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Spatial model predicts dispersal and cell turnover cause reduced intra-tumor heterogeneity
Bartlomiej Waclaw, Ivana Bozic, Meredith E. Pittman, Ralph H. Hruban, Bert Vogelstein, Martin A. Nowak
TL;DR
The paper asks how genetically similar clones can spread through spatially constrained tumors and dominate pre-existing lesions. It uses a three-dimensional spatial model with cell dispersal, turnover, and evolving genetic alterations, predicting rapid mixing, clinically relevant clonal replacement, and rapid treatment resistance. The model also suggests that short-range cellular movement can strongly reshape tumor growth.
Problem
Existing tumor-evolution models generally omit spatial constraints or simplify mutation and growth dimensions, leaving clonal expansion within large three-dimensional tumors unexplained.
Method
The authors model cells on a three-dimensional lattice, tracking genetic alterations, replication, death, dispersal, tumor growth, and therapy-associated resistance.
Results
Short-range dispersal changes tumors from T^3 growth to exponential growth, while small driver advantages reduce heterogeneity and resistant clones rapidly regrow after therapy.
Takeaways & Limitations
Short-range cellular movement and turnover can rapidly mix tumor cells, enable fitter clones to replace precursor masses, and reshape tumor growth.
Takeaways & Limitations
The simulation algorithm is not an exact fully stochastic kinetic Monte Carlo method, and the model relies on assumptions about dispersal and tumor-cell behavior.
Abstract
from arXiv · showhide
Most cancers in humans are large, measuring centimeters in diameter, composed of many billions of cells. 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 their homogeneity - virtually every neoplastic cell within a large cancer contains the same core set of genetic alterations, with heterogeneity confined to mutations that have emerged after the last clonal expansions. How such clones expand within the spatially-constrained three dimensional architecture of a tumor, and come to dominate a large, pre-existing lesion, has never been explained. We here describe a model for tumor evolution that shows how short-range migration and cell turnover can account for rapid cell mixing inside the tumor. With it, we show that even a small selective advantage of a single cell within a large tumor 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 novel insights into spatial and temporal aspects of tumor growth but also suggests that targeting short range cellular migratory activity could have dramatic effects on tumor growth rates.
1. Introduction
The paper addresses a gap in tumor-evolution models by combining spatial constraints with genetic evolution to explain clonal composition, primary tumors, metastases, and treatment resistance.
- 30%-80% of key driver mutations are present in primary tumors and their metastatic lesions.
- Most existing models either assume well-mixed cells, include few mutations, or simplify tumor growth to one or two dimensions.
- The proposed model combines spatial growth and genetic evolution to describe primary tumors, metastases, and resistance to therapeutic agents.
2. Metastasis
The model predicts that even limited cell dispersal changes tumors from slowly expanding spheres into rapidly growing conglomerates of microlesions.
- 8 years without dispersal versus less than 2 years with dispersal are predicted for growth from one cell to one billion cells.
- For M>0, tumors form separated cellular balls resembling metastatic lesions and grow exponentially rather than as T^3.
- For M≈0, tumors become roughly spherical and grow approximately as N∼T^3.
- Short-range dispersal strongly affects tumor size and shape even when cancer cells divide and die at identical rates.
3. Treatment and the evolution of resistance
The spatial model predicts that resistant clones can emerge and regrow rapidly after targeted therapy, particularly in clinically detectable lesions.
- 10^-7 is the assumed probability of a resistant mutation, and one resistant cell is sufficient for metastatic-tumor regrowth.
- A simulated lesion shrinks rapidly after therapy, but resistant clones begin proliferating one month later.
- Resistant subclones are predicted to be nearly always present in lesions visible with clinical imaging.
4. Primary tumors
The primary-tumor model shows that small fitness advantages from new driver mutations can reduce heterogeneity and make alterations common across large tumors.
- Each additional driver mutation reduces the modeled death rate according to d=b(1-s)^k, where s is the average fitness advantage per driver.
- A 1% fitness advantage markedly increases homogeneity compared with no fitness advantage.
- For s=1%, many more genetic alterations are present in a macroscopic fraction exceeding 50% of tumor cells than for s=0%.
- The model predicts that virtually all cells in a large tumor share at least one new driver mutation.
5. Conclusion
The model accounts for clinically and experimentally observed cancer features and shows that small cellular movements can reshape tumors and alter growth rates.
- The spatial cancer-evolution model accounts for many clinically and experimentally observed tumor features.
- Small cellular movements can dramatically reshape tumors, beyond the large-distance migration classically associated with late cancer progression.
- Changing cancer-cell dispersal can substantially alter tumor growth rates without changing cellular doubling times or net growth rates.
- Some model predictions could be tested experimentally using new cell-labelling techniques.
6. Methods
The paper develops a three-dimensional spatial tumor model with mutation, driver, resistance, dispersal, growth, and death processes, then evaluates its robustness and biological scope.
- 6. Methods: The model represents tumors as non-overlapping three-dimensional microlesions on a lattice, with cells carrying passenger, driver, and resistance-associated alterations.
- 6. Methods: Mutations are assigned to daughter cells using Poisson-distributed counts, while drivers increase net growth through altered birth or death rates.
- 6. Methods: Short-range dispersal moves offspring to nearby positions and can produce conglomerates of cell balls separated by stromal cells and extracellular matrix.
- 6. Methods: The model uses initial birth rate b=ln 2≈0.69 days-1 and treatment rates b=0.35 days-1 and d=0.69 days-1, yielding regrowth of about 6 months.
- 6. Methods: The authors describe the model as deliberately oversimplified, while arguing that several assumptions can be experimentally justified or do not qualitatively affect heterogeneity.
- 6. Methods: Cell dispersal is modeled as movement or nutrient-access enhancement, while nearby microlesions may merge despite the model’s independent-ball assumption.
1 Computer algorithm
The algorithm simulates stochastic cell replication, mutation, dispersal, death, and time evolution in a spatial tumor, with alternative rules for replication, death, and mechanical displacement.
- 1 Computer algorithm: Each simulation step randomly selects a tumor cell, determines replication or death, and updates elapsed time.
- 1 Computer algorithm: Replication creates a neighboring daughter with probability proportional to the cell’s birth rate and available empty sites.
- 1 Computer algorithm: Daughter cells independently receive Poisson-distributed new mutations, with different mutation counts possible for the two daughters.
- 1 Computer algorithm: With probability M, an offspring migrates to the source ball’s nearby surface, forming a new microlesion.
- 1 Computer algorithm: Time advances by dt = 1/(bmaxN), producing exponential population growth approximated by N(t) ∼= exp((b−d)t).
- 1 Computer algorithm: The simulation starts from one selectively advantaged cell and does not model tumor initiation.
- 1 Computer algorithm: Alternative models make replication independent of empty-neighbor count, force replication with mechanical displacement, or scale death with available space.
2 Model parameters
The model specifies birth, death, driver-selection, mutation, and dispersal parameters for primary and metastatic lesions. Driver mutations alter net growth through either death or birth rates, while dispersal is explored over a broad range.
- Metastatic lesions use genotype-independent birth and death rates, with d/b = 0.5 . . . 0.9 and lower death rates representing more aggressive growth.
- Driver mutations can reduce death or increase birth, and model behavior is determined by the resulting difference between birth and death rates.
- Primary-tumor simulations use selective advantages s = 0.5% . . . 5%, with most main-text results using s = 1%.
- Driver and resistant mutation probabilities are set to γd = 4 · 10−5 and γr = 10−7, respectively.
- Dispersal probability M is explored from 10−7 to 10−2; M = 10−7 . . . 10−6 typically agrees reasonably with clinically observed metastatic growth times.
3 Tumor growth rate for the neutral model
Without dispersal, neutral tumors expand roughly as spheres through surface growth, producing power-law growth. Any positive dispersal changes the asymptotic behavior to exponential growth once the tumor is sufficiently large.
- Without driver mutations and dispersal, tumors become roughly spherical and grow approximately as N ∼ T^3.Only the outer surface expands because the interior remains near replication–death equilibrium.
- For M > 0, the model transitions to exponential growth when tumor size N becomes much larger than 1/M.
- The neutral model represents tumors as noninteracting cell balls whose size distribution evolves through dispersal and replication.The Master equation tracks balls of size n, with B(n) giving the rate of growth from n to n + 1.
- The analytic treatment uses the large-size approximation B(n) ∼ n^2/3 because B(n) is difficult to determine exactly for large, irregular cell configurations.Finite-size corrections can be strong, but are neglected in the asymptotic calculation.
- The exponential growth rate Bexp is the largest eigenvalue of the Master-equation operator, while p(n) is the stationary distribution of ball sizes.
- For small M, the asymptotic analysis predicts Bexp proportional to M^0.33, while simulations for M > 10−6 show slower scaling of approximately M^0.25...0.3.The slower simulated scaling is attributed to finite-size deviations from B(n) ∼ n^2/3.
4 Reseeding and long-range migration
The model can extend short-range dispersal to long-range reseeding, preserving whole-tumor growth predictions while changing the growth rate through the combined migration probability.
- Short-range dispersal produces exponential tumor growth with rate approximately proportional to M^1/3.M is the dispersal probability.
- Adding long-range reseeding with probability R increases the total-mass growth rate to approximately proportional to (M + R)^1/3.The reseeding model preserves predictions unrelated to the spatial distribution of genetic alterations.
5 Accumulation of driver mutations
Driver mutations accumulate approximately linearly in time because successive clones expand with similar spatial growth laws and arise at roughly equally spaced times.
- The number of drivers per cell increases approximately linearly in time, including in simulations of tumors up to 10^9 cells.This differs from exponential driver accumulation predicted for well-mixed, exponentially growing tumors.
- When successive drivers have the same small selective advantage, each new clone spreads at approximately the tumor’s expansion speed.The resulting expansion resembles a traveling front.
- Successive driver clones follow similar growth laws but begin at later, approximately equally spaced times, producing linear accumulation over time.The rate at which new drivers arise increases only slowly with the driver index.
6 Alternative simulation method: kinetic Monte Carlo (KMC) algorithm
The authors compare their efficient simulation algorithm with an exact kinetic Monte Carlo formulation to assess algorithmic sensitivity, while noting that neither stochastic approach is fully realistic.
- The simulation algorithm is not an exact kinetic Monte Carlo method because real cell division is not fully stochastic and exact stochastic simulation is slower.The authors identify computational speed and biological realism as the reasons for using the alternative algorithm.
- A kinetic Monte Carlo implementation produces only small differences from the main algorithm’s results.The comparison is shown in Figure 14.