As mentioned in the background, recently introduced PK/PD-based breakpoint estimation was put forward to overcome drawbacks of threshold criteria, namely MIC, which determines in vitro antimicrobial efficacy. However, these PK/PD-based methods use drug exposure mainly through the AUC value (the amount of drug absorbed), whereas the variability in drug concentration time course is not integrated. This variability turns out to be an important factor in drug efficacy, as widely reported 27. In bioequivalence studies, for example, it is common to combine AUC and Cmax to compare PK profiles and thus indirectly assess the expected drug efficacy. Therefore, to rely solely on the use of these PK parameters may not be sufficient for drawing reliable conclusions on drug efficacy. To employ these parameters efficiently and optimize their use for specific purposes, we need to adapt them by adding more information on drug PK/PD properties.

AUC-based drug efficacy is generally assessed through statistical methods such as scatter plots. Since PK/PD properties are not fully exploited, the relationship between drug efficacy and PK parameters only partially reflects the pharmacological properties. If additional PK/PD properties can be accounted for, the capacity of the actual empirical PK/PD-based breakpoint estimation is likely to be improved. Ideally, when a PK/PD relationship can be determined in vivo, the power of drug efficacy prediction can be maximized. However, exact dose-response relationships under in vivo situations are not easily accessible. This is the main restriction that prevents full exploration of drug efficacy prediction. Alternatively, combining the in vitro efficacy function (E) – the PK/PD relationship measurable in the laboratory – with AUC provides a better relationship (being more information-loaded) than that of drug efficacy in terms of AUC. As we will see, this combination can be considered an extension of the definition of AUC, thus relating to specific information on drug response.

In the case of antibiotics, dose-response or concentration-response curves against a microbial agent, also called killing or growth inhibition curves, can more easily be established under in vitro conditions. Several functions, such as linear, sigmoid or logistic, can be used to describe drug efficacy 28293031. For example, consider drug efficacy E as a probability function expressing inhibition of bacterial growth in response to antibiotic concentrations. It can be modeled as:

where Emax is the maximum effect (normalized to one in this paper), EC₅₀ the drug concentration that attains 50% of Emax, and H is the Hill constant 31. Since this efficacy function carries rich information about the response in terms of concentration, it should and could be translated under in vivo conditions. In fact, the in vivo situation can be considered as a composite of many "local" in vitro cases. "Locally in vitro" here means that once the antibiotic reaches a certain site in the body (a target organ for example), it behaves in a similar way as in vitro. In the following, we will include this efficacy function E in our approach to predicting the drug's therapeutic performance and apply it to the case of concentration-dependent antibiotics.

To evaluate the performance of a PK profile, we chose to measure it by the expression efficiency, Eff, defined as follows:

where again E is a function related to drug efficacy, T is the therapeutic duration used as a reference period and n = 0, 1, ...

Compared to AUC, E here plays the role of a weighting function. We use it to include the information on the PK/PD relationship as an integral part of drug efficiency measurement expressed through Eff. This information can always be updated and integrated for this purpose. For the particular case of E = 1 and n = 1, we obtain the usual AUC definition, thus making our newly introduced efficiency function a direct extension of AUC.

As an illustration, we will show how the newly introduced efficiency function can differentiate between PK profiles with the same AUC. In Figure 1, the two PK profiles share the same AUC but noticeably different AUC_W. In fact, this additional information level allows drug evaluation and assessment of therapeutic performance to be refined.

As mentioned, the effects of concentration-dependent antimicrobial agents are known to be proportional to concentration. Their efficacy is generally assessed through pharmacokinetic parameters, namely AUC or Cmax. To characterize the efficiency of concentration-dependent drugs, we propose to use the first order version of the efficiency Eff:

We notice that Eff₁ contains information related to both AUC and concentration variation levels. In this newly proposed formula, these two elements are well integrated to reflect their contributions to the evaluation of drug performance. Eff₁ can thus be considered an extension of the classical approach 14. We refer to Eff₁ as the weighted AUC and denote it by AUC_W.

The efficacy of a time-dependent drug depends on the percentage of time during which the concentration exceeds a specific value C_BP, generally called the breakpoint. C_BP acts as a threshold value: the drug is considered to be fully effective when its concentration is over this value, but non-effective otherwise (Figure 2). For time-dependent drugs, we formulate the efficiency as:

where E = χ is the indicative function. We recall here that the indicative function χ_A of a set A is defined as: χ_A (t) = 1 if t belongs to A; 0 otherwise. Hence, expressed in this way, χ will be 1 if C(t) > C_BP and will be 0 otherwise. We notice that Eff₀ is simply the cumulative time during which C(t) remains above the specific concentration value C_BP, which turns out to be exactly the same classic definition for evaluating time-dependent efficacy. However, expressing it in this way, with explicit reference to the zero-order general efficiency Eff_n function proposed above, helps us to understand the direct relationship of efficiency for different drug groups.

In pharmacology, estimation of drug efficacy is important for optimizing a drug regimen such that the best therapeutic outcome can be achieved. Generally, this estimation should be performed under in vivo conditions. Since drug concentrations within the body are unavoidably variable, and in vivo-induced randomness may also be superposed, in vivo estimation of drug efficacy is a complex problem. Microbiologists use in vitro-based methods for estimating antibiotic efficacy. These well-controlled in vitro studies can result in useful partial predictors for the in vivo potency of drug-microorganism interactions. Very often, efficacy-drug concentration curves are well established in vitro. This information makes it possible to establish a certain rule for efficacy equivalence between different real PK profiles such that the efficacy of a drug regimen can be objectively judged. Based on the efficiency function introduced above, two PK profiles are ascertained as efficiency-equivalent, i.e.PK₁ ⇔ PK₂ in efficiency, if and only if they verify the condition:

More precisely, for concentration-dependent drugs, we can try to find a corresponding in vitro (constant) equivalent concentration C^e that is likely to produce the same efficiency provided by a given PK profile. In this case, for a given PK profile C(t), the corresponding equivalent in vitro concentration C^e is the solution of the equation:

For time-dependent drugs, the situation is different. The efficiency is the percentage of time during which the concentration remains above a specific value C_BP. As in vitro concentrations can only have binary efficiency, we have to determine the threshold of time percentage for an effective drug regime. The efficiency of a PK profile can be compared with this threshold to measure its efficacy.

As an application of the AUC_W method, we will consider the case of variability in PK profile generated by irregular drug intake. It is common sense that a deviation between real drug intake and the ideal prescribed dosing regimen is likely to have an impact on the pharmacokinetic profile and eventually the drug response. Non-compliance characteristics can be translated into some derived PK/PD parameters and pharmacological indices.

In a previous study, we investigated the impact of animal feeding behaviour on the pharmacokinetics of chlortetracycline (CTC), a widely-used antibiotic usually given through animal feed 32. We modeled a widely reported animal feeding behaviour and associated it with the CTC disposition model to obtain a feeding behaviour-PK (FBPK) model. Using this model, we revealed the PK variability induced by random drug intake and assessed its main characteristics 32.

In the present paper, we will focus on the estimation of efficiency of CTC in this particular context of irregular drug intake. We have to mention that similar reasoning and analysis can be accomplished using other sources of variability impacting pharmacokinetics. Since CTC is a concentration-dependent antibiotic widely used in collective medical therapy, we will base our analysis on the method we propose for this antibiotic class. For the purpose of illustration, we will use the individual FBPK we previously developed 32. The case of an animal population can be developed by adding the inter-variability to the associated PK parameters. In the following, we will answer the following questions: Can the "efficacy performance" of PK profile be characterized uniquely by its average concentration value? What is the extent of in vitro equivalent concentrations that an average concentration can reach? Since we only have access to the drug concentration in feed, what can we say about the potential efficacy of various drug concentrations in feed compared to that of MIC?

In veterinary medicine, the problem of optimal use can arise for drugs administered through feed, a widely-used practice for therapeutic, metaphylactic or prophylactic treatment of bacterial infections 33. As a consequence, animal feeding behaviour directly influences systemic exposure to drugs. However, variation in the feeding behaviour of animals medicated through feed has been overlooked for more than 50 years, during which feed antibiotic therapy remained empirical. Using widely-reported descriptions of swine feeding behaviour, we have mathematically formulated and integrated this behaviour model into a PK model (FBPK) in order to analyze its influence on systemic exposure to drugs quantitatively 32. We include here a brief review of the FBPK model. Complete details about the model and its analysis can be found in 32.

The feeding behaviour model consists of two typical daily feeding activities: routine peak periods complemented by inter-peak periods of free access to feed. The routine peak periods correspond to intense feeding activities generally referred to as morning and afternoon peaks. Meals consumed between peak periods are referred to as inter-peak meals. The time intervals between two successive inter-peak meals are reported to follow a Weibull distribution. Since the animal consumes the feed in a quasi-continuous manner during the peak periods, and considering the low elimination rate of CTC, we have modeled the feeding activity during these periods as an infusion process, which gives rise to the following concentration time-course:

where [T_s, T_e] is the duration of the peak period, DOSE is the drug concentration mixed in the feed, with units in ppm, is the average ingestion rate, F is the bioavailability, K_a and K_e are the absorption and elimination rates respectively, and V is the volume of distribution.

Inter-peak meals are modeled as individual boluses entering the gastrointestinal tract because their durations are relative short compared to the inter-meal intervals. A two-parameter Weibull distribution is used to account for these irregular feeding events of free access to feed. Figure 3 illustrates a typical PK profile of an animal receiving 500 ppm of drug mixed through feed.

By definition, MIC breakpoints refer to critical drug concentrations that characterize specific antibacterial activities. The values of these MIC breakpoints are highly pertinent to the pharmacokinetic properties as well as to the pharmacodynamic killing capacities of these drugs with respect to particular bacterial strains. In the clinical setting, MIC is considered an important reference index in choosing effective dose regimens. However, because of the evident large variation in concentration time course and the unavoidable pharmacokinetic variability under the in vivo situation, the true PK/PD relationship is generally more complex. Using a single static value of MIC for the decision process is dubious or even misleading. Therefore we have to take account of dynamic in vivo properties when estimating drug efficacy.

In the following, we will use the above-developed feeding behaviour-PK model to show how one can obtain breakpoint information, and of what kind, for an in vivo situation.

To do this, we adopt a Monte Carlo approach to generate, for an animal X, possible drug inputs prior to drug disposition. The corresponding concentration time courses are then produced with these drug inputs. To explain our approach, we need to introduce some new concepts and their notations.

• DOSE: drug concentration mixed in feed, with units of ppm.

• : average over a time duration T of one concentration time course generated by Monte Carlo; it is AUC-based.

• : global mean of all average concentrations .

• : 95% higher mean concentration where 95% of are below this concentration.

• : 95% lower mean concentration where 95% of are above this concentration.

• : in vitro equivalent concentration (Eq. 2) of C_i(t), where 0 ≤ t ≤ T; it is AUC_W-based.

• : global mean of all in vitro equivalent concentrations .

• 95% higher in vitro equivalent concentrations , where 95% of are below this concentration

• : 95% lower in vitro equivalent concentrations , where 95% of are above this concentration.

Using the FBPK model, we can estimate the above concentrations versus DOSE (Figure 4). This figure shows the 95% confidence intervals of in vitro equivalent concentrations and average concentrations in terms of DOSE. For example, given a DOSE = 400 ppm, we obtain [, ] = [0.417 mg/L, 0.450 mg/L] and

[, ] = [0.397 mg/L, 0.435 mg/L].

We can consider that a DOSE is at least 95% efficiency-equivalent to an in vitro concentration C_eff by defining 95% of equivalent in vitro concentrations generated by this DOSE as being above C_eff. In our case, for a given DOSE, we have the relationship C_eff = (Dose) according to this 95% efficiency-equivalence criterion.

However, with each DOSE, we can also associate a 95% confidence interval of average concentrations represented by [ (Dose), (Dose)] as illustrated in the right panel of Figure 4.

Then for each at least 95% efficiency-equivalent in vitro concentration , it corresponds an interval of average concentrations [, ]. This clearly indicates that under in vivo situations, we have an associated uncertainty in average concentrations that may correspond to the same specific PK efficiency value. In other words, the in vivo average concentration when used as a breakpoint to indicate the efficacy of a dosing regimen can only be interpreted probabilistically. This result is reported in Figure 5.

To a given target value Ce (in vitro target), there corresponds a DOSE that gives an interval of equivalent concentrations (hence equivalent efficacy) lying above Ce.

However, a given average concentration , which is in fact measured theoretically (using AUC for example), may be the result of many different DOSEs. We can write this corresponding interval as [DOSElow, DOSEhigh] as a function of . For DOSElow, the lowest in vitro equivalent concentration that can be attained by in the sense of 95% probability will be given by (DOSElow). The same applies to DOSEhigh, where the highest in vitro equivalent concentration that can be attained by in the sense of 95% probability is given by (DOSEhigh). Hence, for each , there is a corresponding whole interval of possible in vitro equivalent concentrations given by these two extreme values and denoted by [ (DOSElow), (DOSEhigh)]. This result is reported in Figure 6. The illustrated (one-to-one) relationship between and DOSE highlights the possibility (need) to dissociate between the average concentration and efficacy, thus questioning the general practice of evaluating efficacy through average concentrations.

To answer the third question, we consider a MIC = 0.5 mg/L, which is the breakpoint normally used in practice for the evaluation of CTC efficacy. For different values of DOSE, we estimate the probability of the in vitro equivalent concentrations with values above MIC. A plot of these probabilities versus DOSE is given in Figure 7. We can see that for low DOSE values, it is almost certain that the therapy is non-efficient while the opposite is the case for high DOSE where success is almost secured. However, there is a critical zone of drug concentration in feed (DOSE) within which a given DOSE has a certain potential of success or failure.

Here, we will explain and illustrate some advantageous properties of AUC_W compared to AUC. In its integration formula, the AUC_W method incorporates the in vitro efficacy function E, thus penalising lower drug concentrations in an appropriate way. Hence, AUC_W constitutes an improvement over AUC since the nonlinearity principle in drug efficiency is respected (Figure 8, right panel). Also, AUC_W proves to be robust in terms of the efficacy function E, which represents an important feature when it comes to application. Indeed, we have generated AUC_W for three efficacy functions, namely the linear, Emax and logistic functions. These functions along with the corresponding AUC_W are plotted in Figure 8, left and right panels respectively. For sake of comparison, the AUC is also depicted on the right panel. This figure shows that the difference between AUC and AUC_W is more noticeable than that of the three generated AUC_Ws.