Nature  Letter
A spatial model predicts that dispersal and cell turnover limit intratumour heterogeneity
 Bartlomiej Waclaw^{1}^{, }
 Ivana Bozic^{2, 3}^{, }
 Meredith E. Pittman^{4}^{, }
 Ralph H. Hruban^{4}^{, }
 Bert Vogelstein^{4, 5}^{, }
 Martin A. Nowak^{2, 3, 6}^{, }
 Journal name:
 Nature
 Volume:
 525,
 Pages:
 261–264
 Date published:
 DOI:
 doi:10.1038/nature14971
 Received
 Accepted
 Published online
Most cancers in humans are large, measuring centimetres in diameter, and composed of many billions of cells^{1}. An equivalent mass of normal cells would be highly heterogeneous as a result of the mutations that occur during each cell division. What is remarkable about cancers is that virtually every neoplastic cell within a large tumour often contains the same core set of genetic alterations, with heterogeneity confined to mutations that emerge late during tumour growth^{2, 3, 4, 5}. How such alterations expand within the spatially constrained threedimensional architecture of a tumour, and come to dominate a large, preexisting lesion, has been unclear. Here we describe a model for tumour evolution that shows how shortrange dispersal and cell turnover can account for rapid cell mixing inside the tumour. We show that even a small selective advantage of a single cell within a large tumour allows the descendants of that cell to replace the precursor mass in a clinically relevant time frame. We also demonstrate that the same mechanisms can be responsible for the rapid onset of resistance to chemotherapy. Our model not only provides insights into spatial and temporal aspects of tumour growth, but also suggests that targeting shortrange cellular migratory activity could have marked effects on tumour growth rates.
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At a glance
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Author information
Affiliations

School of Physics and Astronomy, University of Edinburgh, JCMB, Peter Guthrie Tait Road, Edinburgh EH9 3FD, UK
 Bartlomiej Waclaw

Program for Evolutionary Dynamics, Harvard University, One Brattle Square, Cambridge, Massachusetts 02138, USA
 Ivana Bozic &
 Martin A. Nowak

Department of Mathematics, Harvard University, One Oxford Street, Cambridge, Massachusetts 02138, USA
 Ivana Bozic &
 Martin A. Nowak

The Sol Goldman Pancreatic Cancer Research Center, Department of Pathology, Johns Hopkins University School of Medicine, 401 North Broadway, Weinberg 2242, Baltimore, Maryland 21231, USA
 Meredith E. Pittman,
 Ralph H. Hruban &
 Bert Vogelstein

Ludwig Center and Howard Hughes Medical Institute, Johns Hopkins Kimmel Cancer Center, 1650 Orleans Street, Baltimore, Maryland 21287, USA
 Bert Vogelstein

Department of Organismic and Evolutionary Biology, Harvard University, 26 Oxford Street, Cambridge, Massachusetts 02138, USA
 Martin A. Nowak
Contributions
B.W., M.A.N., I.B. and B.V. designed the study. B.W. wrote the computer programs and made simulations. B.W., I.B. and M.A.N. made analytic calculations. M.E.P. and R.H.H. carried out experimental work. All authors discussed the results. The manuscript was written primarily by B.W., M.A.N., I.B. and B.V., with contributions from M.E.P. and R.H.H.
Competing financial interests
The authors declare no competing financial interests.
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Bartlomiej Waclaw
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Ivana Bozic
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Meredith E. Pittman
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Ralph H. Hruban
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Bert Vogelstein
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Extended data figures and tables
Extended Data Figures
 Extended Data Figure 1: Details of the model. (289 KB)
a, A sketch showing how dispersal is implemented: (1) A ball of cells of radius R_{i}, in which the centre is at X_{i}, is composed of tumour cells and normal cells (blue and empty squares in the zoomedin rectangle (2)). A cell at position x_{i} with respect to the centre of the ball attempts to replicate (3). If replication is successful, the cell migrates with probability M and creates a new microlesion (4). The position X_{j} of this new ball of cells is determined as the endpoint of the vector that starts at X_{i} and has direction x_{i} and length R_{i}. b, Overlap reduction between the balls of cells. When a growing ball begins to overlap with another ball (red), they are both moved apart along the line connecting their centres of mass (green line) by as much as necessary to reduce the overlap to zero. The process is repeated for all overlapping balls as many times as needed until there is no overlap. c, Summary of all parameters used in the model. If, for a given parameter, many different values have been used in different plots, a range of values used is shown. Birth and death rates can also depend on the number of driver mutations, see Methods. Asterisk, parameter estimated from other quantities available in the literature.
 Extended Data Figure 2: Simulation snapshots. (429 KB)
a, b, A few snapshots of tumour growth for no dispersal, and d = 0 (a) and d = 0.9b (b). To visualize clonal sectors, cells have been colourcoded by making the colour a heritable trait and changing each of its RGB components by a small random fraction whenever a cell mutates. The initial cell is grey. Empty space (white) are noncancer cells mixed with extracellular matrix. Note that images are not to scale. c, Tumour shapes for N = 1 × 10^{7}, d = 0.9b, and different dispersal probability M. Images not to scale; the tumour for M = 1 × 10^{−5} is larger than the one for M = 0.
 Extended Data Figure 3: Tumour size as a function of time. (133 KB)
a, Growth of a tumour without dispersal (M = 0), for d = 0.8b. For large times (T), the number of cells grows approximately as const × T ^{3}. The tumour reaches size N = 1 × 10^{9} cells (horizontal line) after about 100 months (8 years) of growth. b, The same data are plotted in the linear scale, with N replaced by ‘linear extension’ N^{1/3}. c, Tumour size versus time when drivers affect the dispersal probability. In all cases, d = 0.9b, and (1, black) drivers increase the dispersal rate tenfold (q = 9) but have no effect on the net growth rate; (2, red) drivers increase both the net growth rate (s = 10%) and M; (3, green) drivers either (with probability 1/2) increase M tenfold (q = 9) or increase the net growth rate by s = 10%; (4, blue) drivers increase only the net growth rate by s = 10%; and (5, black dashed line) neutral case with M = 1 × 10^{−7}, which is indistinguishable from (1). In all cases (1–3) the initial dispersal probability M = 1 × 10^{−7}. Points represent average value over 40–100 simulations per data point, error bars are s.e.m.
 Extended Data Figure 4: Simulation of targeted therapy. (228 KB)
a–c, The total number of cells in the tumour (black) and the number or resistant cells (red) versus time, during growth (T < 0) and treatment (T > 0), for ~100 independent simulations, for d = 0.5b for T < 0. Therapy begins when N = 1 × 10^{6} cells. After treatment, many tumours die out (N decreases to zero) but those with resistant cells will regrow sooner or later. a, M = 0 for all cells at all times. b, M = 0 for all cells for T < 0 and M = 10^{−4} for resistant cells for T > 0. c, M = 0 for nonresistant and M = 10^{−5} for resistant cells at all times. In all three cases, P_{regrowth} is very similar: 36 ± 5% (mean ± s.e.m.) (a), 25 ± 4% (b), and 27 ± 6% for (c). d–g, Regrowth probability for four treatment scenarios not discussed in the main text. Data points correspond to three dispersal probabilities: M = 0 (red), M = 1 × 10^{−5} (green), and M = 1 × 10^{−4} (blue). The probability is plotted as a function of tumour size N just before the therapy commences. d, Before treatment, cells replicate only on the surface. Cells in the core are quiescent and do not replicate. Therapy kills cells on the surface and cells in the core resume proliferation when liberated by treatment. e, As in d, but drug is cytostatic and does not kill cells but inhibits their growth. The results for P_{regrowth} are identical if the drug is cytotoxic and the tumour has a necrotic core (cells die inside the tumour and cannot replicate even if the surface is removed). f, Before treatment, cells replicate and die on the surface. The core is quiescent. Therapy kills cells on the surface (cytotoxic drug). g, As in f, but therapy only inhibits growth (cytostatic drug). In all cases (d–g) error bars represent s.e.m. from 8–1,000 simulations per point.
 Extended Data Figure 5: Accumulation of driver and passenger genetic alterations. (359 KB)
a–c, The number of drivers per cell in the primary tumour plotted as a function of time (10–100 simulations per point, error bars denote s.e.m.). a, M = 0 and three different driver selective advantages. For s = 1%, cells accumulate on average one driver mutation within 5 years. The time can be significantly lower for very strong drivers (s > 1%). b, The rate at which drivers accumulate depends mainly on their selective advantage and not on whether they affect death or birth rate. c, Dispersal does not affect the rate of driver accumulation. d, e, The number of passenger mutations (PMs) per cell versus the number of driver mutations per cell. More passenger mutations are present for smaller driver selective advantage (d), and this is independent of the dispersal probability M (e) in the regime of small M. Data points correspond to independent simulations.
 Extended Data Figure 6: Genetic diversity in a single lesion for different models. (627 KB)
a–d, Representative simulation snapshots, with genetic alterations colourcoded as in Fig. 4. Top: s = 0, bottom: s = 1%. a, Model A from the main text in which cells replicate with rates proportional to the number of empty nearby sites. b, Model B, the replication rate is constant and nonzero if there is at least one empty site nearby, and zero otherwise. c, Model C, cells replicate at a constant rate and push away other cells to make space for their progeny. d, Model D, cells replicate/die only on the surface, the interior of the tumour (‘necrotic core’) is static. In all cases, N = 1 × 10^{7}, d = 0.99b. e, Number of genetic alterations present in at least 50% of cells for identical parameters as in a–d. In all cases except surface growth (d), drivers increase genetic homogeneity, as measured by the number of most frequent genetic alterations. Results averaged over 50–100 simulations, error bars denote s.e.m. f, Model D, with γ_{d} = 2 × 10^{−4} instead of 4 × 10^{−5}, that is, drivers occur five times more often. In this case, driver mutations arise earlier than in d, and the tumour becomes more homogeneous.
 Extended Data Figure 7: The offlattice model. (1,316 KB)
a, Summary of all parameters used in the model. Asterisk, typical value, varies between different types of tissues; dagger symbol, equivalent to 24 h minimal doubling time; double dagger symbol, based on the assumption that macroscopic elastic properties of tissues such as liver, pancreases or mammary glands are primarily determined by the elastic properties of stroma. b, Simulation snapshot of a normal tissue before the invasion of cancer cells. c, Two balls of cancer cells in two nearby ducts repel each other as they grow as a consequence of mechanical forces exerted on each other. d, The balls coalesce if growth is able to break the separating extracellular matrix. e, If the balls are not encapsulated, they quickly merge. f, Isolated balls of cells are not required to speed up growth; migration (left) can cause the tumour to expand much faster even if individual microlesions merge together, as opposed to the case with no migration (right).
 Extended Data Figure 8: Genetic diversity quantified. (323 KB)
a, Tumours are much more genetically heterogeneous in the absence of driver mutations (s = 0) (see Fig. 4). The plot shows the fraction G(r) of genetic alterations (GAs) shared between the cells as function of their separation (distance r) in the tumour. The fraction quickly decreases with increasing r. The distance in the figure is normalized by the average distance <r> between any two cells in the tumour. For a spherical tumour, <r> is approximately equal to half of the tumour diameter. b, Fraction of shared genetic alterations for s = 1% and s = 0%, N = 1 × 10^{7}, and M = 1 × 10^{−7}. In the presence of drivers, G(r) decays slower, indicating more homogeneous tumours. c, The exact value of the selective advantage of driver mutations is not important (all curves G(r) look the same, except for s = 0) as long as s > 0. d–f, Number of genetic alterations present in at least 50% of cells for identical parameters as in a–c, correspondingly. Drivers substantially increase the level of genetic homogeneity. In all panels the results have been averaged over 30–100 simulations, with error bars as s.e.m.
 Extended Data Figure 9: Growth curves for the 26nearest neighbours (26n, red curves) and the 6nearest neighbours (6n, green curves) models. (333 KB)
a, Model A (as in the main text), no death. The tumour grows about twice as slow in the 6n model. Pictures show tumour snapshots for both models; there is no visible difference in the shape. b, Model A, death d = 0.8b. The additional blue curve is for the 6n model, with modified replication probability to account for missing neighbours as explained in the Supplementary Information. c, Model A, with death d = 0.95b, and drivers s = 5%. There is very little difference in the growth curves between the 6n and 26n models. A small asymmetry in the shape is caused by fastergrowing cells with driver mutations. d,ModelC(exponential growth). Growth is the same in both 6n and 26n models, but the shape is more aspheric for the 6n model. This is probably caused by shifting cells along the coordinate axes and not along the shortest path to the surface when making space for new cells. All plots show the mean (average over 50–100 simulations) and s.e.m.
Extended Data Tables
Supplementary information
Video
 Video 1: Simulation of growth and treatment of a small tumor (1e7 cells) (4.61 MB, Download)
 Treatment begins at T=0. Different colors correspond to cells with different GAs.
 Video 2: Simulation of growth of a small tumor (1e7 cells) with no migration and death rate d=0.5. (1.71 MB, Download)
 Different colors correspond to cells with different GAs.
 Video 3: Simulation of growth of a small tumor (1e7 cells) with low migration M=1e6 and death rate d=0.5. (1.54 MB, Download)
 Different colors correspond to cells with different GAs.
 Video 4: Simulation of growth of a small tumor (1e7 cells) with no migration and selective advantage of driver mutations s=5%. (6.42 MB, Download)
 Different colors correspond to cells with different driver mutations.
 Video 5: Simulation of growth of a small tumor (1e7 cells) with no migration and selective advantage of driver mutations s=1%. (10.44 MB, Download)
 Only three mostabundant driver mutations have been colored as in Fig. 4
 Video 6: Simulation of growth and treatment of a small tumor (1e7 cells) for M=1e6. (1.3 MB, Download)
 Cells replicate and die only on the surface. Treatment begins at T=0. Different colors correspond to cells with different GAs.
 Video 7: Simulation of growth and treatment of a small tumor (1e7 cells) for no migration (M=0). (1.98 MB, Download)
 Cells replicate and die only on the surface. Treatment begins at T=0. Different colors correspond to cells with different GAs.
 Video 8: Simulation of the offlattice model, normal tissue with ducts. (16.78 MB, Download)
 This video shows the simulation of the offlattice model, normal tissue with ducts.
 Video 9: Simulation of the offlattice model, two balls of cells growing in nearby ducts (14.95 MB, Download)
 This video shows the simulation of the offlattice model, two balls of cells growing in nearby ducts. ECM does not break.
 Video 10: Simulation of the offlattice model, two balls of cells growing in nearby ducts (13.06 MB, Download)
 This video shows the simulation of the offlattice model, two balls of cells growing in nearby ducts. ECM can replicate.
 Video 11: Simulation of the offlattice model, two balls of cells growing in nearby ducts (10.44 MB, Download)
 The video shows the simulation of the offlattice model, two balls of cells growing in nearby ducts. ECM breaks when stretched too much.
 Video 12: Simulation of the offlattice model, two balls of cells growing in a layer of epithelial tissue merge quickly together. (20.81 MB, Download)
 This video shows the simulation of the offlattice model, two balls of cells growing in a layer of epithelial tissue merge quickly together.
 Video 13: Simulation of the offlattice model, fast growth in the presence of migration. (3.32 MB, Download)
 This video shows the simulation of the offlattice model, fast growth in the presence of migration.
 Video 14: Simulation of the offlattice model, slow growth in the absence of migration. (7.44 MB, Download)
 This video shows the simulation of the offlattice model, slow growth in the absence of migration.
PDF files
 Supplementary Information (375 KB)
This file contains Supplementary Text and Data 18.
Additional data

Extended Data Figure 1: Details of the model.Hover over figure to zoom

Extended Data Figure 2: Simulation snapshots.Hover over figure to zoom

Extended Data Figure 3: Tumour size as a function of time.Hover over figure to zoom

Extended Data Figure 4: Simulation of targeted therapy.Hover over figure to zoom

Extended Data Figure 5: Accumulation of driver and passenger genetic alterations.Hover over figure to zoom

Extended Data Figure 6: Genetic diversity in a single lesion for different models.Hover over figure to zoom

Extended Data Figure 7: The offlattice model.Hover over figure to zoom

Extended Data Figure 8: Genetic diversity quantified.Hover over figure to zoom

Extended Data Figure 9: Growth curves for the 26nearest neighbours (26n, red curves) and the 6nearest neighbours (6n, green curves) models.Hover over figure to zoom

Video 1: Simulation of growth and treatment of a small tumor (1e7 cells)

Video 2: Simulation of growth of a small tumor (1e7 cells) with no migration and death rate d=0.5.

Video 3: Simulation of growth of a small tumor (1e7 cells) with low migration M=1e6 and death rate d=0.5.

Video 4: Simulation of growth of a small tumor (1e7 cells) with no migration and selective advantage of driver mutations s=5%.

Video 5: Simulation of growth of a small tumor (1e7 cells) with no migration and selective advantage of driver mutations s=1%.

Video 6: Simulation of growth and treatment of a small tumor (1e7 cells) for M=1e6.

Video 7: Simulation of growth and treatment of a small tumor (1e7 cells) for no migration (M=0).

Video 8: Simulation of the offlattice model, normal tissue with ducts.

Video 9: Simulation of the offlattice model, two balls of cells growing in nearby ducts

Video 10: Simulation of the offlattice model, two balls of cells growing in nearby ducts

Video 11: Simulation of the offlattice model, two balls of cells growing in nearby ducts

Video 12: Simulation of the offlattice model, two balls of cells growing in a layer of epithelial tissue merge quickly together.

Video 13: Simulation of the offlattice model, fast growth in the presence of migration.

Video 14: Simulation of the offlattice model, slow growth in the absence of migration.

Extended Data Table 1: Experimental results for the percentage of proliferating cells in the centre versus the edge of solid tumoursHover over figure to zoom