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PubMed2003

Mathematical models of cell colonization of uniformly growing domains.

During the development of vertebrate embryos, cell migrations occur on an underlying tissue domain in response to some factor, such as nutrient. Over the time scale of days in which this cell migration occurs, the underlying tissue is itself growing. Consequently cell migration and colonization is strongly affected by the tissue domain growth. Numerical solutions for a mathematical model of chemotactic migration with no domain growth can lead to travelling waves of cells with constant velocity; the addition of domain growth can lead to travelling waves with nonconstant velocity. These observations suggest a mathematical approximation to the full system equations, allowing the method of characteristics to be applied to a simplified chemotactic migration model. The evolution of the leading front of the migrating cell wave is analysed. Linear, exponential and logistic uniform domain growths are considered. Successful colonization of a growing domain depends on the competition between cell migration velocity and the velocity and form of the domain growth, as well as the initial penetration distance of the cells. In some instances the cells will never successfully colonize the growing domain. These models provide an insight into cell migration during embryonic growth, and its dependence upon the form and timing of the domain growth.

Animals↗
PubMed1997

The necessity of cofactors in the pathogenesis of AIDS: a mathematical model.

Current arguments for the role of cofactors in the initiation of a chronic HIV infection and progression of AIDS are given. The natural history of an HIV infection as affected by cofactors which provide additional stimulatory signals is explored through a mathematical model. The model demonstrates that "antigen load" plays a role in determining susceptibility to an HIV infection. It also suggests that certain individuals may not be able to be infected by small doses of HIV and that the identification and treatment of existing cofactors may be useful in treating early stages of HIV infection. Prevention of cofactor exposures may also protect against HIV infection.

Acquired Immunodeficiency Syndrome↗
PubMed2005

Interpretation of relevance of sodium-calcium exchange in action potential of diabetic rat heart by mathematical model.

Sarcolemmal Na+-Ca2+ exchange plays a central role in ion transport of the myocardium and the current carried with it contributes to the late phase of the action potential (AP) besides the contribution of outward K+-currents. In this study, the mathematical model for AP of the diabetic rat ventricular myocytes [34] was modified and used for the diabetic rat papillary muscle. We used our experimentally measured values of two K+-currents; transient outward current, Ito and steady-state outward current, Iss, as well as L-type Ca2+-current, I(CaL), then compared with the simulated values. We have demonstrated that the prolongation in the AP of the papillary muscle of the diabetic rats are not due to the alteration of I(CaL) but mainly due to the inhibition of the K+-currents and also the Na+-Ca2+ exchanger current, I(Na-Ca). In combination with our experimental data on sodium-selenite-treated diabetic rats, our simulation results provide new information concerning plausible ionic mechanisms, and second a possible positive effect of selenium treatment on the altered I(Na-Ca) for the observed changes in the AP duration of streptozotocin-induced diabetic rat heart.

Action Potentials↗
PubMed1994

A mathematical model of detection and dynamics of porcine transmissible gastroenteritis.

Transmissible gastroenteritis (TGE) is a viral disease causing dehydration, diarrhoea and death in pigs. The disease is widespread in pig-producing areas of the world but does not occur in Australia. A mathematical model of TGE spread within a pig herd is proposed and calibrated by reference to published data. The model is then applied to two situations of special interest; first to estimate the delay before detection of TGE (6 to over 30 days) when infection is first introduced into a herd of domestic or feral pigs, and second the effect of the disease in a population of feral pigs (could become endemic if transmission is high).

Animals↗
PubMed2006

Cancer immunotherapy by interleukin-21: potential treatment strategies evaluated in a mathematical model.

The newly characterized interleukin (IL)-21 plays a central role in the transition from innate immunity to adaptive immunity and shows substantial tumor regression in mice. IL-21 is now developed as a cancer immunotherapeutic drug, but conditions for efficacious therapy, and the conflicting immunostimulatory and immunoinhibitory influence of the cytokine, are yet to be defined. We studied the effects of IL-21 on tumor eradication in a mathematical model focusing on natural killer (NK) cell-mediated and CD8+ T-cell-mediated lysis of tumor cells. Model parameters were estimated using results in tumor-bearing mice treated with IL-21 via cytokine gene therapy (CGT), hydrodynamics-based gene delivery (HGD), or standard interval dosing (SID). Our model accurately retrieved experimental growth dynamics in the nonimmunogenic B16 melanoma and the immunogenic MethA and MCA205 fibrosarcomas, showing a strong dependence of the NK-cell/CD8+ T-cell balance on tumor immunogenicity. Moreover, in melanoma, simulations of CGT-like dosing regimens, dynamically determined according to tumor mass changes, resulted in efficient disease elimination. In contrast, in fibrosarcoma, such a strategy was not superior to that of fixed dosing regimens, HGD or SID. Our model supports clinical use of IL-21 as a potent stimulator of cellular immunity against cancer, and suggests selecting the immunotherapy strategy according to tumor immunogenicity. Nonimmunogenic tumors, but not highly immunogenic tumors, should be controlled by IL-21 dosing, which depends on tumor mass at the time of administration. This method imitates, yet amplifies, the natural anticancer immune response rather than accelerates only one of the response arms in an unbalanced manner.

Animals↗
PubMed2005

Mathematical model of paracrine interactions between osteoclasts and osteoblasts predicts anabolic action of parathyroid hormone on bone.

To restore falling plasma calcium levels, PTH promotes calcium liberation from bone. PTH targets bone-forming cells, osteoblasts, to increase expression of the cytokine receptor activator of nuclear factor kappaB ligand (RANKL), which then stimulates osteoclastic bone resorption. Intriguingly, whereas continuous administration of PTH decreases bone mass, intermittent PTH has an anabolic effect on bone, which was proposed to arise from direct effects of PTH on osteoblastic bone formation. However, antiresorptive therapies impair the ability of PTH to increase bone mass, indicating a complex role for osteoclasts in the process. We developed a mathematical model that describes the actions of PTH at a single site of bone remodeling, where osteoclasts and osteoblasts are regulated by local autocrine and paracrine factors. It was assumed that PTH acts only to increase the production of RANKL by osteoblasts. As a result, PTH stimulated osteoclasts upon application, followed by compensatory osteoblast activation due to the coupling of osteoblasts to osteoclasts through local paracrine factors. Continuous PTH administration resulted in net bone loss, because bone resorption preceded bone formation at all times. In contrast, over a wide range of model parameters, short application of PTH resulted in a net increase in bone mass, because osteoclasts were rapidly removed upon PTH withdrawal, enabling osteoblasts to rebuild the bone. In excellent agreement with experimental findings, increase in the rate of osteoclast death abolished the anabolic effect of PTH on bone. This study presents an original concept for the regulation of bone remodeling by PTH, currently the only approved anabolic treatment for osteoporosis.

Animals↗
PubMed1986

The case for mathematical modelling of schistosomiasis.

Large problems, such as the large-scale transmission of parasitic disease, are complex and difficult to predict. Understanding them, in order to make cost-efficient choices about possible control interventions, requires knowledge from a very wide range of specialists. Modelling the system can help to do this, but must not ignore the specific requirements of administrators and executives who have to work with high-level decisions about disease control. To many, the modelling approach seems arcane and divorced from reality, unable to answer the decision-makers. But techniques are improving, and in this article, Norman Bailey puts the case for the mathematical modelling of schistosomiasis.

Journal Article↗
PubMedDate not supplied

Ex vivo biomechanical behavior of abdominal aortic aneurysm: assessment using a new mathematical model.

Knowledge of the biomechanical behavior of abdominal aortic aneurysm (AAA) as compared to nonaneurysmal aorta may provide information on the natural history of this disease. We have performed uniaxial tensile testing of excised human aneurysmal and nonaneurysmal abdominal aortic specimens. A new mathematical model that conforms to the fibrous structure of the vascular tissue was used to quantify the measured elastic response. We determined for each specimen the yield (sigma y) and ultimate (sigma u) strengths, the separate contribution to total tissue stiffness by elastin (EE) and collagen (EC) fibers, and a collagen recruitment parameter (A), which is a measure of the tortuosity of the collagen fibers. There was no significant difference in any of these mechanical properties between longitudinal and circumferential AAA specimens, nor in EE and EC between longitudinally oriented aneurysmal and normal specimens. A, sigma y, and sigma u were all significantly higher for the normal than for the aneurysmal group: A = 0.223 +/- 0.046 versus A = 0.091 +/- 0.009 (mean +/- SEM; p < 0.0005), sigma y = 121.0 +/- 32.8 N/cm2 versus sigma y = 65.2 +/- 9.5 N/cm2 (p < 0.05), and sigma u = 201.4 +/- 39.4 N/cm2 versus sigma u = 86.4 +/- 10.2 N/cm2 (p < 0.0005), respectively. Our findings suggest that the AAA tissue is isotropic with respect to these mechanical properties. The observed difference in A between aneurysmal and normal aorta may be due to the complete recruitment and loading of collagen fibers at lower extensions in the former. Our data indicate that AAA rupture may be related to a reduction in tensile strength and that the biomechanical properties of AAA should be considered in assessing the severity of an individual aneurysm.

Aged↗
PubMed1997

[Fourier analysis as a mathematical model for evaluating and presenting postoperative corneal topography data after non-mechanical perforating keratoplasty].

BACKGROUND: Videokeratography has given the possibility to obtain information in curvature from a much larger region of the cornea than that covered by keratometry. Fourier analysis as a mathematical model can be used to represent real physical attributes of the cornea and to divide corneal topography in its basic components: the zero-frequency component as the mean ring power, the one-cycle component as a representation of decentration and the two-cycle component as a representation of regular corneal toricity. The purpose of this study was the reconstruction of the corneal refraction after penetrating keratoplasty with a small number of characteristic parameters and the evaluation of the time course of the fourier coefficients as indices for a regular astigmatic cornea in the postkeratoplasty period including suture removal. PATIENTS AND METHODS: Fourty patients (group 1: 20 primary dystrophies, group 2: 20 keratoconus) underwent nonmechanical trephination (excimer laser MEL60, Aesculap-Meditec, Heroldsberg, Germany) in penetrating keratoplasty. All procedures (7.5 mm in dystrophies, 8.0 mm in keratoconus, 8 orientation teeth, double-running 10-0 nylon suture) were performed by one surgeon. At a postoperative gate of 6 weeks, 6 months, before partial suture removal and after complete suture removal, corneal topography (TMS1, Tomey, Tennenlohe, Germany), keratometry, visual acuity and subjective refraction were assessed. Radial approximation with a 5th order polynomial fit of the refractive data on 25 non-centric rings of the TMS, within 256 hemimeridians was performed to get data at equally spaced concentric rings. Fast Fourier transformation of the data sets in the mid periphery (1.4-1.8 mm apical distance) was done to get DC-, one-cycle and two-cycle component. Fourier coefficients were correlated with keratometric readings, subjective refractive values and visual acuity. RESULTS: Spherical equivalent was fairly constant in the postoperative interval before suture removal. After suture removal, a corneal flattening of about 3 diopters occurred. The one-cycle component before suture removal tended to be higher in group 1 compared to group 2 before suture removal. This difference was statistically significant (p = 0.01) after suture removal. Comparing keratometry, calculated meridians by the Tomey software, two-cycle component of the fourier decomposition and subjective refraction, the best correlation (p = 0.02) was observed between two-cycle-component and the refractive cylinder in amplitude and axis after suture removal. Best corrected visual acuity was inversely correlated with the amount of energy of higher harmonics compared to the whole energy before (p = 0.04) and after (p = 0.01) suture removal. CONCLUSIONS: Fourier transformation renders reconstruction of corneal topography data with a marked data reduction and a small error. From fourier coefficients conclusions may be drawn concerning potential best-corrected visual acuity and amplitude/axis of subjective cylinder, even in corneas with severe local irregularities.

Adult↗
PubMedDate not supplied

[Mathematical modeling of population dynamics of unstable plasmid-containing bacteria during continuous cultivation in a chemostat].

A structural approach to studying the regularities of the population dynamics of unstable recombinant bacterial strains in a chemostat was elaborated. The approach is based on the mathematical modeling of cell distribution in a population with different numbers of plasmid copies. The effect of decreased selective preference of plasmidless variants of the recombinant strain in the chemostat, which is related to a decrease in the number of plasmid copies in cells upon long-term incubation was analyzed. It is shown that the time of half-elimination of plasmids from the bacterial population in the steady state in the chemostat T1/2 does not depend on the maximum number of plasmid copies in cells N but is determined only by the mean time of generation g and the probability of the loss of one plasmid copy tau. The dependence of the preference of bacterial plasmidless variants on the efficiency of expression of genes cloned into plasmids in chemostat was analyzed using the recombinant strain E. coli Z905, whose plasmids pPHL-7 contain cloned genes for the luminescence system of marine luminescing bacteria Photobacterium leiognathi.

Bioreactors↗
PubMed1979

A mathematical model for the epidemiologic study of infectious diseases.

Many infectious diseases have been hypothesized to represent common virus infections in which only small proportions of cases result in clinically recognizable disease. In order to find a method of studying this class of diseases, a mathematical model of the age distribution of clinical disease was developed using poliomyelitis as a prototype. The model is shown to fit the age distribution of reported poliomyelitis in a variety of localities before the use of artificial immunization. The true yearly rate of infection is easily estimated and ranges from .11 in rural Sweden to 1.20 in Chile. The model accounts for several major features of poliomyelitis epidemiology, including the shift to older ages and the high rate of clinically apparent disease which were frequently observed in populations which could be expected to have a comparatively low rate of spread. An examination of the age distribution of other diseases by these methods may provide a method of identifying other common infections which only occasionally result in clinically apparent disease.

Age Factors↗
PubMed1981

[Mathematical modelling of glycolysis and of adenine nucleotide metabolism of human erythrocytes. II. Simulation of adenine nucleotide breakdown following glucose depletion].

The breakdown of adenine nucleotides in human erythrocytes in physiological and alkaline medium at 37 degrees C after glucose depletion is simulated by a mathematical model of energy metabolism. The simulation consists of time dependent solutions of a system of 16 differential equations derived from the stoichiometry of chemical pathways and kinetic properties of enzymes. Starting with the stationary characteristics of the model (M. Schauer et al.: Acta biol. med. germ. 40, 1659, 1981) the results of the simulation were analysed as a function of 1) the share of adenylate deaminase in the degradation of adenine nucleotides and 2) the interaction between adenylate kinase equilibrium and magnesium ions. The experimental data and the simulated concentration changes are in good accord, provided that the assumed activity of adenylate deaminase is very low so that the degradation of adenine nucleotides proceeds mainly via AMP-hydrolysis. The extensive activation of adenylate deaminase during incubation is explained by its sigmoid kinetics with respect to AMP. To improve the results of simulation changes in the adenylate kinase equilibrium have to taken into consideration. These have been measured during incubation of erythrocytes; they can be attributed only partly to the increasing concentration of magnesium ions and to differences among the constants of magnesium association to adenine nucleotides.

Adenine Nucleotides↗
PubMed1990

A mathematical model of erythropoiesis in mice and rats. Part 4: Differences between bone marrow and spleen.

In a preceding analysis we hypothesized that the most important parameter controlled by erythropoietic regulation in vivo is the degree of amplification (number of cell divisions) in the CFU-E and erythroblast cell stages. It was concluded that erythropoietic amplification in vivo is controlled according to a sigmoidal dose-response relationship with respect to the control parameter which is the haematocrit (or haemoglobin concentration). Here, this hypothesis is extended to include the differences in murine bone marrow and splenic erythropoiesis that are described and quantified by different dose-response relationships. Comparing several sets of experimental data with mathematical model simulations, this approach leads to the following conclusions: (i) in the unperturbed normal steady state at least one extra erythropoietic cell division takes place in the spleen compared with the bone marrow; (ii) a strong erythropoietic stimulus, such as severe bleeding or hypoxia, can induce five to six additional cell divisions in the spleen but only two to three additional divisions in the bone marrow; this results in a considerable increase in the spleen's contribution to erythropoiesis from about 10% in normal animals to over 40% during strong stimulation; (iii) under erythropoietic suppression, such as red cell transfusion, a similar number of cell divisions is skipped in both organs and the splenic contribution to erythropoiesis remains unchanged. In conclusion, the concept that bone marrow and spleen microenvironments differ in the dose-response relationship for erythropoietic regulation provides an explanation for the changing contribution of splenic murine erythropoiesis following a variety of experimental treatments.

Animals↗
PubMedDate not supplied

[A method of assessing actual nutrition of organized collectives based on mathematical modelling].

Subjects completely adapted to a complex of factors of their working and natural environment, besides a stable emotional sphere and physical working capacity, develop a certain stereotype of food consumption, that objectively determines relatively constant levels of this or that component in the ration they receive. The authors have developed a method of mathematical modelling of the expected consumption of food substances basing on the calculation of the ration chemical composition according to the menu, and on the further comparing with the level of actual nutrient consumption. A step-by-step regression analysis has enabled the authors to detect significant (by F-criterion) groups of daily ration food products for actual nutrient consumption, and to present the regression models for calculation the expected consumption of mineral substances in daily rations with the use of regression equation.

Eating↗
PubMed2004

Use of a quantitative gene expression assay based on micro-array techniques and a mathematical model for the investigation of chlamydial generation time.

Chlamydia is an important pathogen which possesses a unique developmental cycle. We used real-time PCR technology to measure gene transcript levels in Chlamydia trachomatis strain L2. By measuring 16S rRNA transcript levels, and developing a mathematical model of the chlamydial developmental cycle fitting the data, we predict an average generation time of approximately 2.6 h. Additionally, potentially this modelling also provides the foundation for the application of emerging micro-array technology in which identification of the gene signals that trigger a chlamydial body to start replicating or transform to its infectious form can be made possible.

Chlamydia trachomatis↗
PubMed1981

Mathematical models of diuresis after oral administration of tizolemide, furosemide and placebo to healthy adults.

Urine volumes and flows as functions of time after oral administration of placebo, furosemide 80 mg and tizolemide 100 mg to sixteen healthy male adults at 08.00 a.m. are described through mathematical models. Accumulated urine volumes after dosing, V, as functions of time, t, are accounted for by sigmoid functions and urine flows are described as dV/dt. No significant differences were found between the 24-hour urine volume mean values after furosemide (2457 ml) and tizolemide (2283 ml) and both were significantly higher than the corresponding mean value after placebo (1369 ml) ( p less than 0.001). Peak diuresis after dosing occurs about 1.5 hours for furosemide, 4 hours for tizolemide and 6 hours for placebo. Fourteen hours after dosing, urine volume amounts to 90% of the 24-hour urine volume after furosemide and to 76% after tizolemide, so that both substances may be regarded as diurnal diuretics.

Administration, Oral↗
PubMed2005

A mathematical model for the effect of a false-negative sentinel node biopsy on breast cancer mortality: a tool for everyday use.

One of the concerns of using sentinel node biopsy (SNB) is the risks of a false-negative result (FNR). We have created a mathematical model to estimate the effects of FNR on mortality because of excess local recurrence and adjuvant therapy inappropriately withheld. With a FNR of 9.7%, the absolute effect on 10-year mortality is estimated to be less than 0.6% for all patients with tumours <2 cm in size. Since the impact of FNR on mortality is small and FNR rates do not improve with training, we suggest that detection rate alone is an adequate criterion for judging competence in SNB.

Breast Neoplasms↗
PubMed2003

Validation and analysis of a mathematical model of a replication-competent oncolytic virus for cancer treatment: implications for virus design and delivery.

Motivated by the rapid expansion in the development of replication-competent viral agents for the treatment of solid tumors, we formulated and analyzed a three-dimensional mathematical model of a tumor that is infected by a replication-competent virus. We initially considered three patterns of intratumoral injection in which a fixed fraction of cells are initially infected with the virus throughout (a) the entire tumor, (b) the tumor core, and (c) the tumor rim, respectively. For each injection pattern, an approximate analysis of the model provides a simple and accurate condition for whether the virus will eradicate the tumor. The model was then generalized to incorporate nutrient-limited necrosis and an innate immune response against virus-infected tumor cells. Recent preclinical and clinical data were used to validate the model and estimate key parameter values. Our analysis has the following implications: even in the absence of an immune response, tumor eradication requires widespread distribution of the virus within the tumor at the time of infection; core or rim injections alone may result in tumor escape, particularly in a well-vascularized tumor; the more rapidly a virus lyses infected cells the more effective it will be at controlling the tumor; and the innate immune response to the virus can potentially prevent the virus from controlling the tumor, even with repeat injections. Therefore, in addition to diffuse intratumoral infection, tumor eradication by oncolytic adenovirus will probably require potent suppression of innate immune clearance mechanisms (e.g., by replacement of adenovirus E3 genes), combinations with traditional (chemotherapy, radiotherapy) treatments, and/or concomitant therapeutic gene expression with resultant bystander effects.

Models, Biological↗