Modelling the interactions between tumour cells 
and a blood vessel in a microenvironment within 
a vascular tumour

Breward, Christopher; Byrne, Helen M.; Lewis, Claire E.

doi:10.1017/s095679250100448x

Cited by 45 publications

(28 citation statements)

References 24 publications

(31 reference statements)

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“…These boundary conditions are written under the assumption that oxygen diffusion through the blood vessel wall is fast with respect to the growth of tumour. A more elaborate approach to describe tumour-vessel interaction is suggested in (Breward et al, 2001). For remote boundaries we assume that they stay undisturbed by the growth and the flux of nutrients through them is zero.…”

Section: Boundary Conditionsmentioning

confidence: 99%

Mathematical modelling of the Warburg effect in tumour cords

Astanin

Preziosi

2009

Journal of Theoretical Biology

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The model proposed here links together two approaches to describe tumours: a continuous medium to describe the movement and the mechanical properties of the tissue, and a population dynamics approach to represent internal genetic inhomogeneity and instability of the tumour. In this way one can build models which cover several stages of tumour progression. In this paper we focus on describing transition from aerobic to purely glycolytic metabolism (the Warburg effect) in tumour cords. From the mathematical point of view this model leads to a free boundary problem where domains in contact are characterized by different sets of equations. Accurate stitching of the solution was possible with a modified ghost fluid method. Growth and death of the cells and uptake of the nutrients are related through ATP production and energy costs of the cellular processes. In the framework of the bi-population model this allowed to keep the number of model parameters relatively small.Keywords: tumour growth, tumour metabolism, Warburg effect, mathematical model, ghost fluid method, population dynamics Metabolic processes in normal tissues require oxygen. Specifically, 6 molecules of oxygen are consumed per oxidated molecule of glucose with the yield of approximately 32 molecules of ATP (Nelson & Cox, 2000) (36 according to , 29.85 according to (Rich, 2003)). In many cancers intensive proliferation exceeds available oxygen supply, which leads to hypoxia. In such hypoxic conditions cells may rely only on glycolysis, the first step of glucose oxydation, to cover their energy needs. This process gives a smaller amount of ATP, 2 molecules of ATP per molecule of glucose, but it is possible in hypoxic conditions. In fact, most tumours are known to rely on glycolytic metabolism even in non-hypoxic conditions. This effect is known as Warburg effect or aerobic glycolysis (Warburg, 1956;Kim & Dang, 2006) and is one of the hallmarks of cancer (Hannahan & Weinberg, 2000). Glycolytic catabolism has the important side effect of tissue acidification. Lower pH is toxic to most normal cells while altered tumour cells are likely to be resistant to it and achieve another invasion advantage (Gatenby et al., 2006). In the same time, there are several therapeutic strategies which allow for targeting tumours with glycolytic metabolism (Kim & Dang, 2006;Mathupala et al., 2007).We want to describe phenomenologically the transition of tumours from normal to glycolytic metabolism in the framework of spatio-temporal model of tumour growth. Well aware that there are several mechanisms which contribute to the Warburg effect, we assumed that this switch in metabolism happens as an all or nothing event, after which cells rely only on glycolytic metabolism even with adequate oxygen levels. The switch is assumed to happen in hypoxic conditions. This problem received a lot of attention from the mathematical modelling community in the recent years. Such works as (Gatenby et al., 2006;Gatenby et al., 2007;Smallbone et al., 2008) have pointed to the possibility o...

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Section: Boundary Conditionsmentioning

confidence: 99%

Mathematical modelling of the Warburg effect in tumour cords

Astanin

Preziosi

2009

Journal of Theoretical Biology

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“…If we extended our model in to two dimensions and allowed for directionality of blood flow so that p γ = p γ (x, t), we would lose the obvious symmetry seen in the experiments. We note also that we do not consider details of vessel compliance since we believe these are important only on the vessel's microscale (see Breward et al (2001)…”

Section: Stress Tensorsmentioning

confidence: 99%

“…On the microscale, several models based on the Krogh Cylinder model (Krogh, 1919) have been proposed (Baish et al, 1996;Maseide and Rofstad, 2000) whilst Breward et al (2001) followed the approach of Ward and King (1997). These papers focus on the interactions between a blood vessel supplying oxygen to a region of tissue within a tumour; the latter paper incorporating the effect of blood vessel compliance on the subsequent remodelling of the tumour tissue.…”

Section: Introductionmentioning

confidence: 99%

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A Multiphase Model Describing Vascular Tumour Growth

Breward¹,

Byrne

Lewis

2003

Bulletin of Mathematical Biology

145

126

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In this paper we present a new model framework for studying vascular tumour growth, in which the blood vessel density is explicitly considered. Our continuum model comprises conservation of mass and momentum equations for the volume fractions of tumour cells, extracellular material and blood vessels. We include the physical mechanisms that we believe to be dominant, namely birth and death of tumour cells, supply and removal of extracellular fluid via the blood and lymph drainage vessels, angiogenesis and blood vessel occlusion. We suppose that the tumour cells move in order to relieve the increase in mechanical stress caused by their proliferation. We show how to reduce the model to a system of coupled partial differential equations for the volume fraction of tumour cells and blood vessels and the phase averaged velocity of the mixture. We consider possible parameter regimes of the resulting model. We solve the equations numerically in these cases, and discuss the resulting behaviour. The model is able to reproduce tumour structure that is found in vivo in certain cases. Our framework can be easily modifed to incorporate the effect of other phases, or to include the effect of drugs.

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“…While there is a rapidly expanding literature devoted to modelling the different aspects of solid tumour growth, including avascular growth [3,9,30], angiogenesis [1], vascular tumour growth [2,4] and the responses of tumours to chemotherapy [12,21,22], the role of macrophages has not been widely studied. The model developed in this paper complements work by Owen and Sherratt [26] who studied the inhibitory effect of normal, non-engineered macrophages caused by their cytolytic activity.…”

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confidence: 99%

Macrophage-tumour interactions: In vivo dynamics

Byrne¹,

Cox²,

Kelly³

2003

DCDS-B

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Abstract. Experimentalists are developing new therapies that exploit the tendency of macrophages, a type of white blood cell, to localise within solid tumours. The therapy studied here involves engineering macrophages to produce chemicals that kill tumour cells. Accordingly, a simple mathematical model is developed that describes interactions between normal cells, tumour cells and infiltrating macrophages. Numerical and analytical techniques show how the ability of the engineered macrophages to eliminate the tumour changes as model parameters vary. The key parameters are m * , the concentration of engineered macrophages injected into the vasculature, and k 1 , the rate at which they lyse tumour cells. As k 1 or m * increases, the average tumour burden decreases although the tumour is never completely eliminated by the macrophages. Also, the stable solutions are oscillatory when k 1 and m * increase through well-defined bifurcation values. The physical implications of our results and directions for future research are also discussed.

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Modelling the interactions between tumour cells and a blood vessel in a microenvironment within a vascular tumour

Cited by 45 publications

References 24 publications

Mathematical modelling of the Warburg effect in tumour cords

Mathematical modelling of the Warburg effect in tumour cords

A Multiphase Model Describing Vascular Tumour Growth

Macrophage-tumour interactions: In vivo dynamics

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