EXAMPLES OF POPULATION PHARMACOKINETIC MODELING IN DRUG DEVELOPMENT USING NONMEM

The population approach to pharmacokinetic analysis, and its application to the identification of patient characteristics that affect a drug's pharrnacokinetic parameters, is achieving greater prominence in the drug development process. Specifically, population analyses are a way to gather information that might be difficult to capture in some subpopulations. In the fall of 1997, the Food and Drug Administration proposed new legislation, commonly known as the ''Pediatric Rule". This new legislation required pharmaceutical companies to collect pediatric data for drugs with indications applicable to children before the compound would be approved. Other than conducting traditional pharrnacokinetic clinical trials, another way to collect this information would be to perform a population pharrnacokinetic analysis. Two different examples ofthis approach are presented. The first study was conducted on traditional pharrnacokinetic data (intense sampling) pooled from four pediatric trials. The second study is an example of the ability of the population approach to take advantage of sparse data obtained as a secondary objective of a clinical study. A population pharrnacokinetic analysis was conducted for azithromycin on data from pediatric patients enrolled in four separate clinical trials. A two compartment model with parallel zeroand first-order absorption was found to best fit the data. Potential covariates were assessed for oral clearance (CL/F), oral volume of distribution in the peripheral compartment (V2/F), intercompartmental oral clearance (Q/F), and the firstorder absorption rate constant (ka). Weight was found to be a significant covariate for both CUF and V2/F. No covariates were found to be significant for Q/F or ka. A population pharrnacokinetic analysis was conducted for prednisolone on data from thoracic organ transplant patients. A one compartment model with a fixed first order rate of absorption was found to best fit the data. Potential covariates were assessed for oral clearance (CUF) and oral volume of distribution (V/F). Sex and concomitant ciprofloxacin use were found to be significant covariates for CUF. No covariates were found to be significant for V IF. Data was also available on plasma concentrations of prednisolone' s metabolite, prednisone. It was not possible to derive a robust and clinically meaningful model that incorporated the metabolite data.


MANUSCRIPT ID
The need to modify the usual dose of a drug in certain populations is determined by comparing the pharmacokinetics of the sub-group to the population as a whole. There may be so many different sub-populations that it is often unrealistic to run a separate clinical trial for each group. Many times, dosages are adjusted empirically across subpopulations; using either clinician experience or assuming dose proportionality with either body weight or age. These empirical approaches increase the tendency for serious adverse events or sub-therapeutic concentration levels (2;3). In order to address the inadequate dosing information, many researchers have been focusing on new approaches to pharmacokinetic analyses and model building.
Originally, pharmacokinetic modeling concentrated on the individual For example, in a traditional standard two stage analysis (STS), a clinical trial is comprised of a small number of subjects from whom a large (12)(13)(14)(15)(16)(17)(18)(19)(20) number of serial blood samples are collected over a dosing interval. Trials are restricted to representative subjects from a particular population to limit variability between subjects. An analysis of this type of data is done in two stages. For the first stage, plasma concentration time data are modeled using nonlinear regression to produce estimates of the pharmacokinetic parameters. For the second stage, the individual pharmacokinetic parameters are combined and descriptive summary statistics are computed (e.g. group mean and group variance). Analysis of the dependencies between the parameter and any covariates use a classical statistical approach (stepwise linear regression, cluster analysis, etc.) (1 ;4;5).
This type of analysis moves from an individual (unit of analysis) out to the population, and as a result, the parameter estimates are unbiased and the random effects are overestimated (I). There are several logistical issues associated with this approach, primarily revolving around the need to perform extensive blood sampling and homogeneity of the population ( 1 ;6). These reasons have led to an alternative approach known as nonlinear mixed effects modeling.
A second approach, nonlinear mixed effects modeling, is a way to directly study the population' s pharmacokinetics. Nonlinear mixed effects modeling is less stringent than a STS analysis; it allows for the use of sparse data (2 or more samples not necessarily from the same dosing interval per patient) from a large number ofrepresentative patients in the population (I ; 4;5;7). The population method pools aU data collected and calculates 2 population pharmacokinetic parameters (e.g. volume of distnbution). Additionally the focus of the analysis is on the source and correlation of variability in drug concentrations among individuals in the population. Thus, population pharmacokinetics focuses on the target population (unit of analysis) and moves out to the individual. Population analyses also provide quantitative estimates of both the interindividual and intraindividual variabilities of the population (4;5). Interindividual variability may be accounted for by adding specific patient characteristics into the population model. Patient characteristics that cause changes in the dose-concentration relationship can be identified and assessed and then appropriate dosing modifications can be determined (1).
Nonlinear mixed effects modeling will be performed using a software package called NONiinear Mixed Effect Model (NONMEM) version 5 level I. I. NONMEM is suitable to analyze these types of data and has been extensively utilized by others (8;9). Both fixed and random effects are modeled using NONMEM. Fixed effects (e.g. time or dose) structure the actual pharmacokinetic parameters (structural portion of model).
Random effects are comprised of random interindividual variability (unexplainable error produced by each individual's variability not accounted by the fixed effects) and intraindividual variability (explainable error accounting for the difference between actual and predicted concentration values) in the pharmacokinetic parameters (statistical portion of model). NONMEM provides estimates of both inter-and intraindividual (i.e. residual random error) variabilities in the pharmacokinetic parameters (4;7;9).

HYPOTHESIS TESTED
To date, there are no published population pharmacokinetic models for azithromycin in the pediatric population and prednisolone in organ transplant patients. For azithromycin, small clinical trials have been conducted in pediatric patients to determine alternative dosing regimens. The drug's label includes results from some of these trials and only has information on adjusting dose by weight (kg). For prednisolone, it appears that a standard dose produces a large variability in prednisolone concentrations. There is little information on the cause of this variability and on what adjustments should be made to doses in certain sub-populations. For the use of prednisolone in organ transplant patients, it is critical that an optimum prednisolone concentration be achieved. It has been shown that a patient with a higher prednisolone clearance is more likely to suffer an allograft loss, while a patient with high prednisolone concentration levels (i.e. low prednisolone clearance) is more likely to suffer from adverse events (2;10) .
The hypothesis to be tested in this investigation is that the population pharmacokinetic modeling approach can be used to evaluate and describe the concentration time data collected in the azithromycin and prednisolone clinical trials. Using this approach, precise est·imates of the pharmacokinetic parameters and their variability will be quantifiable and significant covariates will be identified.

OBJECTIVES
The specific aims ofthis dissertation are as follows:

Post 1994 Regulations to Present
FDA states clinical studies in the pediatric population have been conducted for only a small fraction of drugs currently on the market. The labeling of many of these drugs contain limited, if any, information on either the use of the drug in the pediatric patient or on specific dosing requirements for the different pediatric age groups. The FDA conducted a survey and found that although there was adequate pediatric labeling for vaccines and antibiotics, the labeling for many drugs used to treat common childhood illnesses and other more serious conditions, contained little information fo r pediatrics.
From data collected by IMS America, Ltd. regarding prescription drug usage, FDA compiled the I 0 most prescribed drugs in pediatric patients, on an outpatient basis (Table   I). For these I 0 drugs, FDA claims the label either lacked information for the subpopulation for which the drug was being prescribed, or the information was inadequate (Proposed Rule, Section I (1)). PhRMA responded to these claims by noting that the data was obtained in 1994 and therefore out-dated. After 1994, the manufacturers claim that either they have provided additional pediatric information within the label or that there is no need for additional labeling information-particularly for Ampicillin and Auralgan (2). The Center for Drug Evaluation and Research (CDER) identified the top ten drugs prescribed (on both an inpatient and outpatient basis) in the pediatric population and asked the companies that market these drugs to voice their concerns over the proposed changes in regulations (Proposed Rule, Section 11 (I)).
FDA claims that physicians have to either guess on an appropriate dosage (causing a potential for subtherapeutic levels or adverse events due to toxicity) or prescribe only those drugs with which they have had experience prescribing in the pediatric population (causing a potential for a less effective form of therapy) (Proposed Rule, Section I (I)).
An informal study by the American Academy of Pediatrics in 1990 found that only 20% of the new molecular entities (NME's) approved between 1984-1990 had pediatric information (not all of the NME' s had potential use in the pediatric population) and that 56% of the NME's approved in 1991 that had potential use in the pediatric population had some pediatric labeling at the time of approval. In 1996 (2 years after the passing of the 1994 regulations regarding pediatric use labeling) only 37% of the NME's that had potential use in the pediatric population had some pediatric labeling at the time of approval. The pediatric labeling that was present on the NME's in 1991 and 1996 may not have been adequate for all groups within the pediatric population (Proposed Rule Section lII (1)). PhRMA states that 20 of the approved drugs in 1996 would have potential use in the pediatric population. Of these 20, 19 have been studied or will be studied in pediatric patients, showing an improvement in industry' s response to the 1994 regulations (2).

Description of the Proposed Rule
The proposed rule would be intended for new chemical entities and new biological drug products. A new chemical entity is defined as "a drug that contains no previously approved active moiety." There are three main points to this proposed rule: 1) before approval, a new chemical entity must have safety and efficacy information on relevant pediatric age groups for the claimed indication, 2) drugs already marketed will need to 11 provide more pediatric information if the label is lacking in relevant information, and 3) FDA can call for meetings to discuss the need for pediatric studies early in the development process and postmarketing. FDA bas broken down the pediatric population into 4 subgroups: I) neonates-birth to one month of age, 2) infant-one month to two years of age, 3) children -two years to twelve years of age, and 4) adolescent -twelve years to sixteen years of age. A safety and efficacy assessment would be required for pediatric patients, in all age groups, for the claimed indication. A manufacturer. would not be responsible for providing information for any off-label indications. Companies would not need to provide new information for any supplements filed for new indications (Proposed Rule Section V.A (l)).
Pediatric formulations would be required in the studies to ensure bioavailability and the consistency of the dosing. By using a pediatric formulation in a study, data will be more meaningful and an accurate analysis can be made for safety and effectiveness in the pediatric population. If a manufacturer were unable to produce an appropriate pediatric formulation for a given age group, then a waiver would be granted. FDA was seeking comments on using cost of generating a formulation to be grounds for a waiver (Proposed Rule Section V.E (!)).

Waivers
Pediatric studies would not be necessary if FDA granted a full or partial waiver (Proposed Rule Section V.B.4 (!)). Pediatric assessments are not necessary if I) the product will not be a meaningful therapeutic benefit over already existing treatments and 12 if it will not be widely used in the pediatric population, 2) if studies would be impossible or impractical to carry out, and 3) ifthe compound would pose undue risk to the pediatric patients. A full waiver would be granted if one or more of the conditions above applied to the entire pediatric population. A partial waiver would be granted if there was a need to avoid studies in a specific age group within the pediatric population. FDA was seeking comments regarding whether there should be other situations that might merit a waiver -e.g. costs.
One of the questions that FDA faced was how to quantify "meaningful therapeutic advances" . FDA addressed this issue by deciding that it would be meaningful if a substantial number of patients were to use this new compound. The proposal discusses two different methods for determining a substantial number of patients. The first method would be to assess the number of times the drug would be used annually within the pediatric population. If it is estimated that I 00,000 or more prescriptions may be written for patients within the pediatric population, then the drug would qualify as being given to a substantial number in all age groups. A partial waiver would be granted if fewer than I 5,000 prescriptions were to be written for a specific age group. The second method would assess the number of patients affected by the disease or condition that the drug is designed to treat. If I 00,000 pediatric patients were affected, then the compound would be used in a substantial number of pediatric patients. A partial waiver would be granted if fewer than I 5,000 patients comprised a particular age group (Proposed Rule Section V.B.4 (I)). PhRMA argues over the true representation of the diseased population by using prescription numbers as a basis for calculating a drug as being used in a substantial 13 number of patients. For many diseases (e.g. asthma) multiple prescriptions are refilled several times in a given year for a single patient. PhRMA believes that there would be potential for gross exaggerations of diseased children for certain diseases. PhRMA recommends deciding a meaningful therapeutic advance by unmet medical needs and not by arbitrarily decided numbers which may not be a true measurement of the diseased population (2) .
There may be instances where the deferral of submissions of pediatric studies may be necessary (e.g. if the New Drug Application (NOA) submission or approval is ready for adults before pediatric testing is complete). It may be inappropriate to begin pediatric testing before the safety and efficacy data in adults has been collected. The deferred submission would need to be provided not more than 2 years after the date of the initial approval. Applicants would need to provide pediatric information in their Annual Progress Reports (APRs) to show compliance (Proposed Rule Section V.B.3 (1)).

Legal Ramifications for Inadequate Pediatric Labeling Information
In the proposed rule, FDA states "Denying or withdrawing approval of an otherwise safe and effective drug or biological product is not a satisfactory remedy, because removal of a product from the marketplace could deprive other patients of the benefits of a useful medical product." Therefore, FDA is looking into injunctive actions against companies that fail to provide the necessary pediatric information. Violation of an injunctive action could result in the manufacturer being fined (Proposed Rule Section V.G (1)).
14 What legal authority would FDA have over enforcing this proposal? The proposal cites provisions that apply to FDA's authority over enforcing this proposal. For example, FDA has authority to classify a drug as misbranded if the label is false or misleading, dangerous to health when prescribed, recommended, or suggested in its labeling, or fails to provide adequate directions for each intended use. There are other legal reasons cited in Section VI of the regulations. Still, industry questions whether FDA has any legal authority over forcing manufucturers to provide this data (2).

Analysis of Impact
An assessment of the impact of the proposed regulation is difficult to ascertain. The FDA has estimated the number of additional studies and the cost that would have accrued had these regulations been in place over the period 1991-1995. The drugs approved over this period were categorized according to their potential use in the pediatric population. The drugs were divided into 3 categories: 1) therapeutically important, 2) other approvals, and 3) all other approvals. The "therapeutically important" drug category was composed of those drugs that would have a potential use in the pediatric population. The "other approval" category comprised drugs that would have a potential to be used extensively in the pediatric population. The final category, "all other approvals", consisted of drugs that would not be used for a pediatric patient.
This data was tabulated in Table 2 FDA has also noted that not all compounds will be approved.
To account for the additional pediatric studies that will occur for drugs that will not ultimately be approved, FDA has further padded the numbers by increasing the estimate by 30%, or 14 drugs and 1,850 pediatric patients per year (Proposed Rule Section X.B (I)).
Costs of studies vary proportionately with the complexity of the clinical trial. FDA hired a private consulting firm to estimate the costs of Phase IV trials. The firm estimates that 16 for a fever or vaccine trial, the cost would range from $300-$500 per patient, for renal disease the cost would be $3600 per patient, and for epilepsy the cost would be $5,000 per patient. Many researchers estimate $1 ,500-$3,400 per patient as an average cost.
To include all costs incurred during a study, FDA has estimated the cost to be $5,000-$9,000 per patient. Based on this estimate, the annual cost to conduct the additional studies for the 1,850 patients in any given year would have cost the industry $9 .25 million-$16.65 million per year. This estimate does not include any additional expenditure for the manufacturing of the pediatric formulation . FDA estimates that the cost of the additional formulations will not cost more than $1 million per year for each drug (estimating that a total of 4 drugs per year will need additional formulations bringing the total to $4 million for additional formulations) . There will also be additional paperwork due to the increased regulations and FDA estimates these costs at $220,000 per year. The total estimation comes in at $13.5 million -$20.9 million per year (Proposed Rule Section X.C.(l)). Delays in the submittal ofa NDA might result in a further potential impact for the manufucturer due to extended drug development time lines. These estimates do not include additional staff that will be needed by FDA to process the supplements to already existing NDAs, increases in the number of studies included in future NDAs, and the additional meetings being held during the development process to ensure adequate pediatric trials (2).

Benefits of Regulations
These regulations address providing adequate dosing information in the label for pediatric age groups. This information will be used to avoid adverse drug reactions and undertreatment in this population. Additionally, the information should increase the availability ofnewer medications to the pediatric population. FDA compiled a list of the top 25 NME's responsible for the highest number of adverse events in pediatric patients.
Eight (8) of these NME's had no pediatric labeling information (1 ,273 adverse events) and 5 lacked label information for children under 12 (434 adverse events). Out of these 13 NME's, 11 would have been required to submit further pediatric labeling information under these proposed regulations (Proposed Rule Section X.E. (1 )).

DISCUSSION
What will these proposals mean for industry? Obviously, there will be more initial cost for manufacturers in that will need to run more clinical studies and create new pediatric formulations. What about compounds that are off patent or are unpatentable drugs?
What sort of incentive is there for companies to spend additional resources on drugs that they no longer have patent protection? Some of the smaller manufacturers will simply not be able to fund these additional requirements. It is argued that the FDA estimates of the cost of additional studies and the creation of new formulations are far too low.
PhRMA believes that there could be potential delays in drug development time and NDA approvals. Additionally, the issue oflegal consent will be hotly disputed. Will parental consent be considered enough? Industry is concerned that children recruited in these studies will possibly be injured and subsequently sue for unlawful consent (2).
PhRMA has suggested that FDA follow European countries, Canada, and Japan when looking to create additional regulations regarding pediatric clinical trials. In Europe, 18 pediatric studies begin after the completion of Phase III trials and their age groups are different from those assigned by FDA. Canada tests drugs in children after safety and efficacy has been determined in the adult population. Japan excludes children from Phase I and II trials and conducts trial in neonates and infants only after older children have been studied (2).
Clearly, there is much debate over whether these regulations should be passed, by FDA and industry, as well as health care providers and parents. There is a need for further discussion on this matter both from the viewpoint of the child and also the realistic requirements that can be placed on the pharmaceutical industry. It has been recommended that a committee be formed, comprised of these individuals, to address these many difficult questions. Until then, there are too many unresolved issues to proceed with further implementation of the current proposed regulations as they stand. 19

ABSTRACT
A population pharmacokinetic analysis was conducted for azithromycin on data from pediatric patients enrolled in four separate clinical trials. The data, which consisted of 526 serum concentrations from 58 patients administered one to five daily oral doses of azithromycin ranging from 5 mg/kg to 12 mg/kg per day, was analyzed in NONMEM. A two compartment model with parallel zero-order and first-order absorption was found to best fit the data. Potential covariates were assessed for oral clearance (CL/F), oral volume of distribution in the peripheral compartment (V2/F), intercompartmental oral clearance (Q/F), and the first-order absorption rate constant (ka). Models were initially developed using the first order (FO) method and subsequently refined using the first order conditional estimation (FOCE) method. Weight was found to be a significant covariate for both CL/F and V2/F. Neither age, gender, the presence of anemia, cancer, pneumonia, nausea, colitis, nor the concomitant usage of albuterol, amikacin, captopril, ceftazidime, ceftriaxone, digoxin, diphenhydramine, dopamine, fentanyl, furosemide, midazolarn, morphine, nystatin, ranitidine, sulfamethotrexate, ticarcillin, nor vancomycin appeared significant for any pharmacokinetic parameter.

INTRODUCTION
Azithromycin is an azalide antibiotic (a subset of macrolide antibiotics), is active against a wide spectrum of microorganisms, and bas a low side effect profile (1)(2)(3)(4). Azithromycin is indicated for pediatric usage for the treatment of acute otitis media, community-acquired pneumonia, and pharyngitis/tonsillitis (5). Following administration, azithromycin undergoes extensive and rapid distribution in tissue. Thereafter, distribution from tissue is the rate limiting process for elimination of azithromycin, thus leading to a long terminal half-life (around 55-70 hours for both adult and pediatric populations) (1 ;2;4-6). Because of these properties, azithromycin is administered once daily and for a shorter duration than other macrolide antibiotics ( 1 ;2;7-10). Appendix I provides a more extensive overview of the pharmacokinetics of azithromycin.
Clinical trials have been conducted in pediatric patients to determine the pharmacodynarnic and pharmacokinetic characteristics in this group compared to the adult population (9)(10)(11). These studies, and other safety clinical trials, showed that once daily dosing was well tolerated and efficacious in pediatric patients (7)(8)(9)(10)(11)(12). Results from two of these studies have been used for the development of dosing guidelines in pediatrics (5;10;1 l). Current recommended dosages for pediatric patients are determined by indication. For the indications of otitis media and community-acquired pneumonia, pediatric recommendations are for I 0 mg/kg on day 1 and 5 mg/kg doses on days 2-5. A higher dose of 12 mg/kg for days 1-5 is recommended for children with pharyngitis/tonsillitis (5). In contrast, a 500 mg single dose given on day 1, followed by 24 250 mg single doses on days 2-5 is recommended for adults with these indications (5;13).
Otitis media and pharyngitis are very common infections, especially in the younger pediatric population (8; 12) and an understanding of the contribution ofage or other factors that may explain interpatient variability in clearance may prove beneficial. The 25 lack of information on variability may be addressed by utilizing nonlinear mixed effects modeling (i.e. population pharmacokinetic models) for azithromycin in the pediatric population.
Nonlinear mixed effects modeling permits the use of sparse data (2 or more plasma concentration samples not necessarily from the same dosing interval per patient) from a large number ofrepresentative patients in the population (16)(17)(18). The population method pools all data collected and calculates population pharmacokinetic parameters (e.g. CL/F). Additionally the focus of the analysis is on the source and correlation of variability in pharmacokinetic parameters among individuals in the population (19) .
Thus, population pharmacokinetics focuses on the study population (unit of analysis) initially and moves out to the individual. Population analyses also provide quantitative estimates of both the interindividual and intraindividual (i.e. residual) variabilities of the population (l 7;18). lnterindividual variability may be accounted for by adding specific patient characteristics (e.g. demographic information, concomitant medication usage, etc.) into the population model. Patient characteristics that cause changes in the doseconcentration relationship can be identified, assessed, and then appropriate dosing modifications can be determined to enhance efficacy or to reduce the chance ofadverse events (16;17).
The purpose of this investigation was to evaluate whether a population pharmacokinetic modeling approach could be used to develop a model for data combined from four 26 pediatric trials and to determine if any patient characteristics could be identified that might provide useful information when selecting a dose of azithromycin in children.

METHODS AND MATERIALS
Patients. Plasma concentration-time data were obtained from pediatric patients enrolled in four Phase 1 clinical studies (see Table 1). These four clinical trials were conducted to evaluate safety, efficacy, and pharmacokinetics after oral administration ofazithromycin in pediatric patients. Results for three of the four studies (Protocols 054, 136, and 172) have been reported elsewhere (9)(10)(11). The fourth study (Protocol 043) was terminated early due to difficulties with patient enrollment. Protocol 043, 054, 136, and 172 were conducted during 1993, 1993, 1991, and 1992-1993 respectively. The appropriate institutional review boards approved all protocols. The patient's parent or a legal guardian gave written informed consent prior to inclusion in the study.
This retrospective combined data analysis was conducted on all pediatric patients with measurable azithromycin concentration-time data collected in the four Phase I clinical trials. A random selection of20% of the patients from the combined dataset was reserved to assess the predictive performance of the model, i.e. internal validation of the final model. The data from the remaining 80% of the patients was used for the model development. Two patients in Protocol 054 (I male 2 year old weighting l 3kg and concomitantly medicated with captopril, furosemide, and morphine; I male I year old weighting 9kg with colitis and concomitantly medicated with captopril, digoxin, diphenhydramine, dopamine, fentanyl, furosemide, morphine, and nystatin) were 27 excluded from the combined data analysis because they had no measurable azithromycin concentration levels at any time point. Thus, there were a total of 58 pediatric patients used for the combined data analysis: 46 were included in the model development dataset and 12 were included in the validation dataset. Characteristics of the pediatric patients included in the dataset are presented in Table 2.
An approach proposed by  was used for the data analysis: 1) a base model was developed for the population, 2) the estimates found during step I were used to explore potential covariates with the base model, and 3) a mixed effects model was developed to describe the relationship between the covariates and pbarmacokinetic parameters (26). 1n this analysis, forward addition ofcovariates was used to generate the full model, while a backwards elimination approach from the full model was used to determine the final model.
Pharmacokinetic and statistical models were evaluated to determine the model that best described the model development dataset (n=46 patients). To discriminate between models, the following criteria were used: 1) a decrease in the objective function value (which is proportional to minus twice the log-likelihood of the data) of3.84 <i distribution, df=l , p< 0.05) or greater following the addition ofa single parameter was deemed statistically significant; 2) diagnostic plots (e.g. predicted concentration versus Phannacokinetic Model. To compare adult and pediatric models and estimates, a population model was initially developed using data from a traditional pbarmacokinetic study conducted in healthy normal adult male subjects (age=27-54 years, weight=63-90 30 kg) (28). The subjects had no evidence of a history of disease, were taking no concomitant medications, and were emolled in a study to evaluate azithromycin phannacokinetics after single oral and intravenous doses. An intensive blood sampling regimen was used. A total of twelve subjects started the study but only ten completed both anns of the study. Two subjects dropped out after the first arm (one subject in each cohort), leaving eleven subjects that completed each arm. The eleven subjects from the oral azithromycin administration cohort contributed 120 concentration records to the modeling dataset; blood samples were collected at 0 Gust prior to dosing), 0. 5, I, 2, 3, 4, 6, 8, 12, 24, 48, and 72 hours post dose on day I. Several pharrnacokinetic models were evaluated to fit the adult data: one-compartment and two-compartment models with zero-order, first-order, and a combination zero-and first-order absorption. A twocompartment model with a combination parallel zero-order and first-order absorption best described the data. The two-compartment model with both absorption terms was parameterized as oral clearance (CL/F), oral volume of distribution in the central compartment (V l/F), oral volume of distribution in the peripheral compartment (V2/F), intercompartmental oral clearance (Q/F), the first-order absorption rate constant (ka), and the zero-order rate constant (R) (NONMEM subroutines ADV AN4 TRANS4).
As with the adult dataset, several pharmacokinetic models were used to evaluate the pediatric data: one, two, and three compartment models with zero-, first-, and combination parallel zero-and first-order absorption terms (14;29;30). For these models, the final parameter estimates from the adult model were used as the initial estimates for the modeling of the pediatric data The two compartment model with parallel zero-and first-order absorption rates best fit the pediatric data and was used as the base model. where T\i. 9 is a random variable distributed with a zero mean and variance of ro 2 9 and TV® is the population mean value for 0.

Statistical Model
Residual variability was modeled using a proportional-error model and an additive and proportional error model (  adverse event, or disease status in fewer than four of the patients was not tested. 33 Covariates that were found to be statistically significant from the initial screening in S-Plus were then evaluated in the base model using NONMEM . Each covariate was added one at a time into the base model. Covariates were deemed as statistically significant in NONMEM as outlined above; i.e. a change of >3.84 in the objective function value, diagnostic plots, reductions in variability, and the AIC (I 6;21 ;26;27). A large number of covariates were found to be statistically significant in both S-Plus and NONMEM.
Because of the number of significant variables, the model development was done in a forward stepwise manner, in a manner similar to that published by Lee et. al.(33).
To generate the model in a forward stepwise manner, the change in the objective function value was used as the initial criteria for a covariate' s inclusion into the model.
The list of covariates that generated a change in the objective function value of greater than 3.84 for any pharmacokinetic parameter was sorted in descending order. The 95% confidence interval was calculated for the covariate parameter estimate that generated the largest change in the objective function value. If the 95% confidence interval did not include the null value, this parameter was then added to the base model. If the 95% confidence interval did include the null value, the parameter was not added and the covariate that generated the next largest change in the objective function value was then evaluated. Once the initial covariate was identified, the other covariates in the list (whose change in objective function value were greater than or equal to 3.84) were added individually. Any covariate that did not generate a further change in the objective function value of3.84 or greater in this second run was discarded from the model building process. Again, the covariates were sorted by magnitude of the change in the 34 objective function value. The top covariate whose 95% confidence interval did not include the null value then became the second covariate to be added to the model. This process continued until there were no more covariates whose addition into the model would generate an objective function value change of greater than 3.84 and whose parameter estimate value would not include the null value, thus the full model was The data output from NONMEM was then exported to Microsoft Excel (version 98).
Bias (mean prediction error) and precision (root mean square error) of the predicted concentration values were calculated to describe the predictive performance of the model (33)(34)(35): where pe;= the difference between the ith measured and predicted azithromycin concentration value at a given time and N=the number of pairs of predicted and observed azithromycin concentrations. Ideally, a value of zero is desired for both precision and bias; the smaller the magnitude of the residual, the lower the magnitude of the value of precision and bias (35). The 95% confidence intervals of precision and bias were also calculated by using the following equation (33)(34)(35)): x., ± lo.97l, N -I . SE(X .. ) As another form of validation, the NONMEM analysis using the final model was conducted on I 00% of the data. The estimates of the pharmacok:inetic and statistical parameters were compared to those obtained with the development of the final model dataset (80% of the data) (16; 36;37).
Finally, the analysis was performed using the first order conditional estimation (FOCE) method using 100% of the data (

Adult Phannacokinetics:
The adult dataset was best described by using a two-compartment model with a zeroorder rate of absorption. For the adult model, interindividual variability was described with an exponential error term on CUF. The base model equations, parameter estimates, percent relative standard error (%RSE), and 95% confidence intervals for the adult dataset are given in Table 3 (mean weight=73.6 kg). A previous study modeled a similar dataset using a two-compartment model with zero-order absorption (14). The previous study found that a zero-order absorption rate was a superior fit for the data when compared to a first-order absorption rate model. The pharmacokinetic parameter values reported from the previous study were similar in values for oral clearance and oral volume of distribution to this analysis.

Pediatric Phannacokinetics (FO Method) for Base Model-80% data:
In contrast with the adult data, a two-compartment model with zero-and first-order absorption rate constants best fit the pediatric model development dataset. The parameter estimates calculated from the adult model were used as the initial estimates for the pediatric model development dataset. During model development, interindividual error terms on CL/F, V2/F, ka, and Q/F significantly improved the model, i.e. a decrease in the objective function of3.84 Ci distribution, df=I , p< 0.05). lnterindividual error 37 terms on Vl/F and R did not significantly improve the fit of the model and were excluded from further model development. Interindividual variability was best described by an exponential error model. Residual variability was best described by using a proportional error model. The base model equations, parameter estimates, percent relative standard error (%RSE), and 95% confidence intervals for the pediatric dataset are given in Table 4 (mean weight=26.5kg). When standardized by the mean weight, the parameter values generally compare well for the pediatric and adult populations (see Table 5 Individual covariate testing (FO Method) for Full/Final Model -80% data: A summary of the forward stepwise model development for inclusion of covariates for pharmacokinetic parameters is provided in Table 6. The full model consisted of albuterol and weight as covariates for CL/F, ceftriaxone and weight as covariates for V2/F, dopamine as a covariate for Q/F, and morphine as a covariate for ka (Table 7). 38 Backwards elimination was then performed to generate the final model. Each covariate was removed individually from th. e model. A covariate was retained in the final model if there was a significant decrease in the goodness of fit (i.e., objective function value decreased by 7.88 Ci distnlmtion, df=I, p< 0.005). Following the backwards elimination procedure, only albuterol and weight were identified as significant covariates for CUF and only ceftriaxone and weight for V2/F. Additionally, the parameter ro 2 V2/F was removed from the model since its 95% confidence interval included the null value.
The final model equations, parameter estimates, percent relative standard error (%RSE), and 95% confidence intervals are given in Validation dataset (FO Method) -20% data: The predicted performance of the validation dataset is shown in Table 9 . There are both bias and imprecision in the model between the observed and predicted azithromycin concentrations as shown with the 95% confidence intervals not including the null value.
The observed serum concentration versus predicted serum concentration values and weighted residual versus predicted serum concentration values plots are shown in Figures 8 and 9 respectively. In Figure 8, it appears that the model still had difficulty estimating the larger concentration values. ln Figure 9, larger weighted residuals are seen for a few smaller predicted concentration values.
Final Model (FO Method) -100% data: The final model equations, pharrnacokinetic and statistical parameter estimates, percent relative standard error (%RSE), and 95% confidence intervals generated using I 00% of the data are shown in Table I Figure 13 shows that the weighted residuals are not biased, that is, there is a scatter of weighted residual values over the entire predicted concentration range. When I 00% of the pediatric data 40 was used, weight and albuterol were retained as covariates for CL/F and weight and ceftriaxone for V2/F.

Final Model (FOCE Method) -100% data:
The final model equations, pharmacokinetic and statistical parameter estimates, percent relative standard error (%RSE), and 95% confidence intervals generated using the FOCE method on 100% of the data are shown in Table I  There was a more uniform distribution of data points spread over the line of identity. Figure 16 shows that the residual plots do not differ much between the two models. Figure I 7 shows that the weighted residuals are biased for smaller predicted concentration values. When 100% of the pediatric data was used with the FOCE method, weight remained as a covariate for CUF and V2/F in the final model.

41
Validation dataset (FOCE Method) -20% data: The predicted performance of the validation dataset is shown in Table 12, where it can be observed that bias and imprecision were present between the observed and predicted azithromycin concentrations: the 95% confidence intervals did not include the null value.
The values for precision and bias are similar to the values obtained using the FO method.
The observed serum concentration versus predicted serum concentration values and weighted residual versus predicted serum concentration values plots are shown in  respectively. In Figure 18, the model still had difficulty estimating the higher concentration values. In Figure 19, larger weighted residuals are seen for a few smaller predicted concentration values. Figure 20 shows the overall fit of the model by

DISCUSSION
The study demonstrated that a population pharmacokinetic modeling approach could be used to model azithromycin concentration-time data from four pediatric clinical trials.
Additionally, the study demonstrated that it was possible to identify covariates to explain variability in the pharmacokinetic parameters. When the FOCE method was used, weight was found to be a significant covariate for CUF and V2/F.
Weight was an anticipated covariate for both CL/F and V2/F and supports the current dosing recommendations for azithromycin based on weight (5). Age has been proposed as a potential significant covariate since a higher oral clearance has been seen in children 0-5 years of age as compared to children that are 6-15 years of age (9) . However, the population analysis found that while age created a large difference in the objective function value when added individually to the base model, the 95% confidence interval of age's parameter estimate included the null value; age was not considered statistically significant. Additionally, weight and age were highly correlated covariates. Therefore age was not evaluated in further model development.
When the FO method was used in the analysis, two of the covariates identified as significant subsequently become insignificant when the FOCE method was used.
Albuterol was identified as a statistically significant covariate for CUF and ceftriaxone 43 for V2/F. Albuterol was given to 22% (13 out of 58) of the entire population, while ceftriaxone was given to 14% (8 out of 58). No plausible explanation for these effects could be found in the literature. However, it is interesting to note that many of the patients taking albuterol and ceftriaxone were also taking many concomitant medications. Additionally, all but one of the patients taking these medications were from the 054 study. The patients in study 054 were different from the children in the other three studies in that they received multiple concomitant medications and had more acute and chronic illnesses (all children were hospitalized); 12 of the 26 patients were cancer patients. Children in studies 043, 136, 172, and the non cancer patients in 054 were enrolled in their respective protocols for otitis media or pharyngitis (9)(10)(11).
Consequently, all concomitant medications were taken by approximately half of the patients in study 054 (12 out of58 patients, or 21%). Thus, this analysis may not have had sufficient power to fully evaluate the interacting potential of many of these drugs.
When the FOCE method was used, both albuterol and ceftriaxone ceased to be significant in the model.
A previous analysis found a large interpatient variability in the parameter estimates for study 054 (9). In this study analysis, there were statistical differences (p<0.0001) in oral clearance in children <=5 years old (CL/F=4.27 Uhr/kg) compared to the group of children 6 years of age (2.27 Uhr/kg) and greater. In contrast the present analysis did not find age to be a statistically significant covariate for oral clearance. 44 Model misspecification may have led to the poor fit of the model for large concentration values. A two compartment model was found to best fit the data However, more recent studies indicate that azithromycin follows three compartmental pharmacokinetics (29;30). This could possibly explain the under-prediction of concentrations at later times. In the previous analysis of study 054, a zero-order input was used to model drug absorption (9). In the present study, the data was best fit using both zero order input and simultaneous first order input. While the diagnostic plots and also the change in the objective function value signaled a better fitting model when both inputs were used, the plots still showed that there was bias and imprecision when calculating the predicted concentration values with this model. Additionally, the bias and imprecision in the model was seen with the results of the predictive performance using the validation dataset. In Figure 20, the predicted concentration values were consistently larger than the observed values. This finding would be indicative of a three compartment model being a potentially better fit. Further work could be done evaluating a three compartment model with various error models using the FOCE method.
Our models were generated using both FO and FOCE estimation methods in NONMEM.

General steroid information
Corticosteroids are prescribed for their imrnunosuppressive and anti-inflammatory effects. These effects are produced by the binding of the steroid to cytosolic receptors in many different tissues. These activated receptors then go on to the cell nucleus and increase the transcription of certain genes that regulate the synthesis of specific proteins, second messengers, or enzymes (2). Some effects are seen immediately (e.g. changes in cortisol plasma concentrations) and appear directly related to the pharmacokinetics of the steroid; other effects (e.g. eosinophil counts), have a slow onset (6-8 hours) and slow dissipation of the response (24-36 hours) back to baseline (2;4).
Because steroids are nonselective immunosuppressants (i.e. they affect many genemediated responses simultaneously), these drugs may predispose patients to a greater risk of infections and other side effects (5). These adverse events may include: cushingoid features, hemorrhage, psychoses, myopathy, osteoporosis, cataracts, hyperlipidemia, growth retardation in children, and hypertension (2;6-8). Often the 90 frequency of side effects are increased in patients undergoing chronic therapy, patients with low serum albumin concentrations, and patients receiving certain concomitant medications (e.g. oral contraceptives) that affect the protein binding and metabolism of prednisolone; all factors that increase a patient's steroid exposure. Patients with low serum albumin concentrations may have greater steroid exposure due to altered protein binding and/or a reduced hepatic function (2;9). A study of240 medical inpatients receiving prednisone showed a correlation between the frequency of side effects, the mean daily prednisone dose, and the serum-albumin levels. Side effects were more common with those patients that received higher prednisone doses and in patients with low serum-albumin concentrations ( 4;9).

Prednisolone Pharmacokinetics
For immunosuppression of organ transplants, prednisolone is administered either orally or intravenously. When prednisolone is administered orally, it is administered as prednisolone or as its prodrug prednisone, which is metabolized to active prednisolone (2;10;11).

Absorption
Oral prednisolone has a bioavailability (F) of60-100% (2;11-14). The lower bioavailability has been seen with higher steroid doses. Patients who exhibit subtherapeutic responses with prednisolone often experience poor absorption of drug ( 4).

Distribution
The reversible binding of drug to proteins follows the law of mass action: (Dj+(P) :: (DP) where D= the molar concentration of unbound drug, P= the unoccupied protein, DP= the drug protein complex, k 1 = the forward rate constant, and k2= the reverse rate constant. The ratio ofkl/k2 is known as the equilibrium association constant or affinity constant (Ka) (18;19). Ka provides information as to the affinity between the drug and its binding site on the protein; drugs that are strongly protein bound have large values of Ka (19). The inverse of Ka (i.e. I/Ka) is known as the equilibrium dissociation constant (Kd).
A drug' s extent and ability to bind to proteins will affect its pharmacokinetic parameters, specifically clearance and volume of distribution ( 18; 19). In the typical therapeutic concentration range for most drugs, the fraction unbound remains constant; only a small fraction of the binding sites on proteins are occupied. For a given concentration of protein, the fraction unbound is constant. Consequently, the pharmacokinetic parameters of most drugs are independent of dose (19) . However, for some drugs, protein binding varies with concentration level and thus these drugs exhibit concentration dependent pharmacokinetics. 92 Plasma protein binding of prednisolone appears to be dose related, resulting in nonlinear pharmacok:inetics (2;4;10;16;17;20-24). The nonlinearity is attributed to prednisolone binding to two different proteins; transcortin (i.e. corticosteroid binding globulin) which exhibits a low capacity and high affinity for prednisolone and albumin which exhibits a high capacity and low affinity for prednisolone (2;4;9;20;23-27). Prednisolone protein binding can be expressed: where Db is the concentration of prednisolone bound to both transcortin and albumin sites, N, is the number of binding sites for transcortin, K. is the affinity constant for transcortin, P, is the molar concentration oftranscortin protein, N, is the number of binding sites for albumin, K, is the affinity constant for albumin, P, is the molar concentration of albumin, and Dr is the unbound concentration of prednisolone (16; 23;24;28;29). Assuming K,D,<< l , there is one prednisolone binding site per molecule of albumin, and prednisolone only binds to transcortin and albumin (16; 23;24;28;29), then bound concentration ofprednisolone can be reduced to:

D• = N, P,K,DJ +PaKaDr (1 + K;Dr)
Nonlinear prednisolone protein binding occurs because of limited concentrations of transcortin in plasma (24). At low concentrations of prednisolone, binding to transcortin is maximal at 90-95%, but at large concentrations, saturation occurs thus producing only 60% transcortin binding (2;4;10;24;30). Therefore at low doses, the increased fraction bound of prednisolone to transcortin makes less prednisolone available to distribute to 93 receptor sites (2). The binding capacity (N1P1) and Ka of prednisolone to transcortin have been reported to be (5 .45-8.00)xl0-7 Mand (l . 40-3.39)xl 0 7 UM respectively (2;9;18;28;3 l). The normal concentration of transcortin in plasma is approximately 0. 7 µM and falls in proportion with serum-albumin levels (9). It is thought that only unbound prednisolone is biologically active ( 4; 10;22). Cortisol also complicates the binding ofprednisolone since it competes with prednisolone for binding sites to transcortin (31 ). There appears to be a circadian cycle affecting the binding capacity of transcortin to prednisolone; binding is least at 8a.m. when cortisol levels are high and greatest at midnight when cortisol levels are low (2;32).
At oral prednisolone doses of l 5mg and 50mg, the protein binding of prednisolone has been reported at 87% and 74% respectively (19). The dose dependency ofprednisolone pharmacokinetics has been primarily attributed to nonlinear protein binding ( 4). The apparent steady state volume of distribution based on total and unbound prednisolone concentrations were reported to be 35-114 Land 323-530 L respectively for doses ranging from 1.25 mg eight times daily to I 00 mg once daily; larger doses produced larger values total prednisolone volume of distribution, while the free prednisolone 94 volume of distnbution was not dose dependent (17). The apparent volume of distribution for total prednisone was reported to be 0.97 L/kg (33).
As discussed earlier, the bioavailability of prednisolone is high, thus there is limited presystemic metabolism of prednisolone (36). Prednisolone displays restrictive clearance; clearance is sensitive to fraction unbound in the plasma and the activity of the drug metabolizing enzymes (2; 19;34).
Prednisolone undergoes reversible metabolism (interconversion) to prednisone. The enzyme 11-~-hydroxydehydrogenase is responsible for the interconversion process (IO;13 ;20). Prednisone is also reconverted back to prednisolone. The interconversion of prednisolone and prednisone has been reported to be a nonlinear process. This nonlinearity can be seen in the area under the concentration versus time curve (AUC) for prednisolone and prednisone. The ratio of AUC prednisolone/ AUC prednisone increases with increasing doses ofprednisolone (2;11;20;24). lfthe interconversion were linear, 95 the ratio of the AUCs would remam constant with increasing doses; as the concentration of one of the steroids increases, the other steroid would increase in a proportional manner. Prednisolone dominates the interconversion process and the prednisolone concentrations can be as much as 10 times the prednisone concentrations (2;4;11;21;24).
Conventional linear pharmacokinetic parameter calculation methods that assume no interconversion underestimate clearance and overestimate volume of distribution (37).
The absence of an intravenous formulation of prednisone for humans makes the exact assessment of the interconversion process difficult (I 0).
Garg et.al. preformed an extensive analysis on the interconversion ofprednisolone and prednisone (11). They performed a two-way crossover study between two treatments: oral prednisone tablets and intravenous prednisolone sodium phosphate. They found the 96 irreversible elimination clearances ofprednisone and prednisolone to be 53 .9 mVmin and 196 mVmin respectively. The clearance for the conversion ofprednisolone to prednisone was reported as 836 mVmin. The clearance for the conversion of prednisone to prednisolone was reported as 8822 mVrnin; I 0 times higher then the clearance of prednisolone to prednisone. Since the clearance of prednisone not reconverted to prednisolone was relatively small and the clearance of prednisone reconverted to prednisolone was relatively high, this implied that most prednisone was converted to prednisolone. The recycled fraction (RF), the probability of a molecule being converted to its metabolite and back at least once, was reported to be 0.76 for prednisolone. A large RF indicates a greater role of interconversion between a drug and its metabolite. A RF of0.76 suggests that a large interconversion process occurs between prednisolone and prednisone (11).
As reported by Jusko et. al. , in a study with six normal male subjects dosed 5mg, 20mg, and 50mg oforal prednisone, the oral clearance of total prednisolone was around 8, 12, and 16 L/h respectively ( 4). The increase in oral clearance was statistically significant.
Rohatagi et. al. found similar results for oral clearance of total and unbound prednisolone to be 6-19 L/h and 64-128 L/h respectively across oral prednisolone doses of 1.25 mg eight times daily-I 00 mg once daily (17). These investigators found dose dependent increases in clearance based on total but not unbound concentrations. The mean oral clearance of prednisone was reported to be 0.216 Lib/kg (39). It has been reported that prednisone does not exhibit nonlinear protein binding (I 0).

Nonlinearity of Prednisolone Pharmacokinetics
Total prednisolone concentrations exhibit both dose-dependent clearance and dose dependent steady state volume of distribution (2;4;1O;16;17;24;37). Saturation of prednisolone binding to plasma transcortin, a saturation of the interconversion processes, saturation of elimination pathways, concentration dependent clearance of unbound prednisolone concentrations, and tissue-binding sites may all be responsible for the dose dependency (10;37). Apparent clearance and steady state volume of distribution increase two fold between 5-40mg ofprednisolone (2;16;20;24;40). Because of the nonlinearity seen with total prednisolone concentrations, it has been recommended that unbound prednisolone concentrations be measured (2 ;24). When using unbound prednisolone concentrations, both clearance and steady-state volume of distribution become more constant with dose (l 7;23). However, some investigators report that some nonlinearity still exists. The remaining nonlinearity has been proposed to be caused by nonlinear renal clearance, dose dependency of the interconversion process, and differences in the degree of nonlinearity in the disposition of prednisolone and prednisone (4;10;11;20;24;35;40).
The half-life of unbound prednisolone appears to remain constant (range= 2.3-3.5 hours; mean=2.9 hours) over different doses. This is probably because the volume of distribution and clearance are equally affected by nonlinear effects (2;4; 11; 17). Table I provides a comparison of some population pharmacokinetic parameters.

Age
One study reported tbat total prednisolone clearance was not different between children and adults (41), while another study found a 49% higher clearance (on a per kg basis) in children younger than 12 years of age than children over 12 years ( 42). No information was found regarding volume of distribution in children. Compared to young adults, elderly patients have a higher frequency of adverse events, lower unbound prednisolone clearance (Table I) and smaller unbound prednisolone steady-state volume of distribution (14). These differences have been attributed to a decrease in both renal and nonrenal clearances. The clearance of 6-P-hydroxyprednisolone decreases linearly with the nonrenal clearance of unbound prednisolone, thus indicating the activity of liver enzymes responsible for prednisolone metabolism diminishes in the elderly (24).

Gender
While gender has been found to alter the pharmacokinetics of prednisolone, there have been reported differences in the effect. In two studies, both unbound and total prednisolone clearances in adults were reported as being 20% greater in females than males (Table I) (40;43). No statistically significant differences in gender were found in volume of distribution for unbound and total prednisolone ( 40). In another study, Magee et.al. reported tbat unbound prednisolone clearance normalized to total body weight was approximately 20% higher in white males and 40% higher in black males as 99 compared to females ( 44). The unbound prednisolone apparent volume of distribution normalized to total body weight was approximately 30% higher in white males and 40% higher in black males than females ( 44).

Concomitant Medications Inducers
It has been reported that the metabolism of prednisolone increased when administered concomitantly with anticonvulsants or rifampicin (2;4; I 0;22). Phenytoin increased both the total clearance ( 48%) and nonrenal clearance (77% females; 65% males) of unbound and total prednisolone (Table I} ( 2;4;43;45). The increase in total clearance was due to the increase in nonrenal clearance. The urinary excretion of 6-P-hydroxyprednisolone was greater post phenytoin dosing ( 43). Phenytoin does not affect prednisolone' s volume of distribution, protein binding, or renal clearance (24;43).

Inhibitors
The metabolism of prednisolone was inhibited when administered concomitantly with oral contraceptives. Oral contraceptives cause: I) a decrease in unbound prednisolone clearance and steady-state volume of distribution, 2) an increase in serum transcortin concentrations, 3) an increase in half-life, and 4) lower affinity constants for both prednisolone-albumin and prednisolone-transcortin complexes (2;4;15 ;22;27;29;30;38;46). Three studies compared the effects of oral contraceptive use with different doses of prednisolone. Total prednisolone clearance, unbound prednisolone clearance, and total prednisolone volume of distribution were all lower in 100 oral contraceptive users than the control cohort ( Table 1). The lower values for total body clearance were attributed to a reduction in nonrenal clearance and increased cortisol binding to transcortin (27;29;30). The reduction in nonrenal clearance has been attributed to a reduction in the activity of hepatic 6~-hydroxylase (27). Plasma cortisol concentrations have been reported as being twice as high in oral contraceptive users (30;46). At lower doses of prednisolone; cortisol displaces prednisolone from transcortin binding sites but not albumin binding sites ( 40;46). Oral contraceptive users have decreased unbound prednisolone clearances at low doses ofprednisolone as compared to high doses (30;40).
Inhibition of predniso lone metabolism has also been seen with concomitant administration of other medications. Concomitant administration of diltiazern resulted in a reduction of the total clearance ofprednisolone (Table I), while naproxen and indomethacin reduced the clearance of unbound prednisolone by 35% and 40% respectively (2;4 7). Two studies found minor or no changes in AUC and half-life of prednisolone when given concomitantly with itraconazole (36;48). Both analyses concluded that CYP3A4 was a subsidiary pathway for prednisolone metabolism.
Zurcher et.al. found that ketoconazole, a potent inhibitor ofCYP3A4, decreased the total body clearance and volume of distribution of both unbound and total prednisolone.
The AUC of unbound prednisolone increased by 50% with concurrent ketoconazole use.
It was proposed that ketoconazole decreases renal clearance by impaired tubular secretion and nonrenal clearance by inhibited 6~-hydroxylase activity. Since the unbound prednisolone volume of distribution decreased while transcortin and albumin levels IOI remained constant, they concluded that altered protein binding was not the reason for the reduction in volwne of distribution; the mechanism that reduced the volume of distnbution was not known (13). Contrary to Zurcher's findings, Yamashita et.al. found no significant inhibition with concomitant ketoconazole use ( 49). In summary, the role of the CYP450 enzyme system with prednisolone pharmacokinetics remains unknown.

Cystic Fibrosis (CF)
Prednisolone clearance and steady-state volwne of distribution were approximately 50% higher in adolescent CF patients compared to a control cohort of age matched adolescent asthmatic patients (50). It is believed that enhanced biotransformation is the underlying reason for the differences seen in clearance; more frequent steroid doses may be necessary in the treatment of CF patients (50) .
Additional differences in pharmacokinetic parameters were seen comparing CF patients with normal subjects. Dove et.al. found that the total prednisolone nonrenal clearance and unbound fraction of prednisolone were larger in CF patients. Albumin and total protein serum concentrations for CF are low; therefore increased volume of distribution could be related to decreased protein binding (50).

Menopause
After a 25mg intravenous and 30mg oral dose of prednisolone in premenopausal and postrnenopausal women, Harris et.al. found that total and unbound prednisolone clearances were smaller and half-lives were larger in postmenopausal women (Table 1 ).
There were n<i observed significant differences in volume of distribution, protein binding, or bioavailability ofprednisolone between these groups of women. They proposed that a change in the activity ofat least one enzyme system involved in the metabolism of prednisolone occurs in postmenopausal women (34).

Prednisolone specifics in organ transplantation patients
Rejection levels have been shown to be similar between high and low prednisolone clearance groups. While rejection levels were similar, an increased frequency ofrejection and corresponding allograft loss was found in high prednisolone clearance patients; the number of rejection episodes has been shown to be an important risk factor for allograft failure (1;2;5;10;51;52). Combination drug therapy (e.g. cyclosporine) is typically used for adequate immunosuppression and to minimize adverse events (38;52;53). Bergrem el. al. evaluated cushingoid versus non-cushingoid transplant patients taking I Omg oral prednisolone and found that cushingoid patients had lower total and unbound prednisolone clearances (Table I). The cushingoid patients had a poorer transplant function than the non-cushingoid patients, as determined by creatinine clearance ( 15). In a renal transplant study conducted by Ost et.al., the total prednisolone clearance in cushingoid patients did not differ from non-cushingoid patients (I).

Specific Aims of this Research
Ultimately, the goal of immunosuppression is to taper the dose of prednisolone and eventually switch a patient to either the lowest efficacious dose or an alternate-day therapy regimen while not compromising a patient's therapeutic response (2; I 0). Steroid dosage is tapered as rapidly as possible after transplantation, although without an objective guide to safe steroid withdrawal, this can hasten recurrent rejection (5). The overall patient survival rate is linearly correlated with frequency of rejection episodes; i.e. the more episodes, the less likely a patient is to survive (5;51;52). Ideally, a reduction in the maintenance dose is warranted if the disease symptoms are under control or if transplanted organ function is suitable (2). It is desirable to develop an individualized dosing regimen for prednisolone based on measurable parameters (23).  Table 2.
Prcdnisolonc Administration. Ltd (Wrexham, Clwyd). The dosing interval was increased to 24 hours once a daily dose of l 5-20nig was achieved. Figure I shows the distribution of doses by plotting the percent of concentration samples versus the dose given.
A patient continued on the lowest daily maintenance dose unless they started to reject their transplant. Rejection episodes were treated with high intravenous doses (500-1000 mg/day) ofmethylprednisolone over a period of three consecutive days. Ifa dose of methylprednisolone was given the day prior, or on the day of the sample collection, then the sample collection was not used in this analysis (64 records).

Blood Collection and Sample Analysis.
Plasma samples were collected from patients approximately at the end of weeks I , 2, 3, and 4 and at the end of months 3 (week=12), 6 (week=26), 9 (week=38), and 12 (week=56). The primary objective for the study was to collect cyclosporine concentrations at these visits for each patient. If enough sample remained, then an additional assay was conducted for total prednisolone, total prednisone, and total cortisol. Measurements of total prednisolone, total prednisone and total cortisol were made using a fully validated high-performance liquid chromatography (HPLC) technique as descnlled in a previous publication (54).
Of the concentration records that were obtained, if the values ofprednisolone, prednisone, and cortisol were all equal to 0, then that record was removed. Thus there remained a total of 496 prednisolone and 496 prednisone concentrations (n=992 concentration values). The lower limit of detection (LLD) and lower limit of quantification (LLQ) for both prednisone and prednisolone were 2.1 µ g/L (signal-tonoise ratio no less than 3) and 7 µg/L (signal-to-noise ratio no less than I 0) respectively Prednisolone has a reported half-life of2.3-3.5 hours (4). Since it was not possible to ascertain that steady state bad been achieved, the dosing history for the five days prior to an observed concentration record was included in the database. A covariate, "dose", was generated to represent the dose that a patient was taking in relation to their corresponding plasma concentrations at hours 0, 2, and 6 post dose.
Total prednisolone concentrations were assayed in this study. The unbound fraction was estimated based on patients' albumin concentrations and a published algorithm, which included values for transcortin and albumin binding capacities and affinity constants 108 (9;24;3 l ). The first step was to convert the total prednisolone concentrations (ng/ml) to molar concentrations using a molecular weight of360.4 for prednisolone (21). The values used for the binding capacity (N,P,) and affinity constant (Kt) for transcortin were 5.69xl0-7 M and 3.0lxl0 7 UM respectively (24). The values used for the albumin binding capacity (N.P.) were calculated for each patient using their molar albumin concentrations (molecular weight of albumin=66,300) (18). The value used for the affinity constant (K.) for albumin was 2 .05xl0 3 LIM (24). Again, assuming that K.Di<<l, there is one prednisolone binding site per molecule of albumin (N.= l ), and that prednisolone only binds to transcortin and albumin (23;24;28;29), excel solver can be used to solve for Dr in the following equation: where D, = total prednisolone concentration and DF unbound prednisolone concentration (16). The unbound fraction (fu) was then calculated as:

fu=Df Dt
Unbound prednisolone plasma concentration (ng/ml) was then calculated as: Cp unbound prednisolone= Cp total prednisolone·fu.
The AUC for total prednisolone, unbound prednisolone, and total prednisone were determined by the trapezoidal rule (58).
An approach proposed by Mandema, et.al. (1992) (64) was used for the data analysis: 1) a base model was developed for the population, 2) the estimates found during step 1 were used to explore potential covariates with the base mode~ and 3) a mixed effects model was developed to describe the relationship between the covariates and pharmacokinetic parameters.
Pharmacokinetic and statistical models were evaluated to determine the model that best fit the model dataset. To discriminate between models, the following criteria were used: 1) a decrease in the objective function value (which is proportional to minus twice the log-likelihood of the data) of3.84 (i distribution, df=l, p< 0.05) or greater following the addition of a single parameter was deemed statistically significant; 2) diagnostic plots (e.g. predicted concentration versus observed concentration data, predicted concentrations overlaying' all concentration data, weighted residuals versus predicted concentration values), 3) minimization of variances: reduction ofinterindividual variances and residual variability, and 4) the Akaike Information Criterion (AIC) (65;66).

Prednisolone Phannacokinetic Base Model
Initially, a population base model was developed using only the prednisolone data. A one compartment model with first order absorption was used to fit the unbound prednisolone concentration data in order to determine prednisolone's pharmacokinetic parameters prior to the inclusion of the metabolite (prednisone) data. There were not enough concentrations captured during the absorption phase to adequately model the absorption rate constant (k.). Instead, k. was determined by using a range of values and finding which model had the lowest objective function value. The range of values fork.
was found by using Excel solver to solve fork. based on the following equation: where T max= time at which the peak concentration occurs, k.= absorption rate constant, and k= the elimination rate constant (57) The elimination rate constant, k, can be represented as: k=0.693/t112 where t 112 = prednisolone's half-life (57). Based on literature values, the Tmax and halflife ranges for prednisolone are 1-2 and 1.8-3.41 hours respectively (4;11;24;28). A range of absorption rate constants were determined by using combinations of the minimum and maximum Tmax and half-life values. Models were generated using each absorption rate constant. The model that produced the smallest objective function value was determined to be the best fit: a one-compartment model with a first-order absorption rate of2.84 hr"' . The one-compartment model was parameterized as oral Ill clearance (CL/F), apparent volume of distribution 01 IF), and the first-order absorption rate constant (k.) (NONMEM subroutines ADV AN2 TRANS2).

Analysis of Covariates with Prednisolone Base Model
Once the base prednisolone phannacokinetic model was obtained, the posthoc Bayesian estimation (FO method) was implemented to obtain the individual parameter estimates to evaluate potential influences of covariates. An exponential error model for interindividual variability and a combined additive and proportional error model for the residual variability were initially assumed for the covariate analysis. For pharmacokinetic parameters CLIF and V/F, the potential influence of covariates on the individual pharmacokinetic parameter estimates were evaluated. Age, weight, and time post transplant were treated as continuous variables. Type of transplant was treated as a categorical variable (O=single lung transplant, 1 =double lung and heart transplant, 2=double lung transplant). Gender was treated as an indicator variable (O=female, I =male). A concomitant medication was considered as present (O= not present, I =present) if it was taken at any point within five days or on the same day as a concentration value. The following concomitant medications were evaluated as covariates: flucloxacillin, cefotaxime, ceftazidime, imipenem, ciprofloxacin, acyclovir, ganciclovir, amphotericin, itraconazole, lyposomal amphotericin, and septrin. The presence of cystic fibrosis was evaluated as a categorical covariate (O=not present, I =present). Because menopausal status was not collected in this study, a variable was created that was a marker for women over and under the age of fifty with O=under 50 and I =over fifty. Creatinine clearance and cystatin C were evaluated as continuous covariates and included in the model as markers for renal function. Creatinine clearance, cystatin C, time post transplant, and all concomitant medications were only evaluated on CUF. Cortisol was evaluated as a continuous variable and was only evaluated on VIF.
All other covariates were evaluated on both CLIF and V IF.
Each covariate was added individually to the base model If the inclusion of the covariate caused a decrease of the objective function value of at least 3.84, the covariate was deemed as being statistically significant. Covariates that were found to be statistically significant from the initial screening were then added simultaneously to the base model to generate the full model A backward elimination procedure was then performed on the full model. Each covariate was removed one at a time from the full model. If the objective function value decreased by a more conservative value of7.88 Ci distribution, df=l , p< 0.005), the parameter was included in the final model.

Prednisolone and Predoisone Pharmacokinetic Base Model
The prednisone and prednisolone concentration data was simultaneously modeled. The values for both V IF terms (VPIF=apparent volume of distribution for prednisolone and VM/F=apparent volume of distribution for prednisone) were fixed and therefore not estimated. Since there were no statistically significant covariates found on V IF in the covariate analysis, VPIF was fixed to the population average value determined in the prednisolone base model based on unbound prednisolone concentrations (VP/F=420 L).
VM/F was fixed at 55 L based on literature values (39). There were a total of four possible elimination processes that could be modeled: clearance of prednisolone that is not metabolized to prednisone (CPR), clearance of prednisolone that is metabolized to prednisone (CPM), clearance of prednisone that is reconverted to prednisolone (CMM), and clearance ofprednisone that is not reconverted to prednisolone (CMR). Model I modeled all four rates simultaneously (Figure 2). Because of the difficulty in modeling all four rates, several simplified versions of Model 1 were used: . A summary of the model variations follow. The conversion ofprednisone to prednisolone was removed from the model; thus all prednisolone was converted to the metabolite prednisone (CPM) and then eliminated by means other than reconversion to prednisolone (CMR). CPM and CMR were modeled linearly. This model was attempted to see the effect of forcing all prednisolone to be metabolized to prednisone. (CMM) has been reported to be I 0 times the rate of conversion of prednisolone to prednisone (CPM) (11). CMM was fixed at IO times the rate ofCPM. CPR and CPM were modeled linearly.

Model 6: Nonlinear metabolism of prednisolone to prednisone (Figure 7)
Model 6 was a subset of Model 4. The conversion of prednisolone to prednisone was treated as a Michaelis-Menton process. CMM and CPR were modeled linearly. It has been proposed that the conversion of prednisolone to prednisone follows a non-linear process, so this model explores the possibility of CPM being nonlinear (2;24 ).

Model 7: Nonlinear reconversion of prednisone to prednisolone (Figure 8)
Model 7 was a subset of Model 4. The conversion of prednisone to prednisolone was treated as a Michaelis-Menton process. CPM and CPR were modeled linearly. It has been proposed that the reconversion of prednisone to prednisolone may follow a nonlinear process, so this model explores the possibility of CMM being nonlinear (2;24 ).

Model 8: Nonlinear elimination of all prednisolone not metabolized to prednisone
Model 8 was a subset of Model 4. Prednisolone that was not metabolized to prednisone (CPR) was modeled as a Michaelis-Menton process. In this model, the conversion and reconversion ofprednisolone and prednisone (CPM and CMM respectively) were modeled linearly. It has been proposed that the renal clearance of prednisolone may follow a non-linear process, so this model explores the possibility of CPR being nonlinear (24).

Statistical Model
An exponential-error model was used to describe the interindividual variability of the pharmacokinetic parameters. For example: Exponential model: ®j=TV®*EXP(TJ;.e) where 11;.e is a random variable distributed with a zero mean and variance of ro 2 9 and TV® is the population mean value for® .
Residual variability was modeled separately for prednisolone and prednisone using an additive and proportional-error model: where Cii is the observed plasma concentration value for the jth individual at time=i, ~.ii is the model predicted plasma concentration for the jth individual at time=i, &Iii is a randomly distributed variable with a zero mean and variance of cr 2 1, and & 2;i is a randomly distributed variable with a zero mean and variance of cr 2 2.

Prednisolone and Prednisone Plasma Concentration-Time Data:
The fraction of unbound prednisolone at the various total prednisolone concentrations was estimated using a patient's specific albumin concentrations. The results are shown in Figure I 0. The calculated fraction of unbound prednisolone follows an expected curve for saturable protein binding. Figures 11 and 12 show the observed plasma concentration time data for total and unbound prednisolone respectively. In Figure 13, the total prednisone curve shows that prednisone has a slower elimination since the slope is much shallower between the 2 and 6 hour post dose values than the slope seen in the prednisolone plots. The dose normalized AUC of total prednisolone and unbound prednisolone were plotted versus dose in Figures 14 and 15 respectively. The AUC of total prednisone normalized by prednisolone dose versus dose was plotted in Figure 16. A negative slope was seen for the dose normalized AUC of total prednisolone and total prednisone when plotted versus dose (Figures 14 and 16). A slope of zero was seen for the dose normalized AUC of unbound prednisolone versus dose ( Figure 15). The AUC total prednisolone/ AUC prednisone and AUC unbound prednisolone/AUC prednisone versus dose were plotted in Figures 17 and 18 respectively. There was a negative slope when AUC total prednisolone/AUC prednisone was plotted versus dose, and a positive slope when AUC unbound prednisolone/AUC prednisone was plotted versus dose. The AUC unbound prednisolone/AUC prednisone versus AUC unbound prednisolone/AUC total prednisolone was plotted and exhibited a positive slope ( Figure 19).

Prednisolone Model:
The plasma prednisolone concentration time data was best described using a onecompartment model with a first-order absorption rate of2.84 hr·'. lnterindividual variability was described with exponential error terms for CL/F and V IF. Residual variability was described with a combined additive and proportional-error model.
Originally, a base model was developed using total prednisolone concentrations. The  Table 4. In Figure 22, observed unbound prednisolone concentrations were more clustered around the line of identity though still underpredicted. The weighted residual plot showed that larger values of predicted unbound prednisolone concentrations are no longer overpredicted, but the smaller concentrations still had large variability ( Figure   23). The base model results for the unbound prednisolone dataset are given in Table 4.
CL/F and V/F were found to be 17.2 L/h (0.302 L/h/kg) and 416 L (7.298 L/kg) respectively.  Figure 24).

Analysis of Covariates with Prednisolonc Base Model
No trends were seen in the normalized clearance data; time post transplant was determined to be insignificant. At-test assuming equal variances for independent samples was also performed to further evaluate itraconazole use. There was no statistical difference seen between the mean dose normalized unbound AUC of prednisolone between itraconazole and non-itraconazole users.
All significant covariates were then combined to make the full model, which consisted of sex, ciprofloxacin, septrin, amphotericin, imipenem, and cystic fibrosis as covariates for CL/F. Backwards elimination was then performed to generate a reduced model.
Because there is a limit to the number of characters that can be used in a single line of Fortran code, the backwards elimination analysis was done in 2 steps. Each covariate was removed individually from the first full model (sex, ciprofloxacin, septrin, and amphotericin for CL/F). A covariate was retained in the reduced model if there was a significant decrease in the goodness of fit (i.e., objective function value decreased by 7.88 <i distribution, df=l , p< 0.005). Following the first backwards elimination procedure, only sex and ciprofloxacin were identified as significant covariates for CUF.
The second full model consisted of sex, ciprofloxacin, imipenem, and cystic fibrosis as covariates for CLIF. Again, backwards elimination was performed in the same manner to generate the final model. Following the second backwards elimination procedure, only sex and ciprofloxacin were identified as significant covariates for CLIF in the final model ( Table 6). 14.7% respectively) and the variability estimates, m

Prednisolone and Prednisone Pharmacokinetic Base Model
As explained in the Methods, a pharmacokinetic model for prednisolone and its metabolite, prednisone was developed based on the pharmacokinetic characteristics of these species (Figure 2). Since the amount of prednisone formed from prednisolone was unknown it was not possible to use this complete model. Several simplified versions of the complete model were evaluated as described below.
Model 2: No reconversion ofprednisone to prednisolone ( Figure 3) In this model, the reconversion of prednisone to prednisolone (CMM) was ignored. All other elimination processes, the clearance of prednisolone not metabolized to prednisone (CPR), the clearance of prednisolone metabolized to prednisone (CPM), and the clearance of prednisone not reconverted to prednisolone (CMR) were modeled linearly.
This model was highly dependent on initial estimates and structurally unstable. With one set of initial estimates, the clearance of prednisolone was pushed through processes not involving prednisone. Thus CPM and CMR were very small. When the initial estimates were slightly changed, the model "flipped" and virtually all prednisolone was then eliminated through metabolism to prednisone (CPM). Essentially, the model was unable to distinguish between these clearances. The model was unable to account for the interconversion of prednisolone and prednisone.
Model S: Relative value of the interconversion clearances were fixed ( Figure 6) According to the literature, CMM is estimated to be about I 0 times CPM ( 11 ). Thus in this model, CPM was estimated but CMM was fixed to IO times the value ofCPM. All processes were modeled linearly. This model did not adequately describe the metabolite data as seen in diagnostic plots ( Figure 27). The predicted prednisone concentration values were not distributed around the line of identity and were underpredicted. 123 The plots with the AUC for total prednisone indicated that their nonlinearity might be associated with the interconversion of prednisolone and prednisone (Figures 16-19).
Models 6 and 7 were attempted to address these concerns. In Models 6 and 7, the clearance of prednisone was assumed to occur only through its reconversion to prednisolone (i.e. CMR was set to zero).

Model 6: Nonlinear metabolism of prednisolone to prednisone (Figure 7)
The conversion of prednisolone to prednisone was assumed to follow Michaelis-Menton kinetics. This model was structurally unstable and it was not possible to obtain estimates for CPR.

Model 7: Nonlinear reconversion of prednisone to prednisolone (Figure 8)
The reconversion ofprednisone to prednisolone was modeled using Michaelis-Menton kinetics. This model was structurally unstable. Estimates were obtained for the two prednisolone clearances (CPR and CPM). Initially, the model was able to obtain the Michaelis-Menton estimates for the reconversion of prednisone to prednisolone, but again these estimates were highly sensitive to the initial estimates. In this model, the clearance of prednisolone, by processes other than metabolism to prednisone, was modeled as a nonlinear process. The conversion and reconversion of prednisolone and prednisone (CPM and CMM respectively) were modeled linearly.

124
This model was structurally unstable. This model bad equivalent values for CPM and CMM. Essentially, the model was unable to distinguish between these elimination processes.

DISCUSSION
The study demonstrated that a population pharmacokinetic modeling approach could be used to model prednisolone concentration-time data from a thoracic organ transplant clinical trial. Additionally, the study demonstrated that it was possible to identify covariates to explain variability in the pharmacokinetic parameters. Sex and ciprotloxacin were found to be significant covariates for CL/F. It was not possible to model prednisolone and prednisone concentration-time data simultaneously.
The data used for this study consisted of total prednisolone concentrations. A negative slope in the dose normalized AUC of total prednisolone versus dose was observed ( Figure 14); as dose increased the dose normalized AUC decreased. If total prednisolone concentrations exhibited linear pharmacokinetics, the slope of Figure 14 would have been zero (CL=F·Dose/AUC). The negative trend could be explained by one of two reasons; either CL increased with dose or F decreased with dose. Previous studies have suggested that the nonlinearity in prednisolone's pharmacokinetics is due to saturable protein binding. Furthermore Rose et.al. (24) Base models were generated for both total and estimated unbound prednisolone concentrations. In addition to the nonlinearity seen in Figure 14, it appeared that the base model using total prednisolone concentrations was nonlinear based on the negative slope seen in the weighted residuals plot ( Figure 21  cyclosporine inhibits the metabolism of prednisolone (38;67;68;70). It has been suggested that cyclosporine is an inhibitor of CYP3A4 (72), though references as to the mechanisms of the inhibition have not been found in a literature search (73). In two separate studies, Ost and Langhoff et.al. found that patients taking concomitant cyclosporine had lower total prednisolone clearances than those patients taking concomitant azathioprine (67;68). Frey et.al. found no differences between concomitant cyclosporine and azathioprine users in total or unbound prednisolone clearances (38). Rocci et.al. found that concomitant cyclosporine use did not affect the total or unbound prednisolone clearances (70). Both Frey et.al. and Rocci et.al. attributed their different results to the fact that they measured unbound prednisolone concentrations, conducted repeated measurements over a longer interval, and used both intravenous prednisolone and oral prednisone to eliminate confounding from the interconversion process (38;70).
The CLIF values found in the present analysis were about 50% of the healthy volunteers and the unbound prednisolone VIF value of7.298 L/kg were considerably larger than the 127 reported values for cyclosporine and non-cyclosporine users (1.480-1.600 and 1.460-2.100 Ukg respectively), and about five times larger than the healthy volunteers. There was difficulty in estimating V IF in the present study. Data was only collected at approximately three time points for all patients (0, 2, and 6 hours post dose). Thus, a complete concentration-time profile for the entire population was not captured.
Furthermore, owing to the paucity of information in the early period following the dose, assessment of both volume of distribution and k. were extremely difficult. The k. was fixed in this study.
However, it is possible that the study did not have sufficient statistical power to 129 adequately probe the effects of itraconazole. Only seven subjects were talcing itraconazole at some point during the study period and these seven subjects provided a total of only fifteen samples during concomitant itraconazole use.
Concomitant ciprofloxacin was also identified as a significant covariate for CL/F; ciprofloxacin use reduced the unbound prednisolone oral clearance (by 48% and I 0% in females and males respectively). Unbound clearance values for patients using ciprofloxacin were 0.125 L/h/kg and I.JOO L/h/kg for females and males respectively.
Ciprofloxacin is a known inhibitor ofCYPIA2 and CYP3A4 activity (75)(76)(77). A literature search found no reports of interaction studies for prednisolone and CYP I A2 inhibitors. The results from the present study suggest that CYP I A2 may be a pathway for prednisolone metabolism and that CYPIA2 interaction studies may be warranted.
Other studies have reported that that cystic fibrosis patients have increased total prednisolone clearances (50). In the present study cystic fibrosis was not a significant covariate for CL/Fin the final model. However, all of the cystic fibrosis patients (n=6) were male. Thus, it may be impossible to separate the effects of cystic fibrosis and gender in this data set. It is also possible that effects of cystic fibrosis may have elevated the estimates for CL/F in males in the present study. It is interesting to note that the weight normalized values of CL/F tended to be larger in the male cystic fibrosis patients compared to the other male patients ( Figure 28).
130 It proved to be challenging to model prednisolone and prednisone simultaneously. When a complete model that included all elimination processes was used, the model was structurally unstable (Model 1; Figure 2). The amount of prednisone formed from prednisolone was unknown; consequently the estimates of the parameters used in this model were highly unstable. Many variations of Model 1 were used (Models 2-8;  but they were either structurally unstable or did not adequately describe the data. It is believed that part of the difficulty in modeling the data was due to previously noted nonlinearity in the pharmacokinetics of prednisolone and prednisone. Figure 16 showed that the AUC of total prednisone normalized by prednisolone dose decreased with increasing dose (negative slope), indicating the presence of nonlinear pharmacokinetics. If all clearances of prednisolone had been equally affected by protein binding, increasing the dose ofprednisolone simply would have produced a proportional increase in the conversion of prednisone. Assuming linear pharmacokinetics of prednisone, this scenario would have produced a slope of zero in Figure 16.
Alternatively, if the conversion ofprednisolone to prednisone was more sensitive to protein binding than the other clearances, then as the dose of prednisolone increased, the fraction converted to prednisone would have increased disproportionally. Again assuming linear pharmacokinetics of prednisone, a positive slope would have been expected in Figure 16. Instead, the relationship seen (negative slope) could occur through two possible mechanisms: the conversion of prednisolone to prednisone may be a saturable process and/or the elimination of prednisone may be nonlinear. The reason for the saturable metabolism of the conversion ofprednisolone to prednisone is unknown, though it has been proposed that it may be attributable in part to a saturation of 11-~-hydroxydehydrogenase, the enzyme responsible for the interconversion of prednisolone to prednisone (I 0;20). Nonlinearity with prednisone was noted in Figure   18. AUC total prednisolone/AUC prednisone versus dose produced a negative slope ( Figure 17). When correcting for prednisolone's nonlinearity by using unbound prednisolone concentrations instead of total prednisolone concentrations, AUC unbound prednisolone/ AUC prednisone versus dose produced a positive slope ( Figure 18).
Knowing that unbound prednisolone exhibited linear pharrnacokinetics, it was the nonlinearity of total prednisone driving the increase in slope on this plot.
The concentration-dependent binding of prednisolone may affect the interconversion process. If the interconversion ofprednisone and prednisolone depended solely on the concentration of unbound prednisolone, the ratio of unbound prednisolone to prednisone would remain constant, regardless of the unbound fraction ofprednisolone in plasma (slope=zero ). Instead, a positive slope was seen in Figure 19; interconversion is not so lely dependent on protein binding, other factors (e.g enzyme inhibition) must influence the interconversion process (24).  * 95% confidence interval of parameter estimate includes the nuU value. ** The addition of cortisol as a covariate in the base model caused model instability.
The diagnostic plots showed that cortisol did not add any further improvement in the fit of the model and therefore was not included in any further model development.
Abbreviations: CLIF = oral clearance, V /F = oral volume of distribution, type=type of transplant. A one compartment model with a fixed first order absorption rate constant (k.=2.84 hr-1), exponential interindividual variability, and a combined proportional and additive residual error were used.    2) Clearance of prednisolone that is not metabolized to prednisone =CPR 3) Clearance elimination of prednisolone that is metabolized to prednisone = CPM 4) Clearance of prednisone that is reconverted to preclnisolone =CMM 160    Abbreviations: CL/F = oral clearance A one compartment model with a fixed first order absorption rate constant (k.=2.84 hr" 1 ) , exponential interindividual variability, and a combined proportional and additive residual error were used.

SUMMARY OF CONCLUSIONS
The population approach to pharmacokinetic analysis has become a common tool in the drug development process. Two major advantages of this type of analysis are I) the ability to pool data from a population from which it might otherwise be difficult to collect information and 2) the ability to model sparse data. Thus, this approach can be used to reduce the number of clinical trials that need to be conducted in order to obtain alternate dosing information for sub-populations.

Azithromycin Model:
The azithromycin model was an example of an analysis that pooled data from multiple clinical trials. Dosing information in various sub-populations of the pediatric patients was analyzed without having to conduct more clinical trials. The objective of this analysis was to develop a population pharmacokinetic model for 58 pediatric patients taking azithromycin in four separate clinical trials. A two compartment model with parallel zero-order and first-order absorption was found to best fit the data. When The final azithromycin model found in this analysis supports the current weight adjusted dosing guidelines for azithromycin.

Prednisolone Model:
The prednisolone model was an example of an analysis that utilized sparse data collected as a secondary endpoint in a clinical trial . The objective of this analysis was to develop a population pharmacokinetic model from 41 thoracic organ transplant patients dosed with prednisolone. Unbound prednisolone concentrations were estimated and found to follow linear pharrnacokinetics. A one compartment model with a fixed absorption rate constant of2.84 hr"' was found to best fit the unbound prednisolone concentration time data. Sex and concomitant ciprofloxacin use were found to be significant covariates for CL/F. The final model was an improvement over the base model as seen by reductions in %RSE on parameter estimates, a reduction in the residual variability, and an improvement in the diagnostic plots. The prednisolone and prednisone concentration data were simultaneously modeled using final parameter estimates from the prednisolone alone model and literature values for the V IF terms. Many models were developed, but they all proved to be inadequate because of lack of robustness or lack of clinical meaningfulness with our understanding of prednisolone/prednisone pharrnacokinetics.

APPENDIX A
The following text provides additional information on the pharmacokinetics of azithromycin.
The following figures provide additional information on the modeling process that was employed in Manuscripts II and ill.

Azithromycin Pharmacokinetics
Absorption Oral azithromycin has a bioavailability (F) of approximately 37% (1). In a study of twelve patients that had ileostomies, it was found that slow or incomplete absorption was the most important limitation on the bioavailability of azithromycin, as opposed to acid degradation or extensive first-pass metabolism (2).

Distribution
Azithromycin exhibits a rapid distribution into tissues. Azithromycin is actively transported into cells and then slowly released into the extracellular fluid compartments ( 4). There are significantly higher azithromycin concentrations in tissues than in plasma or serum (10-to 100-fold) (I ; [4][5][6]. Serum protein binding is low and variable. A bound fraction of0.5 has been observed for serum concentration ranges of0.02-0.05 mg/L, and 0.7-0.12 for ranges of0.5-2.0 mg/L; lower concentrations of azithromycin exhibit greater protein binding (I ;4;5).
The whiskers of the plot are the nearest value not beyond a standard span from the quartiles. The standard span is calculated as I .5·interquartile range.
Lines outside of the box and whiskers are outliers. 201