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PubMed1989

Mathematical models and individualized outcome estimates in multiple sclerosis.

There is an urgent need for individualized outcome estimates in multiple sclerosis (MS). This is a prerequisite for selecting appropriate therapies in a disease the outcome of which may vary between malignant and benign forms. This question was addressed by using probabilistic mathematical models. The disease course is described by movements of the patient's condition through well-defined disease states. The data used are based upon 278 reports of definite and probable MS cases collected over a 20-year period (1957-1976) at l'Hôpital Neurologique de Lyon. A Markov model was first prepared. However, it only provided an overall presentation of the disease which was not very meaningful for the individual patient, and limited studies of potential prognostic variables to dichotomous variables and univariate analyses. A stochastic survival model was then elaborated. It was complex at the theoretical level but in practice, easy to use. Personalized prognosis of a patient could be ascertained by combining several single potential prognostic variables and using their actual value in the case of quantitative variables. Although these models still need to be refined and have to be validated from the present state of the patients and other patient cohorts, they provide for the first time a global description of disease course likelihood and prognosis. The survival model may be adapted to each patient.

Age Factors↗
PubMed2004

Steady-state analysis of a mathematical model for capillary network formation in the absence of tumor source.

This paper extends the work done in [S. Pamuk, Ph.D. Thesis, Iowa State University, 2000; Bull. Math. Biol. 63 (5) (2001) 801] in that we investigate the condition that is needed for the degradation of basement membrane in a mathematical model for capillary network formation. To do this, the steady-state behavior of tumor angiogenesis factor is studied under restricted assumptions, and the tumor angiogenesis factor threshold that activates the transport equations in the capillary is estimated using this steady state. Therefore, once the concentration of the tumor angiogenesis factor in the inner vessel wall reaches this threshold value, endothelial cells begin to move into the extracellular matrix for the start of angiogenesis. Furthermore, we do believe that the result we obtain in this paper provides an underlying insight into mechanisms of cell migration which are crucial for tumor angiogenesis.

Algorithms↗
PubMed2001

A mathematical model of the ascending colon of the horse.

In this study we present a geometric model of the ascending colon of the horse, especially the left ventral colon and the right ventral colon, the left dorsal colon and the right dorsal colon and the pelvic flexure. We also present a mathematical model of the cross sections of these ascending colon parts with the exceptions of the pelvic flexure. We show that these cross-sections correspond to the closed algebraic curves known as epitrochoid.

Animals↗
PubMed2004

A mathematical model of baculovirus infection on insect cells at low multiplicity of infection.

The expression efficiency of the insect cells-baculovirus system used for insecticidal virus production and the expression of medically useful foreign genes is closely related with the dynamics of infection. The present studies develop a model of the dynamic process of insect cell infection with baculovirus at low multiplicity of infection (MOI), which is based on the multi-infection cycles of insect cell infection at low MOI described. A mathematical model for the amount of viruses released from primary infected cells and the amount of free viruses before secondary infected cells release viruses has been developed. Comparison of the simulation results with the experimental data confirms qualitatively that this model is highly reasonable before secondary infected cells release viruses. This model is considered as a base for further modeling the entire complicated infection process.

Animals↗
PubMed1997

[A mathematical model of heat exchange between astronaut and environmental medium on the Lunar surface].

To maintain thermal balance of astronaut, and avoid injuries by heats of the solar radiation and radiation from the Moon, a detailed analysis of heat exchange between the astronaut and the environment medium was made and a mathematical model was established. It indicates that the Lunar surface temperature and the thermal current transmitted to the astronaut change with the incident angle of the solar radiation. The thermal balance of the astronaut is affected by absorption coefficient, radiation coefficient and thermal resistance.

Astronauts↗
PubMed1985

Relationship of uteroplacental blood flow to placental clearance of maternal plasma C-19 steroids: evaluation of mathematical models.

The concept that the placental clearance of maternal plasma dehydroepiandrosterone sulfate through estradiol formation is a function of uteroplacental blood flow in women has been disputed. We obtained data on the clearance of maternal plasma dehydroepiandrosterone through placental estradiol formation in the baboon and used these data to evaluate some mathematical models of placental clearance. Our evaluation shows that the placental clearance of dehydroepiandrosterone is proportional to uteroplacental blood flow in the baboon.

Animals↗
PubMed2003

A mathematical model for self-limiting brain tumors.

It is puzzling that certain brain tumors exhibit arrested exponential growth. We have observed in pediatric low-grade astrocytomas (LGA) at a certain volume approximately 100-150 cm(3) that the tumor ceases to grow. This observation led us to develop a macroscopic mathematical model for LGA growth kinetics that assumes the flow through the surface of the astrocytoma of a triggering agent or "promoter" that is uniformly distributed throughout the tumor, thereby providing relatively homogeneous growth. The model relates the transport of the promoter by the electrochemical potential associated with the tumor and diffusion effects through the surface of the growth and its consumption throughout the tumor volume via a pair of ordinary differential equations. The model assumes a constant growth rate, if the promoter density is above some threshold, and is zero otherwise. We also develop equations for an electrochemical (Nernst) transport mechanism for the promoter, and describes the microscopic basis for the macroscopic evolution to the equilibrium state at a well-defined and universal size. The latter description is unstable to asymmetric perturbations and provides a "star-like" shape for emergent tumors and a spheroidal shape for fully developed ones. The underlying assumption in our hypothesis would also result in tumor growth remission beginning from the periphery and proceeding inwards, a feature that has now been validated clinically.

Astrocytoma↗
PubMed2006

Mathematical modeling of the biomechanics of the lamina cribrosa under elevated intraocular pressures.

Comprehensive understanding of the biomechanical performance of the lamina cribrosa (LC) and the optic nerve head is central to understanding the role of elevated intraocular pressures (IOP) in chronic open angle glaucoma. In this paper, six closed-from mathematical models based on different idealizations of the LC are developed and compared. This approach is used to create further understanding of the biomechanical behavior by identifying the LC features and properties that have a significant effect on its performance under elevated IOP. The models developed are based on thin circular plate and membrane theories, and consider influences such as in-plane pretension caused by scleral expansion and large deflections. Comparing the results of the six models against a full ocular globe finite element model suggests the significance of the in-plane pretension and the importance of assuming that the sclera provides the LC with a clamped edge. The model that provided the most accurate representation of the finite element model was also used to predict the behavior of a number of LC experimental tests presented in the literature. In addition to the deflections under elevated IOP, the model predictions include the distributions of stress and strain, which are shown to be compatible with the progression of visual field loss experienced in glaucoma.

Biomechanical Phenomena↗
PubMed1979

Dialysis continuous process for ammonium-lactate fermentation: improved mathematical model and use of deproteinized whey.

Separate terms for substrate limitation and product inhibition were incorporated into an equation describing the rate of cell growth for the steady-state fermentation of lactose to lactic acid with neutralization to a constant pH by ammonia. The equation was incorporated into a generalized mathematical model of a dialysis continuous process for the fermentation, developed previously, in which the substrate is fed into the fermentor and the fermentor contents are dialyzed through a membrane against water. The improved model was used to simulate the fermentation on a digital computer, and the results agreed with previous experimental tests using whole whey as the substrate. Further simulations were then made to guide experimental tests using deproteinized whey as the substrate. Dried cheese-whey ultrafiltrate was rehydrated with tap water to contain 242 mg of lactose per ml, supplemented with 8 mg of yeast extract per ml, charged into a 5-liter fermentor without sterilization, adjusted in pH (5.5) and temperature (44 degrees C), and inoculated with an adapted culture of Lactobacillus bulgaricus. The fermentor and dialysate circuits were connected, and a series of steady-state conditions was managed nonaseptically for 71 days. The fermentation of deproteinized whey relative to whole whey, with both highly concentrated, resulted in similar extents of product accumulation but at a lesser rate.

Journal Article↗
PubMed1988

The cryptoendolithic microbial environment in the Ross Desert of Antarctica: mathematical models of the thermal regime.

Microbial activity in the Antarctic cryptoendolithic habitat is regulated primarily by temperature. Previous field studies have provided some information on the thermal regime in this habitat, but this type of information is limited by the remoteness of the site and the harsh climatic conditions. Therefore, a mathematical model of the endolithic thermal regime was constructed to augment the field data. This model enabled the parameters affecting the horizontal and altitudinal distribution of the community to be examined. The model predicts that colonization should be possible on surfaces with zenith angle less than 15 degrees. At greater zenith angles, colonization should be restricted to surfaces with azimuth angles less than 135 degrees or greater than 225 degrees. The upper elevational limit of the community should be less than 2,500 m. The thermal regime probably does not influence the zonation of the community within a rock.

Algorithms↗
PubMed2005

Mathematical Model analysis of Wallstent and Aneurx: dynamic responses of bare-metal endoprosthesis compared with those of stent-graft.

We performed this study in order to analyze the mechanical properties of bare-metal Wallstent endoprostheses and of AneuRx stent-grafts and to compare their responses to hemodynamic forces. Mathematical modeling, numerical simulations, and experimental measurements were used to study the 2 structurally different types of endoprostheses. Our findings revealed that a single bare-metal Wallstent endoprosthesis is 10 times more flexible (elastic) than is the wall of the aneurysmal abdominal aorta. Graphs showing the changes in the diameter and length of the stent when exposed to a range of internal and external pressures were obtained. If the aorta is axially stiff and resists length change, a force as large as 1 kg can act in the axial direction on the aortic wall. If the stent is not firmly anchored, it will migrate. In contrast, a fabric-covered, fully supported, stent-graft such as the AneuRx is significantly less compliant than the aorta or the bare-metal stent. During each cardiac cycle, the stent frame tends to move due to its higher elasticity, while the fabric resists movement, which might break the sutures that join the fabric to the frame. Elevated local transmural pressure, detected along the prosthesis graft, can contribute to material fatigue.

Alloys↗
PubMed1992

Growth factors and cell kinetics: a mathematical model applied to Il-3 deprivation on leukemic cell lines.

We assume the existence of a specific G1 protein which is an initiator of DNA replication. This initiator is supposed to be synthesized according to Michaelis-Menten kinetics. In order to start DNA replication, it is assumed that this G1 specific protein must be produced in a required amount. Intra-cellular growth inhibitors and extra-cellular growth factors control the production of the initiator. This model allows to calculate the average G1 phase time as a function of the various chemical concentrations of nutrients, enzymes, growth inhibitors and growth factors. This model is compared to cell kinetics experiments on a leukemic cell line responding to Interleukin 3 deprivation. The curves giving the experimental average G1 phase times with respect to Interleukin-3 concentrations are fitted by the mathematical model with a quite good agreement.

Cells, Cultured↗
PubMed1992

Dynamic mathematical model to predict microbial growth and inactivation during food processing.

Many sigmoidal functions to describe a bacterial growth curve as an explicit function of time have been reported in the literature. Furthermore, several expressions have been proposed to model the influence of temperature on the main characteristics of this growth curve: maximum specific growth rate, lag time, and asymptotic level. However, as the predictive value of such explicit models is most often guaranteed only at a constant temperature within the temperature range of microbial growth, they are less appropriate in optimization studies of a whole production and distribution chain. In this paper a dynamic mathematical model--a first-order differential equation--has been derived, describing the bacterial population as a function of both time and temperature. Furthermore, the inactivation of the population at temperatures above the maximum temperature for growth has been incorporated. In the special case of a constant temperature, the solution coincides exactly with the corresponding Gompertz model, which has been validated in several recent reports. However, the main advantage of this dynamic model is its ability to deal with time-varying temperatures, over the whole temperature range of growth and inactivation. As such, it is an essential building block in (time-saving) simulation studies to design, e.g., optimal temperature-time profiles with respect to microbial safety of a production and distribution chain of chilled foods.

Bacteria↗
PubMed2002

A mathematical model of a biological arms race with a dangerous prey.

In a recent paper, Brodie and Brodie provide a very detailed description of advances and counter-measures among predator-prey communities with a poisonous prey that closely parallel an arms race in modern society. In this work, we provide a mathematical model and simulations that provide a theory as to how this might work. The model is built on a two-dimensional classical predator-prey model that is then adapted to account for the genetics and random mating. The deterministic formulation for the genetics for the prey population has been developed and used in other contexts. Adapting the model to allow for genetic variation in the predator is much more complicated. The model allows for the evolution of the poisonous prey and for the evolution of the resistant predator. The biological paradigm is that of the poisonous newt and the garter snake which has been studied extensively although the models are broad enough to cover other examples.

Animals↗
PubMed1997

A mathematical model of the cytosolic-free calcium response in endothelial cells to fluid shear stress.

Important among the responses of endothelial cells to flow stimuli are cytosolic-free calcium transients. These transients are mediated by several factors, including blood-borne agonists, extracellular calcium, and fluid-imposed shear forces. A mathematical model has been developed describing the recognition and transduction of shear stress to the second messenger cytosolic calcium. Shear stress modulates the calcium response via at least two modalities. First, mass transfer of agonist to the cell surface is enhanced by perfusion and is thus related to shear stress. Second, the permeability of the cell membrane to extracellular calcium increases upon exposure to shear stress. A mass balance for agonist in the perfusate is coupled to a previously published calcium dynamics model. Computations indicate a flow region where the transient moves from transport limited to kinetically limited. Parametric studies indicate distinct contributions to the time course by each step in the process. These steps include the time to develop the concentration boundary layer of agonist, receptor activation, and the mobilization of calcium from intracellular stores. Exogenous calcium is presumed to enter the cell via shear stress-gated ion channels. The model predicts a sigmoidal dependence of calcium influx upon shear stress. The peak value of the transient is determined largely by the agonist pathway, whereas the plateau level is governed by calcium influx. The model predicts the modulation of the calcium transient in the physiologically relevant range of flow and the associated shear stress. This implies that hemodynamics is important in regulating endothelial biology.

Animals↗
PubMed2005

B cell development in aging mice: lessons from mathematical modeling.

Previous studies have not completely clarified the precise defect that characterizes B cell development in aged animals. The question of which developmental mechanism is actually deficient in aging remains controversial. The goal of this study was to elucidate the effects of aging on bone marrow B cell population dynamics. We used mathematical modeling to predict the outcome of the different possible effects, and then compared these predictions to experimental data, to find the most plausible effects. Our model shows that the three main differences between B cell development in young and old mice are a decrease in the maximum number of cells in the pre-B compartment and increases in the rate of transition from cycling pre-B cells to resting pre-B cells and in the fractions of static cells included in the immature B cell subset.

Aging↗
PubMed1995

Mathematical model of oxygen transport: a teaching aid for normal physiology adaptable to extracorporeal oxygenation circuits.

The ultimate aim of most intensive care therapies is to improve tissue oxygen delivery; consequently, a detailed knowledge of this area of physiology is important to a wide range of Critical Care Staff. We describe a simple mathematical model of oxygen transport that was initially written as a training aid for extracorporeal oxygenation training. The model has subsequently proved useful for explaining the determinants of oxygen transport to a broader audience. It is based on simple linear equations and is easily displayed with a standard computer spreadsheet. Apart from its teaching value, the model can also generate a graph of oxygen saturation vs. inspired oxygen fraction for different degrees of pulmonary shunt. This provides a noninvasive method for determining the magnitude of pulmonary venous admixture and may also prove to have some clinical value.

Biological Transport, Active↗