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Mathematical Biosciences & Engineering

2006 , Volume 3 , Issue 4

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Visualisation of the numerical solution of partial differential equation systems in three space dimensions and its importance for mathematical models in biology
Heiko Enderling, Alexander R.A. Anderson, Mark A.J. Chaplain and Glenn W.A. Rowe
2006, 3(4): 571-582 doi: 10.3934/mbe.2006.3.571 +[Abstract](293) +[PDF](1589.3KB)
Abstract:
Numerical analysis and computational simulation of partial differential equation models in mathematical biology are now an integral part of the research in this field. Increasingly we are seeing the development of partial differential equation models in more than one space dimension, and it is therefore necessary to generate a clear and effective visualisation platform between the mathematicians and biologists to communicate the results. The mathematical extension of models to three spatial dimensions from one or two is often a trivial task, whereas the visualisation of the results is more complicated. The scope of this paper is to apply the established marching cubes volume rendering technique to the study of solid tumour growth and invasion, and present an adaptation of the algorithm to speed up the surface rendering from numerical simulation data. As a specific example, in this paper we examine the computational solutions arising from numerical simulation results of a mathematical model of malignant solid tumour growth and invasion in an irregular heterogeneous three-dimensional domain, i.e., the female breast. Due to the different variables that interact with each other, more than one data set may have to be displayed simultaneously, which can be realized through transparency blending. The usefulness of the proposed method for visualisation in a more general context will also be discussed.
Noise-sensitive measure for stochastic resonance in biological oscillators
Ying-Cheng Lai and Kwangho Park
2006, 3(4): 583-602 doi: 10.3934/mbe.2006.3.583 +[Abstract](263) +[PDF](1191.7KB)
Abstract:
There has been ample experimental evidence that a variety of biological systems use the mechanism of stochastic resonance for tasks such as prey capture and sensory information processing. Traditional quantities for the characterization of stochastic resonance, such as the signal-to-noise ratio, possess a low noise sensitivity in the sense that they vary slowly about the optimal noise level. To tune to this level for improved system performance in a noisy environment, a high sensitivity to noise is required. Here we show that, when the resonance is understood as a manifestation of phase synchronization, the average synchronization time between the input and the output signal has an extremely high sensitivity in that it exhibits a cusp-like behavior about the optimal noise level. We use a class of biological oscillators to demonstrate this phenomenon, and provide a theoretical analysis to establish its generality. Whether a biological system actually takes advantage of phase synchronization and the cusp-like behavior to tune to optimal noise level presents an interesting issue of further theoretical and experimental research.
Lyapunov functions for tuberculosis models with fast and slow progression
C. Connell Mccluskey
2006, 3(4): 603-614 doi: 10.3934/mbe.2006.3.603 +[Abstract](568) +[PDF](222.0KB)
Abstract:
The spread of tuberculosis is studied through two models which include fast and slow progression to the infected class. For each model, Lyapunov functions are used to show that when the basic reproduction number is less than or equal to one, the disease-free equilibrium is globally asymptotically stable, and when it is greater than one there is an endemic equilibrium which is globally asymptotically stable.
Mathematical modeling of biowall reactors for in-situ groundwater treatment
Donna J. Cedio-Fengya and John G. Stevens
2006, 3(4): 615-634 doi: 10.3934/mbe.2006.3.615 +[Abstract](224) +[PDF](408.8KB)
Abstract:
In this paper we develop a comprehensive model for the remediation of contaminated groundwater in a passive, in-ground reactor, generally known as a biowall. The model is based on our understanding of the component transport and biokinetic processes that occur as water passes through a bed of inert particles on which a biofilm containing active microbial degraders, typically aerobic bacteria, is developing. We give a detailed derivation of the model based on accepted engineering formulations that account for the mass transport of the contaminant (substrate) to the surface of the biofilm, its diffusion into the biofilm to the proximity of a microbe, and its subsequent destruction within that degrader. The model has been solved numerically and incorporated in a robust computer code. Based on representative input values, the results of varying key parameters in the model are presented. The relation between biofilm growth and biowall performance is explored, revealing that the amount of biomass and its distribution within the biowall are key parameters affecting contaminant removal.
Modeling shrimp biomass and viral infection for production of biological countermeasures
H. Thomas Banks, V. A. Bokil, Shuhua Hu, A. K. Dhar, R. A. Bullis, C. L. Browdy and F.C.T. Allnutt
2006, 3(4): 635-660 doi: 10.3934/mbe.2006.3.635 +[Abstract](351) +[PDF](720.5KB)
Abstract:
In this paper we develop a mathematical model for the rapid production of large quantities of therapeutic and preventive countermeasures. We couple equations for biomass production with those for vaccine production in shrimp that have been infected with a recombinant viral vector expressing a foreign antigen. The model system entails both size and class-age structure.
Modelling the human immune response mechanisms to mycobacterium tuberculosis infection in the lungs
Gesham Magombedze, Winston Garira and Eddie Mwenje
2006, 3(4): 661-682 doi: 10.3934/mbe.2006.3.661 +[Abstract](262) +[PDF](696.4KB)
Abstract:
This work elaborates on the effects of cytotoxic lymphocytes (CTLs) and other immune mechanisms in determining whether a TB-infected individual will develop active or latent TB. It answers one intriguing question: why do individuals infected with Mycobacterium tuberculosis (Mtb) experience different clinical outcomes? In addressing this question, we have developed a model that captures the effects of CTLs and the combined effects of CD4+ helper T cells (Th1 and Th2) immune response mechanisms to TB infection. The occurrence of active or latent infection is shown to depend on a number of factors that include effector function and levels of CTLs. We use the model to predict disease progression scenarios, including primary, latency or clearance. Model analysis shows that occurrence of active disease is much attributed to the Mtb pathogen ability to persist outside the intracellular environment and that high levels of CTLs result in latent TB, while low levels of CTLs result in active TB. This is attributed to the CTLs’ ability to directly kill infected macrophages and the bacteria inside the infected macrophages. The study suggests directions for further basic studies and potential new treatment strategies.
Monte carlo simulation of heterotypic cell aggregation in nonlinear shear flow
Jiakou Wang, Margaret J. Slattery, Meghan Henty Hoskins, Shile Liang, Cheng Dong and Qiang Du
2006, 3(4): 683-696 doi: 10.3934/mbe.2006.3.683 +[Abstract](374) +[PDF](210.3KB)
Abstract:
In this paper, we develop a population balance model for cell aggregation and adhesion process in a nonuniform shear flow. Some Monte Carlo simulation results based on the model are presented for the heterotypic cell-cell collision and adhesion to a substrate under dynamic shear forces. In particular, we focus on leukocyte (PMN)-melanoma cell emboli formation and subsequent tethering to the vascular endothelium (EC) as a result of cell-cell aggregation. The simulation results are compared with the results of experimental measurement. Discussions are made on how we could further improve the accuracy of the population balance type modelling.
Complex spatio-temporal features in meg data
Francesca Sapuppo, Elena Umana, Mattia Frasca, Manuela La Rosa, David Shannahoff-Khalsa, Luigi Fortuna and Maide Bucolo
2006, 3(4): 697-716 doi: 10.3934/mbe.2006.3.697 +[Abstract](324) +[PDF](4653.2KB)
Abstract:
Magnetoencephalography (MEG) brain signals are studied using a method for characterizing complex nonlinear dynamics. This approach uses the value of $d_\infty$ (d-infinite) to characterize the system’s asymptotic chaotic behavior. A novel procedure has been developed to extract this parameter from time series when the system’s structure and laws are unknown. The implementation of the algorithm was proven to be general and computationally efficient. The information characterized by this parameter is furthermore independent and complementary to the signal power since it considers signals normalized with respect to their amplitude. The algorithm implemented here is applied to whole-head 148 channel MEG data during two highly structured yogic breathing meditation techniques. Results are presented for the spatiotemporal distributions of the calculated $d_\infty$ on the MEG channels, and they are compared for the different phases of the yogic protocol. The algorithm was applied to six MEG data sets recorded over a three-month period. This provides the opportunity of verifying the consistency of unique spatio-temporal features found in specific protocol phases and the chance to investigate the potential long term effects of these yogic techniques. Differences among the spatio-temporal patterns related to each phase were found, and they were independent of the power spatio-temporal distributions that are based on conventional analysis. This approach also provides an opportunity to compare both methods and possibly gain complementary information.
On the stabilizing effect of cannibalism in stage-structured population models
Bruno Buonomo and Deborah Lacitignola
2006, 3(4): 717-731 doi: 10.3934/mbe.2006.3.717 +[Abstract](256) +[PDF](328.8KB)
Abstract:
In this paper we give a contribution to the systematic investigation of cannibalism in predator-prey models commenced since the publication of the paper by Kohlmeier and Ebenhöh in 1995. We present a stage-structured predator-prey model and study its dynamics. We use a Hopf bifurcation analysis to prove that cycles are possible and that cannibalism suppresses these cycles; that is, when cannibalism attack rate is increased so that it passes a critical value, the coexistence steady state changes from being unstable to being stable. Numerical simulations are provided together with the mathematical analysis. Our modelling approach is based on balance arguments and a comparison with some early models which predict that a destabilizing effect of cannibalism is performed. Our results agree with the output of growth simulation for some cannibalistic copepods.

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