Why the Timing of Rebound Matters
A tumor that grows again after an initial response presents a mechanistic question, not just a question of whether treatment “worked.” Continued growth during treatment may indicate inadequate net tumor control. Regrowth during ongoing treatment after an initial decline raises concern about a resistant population. Growth that becomes apparent mainly after treatment is removed, followed by renewed sensitivity on rechallenge, is consistent with drug-tolerant persister cells. These patterns guide investigation, but none proves a mechanism by itself.
The diagnostic signal is the sequence of tumor burden across treatment, withdrawal, and, where available, retreatment. A single measurement at rebound cannot reveal whether growth began during therapy or after drug removal. Even a detailed volume trajectory measures the aggregate tumor, not the phenotypic state of each cell. Mechanistic interpretation therefore depends on the treatment schedule, adequate drug exposure, measurement quality, and corroborating biological evidence.
A Minimal Trajectory Framework
Let \(V(t)\) denote measured tumor volume at time \(t\), and let \(C(t)\) denote drug concentration or, more generally, the treatment exposure driving the response. A simple exponential description over a short interval is \(dV/dt=rV\), where \(r\) is the interval’s net growth rate. If \(V\) changes from \(V_1\) to \(V_2\) over \(\Delta t\), the estimated rate is \(r=\ln(V_2/V_1)/\Delta t\). Negative \(r\) indicates a decline in volume; positive \(r\) indicates growth. This is an interval summary, not a claim that real tumors follow exponential growth indefinitely.
A useful conceptual model separates sensitive cells \(S\), resistant cells \(R\), and persister cells \(P\), with total burden approximated by \(V=S+R+P\). During treatment, a sensitive population might follow \(dS/dt=[g_S-k_S(C)]S\), where \(g_S\) is its untreated growth rate and \(k_S(C)\) is the drug-associated loss rate. A resistant population may have a smaller drug effect \(k_R(C)\). Persister cells may have low net growth while drug is present and resume growth after withdrawal. Transitions between states can also occur. These equations are conceptual: volume data alone usually do not provide enough information to estimate all rates and transitions.
The simplifying assumptions behind interval-rate comparisons should be explicit: volume is measured consistently and is a usable proxy for burden; the interval is short enough that a single average rate is informative; treatment timing and exposure are known; and large changes in measurement method or tumor composition do not dominate the observed change. If tumors are measured with error, several observations within each treatment phase are more informative than a slope based on only two measurements.
Worked Example: Tumor Growth Despite Treatment
In a hypothetical study, a tumor measures 100 mm3 at treatment initiation and 130 mm3 after 14 days of continuous dosing. A separate untreated reference group has an estimated growth rate of 0.045 day−1. Assume measurements are comparable, exposure is confirmed, and the treatment interval can reasonably be summarized by an average exponential rate.
The treated tumor’s interval rate is
The tumor grew during treatment, so the observed net rate was positive. Compared with the untreated reference rate, the net growth rate was lower by \(0.045-0.01874=0.02626\ \mathrm{day}^{-1}\), a reduction of about \(0.02626/0.045=58.4\%\). This comparison supports partial growth suppression, not tumor regression and not a direct estimate of the fraction of cells killed. Inadequate net control is a reasonable description of this trajectory. It does not establish whether the cause is insufficient exposure, weak drug effect, or an insensitive cell population.
What Treatment and Withdrawal Patterns Can Suggest
The first diagnostic distinction is whether rebound begins while drug exposure continues or only after exposure falls. If the tumor never declines on treatment, or grows from the outset, inadequate net effect is a leading interpretation. Before proposing a biological explanation, check dosing records, exposure, adherence, measurement timing, and whether the treatment is expected to reduce burden at the observed exposure. A positive tumor-volume rate does not by itself mean the drug has no effect: the rate may still be lower than the untreated rate, as in the example above.
An initial decline followed by growth during continued adequate exposure suggests that the balance of tumor populations or their drug response may have changed. A population that grows under the same exposure that previously produced decline is consistent with resistance. “Acquired” should be used cautiously: resistance can emerge through a new change during treatment, but selection of a rare resistant population that was already present can produce a similar volume pattern. Lineage or molecular evidence is needed to distinguish those histories reliably.
A different pattern is a marked initial decline, low or nearly stable burden while treatment continues, and growth after treatment is withdrawn. If a subsequent course of the same drug again produces substantial decline at adequate exposure, reversible drug tolerance or persister-cell regrowth becomes more plausible. The inference is stronger if viable cells collected during treatment show a tolerant state and later regain sensitivity. Withdrawal-related growth alone is not specific: ordinary residual sensitive cells can grow when drug pressure is removed, and rebound may also reflect pharmacokinetic washout or changes in the tumor environment.
Worked Example: Regrowth During Continued Treatment
In a hypothetical experiment, tumor volume falls from 100 mm3 to 50 mm3 during the first 10 days of dosing, then rises to 100 mm3 by day 30 while dosing continues. Assume exposure is verified throughout and measurement procedures are unchanged.
The early rate is
The later rate is
The sign change—from decline to growth under continued treatment—is a warning that the initial response no longer describes the whole trajectory. It is consistent with resistance or a changing mixture of cell states, but does not identify whether resistance was newly acquired or already present at low frequency. Suppose, additionally, that after a 14-day interruption the tumor is 100 mm3, and a 10-day retreatment reduces it only to 95 mm3. The retreatment rate is \(\ln(95/100)/10=-0.00513\ \mathrm{day}^{-1}\), a small decline compared with the initial response. This supports reduced responsiveness, provided exposure is again adequate; it still does not establish the cellular cause.
A Justification for Using Aggregate Slopes Carefully
Even if each subpopulation has a simple exponential rate over an interval, the total tumor need not have a constant rate. For two populations, let \(V=S+R\), with \(dS/dt=r_SS\) and \(dR/dt=r_RR\). Differentiating the total and dividing by total burden gives
Thus, the observed instantaneous rate is a burden-weighted average of the subpopulation rates. If treatment makes \(r_S<0\) but a resistant population has \(r_R>0\), total tumor volume can first decline and later rise as the fraction \(f_R\) increases. For example, with \(r_S=-0.10\) day−1, \(r_R=0.04\) day−1, and \(f_R=0.10\), the aggregate rate is \(0.90(-0.10)+0.10(0.04)=-0.086\) day−1. If \(f_R\) later reaches 0.80, the rate becomes \(0.20(-0.10)+0.80(0.04)=0.012\) day−1. The tumor can therefore switch from declining to growing without any change in the sensitive-cell rate. This explains why a rebound trajectory is informative about changing aggregate behavior but does not, by itself, reveal the underlying population fractions or prove resistance.
Worked Example: Rebound After Withdrawal With Renewed Response
In a hypothetical study, a tumor declines from 100 mm3 to 25 mm3 over 10 days of treatment, remains approximately stable at 24 mm3 through day 30 on treatment, and grows to 48 mm3 during the 14 days after withdrawal. A subsequent 10-day treatment course reduces it from 48 mm3 to 12 mm3. Assume comparable measurements and adequate exposure during both treatment courses.
The initial on-treatment rate is \(\ln(25/100)/10=\ln(0.25)/10=-0.1386\ \mathrm{day}^{-1}\). The post-withdrawal rate is \(\ln(48/24)/14=\ln(2)/14=0.04951\ \mathrm{day}^{-1}\). On retreatment, the rate is \(\ln(12/48)/10=\ln(0.25)/10=-0.1386\ \mathrm{day}^{-1}\). The tumor declined during both treatment periods, stayed low during continued dosing, and grew after withdrawal. This sequence is consistent with a reversible tolerant reservoir, including persister-cell regrowth, rather than stable resistance to the tested drug. It is not proof: a volume trajectory cannot establish that the cells that regrew were persisters or that their sensitivity changed reversibly.
Diagnostics, Alternatives, and Limits
A practical diagnosis compares phases rather than labeling a single rising measurement. Plot tumor burden against time with dosing and withdrawal periods marked. Inspect for an initial response, a change in slope during continued treatment, a delayed rise after withdrawal, and the response to rechallenge. Calculate interval rates only over periods with interpretable exposure and measurement conditions. If the trajectory is irregular, report the observed pattern rather than forcing one exponential rate across it.
Useful supporting measurements include drug concentration or target engagement, viability or pathology assessments, and paired samples taken before treatment, during response, and at rebound. A persistent change in drug sensitivity in cells recovered at rebound supports resistance; a return of sensitivity after a drug-free interval supports reversible tolerance. Clonal or lineage information can help distinguish emergence of a new resistant state from selection of a pre-existing clone. Such assays also have limitations: a biopsy samples only part of a heterogeneous tumor, and ex vivo behavior may not reproduce the in vivo environment.
| Observed pattern | Interpretation it supports | Key check |
|---|---|---|
| No decline, or growth from treatment start | Inadequate net tumor control | Verify exposure, expected drug activity, and measurement consistency |
| Initial decline, then growth under continued treatment | Resistance or a changing tumor-cell mixture | Assess exposure at rebound; distinguish newly acquired from pre-existing resistance with additional evidence |
| Decline on treatment, growth after withdrawal, renewed decline on retreatment | Reversible tolerance or persister-cell regrowth is plausible | Confirm drug washout and test whether sensitivity returns after withdrawal |
The principal alternatives to trajectory-based diagnosis are complementary rather than interchangeable. Direct drug-exposure and target-engagement measurements test whether adequate treatment reached the tumor. Cell-based sensitivity assays test response under controlled exposure. Molecular and lineage analyses address mechanisms and clonal history. Serial volume measurements establish when aggregate growth changes, but cannot replace those mechanistic assessments. A model with separate sensitive, resistant, and persister compartments can organize hypotheses and generate predictions, yet adding compartments does not make them identifiable from volume data alone.
A common pitfall is to reason, “The tumor grew after treatment was stopped, so the tumor must have become resistant.” That conclusion ignores the timing and reversibility of the response. Growth after withdrawal is compatible with surviving drug-sensitive or drug-tolerant cells resuming expansion when treatment pressure is removed. Conversely, a response on rechallenge does not prove that no resistant subpopulation exists; it may be too small to dominate the measured burden. Treat trajectory patterns as evidence that changes the plausibility of mechanisms, not as labels that uniquely identify them.
Check Your Understanding
Answer each question, then compare your reasoning with the solutions below.
- A tumor rises from 80 mm3 to 100 mm3 over 10 days of treatment. What is its interval rate, and what does the sign imply?
- A tumor responds initially but later grows while dosing continues. What mechanism is suggested, and what important distinction cannot be resolved from volume alone?
- A tumor grows after withdrawal and declines again on rechallenge. Why is this compatible with persister-cell regrowth but not proof of it?
- With \(r_S=-0.08\) day−1, \(r_R=0.04\) day−1, and resistant fraction \(f_R=0.25\), calculate the aggregate rate using the two-population expression.
Solutions. (1) \(r=\ln(100/80)/10=\ln(1.25)/10=0.02231\) day−1. The rate is positive, so measured burden grew during the interval; this alone does not show that treatment had no effect relative to untreated growth. (2) Resistance or a changing population mixture is suggested. Volume alone cannot reliably distinguish newly acquired resistance from selection of a resistant population that was present before treatment. (3) Withdrawal-associated growth followed by renewed response is consistent with viable cells resuming growth off drug while retaining sensitivity. The aggregate trajectory does not show the identity or state history of the cells, so biological measurements are needed to establish persister-cell behavior. (4) The sensitive fraction is \(1-0.25=0.75\), so \(r=0.75(-0.08)+0.25(0.04)=-0.060+0.010=-0.050\) day−1. The total tumor is predicted to decline over that instant despite a growing resistant subpopulation.