Evolutionary dynamics of cancer in response to targeted combination therapy
- Harvard University, United States
- Institute of Science and Technology Austria, Austria
- Emmanuel College, United States
- Edinburgh University, United Kingdom
- Harvard College, United States
- Memorial Sloan-Kettering Cancer Center, United States
- Johns Hopkins University School of Medicine; The Sidney Kimmel Comprehensive Cancer Center at Johns Hopkins, United States
- Johns Hopkins Kimmel Cancer Center, United States
Figures
Figure 1
Variability in treatment response to monotherapy among six patients.
Patients were treated with the BRAF inhibitor vemurafenib. Patients P1 and P2 achieved a complete response. Patient P3 had stable disease. Patients P4, P5, and P6 had partial responses. The minimal …
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Figure 1—source data 1
Response to vemurafenib.
- https://doi.org/10.7554/eLife.00747.004
Figure 2
Tumor response to mono and dual therapy.
The tumor grows exponentially until a certain detection size, M, is reached, at which point treatment is initiated. The number of point mutations that could in principle confer resistance to …
Figure 3
Probability of tumor eradication for two-drug combination therapy.
A single mutation conferring cross-resistance to both drugs (n12 = 1) can prohibit any hope for a successful dual therapy. Solid curves show analytical results for dual therapy and dashed curve …
Figure 4
Treatment response dynamics to monotherapy and dual therapy in two patients.
(A) Depiction of all 18 detectable metastases in patient N1, who had a particularly heavy tumor burden (scale 1:4). (B) Simulated treatment of patient N1, comparing monotherapy with n = 50 …
Figure 5 with 1 supplement
Sequential vs simultaneous therapy with two drugs.
(A) If there is even a single mutation that confers cross-resistance to both drugs (n12 = 1), then sequential therapy will fail in all cases. In 73.7% of the cases, this failure is due to the …
Figure 5—figure supplement 1
Examples for the evolution of resistance during sequential therapy.
Two drugs are available for treatment where n1 = 50 and n2 = 50 point mutations confer resistance to each drug individually and one mutation confers resistance to both drugs simultaneously (n12 = …
Tables
Table 1
Probability of treatment failure for combination therapy in patients
| Patient | Primary tumor type | Number of metastases | Total tumor burden (number of cells) | Probability of treatment failure | ||
|---|---|---|---|---|---|---|
| Monotherapy | Dual therapy: n12 = 1 | Dual therapy: n12 = 0 | ||||
| N1 | Pancreas | 18 | 2.6 × 1011 | 1 | 1 | 0.283 |
| N2 | Colon | 25 | 2.3 × 1011 | 1 | 1 | 0.26 |
| N3 | Melanoma | 26 | 1.7 × 1011 | 1 | 1 | 0.203 |
| N4 | Melanoma | 30 | 1.4 × 1011 | 1 | 1 | 0.172 |
| N5 | Colon | 21 | 1.0 × 1011 | 1 | 1 | 0.128 |
| N6 | Melanoma | 8 | 9.8 × 1010 | 1 | 1 | 0.12 |
| N7 | Colon | 25 | 9.1 × 1010 | 1 | 1 | 0.112 |
| N8 | Pancreas | 8 | 7.4 × 1010 | 1 | 1 | 0.092 |
| N9 | Pancreas | 23 | 6.4 × 1010 | 1 | 1 | 0.08 |
| N10 | Pancreas | 5 | 5.5 × 1010 | 1 | 1 | 0.069 |
| N11 | Colon | 14 | 5.4 × 1010 | 1 | 1 | 0.068 |
| N12 | Rectal | 23 | 4.8 × 1010 | 1 | 1 | 0.061 |
| N13 | Melanoma | 9 | 4.1 × 1010 | 1 | 1 | 0.052 |
| N14 | Pancreas | 13 | 4.1 × 1010 | 1 | 1 | 0.051 |
| N15 | Pancreas | 8 | 3.3 × 1010 | 1 | 1 | 0.042 |
| N16 | Melanoma | 7 | 2.2 × 1010 | 1 | 1 | 0.028 |
| N17 | Melanoma | 10 | 2.1 × 1010 | 1 | 1 | 0.027 |
| N18 | Colon | 4 | 2.0 × 1010 | 1 | 1 | 0.026 |
| N19 | Melanoma | 9 | 1.8 × 1010 | 1 | 1 | 0.023 |
| N20 | Colon | 3 | 1.6 × 109 | 1 | 0.881 | 0.002 |
| N21 | Melanoma | 21 | 1.3 × 109 | 1 | 0.828 | 0.002 |
| N22 | Pancreas | 1 | 8.5 × 108 | 1 | 0.677 | 0.001 |
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For monotherapy, we assume that 50 point mutations (n = 50) can in principle confer resistance to the drug. With dual therapy, we assume that 50 point mutations can in principle confer resistance to each drug individually (n1 = n2 = 50). Two scenarios are modeled: in the first, there is one mutation that can in principle confer resistance to both drugs (i.e., cross-resistance, n12 = 1). In the other case, there are no possible mutations that can confer resistance to both drugs (n12 = 0). Parameter values: birth rate, b = 0.14, death rate, d = 0.13, death rate for sensitive cells during treatment, d′ = 0.17, point mutation rate u = 10−9.
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Colon: colonic adenocarcinoma; Rectal: rectal adenocarcinoma; Pancreas: pancreatic ductal adenocarcinoma.
Table 2
Simulation results for the probability of treatment failure when resistance is costly
| Dual therapy: | Number of cells | Birth rate | Probability of treatment failure | ||||
|---|---|---|---|---|---|---|---|
| n1 = n2 | n12 | c = 0% | c = 1% | c = 5% | c = 10% | ||
| 50 | 0 | 109 | 0.14 | 0.0 | 0.0 | 0.0 | 0.0 |
| 50 | 0 | 109 | 1 | 0.01 | 0.01 | 0.01 | 0.0 |
| 50 | 1 | 109 | 0.14 | 0.74 | 0.73 | 0.72 | 0.7 |
| 50 | 1 | 109 | 1 | 0.74 | 0.74 | 0.72 | 0.7 |
| 50 | 0 | 1011 | 0.14 | 0.12 | 0.11 | 0.08 | 0.06 |
| 50 | 0 | 1011 | 1 | 0.53 | 0.51 | 0.42 | 0.32 |
| 50 | 1 | 1011 | 0.14 | 1.0 | 1.0 | 1.0 | 1.0 |
| 50 | 1 | 1011 | 1 | 1.0 | 1.0 | 1.0 | 1.0 |
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Each resistance mutation reduces the net growth rate by a factor c via a decrease of the birth rate b. Parameter values are death rate, d = b − 0.01, death rate for sensitive cells during treatment, d’ = b + 0.03, point mutation rate, u = 10−9. The simulation results are averages over 106 runs per parameter combination.
Additional files
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Supplementary file 1
Mathematical proofs.
- https://doi.org/10.7554/eLife.00747.012
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Supplementary file 2
Lesion sizes of patients who failed conventional treatments.
- https://doi.org/10.7554/eLife.00747.013
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Supplementary file 3
Probability of combination therapy failure in patients.
- https://doi.org/10.7554/eLife.00747.014
