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PubMed2003

Development of a mathematical method for classifying and comparing tree architecture using parameters from a topological model of a trifurcating botanical tree.

This paper describes a model for the topological mapping of trifurcating botanical trees. The model was based on a system of modular units that represented the interconnectivity of shoot meristems (terminal segments) and internodes (internal segments) within whole plant canopies, organized with increasing centrifugal ordering. The model was capable of describing the dynamics of plant growth as expressed by changes in topological parameters over time. Preliminary calculations for experimental trees indicated that the model represents growth in a biologically sound manner. Methods are described for the calculation of the architecture parameters size, size-complexity, structural complexity, and tree asymmetry index (TAI). Parameter calculations were based on the mathematical principles developed for the classification of bifurcating dendrite trees, and were designed to both extract structural information, and to enable statistical comparison between trees of different size. Parameters were mathematically adjusted for trifurcation, and appeared to be able to represent quantitatively the architectural properties of tree structures. In addition to the calculation of the TAI for trifurcating trees, new methods were developed to enable comparisons to be made of the architectural complexity of trifurcating trees of differing size. These were based on the principle of the pair-wise comparison of the mean centrifugal order number (MCON) with respect to segments against highest order number. We argue and illustrate that this principle can be more informative than that of pair-wise comparison of the MCON against tree degree (topological size). Further improvements to this method were made by examining branching points (vertices) rather than segments (links) to calculate the MCON.

Models, Biological↗
PubMed2002

Mathematical modeling of plant metabolic pathways.

The understanding of the control of metabolic flux in plants requires integrated mathematical formulations of gene and protein expression, enzyme kinetics, and developmental biology. Plants have a large number of metabolically active compartments, and non-steady-state conditions are frequently encountered. Consequently steady-state metabolic flux balance and isotopic flux balance modeling approaches have limited utility in probing plant metabolic systems. Transient isotopic flux analysis and kinetic modeling are powerful proven techniques for the quantification of metabolic fluxes in compartmentalized, dynamic metabolic systems. These tools are now widely used to address metabolic flux responses to environmental and genetic perturbations in plant metabolism. Continued developments in isotopic and kinetic modeling, quantifying metabolite exchange between compartments, and transcriptional and posttranscriptional regulatory mechanisms governing enzyme level and activity will enable simulation of large sections of plant metabolism under non-steady-state conditions. Metabolic control analysis will continue to make substantial contributions to the understanding of quantitative distribution of control of flux. From the synergy between mathematical models and experiments, creative methods for controlling the distribution of flux by genetic or environmental means will be discovered and rationally implemented.

Forecasting↗
PubMed2001

The "practical mathematics" of recording three-dimensional eye position using scleral coils.

The "gold standard" for recording the three-dimensional rotation of the eye involves placing two coils of wire, embedded in a soft plastic ring, on the sclera of the eye, then placing the subject inside a set of orthogonal oscillating magnetic fields, and using the currents induced in the eye coils to deduce the position of the coil, and hence of the eye, in space. Eye movements are actually eye rotations, which can be described mathematically by a special class of matrices, rotation matrices, or, alternatively, by a rotation vector related to the axis of the rotation. This article deals with the mathematical tools needed to implement the signal processing from such a multifield, dual-coil system and compute the precise rotational movement of the eye. One reason for making such careful measurements is to study an interesting constraint on eye movements, called Listing's law, which expresses ocular torsion, or rotation of the eye about its line of sight, in terms of the direction of gaze. Techniques for experimentally quantitating these constraints are also presented. Following a treatment of the "ideal" case, with coils and eye in perfect alignment, the additional techniques for dealing with various departures from ideality that are almost always encountered experimentally are examined. A final section deals with developing a validation protocol for eye movement analysis techniques using mechanical and computer simulations of eye movements.

Algorithms↗
PubMed1997

A mathematical model of production, distribution, and metabolism of melatonin in mammalian systems.

Melatonin is a neuroendocrine hormone which is currently receiving considerable attention as a treatment for jet lag, a treatment for insomnia and, by some, a possible "magic bullet" for delaying the effects of aging and preventing cancer. Production of melatonin is focused primarily in the pineal gland with very wide daily shifts in production controlled by the day/night cycle. The potential for increased disease as a consequence of lower or higher than average production of this hormone has not been well studied, although potential environmental agents may modulate circulating levels (e.g., electric and magnetic fields). In this manuscript, a physiologically realistic mathematical model for the production, distribution, and metabolism of melatonin is developed as a precursor to a future study of the role of chemicals and environmental agents in altering this system. Values for key aspects of the system (e.g., diurnal rates of production of the hormone in the pineal gland) were obtained from the literature and the model was validated against data on circulating levels. The mathematical equations and model parameters are presented.

Animals↗
PubMed1998

Distributing working versions of published mathematical models for biological systems via the Internet.

Mathematical models are useful tools for investigating complex systems. By representing physiological systems as models, theories can be tested quantitatively against data from the system. Models can be used to explore new theories prior to experimentation and to design studies to optimize experimental resources. They can also be used as teaching tools to illustrate physiochemical principles. In spite of their usefulness and the time invested in developing models, published models are often underused due to the difficulty in obtaining working versions of the model. To address this problem we have designed a library for mathematical models of biological systems on the Internet. The library contains published models of biological systems in formats compatible with several modeling packages, from the fields of physiology, metabolism, endocrinology, biochemistry, and chemistry. The models can be viewed graphically, model solutions can be viewed as plots against data, and models can be downloaded to be run with software on the user's own system. The address of the library is: http://biomodel.georgetown.edu/model/ Investigators are invited to submit working versions of published models to the library. Models can be submitted electronically at the time a manuscript is accepted for publication. As journals go online, articles containing models can be linked to working versions of the models in the library. By increasing access to working versions of models, more of the investment in kinetic studies and model development can be realized.

Computer Simulation↗
PubMed2003

The ICP-dependency of resistance to cerebrospinal fluid outflow: a new mathematical method for CSF-parameter calculation in a model with H-Tx rats.

The international accepted calculation methods concerning the cerebrospinal fluid dynamics proceed from a pressure independent resistance to cerebrospinal fluid outflow. In a new model we focus our attention on the pressure dependency of resistance. In our calculation model we are monitoring the complete pressure course p(t) over the time t during and after the infusion. The comparison of the pressure rise On(p) during the infusion and the descent Off(p) after the infusion in the same pressure level allows to construct all formulas for the compliance C(p) and resistance R(p). The simultaneous measurement of the resistance and complications during a single investigation allows minimizing the patient's exertion. In contrast to the classical methods it is not necessary that the ICP reach a plateau. Our mathematical method diverges with the description of a pressure dependent slope of the function for the resistance from the static examination models. We proved our mathematical method by the use of a dynamic infusion test in ten H-Tx rats without hydrocephalus and five hydrocephalic H-Tx rats. For that we are able to take the non-linearity of the CSF resorption into consideration.

Animals↗
PubMed1988

Biological exposure index of styrene suggested by a physiologico-mathematical model.

We used a physiologico-mathematical model to study the biological exposure index of styrene correlated to the Threshold Limit Value (TLV) suggested by the ACGIH for 1986-87. This model allows the solvent concentrations in blood, alveolar air, fat tissue, and in other biological media to be estimated and simultaneously the kinetics of its metabolites to be followed when a specific exposure is settled. The comparison between the results obtained from the mathematical model and the numerous research projects documented in the literature suggests a reciprocal validation. Moreover, some biological parameters (particularly the alveolar ventilation) can explain the variability of results obtained from studies concerning the solvent pollution of the factories, which used biological monitoring. The ranges of styrene concentrations in blood and alveolar air and the urinary concentrations of its metabolites (mandelic and phenylglioxylic acids) are discussed in connection with the exposure at 215 mg/m3. Important differences correlated to the definition of set-levels of TLV and Biological Exposure Index (BEI) have been found: particularly the TLVs lead to different solvent uptakes according to some biological parameters; the BEI can better explain the individual solvent uptake and body burden.

Environmental Exposure↗
PubMed1991

A physiologically based mathematical model for the human inhalation pharmacokinetics of 1,1,2-trichloro-1,2,2-trifluoroethane.

A physiologically based mathematical model is described for the human inhalation pharmacokinetics of 1,1,2-trichloro-1,2,2-trifluoroethane (FC113). Physiological parameters for the model are derived from the scientific literature. Partition coefficients are determined from in vitro measurements. Predictions of the resulting model for breath and blood concentrations compare well with results of a human volunteer study described in a companion paper (Woollen et al. 1990). Using this data some alternative models are also examined with different choices of physiological parameters and partition coefficients. The mathematical model is used to examine the consequences of metabolic elimination of FC113. A value for metabolic clearance is estimated using the during-exposure breath concentration data; however, the concentrations of FC113 in breath or blood during and after exposure are shown to be insensitive to metabolic clearance. Consequently, no firm conclusion can yet be drawn as to whether FC113 is metabolised by man.

Air Pollutants, Occupational↗
PubMed1993

Kinetic analysis of AUC-dependent saturable clearance of liposomes: mathematical description of AUC dependency.

The objective of this study was to examine the AUC dependency of saturable hepatic clearance (CLh) of liposomes and to postulate a mathematical model to describe the characteristics. The AUC dependency of saturable CLh was examined under intravenous rapid administration at various doses. The CLh increased with increasing blood concentration but decreased with the increase of AUC at each dose. In addition, the relationship between AUC and CLh was consistent with that observed in previously reported infusion studies. These experimental data confirm the AUC dependency of saturable CLh of liposomes. A mathematical model was developed for this AUC dependency. The decrease of CLh was described by the uptake amount (X) as follows: CLh = CLm(1-X/Xm), where CLm and Xm represent the maximum uptake clearance and the maximum uptake amount, respectively. The rate equation for uptake was analytically solved as CLh = X/AUC = Xm/AUC(1-exp(CLm/XmAUC)). Uptake clearance can be described by CLm, Xm, and AUC, and so uptake clearance is constant if AUC is constant. These experimental analyses and theoretical considerations show the validity of the AUC-dependent saturable CLh of liposomes.

Animals↗
PubMed1995

Apnea testing in suspected brain dead children--physiological and mathematical modelling.

OBJECTIVE: To study the validity and safety of the traditional apnea test in children, and to evaluate a mathematical equation estimating the hemodynamic response to the apnea test. DESIGN: A prospective clinical study. SETTING: Pediatric ICU. PATIENTS AND PARTICIPANTS: 38 pediatric patients suffering severe brain injury aged 2 months to 17 years, undergoing apnea testing for brain death. MEASUREMENTS AND RESULTS: Apnea tests were performed 61 times (once in 19 patients, twice in 15, and 3 times in 4 patients). Mean PaCO2 was 41.1 +/- 10.6 mmHg before apnea and increased to 68.0 +/- 17.6 at 5 min. PaCO2 increased to 81.8 +/- 20.1 and 86.0 +/- 25.6 at 10 and 15 min, respectively. There was a mean PaCO2 increase by 5.38 +/- 1.4 mmHg/min in the first 5 min, and 2.75 +/- 0.5 mmHg/min during the next 5 min. We found a statistically significant (p < 0.05) linear relationship between the natural logarithm of PaCO2, time, and the logarithm of the initial level of PaCO2. An inverse linear relationship (p < 0.05) was found between systemic mean arterial pressure (MAP) and initial level of PaCO2 presented as mathematical correlations and nomograms. CONCLUSIONS: By using our model for predicting MAP and PCO2 prior to apnea testing, hemodynamic embarrassment can be anticipated and prevented, thus allowing a safer procedure in the detection of brain death. Despite the fact that continuous cardiorespiratory monitoring is important, hemodynamic disturbances can be estimated before the apnea test, thus allowing a safer approach to brain death detection.

Adolescent↗
PubMed1992

A mathematical approach to benzo[a]pyrene-induced hematotoxicity.

Benzo[a]pyrene (BaP) has been reported to exert a differential effect on murine hematopoiesis that is mouse strain specific. Interpretation of these results based solely on experimental data is restricted and leaves important questions unanswered. Therefore, a mathematical model of murine hematopoiesis was applied in order to: (1) identify the targets of BaP, (2) quantify the damage to target cells and (3) based on these results, interpret differences in strain susceptibility. Model analysis of the hematopoietic response of D2 and BDF1 mice to a daily oral administration of 125 mg/kg BaP showed that proliferating hematopoietic cells are the targets of BaP. Within this group it was found that: (a) erythropoietic cells were the most susceptible to BaP, (b) granulopoietic cells showed a susceptibility half that of erythropoietic cells and (c) the susceptibility of stem cells ranged between that of erythropoietic and granulopoietic cells. This damage pattern was the same for both strains, indicating that the difference between the strains was quantitative. As cell destruction rates were about 3-fold higher for D2 than BDF1 mice, it was concluded that D2 mice were about three times as susceptible to BaP as BDF1 mice. The study showed that the mathematical model, in addition to experimental methods, provided an efficient tool for the analysis of BaP hematotoxicity.

Administration, Oral↗
PubMed1998

A mathematical model for micturition gives new insights into pressure measurement and function.

Our objective was to analyze the factors contributing to the development of detrusor pressure during micturition in the female with reference to a mathematical model. One hundred patients with predominantly stress incontinence were investigated with micturition pressure studies. Frictional and dynamic losses were estimated at various flow rates using a mathematical model. Almost 25% of patients recorded a micturition pressure below 11 cmH2O at peak flow (mean 23 cmH2O, range 0-91). Large inter- and intrapatient variations in micturition pressures were recorded on retesting. The low pressures were explained by a recently described external opening mechanism, backward stretching of the vagina during micturition by the muscles of the pelvic floor. This opened out the outflow tract and created the potential for a falsely high P(abd). The large variability in micturition pressures on retesting was attributed to changes in urethral radius being magnified to the fourth power. It was concluded that, micturition itself, and the components for pressure generation, are complex non-linear entities which appear to be greatly modified by the external striated pelvic floor opening mechanism. Addressing anatomical defects in this mechanism may be a fruitful route of future enquiry in females with emptying problems.

Adult↗
PubMed1997

Physiology of rectal sensations: a mathematic approach.

PURPOSE: The first awareness of balloon inflation (first sensation (FS)), flatus sensation (constant sensation (CS)), urge to defecate (UD), and maximum tolerated threshold (MTT) are the four commonly evaluated rectal sensations. The traditional view that these sensations are attributable to pelvic floor mechanoreceptor stimulation is challenged by current evidence in favor of rectal wall mechanoreceptors. The aim of this study was to determine the physiology of these sensations, using a dynamic mathematic model of the rectum. METHODS: In a group of 15 healthy adult volunteers (11 female and 4 male; median age, 51.5 (range, 31-74) years), the polynomial behavior of the two smooth muscle components of a dynamic mathematic model of the rectum was analyzed to find strain levels of smooth muscle activity in relation to corresponding strain levels of each of the four "rectal" sensations. RESULTS: Longitudinal and circular smooth muscle relaxation appeared to be the rate detection and signaling mechanisms, respectively. The latter triggered sensations of CS, UD, and MTT. FS was an anal canal sensation, related temporally with onset of rectoanal inhibitory reflex. In vitro validation of the model suggested MTT to be a physiologic protective mechanism associated probably with tetanic smooth muscle contraction. CONCLUSIONS: Evaluation of rectal sensations should be confined to CS and UD because MTT is painful and does not contribute any additional information, and FS is not a true rectal phenomenon.

Adult↗
PubMed1996

Submicroscopic mathematical evaluation of spermatozoa in assisted reproduction. 2. In vitro fertilization. (Notulae seminologicae. 7)

This paper belongs to a series of applications of the Baccetti et al. formula (1) to the submicroscopical mathematical examination of human spermatozoa used for assisted reproduction. The present experiment concerns IVF, a technique requiring careful evaluation of sperm quality to predict the success of the program. Our results demonstrate that the sperm submicroscopic characters introduced in the formula are clearly correlated with the result of IVF. In fact the two numbers concerning sperm quality (percentage of spermatozoa free from structural defects and total number in the ejaculate of spermatozoa free from defects) obtained in successful and unsuccessful IVF groups, showed a large difference. The t distribution in both cases reached a significance of 0.005. The synthetic parameters obtained are therefore a good tool in the prediction of sperm power in in vitro insemination techniques. The most important characteristics seem to be the quality of the acrosomal complex, the status of the chromatin, the shape of mitochondria, the axonemal pattern, and the membrane integrity. All these characteristics are expressed with largely different means in successful and unsuccessful ejaculates (t distribution significant at 0.005). All these data confirm that submicroscopic mathematical diagnosis offers a convincing evaluation of sperm structure and function, involving all organelles, including acrosome function and cell motility. It is also demonstrated that sperm quality is a major factor in the success of IVF and that it is clearly revealed by the integrity of the majority of the sperm organelles.

Acrosome↗
PubMed1993

The Glasgow Coma Scale: a mathematical critique.

The Glasgow Coma Scale permits 120 possible mathematical combinations of eye, verbal and motor scores. Out of these only about 15 are clinically valid and useful in the assessment of altered consciousness. A mathematical analysis of this pruned scale shows a predominant skew towards the motor response. Without clinically altering the scale. The numerical values can be modified to produce a more equitable dominance by each of the factors and greater precision. This is also necessary as the value of a unit is the same in the sum score, whether contributed by the eye, verbal or motor elements.

Coma↗
PubMed1991

A mathematical model of the relationship between cerebral blood volume and intracranial pressure changes: the generation of plateau waves.

The relationship between intracranial pressure (ICP), cerebral blood volume (CBV), cerebrospinal fluid dynamics, and the action of cerebral blood-flow (CBF) regulatory mechanisms is examined in this work with the help of an original mathematical model. In building the model, particular emphasis is placed on reproducing the mechanical properties of proximal cerebral arteries and small pial arterioles, and their active regulatory response to perfusion pressure and cerebral blood flow changes. The model allows experimental results on cerebral vessel dilatation and cerebral blood-flow regulation, following cerebral perfusion pressure decrease, to be satisfactorily reproduced. Moreover, the effect of cerebral blood volume changes--induced by autoregulatory adjustments--on the intracranial pressure time pattern can be examined at different levels of arterial hypotension. The results obtained with normal parameter values demonstrate that, at the lower limits of autoregulation, when dilatation of small arterioles becomes maximal, the increase in cerebral blood volume can cause a significant, transient increase in intracranial pressure. This antagonism between intracranial pressure and autoregulatory adjustments can lead to instability of the intracranial system in pathological conditions. In particular, analysis of the linearized system "in the small" demonstrates that an impairment in cerebrospinal fluid (CSF) reabsorption, a decrease in intracranial compliance and a high-regulatory capacity of the cerebrovascular bed are all conditions which can lead the system equilibrium to become unstable (i.e., the real part of at least one eigenvalue to turn out positive). Accordingly, mathematical simulation "in the large," in the above-mentioned conditions, exhibits intracranial pressure periodic fluctuations which closely resemble, in amplitude, duration, frequency and shape, the well-known Lundberg A-waves (or plateau waves).

Biomechanical Phenomena↗
PubMedDate not supplied

[Mathematical models of dose fractionation based on LQ function. (population-tissue models)].

A mathematical model was developed to calculate the probability of tumor tissue sterilization. It is assumed that tumor tissue contains normal and radio-resistant tumor cells and the survival of both types of tumor cells can be described by LQ functions. A package of programmes was created to solve the extreme problems in the determination of the parameters of LQ functions and the relative count of radio-resistant cells in the volume of tumor tissue. A programme complex was devised to solve practical tasks in radiological care. The series of tasks, which illustrate various aspects of determination of the parameters of the mathematical model by using clinical data and calculating the probability of tumor tissue radiation sterilization.

Carcinoma, Squamous Cell↗
PubMed1984

Mathematical modeling and numerical solutions for functionally dependent bone remodeling.

The phenomenon of bone remodeling is a complex biological process which is dependent on genetic, hormonal, metabolic, and age-related factors as well as functional requirements. The possibility of successfully developing a mathematical model to describe and predict the adaptive response of bone to load will be significantly increased after identification of the nature of the transducer(s) which senses functional requirements and provides signals for the cellular processes responsible for bone synthesis and bone removal. In spite of the present limitations in knowledge about the functional dependence of bone remodeling, a phenomenological model has been developed that assumes that the output signal from the (as yet unspecified) transducer is a remodeling potential that can be modulated by genetic, hormonal, and metabolic factors. An attempt has been made to cast the mathematical model in such a form that the constants and variables appearing in the equations are not mere abstractions, but can be related to biological parameters. In order to use the adaptive hypothesis with specific structural model examples, a numerical procedure has been developed to determine the strain distribution, predict the remodeling (assuming that the remodeling rate is related to the strain history), and update the model by changing the geometry and material properties in response to the remodeling. This numerical procedure is repeatedly iterated to determine the structural architecture at subsequent times. The numerical approach allows use of the remodeling concepts with models of irregular geometry, inhomogeneous material distribution, and anisotropic material properties.

Adaptation, Physiological↗