Research ArticleCANCER

# A mathematical model of ctDNA shedding predicts tumor detection size

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Vol. 6, no. 50, eabc4308

### Figures

• Fig. 1 Evolutionary dynamics of ctDNA shed by a growing cancer.(A) Mathematical model of cancer evolution and ctDNA shedding. Tumor cells divide with birth rate b and die with death rate d per day. During cell apoptosis, cells shed ctDNA into the bloodstream with probability qd. ctDNA is eliminated from the bloodstream with rate ε per day according to the half-life time of ctDNA, t1/2 = 30 min. (B) hGE per plasma ml correlate with tumor volume with a slope of 0.9997 in 176 patients with lung cancer (R2 = 0.32; P = 2.6 × 10−16). Red shaded region depicts 95% confidence interval (CI). Linear regression predicts 0.21 hGE per plasma ml for a tumor volume of 1 cm3, leading to a shedding probability of qd = 1.4 × 10−4 hGE per cell death (Materials and Methods). (C) A tumor starts to grow at time zero with a growth rate of r = b − d = 0.4% (b = 0.14, d = 0.136), typical for early-stage lung cancers (tumor doubling time of 181 days). Tumor sheds ctDNA into the bloodstream according to the product of the cell death rate d and the shedding probability per cell death qd. (D) Distribution of ctDNA hGE in the entire bloodstream at the time [purple dashed line in (C)] when the tumor reaches 1 billion cells [≈1 cm3; gray dashed line in (C)], leading to a mean of 0.21 hGE per plasma ml and a tumor fraction of 0.022% at a mean DNA concentration of 6.3 ng per plasma ml. (E) Probability distribution of ctDNA mutant fragments present in a liquid biopsy of 15 ml of blood when the tumor reaches 1 billion cells (assuming that the covered somatic heterozygous mutation is present in all cancer cells).
• Fig. 2 Tumor growth rate and cell turnover strongly affect the amount of ctDNA.(A to C) ctDNA hGE present in the entire bloodstream when a lung tumor reaches a given size. Bars illustrate distribution of hGE of ctDNA based on 10,000 simulation realizations. Full lines illustrate asymptotic results (Eq. 1; note S1) and perfectly agree with simulation results (bars). (D to F) ctDNA mutant fragments present in a 15-ml blood sample. (A) Tumors with half the cells (0.5 versus 1 billion cells) lead to half the ctDNA level in the bloodstream (birth rate b = 0.14 per cell per day and death rate d = 0.136 per cell per day). (B) Assuming that the growth rate proportionally changes the birth and death rates, fast-growing tumors lead to a lower level of ctDNA when they reach a size of 1 billion cells (≈1 cm3) because fewer cell deaths decrease the amount of released ctDNA (slow growth: b = 0.14 and d = 0.139; fast growth: b = 0.1595 and d = 0.1195). (C) Higher cell turnover rates lead to a higher level of ctDNA at a given tumor size (1 billion cells) compared with lower cell turnover rates because of the increased rate of cells undergoing apoptosis (if the underlying shedding probability per cell death is the same; high cell turnover: b = 0.25 and d = 0.246; low cell turnover: b = 0.1 and d = 0.096). Parameter values: ctDNA half-life time t1/2 = 30 min; ctDNA shedding probability per cell death qd = 1.4 × 10−4 hGE.
• Fig. 3 Expected tumor relapse detection size and lead time compared with current clinical relapse detection.(A) ROC curves for tumors with 100 million cells (≈0.1 cm3; blue line), 200 million cells (≈0.2 cm3; orange line), or 500 million cells (≈0.5 cm3; red line) when 20 clonal tumor-specific mutations are tracked for relapse detection and one of these 20 mutations needs to be called for a positive test. (B to G) For better comparability, positive detection test thresholds were set such that if the test is repeated multiple times, a combined FPR of 5% is obtained over all tests per year. (B) Expected tumor detection size distributions for monthly and quarterly repeated relapse detection tests (sequencing panel covers 20 mutations). Ø indicates diameter of spherical tumor. (C) Expected lead time distributions compared with imaging-based approaches applied at the same frequency with a detection limit of 1 cm3 for monthly and quarterly repeated relapse detection tests (sequencing panel covers 20 clonal mutations). (D to G) Median tumor detection sizes over the number of clonal mutations covered by the sequencing panel (D), the testing frequency (E), the sequencing error rate (F), and the sampled blood amount (G). Parameter values (if not differently specified): birth rate b = 0.14 per cell per day; death rate d = 0.13 per cell per day; ctDNA half-life time t1/2 = 30 min; ctDNA shedding probability per cell death qd = 1.4 × 10−4 hGE; sequencing efficiency of 50%; sequencing error rate per base pair 10−5; 15 ml of blood sampled per test; DNA median concentration 5.2 ng per plasma ml. In all scenarios, at least one mutation needs to be called for a positive test. ROC, receiver operating characteristic; AUC, area under the curve.
• Fig. 4 Expected tumor detection size and lead time distributions for screening with different sequencing panels.(A) Tumor detection size distribution for annually repeated virtual screening tests with a 2000–base pair sequencing panel covering one somatic mutation per lung cancer. (B and E) Lead time distributions for annually repeated virtual screening tests compared with current clinical diagnosis times calculated from detection sizes in the SEER database, assuming an early-stage lung cancer growth rate of r = 0.4% per day. (C and F) PPV (positive predictive value) for a cancer early detection test in populations with different lung cancer incidence rates. (D) Tumor detection size distribution for annually repeated virtual screening tests with a 300,000–base pair sequencing panel covering 10 somatic mutations per lung cancer of a smoker (table S3). Parameter values typical for early-stage lung cancer: birth rate b = 0.14 per cell per day; death rate d = 0.136 per cell per day; ctDNA half-life time t1/2 = 30 min; ctDNA shedding probability qd = 1.4 × 10−4 hGE per cell death; sequencing error rate per base pair 10−5; sequencing efficiency 50%; 15 ml of blood sampled per test; plasma normal DNA median concentration 5.2 ng per ml; 99% test specificity.

# A mathematical model of ctDNA shedding predicts tumor detection size

Stefano Avanzini, David M. Kurtz, Jacob J. Chabon, Everett J. Moding, Sharon Seiko Hori, Sanjiv Sam Gambhir, Ash A. Alizadeh, Maximilian Diehn, Johannes G. Reiter

The PDF file includes:

• Materials and Methods
• Figs. S1 to S13
• Tables S1 to S3
• Note S1
• References

Other Supplementary Material for this manuscript includes the following:

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