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

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.

INTRODUCTION

Tumor resistance to chemotherapy and targeted drugs is a major cause of treatment failure. Both molecular and microenvironmental factors have been implicated in the development of drug resistance.[1] As an example of molecular resistance, upregulation of drug efflux transporters can prevent sufficiently high intracellular drug accumulation, which limits treatment efficacy.[2] Other molecular causes of drug resistance include modification of drug targets, enhanced DNA damage repair mechanisms, dysregulation of apoptotic pathways, and the presence of cancer stem cells.[1-5] Irregular tumor vasculature that results in inconsistent drug distribution and hypoxia is an example of a microenvironmental factor that impacts drug resistance.[6] Other characteristics of the tumor microenvironment influencing drug resistance include regions of acidity, immune cell infiltration and activation, and the tumor stroma.[1,6-10] Experimental and clinical research continues to shed light on the multitude of factors that contribute to cancer drug resistance. Mathematical modeling studies have also been used to explore both broad and detailed aspects of cancer drug resistance, as reviewed previously.[11-13]

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.

Resistance to cancer drugs can be classified as either pre-existing or acquired.[1] Pre-existing—or intrinsic—drug resistance describes the case in which a tumor contains a subpopulation of drug-resistant cells at the initiation of treatment, which makes therapy eventually ineffective as a result of resistant cell selection.[1] As examples, pre-existing BCR-ABL kinase domain mutations confer resistance to the tyrosine kinase inhibitor imatinib in patients with chronic myeloid leukemia,[14,15] and pre-existing MEK1 mutations confer resistance to BRAF inhibitors in patients with melanoma.[16] Many mathematical models have considered how the presence of such pre-existing resistant cells impacts cancer progression and treatment.[17-40]

Acquired drug resistance broadly describes the case in which drug resistance develops during the course of therapy from a population of cells that were initially drug sensitive.[1] The term acquired resistance is really an umbrella term for two distinct phenomena, which complicates the study of acquired resistance. On the one hand, there is resistance that is spontaneously—or randomly—acquired during the course of treatment, be it as a result of random genetic mutations or stochastic nongenetic phenotype switching.[41] This spontaneous form of acquired resistance has been considered in many mathematical models.[18-22,27-29,31,32,35,40,42-49] On the other hand, drug resistance can be induced (ie, caused) by the drug itself.[41,50-52]

The question of whether resistance is an induced phenomenon or predates treatment was first famously studied by Luria and Delbrück[53] in the context of bacterial (Escherichia coli) resistance to a virus (T1 phage). In particular, Luria and Delbrück hypothesized that if selective pressures imposed by the presence of the virus induce bacterial evolution, then the number of resistant colonies formed in their plated experiments should be Poisson distributed and thus have an approximately equal mean and variance. What Luria and Delbrück found instead was that the number of resistant bacteria on each plate varied drastically, with variance being significantly larger than the mean. As a result, the authors concluded that bacterial mutations predated the viral challenge.[53]

In the case of cancer, there is strong evidence that at least some drugs have the ability to induce resistance, as genomic mutations can be caused by cytotoxic cancer chemotherapeutics.[54,55] For instance, nitrogen mustards can induce base substitutions and chromosomal rearrangements, topoisomerase II inhibitors can induce chromosomal translocations, and antimetabolites can induce double-stranded breaks and chromosomal aberrations.[54] Such drug-induced genomic alterations would generally be nonreversible. Drug resistance can also be induced at the epigenetic level.[41,50,56] For example, expression of multidrug resistance 1 (MDR1), an ABC-family membrane pump that mediates the active efflux of the drug, can be induced during treatment.[1,41] In another recent example, the addition of a chemotherapeutic agent is shown to induce, through a multistage process, epigenetic reprogramming in patient-derived melanoma cells.[56] Resistance developed in this way can occur quite rapidly and can often be reversed.[41,52,57]

Despite these known examples of drug-induced resistance, differentiating between drug-selected and drug-induced resistance is nontrivial. For example, what appears to be drug-induced acquired resistance may simply be the rapid selection of a small number of pre-existing resistant cells or the selection of cells that spontaneously acquired resistance.[41,44] In pioneering work by Pisco and colleagues,[41] the relative contribution of resistant cell selection versus drug-induced resistance was assessed in an experimental system that involved HL60 leukemic cells that were treated with the chemotherapeutic agent vincristine. After 1 to 2 days of treatment, expression of MDR1 was demonstrated to be predominantly mediated by cell-individual induction of MDR1 expression and not by the selection of MDR1-expressing cells.[41,58] In particular, these cancer cells exploit their heritable, nongenetic phenotypic plasticity—by which one genotype can map onto multiple stable phenotypes—to change their gene expression to a temporarily more resistant state in response to treatment-related stress.[41,58]

Although there is a wealth of mathematical research that addresses cancer drug resistance, relatively few models have considered drug-induced resistance. Of the models of drug-induced resistance that have been developed, many do not explicitly account for the presence of the drug. Instead, it is assumed that these models apply only under treatment,[41,59-62] with the effects of the drug implicitly captured in model terms. As these models of resistance induction are dose independent, they are unable to capture the effects that the alteration of the drug dose has on resistance formation. To our knowledge, there have been less than a handful of mathematical models developed in which resistance is induced by a drug in a dose-dependent fashion.[33,34,63], In Gevertz et al[33] and follow-up work in Shah, Rejniak, and Gevertz[38] and Perez-Velazquez et al,[64] duration and intensity of drug exposure determines the resistance level of each cancer cell. This model allows for a continuum of resistant phenotypes, but is computationally complex as it is a hybrid discrete-continuous, stochastic spatial model. While interesting features about the relationship between induced resistance and the microenvironment have been deduced from this model, its complexity does not allow for general conclusions to be drawn about dose-dependent resistance induction.

Another class of models that addresses drug-induced resistance is that in Chisholm et al.[63] These models are distinct in that they are motivated by in vitro experiments in which a cancer drug transiently induces a reversible resistant phenotypic state.[52] The individual-based and integro-differential equation models developed consider rapidly proliferating drug-sensitive cells, slowly proliferating drug-resistant cells, and rapidly proliferating drug-resistant cells. An advection term—with the speed depending on drug levels—is used to model drug-induced adaptation of the cell proliferation level, and a diffusion term for both the level of cell proliferation and survival potential (response to drug) is used to model nongenetic phenotype instability.[63] Through these models, the contribution of nongenetic phenotype instability (both drug induced and random), stress-induced adaptation, and selection can be quantified.[63]

Finally, the work in Liu et al[34] models the evolutionary dynamics of the tumor population as a multitype nonhomogeneous continuous time birth-death stochastic process. This model accounts for the ability of a targeted drug to alter the rate of resistant cell emergence in a dose-dependent manner. The authors specifically considered cases in which the rate of mutation that gives rise to a resistant cell: (1) increases as a function of drug concentration, (2) is independent of drug concentration, and (3) decreases with drug concentration. Interestingly, this model led to the conclusion that the optimal treatment strategy is independent of the relationship between drug concentration and the rate of resistance formation. In particular, the authors found that resistance is optimally delayed using a low-dose continuous treatment strategy coupled with high-dose pulses.

As in vitro experiments have demonstrated that treatment response can be affected by drug-induced resistance,[41,52] in the current work we seek to understand this phenomenon further using mathematical modeling. The initial mathematical model that we have developed—and that will be analyzed herein—is a system of two ordinary differential equations with a single control representing the drug. We intentionally chose a minimal model that would be amenable to analysis, as compared with previously developed models of drug-induced resistance which are significantly more complex.[33,38,63,64] Despite the simplicity of the model, it incorporates both spontaneous and drug-induced resistance.

In addition to drug-induced resistance, the other characteristic of cancer dynamics we explore is that of traditional, maximally tolerated dose (MTD) treatment protocols compared with high-frequency, low-dose so-called metronomic therapies. Indeed, the differential response between these therapies is fundamentally related to intratumoral heterogeneity/competition, and is explicitly considered in our model. Furthermore, results presented in this work support recent evidence that promotes the adoption of metronomic therapy in many circumstances,[65-69] and a main objective of this work is to relate competition and drug-induced resistance to therapy design.

This work is organized as follows. We begin by introducing a mathematical model to describe the evolution of drug resistance during treatment with a theoretical resistance-inducing—and noninducing—drug. We use this mathematical model to explore the role played by the drug’s resistance induction rate in treatment dynamics. We demonstrate that the induction rate of a theoretical cancer drug could have a nontrivial impact on the qualitative responses to a given treatment strategy, including tumor composition and the time horizon of tumor control. In our model, for a resistance-preserving drug—that is, a drug that does not induce resistance—better tumor control is achieved using a constant therapeutic protocol versus a pulsed one. Conversely, in the case of a resistance-inducing drug, pulsed therapy prolongs tumor control longer than constant therapy as a result of sensitive/resistant cell competitive inhibition. Once the importance of induced resistance has been established, we demonstrate that all parameters in our mathematical model are identifiable, meaning that it is theoretically possible to determine the rate at which drug resistance is induced for a given treatment protocol. As this theoretical result does not directly lend itself to an experimental approach for quantifying the ability of a drug to induce resistance, we also describe a potential in vitro experiment for approximating this ability utilizing constant therapies. We end with some concluding remarks and a discussion of potential extensions of our analysis, such as a model that differentiates between reversible and nonreversible forms of resistance.

MATERIALS AND METHODS

Here we introduce a general modeling framework to describe the evolution of drug resistance during treatment. Our model captures the fact that resistance can result from random events or can be induced by the treatment itself. Random events that can confer drug resistance include genetic alterations—for example, point mutations or gene amplification—and phenotype switching.[41] These spontaneous events can occur either before or during treatment. Drug-induced resistance is resistance specifically activated by the drug and, as such, depends on the effective dose encountered by a cell. Such a formulation allows us to distinguish the contributions of both drug-dependent and drug-independent mechanisms, as well as any dependence on pre-existing—that is prior to treatment—resistant populations.

We consider the tumor to be composed of two types of cells, sensitive (S) and resistant (R). Sensitive (or wild-type) cells are fully susceptible to treatment, whereas treatment affects resistant cells to a lesser degree. To analyze the role of both random and drug-induced resistance, we use a system of two ordinary differential equations to describe the dynamics between the S and R subpopulations:

All parameters are non-negative. In the absence of treatment, we assume that the tumor grows logistically, with each population contributing equally to competitive inhibition. Phenotypes S and R each possess individual intrinsic growth rates, and we make the assumption in the remainder of the work that . This simply states that resistant cells grow slower than nonresistant cells, an assumption that is supported by experimental evidence.[70-72]

The transition to resistance can be described with a net term of the form εS + αu(t)S. Mathematically, the drug-induced term αu(t)S, where u(t) is the effective applied drug dose at time t, describes the effect of treatment on promoting the resistant phenotype. For example, this term could represent the induced overexpression of the P-glycoprotein gene, a well-known mediator of multidrug resistance, by the application of chemotherapy.[1,73]

Spontaneous evolution of resistance is captured in the ϵS term, which permits resistance to develop even in the absence of treatment. Note that ε is generally considered small,[74] although recent experimental evidence regarding error-prone DNA polymerases suggests that cancer cells may have increased mutation rates as a result of the overexpression of such polymerases.[75-77] For example, in Krutyakov,[75] mutation rates as a result of such polymerases are characterized by probabilities as high as 7.5 × 10−1 per base substitution, and it is known that many point mutations in cancer arise from these DNA polymerases.[77] For this work, we adopt the notion that random point mutations that lead to drug resistance are rare, and that drug-induced resistance occurs on much quicker time scales[41]; therefore, we will assume that α > ε with u = O(1) in our analysis of Equations 1 and 2.

We model the effects of treatment by assuming the log-kill hypothesis,[78] which states that a given dose of chemotherapy eliminates the same fraction of tumor cells regardless of tumor size. We allow for each cellular compartment to have a different drug-induced death rate (d, d); however, to accurately describe resistance it is required that 0 ≤ d < d. Our analysis presented herein will be under the simplest assumption that the drug is completely ineffective against resistant cells, so that d = 0.

The last term in the equations, γR, represents the resensitization of cancer cells to the drug. In the case of nonreversible resistance, γ = 0; otherwise γ > 0. Our subsequent analysis will be done under the assumption of nonreversible resistance. For a discussion of the effect of reversibility on the presented model, see the Appendix.

Finally, we note that the effective drug concentration u(t) can be thought of as a control input. For simplicity, in this work we assume that it is directly proportional to the applied drug concentration; however, pharmacodynamic/pharmacokinetic considerations could be incorporated to more accurately describe the uptake/evolution of the drug in vivo or in vitro—for example, as in Bender, Schindler, and Friberg,[79] Wu et al,[80] and Fetterly et al.[81]

To understand the above system of drug resistance evolution, we reduce the number of parameters via nondimensionalization. Rescaling S and R by their (joint) carrying capacity K, and time t by the sensitive cell growth rate,

Equations 1 and 2 (with γ = d = 0) can be written in the form,

For convenience, we have relabeled , and t to coincide with the nondimensionalization so that the parameters ε, α, and d must be scaled accordingly (by 1/r). As r was assumed to satisfy 0 ≤ r < r, the relative resistant population growth rate p satisfies 0 ≤ p < 1.

One can show (Appendix) that asymptotically, under any treatment regimen u(t) ≥ 0, the entire population will become resistant:

However, tumor control is still possible where one can combine therapeutic efficacy and clonal competition to influence transient dynamics and possibly prolong patient life. Indeed, the modality of adaptive therapy has shown promise in using real-time patient data to inform therapeutic modulation aimed at increasing quality of life and survival times.[67] This work will focus on such dynamics and controls.

RESULTS

Effect of Induction on Treatment Efficacy

We investigate the role of the induction capability of a drug (parameter α in Eqs 4 and 5) on treatment dynamics. Specifically, the value of α may have a substantial impact on the relative success of two standard therapy protocols—constant dosage and periodic pulsing.

Treatment Protocol

To quantify the effects of induced resistance, a treatment protocol must be specified. We adopt a clinical perspective over the course of the disease, which is summarized in Figure 1. We assume that the disease is initiated by a small number of wild-type cells:

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), where treatment is considered to have failed. (B) Illustrative constant and pulsed treatments, both initiated at t = t.

where 0 < S < 1. Denote the tumor volume at time t by V(t):

The tumor then progresses untreated until a specific volume V is detected—or, for hematologic tumors, via appropriate blood markers—which using existing nuclear imaging techniques corresponds to a tumor with diameter on the order of 10 mm.[82] Time to reach V is denoted by t, which in general depends on all parameters that appear in Equations 4 and 5. Note that, assuming ϵ > 0, a nonzero resistant population will exist at the onset of treatment. Therapy, represented through u(t), is then applied until the tumor reaches a critical size V, which we equate with treatment failure. Because the (S,R) = (0,1) state is globally asymptotically stable in the first quadrant, V < 1 is guaranteed to be obtained in finite time. Time until failure, t, is then a measure of efficacy of the applied u(t).

Although a diverse set of inputs u(t) may be theoretically applied, presently we consider only strategies as illustrated in Figure 1B. The blue curve in Figure 1B corresponds to a constant effective dosage u(t) initiated at t—administered approximately using continuous infusion pumps and/or slow-release capsules—whereas the black curve represents a corresponding pulsed strategy u(t), with fixed treatment windows (Δton) and holidays (Δtoff). In general, we may allow for different magnitudes, uon, and uon,, for constant and pulsed therapies respectively—for example, to relate the total dosage applied per treatment cycle (area under the drug concentration-time curve [AUC][83]). However, for simplicity we assume the same magnitude in the subsequent section (although see the Appendix for a normalized comparison). While these represent idealized therapies, such u(t) may form an accurate approximation to in vitro and/or in vivo kinetics. Note that the response V(t) illustrated in Figure 1A will not be identical, or even qualitatively similar, for both presented strategies, as will be demonstrated numerically.

Constant Versus Pulsed Therapy Comparison

To qualitatively demonstrate the role that induced resistance plays in the design of therapy schedules, we consider two drugs with the same cytotoxic potential—that is, the same drug-induced death rate d—each possessing a distinct level of resistance induction (parameter α). A fundamental question, then, is whether there exist qualitative distinctions between treatment responses for each chemotherapy. More specifically, how does survival time compare when scheduling is altered between constant therapy and pulsing? Does the optimal strategy—in this case, optimal across only two schedulings—change depending on the extent to which the drug induces resistance?

We fix two values of the induction parameter α:

Recall that we are studying the nondimensional model Equations 4 and 5, so no units are specified. Parameter α = 0 corresponds to no therapy-induced resistance (henceforth denoted as phenotype preserving), and therefore considering this case allows for a comparison between the classic notion of random evolution toward resistance (α = 0) and drug-induced resistance (α > 0). For the remainder of the section, parameters are fixed as in Table 1. Critically, all parameters excluding α are identical for each drug, which enables an unbiased comparison. Treatment magnitudes uon, and uon, are selected to be equal: uon, = uon, = uon.

TABLE 1.

Parameters Used For Comparison of Treatment Efficacy for Phenotype-Preserving Drugs and Resistance-Inducing Drugs

Note that selecting parameter V = 0.1 implies that the carrying capacity has a diameter of 100 mm, as V corresponds to a detectable diameter of 10 mm. Assuming each cancer cell has volume 10−6 mm3, tumors in our model can grow to a carrying capacity of approximately 12.4 cm in diameter, which is in qualitative agreement with the parameters estimated in Chignola and Foroni[84] (≈12.42 cm, assuming a tumor spheroid).

By examining Figures 2A and 2B, we clearly observe an improved response to constant therapy when using a phenotype-preserving drug, with treatment success time t nearly seven times as long compared with pulsed therapy (t ≈ 90 for constant, compared with t ≈ 14 for pulsed). It can be observed that the tumor composition at treatment conclusion is different for each therapy—not shown for this simulation, but see a comparable result in Appendix Figures A2B and A2D—and it seems that pulsed therapy was not sufficiently strong to hamper the rapid growth of the sensitive population. Indeed, treatment failed quickly as a result of insufficient treatment intensity in this case, as the population remains almost entirely sensitive. Thus, for this patient under these specific treatments, assuming drug resistance only arises via random stochastic events, constant therapy would be preferred. One might argue that pulsed, equal-magnitude treatment is worse when α = 0 simply because less total drug—that is, AUC—is applied. However, we see that even in this case, intermediate doses may be optimal (Figs 3A and 4A). Thus, it is not the larger total drug, per se, that is responsible for the superiority of the constant protocol in this case, a point that is reinforced by the fact that the results remain qualitatively unchanged even if the total drug is controlled for (Appendix).

FIG 2.

Comparison of treatment efficacy for phenotype-preserving drugs (α = 0) and resistance-inducing drugs (α = 10−2). 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 to treatment regimens in panel A. (C) Constant and pulsed therapies after tumor detection for α = 10−2. (D) Responses corresponding to treatment regimens in panel C.

FIG A2.

Treatment dynamics for phenotype-preserving drugs (α = 0). 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. (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.

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 = 10−2, with all 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 that induces resistance. Note that time T(d) increases with drug sensitivity; compare with Figure 4B for purely random resistance evolution.

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 various drug sensitivities d; comparable to Figure 3A, with α = 0. (B) Maximum critical time T0(d). Note that the curve is essentially constant.

Compare this with Figures 2C and 2D, which consider the same patient and cytotoxicity, but for a highly inductive drug. Results are strikingly different and suggest that pulsed therapy is now not worse but in fact substantially improves patient response (t ≈ 61 for pulsed, compared with t ≈ 45 for constant). In this case, both tumors are now primarily resistant (Figs A3B and A3D), but the pulsed therapy allows for prolonged tumor control via sensitive/resistant competitive inhibition. Furthermore, treatment holidays reduce the overall flux into resistance as the application of the drug itself promotes this evolution. The total amount of drug (AUC) is also less for pulsed therapy (22.5 compared with ), so that pulsed therapy is both more efficient in terms of treatment efficacy and less toxic to the patient as adverse effects are typically correlated with the total administered dose, which is proportional to the AUC. This is further consistent with recent experimental and clinical evidence that supports metronomic therapy as a superior alternative to classic chemotherapy regimens. The results presented in Figure 2 suggest that it may be advantageous to apply a smaller amount of drug more frequently; however, we also note that the results depend on patient-specific parameters, so that therapy would ideally be personalized to individual patients. Of note, we do not claim that these results hold for all parameter values—both patient and treatment specific—but instead emphasize that the rate of induction may play a large role in the design of therapies for specific patients.

FIG A3.

Treatment dynamics for resistance-inducing drugs (α = 10−2). 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.

For these specific parameter values, differences between constant and pulsed therapy for the inductive drug are not as extensive as in the phenotype-preserving case; however, recall that time has been nondimensionalized and, hence, the scale may indeed be clinically relevant. Such differences can be further amplified, and, as exact parameters are difficult or even (currently) impossible to measure, qualitative distinctions are paramount. Thus, at this stage, ranking of therapies, rather than their precise quantitative efficacy, should act as the more important clinical criterion.

From these results, we observe a qualitative difference in the treatment strategy to apply based entirely on the value of α, the degree to which the drug itself induces resistance. Thus, in administering chemotherapy, the resistance-promotion rate α of the treatment is a clinically significant parameter. In the next section, we use our model and its output to propose in vitro methods for experimentally measuring a drug’s α parameter.

Identifying the Rate of Induced Drug Resistance

The effect of treatment on the evolution of phenotypic resistance may have a significant impact on the efficacy of conventional therapies. Thus, it is essential to understand the value of the induction parameter α before administering therapy. In this section, we briefly discuss both the theoretical possibility and practical feasibility of determining α from different input strategies u(t). For more details, see the Appendix.

Theoretical Identifiability

We first study the structural identifiability of Equations 4 and 5, a prerequisite for analyzing practical methods for determining parameter values. Structural identifiability is the process of determining model parameters—for example, α—from a set of control experiments. Here, we assume that the only measurable quantity is the tumor volume V = S + R, along with its derivatives, in time. Using four different controls, we show that all model parameters, including the induction rate α, may be determined by precisely measuring the corresponding volume-response curves. For more details, see the Appendix.

An In Vitro Experimental Protocol to Distinguish Spontaneous and Drug-Induced Resistance

As structural identifiability was established in the previous section, we focus on practical qualitative differences exhibited by Equations 4 and 5 as a function of the resistance-induction rate α. Utilizing only constant dosages, we investigate the dependence of t on dose u, cytotoxicity d, and α. Defining the supremum over doses of the response time (Eq 8),

we plot the results for different α values in Figures 3 and 4.

Comparing Figures 3B and 4B, we observe a clear qualitative difference in maximum response times. In the case of a phenotype-preserving drug, the proposed in vitro experiment would produce a flat curve, whereas a resistance-inducing drug (α > 0) would yield an increasing function T (d). Additional comparisons are presented in Figure 5, where the α dependence is more closely analyzed. For more details and analysis, see the Appendix.

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 are given as in Table 1. (A) Plot of T(d) for α near 10−2. (B) Analogous to panel A, where α is now varied over several orders of magnitude. Nonmutagenic case (α = 0) is included for reference.

FIG A4.

Dose yielding maximal response time T(d) computed numerically, as well as the approximation given by Equation A24. All parameters appear as in Table 1, and α = 10−2. The numerical maximum is computed over a discretization of constant dosage procedures u(t) ≡ u, for u ∈ [0,5], with a mesh size Δu = 0.005.

DISCUSSION

In the current work, we analyzed two distinct mechanisms that can result in drug resistance. Specifically, a mathematical model is proposed which describes both the spontaneous generation of resistance and drug-induced resistance. Using this model, we contrasted the effect of standard therapy protocols and demonstrate that contrary to the work in Liu et al,[34] the rate of resistance induction may have a significant effect on treatment outcome. Thus, understanding the dynamics of resistance evolution with regard to the applied therapy is crucial.

To demonstrate that one can theoretically determine the induction rate, we performed an identifiability analysis on the parameter α and demonstrated that it can be obtained via a set of appropriate perturbation experiments on u(t). Furthermore, we presented an alternative method, using only constant therapies, for understanding the qualitative differences between purely spontaneous and induced cases. Such properties could possibly be used to design in vitro experiments on different pharmaceuticals, which allows one to determine the induction rate of drug resistance without an a priori understanding of the precise mechanism. We do note, however, that such experiments may still be difficult to perform in a laboratory environment, as engineering cells with various drug sensitivities d may be challenging. Indeed, this work can be considered as a thought experiment to identify qualitative properties that the induction rate α yields in our modeling framework.

Our simple model allows significant insight into the role of random versus induced resistance. Of course, more elaborate models can be studied by incorporating more biologic detail. For example, while our two-equation model classifies cells as either sensitive or resistant, not all resistance is treated equally. Some resistant cancer cells are permanently resistant, whereas others could transition back to a sensitive state.[41] This distinction may prove to be vitally important in treatment design. A possible extension of our model is one in which we distinguish between sensitive cells S, nonreversible resistant cells R, and reversible resistant cells R (Eqs 9-12):

Here, V denotes the entire tumor population—that is,

In this version of the model, nonreversible resistant cells R can be thought of as resistant cells that form via genetic mutations. Under this assumption, ε represents the rate at which spontaneous genetic mutations give rise to resistance, and α is the drug-induced resistance rate. This situation can be classified as nonreversible as it is incredibly unlikely that genomic changes that occur in response to treatment would be reversed by an "undoing" mutation. Therefore, once cells confer a resistant phenotype via an underlying genetic change, we assume that they maintain that phenotype. This term could also be thought of as describing resistance that forms via stable epigenetic alterations or resistance that forms by some combination of genetic and stable epigenetic changes.

Conversely, reversible resistant cells R denote resistant cells that form via phenotype switching, as described in Pisco et al.[41] Random phenotype switching in the absence of treatment is captured in the ε term. This is consistent—and indeed necessary—to understand the experimental results in Pisco et al,[41] where a stable distribution of MDR1 expressions is observed even in the absence of treatment. The α(t)S term represents the induction of a drug-resistant phenotype. Phenotype switching is often reversible, and therefore we allow a back transition from the R compartment to the sensitive compartment at a nonnegligible rate γ[70] (see Appendix). Formulated in this way, the model can be calibrated to experimental data and we can further consider the effects of the dosing strategy on treatment response. We plan to further study this model in future work.

Other extensions which include different clinical scenarios are also being investigated. In practice, chemotherapies are rarely applied in isolation. Multiple therapies are often administered simultaneously to improve efficacy. The inclusion of multiple drugs, including targeted therapies that act primarily on resistant subpopulations, yields natural control questions that are clinically relevant. Similarly, immune cells, together with immunotherapies, may also be incorporated to more accurately mimic the cancer microenvironment.

Overcoming drug resistance is crucial for the success of both chemotherapy and targeted therapy. Furthermore, the added complexity of induced drug resistance complicates therapy design, as the simultaneous effects of tumor reduction and resistance propagation confound one another. Mathematically, we have presented a clear framework for differentiating random and drug-induced resistance, which will allow for clinically actionable analysis on a biologically subtle, yet important, issue.