Predicting Relapse From Two-Clone Tumor Dynamics

Why Predict Relapse From Both Clones?

A tumor can contain drug-sensitive cells that decline while treatment is active and resistant cells that persist or grow. A threshold-based relapse prediction must therefore account for the sum of the clone burdens, not just the clone that is growing fastest or the clone that is most abundant at one time. The prediction depends on each clone’s burden at the start of the forecast and its net growth rate during the forecast.

The calculation below is a deterministic model over a specified time window. Its result is the first modeled time that total burden reaches a chosen threshold; it is not, by itself, a clinical definition of relapse or a guarantee that a patient will relapse at that time.

Two-Clone Model and Relapse Time

Let \(S(t)\) and \(R(t)\) denote the sensitive- and resistant-clone burdens at time \(t\). Use the same burden unit for both clones, such as millions of cells. At \(t=0\), their explicit initial burdens are \(S_0\) and \(R_0\). Assume that each clone has a constant net growth rate over the prediction window: \(g_S\) for the sensitive clone and \(g_R\) for the resistant clone. Rates have units of inverse time, such as day\(^{-1}\); a negative rate represents net decline.

This model follows from the clone-specific differential equations \(dS/dt=g_SS\) and \(dR/dt=g_RR\), with initial conditions \(S(0)=S_0\) and \(R(0)=R_0\). It assumes constant rates and no exchange between clones during the forecast. The rates are net rates: if proliferation and drug-mediated killing are represented separately in a more detailed model, their net effect must be reflected in \(g_S\) and \(g_R\) for this calculation.

The threshold and clone burdens must use the same measurement scale. The two initial burdens must refer to the same time and tumor, and the rates must apply to the same forecast conditions. For example, a rate estimated under one treatment condition cannot be used without justification for a forecast under a different condition.

When Does the Total Burden Rise?

The clone trajectories are exponential, but their sum is generally not a single exponential. Differentiating the total burden gives the instantaneous direction of change:

If both rates are nonnegative and at least one clone with positive burden has a positive rate, total burden increases. If both rates are nonpositive, total burden does not increase, so a threshold above the initial total cannot be reached. With one negative and one positive rate, the total may first fall and later rise, or first rise and later fall. The signs of the two rates alone do not establish the direction of the total at every time.

For the common mixed case \(g_S<0<g_R\), a turning point occurs when the declining sensitive contribution to the derivative balances the growing resistant contribution. Setting the derivative to zero and solving gives:

This time is relevant only if it is nonnegative. The derivative changes from negative to positive there, because the ratio of the positive derivative contribution to the magnitude of the negative one grows as \(e^{(g_R-g_S)t}\). Thus, when \(t_{\min}>0\), the total has a minimum. A threshold search should not assume the total rises from \(t=0\): the first upward crossing may occur only after that minimum.

Finding a Crossing Numerically

Except in special cases, the equation \(S_0e^{g_St}+R_0e^{g_Rt}=B_{\mathrm{rel}}\) has no simple closed-form solution for \(t\). Define \(f(t)=B(t)-B_{\mathrm{rel}}\). A numerical root is appropriate once a bracket is found: times \(t_L\) and \(t_U\) for which \(f(t_L)<0\) and \(f(t_U)\geq0\) locate an upward crossing in that interval. Bisection repeatedly halves the bracket and preserves the sign change; it is reliable for a continuous function on the bracket. A numerical solver should still be used with care if there may be more than one crossing, since a bracket for a later crossing does not establish that it is the first one.

A practical approach is to evaluate the model at \(t=0\), inspect the rate signs and derivative, and then increase time in sensible increments until the total crosses the threshold. If mixed rates give an initial decline, evaluate around the possible minimum before looking for the upward crossing. Once a bracket is obtained, refine it with bisection. Newton’s method can converge faster when started near a root, but it uses the derivative and may leave the relevant interval or converge to a different crossing.

Assumptions, Diagnostics, and Limits

Before interpreting a calculated time, verify that the inputs and model match the intended prediction:

• Units and scale: \(S_0\), \(R_0\), and \(B_{\mathrm{rel}}\) use the same burden units, and each growth rate is in inverse units of the time axis.
• Starting condition: if \(B(0)\geq B_{\mathrm{rel}}\), the threshold is already met. A positive-time relapse prediction from below threshold is not appropriate without redefining the forecast start.
• Crossing direction: check that the reported root is an upward crossing when relapse is defined as reaching the threshold from below. A root that merely touches the threshold at a turning point may not represent an upward crossing.
• Numerical bracket: confirm that the computed burden is below threshold at the lower endpoint and at or above it at the upper endpoint. Recalculate the total from both clone contributions at the reported time.
• Rate applicability: check that the rates are plausible and apply over the entire forecast interval. A constant rate can be a poor approximation if treatment, tumor environment, or clone behavior changes.

A common mistake is to calculate the resistant clone’s threshold time and call it the tumor relapse time. That ignores the sensitive burden, which may still contribute substantially to total burden. Another error is to average the two clone-specific rates and apply that average to the initial total. In general, \(S_0e^{g_St}+R_0e^{g_Rt}\) is not equal to \((S_0+R_0)e^{\bar g t}\) for a fixed average rate \(\bar g\). Such a one-exponential approximation can misstate both the threshold time and whether total burden initially rises or falls.

The two-clone calculation is transparent and useful when initial burdens and rates are specified, but it treats those inputs as fixed and the trajectories as deterministic. It does not represent uncertainty in measurements or rates, stochastic loss of a small clone, switching between phenotypes, new mutation, interactions between clones, or capacity-limited tumor growth. A threshold time should therefore be interpreted as a conditional model prediction: it follows if the specified initial burdens, rates, threshold, and assumptions hold over the forecast window.