Cancer: A challenge for control theory and computer modelling

Düchting, W.; Ulmer, W.; T, Ginsberg

doi:10.1016/0959-8049(96)00075-5

Cited by 49 publications

(26 citation statements)

References 9 publications

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“…Many different types of discrete models, such as coupled map lattice models, fractal models, diffusion limited aggregation models and L-systems, have already been developed to model general branching processes (including angiogenesis) in a qualitative and phenomenological way (Bell et al, 1979;Gottlieb, 1990Gottlieb, , 1991aDüchting, 1990aDüchting, , 1990bDüchting, , 1992Prusinkiewicz and Lindenmayer, 1990;Kiani and Hudetz, 1991;Landini and Misson, 1993;Indermitte et al 1994;Düchting et al, 1996;Nekka et al, 1996). These discrete models may be considered as particular examples of a wider class of discrete models, referred to generically as cellular automata models, which have been applied to a wide range of problems in many areas of applied mathematics.…”

Section: The Discrete Mathematical Modelmentioning

confidence: 97%

Continuous and Discrete Mathematical Models of Tumor-induced Angiogenesis

Anderson

Chaplain

1998

Bulletin of Mathematical Biology

1,002

1,062

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Angiogenesis, the formation of blood vessels from a pre-existing vasculature, is a process whereby capillary sprouts are formed in response to externally supplied chemical stimuli. The sprouts then grow and develop, driven initially by endothelial-cell migration, and organize themselves into a dendritic structure. Subsequent cell proliferation near the sprout tip permits further extension of the capillary and ultimately completes the process. Angiogenesis occurs during embryogenesis, wound healing, arthritis and during the growth of solid tumors. In this paper we present both continuous and discrete mathematical models which describe the formation of the capillary sprout network in response to chemical stimuli (tumor angiogenic factors, TAF) supplied by a solid tumor. The models also take into account essential endothelial cell-extracellular matrix interactions via the inclusion of the matrix macromolecule fibronectin. The continuous model consists of a system of nonlinear partial differential equations describing the initial migratory response of endothelial cells to the TAF and the fibronectin. Numerical simulations of the system, using parameter values based on experimental data, are presented and compared qualitatively with in vivo experiments. We then use a discretized form of the partial differential equations to develop a biased random-walk model which enables us to track individual endothelial cells at the sprout tips and incorporate anastomosis, mitosis and branching explicitly into the model. The theoretical capillary networks generated by computer simulations of the discrete model are compared with the morphology of capillary networks observed in in vivo experiments.

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Section: The Discrete Mathematical Modelmentioning

confidence: 97%

Continuous and Discrete Mathematical Models of Tumor-induced Angiogenesis

Anderson

Chaplain

1998

Bulletin of Mathematical Biology

1,002

1,062

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“…Ward andKing (1999) extended Greenspan's (1974) model to study the growth of radially symmetric tumours containing multiple cell populations. More recently, Anderson and Chaplain (1998), Baum et al (1999) and Duchting et al (1996) have developed stochastic models which reproduce many of the detailed features of tumour angiogenesis and tumour invasion.…”

Section: Introductionmentioning

confidence: 99%

Estimating the Selective Advantage of Mutant p53 Tumour Cells to Repeated Rounds of Hypoxia

Gammack

Byrne

Lewis

2001

Bulletin of Mathematical Biology

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The tumour suppressor gene, p53, plays an important role in tumour development. Under low levels of oxygen (hypoxia), cells expressing wild-type p53 undergo programmed cell death (apoptosis), whereas cells expressing mutations in the p53 gene may survive and express angiogenic growth factors that stimulate tumour vascularization. Given that cells expressing mutations in the p53 gene have been observed in many forms of human tumour, it is important to understand how both wild-type and mutant cells react to hypoxic conditions. In this paper a mathematical model is presented to investigate the effects of alternating periods of hypoxia and normoxia (normal oxygen levels) on a population of wild-type and mutant p53 tumour cells. The model consists of three coupled ordinary differential equations that describe the densities of the two cell types and the oxygen concentration and, as such, may describe the growth of avascular tumours in vitro and/or in vivo. Numerical and analytical techniques are used to determine how changes in the system parameters influence the time at which mutant cells become dominant within the population. A feedback mechanism, which switches off the oxygen supply when the total cell density exceeds a threshold value, is introduced into the model to investigate the impact that vessel collapse (and the associated hypoxia) has on the time at which the mutant cells become dominant within vascular tumours growing in vivo. Using the model we can predict the time it takes for a subpopulation of mutant p53 tumour cells to become the dominant population within either an avascular tumour or a localized region of a vascular tumour. Based on independent experimental results, our model suggests that the mutant population becomes dominant more quickly in vivo than in vitro (12 days vs 17 days).

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“…Moreover, as explained in [25] and demonstrated for spheroids in [26], nutrient-deprived cells are less apt to undergo mitosis, and the necrotic debris is eventually removed; consequently, the net repopulation rate increases as the tumor shrinks. Unfortunately, the LQ model -which typically assumes that the sensitivity and repopulation rate of a tumor are constant throughout the course of therapy -does not appear to be capable of tackling these issues, except within the context of a large simulation model [27]. In a related paper, O'Donoghue [28] ignores incomplete repair but develops a nonspatial model where the tumor grows exponentially when it is small and Gompertzian when it is big.…”

Section: Weinmentioning

confidence: 99%

Dynamic optimization of a linear–quadratic model with incomplete repair and volume-dependent sensitivity and repopulation

Wein

Cohen

2000

International Journal of Radiation Oncology*Biology*Physics

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Purpose: The linear-quadratic model typically assumes that tumor sensitivity and repopulation are constant over the time course of radiotherapy. However, evidence suggests that the growth fraction increases and the cell loss factor decreases as the tumor shrinks. We investigate whether this evolution in tumor geometry, as well as the irregular time intervals between fractions in conventional hyperfractionation schemes, can be exploited by fractionation schedules that employ time-varying fraction sizes. Methods:We construct a mathematical model of a spherical tumor with a hypoxic core and a viable rim, and embed this model into the traditional linear-quadratic model by assuming instantaneous reoxygenation. Dynamic programming is used to numerically compute the fractionation regimen that maximizes the tumor control probability (TCP) subject to constraints on the biologically effective dose of the early and late tissues. Results:In a numerical example that employs 10 fractions per week, optimally varying the fraction sizes increases the TCP from 0.7 to 0.966, and the optimal regimen incorporates large Friday afternoon doses that are escalated throughout the course of treatment, and larger afternoon doses than morning doses.Conclusions: Numerical results suggest that a significant increase in tumor cure can be achieved by allowing the fraction sizes to vary throughout the course of treatment. Several strategies deserve further investigation: using larger fractions before overnight and weekend breaks, and escalating the dose (particularly on Friday afternoons) throughout the course of treatment.

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Cancer: A challenge for control theory and computer modelling

Cited by 49 publications

References 9 publications

Continuous and Discrete Mathematical Models of Tumor-induced Angiogenesis

Continuous and Discrete Mathematical Models of Tumor-induced Angiogenesis

Estimating the Selective Advantage of Mutant p53 Tumour Cells to Repeated Rounds of Hypoxia

Dynamic optimization of a linear–quadratic model with incomplete repair and volume-dependent sensitivity and repopulation

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