Fig 1.
Sample realisation of the model obtained by simulations.
The primary tumor grows according to a deterministic exponential function n(t)—depicted by the blue line. It initiates distant metastases at rate νn(t), and each of them grows as an independent branching process (only the first five are plotted). The first time τ that any of these metastases reaches a minimal detectable size M is defined as the time to cancer relapse. Also, the primary tumor is surgically removed at a given time T, when it is made of N = n(T) cells. In the realisation shown, the third established metastases (green curve) is the first to reach detectable size, and hence determines the time to cancer relapse τ. Based on clinical data (summarized in Table 1), we estimated model parameters (summarized in Table 2), and here we use those for colorectal cancer, with N = 2 × 1011. Note that a similar illustration for metastasis formation appears in [30].
Fig 2.
Relapse time densities computed from Eq 4 for logistic and exponential primary growths and ν = 10−10, 10−11, 10−12, 10−13, 10−14 cells/day from left to right.
Using parameter estimates for colorectal cancer (see Table 2), the logistic densities (dashed lines) converge to the corresponding exponential ones as the initiation rate increases. Furthermore, in the exponential case and for all the above values of ν, the densities derived from Eq 4 and their approximation obtained from Eq 6 are indistinguishable.
Fig 3.
Relapse time distribution for an exponentially growing primary tumor using parameter estimates for colorectal cancer (see Table 2).
Symbols represent simulation results for a deterministic (diamonds) or a stochastic primary growth (circles, see Discussion at the end of the parameter estimation section), while solid lines correspond to the theory in the small ν—large M limit. On the left, each starred dot denotes the mean of 1000 simulations, while lines represent the theoretical expectation given by Eq 7. These match the simulated means well for values of ν = 10−6 or less. On the right, the relapse time densities derived from Eq 14 yield a good approximation of the simulated data (10000 simulations per curve) for M = 100 or greater for both deterministic and stochastic primary growth.
Fig 4.
Relapse time densities fτ(t∣KT ≥ 1) conditioned on at least one metastasis initiated by the time of resection T.
For different values of T, marked with ticks of corresponding colors, these densities are computed by differentiating Eq 8. As T becomes larger, the probability of metastases being established before resection (see Eq 1) increases and the conditional relapse time densities converge to the red limit one. Here we have used parameter estimates for colorectal cancer (see Table 2), n(t) = eδt and 7 equally spaced resection times between 0.25y and 16.15y. The curves for T > 15y look identical to the limit density.
Table 1.
Typical ranges of volume doubling times for the primary tumor (DTpt) and metastasis (DTm), tumor potential doubling time (Tpot) and tumor diameter at resection (dpt) for breast, colorectal, headneck, lung and prostate cancer.
Table 2.
Parameter estimates for the primary net growth rate δ, the metastatic net growth rate λ, the initiation rate ν, the extinction probability q, the primary tumor size at resection N and the minimal detectable size M.
Fig 5.
Probability of extant metastases, P(KT ≥ 1), dashed curve computed from Eq 1, and synchronous metastases, P(ST ≥ 1), solid curve computed from Eq 4.
These probabilities are plotted as functions of the resection time T for five different cancer types. The primary tumor size at resection is N = eδT and thus depends on the primary net growth rate. These resection sizes are discussed in Table 3. For each cancer type, the shaded areas highlight resection time intervals leading to a probability higher than 85% of established and all undetectable metastases. Using the parameter estimates from Table 2, the widths of these intervals are 3.41, 3.17, 1.92, 0.94, 1.19 years for breast, colorectal, headneck, lung and prostate cancer respectively.
Table 3.
Resection sizes of the primary tumor which yield a 1% and 99% probability of synchronous metastases, respectively.
For each cancer type considered, these sizes are computed with the parameter values in Table 2 and expressed both in terms of number of cells, N, and tumor diameter, d.
Fig 6.
Probability of established and all metachronous metastases—P(UT), as given by Eq 12—plotted as a function of T and δ (left panel) and of T and ν (right panel).
The parameter estimates used are those for colorectal cancer reported in Table 2. The plots show that the width of the high-risk interval—the range of resection times such that P(UT) is high—stays roughly constant for most parameter values. This width (about 3 years) shrinks only for metastases growing significantly faster than the primary tumor that initiated them.
Fig 7.
Expected relapse time measured from resection, conditioned on extant but all undetectable metastases (blue curve).
The dashed line and the light blue shaded area show P(UT) and how spread is the conditional relapse time distribution, respectively. The parameter estimates used are those for colorectal cancer reported in Table 2. For resection times close to zero this conditional expectation coincides with that of the Gumbel distribution given by Eq 14, at about 5 years. As T starts to increase E[τ − T∣UT] reflects the convergence highlighted for Fig 4, first slightly decreasing and then staying constant around 4.4 years. Finally, when the resection time falls into the high-risk window, the expected relapse time drops to zero. This suggests that the bigger the primary tumor size is at resection, the faster relapse will occur.
Table 4.
Typical ranges of P(ST ≥ 1) and E[τ − T∣UT], predicted value from the model and literature references for each cancer type.
Fig 8.
Disease-free curves for different resection times.
The earlier the primary tumor is resected the higher is the probability that no metastases will arise, or cure probability, represented by the value of the final plateaus. The resection times are chosen so that P(KT = 0) = 0.75, 0.6, 0.45, 0.3, 0.15, 0.001 respectively. With the parameter estimates for colorectal cancer (see Table 2) these times range from 12.28 to 14.48 years, corresponding to sizes between 5.12 × 107 and 1.23 × 109 cells (diameter 0.46 − 1.33cm), respectively.
Fig 9.
Probability of the first metastasis being initiated during surgery delay.
This probability P(KT+ΔT ≥ 1, KT = 0)—where T is the set time of resection and ΔT the surgery delay—is plotted as a function of the resection size N = eδT (x-axis) and of the delay ΔT (y-axis). With the parameter estimates for colorectal cancer (see Table 2) we see that if a primary tumor is resected at a critical size (around 2 × 108 cells, diameter ≈ 0.725cm), surgery delays of 2-3 months can decrease the cure probability of more than 10%.