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PubMed1996

Use of mathematically enhanced spectral analysis and spectral contrast techniques for the liquid chromatographic and capillary electrophoretic detection and identification of pharmaceutical compounds.

The use of mathematically enhanced ultraviolet/visible (UV/VIS) absorbance spectral analysis and spectral contrast software techniques in high performance liquid chromatography (HPLC) and micellar electrokinetic capillary electrophoresis (MECC) as an aid for the determination of peak homogeneity, identification, and tracking during method development was investigated. Various structurally similar pharmaceutical compounds, and compounds present as either cis/trans isomers, diastereomers, or enantiomers were used as test compounds to probe the limits of this technique. Two tricyclic antidepressants, nortriptyline and imipramine, were employed to study the effects of HPLC mobile phase composition and pH on the ability to identify and track peaks during method development. It was found that method changes altered the spectral matches used for identification, but not enough to cause incorrect peak identification. It was also shown using HPLC that the cis/trans isomers of doxepin and the diastereomers ephedrine and pseudoephedrine could be distinguished. The mathematically enhanced spectral analysis and spectral contrast software techniques were also employed with MECC. Peaks tracking during method development as pH and the concentration of surfactant changes is shown for a separation of various penicillin type antibiotics. It was shown that during chiral MECC (CMECC) analyses ephedrine/pseudoephedrine diastereomers as well as ephedrine enantiomers could be distinguished. The determination of enantiomers is possible in CMECC since enantiomers are eluted as diastereomeric complexes, as opposed to HPLC where they are eluted in their native state.

Antidepressive Agents, Tricyclic↗
PubMed1995

Mathematical determination of the tibial insertion of the patellar tendon using computed tomography images.

Misalignment of the extensor apparatus is an essential factor in impairment of the patellar-femoral joint. This may be partly or entirely responsible for patellar dislocation or lateral patellar-femoral arthrosis. One surgical method to correct the pathology is medial transposition of the patellar ligament on the tibial tuberosity (anteriorly or posteriorly, distally or ventrally). These interventions correct misalignment of the extensor apparatus relative to statistical norms. We propose a mathematical method based on the use of computed tomography (CT) images to determine the ideal tibial insertion for the patellar tendon. This method is based on biomechanical modeling and the use of equipressure criteria. It is the first step in allowing the use of mathematics to model correctly tibial insertion of the patellar ligament, an entirely new development. This is important because it will allow surgeons greater accuracy in distal correction of extensor apparatus misalignments.

Biomechanical Phenomena↗
PubMed1987

The frequency of cervical cancer screening. Comparison of a mathematical model with empirical data.

The results of a mathematical model used to analyze the frequency of the Pap smear are compared with a recently published independent empirical study of data from large screening programs in Europe and North America. The model's predictions of the reduced incidence of invasive cervical cancer achieved with different screening frequencies match the empirical results closely--the predictions were within 1% of the empirical results for screening frequencies ranging from 1 to 10 years. The data indicate that compared with annual screening, screening every 2, 3, 5, and 10 years retains 99%, 97%, 89%, and 69%, respectively, of the effectiveness measured as a reduction in frequency of invasive cancer. The mathematical model underestimated the effectiveness of screening every 3 years, compared with screening every year.

Europe↗
PubMed2002

Mathematical models of HIV pathogenesis and treatment.

We review mathematical models of HIV dynamics, disease progression, and therapy. We start by introducing a basic model of virus infection and demonstrate how it was used to study HIV dynamics and to measure crucial parameters that lead to a new understanding of the disease process. We discuss the diversity threshold model as an example of the general principle that virus evolution can drive disease progression and the destruction of the immune system. Finally, we show how mathematical models can be used to understand correlates of long-term immunological control of HIV, and to design therapy regimes that convert a progressing patient into a state of long-term non-progression.

Algorithms↗
PubMed2005

Mathematical model of influenza A virus production in large-scale microcarrier culture.

A mathematical model that describes the replication of influenza A virus in animal cells in large-scale microcarrier culture is presented. The virus is produced in a two-step process, which begins with the growth of adherent Madin-Darby canine kidney (MDCK) cells. After several washing steps serum-free virus maintenance medium is added, and the cells are infected with equine influenza virus (A/Equi 2 (H3N8), Newmarket 1/93). A time-delayed model is considered that has three state variables: the number of uninfected cells, infected cells, and free virus particles. It is assumed that uninfected cells adsorb the virus added at the time of infection. The infection rate is proportional to the number of uninfected cells and free virions. Depending on multiplicity of infection (MOI), not necessarily all cells are infected by this first step leading to the production of free virions. Newly produced viruses can infect the remaining uninfected cells in a chain reaction. To follow the time course of virus replication, infected cells were stained with fluorescent antibodies. Quantitation of influenza viruses by a hemagglutination assay (HA) enabled the estimation of the total number of new virions produced, which is relevant for the production of inactivated influenza vaccines. It takes about 4-6 h before visibly infected cells can be identified on the microcarriers followed by a strong increase in HA titers after 15-16 h in the medium. Maximum virus yield Vmax was about 1x10(10) virions/mL (2.4 log HA units/100 microL), which corresponds to a burst size ratio of about 18,755 virus particles produced per cell. The model tracks the time course of uninfected and infected cells as well as virus production. It suggests that small variations (<10%) in initial values and specific rates do not have a significant influence on Vmax. The main parameters relevant for the optimization of virus antigen yields are specific virus replication rate and specific cell death rate due to infection. Simulation studies indicate that a mathematical model that neglects the delay between virus infection and the release of new virions gives similar results with respect to overall virus dynamics compared with a time delayed model.

Animals↗
PubMed2005

Segmentation of skull and scalp in 3-D human MRI using mathematical morphology.

We present a new technique for segmentation of skull and scalp in T(1)-weighted magnetic resonance images (MRIs) of the human head. Our method uses mathematical morphological operations to generate realistic models of the skull, scalp, and brain that are suitable for electroencephalography (EEG) and magnetoencephalography (MEG) source modeling. We first segment the brain using our Brain Surface Extractor algorithm; using this, we can ensure that the brain does not intersect our skull segmentation. We next generate a scalp mask using a combination of thresholding and mathematical morphology. We use the scalp mask in our skull segmentation procedure, as it allows us to automatically exclude background voxels with intensities similar to those of the skull. We find the inner and outer skull boundaries using thresholding and morphological operations. Finally, we mask the results with the scalp and brain volumes to ensure closed and nonintersecting skull boundaries. Visual evaluation indicated accurate segmentations of the cranium at a gross anatomical level (other than small holes in the zygomatic bone in eight subjects) in all 44 MRI volumes processed when run using default settings. In a quantitative comparison with coregistered CT images as a gold standard, MRI skull segmentation accuracy, as measured using the Dice coefficient, was found to be similar to that which would be obtained using CT imagery with a registration error of 2-3 mm.

Algorithms↗
PubMed1994

Mathematical modeling of water permeability of surgical fabrics for vascular use.

The purpose of this study was to derive new mathemtic formulae that could be used reliably to predict water permeability of surgical fabrics before they are made and tested for water permeability. Such a theoretical prediction of water permeability, Qprd, of surgical fabrics is needed for not only timely characterization but also for assisting in more efficient future design and development of better surgical fabrics. Two mathematic formulae, Qw and Qk, were derived from the Buckingham Pi Theorem, in which relevant fiber and fabric parameters were placed into dimensionless pi groups and computed for 25 commercial and experimental vascular fabrics. Linear regression analysis of the relationship between these pi groups with water permeability on the woven and knitted grafts yielded coefficients for the corresponding pi groups that were required for constructing appropriate mathematic formulae to predict water permeability of vascular fabrics. When proper sources of the experimentally determined water permeability, Qexp, were chosen for comparison, we found that 86% of woven fabrics (6 of 7) and 77% of knitted fabrics (14 of 18) had their Qprd within 10% of their Qexp. This high percentage of close matching (within 10%) between Qprd and Qexp should be considered satisfactory because the experimental error for obtaining Qexp is generally higher than 10%. The difference between Qprd and Qexp ranged from as small as 0.27% to as high as 74.2%, depending on the type of fabrics and source of Qexp.(ABSTRACT TRUNCATED AT 250 WORDS)

Models, Theoretical↗
PubMed1994

Description of the mathematical law that defines the relaxation of bovine pericardium subjected to stress.

A material subjected to traction stress increases in length; if we maintain the elongation constant, the stress varies over a period of time. This phenomenon has been referred to as relaxation. The purpose of this study was to define a mathematical law that relates the variation in stress to time when elongation remains constant in bovine pericardium. The mathematical function obtained after assaying 34 samples to the point of relaxation, subjected to initial stresses ranging from 0.17-10.07 MPa, responds to the following equation: y = -0.0252 + 0.953 alpha - (0.0165 + 0.015 alpha)lnt, where y is the stress withstood at an instant in time, t, after initial stress alpha. A normogram, validated by assays of up to 6,340 min duration (4.40 days), is presented for graphic calculation, permitting the computation of the loss of stress due to relaxation of this biomaterial, with initial stresses ranging from 1-10 MPa.

Animals↗
PubMed1988

The morphogenesis of molluscan shells: a mathematical account using biological parameters.

In attempts to account mathematically for the morphogenesis of biological structures it is important that the parameters chosen for the purpose should be "biological," that is, they should refer directly to the growth processes through which the structure is formed. Molluscan shells are formed by accretional growth at the mantle edge, and the parameters used for the mathematical description of their formation should therefore refer to events taking place there. In the best of previous attempts to solve this problem, two out of three form parameters have met with this demand, but it was not possible to eliminate beta, half the apical angle of the cone on which lie the centers of successive whorls. It is shown here that beta may be eliminated for many forms of molluscan shells, and when this is not the case beta may indeed be referred to processes taking place at the mantle edge.

Animals↗
PubMed2003

Mathematical model to simulate the cellular dynamics of infection with human herpesvirus-6 in EBV-negative infectious mononucleosis.

Acute infection with human herpesvirus-6 induces physiological cell proliferation in persons without major immune deficiency. It thus can serve as a parameter to validate a mathematical model designed to simulate cell proliferation under physiological and pathological conditions. Such a mathematical model is presented to simulate various cell changes of the T-cell immune system during the course of HHV-6 infection. Model development follows several steps, beginning with a basic model containing physiological T-cell pools to the introduction of infectious stimuli in the final model. A search algorithm designed to optimize the system parameters, as well as initial variables of the model, is presented. The results of simulation runs for acute HHV-6 infection of the final computational model correspond well to the data, as documented in human patients; they suggest that the computational model presented for the simulation of T-cell levels in a given viral infection may well serve as a tool for similar studies of other viral infections, including those that lead to cellular aplasia or neoplasia.

Adolescent↗
PubMed2002

Examination of the flow rheological and textural properties of polymer gels composed of poly(methylvinylether-co-maleic anhydride) and poly(vinylpyrrolidone): rheological and mathematical interpretation of textural parameters.

The purpose of this study was to mathematically characterize the effects of defined experimental parameters (probe speed and the ratio of the probe diameter to the diameter of sample container) on the textural/mechanical properties of model gel systems. In addition, this study examined the applicability of dimensional analysis for the rheological interpretation of textural data in terms of shear stress and rate of shear. Aqueous gels (pH 7) were prepared containing 15% w/w poly(methylvinylether-co-maleic anhydride) and poly(vinylpyrrolidone) (PVP) (0, 3, 6, or 9% w/w). Texture profile analysis (TPA) was performed using a Stable Micro Systems texture analyzer (model TA-XT 2; Surrey, UK) in which an analytical probe was twice compressed into each formulation to a defined depth (15 mm) and at defined rates (1, 3, 5, 8, and 10 mm s(-1)), allowing a delay period (15 s) between the end of the first and beginning of the second compressions. Flow rheograms were performed using a Carri-Med CSL(2)-100 rheometer (TA Instruments, Surrey, UK) with parallel plate geometry under controlled shearing stresses at 20.0 degrees +/- 0.1 degrees C. All formulations exhibited pseudoplastic flow with no thixotropy. Increasing concentrations of PVP significantly increased formulation hardness, compressibility, adhesiveness, and consistency. Increased hardness, compressibility, and consistency were ascribed to enhanced polymeric entanglements, thereby increasing the resistance to deformation. Increasing probe speed increased formulation hardness in a linear manner, because of the effects of probe speed on probe displacement and surface area. The relationship between formulation hardness and probe displacement was linear and was dependent on probe speed. Furthermore, the proportionality constant (gel strength) increased as a function of PVP concentration. The relationship between formulation hardness and diameter ratio was biphasic and was statistically defined by two linear relationships relating to diameter ratios from 0 to 0.4 and from 0.4 to 0.563. The dramatically increased hardness, associated with diameter ratios in excess of 0.4, was accredited to boundary effects, that is, the effect of the container wall on product flow. Using dimensional analysis, the hardness and probe displacement in TPA were mathematically transformed into corresponding rheological parameters, namely shearing stress and rate of shear, thereby allowing the application of the power law (eta = kgamma(n)) to textural data. Importantly, the consistencies (k) of the formulations, calculated using transformed textural data, were statistically similar to those obtained using flow rheometry. In conclusion, this study has, firstly, characterized the relationships between textural data and two key instrumental parameters in TPA and, secondly, described a method by which rheological information may be derived using this technique. This will enable a greater application of TPA for the rheological characterization of pharmaceutical gels and, in addition, will enable efficient interpretation of textural data under different experimental parameters.

Adhesiveness↗
PubMed2003

Evaluation of nose-only aerosol inhalation chamber and comparison of experimental results with mathematical simulation of aerosol deposition in mouse lungs.

In vivo small rodent efficacy testing of new synthetic and biological molecules for the pulmonary route requires an efficient delivery device. For this purpose, a nose-only inhalation chamber was used to deliver aerosolized aqueous compounds to the respiratory tract of mice. The aim of the study was to determine the efficiency of dose delivery and deposition in the lungs of the mice using this chamber. A secondary goal was to compare the experimental lung deposition results with values predicted from mathematical simulation. Experimental tests were conducted by generating aerosols of a radiolabeled formulation of human serum albumin (HSA) with a mass median aerodynamic diameter (MMAD) of 3.9 +/- 0.5 microm and a geometric standard deviation (GSD) of 1.43 +/- 0.05 using PARI LC STAR jet nebulizers. Based on the total activity placed in the nebulizer, the chamber delivered 0.108 +/- 0.027% to the mice and 0.0087 +/- 0.0021% to the lungs of the mice. In vivo lung deposition was found to be 8.19 +/- 3.56% of total activity deposited in the mouse. Mathematical simulation predictions ranged between 5.89 and 4.40% for various breathing patterns, and did not differ significantly from the in vivo results (p > 0.10). These results provide important quantitative information relevant to aerosol delivery experiments in mouse models. Our results also suggest that the nose-only inhalation chamber would benefit from significant changes to increase the efficiency of deposition in mice such that it can be used for nebulization of expensive therapeutic drugs.

Administration, Inhalation↗
PubMed2006

A mathematical model to predict the size of the pellets formed in freeze pelletization techniques: parameters affecting pellet size.

A mathematical model was developed based on the theory of drop formation to predict the size of the pellets formed in the freeze pelletization process. Further the model was validated by studying the effect of various parameters on the pellet size such as viscosity of the pellet forming and column liquids, surface/interfacial tension, density difference between pellet forming and column liquids; size, shape, and material of construction of the needle tips and temperatures maintained in the columns. In this study, pellets were prepared from different matrices including polyethylene glycols and waxes. The column liquids studied were silicone oils and aqueous glycerol solutions. The surface/interfacial tension, density difference between pellet forming and column liquids and needle tip size were found to be the most important factors affecting pellet size. The viscosity of the column liquid was not found to significantly affect the size of the pellets. The size of the pellets was also not affected by the pellet forming liquids of low viscosities. An increase in the initial column temperature slightly decreased the pellet size. The mathematical model developed was found to successfully predict the size of the pellets with an average error of 3.32% for different matrices that were studied.

Diglycerides↗
PubMed1991

A mathematical computer stimulation model for the development of colonic polyps and colon cancer.

Currently known information about the development and progression of colon polyps and cancer is summarized and organized into a mathematical computer simulation model that successfully predicts the natural history of colon polyp and cancer development for an average patient with (1) familial polyposis coli (2) genetic susceptibility as measured by a positive family history, and (3) negative family history with a high fat diet. The mathematical model uses four distinct types of cells (normal, transformed, polypoid, and cancerous) and two kinetic processes (mutation and promotion). Arachidonic acid metabolites play a role in the model in the promotion of cancer from polyps, and account for that promotion through: (1) their effect on encouraging more polypoid cells in mitosis to move toward cancer; and (2) their immunosuppressive effect over time. The model also shows that one defect in allowing more cells to mutate to the transformed state is sufficient to account for the chain of events leading to the clinical sequelae of familial polyposis coli. A second genetic effect at another point in the process is unnecessary. The mechanism of action of Sulindac on colon polyps is explained by the model through inhibition of production of arachidonic acid metabolites, most notably prostaglandin E.

Adenomatous Polyposis Coli↗
PubMed2007

The effects of intense pulsed light (IPL) on blood vessels investigated by mathematical modeling.

BACKGROUND AND OBJECTIVES: Intense pulsed light (IPL) sources have been successfully used for coagulation of blood vessels in clinical practice. However, the broadband emission of IPL hampers the clinical evaluation of optimal light parameters. We describe a mathematical model in order to visualize the thermal effects of IPL on skin vessels, which was not available, so far. STUDY DESIGN/MATERIALS AND METHODS: One IPL spectrum was shifted towards the near infrared range (near IR shifted spectrum: NIRSS) and the other was heavily shifted toward the visible range (visible shifted spectrum: VSS). The broadband emission was separated in distinct wavelengths with the respective relative light intensity. For each wavelength, the light and heat diffusion equations were simultaneously solved with the finite element method. The thermal effects of all wavelengths at the given radiant exposure (15 or 30 J/cm2) were added and the temperature in the vessels of varying diameters (60, 150, 300, 500 microm) was calculated for the entire pulse duration of 30 milliseconds. RESULTS: VSS and NIRSS both provided homogeneous heating in the entire vessel. With the exception of the small vessels (60 microm), which showed only a moderate temperature increase, all vessels exhibited a temperature raise within the vessel sufficient for coagulation with each IPL parameter. The time interval for effective temperature raise in larger vessels (diameter >60 microm) was clearly shorter than the pulse duration. In most instances, the vessel temperature was higher for VSS when compared to NIRSS. CONCLUSIONS: We presented a mathematical model capable of calculating the photon distribution and the thermal effects of the broadband IPL emission within cutaneous blood vessels.

Blood Vessels↗
PubMed2002

A mathematical model that predicts the force-frequency relationship of human skeletal muscle.

In previous work we developed and validated a mathematical model that predicted force output from skeletal muscles subjected to six-pulse stimulation trains under isometric condition. The current study investigated the model's ability to predict force responses to longer stimulation trains under both nonfatigued and fatigued conditions. Using the six-pulse train model to predict the force produced by longer stimulation trains showed that the model was successful, but a modified parameter identification scheme was required. For most of the trains tested the model accounted for 95% of the variance in the experimental forces produced by stimulation trains, with mean frequencies from 12.5 to 100 HZ, train durations from 485 to 1000 ms, and number of pulses from 14 to 50 for both nonfatigued and fatigued muscles. The success of our mathematical model in predicting forces produced by stimulations with a wide range of frequencies, durations, and number of pulses implies great potential of the model for the identification of optimal activation patterns that should be used during functional electrical stimulation.

Actins↗
PubMed2001

Mathematical modeling of capillary formation and development in tumor angiogenesis: penetration into the stroma.

The purpose of this paper is to present a mathematical model for the tumor vascularization theory of tumor growth proposed by Judah Folkman in the early 1970s and subsequently established experimentally by him and his coworkers [Ausprunk, D. H. and J. Folkman (1977) Migration and proliferation of endothelial cells in performed and newly formed blood vessels during tumor angiogenesis, Microvasc Res., 14, 53-65; Brem, S., B. A. Preis, ScD. Langer, B. A. Brem and J. Folkman (1997) Inhibition of neovascularization by an extract derived from vitreous Am. J. Opthalmol., 84, 323-328; Folkman, J. (1976) The vascularization of tumors, Sci. Am., 234, 58-64; Gimbrone, M. A. Jr, R. S. Cotran, S. B. Leapman and J. Folkman (1974) Tumor growth and neovascularization: an experimental model using the rabbit cornea, J. Nat. Cancer Inst., 52, 413-419]. In the simplest version of this model, an avascular tumor secretes a tumor growth factor (TGF) which is transported across an extracellular matrix (ECM) to a neighboring vasculature where it stimulates endothelial cells to produce a protease that acts as a catalyst to degrade the fibronectin of the capillary wall and the ECM. The endothelial cells then move up the TGF gradient back to the tumor, proliferating and forming a new capillary network. In the model presented here, we include two mechanisms for the action of angiostatin. In the first mechanism, substantiated experimentally, the angiostatin acts as a protease inhibitor. A second mechanism for the production of protease inhibitor from angiostatin by endothelial cells is proposed to be of Michaelis-Menten type. Mathematically, this mechanism includes the former as a subcase. Our model is different from other attempts to model the process of tumor angiogenesis in that it focuses (1) on the biochemistry of the process at the level of the cell; (2) the movement of the cells is based on the theory of reinforced random walks; (3) standard transport equations for the diffusion of molecular species in porous media. One consequence of our numerical simulations is that we obtain very good computational agreement with the time of the onset of vascularization and the rate of capillary tip growth observed in rabbit cornea experiments [Ausprunk, D. H. and J. Folkman (1977) Migration and proliferation of endothelial cells in performed and newly formed blood vessels during tumor angiogenesis, Microvasc Res., 14, 73-65; Brem, S., B. A. Preis, ScD. Langer, B. A. Brem and J. Folkman (1997) Inhibition of neovascularization by an extract derived from vitreous Am. J. Opthalmol., 84, 323-328; Folkman, J. (1976) The vascularization of tumors, Sci. Am., 234, 58-64; Gimbrone, M. A. Jr, R. S. Cotran, S. B. Leapman and J. Folkman (1974) Tumor growth and neovascularization: An experimental model using the rabbit cornea. J. Nat. Cancer Inst., 52, 413-419]. Furthermore, our numerical experiments agree with the observation that the tip of a growing capillary accelerates as it approaches the tumor [Folkman, J. (1976) The vascularization of tumors, Sci. Am., 234, 58-64].

Angiogenesis Inducing Agents↗
PubMed2002

Lipoprotein oxidation and its significance for atherosclerosis: a mathematical approach.

Atherosclerosis is a chronic disease which involves the build up of cholesterol and fatty deposits within the arterial wall. This results in the narrowing of the vessel lumen, which eventually restricts blood flow to vital organs such as the heart and lungs. These events may culminate in a heart attack or stroke, the commonest causes of death in the U.K. population. In this paper we study the early stages of atherosclerosis which include the build up of cholesterol within subendothelial cells to form what is known as a fatty streak, the earliest identifiable evidence of atherosclerosis. The deposition of cholesterol is believed to be a consequence of oxidation of circulating cholesterol-rich lipoproteins, in particular low density lipoproteins (LDLs). Via a mathematical model we investigate this process of oxidation within the context of an in vitro framework. We first recreate existing experimental results and then extend the model to investigate phenomenon not studied by current experimental protocols. We find that the model displays hysteresis which reveals some interesting insights into possible in vivo events. Mathematical analysis of this behaviour predicts that vitamin E supplementation is not as beneficial as high density lipoproteins (HDLs) and vitamin C. Furthermore, the scavenging of oxidants by HDL can provide an important first line of defence against LDL oxidation.

Arteriosclerosis↗