1. Evolutionary Biology
  2. Cancer Biology
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Evolutionary dynamics of cancer in response to targeted combination therapy

  1. Ivana Bozic
  2. Johannes G Reiter
  3. Benjamin Allen
  4. Tibor Antal
  5. Krishnendu Chatterjee
  6. Preya Shah
  7. Yo Sup Moon
  8. Amin Yaqubie
  9. Nicole Kelly
  10. Dung T Le
  11. Evan J Lipson
  12. Paul B Chapman
  13. Luis A Diaz Jr
  14. Bert Vogelstein
  15. Martin A Nowak  Is a corresponding author
  1. Harvard University, United States
  2. Institute of Science and Technology Austria, Austria
  3. Emmanuel College, United States
  4. Edinburgh University, United Kingdom
  5. Harvard College, United States
  6. Memorial Sloan-Kettering Cancer Center, United States
  7. Johns Hopkins University School of Medicine; The Sidney Kimmel Comprehensive Cancer Center at Johns Hopkins, United States
  8. Johns Hopkins Kimmel Cancer Center, United States
Research Article
Cite this article as: eLife 2013;2:e00747 doi: 10.7554/eLife.00747
5 figures, 2 tables and 3 additional files

Figures

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 detection size (indicated by discontinuous red line) was assumed to be ≈63 × 106 cells.

https://doi.org/10.7554/eLife.00747.003
Figure 1—source data 1

Response to vemurafenib.

https://doi.org/10.7554/eLife.00747.004
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 monotherapy is n = 50. For dual therapy, the number of point mutations that could confer resistance to drugs 1 and 2 separately is given by n1 = 50 and n2 = 50. The number of point mutations that could confer resistance to both drugs simultaneously is given by n12. The point mutation rate was assumed to be u = 109 and the rate of cell division b = 0.14 per day and is unaffected by treatment. The rate of cell death before treatment is d = 0.13 per day; it is increased to d’ for sensitive cells during treatment. (A)–(C) For clinically detectable sizes (M = 1010, 109, 108, depending on the location of the tumors and the detection methods used), monotherapy leads to a temporary shrinkage of the tumor but is always followed by tumor regrowth. (D) Due to stochastic fluctuations the few resistant cells present at the start of treatment go extinct and the lesion is eradicated. (E) Treatment leads to a temporary shrinkage of the tumor followed by regrowth. (F) The tumor decreases slowly in response to dual therapy, but resistant cells eventually evolve and cause treatment failure.

https://doi.org/10.7554/eLife.00747.005
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 shows analytical results for a typical monotherapy, both are calculated using equation (1). Markers (square, triangle, circle, diamond) indicate simulation results (averages of 106 runs). 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.

https://doi.org/10.7554/eLife.00747.006
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 resistance mutations and dual therapy with n1 = n2 = 50 resistance mutations to the individual drugs and one (n12 = 1) or no (n12 = 0) cross-resistance mutations to both drugs. (C) Depiction of all 14 detectable metastases in patient N11, who had a more typical tumor burden (scale 1:4). (D) Simulated treatment of patient N11. Parameter values for simulations in (B) and (D): 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.

https://doi.org/10.7554/eLife.00747.008
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 exponential growth of fully resistant cells that were present at the start of treatment. In the remaining 26.3% of cases, the failure is due to resistance mutations that developed during therapy with the first drug. (B) With simultaneous therapy, 26.3% of patients can be cured under the same circumstances. In the remaining patients (73.7%), cross-resistant mutations existed prior to the therapy and their expansive growth will ensure treatment failure whether treatment is simultaneous or sequential (see Figure 5—figure supplement 1 for further details). (C) and (D) If the two drugs have no resistance mutations in common (n12 = 0), then simultaneous therapy is successful with a probability of 99.9% while sequential therapy still fails in all cases.

https://doi.org/10.7554/eLife.00747.009
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 = 1). (A) A typical example of a tumor relapsing in the second wave of panel (A) in Figure 5. The few fully resistant cells go extinct due to stochastic fluctuations at the start of treatment. The cells resistant only to drug 1 produce cross-resistant cells during the treatment with the first drug. The cells resistant to both drugs received sequentially two mutations. (B) A typical example of a tumor relapsing in the first wave of panel (A) in Figure 5. The fully resistant cells are already present at the start of treatment. These cells received the cross-resistance mutation and are therefore immediately resistant to both drugs. The exponential growth of these fully resistant cells cause the relapse; their growth is unaffected by whether treatment is simultaneous or sequential. 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, detection size of tumor (for start of treatment and relapse) M = N = 109 cells.

https://doi.org/10.7554/eLife.00747.010

Tables

Table 1

Probability of treatment failure for combination therapy in patients

https://doi.org/10.7554/eLife.00747.007
PatientPrimary tumor typeNumber of metastasesTotal tumor burden (number of cells)Probability of treatment failure
MonotherapyDual therapy: n12 = 1Dual therapy: n12 = 0
N1Pancreas182.6 × 1011110.283
N2Colon252.3 × 1011110.26
N3Melanoma261.7 × 1011110.203
N4Melanoma301.4 × 1011110.172
N5Colon211.0 × 1011110.128
N6Melanoma89.8 × 1010110.12
N7Colon259.1 × 1010110.112
N8Pancreas87.4 × 1010110.092
N9Pancreas236.4 × 1010110.08
N10Pancreas55.5 × 1010110.069
N11Colon145.4 × 1010110.068
N12Rectal234.8 × 1010110.061
N13Melanoma94.1 × 1010110.052
N14Pancreas134.1 × 1010110.051
N15Pancreas83.3 × 1010110.042
N16Melanoma72.2 × 1010110.028
N17Melanoma102.1 × 1010110.027
N18Colon42.0 × 1010110.026
N19Melanoma91.8 × 1010110.023
N20Colon31.6 × 10910.8810.002
N21Melanoma211.3 × 10910.8280.002
N22Pancreas18.5 × 10810.6770.001
  1. 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.

  2. Colon: colonic adenocarcinoma; Rectal: rectal adenocarcinoma; Pancreas: pancreatic ductal adenocarcinoma.

Table 2

Simulation results for the probability of treatment failure when resistance is costly

https://doi.org/10.7554/eLife.00747.011
Dual therapy:Number of cellsBirth rateProbability of treatment failure
n1 = n2n12c = 0%c = 1%c = 5%c = 10%
5001090.140.00.00.00.0
50010910.010.010.010.0
5011090.140.740.730.720.7
50110910.740.740.720.7
50010110.140.120.110.080.06
500101110.530.510.420.32
50110110.141.01.01.01.0
501101111.01.01.01.0
  1. 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

Supplementary file 1

Mathematical proofs.

https://doi.org/10.7554/eLife.00747.012
Supplementary file 2

Lesion sizes of patients who failed conventional treatments.

https://doi.org/10.7554/eLife.00747.013
Supplementary file 3

Probability of combination therapy failure in patients.

https://doi.org/10.7554/eLife.00747.014

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