A Pharmacokinetic Simulation Tool for Inhaled Corticosteroids

Abstract

The pharmacokinetic (PK) behavior of inhaled drugs is more complicated than that of other forms of administration. In particular, the effects of certain physiological (mucociliary clearance and differences in membrane properties in central and peripheral (C/P) areas of the lung), formulation (as it relates to drug deposition and particle dissolution rate), and patient-related factors (lung function; effects on C/P deposition ratio) affect the systemic PKs of inhaled drugs. The objectives of this project were (1) to describe a compartmental model that adequately describes the fate of inhaled corticosteroids (ICS) after administration while incorporating variability between and within subjects and (2) based upon the model, to provide a freely available tool for simulation of PK trials after ICS administration. This compartment model allows for mucociliary removal of undissolved particles from the lung, distinguishes between central and peripheral regions of the lung, and models drug entering the systemic circulation via the lung and the gastrointestinal tract. The PK simulation tool is provided as an extension package to the statistical software R (‘ICSpkTS’). It allows simulation of PK trials for hypothetical ICS and of four commercially available ICS (budesonide, flunisolide, fluticasone propionate, and triamcinolone acetonide) in a parallel study design. Simulated PK data and parameters agreed well with literature data for all four ICS. The ICSpkTS package is especially suitable to explore the effect of changes in model parameters on PK behavior and can be easily adjusted for other inhaled drugs.

Appendices

Appendix

Laplace transformation (17) of Eqs. 1–7 and rearrangements yields

$$ {{\overline{\mathrm{LC}}}_1}=\frac{{\mathrm{L}{{\mathrm{C}}_{1,0 }}}}{{\left( {s+\kappa } \right)}} $$

(10)

$$ {{\overline{\mathrm{LP}}}_1}=\frac{{\mathrm{L}{{\mathrm{P}}_{1,0 }}}}{{\left( {s+{k_{\mathrm{diss}}}} \right)}} $$

(11)

$$ {{\overline{\mathrm{LC}}}_2}=\frac{{{k_{\mathrm{diss}}}*\mathrm{L}{{\mathrm{C}}_{1,0 }}}}{{\left( {s+{k_{{\mathrm{pul},\ C}}}} \right)\left( {s+\kappa } \right)}} $$

(12)

$$ {{\overline{\mathrm{LP}}}_2}=\frac{{{k_{\mathrm{diss}}}*\mathrm{L}{{\mathrm{P}}_{1,0 }}}}{{\left( {s+{k_{{\mathrm{pul},\ P}}}} \right)\left( {s+{k_{\mathrm{diss}}}} \right)}} $$

(13)

$$ \overline{A}={F_{\mathrm{BA}}}*\left( {\frac{{{k_{\mathrm{muc}}}*\mathrm{L}{{\mathrm{C}}_{1,0 }}}}{{\left( {s+{k_a}} \right)\left( {s+\kappa } \right)}}+\frac{A_0 }{{\left( {s+{k_a}} \right)}}} \right) $$

(14)

$$ \overline{X}=\frac{{{k_{21 }}*\overline{P}+{k_{{\mathrm{pul},\ C}}}*{{{\overline{\mathrm{LC}}}}_2}+{k_{{\mathrm{pul},\ P}}}*{{{\overline{\mathrm{LP}}}}_2}+{k_a}*\overline{A}}}{{\left( {s+{k_{10 }}+{k_{12 }}} \right)}} $$

(15)

$$ \overline{P}=\frac{{{k_{12 }}*\overline{X}}}{{s+{k_{21 }}}} $$

(16)

where s represents the Laplace operator, F _BA was defined above, and LC_1,0, LP_1,0, and A ₀ denote the initial amount of drug in the dissolution compartments of the central and peripheral lung, and the GI absorption compartment, respectively. Particularly,

$$ \mathrm{L}{{\mathrm{C}}_{1,0 }}=\mathrm{Dose}*{F_{\mathrm{Lung}}}*{F_C} $$

(17)

$$ \mathrm{L}{{\mathrm{P}}_{1,0 }}=\mathrm{Dose}*{F_{\mathrm{Lung}}}*\left( {1-{F_C}} \right) $$

(18)

$$ {A_0}=\left( {1-{F_{\mathrm{Lung}}}} \right)*\mathrm{Dose} $$

(19)

where Dose is the by-the-inhaler-emitted dose, F _Lung is the fraction of the emitted dose that is deposited in the lung, and F _C is the fraction of the lung dose that is deposited in central lung regions. Substitution of Eq. 10–14 and Eq. 16 into Eq. 15 and defining α * β = k ₁₀ * k ₂₁ and α + β = k ₁₀ + k ₁₂ + k ₂₁ yields

$$ \overline{X}={{\overline{X}}_1}+{{\overline{X}}_2}+{{\overline{X}}_3}+{{\overline{X}}_4} $$

(20)

where

$$ {{\overline{X}}_1}=\frac{{{k_{{\mathrm{pul},\ C}}}*{k_{\mathrm{diss}}}*\mathrm{L}{{\mathrm{C}}_{1,0 }}*\left( {s+{k_{21 }}} \right)}}{{\left( {s+\alpha } \right)\left( {s+\beta } \right)\left( {s+{k_{{\mathrm{pul},\ C}}}} \right)\left( {s+\kappa } \right)}} $$

(21)

$$ {{\overline{X}}_2}=\frac{{{k_{{\mathrm{pul},\ P}}}*{k_{\mathrm{diss}}}*\mathrm{L}{{\mathrm{P}}_{1,0 }}*\left( {s+{k_{21 }}} \right)}}{{\left( {s+\alpha } \right)\left( {s+\beta } \right)\left( {s+{k_{{\mathrm{pul},\ P}}}} \right)\left( {s+{k_{\mathrm{diss}}}} \right)}} $$

(22)

$$ {{\overline{X}}_3}=\frac{{k_a *{\text F_{\mathrm{BA}}}*{A_0}*\left( {s+{k_{21 }}} \right)}}{{\left( {s+\alpha } \right)\left( {s+\beta } \right)\left( {s+{k_a}} \right)}} $$

(23)

$$ {{\overline{X}}_4}=\frac{{k_a *{k_{\mathrm{muc}}}*{\text F_{\mathrm{BA}}}*\mathrm{L}{{\mathrm{C}}_{1,0}}*\left( {s+{k_{21 }}} \right)}}{{\left( {s+\alpha } \right)\left( {s+\beta } \right)\left( {s+{k_a}} \right)\left( {s+\kappa } \right)}} $$

(24)

Anti Laplace transformation (17) of Eq. 20 yields

$$ X={X_1}+{X_2}+{X_3}+{X_4} $$

(25)

where

$$ {X_1}={B_{11 }}*{e^{{-\alpha *t}}}+{B_{12 }}*{e^{{-\beta *t}}}+{B_{13 }}*{e^{{-{k_{{\mathrm{pul},\ C}}}*t}}}-\left( {{B_{11 }}+{B_{12 }}+{B_{13 }}} \right)*{e^{{-\kappa *t}}} $$

(26)

$$ {X_2}={B_{21 }}*{e^{{-\alpha *t}}}+{B_{22 }}*{e^{{-\beta *t}}}+{B_{23 }}*{e^{{-{k_{{\mathrm{pul},\ P}}}*t}}}-\left( {{B_{21 }}+{B_{22 }}+{B_{23 }}} \right)*{e^{{-{k_{{\mathrm{diss},\ P}}}*t}}} $$

(27)

$$ {X_3}={B_{31 }}*{e^{{-\alpha *t}}}+{B_{32 }}*{e^{{-\beta *t}}}-\left( {{B_{31 }}+{B_{32 }}} \right)*{e^{{-{k_a}*t}}} $$

(28)

$$ {X_4}={B_{41 }}*{e^{{-\alpha *t}}}+{B_{42 }}*{e^{{-\beta *t}}}+{B_{43 }}*{e^{{-{k_a}*t}}}-\left( {{B_{41 }}+{B_{42 }}+{B_{43 }}} \right)*{e^{{-\kappa *t}}} $$

(29)

and

$$ {B_{11 }}=\frac{{{k_{{\mathrm{pul},\ C}}}*{k_{\mathrm{diss}}}*\mathrm{L}{{\mathrm{C}}_{1,0 }}*\left( {{k_{21 }}-\alpha } \right)}}{{\left( {\beta -\alpha } \right)\left( {{k_{{\mathrm{pul},\ C}}}-\alpha } \right)\left( {\kappa -\alpha } \right)}} $$

(30)

$$ {B_{12 }}=\frac{{{k_{{\mathrm{pul},\ C}}}*{k_{\mathrm{diss}}}*\mathrm{L}{{\mathrm{C}}_{1,0 }}*\left( {{k_{21 }}-\beta } \right)}}{{\left( {\alpha -\beta } \right)\left( {{k_{{\mathrm{pul},\ C}}}-\beta } \right)\left( {\kappa -\beta } \right)}} $$

(31)

$$ {B_{13 }}=\frac{{{k_{{\mathrm{pul},\ C}}}*{k_{\mathrm{diss}}}*\mathrm{L}{{\mathrm{C}}_{1,0 }}*\left( {{k_{21 }}-{k_{{\mathrm{pul},\ C}}}} \right)}}{{\left( {\alpha -{k_{{\mathrm{pul},\ C}}}} \right)\left( {\beta -{k_{{\mathrm{pul},\ C}}}} \right)\left( {\kappa -{k_{{\mathrm{pul},\ C}}}} \right)}} $$

(32)

$$ {B_{21 }}=\frac{{{k_{{\mathrm{pul},\ P}}}*{k_{\mathrm{diss}}}*\mathrm{L}{{\mathrm{P}}_{1,0 }}*\left( {{k_{21 }}-\alpha } \right)}}{{\left( {\beta -\alpha } \right)\left( {{k_{{\mathrm{pul},\ P}}}-\alpha } \right)\left( {{k_{\mathrm{diss}}}-\alpha } \right)}} $$

(33)

$$ {B_{22 }}=\frac{{{k_{{\mathrm{pul},\ P}}}*{k_{\mathrm{diss}}}*\mathrm{L}{{\mathrm{P}}_{1,0 }}*\left( {{k_{21 }}-\beta } \right)}}{{\left( {\alpha -\beta } \right)\left( {{k_{{\mathrm{pul},\ P}}}-\beta } \right)\left( {{k_{\mathrm{diss}}}-\beta } \right)}} $$

(34)

$$ {B_{23 }}=\frac{{{k_{{\mathrm{pul},\ P}}}*{k_{\mathrm{diss}}}*\mathrm{L}{{\mathrm{P}}_{1,0 }}*\left( {{k_{21 }}-{k_{{\mathrm{pul},\ P}}}} \right)}}{{\left( {\alpha -{k_{{\mathrm{pul},\ P}}}} \right)\left( {\beta -{k_{{\mathrm{pul},\ P}}}} \right)\left( {{k_{\mathrm{diss}}}-{k_{{\mathrm{pul},\ P}}}} \right)}} $$

(35)

$$ {B_{31 }}=\frac{{k_a *{\text F_{\mathrm{BA}}}*{A_0}*\left( {{k_{21 }}-\alpha } \right)}}{{\left( {\beta -\alpha } \right)\left( {k_a -\alpha } \right)}} $$

(36)

$$ {B_{32 }}=\frac{{k_a *{\text F_{\mathrm{BA}}}*{A_0}*\left( {{k_{21 }}-\beta } \right)}}{{\left( {\alpha -\beta } \right)\left( {k_a -\beta } \right)}} $$

(37)

$$ {B_{41 }}=\frac{{k_a *{k_{\mathrm{muc}}}*{\text F_{\mathrm{BA}}}*\mathrm{L}{{\mathrm{C}}_{1,0}}^A*\left( {{k_{21 }}-\alpha } \right)}}{{\left( {\beta -\alpha } \right)\left( {k_a -\alpha } \right)\left( {\kappa -\alpha } \right)}} $$

(38)

$$ {B_{42 }}=\frac{{k_a *{k_{\mathrm{muc}}}*{\text F_{\mathrm{BA}}}*\mathrm{L}{{\mathrm{C}}_{1,0}}*\left( {{ { \it k}_{21 }}-\beta } \right)}}{{\left( {\alpha -\beta } \right)\left( {k_a -\beta } \right)\left( {\kappa -\beta } \right)}} $$

(39)

$$ {B_{43 }}=\frac{{k_a *{k_{\mathrm{muc}}}*{\text F_{\mathrm{BA}}}*\mathrm{L}{{\mathrm{C}}_{1,0}}*\left( {{ {\it k}_ {21 }}-{ { \it k } _a}} \right)}}{{\left( {\alpha -{k_a}} \right)\left( {\beta -{k_a}} \right)\left( {\kappa -{k_a}} \right)}} $$

(40)

ICSpkTS R EXTENSION PACKAGE—HANDS-ON EXAMPLES

The structure and functions of the ICSpkTS extension package are briefly explained in form of two hands-on examples. In the first example, the ICS module of the ICSpkTS package is introduced by a case study comparing the AUC and C _max in healthy subjects and asthmatic patients. In the second example, the effect of having two FP formulations that differ in their pulmonary dissolution rate constant on the PK behavior is used to explain the FP module. Further details and functions of the ICSpkTS package can be found in the official documentation (available via http://www.cop.ufl.edu/pc/research/areas-of-research/inhaled-glucocorticoids/icspkts-r-extension/) and/or by accessing the official R help files.

HANDS-ON EXAMPLE 1: ICS

Running the following code in R will simulate a PK trial when 500 mcg of an hypothetical ICS are administered to 25 subjects per treatment group (healthy subjects (situation A) vs. asthmatic patients (situation B)) and plasma samples are obtained at 0.17, 0.33, 0.5, 1, 1.5, 2, 3, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, and 24 h after administration. The asthmatic patients are modeled by increasing the fraction of drug that is deposited in the central regions of the lung and by lowering the mucociliary clearance rate constant.

#####################################################

#Number of Subjects (n) per Group

#####################################################

n.subjects = 25

#####################################################

#Time points (h) where plasma samples are obtained

#####################################################

Time = c(0.17,0.33,0.5,1,1.5,2,3,4,6,8,10,12,14,16,18,20,22,24)

#####################################################

#Situation A-Model Parameters-Typical Values (TV) and

#Between-Subject Variability (BSV)

#####################################################

Dose.A = 500

TV.FLung.A = 0.2

TV.FC.A = 0.5

TV.FBA.A = 0.1

TV.kdiss.A = 0.3

TV.kmuc.A = 0.5

TV.kpulC.A = 0.4

TV.kpulP.A = 0.4

TV.ka.A = 0.65

TV.CL.A = 49

TV.VC.A = 87

TV.k12.A = 0.1

TV.k21.A = 0.05

BSV.FLung.A = 0.2

BSV.FC.A = 0.2

BSV.FBA.A = 0.2

BSV.kdiss.A = 0.2

BSV.kmuc.A = 0.2

BSV.kpulC.A = 0.2

BSV.kpulP.A = 0.2

BSV.ka.A = 0.2

BSV.CL.A = 0.2

BSV.VC.A = 0.2

BSV.k12.A = 0.2

BSV.k21.A = 0.2

#####################################################

#Situation B-Model Parameters-Typical Values (TV) and

#Between-Subject Variability (BSV)

#####################################################

Dose.B = 500

TV.FLung.B = 0.2

TV.FC.B = 0.8

TV.FBA.B = 0.1

TV.kdiss.B = 0.3

TV.kmuc.B = 0.25

TV.kpulC.B = 0.4

TV.kpulP.B = 0.4

TV.ka.B = 0.65

TV.CL.B = 49

TV.VC.B = 87

TV.k12.B = 0.1

TV.k21.B = 0.05

BSV.FLung.B = 0.2

BSV.FC.B = 0.2

BSV.FBA.B = 0.2

BSV.kdiss.B = 0.2

BSV.kmuc.B = 0.2

BSV.kpulC.B = 0.2

BSV.kpulP.B = 0.2

BSV.ka.B = 0.2

BSV.CL.B = 0.2

BSV.VC.B = 0.2

BSV.k12.B = 0.2

BSV.k21.B = 0.2

#####################################################

#Within Subject Variability (WSV)

#####################################################

WSV = 0.3

#####################################################

#PK Trial Simulation

#####################################################

ICS(plots = FALSE,tables = FALSE)

The following output displaying the AUC and C _max for both healthy subjects and asthmatic patients and 90% confidence intervals of the geometric means ratios (healthy/asthmatic) for both AUC and C _max is generated by the ICSpkTS package.

Simulation was successful

AUC-Means (Arithmetic Means):

Situation A-Situation B

[1] 2.23 1.96

Cmax-Means (Arithmetic Means):

Situation A-Situation B

[1] 0.43 0.34

AUC-90% Confidence Interval (Geometric Mean Ratio):

[1] 0.99 1.28

Cmax-90% Confidence Interval (Geometric Mean Ratio):

[1] 1.04 1.39

Furthermore, a graph showing the average plasma concentration time profiles for both healthy subjects and asthmatic patients is created (Fig. 4).

Fig. 4

Hands-on example 1, ICS module, simulation of a PK trial after administration of 500 mcg of a hypothetical ICS to 25 subjects per treatment group (healthy subjects (situation A) vs. asthmatic patients (situation B)), and plasma samples are obtained at 0.17, 0.33, 0.5, 1, 1.5, 2, 3, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, and 24 h after administration. Asthmatic patients are modeled by increasing the fraction of drug that is deposited in the central regions of the lung and by lowering the mucociliary clearance rate constant

HANDS-ON EXAMPLE 2: FP

Running the following code in R will simulate a PK trial when 500 mcg FP are administered to 35 subjects per formulation group (formulation A and B differ in their dissolution rate constants, B dissolves 3-fold faster) and plasma samples are obtained at 0.17, 0.33, 0.5, 1, 1.5, 2, 3, 4, 6, 8, 10, 12, 16, 20, and 24 h after administration.

#####################################################

#Number of Subjects (n) per Group

#####################################################

n.subjects = 35

#####################################################

#Time points (h) where plasma samples are obtained

#####################################################

Time = c(0.17,0.33,0.5,1,1.5,2,3,4,6,8,10,12,16,20,24)

#####################################################

#Formulation A-Model Parameters-Typical Values (TV) # and Between-Subject Variability (BSV)

#####################################################

Dose.A = 500

TV.FLung.A = 0.16

TV.FC.A = 0.5

TV.kdiss.A = 0.302

TV.kmuc.A = 0.938

TV.kpulC.A = 10

TV.kpulP.A = 20

BSV.FLung.A = 0.2

BSV.FC.A = 0.2

BSV.kdiss.A = 0.2

BSV.kmuc.A = 0.2

BSV.kpulC.A = 0.2

BSV.kpulP.A = 0.2

BSV.CL.A = 0.2

BSV.VC.A = 0.2

BSV.k12.A = 0.2

BSV.k21.A = 0.2

#####################################################

#Formulation B-Model Parameters-Typical Values (TV) and

#Between-Subject Variability (BSV)

#####################################################

Dose.B = 500

TV.FLung.B = 0.16

TV.FC.B = 0.5

TV.kdiss.B = 0.9

TV.kmuc.B = 0.938

TV.kpulC.B = 10

TV.kpulP.B = 20

BSV.FLung.B = 0.2

BSV.FC.B = 0.2

BSV.kdiss.B = 0.2

BSV.kmuc.B = 0.2

BSV.kpulC.B = 0.2

BSV.kpulP.B = 0.2

BSV.CL.B = 0.2

BSV.VC.B = 0.2

BSV.k12.B = 0.2

BSV.k21.B = 0.2

#####################################################

#Within-Subject Variability (WSV)

#####################################################

WSV = 0.3

#####################################################

#PK Trial Simulation

#####################################################

FP(plots = FALSE,tables = FALSE)

The following output displaying the AUC and C _max for both formulations and 90% confidence intervals of the geometric means ratios (A/B) for both AUC and C _max is generated by the ICSpkTS package.

Simulation was successful

AUC-Means (Arithmetic Means):

Formulation A-Formulation B

[1] 0.71 0.73

Cmax-Means (Arithmetic Means):

Formulation A-Formulation B

[1] 0.20 0.38

AUC-90% Confidence Interval (Geometric Mean Ratio):

[1] 0.85 1.09

Cmax-90% Confidence Interval (Geometric Mean Ratio):

[1] 0.45 0.60

Moreover, a graph showing the average plasma concentration time profiles for both formulations is generated (Fig. 5).

Fig. 5

Hands-on example 2, FP module, simulation of a PK trial when 500 mcg FP are administered to 35 subjects per formulation group (formulations A and B differ in their dissolution rate constants; B dissolves 3-fold faster), and plasma samples are obtained at 0.17, 0.33, 0.5, 1, 1.5, 2, 3, 4, 6, 8, 10, 12, 16, 20, and 24 h after administration