Frank Ernesto Alvarez

Yannick Viossat

Tumor containment for Norton-Simon models

Keywords: Adaptive Therapy, tumor containment, Norton-Simon model, mathematical oncology

Some clinical and pre-clinical data suggests that treating some tumors at a mild, patient-specific dose might delay resistance to treatment and increase survival time. A recent mathematical model with sensitive and resistant tumor cells identified conditions under which a treatment aiming at tumor containment rather than eradication is indeed optimal. This model however neglected mutations from sensitive to resistant cells, and assumed that the growth-rate of sensitive cells is non-increasing in the size of the resistant population. The latter is not true in standard models of chemotherapy. This article shows how to dispense with this assumption and allow for mutations from sensitive to resistant cells. This is achieved by a novel mathematical analysis comparing tumor sizes across treatments not as a function of time, but as a function of the resistant population size.

Introduction

The dominant paradigm in cancer therapy is to treat tumors aggressively. This makes sense if the tumor is curable, but might be counter-productive otherwise. Indeed, tumors contain a large number of cells, some of which may be resistant to treatment. By killing preferentially the most sensitive cells, an aggressive treatment could free resistant cells from competition with sensitive cells, allowing them to develop quickly: a phenomenon called competitive release in ecology (Gatenby 2009b [START_REF] Robert | A change of strategy in the war on cancer[END_REF], Enriquez-Navas et al., 2016 [START_REF] Enriquez-Navas | Exploiting evolutionary principles to prolong tumor control in preclinical models of breast cancer[END_REF], Cunningham et al. 2019 [START_REF] Cunningham | A call for integrated metastatic management[END_REF]).

This led researchers to suggest that, at least for some tumors, treating at, or close to, the maximal tolerated dose should be replaced by treating at the minimal effective dose; that is, the minimal dose that allows to stabilize tumor size, subject to a sufficient quality of life of the patient. The aim is to slow down the growth of resistant cells, by maintaining competition with sensitive cells.

This idea, which is part of the broader framework of cancer adaptive therapy (Gatenby 2009 [START_REF] Robert A Gatenby | Adaptive Therapy[END_REF]), has been tested in vitro, in mice models and on human patients suffering from metastatic castrate-resistant prostate cancer (Gatenby et al. 2009 [START_REF] Robert A Gatenby | Adaptive Therapy[END_REF], Silva et al. 2012 [START_REF] Silva | Evolutionary Approaches to Prolong Progression-Free Survival in Breast Cancer[END_REF], Enriquez-Navas et al. 2016 [START_REF] Enriquez-Navas | Exploiting evolutionary principles to prolong tumor control in preclinical models of breast cancer[END_REF], Zhang et al. 2017 [START_REF] Zhang | Integrating evolutionary dynamics into treatment of metastatic castrate-resistant prostate cancer[END_REF]: trial NCT02415621, Bacevic and Noble et al. 2017 [START_REF] Bacevic | Spatial competition constrains resistance to targeted cancer therapy[END_REF], Smalley et al. 2019 [START_REF] Smalley | Leveraging transcriptional dynamics to improve braf inhibitor responses in melanoma[END_REF], Strobl et al. 2020 [START_REF] Maximilian | Turnover modulates the need for a cost of resistance in adaptive therapy[END_REF], Bondarenko et al. 2021 [START_REF] Bondarenko | Metronomic chemotherapy modulates clonal interactions to prevent drug resistance in non-small cell lung cancer[END_REF], Wang et al. 2021aWang et al. , 2021b [11, 12] [11, 12], Farrokhian et al. 2022 [START_REF] Farrokhian | Measuring competitive exclusion in non-small cell lung cancer[END_REF]). Other clinical trials are ongoing or starting in prostate cancer (NCT03511196, NCT05393791), melanoma (NCT03543969), rhabdomyosarcoma (NCT04388839) and ovarian cancer (ACTOv/NCT05080556) 1 .

On the theoretical side, several mathematical models of tumor containment have been studied (e.g., [START_REF] Martin | Optimal control of tumor size used to maximize survival time when cells are resistant to chemotherapy[END_REF] [START_REF] Martin | Optimal control of tumor size used to maximize survival time when cells are resistant to chemotherapy[END_REF], Monro and Gaffney 2009 [START_REF] Monro | Modelling chemotherapy resistance in palliation and failed cure[END_REF], Gatenby et al. 2009 [START_REF] Robert A Gatenby | Adaptive Therapy[END_REF], Silva et al. 2012 [START_REF] Silva | Evolutionary Approaches to Prolong Progression-Free Survival in Breast Cancer[END_REF], Carrère 2017 [START_REF] Carrère | Optimization of an in vitro chemotherapy to avoid resistant tumours[END_REF], Zhang et al. 2017 [START_REF] Zhang | Integrating evolutionary dynamics into treatment of metastatic castrate-resistant prostate cancer[END_REF], Bacevic and Noble et al. 2017 [START_REF] Bacevic | Spatial competition constrains resistance to targeted cancer therapy[END_REF], Hansen et al. 2017 [START_REF] Hansen | How to Use a Chemotherapeutic Agent When Resistance to It Threatens the Patient[END_REF], [START_REF] Gallaher | Spatial Heterogeneity and Evolutionary Dynamics Modulate Time to Recurrence in Continuous and Adaptive Cancer Therapies[END_REF] [START_REF] Gallaher | Spatial Heterogeneity and Evolutionary Dynamics Modulate Time to Recurrence in Continuous and Adaptive Cancer Therapies[END_REF], [START_REF] Cunningham | Optimal control to develop therapeutic strategies for metastatic castrate resistant prostate cancer[END_REF] [START_REF] Cunningham | Optimal control to develop therapeutic strategies for metastatic castrate resistant prostate cancer[END_REF], Pouchol 2018 [START_REF] Pouchol | Asymptotic analysis and optimal control of an integro-differential system modelling healthy and cancer cells exposed to chemotherapy[END_REF], Carrère and Zidani 2020 [START_REF] Carrère | Stability and reachability analysis for a controlled heterogeneous population of cells[END_REF], Strobl et al. 2020 [START_REF] Maximilian | Turnover modulates the need for a cost of resistance in adaptive therapy[END_REF], Cunningham et al. 2020 [START_REF] Cunningham | Optimal control to reach eco-evolutionary stability in metastatic castrate-resistant prostate cancer[END_REF]) leading to the first workshop on Cancer Adaptive Therapy Models (CATMo; https://catmo2020.org/). However, many of these models make very specific assumptions, e.g., logistic tumor growth with a specific effect of intra-tumor competition and a specific treatment kill-rate (Zhang et al., 2017 [6], [START_REF] Cunningham | Optimal control to develop therapeutic strategies for metastatic castrate resistant prostate cancer[END_REF] [START_REF] Cunningham | Optimal control to develop therapeutic strategies for metastatic castrate resistant prostate cancer[END_REF], Carrère 2017 [START_REF] Carrère | Optimization of an in vitro chemotherapy to avoid resistant tumours[END_REF], Strobl et al. 2020 [START_REF] Maximilian | Turnover modulates the need for a cost of resistance in adaptive therapy[END_REF]). This makes it difficult to generalize their conclusions.

Viossat and Noble (2021) [START_REF] Viossat | A theoretical analysis of tumour containment[END_REF] recently analysed a more general model with two types of tumor cells: sensitive and fully resistant to treatment. The model takes the form: dS dt (t) = S(t)g S (S(t), R(t), L(t)) dR dt (t) = R(t)g R (S(t), R(t))

(Model 1)
where S(t) and R(t) are the total number of sensitive and resistant cells at time t, L(t) is the current dose or treatment level, and g S and g r are per-cell growth-rate functions. They identified qualitative assumptions under which, among other results, containing the tumor at its initial size maximizes the time at which the tumor becomes larger than at the beginning of treatment (for an idealized form of containment) or is close to maximizing it (for a more realistic form). Similarly, an idealized form of containment at a larger threshold size maximizes the time at which tumor size becomes larger than this threshold. By contrast, eliminating all sensitive cells at treatment initiation -an idealized form of an aggressive treatment -leads to the quickest time to progression beyond any threshold size, among all treatments that eliminate sensitive cells before this threshold size is crossed. Some of the assumptions of Viossat and Noble are however debatable. In particular, they assume that the higher the number of resistant cells, the lower the growth rate of sensitive cells. Formally, function g S is nonincreasing in R. This assumption helps to compare the size of sensitive populations across treatments. To see why, assume that sensitive cells hamper the growth of resistant cells (that is, g R is non-increasing in S), and consider two constant dose treatments, with doses L 1 and L 2 > L 1 , respectively, and the same initial conditions. Since treatment 2 is more aggressive, it initially leads to a smaller sensitive population, hence a larger resistant population than treatment 1: for t > 0 small enough, S 2 (t) < S 1 (t) and R 2 (t) ≥ R 1 (t). If the growth-rate of sensitive cells g S is non-increasing in R, the fact that treatment 2 is more aggressive and leads to a larger resistant population both negatively affect the sensitive population under treatment 2, ensuring that the sensitive population remains smaller under treatment 2 than under treatment 1: S 2 < S 1 . This itself ensures that R 2 remains larger than R 1 . The inequalities S 2 < S 1 and R 2 ≥ R 1 thus propagate, and hold for all times t > 0. By contrast, if the growth-rate of sensitive cells g S increases with R, the fact that R 2 ≥ R 1 might boost the growth of sensitive cells under treatment 2, even though treatment 2 is more aggressive. But if the sensitive population becomes larger under treatment 2, the inequality R 2 ≥ R 1 might also cease to hold, and the whole argument of Viossat and Noble seems to break.

Unfortunately, assuming g S non-increasing in R, which may seem a natural consequence of competition between tumor cells, is actually problematic. Indeed, it is not satisfied in the Gompertzian model from [START_REF] Monro | Modelling chemotherapy resistance in palliation and failed cure[END_REF] [START_REF] Monro | Modelling chemotherapy resistance in palliation and failed cure[END_REF] that Viossat and Noble use for simulations:

dS dt (t) = ρ ln(K/N (t)) (1 -L(t))S(t), dR dt (t) = ρ ln(K/N (t)) R(t), ( Model 2)
where

N (t) = S(t) + R(t)
is the total number of tumor cells. More precisely, in the absence of treatment (L(t) = 0), the growth-rate of sensitive cells is decreasing in N , hence in R; however, if the treatment level is high enough (L(t) > 1), the opposite happens, and a large resistant population slows down the regression of the sensitive population. This reflects the fact that chemotherapy typically attacks cells that are actively dividing. For various reasons (e.g., boundary growth), a larger tumor size is thought to be associated with a lower growth-fraction, i.e., a lower proportion of cells actively dividing (Laird 1964 [START_REF] Kane | Dynamics of tumor growth[END_REF]; Norton and Simon 1977 [START_REF] Larry | Tumor size, sensitivity to therapy, and design of treatment schedules[END_REF]; Gerlee 2013 [START_REF] Gerlee | The model muddle: in search of tumor growth laws[END_REF]). Thus the presence of additional resistant cells, by making the tumor larger, makes more sensitive cells quiescent, and shields them against the effect of treatment. As a result, the growth rate of sensitive cells is not always decreasing in R, and the assumptions of Viossat and Noble are not satisfied. The problem occurs for all Norton-Simon models (Norton and Simon 1977 [START_REF] Larry | Tumor size, sensitivity to therapy, and design of treatment schedules[END_REF]), where the growth of the sensitive population takes the form:

dS dt (t) = S(t)g(N (t))(1 -L(t))
for some per-cell growth rate function g. It also occurs for birth-death models with a Norton-Simon treatment kill-rate (Strobl et al. 2020 [START_REF] Maximilian | Turnover modulates the need for a cost of resistance in adaptive therapy[END_REF]):

dS dt (t) = S(t) [b(N (t))(1 -L(t)) -d(N (t))]
where b(N ) and d(N ) are birth-and death-rates in the absence of treatment. Another issue is that Model 1 does not consider mutations from sensitive to resistant cells. This is problematic because one of the theoretical motivations for aggressive treatments is to decrease tumor size in order to limit the number of reproduction events, hence of possible appearance of resistant cells by mutation. Key-contributions to the tumor containment literature analyzed the trade-off between increasing competition (by allowing many sensitive cells to survive) and decreasing the number of mutations from sensitive to resistant cells (Martin et al. 1992[16], Hansen et al. 2017 [START_REF] Hansen | How to Use a Chemotherapeutic Agent When Resistance to It Threatens the Patient[END_REF])

The purpose of our work is to generalize the results of Viossat and Noble to models that encompass Norton and Simon models, and, at least to a certain extent, allow for mutations from sensitive to resistant cells. Mathematically, this is achieved by formulating the model in terms of absolute growth-rates and, more importantly, by replacing a direct analysis of the evolution through time of the number of sensitive and resistant cells, S(t) and R(t), by an analysis of the induced trajectory in what we call the R -N plane, where N = S + R describes the total tumor size. These trajectories describe the evolution of the total size N of the tumor as a function of the size R of the resistant population. This turns out to be an efficient technique, allowing to generalize essentially all results of Viossat and Noble, including the optimality or near-optimality of containment treatments.

The remainder of this article is organized as follows: the model is described in the next section. Results are presented in Section 3, proved in Section 4 and discussed in Section 5. The Appendix elaborates on the extent to which our model allows for mutations from sensitive to resistant cells, and derives the comparison principle on which our results are based.

Model

We consider a model with two types of tumors cells: sensitive to treatment, and fully resistant. Their growth is described by differential equations of the form:

dS dt (t) = φ S (S(t), R(t), L(t)), S(0) = S 0 ≥ 0 dR dt (t) = φ R (S(t), R(t)), R(0) = R 0 > 0 (Model 3)
where φ S and φ r are continuously differentiable absolute growth-rate functions. The quantities φ S (0, R, L) and φ R (S, 0, L) are assumed non-negative to ensure that population sizes cannot become negative. Let N (t) = S(t) + R(t) and N 0 = S 0 + R 0 . We make the following assumptions:

• The patient dies when tumor size reaches a critical size N crit > N 0 .2

• The size of an untreated tumor increases:

φ S (S, R, 0) + φ R (S, R) > 0 if N ≤ N crit .
• The higher the treatment level, the lower the growth-rate of sensitive cells: φ S is non-increasing in L.

• The resistant population keeps growing: φ R (S, R) > 0 whenever R > 0 and N ≤ N crit , so that the tumor is incurable if, as we assume, resistant cells are initially present.

• If R ≥ R 0 and N ≤ N crit , for a given number of resistant cells, the larger the sensitive population, the lower the growth-rate of resistant cells: φ R is non-increasing in S.

This last assumption models competition for resources (space, glucose, oxygen) or some other form of inhibition of resistant cells by sensitive cells (Bondarenko et al, 2021 [10]). It neither forbids nor implies a cost of resistance, i.e., that in the absence of treatment, resistant cells grow slower than sensitive cells. In particular, we do not specify whether resistant cells compete more strongly with sensitive cells or with other resistant cells.

The difference with Viossat and Noble (2021) [START_REF] Viossat | A theoretical analysis of tumour containment[END_REF] is two-fold: first, the model is formulated in terms of absolute growth-rates, allowing for mutations from sensitive to resistant cells and back.

Second, we make no assumption on how the growth-rate of sensitive cells depends on the number of resistant cells. In particular, φ S is not assumed non-increasing in R. This model encompasses many previous models (Silva et al. 2012 [5], Carrère, 2017 [START_REF] Carrère | Optimization of an in vitro chemotherapy to avoid resistant tumours[END_REF], Bacevic and Noble et al. 2017 [START_REF] Bacevic | Spatial competition constrains resistance to targeted cancer therapy[END_REF], Hansen et al. 2017 [START_REF] Hansen | How to Use a Chemotherapeutic Agent When Resistance to It Threatens the Patient[END_REF], Strobl et al. 2020 [START_REF] Maximilian | Turnover modulates the need for a cost of resistance in adaptive therapy[END_REF]), including Model 2, its original formulation with mutations (Monro and Gaffney, 2009 [START_REF] Monro | Modelling chemotherapy resistance in palliation and failed cure[END_REF]), or explicit birth-death models with or without a Norton and Simon treatment effect (Strobl et al. 2020 [START_REF] Maximilian | Turnover modulates the need for a cost of resistance in adaptive therapy[END_REF]).

To analyse Model 3, it is useful to rewrite it in the equivalent form:

dN dt (t) = f N (N (t), R(t), L(t)) dR dt (t) = f R (N (t), R(t)) (Model 4)
where

f N (N, R, L) = φ S (N -R, R, L) + φ R (N -R, R) and f R (N, R) = φ R (N -R, R).
Our main assumptions are then that, on the domain

R 0 ≤ R ≤ N ≤ N crit , f N is non-increasing in L, positive if L = 0
, and f R is positive and non-increasing in N . We also assume that the treatment level cannot be larger than a constant L max (the treatment level corresponding to the maximal tolerated dose). Other assumptions are technical:

• f N and f R are continuously differentiable (on a neighborhood of the relevant domain: R 0 ≤ R ≤ N ≤ N crit and 0 ≤ L ≤ L max ).

• R(t) remains smaller than N (t) (this must be biologically, and follows from our assumption on Model 3 that φ S (0, R, L) is nonnegative).

• The treatment function L(•) is strongly piecewise continuously differentiable (our vocabulary) in the following sense: there exists a positive integer m and times t 0 = 0 < t 1 < ... < t m such that, on each interval [t k , t k+1 ), k ∈ {0, ..., m -1}, and on [t m , +∞), L coincides with a continuously differentiable function defined on a neighborhood of this interval.

This ensures among other things that, for a given initial condition and treatment, there is a unique solution to Model 4. To fix ideas, we assume that the solutions R(t) and N (t) are defined for all times (though they have no clear interpretation once N (t) > N crit ), and that they remain bounded. Both properties can be ensured by modifying growth-rate functions f N and f R on the domain N > N crit . This is without loss of generality since patients are then assumed already deceased.

Outcomes and treatments.

We compare the effect of various treatments on the time at which tumor size becomes larger than a given threshold. Depending on this threshold, this may correspond to:

• time to progression, defined as the time at which tumor size progresses beyond its initial size N 0 .3

• time to treatment failure: the time at which tumor size progresses beyond an hypothetical maximal tolerable tumor size N tol ≥ N 0 , after which the life of the patient is considered at risk or side-effects of the disease are too strong. 4• survival time, defined as the time at which tumor size becomes larger than a critical size N crit ≥ N tol .

Mathematically, results on time to progression and survival time may be obtained through results on time to treatment failure by taking N tol = N 0 , or N tol = N crit , respectively. For this reason, we focus on time to treatment failure.

We consider the following treatments:

• Constant dose treatments, including No treatment (noTreat): L(t) = 0, and Maximal Tolerated Dose (MTD): L(t) = L max throughout.

• Delayed MTD (del-MTD): do not treat until N = N tol for the first time, then treat at L max for ever.

• Containment at N tol (Cont): do not treat until N = N tol and then stabilize tumor size at N tol , as long as possible with a treatment level L(t) ≤ L max . Finally, treat at L max when N > N tol . Formally, during the stabilization phase, the treatment level is chosen so that dN/dt = 0 (e.g., L(t) = N (t)/S(t) in Model 2). Containment treatments are illustrated in Fig. 1, see also Fig. 1 of Viossat and Noble. 5 • Intermittent containment (Int), as in the prostate cancer clinical trial of Zhang et al. (2017): do not treat until N = N tol , then treat at L max until N = N min < N tol , then interrupt treatment until N = N tol , and iterate as long as possible. Finally, treat at L max when N > N tol . This is illustrated by Fig. 2. 6 • An arbitrary treatment, called the alternative treatment (alt): we only assume that 0 ≤ L(t) ≤ L max for all t and L(t

) = L max if N > N tol .
The times to treatment failure under these treatments will be denoted by t noT reat , t M T D , t delM T D , t Cont , t Int , and t alt , respectively.

Following (Martin et al. 2012 [29], Hansen et al. 2017 [START_REF] Hansen | How to Use a Chemotherapeutic Agent When Resistance to It Threatens the Patient[END_REF], Viossat and Noble 2021 [START_REF] Viossat | A theoretical analysis of tumour containment[END_REF]), we also consider idealized treatments, which assume that the sensitive population may be reduced arbitrarily quickly. These treatments are not realistic but are useful theoretical references. Ideal MTD(idMTD) eliminates all sensitive cells instantaneously at the beginning of treatment. Delayed ideal MTD (del-idMTD) lets tumor grow to N tol and then eliminates all sensitive cells. Ideal containment (idCont) lets tumor size grow to N tol , and then stabilizes it as long as some sensitive cells remain. Finally, Ideal intermittent containment (idInt) lets tumor size grow to N tol and then maintains it between N min ≤ N tol and N tol as long as some sensitive cells remain.7 Under these idealized treatments, treatment fails (i.e. tumor size progresses beyond N tol ) when the resistant population reaches size N tol . Sensitive cells have then been fully eliminated. Times to treatment failure are denoted by t idM T D , t del-idM T D , t idCont , and t idInt , respectively. 8To make our life easy, we assume that all treatments we consider may be implemented through a piecewise continuously differentiable treatment level function L(t) (up to possible downward jumps in the sensitive population for idealized treatments), instead of deriving this result from the implicit function theorem and appropriate regularity assumptions.

Results

We show that, up to natural additional assumptions for comparisons of sensitive cell populations, all results of Viossat and Noble on Model 1 still hold on Model 3 (or equivalently Model 4), in spite of our less restrictive assumptions. The results are described below and proved in the next section.

The key point is that if treatment level is never larger than a given constant for treatment 1, and never smaller than the same constant for treatment 2, then the resistant population is no larger under treatment 1 than under treatment 2.

Proposition 1. Consider solutions of Model 4 associated to two treatments L 1 (t) and L 2 (t). 9 If there exists a constant L such that for all t ≥ 0,

L 1 (t) ≤ L ≤ L 2 (t), then R 1 (t) ≤ R 2 (t) for all t ≥ 0.
It follows that for constant dose treatments, lowering the dose or delaying treatment leads to a lower resistant population:

Proposition 2. (constant dose treatments) a) Consider two constant dose treatments L 1 (t) = L 1 and L 2 (t) = L 2 . If L 2 ≥ L 1 , then R 1 (t) ≤ R 2 (t) for all t ≥ 0. 10 b) Assume that L 1 (t) = L > 0 for all t ≥ 0, while L 2 (t) = 0 until N = N start ≥ N 0 , and then L 2 (t) = L. Then R 1 (t) ≤ R 2 (t) for all t ≥ 0.
Proposition 1 also implies that not treating minimizes the resistant population while MTD maximizes it:

Proposition 3. (MTD maximisez resistance) For all t ≥ 0, R noT reat (t) ≤ R alt (t) ≤ R M T D (t).
Of course, not treating is typically not an option, as the number of sensitive cells would explode, but containment is. One of our main results is that containment minimizes the resistant population among all treatments treating at L max after failing.

Proposition 4. (containment minimizes resistance) For all t ≥ 0, R Cont (t) ≤ R alt (t)
It follows that N Cont ≤ N alt +(S Cont -S alt ). Thus, assuming that the tumor is eventually mostly resistant under the containment treatment, tumor size should eventually be smaller, or at least not substantially larger under the containment treatment than under any alternative one. This suggests that, under our assumptions, among treatments that treat at L max when N > N tol , containment should be close to maximizing survival time. Similarly, the fact that the resistant population is larger under MTD that under any alternative treatment suggests that most alternative treatments should eventually lead to a lower tumor size and a longer survival time than MTD.

More precise statements may be made for idealized versions of containment and MTD: ideal containment maximizes time to treatment failure, while ideal MTD minimizes it among all treatments eliminating sensitive cells before failing. Moreover, ideal containment eventually leads to a lower tumor size and ideal MTD to a larger tumor size than any such alternative treatment.

Proposition 5. (comparison with ideal MTD and ideal containment) a) t alt ≤ t idCont . b) Consider an alternative treatment eliminating sensitive cells before failing, that is, such that S alt (t alt ) = 0. Then: b1)

t alt ≥ t idM T D ; b2) for all t ≥ 0, R idCont (t) ≤ R alt (t) ≤ R idM T D (t); b3) for all t ≥ t alt , N idCont (t) ≤ N alt (t) ≤ N idM T D (t).
In particular, survival time is larger with ideal containment and lower with ideal MTD than with any alternative treatment such that S alt (t alt ) = 0.

The next result shows that intermittent containment between N min and N tol > N min leads to outcomes that are intermediate between those of containment at the larger threshold N tol and those of containment at the lower threshold N min (ContNmin). The latter lets tumor size grow until N = N min (or treats at L max until N = N min if N 0 > N min ), and then stabilizes tumor size at N min as long as possible with a treatment level L(t) ≤ L max . In the idealized form, ideal containment at N min , tumor size is stabilized at N min as long as some sensitive cells remain (and initially instantly reduced to the maximum of N min and R 0 , if N min > N 0 ).

Proposition 6. (dose modulation versus treatment vacation) a) For all t ≥ 0, R Cont (t) ≤ R Int (t) ≤ R ContN min (t), and similarly, R idCont (t) ≤ R idInt (t) ≤ R idContN min (t). b) t idContN min ≤ t idInt ≤ t idCont c) For all t ≥ t idInt , N idCont (t) ≤ N idInt (t) ≤ N idContN min (t).
This result suggests that, if the lower threshold N min is close to the larger threshold N tol , there should be little difference between outcomes of containment and intermittent containment, that is, between a continuous low dose treatment based on dose modulation and an intermittent high dose treatment based on treatment vacation. Of course, this disregards many possible differences between these two approaches. For instance, dosemodulation might lead to a more regular vascularization of the tumor, which might be key for an efficient drug delivery (Enriquez-Navas et al., 2016 [START_REF] Enriquez-Navas | Exploiting evolutionary principles to prolong tumor control in preclinical models of breast cancer[END_REF]).

We now compare all reference treatments.

Proposition 7. (comparison between all reference treatments) a) For all t ≥ 0:

a1) R noT reat (t) ≤ R Cont (t) ≤ R Int (t) ≤ R del-M T D (t) ≤ R M T D (t) ≤ R idM T D (t) and a2) R noT reat (t) ≤ R idCont (t) ≤ R idInt (t) ≤ R del-idM T D (t) ≤ R idM T D (t) b) t idM T D ≤ t del-idM T D ≤ t idInt ≤ t idCont c) For all t ≥ t idCont , N idCont (t) ≤ N idInt (t) ≤ N del-idM T D (t) ≤ N idM T D (t).
Sensitive population sizes may also be compared under two mild additional assumptions:

(A1) Not treating maximizes the sensitive population. (A2) The sensitive population decreases if the tumor is treated at L max . These assumptions hold for Model 2, assuming L max ≥ 1, and for most models we are aware of. They lead to the same comparison for sensitive population sizes as in Viossat and Noble, that is, the opposite as for resistant population sizes. 11Proposition 8. (comparison of sensitive populations) Assume that (A1) and (A2) hold. Then for all t ≥ 0:

12 a) S idM T D (t) ≤ S M T D (t) ≤ S alt (t) ≤ S Cont (t) ≤ S noT reat (t) b) S ContN min (t) ≤ S Int (t) ≤ S Cont (t) and S del-M T D (t) ≤ S Int (t) c) S idContN min (t) ≤ S idInt (t) ≤ S idCont (t) d) S idM T D (t) ≤ S del-idM T D (t) ≤ S idInt (t) ≤ S idCont (t) ≤ S noT reat (t)

Proofs

Viossat and Noble's proofs build on their Proposition 1, which gives conditions allowing them to compare the resistant populations or the sensitive populations under two different treatments. The following part of this result is still true in our framework, with the same proof:

Lemma 9. Let 0 ≤ t 0 ≤ t 1 . Consider two solutions (S 1 , R 1 ) and (S 2 , R 2 ) of Model 3, associated to treatment functions L 1 and L 2 , respectively. Assume that: i) R 1 (t 0 ) ≤ R 2 (t 0 ), and ii) S 1 (t 0 ) ≥ S 2 (t 0 ). If: iiia) S 1 (t) ≥ S 2 (t) on [t 0 , t 1 ], or: iiib) N 1 (t) ≥ N 2 (t) on [t 0 , t 1 ], then R 1 (t) ≤ R 2 (t) and S 1 (t) ≥ S 2 (t) on [t 0 , t 1 ].
What is no longer true is that the same conclusions hold if iiia) or iiib) is replaced by iiic): L 1 (t) ≤ L 2 (t) for all t. For instance, in Model 2, if L 1 (t 0 ) = L 2 (t 0 ) > 1, R 1 (t 0 ) < R 2 (t 0 ) and S 1 (t 0 ) = S 2 (t 0 ), then S 2 becomes immediately larger than S 1 . This will slow down the growth of the resistant population under treatment 2. Thus, conceivably, R 2 could later on become smaller than R 1 .

We thus use a new proof technique. Instead of studying directly the evolution of the resistant population R, the sensitive population S, or the total tumor size N as a function of time, we first study, and compare across treatments, the evolution of tumor size N as a function of the number of resistant cells. In other words, we compare trajectories in the R -N plane, that is, the sets of points (R(t), N (t)) for all t ≥ 0.

To be more formal, fix a treatment L, and let R ∞ = lim t→+∞ R(t). Since the resistant population increases continuously, for any r ∈ [R 0 , R ∞ ), there exists a unique time t(r) at which the resistant population has size r, that it, R(t(r)) = r. Denote by S(r), Ñ (r) = S(r) + r, and L(r), the number of sensitive cells, the total number of tumor cells, and the treatment level at time t(r), that is, when the resistant population reaches size r. All these functions may be shown to be piecewise continuously differentiable, and S and Ñ are also continuous. The graph of function Ñ coincides with the trajectory of the solution in the R -N plane. It may be analyzed by noting that function Ñ satisfies the differential equation:

d Ñ dr = G( Ñ , r) where G( Ñ , r) = f N ( Ñ , r, L(r)) f R ( Ñ , r)
Trajectories in the R -N plane, and their connections to the evolution of tumor size and of the resistant population as a function of time are illustrated in Figure 1.

Key lemmata

Our first result shows that if, for any resistant population level r, tumor size is larger under treatment 1 than under treatment 2, then at any time t, the resistant population is smaller under treatment 1 than under treatment 2.

The intuition is the following: at the time t i (r) when the resistant population reaches size r under treatment i, the speed at which the resistant population increases is given by:

dR i dt (t i (r)) = f R (N i (t i (r)), R i (t i (r)) = f R ( Ñi (r), r)
Since f R is non-increasing in N , it follows that if N 1 (r) ≥ N 2 (r), the resistant population will increase quicker from r to r + dr under treatment 2 than under treatment 1 (dr is a small positive increment). If this holds for all resistant population sizes r, then R 2 will remain no-smaller than R 1 at all times t ≥ 0.

Lemma 10. Let L 1 (t) and L 2 (t) be two different treatments. Consider solutions (N 1 , R 1 ) and (N 2 , R 2 ) of Model 4 associated to these treatments such that R

1 (0) = R 2 (0) = R 0 . If Ñ1 (r) ≥ Ñ2 (r) for all r in [R 0 , min{R ∞ 1 , R ∞ 2 }), then R ∞ 1 ≤ R ∞ 2 and R 1 (t) ≤ R 2 (t) for all t ≥ 0. Moreover, if on an interval [r 1 , r 2 ], S1 (r) is non-increasing or S2 (r) is non-increasing, then S 1 (t) ≥ S 2 (t) for all t in [t 1 (r 1 ), t 2 (r 2 )]. Proof. Consider a time t ≥ 0 such that R 1 (t) < R ∞ 2 , so that Ñ2 (R 1 (t)
) is well defined. Since N (t) = Ñ (R(t)), Ñ1 ≥ Ñ2 and f N is non-increasing in N , we obtain:

dR 1 dt (t) = f R (N 1 (t), R 1 (t)) = f R ( Ñ1 (R 1 (t)), R 1 (t)) ≤f R ( Ñ2 (R 1 (t)), R 1 (t)) =:F (R 1 (t))
the second from the fact that S1 ≥ S2 . If it is S2 which is non-increasing (which implicitly requires r 2 < R ∞ 2 , so that S2 is well-defined on [r 1 , r 2 ]), then:

S 1 (t) = S1 (R 1 (t)) ≥ S2 (R 1 (t)) ≥ S2 (R 2 (t)) = S 2 (t).
The first inequality follows from the fact that S1 ≥ S2 , the second from the fact that S2 is non-increasing on [r 1 , r 2 ].

Assume now that for any resistant population level r, the treatment level when the resistant population reaches size r is lower for treatment 1 than for treatment 2. Our second result shows that the tumor size when the resistant population reaches size r is then always larger for treatment 1 than for treatment 2. By the previous lemma, this implies that the resistant population is always smaller under treatment 1 than under treatment 2.

Lemma 11. Let L 1 (t) and L 2 (t) be two different treatments such that

L 1 (t) ≤ L ≤ L 2 (t),
for a certain positive number L, or more generally such that L1 (r

) ≤ L2 (r) for all r in [R 0 , R * ) where R * = min{R ∞ 1 , R ∞ 2 }. Consider solutions of Model 4 associated to these treatments such that R 1 (0) = R 2 (0) and N 1 (0) ≥ N 2 (0). Then Ñ1 (r) ≥ Ñ2 (r) for all r in [R 0 , R * ). Therefore by Lemma 10, R ∞ 1 ≤ R ∞ 2 and R 1 (t) ≤ R 2 (t) for all t ≥ 0.
Proof. Since f N is non-increasing in L, and for all r in [R 0 , R * ), L1 (r) ≤ L2 (r), we get:

d Ñ1 dr = f N ( Ñ1 , r, L1 (r)) f R ( Ñ1 , r) ≥ f N ( Ñ1 , r, L2 (r)) f R ( Ñ1 , r) =: G 2 ( Ñ1 , r) while d Ñ2 dr = G 2 ( Ñ2 , r). Moreover, Ñ2 (R 0 ) = N 2 (0) ≤ N 1 (0) = Ñ1 (R 0 ).
Therefore, by the comparison principle (Proposition 14, item a), in Appendix B), Ñ1 (r) ≥ Ñ2 (r) for all r in [R 0 , R * ). Then apply Lemma 10.

Proof of propositions 1 to 8

Proposition 1 follows from Lemma 11, and Propositions 2 and 3 from Proposition 1.

Proof of Proposition 4. For later purposes, let us prove a more general result:

For all t ≥ 0,

R noT reat (t) ≤ R Cont (t) ≤ R alt (t) ≤ R M T D (t) ≤ R idM T D (t). (1)
This follows from Lemma 10 and the fact that, whenever these comparisons make sense:

ÑidMT D (r) ≤ ÑMT D (r) ≤ Ñalt (r) ≤ ÑCont (r) ≤ ÑnoT reat (r) (2)
To prove (2), note that for any alternative treatment, ÑidMT D (r) = r ≤ Ñalt (r), in particular, ÑidMT D (r) ≤ ÑMT D (r), and by Lemma 11 with L = 0, Ñalt (r) ≤ ÑnoT reat (r), in particular ÑCont (r) ≤ ÑnoT reat (r). Moreover, under the constraint L alt (t) ≤ L max , it follows from Lemma 11 with L = L max that ÑMT D (r) ≤ Ñalt (r). It remains to prove that Ñalt (r) ≤ ÑCont (r) for all r ∈ [R 0 , R * ), where

R * = min{R ∞ alt , R ∞ Cont }.
The notation we introduce is illustrated in Fig. 4. Let r 1 = min{r ≥ R 0 , ÑCont (r) = N tol }. When r ≤ r 1 , ÑCont (r) = ÑnoT reat (r) ≥ Ñalt (r) as explained above. Moreover, for all r ≥ r 1 , ÑCont (r) ≥ N tol . Thus, assuming by contradiction that there exists r 2 ≥ r 1 such that ÑCont (r 2 ) < Ñalt (r 2 ), it follows that Ñalt (r 2 ) > N tol . Let

r max = max{r ≤ r 2 , Ñalt (r) ≤ N tol }.
Note that since Ñalt (r 1 ) < ÑCont (r 1 ) = N tol , we must have r max ≥ r 1 . Therefore, Ñalt (r max ) = N tol ≤ ÑCont (r max ). Moreover, on (r max , r 2 ), Ñalt (r) > N tol , hence Lalt (r) = L max ≥ LCont (r). By a variant of Lemma 11 (comparing treatments starting when the initial resistant population size is r max rather than R 0 ), it follows that Ñalt (r 2 ) ≤ ÑCont (r 2 ), a contradiction. Proof of Proposition 5. Proof of a): Let

t 1 = min{t ≥ 0, N Cont (t) = N tol }. For t ≤ t 1 , R idCont (t) = R noT reat (t) ≤ R alt (t) (the inequality follows from Proposition 3). If t alt ≥ t 1 then, as in Viossat and Noble, on [t 1 , t alt ], N idCont (t) ≥ N tol ≥ N alt (t) so R idCont (t) ≤ R alt (t) by Lemma 9. Thus, R idCont (t alt ) ≤ R alt (t alt ) ≤ N alt (t alt ) = N tol . It follows that t idCont ≥ t alt , since ideal containment fails when R idCont = N tol .
Proof of b): assume now that S alt (t alt ) = 0 (which only makes sense for idealized alternative treatments). Then R alt (t alt ) = N alt (t alt ) = N tol . Thus, as in Viossat and Noble:

N tol = N alt (t alt ) = R alt (t alt ) ≤ R idM T D (t alt ),
so t idM T D ≤ t alt . This proves b1). We now prove the results on ideal containment. On [t alt , t idCont ],

R idCont (t) ≤ N idCont (t) ≤ N tol = N alt (t alt ) = R alt (t alt ) ≤ R alt (t) = N alt (t)
where the last inequality comes from the fact that for all t ≥ t alt , S alt (t) = 0. Moreover, after treatment failure, R alt and R idCont both satisfy the autonomous equation dR/dt = f R (0, R). By invariance of solutions of autonomous equations through translation in time, this implies that for all

t ≥ t idCont , R idCont (t) = R alt (t -[t idCont -t alt ]) ≤ R alt . For t ≤ t idCont , the inequality R idCont (t) ≤ R alt (t)
was derived in the proof of a). Therefore, R idCont (t) ≤ R alt (t) for all t ≥ 0. Finally, for all t in [t alt , t idCont ],

N idCont (t) ≤ N tol = R alt (t alt ) ≤ R alt (t) = N alt (t), while for all t ≥ t idCont , N idCont (t) = R idCont (t) ≤ R alt (t) = N alt (t)
. This completes the proof.

Proof of Proposition 6. Proof of a): The inequalities R Cont ≤ R Int and R idCont ≤ R idInt follow from the proof of Proposition 4 (see Eq. ( 1)) and from Proposition 5. The fact that R Int ≤ R ContN min follows from Lemma 10 and the fact that, as shown below: for all r, ÑContNmin (r) ≤ ÑInt (r).

To prove this, note that for r ≤ r min := min{r ≥ R 0 , ÑnoT reat (r) = N tol }, both treatments coincide so ÑContNmin (r) = ÑInt (r). For r ≥ r min , the argument is as in the proof of ÑCont (r) ≥ Ñalt (r) for r ≥ r 1 in Proposition 4. Similarly, it is easily seen that ÑidContNmin (r) ≤ ÑidInt (r) for all r, so R idInt ≤ R idContN min by Lemma 10. Proof of b):

N tol = N idInt (t idInt ) = R idInt (t idInt ) ≤ R idContN min (t idInt
) by a), hence t idContN min ≤ t idInt . The second inequality follows from item a) of Proposition 5.

Proof of c): Using a), for t ≥ t idInt , N idInt = R idInt ≤ R idContM in = N idContM in , and the first inequality follows from item b3) of Proposition 5.

Proof of Proposition 7. Proof of a1): to see that R Int (t) ≤ R del-M T D (t), note that as long as tumor size is lower than N tol , both treatments coincide, then apply Proposition 3 from that point on. The other inequalities have already been proved. The proof of a2) is similar.

Proofs of b) and c): in b), the inequality t del-idM T D ≤ t idInt follows from item b1) of Proposition 5, applied from the (common) time when tumor size reaches N tol under both treatments, other inequalities were shown already. The proof of c) is as in Viossat and Noble.

Proof of Proposition 8. We first need a lemma. Lemma 12. a) Let N * ≥ 0. Consider a solution (N, R) of Model 4 under a treatment such that L(t) = L max whenever N (t) > N * . Let t ≥ 0 be such that N ( t) ≥ N tol . If the sensitive population decreases when treated at L max , then for all t ≥ t, S(t) ≤ S( t). If moreover N (t) ≥ N * for all t ≥ t, then S is non-increasing on [ t, +∞).

b) Under containment (respectively, containment at N min ), once tumor size reaches N tol for the first time (respectively, N min ), the sensitive population is non-increasing.

Proof. a) The idea is that when N > N * , S is non-increasing by assumption, and when N (t) ≤ N * for t > t, the sensitive population must have decreased since time t because the resistant population increased (by assumption) and total tumor size did not. Formally, let t ≥ t. If for all τ in ( t, t), N (τ ) > N * , hence L(τ ) = L max , then S is non-increasing on [ t, t] by assumption, therefore S(t) ≤ S( t). Otherwise, let t max = max{τ ≤ t, N (τ ) ≤ N * }. The previous argument implies that S(t) ≤ S(t max ). Moreover, since R is increasing,

S(t max ) = N (t max ) -R(t max ) ≤ N tol -R(t max ) ≤ N ( t) -R( t) = S( t)
Therefore, S(t) ≤ S( t). Finally, if for any t 1 ≥ t, N (t 1 ) ≥ N * , then the previous result applied from t 1 on shows that for any t 2 ≥ t 1 , S(t 2 ) ≤ S(t 1 ), hence S is non-increasing on [ t, ∞).

b) For containment, this follows from a) with N * = N tol and the fact that once tumor size reaches N tol under containment, it never becomes smaller. The proof for containment at N min is the same with N min replacing N tol .

We now prove Proposition 8. Proof of a): the first inequality is trivial since S idM T D = 0 (we only mentioned it to show that all inequalities from Eq. 1 are reversed). The inequality S M T D (t) ≤ S alt (t) follows from Lemma 10, the fact that ÑMT D (r) ≤ Ñalt (r) (see Eq. ( 2)), and the fact that SMT D (r) is non-increasing by Assumption (A2). The last inequality follows from Assumption (A1), or, independently of (A1), from Lemma 10, the fact that ÑCont (r) ≤ ÑnoT reat (r) (see Eq. ( 2)), and that once Containment starts treating, S Cont is non-increasing (Lemma 12, item b)).

Let us now prove that S alt (t) ≤ S Cont (t). Let r 1 = min{r ≥ R 0 , ÑCont (r) = N tol }. For t ≤ t Cont (r 1 ), containment does not treat so S Cont (t) = S noT reat (t) ≥ S alt (t) by Assumption (A1). Moreover, on [r 1 , R ∞ Cont ), ÑCont (r) ≥ Ñalt (r) and ÑCont (r) ≥ N tol , therefore SCont (r) is non-increasing by Lemma 12. Thus, by Lemma 10,

S Cont (t) ≥ S alt (t) for any t in [t Cont (r 1 ), t alt (R ∞ Cont ) ). Finally, let t ≥ max(t Cont (r 1 ), t alt (R ∞ Cont )), that is, such that R Cont (t) ≥ r 1 and R alt (t) ≥ R ∞ Cont . Since N Cont (t) ≥ N tol for all t ≥ t(r 1 ), it follows from Lemma 12 that S Cont is non-increasing on [t(r 1 ), +∞[, so S Cont (t) ≥ S ∞
Cont . Thus, it suffices to show that S alt (t) ≤ S ∞ Cont . There are two cases. Case 1: If Ñalt (R ∞ Cont ) ≥ N tol , then by Lemma 12, for all t ≥ t alt (R ∞ Cont ),

S alt (t) ≤ S alt (t alt (R ∞ Cont )) = Salt (R ∞ Cont ) ≤ S ∞

Cont

where the last inequality follows from the fact that for r < R ∞ Cont , Salt (r) ≤ SCont (r) due to Eq. ( 1), so that

Salt (R ∞ Cont ) = lim r→R ∞ Cont Salt (r) ≤ lim r→R ∞ Cont SCont (r) = lim t→+∞ S Cont (t) = S ∞ Cont Case 2: If Ñalt (R ∞ Cont ) < N tol , then as long as N alt (t) ≤ N tol , S alt (t) ≤ N tol -R alt (t) ≤ N tol -R ∞ Cont ≤ S ∞

Cont

Moreover, if at some time t, N alt ( t) = N tol (which must indeed happen), then S alt ( t) ≤ S ∞ Cont by the previous argument, and for all t ≥ t, by Lemma 12, S alt (t) ≤ S alt ( t) ≤ S ∞ Cont . This concludes the proof of a). Proof of b): The inequality S ContN min (t) ≤ S Int (t) follows from Lemma 10, the fact that ÑcontNmin (r) ≤ ÑInt (r), and the fact that once tumor size reaches N min , S ContN min is non-increasing (Lemma 12, item b)). The proof of S Int (t) ≤ S Cont (t) is as the proof of S alt (t) ≤ S Cont (t) (except that Assumption (A1) is not needed). Finally, the inequality S del-M T D ≤ S Int follows from S M T D ≤ S alt applied from the time at which tumor size reaches N tol .

Proof of c): we first prove S idContN min ≤ S idInt . Before tumor size reaches N min , both treatments coincide, then until t idContN min , N idContN min = N min ≤ N idInt while R idContN min ≥ R idInt , so S idContN min ≤ S idInt . Finally, for t ≥ t idContN min , S idContN min (t) = 0 ≤ S idInt (t). The proof of S idInt ≤ S idCont is similar.

Proof of d): the first two inequalities are trivial, the third one was proved in c). The last inequality follows from (A1) but also, independently of (A1), from the following argument: for t ≤ t idCont , N idCont ≤ N noT reat while R idCont ≥ R noT reat by Proposition 7, so S idCont ≤ S noT reat , and for t ≥ t idCont , S idCont = 0.

Discussion

Viossat and Noble [START_REF] Viossat | A theoretical analysis of tumour containment[END_REF] provided qualitative conditions ensuring that a strategy aiming at containment, not elimination, minimizes resistance to treatment and is close to maximizing time to treatment failure. Some of these conditions were however debatable. In particular, their analysis did not allow for mutations from sensitive to resistant cells, a major concern of some key contributions to the field (Martin et al. 1992 [16], Hansen et al. 2017 [START_REF] Hansen | How to Use a Chemotherapeutic Agent When Resistance to It Threatens the Patient[END_REF]), and did not apply to Norton-Simon models [START_REF] Larry | Tumor size, sensitivity to therapy, and design of treatment schedules[END_REF], which are standard to model chemotherapy. We showed how a refined analysis allows to handle these two issues. This suggests that containment strategies are likely to perform well in more general situations than was previously known.

While Viossat and Noble compared across treatments the values of resistant and sensitive populations as a function of time, we first compare the induced trajectories in the R-N plane, that is, tumor sizes not at a given time, but when the resistant population reaches a given size. We made the additional assumption that the resistant population keeps increasing. This is consistent with the assumption that in the presence of fully resistant cells, the tumor is incurable, and is technically helpful (as the trajectory in the R-N plane is then the graph of a function), but we conjecture that our results hold without this assumption.

What is crucial is that, all else being equal, a larger sensitive population leads to a lower resistant population growth-rate. For this reason, our analysis only allows for mutations from sensitive to resistant cells if an increase in the sensitive population size is more detrimental to the growth of the resistant population (through competition, or some other form of inhibition of resistant cells by sensitive cells) than it is beneficial (through mutations from sensitive to resistant cells). We show in Appendix A that this is typically the case for Gompertzian growth, or power-law models, at least in the absence of a strong resistance cost. 13There are however many other concerns with containment. Mutations, or phenotypic switching, could be modeled in other ways, and the fact that maintaining a relatively large tumor burden may lead to an accumulation of driver mutations remains a concern. Modeling patient death as occurring when the tumor reaches a critical size favors containment, and models in which the probability of death increases continuously with tumor size may lead to the conclusion that the expected survival time is lower under containment strategies than under more aggressive treatments (Mistry 2020 [START_REF] Hitesh | Evolutionary Based Adaptive Dosing Algorithms: Beware the Cost of Cumulative Risk[END_REF]). Considering only two types of tumors cells is restrictive, and even with only two types, if resistant cells are only partially resistant, the logic changes, as the growth of resistant cells may be slowed down not only indirectly, through competition with sensitive cells, but also directly, through treatment effect. The impact of a containment strategy on the development of new metastases is also unclear. On the other hand, we did not consider additional benefits of containment, such as reduced treatment toxicity, less drug-induced mutations (Kuosmanen et al. 2021 [31]) or a possible stabilization of tumor vasculature that could increase the efficiency of drug delivery (Enriquez-Navas et al. 2016 [START_REF] Enriquez-Navas | Exploiting evolutionary principles to prolong tumor control in preclinical models of breast cancer[END_REF]).

This article should not be seen as providing unambiguous support for containment strategies, but as part of a wider research program aiming at clarifying the conditions under which a strategy aiming at tumor stabilization is likely to perform better than a more aggressive treatment. Data allowing to fine-tune models is still scarce, but as new competition experiments are run, and new clinical trials open (NCT05393791, ACTOv/NCT05080556), more data should become available, allowing the community to reach more definite conclusions.

(4) into (3) and using that τ 1 and τ 2 are much smaller than 1 leads to:

ln N 0 + 1 ≥ - g(N 0 ) N 0 g (N 0 ) . (5)
Let us now consider various growth-models. Case 1 (power-law model): g(N ) = ρN -γ with 0 < γ < 1. Eq. ( 5) becomes:

ln

N 0 + 1 ≥ 1/γ.
Typical choices for γ are γ = 1/3 or γ = 1/4 (Gerlee, 2013 [START_REF] Gerlee | The model muddle: in search of tumor growth laws[END_REF]; Benzekry et al., 2014[33]; our γ corresponds to 1 -γ in these references). The condition then holds by a huge margin for any detectable tumor size. Case 2 (Gompertzian growth): g(N ) = ρ ln(K/N ). Eq (5) becomes:

ln N 0 + 1 ≥ ln(K/N 0 ), which is satisfied if K ≤ eN 2 .
Standard values of the carrying capacity in Gompertzian models are in the range 10 12 -10 13 (e.g., K = 2 × 10 12 in Monro and Gaffney (2009) [START_REF] Monro | Modelling chemotherapy resistance in palliation and failed cure[END_REF]). Eq. ( 5) is then satisfied for any detectable tumor size. Case 3 (logistic growth) : g(N ) = ρ(1 -K/N ). Eq (5) becomes:

ln N 0 + 1 ≥ K N 0 -1.
This condition need not be satisfied, depending on the interpretation of the model and parameter choices. For instance, [START_REF] Monro | Modelling chemotherapy resistance in palliation and failed cure[END_REF] [START_REF] Monro | Modelling chemotherapy resistance in palliation and failed cure[END_REF] take N 0 = 10 10 . Then ln N 0 23 and the condition is roughly K ≤ 2.5 × 10 11 , which is not satisfied for standard values of the carrying capacity K. 14 The condition would however be satisfied for larger initial tumor sizes, modeling late-stage treatments. Actually, when logistic growth models are used in the adaptive therapy literature, the initial tumor size is often assumed to be a large fraction of the carrying capacity (e.g., Zhang et al. 2017 [START_REF] Zhang | Integrating evolutionary dynamics into treatment of metastatic castrate-resistant prostate cancer[END_REF], Strobl et al. 2020 [START_REF] Maximilian | Turnover modulates the need for a cost of resistance in adaptive therapy[END_REF], Mistry 2020 [START_REF] Hitesh | Evolutionary Based Adaptive Dosing Algorithms: Beware the Cost of Cumulative Risk[END_REF]). This may be interpreted as modeling late-stage treatments, or as a model of local growth. In the latter case, the carrying capacity should be seen as the maximal number of cells for the current tumor volume (or equivalently, the variables N , S, R, K should be interpreted as densities). Assuming 10 10 tumor cells at tumor initiation, the estimate x r /τ 1 23 would still be valid, and Eq. (3) would become K/N ≤ 25, which is bound to be satisfied in a model of local growth.

Let us now consider three variants of Model 5.

Variant 1: birth-death model. In Model 5, the number of mutations is assumed proportional to the net growth-rate of the tumor. It would be natural to assume that the number of mutations is proportional to the net birth-rate. This would increase the effective mutation rate (that is, the average number of mutations relative to a given increase of tumor size). 15However, since the condition we found is insensitive to the precise mutation rates τ 1 and τ 2 , this is unlikely to affect the previous analysis.

Variant 2: late-stage treatment. The previous analysis is better suited for a first line treatment than a second or third line treatment, especially if resistance to the first treatment may be associated with resistance to ulterior ones. However, in such a situation (late-stage treatment), the initial resistant population is likely to be larger than the one given by ( 4), and so condition (3) is more likely to be satisfied.

Variant 3: cost of resistance in the baseline growth-rate. Consider the following variant of Model 5, with a different growth-rate parameter for sensitive cells and for resistant cells:

dS dt = ρ s g(N ) (1 -L)S -τ 1 ρ s g(N )S + τ 2 ρ r g(N )R, dR dt = ρ r g(N ) R + τ 1 ρ r g(N )S -τ 2 ρ r g(N )R. ( Model 6)
(the terms g(N ) in Model 5 correspond here to terms of the form ρg(N ), with ρ = ρ s or ρ = ρ r .) The absolute growth-rate of resistant cells is now

φ R (S, R) = ρ r g(N )R + τ 1 ρ s g(N )S -τ 2 ρ r g(N )R.
The condition for ∂φ R /∂S to be nonpositive becomes:

x r τ 1 (ρ r /ρ s -τ 2 ρ r /ρ s -τ 1 ) + 1 ≥ - g(N ) N g (N ) . (6)
Moreover, if ρ r is substantially smaller than ρ s , then the resistant fraction at treatment initiation is no longer given by ( 4) but approximately by (see Viossat and Noble, 2020 [START_REF] Viossat | The logic of containing tumors[END_REF], Section 7):

x r = τ 1 1 -ρ r /ρ s . (7)
Injecting ( 7) into (6) and using that τ 1 and τ 2 are much smaller than 1 leads to the condition:

1 1 -ρ r /ρ s ≥ - g(N ) N g (N ) . (8)
For a Power-law model, the condition becomes ρ r /ρ s ≥ 1 -γ. For γ = 1/3, this is satisfied if and only if ρ r /ρ s ≥ 2/3. that is, if and only if the resistance cost is not too large. For a Gompertzian model, the right-hand side is ln(K/N ) and Eq. ( 8) may be written: ρ r /ρ s ≥ 1 -1/ ln(K/N ). With Monro and Gaffney's (2009) [START_REF] Monro | Modelling chemotherapy resistance in palliation and failed cure[END_REF] values: N 0 = 10 10 , K = 2 × 10 12 , this is satisfied if ρ r /ρ s ≥ 0.81.

With logistic growth, the right-hand side is K/N -1 and the condition may be written as ρ r /ρ s ≥ (K -2N )/(K -N ). 16 Assuming for instance ρ r /ρ s = 4/5, this boils down to K ≤ 6N . This would not be satisfied at treatment initiation if N and K represent total numbers of cells in the whole tumor (except possibly for a late-stage treatment), but seems likely to be satisfied in a model of local growth.

We conclude that in the absence of resistance costs, our analysis applies to several standard models of tumor growth with mutations, such as Power-law models or Gompertzian growth, and possibly to logistic growth, at least when it models local growth. However, if the baseline growth-rate of resistant cells is substantially smaller than the baseline growth-rate of sensitive cells, our assumptions become more restrictive and might fail even for Gompertzian growth. This is in line with Hansen et al.'s (2017) [START_REF] Hansen | How to Use a Chemotherapeutic Agent When Resistance to It Threatens the Patient[END_REF] finding that, contrary to common wisdom, a resistance cost in the baseline growth rate may make it less likely that containment strategies outperform more aggressive treatments. Note however that the fact that we can no longer prove that containment outperforms MTD does not mean that it would not do so. Moreover, the analysis of Viossat and Noble (2020) [START_REF] Viossat | The logic of containing tumors[END_REF], Section 7 of the supplementary material, suggests that the mutation effect could only make MTD marginally superior to containment.

B Comparison principles

The following comparison principle is standard: Proposition 13. Let Ω be a nonempty open subset of R 2 . Let f : Ω → R be continuously differentiable. Consider the ordinary differential equation x (t) = f (t, x(t)). Let t 1 ≥ t 0 . Let u : [t 0 , t 1 ] → R be solution of this ODE and let v : [t 0 , t 1 ] → R be a subsolution. That is, v is continuous, almost everywhere differentiable, (t, v(t)) ∈ Ω on [t 0 , t 1 ], and, almost everywhere, v (t) ≤ f (t, v(t)). If furthermore v(t 0 ) ≤ u(t 0 ), then v(t) ≤ u(t) for all t ∈ [t 0 , t 1 ].

We want to apply a variant of this result to two equations: first, where L2 and Ñ2 are not continuously differentiable (otherwise Proposition 13 would directly apply), but piecewise C 1 in the strong sense we defined in Section 2. For instance, for L2 , which is defined on [R 0 , R ∞ 2 ), there exist values r 0 = R 0 < r 1 < ... < r n = R ∞ 2 such that for each i in {1, .., n -1}, L2 coincides on [r i , r i+1 ) with a continuously differentiable function Li 2 defined on a neighborhood of [r i , r i+1 ).

We thus need variants of Proposition 13 where f is slightly less regular. The proof of these variants consists in repeated applications of Proposition 13.

Proposition 14. The conclusion u(t) ≤ v(t) on [t 0 , t 1 ] of Proposition 13 still holds in the following cases: a) if f takes the form f (t, x) = ψ(t, x, φ(t)), where ψ is continuously differentiable and φ is strongly piecewise C 1 . b) if f is a function of x only (f : I → R, where I is a nonempty interval) which is strongly piecewise C 1 , and u and v are strictly increasing.

Proof. Proof of a): by assumption, there exists an integer n and real numbers (τ k ) 0≤k≤n satisfying t 0 = τ 0 < τ 1 < ... < τ n = t 1 such that on A k = [t k , t k+1 ) × R, f coincides with a continuously differentiable function f k defined on a neighborhood of A k . Assuming u(τ k ) ≤ v(τ k ), Proposition 13 implies that u(t) ≤ v(t) on [t k , t k+1 ) and this is also true at time t k+1 by continuity of u and v. An induction argument then gives the result.

Proof of b): By assumption, there exist a finite number of values x 0 ,...,x n such that f coincides on [x k , x k+1 ) with a continuously differentiable function φ k defined on a neighborhood of [x k , x k+1 ), and we may assume (up to adding an artificial point) that x 0 ≤ u(t 0 ). Since u and v are strictly increasing, there exists an integer q ≤ 2(n + 1) and a sequence of times τ 0 = t 0 < τ 1 < ... < τ q = t 1 such that on (τ j , τ j+1 ), u(t) and v(t) are never equal to one of the x k . Assuming u(τ j ) ≤ v(τ j ), then either: for all t in [τ j , τ j+1 ), u(t) and v(t) belong to the same interval [x k , x k+1 ) and Proposition 13 implies that u(t) ≤ v(t) on [τ j , τ j+1 ); or there exists k such that for all t in [τ j , τ j+1 ), u(t) ≤ x k ≤ v(t) and the same inequality holds trivially. In both cases, u(τ k+1 ) ≤ v(τ k+1 ) by continuity of u and v. An induction argument then gives the result.

Figure 1 :

 1 Figure 1: Number of resistant cells and number of tumor cells for different treatments. Top row: number of resistant cells and number of tumor cells as a function of time under MTD (left), Containment at the initial size N 0 (center), and Containment at the maximal tolerable size N tol (right). Bottom-row: number of tumor cells under these three treatments as a function of time (left) or as a function of the number of resistant cells (right).

Figure 2 :

 2 Figure 2: Total size under Intermittent treatment.

Figure 4 :

 4 Figure 4: Comparison of N Cont (r) with an hypothetical curve which satisfies N alt (r 2 ) > N Cont (r 2 ) for some point r 2 ≥ r 1 .

  r, Ñ (r)) where G(r, Ñ ) = f N ( Ñ , r, L2 (r)) f R ( Ñ , r) ,(here Ñ (r) plays the role of u(t), and G the role of f in Proposition 13). Second,dR dt = H(R(t)) where H(R) = f R ( Ñ2 (R), R),

The initial prostate cancer trial has been debated (Mistry

[START_REF] Hitesh | On the reporting and analysis of a cancer evolutionary adaptive dosing trial[END_REF], Zhang et al. 2021[START_REF] Zhang | Response to mistry[END_REF])

This assumption is standard but debatable (Mistry, 2020): this will be the topic of some other work.

In the response evaluation criteria in solid tumors (RECIST), progressive disease is defined by a 20% increase in the sum of the largest diameters (LD) of target lesions, compared to the smallest LD sum recorded since the beginning of treatment. However, comparing to the smallest LD sum recorded would not be fair to aggressive treatments, and the 20% margin makes sense in medical practice, to take into account imperfect monitoring and imperfect forecast of treatment effect, but not for our deterministic mathematical model.

The assumption N tol ≥ N0 in without loss of generality in the following sense: if the initial size is larger than the maximal tolerable size, then all treatments we consider would treat at Lmax until tumor size becomes tolerable (N = N tol ), and we could apply our analysis from that point on.

If, after crossing N tol , tumor size comes back to N tol , then the containment treatment stabilizes tumor size at N tol again, as long as possible. Similar remarks apply to intermittent containment or other variants of containment.

[START_REF] Zhang | Integrating evolutionary dynamics into treatment of metastatic castrate-resistant prostate cancer[END_REF] The above description is to fix ideas: our results are still valid for any other way of maintaining tumor size between Nmin and N tol , as long as this may be done with a dose L(t) ≤

Lmax.[START_REF] Bacevic | Spatial competition constrains resistance to targeted cancer therapy[END_REF] Viossat and Noble assumed to fixed ideas and for simulations that, each time tumor size reaches N tol , it drops instantaneously to Nmin, or to R(t) if R(t) > Nmin, but this is not needed.

When comparing idealized treatments to an alternative treatment, for the comparison to be fair, we do not restrict treatment level under the alternative treatment either, and allow it to eliminate sensitive cells arbitrarily quickly.

Unless mentioned otherwise, when comparing two treatments, we assume the same initial conditions: R1(0) = R2(0) and N1(0) = N2(0).

Exceptionally, we assume that even when Ni > N tol , the dose stays equal to Li and is not increased to Lmax. A similar remark applies to b)

For Model 2, (A2) holds obviously if Lmax ≥ 1. To see that (A1) holds, note that dN/dt = ρ ln(K/N )N -ρ ln(K/N )LS ≥ ρ ln(K/N )N with equality for L = 0. The comparison principle thus implies that not treating maximizes tumor size. Since not treating also minimizes the resistant population size (Proposition 3), it follows that it maximizes the sensitive population size.

The only inequality that uses (A1) is S alt ≤ SCont.

For logistic growth, our assumptions are likely to be valid if the variables N , R, S are interpreted as densities, as in Strobl et al. 2020[START_REF] Maximilian | Turnover modulates the need for a cost of resistance in adaptive therapy[END_REF], but not necessarily if they are interpreted as numbers of cells in the whole tumor, see Appendix A.

The choice of carrying capacity may differ for a Gompertz or a logistic growth model. However, in[START_REF] Monro | Modelling chemotherapy resistance in palliation and failed cure[END_REF][START_REF] Monro | Modelling chemotherapy resistance in palliation and failed cure[END_REF], the lethal tumor size is taken to be 5 × 10 11 so in a logistic growth version of their model, K would have to be at least that large and Eq. (5) would not be satisfied.

This is the exact effect when the turnover rate b(N )/(b(N ) -d(N )) is constant, where b(N ) and d(N ) are the birth and death rates.

This condition is approximately correct only if ρr/ρs is substantially different from 1, which explains that taking the limit ρr/ρs → 1 does not lead to the condition obtained in the absence of a resistance cost.

Acknowledgments

This program has received funding from the European Union's Horizon 2020 research and innovation programme under the Marie Sk lodowska-Curie grant agreement No 754362.

Since R 1 (0) = R 2 (0), the comparison principle (Proposition 14, item b), in Appendix B) implies that for all times t ≥ 0 such that R 1 (t) < R ∞ 2 , we have R 1 (t) ≤ R 2 (t) (See Figure 3). We now show that the inequality R 1 (t) < R ∞ 2 , hence the conclusion R 1 (t) ≤ R 2 (t), holds at all times t ≥ 0. Indeed, otherwise there is a first time

2 , a contradiction. We now prove the result on sensitive cells. Assume that on [r 1 , r 2 ], S1 (r) is non-increasing (which implicitly requires

Moreover, the assumption Ñ1 (r) ≥ Ñ2 (r) is equivalent to S1 (r) ≥ S2 (r). Thus we obtain:

The first inequality follows from the fact that S1 is non-increasing on [r 1 , r 2 ],

A Mutations from sensitive to resistant cells

The analysis in this appendix is related to the work of [START_REF] Martin | Optimal control of tumor size used to maximize survival time when cells are resistant to chemotherapy[END_REF] [START_REF] Martin | Optimal control of tumor size used to maximize survival time when cells are resistant to chemotherapy[END_REF] and [START_REF] Hansen | How to Use a Chemotherapeutic Agent When Resistance to It Threatens the Patient[END_REF] [START_REF] Hansen | How to Use a Chemotherapeutic Agent When Resistance to It Threatens the Patient[END_REF]. Consider a basic Norton-Simon model with mutations (Norton and Simon 1977 [START_REF] Larry | Tumor size, sensitivity to therapy, and design of treatment schedules[END_REF], Goldie and Coldman 1979 [START_REF] Goldie | A mathematic model for relating the drug sensitivity of tumors to their spontaneous mutation rate[END_REF], Monro and Gaffney 2009 [START_REF] Monro | Modelling chemotherapy resistance in palliation and failed cure[END_REF]):

where τ 1 and τ 2 are mutation and backmutation rates. Taking g(N ) = ρ ln(K/N ) leads to a version of Model 2 with mutations: the original Monro and Gaffney model (Monro and Gaffney, 2009[17]).

If the growth-rate function g is decreasing in N , an increase in the size of the sensitive population leads to two opposite effects: it slows down the development of existing resistant cells (the competition effect), but usually increases the number of mutations from sensitive to resistant cells (the mutation effect). This trade-off has been studied by Martin et al. (1992[16]) and Hansen et al. (2017[19]). Here, we study whether such a model is compatible with our assumption that, during treatment, a larger sensitive population leads globally to a lower growth-rate of resistant cells. To do so, let φ R denote the growth-rate function of resistant cells:

Denoting by x r = R/N the resistant fraction, it is easily checked that ∂φ R /∂S ≤ 0 if and only if:

Since the resistant fraction increases during treatment, this condition is bound to be hardest to satisfy at treatment initiation. The resistant fraction obtained from Model 5 for the initial condition S = 1, R = 0 is then (Goldie and Coldman, 1979[32]):

where we used the approximation N -τ 1 -τ ln N for τ small. Injecting