Antibiotic Cycling and Antibiotic Mixing: Which One Best Mitigates Antibiotic Resistance?

Abstract

Can we exploit our burgeoning understanding of molecular evolution to slow the progress of drug resistance? One role of an infection clinician is exactly that: to foresee trajectories to resistance during antibiotic treatment and to hinder that evolutionary course. But can this be done at a hospital-wide scale? Clinicians and theoreticians tried to when they proposed two conflicting behavioral strategies that are expected to curb resistance evolution in the clinic, these are known as "antibiotic cycling" and "antibiotic mixing." However, the accumulated data from clinical trials, now approaching 4 million patient days of treatment, is too variable for cycling or mixing to be deemed successful. The former implements the restriction and prioritization of different antibiotics at different times in hospitals in a manner said to "cycle" between them. In antibiotic mixing, appropriate antibiotics are allocated to patients but randomly. Mixing results in no correlation, in time or across patients, in the drugs used for treatment which is why theorists saw this as an optimal behavioral strategy. So while cycling and mixing were proposed as ways of controlling evolution, we show there is good reason why clinical datasets cannot choose between them: by re-examining the theoretical literature we show prior support for the theoretical optimality of mixing was misplaced. Our analysis is consistent with a pattern emerging in data: neither cycling or mixing is a priori better than the other at mitigating selection for antibiotic resistance in the clinic.

Key words: : antibiotic cycling, antibiotic mixing, optimal control, stochastic models.

Fig. 1

An illustration of the toy scenario. A patient seeks treatment and one of two antibiotics can be administered. If the probability of resistant infection to either drug, p and q respectively, can be estimated, optimal behavior maximizes the likelihood of appropriate therapy. This entails finding a that maximizes $1-q+a(p-q)$ where $0\le a\le 1$ but since this expression is linear in a, the maximum occurs when a is zero or one. However, the worst possible drug deployment protocol comes from minimizing this expression and this also arises when a is zero or one.

Fig. 2

(top panel) Two performance histograms (orange and yellow, respectively) illustrate that reactive cycling can outperform all the scheduled cycling protocols. (bottom panel) Two timeseries from the stochastic version of (7) using (1, left) a protocol that cycles reactively between antibiotics and (2, right) optimal mixing: protocol (1) outperforms optimal mixing (2) as can be seen in the top panel. Variables in the legend of (2) are defined in equation (7) and performance here is the number of observed infected patients days.

Fig. 3

The structure of performance histograms of all cycling and all mixing protocols for a typical theoretical model. The shaded regions, each with unit area, illustrate that the range of performances of the cycling protocols is at least as wide as that of the mixing protocols and a series of numerical examples in the text illustrates this for specific models. Note that the “optimal cycling” and “optimal mixing” performance can coincide in this figure whereupon the blue and yellow histograms would have identical ranges. Note also, when illustrating this figure using mathematical model simulations in the text, we do not plot the performance histogram for all the mixing protocols, rather just optimal mixing (colored green), worst mixing (red), and random mixing (blue) are indicated.

Fig. 4

Top and bottom panels show two sets of performance histograms for (6) for cycling (10⁷ simulations in total) and three mixing protocols (10,000 simulations each). Consistent with Figure 3, both panels exhibit complete overlap in performances as indicated by the horizontal bars. While both use simulation data from equation (6), the top panel has different costs of resistance from the bottom panel that uses “symmetric” parameter values that bias in favor of random mixing. In the latter, as the bottom panel shows, there is a low probability of finding a cycling protocol that can approach the performance of random mixing, although some do. Note, this figure does not show the performance histogram for all mixing protocols in the manner done in Figure 3, rather histograms determined from stochastic simulations of (6) are shown for optimal mixing (green), the worst possible mixing (red), and random mixing (blue). Here, (bottom) the blue and green histograms coincide so only the green one is visible.

Fig. 5

(top) This illustrates Figure 3 using performance histograms determined from equation (7), although only three mixing histograms are shown (random, optimal, and worst). The range of performances, from the best to worst mixings, and the range of the cyclings are shown as horizontal bars: note how they overlap consistent with Figure 3. (bottom) The random mixing protocol (indicated with a 2 in top and bottom panels) is deployed into a stochastic version of (7) and it outperforms one cycling protocol (label 3) but it under performs another cycling (label 1). This ordering property is a general feature of theoretical models like (7).

Fig. 6

A schematic of impossible performance histograms. If this outcome were possible in a theoretical model, we would then be certain that mixing antibiotics outperforms cycling in a mathematical model, but it has been shown (Beardmore and Pena-Miller 2010b,a) that this arrangement of histograms cannot arise in equations of the form (6) and (7).

Fig. 7

Illustrating the individual-based model. (a) This shows “health state dynamics” of a patient during a sequential antibiotic treatment: blue and green colored areas show the drug used and the subsequent load of drug-resistant and drug-susceptible pathogens in the host through time. The black line is the density of a commensal bacterium that competes with the pathogen for resources. The patients is deemed “recovered” when the commensals have outcompeted the pathogen. (b) This shows treatments in a ward of five beds where a queue of seven patients (labeled $\{1,2,3,4,5,6,7\}$) is to be treated. Colored boxes indicate the protocol where green and blue boxes, respectively, denote patients only treated with one of the two available drugs, a red outline denotes the discharge of a recovered patient and the arrival of a new patient. (c) An illustration of the spatial structure in the ward resulting from one realization of the model using a queue of 20 patients and five beds where a seven-day cycling protocol has been implemented, blue and green colors represent different drugs used. The relative frequency of each of the two single-drug resistant pathogens in each patient is shown in the right-most skyscraper illustrating that drug resistance is correlated with the drug usage policy. Gray regions are hosts carrying equal fractions of two different single-drug resistant pathogens, blue and green colors indicate that one of the drug resistant strains dominates the infection of the host in that bed.

Fig. 8

Individual model LoS statistics with no treatment and empirical treatment. (a) The mean length-of-stay distribution (LoS) when no patient is treated with antibiotics is a normal distribution with mean close to 16 days for the parameter values implemented (see supplementary text, Supplementary Material online). (b) The LoS distribution when treating empirically follows a log-normal distribution in the same conditions. (c) We grouped patients treated with empirical therapy into two a posteriori classes: appropriate and inappropriate, according to the drug administered. Accordingly, the LoS distribution when drug use is inappropriate in empirical therapies have both higher mean and variance. Inappropriate drug allocation is therefore responsible for the large variance of the empirical treatment strategy in (b). (d) For illustrative purposes: a Kaplan-Meier plot illustrating the LoS for the two classes from (c).

Fig. 9

The optimal cycling period for this one particular instance of the individual patient model is very short at about 3 days. That it is so small relative to real-world clinical trials is likely due to the rapid turnover of patients in the model and the fact that there is no evolution of resistance among pathogens in the community that supplies patients to the model.

Fig. 10

We can improve upon mixing and cycling cadences by rapidly diagnosing the pathogen genotype responsible for infection. LoS distributions for different strategies show the optimal treatment exploits most of the available information (the rapid, DNA-based, diagnosis strategy). Empirical treatment, using no information at all, is the worst performing strategy here, although it is preferable to cycling with very long cycling times (see fig. 9).

Fig. 11

Delaying the availability of pathogen genotype information leads to deterioration in performance of the patient-specific treatment. (a) Delaying the availability of pathogen drug-susceptibility information decreases drug appropriateness to about 80% of patients with a 2-day delay and to no better than random (i.e. empirical treatment) with very long delays. (b) Consistent with this, long delays increase the expected LoS and give this protocol performances close to empirical treatment (shown are LoS means ± 95% CI from 1000 model simulations).