Modeling the Effect of Spatial Structure on Solid Tumor Evolution and Circulating Tumor DNA Composition

Abstract

Circulating tumor DNA (ctDNA) monitoring, while sufficiently advanced to reflect tumor evolution in real time and inform cancer diagnosis, treatment, and prognosis, mainly relies on DNA that originates from cell death via apoptosis or necrosis. In solid tumors, chemotherapy and immune infiltration can induce spatially variable rates of cell death, with the potential to bias and distort the clonal composition of ctDNA. Using a stochastic evolutionary model of boundary-driven growth, we study how elevated cell death on the edge of a tumor can simultaneously impact driver mutation accumulation and the representation of tumor clones and mutation detectability in ctDNA. We describe conditions in which invasive clones are over-represented in ctDNA, clonal diversity can appear elevated in the blood, and spatial bias in shedding can inflate subclonal variant allele frequencies (VAFs). Additionally, we find that tumors that are mostly quiescent can display similar biases but are far less detectable, and the extent of perceptible spatial bias strongly depends on sequence detection limits. Overall, we show that spatially structured shedding might cause liquid biopsies to provide highly biased profiles of tumor state. While this may enable more sensitive detection of expanding clones, it could also increase the risk of targeting a subclonal variant for treatment. Our results indicate that the effects and clinical consequences of spatially variable cell death on ctDNA composition present an important area for future work.

Keywords: ctDNA; spatial evolution; tumor DNA shedding; tumor evolution; tumor growth model.

Figure 1

(A) Illustration of the model. Tumors grow to a clinically detectable size (a 2D cross-section of a 3-billion-cell tumor) and are then partially exposed to a new environment, where the cells die with rate $d_2$. The growth rate in the new environment determines the invasive potential of a clone. If the death rate $d_2$ is higher than the initial birth rate, only clones with mutations increasing the growth rate to a positive number can grow in the new environment, so invasion is driver-dependent. Otherwise, it is driver-independent. Tumor growth can be proliferative or quiescent. In the former, cells divide when they have an empty neighbor on the lattice and die at a rate independent of their neighbors. In the latter, cells also divide when they have an empty neighbor on the lattice; however, cell death also requires empty neighbors. The shedding rate of DNA into the blood is assumed to be proportionate to the death rate. (B) Example trajectories, driver-dependent invasion. Trajectories of clone fractions and total population size for driver-dependent invasion, with 2D visualizations of the tumor at selected timepoints. Each color corresponds to a unique clone, also shown in the trajectory plot. (C) Example trajectories, driver-independent invasion. Trajectories of clone fractions and total population size for driver-independent invasion, with 2D visualizations of the tumor at selected timepoints. For both cases, $\mu =0.001,s=0.1,d_1=0.1,b=0.7$. For driver-dependent invasion, $d_2=0.9$. For driver-independent invasion, $d_2=0.69$.

Figure 2

Clone fraction differences between blood and tissue: (A–D) Each plot shows the results of 50 simulation runs, where each point represents the difference between clonal frequencies estimated from the blood versus those present in the tumor, for a single clone, with the color showing the age of the clone relative to the total simulation time. Tumors were grown from a single cell until reaching a 2D cross-section of a 10-billion-cell tumor. For all simulations, $\mu =0.001,s=0.1,d_1=0.1,b=0.7$. For driver-dependent invasion, $d_2=0.9$. For driver-independent invasion, $d_2=0.69$. The orange and blue lines show the average positive and negative clone fraction difference, respectively. Only clones comprising at least 10% of the tumor were included in the average. Shading is $\pm 1$ s.d. We show the same plots over normalized time in Supplementary Figure S2.

Figure 3

Discrepancies between blood and tissue clonal diversity: (A–D) The subplots show the inverse Simpson diversity index of the clonal frequencies in the blood and tissue for each clone in 50 simulated tumors. Timepoints are normalized by run and then binned and downsampled. Tumors were grown from a single cell until reaching a 2D cross-section of a 10-billion-cell tumor. For all simulations, $\mu =0.001,s=0.1,d_1=0.1,b=0.7$. For driver-dependent regrowth, $d_2=0.9$. For driver-independent regrowth, $d_2=0.69$. Shading represents $\pm 1$ s.d. The figure shows results for proliferative tumors only. For all scenarios, see Supplementary Figure S4.

Figure 4

Influence of spatial bias on limits of detection. (A) Plots of the number of detectable driver mutations starting from the point of relapse for minimum detection frequencies of 1 × 10⁻³ and 1 × 10⁻² for proliferative and quiescent tumors relapsing at ∼10⁸ and ∼10⁹ cells. Mutations were detectable if the estimated VAF exceeded the detection limit. VAFs were estimated based on a tumor fraction of 1% for a 3-billion-cell tumor with death rate of 0.1 (see Methods). (B) Percent change in number of detectable drivers when the VAFs in (A) are compared to VAFs computed assuming the tumor sheds all clones at the same rate for the same detection limits see Methods). (C–F) Scatter plots of mean spatially biased VAFs (green) and unbiased VAFs (blue) at the size where the average spatial bias over all replicates is maximal (marked with the corresponding letter in (B)) Each plot shows all mutations for 50 replicates of the corresponding simulation scenario. The x-axis is the mean distance of the mutation from the tumor’s center. Black points are clonal mutations, which show perfect overlap between the blood and tissue. The vertical line marks the end of the sanctuary region.