External evaluation of population pharmacokinetic models for ciclosporin in adult renal transplant recipients

Abstract

Aims

Several population pharmacokinetic (popPK) models for ciclosporin (CsA) in adult renal transplant recipients have been constructed to optimize the therapeutic regimen of CsA. However, little is known about their predictabilities when extrapolated to different clinical centres. Therefore, this study aimed to externally evaluate the predictive ability of CsA popPK models and determine the potential influencing factors.

Methods

A literature search was conducted and the predictive performance was determined for each selected model using an independent data set of 62 patients (471 predose and 500 2‐h postdose concentrations) from our hospital. Prediction‐based diagnostics and simulation‐based normalized prediction distribution error were used to evaluate model predictability. The influence of prior information was assessed using Bayesian forecasting. Additionally, potential factors influencing model predictability were investigated.

Results

Seventeen models extracted from 17 published popPK studies were assessed. Prediction‐based diagnostics showed that ethnicity potentially influenced model transferability. Simulation‐based normalized prediction distribution error analyses indicated misspecification in most of the models, especially regarding variance. Bayesian forecasting demonstrated that the predictive performance of the models substantially improved with 2–3 prior observations. The predictability of nonlinear Michaelis–Menten models was superior to that of linear compartmental models when evaluating the impact of structural models, indicating the underlying nonlinear kinetics of CsA. Structural model, ethnicity, covariates and prior observations potentially affected model predictability.

Conclusions

Structural model is the predominant factor influencing model predictability. Incorporation of nonlinear kinetics in CsA popPK modelling should be considered. Moreover, Bayesian forecasting substantially improved model predictability.

What is Already Known About this Subject

• Several ciclosporin population pharmacokinetic models for adult renal transplant recipients have been established to facilitate dose individualization.

• Body weight, postoperative days, age and haematocrit, among other factors, have been identified as contributors to the large variability in ciclosporin pharmacokinetics.

What this Study Adds

• The transferability of relevant models was systematically evaluated using an independent data set.

• Structural model was the predominant factor that influenced model predictability.

• Incorporating nonlinear kinetics into modelling might be a promising approach to improving model transferability.

Introduction

Ciclosporin (CsA), a cyclic polypeptide with potent immunosuppressant properties, is widely used to prevent graft‐vs.‐host disease after renal transplantation 1. As the first clinically used calcineurin inhibitor, CsA greatly improves the survival rate of transplant patients, especially the short‐term outcome, and has been the most important component of combined immunosuppressive regimens 2.

CsA pharmacokinetics (PK) have been extensively evaluated. Because of its erratic gastrointestinal absorption and the combined activity of cytochrome P450 (CYP) 3A and P‐glycoprotein (P‐gp) 1, the oral bioavailability of CsA is approximately 25%; however, it varies from 10 to 89% 3. To reduce the variability in oral absorption, a microemulsion of CsA (Neoral^®, Novartis) has been formulated 4. CsA is extensively distributed in peripheral tissues and binds to erythrocytes and plasma proteins 5, 6, 7. Further, CsA is primarily metabolized by CYP3A4 and CYP3A5 8 and then eliminated in the bile 9, 10. Because of its narrow therapeutic range and the marked variability in its PK 11, close therapeutic drug monitoring (TDM) has been suggested to support CsA dosage adaption.

The use of population pharmacokinetic (popPK) models combined with Bayesian estimations to guide CsA dose adaption is more precise than depending only on the personal experiences of transplant physicians 12. Currently, various popPK models have been developed to quantitatively describe the PK characteristics of CsA 13. Compared to conventional PK approaches, popPK is a powerful tool used to analyse sparse TDM data from highly diversified patient populations, estimate intra‐ and interindividual variabilities, and identify the impact of available covariates to evaluate the sources of variability in drug exposure.

Although various popPK models have been built to quantify sources of variability, little is known about their clinical application since full evaluation procedures are lacking 13, 14. Only one‐third of popPK studies on CsA have evaluated the models 13. Moreover, when models are extrapolated to other clinical centres, more rigorous evaluations need to be undertaken to assess model transferability 15, 16. Assessing model transferability may help identify potential centre‐based factors that influence model predictability and determine whether published popPK models are appropriate for use at any one centre. In addition, adopting the most appropriate popPK model for optimizing dosage regimens using a relatively small number of subjects is more cost‐effective than performing a complete popPK study.

In the present study, we hypothesized that all published models represent the PK characteristics of study cohorts. The predictabilities of published popPK models for CsA in adult renal transplants patients were systematically evaluated using an independent data set collected at Huashan Hospital (Shanghai, China). Additionally, potential factors influencing model predictability were investigated.

Materials and methods

Review of published popPK studies on CsA

A systematic literature search for data published on popPK models for CsA until 31 December 2016 was conducted from the following electronic databases: PubMed, Web of Science, Embase, China National Knowledge Infrastructure (www.cnki.net), China Biological Medicine (www.sinomed.ac.cn), and Wanfang (www.wanfangdata.com.cn). The language was limited to English and Chinese. The reference lists of identified reports were also screened. Articles were included in this study if they contained data analysing PK parameters for CsA in adult kidney transplant recipients using the popPK approach. Furthermore, studies were excluded if they were not original, if they focused on patients receiving an emulsion formulation of CsA, if the model parameters were not available for external evaluation, if the data sets overlapped or if the articles were duplicated.

External evaluation data

Patients

Sixty‐two adults (44 men and 18 women) who underwent their first renal transplantation at Huashan Hospital from July 2003 to May 2011 were eligible for this study. Demographic and pathophysiological data were retrospectively collected for further evaluation. Patient follow‐up was conducted up to 90 days after the surgery, and those with more than five predose concentration (C₀) and five 2‐h postdose concentration (C₂) records were included in the study. A total of 971 CsA whole‐blood measurements were collected, including 471 C₀ measurements and 500 C₂ measurements. Patients undergoing dialysis treatment were excluded from this study. The study protocols were approved by the Ethics Committee of Huashan Hospital and written informed consent was obtained from all the subjects.

Immunosuppressive therapy

Each patient received triple immunosuppressive therapy comprising a microemulsion formulation of CsA (Neoral^®, Novartis Pharma Schweiz AG, Emberbach, Germany), mycophenolate mofetil (MMF, CellCept^®; Roche Pharma Ltd, Shanghai, China) and corticosteroids. The initial dosage of CsA was 5 mg kg⁻¹ day⁻¹, which was administered as two doses under fasting conditions immediately after surgery. The CsA C₀ and C₂ were regularly monitored to adjust the dosage and achieve the target concentrations, which were based on guidelines for Chinese patients 17. Target C₀ values were 200–350 ng ml⁻¹ in the 1^st month, 150–300 ng ml⁻¹ during months 1–3, 100–250 ng ml⁻¹ during months 3–12 and 50–100 ng ml⁻¹ thereafter; target C₂ values were 1000–1500 ng ml⁻¹ in the 1^st month, 800–1200 ng ml⁻¹ during months 1–3, 600–1000 ng ml⁻¹ during months 3–12 and 400–600 ng ml⁻¹ thereafter.

MMF was administered based on body weight (WT) and postoperative days (POD). As such, patients weighing <50 kg were administered 1 g day⁻¹ for the first postoperative month and then 0.75 g day⁻¹ thereafter, patients weighing 50–70 kg were administered 1.5 g day⁻¹ for the first postoperative month and then 1 g day⁻¹ thereafter, and patients weighing >70 kg were administered 2 g day⁻¹ for the first postoperative month and then 1.5 g day⁻¹ thereafter.

Intravenous methylprednisolone was administered at 1 g day⁻¹ during the operation, followed by 0.5 g day⁻¹ for the next 3 days. On the 4^th day, treatment was switched to oral prednisolone (80 mg day⁻¹) and the dosage was decreased gradually by 10 mg day⁻¹ until it reached 20 mg day⁻¹ on day 10. The dosage was further tapered to 15, 10 and 5 mg day⁻¹ by months 1, 3 and 6, respectively. However, the rate of tapering differed among the patients.

Bioassay

Whole blood samples were treated with the anticoagulant ethylene diamine tetra‐acetic acid and analysed with a well‐validated fluorescence polarization immunoassay (FPIA) via an immunochemical analyser (AxSYM^®; Abbott Diagnostics, Chicago, IL, USA). The limit of detection was 21.8 ng ml⁻¹, and the calibration range was 40–800 ng ml⁻¹. When the CsA concentration was >800 ng ml⁻¹, manual dilution was performed prior to testing 18.

The bioassay methods used in the published studies were diverse. Some methods used reagents that are known to cross‐react with CsA metabolites, and systematic biases existed 19; therefore, the following formulas obtained from large clinical studies were used to convert C₀ values using the corresponding analytical methods before further analyses were performed.

$$\mathrm{TDx}=1.23\times \text{AxSYM}+3.46$$ (1)

18

$$\text{HPLC}=0.96\times \text{AxSYM}-17.41$$ (2)

18

$$\mathrm{RIA}=1.06\times \text{AxSYM}-5$$ (3)

20

$$\text{EMIT}=0.86\times \text{AxSYM}+1.98$$ (4)

19

$$\text{CEDIA}=0.88\times \text{AxSYM}+4.47$$ (5)

19

where TDx and AxSYM represent the FPIA performed using TDx^® (or TDxFLx^®) and AxSYM^® analysers, respectively; HPLC represents high‐performance liquid chromatography; and RIA, EMIT and CEDIA represent radioimmunoassay, enzyme multiplied immunoassay technique and cloned enzyme donor immunoassay, respectively.

At 2 h post dosing, most of the CsA administered to a patient exists as the parent drug. In addition, few CsA metabolites cross‐react with the antibodies used in immunoassays 1, 21; therefore, C₂ values were not converted for the different bioassays.

Genotyping

Four single nucleotide polymorphisms: CYP3A5*3, MDR1 C1236T, G2677 T/A, and C3435T were genotyped by an independent external contractor (GeneCore Biotechnology Co., Ltd., Shanghai, China) via a DNA sequencing apparatus (Applied Biosystems 3730; Thermo Fisher Scientific, Waltham, MA, USA). The results were then compared with standard allele frequencies using the Hardy–Weinberg equilibrium. More details are provided in Text S1.

External evaluation

An external evaluation was conducted using the NONMEM^® software package (version 7.3; ICON Development Solutions, Ellicott City, MD, USA) in conjunction with Intel Fortran XE 2011 Update 13 (Intel Corp, Santa Clara, CA, USA). Post processing of the NONMEM output was programmed using R software (version 3.3.0, http://www.r‐project.org/). PopPK models were rebuilt and parameters were fixed as reported by each study. Prediction‐ and simulation‐based diagnostics and Bayesian forecasting were then used to evaluate the predictive performance of the candidate models.

Prediction‐based diagnostics

Population predicted concentrations (PRED) were estimated, and the prediction error [PE%, equation (6)] for each patient was calculated by comparing the PRED to the corresponding observations (OBS).

$$\mathrm{PE}\left(\%\right)=\left(\frac{\mathit{\text{PRED}}-\mathit{OBS}}{\mathit{OBS}}\right)\times 100$$ (6)

The accuracy and precision of each candidate model were investigated using median prediction error (MDPE) and median absolute prediction error (MAPE), respectively 22. The percentage of |PE|% within 20% (F₂₀) and 30% (F₃₀) was also calculated. The metrics of a candidate model were arbitrarily considered to be clinically acceptable when the standards of MDPE ≤ ± 15%, MAPE ≤30%, F₂₀ > 35% and F₃₀ > 50% were reached, as previously reported 23.

Simulation‐based diagnostics

By comparing the appropriate statistics from the simulated and evaluation data, the predictability of each candidate model was assessed via simulation‐based diagnostics 24, 25. The normalized prediction distribution error (NPDE) was determined using an add‐on R package (NPDE, version 2.0, www.npde.biostat.fr) 26. Overall, 2000 data sets were simulated according to the reported final model parameters. Based on the null hypothesis that a candidate model is robust enough to describe the evaluation data, diagnostic graphs and statistic tests were conducted to examine whether the NPDE data followed a standard normal distribution. The Wilcoxon signed‐rank test was used to determine whether the mean NPDE was significantly different from 0, and Fisher's test was used to determine whether the variance was significantly different from 1. Additionally, the Shapiro–Wilks test was used to determine whether the observed distribution was significantly different from a normal distribution 26. The global test, which considers the above three tests using the Bonferroni correction, was also used in the analysis 27.

Bayesian forecasting

To assess the influence of prior observed concentrations on model predictability, maximum a posteriori Bayesian (MAPB) forecasting was conducted using NONMEM. The relative difference of the last observation, denoted by the individual PE% [IPE%, equation (7)], was calculated for this analysis. The all data method, in which forecasting is based on all available prior observations, was adopted for the estimation 28. The individual prediction (IPRED) of the last observation was estimated based on the last one, two, three and four prior observations. More details are included in Text S2.

$$\mathrm{IPE}\left(\%\right)=\left(\frac{\mathit{\text{IPRED}}-\mathit{OBS}}{\mathit{OBS}}\right)\times 100$$ (7)

Median IPE%, median absolute IPE%, and F₂₀ and F₃₀ of IPE% (IF₂₀ and IF₃₀, respectively) were used to evaluate the predictability as prior information increased.

The impact of structural models

Different modelling strategies were used in the selected studies that may have affected the predictive performance of the models. Therefore, to explore the impact of modelling strategy, the predictability of different structural models with or without the most involved covariates was evaluated. The evaluation approaches consisted of prediction‐ and simulation‐based diagnostics and Bayesian forecasting, as described above.

Nomenclature of targets and ligands

Key protein targets and ligands in this article are hyperlinked to corresponding entries in http://www.guidetopharmacology.org, the common portal for data from the IUPHAR/BPS Guide to PHARMACOLOGY 29, and are permanently archived in the Concise Guide to PHARMACOLOGY 2015/16 30.

Results

Review of published popPK studies on CsA

After the literature search (Figure 1 and Text S3), 17 popPK studies on CsA 12, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46 were identified for external evaluation. Ten of the studies were conducted in East Asian countries (six in China 33, 36, 38, 41, 43, 44, two each in Japan 32, 34 and Korea 45, 46), seven in Europe (three in France 35, 37, 40, one each in Norway 12 and the Netherlands 42 and two in Spain 31, 39). Additionally, 12 of the studies were conducted using a small sample size of fewer than 100 subjects 12, 31, 32, 35, 36, 37, 39, 40, 42, 43, 45, 46. Six bioassay methods were used in 17 studies, FPIA was used in 12 studies, and HPLC, EMIT, CEDIA and RIA were used in the remaining five studies (Table 1).

Figure 1.

Overview of the strategy used in the literature search. n: number of articles returned by search

Table 1.

Summary of published population pharmacokinetic studies of ciclosporin in adult renal transplant recipients

Study (publication year)	Country (Single/ multiple sites)	Number of patients (Male/ Female)	Sampling schedule (Number of samples)	Postoperative time mean ± SD/ median (range)	Bioassay	Structural model		PK parameters and formula	BSV (%)	Residual error
Porta et al. (1999) 32	Spain (Single)	20 (9/11)	C12–14 (NA)	(< 60) days	TDx	MM	Km	5.57 × (1‐e‐λ × POD) × WT	50.0	0.22 mg kg−1 day−1
							Vm	9.92 × WT	18.0
							λ	0.0236	5.92
Yoshida et al. (2001) 33	Japan (Single)	69 (42/27)	IS (483): C0/C1/C2/C3/C4/C6/C12	NA	TDx	2CMT	CL/F	12.8 + 0.259 × WT	26.9	16.0 ng ml−1
							Vc/F	57.9	98.9
							Q/F	23.3	26.4
							Vp/F	426	266.8
							Ka	3.59	141.4
Rui et al. (2002) 34	China (Single)	745 (540/205)	C0 (1468)	609 ± 632 (21–4448) days	TDx	MM	Km	2120 × (DOSE/235)‐1.67 × POD‐0.651 × [(1.17, if concomitant Tabellae Multiglycosidorum Triptergii) or (1.13, if concomitant Baoshenpian)]	24.1	13.4%
							Vm	601 × POD‐ 0.210 × WT0.139 × (1.078, if male)
									11.7
Tokui et al. (2004) 35	Japan (Single)	125 (78/47)	IS (750): C0/C1/C2/C3/C4/C5	16 ± 2.1 (6–38) months	TDx	2CMT	Kel	0.547	6.17	10.2%
							K12	0.582	32.8
							K21	0.242	53.6
							Vd/F	25.4 × (1 + 0.0125 × WT)	22.3
							Ka	2.16	152
Rousseau et al. (2004) 36	France (Single)	10 (6/4)	ISa (100)	16 ± 19.3 (3–119) months	LC/MS	2CMT	CL/F	26.3	30.0	4.8% 27 ng ml−1
							Vc/F	75.9	47.6
							Q/F	23.9	72.4
		60 (NA)	SSa (240)		EMIT		Vp/F	119	/
							n	5	/
							Ktr	7.31	32.6
Wu et al. (2005) 37	China (Single)	99 (46/53)	C0/C2 (2141)	47 (9–202) days	TDx	1CMT	CL/F	28.5− 1.24 × POD − 0.252 × (TBIL−11) + 0.188 × (WT−58) − 0.191 × (AGE−42) − 0.212 × (HCT−28)a − (2.45, if concomitant diltiazem or verapamil)	19.7	30.8% 42.4 ng ml−1
							Vc/F	133	/
							Ka	1.28 (fixed)	179
Bourgoin et al. (2005) 38	France (Multiple)	84 (62/22)	ISb (200) + SSb (354)	664 (10−3923) days	EMIT	2CMT	CL/F	30.1	33.8	13.7%
							Vc/F	79.8	35.5
							Q/F	18.5	31.7
							Vp/F	163	/
							D1	0.879	26.0
							Tlag	0.12	86.1
Liang et al. (2005) 39	China (Single)	221 (146/75)	C0 (323)	1 month (≥ 3) days	TDx	1CMT	CL/F	13.46−0.063×AGE + 0.08×WT	24.4	8.1%
							Vc/F	228.2	/
							Ka	1.28 (fixed)	/
Lukas et al. (2005) 40	Spain (Single)	11 (NA)	C0 (167) + C2 (92) + C3 (56)	(7–85) days	TDxFlx	1CMT	CL/F	17	27.0	23.0%
							Vc/F	134	27.0
							Ka	4 (fixed)	/
							F	1–0.62 × e‐λ × POD	/
							λ	0.037	120
Marcux et al. (2006) 41	France (Multiple)	24 (NA)	ISc (481)	days 3,7,30	EMIT	2CMT	CL/F	32.8 + 0.2 × (WT‐58.1)	35.0	10.8%
					AxSYM		Vc/F	87.1 + 0.9 × (WT‐58.1)	64.0	37.4 ng ml−1
					LC/MSd		Q/F	32.3	85.0
		20 (NA)	ISd (470)	525 ± 305 (90–1300) days	AxSYM		Vp/F	129.4	86.2
					LC/MSd		n	5	/
							Ktr	5.7 + 0.4 × POD	26.0
Wang et al. (2006) 42	China (Single)	411 (266/145)	C0 (1122)	869.5 ± 871 (6–4975) days	TDx	MM	Km	22.2 × (DOSE/200)‐2.64 × (POD/800)‐0.634 × (WT/60)1.08 × (AGE/42)‐0.922 × (Scr/110)‐0.101 × (ALT/20)‐0.0881	12.3 ng ml−1	18.7 mg day−1
							Vm	222 × (POD/800)‐0.154 × (WT/60)0.411 × (AGE/42)‐0.295 × (ALB/40)‐0.0787 × (DBIL/5)‐0.0315 × (ALT/20)‐0.0384 × (TG/1.8)‐0.0334 × (0.958, if AST > 50)	24.8 mg day−1
Asberg et al. (2010) 12	Norway (Single)	49 (33/16)	ISe (916)	5.1 ± 3.6 (2−17) weeks	CEDIA	2CMT	CL/F	38.7–0.188 × AGE	27.3	22.1%
							Vc/F	20.7/(AGE/55) + 0.322 × (WT‐78.5)	8.7	14.2 ng ml−1
							Q/F	18.5	14.6
							Vp/F	1360	67.7
		20 (13/7)	C0/C2 (NA)	(47–100) days			Ka	1.02 × (POD/35)0.211 + 0.00101×(AGE−55)− 0.00415 × WT	49.6
							Tlag	0.478	12.5
Press et al. (2010) 43	Netherland (Single)	33 (26/7)	IS (NA) + C0/C2/C3 (NA)	week 2,6,12,26,52 week 4,8,10,17,21,39	AxSYM	2CMT	CL	15 × (WT/76)0.75	17.3	26.5%
							Vc	56 × (WT/76)	34.6
							Q	14	/
							Vp	125	/
							Ka	2 × (0.45, if concomitant prednisolone ≥20 mg)	30.0
							MTT	1	/
							n	1	/
							Ktr	2	/
							F	0.5×(0.78, if concomitant prednisolone ≥20 mg)	14.1
Zhou et al. (2011) 44	China (Single)	62 (43/19)	C0 (160)	21.7 (7−58) days	TDxFlx	1CMT	CL/F	30.5 × [1 + 0.0105 × (WT‐61.36)] × [1−1.15×(HCT−0.289)] × [1−0.0125×(TBIL×9.26)]	10.9	27.4%
							Vc/F	3.85 × WT	/
							Ka	1.28 (fixed)	/
Chen et al. (2011) 45	China (Single)	146 (87/59)	C0/C2 (1577)	(3–146) days	AxSYM	1CMT	CL/F	49.5 × (WT/56.5)0.46 × (TBIL/15.1)‐0.11 × (1–0.053 × MDR1)f ×[(POD−0.18, if POD ≤ 30) or (0.54, if POD > 30)] × (1.09, if female)	16.1	33.6%
							Vc/F	226 × (1−0.062 × MDR1) × (1.2, if female)	28.6
							Ka	1.25 (fixed)	/
Song et al. (2012) 46	Korea (Single)	69 (43/26)	C0 (2034)	(<450) days	RIA	1CMT	CL/F	3.32 × [(15.18, if CYP3A5*3/*3 carrier) or (17.99, if CYP3A5*1 carrier)] × [(POD−0.00002, if POD ≤ 30) or (e0.000395 × POD, if POD > 30)]	23.0	35.4%
							Vc/F	5670 × [(1.412,if male) or (1.326, if female)] × [(POD/100, if POD ≤ 30) or (POD/1000, if POD >30)	51.5
							Ka	1.28 (fixed)	/
Baek et al. (2014) 47	Korea (Single)	69 (38/31)	IS (1640) C0/C1/C2/C3/C4/C6/ C8/C12	day 2, 3, 7	TDxFlx	2CMT	CL	0.254 × Vc	/	23.0%
							Vc	142 × (WT/60)0.689 × (AGE/41.2)‐0.227	13.4h
							Q	24.9	35.4
							Vp	236	67.1h
							Ka	(Ka‐d2, if POD = 2) or (Ka‐d2×Ka‐d3, if POD=3) or (Ka‐d2×Ka‐d3×Ka‐d7, if POD≥7)
							Ka‐d2	1.09	136h
							Ka‐d3	4.56	174h
							Ka‐d7	1 (fixed)	268
							Tlag	(0.811, if POD = 2) or (0.933, if POD = 3)
							F	(1, if POD = 2) or (F1‐d3, if POD = 3) or (F1‐d3 × F1‐d7, if POD≥7)
							F1‐d3	1.09	12.8
							F1‐d7	0.807× (AGE/41.2)‐0.293	21.2

ALB, albumin (g l⁻¹); ALT, alanine aminotransferase(U l⁻¹); AST, aspartate transferase (U l⁻¹); AxSYM, FPIA using AxSYM® analyzers; BSV, between subject variability; C_n, concentration at n h postdose; CEDIA, cloned enzyme donor immunoassay; CL/F, apparent clearance (l h⁻¹); CMT, compartment; D1, duration of absorption (h⁻¹); DBIL, direct bilirubin (μmol l⁻¹); DDPR, prednisolone daily dose (mg); DOSE, CsA daily dose (mg); EMIT, enzyme multiplied immunoassay technique; FPIA, fluorescence polarization immunoassay; HCT, haematocrit (%); IS, intensive sampling strategy; K₁₂, rate constant from the central compartment to the peripheral compartment (h⁻¹); K₂₁, rate constant from the peripheral compartment to the peripheral compartment (h⁻¹); K_a, absorption rate constant (h⁻¹); K_el, elimination rate constant from the central compartment (h⁻¹); K_tr, transfer rate constant (h⁻¹); LC/MS, liquid chromatography/mass; MDR1, multidrug resistance 1 transporter genotype; MM, Michaelis–Menten pharmacokinetic model; n, number of sequential compartments; NA, not available; POD, postoperative days (day); Scr, serum creatinine (μmol l⁻¹); SS, sparse sampling; TDx, FPIA using TDx^® analysers; TDxFLx, FPIA using TDxFLx^® analysers; V_c/F, apparent volume of distribution of central compartment (l); V_p/F, apparent volume of distribution peripheral compartment (l); WT, bodyweight (kg); Q/F, apparent inter‐compartmental clearance (l h⁻¹); RIA, radioimmunoassay; T_lag, lag time (h); TBIL, total bilirubin (μmol l⁻¹); TG, triacylglycerol (mmol l⁻¹)

Intensive samples (10 patients) were collected before and 0.33, 0.67, 1, 1.5, 2, 3, 4, 6, 9 h after dosing, sparse samples (60 patients) were collected at 0, 2 h and postdose 2 other sampling times listed above.

^aPOD = 1 when postoperative days = 0–7, POD = 2 when postoperative day 8–14, POD = 3 when postoperative day 15–21, POD = 4 when postoperative day 22–60, POD = 5 when postoperative day 61–150, POD = 6 when postoperative day >150.

^bIntensive samples (20 patients) were collected before and 0.33, 0.67, 1, 1.5, 2, 3, 4, 6, 9 h after dosing, sparse samples (64 patients) were collected at predose 2 h postdose and other sampling times selected randomly.

^cIntensive samples were collected before and 0.33, 0.66, 1, 1.5, 2, 3, 4, 6, 9, 12 h postdose in 24 de novo patients and collected before and 0.33, 0.66, 1, 1.5, 2, 3, 4, 6, 9 h postdose in 20 stable patients.

^dLiquid chromatography/mass spectrometry was considered as reference to each assay in modelling.

^e38 of 49 patients conducted 12‐h pharmacokinetic investigations including predose and 0.25, 0.5, 1, 1.5, 2, 3, 4, 6, 8, 10, 12 h postdose.

^fMDR1 was set to 0–5 for diplotypes CGC/CGC, CGC/Other, Other/Other, CGC/TTT, Other/TTT respectively.

Correlations are V_c ~ V_p: 0.523; Ka‐d2 ~ Ka‐d3: −0.571.

For the 14 compartmental models, the absorption phase was described most frequently by first‐order kinetics 12, 32, 34, 36, 38, 39, 43, 44, 45, 46. Zero‐order kinetics 37, transit 42 and Erlang 35, 40 models were also used in some of the studies. CsA disposition was described using one‐compartmental models in six studies with sparse sampling designs 36, 38, 39, 43, 44, 45. Two‐compartmental models were used in all eight of the studies that included intense sampling 12, 32, 34, 35, 37, 40, 42, 46. All compartmental models were characterized using linear kinetics and first‐order elimination.

In addition to the linear compartmental model, a nonlinear Michaelis–Menten (MM) empirical formula [equation (8)] was used to quantify the relationship between daily dose and C₀ in three studies 31, 33, 41 as follows:

$$\text{Daily Dose}\left(\mathrm{mg}\right)=\frac{\mathrm{Vm}\left(\mathrm{mg}\right)\times \mathrm{C}\left(\mathrm{ng}{\mathrm{ml}}^{-1}\right)}{\mathrm{Km}\left(\mathrm{ng}{\mathrm{ml}}^{-1}\right)+\mathrm{C}\left(\mathrm{ng}{\mathrm{ml}}^{-1}\right)}$$ (8)

where V_m denotes the maximum dose rate (daily dose), K_m denotes the steady‐state C₀ at half‐maximal dose rate and C represents the C₀ of CsA.

Most of the studies were carried out using triple immunosuppressive therapy; however, the studies by Yoshida et al. 32, Rousseau et al. 35 and Lukas et al. 39 did not provide this information. The dose ranges of CsA for linear compartmental and nonlinear MM models were 50–1200 mg day⁻¹ and 50–700 mg day⁻¹, respectively. The details are summarized in Table S1.

Overall, 15 different covariates were incorporated into the final models used in the studies. WT, POD and age were the three most frequently identified covariates in the final models and were reported in 12, six and five studies, respectively (Figure S1). Moreover, CYP3A5*3 and MDR1 polymorphisms were screened in five and three studies, respectively; however, they were both reported in one study.

External evaluation

Patients

The demographic and pathophysiological data, as well as concomitant medications collected for the evaluation are summarized in Table 2 according to the incorporated covariates. No patient was coadministered a calcium channel blocker, verapamil or diltiazem in our data set. All the allele frequencies of CYP3A5*3 and MDR1 genetic polymorphisms were in Hardy–Weinberg equilibrium (Table 3). There were three observations (C₀) below the limit of detection; however, they are included in the analysis as the original reported values.

Table 2.

Characteristics of external evaluation data set

Characteristics	Number or mean ± SD	Median (range)
No. of patients (male/female) a	62 (44/18)	/
No. of samples (C 0 /C 2 ) b	971 (471/500)	/
Age (years)	40.2 ± 10.2	41 (18–58)
Height (cm)	168.0 ± 7.0	168.0 (155.0–188.0)
Weight (kg)	58.2 ± 9.4	58.0 (39.0–85.0)
Postoperation days	22.5 ± 19.8	17 (1–90)
CsA daily dose (mg)	371.2 ± 70.4	325 (100–500)
C 0 (ng ml −1 )	198.2 ± 99.2	181.1 (5.7–587.4)
C 2 (ng ml −1 )	938.0 ± 382.0	904.4 (107.6–2572.8)
Total bilirubin (μmol l −1 )	9.4 ± 5.1	8.0 (1.0–43.2)
Haematocrit (%)	29.4 ± 5.7	29.5 (10.5–47.7)
Alanine aminotransferase (U l −1 )	34.0 ± 42.5	24.0 (5.0–390.0)
Aspartate transferase (U l −1 )	22.4 ± 15.8	19.0 (6.0–139.0)
Albumin (g l −1 )	35.0 ± 5.2	35.0 (21.0–49.0)
Total protein (g l −1 )	60.2 ± 7.2	60.0 (46.0–82.0)
Serum creatinine (μmol l −1 )	133.2 ± 124.3	104.0 (48.0–1052.0)
Creatinine clearance (ml min −1 ) c	65.4 ± 23.3	65.2 (6.2–151.3)
Concomitant calcium channel blocker a	15	/
Felodipine	7	/
Nifedipine	4	/
Perdipine	4	/

C₀, predose concentration; C₂, 2‐h postdose concentration

^aData are expressed as number of patients.

^bData are expressed as number of samples.

^cCalculated following the Cockcroft–Gault formula: Ccr = [(140 – age(year)) × weight(kg)]/ (0.818 × Scr (μmol l⁻¹)) × (0.85 for female).

Table 3.

Allele frequencies of genetic polymorphisms in CYP3A5 and ABCB1 genes of external data set

Single nucleotide polymorphisms	Number of recipients	Frequency (%)
CYP3A5*3 (A6986G, rs776746)
AA (*1/*1)	3	4.84
GA (*1/*3)	21	33.87
GG (*3/*3)	38	61.29
ABCB1‐C1236T (rs1128503)
CC	13	20.97
CT	24	38.71
TT	25	40.32
ABCB1‐G2677 T/A (rs2032582)
AA	6	9.68
GG	12	19.35
GA	7	11.29
TT	9	14.52
TG	24	38.71
TA	4	6.45
ABCB1‐C3435T (rs1045642)
CC	25	40.32
CT	30	48.39
TT	7	11.29

The allele frequencies are found to be in Hardy–Weinberg equilibrium (P > 0.05)

Prediction‐based diagnostics

C₀ was used to evaluate the predictability of models that were developed using linear compartmental or nonlinear MM structural models. In contrast, C₂ was used to evaluate predictability based on compartmental structural models. For C₀, prediction‐based diagnostics showed that the nonlinear MM models by Rui et al. 33 and Wang et al. 41, as well as the linear compartmental model by Zhou et al. 43 met the aforementioned criteria (MDPE ≤ ± 15%, MAPE ≤30%, F₂₀ > 35% and F₃₀ > 50%). The model by Zhou et al. 43 (MDPE, −7.31%; MAPE, 24.36%; F₂₀, 41.61%; F₃₀, 57.96%) was the best‐performing linear compartmental model of those examined. Predictability of the nonlinear MM model developed by Wang et al. 41 (MDPE, −7.46%; MAPE, 10.15%; F₂₀, 82.17%; F₃₀, 89.38%) was obviously better than that of the other models. All prediction‐based metrics from the three nonlinear MM models met the criteria, except the MDPE from the model developed by Porta et al. 27, which was >15%.

For C₂, five linear compartmental models met the criteria and the model developed by Wu et al. 36 (MDPE, 4.04%; MAPE, 25.83%; F₂₀, 40.00%; and F₃₀, 57.60%) performed the best. Furthermore, all the models showing superior performance for C₀ and C₂ were conducted in Chinese patients. These results are presented in Figure 2 and Table S2.

Figure 2.

Box plots of the prediction error for 17 published population pharmacokinetic models. (A) Predose concentration (C₀) and (B) 2‐h postdose concentration (C₂) are shown. Black solid lines and blue dotted lines are reference lines indicating PE% of 0% and ±20%, respectively. Models with an asterisk (*) were developed based on the Michaelis–Menten model

Simulation‐based diagnostics

Regarding the simulation‐based NPDE diagnostics, the mean value and Wilcoxon signed‐rank test for C₀ performed relatively well, while the distribution of NPDE was uniform, indicating an inconsistency between the evaluation and simulated data, especially for variance. Only the global test P‐value based on the linear compartmental model developed by Press et al. 42 was >0.01, indicating the simulation was appropriately applied. None of the NPDE distributions of nonlinear MM models satisfied the expected distribution N (0, 1). Similar to the results observed for C₀, the linear compartmental models did not perform well for C₂ (Table S3). In contrast to the prediction‐based diagnostic results, the nonlinear MM models underperformed in the C₀ analysis and may not be appropriate for simulation‐based applications. As shown in Figure 3, the NPDE of the MM model by Wang et al. 41, which had the best predictive performance in the PE% test, did not follow a normal distribution. The NPDE graphics and statistical tests for all the models are presented in Figure S2 and Table S3.

Figure 3.

Normalized prediction distribution error (NPDE) plots of the model by Wang et al. 42. (A) Quantile–quantile plot of the distribution of the NPDE against theoretical distribution (semitransparent blue fields), (B) histogram of the distribution of the NPDE against theoretical distribution (semitransparent blue fields), (C) NPDE vs. postoperative time (h), and (D) NPDE vs. predicted concentrations are shown. In plots C and D, the solid red line represents the median NPDE of the observations and the semitransparent red field represents a simulation‐based 95% confidence interval (CI) for the median. Solid blue lines represent the NPDE of the observed 5^th percentiles, and semitransparent blue fields represent a simulation‐based 95% CI for the corresponding model‐predicted percentiles. The NPDE of the observations are represented by blue circles

Bayesian forecasting

Bayesian forecasting demonstrated that prior knowledge substantially improves the prediction accuracy of all models, including poor prediction models 32, 34, 38, 46, indicating that a popPK model combined with Bayesian estimation is useful for CsA dosage adjustments. After two or three prior observations were available, model predictability reached a stable state and additional prior observations did not necessarily imply further improvement. Moreover, nonlinear MM models performed better than linear compartmental models did, and most models had IF₂₀ values of ≥80% vs. 35% after information from one prior observation was available. The metric results and box plots are presented in Figure 4 and Table S4, respectively.

Figure 4.

Box plots of individual prediction error with Bayesian forecasting for 17 published population pharmacokinetic models in different scenarios (0 represents predictions without prior information and 1–4 represent predictions with one to four prior observations, respectively). In scenario n, prior n observations were used to estimate the individual prediction and it was then compared with the corresponding observation. (A) Predose concentration (C₀) and (B) 2‐h postdose concentration (C₂) are shown

The impact of structural models

The nonlinear MM model performed significantly better than the prediction‐based evaluations of one‐compartmental and two‐compartmental models (Figure 5 and Table S2). The IF₂₀ of the nonlinear MM base model showed a remarkable improvement over that of the other two linear compartmental base models for C₀ (69.21% vs. 34.39% and 33.12%) and C₂ (63.00% vs. 36.60% and 36.40%). After incorporating the three most identified covariates (WT, POD and age) into the model, the IF₂₀ of the nonlinear MM model and the other two linear compartmental models for C₀ were 82.59%, 33.76% and 32.48%, respectively, and for C₂ were 79.20%, 39.40% and 39.20%, respectively. Improvements were more significant in nonlinear MM models than those in linear compartmental models. Moreover, the predictive performance of linear compartmental models was much worse than that of MM base models, even after the inclusion of covariates (< 40% vs. >63%). These results imply that improvements resulting from nonlinear MM models were considerably more than those resulting from incorporating covariates.

Figure 5.

Box plots of the prediction error (PE%) for structural models with and without covariates. Black solid line and blue dotted lines are reference lines indicating PE% of 0% and ±20%, respectively. 1CMT, one‐compartmental model; 2CMT, two‐compartmental model; C₀: predose concentration; C₂: 2‐h postdose concentration; MM, Michaelis–Menten model; POD, postoperative days; WT, body weight. Models with an asterisk (*) were developed based on the Michaelis–Menten model

Normality for the prediction discrepancies of the MM model was unsatisfied in the simulation‐based diagnostics, which is consistent with the results of the above final models (Table S3 and Figure S3). The parameter estimates are listed in Table S5.

The predictive performance of the structural models improved for both C₀ and C₂ with MAPB. The IF₂₀ values for the MM base models after Bayesian forecasting reached ≥85%, demonstrating that these models are superior to the linear compartmental base models, which reached ≤45%. Box plots are presented in Figure 6 and results are displayed in Table S4.

Figure 6.

Box plots of individual prediction error (IPE%) with Bayesian forecasting for structural models with and without covariates in different scenarios (0 represents predictions without prior information and 1–4 represent predictions with one to four prior observations, respectively). 1CMT, one‐compartmental model; 2CMT, two‐compartmental model; MM, Michaelis–Menten model; POD, postoperative days; WT, body weight. (A) Predose concentration (C₀) and (B) 2‐h postdose concentration (C₂) are shown. Models with an asterisk (*) were developed based on the Michaelis–Menten model

Discussion

Although nearly 20 CsA popPK studies have been conducted in adult renal transplant patients, most of these studies were single centre‐based and predictability was not fully evaluated. Therefore, it is necessary to assess the accuracy and robustness of a model before it is extrapolated to another clinical centre 47.

To the best of our knowledge, this is the first study to evaluate published popPK models of CsA systematically in adult renal transplant patients using an independent data set. Although the data set used in this study was taken from one centre, the conclusions could be helpful when further investigating CsA.

To minimize discrepancies between studies, models developed using adult renal transplant recipients who received Neoral‐based triple immunosuppressive therapy were evaluated. Large variabilities in predictive performance exist in the published models, as demonstrated by conducting evaluations using our data set. After comparing the impact of structural models on predictability, we concluded that the predictability of nonlinear MM models is superior to that of linear compartmental models, indicating that the PK of CsA is nonlinear.

CsA is believed to follow linear PK when administered at its normal dose range. This observation is based on a clinical PK study of stable renal transplant patients who were intravenously administered CsA at three dosage levels (1.25, 2.5 and 5 mg kg⁻¹ day⁻¹) 48. However, the influence of absorption on CsA PK was not investigated. The same research group reported nonlinear PK in renal transplant patients who were taking oral CsA in 1993, as determined by popPK analyses 49.

CsA is a highly lipophilic cyclic polypeptide; therefore, its absorption is problematic 50. The nonlinearity of CsA absorption and disposition was shown by comparing PK parameters across a dose range of 350–1400 mg day⁻¹ in 12 healthy volunteers 51. A new oral microemulsion of CsA (Neoral^®) was formulated to reduce the variability and improve the dose linearity of CsA 4. Mueller et al. 52 compared the PK of Neoral with those of a conventional CsA formulation and concluded that the linearity of Neoral is better. However, it should be noted that this study was conducted in healthy volunteers who were administered a single oral dose of CsA; therefore, inconsistencies in the pathophysiological features between transplant patients and healthy volunteers may have influenced the PK behaviour of CsA.

The nonlinearity of CsA PK in renal transplant patients could be attributed to many factors. During the early postoperative period, recovery of gastrointestinal function may increase CsA bioavailability with POD, resulting in nonlinear kinetics 36, 40, 44, 53. Further, the inhibitory effects of CsA on CYP3A4 and P‐gp can lead to autoinhibition and dose‐dependent PK, which may also be partially attributed to the nonlinearity 54, 55, 56. In addition, an important steroid X receptor regulates CYP3A and MDR1 gene expression; therefore, tapering steroid doses during immunosuppressive therapy may reduce the expression of CYP3A and MDR1. However, this could result in increased absorption and accelerated metabolism of CsA 57, 58.

The studies conducted by Reymond et al. 51 and Mueller et al. 52 show that the terminal half‐life of CsA increases as the dosage is increased, indicating a nonlinear disposition. Saturation of hepatic metabolism would partly be responsible for this phenomenon 51. Moreover, since CsA is a high‐extraction drug, recovery of hepatic function after transplantation may decrease its bioavailability 46.

TDM has shown that individuals with higher drug clearance rates may have lower drug concentrations, resulting in a tendency for clinicians to prescribe higher doses. When a correlation between the clearance and daily dose of CsA is observed, it may be misinterpreted as nonlinearity 59. However, when patients are sampled at more than three or four dose levels, the effects of TDM described above may not influence the judgement of nonlinearity 59.

Compared to linear compartmental models, nonlinear MM models are data‐driven empirical models that are not based on hypotheses of one or two compartments. Therefore, such models can only be used to describe CsA daily dose and steady‐state concentration relationships and cannot be used to describe the full concentration–time course of CsA. Development of all three MM models was based on C₀, and they can therefore be used to predict C₀; however, these MM models cannot be used to appropriately extrapolate C₂.

In addition to the structural model discussed above, study design, ethnic differences and retained covariates were all potential centre‐based factors that may have resulted in interstudy variability and influenced model predictability 60. Minor differences in predictive performance were observed between sparse and intensive sampling design studies, which may be attributable to insufficient information from the sparse samples to precisely estimate apparent clearance (CL/F) 61.

In the present study, models that performed best for C₀ and C₂ were all developed in Chinese patients, indicating that ethnicity potentially influences model transferability. It is believed that similarities in the ethnic background of patients may result in similar genetics, prescribing, and dietary habits. However, this was not observed for tacrolimus (TAC), another commonly used calcineurin inhibitor 23, which is also a substrate of CYP3A4/5 and the P‐gp transporter. The failure of ethnicity to influence the PK of TAC may be attributed to the CYP3A5*3 polymorphism, which was identified as an important covariate for the CL/F of TAC 62. However, no widely acceptable genetic polymorphisms have been identified as an influential factor in the PK of CsA.

The linear compartmental model developed by Liang et al. 38 using a Chinese population did not perform well with a typical estimated CL/F of 15.3 l h⁻¹, which was less than the 17.0–50.4 l h⁻¹ reported for 13 other published compartmental models; however, the characteristics of the study cohort were comparable to those of the other studies conducted in Chinese patients. The reason for this is unknown; however, it is noteworthy that, Liang et al. 38 used the SAS program to perform popPK analyses, whereas NONMEM was used in the other studies.

As most covariates were screened based on pharmacostatistical considerations, there was no consensus on whether the involved covariates had been achieved in published models. WT, POD and age were the three most frequently identified covariates influencing the CL/F of CsA. It is noteworthy that the covariates identified in published studies were dependent on the study population and may not necessarily reflect the external evaluation population.

Although some linear compartmental models performed unsatisfactorily in the prediction‐based diagnostics, Bayesian forecasting indicated that popPK modelling combined with MAPB estimation is a useful tool to guide clinical TDM 63. With the original population parameter estimates being prior distributions, personalised dosage regimens could be designed using Bayesian forecasting and the available concentrations 64, 65. Predictability became stable when 2–3 prior observations were available, which is consistent with the results obtained for TAC 23. Additional prior observations did not necessarily imply further improvement.

Undeniably, there are some limitations to our study. Bioassay type is an important factor that may influence model transferability 19, 60. In the present study, the bioassay methods used in the published studies were diverse. To reduce systematic bioassay biases, various equations based on the assay methods were used 18, 19, 20. After conversion, no obvious bioassay‐related trends were found in the predictive performance analyses. In addition, most F₂₀ values for published models changed <5% when the sensitivity analysis was conducted by changing the slope of conversion formulas by ±10%. Moreover, as external evaluation data were retrospectively collected during routine TDM, patient compliance could not be ascertained. Although sparse samples could be used to estimate CL/F precisely 61, no intensive samples were available to evaluate the biological plausibility of absorption models such as the transit and Erlang models.

In conclusion, published CsA popPK models in adult renal transplant recipients were externally evaluated using an independent data set. The prediction‐based diagnostics show that ethnicity is a potential factor that may influence model transferability. The simulation‐based diagnostics showed variance inconsistencies in most of the published models. By investigating the impact of structural models, we demonstrated that the predictability of nonlinear MM models is superior to that of linear compartmental models, indicating the underlying nonlinear kinetics of CsA. Moreover, popPK modelling combined with MAPB estimation was found to be a useful tool to guide CsA dose adjustments.

Competing Interests

All authors have completed the United Competing Interest form at http://www.icmje.org/coi_disclosure.pdf (available on the request from the corresponding author). Z.J. was supported by grants from the National Natural Science Foundation of China (No. 81573505 and 81 072 702), the “2016 Key Clinical Program of Clinical Pharmacy”, and “Weak Discipline Construction Project” (No. 2016ZB0301–01) of Shanghai Municipal Commission of Health and Family Planning. X.‐Y.Q. received support from the National Natural Science Foundation of China (No. 81302854), which promoted this study. There are no financial relationships with any organizations that might have an interest in the submitted work in the previous 3 years, and no other relationships or activities that could appear to have influenced the submitted work. The other authors have no conflicts of interest to declare.

The authors would like to sincerely thank Dr Gilles Paintaud and Dr Chantal Barin‐Le Guellec of Université François Rabelais (France), Dr Bo Sun of Shanghai General Hospital of Shanghai Jiao Tong University (China), Dr Bing Chen of Ruijin Hospital of Shanghai Jiao Tong University School of Medicine (China), Dr Ke‐Hua Wu of Peking University (China), and Dr Jian‐Zhong Rui of Nanjing General Hospital (China) for active discussions on research and modelling.

Contributors

J.J.M., Z.J., H.Y.Y. and C.Y.Z. participated in the research design. X.Y.Q., M.K.Z. and H.C.C. helped acquire the evaluated data. J.J.M. and C.Y.Z. performed the research and analysed the data. J.J.M., Z.J. and H.Y.Y. drafted the manuscript, which was revised and approved by all the authors.

Supporting information

Supplementary Text S1 Genotyping of CYP3A5*3, MDR1 C1236T, G2677 T/A, and C3435T single‐nucleotide polymorphisms

Supplementary Text S2 Detailed Bayesian forecasting process

Supplementary Text S3 Detailed literature search process

Table S1 Demographics of our data set and 17 investigated published studies

Table S2 Results of the prediction‐based metrics

Table S3 Statistic test results of normalized prediction distribution error (NPDE) diagnostics

Table S4 Results of Bayesian forecasting

Table S5 Parameter estimates of three structural models with and without covariates

Figure S1 Stacked bar plot of the frequency of covariates. Light blue bar plots present the frequency of the covariates screened for 17 candidate models, dark blue bar plots present the frequency of the covariates identified as influencing CsA PK. CYP3A5, CYP3A5 genotype; GEND, gender; HCT, haematocrit; MDR1, multidrug resistance 1 transporter genotype; POD, postoperative days; TBIL, total bilirubin; WT, bodyweight

Figure S2 Normalized prediction distribution error (NPDE) plots of the 17 published models. (A, E) Quantile–quantile plot of the distribution of the NPDE against theoretical distribution; (B, F) histogram of the distribution of the NPDE against theoretical distribution; (C, G) NPDE vs. postoperative time (h); (D, H) NPDE vs. predicted concentrations. Models with an asterisk (*) were developed based on the Michaelis–Menten model

Figure S3 Normalized prediction distribution error (NPDE) plots of structural models with and without covariates. (A, E) quantile–quantile plot of the distribution of the NPDE against theoretical distribution; (B, F) histogram of the distribution of the NPDE against theoretical distribution; (C, G) NPDE vs. postoperative time (h); (D, H) NPDE vs. predicted concentrations. 1CMT, one‐compartmental model; 2CMT, two‐compartmental model; MM, Michaelis–Menten model; POD, postoperative days; WT, body weight. Models with an asterisk (*) were developed based on the Michaelis–Menten model

Mao, J.‐J. , Jiao, Z. , Yun, H.‐Y. , Zhao, C.‐Y. , Chen, H.‐C. , Qiu, X.‐Y. , and Zhong, M.‐K. (2018) External evaluation of population pharmacokinetic models for ciclosporin in adult renal transplant recipients. Br J Clin Pharmacol, 84: 153–171. doi: 10.1111/bcp.13431.