The role of acidity in solid tumour growth and invasion

Acidic pH is a common characteristic of human tumours. It has a significant impact on tumour progression and response to therapies. In this paper, we develop a simple model of three-dimensional tumour growth to examine the role of acidosis in the interaction between normal and tumour cell populations. Both vascular and avascular tumour dynamics are investigated, and a number of different behaviours are observed. Whilst an avascular tumour always proceeds to a benign steady state, a vascular tumour may display either benign or invasive dynamics, depending on the value of a critical parameter. Analysis of the model allows us to assess novel therapies directed towards changing the level of acidity within the tumour.     

Mathematical modelling of radiotherapy strategies for early breast cancer

Targeted intraoperative radiotherapy (Targit) is a new concept of partial breast irradiation where single fraction radiotherapy is delivered directly to the tumour bed. Apart from logistic advantages, this strategy minimizes the risk of missing the tumour bed and avoids delay between surgery and radiotherapy. It is presently being compared with the standard fractionated external beam radiotherapy (EBRT) in randomized trials. In this paper we present a mathematical model for the growth and invasion of a solid tumour into a domain of tissue (in this case breast tissue), and then a model for surgery and radiation treatment of this tumour. We use the established linear-quadratic (LQ) model to compute the survival probabilities for both tumour cells and irradiated breast tissue and then simulate the effects of conventional EBRT and Targit. True local recurrence of the tumour could arise either from stray tumour cells, or the tumour bed that harbours morphologically normal cells having a predisposition to genetic changes, such as a loss of heterozygosity (LOH) in genes that are crucial for tumourigenesis, e.g. tumour suppressor genes (TSGs). Our mathematical model predicts that the single high dose of radiotherapy delivered by Targit would result in eliminating all these sources of recurrence, whereas the fractionated EBRT would eliminate stray tumour cells, but allow (by virtue of its very schedule) the cells with LOH in TSGs or cell-cycle checkpoint genes to pass on low-dose radiation-induced DNA damage and consequently mutations that may favour the development of a new tumour. The mathematical model presented here is an initial attempt to model a biologically complex phenomenon that has until now received little attention in the literature and provides a 'proof of principle' that it is possible to produce clinically testable hypotheses on the effects of different approaches of radiotherapy for breast cancer.     

Mathematical modelling of the Warburg effect in tumour cords

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.     

Metabolic changes during carcinogenesis: potential impact on invasiveness

Successful adaptation to varying microenvironmental constraints plays a crucial role during carcinogenesis. We develop a hybrid cellular automation approach to investigate the cell-microenvironmental interactions that mediate somatic evolution of cancer cells. This allows investigation of the hypothesis that regions of premalignant lesions develop a substrate-limited environment as proliferation carries cells away from blood vessels which remain separated by the intact basement membrane. We find that selective forces in tumoural regions furthest from the blood supply act to favour cells whose metabolism is best suited to respond to local changes in oxygen, glucose and pH levels. The model predicts three phases of somatic evolution. Initially, cell survival and proliferation is limited due to diminished oxygen levels. This promotes adaptation to a second phase of growth dominated by cells with constitutively up-regulated glycolysis, less reliant on oxygen for ATP production. Increased glycolysis induces acidification of the local environment, limiting proliferation and inducing cell death through necrosis and apoptosis. This promotes a third phase of cellular evolution, with emergence of phenotypes resistant to acid-induced toxicity. This emergent cellular phenotype has a significant proliferative advantage because it will consistently acidify the local environment in a way that is toxic to its competitors but harmless to itself. The model's results suggest this sequence is essential in the transition from self-limited premalignant growth to invasive cancer, and, therefore, that this transition may be delayed or prevented through novel strategies directed towards interrupting the hypoxia-glycolysis-acidosis cycle.     

Coupled modelling of tumour angiogenesis, tumour growth and blood perfusion

We propose a mathematical modelling system to investigate the dynamic process of tumour cell proliferation, death and tumour angiogenesis by fully coupling the vessel growth, tumour growth and blood perfusion. Tumour growth and angiogenesis are coupled by the chemical microenvironment and the cell-matrix interaction. The haemodynamic calculation is carried out on the updated vasculature. The domains of intravascular, transcapillary and interstitial fluid flow were coupled in the model to provide a comprehensive solution of blood perfusion variables. An estimation of vessel collapse is made according to the wall shear stress criterion to provide feedback on vasculature remodelling. The simulation can show the process of tumour angiogenesis and the spatial distribution of tumour cells for periods of up to 24 days. It can show the major features of tumour and tumour microvasculature during the period such as the formation of a large necrotic core in the tumour centre with few functional vessels passing through, and a well circulated tumour periphery regions in which the microvascular density is high and associated with more aggressive proliferating cells of the growing tumour which are all consistent with physiological observations. The study also demonstrated that the simulation results are not dependent on the initial tumour and networks, which further confirms the application of the coupled model feedback mechanisms. The model enables us to examine the interactions between angiogenesis and tumour growth, and to study the dynamic response of a solid tumour to the changes in the microenvironment. This simulation framework can be a foundation for further applications such as drug delivery and anti-angiogenic therapies.     

Mathematical modelling of fibre-enhanced perfusion inside a tissue-engineering bioreactor

We develop a simple mathematical model for forced flow of culture medium through a porous scaffold in a tissue-engineering bioreactor. Porous-walled hollow fibres penetrate the scaffold and act as additional sources of culture medium. The model, based on Darcy's law, is used to examine the nutrient and shear-stress distributions throughout the scaffold. We consider several configurations of fibres and inlet and outlet pipes. Compared with a numerical solution of the full Navier-Stokes equations within the complex scaffold geometry, the modelling approach is cheap, and does not require knowledge of the detailed microstructure of the particular scaffold being used. The potential of this approach is demonstrated through quantification of the effect the additional flow from the fibres has on the nutrient and shear-stress distribution.     

Evolution of cell motility in an individual-based model of tumour growth

Tumour invasion is driven by proliferation and importantly migration into the surrounding tissue. Cancer cell motility is also critical in the formation of metastases and is therefore a fundamental issue in cancer research. In this paper we investigate the emergence of cancer cell motility in an evolving tumour population using an individual-based modelling approach. In this model of tumour growth each cell is equipped with a micro-environment response network that determines the behaviour or phenotype of the cell based on the local environment. The response network is modelled using a feed-forward neural network, which is subject to mutations when the cells divide. With this model we have investigated the impact of the micro-environment on the emergence of a motile invasive phenotype. The results show that when a motile phenotype emerges the dynamics of the model are radically changed and we observe faster growing tumours exhibiting diffuse morphologies. Further we observe that the emergence of a motile subclone can occur in a wide range of micro-environmental growth conditions. Iterated simulations showed that in identical growth conditions the evolutionary dynamics either converge to a proliferating or migratory phenotype, which suggests that the introduction of cell motility into the model changes the shape of fitness landscape on which the cancer cell population evolves and that it now contains several local maxima. This could have important implications for cancer treatments which focus on the gene level, as our results show that several distinct genotypes and critically distinct phenotypes can emerge and become dominant in the same micro-environment.     

Modulation of membrane phenotype, matrix adhesion and microinvasiveness of metastatic tumour cells by HUdR

The effect of HUdR, proved to be anti-metastatic in vivo, was studied in vitro on cell proliferation, nucleoside uptake, membrane fluidity, expression of galactosylated glycans and proteoglycans in metastatic HM tumour cells. The observed increase in membrane fluidity and the suppression of nucleoside transport were early events of the HUdR action followed by decrease of galactosylated glycan and HSPG expression. However, these changes did not influence the proliferation capacity of the cells at the concentrations studied. As a consequence of the membrane alterations a reduced adhesiveness and spreading on extracellular matrix components was detected. In addition, the HUdR treated HM cells showed reduced capacity to invade fibroblast monolayers in vitro. Based on these observations, HUdR could be the prototype of new anti-metastatic agents acting at the level of tumour-host interaction.     

Mathematical modelling of the aerobic degradation of two-phase olive mill effluents in a batch reactor

A laboratory-scale study was conducted on the aerobic degradation of two-phase olive mill effluents (TPOME) made up of the mixture of the washwaters derived from the initial cleansing of the olives and those obtained in the washing and purification of virgin olive oil. The process was carried out in a 1-l working volume stirred tank reactor operating in batch mode at room temperature (25 °C). The reactor was operated at influent substrate concentrations of 2.80 g COD/l (TPOME 25%), 5.45 g COD/l (TPOME 50%), 8.18 g COD/l (TPOME 75%) and 10.90 g COD/l (TPOME 100%). After five days of operation time, total and soluble COD removal efficiencies of 64.3% and 66.6% were achieved respectively for the most concentrated influent used (TPOME 100%). A simplified kinetic model for studying the hydrolysis of insoluble organic matter, oxidation of soluble substrate and biomass production was proposed on the basis of the experimental results obtained. The following kinetic constants with their standard deviations were obtained for the above stages in the case of the most concentrated influent used (TPOME 100%): k₁ (kinetic constant for hydrolysis of suspended organic matter): 0.11 ± 0.01 l/(g VSS day); k₂ (kinetic constant for total consumption of soluble substrate): 0.30 ± 0.02 l/(g VSS day); k₃ (endogenous metabolism constant): 0.07 ± 0.01 per day). Finally, the biomass yield coefficient was found to be 0.30 g VSS/g COD_removed. The values of non-biodegradable total and soluble CODs obtained from the model were found to be 3 and 2 g/l, respectively. The kinetic constants obtained and the proposed equations were used to simulate the aerobic degradation process of TPOME and to obtain the theoretical values of non-soluble and soluble CODs and biomass concentration. The small deviations obtained (equal or lower than 10%) between the theoretical and experimental values suggest that the parameters obtained represent and predict the activity of the microorganisms involved in the overall aerobic degradation process of this wastewater.     

Mathematical modelling of intra-aortic balloon pump

Background: Ischemic heart diseases now afflict thousands of Iranians and are the major cause of death in many industrialised countries. Mathematical modelling of an intra-aortic balloon pump (IABP) could provide a better understanding of its performance and help to represent blood flow and pressure in systemic arteries before and after inserting the pump.

Methods: A mathematical modelling of the whole cardiovascular system was formulated using MATLAB software. The block diagram of the model consists of 43 compartments. All the anatomical data was extracted from the physiological references. In the next stage, myocardial infarction (MI) was induced in the model by decreasing the contractility of the left ventricle. The IABP was mathematically modelled and inserted in the model in the thoracic aorta I artery just before the descending aorta. The effects of IABP on MI were studied using the mathematical model.

Results: The normal operation of the cardiovascular system was studied firstly. The pressure–time graphs of the ventricles, atriums, aorta, pulmonary system, capillaries and arterioles were obtained. The volume–time curve of the left ventricle was also presented. The pressure–time curves of the left ventricle and thoracic aorta I were obtained for normal, MI, and inserted IABP conditions. Model verification was performed by comparing the simulation results with the clinical observations reported in the literature.

Conclusions: IABP can be described by a theoretical model. Our model representing the cardiovascular system is capable of showing the effects of different pathologies such as MI and we have shown that MI effects can be reduced using IABP in accordance with the modelling results. The mathematical model should serve as a useful tool to simulate and better understand cardiovascular operation in normal and pathological conditions.     

Mathematical modelling of tissue-engineered angiogenesis

We present a mathematical model for the vascularisation of a porous scaffold following implantation in vivo. The model is given as a set of coupled non-linear ordinary differential equations (ODEs) which describe the evolution in time of the amounts of the different tissue constituents inside the scaffold. Bifurcation analyses reveal how the extent of scaffold vascularisation changes as a function of the parameter values. For example, it is shown how the loss of seeded cells arising from slow infiltration of vascular tissue can be overcome using a prevascularisation strategy consisting of seeding the scaffold with vascular cells. Using certain assumptions it is shown how the system can be simplified to one which is partially tractable and for which some analysis is given. Limited comparison is also given of the model solutions with experimental data from the chick chorioallantoic membrane (CAM) assay.     

Multi-scale computational modelling in biology and physiology

Recent advances in biotechnology and the availability of ever more powerful computers have led to the formulation of increasingly complex models at all levels of biology. One of the main aims of systems biology is to couple these together to produce integrated models across multiple spatial scales and physical processes. In this review, we formulate a definition of multi-scale in terms of levels of biological organisation and describe the types of model that are found at each level. Key issues that arise in trying to formulate and solve multi-scale and multi-physics models are considered and examples of how these issues have been addressed are given for two of the more mature fields in computational biology: the molecular dynamics of ion channels and cardiac modelling. As even more complex models are developed over the coming few years, it will be necessary to develop new methods to model them (in particular in coupling across the interface between stochastic and deterministic processes) and new techniques will be required to compute their solutions efficiently on massively parallel computers. We outline how we envisage these developments occurring.     

Mathematical Talent is Linked to Autism

A total of 378 mathematics undergraduates (selected for being strong at “systemizing”) and 414 students in other (control) disciplines at Cambridge University were surveyed with two questions: (1) Do you have a diagnosed autism spectrum condition? (2) How many relatives in your immediate family have a diagnosed autism spectrum condition? Results showed seven cases of autism in the math group (or 1.85%) vs one case of autism in the control group (or 0.24%), a ninefold difference that is significant. Controlling for sex and general population sampling, this represents a three- to sevenfold increase for autism spectrum conditions among the mathematicians. There were 7 of 1,405 (or 0.5%) cases of autism in the immediate families of the math group vs 2 of 1,669 (or 0.1%) cases in the immediate families of the control group, which again is a significant difference. These results confirm a link between autism and systemizing, and they suggest this link is genetic given the association between autism and first-degree relatives of mathematicians.     

Mathematical modelling of hepatitis C treatment for injecting drug users

Hepatitis C virus (HCV) is a blood-borne infection that can lead to progressive liver failure, cirrhosis, hepatocellular carcinoma and death. In developed countries, the majority of HCV infections are transmitted via injecting drug users (IDUs). Despite effective antiviral treatment for HCV, very few active IDUs are treated. Reluctance to treat is partially due to the risk of reinfection. We develop a mathematical model of HCV transmission amongst active IDUs, and examine the potential effect of antiviral treatment. As most mathematical models of interventions utilise a treatment function proportional to the infected population, but many policy implementations set fixed yearly targets for specific numbers treated, we study the effects of using two different treatment terms: annually treating a proportion of infecteds or a fixed number of infecteds. We examine the behaviour of the two treatment models and find different bifurcation behaviours in each case. We calculate analytical solutions for the treatment level needed for disease clearance or control, and observe that achievable levels of treatment can result in control or eradication across a wide range of prevalence levels. Finally, we calculate the sensitivity of the critical treatment threshold to the model parameters, and find that for a given observed prevalence, the injecting duration and infection risk play the most important role in determining the treatment level needed. By contrast, the sensitivity analysis indicates the presence (or absence) of immunity does not alter the treatment threshold. We conclude by discussing the public health implications of this work, and comment on the importance and feasibility of utilising treatment as prevention for HCV spread amongst IDUs.     

Dual contrasting roles of cysteine cathepsins in cancer progression: apoptosis versus tumour invasion

Cysteine cathepsins have been known for a long time to play an important role in cancer progression and metastasis. Several studies have proposed the concept of anti-cathepsin therapy in cancer treatment. On the other hand, cysteine cathepsins have been recently found to play a role in tumour cell death through mediation of apoptosis. The purpose of this mini-review is therefore to provide an insight into the mechanisms by which cysteine cathepsins modulate apoptosis and/or participate in tumour invasion, and to evaluate the impact of these enzymes on both tumour progression and development of potential strategies for cancer treatment.     

Mathematical modelling of liver regeneration after intoxication with CCl(4)

Liver regeneration is a complex process, having evolved to protect animals from the consequences of liver loss caused by food toxins. In this study, we established a mathematical spatial-temporal model of the liver lobule regenerating after CCl(4) intoxication. The aim of modelling the regeneration process by matching experimental observations with those from a mathematical model is to gain a better understanding of the process and to recognize which parameters are relevant for specific phenomena. In order to set up a realistic minimal model, we first reconstructed a schematised liver lobule after determination of: (i) the mean number of hepatocytes between the central vein and the periphery of the lobule, (ii) the mean size of the hepatocytes and (iii) the mean number of hepatocyte columns in the inner, midzonal and peripheral ring of the lobule. In a next step, we determined the time course of cell death and BrdU incorporation after intoxication of male Sprague Dawley rats with CCl(4), thereby differentiating between inner, midzonal and peripheral hepatocytes. These parameters were used to construct a model. The basic unit of this model is the individual cell. The detailed behaviour of the cells is studied, controlled by the model parameters: (1) probability of cell division at defined positions of the lobule at a given time, (2) "coordinated cell orientation", i.e., the ability of the cells to align during the regeneration process into columns towards the central vein of a liver lobule, (3) cell cycle duration, (4) the migration activity and (5) the polarity of the hepatocytes resulting in polar cell-cell adhesion between them. In a schematised lobule, the model shows that CCl(4) initially induced cell death of a pericentral ring of hepatocytes, followed by a wave of proliferation that starts in the surviving hepatocytes next to the inner ring of dead cells and continues to the peripheral hepatocytes, finally restoring the characteristic micro-architecture of the lobule in a 7-day process. This model was used to systematically analyze the influence of parameters 1-5. Interestingly, coordinated cell orientation and cell polarity were identified to be the most critical parameters. Elimination led to destruction of the characteristic micro-architecture of the lobule and to a high degree of disorder characterized by hexagonal cell structures. Our model suggests that the ability of hepatocytes to realign after cell division by a process of coordinated cell orientation (model parameter 2) in combination with cell polarity (model parameter 5) may be at least as critical as hepatocyte proliferation (model parameter 1) itself.     

The role of mechanical host-tumour interactions in the collapse of tumour blood vessels and tumour growth dynamics

A mathematical model of residual stress evolution in a growing vascular tumour is presented, in an attempt to elucidate the poorly understood phenomenon of vascular collapse. Whereas earlier studies in this area have neglected the effects of mechanical interactions between the tumour and the surrounding host tissue, the significance of these interactions for the long-term development of a tumour is now considered. The model predicts tumour stress distributions which reflect the distinctive patterns of vascular collapse reported in experimental studies. Moreover, while neglecting mechanical host/tumour interactions results in the eventual complete regression of the tumour to its avascular dormant size in the event of vascular collapse, this new model points to the possibility of oscillations in the tumour's size in the long term.     

Cost-effective G-CSF therapy strategies for cyclical neutropenia: mathematical modelling based hypotheses

Using computer simulations of a mathematical model for the regulation of stem cell and neutrophil production in dogs, we have studied the efficacy of four different treatment protocols for cyclical neutropenia involving granulocyte colony stimulating factor (G-CSF). The first treatment scheme is based on the bifurcation analysis of the mathematical model and proposes a daily, phase-dependent, protocol. The second involves alternate day administration of G-CSF. The third triggers G-CSF administration whenever neutrophil levels fall below a predetermined level, and the fourth one follows a random administration protocol. The computer simulations predict that clinically desirable results can be achieved with the three last methods, using far less G-CSF than would be needed with the standard daily treatment. If the results of this modelling are borne out clinically, they will entail a considerable financial savings for patients.