Mathematical Approach to Differentiate Spontaneous and Induced Evolution to Drug Resistance During Cancer Treatment

James M. Greene; Jana L. Gevertz; Eduardo D. Sontag

doi:10.1200/CCI.18.00087

Mathematical Approach to Differentiate Spontaneous and Induced Evolution to Drug Resistance During Cancer Treatment

Greene, James M.; Gevertz, Jana L.; Sontag, Eduardo D. 2019-04-10 00:00:00 original report abstract Mathematical Approach to Differentiate Spontaneous and Induced Evolution to Drug Resistance During Cancer Treatment 1 2 3,4 James M. Greene, PhD ; Jana L. Gevertz, PhD ; and Eduardo D. Sontag, PhD PURPOSE Drug resistance is a major impediment to the success of cancer treatment. Resistance is typically thought to arise from random genetic mutations, after which mutated cells expand via Darwinian selection. However, recent experimental evidence suggests that progression to drug resistance need not occur randomly, but instead may be induced by the treatment itself via either genetic changes or epigenetic alterations. This relatively novel notion of resistance complicates the already challenging task of designing effective treatment protocols. MATERIALS AND METHODS To better understand resistance, we have developed a mathematical modeling framework that incorporates both spontaneous and drug-induced resistance. RESULTS Our model demonstrates that the ability of a drug to induce resistance can result in qualitatively different responses to the same drug dose and delivery schedule. We have also proven that the induction parameter in our model is theoretically identiﬁable and propose an in vitro protocol that could be used to determine a treatment’s propensity to induce resistance. Clin Cancer Inform. © 2019 by American Society of Clinical Oncology Licensed under the Creative Commons Attribution 4.0 License INTRODUCTION drug resistance describes the case in which a tumor contains a subpopulation of drug-resistant cells at the Tumor resistance to chemotherapy and targeted drugs initiation of treatment, which makes therapy even- is a major cause of treatment failure. Both molecular tually ineffective as a result of resistant cell selection. and microenvironmental factors have been implicated As examples, pre-existing BCR-ABL kinase domain in the development of drug resistance. As an example mutations confer resistance to the tyrosine kinase of molecular resistance, upregulation of drug efﬂux inhibitor imatinib in patients with chronic myeloid transporters can prevent sufﬁciently high intracellular 14,15 leukemia, and pre-existing MEK1 mutations drug accumulation, which limits treatment efﬁcacy. confer resistance to BRAF inhibitors in patients with Other molecular causes of drug resistance include 16 melanoma. Many mathematical models have modiﬁcation of drug targets, enhanced DNA damage considered how the presence of such pre-existing repair mechanisms, dysregulation of apoptotic path- resistant cells impacts cancer progression and 1-5 ways, and the presence of cancer stem cells. Ir- 17-40 treatment. ASSOCIATED regular tumor vasculature that results in inconsistent CONTENT Acquired drug resistance broadly describes the case drug distribution and hypoxia is an example of a mi- Appendix in which drug resistance develops during the course of croenvironmental factor that impacts drug resistance. Data Supplement therapy from a population of cells that were initially Other characteristics of the tumor microenvironment Author affiliations drug sensitive. The term acquired resistance is really inﬂuencing drug resistance include regions of acidity, and support an umbrella term for two distinct phenomena, which information (if immune cell inﬁltration and activation, and the tumor complicates the study of acquired resistance. On the 1,6-10 applicable) appear at stroma. Experimental and clinical research con- one hand, there is resistance that is spontaneously—or the end of this tinues to shed light on the multitude of factors that article. randomly—acquired during the course of treatment, be contribute to cancer drug resistance. Mathematical Accepted on February it as a result of random genetic mutations or stochastic modeling studies have also been used to explore both 14, 2018 and nongenetic phenotype switching. This spontaneous broad and detailed aspects of cancer drug resistance, published at form of acquired resistance has been considered in 11-13 ascopubs.org/journal/ as reviewed previously. 18-22,27-29,31,32,35,40,42-49 many mathematical models. On cci on April 10, 2019: Resistance to cancer drugs can be classiﬁed as either the other hand, drug resistance can be induced (ie, DOI https://doi.org/10. 1 41,50-52 1200/CCI.18.00087 pre-existing or acquired. Pre-existing—or intrinsic— caused) by the drug itself. 1 Greene, Gevertz, and Sontag CONTEXT Key Objective Resistance to chemotherapy may arise from Darwinian selection of resistant subclones that either predate therapy or emerge during treatment. In addition, treatment itself may induce genetic or epigenetic variation that catalyzes drug resistance. This work aims to mathematically tease out these various factors. Knowledge Generated A mathematical model is introduced to distinguish the effect of drugs that merely select from those that both create variation and select. The ability of a drug to induce resistance can result in qualitatively different responses on the basis of dose and delivery; constant-infusion regimes are less successful in controlling tumor growth than pulsed therapy for drugs that induce resistance, but the situation is reversed for drugs that act only by selection. Relevance Recent experimental evidence suggests that progression to drug resistance need not occur randomly, but instead may be induced by the treatment itself. Understanding the clinical implications of treatment-induced resistance will help formulate appropriate protocols. The question of whether resistance is an induced phe- drug-induced acquired resistance may simply be the rapid nomenon or predates treatment was ﬁrst famously studied selection of a small number of pre-existing resistant cells or by Luria and Delbruck ¨ in the context of bacterial the selection of cells that spontaneously acquired re- 41,44 41 (Escherichia coli) resistance to a virus (T1 phage). In sistance. In pioneering work by Pisco and colleagues, particular, Luria and Delbruck hypothesized that if selective the relative contribution of resistant cell selection versus pressures imposed by the presence of the virus induce drug-induced resistance was assessed in an experimental bacterial evolution, then the number of resistant colonies system that involved HL60 leukemic cells that were treated formed in their plated experiments should be Poisson with the chemotherapeutic agent vincristine. After 1 to distributed and thus have an approximately equal mean 2 days of treatment, expression of MDR1 was demonstrated and variance. What Luria and Delbruck found instead was to be predominantly mediated by cell-individual induction that the number of resistant bacteria on each plate varied of MDR1 expression and not by the selection of MDR1- 41,58 drastically, with variance being signiﬁcantly larger than the expressing cells. In particular, these cancer cells exploit mean. As a result, the authors concluded that bacterial their heritable, nongenetic phenotypic plasticity—by which mutations predated the viral challenge. one genotype can map onto multiple stable phenotypes— to change their gene expression to a temporarily more In the case of cancer, there is strong evidence that at least 41,58 resistant state in response to treatment-related stress. some drugs have the ability to induce resistance, as ge- nomic mutations can be caused by cytotoxic cancer Although there is a wealth of mathematical research that 54,55 chemotherapeutics. For instance, nitrogen mustards addresses cancer drug resistance, relatively few models can induce base substitutions and chromosomal rear- have considered drug-induced resistance. Of the models of rangements, topoisomerase II inhibitors can induce chro- drug-induced resistance that have been developed, many mosomal translocations, and antimetabolites can induce do not explicitly account for the presence of the drug. double-stranded breaks and chromosomal aberrations. Instead, it is assumed that these models apply only under Such drug-induced genomic alterations would generally be 41,59-62 treatment, with the effects of the drug implicitly nonreversible. Drug resistance can also be induced at the captured in model terms. As these models of resistance 41,50,56 epigenetic level. For example, expression of multi- induction are dose independent, they are unable to capture drug resistance 1 (MDR1), an ABC-family membrane the effects that the alteration of the drug dose has on re- pump that mediates the active efﬂux of the drug, can be sistance formation. To our knowledge, there have been less 1,41 induced during treatment. In another recent example, than a handful of mathematical models developed in which the addition of a chemotherapeutic agent is shown to in- resistance is induced by a drug in a dose-dependent duce, through a multistage process, epigenetic reprog- 33,34,63, 33 fashion. In Gevertz et al and follow-up work in ramming in patient-derived melanoma cells. Resistance 38 64 Shah, Rejniak, and Gevertz and Perez-Velazquez et al, developed in this way can occur quite rapidly and can often duration and intensity of drug exposure determines the 41,52,57 be reversed. resistance level of each cancer cell. This model allows for Despite these known examples of drug-induced resistance, a continuum of resistant phenotypes, but is computa- differentiating between drug-selected and drug-induced tionally complex as it is a hybrid discrete-continuous, resistance is nontrivial. For example, what appears to be stochastic spatial model. While interesting features about 2 © 2019 by American Society of Clinical Oncology Differentiating Spontaneous and Induced Resistance the relationship between induced resistance and the mi- metronomic therapies. Indeed, the differential response croenvironment have been deduced from this model, its between these therapies is fundamentally related to complexity does not allow for general conclusions to be intratumoral heterogeneity/competition, and is explicitly drawn about dose-dependent resistance induction. considered in our model. Furthermore, results presented in this work support recent evidence that promotes the Another class of models that addresses drug-induced re- 65-69 adoption of metronomic therapy in many circumstances, sistance is that in Chisholm et al. These models are and a main objective of this work is to relate competition distinct in that they are motivated by in vitro experiments and drug-induced resistance to therapy design. in which a cancer drug transiently induces a reversible resistant phenotypic state. The individual-based and This work is organized as follows. We begin by introducing integro-differential equation models developed consider a mathematical model to describe the evolution of drug rapidly proliferating drug-sensitive cells, slowly proliferating resistance during treatment with a theoretical resistance- drug-resistant cells, and rapidly proliferating drug-resistant inducing—and noninducing—drug. We use this mathe- cells. An advection term—with the speed depending on matical model to explore the role played by the drug’s drug levels—is used to model drug-induced adaptation of resistance induction rate in treatment dynamics. We the cell proliferation level, and a diffusion term for both the demonstrate that the induction rate of a theoretical cancer level of cell proliferation and survival potential (response drug could have a nontrivial impact on the qualitative re- to drug) is used to model nongenetic phenotype insta- sponses to a given treatment strategy, including tumor bility. Through these models, the contribution of non- composition and the time horizon of tumor control. In our genetic phenotype instability (both drug induced and model, for a resistance-preserving drug—that is, a drug that random), stress-induced adaptation, and selection can does not induce resistance—better tumor control is be quantiﬁed. achieved using a constant therapeutic protocol versus a pulsed one. Conversely, in the case of a resistance- Finally, the work in Liu et al models the evolutionary inducing drug, pulsed therapy prolongs tumor control dynamics of the tumor population as a multitype non- longer than constant therapy as a result of sensitive/ homogeneous continuous time birth-death stochastic resistant cell competitive inhibition. Once the importance of process. This model accounts for the ability of a targeted induced resistance has been established, we demonstrate drug to alter the rate of resistant cell emergence in a dose- that all parameters in our mathematical model are identi- dependent manner. The authors speciﬁcally considered ﬁable, meaning that it is theoretically possible to determine cases in which the rate of mutation that gives rise to a re- the rate at which drug resistance is induced for a given sistant cell: (1) increases as a function of drug concen- treatment protocol. As this theoretical result does not di- tration, (2) is independent of drug concentration, and (3) rectly lend itself to an experimental approach for quanti- decreases with drug concentration. Interestingly, this fying the ability of a drug to induce resistance, we also model led to the conclusion that the optimal treatment describe a potential in vitro experiment for approximat- strategy is independent of the relationship between drug ing this ability utilizing constant therapies. We end with concentration and the rate of resistance formation. In some concluding remarks and a discussion of potential particular, the authors found that resistance is optimally extensions of our analysis, such as a model that differ- delayed using a low-dose continuous treatment strategy entiates between reversible and nonreversible forms of coupled with high-dose pulses. resistance. As in vitro experiments have demonstrated that treatment 41,52 response can be affected by drug-induced resistance, MATERIALS AND METHODS in the current work we seek to understand this phenom- Here we introduce a general modeling framework to de- enon further using mathematical modeling. The initial scribe the evolution of drug resistance during treatment. mathematical model that we have developed—and that will Our model captures the fact that resistance can result from be analyzed herein—is a system of two ordinary differential random events or can be induced by the treatment itself. equations with a single control representing the drug. We Random events that can confer drug resistance include intentionally chose a minimal model that would be ame- genetic alterations—for example, point mutations or gene nable to analysis, as compared with previously developed ampliﬁcation—and phenotype switching. These sponta- models of drug-induced resistance which are signiﬁcantly neous events can occur either before or during treatment. 33,38,63,64 more complex. Despite the simplicity of the model, Drug-induced resistance is resistance speciﬁcally activated it incorporates both spontaneous and drug-induced by the drug and, as such, depends on the effective dose resistance. encountered by a cell. Such a formulation allows us to In addition to drug-induced resistance, the other charac- distinguish the contributions of both drug-dependent and teristic of cancer dynamics we explore is that of traditional, drug-independent mechanisms, as well as any dependence maximally tolerated dose (MTD) treatment protocols on pre-existing—that is prior to treatment—resistant compared with high-frequency, low-dose so-called populations. JCO Clinical Cancer Informatics 3 Greene, Gevertz, and Sontag We consider the tumor to be composed of two types of cells, be under the simplest assumption that the drug is com- sensitive (S) and resistant (R). Sensitive (or wild-type) cells pletely ineffective against resistant cells, so that d =0. are fully susceptible to treatment, whereas treatment af- The last term in the equations, γR, represents the resen- fects resistant cells to a lesser degree. To analyze the role of sitization of cancer cells to the drug. In the case of non- both random and drug-induced resistance, we use a sys- reversible resistance, γ = 0; otherwise γ . 0. Our tem of two ordinary differential equations to describe the subsequent analysis will be done under the assumption of dynamics between the S and R subpopulations: nonreversible resistance. For a discussion of the effect of reversibility on the presented model, see the Appendix. dS S + R r 1 − S − e + αu t S − du t S + γR, (1) dt K Finally, we note that the effective drug concentration u(t) can be thought of as a control input. For simplicity, in this dR S + R work we assume that it is directly proportional to the applied r 1 − R + e + αu t S − d u t R − γR. R R dt K drug concentration; however, pharmacodynamic/phar- (2) macokinetic considerations could be incorporated to more accurately describe the uptake/evolution of the drug in vivo All parameters are non-negative. In the absence of treat- or in vitro—for example, as in Bender, Schindler, and ment, we assume that the tumor grows logistically, with 79 80 81 Friberg, Wu et al, and Fetterly et al. each population contributing equally to competitive in- To understand the above system of drug resistance evo- hibition. Phenotypes S and R each possess individual lution, we reduce the number of parameters via non- intrinsic growth rates, and we make the assumption in dimensionalization. Rescaling S and R by their (joint) the remainder of the work that 0 ≤ r , r.Thissimply carrying capacity K, and time t by the sensitive cell growth states that resistant cells grow slower than nonresistant rate, cells, an assumption that is supported by experimental 70-72 evidence. 1 1 S τ S τ , The transition to resistance can be described with a net K r (3) term of the form «S + αu(t)S. Mathematically, the drug- 1 1 R τ R τ , induced term αu(t)S, where u(t) is the effective applied drug K r dose at time t, describes the effect of treatment on pro- moting the resistant phenotype. For example, this term Equations 1 and 2 (with γ = d = 0) can be written in the could represent the induced overexpression of the form, P-glycoprotein gene, a well-known mediator of multidrug 1,73 dS resistance, by the application of chemotherapy. 1 − S + R S − e + αu t S − du t S, (4) dt Spontaneous evolution of resistance is captured in the eS term, which permits resistance to develop even in the dR absence of treatment. Note that ε is generally considered p 1 − S + R R + e + αu t S. (5) dt small, although recent experimental evidence regarding error-prone DNA polymerases suggests that cancer cells For convenience, we have relabeled S, R, and t to coincide may have increased mutation rates as a result of the with the nondimensionalization so that the parameters ε, α, 75-77 overexpression of such polymerases. For example, in and d must be scaled accordingly (by 1/r). As r was as- Krutyakov, mutation rates as a result of such polymerases sumed to satisfy 0 ≤ r , r, the relative resistant population −1 are characterized by probabilities as high as 7.5 × 10 per growth rate p satisﬁes 0 ≤ p , 1. r r base substitution, and it is known that many point muta- 77 One can show (Appendix) that asymptotically, under any tions in cancer arise from these DNA polymerases. For treatment regimen u(t) ≥ 0, the entire population will be- this work, we adopt the notion that random point mutations come resistant: that lead to drug resistance are rare, and that drug-induced resistance occurs on much quicker time scales ; there- S t t→∞ 0 → . (6) fore, we will assume that α . ε with u = O(1) in our analysis R t of Equations 1 and 2. We model the effects of treatment by assuming the log-kill However, tumor control is still possible where one can hypothesis, which states that a given dose of chemo- combine therapeutic efﬁcacy and clonal competition to in- therapy eliminates the same fraction of tumor cells re- ﬂuence transient dynamics and possibly prolong patient life. gardless of tumor size. We allow for each cellular Indeed, the modality of adaptive therapy has shown promise compartment to have a different drug-induced death rate in using real-time patient data to inform therapeutic mod- (d, d ); however, to accurately describe resistance it is ulation aimed at increasing quality of life and survival times. required that 0 ≤ d , d. Our analysis presented herein will This work will focus on such dynamics and controls. 4 © 2019 by American Society of Clinical Oncology Differentiating Spontaneous and Induced Resistance RESULTS Although a diverse set of inputs u(t) may be theoretically applied, presently we consider only strategies as illustrated Effect of Induction on Treatment Efﬁcacy in Figure 1B. The blue curve in Figure 1B corresponds to We investigate the role of the induction capability of a drug a constant effective dosage u (t) initiated at t —administered c d (parameter α in Eqs 4 and 5) on treatment dynamics. approximately using continuous infusion pumps and/or Speciﬁcally, the value of α may have a substantial impact on slow-release capsules—whereas the black curve represents the relative success of two standard therapy protocols— a corresponding pulsed strategy u (t), with ﬁxed treatment constant dosage and periodic pulsing. windows (Δt )and holidays (Δt ). In general, we may allow on off for different magnitudes, u and u , for constant and on,c on,p Treatment Protocol pulsed therapies respectively—forexample,torelate the To quantify the effects of induced resistance, a treatment total dosage applied per treatment cycle (area under the protocol must be speciﬁed. We adopt a clinical perspective 83 drug concentration-time curve [AUC] ). However, for sim- over the course of the disease, which is summarized in plicity we assume the same magnitude in the subsequent Figure 1. We assume that the disease is initiated by a small section (although see the Appendix for a normalized com- number of wild-type cells: parison). While these represent idealized therapies, such u(t) may form an accurate approximation to in vitro and/or in vivo S 0 S , R 0 0, (7) kinetics. Note that the response V(t)illustrated in Figure 1A will not be identical, or even qualitatively similar, for both where 0 , S , 1. Denote the tumor volume at time t by presented strategies, as will be demonstrated numerically. V (t): Constant Versus Pulsed Therapy Comparison Vt() St() + Rt(). (8) To qualitatively demonstrate the role that induced resistance plays in the design of therapy schedules, we consider two The tumor then progresses untreated until a speciﬁc vol- drugs with the same cytotoxic potential—that is, the same ume V is detected—or, for hematologic tumors, via ap- drug-induced death rate d—each possessing a distinct level propriate blood markers—which using existing nuclear of resistance induction (parameter α). A fundamental imaging techniques corresponds to a tumor with diameter question, then, is whether there exist qualitative distinctions on the order of 10 mm. Time to reach V is denoted by t , d d between treatment responses for each chemotherapy. More which in general depends on all parameters that appear in speciﬁcally, how does survival time compare when sched- Equations 4 and 5. Note that, assuming e . 0, a nonzero uling is altered between constant therapy and pulsing? Does resistant population will exist at the onset of treatment. the optimal strategy—in this case, optimal across only two Therapy, represented through u(t), is then applied until the schedulings—change depending on the extent to which the tumor reaches a critical size V , which we equate with drug induces resistance? treatment failure. Because the (S,R) = (0,1) state is globally asymptotically stable in the ﬁrst quadrant, V , 1is We ﬁx two values of the induction parameter α: guaranteed to be obtained in ﬁnite time. Time until failure, −2 t , is then a measure of efﬁcacy of the applied u(t). α 0, α 10 . c s i AB Constant V u Pulsed on,p V u d on,c t t t t t t d c d c Time Time FIG 1. Schematic of tumor dynamics under two treatment regimes. (A) Tumor volume V in response to treatment initiated at time t . Cancer population arises from a small sensitive population at time t = 0, upon which the tumor grows to detection at volume V . Treatment is begun at t and continues until the tumor reaches a critical size V (at a corresponding time t ), d d c c where treatment is considered to have failed. (B) Illustrative constant and pulsed treatments, both initiated at t = t . JCO Clinical Cancer Informatics 5 Tumor Volume Treatment Greene, Gevertz, and Sontag Recall that we are studying the nondimensional model drug—that is, AUC—is applied. However, we see that Equations 4 and 5, so no units are speciﬁed. Parameter even in this case, intermediate doses may be optimal (Figs α = 0 corresponds to no therapy-induced resistance 3A and 4A). Thus, it is not the larger total drug, per se, that (henceforth denoted as phenotype preserving), and is responsible for the superiority of the constant protocol in therefore considering this case allows for a comparison this case, a point that is reinforced by the fact that the between the classic notion of random evolution toward results remain qualitatively unchanged even if the total resistance (α = 0) and drug-induced resistance (α . 0). drug is controlled for (Appendix). For the remainder of the section, parameters are ﬁxed as Compare this with Figures 2C and 2D, which consider the in Table 1. Critically, all parameters excluding α are same patient and cytotoxicity, but for a highly inductive identical for each drug, which enables an unbiased drug. Results are strikingly different and suggest that comparison. Treatment magnitudes u and u are on,c on,p pulsed therapy is now not worse but in fact substantially selected to be equal: u = u = u . on,c on,p on improves patient response (t ≈ 61 for pulsed, compared with t ≈ 45 for constant). In this case, both tumors are Note that selecting parameter V = 0.1 implies that the c now primarily resistant (Figs A3B and A3D), but the carrying capacity has a diameter of 100 mm, as V cor- pulsed therapy allows for prolonged tumor control via responds to a detectable diameter of 10 mm. Assuming −6 3 sensitive/resistant competitive inhibition. Furthermore, each cancer cell has volume 10 mm , tumors in our model can grow to a carrying capacity of approximately treatment holidays reduce the overall ﬂux into resistance as the application of the drug itself promotes this evolu- 12.4 cm in diameter, which is in qualitative agreement tion. The total amount of drug (AUC) is also less for pulsed with the parameters estimated in Chignola and Foroni therapy (22.5 compared with ≈ 64), so that pulsed (≈12.42 cm, assuming a tumor spheroid). therapy is both more efﬁcient in terms of treatment efﬁ- By examining Figures 2A and 2B, we clearly observe an cacy and less toxic to the patient as adverse effects are improved response to constant therapy when using typically correlated with the total administered dose, a phenotype-preserving drug, with treatment success time which is proportional to the AUC. This is further consistent t nearly seven times as long compared with pulsed with recent experimental and clinical evidence that therapy (t ≈ 90 for constant, compared with t ≈ 14 for c c supports metronomic therapy as a superior alternative to pulsed). It can be observed that the tumor composition at classic chemotherapy regimens. The results presented in treatment conclusion is different for each therapy—not Figure 2 suggest that it may be advantageous to apply shown for this simulation, but see a comparable result in a smaller amount of drug more frequently; however, we Appendix Figures A2B and A2D—and it seems that also note that the results depend on patient-speciﬁc pulsed therapy was not sufﬁciently strong to hamper the parameters, so that therapy would ideally be personal- rapid growth of the sensitive population. Indeed, treat- ized to individual patients. Of note, we do not claim that ment failed quickly as a result of insufﬁcient treatment these results hold for all parameter values—both patient intensity in this case, as the population remains almost and treatment speciﬁc—but instead emphasize that the entirely sensitive. Thus, for this patient under these rate of induction may play a large role in the design of speciﬁc treatments, assuming drug resistance only arises therapies for speciﬁcpatients. via random stochastic events, constant therapy would be For these speciﬁc parameter values, differences between preferred. One might argue that pulsed, equal-magnitude constant and pulsed therapy for the inductive drug are not treatment is worse when α = 0 simply because less total as extensive as in the phenotype-preserving case; how- ever, recall that time has been nondimensionalized and, TABLE 1. Parameters Used For Comparison of Treatment Efﬁcacy for Phenotype- hence, the scale may indeed be clinically relevant. Such Preserving Drugs and Resistance-Inducing Drugs Parameter Biologic Interpretation Value (dimensionless) differences can be further ampliﬁed, and, as exact pa- rameters are difﬁcult or even (currently) impossible to S Initial sensitive population 0.01 measure, qualitative distinctions are paramount. Thus, at R Initial resistant population 0 this stage, ranking of therapies, rather than their precise V Detectable tumor volume 0.1 quantitative efﬁcacy, should act as the more important V Tumor volume deﬁning treatment failure 0.9 clinical criterion. −6 « Background mutation rate 10 From these results, we observe a qualitative difference in d Cytotoxicity of sensitive cells 1 the treatment strategy to apply based entirely on the p Resistant growth fraction 0.2 r value of α, the degree to which the drug itself induces resistance. Thus, in administering chemotherapy, the u Treatment magnitude, constant dose 1.5 on resistance-promotion rate α of the treatment is a clinically Δt Pulsed treatment window 1 on signiﬁcant parameter. In the next section, we use our model Δt Pulsed holiday length 3 off and its output to propose in vitro methods for experimentally NOTE. Parameters used in Figure 2. measuring a drug’s α parameter. 6 © 2019 by American Society of Clinical Oncology 80 60 70 60 70 50 60 Differentiating Spontaneous and Induced Resistance A B Treatment Strategies,  = 0 Tumor Dynamics,  = 0 Constant Constant 1.6 Pulsed Pulsed 0.9 1.4 0.8 1.2 0.7 0.6 0.8 0.5 0.4 0.6 0.3 0.4 0.2 0.2 0.1 0 10203040 0 10 20 30 40 Time Time C D -2 -2 Treatment Strategies,  = 10 Tumor Dynamics,  = 10 Constant Constant 1.6 Pulsed Pulsed 0.9 1.4 0.8 1.2 0.7 0.6 0.8 0.5 0.4 0.6 0.3 0.4 0.2 0.2 0.1 010 20 30 40 0 10203040 Time Time −2 FIG 2. Comparison of treatment efﬁcacy for phenotype-preserving drugs (α = 0) and resistance-inducing drugs (α =10 ). The left column indicates treatment strategy, whereas the right column indicates corresponding tumor volume response. Note that the dashed red line in the right column indicates the tumor volume representing treatment failure, V . (A) Constant and pulsed therapies after tumor detection for α = 0. (B) Responses corresponding −2 to treatment regimens in panel A. (C) Constant and pulsed therapies after tumor detection for α =10 . (D) Responses corresponding to treatment regimens in panel C. Identifying the Rate of Induced Drug Resistance Theoretical Identiﬁability The effect of treatment on the evolution of phenotypic We ﬁrst study the structural identiﬁability of Equations 4 and resistance may have a signiﬁcant impact on the efﬁcacy of 5, a prerequisite for analyzing practical methods for de- conventional therapies. Thus, it is essential to understand termining parameter values. Structural identiﬁability is the the value of the induction parameter α before administering process of determining model parameters—for example, therapy. In this section, we brieﬂy discuss both the theo- α—from a set of control experiments. Here, we assume that the retical possibility and practical feasibility of determining α only measurable quantity is the tumor volume V = S + R,along from different input strategies u(t). For more details, see the with its derivatives, in time. Using four different controls, we Appendix. show that all model parameters, including the induction rate α, JCO Clinical Cancer Informatics 7 u u Tumor Volume S + R Tumor Volume S + R Greene, Gevertz, and Sontag may be determined by precisely measuring the corre- standard therapy protocols and demonstrate that contrary sponding volume-response curves. For more details, see to the work in Liu et al, the rate of resistance induction the Appendix. may have a signiﬁcant effect on treatment outcome. Thus, understanding the dynamics of resistance evolution with An In Vitro Experimental Protocol to Distinguish regard to the applied therapy is crucial. Spontaneous and Drug-Induced Resistance To demonstrate that one can theoretically determine the As structural identiﬁability was established in the previous induction rate, we performed an identiﬁability analysis on section, we focus on practical qualitative differences the parameter α and demonstrated that it can be obtained exhibited by Equations 4 and 5 as a function of the via a set of appropriate perturbation experiments on u(t). resistance-induction rate α. Utilizing only constant dos- Furthermore, we presented an alternative method, using ages, we investigate the dependence of t on dose u, cy- only constant therapies, for understanding the qualitative totoxicity d, and α.Deﬁning the supremum over doses of differences between purely spontaneous and induced the response time (Eq 8), cases. Such properties could possibly be used to design in vitro experiments on different pharmaceuticals, which T d : sup t u, d, α , (9) { } α c allows one to determine the induction rate of drug re- sistance without an a priori understanding of the precise we plot the results for different α values in Figures 3 and 4. mechanism. We do note, however, that such experiments may still be difﬁcult to perform in a laboratory environ- Comparing Figures 3B and 4B, we observe a clear quali- ment, as engineering cells with various drug sensitivities tative difference in maximum response times. In the case of d may be challenging. Indeed, this work can be consid- a phenotype-preserving drug, the proposed in vitro ex- ered as a thought experiment to identify qualitative periment would produce a ﬂat curve, whereas a resistance- properties that the induction rate α yields in our modeling inducing drug (α. 0) would yield an increasing function T framework. (d). Additional comparisons are presented in Figure 5, where the α dependence is more closely analyzed. For Our simple model allows signiﬁcant insight into the role of more details and analysis, see the Appendix. random versus induced resistance. Of course, more elaborate models can be studied by incorporating more DISCUSSION biologic detail. For example, while our two-equation model In the current work, we analyzed two distinct mechanisms classiﬁes cells as either sensitive or resistant, not all re- that can result in drug resistance. Speciﬁcally, a mathe- sistance is treated equally. Some resistant cancer cells are matical model is proposed which describes both the permanently resistant, whereas others could transition spontaneous generation of resistance and drug-induced back to a sensitive state. This distinction may prove to be resistance. Using this model, we contrasted the effect of vitally important in treatment design. A possible extension A B -2 -2 Constant Therapy Dose Response,  = 10 Maximum Critical Time as a Function of Sensitivity,  = 10 d = 0.1 d = 0.2 d = 0.95 140 d = 1.45 d = 2.95 00.5 11.52 2.5 3 3.5 44.5 5 0.5 1 1.5 2 2.5 33.5 4 u d −2 FIG 3. Variation in response time t for a treatment that induces resistance. Constant therapy u(t) ≡ u is applied for t ≤ t ≤ t . Induction rate α =10 , with all c d c other parameters as in Table 1. (A) Time until tumor reaches critical size V for various drug sensitivities d. (B) Maximum response time T (d) for a treatment c α that induces resistance. Note that time T (d) increases with drug sensitivity; compare with Figure 4B for purely random resistance evolution. 8 © 2019 by American Society of Clinical Oncology T (d)  Differentiating Spontaneous and Induced Resistance A B Maximum Critical Time as a Function Constant Therapy Dose Response, = 0 of Sensitivity, = 0  450 450 d = 0.1 400 d = 0.2 400 d = 0.95 350 350 d = 1.45 d = 2.95 300 300 250 250 200 200 150 150 100 100 50 50 00.5 11.522.533.544.55 0.5 11.522.53 u d FIG 4. Change in critical time t for differing drug sensitivities in the case of a phenotype-preserving treatment. (A) Time until tumor reaches critical size V for c c various drug sensitivities d; comparable to Figure 3A, with α = 0. (B) Maximum critical time T (d). Note that the curve is essentially constant. of our model is one in which we distinguish between alterations or resistance that forms by some combination of sensitive cells S, nonreversible resistant cells R , and re- genetic and stable epigenetic changes. versible resistant cells R (Eqs 9-12): Conversely, reversible resistant cells R denote resistant cells that form via phenotype switching, as described in dS V r 1 − S − e + e S − α + α u t S − du t S n r n r Pisco et al. Random phenotype switching in the ab- dt K sence of treatment is captured in the ε S term. This is + γR , consistent—and indeed necessary—to understand the (10) experimental results in Pisco et al, where a stable dis- tribution of MDR1 expressions is observed even in the dR V absence of treatment. The α u(t)S term represents the r 1 − R + e S + α u t S − d u t R , (11) n n n n n n dt K induction of a drug-resistant phenotype. Phenotype switching is often reversible, and therefore we allow a back dR V r transition from the R compartment to the sensitive com- r 1 − R + e S + α u t S − γR − d u t R . r r r r r r r dt K partment at a nonnegligible rate γ (see Appendix). For- mulated in this way, the model can be calibrated to (12) experimental data and we can further consider the effects Here, V denotes the entire tumor population—that is, of the dosing strategy on treatment response. We plan to further study this model in future work. V : S + R + R . (13) n r Other extensions which include different clinical scenarios are also being investigated. In practice, chemotherapies are In this version of the model, nonreversible resistant cells R rarely applied in isolation. Multiple therapies are often can be thought of as resistant cells that form via genetic administered simultaneously to improve efﬁcacy. The in- mutations. Under this assumption, « represents the rate at clusion of multiple drugs, including targeted therapies that which spontaneous genetic mutations give rise to re- act primarily on resistant subpopulations, yields natural sistance, and α is the drug-induced resistance rate. This control questions that are clinically relevant. Similarly, situation can be classiﬁed as nonreversible as it is in- immune cells, together with immunotherapies, may also be credibly unlikely that genomic changes that occur in re- incorporated to more accurately mimic the cancer sponse to treatment would be reversed by an "undoing" microenvironment. mutation. Therefore, once cells confer a resistant pheno- type via an underlying genetic change, we assume that they Overcoming drug resistance is crucial for the success of maintain that phenotype. This term could also be thought of both chemotherapy and targeted therapy. Furthermore, the as describing resistance that forms via stable epigenetic added complexity of induced drug resistance complicates JCO Clinical Cancer Informatics 9  Greene, Gevertz, and Sontag A B Maximum Critical Time for Different Maximum Critical Time for Different Induced Mutation Rates Induced Mutation Rates 180 450 20 0 0.5 1 1.5 2 2.5 0.5 1 1.5 2 2.5 d d = 5.00e-03 = 9.00e-03 = 1.20e-02 -4 -2 = 0 = 10 = 10 = 6.00e-03 = 1.00e-02 = 1.30e-02 -6 -3 -1 = 10 = 10 = 10 = 7.00e-03 = 1.10e-02 = 1.40e-02 -5 = 10 = 8.00e-03 = 1.50e-02 FIG 5. Variation in maximum response time for different induction rates α. For details on computation of T (d), see Appendix Figure A4. All other parameters −2 are given as in Table 1. (A) Plot of T (d) for α near 10 . (B) Analogous to panel A, where α is now varied over several orders of magnitude. Nonmutagenic case (α = 0) is included for reference. therapy design, as the simultaneous effects of tumor re- differentiating random and drug-induced resistance, which duction and resistance propagation confound one another. will allow for clinically actionable analysis on a biologically Mathematically, we have presented a clear framework for subtle, yet important, issue. AUTHOR CONTRIBUTIONS AFFILIATIONS Conception and design: All authors Rutgers University, New Brunswick, NJ Collection and assembly of data: All authors The College of New Jersey, Ewing Township, NJ Northeastern University, Boston, MA Data analysis and interpretation: All authors Harvard Medical School, Cambridge, MA Manuscript writing: All authors Final approval of manuscript: All authors Preprint version available on https://www.biorxiv.org/content/10.1101/ Accountable for all aspects of the work: All authors 235150v2. CORRESPONDING AUTHOR AUTHORS’ DISCLOSURES OF POTENTIAL CONFLICTS OF INTEREST AND DATA AVAILABILITY STATEMENT Eduardo D. Sontag, PhD, Northeastern University, 360 Huntington Avenue, Boston, MA 02115; e-mail: sontag@sontaglab.org. The following represents disclosure information provided by authors of this manuscript. All relationships are considered compensated. Relationships are self-held unless noted. I = Immediate Family Member, AUTHORS’ DISCLOSURES OF POTENTIAL CONFLICTS OF INTEREST Inst = My Institution. Relationships may not relate to the subject matter of AND DATA AVAILABILITY STATEMENT this manuscript. For more information about ASCO's conﬂict of interest Disclosures provided by the authors and data availability statement (if policy, please refer to www.asco.org/rwc or ascopubs.org/jco/site/ifc. applicable) are available with this article at DOI https://doi.org/10.1200/ CCI.18.00087. No potential conﬂicts of interest were reported REFERENCES 1. Holohan C, Van Schaeybroeck S, Longley DB, et al: Cancer drug resistance: An evolving paradigm. Nat Rev Cancer 13:714-726, 2013 2. Gottesman MM: Mechanisms of cancer drug resistance. Annu Rev Med 53:615-627, 2002 3. Dean M, Fojo T, Bates S: Tumour stem cells and drug resistance. Nat Rev Cancer 5:275-284, 2005 4. Woods D, Turchi JJ: Chemotherapy induced DNA damage response: Convergence of drugs and pathways. Cancer Biol Ther 14:379-389, 2013 10 © 2019 by American Society of Clinical Oncology  Differentiating Spontaneous and Induced Resistance 5. Zahreddine H, Borden KL: Mechanisms and insights into drug resistance in cancer. Front Pharmacol 4:28, 2013 6. Tredan ´ O, Galmarini CM, Patel K, et al: Drug resistance and the solid tumor microenvironment. J Natl Cancer Inst 99:1441-1454, 2007 7. Gajewski TF, Meng Y, Blank C, et al: Immune resistance orchestrated by the tumor microenvironment. Immunol Rev 213:131-145, 2006 8. Meads MB, Gatenby RA, Dalton WS: Environment-mediated drug resistance: A major contributor to minimal residual disease. Nat Rev Cancer 9:665-674, 2009 9. Correia AL, Bissell MJ: The tumor microenvironment is a dominant force in multidrug resistance. Drug Resist Updat 15:39-49, 2012 10. McMillin DW, Negri JM, Mitsiades CS: The role of tumour-stromal interactions in modifying drug response: Challenges and opportunities. Nat Rev Drug Discov 12:217-228, 2013 11. Lavi O, Gottesman MM, Levy D: The dynamics of drug resistance: A mathematical perspective. Drug Resist Updat 15:90-97, 2012 12. Brocato T, Dogra P, Koay EJ, et al: Understanding drug resistance in breast cancer with mathematical oncology. Curr Breast Cancer Rep 6:110-120, 2014 13. Foo J, Michor F: Evolution of acquired resistance to anti-cancer therapy. J Theor Biol 355:10-20, 2014 14. Roche-Lestienne C, Preudhomme C: Mutations in the ABL kinase domain pre-exist the onset of imatinib treatment. Semin Hematol 40:80-82, 2003 (suppl 2) 15. Iqbal Z, Aleem A, Iqbal M, et al: Sensitive detection of pre-existing BCR-ABL kinase domain mutations in CD34 cells of newly diagnosed chronic-phase chronic myeloid leukemia patients is associated with imatinib resistance: Implications in the post-imatinib era. PLoS One 8:e55717, 2013 16. Carlino MS, Fung C, Shahheydari H, et al: Preexisting MEK1P124 mutations diminish response to BRAF inhibitors in metastatic melanoma patients. Clin Cancer Res 21:98-105, 2015 17. Jackson TL, Byrne HM: A mathematical model to study the effects of drug resistance and vasculature on the response of solid tumors to chemotherapy. Math Biosci 164:17-38, 2000 18. Komarova NL, Wodarz D: Drug resistance in cancer: Principles of emergence and prevention. Proc Natl Acad Sci USA 102:9714-9719, 2005 19. Michor F, Nowak MA, Iwasa Y: Evolution of resistance to cancer therapy. Curr Pharm Des 12:261-271, 2006 20. Komarova NL, Wodarz D: Stochastic modeling of cellular colonies with quiescence: An application to drug resistance in cancer. Theor Popul Biol 72:523-538, 21. Foo J, Michor F: Evolution of resistance to targeted anti-cancer therapies during continuous and pulsed administration strategies. PLOS Comput Biol 5:e1000557, 2009 [Erratum: PLoS Comput Biol doi:10.1371/annotation/d5844bf3-a6ed-4221-a7ba-02503405cd5e] 22. Foo J, Michor F: Evolution of resistance to anti-cancer therapy during general dosing schedules. J Theor Biol 263:179-188, 2010 23. Silva AS, Gatenby RA: A theoretical quantitative model for evolution of cancer chemotherapy resistance. Biol Direct 5:25, 2010 24. Cunningham JJ, Gatenby RA, Brown JS: Evolutionary dynamics in cancer therapy. Mol Pharm 8:2094-2100, 2011 25. Mumenthaler SM, Foo J, Leder K, et al: Evolutionary modeling of combination treatment strategies to overcome resistance to tyrosine kinase inhibitors in non- small cell lung cancer. Mol Pharm 8:2069-2079, 2011 26. Orlando PA, Gatenby RA, Brown JS: Cancer treatment as a game: Integrating evolutionary game theory into the optimal control of chemotherapy. Phys Biol 9:065007, 2012 27. Bozic I, Reiter JG, Allen B, et al: Evolutionary dynamics of cancer in response to targeted combination therapy. eLife 2:e00747, 2013 28. Foo J, Leder K, Mumenthaler SM: Cancer as a moving target: understanding the composition and rebound growth kinetics of recurrent tumors. Evol Appl 6:54-69, 2013 29. Lavi O, Greene JM, Levy D, et al: The role of cell density and intratumoral heterogeneity in multidrug resistance. Cancer Res 73:7168-7175, 2013 30. Bozic I, Nowak MA: Timing and heterogeneity of mutations associated with drug resistance in metastatic cancers. Proc Natl Acad Sci USA 111:15964-15968, 31. Greene J, Lavi O, Gottesman MM, et al: The impact of cell density and mutations in a model of multidrug resistance in solid tumors. Bull Math Biol 76:627-653, 32. Yamamoto KN, Hirota K, Takeda S, et al: Evolution of pre-existing versus acquired resistance to platinum drugs and PARP inhibitors in BRCA-associated cancers. PLoS One 9:e105724, 2014 33. Gevertz J, Aminzare Z, Norton K-A, et al: Emergence of anti-cancer drug resistance: exploring the importance of the microenvironmental niche via a spatial model, in Jackson T and Radunskaya A (eds): Applications of Dynamical Systems in Biology and Medicine, Volume 158, Berlin, Germany, Springer-Verlag, 2015, pp 1-34 34. Liu LL, Li F, Pao W, et al: Dose-dependent mutation rates determine optimum erlotinib dosing strategies for EGRF mutant non-small lung cancer patients. PLoS One 10:e0141665, 2015 35. Menchon ´ SA: The effect of intrinsic and acquired resistances on chemotherapy effectiveness. Acta Biotheor 63:113-127, 2015 36. Mumenthaler SM, Foo J, Choi NC, et al: The impact of microenvironmental heterogeneity on the evolution of drug resistance in cancer cells. Cancer Inform 14:19-31, 2015 (suppl 4) 37. Waclaw B, Bozic I, Pittman ME, et al: A spatial model predicts that dispersal and cell turnover limit intratumour heterogeneity. Nature 525:261-264, 2015 38. Shah AB, Rejniak KA, Gevertz JL: Limiting the development of anti-cancer drug resistance in a spatial model of micrometastases. Math Biosci Eng 13:1185-1206, 2016 39. Carrere C: Optimization of an in vitro chemotherapy to avoid resistant tumours. J Theor Biol 413:24-33, 2017 40. Chakrabarti S, Michor F: Pharmacokinetics and drug-interactions determine optimum combination strategies in computational models of cancer evolution. Cancer Res 77:3908-3921, 2017 41. Pisco AO, Brock A, Zhou J, et al: Non-Darwinian dynamics in therapy-induced cancer drug resistance. Nat Commun 4:2467, 2013 42. Coldman AJ, Goldie JH: A stochastic model for the origin and treatment of tumors containing drug-resistant cells. Bull Math Biol 48:279-292, 1986 43. Ledzewicz U, Schattler ¨ H: Drug resistance in cancer chemotherapy as an optimal control problem. Discrete Continuous Dyn Syst Ser B 5:129-150, 2006 44. Leder K, Foo J, Skaggs B, et al: Fitness conferred by BCR-ABL kinase domain mutations determines the risk of pre-existing resistance in chronic myeloid leukemia. PLoS One 6:e27682, 2011 45. Hadjiandreou MM, Mitsis GD: Mathematical modeling of tumor growth, drug-resistance, toxicity, and optimal therapy design. IEEE Trans Biomed Eng 61:415-425, 2014 46. Lorz A, Lorenzi T, Hochberg M, et al: Populational adaptive evolution, chemotherapeutic resistance and multiple anti-cancer therapies. ESAIM: Math Model Num Anal 47:377-399, 2013 47. Fu F, Nowak MA, Bonhoeffer S: Spatial heterogeneity in drug concentrations can facilitate the emergence of resistance to cancer therapy. PLOS Comput Biol 11:e1004142, 2015 48. Ledzewicz U, Bratton K, Schattler H: A 3-compartment model for chemotherapy of heterogeneous tumor populations. Acta Appl Math 135:191-207, 2015 JCO Clinical Cancer Informatics 11 Greene, Gevertz, and Sontag 49. Lorz A, Lorenzi T, Clairambault J, et al: Modeling the effects of space structure and combination therapies on phenotypic heterogeneity and drug resistance in solid tumors. Bull Math Biol 77:1-22, 2015 50. Nyce J, Leonard S, Canupp D, et al: Epigenetic mechanisms of drug resistance: Drug-induced DNA hypermethylation and drug resistance. Proc Natl Acad Sci USA 90:2960-2964, 1993 51. Nyce JW: Drug-induced DNA hypermethylation: A potential mediator of acquired drug resistance during cancer chemotherapy. Mutat Res 386:153-161, 1997 52. Sharma SV, Lee DY, Li B, et al: A chromatin-mediated reversible drug-tolerant state in cancer cell subpopulations. Cell 141:69-80, 2010 53. Luria SE, Delbruck ¨ M: Mutations of bacteria from virus sensitivity to virus resistance. Genetics 28:491-511, 1943 54. Szikriszt B, Poti ´ A, Pipek O, et al: A comprehensive survey of the mutagenic impact of common cancer cytotoxics. Genome Biol 17:99, 2016 55. Swift LH, Golsteyn RM: Genotoxic anti-cancer agents and their relationship to DNA damage, mitosis, and checkpoint adaptation in proliferating cancer cells. Int J Mol Sci 15:3403-3431, 2014 56. Shaffer SM, Dunagin MC, Torborg SR, et al: Rare cell variability and drug-induced reprogramming as a mode of cancer drug resistance. Nature 546:431-435, 57. Housman G, Byler S, Heerboth S, et al: Drug resistance in cancer: An overview. Cancers (Basel) 6:1769-1792, 2014 58. Pisco AO, Huang S: Non-genetic cancer cell plasticity and therapy-induced stemness in tumour relapse: ‘What does not kill me strengthens me’. Br J Cancer 112:1725-1732, 2015 59. Goldman A, Majumder B, Dhawan A, et al: Temporally sequenced anticancer drugs overcome adaptive resistance by targeting a vulnerable chemotherapy- induced phenotypic transition. Nat Commun 6:6139, 2015 60. Chapman M, Risom T, Aswani A, et al: A model of phenotypic state dynamics initiates a promising approach to control heterogeneous malignant cell populations.IEEE 55th Conference on Decision and Control, Las Vegas, NV, December 12-14, 2016 (abstr) 61. Axelrod DE, Vedula S, Obaniyi J: Effective chemotherapy of heterogeneous and drug-resistant early colon cancers by intermittent dose schedules: A computer simulation study. Cancer Chemother Pharmacol 79:889-898, 2017 62. Feizabadi MS: Modeling multi-mutation and drug resistance: Analysis of some case studies. Theor Biol Med Model 14:6, 2017 63. Chisholm RH, Lorenzi T, Lorz A, et al: Emergence of drug tolerance in cancer cell populations: An evolutionary outcome of selection, nongenetic instability, and stress-induced adaptation. Cancer Res 75:930-939, 2015 64. Perez-Velazquez J, Gevertz J, Karolak A, et al: Microenvironmental niches and sanctuaries: A route to acquired resistance, in Rejniak K (ed): Systems Biology of Tumor Microenvironment, Volume 936, New York, NY, Springer, 2016, 149-164 65. Kerbel RS, Kamen BA: The anti-angiogenic basis of metronomic chemotherapy. Nat Rev Cancer 4:423-436, 2004 66. Pasquier E, Kavallaris M, Andre´ N: Metronomic chemotherapy: New rationale for new directions. Nat Rev Clin Oncol 7:455-465, 2010 67. Gatenby RA, Silva AS, Gillies RJ, et al: Adaptive therapy. Cancer Res 69:4894-4903, 2009 + + 68. Ghiringhelli F, Menard C, Puig PE, et al: Metronomic cyclophosphamide regimen selectively depletes CD4 CD25 regulatory T cells and restores T and NK effector functions in end stage cancer patients. Cancer Immunol Immunother 56:641-648, 2007 69. Kareva I, Waxman DJ, Lakka Klement G: Metronomic chemotherapy: An attractive alternative to maximum tolerated dose therapy that can activate anti-tumor immunity and minimize therapeutic resistance. Cancer Lett 358:100-106, 2015 70. Lee W-P: The role of reduced growth rate in the development of drug resistance of HOB1 lymphoma cells to vincristine. Cancer Lett 73:105-111, 1993 71. Shackney SE, McCormack GW, Cuchural GJ Jr: Growth rate patterns of solid tumors and their relation to responsiveness to therapy: An analytical review. Ann Intern Med 89:107-121, 1978 72. Brimacombe KR, Hall MD, Auld DS, et al: A dual-ﬂuorescence high-throughput cell line system for probing multidrug resistance. Assay Drug Dev Technol 7:233-249, 2009 73. Thomas H, Coley HM: Overcoming multidrug resistance in cancer: An update on the clinical strategy of inhibiting p-glycoprotein. Cancer Contr 10:159-165, 74. Loeb LA, Springgate CF, Battula N: Errors in DNA replication as a basis of malignant changes. Cancer Res 34:2311-2321, 1974 75. Krutyakov V: Eukaryotic error-prone DNA polymerases: The presumed roles in replication, repair, and mutagenesis. Mol Biol 40:1-8, 2006 76. Makridakis NM, Reichardt JK: Translesion DNA polymerases and cancer. Front Genet 3:174, 2012 77. Lange SS, Takata K, Wood RD: DNA polymerases and cancer. Nat Rev Cancer 11:96-110, 2011 78. Traina TA, Norton L: Log-kill hypothesis, in Schwab M (ed): Encyclopedia of Cancer. Springer, Berlin, Germany, 2011, pp 1713-1714 79. Bender BC, Schindler E, Friberg LE: Population pharmacokinetic-pharmacodynamic modelling in oncology: A tool for predicting clinical response. Br J Clin Pharmacol 79:56-71, 2015 80. Wu Q, Li MY, Li HQ, et al: Pharmacokinetic-pharmacodynamic modeling of the anticancer effect of erlotinib in a human non-small cell lung cancer xenograft mouse model. Acta Pharmacol Sin 34:1427-1436, 2013 81. Fetterly GJ, Aras U, Lal D, et al: Development of a preclinical PK/PD model to assess antitumor response of a sequential aﬂibercept and doxorubicin-dosing strategy in acute myeloid leukemia. AAPS J 15:662-673, 2013 82. Erdi YE: Limits of tumor detectability in nuclear medicine and pet. Mol Imaging Radionucl Ther 21:23-28, 2012 83. Jodrell DI, Egorin MJ, Canetta RM, et al: Relationships between carboplatin exposure and tumor response and toxicity in patients with ovarian cancer. J Clin Oncol 10:520-528, 1992 84. Chignola R, Foroni RI: Estimating the growth kinetics of experimental tumors from as few as two determinations of tumor size: Implications for clinical oncology. IEEE Trans Biomed Eng 52:808-815, 2005 nn n 12 © 2019 by American Society of Clinical Oncology Differentiating Spontaneous and Induced Resistance APPENDIX t→∞ The supplement contains additional results and extensions related to S t , R t → 0, 1 . the model Equations 1 and 2 presented in the main text. Sections include details on the mathematical characteristics of solutions of Proof. From Theorem 1, we have that 0 ≤ S(t)+ R(t) ≤ 1, so that Equations 4 and 5, an extension describing the reversibility of drug Equation A2 implies resistance, treatment regimens with normalized dosages, details on structural identiﬁability, and an expanded discussion on the maximum dR critical time T (d). ≥ 0. dt As 0 ≤ R(t) ≤ S(t)+ R(t) ≤ 1, R(t) must converge, so that there exists Fundamental Solution Properties of Resistance Model 0 ≤ R ≤ 1 such that For convenience, Equations 4 and 5 are reproduced below: t→∞ R t →R . dS 1 − S + R S − e + αu t S − du t S, (A1) dt Deﬁne dR ρ : lim infS t . p 1 − S + R R + e + αu t S. (A2) t→∞ dt Since S(t) 2 [0,1], we know that 0 ≤ ρ ≤ 1. Assume that the inequality is We begin with a standard existence/uniqueness result, as well as the strict on the left: ρ . 0. By deﬁnition, there then exists t . 0 such that dynamical invariance of the triangular region: * T : S, R |S ≥ 0, R ≥ 0, S + R ≤ 1 . (A3) S t ≥ , Note that T represents the region of non-negative tumor sizes less than for all t ≥ t .As S(t)+ R(t) ≤ 1 for all t, Equation A2 implies that for all 1. Biologically, this implies that all solutions remain physical (non- t ≥ t , negative) and bounded above by the carry capacity (non- dimensionalized to 1 here, generally K in Eqs 1 and 2). eρ R t ≥ eS(t)≥ , Theorem 1. For any bounded measurable control u: [0,∞) → [0, which implies that u ], with u , ∞, and (S , R ) 2 T, the initial value problem max max 0 0 Equations A1 and A2, eρ t→∞ R t − R t ≥ ds→ ∞, S 0 S , R 0 R , 0 0 2 tp has a unique solution [S(t), R(t)] deﬁned for all times t 2 R. Fur- a contradiction. Thus, ρ =0. thermore, under the prescribed dynamics, region T is invariant. Deﬁne Proof. Existence and uniqueness of local solutions follow from standard results in the theory of differential equations—for example (Sontag ED: σ : lim inf S t + R t . t→∞ Mathematical Control Theory: Deterministic Finite Dimensional Sys- tems. Springer, 1998). As the vector ﬁeld As above, it is clear that σ 2 [0,1]. Assume that σ , 1. Thus, there exists t , e. 0 such that S(t)+ R(t)≤ 1 − e, 1 for all t ≥ t . Equation p * 1 − S + R S − e + αu t S − du t S A2 then implies that for all t ≥ t , F S, R, t : . p 1 − S + R R + e + αu t S R t ≥ p 1 − S t + R t R . p eR, r r is analytic for (S, R) 2 R , the existence of maximal solutions deﬁned for all t 2 R will follow from boundedness (Sontag ED: Mathematical As in the previous argument, this differential inequality contradicts the Control Theory: Deterministic Finite Dimensional Systems. Springer, boundedness of R. Thus, σ =1. 1998), which we demonstrate below. Since R converges to R , lim inf distributes over the sum of S + R, and Uniqueness implies that solutions remain in the ﬁrst quadrant for all t ≥ we have that 0. Indeed, we ﬁrst note that (0, 0) and (0 ,1) are steady states for any control u(t). As S = 0 implies that S 0, we see that the R-axis in 1 lim inf S t + R t ρ + R . t→∞ t→∞ invariant, with R(t)→ 1. Similarly, R = 0 implies Ṙ ≥ 0, and hence all trajectories with (S , R ) 2 T remain non-negative for all t.As V = S + 0 0 Thus R =1. R satisﬁes the differential equation t→∞ As R(t)→1, there exists t . 0 such that V α t 1 − V − β t , R t ≥ 1 − where α(t):= S(t)+ p R(t), β(t):= du(t)S(t) are both non-negative, V , 10V (t), 1. Thus, if the initial conditions (S , R ) reside in T,we 0 0 0 for all t ≥ t . For such t, Equation A1 implies that Ṡ(t) , 0. Thus, S is are guaranteed that (S(t), R(t)) 2 T for all time t, as desired. eventually decreasing and hence must converge to its lim inf ρ =0. t→∞ Thus, we have that (S(t), R(t))→ (0, 1), as desired. We now prove that, asymptotically, cells will evolve to become entirely resistant. For simplicity, we assume that the tumor is initially below carrying capacity, although a similar result holds Reversible Phenotype Switching for V . 1. In the model analyzed in this work (Eqs 4 and 5), we assumed that Theorem 2. For any bounded measurable control u: [0,∞) → resistance is nonreversible; however, experiments suggest that phenotypic alterations are generally unstable and hence a non-- [0,u ] ,with u , ∞, and initial conditions (S , R ) 2 T, max max 0 0 solutions of Equations A1 and A2 will approach the steady state negligible back transition exists. In this Appendix, we demonstrate (S, R) = (0,1): that an extension of our model to include this phenomenon does not JCO Clinical Cancer Informatics 13 Greene, Gevertz, and Sontag change the qualitative results presented previously, at least for pa- therapy magnitude u is ﬁxed (arbitrarily) at 0.5, which by Equation A12 on,c rameter values that are consistent with experimental data. implies that u =5.Wealso adjust Δt =0.5, Δt = 4.5, and all other on,p on off parameter values remain as in Table 1. To model reversible drug resistance, we include a constant per-capita transition rate from the resistant compartment R back to the wild Results of the simulations are presented in Figures A2 and A3. Note cell-type S. Denoting this rate by γ, we obtain the system that here Figure A2 displays the results for the phenotype-preserving drug (α = 0), whereas Figure A3 represents the resistance-inducing −2 dS drug (α =10 ). Each drug is simulated for the two distinct strategies; 1 − S + R S + γR − e + αu t S − du t S, (A4) each strategy is represented in the left column of the respective ﬁgures. dt The right column illustrates the population response for both the in- dR dividual populations—sensitive S and resistant R cells—as well as for p 1 − S + R R + e + αu t S − γR. (A5) dt the total tumor volume V = S + R. As previously discussed, treatment is continued until a critical tumor size V is obtained, and the corre- We ﬁrst consider an appropriate value of the rate γ. Note that in the sponding time t is used as a measure of treatment efﬁcacy, with absence of treatment (u(t) = 0), the system becomes a larger t indicating a better response. dS Our results are qualitatively in agreement with those presented in the 1 − S + R S + γR − eS (A6) main text (Fig 2), where no preservation of total administered drug dt (area under the drug concentration-time curve) was considered. dR Speciﬁcally, we observe a superior response with constant therapy for p 1 − S + R R + eS − γR. (A7) the phenotype-preserving drug (α = 0), whereas the situation is re- dt −2 versed for the resistance-inducing drug (α =10 ). Note that the tumor volume V(t)= S(t)+ R(t) satisﬁes the equation Identiﬁability V S + p R 1 − V , (A8) We provide details of both the structural and practical identiﬁability of which is nondecreasing (recall Theorem 1). This implies that the control system Equations A1 and A2. Technical details are provided system approaches a steady state (S , R ), which can be easily * * that do not appear in the main text. computed as Theoretical identiﬁability. We ﬁrst show that all parameters in γ e Equations A1 and A2 are identiﬁable using a relatively small set of S , R , . (A9) p p γ + e γ + e controls u(t) via classic methods from control theory. We provide a self- contained discussion. For a thorough review of theory and methods, Note that Equation A9 lies on the line V =1. see a recent article (Sontag ED: PLOS Comput Biol 13:e1005447, 2017) and the references therein. Pisco and colleagues measure a 1% to 2% subpopulation of clonally derived HL60 cells that consistently express high levels of MDR1, Assuming that time and tumor volume are the only clinically ob- which we equate with the resistant population R. Using the 2% upper servable outputs—that is, that one cannot readily determine sensitive bound, this implies that and resistant proportions in a given population—we measure V(t) and its derivatives at time t = t for different controls u(t). For sim- S 0.98, R 0.02. (A10) p p plicity, we assume that t = 0, so that treatment is initiated with a purely wild-type (sensitive) population. Note that this is equivalent Solving Equation A9 with the above values then determines the ratio to assuming an entirely sensitive tumor at treatment initiation. Al- though the results remain valid if t . 0 as the system of equations 49. (A11) gain only a constant, this assumption will simplify the subsequent computations. For a discussion of the practical feasibility of such We now repeat the constant versus pulsed experiments discussed methods, see the next section. in the main text, but for the reversible Equations A4 and A5. Speciﬁcally, consider the Equations A1 and A2 with initial conditions Parameter values are taken again as in Table 1,and γ is de- (Eqn 7). Measuring V(t)= S(t)+ R(t) at time t = 0 implies that we can termined via Equation A11. Results are presented in Figure A1 identify S : and should be compared with that presented in Figure 2.Notethat the same qualitative—and indeed quantitative—conclusions hold: V 0 S 0 + R 0 S : Y , 0 0 constant therapy improves response time compared with pulsing −2 when α = 0, whereas the reverse is true for α =10 . Thus, the where we adopt the notation Y ,i ≥ 0 for measurable quantities. inclusion of instability of the resistant cell subpopulation still Similarly, deﬁne the following for the given input controls: suggests that knowledge of the resistance-induction rate α for a chemotherapy is critical when designing therapies. We note that Y : V 0 , u t ≡ 0, precise agreement of Figures A1 and A2 is aresultofthe small Y : V 0 , u t ≡ 1, values taken for ε and hence γ. Y : V 0 , u t ≡ 0, Y : V 0 , u(t)≡ 1, (A13) Treatment Comparison for Equal Area Under the Drug Y : V 0 , u t ≡ 2, Concentration-Time Curve Y : V 0 , u t ≡ 0, Y : V 0 , u t t. Here, we provide an analogous comparison of treatment outcomes between constant and pulsed therapy as in the main text; however, All quantities Y ,i =0, 1,…, 7 are measurable, as each requires only treatment magnitudes u and u are chosen such that on,c on,p knowledge of V(t) in a small positive neighborhood of t = 0. Note that Z Z the set of controls u(t) is relatively simple, with Y exclusively de- t +Δt +Δt t +Δt +Δt d on off d on off 7 u t dt u t dt, (A12) termined via a nonconstant input. c p t t d d Each measurable Y may also be written in terms of a subset of the which is equivalent to the conservation of total administered dose between parameters d, ε, p ,and α, as all derivatives can be calculated in both strategies over a single pulsing cycle. In Equation A12, u (t)and u (t) terms of the right sides of Equations A1 and A2. For more details, c p denote constant and pulsed therapy schedules, respectively. The constant see the below section. Equating the expressions yields a system of 14 © 2019 by American Society of Clinical Oncology 70 80 50 60 60 70 Differentiating Spontaneous and Induced Resistance A B Treatment Strategies,  = 0 Tumor Dynamics,  = 0 Constant Constant 1.6 Pulsed Pulsed 0.9 1.4 0.8 1.2 0.7 0.6 0.8 0.5 0.4 0.6 0.3 0.4 0.2 0.2 0.1 0 10 20 30 40 0 10 20 30 40 Time Time C D -2 -2 Treatment Strategies,  = 10 Tumor Dynamics,  = 10 Constant Constant 1.6 1 Pulsed Pulsed 0.9 1.4 0.8 1.2 0.7 0.6 0.8 0.5 0.4 0.6 0.3 0.4 0.2 0.2 0.1 0 10 20 30 40 0 10 20 40 Time Time −2 FIG A1. Comparison of treatment efﬁcacy for phenotype-preserving drugs (α = 0) and resistance-inducing drugs (α =10 ), where resistance is reversible. The left column indicates treatment strategy, whereas right indicates corresponding tumor volume response. Note that the dashed red line in the right column indicates the tumor volume representing treatment failure, V . (A) Constant and pulsed therapies after tumor detection for α = 0. (B) Responses −2 corresponding to treatment regimens in panel A. (C) Constant and pulsed therapies after tumor detection for α =10 . (D) Responses corresponding to treatment regimens in panel C. equations for the model parameter, which we are able to solve. 1 1 2 Y − Y + Y − d Y 5 4 3 0 2 2 Carrying out these computations yields the following solution: α . (A17) dY Y − Y 2 1 d − , (A14) Note that in Equations A14 and A17 each quantity is determined by the Y and the parameter values previously listed. We do not 1 3 write the solution in explicit form for the sake of clarity as the Y − Y + Y − 2Y − Y + dY 1 − Y 7 6 5 4 3 0 0 2 2 , (A15) resulting equations are unwieldy. Furthermore, the solution of dY this system relies on the assumption of strictly positive initial 0 0 conditions (S = Y . 0), wild-type drug induction death rate (d), 2 3 Y − 1 − e Y + 3 − e Y − 2Y 3 0 0 0 and background mutation rate (ε), all of which are made in p , (A16) eY 1 − Y 0 0 this work. JCO Clinical Cancer Informatics 15 u u Tumor Volume S + R Tumor Volume S + R Greene, Gevertz, and Sontag A B Treatment Strategy,  = 0 Population Dynamics,  = 0 0.5 1 0.45 0.9 R 0.4 0.8 0.35 0.7 0.3 0.6 0.25 0.5 0.2 0.4 0.15 0.3 0.1 0.2 0.05 0.1 0 20406080 100 120 140 020 40 60 80 100 120 140 Time Time C D Treatment Strategy,  = 0 Population Dynamics,  = 0 5 0.9 4.5 0.8 0.7 3.5 0.6 0.5 2.5 0.4 0.3 1.5 0.2 0.1 0.5 02468 10 12 14 16 18 0246 8 10 12 14 16 18 Time Time FIG A2. Treatment dynamics for phenotype-preserving drugs (α = 0). The left column indicates treatment strategy, whereas the right indicates cor- responding population response. (A) Constant treatment after tumor detection. (B) Response to constant treatment. Note that at the time of treatment failure, the tumor is essentially entirely resistant. (C) Pulsed therapy after tumor detection. (D) Response to pulsed therapy. Note that the treatment fails much earlier versus the constant dose and that the tumor is primarily drug sensitive. Equation A17 is the desired result of our analysis. It demonstrates that phenotype-switching rate α from a class of input controls u(t); the drug-induced phenotype switching rate α may be determined by however, we see that the calculation involved measuring de- a relatively small set of input controls u(t). As discussed in the previous rivatives at the initial detection time t = t . Furthermore, the applied section, the value of α may have a large impact on treatment efﬁcacy controls (Eq A13) are nonconstant and thus require fractional and, therefore, determining its value is clinically signiﬁcant. Our results doses to be administered. Clinically, such strategies and mea- now prove that it is possible to compute the induction rate and, hence, surements may be difﬁcult and/or impractical. In this section, we use this information in the design of treatment protocols. In the next describe an in vitro method for estimating α using constant ther- section, we investigate other qualitative properties that could also be apies only. Speciﬁcally, our primary goal is to distinguish drugs with applied to understand the rate of drug-induced resistance. α = 0 (phenotype preserving) and α . 0 (resistance inducing). Such experiments, described below, may be implemented for a speciﬁc An In Vitro Experimental Protocol to Distinguish drug, even if its precise mechanism of promoting resistance remains Spontaneous and Drug-Induced Resistance uncertain. We have demonstrated that all parameters in Equations 4 and 5 are Before describing the in vitro experiment, we note that we are in- identiﬁable so that it is theoretically possible to determine the terested in qualitative properties for determining α. Indeed, in most 16 © 2019 by American Society of Clinical Oncology Population Population Differentiating Spontaneous and Induced Resistance -2 -2 Treatment Strategy,  = 10 Population Dynamics,  = 10 0.5 0.9 0.45 0.8 0.4 0.7 0.35 0.6 0.3 0.5 0.25 0.4 0.2 0.3 0.15 0.2 0.1 0.1 0.05 R 0 102030405060 0 102030405060 Time Time C D -2 -2 Treatment Strategy,  = 10 Population Dynamics,  = 10 4.5 0.9 4 0.8 3.5 0.7 3 0.6 2.5 0.5 2 0.4 1.5 0.3 1 0.2 0.5 0.1 R 0 102030405060 0 102030405060 Time Time −2 FIG A3. Treatment dynamics for resistance-inducing drugs (α =10 ). The left column indicates treatment strategy, whereas the right indicates corresponding population response. (A) Constant treatment after tumor detection. (B) Response to constant treatment. Note that at the time of treatment failure, the tumor is essentially entirely resistant. Dynamics are similar to Figure 2B, although with a shorter survival time. (C) Pulsed therapy after tumor detection. (D) Response to pulsed therapy. Note that here, in contrast to the case of a phenotype-preserving drug as shown in Figure A2, pulsed therapy exhibits a longer survival time. modeling scenarios, we have little or no knowledge of precise pa- mutation rate and p the relative ﬁtness of resistant cells. We thus rameter values and instead must rely on characteristics that distinguish perform a standard dose-response experiment for each value of the switching rate α independently of quantitative measurements. drug sensitivity d and measure the time t to reach critical size V as c c Furthermore, as a general framework for drug resistance, the only afunctionof d. The response t will then depend on the applied guaranteed clinically observable output variables are the critical tumor dose u—recall that we are only administering constant therapies— volume V and the corresponding time t . For a description of the and the sensitivity of wild-type cells d,aswellasthe induction c c treatment protocol, see above. We cannot expect temporal clonal rate α: subpopulation measurements. Assuming that V is ﬁxed for a given cancer, t is thus the only observable that we consider. t t u, d, α . (A18) c c By examining Equations 4 and 5, we see that the key parameters that dictate progression to the steady state (S, R) = (0,1) are d and α, as these determine the effectiveness and resistance induction of We further imagine that it is possible to adjust the wild-type the treatment, respectively. Recall that ε is the ﬁxed background drug sensitivity d. For example, in the case of multidrug resistance JCO Clinical Cancer Informatics 17 Population Population Greene, Gevertz, and Sontag in which the overabundance of P-glycoprotein affects drug phar- Comparing Figures 3A and 4A, we observe similar properties: small t macokinetics, altering the expression of MDR1 via ABCBC1 or even for small doses, a sharp increase about a critical u ,followedby CDX2 (Koh I, Hinoi T, Sentani K, et al: Cancer Med 5:1546-1555, smooth decrease and eventual horizontal asymptote (for mathe- 2016) may yield a quantiﬁable relationship between wild-type cell matical justiﬁcation, see the below section). However, note that for and d, thus producing a range of drug-sensitive cell types. Figure 3A a resistance-inducing drug (Fig 3A), maximum critical time T (d) exhibits a set of dose-response curves for representative drug increases as a function of d. This is in stark contrast to the constant −2 sensitivities d for the case of a resistance-inducing drug (α =10 ). behavior obtained for α = 0, argued above and demonstrated in Figure 4B. To further understand this phenomenon, we plot T (d)for For each of these cell types, we then deﬁne the supremum response −2 a ﬁxedinductionrate α =10 in Figure 3B. The behavior of this curve time over administered doses: is a result of the fact that the critical dosage u at which T (d)is c α obtained is a decreasing function of d (see Equation A24 and Fig A4 T d : sup t u, d, α . (A19) { } α c in the below section). But as u also controls the amount of resistant cells generated—via the αu(t)S term—resistance growth is impeded Note that in a laboratory setting, only a ﬁnite number of doses will be by a decreasing u . Thus, as a non-negligible amount of resistant administered so that the above supremum is actually a maximum, cells are necessary to yield T (d), more time is required for resistant but for mathematical precision we retain supremum. Thus, we cells to accumulate as d increases. Hence, T (d) increases a function obtain a curve T = T (d) for each value of the induced resistance α α of d. rate α. We then explore the properties of these curves for different α The behavior observed in Figures 3B and 4B is precisely the qualitative values. distinction that could assist in determining the induced resistance rate Consider ﬁrst the case of a phenotype-preserving drug, so that α =0. α. In the case of a phenotype-preserving drug, the proposed in vitro As u(t) ≡ u, we see that the system Equations 4 and 5 depends only experiment would produce a ﬂat curve, whereas a resistance-inducing on the product of u and d. Hence, the dependence in Equation A18 drug (α. 0) would yield an increasing function T (d). Furthermore, we becomes the form t (u$d, 0), and thus the supremum in Equation could use this phenomenon, in principle, to measure the induction rate A19 is instead across the joint parameter D := u$d: from the experimental T (d) curve. For example, Figure 5A displays −2 a range of T (d) for α near 10 . T : sup t D, 0 . (A20) 0 c Figure 5A shows a clear dependence of T (d) on the value of α. Quantitatively characterizing such curves would allow us to reverse Clearly, this is independent of d so that T is simply a horizontal line for engineer the induction rate α; however, we note that, in general, the α = 0. Qualitatively, the resulting curve will have no variation among the precise characteristics will depend on the other ﬁxed parameter engineered sensitive phenotypes, save for experimental and mea- values, such as p , V ,and «. Indeed, only order of magnitude r c surement noise. Figure 4 shows both representative curves (Fig 4A estimates may be feasible. Illustrative sample curves are provided compared with Fig 3A) and a plot of T (d)(Fig 4B), which veriﬁes its in Figure 5B. Two such characteristics are apparent from this independence of d. We make two minor technical notes. First, it is ﬁgure, both related to the slope of T (d). First, as d → 0 ,we important that we assume d. 0 here, as otherwise D = 0, independent observe an increase in the slope of T (d)as α decreases (note that of dose u and the supremum is over a one element set. See below for α in Fig 5B,only d ≥ 0.05 are plotted). This follows from the con- more details and the implications for α identiﬁability. Second, the slight tinuity of solutions on parameters and the fact that T (d) possesses variation for large values of d is a result of numerical error, as the 0 a jump discontinuity at d =0—that is, its distributional derivative is maximum of t occurs at decreasing doses (see the below section and given by Figure A4 for more details). | T d kδ d , (A21) d0 0 ∂d where δ is the Dirac function and k is a positive constant. As discussed Maximum Dosage Yield T (d ) previously (Equation A20 and the subsequent paragraph), T (d)is ﬂat, 3.5 except at d = 0 where the vector ﬁeld contains no u dependence. Numerical Therefore, the set over which the maximum is taken is irrelevant and Theory T (d) is thus proportional to the Heaviside function, which possesses the distributional derivative Equation A21. The constant k is de- termined by the size of the discontinuity of T (d). Continuous de- 2.5 pendence on parameters then implies that as α increases, the resulting derivative decreases away from positive inﬁnity as the corresponding derivative for T (d) with α . 0isdeﬁned in the classic sense for α . 0: 1.5 ∂ ∂ | T d ≤ 0. d0 α ∂α ∂d The above argument implies that measuring the slope of T (d)at d = 0 will give a characterization of the phenotypic alteration rate α of the 0.5 treatment; however, such experiments may be impractical, as ﬁne- tuning the sensitivity of a cell near complete resistance may be difﬁcult. Alternatively, one could analyze the degree of ﬂatness for a relatively 0 0.5 1 1.5 2 2.5 3 3.5 4 large d—so to be sufﬁciently far from d =0—and correlate this measure with α. For example, examining d =2in Figure 5B, we see that the relative slope of T (d) with respect to d should correlate with decreasing α. An argument similar to the above makes this rigorous, for FIG A4. Dose yielding maximal response time T (d) computed nu- d sufﬁciently large. Practical issues still arise, but this second method merically, as well as the approximation given by Equation A24. All provides a more global method for possibly computing α. Indeed, −2 parameters appear as in Table 1, and α =10 . The numerical slopes at a given d can be approximated by a wider range of secant maximum is computed over a discretization of constant dosage approximations as the result holds for a range of d compared with the procedures u(t) ≡ u, for u 2 [0,5], with a mesh size Δu = 0.005. previously discussed case when d is near zero. Furthermore, our focus 18 © 2019 by American Society of Clinical Oncology u Differentiating Spontaneous and Induced Resistance is largely on the qualitative aspects of α determination, such as the cells. Indeed, as u → 0, the dynamics of Equations A1 and A2 differences in Figures 3A and 4A, and determining whether the approach those of treatment itself induces resistance to emerge. dS ˜ ˜ ˜ 1 − S + R S, (A22) Introduction to Identiﬁability Analysis dt In this section, we provide additional details on the theoretical iden- dR tiﬁability of model parameters. As mentioned in the main text, all ˜ ˜ ˜ p 1 − S + R R, (A23) dt higher-order derivatives at initial time t = 0 may be calculated in terms of the initial conditions (S(0), R(0)) and the control function u(t). For for small ﬁnite times, as ε ,,1. Trajectories of Equations A22 and A23 example, for an arbitrary system remain on the curve x˙ t f x t , u t , pr R S S pr with external control u(t), the second derivative ẍ may be calculated using the chain rule: as the solution approaches the line S + R = 1. The critical time t is then determined by the intersection of this curve with S + R = V and, thus, x¨ J x, u f +∇ f x, u u˙, f u has sensitive population S at t given by the unique solution in the ﬁrst c c quadrant of where J (x) is the Jacobian matrix of f, evaluated at state x and control u.If x(0) = x is known, the above expression is a relation among ˙ S + S V . parameters, together with u and u evaluated at time t = 0. An anal- c c p c ogous statement holds for a measurable output y = h(x), but will also involve the Jacobian of h. Concretely, for the model of induced drug As R ,, 1 (as ε ,,1), S ≈ V , as claimed. Time t is thus small for 0 c c c resistance Equations 4 and 5, ﬁrst derivatives of the tumor volume may small u. be calculated as Increasing values of u imply that t also increases as the overall growth rate of sensitive cells is decreased; however, there exists a critical dose V 0 S 0 + R 0 , u such that sensitive cells alone are not able to multiply sufﬁciently to 1 − S + R S − e + αu t S − du t S attain V , so that the critical volume must have non-negligible con- tributions from the resistant fraction. This leads to the bifurcation + p 1 − S + R R + e + αu t S | , r t0 apparent in Figures 3A and 4A. We can even approximate the critical 1 − S S − du 0 S , 0 0 0 dose maximizing t ,as V must be an approximation for the carrying c c capacity of the sensitive cells: for any control u(t) [recall that R(0) = 0]. Similarly, for the second derivative, we compute: S ≈ V . K c V 0 S 1 − S − e + d u 0 1 − 2S − du(0) 0 0 0 Examining the right side of Equation A1 and assuming that the dy- namics of the resistant population are negligible, which is accurate in + S αu 0 + e p − S 1 + p − dS u 0 . 0 r 0 r 0 the early stages of treatment (Figs 2B and 3B), we see that the dose that yields the maximum temporal response should be Using such expressions—or, more precisely, the Lie derivatives of the vector ﬁeld [see Sontag (PLOS Comput Biol 13:e1005447, 2017)] 1 − e − V —for the controls in Equation A13, one is able to obtain a set of u ≈ . (A24) α + d equations between the set of Y ,i =0, 1,…, 7 and the parameters d, «, p , and α. Solving these equations allows us to determine the pa- That is, the dose at which T (d) is obtained is given approximately by rameters with respect to the measurable quantities. The algebraic the expression in Equation A24. For a sample numerical comparison of solution is Equations A14-A17. the predicted formula Equation A24 and a numerical optimization over a range of drug sensitivities d (Figure A4). Note that in actuality, S , Analysis of Critical Time T (d) V , as the resistant dynamics cannot be ignored entirely. Thus, the We provide a qualitative understanding of the properties of T (d), precise value of u will be smaller than that provided in the previous the maximum time, across all constant doses for the tumor to formula as we numerically observe. Lastly, u decreases with in- reach size V . This Appendix is designed to explain the basic creasing values of parameter d and, thus, requires an increasingly ﬁne properties discussedinAnInVitro Experimental Protocol to discretization to numerically locate the maximum value. Hence, some Distinguish Spontaneous and Drug-Induced Resistance in the numerical error is observed in Figure 4B. main text. Lastly, as u is increased further, the dose becomes sufﬁciently large so We ﬁrst note that T (d) is achieved at a medium dose u .More that the inhibition of S via therapy implies that S cannot approach the α c precisely, we describe the qualitative properties of Figures 3A and critical volume V and, hence, V is again reached by an essentially c c 4A.Fix adrugsensitivity d. For small u, the sensitive subpopulation is homogeneous population, but now of resistant cells. As resistant cells not sufﬁciently inhibited and thus expands rapidly to cross the divide at a slower rate (p , 1), the corresponding time t is smaller. For r c threshold V , with an essentially homogenous population of sensitive a schematic of the three regimes described above (Figure A5). JCO Clinical Cancer Informatics 19 Greene, Gevertz, and Sontag u u u < u u > u t t Time FIG A5. Schematic demonstrating dynamics of variation in t on dosage u. Sensitive cell population plotted as a function of time for three representative doses. For u , u , sensitive cells grow and reach V in a short amount of time. As u→u , the sensitive population approaches its approximate carrying capacity of V , but subsequently decreases as a result of the dynamics of resistance. Here, t is maximized as the sensitive population spends a large amount of time near V . For u . u , the sensitive population is eliminated quickly, and c c V is obtained by a primarily resistant population. 20 © 2019 by American Society of Clinical Oncology Volume http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png JCO Clinical Cancer Informatics Wolters Kluwer Health http://www.deepdyve.com/lp/wolters-kluwer-health/mathematical-approach-to-differentiate-spontaneous-and-induced-iy047KcP69

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References (86)

E. Pasquier, M. Kavallaris, N. André (2010)
Metronomic chemotherapy: new rationale for new directions
Nature Reviews Clinical Oncology, 7
T. Traina, L. Norton (2015)
Log-Kill Hypothesis
M. Feizabadi (2017)
Modeling multi-mutation and drug resistance: analysis of some case studies
Theoretical Biology & Medical Modelling, 14
J. Nyce, S. Leonard, D. Canupp, S. Schulz, So Wong (1993)
Epigenetic mechanisms of drug resistance: drug-induced DNA hypermethylation and drug resistance.
Proceedings of the National Academy of Sciences of the United States of America, 90 7
J. Nyce (1997)
Drug-induced DNA hypermethylation: a potential mediator of acquired drug resistance during cancer chemotherapy.
Mutation research, 386 2
Lin Liu, Fei Li, W. Pao, F. Michor (2015)
Dose-Dependent Mutation Rates Determine Optimum Erlotinib Dosing Strategies for EGFR Mutant Non-Small Cell Lung Cancer Patients
PLoS ONE, 10
U. Ledzewicz, H. Schättler (2005)
Drug resistance in cancer chemotherapy as an optimal control problem
Discrete and Continuous Dynamical Systems-series B, 6
B. Bender, E. Schindler, L. Friberg (2013)
Population pharmacokinetic–pharmacodynamic modelling in oncology: a tool for predicting clinical response
British Journal of Clinical Pharmacology, 79
Lurias, M. Delbrock (2003)
MUTATIONS OF BACTERIA FROM VIRUS SENSITIVITY TO VIRUS RESISTANCE’-’
N. Komarova, D. Wodarz (2007)
Stochastic modeling of cellular colonies with quiescence: an application to drug resistance in cancer.
Theoretical population biology, 72 4
(2015)
Preexisting MEK1P124mutations diminish response to BRAF inhibitors in metastatic melanoma patients
F. Ghiringhelli, C. Ménard, P. Puig, S. Ladoire, S. Roux, F. Martin, E. Solary, A. Cesne, L. Zitvogel, B. Chauffert (2007)
Metronomic cyclophosphamide regimen selectively depletes CD4+CD25+ regulatory T cells and restores T and NK effector functions in end stage cancer patients
Cancer Immunology, Immunotherapy, 56
Sreenath Sharma, Diana Lee, Bihua Li, M. Quinlan, F. Takahashi, S. Maheswaran, U. McDermott, N. Azizian, L. Zou, M. Fischbach, Kwok-kin Wong, K. Brandstetter, Ben Wittner, Sridhar Ramaswamy, M. Classon, J. Settleman (2010)
A Chromatin-Mediated Reversible Drug-Tolerant State in Cancer Cell Subpopulations
Cell, 141
T. Jackson, H. Byrne (2000)
A mathematical model to study the effects of drug resistance and vasculature on the response of solid tumors to chemotherapy.
Mathematical biosciences, 164 1
Sydney Shaffer, Margaret Dunagin, Stefan Torborg, E. Torre, Benjamin Emert, C. Krepler, M. Beqiri, Katrin Sproesser, P. Brafford, M. Xiao, Elliott Eggan, Ioannis Anastopoulos, C. Vargas-Garcia, Abhyudai Singh, K. Nathanson, M. Herlyn, A. Raj (2017)
Rare cell variability and drug-induced reprogramming as a mode of cancer drug resistance
Nature, 546
K. Leder, J. Foo, B. Skaggs, M. Gorre, C. Sawyers, F. Michor (2011)
Fitness Conferred by BCR-ABL Kinase Domain Mutations Determines the Risk of Pre-Existing Resistance in Chronic Myeloid Leukemia
PLoS ONE, 6
Margaret Chapman, Tyler Risom, A. Aswani, Roel Dobbe, R. Sears, C. Tomlin (2016)
A model of phenotypic state dynamics initiates a promising approach to control heterogeneous malignant cell populations
2016 IEEE 55th Conference on Decision and Control (CDC)
N. Komarova, D. Wodarz (2005)
Drug resistance in cancer: principles of emergence and prevention.
Proceedings of the National Academy of Sciences of the United States of America, 102 27
F. Michor, M. Nowak, Y. Iwasa (2006)
Evolution of resistance to cancer therapy.
Current pharmaceutical design, 12 3
N. Makridakis, J. Reichardt (2012)
Translesion DNA Polymerases and Cancer
Frontiers in Genetics, 3
R. Gatenby, A. Silva, R. Gillies, B. Frieden (2009)
Adaptive therapy.
Cancer research, 69 11
Lucy Swift, R. Golsteyn (2014)
Genotoxic Anti-Cancer Agents and Their Relationship to DNA Damage, Mitosis, and Checkpoint Adaptation in Proliferating Cancer Cells
International Journal of Molecular Sciences, 15
A. Silva, R. Gatenby (2010)
A theoretical quantitative model for evolution of cancer chemotherapy resistance
Biology Direct, 5
S. Chakrabarti, F. Michor (2017)
Pharmacokinetics and Drug Interactions Determine Optimum Combination Strategies in Computational Models of Cancer Evolution.
Cancer research, 77 14
J. Foo, K. Leder, S. Mumenthaler (2012)
Cancer as a moving target: understanding the composition and rebound growth kinetics of recurrent tumors
Evolutionary Applications, 6
T. Brocato, P. Dogra, E. Koay, Armin Day, Y. Chuang, Zhihui Wang, V. Cristini (2014)
Understanding Drug Resistance in Breast Cancer with Mathematical Oncology
Current Breast Cancer Reports, 6
Z. Iqbal, A. Aleem, M. Iqbal, Mubashar Naqvi, Ammara Gill, A. Taj, Abdul Qayyum, N. ur-Rehman, A. Khalid, I. Shah, M. Khalid, R. Haq, M. Khan, S. Baig, A. Jamil, Muhammad Abbas, M. Absar, Amer Mahmood, M. Rasool, T. Akhtar (2013)
Sensitive Detection of Pre-Existing BCR-ABL Kinase Domain Mutations in CD34+ Cells of Newly Diagnosed Chronic-Phase Chronic Myeloid Leukemia Patients Is Associated with Imatinib Resistance: Implications in the Post-Imatinib Era
PLoS ONE, 8
A. Correia, M. Bissell (2012)
The tumor microenvironment is a dominant force in multidrug resistance.
Drug resistance updates : reviews and commentaries in antimicrobial and anticancer chemotherapy, 15 1-2
S. Menchón (2015)
The Effect of Intrinsic and Acquired Resistances on Chemotherapy Effectiveness
Acta Biotheoretica, 63
A. Lorz, T. Lorenzi, J. Clairambault, A. Escargueil, B. Perthame (2014)
Modeling the Effects of Space Structure and Combination Therapies on Phenotypic Heterogeneity and Drug Resistance in Solid Tumors
Bulletin of Mathematical Biology, 77
Paul Orlando, R. Gatenby, Joel Brown (2012)
Cancer treatment as a game: integrating evolutionary game theory into the optimal control of chemotherapy
Physical Biology, 9
Orit Lavi, J. Greene, D. Levy, M. Gottesman (2013)
The role of cell density and intratumoral heterogeneity in multidrug resistance.
Cancer research, 73 24
A. Pisco, A. Brock, J. Zhou, A. Moor, Mitra Mojtahedi, D. Jackson, Sui Huang (2013)
Non-Darwinian dynamics in therapy-induced cancer drug resistance
Nature communications, 4
T. Gajewski, Yuru Meng, C. Blank, I. Brown, A. Kacha, J. Kline, H. Harlin (2006)
Immune resistance orchestrated by the tumor microenvironment
Immunological Reviews, 213
Wei‐Ping Lee (1993)
The role of reduced growth rate in the development of drug resistance of HOB1 lymphoma cells to vincristine.
Cancer letters, 73 2-3
J. Cunningham, R. Gatenby, Joel Brown (2011)
Evolutionary dynamics in cancer therapy.
Molecular pharmaceutics, 8 6
C. Holohan, S. Schaeybroeck, D. Longley, P. Johnston (2013)
Cancer drug resistance: an evolving paradigm
Nature Reviews Cancer, 13
Orit Lavi, M. Gottesman, D. Levy (2012)
The dynamics of drug resistance: a mathematical perspective.
Drug resistance updates : reviews and commentaries in antimicrobial and anticancer chemotherapy, 15 1-2
I. Bozic, M. Nowak (2014)
Timing and heterogeneity of mutations associated with drug resistance in metastatic cancers
Proceedings of the National Academy of Sciences, 111
H. Zahreddine, K. Borden (2013)
Mechanisms and insights into drug resistance in cancer
Frontiers in Pharmacology, 4
A Lorz, T Lorenzi, M Hochberg (2013)
Populational adaptive evolution, chemotherapeutic resistance and multiple anti-cancer therapies
ESAIM: Math Model Num Anal, 47
D. Axelrod, Sudeepti Vedula, James Obaniyi (2017)
Effective chemotherapy of heterogeneous and drug-resistant early colon cancers by intermittent dose schedules: a computer simulation study
Cancer Chemotherapy and Pharmacology, 79
G. Fetterly, Urvi Aras, D. Lal, M. Murphy, Patricia Meholick, E. Wang (2013)
Development of a Preclinical PK/PD Model to Assess Antitumor Response of a Sequential Aflibercept and Doxorubicin-Dosing Strategy in Acute Myeloid Leukemia
The AAPS Journal, 15
V. Krutyakov (2006)
Eukaryotic error-prone DNA polymerases: The presumed roles in replication, repair, and mutagenesis
Molecular Biology, 40
Qiong Wu, Mengyao Li, Han-qing Li, C. Deng, Liang Li, T. Zhou, W. Lu (2013)
Pharmacokinetic-pharmacodynamic modeling of the anticancer effect of erlotinib in a human non-small cell lung cancer xenograft mouse model
, 34
Douglas McMillin, J. Negri, C. Mitsiades (2013)
The role of tumour–stromal interactions in modifying drug response: challenges and opportunities
Nature Reviews Drug Discovery, 12
J. Foo, F. Michor (2014)
Evolution of acquired resistance to anti-cancer therapy.
Journal of theoretical biology, 355
M. Meads, R. Gatenby, W. Dalton (2009)
Environment-mediated drug resistance: a major contributor to minimal residual disease
Nature Reviews Cancer, 9
U. Ledzewicz, K. Bratton, H. Schättler (2015)
A 3-Compartment Model for Chemotherapy of Heterogeneous Tumor Populations
Acta Applicandae Mathematicae, 135
J. Gevertz, Zahra Aminzare, K. Norton, Judith Pérez-Velázquez, A. Volkening, K. Rejniak (2014)
Emergence of Anti-Cancer Drug Resistance: Exploring the Importance of the Microenvironmental Niche via a Spatial Model
arXiv: Populations and Evolution
D. Jodrell, M. Egorin, R. Canetta, Patricia Langenberg, Ellie Goldbloom, James Burroughs, J. Goodlow, S. Tan, E. Wiltshaw (1992)
Relationships between carboplatin exposure and tumor response and toxicity in patients with ovarian cancer.
Journal of clinical oncology : official journal of the American Society of Clinical Oncology, 10 4
A. Lorz, T. Lorenzi, M. Hochberg, J. Clairambault, B. Perthame (2012)
Populational adaptive evolution, chemotherapeutic resistance and multiple anti-cancer therapies
arXiv: Analysis of PDEs
Marios Hadjiandreou, G. Mitsis (2014)
Mathematical Modeling of Tumor Growth, Drug-Resistance, Toxicity, and Optimal Therapy Design
IEEE Transactions on Biomedical Engineering, 61
Ami Shah, K. Rejniak, J. Gevertz (2016)
Limiting the Development of Anti-Cancer Drug Resistance in a Spatial Model of Micrometastases
bioRxiv
I. Bozic, Johannes Reiter, Benjamin Allen, T. Antal, K. Chatterjee, Preya Shah, Yo Moon, A. Yaqubie, Nicole Kelly, D. Le, E. Lipson, P. Chapman, L. Diaz, B. Vogelstein, M. Nowak (2013)
Evolutionary dynamics of cancer in response to targeted combination therapy
eLife, 2
C. Carrère (2017)
Optimization of an in vitro chemotherapy to avoid resistant tumours.
Journal of theoretical biology, 413
B. Wacław, I. Bozic, M. Pittman, R. Hruban, B. Vogelstein, M. Nowak (2015)
A spatial model predicts that dispersal and cell turnover limit intratumour heterogeneity
Nature, 525
J. Foo, F. Michor (2009)
Evolution of Resistance to Targeted Anti-Cancer Therapies during Continuous and Pulsed Administration Strategies
PLoS Computational Biology, 5
Derek Woods, J. Turchi (2013)
Chemotherapy induced DNA damage response
Cancer Biology & Therapy, 14
B. Szikriszt, Á. Póti, O. Pipek, M. Krzystanek, N. Kanu, J. Molnár, D. Ribli, Z. Szeltner, G. Tusnády, I. Csabai, Z. Szallasi, C. Swanton, Dávid Szüts (2016)
A comprehensive survey of the mutagenic impact of common cancer cytotoxics
Genome Biology, 17
Sabine Lange, K. Takata, R. Wood (2011)
DNA polymerases and cancer
Nature Reviews Cancer, 11
J. Foo, F. Michor (2010)
Evolution of resistance to anti-cancer therapy during general dosing schedules.
Journal of theoretical biology, 263 2
L. Loeb, C. Springgate, N. Battula (1974)
Errors in DNA replication as a basis of malignant changes.
Cancer research, 34 9
M. Gottesman (2002)
Mechanisms of cancer drug resistance.
Annual review of medicine, 53
A. Goldman, B. Majumder, A. Dhawan, S. Ravi, D. Goldman, M. Kohandel, P. Majumder, Shiladitya Sengupta (2015)
Temporally sequenced anticancer drugs overcome adaptive resistance by targeting a vulnerable chemotherapy-induced phenotypic transition
Nature Communications, 6
O. Trédan, C. Galmarini, Krupa Patel, I. Tannock (2007)
Drug resistance and the solid tumor microenvironment.
Journal of the National Cancer Institute, 99 19
J Perez-Velazquez, J Gevertz, A Karolak, K Rejniak (2016)
Microenvironmental niches and sanctuaries: A route to acquired resistance
Systems Biology of Tumor Microenvironment, Volume 936
Feng Fu, M. Nowak, S. Bonhoeffer (2014)
Spatial Heterogeneity in Drug Concentrations Can Facilitate the Emergence of Resistance to Cancer Therapy
PLoS Computational Biology, 11
C. Roche-Lestienne, C. Preudhomme (2003)
Mutations in the ABL kinase domain pre-exist the onset of imatinib treatment.
Seminars in hematology, 40 2 Suppl 2
J. Greene, Orit Lavi, M. Gottesman, D. Levy (2014)
The Impact of Cell Density and Mutations in a Model of Multidrug Resistance in Solid Tumors
Bulletin of Mathematical Biology, 76
Y. Erdi (2012)
Limits of Tumor Detectability in Nuclear Medicine and PET
Molecular Imaging and Radionuclide Therapy, 21
H. Thomas, H. Coley (2003)
Overcoming multidrug resistance in cancer: an update on the clinical strategy of inhibiting p-glycoprotein.
Cancer control : journal of the Moffitt Cancer Center, 10 2
R. Kerbel, B. Kamen (2004)
The anti-angiogenic basis of metronomic chemotherapy
Nature Reviews Cancer, 4
M. Dean, T. Fojo, S. Bates (2005)
Tumour stem cells and drug resistance
Nature Reviews Cancer, 5
Genevieve Housman, Shannon Byler, S. Heerboth, K. Lapinska, Mckenna Longacre, N. Snyder, S. Sarkar (2014)
Drug Resistance in Cancer: An Overview
Cancers, 6
A. Coldman, J. Goldie (1986)
A stochastic model for the origin and treatment of tumors containing drug-resistant cells
Bulletin of Mathematical Biology, 48
S. Mumenthaler, J. Foo, K. Leder, Nathan Choi, D. Agus, W. Pao, P. Mallick, F. Michor (2011)
Evolutionary Modeling of Combination Treatment Strategies To Overcome Resistance to Tyrosine Kinase Inhibitors in Non-Small Cell Lung Cancer
Molecular Pharmaceutics, 8
I. Kareva, D. Waxman, Giannoula Klement (2015)
Metronomic chemotherapy: an attractive alternative to maximum tolerated dose therapy that can activate anti-tumor immunity and minimize therapeutic resistance.
Cancer letters, 358 2
K. Brimacombe, M. Hall, D. Auld, James Inglese, C. Austin, M. Gottesman, K. Fung (2009)
A dual-fluorescence high-throughput cell line system for probing multidrug resistance.
Assay and drug development technologies, 7 3
R. Chisholm, T. Lorenzi, A. Lorz, A. Larsen, Luís Almeida, A. Escargueil, J. Clairambault (2015)
Emergence of drug tolerance in cancer cell populations: an evolutionary outcome of selection, nongenetic instability, and stress-induced adaptation.
Cancer research, 75 6
Judith Pérez-Velázquez, J. Gevertz, A. Karolak, K. Rejniak (2016)
Microenvironmental Niches and Sanctuaries: A Route to Acquired Resistance.
Advances in experimental medicine and biology, 936
A. Pisco, S. Huang (2015)
Non-genetic cancer cell plasticity and therapy-induced stemness in tumour relapse: ‘What does not kill me strengthens me'
British Journal of Cancer, 112
S. Shackney, G. McCormack, G. Cuchural (1978)
Growth rate patterns of solid tumors and their relation to responsiveness to therapy: an analytical review.
Annals of internal medicine, 89 1
S. Mumenthaler, J. Foo, Nathan Choi, N. Heise, K. Leder, D. Agus, W. Pao, F. Michor, P. Mallick (2015)
The Impact of Microenvironmental Heterogeneity on the Evolution of Drug Resistance in Cancer Cells
Cancer Informatics, 14
R. Chignola, R. Foroni (2005)
Estimating the growth kinetics of experimental tumors from as few as two determinations of tumor size: implications for clinical oncology
IEEE Transactions on Biomedical Engineering, 52
Kimiyo Yamamoto, K. Hirota, S. Takeda, H. Haeno (2014)
Evolution of Pre-Existing versus Acquired Resistance to Platinum Drugs and PARP Inhibitors in BRCA-Associated Cancers
PLoS ONE, 9

Publisher: Wolters Kluwer Health
Copyright: (C) 2019 by Lippincott Williams & Wilkins, Inc.
ISSN: 2473-4276
DOI: 10.1200/CCI.18.00087
Publisher site: See Article on Publisher Site

Abstract

original report abstract Mathematical Approach to Differentiate Spontaneous and Induced Evolution to Drug Resistance During Cancer Treatment 1 2 3,4 James M. Greene, PhD ; Jana L. Gevertz, PhD ; and Eduardo D. Sontag, PhD PURPOSE Drug resistance is a major impediment to the success of cancer treatment. Resistance is typically thought to arise from random genetic mutations, after which mutated cells expand via Darwinian selection. However, recent experimental evidence suggests that progression to drug resistance need not occur randomly, but instead may be induced by the treatment itself via either genetic changes or epigenetic alterations. This relatively novel notion of resistance complicates the already challenging task of designing effective treatment protocols. MATERIALS AND METHODS To better understand resistance, we have developed a mathematical modeling framework that incorporates both spontaneous and drug-induced resistance. RESULTS Our model demonstrates that the ability of a drug to induce resistance can result in qualitatively different responses to the same drug dose and delivery schedule. We have also proven that the induction parameter in our model is theoretically identiﬁable and propose an in vitro protocol that could be used to determine a treatment’s propensity to induce resistance. Clin Cancer Inform. © 2019 by American Society of Clinical Oncology Licensed under the Creative Commons Attribution 4.0 License INTRODUCTION drug resistance describes the case in which a tumor contains a subpopulation of drug-resistant cells at the Tumor resistance to chemotherapy and targeted drugs initiation of treatment, which makes therapy even- is a major cause of treatment failure. Both molecular tually ineffective as a result of resistant cell selection. and microenvironmental factors have been implicated As examples, pre-existing BCR-ABL kinase domain in the development of drug resistance. As an example mutations confer resistance to the tyrosine kinase of molecular resistance, upregulation of drug efﬂux inhibitor imatinib in patients with chronic myeloid transporters can prevent sufﬁciently high intracellular 14,15 leukemia, and pre-existing MEK1 mutations drug accumulation, which limits treatment efﬁcacy. confer resistance to BRAF inhibitors in patients with Other molecular causes of drug resistance include 16 melanoma. Many mathematical models have modiﬁcation of drug targets, enhanced DNA damage considered how the presence of such pre-existing repair mechanisms, dysregulation of apoptotic path- resistant cells impacts cancer progression and 1-5 ways, and the presence of cancer stem cells. Ir- 17-40 treatment. ASSOCIATED regular tumor vasculature that results in inconsistent CONTENT Acquired drug resistance broadly describes the case drug distribution and hypoxia is an example of a mi- Appendix in which drug resistance develops during the course of croenvironmental factor that impacts drug resistance. Data Supplement therapy from a population of cells that were initially Other characteristics of the tumor microenvironment Author affiliations drug sensitive. The term acquired resistance is really inﬂuencing drug resistance include regions of acidity, and support an umbrella term for two distinct phenomena, which information (if immune cell inﬁltration and activation, and the tumor complicates the study of acquired resistance. On the 1,6-10 applicable) appear at stroma. Experimental and clinical research con- one hand, there is resistance that is spontaneously—or the end of this tinues to shed light on the multitude of factors that article. randomly—acquired during the course of treatment, be contribute to cancer drug resistance. Mathematical Accepted on February it as a result of random genetic mutations or stochastic modeling studies have also been used to explore both 14, 2018 and nongenetic phenotype switching. This spontaneous broad and detailed aspects of cancer drug resistance, published at form of acquired resistance has been considered in 11-13 ascopubs.org/journal/ as reviewed previously. 18-22,27-29,31,32,35,40,42-49 many mathematical models. On cci on April 10, 2019: Resistance to cancer drugs can be classiﬁed as either the other hand, drug resistance can be induced (ie, DOI https://doi.org/10. 1 41,50-52 1200/CCI.18.00087 pre-existing or acquired. Pre-existing—or intrinsic— caused) by the drug itself. 1 Greene, Gevertz, and Sontag CONTEXT Key Objective Resistance to chemotherapy may arise from Darwinian selection of resistant subclones that either predate therapy or emerge during treatment. In addition, treatment itself may induce genetic or epigenetic variation that catalyzes drug resistance. This work aims to mathematically tease out these various factors. Knowledge Generated A mathematical model is introduced to distinguish the effect of drugs that merely select from those that both create variation and select. The ability of a drug to induce resistance can result in qualitatively different responses on the basis of dose and delivery; constant-infusion regimes are less successful in controlling tumor growth than pulsed therapy for drugs that induce resistance, but the situation is reversed for drugs that act only by selection. Relevance Recent experimental evidence suggests that progression to drug resistance need not occur randomly, but instead may be induced by the treatment itself. Understanding the clinical implications of treatment-induced resistance will help formulate appropriate protocols. The question of whether resistance is an induced phe- drug-induced acquired resistance may simply be the rapid nomenon or predates treatment was ﬁrst famously studied selection of a small number of pre-existing resistant cells or by Luria and Delbruck ¨ in the context of bacterial the selection of cells that spontaneously acquired re- 41,44 41 (Escherichia coli) resistance to a virus (T1 phage). In sistance. In pioneering work by Pisco and colleagues, particular, Luria and Delbruck hypothesized that if selective the relative contribution of resistant cell selection versus pressures imposed by the presence of the virus induce drug-induced resistance was assessed in an experimental bacterial evolution, then the number of resistant colonies system that involved HL60 leukemic cells that were treated formed in their plated experiments should be Poisson with the chemotherapeutic agent vincristine. After 1 to distributed and thus have an approximately equal mean 2 days of treatment, expression of MDR1 was demonstrated and variance. What Luria and Delbruck found instead was to be predominantly mediated by cell-individual induction that the number of resistant bacteria on each plate varied of MDR1 expression and not by the selection of MDR1- 41,58 drastically, with variance being signiﬁcantly larger than the expressing cells. In particular, these cancer cells exploit mean. As a result, the authors concluded that bacterial their heritable, nongenetic phenotypic plasticity—by which mutations predated the viral challenge. one genotype can map onto multiple stable phenotypes— to change their gene expression to a temporarily more In the case of cancer, there is strong evidence that at least 41,58 resistant state in response to treatment-related stress. some drugs have the ability to induce resistance, as ge- nomic mutations can be caused by cytotoxic cancer Although there is a wealth of mathematical research that 54,55 chemotherapeutics. For instance, nitrogen mustards addresses cancer drug resistance, relatively few models can induce base substitutions and chromosomal rear- have considered drug-induced resistance. Of the models of rangements, topoisomerase II inhibitors can induce chro- drug-induced resistance that have been developed, many mosomal translocations, and antimetabolites can induce do not explicitly account for the presence of the drug. double-stranded breaks and chromosomal aberrations. Instead, it is assumed that these models apply only under Such drug-induced genomic alterations would generally be 41,59-62 treatment, with the effects of the drug implicitly nonreversible. Drug resistance can also be induced at the captured in model terms. As these models of resistance 41,50,56 epigenetic level. For example, expression of multi- induction are dose independent, they are unable to capture drug resistance 1 (MDR1), an ABC-family membrane the effects that the alteration of the drug dose has on re- pump that mediates the active efﬂux of the drug, can be sistance formation. To our knowledge, there have been less 1,41 induced during treatment. In another recent example, than a handful of mathematical models developed in which the addition of a chemotherapeutic agent is shown to in- resistance is induced by a drug in a dose-dependent duce, through a multistage process, epigenetic reprog- 33,34,63, 33 fashion. In Gevertz et al and follow-up work in ramming in patient-derived melanoma cells. Resistance 38 64 Shah, Rejniak, and Gevertz and Perez-Velazquez et al, developed in this way can occur quite rapidly and can often duration and intensity of drug exposure determines the 41,52,57 be reversed. resistance level of each cancer cell. This model allows for Despite these known examples of drug-induced resistance, a continuum of resistant phenotypes, but is computa- differentiating between drug-selected and drug-induced tionally complex as it is a hybrid discrete-continuous, resistance is nontrivial. For example, what appears to be stochastic spatial model. While interesting features about 2 © 2019 by American Society of Clinical Oncology Differentiating Spontaneous and Induced Resistance the relationship between induced resistance and the mi- metronomic therapies. Indeed, the differential response croenvironment have been deduced from this model, its between these therapies is fundamentally related to complexity does not allow for general conclusions to be intratumoral heterogeneity/competition, and is explicitly drawn about dose-dependent resistance induction. considered in our model. Furthermore, results presented in this work support recent evidence that promotes the Another class of models that addresses drug-induced re- 65-69 adoption of metronomic therapy in many circumstances, sistance is that in Chisholm et al. These models are and a main objective of this work is to relate competition distinct in that they are motivated by in vitro experiments and drug-induced resistance to therapy design. in which a cancer drug transiently induces a reversible resistant phenotypic state. The individual-based and This work is organized as follows. We begin by introducing integro-differential equation models developed consider a mathematical model to describe the evolution of drug rapidly proliferating drug-sensitive cells, slowly proliferating resistance during treatment with a theoretical resistance- drug-resistant cells, and rapidly proliferating drug-resistant inducing—and noninducing—drug. We use this mathe- cells. An advection term—with the speed depending on matical model to explore the role played by the drug’s drug levels—is used to model drug-induced adaptation of resistance induction rate in treatment dynamics. We the cell proliferation level, and a diffusion term for both the demonstrate that the induction rate of a theoretical cancer level of cell proliferation and survival potential (response drug could have a nontrivial impact on the qualitative re- to drug) is used to model nongenetic phenotype insta- sponses to a given treatment strategy, including tumor bility. Through these models, the contribution of non- composition and the time horizon of tumor control. In our genetic phenotype instability (both drug induced and model, for a resistance-preserving drug—that is, a drug that random), stress-induced adaptation, and selection can does not induce resistance—better tumor control is be quantiﬁed. achieved using a constant therapeutic protocol versus a pulsed one. Conversely, in the case of a resistance- Finally, the work in Liu et al models the evolutionary inducing drug, pulsed therapy prolongs tumor control dynamics of the tumor population as a multitype non- longer than constant therapy as a result of sensitive/ homogeneous continuous time birth-death stochastic resistant cell competitive inhibition. Once the importance of process. This model accounts for the ability of a targeted induced resistance has been established, we demonstrate drug to alter the rate of resistant cell emergence in a dose- that all parameters in our mathematical model are identi- dependent manner. The authors speciﬁcally considered ﬁable, meaning that it is theoretically possible to determine cases in which the rate of mutation that gives rise to a re- the rate at which drug resistance is induced for a given sistant cell: (1) increases as a function of drug concen- treatment protocol. As this theoretical result does not di- tration, (2) is independent of drug concentration, and (3) rectly lend itself to an experimental approach for quanti- decreases with drug concentration. Interestingly, this fying the ability of a drug to induce resistance, we also model led to the conclusion that the optimal treatment describe a potential in vitro experiment for approximat- strategy is independent of the relationship between drug ing this ability utilizing constant therapies. We end with concentration and the rate of resistance formation. In some concluding remarks and a discussion of potential particular, the authors found that resistance is optimally extensions of our analysis, such as a model that differ- delayed using a low-dose continuous treatment strategy entiates between reversible and nonreversible forms of coupled with high-dose pulses. resistance. As in vitro experiments have demonstrated that treatment 41,52 response can be affected by drug-induced resistance, MATERIALS AND METHODS in the current work we seek to understand this phenom- Here we introduce a general modeling framework to de- enon further using mathematical modeling. The initial scribe the evolution of drug resistance during treatment. mathematical model that we have developed—and that will Our model captures the fact that resistance can result from be analyzed herein—is a system of two ordinary differential random events or can be induced by the treatment itself. equations with a single control representing the drug. We Random events that can confer drug resistance include intentionally chose a minimal model that would be ame- genetic alterations—for example, point mutations or gene nable to analysis, as compared with previously developed ampliﬁcation—and phenotype switching. These sponta- models of drug-induced resistance which are signiﬁcantly neous events can occur either before or during treatment. 33,38,63,64 more complex. Despite the simplicity of the model, Drug-induced resistance is resistance speciﬁcally activated it incorporates both spontaneous and drug-induced by the drug and, as such, depends on the effective dose resistance. encountered by a cell. Such a formulation allows us to In addition to drug-induced resistance, the other charac- distinguish the contributions of both drug-dependent and teristic of cancer dynamics we explore is that of traditional, drug-independent mechanisms, as well as any dependence maximally tolerated dose (MTD) treatment protocols on pre-existing—that is prior to treatment—resistant compared with high-frequency, low-dose so-called populations. JCO Clinical Cancer Informatics 3 Greene, Gevertz, and Sontag We consider the tumor to be composed of two types of cells, be under the simplest assumption that the drug is com- sensitive (S) and resistant (R). Sensitive (or wild-type) cells pletely ineffective against resistant cells, so that d =0. are fully susceptible to treatment, whereas treatment af- The last term in the equations, γR, represents the resen- fects resistant cells to a lesser degree. To analyze the role of sitization of cancer cells to the drug. In the case of non- both random and drug-induced resistance, we use a sys- reversible resistance, γ = 0; otherwise γ . 0. Our tem of two ordinary differential equations to describe the subsequent analysis will be done under the assumption of dynamics between the S and R subpopulations: nonreversible resistance. For a discussion of the effect of reversibility on the presented model, see the Appendix. dS S + R r 1 − S − e + αu t S − du t S + γR, (1) dt K Finally, we note that the effective drug concentration u(t) can be thought of as a control input. For simplicity, in this dR S + R work we assume that it is directly proportional to the applied r 1 − R + e + αu t S − d u t R − γR. R R dt K drug concentration; however, pharmacodynamic/phar- (2) macokinetic considerations could be incorporated to more accurately describe the uptake/evolution of the drug in vivo All parameters are non-negative. In the absence of treat- or in vitro—for example, as in Bender, Schindler, and ment, we assume that the tumor grows logistically, with 79 80 81 Friberg, Wu et al, and Fetterly et al. each population contributing equally to competitive in- To understand the above system of drug resistance evo- hibition. Phenotypes S and R each possess individual lution, we reduce the number of parameters via non- intrinsic growth rates, and we make the assumption in dimensionalization. Rescaling S and R by their (joint) the remainder of the work that 0 ≤ r , r.Thissimply carrying capacity K, and time t by the sensitive cell growth states that resistant cells grow slower than nonresistant rate, cells, an assumption that is supported by experimental 70-72 evidence. 1 1 S τ S τ , The transition to resistance can be described with a net K r (3) term of the form «S + αu(t)S. Mathematically, the drug- 1 1 R τ R τ , induced term αu(t)S, where u(t) is the effective applied drug K r dose at time t, describes the effect of treatment on pro- moting the resistant phenotype. For example, this term Equations 1 and 2 (with γ = d = 0) can be written in the could represent the induced overexpression of the form, P-glycoprotein gene, a well-known mediator of multidrug 1,73 dS resistance, by the application of chemotherapy. 1 − S + R S − e + αu t S − du t S, (4) dt Spontaneous evolution of resistance is captured in the eS term, which permits resistance to develop even in the dR absence of treatment. Note that ε is generally considered p 1 − S + R R + e + αu t S. (5) dt small, although recent experimental evidence regarding error-prone DNA polymerases suggests that cancer cells For convenience, we have relabeled S, R, and t to coincide may have increased mutation rates as a result of the with the nondimensionalization so that the parameters ε, α, 75-77 overexpression of such polymerases. For example, in and d must be scaled accordingly (by 1/r). As r was as- Krutyakov, mutation rates as a result of such polymerases sumed to satisfy 0 ≤ r , r, the relative resistant population −1 are characterized by probabilities as high as 7.5 × 10 per growth rate p satisﬁes 0 ≤ p , 1. r r base substitution, and it is known that many point muta- 77 One can show (Appendix) that asymptotically, under any tions in cancer arise from these DNA polymerases. For treatment regimen u(t) ≥ 0, the entire population will be- this work, we adopt the notion that random point mutations come resistant: that lead to drug resistance are rare, and that drug-induced resistance occurs on much quicker time scales ; there- S t t→∞ 0 → . (6) fore, we will assume that α . ε with u = O(1) in our analysis R t of Equations 1 and 2. We model the effects of treatment by assuming the log-kill However, tumor control is still possible where one can hypothesis, which states that a given dose of chemo- combine therapeutic efﬁcacy and clonal competition to in- therapy eliminates the same fraction of tumor cells re- ﬂuence transient dynamics and possibly prolong patient life. gardless of tumor size. We allow for each cellular Indeed, the modality of adaptive therapy has shown promise compartment to have a different drug-induced death rate in using real-time patient data to inform therapeutic mod- (d, d ); however, to accurately describe resistance it is ulation aimed at increasing quality of life and survival times. required that 0 ≤ d , d. Our analysis presented herein will This work will focus on such dynamics and controls. 4 © 2019 by American Society of Clinical Oncology Differentiating Spontaneous and Induced Resistance RESULTS Although a diverse set of inputs u(t) may be theoretically applied, presently we consider only strategies as illustrated Effect of Induction on Treatment Efﬁcacy in Figure 1B. The blue curve in Figure 1B corresponds to We investigate the role of the induction capability of a drug a constant effective dosage u (t) initiated at t —administered c d (parameter α in Eqs 4 and 5) on treatment dynamics. approximately using continuous infusion pumps and/or Speciﬁcally, the value of α may have a substantial impact on slow-release capsules—whereas the black curve represents the relative success of two standard therapy protocols— a corresponding pulsed strategy u (t), with ﬁxed treatment constant dosage and periodic pulsing. windows (Δt )and holidays (Δt ). In general, we may allow on off for different magnitudes, u and u , for constant and on,c on,p Treatment Protocol pulsed therapies respectively—forexample,torelate the To quantify the effects of induced resistance, a treatment total dosage applied per treatment cycle (area under the protocol must be speciﬁed. We adopt a clinical perspective 83 drug concentration-time curve [AUC] ). However, for sim- over the course of the disease, which is summarized in plicity we assume the same magnitude in the subsequent Figure 1. We assume that the disease is initiated by a small section (although see the Appendix for a normalized com- number of wild-type cells: parison). While these represent idealized therapies, such u(t) may form an accurate approximation to in vitro and/or in vivo S 0 S , R 0 0, (7) kinetics. Note that the response V(t)illustrated in Figure 1A will not be identical, or even qualitatively similar, for both where 0 , S , 1. Denote the tumor volume at time t by presented strategies, as will be demonstrated numerically. V (t): Constant Versus Pulsed Therapy Comparison Vt() St() + Rt(). (8) To qualitatively demonstrate the role that induced resistance plays in the design of therapy schedules, we consider two The tumor then progresses untreated until a speciﬁc vol- drugs with the same cytotoxic potential—that is, the same ume V is detected—or, for hematologic tumors, via ap- drug-induced death rate d—each possessing a distinct level propriate blood markers—which using existing nuclear of resistance induction (parameter α). A fundamental imaging techniques corresponds to a tumor with diameter question, then, is whether there exist qualitative distinctions on the order of 10 mm. Time to reach V is denoted by t , d d between treatment responses for each chemotherapy. More which in general depends on all parameters that appear in speciﬁcally, how does survival time compare when sched- Equations 4 and 5. Note that, assuming e . 0, a nonzero uling is altered between constant therapy and pulsing? Does resistant population will exist at the onset of treatment. the optimal strategy—in this case, optimal across only two Therapy, represented through u(t), is then applied until the schedulings—change depending on the extent to which the tumor reaches a critical size V , which we equate with drug induces resistance? treatment failure. Because the (S,R) = (0,1) state is globally asymptotically stable in the ﬁrst quadrant, V , 1is We ﬁx two values of the induction parameter α: guaranteed to be obtained in ﬁnite time. Time until failure, −2 t , is then a measure of efﬁcacy of the applied u(t). α 0, α 10 . c s i AB Constant V u Pulsed on,p V u d on,c t t t t t t d c d c Time Time FIG 1. Schematic of tumor dynamics under two treatment regimes. (A) Tumor volume V in response to treatment initiated at time t . Cancer population arises from a small sensitive population at time t = 0, upon which the tumor grows to detection at volume V . Treatment is begun at t and continues until the tumor reaches a critical size V (at a corresponding time t ), d d c c where treatment is considered to have failed. (B) Illustrative constant and pulsed treatments, both initiated at t = t . JCO Clinical Cancer Informatics 5 Tumor Volume Treatment Greene, Gevertz, and Sontag Recall that we are studying the nondimensional model drug—that is, AUC—is applied. However, we see that Equations 4 and 5, so no units are speciﬁed. Parameter even in this case, intermediate doses may be optimal (Figs α = 0 corresponds to no therapy-induced resistance 3A and 4A). Thus, it is not the larger total drug, per se, that (henceforth denoted as phenotype preserving), and is responsible for the superiority of the constant protocol in therefore considering this case allows for a comparison this case, a point that is reinforced by the fact that the between the classic notion of random evolution toward results remain qualitatively unchanged even if the total resistance (α = 0) and drug-induced resistance (α . 0). drug is controlled for (Appendix). For the remainder of the section, parameters are ﬁxed as Compare this with Figures 2C and 2D, which consider the in Table 1. Critically, all parameters excluding α are same patient and cytotoxicity, but for a highly inductive identical for each drug, which enables an unbiased drug. Results are strikingly different and suggest that comparison. Treatment magnitudes u and u are on,c on,p pulsed therapy is now not worse but in fact substantially selected to be equal: u = u = u . on,c on,p on improves patient response (t ≈ 61 for pulsed, compared with t ≈ 45 for constant). In this case, both tumors are Note that selecting parameter V = 0.1 implies that the c now primarily resistant (Figs A3B and A3D), but the carrying capacity has a diameter of 100 mm, as V cor- pulsed therapy allows for prolonged tumor control via responds to a detectable diameter of 10 mm. Assuming −6 3 sensitive/resistant competitive inhibition. Furthermore, each cancer cell has volume 10 mm , tumors in our model can grow to a carrying capacity of approximately treatment holidays reduce the overall ﬂux into resistance as the application of the drug itself promotes this evolu- 12.4 cm in diameter, which is in qualitative agreement tion. The total amount of drug (AUC) is also less for pulsed with the parameters estimated in Chignola and Foroni therapy (22.5 compared with ≈ 64), so that pulsed (≈12.42 cm, assuming a tumor spheroid). therapy is both more efﬁcient in terms of treatment efﬁ- By examining Figures 2A and 2B, we clearly observe an cacy and less toxic to the patient as adverse effects are improved response to constant therapy when using typically correlated with the total administered dose, a phenotype-preserving drug, with treatment success time which is proportional to the AUC. This is further consistent t nearly seven times as long compared with pulsed with recent experimental and clinical evidence that therapy (t ≈ 90 for constant, compared with t ≈ 14 for c c supports metronomic therapy as a superior alternative to pulsed). It can be observed that the tumor composition at classic chemotherapy regimens. The results presented in treatment conclusion is different for each therapy—not Figure 2 suggest that it may be advantageous to apply shown for this simulation, but see a comparable result in a smaller amount of drug more frequently; however, we Appendix Figures A2B and A2D—and it seems that also note that the results depend on patient-speciﬁc pulsed therapy was not sufﬁciently strong to hamper the parameters, so that therapy would ideally be personal- rapid growth of the sensitive population. Indeed, treat- ized to individual patients. Of note, we do not claim that ment failed quickly as a result of insufﬁcient treatment these results hold for all parameter values—both patient intensity in this case, as the population remains almost and treatment speciﬁc—but instead emphasize that the entirely sensitive. Thus, for this patient under these rate of induction may play a large role in the design of speciﬁc treatments, assuming drug resistance only arises therapies for speciﬁcpatients. via random stochastic events, constant therapy would be For these speciﬁc parameter values, differences between preferred. One might argue that pulsed, equal-magnitude constant and pulsed therapy for the inductive drug are not treatment is worse when α = 0 simply because less total as extensive as in the phenotype-preserving case; how- ever, recall that time has been nondimensionalized and, TABLE 1. Parameters Used For Comparison of Treatment Efﬁcacy for Phenotype- hence, the scale may indeed be clinically relevant. Such Preserving Drugs and Resistance-Inducing Drugs Parameter Biologic Interpretation Value (dimensionless) differences can be further ampliﬁed, and, as exact pa- rameters are difﬁcult or even (currently) impossible to S Initial sensitive population 0.01 measure, qualitative distinctions are paramount. Thus, at R Initial resistant population 0 this stage, ranking of therapies, rather than their precise V Detectable tumor volume 0.1 quantitative efﬁcacy, should act as the more important V Tumor volume deﬁning treatment failure 0.9 clinical criterion. −6 « Background mutation rate 10 From these results, we observe a qualitative difference in d Cytotoxicity of sensitive cells 1 the treatment strategy to apply based entirely on the p Resistant growth fraction 0.2 r value of α, the degree to which the drug itself induces resistance. Thus, in administering chemotherapy, the u Treatment magnitude, constant dose 1.5 on resistance-promotion rate α of the treatment is a clinically Δt Pulsed treatment window 1 on signiﬁcant parameter. In the next section, we use our model Δt Pulsed holiday length 3 off and its output to propose in vitro methods for experimentally NOTE. Parameters used in Figure 2. measuring a drug’s α parameter. 6 © 2019 by American Society of Clinical Oncology 80 60 70 60 70 50 60 Differentiating Spontaneous and Induced Resistance A B Treatment Strategies,  = 0 Tumor Dynamics,  = 0 Constant Constant 1.6 Pulsed Pulsed 0.9 1.4 0.8 1.2 0.7 0.6 0.8 0.5 0.4 0.6 0.3 0.4 0.2 0.2 0.1 0 10203040 0 10 20 30 40 Time Time C D -2 -2 Treatment Strategies,  = 10 Tumor Dynamics,  = 10 Constant Constant 1.6 Pulsed Pulsed 0.9 1.4 0.8 1.2 0.7 0.6 0.8 0.5 0.4 0.6 0.3 0.4 0.2 0.2 0.1 010 20 30 40 0 10203040 Time Time −2 FIG 2. Comparison of treatment efﬁcacy for phenotype-preserving drugs (α = 0) and resistance-inducing drugs (α =10 ). The left column indicates treatment strategy, whereas the right column indicates corresponding tumor volume response. Note that the dashed red line in the right column indicates the tumor volume representing treatment failure, V . (A) Constant and pulsed therapies after tumor detection for α = 0. (B) Responses corresponding −2 to treatment regimens in panel A. (C) Constant and pulsed therapies after tumor detection for α =10 . (D) Responses corresponding to treatment regimens in panel C. Identifying the Rate of Induced Drug Resistance Theoretical Identiﬁability The effect of treatment on the evolution of phenotypic We ﬁrst study the structural identiﬁability of Equations 4 and resistance may have a signiﬁcant impact on the efﬁcacy of 5, a prerequisite for analyzing practical methods for de- conventional therapies. Thus, it is essential to understand termining parameter values. Structural identiﬁability is the the value of the induction parameter α before administering process of determining model parameters—for example, therapy. In this section, we brieﬂy discuss both the theo- α—from a set of control experiments. Here, we assume that the retical possibility and practical feasibility of determining α only measurable quantity is the tumor volume V = S + R,along from different input strategies u(t). For more details, see the with its derivatives, in time. Using four different controls, we Appendix. show that all model parameters, including the induction rate α, JCO Clinical Cancer Informatics 7 u u Tumor Volume S + R Tumor Volume S + R Greene, Gevertz, and Sontag may be determined by precisely measuring the corre- standard therapy protocols and demonstrate that contrary sponding volume-response curves. For more details, see to the work in Liu et al, the rate of resistance induction the Appendix. may have a signiﬁcant effect on treatment outcome. Thus, understanding the dynamics of resistance evolution with An In Vitro Experimental Protocol to Distinguish regard to the applied therapy is crucial. Spontaneous and Drug-Induced Resistance To demonstrate that one can theoretically determine the As structural identiﬁability was established in the previous induction rate, we performed an identiﬁability analysis on section, we focus on practical qualitative differences the parameter α and demonstrated that it can be obtained exhibited by Equations 4 and 5 as a function of the via a set of appropriate perturbation experiments on u(t). resistance-induction rate α. Utilizing only constant dos- Furthermore, we presented an alternative method, using ages, we investigate the dependence of t on dose u, cy- only constant therapies, for understanding the qualitative totoxicity d, and α.Deﬁning the supremum over doses of differences between purely spontaneous and induced the response time (Eq 8), cases. Such properties could possibly be used to design in vitro experiments on different pharmaceuticals, which T d : sup t u, d, α , (9) { } α c allows one to determine the induction rate of drug re- sistance without an a priori understanding of the precise we plot the results for different α values in Figures 3 and 4. mechanism. We do note, however, that such experiments may still be difﬁcult to perform in a laboratory environ- Comparing Figures 3B and 4B, we observe a clear quali- ment, as engineering cells with various drug sensitivities tative difference in maximum response times. In the case of d may be challenging. Indeed, this work can be consid- a phenotype-preserving drug, the proposed in vitro ex- ered as a thought experiment to identify qualitative periment would produce a ﬂat curve, whereas a resistance- properties that the induction rate α yields in our modeling inducing drug (α. 0) would yield an increasing function T framework. (d). Additional comparisons are presented in Figure 5, where the α dependence is more closely analyzed. For Our simple model allows signiﬁcant insight into the role of more details and analysis, see the Appendix. random versus induced resistance. Of course, more elaborate models can be studied by incorporating more DISCUSSION biologic detail. For example, while our two-equation model In the current work, we analyzed two distinct mechanisms classiﬁes cells as either sensitive or resistant, not all re- that can result in drug resistance. Speciﬁcally, a mathe- sistance is treated equally. Some resistant cancer cells are matical model is proposed which describes both the permanently resistant, whereas others could transition spontaneous generation of resistance and drug-induced back to a sensitive state. This distinction may prove to be resistance. Using this model, we contrasted the effect of vitally important in treatment design. A possible extension A B -2 -2 Constant Therapy Dose Response,  = 10 Maximum Critical Time as a Function of Sensitivity,  = 10 d = 0.1 d = 0.2 d = 0.95 140 d = 1.45 d = 2.95 00.5 11.52 2.5 3 3.5 44.5 5 0.5 1 1.5 2 2.5 33.5 4 u d −2 FIG 3. Variation in response time t for a treatment that induces resistance. Constant therapy u(t) ≡ u is applied for t ≤ t ≤ t . Induction rate α =10 , with all c d c other parameters as in Table 1. (A) Time until tumor reaches critical size V for various drug sensitivities d. (B) Maximum response time T (d) for a treatment c α that induces resistance. Note that time T (d) increases with drug sensitivity; compare with Figure 4B for purely random resistance evolution. 8 © 2019 by American Society of Clinical Oncology T (d)  Differentiating Spontaneous and Induced Resistance A B Maximum Critical Time as a Function Constant Therapy Dose Response, = 0 of Sensitivity, = 0  450 450 d = 0.1 400 d = 0.2 400 d = 0.95 350 350 d = 1.45 d = 2.95 300 300 250 250 200 200 150 150 100 100 50 50 00.5 11.522.533.544.55 0.5 11.522.53 u d FIG 4. Change in critical time t for differing drug sensitivities in the case of a phenotype-preserving treatment. (A) Time until tumor reaches critical size V for c c various drug sensitivities d; comparable to Figure 3A, with α = 0. (B) Maximum critical time T (d). Note that the curve is essentially constant. of our model is one in which we distinguish between alterations or resistance that forms by some combination of sensitive cells S, nonreversible resistant cells R , and re- genetic and stable epigenetic changes. versible resistant cells R (Eqs 9-12): Conversely, reversible resistant cells R denote resistant cells that form via phenotype switching, as described in dS V r 1 − S − e + e S − α + α u t S − du t S n r n r Pisco et al. Random phenotype switching in the ab- dt K sence of treatment is captured in the ε S term. This is + γR , consistent—and indeed necessary—to understand the (10) experimental results in Pisco et al, where a stable dis- tribution of MDR1 expressions is observed even in the dR V absence of treatment. The α u(t)S term represents the r 1 − R + e S + α u t S − d u t R , (11) n n n n n n dt K induction of a drug-resistant phenotype. Phenotype switching is often reversible, and therefore we allow a back dR V r transition from the R compartment to the sensitive com- r 1 − R + e S + α u t S − γR − d u t R . r r r r r r r dt K partment at a nonnegligible rate γ (see Appendix). For- mulated in this way, the model can be calibrated to (12) experimental data and we can further consider the effects Here, V denotes the entire tumor population—that is, of the dosing strategy on treatment response. We plan to further study this model in future work. V : S + R + R . (13) n r Other extensions which include different clinical scenarios are also being investigated. In practice, chemotherapies are In this version of the model, nonreversible resistant cells R rarely applied in isolation. Multiple therapies are often can be thought of as resistant cells that form via genetic administered simultaneously to improve efﬁcacy. The in- mutations. Under this assumption, « represents the rate at clusion of multiple drugs, including targeted therapies that which spontaneous genetic mutations give rise to re- act primarily on resistant subpopulations, yields natural sistance, and α is the drug-induced resistance rate. This control questions that are clinically relevant. Similarly, situation can be classiﬁed as nonreversible as it is in- immune cells, together with immunotherapies, may also be credibly unlikely that genomic changes that occur in re- incorporated to more accurately mimic the cancer sponse to treatment would be reversed by an "undoing" microenvironment. mutation. Therefore, once cells confer a resistant pheno- type via an underlying genetic change, we assume that they Overcoming drug resistance is crucial for the success of maintain that phenotype. This term could also be thought of both chemotherapy and targeted therapy. Furthermore, the as describing resistance that forms via stable epigenetic added complexity of induced drug resistance complicates JCO Clinical Cancer Informatics 9  Greene, Gevertz, and Sontag A B Maximum Critical Time for Different Maximum Critical Time for Different Induced Mutation Rates Induced Mutation Rates 180 450 20 0 0.5 1 1.5 2 2.5 0.5 1 1.5 2 2.5 d d = 5.00e-03 = 9.00e-03 = 1.20e-02 -4 -2 = 0 = 10 = 10 = 6.00e-03 = 1.00e-02 = 1.30e-02 -6 -3 -1 = 10 = 10 = 10 = 7.00e-03 = 1.10e-02 = 1.40e-02 -5 = 10 = 8.00e-03 = 1.50e-02 FIG 5. Variation in maximum response time for different induction rates α. For details on computation of T (d), see Appendix Figure A4. All other parameters −2 are given as in Table 1. (A) Plot of T (d) for α near 10 . (B) Analogous to panel A, where α is now varied over several orders of magnitude. Nonmutagenic case (α = 0) is included for reference. therapy design, as the simultaneous effects of tumor re- differentiating random and drug-induced resistance, which duction and resistance propagation confound one another. will allow for clinically actionable analysis on a biologically Mathematically, we have presented a clear framework for subtle, yet important, issue. AUTHOR CONTRIBUTIONS AFFILIATIONS Conception and design: All authors Rutgers University, New Brunswick, NJ Collection and assembly of data: All authors The College of New Jersey, Ewing Township, NJ Northeastern University, Boston, MA Data analysis and interpretation: All authors Harvard Medical School, Cambridge, MA Manuscript writing: All authors Final approval of manuscript: All authors Preprint version available on https://www.biorxiv.org/content/10.1101/ Accountable for all aspects of the work: All authors 235150v2. CORRESPONDING AUTHOR AUTHORS’ DISCLOSURES OF POTENTIAL CONFLICTS OF INTEREST AND DATA AVAILABILITY STATEMENT Eduardo D. Sontag, PhD, Northeastern University, 360 Huntington Avenue, Boston, MA 02115; e-mail: sontag@sontaglab.org. The following represents disclosure information provided by authors of this manuscript. All relationships are considered compensated. Relationships are self-held unless noted. I = Immediate Family Member, AUTHORS’ DISCLOSURES OF POTENTIAL CONFLICTS OF INTEREST Inst = My Institution. Relationships may not relate to the subject matter of AND DATA AVAILABILITY STATEMENT this manuscript. For more information about ASCO's conﬂict of interest Disclosures provided by the authors and data availability statement (if policy, please refer to www.asco.org/rwc or ascopubs.org/jco/site/ifc. applicable) are available with this article at DOI https://doi.org/10.1200/ CCI.18.00087. No potential conﬂicts of interest were reported REFERENCES 1. Holohan C, Van Schaeybroeck S, Longley DB, et al: Cancer drug resistance: An evolving paradigm. Nat Rev Cancer 13:714-726, 2013 2. Gottesman MM: Mechanisms of cancer drug resistance. Annu Rev Med 53:615-627, 2002 3. Dean M, Fojo T, Bates S: Tumour stem cells and drug resistance. Nat Rev Cancer 5:275-284, 2005 4. Woods D, Turchi JJ: Chemotherapy induced DNA damage response: Convergence of drugs and pathways. Cancer Biol Ther 14:379-389, 2013 10 © 2019 by American Society of Clinical Oncology  Differentiating Spontaneous and Induced Resistance 5. Zahreddine H, Borden KL: Mechanisms and insights into drug resistance in cancer. Front Pharmacol 4:28, 2013 6. Tredan ´ O, Galmarini CM, Patel K, et al: Drug resistance and the solid tumor microenvironment. J Natl Cancer Inst 99:1441-1454, 2007 7. Gajewski TF, Meng Y, Blank C, et al: Immune resistance orchestrated by the tumor microenvironment. Immunol Rev 213:131-145, 2006 8. Meads MB, Gatenby RA, Dalton WS: Environment-mediated drug resistance: A major contributor to minimal residual disease. Nat Rev Cancer 9:665-674, 2009 9. Correia AL, Bissell MJ: The tumor microenvironment is a dominant force in multidrug resistance. Drug Resist Updat 15:39-49, 2012 10. McMillin DW, Negri JM, Mitsiades CS: The role of tumour-stromal interactions in modifying drug response: Challenges and opportunities. Nat Rev Drug Discov 12:217-228, 2013 11. Lavi O, Gottesman MM, Levy D: The dynamics of drug resistance: A mathematical perspective. Drug Resist Updat 15:90-97, 2012 12. Brocato T, Dogra P, Koay EJ, et al: Understanding drug resistance in breast cancer with mathematical oncology. Curr Breast Cancer Rep 6:110-120, 2014 13. Foo J, Michor F: Evolution of acquired resistance to anti-cancer therapy. J Theor Biol 355:10-20, 2014 14. Roche-Lestienne C, Preudhomme C: Mutations in the ABL kinase domain pre-exist the onset of imatinib treatment. Semin Hematol 40:80-82, 2003 (suppl 2) 15. Iqbal Z, Aleem A, Iqbal M, et al: Sensitive detection of pre-existing BCR-ABL kinase domain mutations in CD34 cells of newly diagnosed chronic-phase chronic myeloid leukemia patients is associated with imatinib resistance: Implications in the post-imatinib era. PLoS One 8:e55717, 2013 16. Carlino MS, Fung C, Shahheydari H, et al: Preexisting MEK1P124 mutations diminish response to BRAF inhibitors in metastatic melanoma patients. Clin Cancer Res 21:98-105, 2015 17. Jackson TL, Byrne HM: A mathematical model to study the effects of drug resistance and vasculature on the response of solid tumors to chemotherapy. Math Biosci 164:17-38, 2000 18. Komarova NL, Wodarz D: Drug resistance in cancer: Principles of emergence and prevention. Proc Natl Acad Sci USA 102:9714-9719, 2005 19. Michor F, Nowak MA, Iwasa Y: Evolution of resistance to cancer therapy. Curr Pharm Des 12:261-271, 2006 20. Komarova NL, Wodarz D: Stochastic modeling of cellular colonies with quiescence: An application to drug resistance in cancer. Theor Popul Biol 72:523-538, 21. Foo J, Michor F: Evolution of resistance to targeted anti-cancer therapies during continuous and pulsed administration strategies. PLOS Comput Biol 5:e1000557, 2009 [Erratum: PLoS Comput Biol doi:10.1371/annotation/d5844bf3-a6ed-4221-a7ba-02503405cd5e] 22. Foo J, Michor F: Evolution of resistance to anti-cancer therapy during general dosing schedules. J Theor Biol 263:179-188, 2010 23. Silva AS, Gatenby RA: A theoretical quantitative model for evolution of cancer chemotherapy resistance. Biol Direct 5:25, 2010 24. Cunningham JJ, Gatenby RA, Brown JS: Evolutionary dynamics in cancer therapy. Mol Pharm 8:2094-2100, 2011 25. Mumenthaler SM, Foo J, Leder K, et al: Evolutionary modeling of combination treatment strategies to overcome resistance to tyrosine kinase inhibitors in non- small cell lung cancer. Mol Pharm 8:2069-2079, 2011 26. Orlando PA, Gatenby RA, Brown JS: Cancer treatment as a game: Integrating evolutionary game theory into the optimal control of chemotherapy. Phys Biol 9:065007, 2012 27. Bozic I, Reiter JG, Allen B, et al: Evolutionary dynamics of cancer in response to targeted combination therapy. eLife 2:e00747, 2013 28. Foo J, Leder K, Mumenthaler SM: Cancer as a moving target: understanding the composition and rebound growth kinetics of recurrent tumors. Evol Appl 6:54-69, 2013 29. Lavi O, Greene JM, Levy D, et al: The role of cell density and intratumoral heterogeneity in multidrug resistance. Cancer Res 73:7168-7175, 2013 30. Bozic I, Nowak MA: Timing and heterogeneity of mutations associated with drug resistance in metastatic cancers. Proc Natl Acad Sci USA 111:15964-15968, 31. Greene J, Lavi O, Gottesman MM, et al: The impact of cell density and mutations in a model of multidrug resistance in solid tumors. Bull Math Biol 76:627-653, 32. Yamamoto KN, Hirota K, Takeda S, et al: Evolution of pre-existing versus acquired resistance to platinum drugs and PARP inhibitors in BRCA-associated cancers. PLoS One 9:e105724, 2014 33. Gevertz J, Aminzare Z, Norton K-A, et al: Emergence of anti-cancer drug resistance: exploring the importance of the microenvironmental niche via a spatial model, in Jackson T and Radunskaya A (eds): Applications of Dynamical Systems in Biology and Medicine, Volume 158, Berlin, Germany, Springer-Verlag, 2015, pp 1-34 34. Liu LL, Li F, Pao W, et al: Dose-dependent mutation rates determine optimum erlotinib dosing strategies for EGRF mutant non-small lung cancer patients. PLoS One 10:e0141665, 2015 35. Menchon ´ SA: The effect of intrinsic and acquired resistances on chemotherapy effectiveness. Acta Biotheor 63:113-127, 2015 36. Mumenthaler SM, Foo J, Choi NC, et al: The impact of microenvironmental heterogeneity on the evolution of drug resistance in cancer cells. Cancer Inform 14:19-31, 2015 (suppl 4) 37. Waclaw B, Bozic I, Pittman ME, et al: A spatial model predicts that dispersal and cell turnover limit intratumour heterogeneity. Nature 525:261-264, 2015 38. Shah AB, Rejniak KA, Gevertz JL: Limiting the development of anti-cancer drug resistance in a spatial model of micrometastases. Math Biosci Eng 13:1185-1206, 2016 39. Carrere C: Optimization of an in vitro chemotherapy to avoid resistant tumours. J Theor Biol 413:24-33, 2017 40. Chakrabarti S, Michor F: Pharmacokinetics and drug-interactions determine optimum combination strategies in computational models of cancer evolution. Cancer Res 77:3908-3921, 2017 41. Pisco AO, Brock A, Zhou J, et al: Non-Darwinian dynamics in therapy-induced cancer drug resistance. Nat Commun 4:2467, 2013 42. Coldman AJ, Goldie JH: A stochastic model for the origin and treatment of tumors containing drug-resistant cells. Bull Math Biol 48:279-292, 1986 43. Ledzewicz U, Schattler ¨ H: Drug resistance in cancer chemotherapy as an optimal control problem. Discrete Continuous Dyn Syst Ser B 5:129-150, 2006 44. Leder K, Foo J, Skaggs B, et al: Fitness conferred by BCR-ABL kinase domain mutations determines the risk of pre-existing resistance in chronic myeloid leukemia. PLoS One 6:e27682, 2011 45. Hadjiandreou MM, Mitsis GD: Mathematical modeling of tumor growth, drug-resistance, toxicity, and optimal therapy design. IEEE Trans Biomed Eng 61:415-425, 2014 46. Lorz A, Lorenzi T, Hochberg M, et al: Populational adaptive evolution, chemotherapeutic resistance and multiple anti-cancer therapies. ESAIM: Math Model Num Anal 47:377-399, 2013 47. Fu F, Nowak MA, Bonhoeffer S: Spatial heterogeneity in drug concentrations can facilitate the emergence of resistance to cancer therapy. PLOS Comput Biol 11:e1004142, 2015 48. Ledzewicz U, Bratton K, Schattler H: A 3-compartment model for chemotherapy of heterogeneous tumor populations. Acta Appl Math 135:191-207, 2015 JCO Clinical Cancer Informatics 11 Greene, Gevertz, and Sontag 49. Lorz A, Lorenzi T, Clairambault J, et al: Modeling the effects of space structure and combination therapies on phenotypic heterogeneity and drug resistance in solid tumors. Bull Math Biol 77:1-22, 2015 50. Nyce J, Leonard S, Canupp D, et al: Epigenetic mechanisms of drug resistance: Drug-induced DNA hypermethylation and drug resistance. Proc Natl Acad Sci USA 90:2960-2964, 1993 51. Nyce JW: Drug-induced DNA hypermethylation: A potential mediator of acquired drug resistance during cancer chemotherapy. Mutat Res 386:153-161, 1997 52. Sharma SV, Lee DY, Li B, et al: A chromatin-mediated reversible drug-tolerant state in cancer cell subpopulations. Cell 141:69-80, 2010 53. Luria SE, Delbruck ¨ M: Mutations of bacteria from virus sensitivity to virus resistance. Genetics 28:491-511, 1943 54. Szikriszt B, Poti ´ A, Pipek O, et al: A comprehensive survey of the mutagenic impact of common cancer cytotoxics. Genome Biol 17:99, 2016 55. Swift LH, Golsteyn RM: Genotoxic anti-cancer agents and their relationship to DNA damage, mitosis, and checkpoint adaptation in proliferating cancer cells. Int J Mol Sci 15:3403-3431, 2014 56. Shaffer SM, Dunagin MC, Torborg SR, et al: Rare cell variability and drug-induced reprogramming as a mode of cancer drug resistance. Nature 546:431-435, 57. Housman G, Byler S, Heerboth S, et al: Drug resistance in cancer: An overview. Cancers (Basel) 6:1769-1792, 2014 58. Pisco AO, Huang S: Non-genetic cancer cell plasticity and therapy-induced stemness in tumour relapse: ‘What does not kill me strengthens me’. Br J Cancer 112:1725-1732, 2015 59. Goldman A, Majumder B, Dhawan A, et al: Temporally sequenced anticancer drugs overcome adaptive resistance by targeting a vulnerable chemotherapy- induced phenotypic transition. Nat Commun 6:6139, 2015 60. Chapman M, Risom T, Aswani A, et al: A model of phenotypic state dynamics initiates a promising approach to control heterogeneous malignant cell populations.IEEE 55th Conference on Decision and Control, Las Vegas, NV, December 12-14, 2016 (abstr) 61. Axelrod DE, Vedula S, Obaniyi J: Effective chemotherapy of heterogeneous and drug-resistant early colon cancers by intermittent dose schedules: A computer simulation study. Cancer Chemother Pharmacol 79:889-898, 2017 62. Feizabadi MS: Modeling multi-mutation and drug resistance: Analysis of some case studies. Theor Biol Med Model 14:6, 2017 63. Chisholm RH, Lorenzi T, Lorz A, et al: Emergence of drug tolerance in cancer cell populations: An evolutionary outcome of selection, nongenetic instability, and stress-induced adaptation. Cancer Res 75:930-939, 2015 64. Perez-Velazquez J, Gevertz J, Karolak A, et al: Microenvironmental niches and sanctuaries: A route to acquired resistance, in Rejniak K (ed): Systems Biology of Tumor Microenvironment, Volume 936, New York, NY, Springer, 2016, 149-164 65. Kerbel RS, Kamen BA: The anti-angiogenic basis of metronomic chemotherapy. Nat Rev Cancer 4:423-436, 2004 66. Pasquier E, Kavallaris M, Andre´ N: Metronomic chemotherapy: New rationale for new directions. Nat Rev Clin Oncol 7:455-465, 2010 67. Gatenby RA, Silva AS, Gillies RJ, et al: Adaptive therapy. Cancer Res 69:4894-4903, 2009 + + 68. Ghiringhelli F, Menard C, Puig PE, et al: Metronomic cyclophosphamide regimen selectively depletes CD4 CD25 regulatory T cells and restores T and NK effector functions in end stage cancer patients. Cancer Immunol Immunother 56:641-648, 2007 69. Kareva I, Waxman DJ, Lakka Klement G: Metronomic chemotherapy: An attractive alternative to maximum tolerated dose therapy that can activate anti-tumor immunity and minimize therapeutic resistance. Cancer Lett 358:100-106, 2015 70. Lee W-P: The role of reduced growth rate in the development of drug resistance of HOB1 lymphoma cells to vincristine. Cancer Lett 73:105-111, 1993 71. Shackney SE, McCormack GW, Cuchural GJ Jr: Growth rate patterns of solid tumors and their relation to responsiveness to therapy: An analytical review. Ann Intern Med 89:107-121, 1978 72. Brimacombe KR, Hall MD, Auld DS, et al: A dual-ﬂuorescence high-throughput cell line system for probing multidrug resistance. Assay Drug Dev Technol 7:233-249, 2009 73. Thomas H, Coley HM: Overcoming multidrug resistance in cancer: An update on the clinical strategy of inhibiting p-glycoprotein. Cancer Contr 10:159-165, 74. Loeb LA, Springgate CF, Battula N: Errors in DNA replication as a basis of malignant changes. Cancer Res 34:2311-2321, 1974 75. Krutyakov V: Eukaryotic error-prone DNA polymerases: The presumed roles in replication, repair, and mutagenesis. Mol Biol 40:1-8, 2006 76. Makridakis NM, Reichardt JK: Translesion DNA polymerases and cancer. Front Genet 3:174, 2012 77. Lange SS, Takata K, Wood RD: DNA polymerases and cancer. Nat Rev Cancer 11:96-110, 2011 78. Traina TA, Norton L: Log-kill hypothesis, in Schwab M (ed): Encyclopedia of Cancer. Springer, Berlin, Germany, 2011, pp 1713-1714 79. Bender BC, Schindler E, Friberg LE: Population pharmacokinetic-pharmacodynamic modelling in oncology: A tool for predicting clinical response. Br J Clin Pharmacol 79:56-71, 2015 80. Wu Q, Li MY, Li HQ, et al: Pharmacokinetic-pharmacodynamic modeling of the anticancer effect of erlotinib in a human non-small cell lung cancer xenograft mouse model. Acta Pharmacol Sin 34:1427-1436, 2013 81. Fetterly GJ, Aras U, Lal D, et al: Development of a preclinical PK/PD model to assess antitumor response of a sequential aﬂibercept and doxorubicin-dosing strategy in acute myeloid leukemia. AAPS J 15:662-673, 2013 82. Erdi YE: Limits of tumor detectability in nuclear medicine and pet. Mol Imaging Radionucl Ther 21:23-28, 2012 83. Jodrell DI, Egorin MJ, Canetta RM, et al: Relationships between carboplatin exposure and tumor response and toxicity in patients with ovarian cancer. J Clin Oncol 10:520-528, 1992 84. Chignola R, Foroni RI: Estimating the growth kinetics of experimental tumors from as few as two determinations of tumor size: Implications for clinical oncology. IEEE Trans Biomed Eng 52:808-815, 2005 nn n 12 © 2019 by American Society of Clinical Oncology Differentiating Spontaneous and Induced Resistance APPENDIX t→∞ The supplement contains additional results and extensions related to S t , R t → 0, 1 . the model Equations 1 and 2 presented in the main text. Sections include details on the mathematical characteristics of solutions of Proof. From Theorem 1, we have that 0 ≤ S(t)+ R(t) ≤ 1, so that Equations 4 and 5, an extension describing the reversibility of drug Equation A2 implies resistance, treatment regimens with normalized dosages, details on structural identiﬁability, and an expanded discussion on the maximum dR critical time T (d). ≥ 0. dt As 0 ≤ R(t) ≤ S(t)+ R(t) ≤ 1, R(t) must converge, so that there exists Fundamental Solution Properties of Resistance Model 0 ≤ R ≤ 1 such that For convenience, Equations 4 and 5 are reproduced below: t→∞ R t →R . dS 1 − S + R S − e + αu t S − du t S, (A1) dt Deﬁne dR ρ : lim infS t . p 1 − S + R R + e + αu t S. (A2) t→∞ dt Since S(t) 2 [0,1], we know that 0 ≤ ρ ≤ 1. Assume that the inequality is We begin with a standard existence/uniqueness result, as well as the strict on the left: ρ . 0. By deﬁnition, there then exists t . 0 such that dynamical invariance of the triangular region: * T : S, R |S ≥ 0, R ≥ 0, S + R ≤ 1 . (A3) S t ≥ , Note that T represents the region of non-negative tumor sizes less than for all t ≥ t .As S(t)+ R(t) ≤ 1 for all t, Equation A2 implies that for all 1. Biologically, this implies that all solutions remain physical (non- t ≥ t , negative) and bounded above by the carry capacity (non- dimensionalized to 1 here, generally K in Eqs 1 and 2). eρ R t ≥ eS(t)≥ , Theorem 1. For any bounded measurable control u: [0,∞) → [0, which implies that u ], with u , ∞, and (S , R ) 2 T, the initial value problem max max 0 0 Equations A1 and A2, eρ t→∞ R t − R t ≥ ds→ ∞, S 0 S , R 0 R , 0 0 2 tp has a unique solution [S(t), R(t)] deﬁned for all times t 2 R. Fur- a contradiction. Thus, ρ =0. thermore, under the prescribed dynamics, region T is invariant. Deﬁne Proof. Existence and uniqueness of local solutions follow from standard results in the theory of differential equations—for example (Sontag ED: σ : lim inf S t + R t . t→∞ Mathematical Control Theory: Deterministic Finite Dimensional Sys- tems. Springer, 1998). As the vector ﬁeld As above, it is clear that σ 2 [0,1]. Assume that σ , 1. Thus, there exists t , e. 0 such that S(t)+ R(t)≤ 1 − e, 1 for all t ≥ t . Equation p * 1 − S + R S − e + αu t S − du t S A2 then implies that for all t ≥ t , F S, R, t : . p 1 − S + R R + e + αu t S R t ≥ p 1 − S t + R t R . p eR, r r is analytic for (S, R) 2 R , the existence of maximal solutions deﬁned for all t 2 R will follow from boundedness (Sontag ED: Mathematical As in the previous argument, this differential inequality contradicts the Control Theory: Deterministic Finite Dimensional Systems. Springer, boundedness of R. Thus, σ =1. 1998), which we demonstrate below. Since R converges to R , lim inf distributes over the sum of S + R, and Uniqueness implies that solutions remain in the ﬁrst quadrant for all t ≥ we have that 0. Indeed, we ﬁrst note that (0, 0) and (0 ,1) are steady states for any control u(t). As S = 0 implies that S 0, we see that the R-axis in 1 lim inf S t + R t ρ + R . t→∞ t→∞ invariant, with R(t)→ 1. Similarly, R = 0 implies Ṙ ≥ 0, and hence all trajectories with (S , R ) 2 T remain non-negative for all t.As V = S + 0 0 Thus R =1. R satisﬁes the differential equation t→∞ As R(t)→1, there exists t . 0 such that V α t 1 − V − β t , R t ≥ 1 − where α(t):= S(t)+ p R(t), β(t):= du(t)S(t) are both non-negative, V , 10V (t), 1. Thus, if the initial conditions (S , R ) reside in T,we 0 0 0 for all t ≥ t . For such t, Equation A1 implies that Ṡ(t) , 0. Thus, S is are guaranteed that (S(t), R(t)) 2 T for all time t, as desired. eventually decreasing and hence must converge to its lim inf ρ =0. t→∞ Thus, we have that (S(t), R(t))→ (0, 1), as desired. We now prove that, asymptotically, cells will evolve to become entirely resistant. For simplicity, we assume that the tumor is initially below carrying capacity, although a similar result holds Reversible Phenotype Switching for V . 1. In the model analyzed in this work (Eqs 4 and 5), we assumed that Theorem 2. For any bounded measurable control u: [0,∞) → resistance is nonreversible; however, experiments suggest that phenotypic alterations are generally unstable and hence a non-- [0,u ] ,with u , ∞, and initial conditions (S , R ) 2 T, max max 0 0 solutions of Equations A1 and A2 will approach the steady state negligible back transition exists. In this Appendix, we demonstrate (S, R) = (0,1): that an extension of our model to include this phenomenon does not JCO Clinical Cancer Informatics 13 Greene, Gevertz, and Sontag change the qualitative results presented previously, at least for pa- therapy magnitude u is ﬁxed (arbitrarily) at 0.5, which by Equation A12 on,c rameter values that are consistent with experimental data. implies that u =5.Wealso adjust Δt =0.5, Δt = 4.5, and all other on,p on off parameter values remain as in Table 1. To model reversible drug resistance, we include a constant per-capita transition rate from the resistant compartment R back to the wild Results of the simulations are presented in Figures A2 and A3. Note cell-type S. Denoting this rate by γ, we obtain the system that here Figure A2 displays the results for the phenotype-preserving drug (α = 0), whereas Figure A3 represents the resistance-inducing −2 dS drug (α =10 ). Each drug is simulated for the two distinct strategies; 1 − S + R S + γR − e + αu t S − du t S, (A4) each strategy is represented in the left column of the respective ﬁgures. dt The right column illustrates the population response for both the in- dR dividual populations—sensitive S and resistant R cells—as well as for p 1 − S + R R + e + αu t S − γR. (A5) dt the total tumor volume V = S + R. As previously discussed, treatment is continued until a critical tumor size V is obtained, and the corre- We ﬁrst consider an appropriate value of the rate γ. Note that in the sponding time t is used as a measure of treatment efﬁcacy, with absence of treatment (u(t) = 0), the system becomes a larger t indicating a better response. dS Our results are qualitatively in agreement with those presented in the 1 − S + R S + γR − eS (A6) main text (Fig 2), where no preservation of total administered drug dt (area under the drug concentration-time curve) was considered. dR Speciﬁcally, we observe a superior response with constant therapy for p 1 − S + R R + eS − γR. (A7) the phenotype-preserving drug (α = 0), whereas the situation is re- dt −2 versed for the resistance-inducing drug (α =10 ). Note that the tumor volume V(t)= S(t)+ R(t) satisﬁes the equation Identiﬁability V S + p R 1 − V , (A8) We provide details of both the structural and practical identiﬁability of which is nondecreasing (recall Theorem 1). This implies that the control system Equations A1 and A2. Technical details are provided system approaches a steady state (S , R ), which can be easily * * that do not appear in the main text. computed as Theoretical identiﬁability. We ﬁrst show that all parameters in γ e Equations A1 and A2 are identiﬁable using a relatively small set of S , R , . (A9) p p γ + e γ + e controls u(t) via classic methods from control theory. We provide a self- contained discussion. For a thorough review of theory and methods, Note that Equation A9 lies on the line V =1. see a recent article (Sontag ED: PLOS Comput Biol 13:e1005447, 2017) and the references therein. Pisco and colleagues measure a 1% to 2% subpopulation of clonally derived HL60 cells that consistently express high levels of MDR1, Assuming that time and tumor volume are the only clinically ob- which we equate with the resistant population R. Using the 2% upper servable outputs—that is, that one cannot readily determine sensitive bound, this implies that and resistant proportions in a given population—we measure V(t) and its derivatives at time t = t for different controls u(t). For sim- S 0.98, R 0.02. (A10) p p plicity, we assume that t = 0, so that treatment is initiated with a purely wild-type (sensitive) population. Note that this is equivalent Solving Equation A9 with the above values then determines the ratio to assuming an entirely sensitive tumor at treatment initiation. Al- though the results remain valid if t . 0 as the system of equations 49. (A11) gain only a constant, this assumption will simplify the subsequent computations. For a discussion of the practical feasibility of such We now repeat the constant versus pulsed experiments discussed methods, see the next section. in the main text, but for the reversible Equations A4 and A5. Speciﬁcally, consider the Equations A1 and A2 with initial conditions Parameter values are taken again as in Table 1,and γ is de- (Eqn 7). Measuring V(t)= S(t)+ R(t) at time t = 0 implies that we can termined via Equation A11. Results are presented in Figure A1 identify S : and should be compared with that presented in Figure 2.Notethat the same qualitative—and indeed quantitative—conclusions hold: V 0 S 0 + R 0 S : Y , 0 0 constant therapy improves response time compared with pulsing −2 when α = 0, whereas the reverse is true for α =10 . Thus, the where we adopt the notation Y ,i ≥ 0 for measurable quantities. inclusion of instability of the resistant cell subpopulation still Similarly, deﬁne the following for the given input controls: suggests that knowledge of the resistance-induction rate α for a chemotherapy is critical when designing therapies. We note that Y : V 0 , u t ≡ 0, precise agreement of Figures A1 and A2 is aresultofthe small Y : V 0 , u t ≡ 1, values taken for ε and hence γ. Y : V 0 , u t ≡ 0, Y : V 0 , u(t)≡ 1, (A13) Treatment Comparison for Equal Area Under the Drug Y : V 0 , u t ≡ 2, Concentration-Time Curve Y : V 0 , u t ≡ 0, Y : V 0 , u t t. Here, we provide an analogous comparison of treatment outcomes between constant and pulsed therapy as in the main text; however, All quantities Y ,i =0, 1,…, 7 are measurable, as each requires only treatment magnitudes u and u are chosen such that on,c on,p knowledge of V(t) in a small positive neighborhood of t = 0. Note that Z Z the set of controls u(t) is relatively simple, with Y exclusively de- t +Δt +Δt t +Δt +Δt d on off d on off 7 u t dt u t dt, (A12) termined via a nonconstant input. c p t t d d Each measurable Y may also be written in terms of a subset of the which is equivalent to the conservation of total administered dose between parameters d, ε, p ,and α, as all derivatives can be calculated in both strategies over a single pulsing cycle. In Equation A12, u (t)and u (t) terms of the right sides of Equations A1 and A2. For more details, c p denote constant and pulsed therapy schedules, respectively. The constant see the below section. Equating the expressions yields a system of 14 © 2019 by American Society of Clinical Oncology 70 80 50 60 60 70 Differentiating Spontaneous and Induced Resistance A B Treatment Strategies,  = 0 Tumor Dynamics,  = 0 Constant Constant 1.6 Pulsed Pulsed 0.9 1.4 0.8 1.2 0.7 0.6 0.8 0.5 0.4 0.6 0.3 0.4 0.2 0.2 0.1 0 10 20 30 40 0 10 20 30 40 Time Time C D -2 -2 Treatment Strategies,  = 10 Tumor Dynamics,  = 10 Constant Constant 1.6 1 Pulsed Pulsed 0.9 1.4 0.8 1.2 0.7 0.6 0.8 0.5 0.4 0.6 0.3 0.4 0.2 0.2 0.1 0 10 20 30 40 0 10 20 40 Time Time −2 FIG A1. Comparison of treatment efﬁcacy for phenotype-preserving drugs (α = 0) and resistance-inducing drugs (α =10 ), where resistance is reversible. The left column indicates treatment strategy, whereas right indicates corresponding tumor volume response. Note that the dashed red line in the right column indicates the tumor volume representing treatment failure, V . (A) Constant and pulsed therapies after tumor detection for α = 0. (B) Responses −2 corresponding to treatment regimens in panel A. (C) Constant and pulsed therapies after tumor detection for α =10 . (D) Responses corresponding to treatment regimens in panel C. equations for the model parameter, which we are able to solve. 1 1 2 Y − Y + Y − d Y 5 4 3 0 2 2 Carrying out these computations yields the following solution: α . (A17) dY Y − Y 2 1 d − , (A14) Note that in Equations A14 and A17 each quantity is determined by the Y and the parameter values previously listed. We do not 1 3 write the solution in explicit form for the sake of clarity as the Y − Y + Y − 2Y − Y + dY 1 − Y 7 6 5 4 3 0 0 2 2 , (A15) resulting equations are unwieldy. Furthermore, the solution of dY this system relies on the assumption of strictly positive initial 0 0 conditions (S = Y . 0), wild-type drug induction death rate (d), 2 3 Y − 1 − e Y + 3 − e Y − 2Y 3 0 0 0 and background mutation rate (ε), all of which are made in p , (A16) eY 1 − Y 0 0 this work. JCO Clinical Cancer Informatics 15 u u Tumor Volume S + R Tumor Volume S + R Greene, Gevertz, and Sontag A B Treatment Strategy,  = 0 Population Dynamics,  = 0 0.5 1 0.45 0.9 R 0.4 0.8 0.35 0.7 0.3 0.6 0.25 0.5 0.2 0.4 0.15 0.3 0.1 0.2 0.05 0.1 0 20406080 100 120 140 020 40 60 80 100 120 140 Time Time C D Treatment Strategy,  = 0 Population Dynamics,  = 0 5 0.9 4.5 0.8 0.7 3.5 0.6 0.5 2.5 0.4 0.3 1.5 0.2 0.1 0.5 02468 10 12 14 16 18 0246 8 10 12 14 16 18 Time Time FIG A2. Treatment dynamics for phenotype-preserving drugs (α = 0). The left column indicates treatment strategy, whereas the right indicates cor- responding population response. (A) Constant treatment after tumor detection. (B) Response to constant treatment. Note that at the time of treatment failure, the tumor is essentially entirely resistant. (C) Pulsed therapy after tumor detection. (D) Response to pulsed therapy. Note that the treatment fails much earlier versus the constant dose and that the tumor is primarily drug sensitive. Equation A17 is the desired result of our analysis. It demonstrates that phenotype-switching rate α from a class of input controls u(t); the drug-induced phenotype switching rate α may be determined by however, we see that the calculation involved measuring de- a relatively small set of input controls u(t). As discussed in the previous rivatives at the initial detection time t = t . Furthermore, the applied section, the value of α may have a large impact on treatment efﬁcacy controls (Eq A13) are nonconstant and thus require fractional and, therefore, determining its value is clinically signiﬁcant. Our results doses to be administered. Clinically, such strategies and mea- now prove that it is possible to compute the induction rate and, hence, surements may be difﬁcult and/or impractical. In this section, we use this information in the design of treatment protocols. In the next describe an in vitro method for estimating α using constant ther- section, we investigate other qualitative properties that could also be apies only. Speciﬁcally, our primary goal is to distinguish drugs with applied to understand the rate of drug-induced resistance. α = 0 (phenotype preserving) and α . 0 (resistance inducing). Such experiments, described below, may be implemented for a speciﬁc An In Vitro Experimental Protocol to Distinguish drug, even if its precise mechanism of promoting resistance remains Spontaneous and Drug-Induced Resistance uncertain. We have demonstrated that all parameters in Equations 4 and 5 are Before describing the in vitro experiment, we note that we are in- identiﬁable so that it is theoretically possible to determine the terested in qualitative properties for determining α. Indeed, in most 16 © 2019 by American Society of Clinical Oncology Population Population Differentiating Spontaneous and Induced Resistance -2 -2 Treatment Strategy,  = 10 Population Dynamics,  = 10 0.5 0.9 0.45 0.8 0.4 0.7 0.35 0.6 0.3 0.5 0.25 0.4 0.2 0.3 0.15 0.2 0.1 0.1 0.05 R 0 102030405060 0 102030405060 Time Time C D -2 -2 Treatment Strategy,  = 10 Population Dynamics,  = 10 4.5 0.9 4 0.8 3.5 0.7 3 0.6 2.5 0.5 2 0.4 1.5 0.3 1 0.2 0.5 0.1 R 0 102030405060 0 102030405060 Time Time −2 FIG A3. Treatment dynamics for resistance-inducing drugs (α =10 ). The left column indicates treatment strategy, whereas the right indicates corresponding population response. (A) Constant treatment after tumor detection. (B) Response to constant treatment. Note that at the time of treatment failure, the tumor is essentially entirely resistant. Dynamics are similar to Figure 2B, although with a shorter survival time. (C) Pulsed therapy after tumor detection. (D) Response to pulsed therapy. Note that here, in contrast to the case of a phenotype-preserving drug as shown in Figure A2, pulsed therapy exhibits a longer survival time. modeling scenarios, we have little or no knowledge of precise pa- mutation rate and p the relative ﬁtness of resistant cells. We thus rameter values and instead must rely on characteristics that distinguish perform a standard dose-response experiment for each value of the switching rate α independently of quantitative measurements. drug sensitivity d and measure the time t to reach critical size V as c c Furthermore, as a general framework for drug resistance, the only afunctionof d. The response t will then depend on the applied guaranteed clinically observable output variables are the critical tumor dose u—recall that we are only administering constant therapies— volume V and the corresponding time t . For a description of the and the sensitivity of wild-type cells d,aswellasthe induction c c treatment protocol, see above. We cannot expect temporal clonal rate α: subpopulation measurements. Assuming that V is ﬁxed for a given cancer, t is thus the only observable that we consider. t t u, d, α . (A18) c c By examining Equations 4 and 5, we see that the key parameters that dictate progression to the steady state (S, R) = (0,1) are d and α, as these determine the effectiveness and resistance induction of We further imagine that it is possible to adjust the wild-type the treatment, respectively. Recall that ε is the ﬁxed background drug sensitivity d. For example, in the case of multidrug resistance JCO Clinical Cancer Informatics 17 Population Population Greene, Gevertz, and Sontag in which the overabundance of P-glycoprotein affects drug phar- Comparing Figures 3A and 4A, we observe similar properties: small t macokinetics, altering the expression of MDR1 via ABCBC1 or even for small doses, a sharp increase about a critical u ,followedby CDX2 (Koh I, Hinoi T, Sentani K, et al: Cancer Med 5:1546-1555, smooth decrease and eventual horizontal asymptote (for mathe- 2016) may yield a quantiﬁable relationship between wild-type cell matical justiﬁcation, see the below section). However, note that for and d, thus producing a range of drug-sensitive cell types. Figure 3A a resistance-inducing drug (Fig 3A), maximum critical time T (d) exhibits a set of dose-response curves for representative drug increases as a function of d. This is in stark contrast to the constant −2 sensitivities d for the case of a resistance-inducing drug (α =10 ). behavior obtained for α = 0, argued above and demonstrated in Figure 4B. To further understand this phenomenon, we plot T (d)for For each of these cell types, we then deﬁne the supremum response −2 a ﬁxedinductionrate α =10 in Figure 3B. The behavior of this curve time over administered doses: is a result of the fact that the critical dosage u at which T (d)is c α obtained is a decreasing function of d (see Equation A24 and Fig A4 T d : sup t u, d, α . (A19) { } α c in the below section). But as u also controls the amount of resistant cells generated—via the αu(t)S term—resistance growth is impeded Note that in a laboratory setting, only a ﬁnite number of doses will be by a decreasing u . Thus, as a non-negligible amount of resistant administered so that the above supremum is actually a maximum, cells are necessary to yield T (d), more time is required for resistant but for mathematical precision we retain supremum. Thus, we cells to accumulate as d increases. Hence, T (d) increases a function obtain a curve T = T (d) for each value of the induced resistance α α of d. rate α. We then explore the properties of these curves for different α The behavior observed in Figures 3B and 4B is precisely the qualitative values. distinction that could assist in determining the induced resistance rate Consider ﬁrst the case of a phenotype-preserving drug, so that α =0. α. In the case of a phenotype-preserving drug, the proposed in vitro As u(t) ≡ u, we see that the system Equations 4 and 5 depends only experiment would produce a ﬂat curve, whereas a resistance-inducing on the product of u and d. Hence, the dependence in Equation A18 drug (α. 0) would yield an increasing function T (d). Furthermore, we becomes the form t (u$d, 0), and thus the supremum in Equation could use this phenomenon, in principle, to measure the induction rate A19 is instead across the joint parameter D := u$d: from the experimental T (d) curve. For example, Figure 5A displays −2 a range of T (d) for α near 10 . T : sup t D, 0 . (A20) 0 c Figure 5A shows a clear dependence of T (d) on the value of α. Quantitatively characterizing such curves would allow us to reverse Clearly, this is independent of d so that T is simply a horizontal line for engineer the induction rate α; however, we note that, in general, the α = 0. Qualitatively, the resulting curve will have no variation among the precise characteristics will depend on the other ﬁxed parameter engineered sensitive phenotypes, save for experimental and mea- values, such as p , V ,and «. Indeed, only order of magnitude r c surement noise. Figure 4 shows both representative curves (Fig 4A estimates may be feasible. Illustrative sample curves are provided compared with Fig 3A) and a plot of T (d)(Fig 4B), which veriﬁes its in Figure 5B. Two such characteristics are apparent from this independence of d. We make two minor technical notes. First, it is ﬁgure, both related to the slope of T (d). First, as d → 0 ,we important that we assume d. 0 here, as otherwise D = 0, independent observe an increase in the slope of T (d)as α decreases (note that of dose u and the supremum is over a one element set. See below for α in Fig 5B,only d ≥ 0.05 are plotted). This follows from the con- more details and the implications for α identiﬁability. Second, the slight tinuity of solutions on parameters and the fact that T (d) possesses variation for large values of d is a result of numerical error, as the 0 a jump discontinuity at d =0—that is, its distributional derivative is maximum of t occurs at decreasing doses (see the below section and given by Figure A4 for more details). | T d kδ d , (A21) d0 0 ∂d where δ is the Dirac function and k is a positive constant. As discussed Maximum Dosage Yield T (d ) previously (Equation A20 and the subsequent paragraph), T (d)is ﬂat, 3.5 except at d = 0 where the vector ﬁeld contains no u dependence. Numerical Therefore, the set over which the maximum is taken is irrelevant and Theory T (d) is thus proportional to the Heaviside function, which possesses the distributional derivative Equation A21. The constant k is de- termined by the size of the discontinuity of T (d). Continuous de- 2.5 pendence on parameters then implies that as α increases, the resulting derivative decreases away from positive inﬁnity as the corresponding derivative for T (d) with α . 0isdeﬁned in the classic sense for α . 0: 1.5 ∂ ∂ | T d ≤ 0. d0 α ∂α ∂d The above argument implies that measuring the slope of T (d)at d = 0 will give a characterization of the phenotypic alteration rate α of the 0.5 treatment; however, such experiments may be impractical, as ﬁne- tuning the sensitivity of a cell near complete resistance may be difﬁcult. Alternatively, one could analyze the degree of ﬂatness for a relatively 0 0.5 1 1.5 2 2.5 3 3.5 4 large d—so to be sufﬁciently far from d =0—and correlate this measure with α. For example, examining d =2in Figure 5B, we see that the relative slope of T (d) with respect to d should correlate with decreasing α. An argument similar to the above makes this rigorous, for FIG A4. Dose yielding maximal response time T (d) computed nu- d sufﬁciently large. Practical issues still arise, but this second method merically, as well as the approximation given by Equation A24. All provides a more global method for possibly computing α. Indeed, −2 parameters appear as in Table 1, and α =10 . The numerical slopes at a given d can be approximated by a wider range of secant maximum is computed over a discretization of constant dosage approximations as the result holds for a range of d compared with the procedures u(t) ≡ u, for u 2 [0,5], with a mesh size Δu = 0.005. previously discussed case when d is near zero. Furthermore, our focus 18 © 2019 by American Society of Clinical Oncology u Differentiating Spontaneous and Induced Resistance is largely on the qualitative aspects of α determination, such as the cells. Indeed, as u → 0, the dynamics of Equations A1 and A2 differences in Figures 3A and 4A, and determining whether the approach those of treatment itself induces resistance to emerge. dS ˜ ˜ ˜ 1 − S + R S, (A22) Introduction to Identiﬁability Analysis dt In this section, we provide additional details on the theoretical iden- dR tiﬁability of model parameters. As mentioned in the main text, all ˜ ˜ ˜ p 1 − S + R R, (A23) dt higher-order derivatives at initial time t = 0 may be calculated in terms of the initial conditions (S(0), R(0)) and the control function u(t). For for small ﬁnite times, as ε ,,1. Trajectories of Equations A22 and A23 example, for an arbitrary system remain on the curve x˙ t f x t , u t , pr R S S pr with external control u(t), the second derivative ẍ may be calculated using the chain rule: as the solution approaches the line S + R = 1. The critical time t is then determined by the intersection of this curve with S + R = V and, thus, x¨ J x, u f +∇ f x, u u˙, f u has sensitive population S at t given by the unique solution in the ﬁrst c c quadrant of where J (x) is the Jacobian matrix of f, evaluated at state x and control u.If x(0) = x is known, the above expression is a relation among ˙ S + S V . parameters, together with u and u evaluated at time t = 0. An anal- c c p c ogous statement holds for a measurable output y = h(x), but will also involve the Jacobian of h. Concretely, for the model of induced drug As R ,, 1 (as ε ,,1), S ≈ V , as claimed. Time t is thus small for 0 c c c resistance Equations 4 and 5, ﬁrst derivatives of the tumor volume may small u. be calculated as Increasing values of u imply that t also increases as the overall growth rate of sensitive cells is decreased; however, there exists a critical dose V 0 S 0 + R 0 , u such that sensitive cells alone are not able to multiply sufﬁciently to 1 − S + R S − e + αu t S − du t S attain V , so that the critical volume must have non-negligible con- tributions from the resistant fraction. This leads to the bifurcation + p 1 − S + R R + e + αu t S | , r t0 apparent in Figures 3A and 4A. We can even approximate the critical 1 − S S − du 0 S , 0 0 0 dose maximizing t ,as V must be an approximation for the carrying c c capacity of the sensitive cells: for any control u(t) [recall that R(0) = 0]. Similarly, for the second derivative, we compute: S ≈ V . K c V 0 S 1 − S − e + d u 0 1 − 2S − du(0) 0 0 0 Examining the right side of Equation A1 and assuming that the dy- namics of the resistant population are negligible, which is accurate in + S αu 0 + e p − S 1 + p − dS u 0 . 0 r 0 r 0 the early stages of treatment (Figs 2B and 3B), we see that the dose that yields the maximum temporal response should be Using such expressions—or, more precisely, the Lie derivatives of the vector ﬁeld [see Sontag (PLOS Comput Biol 13:e1005447, 2017)] 1 − e − V —for the controls in Equation A13, one is able to obtain a set of u ≈ . (A24) α + d equations between the set of Y ,i =0, 1,…, 7 and the parameters d, «, p , and α. Solving these equations allows us to determine the pa- That is, the dose at which T (d) is obtained is given approximately by rameters with respect to the measurable quantities. The algebraic the expression in Equation A24. For a sample numerical comparison of solution is Equations A14-A17. the predicted formula Equation A24 and a numerical optimization over a range of drug sensitivities d (Figure A4). Note that in actuality, S , Analysis of Critical Time T (d) V , as the resistant dynamics cannot be ignored entirely. Thus, the We provide a qualitative understanding of the properties of T (d), precise value of u will be smaller than that provided in the previous the maximum time, across all constant doses for the tumor to formula as we numerically observe. Lastly, u decreases with in- reach size V . This Appendix is designed to explain the basic creasing values of parameter d and, thus, requires an increasingly ﬁne properties discussedinAnInVitro Experimental Protocol to discretization to numerically locate the maximum value. Hence, some Distinguish Spontaneous and Drug-Induced Resistance in the numerical error is observed in Figure 4B. main text. Lastly, as u is increased further, the dose becomes sufﬁciently large so We ﬁrst note that T (d) is achieved at a medium dose u .More that the inhibition of S via therapy implies that S cannot approach the α c precisely, we describe the qualitative properties of Figures 3A and critical volume V and, hence, V is again reached by an essentially c c 4A.Fix adrugsensitivity d. For small u, the sensitive subpopulation is homogeneous population, but now of resistant cells. As resistant cells not sufﬁciently inhibited and thus expands rapidly to cross the divide at a slower rate (p , 1), the corresponding time t is smaller. For r c threshold V , with an essentially homogenous population of sensitive a schematic of the three regimes described above (Figure A5). JCO Clinical Cancer Informatics 19 Greene, Gevertz, and Sontag u u u < u u > u t t Time FIG A5. Schematic demonstrating dynamics of variation in t on dosage u. Sensitive cell population plotted as a function of time for three representative doses. For u , u , sensitive cells grow and reach V in a short amount of time. As u→u , the sensitive population approaches its approximate carrying capacity of V , but subsequently decreases as a result of the dynamics of resistance. Here, t is maximized as the sensitive population spends a large amount of time near V . For u . u , the sensitive population is eliminated quickly, and c c V is obtained by a primarily resistant population. 20 © 2019 by American Society of Clinical Oncology Volume

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JCO Clinical Cancer Informatics – Wolters Kluwer Health

Published: Apr 10, 2019

Keywords: ABCB1, BCR, ABL1, MTTP, MAP2K1, BRAF, CD5, SGCB

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Mathematical Approach to Differentiate Spontaneous and Induced Evolution to Drug Resistance During Cancer Treatment

Mathematical Approach to Differentiate Spontaneous and Induced Evolution to Drug Resistance During Cancer Treatment

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Mathematical Approach to Differentiate Spontaneous and Induced Evolution to Drug Resistance During Cancer Treatment

Mathematical Approach to Differentiate Spontaneous and Induced Evolution to Drug Resistance During Cancer Treatment

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