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PubMed2002

The value of mathematical modelling in understanding contrast enhancement in CT with particular reference to the detection of hypovascular liver metastases.

OBJECTIVE: Few subjects in body imaging have generated such intensive debate as that surrounding the optimising of strategies for the detection of liver metastases using contrast-enhanced Computed Tomography (CT). Despite this, research on the subject has been almost entirely focused on experimental studies, little attention having been paid to the value of theoretical analyses. The authors' aim was to develop a relatively simple and robust mathematical model which could be implemented on a personal computer. METHODS: Simple differential equations describing the distribution of contrast agent between intra- and extra-vascular spaces during an infusion are set up. An analytic solution is obtained for the plasma/blood concentrations as a fraction of time and a solution for the interstitial concentrations as a fraction of time is obtained by converting the differential equations to difference equations which are solved in a stepwise manner. RESULTS: Vascular and hepatic parenchymal enhancement-time curves are generated which are in close agreement with expectations. The model may be used to compare different infusion regimes, e.g. lower versus higher dose, faster versus slower infusions, monophasic versus biphasic infusions and scan start delay. The implications of the results of the model for clinical protocol design are discussed and the special value of spiral/helical technology indicated. CONCLUSION: A simple mathematical model has been developed to model blood and tissue contrast enhancement in CT during contrast infusion. The model clarifies a number of issues related to contrast enhancement regimes for the study of the liver and these have been discussed.

Contrast Media↗
PubMed1973

The physical interpretation of mathematical models for sodium permeability changes in excitable membranes.

This paper deals with the physical interpretation of existing mathematical models which describe the transient sodium conductance changes in excitable membranes. It is shown that there are clear limitations to the specificity of inferences which may be drawn about physical mechanism from the behavior of abstract models. Within these limitations, it is shown that a pronounced inactivation shift is not necessarily evidence for coupling between the events responsible for the rise and inactivation of the sodium conductance, but that the inactivation shift may be associated with an event whose rate explicitly depends on the rate of continuous voltage change or magnitude of instantaneous voltage change.

Biological Transport↗
PubMed1998

Mathematical modeling of normal pharyngeal bolus transport: a preliminary study.

Dysphagia (difficulty in swallowing) is a common clinical symptom associated with many diseases, such as stroke, multiple sclerosis, neuromuscular diseases, and cancer. Its complications include choking, aspiration, malnutrition, cachexia, and dehydration. The goal in dysphagia management is to provide adequate nutrition and hydration while minimizing the risk of choking and aspiration. It is important to advance the individual toward oral feeding in a timely manner to enhance the recovery of swallowing function and preserve the quality of life. Current clinical assessments of dysphagia are limited in providing adequate guidelines for oral feeding. Mathematical modeling of the fluid dynamics of pharyngeal bolus transport provides a unique opportunity for studying the physiology and pathophysiology of swallowing. Finite element analysis (FEA) is a special case of computational fluid dynamics (CFD). In CFD, the flow of a fluid in a space is modeled by covering the space with a grid and predicting how the fluid moves from grid point to grid point. FEA is capable of solving problems with complex geometries and free surfaces. A preliminary pharyngeal model has been constructed using FEA. This model incorporates literature-reported, normal, anatomical data with time-dependent pharyngeal/upper esophageal sphincter (UES) wall motion obtained from videofluorography (VFG). This time-dependent wall motion can be implemented as a moving boundary condition in the model. Clinical kinematic data can be digitized from VFG studies to construct and test the mathematical model. The preliminary model demonstrates the feasibility of modeling pharyngeal bolus transport, which, to our knowledge, has not been attempted before. This model also addresses the need and the potential for CFD in understanding the physiology and pathophysiology of the pharyngeal phase of swallowing. Improvements of the model are underway. Combining the model with individualized clinical data should potentially improve the management of dysphagia.

Deglutition↗
PubMed2005

Mathematical model to assess the control of Aedes aegypti mosquitoes by the sterile insect technique.

We propose a mathematical model to assess the effects of irradiated (or transgenic) male insects introduction in a previously infested region. The release of sterile male insects aims to displace gradually the natural (wild) insect from the habitat. We discuss the suitability of this release technique when applied to peri-domestically adapted Aedes aegypti mosquitoes which are transmissors of Yellow Fever and Dengue disease.

Aedes↗
PubMed1994

Role of active changes in venous capacity by the carotid baroreflex: analysis with a mathematical model.

To elucidate the role of venous capacity active changes in short-term cardiovascular homeostasis, a mathematical model of the carotid-sinus baroreflex system has been developed. In the model the cardiovascular system is represented as the series arrangement of six lumped compartments, which synthesize the fundamental hemodynamic properties of the systemic arterial, systemic venous, pulmonary arterial, and pulmonary venous circulations as well as of the left and right cardiac volumes. Cardiac outputs from the left and right ventricles are computed as a function of both downstream arterial pressure (afterload) and upstream atrial pressure (preload). Four distinct feedback regulatory mechanisms, working on systemic arterial resistance, heart rate, systemic venous unstressed volume, and systemic venous compliance, are assumed to operate on the cardiovascular system in response to carotid sinus pressure changes. All model parameters, both in the cardiovascular system and in feedback regulatory mechanisms, have been assigned on the basis of physiological data now available. The model is used here to simulate the pattern of the main hemodynamic quantities in the short time period (1-2 min) after acute carotid sinus activation in vagotomized subjects. Simulation results indicate that the model can reproduce experimental data quite well, with reference both to open-loop experiments and to an acute blood hemorrhage performed in closed-loop conditions. Moreover, computer simulations indicate that active changes in venous unstressed volume are of primary importance in regulating cardiac output and systemic arterial pressure during activation of the carotid sinus baroreflex.

Animals↗
PubMed1985

A mathematical model to quantitate histamine release from basophils challenged in vitro with insect venom.

Using a two parameter mathematical model, it was possible to describe clinically obtained data from histamine release (HR) in vitro as a response to basophils challenged by insect venom. The Gompertz function used in this model not only reflects the observed HR data but can be used to predict maximal HR values from initial lower antigen concentrations HR determinations. The model was used to calculate the concentration of venom antigens releasing 25% of total intracellular histamine (HR25). This is clinically significant in that while knowledge of the HR25, or any other reference fraction, facilitates comparisons between allergic patients, it is impossible to empirically administer venom antigen concentrations that will produce exactly HR25 in a given clinical setting.

Adult↗
PubMed2005

Mathematical model for pressure losses in the hemodialysis graft vascular circuit.

Stenosis-induced thrombosis and abandonment of the hemodialysis synthetic graft is an important cause of morbidity and mortality. The graft vascular circuit is a unique low-resistance shunt that has not yet been systematically evaluated. In this study, we developed a mathematical model of this circuit. Pressure losses (deltaPs) were measured in an in vitro experimental apparatus and compared with losses predicted by equations from the engineering literature. We considered the inflow artery, arterial and venous anastomoses, graft, stenosis, and outflow vein. We found significant differences between equations and experimental results, and attributed these differences to the transitional nature of the flow. Adjustment of the equations led to good agreement with experimental data. The resulting mathematical model predicts relations between stenosis, blood flow, intragraft pressure, and important clinical variables such as mean arterial blood pressure and hematocrit. Application of the model should improve understanding of the hemodynamics of the stenotic graft vascular circuit.

Animals↗
PubMed2000

Maxillary distraction osteogenesis: a two-dimensional mathematical model.

Patients with cleft lip and palate with severe maxillary retrusion usually have a mandible with anterior-superior autorotation and subsequent overclosure and loss of the vertical facial dimension. Maxillary distraction osteogenesis can correct the sagittal maxillomandibular relationship and should simultaneously reestablish vertical dimension through maxillary vertical height increase and clockwise rotation of the mandible to restore facial balance. We present a two-dimensional mathematical model in the sagittal plane, which reestablishes sagittal and vertical skeletal deficiencies and proper occlusal alignment for planning maxillary advancement with distraction osteogenesis in patients with cleft lip and palate. The model is illustrated in a case of a 13-year-old boy with a complete bilateral cleft lip and palate and severe maxillary retrusion. The two-dimensional mathematical model described in this article allows the surgeon and orthodontist to calculate in a simple and accurate way the ideal distraction vector to advance the maxilla to its desired position.

Adolescent↗
PubMedDate not supplied

[Mathematical modeling of the dynamics of postradiation injury and recovery of intestinal epithelium].

A mathematical model for the dynamics of the crypt-villus system in irradiated mammals has been developed. The model involves a chalones mechanism of regulation crypt cell reproduction rate and represents a system of four nonlinear differential equations. The simulation results are in a good agreement with the experimental data obtained within a wide range of doses.

Animals↗
PubMed1991

Three-dimensional mathematical model analysis of the patellofemoral joint.

This paper is concerned with a mathematical model analysis of the patellofemoral joint in the human knee, taking into account the articular surface geometry and mechanical properties of the ligament. It was made by the application of a computer-aided design theory (previously studied) and it was possible to express the articular surface geometries in a mathematical formulation and hence elucidate the joint movement mechanics. This method was then applied to a three-dimensional geometrical model of the patellofemoral joint. For the modelling of tendofemoral contact at large angles of knee flexion, the geodestic line theory was adopted. Applying the Newton-Raphson method and the Runge-Kutta Gil method to the model, variables such as patellar attitudes, patellofemoral contact force and tensile force of the patellar ligament for various knee flexion angles were computed. Applying the Hertzian elastic theory, contact stress was also computed. These results showed good agreement with the previously reported experimental results. As an application for the model, some parameter analyses were performed in terms of the contact stress variations and compared with those of the normal knee. The simulation results indicated that both the Q-angle increase and decrease increased contact stress, the patella alta showed undulating variations of stress while the patella infera showed little change of stress, and the tibial tuberositas elevation showed 20-30% reduction of stress.

Aged↗
PubMed1976

A unified mathematical model for diffusion from drug-polymer composite tablets.

The derivation and experimental verification of a unified mathematical model for the estimation of drug release rate from drug-polymer composite tablets are presented. Cylindrical coordinates are utilized in the solution of the diffusion equation for a three-dimensional system. The model is applicable to tablets that range from the shape of a flat disk (radius greater than thickness) to that of a cylindrical rod (radius less than thickness). The general solution for the fraction of drug released at a time t is (see article). This approach to a three-dimensional system, utilizing cylindrical coordinates, presents a comprehensive method for the estimation of drug release rates from sustained release tablets with drug distributed homogeneously throughout a polymer matrix. The calculated and experimental drug diffusion rate of pyrimethamine from pyrimethamine-silicone rubber composite tablets that range in shape from that of a disk to a cylinder, and of hydrocortisone from EVA, polycaprolactone, and PVA terpolymer, are compared.

Caprolactam↗
PubMed1985

Mathematical models of cell variation seen in a heterogeneous malignant cell population.

We established mathematical models for the cell variation seen in a heterogeneous malignant cell population, with the supposition that it occurs as the result of the competition between two types of cells, (A) and (B), leading to a change of stem cells. Models I and II: In the case of differences in the ability of (A) and (B) cells to adapt themselves to an environment, the proportion of cells which are less adaptable to the environment decreases exponentially and eventually disappears. Model III supposes that under certain environmental conditions, the two types of cells exist simultaneously in fixed proportions, and transformations of (B) cell to (A) cell and of (A) cell to (B) cell occur at a certain rate but are independent of each other. This process is considered to follow the Markov's chain theory. Based on this supposition, we established Model III and introduced the concept of "coefficient of cell variation". We found that Model III fits the process of cell variation seen in m cell line and we calculated the coefficients of cell variation seen in this cell line in different environments. The possible mechanism of the cell variation of this cell line is discussed.

Animals↗
PubMedDate not supplied

[Mathematical model of the infection process in diphtheria for determining the therapeutic dose of antitoxic anti-diphtheria serum].

It is known that administration of horse serum against diphtheria toxin can cause autoimmune and allergic complications. Therefore it is important for improvement of serotherapy to develop methods of prediction of disease course and quantity of diphtheria toxin and antitoxic antibodies in a serum. We have developed the mathematical model of diphtheria infection, which consists of six differential equations describing dynamics of diphtheria toxin and antitoxic antibodies in a serum, quantity of infection agent and macrophages in a site of inflammation. This mathematical model allows to predict the course of infectious process, the level of diphtheria toxin and antitoxic antibodies in the sera of people with diphtheria and to calculate the individual therapeutic dose of antitoxic serum for each patient.

Diphtheria↗
PubMed1982

Kinetics of killing Listeria monocytogenes by macrophages: correlation of 3H-DNA release from labeled bacteria and changes in numbers of viable organisms by mathematical model.

Conventional methods of assessing antibacterial activities of macrophages by viable counting are limited by the precision of the statistics and are difficult to interpret quantitatively because of unrestrained extracellular growth of bacteria. An alternative technique based on the release of radioactive DNA from labeled bacteria has been offered as overcoming these drawbacks. To assess it for use with macrophages I have made a correlation with the conventional viable counting method using a mathematical model. Opsonized Listeria monocytogenes labeled with 3H-thymidine were exposed to rat macrophages for periods up to 4 hr. Numbers of viable bacteria determined after sonication increased exponentially in the absence of live cells and this growth rate was progressively inhibited by increasing numbers of macrophages. After a lag period of 30-60 min soluble 3H appeared in the supernatant, the amount increasing with time and numbers of macrophages. To correlate these data I developed a mathematical model that considered that changes in numbers of viable organisms were due to the difference between rates of 1) growth of extracellular bacteria and 2) killing within the macrophage. On the basis of this model curves of best fit to the viable counts data were used to predict the release of radioactivity, assuming that death of a bacterium led to the total release of its label. These predictions and the experimental data agreed well, the lag period of 30-60 min between death of the bacterium and release of radioactivity being consistent with intracellular digestion. Release of soluble radioactivity appears to be an accurate reflection of the number of bacteria killed within the macrophage.

Animals↗
PubMed2006

A mathematical model of pacemaker activity recorded from mouse small intestine.

The pacemaker activity of interstitial cells of Cajal (ICCs) has been known to initiate the propagation of slow waves along the whole gastrointestinal tract through spontaneous and repetitive generation of action potentials. We studied the mechanism of the pacemaker activity of ICCs in the mouse small intestine and tested it using a mathematical model. The model includes ion channels, exchanger, pumps and intracellular machinery for Ca2+ regulation. The model also incorporates inositol 1,4,5-triphosphate (IP3) production and IP3-mediated Ca2+ release activities. Most of the parameters were obtained from the literature and were modified to fit the experimental results of ICCs from mouse small intestine. We were then able to compose a mathematical model that simulates the pacemaker activity of ICCs. The model generates pacemaker potentials regularly and repetitively as long as the simulation continues. The frequency was set at 20 min(-1) and the duration at 50% repolarization was 639 ms. The resting and overshoot potentials were -78 and +1.2 mV, respectively. The reconstructed pacemaker potentials closely matched those obtained from animal experiments. The model supports the idea that cyclic changes in [Ca2+]i and [IP3] play key roles in the generation of ICC pacemaker activity in the mouse small intestine.

Action Potentials↗
PubMed1993

A mathematical model for cell cycle progression under continuous low-dose-rate irradiation.

A mathematical model of the progression of cells through the mitotic cycle under continuous low-dose-rate irradiation is described. The model considers explicitly two special cases: (a) when a fraction of cells disintegrate and disappear after mitosis and (b) when a fraction of cells which have reached mitosis do not progress further but do not disintegrate either. We have established a relationship between the parameters of the model and dose and/or the age of the cell at exposure. This formalism is applied to studies of the effects of dose rate on HeLa cells (Mitchell, Bedford, and Bailey, Radiat. Res. 79, 520-536, 1979; 80, 186-197, 1979). Detailed information on the fraction of cells of a certain biological age at a given chronological time is needed because of the variation in the radioresponse of the cells as a function of age.

Cell Cycle↗
PubMed1986

Mathematical models and natural history in cervical cancer screening.

This paper examines several mathematical models that have been developed to investigate cervical cancer screening. Each model is discussed from the perspective of the inferences that can be drawn about the natural history of cervical cancer. The modelling analyses of Barron and Richart, Coppleson and Brown, Albert, Knox and Habbema and colleagues are examined in an attempt to gain insight about the transition and duration properties of the preclinical stages of cervical cancer. The picture that emerges is one of a complex preclinical natural history. There is a clear indication of regression of carcinoma in situ, as well as an age-dependence of transition probabilities and duration of disease status.

Adult↗
PubMed2005

Prediction of the distance from the skin to the lumbar epidural space in the Greek population, using mathematical models.

BACKGROUND AND OBJECTIVES: The skin to lumbar epidural space distance (SLED) is variable, and therefore the ability to clinically predict the SLED may help increase the success of epidural anesthesia/analgesia. The goal of this study was to determine the relationship between the SLED and demographic/anthropometric variables in the Greek population, and develop a mathematical model for its prediction. METHODS: This prospective randomized study enrolled 406 male and female Greek patients who required an epidural block as part of their anesthetic management. With patients placed in the left lateral and knee-chest position, the lumbar epidural space was located by the loss of resistance to normal saline technique. Statistical analysis was used to identify the relationship between SLED, and the following variables were evaluated: age, weight, height, body mass index, body surface area, intervertebral space used, pregnancy, and geographic origin within Greece. RESULTS: No adverse events or dural punctures occurred. Mean SLED in the general population was 4.98 +/- 0.95 cm, with values significantly higher in males (5.37 +/- 0.88 cm) compared with females (4.83 +/- 0.93 cm). SLED was best associated with weight, body surface area, and body mass index. Mathematical formulae for prediction of SLED in the general population and the female population were derived from linear regression analysis. These formulae were able to predict approximately half of the observed variability in SLED. CONCLUSIONS: While mathematical models of SLED can be a useful tool, they should not be exclusively relied on in the clinical setting, but rather should be used as an adjunct to standardized techniques to improve the safety and efficacy of epidural anesthesia/analgesia.

Journal Article↗