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.
DOI: http://dx.doi.org/10.7554/eLife.00747.003
10.7554/eLife.00747.004Figure 1—source data 1.Response to vemurafenib.DOI: http://dx.doi.org/10.7554/eLife.00747.004

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 = 10⁻⁹ 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.
DOI: http://dx.doi.org/10.7554/eLife.00747.005

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 . 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.
DOI: http://dx.doi.org/10.7554/eLife.00747.006

(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.
DOI: http://dx.doi.org/10.7554/eLife.00747.008

(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 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.
DOI: http://dx.doi.org/10.7554/eLife.00747.009

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 . 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 . 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.
DOI: http://dx.doi.org/10.7554/eLife.00747.010