Quantitative modelling is increasingly important in cancer research, helping to integrate myriad diverse experimental data into coherent pictures of the disease and able to discriminate between competing hypotheses or suggest specific experimental lines of enquiry and new approaches to therapy. Here, we review a diverse set of mathematical models of cancer cell plasticity (a process in which, through genetic and epigenetic changes, cancer cells survive in hostile environments and migrate to more favourable environments, respectively), tumour growth, and invasion. Quantitative models can help to elucidate the complex biological mechanisms of cancer cell plasticity. In this review, we discuss models of plasticity, tumour progression, and metastasis under three broadly conceived mathematical modelling techniques: discrete, continuum, and hybrid, each with advantages and disadvantages. An emerging theme from the predictions of many of these models is that cell escape from the tumour microenvironment (TME) is encouraged by a combination of physiological stress locally (e.g., hypoxia), external stresses (e.g., the presence of immune cells), and interactions with the extracellular matrix. We also discuss the value of mathematical modelling for understanding cancer more generally.

Cancer is a disease involving, initially, abnormal cell growth with the potential to invade locally and—later—spread to other organs. In normal cells, the highly complex processes of cell division and death are controlled by myriad genes. Proliferation, differentiation, and apoptosis of cells are controlled by the activities of those genes and balance normal cell growth while appropriately regulating programmed cell death. Cells become cancerous when mutations accumulate in the various genes that control cell proliferation and the cell cycle, death, and responses to stress, but how these mechanisms operate and the interplay between them continues to be mysteries. Cancer cells appear to behave, in some sense, as “autonomous entities,” growing without control to form tumours. Spreading cancer cells from their primary tumour to other parts of the body—through blood and lymph,—causes cancer metastasis. Once this state has been reached, treatment options become limited, and the disease is usually fatal.

The tumour microenvironment (TME) contains different types of cells and is a key actor in the cascade of local invasion and progression towards metastasis. Apart from malignant cells, it consists of cells of the immune system, tumour vasculature and lymphatics, fibroblasts, and pericytes,

Cell migration can be broadly classified into single cell migration and collective migration modes. These molecular programs are associated with a characteristic structure of the actin cytoskeleton, integrins, matrix-degrading enzymes, cell-cell adhesion, and signalling towards the cytoskeleton [

The tumour microenvironment (TME) promotes cancer cell plasticity because of the effect of ECM stiffness, acidity, hypoxia, and the presence of immune cells in TME. Cancer cells can survive in TME through epigenetic changes and also escape to more favourable environments. To escape from stress within the TME, cancer cells use individual or collective cell migration mechanisms. Two individual cell migration methods are amoeboid and mesenchymal migration. Cancer cells can migrate from the TME by collective cell migration mechanisms, in cluster or multicellular sheets.

In the context of the above discussion, identifying first-order principles and key biological mechanisms in cancer cell plasticity is clearly indispensable to a clear understanding of cancer progression. In particular, these principles are likely to be important in the therapeutic context of blocking or slowing the spread of cancer cells: a major challenge for developing modern cancer treatment therapies. Since biological experiments are expensive, time-consuming, ethically challenging and sometimes downright impossible with existing technology, mathematical models can provide an independent, experiment-free check of hypothesis consistency, ideally focusing (or moderating or altering) hypotheses before experimental work as well as in a feedback loop of model-experiment-model. More specifically, mathematical models can be used to describe cancer at various scales and act as an exploratory tool to complement experimental work. Furthermore, mathematical models can be used as a predictive tool. Quantitative descriptions of cancer-driven mechanisms can lead to the development of new and novel cancer treatment therapies [

The strength of a mathematical model rests in its ability to combine experimental data, consolidating it into a coherent framework, which can be used to predict the overall (or precise) dynamics of a system. Mathematical models are very useful in identifying the parameters that are most sensitive to the system and they allow for logical reasoning beyond the provision of experiments. In the specific context of tumour growth, mathematical models can, for example, quantify the links of three-dimensional tumour-tissue architecture with growth, invasion, and underlying microscale cellular and environmental characteristics. Ideally, these approaches can lead to the design of new, targeted experiments and strategies for cancer treatments [

Mathematical modelling and computer simulation allow us to explore the so-called “what-if” scenarios describing potentially complex biophysical, chemical, and physiological processes that are often beyond the reach of experimental or clinical protocols. This might be due to the protocols being expensive, invasive, hard-to-capture, or highly variable.

Despite oft-held beliefs to the contrary in the biological and medical sciences, mathematical models can in fact be simple and used to capture the “sense” of the system under study, often through phenomenological approaches. They may, of course, also be very complex and this complexity implies that the computer simulation is very intensive, takes a long time to run and/or require substantial resources of another type (e.g., computer memory).

There is also an additional issue: potentially a large number of parameters need to be calibrated against available data. If there are insufficient data, then the model may well be “undetermined”, that is, unable to discriminate between different hypotheses. In addition to the calibration issue, the models need to be validated so that they can be used in predictive settings. Validation can be performed by running computer simulations using just some of the available data and then seeing how well the outputs derived from the simulations match the data that were “held back” when calibrating.

Models may be static (representing known behaviour) at a specific point in time. Models may also evolve in time, for example, describing the action of a drug, or they may evolve in time and space, for example describing the growth and motion of a tumour. Models may also be stochastic in that they try to represent processes, such as diffusion, that are fundamentally stochastic in nature.

In outline, and broadly speaking, there are three mathematical modelling techniques having been so far used to understand cancer dynamics and plasticity: discrete, continuum, and hybrid. Discrete models track and update individual cells according to a set of biological rules [

In the discussion below, we organise the individual models/studies under review along these lines, but, importantly, we also taxonomise the models by their primary findings (and aims). In particular, we distinguish between models whose aim is (a) the understanding of cell plasticity and tumour progression in their own right versus (b) those with clinical implications, that attempt to understand the effect of therapy on tumour progression and/or how to optimise treatment efficacy. We are also not aiming here to cover every existing model or to go into the details of specific models. Rather, our aim is to give the reader a flavour for each type of mathematical approach. We also aim to illustrate the kinds of insight that mathematical models of different kinds can provide with respect to understanding basic cancer cell biology but also with respect to therapy design and optimisation.

A discrete model can address the behaviour of one or more

A substantial advantage of discrete models is that they operate with simple transition rules, e.g., a cell can divide and its daughter will be placed in a neighbouring cell, rather than using constitutive differential equations. This makes them fundamentally accessible to nonexperts, since the rules can be designed and understood by the nonmathematician. Rule-based approaches also are more fundamentally suited to describing biological interactions, which are themselves rule-based, involving individual agents. This is a major advantage but brings with it two drawbacks: firstly, these models generally require much more computational power (less of an issue in the era of high-performance computing than previously) and secondly, because of the absence of equations, these models lend themselves less to drawing simple conclusions about the system under study. For example, it is more difficult to find a simple relationship between a fundamental parameter, e.g., cell motility and a variable of interest, e.g., time to plasticity-induced invasion of local stroma.

Lattice-based models themselves are different from one another. One way to think about this is that there are different conceptual methods can be used to implement the lattice: allowing either exactly one, more than one, or less than one cell per lattice site (in the latter case, basically sites represent subcellular compartments rather than entire cells). More specifically, there are models that use

During approximately the same period, Dutching and colleagues began to develop tumour growth models based on

Kansal et al. [

Some of the most recent work on CA models in tumour growth involves modelling three-dimensional invasive solid tumour growth in heterogeneous microenvironments under chemotherapy [

Ferreira et al. [

Hatzikirou and Deutsch introduced a microscopic modelling method called

The

In lattice-free approaches, cells are at liberty to move in any direction and any distance consistent with the underlying biological, physical, and chemical processes. A complicating feature of these models is, of course, collision detection between the cells since, unlike in lattice-based approaches, the cells are not confined to discrete “voxels.”

Drasdo and Hohme introduced an off-lattice model for tumour growth [

The general form of a continuum model is a partial differential equation (PDE) that describes dynamics in time and space. Typically, we may have a function

Here, _{0}, an initial time, and boundary conditions on the boundary of the domain. This then guarantees the existence of a (unique) solution to the model.

Various mathematical models of tumour growth by diffusion have been developed in the literature, which is rich along these lines. Controlling for the mitosis of tissues, tumour growth was modelled by considering sources such as negative feedback mechanism [

McElwain and Ponzo developed a model for the growth of solid tumours with nonuniform oxygen consumption [

Chaplain and Britton presented a mathematical model for the production of a growth inhibitory factor (GIF) within a multicell spheroid representing a tumour by assuming that the GIF is produced by cells within the spheroid in some prescribed nonlinear, spatially dependent manner [

Anderson and Chaplain described the formation of the capillary sprout network in response to TAF supplied by a solid tumour. Their model takes account of the essential endothelial cell-ECM interactions via the inclusion of the matrix macromolecule fibronectin. It consists of a system of nonlinear partial differential equations (PDEs) describing the initial migratory response of endothelial cells to the TAF and the fibronectin [

Instead of the more complex PDE models described above, which directly represent physical space, tumour growth and treatment can be modelled based on ordinary differential equations (ODEs) that describe the evolution of a system in time only, such as

In some cases, a time-dependent control

Andasari and Chaplain derived a system of ODEs for intracellular modelling of cell-matrix adhesion during cancer cell invasion by applying the law of mass action [

Gerich and Chaplain modified and extended the model developed by Anderson et al. [

Friedman and Reitich developed a model concentrating on the case where at the boundary of the tumour, the birth rate of cells exceeds their death rate [

A free boundary (FB) model is a special type of differential equation model in which we wish not only to find the solution but also where the solution is actively interacting with the spatial medium and, consequently, the domain itself is unknown. FBs arise in biological models when there is an effect from the medium, e.g., the TME affecting the tumour or an area of the spatial domain becomes active from a normal inactive state. Several models of this type have been developed, all of a highly theoretical nature, some aimed at simply proving the existence of solutions to these difficult problems.

Chaplain and Stuart [

Friedman and Reitich [

Bazaliy and Friedman [

Friedman [

Cui and Friedman [

Friedman [

Xu et al. [

Enderling et al. [

Based on the PDE model developed by Anderson and colleagues [

Byrne and Chaplain studied growth of nonnecrotic tumours on the effect of inhibitors using a model with two reaction-diffusion equations that describe the distribution of externally supplied nutrient and inhibitor species and an integrodifferential equation that governs the evolution of outer radius of the tumour [

Hybrid or continuum-discrete models can bridge the gap between the cellular scale and the tumour (or even organism) scale. In the hybrid approach, tumour tissue is modelled using both discrete and continuum elements. In general (but with exceptions), oxygen, nutrient, drugs, growth factors, and certain tissue features are described as continuum fields and cells are described as discrete elements. There are sometimes subtle elements in interfacing the two approaches and to get the model to behave appropriately.

Anderson developed a hybrid mathematical model of solid tumour invasion to study how the geometry of the growing tumour is affected by mutation-driven tumour cell heterogeneity focusing on four key variables: tumour cells, the ECM, matrix-degradative enzymes, and oxygen [

Jiang et al. [

Jeon et al. developed an off-lattice hybrid model for tumour growth and invasion [

We have here reviewed a large number of fairly diverse quantitative models of cancer cell plasticity, falling roughly into three categories: discrete, continuum, and hybrid, each with strengths and weaknesses relative to a specific research question. The schematic relationships of these classes of models to one another are illustrated in Figure

D

The important models reviewed here are organised by modelling approach and a summary of their key finding(s). The list is nonexhaustive.

Reference | Model | Submodel | Key result |
---|---|---|---|

Donaghey [ |
Discrete | CA | Higher proportion of cells will enter the |

Düchting and Vogelsaenger [ |
Discrete | CA | After treatment, undamaged |

Duchting and Dehl [ |
Discrete | CA | Critical initial number of tumour cells of a tumour nucleus is necessary for the growth of a tumour. Additional high-influence variables are the mean life span of a tumour cell and the amount of tumour cell loss. |

Smolle and Stettner [ |
Discrete | CA | Histological tumour patterns depend complexly on the autocrine and paracrine factors. |

Smolle et al. [ |
Discrete | CA | Relative degree of motility to proliferation decreases from benign to primary malignant and metastatic, but the absolute degree of motility is increasing. |

Ferreira et al. [ |
Discrete | CA | Growth patterns of the tumour are compact with gyration radius, surface roughness, and number of peripheral cells. |

Schmitz et al. [ |
Discrete | CA | A tumorous subpopulation is most highly favoured when the interfacial area among strains is maximized. Total volumetric fraction of nonlocalized strains is not important in tumour development. |

Jiao and Torquato [ |
Discrete | CA | Quantitative properties of the host microenvironment can significantly affect tumour morphology and growth dynamics. |

Jiao and Torquato [ |
Discrete | CA | Strong cell-cell adhesion can suppress the invasive behaviour of the tumours growing in soft microenvironments; cancer malignancy can be significantly enhanced by harsh microenvironmental conditions, such as exposure to high pressure levels. |

Xie et al. [ |
Hybrid | CA, diffusion reaction | In chemotherapy, constant dosing is generally more effective in suppressing primary tumour growth than periodic dosing, due to the resulting continuous high drug concentration. |

Hatzikirou et al. [ |
Discrete | LGCA | Width of the travelling front is proportional to the front speed. |

Chopard et al. [ |
Discrete | LGCA | There is a positive effect of fibre track on glioma growth. |

Graner and Glazier [ |
Discrete | CPM | Long-distance cell movement leads to sorting with a logarithmic increase in the length scale of homogeneous cell clusters. |

Jiang et al. [ |
Discrete | CPM | The microenvironmental conditions required for tumour cell survival and growth promoters and inhibitors have diffusion coefficients in the range |

Turner and Sherratt [ |
Discrete | CPM | Increased proliferation rate results in an increased depth of invasion into the extracellular matrix. |

Shirinifard et al. [ |
Discrete | CPM | Simulated avascular tumours form cylinders following the blood vessels, leading to a differential distribution of hypoxic cells within the tumour. |

Anderson and Chaplain [ |
Continuum and discrete (discrete model is the discretized form of the continuum model) | Diffusion-reaction equation | |

Random walk model | Both chemotaxis and haptotaxis are necessary for the formation of a capillary network at large scales. | ||

Anderson et al. [ |
Continuum and discrete (discrete model is the discretized form of the continuum model) | Diffusion-reaction equation | |

Random walk model | ECM structures can aid or hinder the migration of individual cells that have the potential to metastasis. As time increases, small cell clusters can be observed. | ||

Chaplain and Stuart [ |
Continuum | PDE | Possible explanation for anastomosis |

Bazaliy and Friedman [ |
Continuum | PDE | Establish the existence and uniqueness of a solution for some time interval |

Friedman [ |
Continuum | PDE | For the densities of three types of cells: proliferating, quiescent and necrotic, the nutrient concentration, fluid velocity, and pressure have a unique smooth solution, with a smooth free boundary for a small time interval |

Chen et al. [ |
Continuum | Partial integro-differential equations | Stationary solution of the model is linearly asymptotically stable |

Cui and Friedman [ |
Continuum | ODE | Initial value problem has a one-parameter family of solutions and there exists a unique solution to the free boundary problem. |

Zhang and Tao [ |
Continuum | PDE | Prove the global solvability of the model |

Xu and Wu [ |
Continuum | PDE | Prove the existence and stability of the steady-state solutions when the rate at which the tumour attracts blood vessels is constant. |

We now compare, in brief, the strengths and weaknesses of the different types of models reviewed here. One may consider different approaches when one is addressing roughly different degrees of detail. For example, the discrete lattice-free models perhaps include the most detailed information about the relevant biology, but are also the most computationally expensive, restricting their applications to relatively smaller scales or requiring the use of supercomputers. In contrast, discrete lattice-bound models coarse-grain the free space into discrete lattice sites, reducing the computational cost regarding cell movements, often at the cost of losing spatial precision, which is a good compromise depending on the questions being asked. The continuum models further coarse-grain discrete cells into a continuous density, with the aim of enabling such models to simulate large-scale phenomena but, again, at the cost of losing all local spatial information, e.g., local cell-cell interactions. If this loss impairs the modelling approach from answering important questions but we still require the simulation of large-scale phenomena, we may then consider using hybrid modelling.

We now offer some perspectives on the future development of this exciting and highly active area of research. An important question is how such models can be used in a clinical context. In a so-called “personalized medicine” clinical setting, there may be a need to have results available from computer simulations very quickly, to aid, say, invasive operations. Recently, there has been a move to run such simulations on cheap graphical processing units (GPUs) that can be deployed in standard PCs that may speed up simulations by several orders of magnitude, placing this technology at the bedside. The simulations can be combined with advanced visualization techniques.

Models can also be used to enhance either missing data or data that hard to collect. Statistical or machine learning/artificial intelligence techniques can then be used to characterize or learn about important features. This is not possible if data are sparse, since these approaches require large amounts of high-quality data for training the system.

Models can also be used to make sense of data that are highly variable, for example data collected from a cohort of patients in some disease state. This can be done by calibrating not a single model to, say, the mean of the highly variable data but by calibrating a population of models (with the same framework but different parameters) against the data. This population of models can be studied statistically, quantifying the uncertainty, or run forward in time to provide probabilistic outcomes, that is beyond just a single model. Such an ensemble is sometimes called a

Before data collection, a new experimental protocol requires testing and optimisation. To identify what types of data should needed for the experiment, a mathematical model can be used. Therefore, mathematical models represent a natural framework for experimental protocols. To improve the quality and the practical use of mathematical models and to generate qualitative and quantitative predictions, stronger collaboration of mathematicians and biomedical researchers is needed.

The models of tumour cell plasticity we surveyed here illustrate and colour the discussion above. For example, using a discrete model, one can identify biological factors such as heterogeneity and interaction between cells. However, it can be difficult to investigate the model behaviour because many simulation runs are required. Continuum models provide tools to describe movement and aggregation patterns of cell populations under various tumour microenviornmental factors. In this way, for example, the effect of microenviornmental factors for the tumour progression and the mechanism of cancer cell plasticity can be observed. Hybrid models combine discrete and continuum descriptions of cancer biology by bridging the gap between the cellular scale and tumour scales. Hybrid models attempt to combine the best features of discrete models and continuum models.

The challenge, going forward, for mathematical modelling in cancer in general and plasticity specifically, is moving from a descriptive to a prescriptive power, for example, generalizing their findings after appropriate validation so that they can be used to design

Mathematical models can be used to study the different stages of tumour progression such as avascular and vascular tumour growth, angiogenesis, invasion, and metastasis under the effects of TME factors. Because of tumour cell plasticity, cancer cells are able to escape from the TME and, further down the cascade, resistance to therapies as well as immune system evasion arises. By studying quantitative models, the factors and influences leading to cell plasticity can be identified. For example, as many models now confirm, ECM stiffness, hypoxia, nutrient deprivation, hypoxia, and acidity all promote cell plasticity and motility within the TME [

The authors declare that they have no conflicts of interest.

DVN is supported by a Future Fellowship of the Australian Research Council. The authors would like to thank the anonymous reviewers for their comments, which have improved the manuscript.