Computing first passage time statistics

To estimate the class mean first passage time (CMFPT), we simulate the trajectory of random walkers on cell mixing patterns embedded in a square lattice. The mean number of steps, τ_αβ, taken to first arrive at a node of class β when departing from a node of class α, is computed over all starting nodes of the same class α on the graph. In this study we focus on spatial patterns of two classes, mutated (red) and WT cells (yellow), giving rise to four possible CMFPT quantities: τ_rr, τ_ry, τ_yr and τ_yy. Different spatial patterns will differ naturally in magnitude and shape. Thus, to compare the CMFPT of different images we normalise the first passage times for each image to the corresponding quantities derived under a null model, in which the constituent classes in the spatial pattern are reassigned uniformly at random to each lattice point, but maintaining the abundances of each class and the general geometry of the original image. We denote these quantities as the normalised first passage times , , and .

We explore the phase space spanned by and , where and denote the normalised CMFPT for red→yellow and yellow→red transitions respectively. Through our analysis we find that the combination of these quantities enables us to distinguish between spatial patterns which differ in heterogeneity and pattern structure, and separates images containing clusters of different characteristic sizes. Measurements in this phase space have been used in previous analyses involving CMFPT to characterise colour distributions on 2-dimensional lattices [25]. Spatial patterns in which the two colours are distributed uniformly at random will lie at , since this particular pattern exactly resembles the null model used for normalisation. In general, patterns which contain a more ordered, or segregated, distribution of colours give rise to . The CMFPT reflects not only the extent of segregation of the colours, but also the fine-grain details of the patterns. As a result, the size, shape and spatial distribution of clusters will affect the CMFPT, with patterns containing large yellow clusters leading to and those with large red clusters giving rise to .

We define the class ratio, ϕ, of an image as the ratio of red to yellow classes, (1) where N_r and N_y represent the number of red and yellow pixels in an image respectively. Note while ϕ is a direct input in our artificial cell mixing model, it is an outcome of the agent-based simulations, depending on the replicative advantage of the sub-clonal population, the relative time at which it emerges in the simulations, and the strength of cell pushing.

A model of 2-dimensional cell mixing patterns

We first develop a simple model to generate a set of diverse spatial patterns of two colours (red and yellow) in a 2-dimensional lattice background. In particular, we generate three distinct patterns, which are: (1) clusters; (2) large centred cluster and (3) column arrangement (Fig 2a–2f). We run a number of simulations whilst varying class ratio, ϕ, between 0 < ϕ < 1 for each of these patterns and calculate the corresponding normalised CMFPT in the (,) phase space in each case (Fig 2g). Results from these initial calculations demonstrate how pattern structure and class ratio are reflected in the CMFPT, with a clear correspondence between ϕ and value (vertical axis) observed across all patterns. Patterns with generally lie below , and those with fall above , reflecting the increasing segregation of classes at either extreme of class ratio. Moreover, this effect is intensified for specific spatial patterns, with a stronger dependence for fully segregated patterns (i.e. column patterns e, f in Fig 2g), and a weaker dependence for more intermixed patterns (i.e. cluster patterns a, b in Fig 2g).

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Fig 2. Characterising 2-dimensional mixing patterns using CMFPT.

Patterns were generated using the (a, b) clusters; (c, d) centred and (e, f) column models. (g) Location of the patterns in the , phase space obtained for three models with varying class ratio, ϕ.

https://doi.org/10.1371/journal.pcbi.1010952.g002

Our normalised CMFPT measure allows for more precise interrogation of pattern structures than class ratio alone. The separation of each pattern type is shown along the (horizontal) direction of Fig 2g. Being most similar to its corresponding null model used to normalise the CMFPT, the small clusters pattern (Fig 2a) occupies a region of the phase space near to ,. The normalised CMFPT measurement is sensitive to an increase in the cluster size, even when class ratio remains constant (Fig 2b). For both the centred (Fig 2c and 2d) and column (Fig 2e and 2f) patterns, discrete trajectories in the phase space are observed in Fig 2g, with the value of ϕ determining the position of the specific pattern along the curve.

We compared the CMFPT metric to Shannon’s entropy (S1 and S2 Figs), which was previously applied by Baker et al. to colorectal cancer samples analysed with BaseScope [20], and mean shortest distance, a similar measure to the CMFPT (see Methods). Shannon’s entropy, H, ranges between H = 0 for fully segregated patterns, and H = 1 for fully mixed patterns (S1(a) & S1(b) Fig) and, when supplemented with information on ϕ for each pattern, is able to describe some of the geometrical features of the different pattern structures. Crucially, however, we found that it is not sensitive to changes in the column model, and is only weakly sensitive to changes in patterns generated for different centered cluster sizes (S1(i) Fig). More fundamentally, the Shannon’s entropy approach requires one to choose a scale at which to analyse the pattern. This should not affect the analysis of the simple patterns analysed here, since these patterns typically contain structures with only one, known, characteristic length scale, however it limits the efficacy of Shannon’s entropy when applied to more complex data, such as biological tissue. Indeed, more complex data such as that derived from human tissue may exhibit multiple characteristic length scales which are not known a priori, and the size and dimensions of samples cannot be reliably controlled at all stages of the data acquisition process.

Analysis of the same patterns using the mean shortest distance metric produces similar results to those derived using the CMFPT (S3 and S4 Figs). As with the CMFPT, the mean shortest distance is computed for transitions between cell types, such as the yellow to red mean shortest distance, , and is related the mean shortest path, in terms of number of nodes, connecting cells of one type to the other. The quantity itself is normalised, similar to the CMFPT, by the corresponding quantity derived in a null model in which the constituent classes in the spatial pattern are reassigned uniformly at random to each lattice point (see Methods). Although CMFPT and mean shortest distance are similar measures, the main difference between them is in the construction of the path between nodes. Mean shortest distance is a deterministic measure of the distance between cells, whereas CMFPT constructs paths between pairs of cells using a stochastic process, resulting in a distribution of first passage times for each starting cell. As such, CMFPT may provide different information about the texture of the patterns than mean shortest distance by incorporating information about the neighbourhood of each cell. In this study, we summarise the CMFPT distributions using the mean, however one could also make use of higher order moments of the distributions to obtain further information about each cellular neighbourhood.

Analysing simulated tumour sub-clone mixing patterns

We next investigate the efficacy of our method in characterising cell mixing patterns generated in silico. We developed a generic 2-dimensional agent-based simulation model of expanding tumour populations, incorporating cell birth, death and mutation on a lattice (Fig 1b and 1c). Simulations begin with a single WT tumour cell in the centre of the lattice which seeds a growing WT population. After a threshold number of WT cell divisions, specified by the model parameter n_mut, an existing WT cell acquires a mutation which confers a fitness advantage of s (see Methods). We model fitness via a modulation of cell replication rates, such that cells carrying the mutation have a replication rate which is (s × 100)% greater than that of WT cells. Neutral selection, where the mutant sub-clone grows at the same rate as the WT population, corresponds to s = 0. Here, n_mut represents the fraction of the final system size, N_max (a fixed model parameter), at which point the mutant population first appears. Throughout this paper we will refer to this parameter as representing the time of sub-clone emergence, however this interpretation should be adopted with caution. Despite an increase in n_mut implying a later arising mutation, the number of cell replication events per unit time scales with the size of the tumour and thus n_mut should not be interpreted straight-forwardly as a time, but rather an expression of absolute number of cell divisions. For example, for N_max = 10⁵ and n_mut = 0.1, the mutation appears in an existing WT cell once the initially expanding WT population has reached N_max × n_mut = 10⁴ cells. All descendants of the mutated cell also carry the mutation. These simulations result in 2-dimensional spatial patterns of WT cells coexisting with a sub-clonal population of mutated cells. We utilise this framework to explore the impact on the resulting patterns of population mixing when varying three model parameters, i.e. mutant replicative advantage, s, the relative time, n_mut, at which the mutation appears in the growing WT population, and the strength of cell-cell pushing on the lattice, q (see S1 Table).

Before analysing the patterns of the normalised CMFPT as a function of s, n_mut and q, we first characterise its relationship with the class ratio, ϕ, in each of the patterns (Fig 3). Similar to our observation in the artificial cell mixing model, we find a complex non-linear relationship between ϕ and the CMFPT, and a general trend of increasing with class ratio. In the artificial model, we can vary ϕ and the underlying pattern independently, since ϕ is an explicit model parameter. In our agent-based tumour model, however, this ratio has a dependence on s, n_mut and q, being simply a readout quantity rather than a tunable parameter. Thus, class ratio and pattern structure are naturally coupled. Correspondingly, certain underlying pattern structures are observed only for small mutant sub-clones, and therefore small ϕ (Fig 3d), whereas combinations of (s, n_mut, q) which give rise to higher ϕ will also result in substantially different pattern topologies (Fig 3a). As a result, the CMFPT measurements of the simulated patterns do not fall onto distinct curves (Fig 3e), as was observed for different underlying structures in the artificial model.

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Fig 3. Analysis of simulated sub-clonal mixing patterns.

(a-d) Simulated sub-clonal mixing patterns with cell pushing strength of q = 5. (e) Analysis of all simulated mixing patterns simulated with a cell pushing value of q = 5 in the (,) phase space. Data represent simulations for all possible combinations of s and n_mut where s ∈ {0, 0.1, 0.2, 0.5, 1, 2, 3} and n_mut ∈ {0.001, 0.01, 0.03, 0.05, 0.08, 0.1, 0.5}, with q = 5 (approximately 100 simulated patterns for each parameter combination). Point colour corresponds to the relative abundance of red and yellow, ϕ.

https://doi.org/10.1371/journal.pcbi.1010952.g003

Beyond class ratio, we investigate the extent to which the CMFPT reflects spatial patterns generated under different model parameter combinations (Fig 4). When we classify our simulations according to mutant fitness, s, and the arising time of the mutant, n_mut, spatial patterns generated by different parameter combinations occupy distinct regions of the (,) phase space, demonstrating that the CMFPT can be used to quantify distinct spatial heterogeneity. Patterns corresponding to neutral selection, s = 0 (purple points in Fig 4), lie in the leftmost region , and sub-clones with strong selection, 2 ≤ s ≤ 3 (yellow and red points in Fig 4), are associated with larger values of . Due to the coupled effects of s and n_mut, tumours with low s and n_mut (Fig 4, navy & turquoise circles), and those with large s and n_mut (Fig 4, yellow & orange triangles) fall into similar regions of the phase space, especially for q = 0. Here, the CMFPT may struggle to distinguish these different dynamics. Nevertheless we would still expect to see differences in the appearance of the sub-clonal patterns under these different dynamical regimes, however, and it does appear that the mean CMFPT of these patterns subtly reflects these differences. Tumours with the earliest arising sub-clones with the strongest replicative advantage (Fig 4, red & orange circles), representing the most extreme dynamics we tested, occupy a unique area of the phase space for all cell pushing strengths.

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Fig 4. Analysis of simulated sub-clonal mixing patterns for varying sub-clonal dynamics.

Mean CMFPT values for simulated sub-clonal mixing patterns in the phase space (,), with points coloured according to values of model parameter s and point shape depending on parameter n_mut. Data within each panel represent simulations for all possible combinations of s and n_mut where s ∈ {0, 0.1, 0.2, 0.5, 1, 2, 3} and n_mut ∈ {0.001, 0.01, 0.03, 0.05, 0.08, 0.1, 0.5} (approximately 100 simulated patterns for each parameter combination). Inset plots contain the same data as in the main panels, but show individual data points (one per pattern, n ≈ 100 patterns per s and n_mut combination) with point colour corresponding to the class ratio, ϕ. Images are separated depending on their pushing value q = 0 (a), q = 5 (b), q = 10 (c) and q = 20 (d).

https://doi.org/10.1371/journal.pcbi.1010952.g004

Varying the strength of cell pushing, q, has clear qualitative impacts on spatial heterogeneity. At weak pushing strengths cell displacement occurs at short ranges, leading to greater clustering of sub-clones. Conversely, at larger values of q cells are subject to more frequent displacement by nearby dividing cells, so that any sub-clonal clusters are more quickly dispersed. These effects are also reflected in the distribution of CMFPT measurements in the (,) phase space (Fig 4a–4d). For weak pushing strengths (Fig 4a) the simulated spatial patterns follow a path in phase space more similar to that of the centred artificial model patterns (Fig 2c and 2d), with two distinct “arms” demarcating patterns with ϕ < 0.5 and those with ϕ > 0.5. In the limit of zero cell pushing (q = 0), varying the mutant replicative advantage, s, and arising time, n_mut, impacts the final class ratio. Since, however, cell proliferation is limited to the periphery of the system, we observe little variation in the size and spatial heterogeneity of sub-clonal patterns for different realisations of the same system (i.e. the same value of s and n_mut). This lack of variation in sub-clonal pattern structure at weaker q is reflected in the CMFPT, with measured values clustering strongly in the phase space according to the values of s and n_mut and the class ratio of the sub-clonal pattern (Fig 4a, inset).

Conversely, under stronger cell pushing the diversity in the observed sub-clonal pattern structures is greater, resulting in increased variation in measured CMFPT values for different tumours with the same s and n_mut values, and weaker clustering of CMFPT measurements according to the class ratio of the simulated pattern. This is explained by the loss of distinct clusters of closely related cells when cell pushing strength is increased. The increased fragmentation of sub-clones is captured by the CMFPT and represented by a steeper initial decline in / for patterns with low s.

Overall, our analysis of simulated tumour sub-clonal mixing patterns points to an important result. That is, despite carrying little genetic information, different underlying sub-clonal growth dynamics each have their own spatial signature, left behind in the mixing patterns of WT and sub-clonal tumour cell populations, and it is possible to narrow down the potential underlying cell dynamics by quantifying the resulting spatial patterns alone. Moreover, our results suggest that CMFPT provides a promising new route to understanding these spatial patterns by describing the structural features of the patterns over multiple spatial scales.

Quantifying sub-clonal dynamics in human colorectal cancer

Having shown that CMFPT can be used to recover the underlying parameters of generative models by analysing spatial mutation patterns alone, both with a simple artificial model and agent-based simulated tumours, we next explore an application of our method to human colorectal carcinoma samples spatially mapped by BaseScope [20]. We analysed 22 cell mixing patterns produced with BaseScope (S10(a) Fig) which detail the spatial composition of tumour cells, mapping out sub-clonal populations carrying point mutations in one of three genes commonly mutated in colorectal cancer—PIK3CA, BRAF and KRAS. Mutations in these genes have been associated with cancer driver events, poor prognosis and treatment resistance [28–32]. Based on our understanding of how the CMFPT measurement relates to sub-clonal mixing patterns generated with our agent-based model, we aim to reconstruct the underlying dynamics driving the growth of the mutated sub-clones observed with BaseScope, in terms of our computational model parameters. It is important to note that, whilst the mutation time in our 2-dimensional simulations, n_mut, describes the time at which the mutation first appears in the entire system, the interpretation of this parameter changes when applied to the BaseScope samples. It is extremely rare that the same mutation will happen independently in multiple regions of any tumour, yet since the BaseScope images represent 2-dimensional sub-samples of a larger 3-dimensional system, a single spatially continuous sub-clone in the 3-dimensional tumour can lead to the appearance of spatially distinct sub-clonal regions in 2-dimensional samples. Accordingly, any inferred values of n_mut will represent the relative length of time that the WT population was expanding in that localised area prior to the mutant sub-clone. In a few cases, where the BaseScope samples involve multiple spatially discontinuous tumour areas, the distribution of inferred n_mut values across all local areas reveals how fast the mutant population spread in these tumours (details explained below).

Whilst the number of tumour cells contained within the experimental colorectal cancer samples is of the same order of magnitude as the simulated patterns to which they will be compared, the experimental images vary in resolution and contain far more pixels than the simulated patterns. Furthermore, we fix the final number of cells in our simulations, however the number of cells contained within the experimental patterns, and smaller sub-regions which we later analyse, varies across samples. Despite our efforts to parameterise our model such that simulated and experimental patterns contain a similar number of tumour cells, our comparisons may not be optimal in this regard and could possibly be more informative by improving the concordance of tumour cell numbers under comparison. A modified agent-based model, in which the final system size, N_max, is allowed to vary, could allow for better matching of cell numbers. Alternatively, keeping N_max fixed in the simulations, but extracting sub-regions of the simulated patterns, the size of which could be varied, could also achieve this. One of the strengths of the CMFPT, however, is the normalisation of this quantity to corresponding first passage times in a null model, which enables the comparison of geometrical features of patterns of different sizes.

Initial analyses of the BaseScope images, for which we estimated the CMFPT over the totality of each image, highlighted some important additional processing steps that should be applied to the BaseScope images prior to parameter inference. Crucially, the presence of un-highlighted areas in the BaseScope images, representing all other non-cancerous tissue, significantly impacted our estimations of the CMFPT. We do not consider non-cancerous cells in our agent-based model, in order to maintain modest model complexity, however this makes naive comparisons of experimental to simulated sub-clonal patterns less informative. This is evidenced by our initial analysis of the unprocessed BaseScope images where we found some CMFPT values falling far outside of the range of measured times from our simulated patterns, and little correlation between the CMFPT of a BaseScope pattern and it’s class ratio (S11 Fig). To improve comparisons between experimental and simulated sub-clonal patterns, we execute a number of pre-processing steps on the BaseScope images. We first process each image by manually filling in un-highlighted interior areas of colonic crypts in predominantly WT or mutant regions with the relevant colour (S10(b) Fig). To address the remaining un-highlighted areas which correspond to regions of non-cancerous cells (e.g. connective tissue), we separate BaseScope images containing spatially discontinuous tumour regions into smaller sub-sections. These sub-sections are not arbitrarily selected, but instead are manually delineated regions, identified by us, which contain locally connected areas of WT and mutated cells. By analysing these sections individually, we reduce the influence of un-highlighted space separating these regions, enabling a better comparison of experimental and simulated measurements.

Following these pre-processing steps, the results are largely improved compared to our initial analyses of the full BaseScope images, and reveal a similar general trend to that observed with our in silico pattern results, where higher ϕ patterns appear in the region of the phase space, and lower ϕ patterns are situated below (Fig 5e). Only a handful of the analysed BaseScope patterns lie below , a region of the phase space predominantly occupied by simulated patterns with high cell pushing, q, and low mutant selection, s, suggesting cell pushing might be generally weak in vivo.

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Fig 5. Analysing sub-components of BaseScope sub-clonal patterns.

(a-d) Representative examples of sub-components of the BaseScope samples (after pre-processing steps applied) and their best-fit simulated sub-clonal patterns. (d) Marginal posterior distributions of model parameters s, n_mut, and q, comprising of the n = 100 nearest simulated sub-clonal patterns in the 4-dimensional phase space spanned by , , and . Inferred parameter value is indicated by the vertical dashed line in the diagonal panels. 95% credible regions lie within the shaded region in the diagonal panels. (e) CMFPT analysis of all simulated sub-clonal patterns (circles) and connected WT and mutant cell sub-regions within BaseScope images (triangles). Points are coloured according to pattern class ratio, ϕ. (f) Distribution of predicted s, n_mut, and q values inferred from simulated patterns (left) and probability of correct prediction for each parameter value (right).

https://doi.org/10.1371/journal.pcbi.1010952.g005

To describe the sub-clonal dynamics more quantitatively, we performed a grid-search to find the best-fit simulated parameters for each of the analysed sub-components in the BaseScope images (Fig 5a–5d, S12–S27 Figs) (see Methods). Plotting the best-fit simulated patterns demonstrates that our method not only captures the statistical features of the experimental images, but that the best-fit simulated patterns often also share a close visual correspondence to the BaseScope patterns both in terms of class ratio and pattern heterogeneity. BaseScope patterns with high segregation of WT and mutant populations are predicted to have low cell pushing. Highly segregated patterns which, in addition, have a large mutant frequency, ϕ, are most consistent with model patterns simulated with early arising mutant sub-clones endowed with a strong replicative advantage.

We tested the robustness of our inference method by predicting model parameter values for simulated sub-clonal patterns, generated with known values of s, n_mut and q (Fig 5f). In the phase space of first passage times we find that the spread of values for simulated tumours with identical parameters can be reasonably large, and that the distributions obtained for tumours differing only by small changes in their parameters are not always easily distinguishable. Despite this, we find our method of inference to be reasonably accurate for parameters s and q, as evidenced by the agreement between the “ground truth” parameter values and the predicted values. Large values of s were more consistently correctly predicted than lower values, which is likely due to the small differences between values at the low end of the parameter range, resulting in sub-clonal patterns with similar statistical properties which were then misclassified by our algorithm. Of the three model parameters of interest, our method appears to be least sensitive to the timing of the mutation, n_mut. Patterns simulated with intermediate values (n_mut = 0.01, 0.03, 0.05, 0.08 & 0.1) are often predicted to have a n_mut value an increment above or below the true value (e.g. true value of n_mut = 0.05 ascribed a value of n_mut = 0.03 or n_mut = 0.08). These values, however, correspond to differences of 2,000–3,000 WT cell divisions, which are small relative to the size of the whole tumour itself. Accuracy with respect to cell pushing strength, q, was particularly good, with our method appearing to correctly predict the value of q in the majority of cases.

The inferred best-fit parameters for connected WT and mutant regions within our BaseScope images point to a wide range of sub-clonal growth dynamics. Analysis of the majority of the BaseScope samples suggests that the mutated populations experience some degree of replicative advantage, s, over the WT population, with some patterns consistent with mutated cells dividing at up to 4 times the rate of WT cells (s = 3). Approximately 10% of the sub-regions we analysed were consistent with neutral sub-clonal dynamics (s = 0). Inferred values for the relative time of mutant sub-clone invasion, n_mut, within each BaseScope sub-component range between 0.1% and 10% for the majority of the samples, suggesting that in many of the sub-regions we analysed, the mutant sub-clones arose early in the expansion of the WT tumour populations. Consistent with our earlier supposition, the inferred value of cell pushing parameter q is mostly q = 0 or q = 5, however some larger inferred pushing values are predicted.

The newly implemented pre-processing steps reduced the influence of spatial regions of non-tumour tissue on the CMFPT measurements, improving the concordance between experimental and simulated tumours. Yet, not all of the experimental samples are visually consistent with their ascribed best-fit model image, despite having similar CMFPT values. This is due to the variance in the distribution of CMFPT values for certain combinations of s, n_mut and q, meaning that, in some instances, experimental patterns can have CMFPT values consistent with multiple combinations of model parameters (S9 Fig). As a result, we occasionally assign best-fit model patterns and parameters that do not accurately resemble the experimental image (for example, sub-region 1 in Fig 6a), despite being supported statistically within the inference framework.

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Fig 6. Sub-section analysis of BaseScope sample 34.

(a) Best-fit simulated sub-clonal pattern shown next to each sub-section with corresponding model parameters. (b) Marginal distributions of inferred model parameters across all sub-sections.

https://doi.org/10.1371/journal.pcbi.1010952.g006

For three of the most fragmented BaseScope images, samples 02, 28 and 34, we were able to separately analyse several isolated sub-components of tumour cells and construct a distribution of inferred parameter values across the full images (S28 and S29 Figs & Fig 6 respectively). We surmise that the distribution of inferred selection strength, s, and mutation time, n_mut, can offer additional insights into the nature of the early sub-clonal evolution in these tumours. As previously mentioned, n_mut should not be interpreted as the time at which the mutant population first appears in the tumour, but instead it indicates the relative length of time that the WT population was expanding in that area prior to invasion by the expanding mutated sub-clone. Despite this, there will be some correspondence between the distribution of inferred local n_mut values and the time of the original mutational event: with our computational modelling demonstrating that earlier arising mutations tend to lead to higher infiltration of the mutant sub-clone throughout the mature tumour, and less variegation in the mutant sub-clone pattern. Following this rationale, we would expect an earlier mutational event to result in a unimodal distribution of inferred n_mut values, with a low variance. Conversely, later occurring mutations should lead to a wider distribution of inferred n_mut values, which may deviate from unimodality.

In particular, sample 34 (Fig 6) suggests high mutant replicative advantage and an early emerging sub-clone. This sample possesses a narrow distribution of inferred n_mut across the different sub-sampled regions, concentrated towards low n_mut values, indicating early invasion of the mutant sub-clone within the WT population, and pointing to an overall early emerging mutation in the evolution of the tumour. This narrow distribution of n_mut is coupled to a narrow distribution of s, trending towards large values. Given that our inference method is most accurate when predicting large selection strengths (Fig 5f), this trend of high selection across the majority of the sub-regions would suggest that the mutant sub-clone as a whole is indeed subject to strong selection.

These factors point to an early arising, positively selected mutant sub-clone within this tumour. Conversely, samples 02 and 28 (S28 and S29 Figs) contain a larger proportion of sub-regions consistent with a late invading mutant sub-clone (n_mut = 0.5), and a wider distribution of inferred n_mut overall, along with lower inferred selection strength. Despite our inference method appearing to have reduced accuracy for intermediate n_mut values, the results from our inference validation lend support to the wide distribution of sub-clone invasion times which were predicted within samples 02 and 28. Whilst we have limited resolution in these distributions, this might suggest that the mutation itself occurred later in the evolution of samples 02 and 28 than it did in sample 34.

Our inference of cell pushing within samples 02, 28 and 34 pointed to a trend of weak cell pushing. One should take care, however, when drawing conclusions about the cell pushing strength within these samples as a whole based on the sub-region analyses. This is because, of the three model parameters we inferred, the perceived cell pushing strength may be most affected by the sub-sampling strategy employed here, as even highly mixed patterns will begin to appear segregated when viewed with a higher magnification. Validation of our inference method (Fig 5f) would suggest that the majority of sub-regions across these samples were indeed consistent with weak cell pushing, however it is also plausible that stronger cell pushing was acting within these tumours, which led to the observed fragmented patterns of tumour cells, which in turn necessitated the sub-region analysis in these samples.

The observed distributions of inferred values for mutant replicative advantage, s, and cell pushing, q, within the same BaseScope image could be due to a number of reasons. The experimental tissue represents 2-dimensional samples taken from a 3-dimensional system along a randomly oriented plane. As previously discussed, this sampling can have significant impacts on the appearance of the sub-clonal pattern, potentially creating the false appearance of spatially distinct sub-clones. When analysed individually, these small isolated sub-clones are likely to lead to variances in the inferred dynamics. Beyond this, however, such variances in inferred parameter values could reflect local fluctuations in the dynamics of the system. Cell mixing is likely to be a result of a complex combination of cell-intrinsic and extrinsic factors, influenced by physical stresses exerted by the tumour microenvironment. Physical stresses may fluctuate even across small areas of tissue, and inferring fluctuating values of cell pushing in different areas of the tumour may be consistent with this notion. In line with this reasoning, the observed distribution of mutant replicative advantage, s, could reflect the varying propensity of mutant cells to proliferate, which can be expected to fluctuate spatially. Whilst certain genetic alterations may confer a cell-intrinsic increase in proliferation rate, other cell-extrinsic factors are also expected to contribute to the “effective” replication rate of any particular cell, such as the availability of free space, access to nutrients, and proximity to the immune system.

Ethics declaration

Formalin-fixed paraffin embedded (FFPE) tissue blocks were obtained from University College and St Mark’s Hospitals, London, under multi-centre ethical approval (11/LO/1613); the QUASAR2 trial (ISRCTN45133151, ethical approval REC 09/H0606/5+5); the COIN trial (ISRCTN; 27286448); and the FOCUS trial (ISRCTN; 79877428, ethical approval 15/EE/0241). Written informed consent was waived by the relevant RECs due to the retrospective and anonymous nature of this study.

Simulating random walkers and estimating first passage time statistics

To assess the spatial heterogeneity of cell types we use the normalised class mean first passage times between any pair of classes α and β [25]. To estimate this quantity, we first compute the unnormalised class mean first passage time from class α to β by simulating 5,000 random walks, and averaging over the resulting first passage times, for each starting node of class α in the image. To compute the class mean first passage time, we then average over this value across all α nodes. The normalised class mean first passage time, , is obtained by dividing the unnormalised class mean first passage time by its null-model counterpart, . To estimate we simulate random walks on a graph with the same connectivity as the original graph, but with the node classes redistributed uniformly at random, such that for an original graph with class ratio ϕ = x, any arbitrarily selected coloured node will be of class α and β with probabilities P_α = x and P_β = 1 − x respectively. For each pattern, the null model values are obtained over multiple realisations of this colouring process. Since the quantity is normalised by a null-model, it is dimensionless and represents whether the time to arrive at a node of a given class is smaller or larger than the corresponding transition in the null-model. For example, if then the expected number of steps taken before arriving at a node of class β is less in the experimental pattern than in the null-model, and is greater when .

To reduce computational time, we downsample the BaseScope data by binning image pixels. When altering the image resolution, the proportion of pixel classes is preserved in each bin. For example, if the pixels falling into a particular bin consisted of 50% red and 50% yellow pixels, then a walker on the downsampled network will encounter either a red or yellow bin at these coordinates with equal probabilities of . We downsample all BaseScope and simulated sub-clonal patterns to the same resolution, 3000 pixels in total, prior to analysis.

Spatial simulations of sub-clonal evolution in tumours

We generate cell mixing patterns using an algorithm based on the Gillespie algorithm [50] to simulate stochastic birth and death of cells on a 2-dimensional square lattice. Since the BaseScope assay targets and detects specific genetic point mutations, we adopt a binary system in which a particular cell can either be wild-type (WT) or mutated. A simulation begins by seeding a single WT cell on the lattice, which has a birth rate greater than its death rate, and terminates when the entire system (WT and mutated cells) reaches a specified size N_max. During initial growth, the WT clone expands until it reaches a size n = N_max⋅n_mut cells, at which point an existing cell is randomly chosen to acquire a mutation. All subsequent cells related to the newly mutated cell will inherit the mutation, and cells can not revert from the mutated state back to WT.

Cells carrying the mutation experience a relative fitness advantage compared to WT cells. This fitness advantage manifests as an increased birth rate, giving the birth rate of a mutated cell, b_mut, as (2) where s represents the relative fitness advantage conferred by mutations and b_WT is the replication rate of WT cells. Our model assumes that cell cycle times are Poissonian distributed. Whilst in reality cell cycle distributions are more tightly peaked, this assumption has been shown to reduce lattice artefacts in cellular automaton models [51]. Cell death is coupled to cell birth in the simulations. Before a cell divides, it is randomly determined either to continue with the division, or die giving a death rate for cell i, d_i, of (3) where ψ ∈ [0, 1], and is fixed at ψ = 0.3 throughout this study unless stated otherwise. This implies that cells which divide at a faster rate also die at a faster rate, which could reflect the greater rate of apoptosis resulting from faster cell cycling in these cells. When a cell dies it immediately is removed from the lattice leaving behind an empty lattice point.

To model the displacement of neighbouring cells during division, we adapted an algorithm developed by Waclaw et al. [8], which involves the dividing cell searching its neighbourhood and finding a “path” to one of the nearest empty lattice points. In order to create space to divide into two daughter cells, the dividing cell pushes neighbouring cells along this path, filling the nearby empty lattice point and creating a new empty space adjacent to itself. In our simulations we implement these mechanics and extend them by accounting for relative positions of cells during pushing, favouring straight-line pushing over displacement in other directions, in accordance with Newton’s second law of motion. The parameter q determines the maximum allowed size for the constructed path in this algorithm. If no path can be found which is less than or equal to q in length, the cell cannot successfully divide. Setting this parameter to q = 0 leads to surface growth dynamics, in which a cell must be adjacent to an empty lattice point in order to divide successfully.

The range of values used for mutant selective advantage, s ∈ {0, 0.1, 0.2, 0.5, 1, 2, 3}, was chosen so that we were able to investigate spatial signatures of neutral sub-clonal growth, through intermediate positive selection, up to very strong positive selection. These values span the expected range of selective coefficients related to driver point mutations [52, 53]. Our chosen range of n_mut was naturally restricted between 0 < n_mut < 1. We chose to explore values spanning several orders of magnitude ranging n_mut ∈ {0.001, 0.01, 0.03, 0.05, 0.08, 0.1, 0.5} leading to, for N_max = 1 × 10⁵, the earliest emerging mutant populations appearing after just 100 WT divisions (n_mut = 0.001) and the latest appearing after 50,000 WT divisions (n_mut = 0.5). Our motivation for our chosen range of pushing values, q ∈ {0, 5, 10, 20}, was to simulate a spectrum of pushing dynamics ranging from boundary driven growth (q = 0) up to increasingly stronger pushing strengths approaching the volumetric (exponential) growth regime. Whilst our implementation of cell pushing is somewhat abstract, and one could conceive of other approaches to modelling cell movement on the lattice, the mode of growth in human tumours and its relation to tumour type and stage is not well known. This model design allows us to explore the spatial signatures of the different regimes and smoothly transition between different modes of growth using the parameter q.

Model parameters used to generate in silico mixing patterns are listed in S1 Table. Note that, due to decreased probability of sub-clonal survival for (s, n_mut, q) = (0, 0.5, 20), (0.1, 0.5, 20) and (0.2, 0.5, 20), sub-clonal pattern data could not be generated for these parameter combinations.

Bayesian grid-search for best-fit model parameter inference

To perform inference of mutant replicative advantage (s), mutation timing (n_mut) and cell pushing strength (q) for each BaseScope sub-sampled region (S12–S27 Figs), we implement a Bayesian-style grid search. For each experimental pattern, we construct the posterior distribution of s, n_mut and q by finding the n nearest simulated sub-clonal patterns in the 4-dimensional phase space of class mean first passage times , , and , as determined by log-Euclidean distance. We chose a sample size of n = 100 for each experimental pattern, and computed 95% credible regions for our estimations of each parameter value. To select the “best fit” simulated sub-clonal pattern we found the most abundant combination of s, n_mut and q within the posterior sample set and assigned these parameter values as point estimates for the experimental pattern. Information on the mutant frequency ϕ, could be used to supplement the CMFPT measurements during our grid-search inference, however this ratio is highly susceptible to uncertainty introduced by the tissue sampling process, as evidenced by our exploratory 3D simulations which demonstrate that measured ϕ can vary wildly depending on the orientation of the 2D sample acquired from the 3D tumour. Measurements of ϕ could in principle be incorporated into the grid-search analysis, however in this study we were interested in exploring the efficacy of CMFPT measurements alone in quantifying sub-clonal dynamics based on the appearance of the mutant sub-clones.

Shannon’s entropy calculations

We replicated the calculations of Shannon’s entropy from Ref [20] and applied these to our own set of simple 2-dimensional mixing patterns generated using the clusters, centred and column models (S1 Fig). To calculate the Shannon’s entropy of a pattern, we divide the pattern into n non-overlapping quadrats with dimensions L × L cells, and compute the frequency of mutant cells within each quadrat. We then compute Shannon’s entropy, H, as (4) where p_i denotes the frequency of mutant cells in the i^th quadrat. When analysing our 2-dimensional mixing patterns, which themselves have dimensions of 54 × 54 cells, we use quadrats with dimensions 10 × 10. As has been discussed with respect to quadrat-based methods for estimating fractal dimension [54–56], the estimation of Shannon’s entropy using this method can change depending on the placement of the grid in the (x, y) plane. To address this, we compute H for each pattern for a number of x and y grid offsets, specifically from 0 up to L in both orthogonal directions independently. We then take the minimum measured value of H across all x and y offsets.

Mean shortest distance calculations

We compute the mean shortest distance [57] on our simple 2-dimensional mixing patterns generated using the clusters, centred and column models (S3 Fig). As with our CMFPT calculations, we obtain four quantities from our 2-colour patterns, , , and . For example, to compute the mean shortest distance from yellow to red, , we compute the average shortest path length in the adjacency network between cells (in terms of number of edges) between the pair of nodes i and j for i ∈ Y and j ∈ R (5) where Y and R denote the set of all yellow and red nodes, respectively, and d_ij denotes the shortest path length connecting cells i and j. To obtain , 〈d_yr〉 is normalized by , which is the same quantity calculated over a null-model where colors are reshuffled at random.