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Conceived and designed the experiments: PG SN. Performed the experiments: PG. Analyzed the data: PG SN. Wrote the paper: PG SN.

The authors have declared that no competing interests exist.

The brain tumour glioblastoma is characterised by diffuse and infiltrative growth into surrounding brain tissue. At the macroscopic level, the progression speed of a glioblastoma tumour is determined by two key factors: the cell proliferation rate and the cell migration speed. At the microscopic level, however, proliferation and migration appear to be mutually exclusive phenotypes, as indicated by recent

In this work, we develop a spatial mathematical model in order to analyse the growth behavior of the brain tumour glioblastoma. Tumours of this type have a diffuse boundary, with considerable local invasion of surrounding brain tissue, making surgery difficult. At the cellular level, the progression of a glioblastoma is known to depend on the balance between cell division (proliferation) and cell movement (migration). Based on recent evidence, our model assumes that each cell in a glioblastoma tumour resides in either of two mutually exclusive states: proliferating or migrating. From a probabilistic model of switching between these two phenotypes, we go on to derive equations that link cellular phenotypes to disease progression. The model has several possible applications. For instance, it could be used to predict the rate of disease progression in an individual patient, and to improve screening methods.

Cancer progression is the macroscopic outcome of numerous cellular processes, such as elevated proliferation rates, defects in apoptosis regulation and abnormal angiogenesis

The importance of cancer cell migration is perhaps most evident in the common brain tumour glioblastoma, which is characterised by rapid and infiltrative growth into the surrounding brain tissue. In glioblastomas, neoplastic cells are often found at a long distance (several centimeters) from the main tumour mass. This diffuse growth pattern presents a difficult clinical problem, since residual ‘satellite cells’ can mediate rapid recurrence of the disease after surgery

However, the potential of migration as a therapeutic target is complicated by the strong dependency between migration and proliferation phenotypes. Early

In recent experiments,

Here, we focus on the relationship between cell-level phenotypic switching in glioblastoma, and the properties of the tumour as a whole. In particular we elucidate how the growth rate of the tumour and speed of invasion depends on the specific underlying microscopic parameters, such as phenotypic switching rates, rate of apoptosis

In the following we first review previous work in the field of glioblastoma modelling and then proceed by introducing our individual-based (IB) stochastic model of glioma growth. From this model we derive an approximate continuum description of the system, whose properties are compared to the IB-model. We proceed to analyse the continuum model to reveal the influence of the model parameters on the rate of spread of the tumour, and finally discuss our results in the context of other models and experimental results.

The growth of glioblastomas was first modelled by means of a continuum approach, which captures the two main features of glioma cells: proliferation and migration (

The Fisher equation has been of particular interest since it gives rise to traveling wave solution

The above modeling approach rests on the assumption that glioblastoma cells follow a random walk (which at the macroscopic scale corresponds to the diffusion of cells). Recently this assumption has been under scrutiny, and this has led to a number of explorations of non-random migration, i.e. where migration is influenced by biological processes such as cell-cell signaling, oxygen pressure, nutrient availability and phenotype switching. In one line of work, Aubert et al.

The cellular behaviour implied in the ‘go-or-grow’ hypothesis (see Introduction) is also thought to affect migration and growth dynamics of glioblastomas, in a manner that is not captured by the Fisher-Kolmogorov equation. Hatzikirou et al.

The model that we propose draws from these previous models, but is different in some crucial ways. We consider two distinct subpopulations with a stochastic switching in between (as in Fedotov and Iomin, and Lewis and Schmitz), but instead of starting with a continuum description, we begin with an IB-model in which the cells occupy a lattice and obey size exclusion (as in Deroulers et al.), and from that derive a system of PDEs. This allows for an analytical treatment of the IB-model which establishes a connection between cell characteristics and the macroscopic behaviour of the system previously not demonstrated.

The cells are assumed to occupy a

The lack of knowledge of the intra-cellular dynamics and extra-cellular cues that lead to the phenotypic switching behaviour poses a problem, but we will circumvent it by, as a first approximation, considering the switching as a stochastic event. The behaviour of each cell is therefore modelled as a time continuous Markov process where each transition or action occurs with a certain rate, which only depends on the current and not previous states, known as the Markov property. The rates are interpreted in the standard way, i.e. if transition in a variable

Each cell is assumed to be in either of two states: proliferating or migrating, and switching between the states occurs at rates

The stochastic process is depicted schematically in

A living glioma cell can be in either of two states, proliferating (P) or migrating (M), and transitions between the states with rates

We will consider a lattice of linear size

The motility rate is set to

Our concern is the influence of the microscopic cell-level parameters on the growth rate of the tumour as a whole, and we will therefore measure the size the tumour after a fixed time for a given set of initial conditions, as a function of the phenotypic switching rates. More specifically we will measure the tumour mass (the total number of cells), and also later, quantify the rate of spread by measuring the velocity of the tumour interface. The precise initial condition of the model has little impact on the long-term rate of spread (data not shown), but in line with the clonal origin of cancer we initialise the model with a single cell (in the P-state). All simulations of the IB-model are carried out using the commonly employed Gillespie algorithm

Simulation results for

In order to quantify the dependence on the phenotypic switching rates we measured the tumour mass at

(a) The tumour mass at

In an effort to get a deeper understanding of the somewhat unintuitive relationship between tumour growth rate and phenotypic switching rates we will derive a set of two coupled PDEs which will serve as an approximate way of describing the time evolution of the occupancy probability

The derivation is carried out in two steps: firstly, a set of coupled master equations, for the two sub-populations, are derived by considering the processes which alter the occupation probabilities at a given site, and secondly these master equations are approximated by a set of PDEs. In brief, the second step is achieved by identifying the on-lattice master equations with a set of coupled PDEs, which when discretised on the length scale of the lattice spacing, equal the master equations. The full derivation is given in

The first question one might ask about a system of equations that presumably describes tumour growth is if it exhibits tumour invasion and hence travelling wave solutions, and further how the model parameters influence the wave speed, i.e. the velocity of the invading tumour front. The results from the IB-model (

In order to investigate this, we first solved the continuum model (3)–(4) numerically (which actually corresponds to reverting to the master equations eq. (8)–(9)), for a range of parameter values, in the domain

The results can be seen in

Solutions of the 1-dimensional continuum model (equation (3) and (4)) for three different values of the switching rates at

The observation that the numerical solutions are stationary in a moving frame suggests the existence of travelling wave solutions. In order to close in on these solutions, and get an estimate of their velocity, we will make use of the travelling wave ansatz:

Although phase-space analysis does not yield an analytic closed-form expression for the wave speed

In order to test the validity of the wave speed analysis we compared the wave speeds obtained in the continuum and IB models with those from the phase space analysis. For the continuum model an estimate of the wave speed was obtained by, from the initial condition

The wave speed of the propagating tumour margin determined from both phase space analysis (solid line) and numerical simulation (dashed line). In (a) the switch rate to proliferation is fixed at

When it comes to the IB model, we have to take into account the stochastic nature of the model, and therefore need to estimate the average margin velocity from a large number of simulations (100 independent realisations). Each simulation was started with a single P-cell at the center of the lattice and the model was simulated for 100 time steps (cell cycles). In each time step the location of the cells was recorded and from this we calculated the occupation probability

The wave speed of the propagating tumour margin determined from both the individual-based model (dashed line) and phase space analysis of the continuum approximation (solid line). In (a) the switch rate to proliferation is fixed at

Naturally the other model parameters also affect the rate of tumour invasion (see

The wave speed of the propagating tumour margin as a function of (a)

Our model gives considerable insight into the dependency between five cell-level parameters (switching rates

The above reasoning, and our model, do however not take into account the effects of mechanical forces between the cells. In particular it is, in real tumours, possible for cells to push one another and hence to divide and move, although there is no free space. This process will most likely lessen the positive effect of cell migration on tumour growth, but since it has been experimentally established that few cell divisions occur in the core of the tumour due to pressure build-up and hypoxia, we believe that the conclusions of our model still hold to a large extent.

A similar trade-off between proliferation and migration has in fact been observed in the models of Hatzikirou et al. and Fedotov and Iomin

A trade-off between proliferation and migration has also been investigated in relation to cancer stem cells and tumour progression by Enderling et al.

We have also demonstrated that the other parameters in the model affect the speed of invasion. Firstly, the impact of the motility rate and the proliferation rate imply that the wave speed dependence observed in the Fisher equation (1),

Secondly, we observed a second-order phase transition in the velocity with respect to the rate of apoptosis

While data from Farin et al.

In order to gain further experimental support for our model, we plan in future work, to measure the five cell specific parameters directly. Such measurements should be possible by applying live imaging microscopy techniques to primary glioblastoma-derived cell cultures. A first application of such measurements could be exploited to develop the model further, to predict progression for an individual patient based on cell-level phenotyping, and to develop chemical compound screens where the impact of a chemical on the model parameters are observed. This might in turn lead to a strategy to define

The current model is however far from these highly set goals, and there are a number of extensions that would make the model more realistic. In its current form the model does not account for cell-cell adhesion, which could be incorporated letting the motility rate

Despite this we would still expect the results of our model to hold at least with respect to the large-scale behaviour of the tumour. The real situation is also complicated by the fact that cancer cells are selected for based on their phenotype. One hypothesis which emerges from our model is that selection could drive the behaviour of the cells to the optimal balance between

Adding these mechanisms would of course make the model less tractable from an analytical point of view, but this trade-off between simplicity and reality is something that all modellers must deal with.

Let us consider a one-dimensional lattice indexed by the integers. We let

Let us first consider

an existing P-cell can die through apoptosis (with rate

an existing P-cell might switch to an M-cell (with rate

an M-cell residing at the site might switch into becoming a P-cell (with rate

a neighbouring cell might divide placing its offspring in the (empty) considered site (with rate

Summarising all these processes we can write:

In order to simplify the expression and also draw parallels to continuum systems we define a discrete Laplace operator

If we now turn to the motile cells, the following processes affect

an existing M-cell can die through apoptosis (with rate

an existing M-cell might switch to a P-cell (with rate

a P-cell residing at the site might switch into becoming a M-cell (with rate

an existing cell might move away from the considered site (with rate

a neighbouring cell might move into the (empty) considered site (with rate

Taking all these processes into account we can write

The first three terms can be recognised as apoptosis and switching terms, while the fourth and fifth are due to movement out of and into the site. After a bit of algebra and making use of the discrete Laplacian defined in eq. (6) we get

In summary we have that the time evolution of the occupation probabilities are described by the following coupled equations:

Please note that despite the similarity to PDEs, that describe the changes of a quantity in continuous space and time, these equations are defined on the lattice and describe the probability of finding a cell of a specific type in a certain location. In many instances it is natural to proceed by taking the spatial continuum limit of the discrete master equation(s), but in this case, where we are considering expansion via both cell movement and pure cell division (the case

However in order to proceed with the analysis and make use of the toolbox of real analysis we will approximate the above equations with the following PDEs:

The motivation behind this choice is that the master equations (8) and (9) are the (space) discretised versions of (10) and (11). The diffusion constants are given by

For the sake of clarity let us recapitulate the method applied to the Fisher equation (1) in order to calculate its speed of invasion. The travelling wave ansatz (

In our case the travelling wave ansatz transforms the system of PDEs (10)–(11) to the following set of coupled ordinary differential equations (ODE):

The boundary conditions reflect the fixed points of the system, which are

What will help us determine the wave speed

We will now apply the same kind of reasoning of non-negativity as for the Fisher equation in order to obtain a minimal wave speed

The roots of this equation

The authors would like to thank David Basanta and Jacob G. Scott for critical reading of the manuscript, and the referees for their helpful comments and critique.