Towards whole-organ modelling of tumour growth

Multiscale approaches to modelling biological phenomena are growing rapidly. We present here some recent results on the formulation of a theoretical framework which can be developed into a fully integrative model for cancer growth. The model takes account of vascular adaptation and cell-cycle dynamics. We explore the effects of spatial inhomogeneity induced by the blood flow through the vascular network and of the possible effects of p27 on the cell cycle. We show how the model may be used to investigate the efficiency of drug-delivery protocols

A design principle for vascular beds: the effects of complex blood rheology

We propose a design principle that extends Murray's original optimization principle for vascular architecture to account for complex blood rheology. Minimization of an energy dissipation function enables us to determine how rheology affects the morphology of simple branching networks. The behavior of various physical quantities associated with the networks, such as the wall shear stress and the flow velocity, is also determined. Our results are shown to be qualitatively and quantitatively compatible with independent experimental observations and simulations

Biphasic behaviour in malignant invasion

Invasion is an important facet of malignant growth that enables tumour cells to colonise adjacent regions of normal tissue. Factors known to influence such invasion include the rate at which the tumour cells produce tissue-degrading molecules, or proteases, and the composition of the surrounding tissue matrix. A common feature of experimental studies is the biphasic dependence of the speed at which the tumour cells invade on properties such as protease production rates and the density of the normal tissue. For example, tumour cells may invade dense tissues at the same speed as they invade less dense tissue, with maximal invasion seen for intermediate tissue densities. In this paper, a theoretical model of malignant invasion is developed. The model consists of two coupled partial differential equations describing the behaviour of the tumour cells and the surrounding normal tissue. Numerical methods show that the model exhibits steady travelling wave solutions that are stable and may be smooth or discontinuous. Attention focuses on the more biologically relevant, discontinuous solutions which are characterised by a jump in the tumour cell concentration. The model also reproduces the biphasic dependence of the tumour cell invasion speed on the density of the surrounding normal tissue. We explain how this arises by seeking constant-form travelling wave solutions and applying non-standard phase plane methods to the resulting system of ordinary differential equations. In the phase plane, the system possesses a singular curve. Discontinuous solutions may be constructed by connecting trajectories that pass through particular points on the singular curve and recross it via a shock. For certain parameter values, there are two points at which trajectories may cross the singular curve and, as a result, two distinct discontinuous solutions may arise

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

Cancer modelling: Getting to the heart of the problem

Paradoxically, improvements in healthcare that have enhanced the life expectancy of humans in the Western world have, indirectly, increased the prevalence of certain types of cancer such as prostate and breast. It remains unclear whether this phenomenon should be attributed to the ageing process itself or the cumulative effect of prolonged exposure to harmful environmental stimuli such as ultraviolet light, radiation and carcinogens (Franks and Teich, 1988). Equally, there is also compelling evidence that certain genetic abnormalities can predispose individuals to specific cancers (Ilyas et al., 1999). The variety of factors that have been implicated in the development of solid tumours stems, to a large extent, from the fact that ‘cancer’ is a generic term, often used to characterize a series of disorders that share common features. At this generic level of description, cancer may be viewed as a cellular disease in which controls that usually regulate growth and maintain homeostasis are disrupted. Cancer is typically initiated by genetic mutations that lead to enhanced mitosis of a cell lineage and the formation of an avascular tumour. Since it receives nutrients by diffusion from the surrounding tissue, the size of an avascular tumour is limited to several millimeters in diameter. Further growth relies on the tumour acquiring the ability to stimulate the ingrowth of a new, circulating blood supply from the host vasculature via a process termed angiogenesis (Folkman, 1974). Once vascularised, the tumour has access to a vast nutrient source and rapid growth ensues. Further, tumour fragments that break away from the primary tumour, on entering the vasculature, may be transported to other organs in which they may establish secondary tumours or metastases that further compromise the host. Invasion is another key feature of solid tumours whereby contact with the tissue stimulates the production of enzymes that digest the tissue, liberating space into which the tumour cells migrate. Thus, cancer is a complex, multiscale process. The spatial scales of interest range from the subcellular level, to the cellular and macroscopic (or tissue) levels while the timescales may vary from seconds (or less) for signal transduction pathways to months for tumour doubling times The variety of phenomena involved, the range of spatial and temporal scales over which they act and the complex way in which they are inter-related mean that the development of realistic theoretical models of solid tumour growth is extremely challenging. While there is now a large literature focused on modelling solid tumour growth (for a review, see, for example, Preziosi, 2003), existing models typically focus on a single spatial scale and, as a result, are unable to address the fundamental problem of how phenomena at different scales are coupled or to combine, in a systematic manner, data from the various scales. In this article, a theoretical framework will be presented that is capable of integrating a hierarchy of processes occurring at different scales into a detailed model of solid tumour growth (Alarcon et al., 2004). The model is formulated as a hybrid cellular automaton and contains interlinked elements that describe processes at each spatial scale: progress through the cell cycle and the production of proteins that stimulate angiogenesis are accounted for at the subcellular level; cell-cell interactions are treated at the cellular level; and, at the tissue scale, attention focuses on the vascular network whose structure adapts in response to blood flow and angiogenic factors produced at the subcellular level. Further coupling between the different spatial scales arises from the transport of blood-borne oxygen into the tissue and its uptake at the cellular level. Model simulations will be presented to illustrate the effect that spatial heterogeneity induced by blood flow through the vascular network has on the tumour’s growth dynamics and explain how the model may be used to compare the efficacy of different anti-cancer treatment protocols

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

Multiscale modelling of tumour growth and therapy: the influence of vessel normalisation on chemotherapy

Following the poor clinical results of antiangiogenic drugs, particularly when applied in isolation, tumour biologists and clinicians are now turning to combinations of therapies in order to obtain better results. One of these involves vessel normalisation strategies. In this paper, we investigate the effects on tumour growth of combinations of antiangiogenic and standard cytotoxic drugs, taking into account vessel normalisation. An existing multiscale framework is extended to include new elements such as tumour-induced vessel dematuration. Detailed simulations of our multiscale framework allow us to suggest one possible mechanism for the observed vessel normalisation-induced improvement in the efficacy of cytotoxic drugs: vessel dematuration produces extensive regions occupied by quiescent (oxygen-starved) cells which the cytotoxic drug fails to kill. Vessel normalisation reduces the size of these regions, thereby allowing the chemotherapeutic agent to act on a greater number of cells

Beyond technology and finance: pay-as-you-go sustainable energy access and theories of social change

Two-thirds of people in sub-Saharan Africa lack access to electricity, a precursor of poverty reduction and development. The international community has ambitious commitments in this regard, e.g. the UN's Sustainable Energy for All by 2030. But scholarship has not kept up with policy ambitions. This paper operationalises a sociotechnical transitions perspective to analyse for the first time the potential of new, mobileenabled, pay-as-you-go approaches to financing sustainable energy access, focussing on a case study of pay-as-you-go approaches to financing solar home systems in Kenya. The analysis calls into question the adequacy of the dominant, two-dimensional treatment of sustainable energy access in the literature as a purely financial/technology, economics/ engineering problem (which ignores sociocultural and political considerations) and demonstrates the value of a new research agenda that explicitly attends to theories of social change – even when, as in this paper, the focus is purely on finance. The paper demonstrates that sociocultural considerations cut across the literature's traditional two-dimensional analytic categories (technology and finance) and are material to the likely success of any technological or financial intervention. It also demonstrates that the alignment of new payas- you-go finance approaches with existing sociocultural practices of paying for energy can explain their early success and likely longevity relative to traditional finance approaches

Seismic Response of Structures on Soft Foundations

A method of analysis for predicting the earthquake induced response a structure on a flexible foundation soil is presented. The foundation soil is represented by a bed of elasticplastic springs allowing both the strength of the soil and its stiffness to be incorporated in the analysis. Application of the method indicates: (1) the vertical component of the earthquake has a negligible effect on the response of the structure (2) the maximum induced overturning moment depends primarily on the strength of the soil and whether the foundation is free to lift from the soil (3) overturning of tall buildings is unlikely to occur unless the foundation soil suffers a strength loss due to the shaking

Mathematical modelling of angiogenesis and vascular adaptation

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