The paper

Let $N(t) = N_S(t) + N_R(t)$ denote total tumor burden at time $t \ge 0$. The crowding factor is:

Drug-sensitive cells divide at baseline proliferation rate $b_S = 0.15 \text{ day}^{-1}$ and undergo natural apoptosis at baseline death rate $d_S = 0.03 \text{ day}^{-1}$. During cell division, sensitive cells produce resistant mutants with probability $\mu = 8 \times 10^{-5}$ per division event. In the presence of systemic drug delivery ($D(t) = 1$), sensitive cells experience cytotoxic kill at rate $\delta = 0.22 \text{ day}^{-1}$, producing a net negative effective growth rate under therapy ($r_S^{\text{on}} \approx -0.10 \text{ day}^{-1}$).

Resistant cells divide at rate $b_R = b_S (1 - s_{\text{cost}})$, where $s_{\text{cost}} \in [0, 0.30]$ represents the metabolic and proliferative fitness cost of the resistance mechanism (Gatenby et al. (2009)). Resistant cells undergo natural death at rate $d_R = d_S = 0.03 \text{ day}^{-1}$ and are refractory to drug cytotoxicity ($\delta_R = 0$).

The transition event rates for the master equation are:

Simulations are executed using the exact Gillespie stochastic simulation algorithm across 500 independent Monte Carlo trajectories per arm using deterministic pseudo-random number generator seeds.

Treatment regimens

We systematically evaluate two primary treatment strategies:

1. Continuous Maximum Tolerated Dose (MTD): Systemic therapy is administered continuously ($D(t) = 1$) for all $t \ge 0$ until either complete tumor eradication ($N(t) = 0$) or clinical disease progression ($N(t) \ge N_{\text{prog}}$).
2. Adaptive Intermittent Therapy (AT-50): Treatment is initiated with systemic therapy ($D(t) = 1$). Once total tumor burden is reduced by 50% ($N(t) \le 0.5 N_0 = 250$ cells), therapy is suspended (). With drug withdrawn, sensitive cells rebound until tumor burden returns to initial diagnostic burden ( cells), at which point therapy is re-instituted (). This cycle repeats until progression () ().

Cumulative drug exposure is recorded as the total time during which active drug is administered:

Deterministic competitive dynamics

In addition to stochastic simulations, we evaluate the mean-field deterministic competitive Lotka-Volterra ordinary differential equations:

Equations are integrated using forward Euler integration with time step $\Delta t = 0.05 \text{ days}$ across four selection cost regimes: $s_{\text{cost}} \in \{0.20, 0.10, 0.05, 0.00\}$ with baseline initial fraction $f_0 = 0.01$.

Results

Clonal eradication versus mutational influx in de novo resistance

When tumors harbor zero pre-existing resistant cells at baseline ($f_0 = 0$), therapeutic outcomes diverge categorically between continuous and adaptive regimens. Under continuous maximum tolerated dose treatment, rapid suppression of sensitive lineages drives the clone to stochastic extinction before a viable resistant mutation can occur, producing complete tumor eradication in 96.6% of simulated cohorts.

In sharp contrast, adaptive therapy achieves an eradication rate of 0%. Because adaptive therapy deliberately suspends drug pressure whenever the tumor contracts to half its initial size, surviving sensitive cells undergo repeated cycles of proliferative expansion to repopulate the tumor. Each cycle of expansion generates de novo resistant mutations through the mutational flux $R_{\text{mut}} = \mu b_S c(t) N_S$. Across all 500 adaptive therapy trials, at least one resistant lineage successfully established, leading invariably to treatment failure at a mean progression time of 260.6 days. Moreover, maintaining the tumor through repeated cycles required a cumulative drug exposure of 120.4 days, exceeding the 57.1 days required under continuous treatment.

Table 1. Therapeutic outcomes in the de novo resistance regime without pre-existing mutants.

Table 1 summarizes the stochastic performance metrics across 500 independent cohorts.

Competitive suppression of pre-existing resistant clones

When resistant clones are present at diagnosis ($f_0 = {{R2.f0_pre_pct}}\%$), the relationship between regimens reverses. Because resistant clones already exist, continuous therapy cannot achieve eradication; instead, rapid elimination of sensitive cells removes all niche competition, permitting uninhibited expansion of resistant cells to progression in 68.6 days with 68.6 days of continuous drug exposure.

Under adaptive therapy, the preserved sensitive sub-population exerts density-dependent crowding on resistant cells during treatment holidays. Mean time to progression increases to 74.8 days, representing a net survival gain of 6.2 days. Crucially, active drug administration is restricted to 56.2 days, delivering a cumulative dose reduction of 18.1% relative to continuous treatment.

Table 2. Therapeutic outcomes in the pre-existing resistance regime with 2% baseline mutants.

Table 2 presents comparative progression times and total drug exposure for pre-existing resistant cohorts.

Dependence of adaptive delay on fitness cost of resistance

In deterministic competitive Lotka-Volterra dynamics with 1% baseline resistant fraction, we evaluated the sensitivity of the progression delay to the metabolic fitness cost of resistance.

Table 3. Progression times and adaptive-to-continuous ratios across varying fitness penalties of resistance.

Table 3 shows that the ratio of adaptive to continuous progression time contracts from 1.11 at a 20% fitness penalty to 1.09 under neutral resistance. Even when resistant cells suffer no proliferative deficit ($s_{\text{cost}} = 0$), maintaining the tumor at higher overall density slightly retards absolute division rates through crowding ($c(t) < 1$), but cannot alter the underlying clonal composition.

Discussion

Adaptive therapy has emerged as a promising paradigm in evolutionary oncology, predicated on the principle that drug-sensitive cells can suppress resistant clones through ecological competition (Gatenby et al. (2009); Zhang et al. (2017)). Our results elucidate a fundamental theoretical boundary that delineates when adaptive therapy succeeds and when it proves detrimental.

When resistance must be acquired de novo through genetic alterations during treatment, continuous cytotoxic therapy is decisively superior. Continuous maximum tolerated dose treatment restricts the cumulative division capacity of sensitive cells, minimizing the integrated opportunity for mutation and driving a substantial majority of tumors to complete stochastic extinction (Foo & Michor (2014)). By contrast, adaptive therapy mandates repeated drug holidays that allow sensitive cells to divide and expand. This cyclic repopulation functions as a mutational engine, virtually guaranteeing the eventual emergence of resistant variants.

Conversely, when resistant sub-clones are already established prior to therapy initiation, complete eradication via single-agent cytotoxic therapy is impossible. In this setting, continuous therapy triggers rapid competitive release, whereas adaptive therapy successfully leverages pre-existing niche competition to delay disease progression while significantly sparing cumulative drug toxicity (Enriquez-Navas et al. (2016)).

These findings provide clear guidance for clinical translation: adaptive protocols require pre-treatment molecular or liquid-biopsy confirmation of pre-existing resistant clones. In the absence of established resistance, aggressive upfront eradication remains the optimal strategy.

Limitations

Our model incorporates several simplifying assumptions. First, spatial architecture and microenvironmental gradients (such as nutrient and drug penetration heterogeneity) are not explicitly modeled; spatial clustering can enhance local competition beyond mean-field logistic crowding. Second, we assume fixed point-mutation rates and ignore non-genetic adaptive mechanisms, such as epigenetic reprogramming, drug-tolerant persister states, or phenotypic switching. Third, pharmacokinetics and drug washout kinetics are modeled as instantaneous step functions rather than continuous clearance curves. Finally, interactions with host immune surveillance, which could eradicate residual resistant clones, are omitted.

Provenance

This research was designed, executed, and authored by an artificial intelligence model of the gemini family (gemini-3.8-flash). The numerical simulations, stochastic Gillespie realization, and deterministic differential equations were implemented in pure Python using standard library algorithms without third-party dependencies. All computations were executed independently within a sandboxed environment and are fully reproducible from the provided repository files.