Turnover Modulates the Need for a Cost of Resistance in Adaptive Therapy

The evolution of drug resistance is one of the biggest challenges in cancer therapy. In one of the first articles on chemotherapy, Farber and colleagues (1) already reported not only unprecedented remissions in children suffering from acute leukemia, but also that these remissions were temporary. For many patients this pattern still holds true today, and applies to both chemo- as well as targeted therapies.

Research aiming to combat drug resistance has traditionally focused on developing drugs that target either the resistance mechanism or kill the cell through a different route. As an alternative to this molecular approach, a number of authors have proposed that resistance could be delayed, if not averted, through changes in drug scheduling (2–6). Current treatment schedules are designed to maximize cell killing by treating at the maximum tolerated dose (MTD) as frequently as toxicity permits. However, if drug-resistant cells are present a priori or develop during therapy, such aggressive treatment releases these cells from the competition for space and resources and facilitates their growth, a process known as “competitive release” (3, 4, 7). Inspired by approaches used in the management of invasive species and agricultural pests, Gatenby and colleagues (4, 7) proposed “adaptive therapy,” which aims not to eradicate the tumor, but to control it. Therapy is applied to reduce tumor burden to a tolerable level but is subsequently modulated or withdrawn to maintain a pool of drug-sensitive cancer cells. Over the past 10 years, studies have shown that adaptive therapy can extend time to progression (TTP) in vivo in ovarian (7) and breast cancer (8), and most recently in melanoma (9). Moreover, the interim analysis of the first human trial of adaptive therapy, applied to the treatment of metastatic castrate-resistant prostate cancer with androgen deprivation therapy, reported an increase in TTP of at least 10 months and a reduction in cumulative drug usage of 53% (10).

The success of adaptive therapy in prostate cancer has spurred interest in extending the protocol to other cancers, such as thyroid cancer and melanoma (clinicaltrials.gov identifiers NCT03630120 and NCT03543969, respectively). As such, it is urgent to understand the characteristics of a cancer that determine whether it will respond well to adaptive therapy, or not. Bacevic and colleagues (11) showed through in vitro experiments and an agent-based computational model that it is important that cells are spatially constrained, as otherwise the competition is too weak to effectively control the growth of resistant cells. Furthermore, through ordinary differential equation (ODE) modeling they found that the fitness of the resistant population, when the population is rare, is a key determinant of the benefit derived from adaptive therapy (11). These results were corroborated by Gallaher and colleagues (12), who compared two adaptive therapy strategies with MTD using an agent-based model, which modeled resistance as a continuous trait. They further found that the rate of spatial mixing through migration and the rate of acquisition of resistance through mutation were key factors in determining the benefit from adaptive therapy, and that adaptive therapy strategy was most effective. Using a nonspatial game theory model, West and colleagues (13) showed that the stronger the selection against resistance in the absence of drug, the longer adaptive therapy can be expected to extend TTP. Taking a more general approach, Hansen and colleagues (6, 14) demonstrated that the decision of whether to treat aggressively or not, depends on the dominant mechanism of resistant cell production (clonal expansion vs. de novo acquisition), and the initial resistant cell number. By making the simplifying assumption that progression can be predicted purely from the governing equation of the resistant population, they derive an explicit form of the decision boundary for a simple ODE model (6, 14). Most recently, Smalley and colleagues (9) showed that adaptive therapy can be beneficial even in the light of phenotypic plasticity, but for that to achieve optimal results, personalization of the treatment schedule is required.

However, a factor that is still poorly understood is the so-called “cost of resistance.” This refers to the idea that while a resistance adaptation confers a fitness advantage under drug exposure, it may come at a fitness cost in a treatment-free environment (4, 15). Resistance costs have been widely studied in agricultural pests and antibiotic resistance (16) and may be measured in different ways (Fig. 1A–C). An example in the cancer literature can be found in work by Gallaher and colleagues (12), who report that doxorubicin-resistant MCF7Dox breast cancer cells replicate 50% slower compared with their sensitive counterparts, and that in competition experiments, the sensitive cells outcompete the resistant cells in the absence of the drug. This is likely because these cells use P-glycoprotein–related efflux pumps to resist treatment, for which they have to divert energy away from proliferation towards the running of the pumps (12, 17). Similar observations have been made in colorectal, lung, and other cancers in response to chemo- (18, 19) as well as targeted therapies (9, 11, 20). In their original work on adaptive therapy, Gatenby and colleagues (4, 7) used the cost of resistance to motivate adaptive therapy, and most theoretical studies since have made this assumption (11–13).

However, not all resistance mechanisms come at a cost, and the magnitude of any cost depends on a cell's genetic and environmental context (9, 18, 20). Despite using identical experimental protocols, Jensen and colleagues (18), for example, found that some of their colorectal cancer cell lines exhibited a cost, while others did not, and some even showed increased fitness. Similar findings were made by Smalley and colleagues (9) for melanoma. Moreover, when we repeated the experiments by Gallaher and colleagues (12) in 3D spheroids, we found that drug-resistant MCF7 cells grew faster than their sensitive counterparts (Fig. 1D and E). However, when we reduced the glucose concentration in the culture medium, this advantage was lost (Fig. 1F and G; see also Supplementary Fig. S1A and S1B). Finally, it is not clear that the presence of a cost guarantees a quick return to sensitivity if drug is withdrawn. For example, while Smalley and colleagues (9) found that one melanoma cell line returned to sensitivity within 6 weeks after drug withdrawal, another remained resistant for months, an observation also made elsewhere (11, 18, 21). Given this wide range of possibilities for how a resistant population might, or might not, differ from their sensitive counterparts, it is important to clarify the relationship between resistance costs and the success of adaptive therapy.

In this article we use a simple mathematical model to address this question. We divide the tumor into drug-sensitive and drug-resistant cells and model their growth with two ODEs. We compare the TTP under standard-of-care continuous therapy with that of the adaptive therapy algorithm used in the clinical trial by Zhang and colleagues (10), first in the absence, and subsequently in the presence, of a cost. We will show how increased tumor density and small levels of preexisting resistance maximize the benefit of adaptive relative to continuous therapy, as these conditions maximize the effect of competition between sensitive and resistant cell populations. Subsequently, we will demonstrate that turnover is a key factor to consider, not only to understand the impact of a resistance cost, but also to assess the ability to control a tumor with adaptive therapy, more generally. To conclude, we present an analysis of intermittent androgen deprivation therapy in prostate cancer, which provides evidence for our modeling results, and illustrates how our insights can help to improve our understanding of resistance management in a specific disease setting.

Tumors are heterogeneous populations of cells with differential responses to drug, indicating a degree of preexisting resistance in most tumors (22–24). To model this heterogeneity, we assumed two competing cell types: drug-sensitive cells, S(t), and fully resistant cells, R(t), modeled via the following equations:

with initial conditions:

where N₀ = S₀ + R₀. The model is derived on the basis of the following assumptions (Fig. 2):

• (i) In isolation, each population grows logistically with proliferation rates, r_S and r_R, where the fraction of dividing cells decays linearly from 1 to 0 as the population approaches its environmental carrying capacity, K. Furthermore, cells die at a density-independent rate, d_T, where for simplicity, we assumed that this turnover rate is the same for both populations.

• (ii) Cells compete for resources and space according to the Lotka–Volterra competition model. This means that the presence of the competitor reduces a population's growth rate in a fashion that is linearly proportional to the competitor's population density. For simplicity, we assumed that inter- and intraspecies competition coefficients are identical, and equal to one.

• (iii) Only actively dividing cells are killed by the drug. Many chemotherapies induce DNA damage or inhibit the cell division machinery, which induces apoptosis only in cells that attempt to divide. We adopted the classical Norton–Simon model (25), which assumes that cell killing increases linearly with the drug dose, D(t), so that at MTD, a fraction, d_D, of dividing cells is killed. The factor of 2 in the drug response accounts for the fact that if a cell dies during mitosis, not only the potential daughter cells, but also the mother cell is lost.

• (iv) Progression is primarily driven by preexisting resistant cells, and resistance acquisition during treatment is negligible. Viossat and Noble (26) have recently shown that inclusion of genetic mutation does not significantly impact the benefits of adaptive therapy in simple models such as ours. Incorporating phenotypic cell plasticity, which may occur at a much higher rate (e.g., ref. 27), will be subject to future research.

• (v) To illustrate our key results, the main text of the article will assume that the resistance cost manifests itself solely in the growth rate, r_R, as this is the way a cost is typically modeled (e.g., refs. 11, 12, 23). For a discussion of other types of cost, see Supplementary Fig. S2A–S2E and Supplementary Section S2 of the Supplementary Data.

We considered two treatment schedules:

• (i) Continuous therapy at the MTD, D_Max: D(t) = D_Max ∀t.

• (ii) Adaptive therapy, which withdraws treatment once a 50% decrease from the initial tumor size is achieved, and reinstates it once the original tumor size (N₀) is reached:

  

To facilitate numerical simulation of the model, we nondimensionalize Eqs. A–C using r_S as a time scale, and K as a scale for the cell densities (Table 1). For details of the nondimensionalization, the numerical methods, and a steady-state analysis, see Supplementary Sections S3 and S4; Supplementary Table S1; and Supplementary Fig. S3A–S3C of the Supplementary Data, respectively. For completeness, we also considered a model in which birth is constant, but death is density dependent, which we found yields similar results (Supplementary Section S5 and Supplementary Fig. S4A–S4D of the Supplementary Data). The code associated with our analyses is available at: https://github.com/MathOnco/AT_costOfResistance_LVModel.

Given the key role that prostate cancer has played in the development of adaptive therapy, we parametrized our model according to this disease. As such, we adopted the proliferation rate for sensitive cells given in (r_S = 0.027/day; ref. 10) as our time scale and the drug kill parameter, |${\smash{\hskip3pt{\hat{\hskip -3pt d}_D}}$| = 1.5 from (28). For the other parameters, we performed parameter sweeps within their biologically realistic ranges. All parameters, their definitions, and their ranges are summarized in Table 1.

To test our predictions, we fitted our model to publicly available, longitudinal response data from patients with prostate cancer undergoing intermittent androgen deprivation therapy (29). The data were downloaded from http://www.nicholasbruchovsky.com/clinicalResearch.html. Patients who developed a metastasis were excluded, to avoid potentially confounding effects from a change in the number of lesions, yielding data from a total of 67 patients. The model was fitted to each patient by minimizing the root mean squared difference between the normalized prostate specific antigen (PSA) measurements, |$\hskip1pt{\hat{\hskip -1pt y}}$|(t)/|$\hskip1pt{\hat{\hskip -1pt y}}$|(0), where |$\hskip1pt{\hat{\hskip -1pt y}}$|(t) is the PSA measurement at time t, and the normalized predicted cell density, n(t)/n₀. The values for r_S and |$\hskip 2pt{\hat{\hskip -2.5pt d}_D}$| were fixed as in Table 1, and the remaining four parameters (⁠|$\hskip 2pt{\hat{\hskip -2.5pt r}_R}$|⁠, |$\hskip 2pt{\hat{\hskip -2.5pt d}_T}$|⁠, n₀, and f_R) were allowed to vary freely between the following bounds: |$\hskip 2pt{\hat{\hskip -2.5pt r}_R}$| ∈ [0,1], |$\hskip 2pt{\hat{\hskip -2.5pt d}_T}$| ∈ [0,1], n₀ ∈ [0.1,1], and f_R ∈ [1 × 10⁻⁵,0.25]. Fitting was done using the Levenberg–Marquardt algorithm, and repeated 25 times for each patient starting from different initial parameter combinations, with the best fit according to the Akaike information criterion taken forward for analysis. For 15 patients no satisfactory fit could be obtained (r² < 50% or failure to capture later stage cycles), and these were excluded from all further analyses. Classification of patients into relapsing and nonrelapsing was taken from the annotation provided in the data, where relapse is defined as a series of three sequential increases of serum PSA > 4 μg/L despite castrate levels of serum testosterone. An overview of all fits is shown in Supplementary Fig. S5. Fitting was done using the lmfit package in Python (30).

In advanced cancers, curative approaches rarely show durable complete response. Instead, treatment success is often defined by how long therapy can prevent the tumor from progressing beyond a certain size (i.e., TTP). Herein, we compare the model-predicted TTP for both adaptive (TTP_AT) and continuous therapy (TTP_CT), respectively, using RECIST criteria (20% increase in tumor size from the pretreatment baseline). We show that time gained by adaptive therapy depends on the following tumor characteristics: initial tumor density (n₀; see Fig. 3), initial levels of preexisting resistance (f_R; see Fig. 3), cost of resistance (r_R < r_S; see Fig. 4), and the density-independent cell turnover rate (d_T; see Fig. 4). While we focus here on prostate cancer, our theoretical insights apply more broadly. For application of our work to breast cancer, see Supplementary Fig. S6A and S6B.

In the absence of a resistance cost, adaptive therapy still results in significant improvements in TTP over continuous therapy. To illustrate this, in Fig. 3A we show representative simulations of adaptive therapy for different values of tumor density (plots, left to right) and preexisting resistance (top to bottom). Each subpanel shows sensitive (red lines), resistant (green), and total (blue) populations over time under adaptive therapy dosing. Treatment administration is illustrated by black bars at the top of the graphs. This shows that TTP_AT (vertical blue dashed lines) exceeds TTP_CT (vertical yellow line) under a range of conditions, and for the combination of low resistance and high tumor density (top, right). Figure 3B quantifies time gained (TTP_AT − TTP_CT) for a range of tumor density and resistance values. For our parameter sweep, we found that adaptive therapy can extend TTP by between 3 and 104 days when the initial resistance fraction is f_R = 1%, and by between 19 and 211 days when f_R = 0.1%. Finally, we note that in the worst case scenario (f_R = 10%), adaptive therapy becomes indistinguishable from continuous therapy, so that while there is no benefit, the patient has also not progressed faster than under standard of care (Fig. 3A and B, bottom rows).

Moreover, we found that each of the two characteristics of the tumor's initial composition has a distinct impact on the treatment dynamics. As we increase the initial abundance of resistant cells from 0.1% to 10%, we decrease the number of completed adaptive therapy cycles (Fig. 3A). In the most extreme case, at 10% initial resistance, treatment could not decrease the tumor burden sufficiently to trigger any treatment withdrawal in the three tumors. In contrast, increasing the initial tumor density, which biologically corresponds to greater competition for resources, does not alter the adaptive therapy cycle number, but does increase the benefit delivered by each cycle. For example, even though all tumors with 1% initial resistance complete one adaptive therapy cycle, a meaningful benefit in TTP is only achieved in the case when the tumor is 75% saturated (Fig. 3A). The reason for this is that the competition exerted by sensitive cells only has significant impact on the growth of the resistant cells if the tumor is close to K. In summary, for adaptive therapy to provide a benefit, the tumor burden must undergo a sufficient decline to allow for treatment withdrawal (small f_R), and competition within the tumor must be sufficiently strong to noticeably slow the expansion of the resistant population (proximity to K).

How can we explain the benefit of adaptive over continuous therapy in the absence of a cost of resistance? We applied the framework by Hansen and colleagues (6) to derive the following explanation. Progression is primarily driven by the expansion of the resistant population, as drug-sensitive cells are easily depleted by additional treatment. Thus, the more a treatment strategy can inhibit the resistant population growth rate, dR/dt, while also maintaining control of the tumor size overall, the longer is the TTP. To illustrate this, we plotted dr/dτ (the nondimensional form of dR/dt) as a function of the tumor composition in Fig. 3C. In this representation, a tumor lesion is seen as a point in a 2-D space, where its x-position represents the current relative density of sensitive cells, s(τ), and its y-position is the current density of resistant cells, r(τ). Each point is colored according to the resistant population growth rate. This representation clearly illustrates how high tumor densities (see inset gray arrow) are generally associated with lower resistant growth rates. Rewriting Eq. B as:

shows that this slow down occurs due to intra- and interspecific competition. Importantly, while a cost in r_R modulates the strength of growth inhibition through competition, it is not required (as also illustrated in Fig. 3C).

During therapy, the composition of a tumor changes, so that treatment corresponds to a trajectory through the S − R space. In Fig. 3C we show trajectories of continuous (yellow) and adaptive therapy (blue) for two tumors corresponding to tumors 1 and 4 in Fig. 3A. As can be seen, continuous therapy trajectories tend to traverse regions of high resistant growth (dark green shading). In contrast, these same regions are avoided under adaptive therapy regimens, especially for tumor 4. Again, we can formalize this using Eq. B: if S_CT(t) and S_AT(t) denote the density of sensitive cells after t subsequent days of treatment under continuous and adaptive therapy, respectively, then we have that, S_CT(t) ≤ S_AT(t) because continuous therapy does not provide sensitive cells with an opportunity to recover. Assuming that the tumor is still far from progression (R << S) so that the resistant cells primarily compete with sensitive cells, the resistant population growth rate is greater for continuous |( {{{ {{\frac{{dR}}{{dt}}}} |}_{CT}}} )$| than for adaptive therapy |( {{{ {\frac{{dR}}{{dt}}} |}_{AT}}} )$|⁠: |{\rm{\ }}{ {{\frac{{dR}}{{dt}}}} |_{CT}} \approx ( {{r_R} - {d_T}} )R - {\frac{{{r_R}{S_{CT}}}}{K}}R\ \ge ( {{r_R} - {d_T}} )R - {\frac{{{r_R}{S_{AT}}}}{K}}R \approx { {{\frac{{dR}}{{dt}}}} |_{AT}}$|⁠.

This implies that adaptive therapy can provide a benefit even in the absence of a cost of resistance. This benefit increases with increased tumor density as S_AT/K can be maintained large for longer. In addition, this implies that the longest control in this model can be achieved by maintaining the tumor burden as close to its original size as possible, to maximize the effects of interspecific competition. To demonstrate this, we simulate adaptive treatment in which drug is withdrawn after a greater than, and a smaller than, 50% burden reduction (Supplementary Fig. S7A and S7B). This suggests that, assuming the larger burden is tolerable, the longest TTP can be achieved by the algorithm with the smallest burden reduction.

Next, we examined the role of resistance cost. Figure 4A and B compares low density tumors (tumor 1 and 2, repeated from Fig. 3) with high density tumors (tumor 3 and 4). As expected, TTP_AT (blue line) increases with a cost of resistance in all four cases (Fig. 4A). However, an increase of the benefit of adaptive over continuous therapy is only seen when tumors are close to K and resistant cells are rare, as in tumors 1 and 4. In contrast, in highly resistant tumors, a cost of resistance increases the TTP in roughly equal terms for continuous and adaptive therapy. This effect is consistent for a wide range of resistance cost values (Fig. 4B).

So far, we have assumed that the turnover of tumor cells is negligible. However, tumors are subject to resource starvation and immune predation, resulting in continuous tumor turnover (see Supplementary Section S6 for further discussion). In Fig. 4C, we show how such turnover modulates the impact of a resistance cost on the competition between sensitive and resistant cells in the absence of drug. Increasing the cost of resistance (dashed lines) leads to lower levels of resistance in untreated populations. Importantly, while a cost reduces the number of resistant cells, these cells never go extinct in the absence of turnover (Fig. 4C). In contrast, if we introduce intrinsic cellular turnover (dˆ_T > 0), resistant cells go extinct over time (Fig. 4C). Selection against resistance depends not only on the cost of resistance, but also the turnover rate.

How does this insight affect adaptive therapy? In Fig. 4D we show TTP_AT as a function of turnover and cost for tumor 1. If turnover is low, then a large cost of resistance results only in small gains for adaptive therapy, as seen by comparing cases i and ii in Fig. 4D and E. In contrast, if turnover is high, then adaptive therapy provides significant benefits even if the resistance cost is small, or completely absent (case i vs. iii, Fig. 4D and E). We also observe in case iv that, once turnover increases beyond a threshold of (1 − 1.2n₀)rˆ_R, the tumor becomes indefinitely controllable in the model, so that continuous and adaptive therapies maintain the tumor below the size for progression forever (see Supplementary Section S7 for further discussion).

Next, we examined how this result generalizes as we change the initial tumor density and resistance fraction. We repeated the cost–gain relationship analysis from Fig. 4B with a turnover rate of dˆ_T = 25%. As tumors 3 and 4 are indefinitely controllable in this parameter regime (see Supplementary Fig. S8A–S8C), we replaced them by tumors with a slightly lower initial density (n₀ = 50%), denoted by tumors 3* and 4*, respectively. We found that all, but the less dense and highly resistant tumor (tumor 2), benefited from adaptive therapy even in the absence of a resistance cost (Fig. 4F). Interestingly, tumor 3* did not benefit from adaptive therapy in the absence of turnover (Fig. 3A; bottom row, center), but with turnover, adaptive therapy becomes superior to continuous therapy. Moreover, in all cases, a cost of resistance increased the time gained with adaptive therapy. To test the robustness of our findings to the model assumptions, we reformulated our model from a density-dependent birth to a density-dependent death model, which assumes that cells proliferate at a constant rate and die at a rate proportional to the cell density. This further corroborates our main result that turnover increases the effect of competition between sensitive and resistant cells (Supplementary Section S5; Supplementary Fig. S4A–S4D).

Our theoretical analyses indicate that the success of resistance management depends on an interaction between resistance costs and cellular turnover. To test this hypothesis, we fitted our model to publicly available, longitudinal tumor burden data from 67 patients undergoing intermittent androgen deprivation treatment for recurrent, locally advanced prostate cancer (29). After an initial 36-week lead-in period, treatment of these patients was withdrawn whenever PSA levels were below 4 μg/L and was reinstated when PSA levels exceeded 10 μg/L, resulting in treatment cycles similar to adaptive therapy (29). Using the monthly serum PSA measurement as a marker of tumor burden, we fitted the values for cost, turnover, n₀, and f_R for each patient, fixing all other parameters as before (Table 1). We found that even though our model was constructed as a conceptual tool, it can recapitulate individual patient dynamics for a majority of patients, and that it can describe patients who continuously respond, as well as those who eventually relapse (Fig. 5A; for an overview of all patients see Supplementary Fig. S5).

Next, we examined what role turnover and cost play in the response dynamics. Analysis of the fitted parameters shows a value of cost and turnover far from 0 in most patients (Supplementary Fig. S9). In fact, setting turnover or cost, or both, to 0 significantly decreases the goodness-of-fit relative to the available degrees-of-freedom, demonstrating the importance of both in understanding treatment dynamics (Fig. 5B). Moreover, we found that estimates for turnover and cost show a strong negative correlation, with most patients clustering closely along a line in parameter space (Pearson correlation coefficient, 0.77; Fig. 5C). This trade-off, which is not imposed, but emerges from the fitting, supports our hypothesis of the importance of turnover. First, no patient fell into the region in which both turnover and cost, and so response, were small, presumably because such patients would not have achieved PSA level normalization in the lead-in phase required for study inclusion (compare with Fig. 4D). Similarly, no patients were seen in the bottom right corner, where our model predicts long-term tumor control, consistent with the fact that all patients in the study were refractory after initial therapy. Subsequently, we investigated whether differences in turnover or cost are predictive of whether a patient will relapse. While there was no difference in either variable in isolation (Supplementary Fig. S9; P value in Mann–Whitney U tests > 5%), we observed that patients with a relapse tend to have smaller values of cost than patients with a similar level of turnover in whom no relapse occurs (Fig. 5C). Consistent with this observation, we found that the difference in the sum of cost and turnover between patients with and without relapse was statistically significant (Mann–Whitney U test; U = 82; P < 1%). To illustrate this, we show three sets of examples in Fig. 5D. This also shows how different regions in turnover–cost space correspond to different types of patient dynamics, with patients with lower estimates for turnover tending to have shorter drug-off periods, than patients with high values of turnover (transect 1 vs. 3). To sum up, this analysis demonstrates that turnover and cost are important in describing patient response to intermittent androgen deprivation treatment, and that it is their combination that has to be considered to understand outcome.

We deduce that there are three ways in which one may amplify intratumoral competition and improve tumor control with adaptive therapy (Fig. 5E). First, one can lower the threshold for treatment withdrawal, so that treatment is withdrawn earlier and more frequently, assuming the increased tumor burden is acceptable. Second, one can target the environmental carrying capacity, for example, by administering an antiangiogenic drug to reduce the supply of nutrients. Finally, one can increase cell death of both sensitive and resistant cells to reduce the growth of resistant cells and the strength of selection against them if a cost is present. This may be done, for example, by adding low-dose chemotherapy. In Fig. 5E we illustrate each strategy on patient 99 who relapsed during intermittent androgen deprivation treatment. Interestingly, while it was strategy 1 that optimized the gain of adaptive relative to continuous therapy, it was strategy 3 that maximized absolute TTP. This suggests that depending on the clinical aim, different strategies, or combinations of these, may be optimal.

With four clinical trials of adaptive therapy already ongoing (clinicaltrials.gov identifiers: NCT02415621, NCT03511196, NCT03543969, and NCT03630120) and more in preparation (31), it is important to develop criteria to identify which patients might benefit from adaptive therapy over standard MTD approaches. Intuitively, it appears that the presence of a fitness cost of resistance might serve as an inclusion criterion. However, a cost does not guarantee resistance control (11, 12). Moreover, Zhang and colleagues (10) did not report a resistance cost in vitro, yet found that adaptive therapy extends median TTP by over 10 months in patients.

The aim of this article was to consolidate these findings and to develop an understanding of the circumstances under which a resistance cost is required, and under which circumstances it is not. To do so, we developed a simple two-population Lotka–Volterra competition model and treated it according to the adaptive therapy algorithm from Zhang and colleagues (10). We have shown that the time gained in adaptive therapy compared with continuous therapy depends on four key factors: (i) the initial fraction of resistant cells (f_R), (ii) the proximity of the tumor to environmental carrying capacity (n₀), (iii) the resistant cell proliferation rate (r_R), and (iv) the rate of cellular turnover (d_T). While the importance of the first two factors has been recognized previously (6, 11, 12), it is unclear how they interact with the latter two. First, we show that a resistance cost is not necessary for adaptive therapy to extend TTP. If the tumor is close to carrying capacity, and resistance is rare, then adaptive therapy can achieve significant gains even in the absence of a cost. We demonstrate how to use the work by Hansen and colleagues (6) to explain this result mathematically and how it may be visualized for a nonmathematical audience using phase plane techniques. Viossat and Noble (26) have recently applied this approach to a broad class of tumor models, and we recommend their article for a detailed discussion of how some of our results may be generalized. Finally, it is worth pointing out that resistance costs may also manifest themselves in other ways. For example, in the 3D spheroids in Fig. 1, there are significantly more debris around the resistant spheroids than around their sensitive counterparts (see also Supplementary Fig. S1; Supplementary Section S1.4), suggesting a potentially higher turnover in resistant compared with sensitive cells. Our analysis easily extends to costs in the turnover rate and carrying capacity (Supplementary Section S2).

Given its pivotal role, an important question to ask is: what is a tumor's carrying capacity? Carrying capacity is defined as the population size, or density, at which growth saturates (32). As such, it is a multifaceted concept that may be defined at different spatial scales and that may vary over time and space. For example, one can define a systemic carrying capacity for the whole body, or local carrying capacities for individual tumor lesions. Moreover, because of spatial heterogeneity in resource supply, or available space, different lesions, or even different regions within the same tumor, may have different carrying capacities. Importantly, while tumors, especially solid tumors, are likely far away from their fatal, systemic carrying capacity, they may be close to their tissue- or subtissue-level capacity (33).

Furthermore, carrying capacity is a dynamic quantity that changes over time. It is determined by space and resource availability, which will change as the tumor recruits vasculature and stromal support or gains the ability to invade surrounding tissue. Moreover, it is determined by the ratio of birth to death in the tumor. Working with the popular r − K form of the logistic growth equation, this fact is easily overlooked, and results in the misleading perception that r and K are independent. A key contribution of this article is to highlight that not only environmental constraints, but also the cells' proliferation and death rates, determine the impact of intratumoral competition. The shorter a cell's life span, the fewer opportunities it will have to divide, and the greater will be the impact of interference through intra- and interspecific competition. Changes in the proliferation or death rate, due to mutation, immune predation, or treatment, will alter the tumor's carrying capacity and so the competition between tumor cells. The relationship between r and K has been an area of intense debate in ecology, centered in particular around the so-called r − K selection hypothesis (32). Realization of their interdependence has been an important step in consolidating ecological and evolutionary theory (34–36), and may similarly help to improve our understanding of tumor evolution, and treatment response.

An important implication of this dynamic view of carrying capacity is that tissue turnover plays a role in adaptive therapy (Fig. 6). Specifically, our modeling indicates that higher turnover facilitates the control of drug resistance, and that if a resistance cost is present, its benefit depends not on its absolute value, but on its value relative to the turnover rate. We tested this hypothesis by fitting our model to response data from patients with prostate cancer undergoing intermittent androgen deprivation therapy. This shows that both cost and turnover are required to explain the treatment dynamics, as their removal significantly reduces goodness-of-fit, even when a penalty for the additional degrees-of-freedom is applied. Interestingly, and contrary to our expectations, we did not find a difference in turnover when we tested for differences between patients who relapse and those who do not. Possibly, this is because tumors with higher turnover may also harbor more resistance as these tumors will have passed through more generations before detection (37), although we did not detect such a correlation ourselves. However, when we examined both cost and turnover together, we saw that patients who relapse tend to be defined by a smaller combination of cost and turnover than patients without a relapse. While these results should be viewed with some caution, as neither cost nor turnover were measured directly, they are consistent with our hypothesis, and show how our insights may help to understand treatment response.

One reason why it was difficult to test our hypothesis more directly is that relatively little is known about turnover rates in human tumors. Most available data were collected between the 1960s and 1990s based on a method pioneered by Steel (38). This method compares the potential doubling time expected from the fraction of dividing cells as identified by chemical labeling with the actual observed volumetric doubling time. Interestingly, most of these studies found that tumors grew significantly slower than expected, suggesting that the rate of cell death closely matches that of cell production (see Supplementary Section S6 and Supplementary Table S2 for further discussion; refs, 38–40). Yet all of these measurements were indirect. In light of our findings, we advocate that more focus should be put on quantifying the turnover rates in tumors. Not only could the information on turnover rates help to identify suitable candidates for adaptive therapy, but it could also be important in interpreting data from experimental model systems. For example, while in vitro, no resistance cost may be observed, tumor control may still be possible in vivo or in patients as turnover, via greater nutrient deprivation, and immune suppression will intensify competition, and amplify even small fitness differences.

We made a number of simplifications in our model. First, similar to previous studies (6, 11), we assumed that the tumor is not curable. However, as can be seen in Fig. 4E, including a resistance cost and greater turnover also result in fewer cells at the nadir during continuous therapy. This implies that the tumors in which adaptive therapy will be most effective are those in which continuous therapy just misses being curative, an observation also made elsewhere (3, 12). Thus, the very long gains predicted under adaptive therapy have to be viewed with some skepticism, because in a proportion of these patients, continuous therapy would have potentially cured the tumor. That being said, adaptive therapy is intended for advanced settings, in which the available treatment is rarely curative, which suggests that such cases will be rare (4, 41). Moreover, it seems plausible that if one finds that adaptive therapy controls the tumor very well, one could then decide to switch to a curative approach. In fact, adaptive therapy has been shown to have stabilizing effects on tumor vasculature (8), which suggests that a curative approach may be even more effective after an initial period of adaptive therapy.

Second, we made the simplifying assumption that resistance is preexisting and that progression is driven solely by clonal expansion of this subpopulation. However, there is mounting evidence that acquisition of resistance is a multistep process in which genotype- and environment-mediated cell plasticity (e.g., via growth factors; ref. 27 or extracellular matrix stiffness; ref. 42) play a key role. While previous research, for example, in melanoma (9), has shown that adaptive therapy can still succeed despite this plasticity, it does pose additional challenges for resistance management (6, 14, 26). Growth factor–driven interactions can result in spatial, nonlinear evolutionary games between subpopulations, whose dynamics may change with environmental conditions (20). Future research should examine adaptive therapy in the context of this plasticity in more detail.

A further important caveat of our study is that we did not account for the potential health risks associated with the increased tumor burden under adaptive therapy. As a result, our model predicts that the longest TTP can be achieved by keeping the tumor as close to its initial size as possible. This is true for most studies in this area (7, 9, 11, 13, 28). However, a higher tumor level increases the potential for de novo resistance acquisition and metastasis, and under certain circumstances, these costs may outweigh the benefits of adaptive therapy (6, 26, 43). Investigating the risks of adaptive therapy is an important direction of future research.

To facilitate analysis, we neglected the impact of space in this article. However, previous work (11, 12) has demonstrated that the tumor's spatial architecture is an important factor in adaptive therapy. If resistant cells can be “trapped” by surrounding sensitive cells, the tumor may be controlled for a long time (11, 12), as resistant cells are close to their local carrying capacity. This may negate the need for a resistant cost for small values of initial burden, and would define carrying capacity in a local fashion. At the same time, by creating gaps in the layer of surrounding cells, turnover may facilitate escape of trapped resistant cells, so that the benefit derived from high turnover may be smaller than expected from the ODE model. In addition, it would allow us to examine the impact of microenvironmental factors, such as the stiffness of the extracellular matrix, acidity of the microenvironment, or presence of stromal support, which are known to affect drug resistance (42). We have already started to test how our current predictions translate to a spatial model, and have made a preprint of our results available on bioRxiv (44).

To conclude, we want to highlight interesting parallels to the pest and antibiotic resistance management literature, which have investigated resistance costs since the mid-20th century (see ref. 45 for an early review of resistance costs in insecticides). First, also in these fields it is a controversial issue (16, 46–48). For example, Bergelson and colleagues (46) found that of 88 studies examining the cost question in plants, 44 showed a cost, 40 found no difference, and four even observed a benefit to resistance. This is because the fitness effect of a resistance adaptation depends on the resistance mechanism, the genetic, and the environmental context (15, 16, 46). Thus, in developing adaptive therapy we should neither assume the presence of costs nor that they are uniform throughout space and time. Importantly, resistance management can be successful despite this uncertainty. For example, the resistance management scheme for the insecticide-producing Bacillus thuringiensis crop has been successful despite inconclusive evidence regarding the presence of resistance costs (47, 48). The reason for this success lies in integration of multiple treatment modalities (49, 50). As such, we advocate multidrug approaches in which one, or several drugs, are given adaptively. Initial theoretical work on multidrug adaptive therapy has already been carried out (28, 51, 52), and we have illustrated here how by targeting the resource availability or turnover in a tumor with secondary drugs one can greatly enhance tumor control. We also note that while abiraterone was given adaptively in the initial prostate cancer adaptive therapy trial (NCT02415621), patients were concurrently receiving a continuous dose of Lupron, which suppresses systemic testosterone production and reduces the cancer's supply of growth factors. A follow-up study is now giving both abiraterone and Lupron in an adaptive fashion (NCT03511196). Going forward, it will be important to extend this work to develop multimodal strategies that exploit a cost of resistance, yet are robust if this cost disappears due to environmental or genetic compensation. With better understanding of tumor growth, resistance costs, and turnover rates, adaptive therapy can be more carefully tailored to patients who stand to benefit from it the most.

M.A.R. Strobl reports grants from Engineering and Physical Sciences Research Council, UK during the conduct of the study. Y. Viossat reports grants from PGM0 (Gaspard Monge Program for Optimization) during the conduct of the study; this paid for travel, but did not generate any potential conflict of interest. M. Robertson-Tessi reports grants from NIH during the conduct of the study and outside the submitted work. No disclosures were reported by the other authors.

M.A.R. Strobl: Conceptualization, data curation, software, formal analysis, validation, investigation, visualization, writing-original draft. J. West: Conceptualization, formal analysis, investigation, writing-review and editing. Y. Viossat: Formal analysis, methodology, writing-review and editing. M. Damaghi: Data curation, investigation, writing-review and editing. M. Robertson-Tessi: Funding acquisition, methodology, writing-review and editing. J.S. Brown: Writing-review and editing. R.A. Gatenby: Funding acquisition, writing-review and editing. P.K. Maini: Conceptualization, supervision, project administration, writing-review and editing. A.R.A. Anderson: Conceptualization, supervision, visualization, project administration, writing-review and editing.

We acknowledge Prof. Carlo Maley for inspiring Fig. 6 with a sketch he presented at the Meeting on Evolutionary Therapy on May 24, 2018 (https://twitter.com/evoltherapy/status/999665282737205249?s=21). This research was supported by funding from the Engineering and Physical Sciences Research Council (EPSRC) and the Medical Research Council (MRC; grant no., EP/L016044/1 to M.A.R. Strobl). Y. Viossat benefited from the support of the FMJH “Program Gaspard Monge for optimization and operation research and their interactions with data science” and from the support from EDF, Thales, Orange and Criteo. A.R.A. Anderson and R.A. Gatenby gratefully acknowledge funding from both the Cancer Systems Biology Consortium and the Physical Sciences Oncology Network at the NCI, through grants U01CA232382 and U54CA193489, as well as support from the Moffitt Center of Excellence for Evolutionary Therapy.

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