Characterisation of binding kinetics

As described in section 4.3.4, a 3-compartment model with a multiplicative error model was used to describe the plasma PK after IV-administration of FP (90 nmol/kg) and budesonide (167 nmol/kg). The compartmental model equations were given by eqs. 4.3-4.5 and the estimated PK-parameters were presented in table 4.1. The unbound drug concentration in plasma (Cu) was calculated from eq. 4.6, where fu

is the fraction unbound in plasma and C1 is the drug concentration in the central

compartment given below:

)

(

)

(t

f

C

t

C



The receptor density (Bmax) in the spleen has previously been measured and found to

be 31.5 nM [138]. The concentration of the receptor-drug complex (RD) could therefore be calculated from the spleen GR occupancy data, which were favoured for modelling purposes as this organ had a higher ratio of total-to-nonspecific tracer binding and thus allowed for more accurate occupancy estimations [126].

For both compounds a high initial occupancy was observed after IV- administration, which then returned to baseline within 7 and 4 h after dosing for FP and budesonide, respectively. The estimated occupancy at 24 h after dosing of FP was higher than after 7 h, which, most likely, is not reflective of the drug-receptor interaction. Rather, the observation may reflect the dynamics of the glucocorticoid receptor with a drug-induced downregulation of the receptor population, which has been demonstrated to take place after administration of GR agonists both in vitro and

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used for parameter estimation for FP and budesonide, respectively. Within these two time intervals, two receptor occupancy observations had been calculated as negative (-4.0 and -5.3%). Negative receptor occupancy is not possible from a theoretical point of view; however, since the maximum tracer concentration is obtained from a separate group of animals, interindividual variability can explain why these two estimates were slightly lower than 0%. As the receptor occupancy cannot be lower than 0%, these two values were set to zero in the modelling data set.

The binding kinetics was described accordingly:

0 ) 0 ( ), ( )) ( )( ( ) ( max    K C t B RD t K RD t RD dt t dRD off u on (4.7) on off d K K K  , (4.8)

where Kd is the dissociation constant, Kon is the association rate constant and Koff is

the dissociation rate constant. As opposed to the PK model, an additive error model was used for the binding kinetics.

Both the PK- and the binding kinetics models were implemented in Phoenix™ WinNonlin® 6.3.0 (Pharsight, Sunnyvale, CA). “The Naïve pooled engine” was used, i.e. the observations were treated as if they came from one individual. This engine minimises the exact negative log-likelihood by using a quasi-Newton algorithm. The default ODE-solver was used for the PK-model (matrix exponential). A stiff ODE-solver was used for the binding kinetics model. The estimated binding kinetics parameters are presented in table 4.4. The observations and the corresponding model fit are shown in figure 4.5a and 4.5b for FP and budesonide, respectively.

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The estimated correlation matrices for FP and budesonide are presented in table 4.5 and 4.6, respectively. A negative correlation between Koff and Kd was found (-0.76

and -0.74 for FP and budesonide, respectively). The optimisation was restarted several times with different sets of initial estimates to check whether it converged to the same estimates. The optimisation algorithm was found to converge to same solution when a broad range of initial estimates were used.

An exhaustive search was subsequently performed in MATLAB to ensure that the global minimum had been found within the expected parameter space. The sum of squares was evaluated at 360000 combinations of parameter values. For FP, Kon

varied between 1 and 100 L/nmol/h and Koff varied between 0.05 and 5 h-1. For

budesonide, Kon varied between 0.1 and 100 L/nmol/h and Koff varied between 0.05

and 10 h-1. The parameter values were selected from a logarithmic scale to ensure that the search was not biased to the large regions.

In the exhaustive search for FP, the lowest cost function was found at 33.8 L/nmol/h and 0.510 h-1 for Kon and Koff, respectively. For budesonide, the lowest cost

function was found at 1.13 L/nmol/h and 1.31 h-1 _{for K}

on and Koff, respectively.

Hence, the exhaustive search showed that the global minimum had been found for both compounds within the defined parameter space. The two exhaustive searches are graphically illustrated in figures 4.6a and 4.6b.

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Figure 4.5 Concentration of receptor-drug complex in the spleen (RDspleen) after intravenous (IV)

administration of a) fluticasone propionate (90 nmol/kg, n = 3/time point) and b) budesonide (167

nmol/kg, n = 3/time point). Observed data are indicated by circles and the model fit by a solid line.

Figure 4.6 An exhaustive search was performed for a) fluticasone propionate, and b) budesonide by

evaluating the cost function (the sum of squares) at 360000 different combinations of parameter

values. These two exhaustive searches confirmed that the global minimum had been found within the

expected parameter space.

Table 4.4 Estimated binding kinetics parameters for fluticasone propionate (FP) and budesonide

(mean ± SE)

Parameter FP Budesonide

Kd (nM) 0.015 ± 0.0045 1.2 ± 0.34

Koff (h-1) 0.51 ± 0.17 1.3 ± 0.39

Kon (L/nmol/h) 34 ± 20 1.12 ± 0.62

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Table 4.5 Estimated correlation matrix for the parameters estimates obtained from modelling of the

binding kinetics of fluticasone propionate.

Kd Koff

Kd 1 -0.76

Koff -0.76 1

Table 4.6 Estimated correlation matrix for the parameters estimates obtained from modelling of the

binding kinetics of budesonide.

Kd Koff

Kd 1 -0.74

Koff -0.74 1

4.4.2 Sensitivity analysis

A sensitivity analysis was performed by considering the partial derivatives of the output RD with respect to each estimated binding kinetics parameter pi. That is, each

partial derivative was evaluated at the final parameter estimate. Phoenix™ WinNonlin® 6.3.0 computed a numerical approximation of the partial derivatives accordingly        ( _i ) ( _i) i p RD p RD p RD , (4.9)

where pi is Kd or Koff (Kon was a secondary parameter) and Δ is the increment

multiplied by the final parameter estimate (Δ = 0.00001). That is, all other parameters but pi were kept at their nominal values in the calculations.

The plots showing ∂RD/∂Kd over time (figs. 4.7a and 4.7c for FP and budesonide,

respectively) indicate that small changes in Kd would have a profound effect on RD

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plasma concentration. This was especially pronounced for FP. In general, with the exception of a small bump, this parameter became less influential over time. Hence, the sensitivity analysis shows that observations made at early time points had the largest influence on the estimation of Kd. From a mathematical point of view, if the

studies were to be repeated they would benefit from a sampling scheme with more frequent sampling at early time points, given that the purpose was to obtain a good estimate of Kd. Nevertheless, in general the sampling scheme used was informative

with respect to Kd. However, at the last time point included in the modelling data set

of budesonide (t = 4 h), the analysis showed that Kd only had a minor effect on the

output RD.

Figures 4.7b and 4.7d show the ∂RD/∂Koff over time for FP and budesonide,

respectively. Again, the analysis shows that the output RD was sensitive to small changes in the investigated parameter (Koff) directly after dosing. ∂RD/∂Koff had the

same behaviour for both compounds and it changed sign from positive to negative over the investigated time course. For FP, RD was very sensitive to changes in Koff at

approximately t = 3 h, whereas the output would have been less sensitive to changes in the parameter at later time points. For budesonide, the corresponding nadir occurred at approximately t = 1.8 h.

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Figure 4.7 Sensitivity analysis was performed by considering the partial derivatives of the output RD

with respect to the estimated binding kinetics parameters Kd and Koff. This was done for both the study

comprising fluticasone propionate (FP) and budesonide.