The developed PBPK model focuses on the PK profiles of AMP/SBC in plasma and especially in jawbone tissue in the mentioned population in order to evaluate prophylactic treatment in maxillofacial surgery.

As far as we know, this work is the first to present a PBPK model for both AMP and SBC simultaneously in plasma as well as in tissue. The simulations within the model satisfactorily reproduce the clinically measured concentration versus time courses, particularly those in plasma, as can already be seen from visual inspection. This is objectively confirmed by the calculated FEs and AFEs for the PK parameters AUCtEND, Cmax, and t1/2 in the plasma data used, since these are all (with one exception) within the twofold criterion (see Supplementary Table S3 and S4).

Compared with other published PBPK models on aminopenicillins [14, 45], we did not include any hepatic metabolism process in our model, since this excretion, as already mentioned in Sect. 2.3, makes up only a very small part of the overall elimination process. In the PBPK model by Li et al. [14], which was developed using the Simcyp® population-based simulator, the authors deal with the exposure of AMP in fetuses and neonates. In the underlying adult PBPK model for AMP, drug elimination was modeled via additional processes compared with those in the approach presented here. In addition to glomerular filtration and the tubular secretion characterized there in more detail, they also included a bile clearance process and a hepatic metabolism process. As in our model, the Kp of the individual tissues was predicted using the in silico method according to Rodgers and Rowland.

In a sensitivity analysis carried out during PBPK model development via PK-Sim®, the hepatic elimination process was identified as having little influence on various PK parameters, as well as on the magnitude and dynamic shape of the concentration versus time curves of the individual simulations. This observation confirmed our decision to omit the hepatic process for the sake of simplicity. Accordingly, the elimination of the two substances in the present PBPK model consists of the passive process of glomerular filtration and the active process of tubular secretion, in accordance with the known PK properties of AMP/SBC [9, 46].

Published data [47, 48] show that AMP, as a typical representative of β-lactam antibiotics, does not or only negligibly diffuse into tissue intracellular spaces. Owing to the hydrophilic character of the substance, it cannot naturally penetrate plasma membranes. Investigations on the Kp of AMP and other β-lactam antibiotics between plasma and erythrocytes in rats confirm the inability to passively penetrate eukaryotic cells [49]. Therefore, the Kp (intracellular/plasma) was set to 0.01 in our PBPK model (see Sect. 2.3). Owing to the very similar PK behavior of SBC, the same proceeding was used here.

When looking at the generated concentration versus time profiles of the individual tissues (skin, lung, bone), which were part of the development process of the present PBPK model, it is noticeable that the measured mean tissue concentrations show a very high variability. In line with requests for better verification of PBPK models designed to predict tissue concentrations [50], the Kp values of tissues for which measured data were also available were optimized using a fitting tool (Parameter Identification Module of PK-Sim®) as a function of plasma and tissue concentrations. Not just on the basis of plasma measurements. The high variability between the individual measurements (between-subject variability) is probably due to the differences in the quantitative constitution of the individual tissue samples. For example, the skin samples (Wenzel et al. [26]) always contain a certain amount of subcutaneous fat, although the exact amount is not known, since the exact anatomical location of the sample is also unknown. This results in variability with respect to the histological composition between the samples of the individual subjects, which is more pronounced than in plasma samples. In contrast, the tissue sample can be contaminated by adhering blood, which ultimately leads to less accurate and less precise analysis results and predictions of tissue concentration. Contamination of the sample with blood residues usually leads to higher total tissue concentration measurements than actually existent. Furthermore, there are fewer sampling times than in regular PK studies with plasma as the specimen of interest and, in contrast to plasma measurements, the single measurement points in tissue all come from different subjects. Multiple tissue sampling from the same patient is very unusual.

To assess the effectiveness of antimicrobial therapy in the tissue of interest, a concentration versus time profile is inevitably necessary for antibiosis with β-lactams including AMP/SBC, as statements on effectiveness are based on the proportion of time within a dose interval at which the unbound drug concentration exceeds the MIC of a specific microorganism (ƒT>MIC). Although the profiles could be predicted in the present PBPK model for the three tissue types mentioned, it could only be verified with a single time measurement (often taken shortly after application) per subject in contrast to plasma profiles. This increases the uncertainty of the prediction of tissue concentrations, especially regarding the magnitude and shape of the concentration versus time curve in both the initial distribution phase and the elimination phase [50]. Therefore, it remains inconclusive whether the simulated tissue concentration versus time profiles actually resemble thatsuggested by Figs. 2d–f and 3d–f.

When focusing on the bone samples, it must also be noted that these do not represent a homogenous matrix. The differences in the exact composition between the subjects can vary considerably, since the samples consist of a certain proportion of cancellous bone and a certain proportion of cortical bone with different densities and bone penetration properties regarding the antibiotic substances. Since we only used jawbone samples from our own clinical study [24] to develop the model, it can be assumed that cortical bone was present for the most part, since the outer region of the mandible, where most of the samples come from, consists mainly of this type of bone [51]. Therefore, the density of cortical bone was also assumed for the model evaluation (see Sect. 2.1). A much greater variability of the measurements is most likely because both necrotic and vital mandibular tissue were analyzed and incorporated into the PBPK model building. The investigations carried out [24] showed that the antibiotic concentrations between necrotic and neighboring vital bone did not differ significantly from each other, but no distinctions were made with regard to the extent of necrosis (larger or smaller proportion of dead tissue in the sample). The sample was simply denoted as “necrotic” without further subclassification. This could probably have contributed to the high variability, since the exact proportions of necrotic bone in the sample differ between subjects, ultimately resulting in a large number of individual jawbone tissue matrices that are more or less well perfused, depending on the severity of the necrosis. Accordingly, anti-infective agents can be deposited to a greater or lesser extent and subsequently quantified via bioanalysis [52].

Of the tissue studies, PK parameter predictions could only be compared with observed PK parameters and FEs calculated for Wetzel et al. (skin) and Frank et al. (lung). The jawbone study (Straub et al.) did not allow for a reasonable estimation of the PK parameters considered (see Supplementary Table S3 and S4) owing to the tissue sampling over a very short period of time (7–75 min after the end of the infusion administered before incision). Furthermore, the study of Straub et al. was originally designed only to evaluate the concentrations of AMP/SBC in the jawbone at the time of surgery to assess whether sufficiently high jawbone concentrations for effective prophylaxis could be achieved with an established antibiotic regimen.

As can be seen from Supplementary Tables S3 and S4, the predicted-to-observed PK parameter ratios of the tissue examinations mostly fall outside the twofold acceptance range, which may presumably be due to the high inter-individual variability of the measured concentrations in the tissue, as just discussed. Thus far, there is little agreement in the literature on which acceptance range is appropriate. As van der Heijden et al. suggest [38], it may be appropriate to assume a wider range than the commonly applied twofold range. In particular, a less narrow range should be considered for drug exposure scenarios in which a high inter-individual variability prevails, as is also the case with tissue exposures to AMP/SBC considered here.

The goodness-of-fit plots (Fig. 4a, b) indicate that all plasma data have a good fit (within the twofold deviation lines). Furthermore, all predicted lung and skin concentrations are within the twofold limits. All other predictions outside the set limits refer to the concentration measurements in jawbone. It should be noted that the jawbone data points are individual, whereas the other tissues and plasma data points are mean data.

As can be seen in Figs. 2d and 3d, the individual bone samples were taken within a single time interval after the end of the last infusion (point cloud), and not at one point in time. Thus, we did not consider it appropriate to combine the individual measurements into one or more mean values here. Even though this might have led to the predictions of the bone tissue concentrations being within the twofold range. Similar to a previous PBPK study by Garreau et al. [53], which examined the exposure of skin and bone tissue to the reserve antibiotic daptomycin, our study also showed that the largest deviation from the prediction exists for bone tissue. In a further PBPK analysis by de Sutter et al. [50], the predictive performance of a model developed by them was evaluated for cefuroxime exposure in bone in relation to observed bone concentrations from Tottrup et al. [54]. In both the daptomycin and cefuroxime studies, the biological samples were obtained in the form of microdialysis samples. Therefore, the samples represented the interstitial fluid of the corresponding bone tissue, which was examined. As with the other study, the authors observed a higher deviation from the usually applied twofold criterion. This raises the question of whether the twofold criterion might be too strict for predictions for bone tissue and whether, as suggested by de Stutter et al. [50], a threefold range could be acceptable for PBPK predicted bone tissue exposure.

The simulations in plasma show that the selected PK/PD target of 50% ƒT>MIC is exceeded by far with the usual 2 g application of AMP given as a short infusion. Applying the dosing regimen used in clinical routine (2 g/1g AMP/SBC as a single infusion or as an infusion q8h), it can be assumed that even a more conservative target of 100% ƒT>MIC is achieved in all populations. This would also apply to a 1g AMP application over a 15-min infusion period, as suggested by the population simulation for Wildfeuer et al. [22] (see Fig. 2i). However, there is a lack of real-world human data from tissue samples that could support the thesis for the 1 g AMP dosing. The selected PK/PD target of 50% ƒT>MIC is set relatively low compared with other targets proposed in the literature. However, in terms of perioperative infection prophylaxis, it seemed appropriate to the authors, since stricter targets are rather applied for therapeutic/curative purposes, in critically ill patients, or in generally critical patient populations, e.g. populations with low immune system functionality. Furthermore, owing to the prophylactic nature of antibiotic administration in this case, we have chosen a target that is close to the lowest target available in the literature for the treatment of a manifest infection. Since there was no infection in the surgical area at the time of the ONJ surgery, or at least not necessarily, and since we do not consider the ONJ surgical population to be critically ill (intensive care patients), this reinforced our decision to set the target value of 50% ƒT>MIC [12, 13].

If the prediction of AMP in bone tissue is consulted, an exceedance of the MIC for at least 6 h in the population can be safely assumed on this basis alone. The target of 50% ƒT>MIC is achieved in any case. As mentioned earlier in this section, observed data to verify bone tissue concentration predictions are only available over a limited period of time at the beginning of the dosing interval and the simulated courses at later time points are therefore subject to greater uncertainty. Based on the observed data, it is clear that some individuals or a small proportion of the population may not reach the PK/PD target with respect to bone tissue. This holds particularly true if surgery times are extended due to complications or in the presence of extensive necrotic areas. In this case, it may be appropriate to administer an additional dose intraoperatively. The study by Straub et al. [24] used in the present PBPK model only includes concentration measurements from patients whose bone sample was obtained no later than 90 min after administration of the last infusion of 2 g/1g AMP/BSC. Therefore, there were no lengthy surgeries, which is why intraoperative redosing was not necessary. As described elsewhere [55, 56], continuous infusion could possibly provide more consistent plasma levels and thus more stable concentrations in the target tissue, which would lower the risk of not reaching the PK/PD target values. However, it should be noted that this approach is difficult to implement in routine clinical practice, where treatment of ONJ is predominantly provided on an outpatient basis.

Looking at the simulations carried out with populations with renal impairment, it can be deduced from this prediction alone that a less excessive dosing regimen than that commonly used in clinical practice (namely 2 g/1g AMP/SBC as an infusion q8h) will most likely be sufficient to exceed the selected PK/PD target of 50% ƒT>MIC in both plasma and bone tissue. For this reason, the authors recommend a reduced AMP/SBC dose of 1 g/0.5 g, administered as a single infusion or q8h, especially in patients with an eGFR of 30–60 mL/min/1.73 m2. This recommendation is consistent with the results of previous studies [10, 57] that have proposed dosage recommendations on the basis of population pharmacokinetic models. However, when interpreting the results, we would like to point out that the simulations could not be verified with observed data and may therefore be subject to considerable uncertainty.

To the best of our knowledge, the reported work is the first PBPK model to present simultaneous predictions for both of the usually coadministered substances AMP and SBC in plasma and in various tissues. Within this study we were able to circumvent to some extent one of the major criticisms and limitations of previous PBPK models dealing with the prediction of tissue exposure after antibiotic administration by verifying all model predictions with corresponding observed human data from the associated tissue (jawbone, skin, lung). Many of the tissue predictions from previous PBPK models were only verified with human plasma data or animal tissue data [39, 49, 58,59,60], which, however, is largely due to the lack of pharmacometrically exploitable tissue concentration data. Likewise, the observed data from the tissue concentration studies [24,25,26] were also used in model development in addition to the plasma concentrations to optimize corresponding Kp values using the PK-Sim® Parameter Identification module. In previous PBPK models, this was only done with plasma concentrations as input data [50]. The PBPK model developed here should help clinicians to assess whether effective antibiotic prophylaxis with AMP/SBC can be achieved in typical patients undergoing maxillofacial surgery using the conventional administration modalities employed in clinical practice. This is determined on the basis of the model predictions in plasma and especially bone tissue.

Despite some improvements in the modeling process in terms of verification, there are still some limitations, and some questions of interest remain in whole or in part unanswered. Firstly, our PBPK model also assumes that the PK/PD targets associated with the microbial outcomes are valid for both plasma and the tissue types discussed, which is not necessarily the case. In addition, it remains unclear how intensively the investigated substances AMP/SBC bind to bone tissue, since only unbound drugs can have an antimicrobial effect, whereby studies suggest that β-lactam antibiotics hardly bind to bone powder [61]. Data from microdialysis samples, which would represent the unbound substance concentrations in the interstitial fluid, are missing. However, it should be mentioned here that the microdialysis technique for sampling bone tissue also has its disadvantages, such as the necessary hole that must be inserted into the bone. This, can in turn, fill with blood and extracellular fluid, so that measured concentrations can be falsified, since they reflect concentrations in the resulting dead space or the concentration of interstitial fluid from the neighboring distinct tissue [52]. The PBPK model presented here is designed for populations undergoing maxillofacial surgery due to ONJ. Since these patients are almost exclusively middle-aged or elderly, the model was developed using PK data from appropriate studies with compatible demographics. Given that the elimination of AMP/SBC depends largely on renal clearance, elimination in the considered population is naturally reduced when compared with young, healthy populations. Accordingly, the half-life of the substances increases with age [6]. The formulated PK/PD targets are thus already achieved at a lower dose, especially in the elderly population. In young patients, who are rarely treated in the wake of ONJ in practice, there might be a risk that plasma or tissue levels will be below the decisive MIC for a comparatively longer period of time due to the shorter half-life of AMP and SBC. In this case, a shorter dosing interval could be reasonable in the event of longer surgery duration.

The dosing interval would also have to be shortened with regard to the simulated bone tissue concentrations if other, less sensitive germs than the most common SSI germs in the course of maxillofacial surgery, such as Streptococcus spp. and Staphylococcus spp. (with MICs ≤ 0.5mg/L) are involved (e.g. Haemophilus species where the susceptibility MIC break point is ≤ 1mg/L, or in the presence of MSSA with a MIC break point of 2 mg/L) [42, 43]. It is also largely unclear whether a PK/PD target for SBC should be defined, since it is not known at which concentration the antimicrobial spectrum-expanding properties of SBC are lost in vivo in the context of prophylaxis. Assuming a minimal critical concentration (MCC) threshold of 4 mg/L (the value is based on the fact that this concentration is currently used as a break point concentration in susceptibility testing), as in a population PK study by Reeder et al. [10], it becomes apparent (see Fig. 3d) that this concentration is hardly ever reached in bone tissue, regardless of the time after administration of 2 g/1g AMP/SBC. Furthermore, as our investigations of AMP/SBC concentration ratios in tissue demonstrate (see Sect. 3.4), a ratio of 1.0–2.0 between the two substances, which previous studies indicate as the most effective antibacterial activity [57], is only achieved in skin tissue. The ratios are clearly shifted towards AMP, particularly in bone, which raises the question of whether this deviation in concentration ratios can lead to a reduction in prophylactic effectiveness and consequently to a higher susceptibility to SSIs.

What remains at the end, however, is the question of actual, measured concentrations in the (jaw)bone tissue in the middle and terminal part of the dosing interval. It would be desirable to have further measured data available at later time points after the end of the last preoperative infusion. An obvious consideration would be to start the surgery at a later time after the end of the last infusion in order to generate more data in the terminal phase of the profile. Since the operations often take place in an outpatient clinical setting, this is difficult to implement in practice, and artificially prolonging the time between the end of the last preoperative infusion and the end of the surgery is unreasonable for the patient and cannot be justified from a medical and ethical point of view. Thus, the determination of concentration data in the middle and terminal part of the dosing interval will probably also not be possible in the future. Here, PBPK models such as the one presented within this work, can provide valuable assistance. However, the question and the necessity of PK/PD target implementations in the context of perioperative infection prophylaxis [61] is paramount to be able to make reliable statements about the effectiveness of applied prophylaxis schemes with the help of models. With regard to the first question it seems justified to re-dose early when in doubt in order to ensure sufficiently high tissue levels, given the wide therapeutic index and good safety profile of beta-lactam antibiotics even at high doses [62]. This is particularly the case if the start of the operation is delayed or the duration of the operation is extended, for example in the case of extensive necrotic bone areas or large-scale resections. Finally, it is not entirely clear which compartment concentrations are actually relevant for effective infection prophylaxis in ONJ surgery. However, the authors of this manuscript are convinced that plasma concentration, bone concentration, and, to a lesser extent, concentration in skin tissue will be of particular importance here. Since we do not know exactly how the active substances behave in bone (binding to bone tissue/bone components), how high the unbound concentration will be as a result, and which PD effects could be limited as a result, it remains difficult to assess whether falling below the MIC in bone automatically indicates ineffective prophylaxis with the AMP/SBC combination.