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Mathematical Biosciences & Engineering

2009 , Volume 6 , Issue 1

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Control entropy: A complexity measure for nonstationary signals
Erik M. Bollt , Joseph D. Skufca and  Stephen J . McGregor
2009, 6(1): 1-25 doi: 10.3934/mbe.2009.6.1 +[Abstract](43) +[PDF](1094.6KB)
We propose an entropy statistic designed to assess the behavior of slowly varying parameters of real systems. Based on correlation entropy, the method uses symbol dynamics and analysis of increments to achieve sufficient recurrence in a short time series to enable entropy measurements on small data sets. We analyze entropy along a moving window of a time series, the entropy statistic tracking the behavior of slow variables of the data series. We employ the technique against several physiological time series to illustrate its utility in characterizing the constraints on a physiological time series. We propose that changes in the entropy of measured physiological signal (e.g. power output) during dynamic exercise will indicate changes in underlying constraint of the system of interest. This is compelling because CE may serve as a non-invasive, objective means of determining physiological stress under non-steady state conditions such as competition or acute clinical pathologies. If so, CE could serve as a valuable tool for dynamically monitoring health status in a wide range of non-stationary systems.
A lumped model for blood flow and pressure in the systemic arteries based on an approximate velocity profile function
Wouter Huberts , E. Marielle H. Bosboom and  Frans N. van de Vosse
2009, 6(1): 27-40 doi: 10.3934/mbe.2009.6.27 +[Abstract](41) +[PDF](310.5KB)
Previously, by assuming a viscous dominated flow in the boundary layer and an inertia dominated flow in the vessel core, a velocity profile function for a 1D-wave propagation model was derived. Because the time dependent shape of the velocity profile in this boundary layer model depends on the size of the inviscid core and the boundary layer, and thus on the Womersley number, it differs along the arterial tree. In this study we evaluated a lumped model for a vessel segment in which the element configuration is based on physical phenomena described by the boundary layer model and for which all parameters have a physically based quantitative value dependent on the Womersley number. The proposed electrical analog consists of a Womersley number dependent resistor and an inductor arranged in parallel, representing the flow impedance in respectively the vessel core and the boundary layer, in series with a second resistor. After incorporating a capacitor representing the vessel compliance in this rigid tube model, the element configuration resembles the configuration of the four-element windkessel model. For arbitrary Womersley numbers the relative impedance of Womersley theory is approximated with high accuracy. In the limits for small and large Womersley numbers the relative impedances of the proposed lumped model correspond exactly to Womersley theory.
Systemically modeling the dynamics of plasma insulin in subcutaneous injection of insulin analogues for type 1 diabetes
Jiaxu Li and  Yang Kuang
2009, 6(1): 41-58 doi: 10.3934/mbe.2009.6.41 +[Abstract](35) +[PDF](283.9KB)
Type 1 diabetics must inject exogenous insulin or insulin analogues one or more times daily. The timing and dosage of insulin administration have been a critical research area since the invention of insulin analogues. Several pharmacokinetical models have been proposed, and some are applied clinically in modeling various insulin therapies. However, their plasma insulin concentration must be computed separately from the models' output. Furthermore, minimal analytical study was performed in these existing models. We propose two systemic and simplified ordinary differential equation models to model the subcutaneous injection of rapid-acting insulin analogues and long-acting insulin analogues, respectively. Our models explicitly model the plasma insulin and hence have the advantage of computing the plasma insulin directly. The profiles of plasma insulin concentrations obtained from these two models are in good agreement with the experimental data. We also study the dynamics of insulin analogues, plasma insulin concentrations, and, in particular, the shape of the dynamics of plasma insulin concentrations.
Feedback regulation in multistage cell lineages
Wing-Cheong Lo , Ching-Shan Chou , Kimberly K. Gokoffski , Frederic Y.-M. Wan , Arthur D. Lander , Anne L. Calof and  Qing Nie
2009, 6(1): 59-82 doi: 10.3934/mbe.2009.6.59 +[Abstract](49) +[PDF](344.3KB)
Studies of developing and self-renewing tissues have shown that differentiated cell types are typically specified through the actions of multistage cell lineages. Such lineages commonly include a stem cell and multiple progenitor (transit amplifying; TA) cell stages, which ultimately give rise to terminally differentiated (TD) cells. In several cases, self-renewal and differentiation of stem and progenitor cells within such lineages have been shown to be under feedback regulation. Together, the existence of multiple cell stages within a lineage and complex feedback regulation are thought to confer upon a tissue the ability to autoregulate development and regeneration, in terms of both cell number (total tissue volume) and cell identity (the proportions of different cell types, especially TD cells, within the tissue). In this paper, we model neurogenesis in the olfactory epithelium (OE) of the mouse, a system in which the lineage stages and mediators of feedback regulation that govern the generation of terminally differentiated olfactory receptor neurons (ORNs) have been the subject of much experimental work. Here we report on the existence and uniqueness of steady states in this system, as well as local and global stability of these steady states. In particular, we identify parameter conditions for the stability of the system when negative feedback loops are represented either as Hill functions, or in more general terms. Our results suggest that two factors -- autoregulation of the proliferation of transit amplifying (TA) progenitor cells, and a low death rate of TD cells -- enhance the stability of this system.
Computation of traveling wave fronts for a nonlinear diffusion-advection model
M. B. A. Mansour
2009, 6(1): 83-91 doi: 10.3934/mbe.2009.6.83 +[Abstract](37) +[PDF](139.1KB)
This paper utilizes a nonlinear reaction-diffusion-advection model for describing the spatiotemporal evolution of bacterial growth. The traveling wave solutions of the corresponding system of partial differential equations are analyzed. Using two methods, we then find such solutions numerically. One of the methods involves the traveling wave equations and solving an initial-value problem, which leads to accurate computations of the wave profiles and speeds. The second method is to construct time-dependent solutions by solving an initial-moving boundary-value problem for the PDE system, showing another approximation for such wave solutions.
Estimation and identification of parameters in a lumped cerebrovascular model
Scott R. Pope , Laura M. Ellwein , Cheryl L. Zapata , Vera Novak , C. T. Kelley and  Mette S. Olufsen
2009, 6(1): 93-115 doi: 10.3934/mbe.2009.6.93 +[Abstract](44) +[PDF](560.9KB)
This study shows how sensitivity analysis and subset selection can be employed in a cardiovascular model to estimate total systemic resistance, cerebrovascular resistance, arterial compliance, and time for peak systolic ventricular pressure for healthy young and elderly subjects. These quantities are parameters in a simple lumped parameter model that predicts pressure and flow in the systemic circulation. The model is combined with experimental measurements of blood flow velocity from the middle cerebral artery and arterial finger blood pressure. To estimate the model parameters we use nonlinear optimization combined with sensitivity analysis and subset selection. Sensitivity analysis allows us to rank model parameters from the most to the least sensitive with respect to the output states (cerebral blood flow velocity and arterial blood pressure). Subset selection allows us to identify a set of independent candidate parameters that can be estimated given limited data. Analyses of output from both methods allow us to identify five independent sensitive parameters that can be estimated given the data. Results show that with the advance of age total systemic and cerebral resistances increase, that time for peak systolic ventricular pressure is increases, and that arterial compliance is reduced. Thus, the method discussed in this study provides a new methodology to extract clinical markers that cannot easily be assessed noninvasively.
An in vivo intermediate filament assembly model
Stéphanie Portet and  Julien Arino
2009, 6(1): 117-134 doi: 10.3934/mbe.2009.6.117 +[Abstract](36) +[PDF](393.7KB)
A model is developed to study the in vivo intermediate filament organization in terms of repartition between four different structural states: soluble proteins, particles, short, and long filaments. An analysis is conducted, showing that the system has a unique, globally asymptotically stable equilibrium. By means of sensitivity analysis, the influence of parameters on the system is studied. It is shown that, in agreement with biological observations, posttranslational modifications of intermediate filament proteins resulting in filament solubilization are the main regulators of the intermediate filament organization. A high signalling-dependent solubilization of filaments favours the intermediate filament aggregation in particles.
Modeling HIV outbreaks: The male to female prevalence ratio in the core population
Brandy Rapatski and  James Yorke
2009, 6(1): 135-143 doi: 10.3934/mbe.2009.6.135 +[Abstract](32) +[PDF](135.9KB)
What affects the ratio of infected men to infected women in the core population in a heterosexual HIV epidemic? Hethcote & Yorke [5] introduced the term "core" initially to loosely describe the collection of individuals having the most unprotected sex partners. We study the early epidemic during the exponential growth phase and focus on the core group because most infected people were infected by people in the core. We argue that in the early outbreak phase of an epidemic, there is an identity, which we call the "outbreak equation." It relates three ratios that describe the core men versus the core women, namely, the ratio $E$ of numbers of all core men to all core women, the ratio $C$ of numbers of infected core men to core women, and the ratio $M$ of the infectiousness of a typical core man to that of a typical core woman. Then the relationship between the ratios is $E=MC^2$ in the early outbreak phase. We investigate two very different scenarios, one in which there are two times as many core men as core women ($E=2$) and the other in which core men equal core women ($E=1$). In the first case, the HIV epidemic grows at a much faster rate. We conclude that if the female core group was larger, that is, if more women in the total population were promiscuous (or if fewer men were promiscuous) then the HIV epidemic would grow more slowly.
The dynamics of a simple Laissez-Faire model with two predators
Gunog Seo and  Mark Kot
2009, 6(1): 145-172 doi: 10.3934/mbe.2009.6.145 +[Abstract](42) +[PDF](1578.2KB)
In this paper, we study the dynamics of a laissez-faire predator--prey model with both a specialist and a generalist predator. We analyze the stabilities of equilibria by performing linearized stability analyses. We then reexamine the stability of the equilibrium where the prey and predator coexist by constructing a Lyapunov function. If we hold the generalist predator population constant, treating it as a bifurcation parameter, we show that our model can possess multiple (up to three) limit cycles that surround an equilibrium in the interior of the first quadrant. Our model shows rich dynamics including fold, transcritical, pitchfork, Hopf, cyclic-fold, and Bautin bifurcations as well as heteroclinic connections. If we instead vary the generalist predator population slowly across bifurcations, the model exhibits bursting behavior as it alternates between a repetitive spiking phase and a quiescent phase.
Solution of the Michaelis-Menten equation using the decomposition method
Jagadeesh R. Sonnad and  Chetan T. Goudar
2009, 6(1): 173-188 doi: 10.3934/mbe.2009.6.173 +[Abstract](35) +[PDF](317.5KB)
We present a low-order recursive solution to the Michaelis-Menten equation using the decomposition method. This solution is algebraic in nature and provides a simpler alternative to numerical approaches such as differential equation evaluation and root-solving techniques that are currently used to compute substrate concentration in the Michaelis-Menten equation. A detailed characterization of the errors in substrate concentrations computed from decomposition, Runge-Kutta, and bisection methods over a wide range of $s_{0} $:$K_{m}$ values was made by comparing them with highly accurate solutions obtained using the Lambert $W$ function. Our results indicated that solutions obtained from the decomposition method were usually more accurate than those from the corresponding classical Runge-Kutta methods. Moreover, these solutions required significantly fewer computations than the root-solving method. Specifically, when the stepsize was 0.1% of the total time interval, the computed substrate concentrations using the decomposition method were characterized by accuracies on the order of 10$^-8$ or better. The algebraic nature of the decomposition solution and its relatively high accuracy make this approach an attractive candidate for computing substrate concentration in the Michaelis-Menten equation.
Dynamic effects and information quantifiers of statistical memory of MEG's signals at photosensitive epilepsy
R. M. Yulmetyev , E. V. Khusaenova , D. G. Yulmetyeva , P. Hänggi , S. Shimojo , K. Watanabe and  J. Bhattacharya
2009, 6(1): 189-206 doi: 10.3934/mbe.2009.6.189 +[Abstract](48) +[PDF](293.6KB)
The time series analysis of magnetoencephalographic (MEG) signals is very important both for basic brain research and for medical diagnosis and treatment. Here we discuss the crucial role of statistical memory effects (ME) in human brain functioning with photosensitive epilepsy (PSE). We study two independent statistical memory quantifiers that reflect the dynamical characteristics of neuromagnetic brain responses on a flickering stimulus of different colored combinations from a group of control subjects, which are contrasted with those from a patient with PSE. We analyze the frequency dependence of two memory measures for the neuromagnetic signals. The strong memory and the accompanying transition to a regular and robust regime of the signals' chaotic behavior in the separate areas are characteristic for a patient with PSE. This particularly interesting observation most likely identifies the regions of the protective mechanism in a human organism against occurrence of PSE.
Letter to the editors
Shigui Ruan
2009, 6(1): 207-208 doi: 10.3934/mbe.2009.6.207 +[Abstract](36) +[PDF](38.1KB)
Dear Editors:
I request that Mathematical Biosciences and Engineering publish this Letter of Correction to an article published in the journal for which I was the corresponding author, "The Effect of Global Travel on the Spread of SARS'' (2006;3(1):205-218). The goal of this article was to study the effect of global travel on the geographic spread of SARS. A multiregional compartmental model was proposed, mathematically analyzed, and numerically simulated to study how SARS spread out from Guangdong, China. The article consists of six sections: (1) an introduction, (2) a background section on medical geography theory, (3) the mathematical model, (4) mathematical analysis and results, (5) numerical simulations, and (6) discussion. Sections 3, 4, and 5 are the main parts of the article which are all original work.

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