Population Pharmacokinetics of Fusidic Acid: Rationale for Front-Loaded Dosing Regimens Due to Autoinhibition of Clearance

Abstract

The objectives of this analysis were to develop a population pharmacokinetic (PK) model to describe the absorption and disposition of fusidic acid after single and multiple doses and to determine the effect of food on the rate and extent of bioavailability. Plasma PK data from three phase 1 studies (n = 75; n = 14 with and without food) in which healthy subjects received sodium fusidate (500 to 2,200 mg) as single or multiple oral doses every 8 h (q8h) or q12h for up to 7 days were modeled using S-ADAPT (MCPEM algorithm). Accumulation of fusidic acid after multiple doses was more than that predicted based on single-dose data. The PK of fusidic acid was best described using a time-dependent mixed-order absorption process, two disposition compartments, and a turnover process to describe the autoinhibition of clearance. The mean total clearance (% coefficient of variation) was 1.28 liters/h (33%) and the maximum extent of autoinhibition was 71.0%, with a 50% inhibitory concentration (IC₅₀) of 46.3 mg/liter (36%). Food decreased the extent of bioavailability by 18%. As a result of the autoinhibition of clearance, steady state can be achieved earlier with dosing regimens that contain higher doses (after 8 days for 750 mg q12h and 1 day for 1,500 mg q12h on day 1 followed by 600 mg q12h versus 3 weeks for 500 mg q12h). Given that large initial doses autoinhibit the clearance of fusidic acid, this characteristic provides a basis for the administration of front-loaded dosing regimens of sodium fusidate which would allow for effective concentrations to be achieved early in therapy.

INTRODUCTION

Although fusidic acid (or sodium fusidate) has been used for the treatment of staphylococcal infections in patients since the early 1960s outside the United States, an act of Congress was required to resurrect this agent from the “dead drug list” in the United States in order for it to be developed for the treatment of patients with chronic staphylococcal infections. The longstanding use of fusidic acid in other countries, including the United Kingdom, Australia, and Canada, for the treatment of staphylococcal infections has allowed a better understanding of the safety profile for this agent. In addition, clinical studies conducted over the last 2 decades (1–4), including recent phase 1 and 2 studies (5, 6), have shown fusidic acid to be safe and well tolerated.

Fusidic acid is orally bioavailable and extensively metabolized, with the metabolites predominantly eliminated by biliary excretion. While at least three of the seven identified fusidic acid metabolites have antimicrobial activity (7, 8), such activity is less than that of fusidic acid. Given that fusidic acid has never been evaluated according to the requirements of modern drug development, major clinically relevant gaps exist in our understanding of its pharmacokinetics (PK). Early PK studies for fusidic acid were based on data from bioassays (8), and, as such, these findings may be biased given the similarity in MIC values for active metabolites and fusidic acid against Staphylococcus aureus. Additionally, fusidic acid exhibits complex and nonlinear PK. Previous studies have demonstrated decreased apparent total clearance after dosing regimens containing multiple intermediate and high doses (500 mg every 12 h [q12h] or higher) compared to single doses, as assessed by the extent of fusidic acid accumulation (9–11). Such accumulation was evident after low doses (250 mg q12h) (12). Other PK features, such as a longer terminal half-life after multiple intravenous or oral doses than after a single dose, time to steady state exceeding five times the terminal half-life, and an effect of food on the rate and extent of bioavailability (8, 9, 11, 13, 14), are observed with fusidic acid. However, quantitative PK models describing such attributes have not been characterized. A population PK model, which describes the disposition of fusidic acid, together with the application of pharmacokinetic-pharmacodynamic (PK-PD) principles, would be a valuable tool that could be used to support the selection of optimal dosing regimens of sodium fusidate.

Using data from healthy subjects from three phase 1 studies, the objectives of this analysis were 2-fold. The first objective was to develop a population PK model to describe the absorption and disposition of fusidic acid after single and multiple oral doses. The second objective was to determine the effect of food on the rate and extent of fusidic acid bioavailability.

(Parts of this work were presented as poster A1-1932 at the 49th Interscience Conference on Antimicrobial Agents and Chemotherapy, San Francisco, CA, 2009.)

MATERIALS AND METHODS

Subjects and study designs.

Data were pooled from three phase 1 studies, studies CE06-102A (study 102-A), CE06-102B (study 102-B), and CE06-103 (study 103), which were conducted in healthy adult subjects. All subjects gave their written informed consent prior to entering into the respective study. The studies were approved by the responsible Institutional Review Board and followed the Declaration of Helsinki.

Study 102-A was a 3-period, randomized, crossover study in which 28 subjects (23 males, 5 females) receiving single doses of sodium fusidate were evaluated in the fasted state and 14 of the 28 subjects were also evaluated in the fed state. For administration in the fasted state, subjects received 500 mg of sodium fusidate as the test formulation (CEM-102, equivalent to 480 mg fusidic acid) or reference formulation (Fucidin; Leo Pharma) in study periods 1 and 2. In study period 3,500 mg of sodium fusidate was given under fed conditions (14 subjects) or as Fucidin (14 subjects). For dosing in the fed state, subjects consumed a high-fat, high-calorie breakfast within 30 min prior to dosing. While PK data from all 28 subjects in this study were available, data from the reference formulation arm were not used for the analysis described herein.

Study 102-B was a placebo-controlled, randomized study in 24 subjects (17 males, 7 females) who received 13 doses of 500 mg of sodium fusidate or placebo every 8 h (q8h) in the fasted state. Eighteen subjects received sodium fusidate and six subjects received placebo. Of the 24 subjects, one was discontinued from the study due to a macular rash which occurred after a dose of sodium fusidate. The rash was judged as probably related to sodium fusidate by the clinical investigator. Therefore, 17 subjects from this study were available for the population PK analysis.

Study 103 was a double-blind, randomized, placebo-controlled, single- and multiple-dose dose escalation study in 32 subjects (17 males, 15 females) (6). This study was originally designed to contain four dose groups receiving 550, 1,100, 1,650, and 2,200 mg of sodium fusidate in the fasted state (cohorts 1, 2, 3, and 4, respectively). In each dose group, 6 subjects received sodium fusidate as a single dose and 2 subjects received placebo in period 1. After a 7-day washout period, cohorts 1, 2, and 3 received 11 doses of 550, 1,100, or 1,650 mg of sodium fusidate q12h, respectively, in period 2. Data from 1 subject in cohort 3 were not available for PK analysis due to nausea and vomiting in period 2. Given the dose-limiting gastrointestinal intolerance observed after the administration of the 1,650-mg q12h regimen in cohort 3, doses for cohort 4 were reduced in period 2. In this period, cohort 4 subjects received 2 doses of 1,100 mg q12h followed by 13 doses of 550 mg q12h. An additional group of 6 subjects (cohort 5), who received 2 doses of 1,650 mg q12h followed by 13 doses of 825 mg q12h, was studied.

For each of the phase 1 studies described above, film-coated tablets containing 250 or 275 mg of sodium fusidate were administered together with 240 ml of water at ambient temperature. Intake of water or other fluids was not permitted from 2 h before to 2 h after dosing (except for the water given during dosing and the milk given with the high-calorie breakfast). All subjects were nonsmokers, had to abstain from caffeine from 24 h before dosing until the last blood sample for PK analysis in the respective period, and were not allowed to consume any products containing grapefruit from 7 days prior to the first dose to the last blood sample for PK analysis. For all single doses and for the last morning dose of multiple-dose regimens, subjects fasted for at least 10 h prior to and at least 4 h after dosing (with the exception of subjects studied under fed conditions). Lunch was provided approximately 4.5 h after the morning dose. For multiple-dose regimens with q12h dosing, the morning dose was given at least 30 min before breakfast and the evening dose was given at least 2 h after dinner.

PK sampling and analytical methods.

All blood samples were collected using tubes with K₂-EDTA as the anticoagulant. The samples were immediately centrifuged, frozen, and stored at −80 C. Plasma samples were prepared for fusidic acid concentration determination by adding tetrahydrofusidic acid as the internal standard to a volume of 100 μl of plasma. An aqueous solution of 250 μl buffer (0.05 M citric acid, 0.2 M dibasic ammonium phosphate) was also added for extraction. Liquid/liquid extraction was then performed with a mixture of dichloromethane–hexane–methyl tert-butyl ether (1:1:1, vol/vol/vol). After centrifugation, the organic phase was concentrated and reconstituted in water-acetonitrile (1:1, vol/vol). A sample volume of 15 μl was injected into a reverse-phase high performance liquid chromatography (LC) system (HP1100 Series system; Agilent) using a Hydro-RP column (2.0 by 100 mm, 4-μm particle size; Phenomenex) that was maintained at 35°C. Retention times were 2.5 min for fusidic acid and 3.5 min for the internal standard. The mobile phase was nebulized by heated nitrogen gas in a Z-spray source/interface. Ions were detected in negative mode by a tandem quadrupole mass spectrometer (MS/MS) (Quattro Ultima; Micromass). The mass-to-charge ratios were 515 to 455 for fusidic acid and 519 to 59.6 for the internal standard.

The assay was linear over a concentration range from 0.0200 to 50.0 mg/liter. The lower limit of quantification was 0.0200 mg/liter. The applicability of the assay to diluted samples was successfully tested using a 10-fold dilution. Interday precision was 12.3% at 0.060 mg/liter, 8.9% at 2 mg/liter, and 8.5% at 40 mg/liter. Intraday precision ranged from 5.1 to 11.6% over the same concentration range. Accuracy ranged from −4.4 to 12.7%, and recovery was 101.6% for fusidic acid.

Noncompartmental PK analysis.

A noncompartmental PK analysis (NCA) was conducted to provide insights about the structure of the PK model. The NCA was performed, as described previously (15), using WinNonlin Pro (version 5.2.1; Pharsight Corp., Mountain View, CA). Apparent terminal half-lives were determined only if the terminal slope could be reliably estimated for the respective profile. The area under the curve (AUC) was calculated using the linear up/log down trapezoidal method as implemented in WinNonlin. AUC_0-∞ was calculated as the area under the curve from time zero to infinity for a single dose, and AUC_0-τ was calculated as the area under the curve during one dosing interval for a multiple-dose regimen. AUC_Dose was defined as the AUC_0-∞ for a single dose or the AUC_0-τ after the last dose for a multiple-dose regimen. The mean input time was approximated as half of the time of peak concentration for the calculation of the apparent volume of distribution at steady state (V_ss/F) (15).

Population PK analysis. Absorption.

Population PK models considered included models with first-order, parallel first-order and Michaelis-Menten (MM), or multiple first-order absorption processes. As an alternative method to describe complex absorption profiles not adequately described by the aforementioned absorption processes, a previously described semiphysiological absorption model (16), which was based on data for single doses, was adapted for the single- and multiple-dose data described herein. For this absorption model, the rate of drug released from stomach [Rel(t)] was allowed to change over time, as shown in equation 1:

$$\text{Rel}\left(t\right)=\frac{V_{\max }\left(t\right)·A_{\text{Stomach}}}{K_{\text{m}}+A_{\text{Stomach}}}$$ (1)

where A_Stomach represents the amount of drug in the stomach, V_max(t) represents the maximum rate of drug release from stomach at time t, and K_m is the amount of drug associated with a Rel(t) equal to half of the maximal rate of V_max(t) at time t. V_max(t) is described by the Hill-type equation provided in equation 2:

$$V_{\max }\left(t\right)=V_{\max }\left(0\right)·\left[1+\frac{E_{\max }·{\text{TSD}}^{\gamma }}{{\text{TSD}}_{\text{50}}^{\gamma }+{\text{TSD}}^{\gamma }}\right]$$ (2)

where TSD is the time since last dose, TSD₅₀ is the time since last dose associated with a half-maximal change of V_max(t), V_max(0) is the rate of drug release from the stomach at time zero, and γ is the Hill coefficient. The change in V_max(t) over time allows the population PK model to accommodate complex absorption PK profiles. The maximum change of V_max(t) over time is characterized by E_max, which was modeled using a logistic transform with a lower and upper limit of −1 and 9. The lower-limit value of −1 for E_max represents complete inhibition of gastric release, and the upper-limit value of 9 for E_max represents a rapid maximum rate of gastric release [10 times V_max(0)], particularly when TSD is much larger than TSD₅₀. The γ was fixed to 10 to support estimation (16).

Given that the time of peak concentration (T_max) is dose independent for sodium fusidate, as reported by MacGowan et al. (17), the rate of gastric release was assumed to be primarily determined by the stomach content and not by the amount of drug in stomach. To accommodate this, the V_max(0)/dose and K_m/dose were estimated. These model parameters were subsequently multiplied by the administered dose to obtain V_max(0) and K_m in equations 1 and 2.

The differential equations for absorption of fusidic acid, which reflect the change in the amount of drug in the stomach (A_Stomach) and intestine (A_Intestine) before transfer to the central compartment, are provided in equations 3 and 4:

$$\frac{d{A}_{\text{Stomach}}}{dt}=-\text{Rel}\left(t\right)$$ (3)

$$\frac{d{A}_{Intestine}}{dt}=F_{\text{fed}}·\text{Rel}\left(\text{t}\right)-k_{abs}·A_{Intestine}$$ (4)

where F_fed represents the relative bioavailability of fusidic acid under fed compared to fasting conditions (F_fed is 1 in the fasted state) and k_abs is the first-order absorption rate constant from the intestine to the central compartment. The mean absorption time (t_abs) was estimated using the inverse of k_abs (1/k_abs).

Disposition.

Population PK models with one or two disposition compartments and first-order, MM, or parallel first-order and MM elimination were considered. To account for the potential saturation of clearance at high fusidic acid concentrations and increasing accumulation with time, a model with autoinhibition of clearance was evaluated. Given that clearance was not anticipated to change instantaneously with drug concentration, as would be assumed with MM elimination, the time course of the inhibition of fusidic clearance was evaluated by a turnover process.

For the two-compartment disposition model with autoinhibition of clearance, the differential equations for the amount of drug in the central (A1) and peripheral (A2) compartments and extent of inhibition (INH), are provided in equations 5, 6, and 7, respectively:

$$\frac{dA1}{dt}=k_{\text{abs}}·A_{\text{Intestine}}-\left[\left(1-\text{INH}\right)·\text{CL}+\text{CL}d\right]·\text{C}1+\text{CL}d·\text{C}2$$ (5)

$$\frac{dA2}{dt}=\text{CL}d·\left(\text{C}1-\text{C}2\right)$$ (6)

$$\frac{d\text{INH}}{dt}=k_{\text{out}}·\left(\frac{I_{\max }·\text{C}{1}^{\text{Hill}}}{{\text{IC}}_{50}^{\text{Hill}}+C{1}^{\text{Hill}}}-\text{INH}\right)$$ (7)

where C1 and C2 are the drug concentrations in the central and peripheral compartments, respectively, k_out describes the turnover rate constant for the inhibition compartment with a maximum extent of inhibition (I_max), CL is clearance, CLd is distribution clearance, IC₅₀ is the drug concentration causing 50% of I_max, and Hill is an estimated Hill coefficient.

While the unknown extent of bioavailability after oral dosing (F) was not included in the clearance and volume terms in above-described equations, such terms represent apparent clearances and volumes. All initial conditions for each of the compartments in the population PK model were zero.

Parameter variability and residual error model.

The variability of all population PK parameters was described by a log-normal distribution with the exception of E_max and I_max. For these two parameters, a normal distribution on a logistically transformed scale was applied. Between-occasion variability (BOV) (18) was considered for absorption parameters. The residual error was described by a proportional plus additive model. The Beal M3 method (19) was used to fit observations reported to be below the lower limit of quantification (BLQ). Using this approach, the algorithm considers the likelihood distribution of the BLQ value to be normally distributed and to be between negative infinity and the lower limit of quantification.

Covariate effects.

Relationships between all PK parameters and select covariates, such as age, body size, and sex, were visually assessed for trends by plotting the post hoc individual PK parameter estimates or the individual random deviates from the population mean versus continuous or categorical covariates. Covariate evaluations for body size also included the evaluation of an allometric body size model to reduce the between-subject variability (BSV) of clearances and volumes of distribution (20–22). For these evaluations, the clearances and volumes of distribution were normalized using a standard total body weight of 70 kg and fixed allometric exponents of 0.75 and 1 for clearances and volumes of distribution, respectively.

Model discrimination.

Model discrimination was carried out by conducting visual predictive checks and by evaluating standard diagnostic plots, the objective function (−1 · log likelihood), and normalized prediction distribution errors (23), as calculated in S-ADAPT 1.57 beta (24).

Computation.

The population PK analysis was performed in S-ADAPT (versions 1.56 and 1.57 beta) (24) using the parallelized importance sampling Monte Carlo parametric expectation maximization algorithm (PMETHOD = 4 in S-ADAPT). Estimation settings that have been previously qualified for complex population PK-PD models (25) were used. A translator tool (SADAPT-TRAN) was used to facilitate model building, model evaluation, and automated plotting (26). The visual predictive checks evaluating the performance of the model relative to the observed data were performed in NONMEM VI (level 1.2), and deterministic simulations were conducted in Berkeley Madonna (version 8.3.14).

RESULTS

Demographic data, by study and overall, for all three phase 1 studies are provided in Table 1. The NCA PK parameter estimates for fusidic acid by study and dosing regimen are provided in Table 2. The geometric mean C_max normalized to a 500-mg dose of sodium fusidate ranged from 27 to 38 mg/liter after single doses ranging from 500 to 2,200 mg, thus indicating no apparent dose dependency. As shown in Table 2 for study 103, the median T_max, which after morning sodium fusidate doses of 500 to 1,650 mg ranged from 1.5 to 4 h, appeared to be independent of dose. However, the median T_max associated with the morning 2,200-mg dose, which was 6 h, occurred modestly later. Compared to the morning doses of the same amount, the median T_max occurred 1 to 5 h later for the evening doses (3.5 to 8 h). Both the apparent terminal half-life and V_ss/F were variable and showed no apparent dose dependency. There was a greater-than-dose-proportional increase in the AUC_Dose after a single dose in study 103, and the apparent total clearance (CL_T/F) decreased systematically with dose from 1.40 liters/h for 550 mg to 0.846 liters/h for 2,200 mg.

Table 1.

Summary statistics of demographic characteristics for phase 1 subjects included in the analysis data set by study and for all subjects^a

Study and subject group	No. of subjects		Mean (SD)
	Male	Female	Age (yr)	Ht (cm)	Wt (kg)	Body mass index (kg/m2)
Study 102-A	23	5	42.4 (8.61)	173 (8.32)	74.2 (9.66)	24.7 (2.06)
Subjects only fasted	13	1	45.1 (8.33)	175 (7.35)	73.6 (9.11)	24.1 (1.90)
Subjects both fed and fasted	10	4	39.7 (8.29)	172 (9.2)	74.7 (10.5)	25.3 (2.10)
Study 102-B	12	5	39.8 (8.65)	170 (9.84)	72.4 (12.6)	24.8 (2.60)
Study 103	15	15	37.8 (10.5)	166 (8.38)	73.5 (10.5)	26.7 (2.44)
Cohort 1	4	2	35.5 (11.1)	168 (9.97)	70.2 (14.9)	24.8 (3.64)
Cohort 2	3	3	37.7 (6.65)	163 (4.86)	75.5 (7.29)	28.4 (1.33)
Cohort 3	4	2	38.8 (7.99)	168 (9.99)	76.8 (13.4)	26.9 (2.04)
Cohort 4	2	4	40.7 (14.4)	162 (7.56)	71.2 (6.85)	27.1 (1.86)
Cohort 5	2	4	36.5 (13.5)	167 (9.35)	73.7 (10.1)	26.4 (1.93)
All subjects	50	25	40.0 (9.53)	170 (9.23)	73.5 (10.6)	25.5 (2.51)

^aNumbers in parentheses indicate standard deviations.

Table 2.

Noncompartmental PK parameter estimates for fusidic acid^a

Study	Cohort	N	Dose (mg)	Dosing interval	Dose no.b	Geometric mean (% CV)c of PK parameter
						Cmax (mg/liter)	Tmax (h)	t1/2 (h)	AUC0–12 h (mg · h/liter)	AUCDosed (mg · h/liter)	CLT/F (liters/h)	Vss/F (liters)
103	1	6	550	Single	1	31.2 (32)	2.0 (2.0, 3.0)	20.4 (25)	222 (42)	377 (59)	1.40 (59)	22.5 (40)
		6	550	12 h	1	30.7 (35)	2.0 (1.5, 2.0)		193 (41)
		6	550	12 h	2	31.1 (45)	7.0 (1.0, 8.0)		239 (53)
		6	550	12 h	11	105 (40)	3.0 (1.5, 4.0)	13.5 (25)	886 (50)	886 (50)	0.596 (65)	12.8 (60)
	2	6	1,100	Single	1	71.5 (15)	3.5 (1.0, 4.0)	17.7 (9)	561 (15)	1,072 (23)	0.984 (20)	15.5 (8)
		6	1,100	12 h	1	82.7 (18)	1.5 (1.5, 3.0)		536 (7)
		6	1,100	12 h	2	102 (48)	4.0 (3.0, 11.5)		737 (20)
		6	1,100	12 h	11	276 (19)	4.0 (4.0, 8.0)	16.7 (31)	2,483 (16)	2,483 (16)	0.425 (22)	14.2 (16)
	3	6	1,650	Single	1	99.4 (25)	3.0 (2.0, 4.0)	15.1 (15)	808 (28)	1,662 (39)	0.952 (49)	17.8 (18)
		5	1,650	12 h	1	90.8 (20)	4.0 (2.0, 8.0)		682 (12)
		5	1,650	12 h	2	105 (7)	8.0 (2.0, 11.5)		979 (8)
		5	1,650	12 h	11	316 (7)	4.0 (1.5, 4.0)	26.1 (61)	3,304 (4)	3,304 (4)	0.479 (4)	24.6 (42)
	4	6	2,200	Single	1	126 (22)	6.0 (3.0, 8.0)	16.3 (26)	981 (25)	2,493 (39)	0.846 (47)	17.1 (32)
		6	1,100	12 h	1	64.5 (22)e	2.5 (1.5, 4.0)		507 (27)e
		6	1,100	12 h	2	82.7 (31)e	7.0 (2.0, 11.5)		767 (31)e
		6	500	12 h	15	141 (26)e	4.0 (3.0, 6.0)	18.0 (40)	1,370 (25)e	1,370 (25)e	0.385 (21)	10.2 (20)
	5	6	1,650	12 h	1	91.8 (21)	2.5 (2.0, 6.0)		819 (16)
		6	1,650	12 h	2	157 (11)	3.5 (1.5, 8.0)		1,537 (11)
		6	825	12 h	15	259 (12)e	3.5 (3.0, 4.0)	29.3 (35)	2,551 (13)e	2,551 (13)e	0.310 (12)	14.3 (29)
102-A		28	500	Single	Fasted	27.4 (23)	3.0 (1.0, 4.0)	11.8 (22)	182 (26)	314 (31)	1.53 (32)	21.0 (27)
		14f	500	Single	Fasted	28.4 (20)	2.5 (1.0, 4.0)	11.8 (22)	197 (20)	342 (25)	1.40 (24)	19.4 (24)
		14	500	Single	Fed	21.2 (24)	3.5 (1.5, 6.0)	11.5 (26)	152 (25)	281 (35)	1.71 (31)	24.5 (30)
102-B		17	500	8 h	13	146 (25)	3.0 (0.5, 6.0)	18.0 (29)	1,030 (26)g	1,030 (26)	0.466 (25)	13.1 (23)

^aN, number of healthy subjects; Dose no., number of doses administered; Single, single dose; C_max, maximum observed concentration; T_max, time of C_max postdose; t_1/2, apparent half-life during the terminal phase; AUC_0-12h, area under the curve from time zero to 12 h; AUC_Dose, area under the curve from time zero to infinity for a single dose or area under the curve during one dosing interval after the last dose for a multiple-dose regimen; F, (unknown) extent of bioavailability after oral dosing; CL_T/F, apparent total clearance; V_ss/F, apparent volume of distribution at steady state.

^bDose numbers 1, 11, 13, and 15 refer to morning doses; dose number 2 refers to evening doses.

^cMedian (minimum, maximum) is reported for T_max.

^dThe comparison of AUC_Dose after single and multiple doses assumes that steady state was achieved after multiple dosing. If steady state had not been achieved, AUC_Dose would be expected to be even higher and CL/F lower after multiple doses compared to the estimates reported.

^ePlease note that multiple doses were lower than single dose for cohorts 4 and 5. For cohorts 1, 2, and 3, single and multiple doses administered were the same.

^fData for the 14 subjects who received sodium fusidate in the fasted and fed state within study 102-A are reported in this row to support the conclusions for the effect of food on the rate and extent of bioavailability. These 14 subjects are part of the 28 subjects receiving sodium fusidate in the fasted state.

^gArea under the curve from 0 to 8 h.

The impact of food on the PK of fusidic acid, as assessed by NCA using crossover data from 14 subjects who received sodium fusidate in both the fed and fasted state in study 102-A, is also shown in Table 2. Food decreased the C_max of fusidic acid by a mean of 23.2% and the AUC_Dose by a mean of 16.7%.

As part of the assessment of the extent of nonlinearity in the PK of fusidic acid, the ratios of C_max, AUC, t_1/2, and CL_T/F after multiple doses compared to those after single doses for each cohort in study 103 were calculated and are provided in Table 3. The ratios of the peak concentration after multiple doses to that after a single dose observed for sodium fusidate doses of 550, 1,100, and 1,650 mg were 3.35, 3.86, and 3.33, respectively. Substantial accumulation of fusidic acid was evident for all dose groups after multiple relative to single doses, as evidenced by the 2.19- to 2.35-fold increase in AUC_Dose. In addition, the CL_T/F ratio, which ranged from 0.426 to 0.457, indicated that for single and multiple doses ranging from 550 to 1,650 mg, CL_T/F decreased after multiple relative to single doses by approximately 56%. Given that the CL_T/F ratios for these doses were less than 1, this suggested that the clearance of fusidic acid is nonlinear. Thus, the results of these NCA data suggested that a complex structural PK model would be required to adequately describe the PK of fusidic acid.

Table 3.

Ratio of noncompartmental PK analysis parameter estimates after multiple doses to those after single doses for fusidic acid by cohort for study 103^a

Cohort	Dose (mg)	Geometric mean (% CV) ratio of PK parameter estimate for multiple to single doses
		Cmax	AUC0–12 h	AUCDose	t1/2	CLT/F
1 (n = 6)	550	3.35 (28)	3.99 (26)	2.35 (21)	0.663 (36)b	0.426 (23)
2 (n = 6)	1,100	3.86 (13)	4.43 (12)	2.32 (15)	0.945 (33)	0.432 (17)
3 (n = 5)	1,650	3.33 (30)	4.40 (26)	2.19 (46)	1.80 (65)	0.457 (36)

^aSee Table 2 for abbreviations.

^bThe median ratio was 0.791.

The PK of fusidic acid was best described using a time-dependent mixed-order absorption process, two disposition compartments, and a turnover process to describe the autoinhibition of clearance. A schematic of this model is provided in Fig. 1. The use of a time-dependent mixed-order absorption process provided the best fit of the data as evidenced by an objective function that was better than that obtained using an MM absorption model with a time-independent V_max. The use of two disposition compartments and a turnover process to describe the autoinhibition of clearance successfully described the rate and extent of accumulation after multiple doses and the lower clearance at high compared to low single doses. The parameter estimates for the final population PK model for fusidic acid are provided in Table 4.

Fig 1.

Structure of the final population PK model for fusidic acid.

Table 4.

Parameter estimates for the final population PK model for fusidic acid

Definition	Abbreviation	Unit	Estimatea
			Population mean	Between-subject variability	Between-occasion variability
Apparent maximum clearance without autoinhibition	CL/F70kg	liters/h	1.28b	33.4%
Distribution clearance between the central and peripheral compartments	CLd/F70kg	liters/h	0.714b	56.0%
Vol of central compartment	V1/F70kg	liters	12.3b	14.7%
Vol of peripheral compartment	V2/F70kg	liters	6.96b	46.6%
Vol of distribution at steady state	Vss/F70kg	liters	19.2b,c
Concn causing half-maximal autoinhibition of clearance	IC50	mg/liter	46.3	36.5%
Maximum extent of autoinhibition	Imax		0.710c	0.530–0.821d
Imax on transformed scale	Tr_Imax		0.897	0.338e
Hill coefficient	γ		4.61	15% (fixed)f
Mean turnover time of inhibition compartment	toutg	h	0.940	95.1%
Maximum rate of drug release from stomach to intestine at time zero divided by doseh	Vmax(0)/doseh	1/h	0.554e
Fraction of fusidic acid dose associated with 50% of Vmax(0)	Km/doseh		0.0376	59.2%	18.4%
Fastest half-life of gastric release, if the fraction of dose in stomach is ≪Km/dose	Ln(2) · Km/Vmax(0)	min	2.82i	6.8%	88.2%
Mean absorption time from the intestine to the central compartment	tabsg	min	33.3	65.0%
Time past last dose at which Vmax(t) changed by 50%	TSD50	h	0.509c	20.5%	47%
Maximum fold change in Vmax over time	Emax		1.51c	−0.978–8.93d
Emax on transformed scale	Tr_Emax		−1.09	1.99e	1.87e
Relative extent of bioavailability with a high-fat, high-calorie breakfast	Ffed		0.818	11.4%
Ratio of Vmax(0) in fed vs fasted state	cVmaxfed		1.39	2.7%
Ratio of Km in fed vs fasted state	cKmfed		0.124	4.3%
Ratio of TC50 in fed vs fasted state	cTSD50,fed		1.89	4.4%
Emax in fed state on untransformed scale	Emaxfed		9.00c	8.97–9.00d
Difference of Emax in fed vs fasted state on transformed scale	dEmaxfed		12.2	0.156e
Ratio of Vmax(0) in evening vs morning	cVmaxEve		0.324j
Ratio of Km in evening vs morning	cKmEve		1.27j
Ratio of TC50 in evening vs morning	cTSD50,Eve		3.02j
Emax in evening on untransformed scale	EmaxEve		0.733
Difference of Emax in evening vs morning on transformed scale	dEmaxEve		−0.471j
SD of proportional residual error	CVCP		0.143
SD of additive residual error	SDCP	mg/liter	0.319

^aRelative standard errors for all variability terms could not be computed due to the complexity of the model.

^bEstimates apply to a healthy volunteer with a total body weight of 70 kg using an allometric size model.

^cNot an estimated parameter. Value was calculated from the other parameter estimates.

^dRange of individual estimates.

^eEstimate represents the standard deviation on logistically transformed scale.

^fThe between-subject variability of the Hill coefficient was fixed to a small value (15% CV) to allow estimation of the population mean using the MC-PEM algorithm.

^gt_out equals 1/k_out; t_abs equals 1/k_abs.

^hDose is measured in mg of fusidic acid.

ⁱWe estimated the ratio of V_max(0)/K_m which represents the apparent first-order release rate constant if the fraction of drug remaining in stomach is ≪K_m/dose. The half-life of this process is reported in the table since this value is easier to interpret.

^jThese differences in the absorption parameters between the morning and evening doses were estimated within the between-occasion variability model. The mean of the random deviates for the evening dose was allowed to be different from 1 for V_max(0), K_m, and TSD₅₀ and different from 0 for Tr_E_max.

As evidenced by the low value of the mean K_m/dose (0.0376), the absorption of fusidic acid more closely resembled a zero-order rather than a first-order process. The mean (range) maximum extent of autoinhibition of clearance (I_max) was 71.0% (53%, 82.1%), and half-maximal inhibition was achieved by plasma concentrations (IC₅₀) of 46.3 mg/liter. The Hill coefficient for this relationship was 4.61 and near-maximal autoinhibition was predicted to be achieved at plasma concentrations at or above approximately 100 mg/liter. As shown in the Appendix, the final model with autoinhibition of clearance converges to a model with parallel first-order and MM elimination if the turnover (k_out) of the inhibition compartment is rapid and if the Hill coefficient is 1. Inclusion of an allometric body size model for CL_T/F, CLd/F, V1/F, and V2/F reduced the apparent BSV of these parameters by 13%, 17%, 44%, and 18%, respectively.

The effect of food was also evaluated on the PK of fusidic acid using the population PK model. A relative bioavailability of 81.8% (11.4% coefficient of variance [CV] for BSV) was estimated for fusidic acid under fed compared to fasted conditions. The K_m of the absorption process in the fed state was 87.6% lower than that in the fasted state, indicating that administration with food yielded a more pronounced zero-order-like absorption process. It is important to note that the V_max(t) increased by 1.51-fold in the fasted state and by 9.00-fold in the fed state from its initial value [V_max(0)] at 0.509 h post-morning doses. This suggests that administration with food resulted in a higher rate of absorption of fusidic acid. There was a notably lower rate of absorption for evening compared to morning doses in the fasted state, as evidenced by a V_max(0) for the evening dose that was 32.4% the V_max(0) of the morning dose. As shown in Table 2, this result is consistent with the increased T_max observed for the evening relative to the morning doses.

The relationships between the individual and population-fitted concentrations based on the final population PK model and the observed concentrations are shown in Fig. S1 in the supplemental material. The scatter of the plasma concentrations around the line of identity demonstrated that the model fit the data with good precision and without bias, with or without stratification by dose. The normalized prediction distribution errors across doses (see Fig. S2 in the supplemental material) provided support of the good predictive performance of the final model. As shown in Fig. 2, visual predictive checks evaluating the performance of the final population PK model by study were carried out by comparing the model-predicted concentration-time profiles at the 10th, 50th, and 90th percentile to the observed data. As evidenced by these checks, the final model predicted the median concentration-time profiles after single and multiple doses well for all dosing regimens except for study 102-B and cohort 5 in study 103. For these two groups, median concentration-time profiles were slightly underpredicted.

Fig 2.

Visual predictive check for the final population PK model for study 102-A in the fasted (A) and fed (B) states, study 102-B (C), and cohorts 1 to 5 of study 103 (D to H). The insets show the first 72 h of multiple-dose regimens. Ideally, the percentile from the model predictions (solid lines) should match the respective percentile calculated from the observations (dashed lines). Percentiles for observations were not calculated for groups of 6 or fewer subjects.

To illustrate the features of the autoinhibition of clearance of fusidic acid, accumulation ratios were calculated as AUC_0-τ at steady state or after the 11th dose divided by the AUC_0-∞ after a single dose for a range of sodium fusidate q12h dosing regimens. These ratios were generated using the population mean parameter estimates for the final PK model and a PK model with parallel linear and MM elimination (data not shown). The relationship between the accumulation ratio and sodium fusidate dosing regimens, overlaid with the observed accumulation ratios for the 550, 1,100, and 1,650 mg q12h dosing regimens evaluated in study 103, are provided in Fig. 3. The accumulation ratios for the three studied sodium fusidate dosing regimens in study 103 were very well predicted by the model with autoinhibition of clearance implemented but poorly predicted by the model with parallel linear and MM elimination. For dosing regimens of approximately 400 mg q12h or lower, the accumulation ratio was small due to a lack of clearance inhibition at steady state. For dosing regimens of approximately 450 to 500 mg q12h, higher clearances were evident, a finding that was likely due to limited saturation with initial doses and achievement of clearance inhibition only at steady state. Doses of approximately 750 mg q12h or higher led to notable clearance inhibition, both after a single dose and at steady state. The results of the simulations conducted demonstrated that significant accumulation of fusidic acid occurred with the administration of dosing regimens of approximately 500 mg q12h or total daily doses of 1,000 mg or greater, regardless of schedule of administration, due to the inhibition of clearance.

Fig 3.

Relationship between the accumulation ratios for fusidic acid and sodium fusidate dosing regimens. The accumulation ratios were calculated as AUC_0-τ at steady state or after the 11th dose divided by the AUC_0-∞ after a single dose. The thick lines represent the accumulation ratio using the steady-state AUCs, and the thin lines show the accumulation ratio using the AUCs after the 11th dose. The filled diamonds represent the observed accumulation ratios for the 11th dose from study 103 based on the NCA.

To illustrate the effect of front-loaded dosing regimens on the time to steady state, plasma concentration-time profiles for such dosing regimens were simulated using the population mean PK parameters provided in Table 4. Simulated plasma concentration-time profiles for various sodium fusidate q12h dosing regimens without (Fig. 4A) and with (Fig. 4B) front loading, the latter of which involved the administration of two large doses at 0 and 12 h, along with a comparison of the average steady-state concentration (C_ss) expected with each dose (Fig. 4C), are shown in Fig. 4. As shown in Fig. 4A, the simulation results indicated that the time to steady state was dose dependent and was approximately 3 weeks or longer for q12h dosing regimens with intermediate doses of 500 mg. Times to steady state were approximately 8 and 3 days for q12h dosing regimens containing doses of 750 mg or higher and 250 mg or less, respectively. However, as shown in Fig. 4B, the front-loaded dosing regimens of 1,200 mg q12h or greater on day 1 followed by 600 mg q12h allowed for steady state to be achieved rapidly (24 h or less).

Fig 4.

Simulated plasma concentration-time profiles for various doses of sodium fusidate given every 12 h without (A) or with (B) two front-loaded doses at 0 and 12 h. (C) Comparison of the average concentration at steady state predicted by the differential equations of the full model and by the two equations for the low and high clearance states.

The autoinhibition of clearance of fusidic acid resulted in a high clearance state for dosing regimens containing low doses of sodium fusidate (less than approximately 400 mg q12h) and a low clearance state for dosing regimens containing high doses (greater than approximately 550 mg q12h). As shown in Fig. 4C, the high clearance state at low doses resulted in a gradual increase in the C_ss of fusidic acid whereas the low clearance state resulted in a rapid increase in the C_ss. These two clearance states transitioned from one to the other over a relatively narrow dose range due to the high Hill coefficient of 4.61. For smaller Hill coefficients, the dose range between a low and a high clearance state would be wider. For dosing regimens of approximately 400 mg q12h or less, the CL_T/F was approximately 1.27 liters/h with an apparent terminal half-life of 14.0 h. For dosing regimens of approximately 550 mg q12h or higher, the CL_T/F of fusidic acid was approximately 0.370 liters/h with an apparent terminal half-life of 38.9 h. The C_ss of fusidic acid for low concentrations (C1 ≪ IC₅₀) in a high clearance state (CL_high/F) can be approximated and is provided in equation 8:

$$C_{ss}=\frac{\text{Daily}\hspace{0.17em}\text{dose}}{24\hspace{0.17em}\text{h}·{\text{CL}}_{\text{high}}/F}=\frac{\text{Daily}\hspace{0.17em}\text{dose}}{24\hspace{0.17em}\text{h}·1.27\hspace{0.17em}\text{liters}/\text{h}}$$ (8)

The equation for high plasma concentrations (C1 ≫ IC₅₀), which results in an apparent low clearance state (CL_low/F), is provided in equation 9:

$$C_{ss}=\frac{\text{Daily}\hspace{0.17em}\text{dose}}{24\hspace{0.17em}\text{h}·{\text{CL}}_{\text{low}}/F}=\frac{\text{Daily}\hspace{0.17em}\text{dose}}{24\hspace{0.17em}\text{h}·0.370\hspace{0.17em}\text{liters}/\text{h}}$$ (9)

It is important to note that the daily dose is the amount of fusidic acid and the C_ss needs to be multiplied by F_fed for administration in the fed state. The C_ss predicted from equations 8 and 9 are consistent with that predicted by the full model for all doses except 450 to 500 mg of sodium fusidate q12h (Fig. 4C).

DISCUSSION

Using data from healthy subjects from three phase 1 studies, the objectives of this analysis were to develop a population PK model to describe the absorption and disposition of fusidic acid after single and multiple oral doses and to determine the effect of food on the rate and extent of fusidic acid bioavailability. As a result of the wide dose range of the single and multiple oral sodium fusidate doses studied in the fed and fasted state, a robust population PK model that accommodated the PK characteristics of fusidic acid, including complex absorption and nonlinearity in clearance, was successfully developed.

The final population PK model was a semiphysiological model, which described the absorption of fusidic acid using a time-dependent mixed-order absorption process and which contained two disposition compartments and a turnover process to describe the autoinhibition of clearance. The above-described autoinhibition of clearance implemented in the final population PK model is similar to the semiphysiological turnover model (i.e., indirect-response model) approach proposed by Gordi et al. (27, 28) to describe the auto-induction of hepatic metabolism by artemisinin. Both models, however, differ from those for clarithromycin (29) and linezolid (30), the clearance of which was also described to be autoinhibited. While the models for clarithromycin and linezolid describe the time delay between plasma concentrations and inhibition of clearance by drug penetration to an effect compartment, the indirect response model proposed by Gordi et al. (27, 28) and the model for fusidic acid described herein are based on the assumption that this time delay is caused by the turnover rate of hepatic enzymes.

An important implication of the autoinhibition of clearance of fusidic acid is the presence of two clearance states for sodium fusidate dosing regimens containing high and low doses. For dosing regimens containing low doses (250 mg or less q12h), a high clearance state was evident with little to no accumulation. For dosing regimens containing high doses (750 mg q12h or higher), a low clearance state was evident with long terminal half-lives. Given the BSV of fusidic acid, intermediate doses of sodium fusidate (450 to 500 mg q12h) may result in either low or high clearance states.

Another important implication of the autoinhibition of clearance of fusidic acid is that the administration of sodium fusidate q12h dosing regimens containing high doses without front loading result in the achievement of steady state in approximately 8 days compared to approximately 3 weeks for dosing regimens containing intermediate doses. Such a delay in time to achieving steady-state concentrations may increase the risk of failure to therapy and increased emergence of bacterial resistance. Administration of front-loaded dosing regimens, such as 1,500 mg q12h on day 1 followed by 600 mg q12h, which involve delivering a large amount of the total drug exposure early in therapy, would allow for a low clearance state to be achieved. As a result, high fusidic acid exposures would be achieved on day 1. A maintenance dose of 600 mg sodium fusidate every 12 h is proposed to maintain the low clearance state and provide high fusidic acid concentrations throughout the therapy.

The impact of the front-loaded dosing regimens and those without front loading on the bacterial burden of methicillin-resistant Staphylococcus aureus (MRSA) and suppression of the emergence of resistance during therapy was evaluated using a one-compartment in vitro infection model which contained physiologic concentrations of albumin to account for protein binding (31, 32). In this 48-h study, regrowth of MRSA was associated with fusidic acid at 550 mg q12h and 1,100 mg q24h while suppression of bacterial regrowth was achieved by 550-mg and 1,100-mg fusidic acid front-loaded dosing regimens (for which front-loaded doses were 2.3 and 4.4 times the maintenance dose, respectively). Data from a 240-h experiment using a hollow-fiber in vitro infection model (which also contained physiologic concentrations of albumin to account for protein binding) evaluating the activity of three fusidic acid dosing regimens, 600 mg q12h, 1,200 mg q12h on day 1 followed by 600 mg q12h, and 1,500 mg q12h on day 1 followed by 600 mg q12h, against MRSA provided further support for front-loaded dosing regimens. In contrast to the non-front-loaded dosing regimen, both front-loaded dosing regimens were associated with a delay in the emergence of resistant subpopulations over the study period (31, 32). Thus, the autoinhibition of clearance of fusidic acid described herein provides the opportunity to use front-loaded dosing regimens to achieve exposures associated with efficacy earlier in therapy, a strategy that has demonstrated pharmacodynamic benefits using data from in vitro infection models. Subsequent use of parameter estimates from a PK-PD model based on the above-described one-compartment in vitro infection model data and the final population PK model described herein, together with Monte Carlo simulation which allowed for the assessment of the impact of PK variability on achieving relevant bacterial reduction endpoints, provided further support for the use of front-loaded sodium fusidate dosing regimens (31, 32). Recent findings of a phase 2 study in patients with acute bacterial skin and skin structure infections, for which comparable efficacy was observed for sodium fusidate at 1,500 mg q12h on day 1 followed by 600 mg q12h and linezolid 600 mg q12h, each administered for 10 to 14 days, provide further support for the use of front-loaded sodium fusidate dosing regimens (5).

In summary, a population PK model that described the disposition of fusidic acid was successfully developed. This model accommodated the complex PK characteristics of fusidic acid, including complex absorption and nonlinearity in clearance. The autoinhibition of clearance of fusidic acid provides a rationale for the administration of front-loaded dosing regimens for sodium fusidate. Such a population PK model, which can be used with data from nonclinical and clinical PK-PD models, provides an important foundation to support the selection of sodium fusidate dosing regimens for further study.

APPENDIX

This appendix presents the mathematical derivation for the model with autoinhibition of clearance that was incorporated into the final population PK model. The autoinhibition of clearance of fusidic acid described herein represents an expansion of a model with parallel first-order and Michaelis-Menten elimination. As shown below, this is demonstrated for a one-compartment intravenous bolus model with autoinhibition of clearance. This model converges to a parallel first-order and Michaelis-Menten elimination model when k_out is large (i.e., rapid turnover) and if Hill is 1.

The differential equations for the amount of drug in the central compartment (A) and for the inhibition compartment (INH) are as follows (C1 is the drug concentration in the central compartment and IC is the initial condition; see Table 4 for all other parameters):

$$\frac{dA}{dt}=-\left[\text{CL}·\left(1-\text{INH}\right)\right]·\text{C}1\text{IC}:\text{Dose}$$ (A1)

$$\frac{d\text{INH}}{dt}=k_{\text{out}}·\left(\frac{I_{\max }·\text{C}{1}^{\text{Hill}}}{{\text{IC}}_{50}^{\text{Hill}}+C{1}^{\text{Hill}}}-\text{INH}\right)\text{IC}:0$$ (A2)

An equivalent parameterization of this model would be to use CL · INH′ instead of CL · (1 − INH) in equation A1 and to write the differential equation for INH′ as follows:

$$\frac{d\text{INH}\prime }{\text{dt}}=k_{\text{out}}·\left(\left(1-\frac{I_{\max }·\text{C}{1}^{\text{Hill}}}{{\text{IC}}_{50}^{\text{Hill}}+\text{C}{1}^{\text{Hill}}}\right)-\text{INH}\prime \right)\text{IC}:1$$ (A2b)

The I_max can take values from 0 to 1 (bounds included). If the turnover rate constant (k_out) of the inhibition compartment is much faster than the elimination rate constant (k_el = CL/V1), the differential in equation A2 can be set to zero and the steady-state solution for INH becomes:

$$\text{INH}=\frac{I_{\max }·\text{C}{1}^{\text{Hill}}}{{\text{IC}}_{50}^{\text{Hill}}+\text{C}{1}^{\text{Hill}}}$$ (A3)

With the assumption that Hill equals 1 as described above, equations A3 and A1 yield:

$$\text{INH}=\frac{I_{\max }·\text{C}1}{{\text{IC}}_{\text{50}}+\text{C}1}$$ (A4)

$$\frac{dA}{\text{dt}}=-\left[\text{CL}·\left(1-\frac{I_{\max }·\text{C}1}{{\text{IC}}_{\text{50}}+\text{C}1}\right)\right]·\text{C}1\text{IC}:\text{Dose}$$ (A5)

This equation can be reparameterized to yield the equation for a model with parallel first-order (clearance: CL_lin) and Michaelis-Menten elimination as described below. The maximum rate of elimination (V_max) is parameterized as the product of intrinsic clearance (CL_ic) and the Michaelis-Menten constant (K_m):

$$\frac{dA}{\text{dt}}=-\left[{\text{CL}}_{\text{lin}}+\frac{{\text{CL}}_{\text{ic}}·K_m}{K_m+\text{C}1}\right]·\text{C}1\text{IC}:\text{Dose}$$ (A6)

For C1 ≪ K_m and C1 ≪ IC₅₀, the square brackets in equations A5 and A6 yield the maximum achievable total clearance at low drug concentrations:

$$\text{CL}={\text{CL}}_{\text{lin}}+{\text{CL}}_{\text{ic}}$$ (A7)

For C1 ≫ K_m and C1 ≫ IC₅₀, the lowest clearance at high drug concentrations is as follows:

$$\text{CL}·\left(1-I_{\max }\right)={\text{CL}}_{\text{lin}}$$ (A8)

Rearranging equation A8 and inserting equation A7 yields the following:

$$I_{\max }=1-\frac{{\text{CL}}_{\text{lin}}}{\text{CL}}=1-\frac{{\text{CL}}_{\text{lin}}}{{\text{CL}}_{\text{lin}}+{\text{CL}}_{\text{ic}}}$$ (A9)

With equation A9 the square bracket in equation A5 yields the following:

$$\text{CL}·\left(1-\frac{I_{\max }·\text{C}1}{{\text{IC}}_{50}+\text{C}1}\right)=\text{CL}·\left(1-\frac{\left(1-\frac{{\text{CL}}_{\text{lin}}}{\text{CL}}\right)·\text{C}1}{{\text{IC}}_{50}+\text{C}1}\right)=\frac{\text{CL}·\left({\text{IC}}_{50}+\text{C}1\right)-\text{CL}·\text{C}1+{\text{CL}}_{\text{lin}}·\text{C}1}{{\text{IC}}_{50}·\text{C}1}=\frac{\text{CL}·{\text{IC}}_{50}+{\text{CL}}_{\text{lin}}·\text{C}1}{{\text{IC}}_{50}+\text{C}1}$$ (A10)

Inserting equation A7 into equation A10 yields the following:

$$\frac{\text{CL}·{\text{IC}}_{50}+{\text{CL}}_{\text{lin}}\text{·C}1}{{\text{IC}}_{50}·\text{C}1}=\frac{\left({\text{CL}}_{\text{lin}}+{\text{CL}}_{\text{ic}}\right)·{\text{IC}}_{50}+{\text{CL}}_{\text{lin}}·\text{C}1}{{\text{IC}}_{50}+\text{C}1}=\frac{{\text{CL}}_{\text{lin}}·\left({\text{IC}}_{50}+\text{C}1\right)+{\text{CL}}_{\text{ic}}·{\text{IC}}_{50}}{{\text{IC}}_{50}+\text{C}1}={\text{CL}}_{\text{lin}}+\frac{{\text{CL}}_{\text{ic}}·{\text{IC}}_{50}}{{\text{IC}}_{50}+\text{C}1}$$ (A11)

With K_m = IC₅₀, equation A11 is identical to the square bracket for the apparent total clearance in equation A6. Thus, the inhibition compartment model converges to a model with parallel first-order and Michaelis-Menten elimination when k_out is large (i.e., rapid turnover) and the Hill coefficient is 1.