Study design and conduct
Data for the present analyses were obtained from clinical studies of omalizumab in patients with asthma, including long-term adult data from the EXCELS study (Q2948g) and children from study 10 and its extension (Table 1). EXCELS is an ongoing epidemiological study initiated in June 2004 to evaluate the clinical effectiveness and long-term safety of omalizumab in patients with moderate to severe asthma . A substudy of EXCELS was designed to provide a longitudinal assessment of serum IgE concentrations in omalizumab- and non-omalizumab-treated patients. Data as of April 2009 were utilized in the analysis reported here. Study 10 was a phase III, 7-month double-blind, randomized, placebo-controlled trial  with a 5-month open-label extension period  to assess safety and efficacy of omalizumab in children (6–12 years) with allergic asthma requiring daily treatment with inhaled corticosteroids. The extension protocol (10E1) provided continued treatment with omalizumab for up to three further years .
Table 1. Patient numbers and baseline demographic data for studies contributing to the population PK/PD model analysis
|Study||Treatment||Active, placebo||Patient numbers||Demographic data, mean ± SD (range)|
|Treated||Used in analysis||Age (years)||Bodyweight (kg)||Baseline IgE (ng ml−1)|
|10||1 year, washout, 3-year extension then washout||A||225||225||9 ± 2 (5–12)||39 ± 13 (20–79)||841 ± 645 (48–3 071)|
|P||108||80||10 ± 2 (6–12)||39 ± 14 (20–78)||853 ± 710 (75–2 933)|
|10E1*|| ||A||189||189||9 ± 2 (5–12)||38 ± 14 (20–79)||808 ± 624 (48–3 071)|
|EXCELS†||Up to 5 years||A||127||81||51 ± 14 (12–76)||85 ± 21 (46–143)||907 ± 2 281 (44–15 799)|
|IA05||1 year plus washout||A||421||373||9 ± 2 (6–11)||34 ± 11 (19–92)||1 155 ± 846 (65–3 318)|
|P||207||181||8 ± 2 (6–11)||34 ± 12 (20–78)||1 116 ± 795 (70–3 027)|
|8||1 year||A||268||268||39 ± 13 (12–73)||80 ± 20 (39–150)||417 ± 341 (48–2 081)|
|P||257||257||39 ± 14 (12–74)||78 ± 19 (39–136)||451 ± 345 (51–1 699)|
|9||1 year||A||274||271||40 ± 15 (12–76)||77 ± 17 (46–136)||541 ± 411 (51–1 900)|
|P||272||266||39 ± 14 (12–72)||78 ± 18 (40–148)||501 ± 391 (53–1 970)|
|11||32 weeks||A||175||144||43 ± 14 (12–73)||76 ± 18 (41–135)||578 ± 461 (63–2 553)|
|P||164||130||43 ± 14 (12–74)||74 ± 14 (41–115)||613 ± 450 (46–1 902)|
|2306||28 weeks plus washout||A||245||226||42 ± 14 (12–79)||79 ± 20 (45–148)||509 ± 375 (51–1 692)|
|P||232||214||43 ± 13 (14–74)||77 ± 17 (39–143)||479 ± 387 (53–2 173)|
|2204||Single dose,||A||155||152||35 ± 12 (18–64)||71 ± 12 (48–91)||186 ± 124 (47–620)|
|2101||12-week washout||A||180||180||38 ± 13 (18–65)||71 ± 10 (46–90)||204 ± 114 (73–719)|
|Q0673g‡||47 weeks plus washout||A||47||47||31 ± 8 (19–55)||77 ± 13 (51–112)||535 ± 249 (204–1 255)|
In addition to EXCELS and study 10/10E1, the model-based analysis-included data from six other clinical trials, five of which were phase III, randomized, double-blind, placebo-controlled, parallel-group, multicentre trials. Studies 8 [7, 32] and 9 [5, 33] were both phase III studies with 7-month treatment periods and 5-month blinded extension periods that enrolled adolescents and adults with moderate to severe allergic asthma requiring daily treatment with inhaled corticosteroids. Study 11 was a phase III 32-week pilot study to assess the potential for corticosteroid reduction during omalizumab therapy in adolescents and adults with severe allergic asthma requiring daily treatment with high dose-inhaled corticosteroids, with or without oral corticosteroids . Study IA05 was a phase III 1-year study in children (aged 6 to <12 years) with moderate to severe, persistent, inadequately controlled allergic asthma . Study 2306 was a phase III 7-month study of patients with severe atopic (IgE mediated) allergic asthma . Studies 2204  and 2101 were single-dose, parallel-group investigations of omalizumab bioequivalence (150 and 300 mg s.c.) in healthy atopic volunteers with total IgE above normal concentrations (30–300 IU ml−1) at the screening visit. Q0673g was a phase I open-label study investigating the safety, tolerability, pharmacokinetics and IgE pharmacodynamics of high doses of omalizumab in 47 patients with perennial allergic rhinitis, with or without asthma. All studies were approved by Institutional Review Boards and all patients gave informed written consent. The studies were conducted in accordance with the declaration of Helsinki and Good Clinical Practice guidelines.
The basic non-linear model of omalizumab–IgE turnover and binding has been described previously [27, 28]. Briefly, the binding of omalizumab with IgE was written as a system of three differential equations, one for the subcutaneous administration site, one for total omalizumab (free plus complex) and another for total IgE (free plus complex). The equations, in terms of molar masses of omalizumab, IgE and the complex, with time expressed in days, were given by:
where C (complex) is the solution to the quadratic for the equilibrium binding equation:
and S is the amount of omalizumab in the subcutaneous site, XT and ET are molar masses of total omalizumab and IgE; X and E are free omalizumab and IgE, ka is the absorption rate constant, RE is the rate of production or expression of IgE, CLn and Vn are the clearances and volumes of free omalizumab, free IgE and the complex, Kd is the in vivo, apparent equilibrium binding constant and α is the change in the affinity of binding between omalizumab and IgE as a function of the molar ratio of total omalizumab to total IgE. Amounts in the above equations were converted to molar units using the molecular weights of omalizumab (150 kDa) and IgE (190 kDa). It is apparent from the above equation that the production of IgE was assumed to be constant over time, with a fixed rate RE.
The initial observations on the collected data from omalizumab, free and total IgE that prompted the search for time-dependent changes in IgE production was a slight misfit in a previously published model (e.g. see Figure 1 in Lowe et al. ) together with observations from paediatric study 10 that total IgE was decreasing after the initial increase due to the formation of complexes . Further-more, following the patients from study 10E after drug washout, the total IgE values were, on average, lower than at baseline, indicating there to be less IgE in the system. A test was therefore constructed comparing two PK–IgE models, one specifying that IgE production did not change with time and the other allowing changes in turnover. The PK–IgE binding model had two parameters describing IgE turnover: production and clearance. The potential for these to change with time was explored using models where an IgE turnover parameter could change over time from a baseline value to a new state according to an exponential function:
Figure 1. Comparison of individual predicted vs. observed for total IgE data for the competing models. The first row, A, contrasts observed total IgE and individual predictions from models based on data with up to 1 year of treatment. Row B contrasts observed total IgE and individual predictions from models based on data with up to 5 years of treatment. The comparisons are given for observations from the 3–5 year data only. The blue dots are data from paediatric study 10E1, the black triangles from the adult EXCELS study. The line of identity is red. Note that minimization for the control model in row B terminated with errors and should be regarded cautiously. 10E1 (); EXCELS (▵)
Download figure to PowerPoint
where RB denotes the baseline IgE production rate, RN the IgE production rate after reaching new equilibrium and kE the rate of change over time in IgE production. All three parameters, RB, RN and kE, were allowed to vary randomly between patients, with kE, especially, able to be either positive or negative. This was important to enable the IgE production for placebo-control patients (kE,P) to change over time, either up or down. Parameter kE,P, for the placebo patients, was, effectively, a disease progression parameter, estimated separately from patients treated with omalizumab (kE,X). Initially, this model was run on an early data set consisting of phase III studies 8, 9 and 2306, plus bioequivalence data. The objective function was significantly lower, by 2147 points, for the time changing IgE production model compared with the control-fixed production model, with the residual error variance for total IgE decreasing from 25.0% to 21.1% CV. Alternative hypotheses were also investigated. If IgE clearance was allowed to increase, the objective function decreased by 2053 points and total IgE residual variance decreased to 21.6% CV. When changes in both IgE production and clearance were specified, the model became over-parameterized with a singularity in the R matrix and nonsensical values for some of the parameters, such as the projected steady-state IgE clearance (1.78 × 10−11 l day−1) and inter-individual variance in the rate of change of IgE clearance (0.14% when expressed as CV).
The second alternative was a feedback model. This assumed that IgE production is, effectively, under the control of free IgE, through the binding of IgE-allergen complexes to receptors. IgE production can therefore increase or decrease with time. This was implemented by applying a positive feedback such that free IgE controls IgE production through a modulating differential equation, which is shown below together with the updated differential equation for total IgE:
where RB denotes the baseline IgE production rate prior to anti-IgE therapy, RN the IgE production rate after reaching new equilibrium, kE the background rate of change in IgE production, the disease progression parameter, to account for IgE changes in placebo treated patients. Finally, kM is the turnover rate of the modulator, accounting for physiological and biochemical delays between changes in free IgE concentrations and changes in IgE production. The three parameters RB, RN and kE were allowed to vary randomly between patients; when tested on kM, inter-individual variability was very small and not included in the model. In the final model, the value of kE (and associated inter-individual variability) was fixed following a separate fit of an exponential function to IgE data from patients receiving a matched placebo to omalizumab. Two parameters, the absorption rate constant, ka and volume of complex, VC, as well as their associated inter-individual variances, were fixed using improved first order conditional with interaction estimates obtained from a separate analysis of two richly sampled, single-dose bioequivalence studies, using the control (invariant IgE production) model.
The above models were compared using a fixed starting covariate adjustment based on previous modelling of the data. Covariates included age less than 12 years, bodyweight, body mass index (BMI), race (Caucasian, Black, Oriental and other), gender and baseline IgE concentration. The effect of covariates was multiplicative, i.e. for continuous covariates coefficients were exponents whereas for categorical covariates they were interpreted as ratios relative to the reference category. Continuous covariates were normalized relative to historical reference values: 70 kg for bodyweight, 365 ng ml−1 for baseline IgE and 20 kg m−2 for BMI. The starting inter-individual variability structure was also taken from the same previous model. Random effects characterizing inter-individual variability acted multiplicatively on selected model parameters through log-normal distributions. There were multiple criteria for model selection: (i) significant changes in the log-likelihood objective function; (ii) reduced in inter- and/or intra-individual variances; (iii) increased precision of the parameter estimates, whether they be structural or random effects (variances); (iv) diagnostic plots of predicted vs. observed, or time or predicted concentrations vs. residuals which were without overt bias; and, finally, (v) improved ability to predict the total IgE response in the 3–5 year period of time and for rich data from a phase I study q0673g. For the latter, predictions from the control (invariant production) and feedback models were created for total IgE samples from these patients, conditional upon the population parameters from the main analysis, using the post hoc procedure from NONMEM (setting MAXEVAL = 0 in the $EST statement). These were then compared with the original data as weighted residuals. Further covariate searches on the newly introduced parameters, as well as refinement of the interindividual variance structure, were undertaken after identification of the best model in the initial comparison. Random effects acted multiplicatively on the new parameters except for kE where an additive effect was assumed. The criterion for addition or removal of a covariate was a significant change in the objective function (P < 0.05, χ2-test with the appropriate degrees of freedom).
The final model from this analysis is provided as an electronic supplement available from the British Journal of Clinical Pharmacology website.