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Mathematical Biosciences & Engineering

2005 , Volume 2 , Issue 2

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Partial Differential Equations-Based Segmentation for Radiotherapy Treatment Planning
Frédéric Gibou, Doron Levy, Carlos Cárdenas, Pingyu Liu and Arthur Boyer
2005, 2(2): 209-226 doi: 10.3934/mbe.2005.2.209 +[Abstract](716) +[PDF](939.1KB)
Abstract:
The purpose of this study is to develop automatic algorithms for the segmentation phase of radiotherapy treatment planning. We develop new image processing techniques that are based on solving a partial differential equation for the evolution of the curve that identifies the segmented organ. The velocity function is based on the piecewise Mumford-Shah functional. Our method incorporates information about the target organ into classical segmentation algorithms. This information, which is given in terms of a three-dimensional wireframe representation of the organ, serves as an initial guess for the segmentation algorithm. We check the performance of the new algorithm on eight data sets of three different organs: rectum, bladder, and kidney. The results of the automatic segmentation were compared with a manual segmentation of each data set by radiation oncology faculty and residents. The quality of the automatic segmentation was measured with the ''$\kappa$-statistics'', and with a count of over- and undersegmented frames, and was shown in most cases to be very close to the manual segmentation of the same data. A typical segmentation of an organ with sixty slices takes less than ten seconds on a Pentium IV laptop.
A reaction-diffusion system modeling the spread of resistance to an antimalarial drug
Nicolas Bacaër and Cheikh Sokhna
2005, 2(2): 227-238 doi: 10.3934/mbe.2005.2.227 +[Abstract](635) +[PDF](199.1KB)
Abstract:
A mathematical model representing the diffusion of resistance to an antimalarial drug is developed. Resistance can spread only when the basic reproduction number of the resistant parasites is bigger than the basic reproduction number of the sensitive parasites (which depends on the fraction of infected people treated with the antimalarial drug). Based on a linearization study and on numerical simulations, an expression for the speed at which resistance spreads is conjectured. It depends on the ratio of the two basic reproduction numbers, on a coefficient representing the diffusion of mosquitoes, on the death rate of mosquitoes infected by resistant parasites, and on the recovery rate of nonimmune humans infected by resistant parasites.
Spatially Distributed Morphogen Production and Morphogen Gradient Formation
Arthur D. Lander, Qing Nie and Frederic Y. M. Wan
2005, 2(2): 239-262 doi: 10.3934/mbe.2005.2.239 +[Abstract](679) +[PDF](267.1KB)
Abstract:
Partial differential equations and auxiliary conditions governing the activities of the morphogen Dpp in Drosophila wing imaginal discs were formulated and analyzed as Systems B, R, and C in [7][9][10]. All had morphogens produced at the border of anterior and posterior chamber of the wing disc idealized as a point, line, or plane in a one-, two-, or three-dimensional model. In reality, morphogens are synthesized in a narrow region of finite width (possibly of only a few cells) between the two chambers in which diffusion and reversible binding with degradable receptors may also take place. The present investigation revisits the extracellular System R, now allowing for a finite production region of Dpp between the two chambers. It will be shown that this more refined model of the wing disc, designated as System F, leads to some qualitatively different morphogen gradient features. One significant difference between the two models is that System F impose no restriction on the morphogen production rate for the existence of a unique stable steady state concentration of the Dpp-receptor complexes. Analytical and numerical solutions will be obtained for special cases of System F. Some applications of the results for explaining available experimental data (to appear elsewhere) are briefly indicated. It will also be shown how the effects of the distributed source of System F may be aggregated to give an approximating point source model (designated as the aggregated source model or System A for short) that includes System R as a special case. System A will be analyzed in considerable detail in [6], and the limitation of System R as an approximation of System F will also be delineated there.
Incubation-time distribution in back-calculation applied to HIV/AIDS data in India
Arni S.R. Srinivasa Rao and Masayuki Kakehashi
2005, 2(2): 263-277 doi: 10.3934/mbe.2005.2.263 +[Abstract](553) +[PDF](241.3KB)
Abstract:
In this article, HIV incidence density is estimated from prevalence data and then used together with reported cases of AIDS to estimate incubation-time distribution. We used deconvolution technique and maximum likelihood method to estimate parameters. The effect of truncation in hazard was also examined. The mean and standard deviation obtained with the Weibull assumption were 12.9 and 3.0 years, respectively. The estimation seemed useful to investigate distribution of time between report of HIV infection and that of AIDS development. If we assume truncation, the optimum truncating point was sensitive to the HIV growth assumed. This procedure was applied to US data for validating the results obtained from the Indian data. The results show that method works well.
The Dynamics of the Schoener-Polis-Holt model of Intra-Guild Predation
Eric Ruggieri and Sebastian J. Schreiber
2005, 2(2): 279-288 doi: 10.3934/mbe.2005.2.279 +[Abstract](502) +[PDF](670.0KB)
Abstract:
Intraguild predation occurs when one species (the intraguild predator) predates on and competes with another species (the intraguild prey). A classic model of this interaction was introduced by Gary Polis and Robert Holt building on a model of competing species by Thomas Schoener. A global analysis reveals that this model exhibits generically six dynamics: extinction of one or both species; coexistence about a globally stable equilibrium; contingent exclusion in which the first established species prevents the establishment of the other species; contingent coexistence in which coexistence or displacement of the intraguild prey depend on initial conditions; exclusion of the intraguild prey; and exclusion of the intraguild predator. Implications for biological control and community ecology are discussed.
Parameter Estimation in a Coupled System of Nonlinear Size-Structured Populations
Azmy S. Ackleh, H.T. Banks, Keng Deng and Shuhua Hu
2005, 2(2): 289-315 doi: 10.3934/mbe.2005.2.289 +[Abstract](598) +[PDF](1002.1KB)
Abstract:
A least squares technique is developed for identifying unknown parameters in a coupled system of nonlinear size-structured populations. Convergence results for the parameter estimation technique are established. Ample numerical simulations and statistical evidence are provided to demonstrate the feasibility of this approach.
Thresholds for Epidemic Outbreaks in Finite Scale-Free Networks
Dong-Uk Hwang, S. Boccaletti, Y. Moreno and R. López-Ruiz
2005, 2(2): 317-327 doi: 10.3934/mbe.2005.2.317 +[Abstract](652) +[PDF](884.6KB)
Abstract:
We numerically investigate the existence of a threshold for epidemic outbreaks in a class of scale-free networks characterized by a parametrical dependence of the scaling exponent, influencing the convergence of fluctuations in the degree distribution. In finite-size networks, finite thresholds for the spreading of an epidemic are always found. However, both the thresholds and the behavior of the epidemic prevalence are quite different with respect to the type of network considered and the system size. We also discuss agreements and differences with some analytical claims previously reported.
Impulsive Ecological Control Of A Stage-Structured Pest Management System
Guirong Jiang, Qishao Lu and Linping Peng
2005, 2(2): 329-344 doi: 10.3934/mbe.2005.2.329 +[Abstract](699) +[PDF](1045.3KB)
Abstract:
The dynamics of a stage-structured pest management system is studied by means of autonomous piecewise linear systems with impulses governed by state feedback control. The sufficient conditions of existence and stability of periodic solutions are obtained by means of the sequence convergence rule and the analogue of the Poincaré criterion. The attractive region of periodic solutions is investigated theoretically by qualitative analysis. The bifurcation diagrams of periodic solutions are obtained by using the Poincaré map, as well as the chaotic solution generated via a cascade of period-doubling bifurcations. The superiority of the state feedback control strategy is also discussed.
Two-Species Competition with High Dispersal: The Winning Strategy
S.A. Gourley and Yang Kuang
2005, 2(2): 345-362 doi: 10.3934/mbe.2005.2.345 +[Abstract](673) +[PDF](1657.6KB)
Abstract:
This paper is motivated by the following simple question: how does diffusion affect the competition outcomes of two competing species that are identical in all respects other than their strategies on how they spatially distribute their birth rates. This may provide us with insights into how species learn to compete in a relatively stable setting, which in turn may point out species evolution directions. To this end, we formulate some extremely simple two- species competition models that have either continuous or discrete diffusion mechanisms. Our analytical work on these models collectively and strongly suggests the following in a fast diffusion environment: where different species have the same birth rates on average, those that do well are those that have greater spatial variation in their birth rates. We hypothesize that this may be a possible explanation for the evolution of grouping behavior in many species. Our findings are confirmed by extensive numerical simulation work on the models.
A mathematical model for treatment-resistant mutations of HIV
Helen Moore and Weiqing Gu
2005, 2(2): 363-380 doi: 10.3934/mbe.2005.2.363 +[Abstract](547) +[PDF](263.8KB)
Abstract:
In this paper, we propose and analyze a mathematical model, in the form of a system of ordinary differential equations, governing mutated strains of human immunodeficiency virus (HIV) and their interactions with the immune system and treatments. Our model incorporates two types of resistant mutations: strains that are not responsive to protease inhibitors, and strains that are not responsive to reverse transcriptase inhibitors. It also includes strains that do not have either of these two types of resistance (wild-type virus) and strains that have both types. We perform our analysis by changing the system of ordinary differential equations (ODEs) to a simple single-variable ODE, then identifying equilibria and determining stability. We carry out numerical calculations that illustrate the behavior of the system. We also examine the effects of various treatment regimens on the development of treatment-resistant mutations of HIV in this model.
The Ecology and Evolutionary Biology of Cancer: A Review of Mathematical Models of Necrosis and Tumor Cell Diversity
John D. Nagy
2005, 2(2): 381-418 doi: 10.3934/mbe.2005.2.381 +[Abstract](741) +[PDF](469.0KB)
Abstract:
Recent evidence elucidating the relationship between parenchyma cells and otherwise ''healthy'' cells in malignant neoplasms is forcing cancer biologists to expand beyond the genome-centered, ''one-renegade-cell'' theory of cancer. As it becomes more and more clear that malignant transformation is context dependent, the usefulness of an evolutionary ecology-based theory of malignant neoplasia becomes increasingly clear. This review attempts to synthesize various theoretical structures built by mathematical oncologists into potential explanations of necrosis and cellular diversity, including both total cell diversity within a tumor and cellular pleomorphism within the parenchyma. The role of natural selection in necrosis and pleomorphism is also examined. The major hypotheses suggested as explanations of these phenomena are outlined in the conclusions section of this review. In every case, mathematical oncologists have built potentially valuable models that yield insight into the causes of necrosis, cell diversity and nearly every other aspect of malignancy; most make predictions ultimately testable in the lab or clinic. Unfortunately, these advances have gone largely unexploited by the empirical community. Possible reasons why are considered.

2017  Impact Factor: 1.23

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