Biomedical subjects

H M Byrne

Publications and source records attributed to H M Byrne.

At least 19 recordsLinked to original sources

PubMed2006

DNA charge neutralization by linear polymers. II. Reversible binding.

We model the way in which polymers bind to DNA and neutralize its charged backbone by analyzing the dynamics of the distribution of gaps along the DNA. We generalize existing theory for irreversible binding to construct deterministic models which include polymer removal, movement along the DNA, and allow for binding with overlaps. We show that reversible binding alters the capacity of the DNA for polymers by allowing the rearrangement of polymer positions over a longer time scale than when binding is irreversible. When the polymers do not overlap, allowing reversible binding increases the number of polymers adhered and hence the charge that the DNA can accommodate; in contrast, when overlaps occur, reversible binding reduces the amount of charge neutralized by the polymers.

Binding Sites↗

PubMed2006

DNA charge neutralization by linear polymers: irreversible binding.

We develop a deterministic mathematical model to describe the way in which polymers bind to DNA by considering the dynamics of the gap distribution that forms when polymers bind to a DNA plasmid. In so doing, we generalize existing theory to account for overlaps and binding cooperativity whereby the polymer binding rate depends on the size of the overlap. The proposed mean-field models are then solved using a combination of numerical and asymptotic methods. We find that overlaps lead to higher coverage and hence higher charge neutralizations, results which are more in line with recent experimental observations. Our work has applications to gene therapy where polymers are used to neutralize the negative charges of the DNA phosphate backbone, allowing condensation prior to delivery into the nucleus of an abnormal cell.

Binding Sites↗

PubMed2006

Modelling aspects of cancer dynamics: a review.

Cancer is a complex disease in which a variety of factors interact over a wide range of spatial and temporal scales with huge datasets relating to the different scales available. However, these data do not always reveal the mechanisms underpinning the observed phenomena. In this paper, we explain why mathematics is a powerful tool for interpreting such data by presenting case studies that illustrate the types of insight that realistic theoretical models of solid tumour growth may yield. These range from discriminating between competing hypotheses for the formation of collagenous capsules associated with benign tumours to predicting the most likely stimulus for protease production in early breast cancer. We will also illustrate the benefits that may result when experimentalists and theoreticians collaborate by considering a novel anti-cancer therapy.

Animals↗

PubMed2006

Interactions between pattern formation and domain growth.

In this paper we develop a theoretical framework for investigating pattern formation in biological systems for which the tissue on which the spatial pattern resides is growing at a rate which is itself regulated by the diffusible chemicals that establish the spatial pattern. We present numerical simulations for two cases of interest, namely exponential domain growth and chemically controlled growth. Our analysis reveals that for domains undergoing rapid exponential growth dilution effects associated with domain growth influence both the spatial patterns that emerge and the concentration of chemicals present in the domain. In the latter case, there is complex interplay between the effects of the chemicals on the domain size and the influence of the domain size on the formation of patterns. The nature of these interactions is revealed by a weakly nonlinear analysis of the full system. This yields a pair of nonlinear equations for the amplitude of the spatial pattern and the domain size. The domain is found to grow (or shrink) at a rate that depends quadratically on the pattern amplitude, the particular functional forms used to model the local tissue growth rate and the kinetics of the two diffusible species dictating the resulting behaviour.

Body Patterning↗

PubMed2006

Crypt dynamics and colorectal cancer: advances in mathematical modelling.

Mathematical modelling forms a key component of systems biology, offering insights that complement and stimulate experimental studies. In this review, we illustrate the role of theoretical models in elucidating the mechanisms involved in normal intestinal crypt dynamics and colorectal cancer. We discuss a range of modelling approaches, including models that describe cell proliferation, migration, differentiation, crypt fission, genetic instability, APC inactivation and tumour heterogeneity. We focus on the model assumptions, limitations and applications, rather than on the technical details. We also present a new stochastic model for stem-cell dynamics, which predicts that, on average, APC inactivation occurs more quickly in the stem-cell pool in the absence of symmetric cell division. This suggests that natural niche succession may protect stem cells against malignant transformation in the gut. Finally, we explain how we aim to gain further understanding of the crypt system and of colorectal carcinogenesis with the aid of multiscale models that cover all levels of organization from the molecular to the whole organ.

Cell Differentiation↗

PubMed2005

Biological inferences from a mathematical model of comedo ductal carcinoma in situ of the breast.

The growth of a tumour in a duct is examined in order to model ductal carcinoma in situ (DCIS) of the breast, the earliest known stage of breast cancer. Interactions between the expansive forces created by tumour cell proliferation and the stresses that develop in the compliant basement membrane are studied using numerical and analytical techniques. Particular attention focuses on the impact that proteolytic enzymes have on the tumour's progression. As the tumour expands and the duct wall deforms, the tumour cells are subjected to mechanical and nutritional stresses caused by high pressures and oxygen deprivation. Such stresses may stimulate the cells to produce proteolytic enzymes that degrade the duct wall, making it more compliant and prone to penetration by the tumour cells. We use our model to compare these two hypotheses for enzyme production and find that mechanical stress is likely the dominant mechanism, with the wall deforming most at the centre of the duct. We then discuss the biological implications of our theoretical results and suggest possible directions for future work.

Animals↗

PubMed2004

A mathematical model of the effects of hypoxia on the cell-cycle of normal and cancer cells.

The evolution of the cell-cycle is known to be influenced by environmental conditions, including lack of extracellular oxygen (hypoxia). Notably, hypoxia appears to have different effects on normal and cancer cells. Whereas both experience hypoxia-induced arrest of the G1 phase of the cell-cycle (i.e. delay in the transition through the restriction point), experimental evidence suggests that only cancer cells undergo hypoxia-induced quiescence (i.e. the transition of the cell to a latent state in which most of the cell functions, including proliferation, are suspended). Here, we extend a model for the cell-cycle due to Tyson and Novak (J. Theor. Biol. 210 (2001) 249) to account for the action of the protein p27. This protein, whose expression is upregulated under hypoxia, inhibits the activation of the cyclin dependent kinases (CDKs), thus preventing DNA synthesis and delaying the normal progression through the cell-cycle. We use a combination of numerical and analytic techniques to study our model. We show that it reproduces many features of the response to hypoxia of normal and cancer cells, as well as generating experimentally testable predictions. For example our model predicts that cancer cells can undergo quiescence by increasing their levels of p27, whereas for normal cells p27 expression decreases when the cellular growth rate increases.

Animals↗

PubMed2003

A cellular automaton model for tumour growth in inhomogeneous environment.

Most of the existing mathematical models for tumour growth and tumour-induced angiogenesis neglect blood flow. This is an important factor on which both nutrient and metabolite supply depend. In this paper we aim to address this shortcoming by developing a mathematical model which shows how blood flow and red blood cell heterogeneity influence the growth of systems of normal and cancerous cells. The model is developed in two stages. First we determine the distribution of oxygen in a native vascular network, incorporating into our model features of blood flow and vascular dynamics such as structural adaptation, complex rheology and red blood cell circulation. Once we have calculated the oxygen distribution, we then study the dynamics of a colony of normal and cancerous cells, placed in such a heterogeneous environment. During this second stage, we assume that the vascular network does not evolve and is independent of the dynamics of the surrounding tissue. The cells are considered as elements of a cellular automaton, whose evolution rules are inspired by the different behaviour of normal and cancer cells. Our aim is to show that blood flow and red blood cell heterogeneity play major roles in the development of such colonies, even when the red blood cells are flowing through the vasculature of normal, healthy tissue.

Animals↗

PubMed2003

Modelling the early growth of ductal carcinoma in situ of the breast.

The growth of a tumour in a rigid walled cylindrical duct is examined in order to model the initial stages of tumour cell expansion in ductal carcinoma in situ (DCIS) of the breast. A nutrient-limited growth model is formulated, in which cell movement is described by a Stokes flow constitutive relation. The effects on the shape of the tumour boundary of the material properties (i.e. the viscosity) and the extent to which the cells adhere to the duct wall are studied using numerical and asymptotic methods. It is shown how stable, non-planar, interface configurations result and that, during these initial stages, before the duct wall has been breached, few cells die and a nutrient-rich model is usually sufficient to capture the behaviour. Finally, we discuss the relevance of this approach to DCIS and suggest possible avenues for further work.

Apoptosis↗

PubMed2003

Mathematical modelling of comedo ductal carcinoma in situ of the breast.

The growth of a tumour in a cylindrical duct with compliant walls is examined in order to model the early stages of ductal carcinoma in situ (DCIS) of the breast, the earliest known stage of breast cancer. A nutrient-limited growth model is formulated, in which cell movement is described by a Stokes flow constitutive relation. The interactions between the expansive forces created by tumour cell proliferation and the stresses that develop in the compliant basement membrane are studied using asymptotic and numerical methods. In this way we show how the duct wall deforms as the tumour grows and also how the progression of the tumour along the duct depends upon the stiffness of the wall. By varying key parameters we determine how treatment, protease production and the inclusion of the surrounding stroma affect the growth. Finally, we discuss the biological relevance of our results and suggest possible directions for future work.

Basement Membrane↗

PubMed2002

The role of cell-cell interactions in a two-phase model for avascular tumour growth.

A two-phase model is presented to describe avascular tumour growth. Conservation of mass equations, including oxygen-dependent cell growth and death terms, are coupled with equations of momentum conservation. The cellular phase behaves as a viscous liquid, while the viscosity of the extracellular water manifests itself as an interphase drag. It is assumed that the cells become mechanically stressed if they are too densely packed and that the tumour will try to increase its volume in order to relieve such stress. By contrast, the overlapping filopodia of sparsely populated cells create short-range attractive effects. Finally, oxygen is consumed by the cells as it diffuses through the tumour. The resulting system of equations are reduced to three, which describe the evolution of the tumour cell volume fraction, the cell speed and the oxygen tension. Numerical simulations indicate that the tumour either evolves to a travelling wave profile, in which it expands at a constant rate, or it settles to a steady state, in which the net rates of cell proliferation and death balance. The impact of varying key model parameters such as cellular viscosity, interphase drag, and cellular tension are discussed. For example, tumours consisting of well-differentiated (i.e. viscous) cells are shown to grow more slowly than those consisting of poorly-differentiated (i.e. less viscous) cells. Analytical results for the case of oxygen-independent growth are also presented, and the effects of varying the key parameters determined (the results are in line with the numerical simulations of the full problem). The key results and their biological implications are then summarised and future model refinements discussed.

Animals↗

PubMed2001

Estimating the selective advantage of mutant p53 tumour cells to repeated rounds of hypoxia.

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).

Apoptosis↗

PubMed2001

Modeling and analysis of a virus that replicates selectively in tumor cells.

Replication-competent viruses have shown considerable promise in overcoming the inefficient gene transduction experienced by traditional gene therapy approaches to cancer treatment. The viruses infect tumor cells and replicate inside them, eventually causing lysis. Virus particles released during lysis are then able to infect other tumor cells, and, in this way, continuous rounds of infection and lysis allow the virus to spread throughout the tumor. Motivated by this novel cancer treatment, we formulate and analyse a system of partial differential equations that is essentially a radially-symmetric epidemic model embedded in a Stefan problem. We compare three, alternative virus-injection strategies: a fixed fraction of cells pre-infected with the virus are introduced throughout the entire tumor volume, within the tumor core, or within the tumor rim. For all three injection methods, simple and accurate conditions that predict whether the virus will control the tumor are derived.

Animals↗

PubMed2001

The influence of growth-induced stress from the surrounding medium on the development of multicell spheroids.

A mathematical model is adapted to the description of the growth of an avascular tumour spheroid embedded in a deformable medium (gel). Attention is focused on the influence that the mechanical properties of the gel have on the tumour's growth dynamics. Following the work of Landman and Please (2000), the tumour is treated as a two phase material, whereby the tumour cells and the extracellular fluid form distinct phases. Its growth is modelled by combining mass conservation and force balances for each phase with the usual concepts of diffusion limited growth in response to an externally supplied nutrient. The mechanical properties of the gel are characterised by a strain energy function. The stress induced in the gel by the tumour's expansion is incorporated into the force balance equations, thereby linking it to the tumour's growth. Numerical simulations of the model equations show that as the stiffness of the gel increases, the tumour's growth rate and equilibrium size decrease and the time at which necrosis is initiated is delayed (if it occurs at all). Similar results are obtained when the initial size of the tumour is reduced whilst the mechanical properties of the surrounding gel are held fixed. Such results, which are in good qualitative agreement with available experimental data, suggest that mechanical interactions between a tumour and the tissue or medium in which it is located can significantly influence its growth dynamics.

Animals↗

PubMed2000

A mathematical model of the stress induced during avascular tumour growth.

In this paper a mathematical model is developed to describe the effect of nonuniform growth on the mechanical stress experienced by cells within an avascular tumour. The constitutive law combines the stress-strain relation of linear elasticity with a growth term that is derived by analogy with thermal expansion. To accommodate the continuous nature of the growth process, the law relates the rate of change of the stress tensor to the rate of change of the strain (rather than relating the stress to the strain directly). By studying three model problems which differ in detail, certain characteristic features are identified. First, cells near the tumour boundary, where nutrient levels and cell proliferation rates are high, are under compression. By contrast, cells towards the centre of the tumour, where nutrient levels are low and cell death dominant, are under tension. The implications of these results and possible model developments are also discussed.

Humans↗

PubMed2000

A mathematical model to study the effects of drug resistance and vasculature on the response of solid tumors to chemotherapy.

A mathematical model is developed that describes the reduction in volume of a vascular tumor in response to specific chemotherapeutic administration strategies. The model consists of a system of partial differential equations governing intratumoral drug concentration and cancer cell density. In the model the tumor is treated as a continuum of two types of cells which differ in their proliferation rates and their responses to the chemotherapeutic agent. The balance between cell proliferation and death within the tumor generates a velocity field which drives expansion or regression of the spheroid. Insight into the tumor's response to therapy is gained by applying a combination of analytical and numerical techniques to the model equations.

Animals↗

PubMed1999

A weakly nonlinear analysis of a model of avascular solid tumour growth.

In this paper we study a mathematical model that describes the growth of an avascular solid tumour. Our analysis concentrates on the stability of steady, radially-symmetric model solutions with respect to perturbations taken from the class of spherical harmonics. Using weakly nonlinear analysis, previous results are extended to show how the amplitudes of the asymmetric modes interact. Attention focuses on a special case for which the model equations simplify. Analysis of the simplified model equations leads to the identification of a two-parameter family of asymmetric steady solutions, the dimensions of whose stable and unstable manifolds depend on the system parameters. The asymmetric steady solutions limit the basin of attraction of the radially-symmetric steady state when it is linearly stable. On the basis of these numerical and analytical results we postulate the existence of fully nonlinear steady solutions which are stable with respect to time-dependent perturbations.

Cell Division↗

PubMed1999

Extracellular matrix concentration exerts selection pressure on invasive cells.

The mortality associated with cancer depends upon the ability of malignant cells to invade and metastasise into adjacent and distant regions. Such malignant spread is caused by the acquisition of an invasive phenotype which involves variable changes in cell-cell adhesion, proteolysis of adjoining extracellular matrix molecules, and the ability to move in a directed fashion in response to fixed and soluble gradients. Whilst the degree of variability in the pattern of metastasis is large, several cancers show regional predilections for invasion. Tumour cell heterogeneity and extracellular matrix composition have been shown to account for some regional variations. In this study, an invasion assay was used to assess the invasiveness of HT1080 tumour cells migrating through a collagen gel. Our experiments showed that, in the absence of externally imposed chemical gradients, HT1080 invasiveness was related in a biphasic manner to collagen concentration. Using a mathematical model, developed to study this phenomenon, we predicted that tumour cell proliferation may also be related in a biphasic manner to collagen concentration. This model prediction was confirmed using a combination of collagen gel invasion and proliferation assays. Investigation of the mathematical model suggests that interactions between haptotaxis and proliferation of the HT1080 cells may be responsible for the biphasic dependence of the penetration depth and proliferation on collagen gel concentration. In conclusion, we showed how mathematical methods can be combined with experimental work to provide new and valuable insight into important biological issues.

Cell Adhesion↗