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QUANTITATIVE CLINICAL PHARMACOLOGY |
From Pfizer, Inc, Ann Arbor, Michigan (Mr Hutmacher); ZymoGenetics, Seattle, Washington (Dr Nestorov); Globomax ICON, Hanover, Maryland (Dr Ludden); and Amgen, Inc, Thousand Oaks, California (Dr Zitnik, Dr Banfield).
Address for reprints: Address for correspondence: Christopher Banfield, PhD, Amgen, Inc, One Amgen Center Drive, M.S. 38-3-A, Thousand Oaks, CA 91320; e-mail: banfield{at}amgen.com.
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ABSTRACT |
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 TOP ABSTRACT METHODSRESULTSDISCUSSIONAPPENDIXACKNOWLEDGEMENTSREFERENCES |
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Key Words: Population pharmacokinetic/pharmacodynamic (PK/PD) modeling • exposure-response modeling • generalized nonlinear mixed-effects models • logistic regression • NONMEM
Psoriasis is a chronic inflammatory disorder that affects approximately 2% of the world's population.2 The dysregulation of T cell antigen-presenting cell interactions and overexpression of proinflammatory cytokines are hypothesized to play the central role in psoriatic skin lesion pathogenesis.3 T cells and keratinocytes overproduce tumor necrosis factor (TNF), and elevated TNF levels are observed in psoriatic skin lesion relative to uninvolved skin in patients and in normal individuals. Decreased serum and lesional TNF levels have been associated with effective therapies, which have resulted in the clinical improvement of psoriasis.4
Etanercept is a dimeric fusion protein that consists of the extracellular ligand-binding protein of the human 75 kilodalton TNF receptor linked to the Fc portion of human IgG1.5 Etanercept competitively inhibits the interaction of TNF with endogenous cell surface receptors. The TNF-mediated cellular responses are prevented, thereby modulating the activity of other proinflammatory cytokines. Thus, etanercept is expected to mitigate the proinflammatory effects of TNF and reduce the symptoms of psoriatic lesions.
The exposure-time and ER relationships of etanercept have been characterized in several patient populations. A population ER model was developed in adult patients with rheumatoid arthritis and in patients with juvenile rheumatoid arthritis.6,7 The population pharmacokinetics (PK) of etanercept were characterized in patients with ankylosing spondylitis.8 Most recently, an assessment of the dose-exposureresponse relationship for etanercept in psoriasis was initiated. The results of the development of a population PK model have been published in an earlier manuscript.9 The current publication presents the development of a population ER model for the treatment of patients with chronic moderate to severe plaque psoriasis.
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METHODS |
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TOPABSTRACTMETHODSRESULTSDISCUSSIONAPPENDIXACKNOWLEDGEMENTSREFERENCES |
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Study 1 compared placebo and 25 mg of etanercept twice weekly (BIW) for 24 weeks.10 Study 2 compared placebo, 25 mg BIW, and 50 mg BIW for 12 weeks.12 Study 3 compared placebo, 25 mg once weekly (QW), 25 mg BIW, and 50 mg BIW for 24 weeks.11 After 12 weeks, the placebo patients in study 3 were switched to 25 mg BIW in a blinded fashion.
Pharmacokinetic Measurements and Determinations
Serum samples were collected and analyzed for etanercept concentrations according to the following study schedules:
Study 1: Baseline (screening), week 12, week 24
Study 2: Baseline/day 1; weeks 2, 4, 8, and 12; and at early termination
Study 3: Sparse sampling at baseline (0 hours), week 2 (336 hours), week 4 (672 hours), week 8 (1344 hours), week 12 (2016 hours), and week 24 (3864 hours). Semisparse sampling (on a subset of patients) at week 1 (48, 72, 96, and 168 hours), week 12 (1848, 1872, 1920, and 1968 hours), and week 24 (3864, 3888, 3936, 3960, and 4008 hours).
Etanercept serum concentrations were determined using a validated enzyme-linked immunosorbent assay (ELISA).9 The lower limit of quantification (LLOQ) was 0.625 ng/mL, except for study 1, in which the LLOQ was 0.3 ng/mL based on a 1:5 sample dilution developed by Immunex Corporation.14 Serum sample analyses were conducted at Amgen, Inc, Seattle, Washington (study 1); Covance Laboratories, Inc, Madison, Wisconsin (study 2); and MDS-Pharma, St Laurent, Canada (study 3). The accuracy and precision characteristics of the assays have been previously reported.9 Briefly, the accuracy of the assay in study 1 ranged from 1.3% to -24.9%, and its precision ranged from 9.8% to 13.8%. The accuracy of the assay in study 2 ranged from 13% to -13.6%, whereas the precision ranged from 6.5% to 14%. The ranges of the accuracy (6% to -21%) and precision (9% to 14%) of the assay in study 3 were similar to those of study 2.
Pharmacodynamic Determinations
The clinical efficacy measure commonly used in psoriasis is the psoriasis area and severity index (PASI), which is derived as the weighted average of the areas of psoriatic involvement and their respective severities.15 The score ranges from 0 to 72 and can essentially be considered continuous.
The primary clinical endpoint for psoriasis is a binary response indicating the achievement of a 
75% reduction in PASI from baseline. The percent change from baseline was calculated at each visit for every patient. If a 75% reduction was observed for a visit, the patient was considered a "PASI75" responder (PASI75 = 1). Otherwise, PASI75 was set to 0, and the patient was considered a "PASI75" nonresponder.
The PASI scores were determined at baseline/day 1 and weeks 2, 4, 8, 12, 16, 20, and 24 for study 1; at baseline/day 1 and weeks 2, 4, 8, and 12 for study 2; and at baseline/day 1 and weeks 2, 4, 8, 12, 16, 20, and 24 for study 3. Table I displays the number of PASI75 derivations by actual dose regimen and week. Overall, 1333 patients with 7157 data points supported model development.
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Covariate Data
Four continuous covariates (age, baseline PASI [PASB], duration of psoriasis disease [DOPD], and weight) and 3 categorical covariates (prior systemic or photo therapy [PSPT], race, and sex) were considered in the model selection algorithm. Descriptive statistics of the covariates are displayed in Table II. Because of the limited number of non-Caucasians, race was bifurcated into Caucasians (n = 1186 or 89%) and non-Caucasians (n = 147 or 11%). Typically for psoriasis, the majority of the population was male (approximately 2:1 to females).
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Model Development
Exposure Measures
A 1-compartment disposition model with first-order absorption most adequately characterized the etanercept concentration profile.9 Time-dependent disposition parameters were required to model etanercept's atypical accumulation to steady state. Etanercept's multiple-dose profile had 3 stages. The first stage occurred during the first 2 weeks of dosing and exhibited a rapid rise in concentration, which overshot the eventual steady-state levels. This rapid rise was followed by a 2- to 4-week downward correction, characterized by an approximately 30% decrease in average concentration relative to the level at week 2. Thereafter, accumulation resumed and proceeded to steady-state levels by 12 weeks after the first dose. It is hypothesized that the complex cascade of both TNF and etanercept binding and redistribution processes between the blood compartment and the site of action compartment caused this fluctuation. Interindividual and interoccasion (intraindividual) variances (IIV and IOV, respectively) in apparent clearance and apparent volume were modeled.16 Lack of informative sampling during absorption precluded fitting IIV or IOV components on the absorption rate and fitting a lag-time parameter. The overall complexity of this 1-compartment model is reflected by the 13 fixed effects parameters, 3 IIV, and 14 IOV variance components. Ultimately, the PK model was considered to adequately represent the etanercept concentration-time profile.
Three different exposure measures were evaluated for their predictive capabilities of the PASI75 responses in the ER structural model development: cumulative dose (CDOSE), predicted cumulative AUC (PCAUC), and predicted etanercept trough before each PASI75 assessment (PCmin). The predicted exposure measures were generated using the posterior Bayes predictions of the PK parameters based on PK model fit9 that implemented a sequential pharmacokinetic/pharmacodynamic analysis procedure.17 The NONMEM differential equations solver (subroutine DES) was used to numerically compute the integral in the area calculations.18
CDOSE and PCAUC are empirical predictor variables that attempt to aid modeling delay in the response by incorporating the previous exposure trajectory into the current exposure measure. PCmin was assessed for its ability to relate the response to a concentration threshold and does not have an intrinsic delay property.
Model Structure
As PASI75 represents a Bernoulli random variable, the probabilities of the PASI75 responses were modeled as a function of the etanercept exposure using mixed effects logistic regression.19-22 The Laplacian method, which uses a second-order expansion around the empirical Bayes predictions of the interindividual random effects (
s), was implemented in NONMEM and used to approximate the marginal likelihood.
The general functional form of the base model, posited below, is similar to that reported by Zhang et al,17 Sheiner et al,22 and Sheiner21 for post-surgical pain. The logit link function was used to model the probabilities as follows:
 | (1) |
where p(·) represents the probability operator. The intercept parameter, ß (extrapolated to time = 0), reflects the instantaneous logit score of a successful response immediately after the first dose of study medication. The placebo time-effect, fp(tij), was modeled as a linear function of time, that is,
 | (2) |
where 
represents the slope parameter, and tij is the actual time of the jth PASI75 derivation of the ith patient relative to the first dose. Preliminary modeling suggested that more complex models, which allowed the placebo effect to return to 0 at later time points, were overparameterized. The drug effect contribution to equation (1), fd(x(t)ij, tij), was modeled using a nonlinear function of exposure, that is,
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 | (3) |
where Emax is the maximum drug effect, EC50 is the exposure that achieves half of Emax, 
is a shape parameter that reflects the steepness of the concentration response, x(t)ij represents the exposure measure (CDOSE, PCAUC, or PCmin), and keo represents a temporal delay between the exposure and its effect. The placebo and drug components are additive in equation (1) to reflect the assumption that they act independently. The parameter, 
, allows the interpatient variance to change with time. The i represents the random effect for patient i, and its Bayes prediction allows for patient-specific predictions of the probability PASI75ij = 1. The is are assumed to have zero mean and variance 
2.
Exposure Measure Selection
Population mean predictions, individual data predictions, and a posterior predictive check (PPC) were used to evaluate the goodness of fit for each exposure measure. A simple PPC was performed to assess the overall structural and stochastic assumptions of the model.23 Specifically, the check tested the assumption that the s have mean 0 and are normally distributed. Verification of the overall model form, both structural and stochastic, is imperative because logistic mixed-effects models can suffer from severe biases in certain circumstances.24
Covariate Selection Algorithms
The ER model with the best predictive capabilities was subjected to covariate selection. To this end, a model that incorporated all the covariate parameters on the structural parameters of equations (1) through (3) was constructed. This full covariate model was subjected to the Wald approximation method (WAM) and backwards selection algorithm (BASA) to elicit a parsimonious covariate model.
The WAM performed all subset regression and ranked each covariate model by its approximate Schwarz Bayesian criterion (SBC). The 15 models with the greatest approximate SBC were fit using NONMEM, and the NONMEM objective function (OFV) was used to compute the NONMEM-derived SBC. The WAM SBC and NONMEM SBC rankings were then compared.
A backward elimination algorithm was also implemented. The algorithm was terminated when each parameter increased the OFV by more than 8.876 when fixed to its null hypothesis value (no effect). This cutoff value was chosen to emulate the penalty applied to the OFV by the SBC for adding 1 parameter. Thus, the cutoff ensures that each step of the algorithm has a greater SBC than the previous step. The value, 8.876, translates into an 
of 0.00289 (2).1
Covariate Selection Assessment
A nonparametric bootstrap procedure (800 bootstraps) was used to calculate 95% confidence intervals (CIs) for the parameters of the full covariate model. These 95% CIs were also used to assess the performance of the WAM and BASA procedures. The bootstrap was performed on the full model, incorporating all covariates, because it is known that confidence intervals constructed using reduced models can fail to provide full coverage.24
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RESULTS |
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TOPABSTRACTMETHODSRESULTSDISCUSSIONAPPENDIXACKNOWLEDGEMENTSREFERENCES |
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= 0.05 for 2 in the OFV) were fixed to their null hypothesis values.1 The OFV for the initial and reduced base models and the reduced base model parameter estimates are displayed in Table III. The exposure measures generate roughly the same predictions, indicating the population mean has little power to discriminate between models. Again, this goodness-of-fit procedure lacks discriminatory power because the 3 exposure models predicted approximately 96% of the data.
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However, when the PPC was performed, the PCAUC model provided the best concordances between the simulated and observed data overall. As a result, PCAUC was selected as an exposure variable for the ER relationship model. The population mean predictions for PCAUC are displayed in Figure 1 by dose regimen. The PCAUC-based model predicts the central tendency of the data well in general. However, it underpredicts the placebo data at latter time points, possibly due to dropout of patients with inadequate PASI75 response. The PPC results for the PCAUC model are displayed in Figure 2. Generally, the simulated data are centered on the line of concordance.
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parameters), a reduction and stabilization exercise led to the identification of a working full model with the following parameterization:
 | (4) |
 | (5) |
 | (6) |
 | (7) |
 | (8) |
As the WAM and BASA procedures, which rely on the asymptotic distribution of the objective function, yielded contradicting results, the final model was selected based on the nonparametric bootstrap procedure. In total, 734 of 800 bootstrap model fits converged. Models that did not converge suffered from a nonpositive definite Hessian at the 0th iteration; these model runs were not used as they did not produce any parameter estimates. The parameter estimates, asymptotic SEs from NONMEM, the bootstrap mean and standard deviation of the estimates, and the bootstrap 95% CIs are reported in Table IV.
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For model reduction, covariate parameters, which had 95% CIs that did not contain the null hypothesis value of 0, were selected for inclusion in the PASI75 model; this model was denoted model BCIM (for bootstrap confidence interval model). The bootstrap CI suggested that race and sex affected ß, PASB and PSPT affected Emax, and age affected keo.
Inspection of the weight covariate parameter on EC50 in Table IV revealed an estimate far from 0 and with an extremely wide 95% CI. Further inspection revealed that the weight parameter on keo was also of large magnitude and had a broad CI. Both EC50 and keo mediate the PCAUC's contribution to the logit, so it was conjectured that these estimates were highly correlated. In fact, the asymptotic correlation estimate (NONMEM) between those estimates was 0.868; the bootstrap correlation estimate yielded 0.835. Because the weight covariate parameter on EC50 was nearly significant, it was added to the final model.
Thus, the final population pharmacokinetic/pharmacodynamic model for the activity of etanercept in psoriasis is given by
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| (9) |
 | (10) |
 | (11) |
 | (12) |
The parameter estimates and asymptotic SEs from the final model fit are displayed in Table V.
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Data-based diagnostics were performed to evaluate the results of model reduction algorithms. To this end, observed PASI75 percentages were computed from the data after stratifying by the values of the covariates in the final model (equations (9)-(12)). Two examples are discussed below.
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Weight was determined to have the largest effect on PASI75. The magnitude of the effect was a function of the magnitude of the covariate parameter estimate and the covariate data range. The other covariates demonstrated less difference in PASI75 between strata (data not shown).
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DISCUSSION |
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TOPABSTRACTMETHODSRESULTSDISCUSSIONAPPENDIXACKNOWLEDGEMENTSREFERENCES |
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A working full model (equations (4)-(8)) was constructed by considering 7 covariates—namely, age, DOPD, PASB, PSPT, race, sex, and weight. This full model was subjected to the WAM and BASA selection procedures, which yielded contradictive results. Consequently, a nonparametric bootstrap procedure was implemented to validate the selection procedures. The final model (equations (9)-(12)) included race and sex effects on the baseline logit, PASB and PSPT effects on the maximum drug effect, a weight effect on the apparent potency, and an age effect on the rate of drug effect.
The nonparametric bootstrap does not depend on asymptotic cutoff values, and theoretically, it provides some protection from outlying data influencing inference. Often, the bootstrap is only performed on the model resulting from the selection procedures (final model) and is used to "validate" the selection procedure by testing for spurious covariate effects. However, confidence intervals based on model reduction strategies could lack appropriate coverage, thereby failing to guard against spurious effects with the intended degree.25 Overall, the bootstrap increases the information content of the covariate selection validation and may help alleviate the calibration problem in model selection (parameter-specific type I error rates).
The ER model developed herein has 2 inherent limitations. The monotonic, nondecreasing (cumulative) PCAUC would not be able to predict the PASI75 of patients, who were to discontinue etanercept treatment. The PCAUC measure is "irreversible"; the predictions would not return to baseline as etanercept washes out.
The second limitation is also related to the use of PCAUC. The PCAUC allowed the ER model to account for the delay in the PASI75 response relative to the etanercept PK. However, its empirical nature divorced the model from pharmacological interpretation (ie, the response was not linked to etanercept concentration). Hence, interpretation of the efficacy (drugrelated) parameters was clouded in relation to the hypothetical mechanism of action. Differentiation between pharmacokinetic and pharmacodynamic delays in the response was not explicit. For example, EC50 did not (entirely) represent the potency of etanercept, and the keo may not have reflected equilibration of the drug to the effect site. Nevertheless, neither of the limitations affected the primary objective of pursuing an ER model. The model identified covariates predictive of data trends (Figures 3 and 4), and the PPC demonstrated that the model adequately characterized (by simulation) the PASI75 over the entire (clinical trial) design space (Figure 2).
Ultimately, the ER model, in combination with a statistical subgroup analysis, indicated that dose adjustment was not necessary for any patient subpopulation. That is, the statistically significant covariates in the ER model were deemed to be not clinically relevant to the dosing of etanercept. In general, if covariate trends in response data are considered clinically relevant, and dose adjustment for patient subpopulations is desired to promote pharmacodynamic homogeneity, discriminating between drug potency and pharmacokinetic or pharmacodynamic delays would be desirable. In this example, the model was able to identify a trend in PASI75 with weight, but it had difficulty discriminating if the effect influenced EC50 or keo. Although the model predicted an association between PASI75 and weight, statistical correlation analyses of clinical data showed that improvement in PASI score at week 12 was only weakly correlated with body weight at baseline. These analyses demonstrated that a dose adjustment for body weight was not appropriate for etanercept.12 However, for a general hypothetical example, if a dose adjustment could be considered relevant and further analyses (such as subgroup) were inconclusive, determining if a covariate influences the apparent potency or distribution rate to the effect would be important. Patients with extreme values of a covariate, which influenced apparent potency, might require dose adjustments because at the pharmacodynamic equilibrium, the repose for the covariate subgroups would be parallel. In contrast, patients with covariates that influenced the drug distribution rate would not require dose adjustments because the patient covariate groups would eventually achieve the same pharmacodynamic responses as drug levels achieved equilibrium.
Notably, the choice of a binary response variable, such as PASI75, for the ER model by necessity forces a certain degree of empiricism and limits to some extent the mechanistic interpretation of the model components. On the other hand, such model construction is convenient for simulation as it yields prediction of the well-established clinical endpoint for psoriasis directly. The latter was one of the intentions of the model development program.
A model for the continuous PASI scores could promote a clearer mechanistic interpretation but may pose technical difficulties. In fact, in the current exercise, the original pursuit of a continuous PASI model was abandoned by an analyst after encountering several unique data-analytic issues.
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APPENDIX |
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TOPABSTRACTMETHODSRESULTSDISCUSSIONAPPENDIXACKNOWLEDGEMENTSREFERENCES |
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Study 2 Clinical Trial Sites
Academic Medical Centre Amsterdam, Amsterdam, The Netherlands; Atlanta Dermatology, Vein, & Research Center, Alpharetta, Georgia; Bressnick, Gibson, Parker, Dinehart, Sangster Dermatology, Little Rock, Arkansas; Cabinet Medical, Cannes, France; Cabinet Medical, Nice, France; Cabinet Medical, Paris, France; Center for Clinical Studies, PLLC, Houston, Texas; Center for Pharmaceutical Research, Kansas City, Missouri; Central Dermatology, St Louis, Missouri; Centre de Recherche Dermatologique du Quebec, Saint-Foy, QC, Canada; Charite, Berlin, Germany; Charlottesville Medical Research, Charlottesville, Virginia; CHU de Poitiers, Poitiers, France; Dermatology Associates of San Diego County, Encinitas, California; Dermatology Clinic, Moncton, NB, Canada; Dermatology Research Associates, Nashville, Tennessee; Dermatology Specialists, Inc, Vista, California; DermResearch Center of New York, Inc, Stony Brook, New York; DermResearch, Inc, Austin, Texas; George Eliot Hospital, Nuneaton, UK; Hope Hospital, Salford, UK; Hopital de l'Archet II, Nice, France; Hospital Saint-Louis, Universite Paris VII, Paris, France; Innovaderm Research, Inc, Montreal, QC, Canada; International Dermatology Research, Inc, Montreal, QC, Canada; Klinikum der Universitat Frankfurt, Frankfurt, Germany; Klinikum der Universitat Munchen, Munchen, Germany; Mount Sinai Medical Center, New York; MSHJ Research Associates, Halifax, NS, Canada; NewLab Clinical Research, St John's, NF, Canada; North Bay Dermatology Center, North Bay, ON, Canada; Otto-von-Guericke Univesitat Magdeberg, Magdeberg, Germany; Probity Medical Research, Waterloo, ON, Canada; Probity Medical Research, Windsor, ON, Canada; Queen Elizabeth II Health Sciences Centre, Halifax, NS, Canada; Radiant Research-Columbus, Columbus, Ohio; Rockwood Clinic, PS, Spokane, Washington; Royal Free Hospital, London, UK; Ruhr-Universitat Bochum, Bochum, Germany; Solano Clinical Research, Vallejo, California; Technischen Universitat Dresden, Dresden, Germany; Tulane University Medical Center, New Orleans, Louisiana; UMDNJ-Robert Wood Johnson University Hospital, New Brunswick, New Jersey; Univesitair Medisch Centrum St Radboud, Nijmegen, The Netherlands; Universitat zu Koln, Koln, Germany; Universitatsklinikum Kiel, Kiel, Germany; Universitatsklinikum Leipzig, Leipzig, Germany; University Hospital Rotterdam, Dijkzigt, Rotterdam, The Netherlands; University of Utah Medical Center, Salt Lake City, Utah; UTMB Center for Clinical Studies, Houston, Texas; Ziekenhuis Walcheren, Vlissingen, The Netherlands
Study 3 Clinical Trial Sites
Academic Dermatology Associates, Albuquerque, New Mexico; Brown University, Providence, Rhode Island; Case Western University, Cleveland, Ohio; Central Texas Health Research, New Braunfels, Texas; Cleveland Clinic Foundation—Florida, Naples, Florida; Clinical Partners, Johnston, Rhode Island; Clinical Research Specialists, Santa Monica, California; Colorado Medical Research Center, Denver, Colorado; Dermatology Consultants, Nashville, Tennessee; Emory University School of Medicine, Atlanta, Georgia; Hospital of the University of Pennsylvania, Philadelphia, Pennsylvania; Massachusetts General Hospital, Boston, Massachusetts; Medaphase, Inc, Newnan, Georgia; Midwest Arthritis Center, Kalamazoo, Michigan; Minor and James Medical Center, Seattle, Washington; Mount Sinai School of Medicine, New York; New York University Medical Center, New York; Northwestern University, Chicago, Illinois; Oregon Medical Research Center, Portland, Oregon; Physicians Clinic of Spokane, Spokane, Washington; Piedmont Medical Research, Winston-Salem, North Carolina; Probity Medical Research, St Louis, Missouri; Psoriasis Treatment Center of Central New Jersey, East Windsor, New Jersey; Radiant Research, Greer, South Carolina; Radiant Research, Scottsdale, Arizona; Radiant Research, Tucson, Arizona; Radiant Research, West Palm Beach, Florida; Rush Presbyterian St Luke's Medical Center, Chicago, Illinois; Sneeze, Wheeze & Itch Associates, Normal, Illinois; Southern Illinois University School of Medicine, Springfield, Illinois; Stanford University, Stanford, California; Sylvana Research, San Antonio, Texas; Texas Dermatology Research Institute, Dallas, Texas; The Clinic, Lake Charles, Louisiana; The Savin Center, New Haven, Connecticut; UMDNJ-Robert Wood Johnson School of Medicine, New Brunswick, New Jersey; University of Alabama at Birmingham, Birmingham, Alabama; University of Arkansas Medical Sciences, Little Rock, Arkansas; University of California Irvine College of Medicine, Irvine, California; University of Maryland School of Medicine, Baltimore, Maryland; University of Miami, Miami, Florida; University of Michigan Medical Center, Ann Arbor, Michigan; University of Oklahoma, Oklahoma City, Oklahoma; University of Utah Medical Center, Salt Lake City, Utah; Wake Forest University School of Medicine, Winston-Salem, North Carolina; Welborn Clinic, Evansville, Indiana
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ACKNOWLEDGEMENTS |
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REFERENCES |
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