Optimal Drug Synergy in Antimicrobial Treatments

Simple model for the evolution of resistance in multi-drug treatment

We based our model on work by Jumbe et al. [21], which investigated a mouse-thigh P. aeruginosa infection model [22]–[24] and provided a mathematical model that quantitatively described the relationship between exposure to the fluoroquinolone antibiotic levofloxacin, and changes in drug-susceptible and drug-resistant bacterial subpopulations over time. This mathematical model was successful both in reproducing the observed changes in drug-susceptible and –resistant subpopulations over time, and in predicting the dose of levofloxacin needed to suppress amplification of levofloxacin-resistant (efflux-pump-expressing) mutants. To investigate the effect of antibiotic interactions on treatment efficacy and the prevention of multi-drug resistance in a simple scenario, we modified the Jumbe et al. model in four ways: we include a second antibiotic in our model; we assume a constant antibiotic dose; we assume a low hill coefficient, consistent with the mechanisms of a range of antibiotics [25]; and we assume no cost for antibiotic resistance. The consequences of these assumptions are discussed throughout the text.

Our model incorporates treatment with two antibiotics, A and B. It uses a set of ordinary differential equations (ODEs) to follow the population sizes of the drug-sensitive wild-type strain (), the total single-drug resistant population (; we assume symmetry between drugs A and B such that their respective resistant populations are equal, ), and the expected number of multi-drug resistant mutants () arising over time (Fig. 1A; Methods):

(1)

(2)

(3)

Populations are affected by growth, antibiotic killing and mutation, where , and are the growth rate, antibiotic killing rate and frequency of resistance mutations per generation, respectively, and and are the effective doses of antibiotic felt by the wild-type and single-drug resistant mutant populations. We assume for simplicity that antibiotic resistance imposes no fitness cost, so that the growth rates of the sensitive and resistant populations are the same. To account for competition for resources, we assume this growth rate is given by the logistic equation,

(4)

where is the maximal growth rate, is the total population size, and is the maximal carrying capacity (Fig. S1B). This competition for resources was included in the in vivo murine infection model [21], and has been observed in or inferred from a range of infections [26], including S. pneumoniae [27], and Methicillin-Resistant S. aureus (MRSA) [28].

While we assume the growth rates of the wild-type and single-drug resistant mutants are the same, the rates at which they are killed by antibiotic are different and depend on the effective antibiotic dose, , felt by each population:

(5)

where is the maximal killing rate and is the Hill coefficient, which determines the steepness of the killing rate as a function of drug dose. In contrast to Jumbe et al., we set , which is representative of many common antibiotics [25], although different values of give rise to similar overall model behavior (Fig. S2). The effective drug dose, , depends on the dosage of the two drugs and on their interaction (Fig. 1B, Text S1, Fig. S1A). For simplicity we assume both drugs are administered at the same dose, , defined in units of their minimum inhibitory concentration (), the single-drug dose at which the wild-type death rate equals its growth rate at resource-unlimited conditions: . For the wild-type, the effective dose is the sum of the dosage of the two drugs plus their level of interaction : . Values of are positive, zero, or negative for synergy, additivity and antagonism, respectively. While in practice the value of is specific to a given drug pair [29], we treat it as continuous in order to investigate all potential treatment strategies. We assume that single-drug resistant mutants are affected by only one of the drugs, which is reasonable in the case of resistance mechanisms that decrease the intracellular concentration of antibiotic, such as efflux pump expression or enzymatic degradation [11], [13], [30]. The effective dose of single-drug resistant mutants is therefore , and is independent of (Fig. 1B). Except where indicated, we set (general model behavior is robust to changes in drug dosage, Fig. S2A), which for an additive drug pair is consistent with the drug dosage used in Jumbe et al. [21].

Mutations from wild-type to single-drug resistance, or from single- to double-drug resistance, arise at a rate per individual per replication, or per individual per unit time. Since in any effective treatment the number of double mutants arising is smaller than 1, we do not account for the growth or death of this fractional population, but rather define as the integrated number of double mutants generated via mutation during treatment (Eq. 3). Prevention of multi-drug resistance is then defined as .

The model therefore consists of Eqs. 1–5. Parameter values, following Jumbe et al. [21], are given in Table S1. Initial conditions for the model are , , - the population sizes at the onset of treatment. We assume that prior to treatment, the infections have grown from a single cell to the initial population size while mutating; while the overall mutation rate is a function of model parameters , and (Eqs. 2, 3), and are functions of alone: , . No double-drug resistant mutants are present: . We integrate the ODEs with these initial conditions (Methods) and define as the time at which the total population size drops below one ( is defined as infinity if the population reaches a non-zero steady state).

Antibiotic interactions create a saturable tradeoff between treatment efficacy and prevention of multi-drug resistance

To determine the impact of drug interaction on treatment outcome, we first looked at the differences in treatment efficacy () and prevention of multi-drug resistance () over a range of drug interaction values () while holding drug dosage fixed (Fig. 2, ). We observed two distinct and robust (Fig. S2) behaviors, depending on whether falls above or below a critical value, (Fig. 2; for the parameters used, ).

For , we observed a tradeoff between treatment efficacy and prevention of resistance. In this regime, increasing synergy yields greater (Fig. 2, unshaded region); this is expected, as increasing the synergistic interaction between the drugs kills the wild-type more quickly. Despite faster infection clearance, however, greater synergy actually decreases ; namely, it increases the risk of multi-drug resistance. Conversely, more antagonistic drug pairs increase , albeit at the expense of reduced efficacy.

This tradeoff between efficacy and prevention of resistance breaks down at a critical threshold, , above which increasing synergy no longer increases , but still decreases (Fig. 2, shaded region). Above this “synergy ceiling,” further increasing synergy therefore has only undesirable effects, since it increases the risk of multi-drug resistance without increasing efficacy. Optimal drug pairs for treating the given infection must therefore have a level of drug interaction lower than and, due to the tradeoff between and , the optimal value of will depend on the relative importance assigned to these two conflicting goals.

The frequency of resistance mutations determines the “synergy ceiling”

To understand what determines the level of the synergy ceiling, , we asked what causes the transition from tradeoff behavior at , to plateau behavior at (Fig. 2). Due to the sharp biphasic behavior of efficacy () around , we looked to population time courses to determine how the time of clearance, , was affected by drug interactions below, at or above (, respectively; Fig. 3A). For (Fig. 3A, top), the wild-type subpopulation outlives the single-drug resistant mutants and . Since wild-type killing is stronger for more synergistic drug pairs, increasing decreases , explaining why efficacy increases with in this region (Fig. 2, unshaded region). For (Fig. 3A; bottom), however, the wild-type is eliminated before the single-mutant population, and . Because the killing rate of the single-drug resistant mutants is independent of , is effectively independent of , causing to plateau for (Fig. 2, shaded region). is therefore the level of drug interaction for which wild-type and single-mutant populations are cleared simultaneously (; Fig. 3A, middle).

Since represents the level of drug interaction for which , parameters that differentially alter and will alter . While we found that a number of model parameters had some effect on (Fig. S2), the strongest effect was due to changes in the frequency of resistance mutations, . differentially affects and because, although it has virtually no effect on , the single-mutant population size at the onset of treatment increases linearly with , , thereby increasing . For to match this increase in , the wild-type killing rate must decrease; namely, must be reduced. We therefore expected to decrease with increasing and, indeed, increasing the frequency of resistance mutations gave rise to consistent decreases in (Fig. 3B). Interestingly, for high frequencies of resistance () the synergy ceiling falls below zero (representing an antagonistic drug interaction); in this case mildly synergistic and even additive interactions fall in the undesirable regime where the risk of multi-drug resistance increases without any corresponding gain in treatment efficacy.

Competition for resources underlies the tradeoff between treatment efficacy and prevention of multi-drug resistance

Why does synergy, despite clearing the infection faster, increase the risk of multi-drug resistance (Fig. 2)? Since synergistic drug pairs clear the infection more quickly than antagonistic drug pairs, the rate at which they generate double mutants must also be higher. The overall rate at which double mutants arise, (Eqs. 3, 4), is affected by two variables: it increases with the size of the single-mutant population, , and decreases with total population size, , due to the inhibitory effect of resource limitation on growth and mutation (Fig. 4A). The total number of double mutants expected to arise is simply the integral of this instantaneous rate over the treatment course. In order to determine why synergistic treatments increase , we therefore analyzed the trajectories of synergistic and antagonistic treatments through the space of versus (Fig. 4A; , solid line, , dashed line). The initial slopes of these trajectories (Fig. 4A, arrows) are determined by the relative fitness of the wild-type and single-drug resistant populations under antibiotic treatment. The synergistic treatment selects strongly against the wild-type population, producing a trajectory with a steep slope that drives treatment into a region of high (Fig. 4A, red region); this is because the rapid decrease in wild-type population size relieves competition for resources, creating a window of opportunity in which the still-large single-mutant population can rapidly grow and mutate. Conversely, the antagonistic treatment selects only weakly against the wild-type, producing a trajectory with a shallow slope that skirts the high region. Antagonistic drug pairs therefore decrease in a competition-dependent fashion: weak killing of the wild-type maintains competition for resources, limiting growth and mutation of the single-drug resistant population until it is eliminated.

It is important to note that resource competition is significant only at the beginning of treatment, when . If competition for resources is required for the advantage of antagonism over synergy in preventing resistance, then we should expect a decrease in initial population size to decrease this advantage. To test this prediction, we looked at the relative ability of our representative synergistic () and antagonistic () drug pairs to prevent multi-drug resistance, , over a range of (Fig. 4C, circles; sensitivity to other model parameters is minimal, Fig. S2). Indeed, we found that the advantage of antagonistic drug pairs in preventing resistance (, below dashed line) was limited to cases where is close to . In fact, for significantly lower than , the trend reverses and synergy better prevents resistance (, above dashed line). This is because, for low population sizes, resource competition effects are negligible; therefore no longer depends on , and becomes a function of alone. Because synergistic drug pairs better limit by quickly killing the wild-type population from which single mutants arise, they therefore also better limit multi-drug resistance in cases of weak competition. Indeed, this advantage of synergy disappeared entirely when we artificially turned off wild-type to single-mutant mutation, allowing only those single mutants present at the start of treatment to contribute to (Fig. 4C, triangles).

Importantly, when competition for resources is weak ( significantly less than , ), the tradeoff between treatment efficacy and prevention of multi-drug resistance no longer exists (Fig. 4D; compare with Fig. 2). As a result, is no longer useful as a “synergy ceiling” because, although drug pairs with do not further improve , they do improve . For infections with weak competition, the use of maximally synergistic drug pairs therefore represents the best possible treatment strategy.

Model details

Our model consists of 3 ODEs (Eq. 1–3) describing the population sizes of the wild-type and single-drug resistant mutants (, ), as well as the number of double mutants expected to arise during a treatment course (). Parameter values for this model include first-order maximal growth () and death rate () constants, carrying capacity and mutation rate (per individual per generation), which were taken from the in vivo murine model investigated in Jumbe et al. [21] (Table S1). Initial population sizes () were determined by assuming that, prior to treatment, the infections grew from a single cell to the initial population size while mutating, such that and ; unless otherwise indicated, . ODEs were solved in MATLAB (Version 7.1, MathWorks, Natick, MA) using a built-in, numerical ODE solver (ODE45). To avoid artifacts associated with using continuous ODEs to describe finite populations, each step was modified with the assumption that the wild-type or single-drug resistant population is eliminated (size decreases to zero) if its size drops below one.