data for the model were obtained from ten male subjects who adopted three basic positions: Position 1 with the knees located at the inside edge of the sidedeck, Position 2 with the knees at the middle of the sidedeck, and Position 3 with the knees at the outside edge of the sidedeck. Each resultant muscle torque was expressed as a percentage of each subject's maximum voluntary hip flexion or knee extension torque. It was found that where Positions 1 and 2 were equally effective in keeping the boat upright, Position 2 was superior to Position 1 in regard to the per cent of maximum muscle torque required. The superiority of Position 2 over Position 3 depended on the individual's relative muscle strength at the hip and knee joints. The stronger the hip flexors with respect to the knee estensors, the more desirable was Position 2 and vice versa.

Biomechanical Phenomena↗
PubMedDate not supplied

Understanding mathematical models for breast cancer risk assessment and counseling.

Chemoprevention and prophylactic surgery are effective interventions for lowering breast cancer incidence. However, these approaches are associated with risks of their own. Accurate individualized breast cancer risk assessment is an essential component of the risk/benefit analysis that must take place prior to implementing either of these strategies. Several mathematical models for estimating individual breast cancer risk have been proposed over the last decade. The Gail model is the most generally applicable model; however, it neglects family history information in second-degree relatives, treats pre- and postmenopausal breast cancer the same, and ignores personal histories of lobular neoplasia. The Claus model is a better family history model, but it does not assign any special relevance to histories of bilateral breast cancer or ovarian cancer, and neglects all of the nonfamily history information accounted for by the Gail model. BRCAPRO is a Bayesian family history model that calculates individual breast cancer probabilities based on the probability that a family carries a mutation in one of the BRCA genes. Though its treatment of family history information is more thorough than the other models, it neglects the nonfamily history risk factors accounted for by the Gail model and may not appreciate familial clustering unrelated to BRCA gene mutation. A thorough understanding of the principles of risk analysis and the available mathematical models is essential for anyone wishing to perform intervention counseling. This review describes the basic components of risk analysis, explains how the mathematical models work and compares the strengths and weaknesses of the various models. CancerGene is a software tool for running all of these models. It may be obtained without charge at http://www.swmed.edu/home_pages/cancergene.

Breast Neoplasms↗
PubMed2001

A mathematical model for cell density and proliferation in squamous epithelium after single-dose irradiation.

PURPOSE: To establish a mathematical model describing changes in cell density in squamous epithelia induced by single-dose irradiation. Detailed data from previous studies in mouse tongue epithelium have been used for this study. MATERIALS AND METHODS: The major mechanisms of the epithelial regeneration response, i.e. loss of division asymmetry and accelerated proliferation of stem cells, in combination with residual, abortive proliferation of sterilized cells, have been included in a tissue compartment model. These phenomena have been incorporated via three parameters; T(delay), the duration of the cell cycle block; T(min), the minimum stem cell cycle time due to acceleration; and T(stop), the duration of abortive proliferation. The compartments introduced in the model are normal stem cells, S1; sterilized stem cells, S2; and post-mitotic, functional cells, F. The flux rats between the tissue compartments were defined by autoregulation of the stem cell population, and by overall cell numbers. The model was applied to fit experimental data on changes in oral mucosal cell density after single-dose exposure with 13 and 20 Gy. The best-fit sets of parameters were identified by L2 norm error analysis based on the total cell count. RESULTS: For 13 Gy, the best fit was achieved with T(min) = 1.0 days, T(delay) = 1.2 days and T(stop) = 7.5 days. For 20 Gy, the parameters were, T(min) =0.7 days, T(delay)= 1.0 days and T(stop) =9.5 days. In both data sets, T(min) was the most influential parameter. The resulting fluctuations in stem cell numbers were in good accordance with changes in radiation tolerance after 13 Gy. CONCLUSIONS: The model can be used to define dose-dependent parameters describing the morphological response of squamous epithelia to single-dose irradiation. Based on these parameters, post-irradiation fluctuations in radiosensitivity can be predicted. For developing more complex and reliable mathematical models, which could incorporate transit divisions or fractionated radiotherapy, further experimental data at various dose levels are required.

Animals↗
PubMed1992

A new mathematical model to study bone turnover in growing rats.

A new mathematical model for the study of bone turnover in growing rats was developed. The model predicts a linear relationship between bone mineral content (BMC) and biochemical markers (BMK) of bone turnover assuming that rats are growing, bone turnover is profoundly affected by skeletal maturation, and resorption and formation are physiologically balanced. The model validation was performed by measuring galactosyl-hydroxylysine (GHYL) and hydroxyproline (HYP) in urines. This mathematical evidence supports our proposed use of the specific bone resorption marker GHYL to predict bone mineral content. Further studies on bone turnover will be possible by the application of the same approach.

Absorptiometry, Photon↗
PubMed1980

[Construction and study of mathematical models of the dynamics of bacterial biomass growth taking into account the effect of interchangeable metabolic links].

Systemic approach was used as a basis for developing mathematical models for the dynamics of the growth of bacterial biomass with regard to the intracellular substrate pool in case of ramification of metabolic links. The results obtained by calculations with the use of such model were compared with the experimental results of the batch cultivation of E coli M 17. The mathematical models were shown to give a qualitatively correct description characterizing process of the growth of bacterial biomass under conditions of the limited supply of inorganic phosphorus and magnesium.

Bacteria↗
PubMed2004

A mathematical model of factors that influence the performance of accommodative intraocular lenses.

In this work a mathematical model of capsule movement during pseudophakic accommodation is described to allow identification and evaluation of factors that may explain the variation in effect of accommodative intraocular lenses (IOLs) between patients. The model assumes that increasing vitreous pressure pushes the lens capsule forward as a circular diaphragm and that this movement is from a fixed fulcrum. With an IOL in situ, the capsule is taken to have a non-uniform thickness due to the presence of the anterior capsulorhexis. The model assumes a uniform capsular elasticity and ignores contributions from cellular elements such as posterior capsule opacification. Using our model and a regression formula to calculate capsular bag size, taking into account axial length and keratometry values, we are able to predict accommodative effect in individual patients. By simple geometry we have developed a mathematical model to identify variables that are important in pseudophakic accommodation. It provides the basis for the development of a more complex model that would address the movement of a lens taking into account the influence of the zonular system during accommodation.

Accommodation, Ocular↗
PubMed2005

The implantation of every embryo facilitates the chances of the remaining embryos to implant in an IVF programme: a mathematical model to predict pregnancy and multiple pregnancy rates.

BACKGROUND: We aimed to assess the validity of a theoretical mathematical model to predict the pregnancy rate and the multiple pregnancy rate in IVF/oocyte donation programmes on the basis of the implantation rate and the number of transferred embryos. METHODS: A total of 1835 embryo transfers corresponding to three different programmes in two centres with different implantation rates were analysed. Pregnancy and multiple pregnancy rates observed in the aforementioned programmes were compared with those obtained following different mathematical models. Four models were tested: binomial model, ground model, maternal variability model and collaborative model. The goodness of fit was performed by means of the maximum likelihood fit method. RESULTS: The binomial model could not predict the pregnancy rate, and especially the multiple pregnancy rate. The multiple pregnancy rate predicted following the binomial model was much lower than observed, up to 40-fold reduced. Ground model and maternal variability model adjusted to the data with more precision, but were still not accurate. Finally, the collaborative model reproduced with very great accuracy both pregnancy rate and the multiple pregnancy rate. A collaborative parameter of 22% was found, implying that the implantation probability of each embryo is increased by 22% for every embryo previously implanted. CONCLUSIONS: Embryonic implantation does not follow a binomial law, showing that the implantation is not independent from the number of embryos implanted. The best fit to the data is obtained following a collaborative model by which the implantation of one embryo is facilitated by the implantation of other embryo(s). The mathematical formula of the collaborative model predicts very accurately the pregnancy rate and the multiple pregnancy rate in IVF/oocyte donation programmes, based on the implantation rate of this specific programme and the number of embryos transferred up to five embryos. We recommend using the aforementioned formula to quantify the pregnancy rate and the risk of multiple pregnancy in the counselling of the infertile couple at embryo transfer. Such a formula is freely available at www.ifca.unican.es/matorras/mathpreg/.

Embryo Implantation↗
PubMed1987

Profiles of water and solute transport along long-loop descending limb: analysis by mathematical model.

We simulated profiles of water and solute transport along the descending limb of the long-loop nephron by a mathematical model based on mass balance equations for water, sodium, potassium, and urea, using phenomenological coefficients reported for hamsters. We assumed that interstitial concentration of sodium, potassium, and urea increased linearly along the descending limb from 150 to 350, from 5 to 50, and from 5 to 300 mM, respectively. Under this condition an increase in osmolality at the end-descending limb was mainly accounted for by the absorption of water. Considerable amounts of potassium and urea were secreted along the descending limb. Sodium was reabsorbed rather than secreted along the descending limb by both diffusion and solvent drag. The secreted amounts of urea and potassium were comparable to those observed by micropuncture studies. The sodium concentration in the lumen was higher than in the interstitium, with the transmural sodium gradient being 15 meq/liter at the hairpin turn. The potassium mass flow rate at the end-descending limb increased by 2.4 times. Large variations in potassium concentration of the delivered fluid scarcely changed the potassium mass flow rate at the end-descending limb. The secretion of urea and potassium and the reabsorption of sodium were increased as a function of delivered flow rate. An increase in corticomedullary urea gradient decreased the net potassium secretion along the descending limb. When the transport parameters for rabbits were used, both reabsorption of sodium and addition of urea were decreased, but a similar amount of potassium was secreted. These analyses indicate that the mathematical model that takes the species difference and internephron heterogeneity into consideration is useful in illustrating the transport processes along the descending limb of Henle's loop under various physiological and pathophysiological conditions.

Biological Transport, Active↗
PubMed1998

Mathematical modeling of deposition and disposition of drugs administered via the nose.

This articles reviews the mathematical models of deposition and disposition of drugs administered into the human nasal cavity for systemic activity. The modeling of the disposition kinetics includes drug release from carriers, translocation within the nasal cavity and into the gastrointestinal tract, drug decomposition, and drug absorption during the transit through the nose and the gastrointestinal tract. The uses of such mathematical models for design and analysis of nasal delivery systems are illustrated, with particular reference to the new therapeutic materials and excipients.

Journal Article↗
PubMed1996

Building intelligent alarm systems by combining mathematical models and inductive machine learning techniques.

In this article a technique is described to develop knowledge-based alarm systems for ventilator therapy, using mathematical modeling and machine learning. With a mathematical model airway pressure, expiratory gas flow and CO2 concentration at the endotracheal tube are simulated for patients, undergoing volume-controlled ventilation with constant ventilator settings, during normal functioning of the breathing circuit and during breathing circuit mishaps (leaks and obstructions). Simulations were performed for 94 physiologically different 'patients', by varying airway resistance and lung/thorax compliance values in the model. Each simulated breath was described by a set of derived signal features and a label that constituted during which event (normal function or mishap) the breath was recorded. With an inductive machine learning algorithm rules, linking signal feature values to breathing circuit events, were created from data of 54 of the simulated patients. The resulting set of rules was able to classify 99% of events in the data of the remaining 40 patients correctly. Of signals, measured at a ventilated lung simulator, 100% of events were classified correctly.

Airway Resistance↗
PubMed2006

Mathematical modeling of endovenous laser treatment (ELT).

BACKGROUND AND OBJECTIVES: Endovenous laser treatment (ELT) has been recently proposed as an alternative in the treatment of reflux of the Great Saphenous Vein (GSV) and Small Saphenous Vein (SSV). Successful ELT depends on the selection of optimal parameters required to achieve an optimal vein damage while avoiding side effects. Mathematical modeling of ELT could provide a better understanding of the ELT process and could determine the optimal dosage as a function of vein diameter. STUDY DESIGN/MATERIALS AND METHODS: The model is based on calculations describing the light distribution using the diffusion approximation of the transport theory, the temperature rise using the bioheat equation and the laser-induced injury using the Arrhenius damage model. The geometry to simulate ELT was based on a 2D model consisting of a cylindrically symmetric blood vessel including a vessel wall and surrounded by an infinite homogenous tissue. The mathematical model was implemented using the Macsyma-Pdease2D software (Macsyma Inc., Arlington, MA, USA). Damage to the vein wall for CW and single shot energy was calculated for 3 and 5 mm vein diameters. In pulsed mode, the pullback distance (3, 5 and 7 mm) was considered. For CW mode simulation, the pullback speed (1, 2, 3 mm/s) was the variable. The total dose was expressed as joules per centimeter in order to perform comparison to results already reported in clinical studies. RESULTS: In pulsed mode, for a 3 mm vein diameter, irrespective of the pullback distance (2, 5 or 7 mm), a minimum fluence of 15 J/cm is required to obtain a permanent damage of the intima. For a 5 mm vein diameter, 50 J/cm (15W-2s) is required. In continuous mode, for a 3 mm and 5 mm vein diameter, respectively 65 J/cm and 100 J/cm are required to obtain a permanent damage of the vessel wall. Finally, the use of different wavelengths (810 nm or 980 nm) played only a minor influence on these results. DISCUSSION AND CONCLUSION: The parameters determined by mathematical modeling are in agreement with those used in clinical practice. They confirm that thermal damage of the inner vein wall (tunica intima) is required to achieve the tissue alterations necessary in order to lead the vein to permanent occlusion. However, in order to obtain a high rate of success without adverse events, the knowledge of the vein diameter after tumescent anesthesia is recommended in order to use the optimal energy. As clearly demonstrated by our calculations, both pulsed and continuous mode operations of the laser can be efficient. An interesting observation in our model is that less amount of energy is required in pulsed mode than in continuous mode. Damaging the vein sequentially along its entire length may lead to permanent occlusion. However, the pulsed mode requires a very precise positioning of the fiber after each pullback and the duration of the treatment is much longer. For these reasons, continuous irradiation seems to be preferred by most clinicians. This model should serve as a useful tool to simulate and better understand the mechanism of action of the ELT.

Animals↗
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↗
PubMed1989

[Biomathematical aspects of numerical evaluation and mathematical modelling of non-monotonous exponential measuring processes].

Measurements of endocrinological and pharmacological processes often yield courses of time series with exponentially saturated increasing first part followed by an exponentially decreasing part. Such measured courses may be mathematically modelled by the so-called BATEMAN function type, an expression consisting of 2 e-function terms. In this paper, the method of locally adjusted functional approximation for model-free quantitative evaluation of measured time series is sketched. By means of 2 real examples of measured data, it will be demonstrated how the results of the model-free evaluation may serve for internal regression to estimate starting parameter values for an iterative fitting of a BATEMAN function to measured data courses. Furthermore, it is shown that the model-free approach of data evaluation may give substantial hints for the mathematical model building process and for model verification.

Animals↗
PubMed1982

Kinetics of protein and DNA synthesis studied by mathematical modelling of flow cytometric protein and DNA histograms.

Mathematical models for histograms of cellular protein content as measured by flow cytometry were developed, based on theoretical protein distributions. These were derived from the age distribution of cells and the accumulation function for cellular protein content as a function of age within the cell cycle. A model assuming an exponential age distribution and an exponential protein accumulation function was found to give the best representation of protein histograms of exponentially growing NHIK 3025 cells. This is in good agreement with the known kinetic behaviour of such cells. By the combined use of the protein histogram model and a similar model for DNA content, and assuming linear DNA accumulation during S, the fraction of cells in S, as a function of cellular protein content, was simulated. This function showed good agreement with values of the [3H]TdR labelling index scored in cells sorted by flow cytometry from 5-channel intervals of the protein histogram. The protein and DNA histogram models were combined into a two-dimensional model for correlated protein/DNA measurements. Comparison between simulated data and experimentally derived two-dimensional protein/DNA histograms gave further support to the cell kinetic assumptions underlying the models, but also identified some minor deviations which could not be recognized in the analysis of the one-dimensional histograms.

Cell Cycle↗
PubMed2001

The effect of mathematical modeling on critical velocity.

The purpose of this investigation was to examine the effects of mathematical modeling on critical velocity (CV) estimates and the oxygen consumption (VO2), heart rate (HR), and plasma lactate values that corresponded to the five CV estimates. Ten male subjects performed a maximal, incremental treadmill test to determine maximal VO2, and four randomly ordered treadmill runs for the estimation of CV. Two linear, two nonlinear, and one exponential mathematical models were used to estimate CV. Regression analyses were used to determine the VO2, HR, and plasma lactate values that corresponded to the five CV estimates from the relationships for VO2, HR, and plasma lactate versus running velocity from the maximal, incremental test. The nonlinear, three-component model (Nonlinear-3) resulted in a mean CV that was significantly (P < 0.05) less than the mean values derived from the other four models, and was the lowest CV estimate for each subject. The percent of maximal VO2, HR, and plasma lactate values that corresponded to the Nonlinear-3 model were 89%, 93%, and 63%, respectively. These findings indicate that CV estimates differ by as much as 20% depending upon the model used to determine the characteristics of the velocity/time relationship. Future studies are needed to determine which model provides the most valid estimate of the demarcation point between heavy and severe exercise.

Adult↗