**DOI:**10.1128/AAC.00398-07

## ABSTRACT

The echinocandins potentially have an important role in treatment of infections caused by *Candida* spp. and *Aspergillus* spp. in immunocompromised children. However, there are no population pharmacokinetic models of the echinocandins for pediatric patients. The safety and descriptive pharmacokinetics of micafungin in children were recently reported. However, a population pharmacokinetic model in children is needed in order to accurately determine the dosage of micafungin that produces an equivalent magnitude of drug exposure to that observed in adults. In order to explore the effect of weight on micafungin pharmacokinetics, a standard two-compartment pharmacokinetic model, a linear model, and an allometric power model were developed. For all three models, the fit to the data was excellent, with comparable measures of precision and bias. However, the superior log-likelihood value of the allometric power model suggested that it best reflected the data and was therefore chosen for a more detailed analysis of the magnitude and pattern of drug exposure which develop following the administration of micafungin. The allometric power model suggested that clearance in smaller children is higher than that predicted on the basis of weight alone. Consequently, a degree of dosage increase is required in smaller children to ensure comparable levels of drug exposure to those observed in larger children and adults. The allometric power model developed in this study enables identification of pediatric dosage regimens of micafungin which, based upon Monte Carlo simulations, result in equivalent drug exposures to those observed in adults, for which antifungal efficacy has been established.

Invasive fungal infections in pediatric patients are associated with serious morbidity and mortality. Children with leukemia, hematopoietic stem cell transplantation, solid organ transplantation, low birth weight, and certain primary immunodeficiencies, such as chronic granulomatous disease, are at increased risk of invasive fungal infections (2, 5, 8, 12, 19).

The echinocandins represent a novel class of antifungal agents with in vitro, in vivo, and clinical activities against the medically important opportunistic fungal pathogens *Candida* spp. and *Aspergillus* spp. (10). The pharmacokinetics of echinocandins in pediatric patients have been studied in phase I trials (4, 13, 16, 17). However, there are no population pharmacokinetic models of the echinocandins for pediatric patients. The safety and descriptive pharmacokinetics of micafungin in neonates and children have recently been established in two phase I trials conducted in the United States (13, 16).

Important pharmacokinetic questions remain unresolved for the dosing of micafungin in pediatric patients. Specifically, what is the dosage of micafungin which produces an equivalent magnitude of drug exposure to that observed in adults, and what is the degree of expected pharmacokinetic variability? In order to address these questions, we first described the population pharmacokinetics of micafungin in children aged 2 to 17 years. We then compared three structural pharmacokinetic models, two of which explicitly incorporated weight as a covariate. These data provide further valuable information regarding appropriate pediatric dosing of micafungin and serve as a critical step in the optimization of antifungal therapy for invasive fungal diseases in this vulnerable population.

(Presented, in part, at the 46th Interscience Conference on Antimicrobial Agents and Chemotherapy, San Francisco, CA, 27 to 30 September 2006, abstr. M-299.)

## MATERIALS AND METHODS

Patient population and pharmacokinetic data.The safety and pharmacokinetic data following the administration of micafungin to children aged 2 to 17 years were collected in seven institutions within the United States and have been reported elsewhere (16). Briefly, micafungin was administered at dosages of 0.5, 1.0, 1.5, 2.0, 3.0, and 4.0 mg/kg of body weight/day as early empirical therapy for febrile patients with neutropenia. Micafungin was infused over 1 hour. Plasma was collected on days 1 and 4 at time zero (predose), 60 min (end of infusion), 90 min, and 2, 4, 6, 8, 10, and 24 h post-initiation of infusion.

Pharmacokinetic models and modeling.All data were examined using a population methodology with the use of the Big and Little population pharmacokinetic nonparametric adaptive grid with adaptive γ (NPAG) programs developed by Leary, Schumitzky, Jelliffe, and van Guilder (15). The inverse of the estimated assay variance was used as the weighting throughout the modeling process.

Three structural pharmacokinetic models were used in this study. The first represents a standard two-compartment pharmacokinetic model with time-delimited zero-order intravenous infusion and first-order elimination from the central compartment. The model is described by the differential equations 1a and 1b, below.
$$mathtex$$\[\frac{dX(1)}{dt}{=}R(1){-}\ \left(K_{\mathrm{cp}}{+}\frac{SCL}{V_{c}}\right){\cdot}X(1){+}[K_{\mathrm{pc}}{\cdot}X(2)]\]$$mathtex$$(1a)
$$mathtex$$\[\frac{dX(2)}{dt}{=}K_{\mathrm{cp}}{\cdot}X(1){-}K_{\mathrm{pc}}{\cdot}X(2)\]$$mathtex$$(1b) where *X*(1) and *X*(2) represent the amount of micafungin (in milligrams) in the central (*c*) and peripheral (*p*) compartments, respectively. R(1) represents the rate of infusion of drug into the central compartment (in milligrams per hour [not milligrams per hour per kg]). The central compartment has volume *V*_{c} (in liters), from which there is clearance SCL (in liters per hour). The central and peripheral compartments are connected by the first-order rate constants *K*_{cp} and *K*_{pc} (in hours^{−1}).

The second (equations 2a and 2b) and third (equations 3a and 3b) models, described below, were developed in order to further elucidate the effect of size (measured in terms of weight) on the pharmacokinetics of micafungin. The initial step in the development of these models involved an examination of the Bayesian estimates for clearance obtained from the standard model. These estimates were then plotted against weight, using both linear and logarithmic scales. Since both linear and logarithmic relationships appeared tenable (see Fig. 2, below), linear and power models which explicitly incorporated weight as a covariate were developed. The linear model took the following form:
$$mathtex$$\[\frac{dX(1)}{dt}{=}R(1){-}\ \left(K_{\mathrm{cp}}{+}\frac{SCL_{\mathrm{int}}{+}SCL_{\mathrm{slope}}{\cdot}\mathrm{weight}}{V_{c}}\right){\cdot}X(1){+}[K_{\mathrm{pc}}{\cdot}X(2)]\]$$mathtex$$(2a)
$$mathtex$$\[\frac{dX(2)}{dt}{=}K_{\mathrm{cp}}{\cdot}X(1){-}K_{\mathrm{pc}}{\cdot}X(2)\]$$mathtex$$(2b) where SCL_{int} and SCL_{slope} represent the intercept and slope of the linear relationship between weight and clearance, respectively; the other terms and relationships are the same as were described for the standard model.

Since a relationship between the log-transformed values of weight and clearance also was apparent, with the slope of the regression being 0.744 (see Fig. 2B, below), the performance of an allometric power model was also investigated. Such models have been widely used to investigate the effect of size on the pharmacokinetics of a variety of compounds, including analgesics and antimicrobial agents (3, 14, 18). Since the slope of the regression line of the Bayesian estimates of clearance and weight depicted in Fig. 2B, below, approximated 0.75 and this exponent has been extensively used to scale for size (7), the allometric scaling parameter in equation 3a assumed a value of 3/4. Furthermore, the pharmacokinetic parameters influenced by size (clearance and volume) were standardized to a 70-kg adult, as described elsewhere (1, 6). The differential equations describing this allometric model are as follows:
$$mathtex$$\[\frac{dX(1)}{dt}{=}R(1){-}\ \left[K_{\mathrm{cp}}{+}\frac{\mathrm{SCL}_{\mathrm{std}}{\cdot}\left(\frac{\mathrm{weight}}{70}\right)^{0.75}}{V_{\mathrm{std}}{\cdot}\frac{\mathrm{weight}}{70}}\right]{\cdot}X(1){+}[K_{\mathrm{pc}}{\cdot}X(2)]\]$$mathtex$$(3a)
$$mathtex$$\[\frac{dX(2)}{dt}{=}K_{\mathrm{cp}}{\cdot}X(1){-}K_{\mathrm{pc}}{\cdot}X(2)\]$$mathtex$$(3b) where SCL_{std} and *V*_{std} represent the normalized estimates for clearance and volume in a 70-kg individual, respectively; the other parameters are as previously described. One advantage of expressing pharmacokinetic parameters in this manner is that SCL_{std} and *V*_{std} should approximate adult values, thus providing a check for the model-fitting process (6). This allometric model was implemented and fitted using the Big NPAG population pharmacokinetic program of Leary et al. (15).

Model evaluation, comparison, and performance.For all models, the Bayesian estimates for each patient were obtained using the “population of one” utility within NPAG and Big NPAG. The population mean, median, and modal values were evaluated in the maximum a posteriori Bayesian analysis. For each model, scatter plots of the observed-predicted relationships for each individual patient and the population as a whole were examined. Goodness of fit was evaluated on the basis of a visual inspection and coefficient of determination of the observed-predicted scatter plot after the Bayesian step, as well as the log-likelihood values for each model. Models were compared with reference to the respective log-likelihood values, and statistical comparisons were made using the likelihood ratio test, where twice the likelihood difference was evaluated against a χ^{2} distribution with the appropriate number of degrees of freedom. Predictive performance was based upon the weighted mean error and the bias-adjusted weighted mean squared error.

Monte Carlo simulations.Monte Carlo simulations were performed using the allometric power model. The structural models were implemented within the simulation module of the pharmacokinetic program ADAPT II (9), and the full covariance matrix was inserted into subroutine PRIOR of ADAPT II. A subroutine within ADAPT II (courtesy of David D'Argenio, University of Southern California) enabled micafungin to be administered to each of the 9,999 simulated patients on a weight basis (in milligram per kg). For each simulated patient, the weight-based dose of drug in milligrams per kg was converted internally to an absolute dose of micafungin (in milligrams) by multiplying by the simulated weight. Thus, the simulation process mimicked drug administration as it occurred at the bedside in the original clinical trial, in which the dose of micafungin was planned on a mg per kg basis, but the absolute amount of drug administered to each patient was determined with reference to weight. To ensure consistency with the clinical trial, micafungin was infused over 1 h to simulated children. Both normal and log-normal parameter distributions were explored and discriminated on the basis of their ability to recapitulate the original parameter means and their dispersions. All simulations were performed at steady state between days 13 and 14 post-initiation of therapy. The area under the concentration-time curve from 0 to 24 h (AUC_{0-24}) was determined by integration.

## RESULTS

The mean ± standard deviation weight for children aged 2 to 17 years was 35.65 ± 18.66 kg. The pharmacokinetic data from the 72 children were initially examined using the standard model. The estimates for the mean and median values for each parameter and their standard deviations from this analysis are shown in Table 1, and the observed-predicted micafungin concentrations after the Bayesian step are shown in Fig. 1A. The fit of the standard model to the data was excellent, with an *r*^{2} of 0.960, along with acceptable measures of precision and bias (Table 2).

The relationship between the Bayesian estimates of clearance, obtained using the mean population parameter values from the standard model, and weight is shown in Fig. 2. Here, both linear and logarithmic relationships between clearance and weight appeared tenable, and these observations formed the basis for the development and comparison of the linear and allometric power models in which the effect of the incorporation of weight as a covariate on model performance was assessed. Importantly, the slope of the regression line in Fig. 2B approximated 0.75, which has been extensively used as an allometric scaling exponent.

The estimates for the mean and median values for the parameters from the linear and allometric power models are summarized in Table 1. For both models, the fit to the data was excellent (*r*^{2} of 0.953 to 0.9600) (Fig. 1B and C), with measures of precision and bias which were comparable to the standard model (Table 2). While the differences were relatively small, the better (more positive) log-likelihood values of the linear and allometric power models compared with the standard model suggested that the incorporation of weight as a covariate added explanatory power to the models. While the predictive performances of each of the three models were comparable, the allometric power model had the largest log-likelihood value, suggesting that it best accounted for the data; for this reason this model was used in the Monte Carlo simulations. The full covariance matrix used for these analyses is shown in Table 3. The mean parameter values and their standard deviations could be recapitulated with a 9,999-patient Monte Carlo simulation in which a log-normal distribution was used for each model parameter (Table 4). The validity of the allometric power model was further confirmed when the values for the constants normalized to a 70-kg adult from the allometric power model (*V*_{std} and SCL_{std}) fitted to the pediatric data set closely approximated estimates for the volume of distribution and clearance obtained from a separate adult data set (10.43 ± 5.60 liters and 1.17 ± 0.38 liters/h, respectively; T. Gumbo, submitted for publication).

The extent of the predicted variability in serum micafungin concentrations and the resultant AUCs within a simulated human population receiving 2 mg/kg at steady state are shown in Fig. 3. Micafungin concentrations varied approximately fourfold. The simulated AUCs conformed to a log-normal distribution (Fig. 3B).

Figure 4 demonstrates that as weight declined, progressively higher dosages were required to ensure an equivalent magnitude of drug exposure as that observed in larger children and adults. To achieve an equivalent mean AUC to adults receiving 100, 150, and 200 mg/day, pediatric patients required a dosage (in mg/kg) of 3.38 × weight^{−0.25}, 5.07 × weight^{−0.25}, and 6.77 × weight^{−0.25}, respectively. The requirement for higher weight-adjusted dosages was most apparent in children with weights of 10 to 20 kg: in order to achieve the same mean AUC as in larger children and adults, these children needed nearly double the dosage of micafungin.

## DISCUSSION

Population pharmacokinetics differ from conventional pharmacokinetic approaches by allowing true interpatient pharmacokinetic variability to be modeled. The behavior of a drug within a population can be summarized using measures of central tendency and dispersions for each model parameter (11). Using simulation techniques, this information can be used to provide predictions as to the likely behavior of a drug when administered to a large number of patients.

Population modeling of pediatric patients presents a number of challenges. Most important is the effect of size on drug disposition and elimination. The sizes of pediatric patients vary by more than an order of magnitude. To enable a more complete understanding of the importance of weight as a covariate, we modeled absolute (rather than weight-adjusted) dosages, volumes, and clearances. The improved log-likelihood values of the two models in which weight was incorporated as a covariate suggested that it accounted for a portion of the residual interpatient variability observed using the standard model (Table 2).

Allometry is the field of study which relates bodily function and morphology to body size (7). An understanding of allometric relationships is important, as adult pharmacokinetic models cannot necessarily be used to predict appropriate dosing regimens for children. An abundance of data suggests that organisms do not exhibit simple geometric scaling; this is a consequence of powerful biological constraints on structure and function that do not allow organisms to maintain the same geometric relationships as size changes (7). The 3/4 power law has been extensively used to model the relationship between physiological functions, such as metabolic rate and clearance, and size (14, 18). Body surface area represents a measure of size and has been used to guide pediatric dosing. Body surface area can be related to weight by using a 2/3 power scaling exponent (14). Predictions of clearance based upon body surface area and the 3/4 power model are similar and certainly superior to assuming that clearance is directly proportional to weight (14). In the current study, the decision to use a 3/4 power model, rather than a surface area model, was based upon two observations: first, many body functions scale predictably with an allometric 3/4 power model rather than with body surface area (7); second, the slope of the regression line in Fig. 2B better approximated 3/4 rather than 2/3. Furthermore, there is a theoretical basis for the allometric 3/4 power law which provides for a deeper understanding for the observation that clearance is proportionally higher in smaller children. There appear to be inherent limitations in the efficiency with which a progressively larger mass of tissue (in this case the liver) can be supplied with nutrients (or in the case of drug clearance, with the drug itself). Consequently, the rate of clearance in larger organisms is slower than predicted on the basis of tissue mass alone (conversely, the rate of clearance in smaller children is higher than predicted from data derived in adults). These concepts are presented in detail elsewhere (7). The nonlinear relationship between weight and clearance (resulting from inherent physiological constraints) means that as weight decreases, progressively higher dosages of micafungin (on a mg/kg basis) are required to achieve equivalent drug exposure; these increased dosages are higher than predicted on the basis of weight alone and are especially so in children weighing <10 to 15 kg.

A reasonable aim of pediatric dosing is to ensure levels of drug exposure which are comparable to those achievable in adults and which approximate those for which antifungal efficacy has been established. The current study demonstrates that smaller children require higher dosages of micafungin to ensure mean levels of drug exposure equivalent to those observed in larger children and adults. This can be achieved by increasing the dosage of micafungin at some (arbitrarily chosen) cutoff weight. In the current study, however, we were able to describe the continuous relationship between dose and weight and use this to precisely define antifungal dosing in pediatric patients. The Monte Carlo simulations also demonstrated that there is a degree of interpatient variability in serum drug concentrations and exposures following the administration of micafungin (Fig. 4); this observation may prompt an increase in dosage in the circumstance of a suboptimal therapeutic response despite the administration of a seemingly appropriate dose of micafungin.

In summary, the population pharmacokinetics of micafungin in children aged 2 to 17 years demonstrate the importance of considering and incorporating weight as a covariate in order to adequately describe drug behavior. The allometric power model developed in this study enables the identification of pediatric dosage regimens of micafungin that, based upon Monte Carlo simulations, result in drug exposure that is equivalent to that observed in adults, for whom antifungal efficacy has been established.

## ACKNOWLEDGMENTS

This study was supported by the intramural research program of the National Cancer Institute, Bethesda, MD. William Hope was funded by an unrestricted educational grant from Astellas Pharma Inc.

## FOOTNOTES

- Received 23 March 2007.
- Returned for modification 4 May 2007.
- Accepted 9 July 2007.
↵▿ Published ahead of print on 16 July 2007.

- American Society for Microbiology