2.1 Study Design

This study is a refinement made in a subset of our previous study [5], in which 108 patients with diffuse large B-cell NHL (DLBCL) were included in the multicenter trials LNH2007-3B (NCT00498043) and GOELAMS 02.03 (NCT00841945) [15], 118 patients with CLL were included in the CLL 2010 FMP (NCT01370772) multicenter trial [16], and a registered cohort (University Hospital of Tours, France) of 90 patients with RA reported elsewhere (patients from the AIR-PR cohort, French Society of Rheumatology) [17, 18]. All of these studies were approved by ethics committees. All data sets were anonymized before data analysis.

2.1.1 Rituximab Treatment

Details of protocols have been described elsewhere [5, 6, 15,16,17,18]. In brief, patients with NHL received intravenous rituximab 375 mg/m2 every other week with or without an induction protocol (supplemental doses of 275 mg/m2 at days 1 and 4); patients with CLL received six intravenous rituximab doses of 500 mg/m2, with or without an induction protocol (500 mg on day 0 and 2000 mg on days 1, 8, and 15); and patients with RA received two intravenous rituximab doses of 1000 mg, 2 weeks apart.

2.1.2 Rituximab Concentrations

During rituximab treatment, blood samples were collected before and 2 h after each injection. In addition, in LNH2007-3B, a supplemental sample was drawn 5 days after each injection. At the end of rituximab treatment, no supplemental sample was drawn in patients with NHL, two samples were collected at months 3 and 6 in patients with CLL, and three samples were collected at months 3, 6, and 10 in patients with RA. Rituximab serum concentrations were measured using a validated enzyme-linked immunosorbent assay [19] with a lower limit of quantitation of 0.25 mg/L. Among the 3224 concentration measurements available in the 316 patients, 53 (1.6%) were inferior to the lower limit of quantitation and were left-censored during population pharmacokinetic analysis.

2.1.3 Other Measurements

Counts of CD19 cells were conducted locally in each participating center using flow cytometry. Baseline metabolic tumor volume (MTV) in patients with DLBCL was measured using Imagys (Keosys, Saint-Herblain, France), a semi-automatic software approved by the US Food and Drug Administration, as previously described [15].

2.2 Pharmacokinetic Modeling Strategy

Rituximab concentration–time data were described using nonlinear mixed-effect modeling (population approach) with MONOLIX Suite 2022 (Lixoft®, Antony, France). A large number of iterations with the SAEM algorithm (600 and 200 iteration kernels 1 and 2, respectively) was used for each run. Fisher information matrix and -2.ln-likelihood (-2LL) were estimated using stochastic approximation and importance sampling, respectively. The modeling strategy consisted of building the structural model and then the statistical models (interindividual, residual, covariates).

2.3 Structural pharmacokinetic model design

The ultimate goal of this study was to investigate the potential role of FcγR and FcRn in the distribution and clearance of rituximab. The pharmacokinetic model was developed on the basis of our previously published TMDD model [3], which included a two-compartment model and in which target-mediated elimination was described using the irreversible binding approximation [6, 16, 20]. The irreversible binding elimination term is written as:

where C is the concentration of rituximab unbound to CD20, qualified as “Fab-unbound rituximab”, R is the target (CD20) level unbound to rituximab, and kdeg is the second-order target-mediated elimination rate constant. The kinetics of R are:

$$\fracR}t}=_}-_}.R-_}. C.R,$$

where kin and kout are target zero-order input and first-order output rate constants, respectively. Target levels were not measured but latent, an approach that did not hamper the estimation of parameters, notably baseline target amount (R0 = kin/kout).

This model was refined by describing the influence of FcγR and FcRn on rituximab distribution and elimination through four steps. First, we re-wrote the two-compartment pharmacokinetic model for Fab-unbound rituximab to describe its FcRn-regulated non-specific distribution and elimination using modeling strategies derived from previous studies [21, 22]. Second, we described the potential role of FcγR using strategies derived from TMDD modeling [23, 24] with no FcγR-mediated output. Third, we investigated the potential elimination of Fc-bound rituximab following FcγR-mediated output in a continuation of the second step. Finally, we described the potential role of FcRn expressed by cells involved in FcγR-mediated disposition using modeling strategies similar to those in the first step.

2.3.1 FcRn-Regulated Non-specific Distribution and Elimination of Fab-Unbound Rituximab

The goal of this step was to rewrite the two-compartment model describing the FcRn-mediated distribution and elimination (protection) kinetics of Fc-unbound rituximab. Previous works [21, 22] proposed to describe non-specific elimination of IgG unbound to FcRn as:

where C is Fab-unbound rituximab serum concentration over time corresponding to measured serum concentration, kup is first-order rate constant of IgG cell-mediated uptake, and FU is IgG fraction unbound to FcRn. As in a previous work [25], we assumed that elimination of FcRn-unbound rituximab occurs during its bidirectional transfer between central and peripheral compartments as:

$$\fracC}_}}t}=-_}._}+_}._}\left(1-_2}\right)$$

$$\fracC}_}}t}=-_}._}+_}._}\left(1-_1}\right),$$

where CC and CP are Fab-unbound rituximab concentrations in central (corresponding to measured serum concentration) and peripheral compartments, respectively, and FU1 and FU2 are rituximab fractions unbound to FcRn that are cleared during central-to-peripheral and peripheral-to-central transfers, respectively. Then, FU1 and FU2 were rewritten in function of the amount of FcRn binding rituximab at each transfer cycle (QFN) and the dissociation constant of rituximab and FcRn (KD,FN). Similar to previous works [21, 22], assuming quasi-equilibrium for rituximab-FcRn binding, the central and peripheral concentration of rituximab unbound to FcRn can be written as:

$$_}=\frac\left[\left(_}-_}-_,\text}\right)+\sqrt_}-_}-_,\text}\right)}^-4._,\text}._}}\right]$$

$$_}=\frac\left[\left(_}-_}-_,\text}\right)+\sqrt_-_}-_,\text}\right)}^-4._,\text}._}}\right],$$

where CCU and CPU are central and peripheral concentrations of rituximab unbound to FcRn, respectively. Then, as in Fronton et al. [26], FU1 and FU2 can be calculated as:

$$_1}=\frac_,\text}}_}+_,\text}}$$

$$_2}=\frac_}_}+_,\text}}.$$

Except for TMDD parameters, this model includes four structural parameters (i.e., the central volume of distribution [VC], kup, QFN, and KD,FN) and was compared to our previous two-compartment TMDD model [5].

2.3.2 FcγR-Mediated Disposition of Rituximab

Fab-unbound rituximab was assumed to interact with FcγR-expressing cells with a reversible stoichiometric binding of the rituximab Fc portion to FcγR. The FcγR-mediated disposition of Fab-unbound rituximab was described using TMDD modeling strategies based on quasi-steady-state (QSS) or Wagner approximations [23, 24]. First attempts to detect serum-FcγR binding in central and/or peripheral compartments were made using a turnover model assuming no elimination of rituximab-FcγR complexes:

$$\frac_}t}=_,X}-_,X}. \left(_-_+C\right),$$

where X is level of FcγR unbound to rituximab, XT is total (i.e., unbound + rituximab-FcγR complexes), C is concentration of rituximab unbound to FcγR, qualified as “FcγR-unbound rituximab”, CT is total rituximab concentration (i.e., unbound to FcγR + FcγR rituximab-FcγR complexes), kin,X and kout,X are zero-order input and first-order output of FcγR, respectively, with baseline FcγR amount being X0 = kin,X/kout,X. Concentration of FcγR-unbound rituximab was written as:

$$C=\frac\left[\left(_-_-_\right)+\sqrt_-_-_\right)}^-4._._}\right],$$

where KSS is the steady-state dissociation constant of rituximab–FcγR complexes. Models describing rituximab serum–FcγR binding in central (XC) and/or peripheral (XP) compartments (with respective KSS,C and KSS,P), with CCT and CPT in central and peripheral compartments, respectively, were compared with pharmacokinetic models with no or one less binding compartment.

2.3.3 FcγR-Mediated Elimination of Rituximab

To investigate the influence of FcγR on rituximab clearance, we tested two models of FcγR-mediated elimination of Fab-unbound rituximab.

First, the first-order elimination rate constant of rituximab-FcγR complexes (kint):

$$\frac_}t}=_,X}-_,X}.\left(_-_+C\right)-_.\left(_-_\right).$$

Second, the first-order elimination rate constant of unbound and bound FcγR that assumed equal constants (kout,X = kint, Wagner model), resulting in dXT/dt = 0 and XT = X0.

These models were compared with the turnover model assuming no elimination of complexes obtained in step 2.

2.3.4 FcRn-Regulated Distribution and Elimination of Rituximab by FcγR-Expressing Cells

Because most of the FcγR-expressing immune cells are also involved in FcRn-mediated IgG recycling, we tested whether FcγR binding compartment(s) was (were) associated with rituximab FcRn-mediated disposition using the same model as described in Sect. 2.3.2. We assumed that the amount of FcRn in central (QFN,Xc) and/or peripheral (QFN,Xp) compartments were proportional to baseline FcγR amount (X0) in central (XC0) and/or peripheral (XP0) compartments, respectively: QFN,Xc = αC.XC0 and/or QFN,Xp = αP.XP0, where αC and αP, are proportionality factors to be estimated. The dissociation constant of rituximab and FcRn in central and/or peripheral FcγR binding compartments was assumed to be equal to KD,FN, whereas uptake rate constants in binding compartments (kup,X) were assumed to be different from kup.

2.4 Design of Statistical Models2.4.1 Interindividual and Error Models

The statistical model for interindividual variability was assumed to be exponential for all structural parameters. Interindividual standard deviations were fixed to 0 if impossible to estimate properly (i.e., high shrinkage and/or poor relative standard error). Additive, proportional, and mixed additive-proportional error models were tested.

2.4.2 Covariate Analysis

Underlying disease (DIS, i.e., CLL, NHL, or RA) and sex (male or female) were tested as categorical covariates. Reference categories were RA and female, respectively. Body surface area (BSA, m2), baseline CD19 count (mm-3), and baseline MTV (MTV0, cm3) were tested as continuous covariates and were coded as power functions; BSA and CD19 were centered to their median, whereas MTV0 was centered to 1. Associations between covariates with structural parameters were tested only for parameters with estimable interindividual variability.

2.5 Model Evaluation2.5.1 Model Comparison

Models were compared on the basis of objective functions (i.e., the -2LL) and the Bayesian information criterion (BIC). Structural models were compared using BIC. Lumping of structural parameters KSS and kout was conducted if no increase in BIC was observed. Statistical (interindividual, error, covariate) models were compared using the likelihood ratio test (LRT) assuming that the difference in – 2LL of two nested models followed a χ2 distribution. The association between covariates and structural parameters was assessed in two steps. During the univariate step, the association of each covariate was tested separately on each structural parameter distribution, with an alpha risk of 0.1. During the multivariate step, forward (alpha risk of 0.05) then backward (alpha risk of 0.02) stepwise, covariates significantly associated with parameters during the univariate step were sequentially added in the model (full model) and then sequentially removed (final model).

2.5.2 Model Goodness of Fit

As we developed models involving a considerable number of parameters to be estimated, we evaluated the correlation matrix of estimates and the highest/lowest eigen value ratio (condition number) at each step. We also graphically evaluated models using goodness-of-fit diagnostic plots (i.e., observed vs. population [PRED] or individual [IPRED] predictions; population [WRES] and individual [IWRES] weighted residuals vs. PRED and IPRED, respectively); normalized prediction distribution errors and visual predictive checks were performed by simulating 1000 replicates from the final model fixed and random parameters.

2.5.3 Quantitative Contribution of Target and FcγR to Rituximab Disposition

The objectives of simulations were to evaluate the influence of Fab-target binding and Fc-FcγR and Fc-FcRn binding on rituximab concentration decay over time. The simulated dosing regimen consisted of a unique intravenous 1000 mg dose of rituximab; typical parameters estimated in the final model were used. First, the influence of each compartment on concentration decay was evaluated by setting input rates of target (R), of FcγR in central (XC) compartments or of (XP) compartments to “on” (i.e., using estimates of R, XC, and XP) or “off” (i.e., setting these values to 0) as follows:

1.

Linear pharmacokinetics, no influence of antigen or FcγR (R input = 0, XC input = 0 and XP input = 0).

2.

Influence of antigen, no influence of FcγR (XC input = 0 and XP input = 0).

3.

Influence of FcγR in central compartment (R input = 0 and XP input = 0).

4.

Influence of FcγR in peripheral compartment (R input = 0 and XC input = 0).

5.

Influence of FcγR in central and peripheral compartments (R input = 0 and XC input = 0).

6.

Influence of antigen and of FcγR in central and peripheral compartments (no input rate set to 0).

Then, in each of these conditions, we computed the area under the concentration–time curve towards infinity (AUC∞) and terminal elimination half-life (t½-∞). Finally, the influence of each covariate on rituximab concentration decay in the final model was evaluated by setting each covariate to its 5th, 50th, and 95th values in the population (see Table 1).

Table 1 Baseline characteristics of the patients included in the analyses