Weibel's Model Research Articles

Nonneutral Weibel model plasma in the non-minimal CPT-odd coupling

Inspired by the extension of the Standard Model, we investigate the effects of the space-time anisotropy caused by Lorentz symmetry violation (LSV) on a plasma column confinement configuration. The model of Plasma taken into account is the z-pinch model that was in the earliest efforts in fusion power research. This model comprises particles in a nonequilibrium cylindrical distribution, which remains stationary in the absence of collisions. We propose a disturbance in the distribution by a Lorentz violation environment. As proposed by Carroll, Field, and Jackiw, in a scenario of (LSV), a background field vector could couple with the electromagnetic field, modifying the classical behavior of the electromagnetic field. As reported here, considering the presence of a background field vector, the intensities of the fields and particle densities would be disturbed by the influence of the LSV. For different values of the background field vector coupling constant, the contribution of the background vector field could modify the intensity of the electromagnetic fields, and concentrate even more the electrons densities in the edge of the plasma column, evidencing a behavior similar to a skin effect in this plasma column.

Numerical study on normal lung sounds in bronchial airways under different breathing intensities

BackgroundDue to the complexity of airways and the limitation of experiments, the production mechanism of the lung sounds in airways has not been fully understood, which often confuses diagnosis. MethodA 3D geometrical model of human airways (G5-G8) has been developed based on Weibel's model. Simulation on transient airflow and the noise production during exhalation under different breathing intensities (Q = 15, 30, 45, 60, 75, 90 L/min) has been carried out with Direct Noise Computation (DNC) and Ffowcs Williams-Hawkings (FW-H) method. Results(1) The junctions between airways are most likely to produce lung sounds, and the peak value is located in the junction between G7 and G6 at the middle of exhalation (about 0.75 s). (2) With the increase in breathing intensity, the average sound pressure level first increases, reaches the peak value at 70–75 L/min, and then drops. (3) Higher breathing intensity is helpful to produce the feature of wheezing, namely a comparatively higher sound pressure level in the range of 200–500 Hz. Moreover, this feature is prominent with the increase in breathing intensity.

Fine Particulate Matter Deposition in 3D Out-of-Plane Bifurcation Lung Airway

Fine particulate matter (PM2.5) produced from traffic-loaded urban areas, combustion processes during industrial processes, and dust resuspension during mechanical disturbances may cause considerable health hazards when it is inhaled, owing to an enhanced accumulation of this particulate material in the lung. While the in-plane airway structures of human lungs are usually used in numerical models, a 3D out-of-plane layout that can geometrically represent triple bifurcations airways of the so-called Weibel model is developed in the present study with the presence of fine particles inside. For in-plane airways, the centerlines of all generations always lie on the same plane. In contrast, 3D out-of-plane is typical of the centerline of each descendant generation that rotates 90° to the centerline of its grandmother generation. Given three different breathing conditions, ranging between 15 L‧m-1 and 60 L‧m-1, the sizes of the deposition particles considered herein vary from 0.3 μm to 0.75 μm. The numerical results are discussed in terms of the airflow patterns (e.g., streamlines and velocity contours and vectors), the particle deposition patterns, deposition fractions (DFs) in different sections of the lung, the correlation between Stokes number and total DFs, and the correlation between the total DFs and the PM2.5 diameters. The predictions in Generations 8 to 12 (G8-G12) reveal that deposition fractions (DFs) increase with respiration frequency and intensity. It is also observed that the more anterior the bronchus is, the more significant the particle deposition is, regardless of the respiratory state. Also, the total DF tends to increase as the particle size decreases. This will lead to the development of subsequent tracheal atrophy.

Fiber optic endoscopic optical coherence tomography (OCT) to assess human airways: The relationship between anatomy and physiological function during dynamic exercise.

Airway luminal area (Ai) influences respiratory mechanics during dynamic exercise; however, previous studies have investigated the relationship between airway anatomy and physiological function in different groups of individuals. The purpose of this study was to determine the effect of Ai on respiratory mechanics by making in vivo measures of airway dimensions and work of breathing (Wb) in the same individuals. Healthy participants (3F/2M; 23–45 years) completed a cycle exercise test to exhaustion. During exercise, Wb was assessed using an esophageal balloon catheter, while simultaneously assessing minute ventilation (V˙ E). On a separate day, subjects underwent a bronchoscopy procedure to capture optical coherence tomography (OCT) measures of three airways in the right lung. Each participant's Wb‐V˙ E data were fit to a non‐linear regression equation (Wb = a V˙ E 3 + b V˙ E 2) that partitions Wb into its turbulent resistive (a) and viscoelastic (b) components. Measures of Ai and luminal diameter were made for the 4th–6th airway generations. A composite index of airway size was calculated as the sum of the Ai for each generation and the total area of the 4th–6th generation was calculated based on Weibel's model. Constant a was significantly correlated to the Weibel model total airway area (r = −0.94, p = 0.017) and index of airway size (r = −0.929, p = 0.023), whereas constant b was not associated with either measure (both p > 0.05). We found that individuals who had the smallest Ai had the highest resistive Wb and our findings provide the basis for further study of the relationship between airway size and respiratory mechanics during exercise.

Controlled Release of Magnetic Particles for Drug Delivery in the Human Lung

Improving drug delivery to a specific site of body is an important issue. Magnetic drug targeting is an appropriate technique for drug delivery to a specified area. This method has been adopted in drug delivery to the lung tissues due to widespread outbreak of the lung cancer in the recent decades. Therefore, increasing the efficiency of drug delivery to the lung tissues is of paramount importance. In this article, controlled particle release from a determined area and magnetic drug targeting methods are used to deliver the drug to a tumor in the generation G2 of the human lung. Weibel model is used for modeling of the human lung airways, and particles are made of iron oxide magnetic cores and poly lactic coglycolic acid shell. Herein, two different methods of drug delivery are tested and compared. In the first method, controlled particle release in the absence of magnetic field is accomplished and in the second method, controlled particle release in the presence of magnetic field is performed for the first time. A coil is used to produce magnetic field. Comparison of the results obtained in this article with the previous results of the drug delivery to tumors by random particle release method (in the presence and absence of magnetic field) showed that controlled particle release method improves the deposition efficiency on the tumor surface and increases the efficiency from 0.09% (random particle release in the absence of magnetic field) and 5.69% (random particle release in the presence of magnetic field) for tumor with r/R = 1.5 to 87% and 100% in the controlled particle release without and with the presence of magnetic field, respectively. Consequently, the best method of drug targeting is controlled particle release in the presence of the magnetic field. Using the mentioned method, deposition efficiency reaches 100% for tumors with r/R values of 0.5, 1, and 1.5 in this article.

Implementation of a specific boundary condition for a simplified symmetric single-path CFD lung model with OpenFOAM

CFD modeling research about the lung airflow with a complete resolution and an adequate accuracy at all scales requires a great amount of computational resources due to the vast number of necessary grid elements. As a result, a common practice is to conduct simplifications that allows to manage it with ordinary computational power. In this study, the implementation of a special boundary condition in order to develop a simplified single conductive lung airway model, which exactly represents the effect of the removed airways, is presented. The boundary condition is programmed in the open-source software OpenFOAM®, and the developed source code is presented in the proper syntax. After this description, modeling accuracy is evaluated under different flow rate conditions typical of human breathing processes, including both inspiration and expiration movements. Afterward, a validation process is conducted using results of a Weibel's model (0-4 generations) simulation for a medium flow rate of 50 L/min. Finally, a comparison against the proposed boundary condition implemented in the commercial code ANSYS Fluent is made, which highlights the benefits of using the free code toolbox. The specific contribution of this paper will be to show that OpenFOAM® developed model can perform even better than other commercial codes due to a precise implementation and coupling of the default solver with the in-house functions by virtue of the open-source nature of the code.

Numerical simulation of magnetic drug targeting to a tumor in the simplified model of the human lung

BackgroundMagnetic drug targeting improves effectiveness of medicine application and reduces its side effects. In this method, drugs with magnetic core are released in the lung and they are steered towards the tumor by applying an external magnetic field. A number of researchers utilized numerical methods to study particle deposition in the lung, but magnetic drug delivery to the tumors in the human lung has not been addressed yet. MethodIn the present study, Weibel model is used for human airway geometry from generation G0–G3. Moreover, a tumor is considered in the lung, which is located in G2. Particles are made of iron oxide magnetic cores and poly lactic coglycolic acid shells. Fluid flow is assumed laminar and particles are coupled with the fluid by one-way method. The magnetic field is produced by a coil with law current intensities instead of a wire with high current intensities. Influences of various parameters such as particle diameter, magnetic source position, current intensity, and inlet mass flow rate and tumor size on the deposition efficiency on the tumor surface are reported. ResultsResults show that magnetic drug targeting enhances deposition efficiency on the tumor surface Furthermore, when the current intensity rises from 10 (A) to 20 (A), tumor enlarging, and increasing particle diameter, lead to deposition efficiency enhancement, but efficiency decreases by increasing mass flow rate. However, when current intensity is 20 (A), deposition efficiency decreases in two situations. The first situation is when mass flow rate is 7 (L/min) and particle diameter is 9 (µm), and the second one is in 10 (L/min) mass flow rate and 9 (µm) diameter. ConclusionThe results demonstrated that magnetic drug targeting is applicable and suitable for all tumors specially for small tumors (r/R = 0.5 in this case) that efficiency increase from 0% in the absence of magnetic field to more than 2% in the presence of magnetic field.

Magnetic aerosol drug targeting in lung cancer therapy using permanent magnet

Primary bronchial cancer accounts for almost 20% of all cancer death worldwide. One of the emerging techniques with tremendous power for lung cancer therapy is magnetic aerosol drug targeting (MADT). The use of a permanent magnet for effective drug delivery in a desired location throughout the lung requires extensive optimization, but it has not been addressed yet. In the present study, the possibility of using a permanent magnet for trapping the particles on a lung tumor is evaluated numerically in the Weibel's model from G0 to G3. The effect of different parameters is considered on the efficiency of particle deposition in a tumor located on a distant position of the lung bronchi and bronchioles. Also, the effective position of the magnetic source, tumor size, and location are the objectives for particle deposition. The results show that a limited particle deposition occurs on the lung branches in passive targeting. However, the incorporation of a permanent magnet next to the tumor enhanced the particle deposition fraction on G2 to up to 49% for the particles of 7 µm diameter. Optimizing the magnet size could also improve the particle deposition fraction by 68%. It was also shown that the utilization of MADT is essential for effective drug delivery to the tumors located on the lower wall of airway branches given the dominance of the air velocity and resultant drag force in this region. The results demonstrated the high competence and necessity of MADT as a noninvasive drug delivery method for lung cancer therapy.

Revisiting atelectasis in lung units with low ventilation/perfusion ratios.

Patients on high inspired O2 concentrations are at risk of atelectasis, a problem that has been quantitatively assessed using analysis of ratio of ventilation to perfusion (V̇a/Q̇) equations. This approach ignores the potential of the elastic properties of the lung to support gas exchange through "apneic" oxygenation in units with no tidal ventilation, and is based on an error in the conservation of mass equations. To fill this gap, we correct the error and compare the pressure drops associated with apneic gas exchange with the pressure differences that can be supported by lung recoil. We analyze a worst case scenario: a small test unit in the Weibel model A tree structure with zero tidal ventilation, 100% inspired O2, the rest of the lung being normally ventilated tidally. We first computed the gas flux to the (unventilated) test unit and estimated the associated pressure drops. We then computed the difference in local gas pressure relative to the surrounding lung that would cause the unit to collapse. We compared these two, and finally computed the degree of airway narrowing that would effect change from the stable (apneic gas exchange) regime to the unstable regime leading to collapse. We find that except under extreme conditions of loss of airway caliber exceeding roughly 90%, lung recoil is sufficient to maintain oxygenation through convective transport alone. We further argue that the fundamental V̇a/Q̇ equations are invalid in these circumstances, and that the issue of atelectasis in low V̇a/Q̇ will require modifications to account for this additional mode of gas exchange. NEW & NOTEWORTHY Breathing high concentrations of oxygen increases the likelihood of atelectasis because of oxygen absorption, which is thought to be inevitable in regions with relatively low ventilation/perfusion ratios. However, airspaces of the lung resist collapse because of the forces of interdependence, and can, with low or even zero active tidal ventilation, draw in an inspiratory flow of oxygen sufficient to replace the oxygen consumed, thus preventing collapse of airspaces served by all but the most narrowed airways.

New insights into the mechanisms controlling the bronchial mucus balance.

In this work, we aim to analyze and compare the mechanisms controlling the volume of mucus in the bronchial region of the lungs of a healthy human adult, at rest and in usual atmospheric conditions. This analysis is based on a balance equation for the mucus in an airway, completed by a computational tool aiming at characterizing the evaporation, during respiration, of the water contained in the bronchial mucus. An idealized representation of the lungs, based on Weibel’s morphometric model, is used. The results indicate that the mechanisms controlling the volume of mucus in an airway depend on the localization of the airway in the bronchial region of the lungs. In the proximal generations, the volume of mucus in an airway is mainly controlled by the evaporation of the water it contains and the replenishment, with water, of the mucus layer by epithelial cells or the submucosal glands. Nevertheless, cilia beating in this part of the bronchial region remains of fundamental importance to transport the mucus and hence to eliminate dust and pathogens trapped in it. On the other hand, in the distal generations of the bronchial region, the volume of mucus in an airway is mainly controlled by the mucociliary transport and by the absorption of liquid by the epithelium. This absorption is a consequence of the mucus displacement by the cilia along generations with an interface between the epithelium and the airway surface layer of decreasing area. The numerical results obtained are in good agreement with previously published experimental data, thus validating our approach. We also briefly discuss how our results can improve the understanding and, possibly, the treatment of pulmonary diseases.

Penetration of inhaled aerosols in the bronchial tree

It has long been recognized that the pattern of particle deposition in the respiratory tree affects how far aerosols penetrate into the deeper zones of the arterial tree, and hence contribute to either their pathogenic potential or therapeutic benefit. In this paper, we introduce an anatomically-inspired model of the human respiratory tree featuring the generations 0–7 in the Weibel model of respiratory tree (i.e., the conducting zone). This model is used to study experimentally the dynamics of inhaled aerosol particles (0.5–20µm aerodynamic diameter), in terms of the penetration fraction of particles (i.e., the fraction of inflowing particles that leave the flow system) during typical breathing patterns. Our study underline important modifications in the penetration patterns for coarse particles compared to fine particles. Our experiments suggest a significant decrease of particle penetration for large-sized particles and higher respiratory frequencies. Dimensionless numbers are also introduced to further understand the particle penetration into the respiratory tree. A decline is seen in the penetration fraction with decreasing Reynolds number and increasing Stokes number. A simple conceptual framework is presented to provide additional insights into the findings obtained.

An adaptable model for growth and/or shrinkage of droplets in the respiratory tract during inhalation of aqueous particles

The site of deposition of pulmonary delivered aerosols is dependent on the aerosol׳s droplet size distribution, which may change during inhalation. The aim of this study was to develop a freely accessible and adaptable model that describes the growth (due to condensation) and shrinkage (due to evaporation) of inhaled droplets as a function of the distance from the airway wall during various inhalation conditions, for a laminar flow scenario. This was achieved by developing a model with which the evaporation of water from a droplet surface or condensation of water onto the droplet surface can be calculated. This model was then applied to a second model that describes the heat and mass transfer from the airway wall to the inhaled aerosol. The latter was based on the Weibel model. It was found that the growth and shrinkage of inhaled droplets markedly differs, depending on the distance from the airway wall. Droplets near the wall start to grow immediately due to fast water vapor transfer from the wall to the cold inhaled air. This growth continues until the air reaches body temperature and is fully saturated. However, droplets in the center of the airway first evaporate partly, due to a delay in water vapor transfer from the airway wall, before they start to grow. Depending on the conditions during inhalation, the droplet size distribution can widen considerably, which may affect the lung deposition and thereby the efficacy of the inhalation therapy. In conclusion, the model was able to show the effect of the conditions in the respiratory tract on the growth and shrinkage of inhaled droplets during standard inhalation conditions. Future developments can be aimed at expanding the model to include turbulent flow and hygroscopic growth, to improve the accuracy of the model and make it applicable to both droplets of solutions and dry particles.

Association of Visceral Obesity and Glomerulomegaly Among Kidney Donors.

OBJECTIVE: Due to increasing demand and shortage of kidneys for donation, there has been a liberalization of criteria with obese donors now being increasingly accepted. However, obesity causes glomerulomegaly, which is an established risk factor for future chronic kidney disease. Identifying a non-invasive obesity pattern most closely associated with glomerulomegaly may help inform the living kidney donor selection. METHODS: Implantation renal biopsy sections were reviewed on 1075 kidney donors (between 1999 and 2012), and the mean non-sclerotic glomerular volumes determined using the Weibel and Gomez stereological model. The 20 kidney donors with the largest glomerular volume were age and gender matched to the 20 kidney donors with the smallest glomerular volume. Abdominal obesity patterns were characterized from pre-donation abdominal CT images. The subcutaneous and visceral fat areas were measured at the umbilicus. Peri-renal (deep to the fascia but superficial to the capsule and para-renal (superficial to the fascia) fat for both kidneys were measured on cross-sectional scans at ¼, mid, and ¾ of kidney length, and averaged. The sinus fat was measured at the renal hilum for both kidneys and averaged. The waist circumference was assessed at the umbilicus. RESULTS: Mean age was 44 years and 70% were female in both groups. The mean ± SD glomerular volume was 0.0010 ± 0.0003 mm3 in the low and 0.0060 ± 0.0011 mm3 in the high group (p <0.001). The table compares fat distribution between these two groups.Table: No Caption available.CONCLUSION: In this pilot study, the proportional increase in visceral obesity with glomerulomegaly was greater than the other indices of obesity. Since contrast computed tomographic imaging is routinely performed during predonation kidney evaluations, visceral fat quantification could potentially be used to better inform the kidney donor selection process.

Numerical Simulations of High-Frequency Respiratory Flows in 2D and 3D Lung Bifurcation Models

To better understand the human pulmonary system and optimize the high-frequency oscillatory ventilation (HFOV) design, numerical simulations were conducted under normal breathing frequency and HFOV condition using a CFD code Ansys Fluent and its user-defined C programs. 2D and 3D double bifurcating lung models were created, and the geometry corresponds to fifth to seventh generations of airways with the dimensions based on the Weibel's pulmonary model. Computations were carried out for different Reynolds numbers (Re = 400 and 1000) and Womersley numbers (α = 4 and 16) to study the air flow fields, gas transportation, and wall shear stresses in the lung airways. Flow structure was compared with experimental results. Both 2D and 3D numerical models successfully reproduced many results observed in the experiment. The oxygen concentration distribution in the lung model was investigated to analyze the influence of flow oscillation on gas transport inside the lung model.

Simulation of apneas at the end of argon inhalation, using seven parallel regions of the lung

Jeng et al. (Ann. Biomed. Eng, 28:453, 2000) could not simulate apneas after argon (Ar) inhalation because the convection‐diffusion equation, solved by diluting the inhaled Ar with FRC, obtained too much Ar in the alveoli. Using the model of 7 parallel regions to simulate a normal breath of O2, and CO2, new concepts were introduced in mixing the inhaled gas with the expiratory reserve volume (ERV), by the mechanism of convection‐diffusion during inspiration. During the time of expiration, gases diffused from RV to ERV for CO2 and from ERV to RV for O2 (Caucha et al. Am. J. Respir. Crit. Care Med. 183: A5180, 2011). These concepts are now applied to forced inspiratory (2s) and expiratory (3s) maneuvers by inhaling 1.9 L of 21% O2, 79% Ar from FRC and exhaling to RV. The breath was held at the end of inspiration for 5s, 10s and 20s. The simulated expired curves are shown in the Figure. The last 8, 9 and 10 generations of Weibel's anatomical model were used at the base, middle and apical regions, respectively, at 5s a 10s of apnea. For 20s it was necessary to use the last 10th and 11th generations for the base and apical regions. In conclusion, the model described is capable of simulating the experiments of Cruz (Respir. Physiol. 86:1, 1991).

Computational Fluid Dynamics Study of Airflow and Particle Transport in Third to Sixth Generation Human Respiratory Tract

The computational fluid dynamics (CFD) technique is used to simulate airflow and aerosol-particle deposition in human respiratory tract. The three dimensional respiratory tract is extended from third to six generation bronchus based on Weibel model (1963). The computation is performed for laminar flow condition. Discrete phase modeling (DPM) is used to study the two phase flow. The aerosol-particles of three different diameters 1, 5 and 10 μm are considered to obtain the effect of aerosol-particle size on deposition efficiency. The results of velocity contours, aerosol-particle deposition efficiency and aerosol-particle trapping process are determined at different locations of the respiratory tract. It is obtained that deposition efficiency increases with the increase of aerosol-particle size. It is also found that more are deposited at the bifurcation junctions because of the direct inertial impaction at these locations. Computational results are found useful to correlate the medical and engineering aspects.

Particle transport modeling in pulmonary airways with high-order elements

Inhaled particles can be either harmful (e.g., smoke, exhaust, viruses) or beneficial (e.g., a therapeutic drug). The accurate and computationally efficient simulation of particle transport and deposition remains a challenge because it requires the simultaneous solution of the Navier–Stokes equations and multiple advection–diffusion mass transport equations when the particles are modeled as multiple mono-dispersed populations. The solution of these equations requires that multiple length scales be resolved since the ratio of advection to diffusion varies among the different equations. Here, the spectral element method is examined because the high-order approximation provides greater flexibility in resolving multiple length scales. The problem geometry is based on the Weibel model A of the human airway for convergence tests and the first three generations of a typical rat airway for experimental validation. Particles in the size range 1 to 100 nm are simulated for deposition results. The particle concentration and flux were determined using meshes of varying coarseness to represent the geometry along with basis polynomials of order 5 to 11. The higher-order elements accurately propagate the short wavelengths contained in the advection–diffusion solution without sacrificing efficiency for the more computationally expensive Navier–Stokes solution. As the diffusion coefficient in the advection–diffusion equation decreases (i.e., particle size increases) the advantages of the spectral elements become apparent for the coupled system.

Numerical analysis of respiratory flow patterns within human upper airway

A computational fluid dynamics (CFD) approach is used to study the respiratory airflow dynamics within a human upper airway. The airway model which consists of the airway from nasal cavity, pharynx, larynx and trachea to triple bifurcation is built based on the CT images of a healthy volunteer and the Weibel model. The flow characteristics of the whole upper airway are quantitatively described at any time level of respiratory cycle. Simulation results of respiratory flow show good agreement with the clinical measures, experimental and computational results in the literature. The air mainly passes through the floor of the nasal cavity in the common, middle and inferior nasal meatus. The higher airway resistance and wall shear stresses are distributed on the posterior nasal valve. Although the airways of pharynx, larynx and bronchi experience low shear stresses, it is notable that relatively high shear stresses are distributed on the wall of epiglottis and bronchial bifurcations. Besides, two-dimensional fluid–structure interaction models of normal and abnormal airways are built to discuss the flow-induced deformation in various anatomy models. The result shows that the wall deformation in normal airway is relatively small.

New method of calculating alveolar CO2 transport

Cruz proposed that the expirogram of CO2 does not represent the process of gas exchange (FASEB J. 21:609.4, 2007). It represents a simple alveolar gas dilution by the tidal volume that does not reach the residual volume (RV). Caucha et al. showed that phase II is formed by the distribution of regional dead spaces (FASEB J. 22:763.6, 2008). However, this latter model could not predict which anatomic Weibel's model generations were involved. Thus, here, we modify Caucha's model by separating the convection‐diffusion process (method 1) in three different steps (method 2): 1. Convection. 2. Convection‐diffusion and 3. Diffusion. As a result, new alveolar CO2 concentrations at the apex and base regions of the lung were found (see figure, interrupted and continuous lines for method 1 and 2, respectively). This picture was obtained after searching the generations involved: the first 8 generations (step 1), generations 9th‐19th (step 2) and the last 5 generations (step 3), which produced the simulated expirogram similar to the experimental curve. In conclusion, the analysis presented follows our theory of alveolar gas mixing; during inspiration, the alveolar CO2 is diluted by the tidal volume (figure), and during expiration, the uneven emptying of parallel regions of the lung generates the expirogram.Supported in part by ONG ‐ Centro de Enseñanza Investigación y Servicios.

Modeling of Aerosol Transport and Deposition in the Human Respiratory Tract: A Mesoscopic Approach

A dynamical model of respiratory deposition is developed using a mesoscopic approach, which is based on a stochastic description defined by a discrete master equation. The model simulates detailed lung deposition along all generations of the human respiratory tract by solving numerically the convection-diffusion equation with an adsorption and coagulation terms. This continuity equation is solved in onedimensional using a finite difference method. To describe the respiratory tract geometry, the classical Weibel's morphometric model is used. The simulation results show that the model compares well with the existing models and have fewer parameters. African Journal of Educational Studies in Mathematics and Sciences Vol. 5 2007: pp. 61-69