When does high-dose antimicrobial chemotherapy prevent the evolution of resistance?

High-dose chemotherapy has long been advocated as a means of controlling drug resistance in infectious diseases but recent empirical and theoretical studies have begun to challenge this view. We show how high-dose chemotherapy engenders opposing evolutionary processes involving the mutational input of resistant strains and their release from ecological competition. Whether such therapy provides the best approach for controlling resistance therefore depends on the relative strengths of these processes. These opposing processes lead to a unimodal relationship between drug pressure and resistance emergence. As a result, the optimal drug dose always lies at either end of the therapeutic window of clinically acceptable concentrations. We illustrate our findings with a simple model that shows how a seemingly minor change in parameter values can alter the outcome from one where high-dose chemotherapy is optimal to one where using the smallest clinically effective dose is best. A review of the available empirical evidence provides broad support for these general conclusions. Our analysis opens up treatment options not currently considered as resistance management strategies, and greatly simplifies the experiments required to determine the drug doses which best retard resistance emergence in patients. Significance Statement The evolution of antimicrobial resistant pathogens threatens much of modern medicine. For over one hundred years, the advice has been to ‘hit hard’, in the belief that high doses of antimicrobials best contain resistance evolution. We argue that nothing in evolutionary theory supports this as a good rule of thumb in the situations that challenge medicine. We show instead that the only generality is to either use the highest tolerable drug dose or the lowest clinically effective dose; that is, one of the two edges of the therapeutic window. This approach suggests treatment options not currently considered, and greatly simplifies the experiments required to identify the dose that best retards resistance evolution.

suppress even then HLR strain. 126 We can now provide a precise definition of high-level resistance (HLR). Although With the above formalism, we focus on resistance emergence, defined as the replication of 134 resistant microbes to a high enough density within a patient to cause symptoms and/or to 135 be transmitted (19). In the analytical part of our results this is equivalent to the resistant 136 strain not being lost by chance while rare. λ ∇ x π · x c + ∂π ∂c ds + n 1 − π ∇ x π 0 · x 0 c + ∂π 0 ∂c standing hazard (4) where π 0 = π[x(0; c), c], x 0 = x(0; c), and subscripts denote differentiation. Equation (4) is 154 partitioned in two different ways to better illustrate the effect of increasing dose. The first 155 is a partitioning of its effect on mutation and replication. The second is a partitioning of 156 its effect on the de novo and standing hazards. We have also indicated the terms that 157 represent competitive release in blue (as explained below).

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The first term in equation (4) represents the change in de novo mutation towards the HLR 159 strain that results from an increase in dose. The term (∂λ/∂p)(∂p/∂c) is the change in 160 mutation rate, mediated through a change in wild type density; ∂λ/∂p specifies how 161 mutation rate changes with an increase in the wild type density p (positive) while ∂p/∂c 162 specifies how the wild type density changes with an increase in dose (typically negative for 163 much of the duration of treatment). Thus the product, when integrated over the duration 164 of treatment, is expected to be negative. The term ∂λ/∂c is the change in mutation rate 165 that occurs directly as a result of an increased dose (e.g., the direct suppression of wild 166 type replication, which suppresses mutation). This, is expected to be non-positive in the 167 simplest cases and is usually taken as such by proponents of high-dose chemotherapy. 168 Therefore high-dose chemotherapy decreases the rate at which HLR mutations arise during 169 treatment. Note, however, that if the drug itself causes a higher mutation rate (e.g., 21), 170 then it is possible for an increased dose to increase the rate at which resistance appears.

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The second term in equation (4) represents replication of HLR strains once they have 172 appeared de novo during the course of treatment. The term ∇ x π · x c is the indirect 173 increase in escape probability, mediated through the effect of within-host state, x.

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Specifically, x c is a vector whose elements give the change in each state variable arising 175 from an increased dosage (through the removal of the wild type). These elements are 176 typically expected to be positive for much of the duration of treatment because an increase 177 in dose causes an increased rebound of the within-host state through a heightened removal 178 of wild type microbes. The quantity ∇ x π is the gradient of the escape probability with 179 respect to host state x, and its components are expected to be positive (higher state leads 180 to a greater probability of escape). The integral of the dot product ∇ x π · x c is therefore the 181 competitive release of the HLR strain in terms of de novo hazard (19). This will typically 182 be positive. The term ∂π/∂c is the direct change in escape probability of de novo mutants 183 as a result of an increase in dosage (i.e., the extent to which the drug suppresses even the 184 HLR strain). This term is negative at all times during treatment but, by the definition of 185 HLR, this is small. Therefore, high-dose chemotherapy increases the replication of any HLR 186 mutants that arise de novo during treatment. 187 Finally, the third term in equation (4) represents the replication of any HLR strains that 188 are already present at the start of treatment. The term n 1 − π (∇ x π 0 · x 0 c ) is the indirect  that it is the relative balance among these opposing processes that determines whether 211 high-dose chemotherapy is the optimal approach. We will present a specific numerical 212 example shortly that illustrates these points, but first we draw two more general 213 conclusions from the theory.

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We begin by considering a situation in which the maximum tolerable drug concentration c U 252 causes significant suppression of the resistant strain ( Figure 2a). We stress however that if 253 this were true then, by definition, the resistant strain is not really HLR and thus there 254 really is no resistance problem to begin with. We include this extreme example as a 255 benchmark against which comparisons can be made.

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Not surprisingly, under these conditions a large dose is most effective at preventing 257 resistance (compare Figure 2b with 2c). This is a situation in which the conventional 'hit 258 hard' strategy is best.

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Now suppose that the maximum tolerable drug concentration c U is not sufficient to directly  Under these conditions we see that a small dose is more effective at preventing resistance 265 emergence than a large dose (compare Figure 3b with 3c). This is a situation in which the 266 conventional or orthodox 'hit hard' strategy is not optimal.
Equation (4) provides insight into these contrasting results. The only difference between 268 the models underlying Figures 2 and 3 is that ∂π/∂c and ∂π 0 /∂c are both negative for 269 Figure 2 whereas they are nearly zero for Figure 3 (that is, at tolerable doses, the drug has 270 negligible effects on resistant mutants). As a result, the negative terms in equation (4) 271 outweigh the positive terms for Figure 2 whereas the opposite is true for Figure 3.

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These results appear to contradict those of a recent study by Ankomah and Levin (12).

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Although their model is more complex than that used here, equation (4)  is really a measure of the occurrence of resistance mutations rather than emergence per se.

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In comparison, we consider emergence to have occurred only once the resistant strain 281 reaches clinically significant levels; namely, a density high enough to cause symptoms or to 282 be transmitted. There are two process that must occur for de novo resistant strains to 283 reach clinically relevant densities. First, the resistant strain must appear by mutation, and 284 both our results ( Figure 3d) and those of Ankomah and Levin (12) show that a high dose 285 better reduces the probability that resistance mutations occur (this can also be seen in 286 equation 4). Second, the resistant strain must replicate to clinically significant levels. 287 Ankomah and Levin (12) did not account for this effect and our results show that a high 288 concentration is worse for controlling the replication of resistant microbes given a resistant 289 strain has appeared (Figure 3d). This is because higher doses maximally reduce 290 competitive suppression. In Figure 3 the latter effect overwhelms the former, making 291 low-dose treatment better. In Figure 2 these opposing processes are also acting but in that 292 case the drug's effect on controlling mutation outweighs its effect on increasing the 293 replication of such mutants once they appear.

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More generally, Figure 4 illustrates the relationship between drug concentration and the 295 maximum size of the resistant population during treatment, for the model underlying 296 Figure 3. In this example a high concentration tends to result in relatively few outbreaks of 297 the resistant strain but when they occur they are very large. Conversely, a low 298 concentration tends to result in a greater number of outbreaks of the resistant strain but 299 when they occur they are usually too small to be clinically significant. None of these factors alters the general finding that the optimal strategy depends on the 308 balance between competing evolutionary processes. competition determines the optimal resistance management strategy (13,19). Increasing 313 the drug concentration reduces mutational inputs into the system but it also unavoidably 314 reduces the ecological control of any HLR pathogens that are present. These opposing 315 forces generate an evolutionary hazard curve that is unimodal. Consequently, the worst 316 approach is to treat with intermediate doses (Figure 1) as many authors have recognized 317 (5)(6)(7)9). The best approach is to administer either the largest tolerable dose or the 318 smallest clinically effective dose (that is, the concentration at either end of the therapeutic 319 window). Which of these is optimal depends on the relative positions of the hazard curve 320 and the therapeutic window ( Figure 1). Administering the highest tolerable dose can be a 321 good strategy (Figure 1c,d) but it can also be less than optimal (Figure 1b) or even the 322 worst thing to do ( Figure 1a). Thus, nothing in evolutionary theory supports the 323 contention that a 'hit-hard' strategy is a good rule of thumb for resistance management. probability that microbes with two or more resistance mutations will appear. Considerable 353 effort has been put into estimating the MPC for a variety of drugs and microbes (4).

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The relationship between these ideas and the theory presented here is best seen using the 355 extension of equation (4)  (although see Figure C3 of Appendix C for a counterexample). If, however, the MPC is less suggests that the hazards need be estimated only at the bounds of the therapeutic window.

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These bounds are typically well known because they are needed to guide clinical practice.

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Estimating the resistance hazard experimentally can be done in vitro and in animal models 415 but we note that since the solution falls at one end of the therapeutic window, they can 416 also be done practically and ethically in patients. That will be an important arena for 417 testing, not least because an important possibility is that, as conditions change, the 418 optimal dose might change discontinuously from the lowest effective dose to the highest 419 tolerable dose or vice versa. There is considerable scope to use mathematical and animal 420 models to determine when that might be the case and to determine clinical predictors of 421 when switches should be made.

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Managing resistance in non-targets 423 Our focus has been on the evolution of resistance in the pathogen population responsible it is important to determine the level of caution that is clinically warranted rather than 449 simply perceived.

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For many years, physicians have been reluctant to shorten antimicrobial courses, using long 451 courses on the grounds that it is better to be safe than sorry. It is now increasingly clear 452 from randomized trials that short courses do just as well in many cases (e.g., 56-58) and 453 they can reduce the risk of resistance emergence (56,59,60 generally true this is for other pathogens, or pathogens of other hosts, remains to be seen. 458 We also note that our arguments about the evolutionary merits of considering the lowest 459 clinically useful doses have potential relevance in the evolution of resistance to cancer 460 chemotherapy as well (62).           In the absence of treatment we model the within-host dynamics using a system of where P is the density of the wild type and X is a vector of variables describing the 4 within-host state (e.g., RBC count, densities of different immune molecules, etc). The 5 initial conditions are P (0) = P 0 X(0) = X 0 . At some point, t * , drug treatment is 6 introduced. Using lower case letters to denote the dynamics in the presence of treatment, 7 we then have with initial conditions p(0; c) = P (t * ) and x(0; c) = X(t * ), and where c is the dosage. For 9 simplicity, here we assume that a constant drug concentration is maintained over the 10 course of the infection. Appendix E considers the pharmacokinetics of discrete drug dosing. The notation p(t; c) and x(t; c) reflects the fact that the dynamics of the wild type and the 12 host state will depend on dosage. For example, if the dosage is very high p will be driven to 13 zero very quickly.
14 As the drug removes the wild type pathogen, resistant mutations will continue to arise

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There are also other plausible forms for the mutation rate as well, and therefore we simply any mutant will depend on the host state at the time of its appearance, x(t; c), and it will 31 therefore depend indirectly on c. Note that π will also depend directly on c, however, 32 because drug dosage might directly suppress resistant strains as well if the dose is high 33 enough. Therefore we use the notation π[x(t; c), c], and assume that π is an increasing 34 function of x and a decreasing function of c.

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With the above assumptions the host can be viewed as being in one of two possible states 36 at any point in time during the infection: (i) resistance has emerged (i.e., a resistant strain 37 has appeared and escaped), or (ii) resistance has not emerged. We model emergence as an 38 inhomogeneous birth process, and define q(t) as the probability that resistance has emerged 39 by time t. A conditioning argument gives where λ∆t is the probability that a mutant arises in time ∆t, and π is the probability that 41 such a mutant escapes. Re-arranging and taking the limit ∆t → 0 we obtain with initial condition q(0) = q 0 . Note that q 0 is the probability that emergence occurs as a

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The solution to the above differential equation is If a is the time at which treatment is stopped, and Q is the probability of emergence already presents at the start of treatment. 54 Given the expression for Q, all else equal, resistance management would seek the treatment 55 strategy, c that makes Q as small as possible. Since Q is a monotonic function of D + S, 56 we can simplify matters by focusing on these hazards instead. Thus we define which is the 'total hazard' during treatment. Equation (4) is then obtained by 58 differentiating the the total hazard H with respect to c.  The calculations in Appendix A can again be followed. We obtain an equation identical to 85 equation (4) except that the first term is replaced by where subscripts denote differentiation with respect to that variable. The difference is that 87 (∂λ/∂p)(∂p/∂c) in equation (4) is replaced with ∇ p λ · p c . The quantity p c is a vector whose elements of ∇ p λ). Either way, however, this does not alter the salient conclusion that the 97 optimal resistance management dose will depend on the details.

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In an analogous fashion we might also alter the derivation in Appendix A to account for where y is a vector of commensal microbe densities. We might then model λ as   Although this is sometimes the case (Day, unpubl. results) the opposite is possible as well.

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As an example, Figure C2 presents results for the probability of emergence as a function of 166 dose, for three different levels of resistance frequency in the initial infection.
where P m1 is the density of the mutant strain with intermediate resistance and P m2 is the 185 strain with HLR. Also, r(·), r m1 (·), and r m2 (·) are the growth rates of the wild type and  Figure D1a). There is an important structure to these failures, however, that can be better inter-dose interval, etc but our focus on T will be sufficient to see how one would deal with 260 these other factors as well.

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To allow for more general pharmacokinetics we must model the dynamics of drug 262 concentration explicitly. Once treatment has begun the model becomes The third equation accounts for the pharmacokinetics of the drug and allows for the 264 treatment protocol to vary through time. These equations must also be supplemented with 265 an initial condition specifying the values of the variables at the start of treatment.

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After time T has elapsed treatment is stopped and the dynamics then follow a different set 267 of equations given by The tildes reflect the fact that the functional form of the dynamical system might change where we have simplified the notation by using a tilde above a function to indicate that the 277 function is evaluated along the variables with a tilde. Differentiating with respect to T gives By the continuity of the state variables the first two terms cancel and therefore we have We can see that this has a form that is identical to de novo part of equation (4) except that now the drug concentration is no longer directly under our control. Instead, changes in T affect resistance emergence by how they affect changes in drug concentration. More generally, the very same potentially opposing processes as those in equation 4 will arise regardless of how we alter the drug dosing regimen because any such alteration must ultimately be mediated through its affect on the drug concentration at each point in time during an infection.   3)))) and the resistant strain in red (r m (c) = 0.59(1 − tanh(15(c − 0.6)))) as well as thetherpeutic window in green. Dots indicate the probability of resistance emergence. Probability of resistance emergence is defined as the fraction of 5000 simulations for which resistance reached a density of at least 100 (and thus caused disease). Parameter values are P (0) = 10, I(0) = 2, α = 0.05, δ = 0.05, κ = 0.075, µ = 10 −2 , and γ = 0.01. Bar graphs: the probability that a resistant strain appears by mutation is indicated by the left-hand grey bars for each drug concentration (the right-hand grey bar is the probability that a resistant strain does not appear). The probability of treatment failure for a specific drug dose is the sum of the red bars for that dose. (b) Same as panel (a) but with mutation rate decreased to µ = 10 −3 .