# American Institute of Mathematical Sciences

ISSN:
1551-0018

eISSN:
1547-1063

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## Mathematical Biosciences & Engineering

2005 , Volume 2 , Issue 3

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2005, 2(3): i-ii doi: 10.3934/mbe.2005.2.3i +[Abstract](1010) +[PDF](35.6KB)
Abstract:
The idea for this volume came from the session ''Control and Dynamics of Biomedical Systems,'' which we organized at the Fourth World Congress of Nonlinear Analysts held in Orlando, Florida, on June 30–July 7, 2004. We invited a number of speakers, and although not all could attend and give a talk in person, many expressed interest in contributing a paper. Following an invitation from Yang Kuang to put together a volume on the topic as a special issue of Mathematical Biosciences and Engineering, some papers presented at the session were submitted immediately, and others were contributed after the Congress.

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2005, 2(3): 421-436 doi: 10.3934/mbe.2005.2.421 +[Abstract](1213) +[PDF](201.3KB)
Abstract:
In light of recent clinical developments, the importance of mathematical modeling in cancer prevention and treatment is discussed. An existing model of cancer chemotherapy is reintroduced and placed within current investigative frameworks regarding approaches to treatment optimization. Areas of commonality between the model predictions and the clinical findings are investigated as a way of further validating the model predictions and also establishing mathematical foundations for the clinical studies. The model predictions are used to propose additional ways that treatment optimization could enhance the clinical processes. Arising out of these, an expanded model of cancer is proposed and a treatment model is subsequently obtained. These models predict that malignant cells in the marrow and peripheral blood exhibit the tendency to evolve toward population levels that enable them to replace normal cells in these compartments in the untreated case. In the case of dose-dense treatment along with recombinant hematopoietic growth factors, the models predict a situation in which normal and abnormal cells in the marrow and peripheral blood are obliterated by drug action, while the normal cells regain their growth capabilities through growth-factor stimulation.
2005, 2(3): 437-443 doi: 10.3934/mbe.2005.2.437 +[Abstract](934) +[PDF](169.1KB)
Abstract:
This work is concerned with the analysis of the possibility for eradicating a disease in an infected population. The epidemiological model under study is of SI type with diffusion. We assume the policy strategy acting on the infected individuals over a subset of the whole spatial territory. Using the framework of nonlinear reaction-diffusion equations, and spectral theory of linear differential operators, we give necessary conditions and sufficient conditions of eradicability.
2005, 2(3): 445-460 doi: 10.3934/mbe.2005.2.445 +[Abstract](1382) +[PDF](405.0KB)
Abstract:
This work illustrates the behavior of the interstitial pressure and of the interstitial fluid motion in tumor cords (cylindrical arrangements of tumor cells growing around blood vessels of the tumor) by means of numerical simulations on the basis of a mathematical model previously developed. The model describes the steady state of a tumor cord surrounded by necrosis and its time evolution following cell killing. The most relevant aspects of the dynamics of extracellular fluid are by computing the longitudinal average of the radial fluid velocity and of the pressure field. In the present paper, the necrotic region is treated as a mixture of degrading dead cells and fluid.
2005, 2(3): 461-472 doi: 10.3934/mbe.2005.2.461 +[Abstract](1406) +[PDF](255.0KB)
Abstract:
A simple model of avascular solid tumor dynamics is studied in the paper. The model is derived on the basis of reaction-diffusion dynamics and mass conservation law. We introduce time delay in a cell proliferation process. In the case studied in this paper, the model reduces to one ordinary functional-differential equation of the form that depends on the existence of necrotic core. We focus on the process of this necrotic core formation and the possible influence of delay on it. Basic mathematical properties of the model are studied. The existence, uniqueness and stability of steady state are discussed. Results of numerical simulations are presented.
2005, 2(3): 473-485 doi: 10.3934/mbe.2005.2.473 +[Abstract](1220) +[PDF](392.0KB)
Abstract:
The cell-division cycle and apoptosis are key cellular processes deregulated during carcinogenesis. Recent work of Aguda and Algar suggests a modular organization of regulatory molecular pathways linking the cellular processes of division and apoptosis. We carry out a detailed mathematical analysis of the Aguda-Algar model to unravel the dynamics implicit in the model structure. In addition, we further explore model parameters that control the bifurcations corresponding to the aforementioned cellular state transitions. We show that this simple model predicts interesting behavior, such as hysteretic oscillations and different conditions in which apoptosis is triggered.
2005, 2(3): 487-498 doi: 10.3934/mbe.2005.2.487 +[Abstract](1147) +[PDF](191.7KB)
Abstract:
This paper reviews the state-of-the-art knowledge concerning the relationship between Neanderthals and Upper Paleolithic modern humans. The branching-process method is applied to infer the upper limit of hypothetical Neanderthal admixture, consistent with the evidence based on mitochondrial DNA sequences of contemporary modern humans, as well as Neanderthal and early modern European H. sapiens fossils. As a result, a maximum value of 15% admixture is obtained. This estimate is discussed in the context of its consequences for the two competing theories of modern human origin.
2005, 2(3): 499-510 doi: 10.3934/mbe.2005.2.499 +[Abstract](1467) +[PDF](202.4KB)
Abstract:
We investigate mathematical models for the dynamics between tumor cells, immune-effector cells, and cytokine interleukin-2 (IL-2). To better determine under what circumstances the tumor can be eliminated, we implement optimal control theory. We design two control functionals, the first functional having one control and the second having two controls, to maximize the effector cells and interleukin-2 concentration and to minimize the tumor cells. Next, we show that bang-bang optimal controls exist for each problem. Then, we characterize our optimal controls in terms of the solutions to the optimality system, which is the state system coupled with the adjoint system. Finally, we analyze the various optimal controls and optimality systems using numerical techniques.
2005, 2(3): 511-525 doi: 10.3934/mbe.2005.2.511 +[Abstract](1385) +[PDF](237.8KB)
Abstract:
We perform critical-point analysis for three-variable systems that represent essential processes of the growth of the angiogenic tumor, namely, tumor growth, vascularization, and generation of angiogenic factor (protein) as a function of effective vessel density. Two models that describe tumor growth depending on vascular mass and regulation of new vessel formation through a key angiogenic factor are explored. The first model is formulated in terms of ODEs, while the second assumes delays in this regulation, thus leading to a system of DDEs. In both models, the only nontrivial critical point is always unstable, while one of the trivial critical points is always stable. The models predict unlimited growth, if the initial condition is close enough to the nontrivial critical point, and this growth may be characterized by oscillations in tumor and vascular mass. We suggest that angiogenesis per se does not suffice for explaining the observed stabilization of vascular tumor size.
2005, 2(3): 527-534 doi: 10.3934/mbe.2005.2.527 +[Abstract](1446) +[PDF](168.9KB)
Abstract:
This paper concerns the problem of fitting of mathematical models of cell signaling pathways. Such models frequently take the form of a set of nonlinear ordinary differential equations. While the model is continuous-time, the performance index, used in the fitting procedure, involves measurements taken only at discrete-time moments. Adjoint sensitivity analysis is a tool that can be used for finding a gradient of a performance index in the space of the model’s parameters. The paper uses a structural formulation of sensitivity analysis, especially dedicated for hybrid, continuous/discrete-time systems. A numerical example of fitting of the mathematical model of the NF-kB regulatory module is presented.
2005, 2(3): 535-560 doi: 10.3934/mbe.2005.2.535 +[Abstract](1285) +[PDF](528.8KB)
Abstract:
The immune response to Mycobacterium tuberculosis infection (Mtb) is the formation of unique lesions, called granulomas. How well these granulomas form and function is a key issue that might explain why individuals experience different disease outcomes. The spatial structures of these granulomas are not well understood. In this paper, we use a metapopulation framework to develop a spatio-temporal model of the immune response to Mtb. Using this model, we are able to investigate the spatial organization of the immune response in the lungs to Mtb. We identify both host and pathogen factors that contribute to successful infection control. Additionally, we identify specific spatial interactions and mechanisms important for successful granuloma formation. These results can be further studied in the experimental setting.
2005, 2(3): 561-578 doi: 10.3934/mbe.2005.2.561 +[Abstract](1277) +[PDF](286.3KB)
Abstract:
Mathematical models for cancer chemotherapy as optimal control problems are considered. Results on scheduling optimal therapies when the controls represent the effectiveness of chemotherapeutic agents, or, equivalently, when the simplifying assumption is made that drugs act instantaneously, are compared with more realistic models that include pharmacokinetic (PK) equations modelling the drug's plasma concentration and various pharmacodynamic (PD) models for the effect the concentrations have on cells.
2005, 2(3): 579-590 doi: 10.3934/mbe.2005.2.579 +[Abstract](1033) +[PDF](543.6KB)
Abstract:
I describe several models of population dynamics, both unstructured and gender structured, that include groups of individuals who do not reproduce. I analyze the effect that the nonreproductive group may have on the dynamics of the whole population in terms of the vital rates and the proportion of nonreproductive individuals, and we provide specific examples for real populations.
2005, 2(3): 591-611 doi: 10.3934/mbe.2005.2.591 +[Abstract](1184) +[PDF](243.5KB)
Abstract:
In this paper, a general periodic vaccination has been applied to control the spread and transmission of an infectious disease with latency. A $SEIRS^1$ epidemic model with general periodic vaccination strategy is analyzed. We suppose that the contact rate has period $T$, and the vaccination function has period $LT$, where $L$ is an integer. Also we apply this strategy in a model with seasonal variation in the contact rate. Both the vaccination strategy and the contact rate are general time-dependent periodic functions. The same SEIRS models have been examined for a mixed vaccination strategy composed of both the time-dependent periodic vaccination strategy and the conventional one. A key parameter of the paper is a conjectured value $R^c_0$ for the basic reproduction number. We prove that the disease-free solution (DFS) is globally asymptotically stable (GAS) when $R^{"sup"}_0 < 1$. If $R^{"inf"}_0 > 1$, then the DFS is unstable, and we prove that there exists a nontrivial periodic solution whose period is the same as that of the vaccination strategy. Some persistence results are also discussed. Necessary and sufficient conditions for the eradication or control of the disease are derived. Threshold conditions for these vaccination strategies to ensure that $R^{"sup"}_0 < 1$ and $R^{"inf"}_0 > 1$ are also investigated.
2005, 2(3): 613-624 doi: 10.3934/mbe.2005.2.613 +[Abstract](1640) +[PDF](371.8KB)
Abstract:
We introduce a model for describing the dynamics of large numbers of interacting cells. The fundamental dynamical variables in the model are subcellular elements, which interact with each other through phenomenological intra- and intercellular potentials. Advantages of the model include i) adaptive cell-shape dynamics, ii) flexible accommodation of additional intracellular biology, and iii) the absence of an underlying grid. We present here a detailed description of the model, and use successive mean-field approximations to connect it to more coarse-grained approaches, such as discrete cell-based algorithms and coupled partial differential equations. We also discuss efficient algorithms for encoding the model, and give an example of a simulation of an epithelial sheet. Given the biological flexibility of the model, we propose that it can be used effectively for modeling a range of multicellular processes, such as tumor dynamics and embryogenesis.
2005, 2(3): 625-642 doi: 10.3934/mbe.2005.2.625 +[Abstract](1008) +[PDF](491.8KB)
Abstract:
The regulation of the cell cycle clock is examined using a theoretical model for the embryonic cell cycle, where the clock is described as a single-limit cycle [1]. By taking the coefficient of the autocatalytic reaction as proportional to the deviation of the system from its equilibrium state, we show how such clocks can be adjusted to function on several time scales. This feedback control, causing a periodic change in the sign of the autocatalytic reaction, may be interpreted as a periodic change in the ratio of cdc25/wee1 activity. Its introduction results in the appearance of a double limit cycle, signifying the acquisition of the G1 phase and the G2 phase, during embryonic development. Following the loss of stability of the double cycle, through a period-doubling bifurcation, another limit set—a strange attractor—is born. The complicated geometry of this strange attractor can be viewed as an unlimited reservoir of periods in the phase space.

We hypothesize that the existence of such a reservoir is advantageous in morphogenetic tissues, such as the bone marrow, as it enables time- and site-specific selection of the optimal cell-cycle period for any specific micro- environment. This can be obtained by the addition of a time delay in the autocatalytic reaction, reflecting, for example, the influence of external molecular signals on cell-cycle progression.
2005, 2(3): 643-655 doi: 10.3934/mbe.2005.2.643 +[Abstract](1851) +[PDF](1188.9KB)
Abstract:
Interactions between tumor cells and their environment lead to the formation of microregions containing nonhomogeneous subpopulations of cells and steep gradients in oxygen, glucose, and other metabolites. To address the formation of tumor microregions on the level of single cells, I propose a new two-dimensional time-dependent mathematical model taking explicitly into account the individually regulated biomechanical processes of tumor cells and the effect of oxygen consumption on their metabolism. Numerical simulations of the self-organized formation of tumor microregions are presented and the dynamics of such a process is discussed.
2005, 2(3): 657-670 doi: 10.3934/mbe.2005.2.657 +[Abstract](1296) +[PDF](201.8KB)
Abstract:
This paper presents analysis and biomedical implications of a certain class of bilinear systems that can be applied in modeling of cancer chemotherapy. It combines models that so far have been studied separately, taking into account both the phenomenon of gene amplification and drug specificity in chemotherapy in their different aspects. The methodology of analysis of such models, based on system decomposition, is discussed. The mathematical description is given by an infinite dimensional state equation with a system matrix, the form of which allows decomposing the model into two interacting subsystems. While the first one, of a finite dimension, can have any form, the second one is infinite-dimensional and tridiagonal. Then the optimal control problem is defined in $l^1$ space. To derive necessary conditions for optimal control, the model description is transformed into an integrodifferential one.
2005, 2(3): 671-671 doi: 10.3934/mbe.2005.2.671 +[Abstract](987) +[PDF](54.0KB)
Abstract:
In the paper ''Spatially Distributed Morphogen Production and Morphogen Gradient Formation" by Arthur D. Lander, Qing Nie, and Frederic Y. M. Wan (Mathematical Biosciences and Engineering, Volume 2, Number 2, April 2005), the following correction should be made:

In the statement of Lemma 10, "(0,1)" should hav been "($-x_m$, 1)."

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