Fig 1.
A) Basic algorithm (for implementation purposes). Initialization requires creating a 3D grid, specifying the final simulation time, defining subpopulation traits and setting the initial state. Temporal iterations are then carried out until the end time is reached. At each time step and each voxel, all clonal populations are updated. This updating involves calculating how many cells will proliferate, migrate, mutate or die. When all populations at all voxels have been evaluated, they are updated synchronously. B) Two-dimensional example of model behavior. Synchronous updating results in population increasing, diversifying and spreading at each time step, with probabilities computed according to the biological rules. Compartment color indicates occupancy. Cell color indicates cell type. C) Bottom image is a slice of an actual simulation, with colors indicating occupation. Each voxel contains a variable number of cells and subtypes as depicted above.
Fig 2.
Mutation tree used in this paper.
Relationships between the eight possible genotypes, according to the three alterations selected. Each clonal type can emerge from several ancestors by various alterations. Depending on the mutational history, tumors follow different paths on the mutation tree.
Table 1.
Parameter values for simulations of GBM.
Basal rates refer to cells with no mutations. Weights specify how each mutation affects the basal rate.
Fig 3.
Longitudinal tumor growth dynamics.
A) Time dynamics of the number of newborn cells at the central voxel (surrogate for tumor activity). B) Number of cells at the central voxel: total (dashed red), active (blue) and necrotic (grey). C,D) Tumor volume and MSR during the initial stages, starting from a single cell at the center of the lattice until tumor reaches 1 cm3. E) Dynamics of MSR for 100 simulations. Median run is shown in red. Time span shown starts when tumor reaches 1 cm3 in volume (equivalent to 6.2 mm of MSR) and ends after reaching 100 cm3. F) Example of tumor dynamics of the MSR and rendered 3D tumor shape for three different times (8.5, 10, 11.5 months). Basal rate parameters for simulation in this figure are τrep = 216.5 h, τdeath = 112.7 h, τmut = 200.4 h, . G) Dynamics of tumor volume for 100 simulations. Black lines represent different fits of the median run (solid red line): Exponential (solid), power law with β = 1.2 (dotted), Gompertz (dashed) and radial (dashed-dotted). H) Root-mean-square error (RMSE) of each fit.
Fig 4.
Tumor slice and simulated profiles.
A) Two-dimensional slice of a T1-weighted post-contrast MRI scan of an actual GBM. B) Two-dimensional slices of different simulated tumors. Each simulated tumor corresponds to the final state (100 cm3) of a different running of the model. Basal parameters for each simulation are: 1) τrep = 242.6 h, τdeath = 213.27 h, τmut = 221.6 h, , 2) τrep = 206.7 h, τdeath = 90.5 h, τmut = 219.5 h,
, 3) τrep = 184.2 h, τdeath = 325.4 h, τmut = 186.5 h,
, 4) τrep = 233.5 h, τdeath = 143.0 h, τmut = 175.9 h,
, 5) τrep = 201.2 h, τdeath = 295.5 h, τmut = 132.3 h,
, 6) τrep = 219.8 h, τdeath = 177.2 h, τmut = 87.33 h,
. Other parameters are listed in Table 1.
Fig 5.
Example of the dynamics of the tumor’s clonal composition.
A) Evolution of the eight clonal populations included in the model (one per row). Total tumor volume is shown as a light blue background. Times correspond to 8.5, 10 and 11.5 months. Parameters for this simulation are τrep = 179.1 h, τdeath = 292.5 h, τmut = 222.7 h, . Other parameters are as in Table 1. B) Tumor central slice showing in white high tumor cell density. C) Three-dimensional subtype composition of the tumor. D) Reconstruction of the phylogeny of the tumor. Each bifurcation represents a mutation. Bifurcations occurring first in each branch represent mutations appearing earlier in time.
Fig 6.
Dynamical behavior of tumor heterogeneity.
A) Evolution of Shannon and Simpson’s indexes for a typical run. Basal rates for this run are τrep = 242.6 h, τdeath = 213.3 h, τmut = 221.6 h, . The other parameters are those of Table 1. B) Abundance of each subtype in logarithmic scale as a function of time. C) Superposition of final subtype abundance for all simulated tumors. D) Final abundance per subtype per simulation. Each row corresponds to one subtype, in the same order as above. Color indicates end-point abundance.
Fig 7.
Dynamics of prognostic measures obtained from the model.
A,B) Time evolution of spherical rim width and surface regularity for 100 simulations with parameters from Table 1. The solid line is the average value and the dashed line the standard deviation. 2D reconstructions correspond to characteristic upper and lower values of each variable. Basal parameters, measured in hours, for each subplot are: A1) τrep = 94.2 h, τdeath = 170.8 h, τmut = 99.4 h, , A2) τrep = 184.9 h, τdeath = 230.1 h, τmut = 120.0 h,
, B1) τrep = 104.8 h, τdeath = 323.5 h, τmut = 222.7 h,
, B2) τrep = 158.7 h, τdeath = 54.2 h, τmut = 197.7 h,
.
Fig 8.
Prognostic measures obtained from the mathematical model recapitulate the behavior of those obtained from MRI and PET images of GBMs.
A). Kaplan-Meier curves for the population splitting using the spherical rim width taking a threshold equal to 1.93 mm. Median survival difference between groups was 0.62 months. B). Kaplan-Meier curves for the population splitting using the surface regularity taking a threshold equal to 0.84. Median survival difference between groups was 0.58 months.
Fig 9.
Elapsed time of typical model simulations.
A). Elapsed simulation time for a set of 100 simulations ran independently. B). Comparison between elapsed simulation time and tumor cell number at different time steps for a cohort of 100 simulations. Cell number does not include dead cells.