Cell kinetics in a tumour cord

Computational modelling of drug delivery to solid tumour: Understanding the interplay between chemotherapeutics and biological system for optimised delivery systems

Drug delivery to solid tumour involves multiple physiological, biochemical and biophysical processes taking place across a wide range of length and time scales. The therapeutic efficacy of anticancer drugs is influenced by the complex interplays among the intrinsic properties of tumours, biophysical aspects of drug transport and cellular uptake. Mathematical and computational modelling allows for a well-controlled study on the individual and combined effects of a wide range of parameters on drug transport and therapeutic efficacy, which would not be possible or economically viable through experimental means. A wide spectrum of mathematical models has been developed for the simulation of drug transport and delivery in solid tumours, including PK/PD-based compartmental models, microscopic and macroscopic transport models, and molecular dynamics drug loading and release models. These models have been used as a tool to identify the limiting factors and for optimal design of efficient drug delivery systems. This article gives an overview of the currently available computational models for drug transport in solid tumours, together with their applications to novel drug delivery systems, such as nanoparticle-mediated drug delivery and convection-enhanced delivery.

Computational Modelling of Cancer Development and Growth: Modelling at Multiple Scales and Multiscale Modelling

In this paper, we present two mathematical models related to different aspects and scales of cancer growth. The first model is a stochastic spatiotemporal model of both a synthetic gene regulatory network (the example of a three-gene repressilator is given) and an actual gene regulatory network, the NF-[Formula: see text]B pathway. The second model is a force-based individual-based model of the development of a solid avascular tumour with specific application to tumour cords, i.e. a mass of cancer cells growing around a central blood vessel. In each case, we compare our computational simulation results with experimental data. In the final discussion section, we outline how to take the work forward through the development of a multiscale model focussed at the cell level. This would incorporate key intracellular signalling pathways associated with cancer within each cell (e.g. p53-Mdm2, NF-[Formula: see text]B) and through the use of high-performance computing be capable of simulating up to [Formula: see text] cells, i.e. the tissue scale. In this way, mathematical models at multiple scales would be combined to formulate a multiscale computational model.

Oxygen and placental development; parallels and differences with tumour biology

Human placentation involves the invasion of the conceptus into the wall of the uterus, and establishment of a blood supply from the maternal spiral arteries. The placenta has therefore been likened to a malignant tumour, albeit a highly regulated one. Oxygen plays an important role in controlling both placental development and tumour behaviour. In the placenta, early development takes place in a physiological low oxygen environment, which undergoes a transition with onset of the full maternal arterial circulation towards the end of the first trimester. By comparison, in tumours there is often a progressive hypoxia as the mass outgrows its blood supply. Both early placental tissues and tumour cells show high rates of proliferation, and the energy required to support these comes principally from glycolysis. Glycolysis is maintained in placental tissues by reoxidation of pyridine nucleotides through the polyol pathways, whereas in tumours there is fermentation to lactate, Warburg metabolism. In both cases, the reliance on glycolysis rather than oxidative phosphorylation preserves carbon skeletons that can be utilised in the synthesis of nucleotides, cell membranes and organelles, and that would otherwise be excreted as carbon dioxide. In the placenta, this reliance may also protect the embryo from free radical-mediated teratogenesis. Local oxygen gradients within both sets of tissues may influence the cell behaviour. In particular, they may induce an epithelial-mesenchymal transition, promoting extravillous trophoblast invasion in the placenta and metastasis in a tumour. Further investigations into the two scenarios may provide new insights of benefit to these contrasting, but similar, fields of cellular biology.

Progression and metastasis of lung cancer

Metastasis in lung cancer is a multifaceted process. In this review, we will dissect the process in several isolated steps such as angiogenesis, hypoxia, circulation, and establishment of a metastatic focus. In reality, several of these processes overlap and occur even simultaneously, but such a presentation would be unreadable. Metastasis requires cell migration toward higher oxygen tension, which is based on changing the structure of the cell (epithelial-mesenchymal transition), orientation within the stroma and stroma interaction, and communication with the immune system to avoid attack. Once in the blood stream, cells have to survive trapping by the coagulation system, to survive shear stress in small blood vessels, and to find the right location for extravasation. Once outside in the metastatic locus, tumor cells have to learn the communication with the “foreign” stroma cells to establish vascular supply and again express molecules, which induce immune tolerance.

Understanding cancer mechanisms through network dynamics

Cancer is a complex, multifaceted disease. Cellular systems are perturbed both during the onset and development of cancer, and the behavioural change of tumour cells usually involves a broad range of dynamic variations. To an extent, the difficulty of monitoring the systemic change has been alleviated by recent developments in the high-throughput technologies. At both the genomic as well as proteomic levels, the technological advances in microarray and mass spectrometry, in conjunction with computational simulations and the construction of human interactome maps have facilitated the progress of identifying disease-associated genes. On a systems level, computational approaches developed for network analysis are becoming especially useful for providing insights into the mechanism behind tumour development and metastasis. This review emphasizes network approaches that have been developed to study cancer and provides an overview of our current knowledge of protein-protein interaction networks, and how their systemic perturbation can be analysed by two popular network simulation methods: Boolean network and ordinary differential equations.

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.

Angiogenesis and vascular remodelling in normal and cancerous tissues

Vascular development and homeostasis are underpinned by two fundamental features: the generation of new vessels to meet the metabolic demands of under-perfused regions and the elimination of vessels that do not sustain flow. In this paper we develop the first multiscale model of vascular tissue growth that combines blood flow, angiogenesis, vascular remodelling and the subcellular and tissue scale dynamics of multiple cell populations. Simulations show that vessel pruning, due to low wall shear stress, is highly sensitive to the pressure drop across a vascular network, the degree of pruning increasing as the pressure drop increases. In the model, low tissue oxygen levels alter the internal dynamics of normal cells, causing them to release vascular endothelial growth factor (VEGF), which stimulates angiogenic sprouting. Consequently, the level of blood oxygenation regulates the extent of angiogenesis, with higher oxygenation leading to fewer vessels. Simulations show that network remodelling (and de novo network formation) is best achieved via an appropriate balance between pruning and angiogenesis. An important factor is the strength of endothelial tip cell chemotaxis in response to VEGF. When a cluster of tumour cells is introduced into normal tissue, as the tumour grows hypoxic regions form, producing high levels of VEGF that stimulate angiogenesis and cause the vascular density to exceed that for normal tissue. If the original vessel network is sufficiently sparse then the tumour may remain localised near its parent vessel until new vessels bridge the gap to an adjacent vessel. This can lead to metastable periods, during which the tumour burden is approximately constant, followed by periods of rapid growth.

Asymptotic behaviour of solutions to abstract logistic equations

We analyze the asymptotic behaviour of solutions of the abstract differential equation u'(t)=Au(t)-F(u(t))u(t)+f. Our results are applicable to models of structured population dynamics in which the state space consists of population densities with respect to the structure variables. In the equation the linear term A corresponds to internal processes independent of crowding, the nonlinear logistic term F corresponds to the influence of crowding, and the source term f corresponds to external effects. We analyze three separate cases and show that for each case the solutions stabilize in a way governed by the linear term. We illustrate the results with examples of models of structured population dynamics -- a model for the proliferation of cell lines with telomere shortening, a model of proliferating and quiescent cell populations, and a model for the growth of tumour cord cell populations.

An immersed boundary framework for modelling the growth of individual cells: an application to the early tumour development

A biomechanical approach in modelling the growth and division of a single fully deformable cell by using an immersed boundary method with distributed sources is presented, and its application to model the early tumour development is discussed. This mathematical technique couples a continuous description of a viscous incompressible cytoplasm with the dynamics of separate elastic cells, containing their own point nuclei, elastic plasma membranes with membrane receptors, and individually regulated cell processes. This model enables one to focus on the biomechanical properties of individual cells and on communication between cells and their microenvironment, simultaneously allowing for the formation of clusters or sheets of cells that act together as one complex tissue. Several examples of early tumours growing in various geometrical configurations and with distinct conditions of their initiation and progression are also presented to show the strength of our approach in modelling different topologies of the growing tissues in distinct biochemical conditions of the surrounding media.

A structured-population model of Proteus mirabilis swarm-colony development

In this paper we present continuous age- and space-structured models and numerical computations of Proteus mirabilis swarm-colony development. We base the mathematical representation of the cell-cycle dynamics of Proteus mirabilis on those developed by Esipov and Shapiro, which are the best understood aspects of the system, and we make minimum assumptions about less-understood mechanisms, such as precise forms of the spatial diffusion. The models in this paper have explicit age-structure and, when solved numerically, display both the temporal and spatial regularity seen in experiments, whereas the Esipov and Shapiro model, when solved accurately, shows only the temporal regularity. The composite hyperbolic-parabolic partial differential equations used to model Proteus mirabilis swarm-colony development are relevant to other biological systems where the spatial dynamics depend on local physiological structure. We use computational methods designed for such systems, with known convergence properties, to obtain the numerical results presented in this paper.

Statistical and fractal analyses of rat prostate cancer cell motility in a direct current electric field: comparison of strongly and weakly metastatic cells

The problems addressed here comprised (1) possible differences in galvanotactic properties of strongly versus weakly metastatic rat prostate cancer cells, with MAT-LyLu and AT-2 as examples, respectively; (2) quantitative description of the responses of the MAT-LyLu cells to direct current (dc) electric fields (EFs) of physiological strength (0.3-3 V/cm); and (3) voltage and time dependency of the cells' responses to the dcEFs. These issues were studied by application of statistical and fractal analyses of the cells' trajectories. The results showed that the MAT-LyLu cells responded strongly to the applied dcEFs by migrating towards the cathode. On the other hand, the galvanotactic response of the AT-2 cells was weak and towards the anode. Further studies of the MAT-LyLu cell motility in dcEFs of increasing strength showed that their response consisted of two voltage domains. Weaker fields (approximately 0.6 V/cm) induced "straightening" of the cells' trajectories without the cells showing a clear tendency to move along the applied field. Stronger fields (>0.6 V/cm) made the cells' movement oriented with respect to the direction of the applied field, without further changing the trajectories' structure. The results also showed that the cells do not perform a directed movement instantaneously after switching on a dcEF of 3 V/cm; approximately 30 min lapsed before the cells were able to fully follow the direction of the applied field. Possible biophysical bases and pathophysiological significance of the results obtained are discussed.

Cell kinetics in tumour cords studied by a model with variable cell cycle length

A mathematical model is developed that describes the proliferative behaviour at the stationary state of the cell population within a tumour cord, i.e. in a cylindrical arrangement of tumour cells growing around a blood vessel and surrounded by necrosis. The model, that represents the tumour cord as a continuum, accounts for the migration of cells from the inner to the outer zone of the cord and describes the cell cycle by a sequence of maturity compartments plus a possible quiescent compartment. Cell-to-cell variability of cycle phase transit times and changes in the cell kinetic parameters within the cord, related to changes of the microenvironment, can be represented in the model. The theoretical predictions are compared against literature data of the time course of the labelling index and of the fraction of labelled mitoses in an experimental tumour after pulse labelling with 3H-thymidine. It is shown that the presence of cell migration within the cord can lead to a marked underestimation of the actual changes along cord radius of the kinetics of cell cycle progression.

Competition effects in the dynamics of tumor cords

A general feature of cancer growth is the cellular competition for available nutrients. This is also the case for tumor cords, neoplasms forming cylindrical structures around blood vessels. Experimental data show that, in their avascular phase, cords grow up to a limit radius of about 100 microm, reaching a quasi-steady-state characterized by a necrotized area separating the tumor from the surrounding healthy tissue. Here we use a set of rules to formulate a model that describes how the dynamics of cord growth is controlled by the competition of tumor cells among themselves and with healthy cells for the acquisition of essential nutrients. The model takes into account the mechanical effects resulting from the interaction between the multiplying cancer cells and the surrounding tissue. We explore the influence of the relevant parameters on the tumor growth and on its final state. The model is also applied to investigate cord deformation in a region containing multiple nutrient sources and to predict the further complex growth of the tumor.

Diffusion with evolving sources and competing sinks: development of angiogenesis

Tumors ensure their long-time growth by emitting molecular messengers that induce cellular modifications in neighboring capillaries. These modifications are conducive to the enlargement of the vascular system feeding the tumor. This phenomenon, termed angiogenesis, is controlled by the diffusion and competitive trapping of nutrients and molecular messengers by several cell species. The number, location, and properties of these traps change continuously. The angiogenic process also implies that nutrient sources are time dependent. Starting from assumptions at the cellular level, we formulate a mathematical model that predicts the evolution of angiogenesis and the increase in the blood flow to the tumor. The model also predicts the emergence of directed growth and the possibility of therapeutical synergy. Simulations permit a careful analysis of the influence of the main parameters.