J Pharmacokinet Pharmacodyn (2015) 42:33–43 DOI 10.1007/s10928-014-9396-7
ORIGINAL PAPER
In vitro simulation of in vivo pharmacokinetic model with intravenous administration via flow rate modulation Yuan-cheng Chen • Wang Liang • Jia-li Hu Gao-li He • Xiao-jie Wu • Xiao-fang Liu • Jing Zhang • Xue-qian Hu
•
Received: 20 July 2014 / Accepted: 21 October 2014 / Published online: 30 October 2014 Ó Springer Science+Business Media New York 2014
Y. Chen W. Liang J. Hu G. He X. Wu X. Liu J. Zhang Key Laboratory of Clinical Pharmacology of Antibiotics, National Population and Family Planning Commission, Shanghai 200040, China
Then an experiment was performed to confirm the feasibility of flow rate modulation method using levo-ornidazole as an example. The accuracy and precision of PK simulations were evaluated using average relative deviation (ARD), mean error and root mean squared error. In vitro model with constant flow rate could mimic one-compartment model, while the in vitro model with decreasing flow rate could simulate the linear mammillary model with multiple compartments. Zero-order model could be simulated using the in vitro model with elevating flow rate. In vitro PK model with gradually decreasing flow rate reproduced the two-compartment kinetics of levo-ornidazole quite well. The ARD was 0.925 % between in vitro PK parameters and in vivo values. Results suggest that various types of PK model could be simulated using flow rate modulation method without modifying the structure. The method provides uniform settings for the simulation of linear mammillary model and zero-order model based on in vitro one-compartment model, and brings convenience to the pharmacodynamic study.
Present Address: W. Liang Gene Tech (Shanghai) Company Limited, Shanghai 200241, China
Keywords In vitro model Pharmacokinetic simulation Flow rate modulation Equation method Intravenous injection
Present Address: J. Hu Department of Pharmacy, Shanghai Ruijin Hospital, Shanghai 200025, China
Introduction
Abstract The aim of this paper was to propose a method of flow rate modulation for simulation of in vivo pharmacokinetic (PK) model with intravenous injection based on a basic in vitro PK model. According to the rule of same relative change rate of concentration per unit time in vivo and in vitro, the equations for flow rate modulation were derived using equation method. Four examples from literature were given to show the application of flow rate modulation in the simulation of PK model of antimicrobial agents in vitro.
Y. Chen W. Liang J. Hu G. He X. Wu X. Liu J. Zhang (&) Institute of Antibiotics, Huashan Hospital, Fudan University, Shanghai 200040, China e-mail: [email protected]
Present Address: X. Liu Department of Pharmacy, Shanghai Tongji Hospital, Shanghai 200065, China X. Hu Department of Pharmacy, Tongren Hospital Affiliated to Medical College of Shanghai Jiaotong University, Shanghai 200336, China
Pharmacokinetic/pharmacodynamic (PK/PD) is an important concept in the clinical pharmacology that proposes to take PK process into account when evaluating PD properties [1, 2]. The primary data for PK/PD analysis are often derived from clinical studies and animal models, which are always limited by the ethics or species difference in PK process. The method of in vitro PK/PD model could overcome these limits by mimicking the PK/PD process in
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an in vitro setting. Such an approach is usually adopted in the PD studies of antibiotics because the antimicrobial effects are mainly associated with the interaction between antimicrobial agents and pathogenic microorganisms [3] and most of the pathogenic microorganisms are easy to grow in the in vitro circumstances. The in vitro PK/PD models have been proved successful in the pharmacological study of antimicrobial agents [4, 5]. The basic structure of an in vitro PK/PD model mainly includes a fresh medium reservoir, a central chamber and a waste collector which are usually connected using silicon tubes in tandem [6, 7]. The medium is driven by a peristaltic pump from the fresh medium reservoir to the central chamber then flows out to the waste collector under pressure. According to the structure of central chamber, the in vitro PK/PD models are classified into dilution, dialysis and filtration models [5]. The last two are the modified forms of dilution model with the addition of a cellulose ester membrane or a hollow fiber for preventing bacteria from being washed out by the medium flow [5, 8–10]. It’s no matter which form is used in the model, the mimicking of the PK process is the same, which is realized by setting the medium volume in the central chamber and the medium flow rate. With a fixed flow rate and given a specific medium volume, the one-compartment model with intravenously dosing could be easily mimicked. This is the most popular form in many in vitro PK/PD studies [11–13]. The two- or three-compartment model and the zero-order model are more complex in nature than one-compartment model, which carries difficulties for simulation of PK process in in vitro models. Sevillano D and his colleagues had proximately mimicked the two-compartment model by dividing flow rate profile into two one-compartment PK processes [14], which is a breakthrough for the complex model simulation in vitro. However, the shortcoming of Sevillano D’s method is the inconvenience to extrapolate. This method inspired us to hypothesize that the complex PK processes could be accurately mimicked in a simple model setting by intensively or continuously modulating the flow rate. Here, we described a method for calculation of flow rate scheme, which is applicable for linear mammillary models and zero-order model.
Materials and methods In vitro PK/PD model Our method was based on a basic in vitro PK/PD model which contained a fresh medium reservoir, central chamber and waste collector connected in tandem by silicon tubes. A smart peristaltic pump drives the medium from the reservoir to the central chamber, and with the effect of
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Fig. 1 Schematic illustration of the in vitro one-compartment model. The broth in the chamber and reservoir contained drug and drug-free medium, respectively
pressure an identical volume of medium flows out from the central chamber to the waste collector. The central chamber could be any form by means of dilution, dialysis or filtration. Figure 1 illustrates the structure of a dilutionbased chamber model. When it was used in an in vitro PK/ PD study, a cellulose ester membrane (0.45-lm pore size) was placed at the bottom of the central chamber to prevent bacteria loss [15]. Equation method for flow rate modulation The following criteria should be reached when performing in vitro simulation of in vivo PK: (1) the drug concentration in the central chamber (C) is identical to that in vivo (c); (2) the elimination rate of the drug in vitro (dC/dt) should be the same as that in vivo (dc/dt). Therefore, the equation (Eq) 1 could be obtained as follows: dCðmg=LÞ=dtðhÞ dcðmg=LÞ=dtðhÞ ¼ Cðmg=LÞ cðmg=LÞ
ð1Þ
where (dC/dt)/C and (dc/dt)/c indicate the relative change rate of drug concentration per unit time in vitro and in vivo. The symbol ‘Krel’ and ‘krel’ were used to denote (dC/dt)/ C and (dc/dt)/c, respectively. As for the in vitro model (Fig. 1), the change rate of drug concentration in the central chamber (C) was in positive proportion to the flow rate (F), while it also had inverse correlation with the broth volume in the chamber (VC), which could be described as follows: dCðmg=LÞ FðL=hÞ ¼ Cðmg=LÞ; dtðhÞ VC ðLÞ
C0 ðmg=LÞ ¼
DðmgÞ VC ðLÞ ð2Þ
where D is the drug dose in vitro calculated as c0 9 VC. Hence, the relative change rate of concentration in vitro could be derived by the transformation of Eq. (2):
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Table 1 Concentration–time equation, krel formula for in vivo PK model and flow rate (F) for in vitro PK simulation following single dose Type
Concentration-time equation
Linear mammillary model
c¼
l P
Ai eki t
F
Krel l P
(i)
krel ¼ i¼1P l
i¼1
l P
(vi)
Ai ki eki t Ai eki t
i¼1
One-compartment modela Two-compartment model
a
Three-compartment modela Zero-order model
(ii)
c = Ae-at ? Be-bt
(iii)
krel ¼
c = Ae-at ? Be-bt ? Re-ct
(iv)
þ Bbe þ Rce krel ¼ Aae Aeat þ Bebt þ Rect
c = c0 - ket
(v)
krel = - ke
krel ¼
Ai eki t
i¼1
c ¼ c0 eke t
(vii)
at þ Bbebt Aae Aeat þ Bebt at
(xi)
Ai ki eki t
F ¼ VC i¼1 l P
bt
ct
ke c0 k et
F = ke VC
(xii)
Aaeat þBbebt Aeat þBebt
(viii)
F ¼ VC
(ix)
þBbe þRce F ¼ VC Aae Aeat þBebt þRect
(x)
at
F¼
bt
(xiii) ct
VC ke c0 ke t
(xiv) (xv)
krel: relative change rate of concentration per unit time in vivo calculated as (dc/dt)/c a
One-, two- and three-compartment model are the common types of linear mammillary model
Krel ð1=hÞ ¼
FðL=hÞ : VC ðLÞ
ð3Þ
The expression of relative change rate of in vivo concentration was derived based on the type of PK model. Making that the relative change rate of concentration in vitro equal to that in vivo, the formula for the flow rate modulation could be deduced as the following:
Formula of flow rate modulation following single dosing For the one-compartment model with iv injection in vivo, the concentration (c) could be described as Eq.(ii) in Table 1, where ke indicates the elimination rate constant. The relative change rate of concentration could be derived shown as Eq.(vii) (Table 1). Making this equation identical to Eq. (3), the formula of flow rate could be obtained as Eq.(xii) (Table 1). For the two-compartment model with iv injection in vivo, making the relative change rate of concentration defined as Eq.(viii) (Table 1) identical to Eq. (3), the formula of flow rate modulation could be obtained as Eq.(xiii) (Table 1). In this equation, A and B are the intercept of log concentration–time (c-t) curve in the distribution and elimination phases, while a and b are the corresponding slopes. The formula of flow rate modulation for simulating three-compartment model with iv injection could be derived using the same method, and the result was defined as Eq.(xiv) in Table 1. Similarly, the flow rate equation for simulating linear mammillary model with iv injection could be derived as Eq.(xi) (Table 1). In this formula, Ai and ki indicate the intercept and slope of log c-t curve in the ith phase, and l means the total number of phases (i.e., compartments). For the simulation of zero-order model with iv injection, the flow rate could be derived using the same method. The result was shown as Eq.(xv) (Table 1), where c0 represents the initial concentration, and ke is the slope of c–t curve.
Fig. 2 Flow chart representing the application of flow rate modulation method in the in vitro PK simulation. CA, compartment analysis; NCA, non-compartment analysis; PK, pharmacokinetic
Verification of the formula for flow rate modulation Substituting the formula of flow rate modulation into the differential equation of concentration in the chamber (Eq. (2)), then transformed the expression and performed the definite integral of the equation. The formula verification was successful if the integral formula had the same form as the expression in vivo. Detail of the verification was provided in the Appendix. Validation of the method using literature data Four examples were given to show the application of flow rate modulation in the simulation of PK model of antimicrobials in vitro. The process was summarized in Fig. 2. Specifically, the in vivo mean concentration was obtained from the literature using a function of graphical input from mouse in Matlab software (Ver7.0, Mathworks Inc., USA), then the in vivo PK analysis was performed. The flow rate at each time point was calculated according to the specified PK parameters. Predicted concentration in the central
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Table 2 Estimated and setting values of PK parameters for one-compartment model and zero-order model Ke (1/h)
Vd (L)
0.232
14.8
Estimated value
0.233
14.8
Setting value
0.08
60
Estimated value
0.08
60.0
Parameter One-compartment model Zero-order model
Setting valuea
ARDb (10-1 %)
MEb (10-1)
RMSEb (10-1)
1.16
-0.00283
0.00382
-0.0001
-4.51 9 10-6
-6.51 9 10-6
a
Setting values were estimated from the literature data of imipenem [16] P P P Calculation according to: ARD(%) = 100 9 (yi’/yi-1)/m, ME = (yi-yi’)/m, RMSE = [ (yi-yi’)2/m]0.5, where yi and yi’ are the setting value and estimated value, m is the number of data
b
Table 3 Estimated and setting values of ceftazidime PK parameters for two- and three- compartment model with iv injection a (1/h)
Parameter Two-compartment model Three-compartment model
c (1/h)
A (lg/mL)
B (lg/mL)
R (lg/mL)
Vd (L)
Setting valuea
3.22
0.324
\
115
47.5
\
6.14
Estimated value
3.26
0.325
\
117
45.5
\
6.14
Setting value
a
Estimated value a
b (1/h)
3.65
0.286
0.0554
44.4
36.5
2.85
11.9
3.68
0.286
0.0556
45.5
35.4
2.75
11.9
ARD (%)
ME (10-3)
RMSE
-0.173
-0.185
1.27
-0.340
-4.81
0.602
Setting values were estimated from the literature data [17]. The equations for calculating ARD, ME and RMSE are shown in Table 2
chamber was obtained by substituting flow rate values into a recurrence formula (Eq. (4)) derived from Eq. (2) using forward difference method. The computing time gradient (ti?1-ti) was fixed at 0.02 h. Ciþ1 ¼ Ci ½1 Fi ðtiþ1 ti Þ=VC :
ð4Þ
in Table 1. Then the simulated concentration in the central chamber was calculated according to Eq. (4). Simulation of ceftazidime time-profile in healthy subjects as two-compartment model
The simulation ratio of in vitro concentration (C) to the fitting values of mean concentration in vivo (c) was calculated. Then the PK parameters of in vitro model were estimated based on simulated concentration data and compared with the estimated values in vivo. The accuracy and precision of the PK simulations were evaluated using average relative deviation (ARD), mean error (ME) and root mean squared error (RMSE). All these procedures were implemented in the Excel software (Microsoft Co. Ltd, Redmond, WA).
The in vivo concentration was obtained from the mean serum data of healthy volunteers following iv bolus administration of 1 g ceftazidime [17]. The estimated values of PK parameters (a, b, A, B and Vd) were shown in Table 3. The VC was 250 mL. The drug dose in vitro was calculated using the same method as example 1 (40.69 mg). The flow rate was calculated according to Eq.(xiii) (Table 1). Simulated concentrations in vitro were obtained by substituting flow rate values into Eq. (4).
Simulation of imipenem profile in patients as onecompartment model
Simulation of ceftazidime time-profile in patients as threecompartment model
The in vivo concentration of imipenem was obtained from the mean plasma data in the patients with renal insufficiency following iv bolus injection of imipenem-cilastatin (500 mg each) [16]. The estimated value of PK parameters [distribution volume (Vd) and ke] were presented in Table 2. The volume of broth in the central culture chamber (VC) was 250 mL. The drug dose in the central chamber (D) was calculated according to the rule of same initial concentration in vitro as that in vivo (Dose 9 VC/Vd), therefore D was 8.433 mg. The flow rate was calculated according to Eq.(xii)
The in vivo concentration was obtained from the mean serum data of patients with liver cirrhosis following iv bolus injection of 1 g ceftazidime [17]. The estimated values of PK parameter (a, b, c, A, B, R and Vd) were indicated in Table 3. VC was the same as that in example 1. The in vitro drug dose was 20.9 mg, which was calculated using the same method as that in example 1. The flow rate value was calculated according to Eq.(xiv) in Table 1. Simulated concentration was obtained by substituting flow rate value into Eq. (4).
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Simulation of zero-order model The rate of concentration decrease over time (ke) was 0.08 mg/(hL) (Table 2). The Dose and Vd were 240 mg and 60 L, while VC was the same as that in example 1. The drug dose in vitro was 1 mg. The flow rate was calculated according to Eq.(xv) in Table 1. The predicted concentration was obtained by substituting the flow rate values into Eq. (4). Experiment validation using levo-ornidazole as an example The pharmacokinetics of levo-ornidazole (a 5-nitroimidazole drug) was investigated following single infusion of 500 mg for 1 h in twelve healthy Chinese volunteers. PK analysis was performed using WinNonlin software (Ver6.0, Pharsight Co.Ltd, USA), including non-compartment analysis and compartment analysis. During the compartment analysis, model selection was based on the Akaike Information Criterion (AIC), and goodness of fit was assessed by the weighted sum of square of error (the weight was 1/C). Results showed that the mean concentration curve of levo-ornidazole was best described using two-compartment model (AIC was -6.35 for one-compartment model and -21.7 for two-compartment model). The in vitro one-compartment model (Fig. 1) was employed to simulate the mean time profile of levo-ornidazole following the end of infusion. The components of in vitro model system include: reservoir; MasterFlex L/SÒ pump and WinLIN software (Ver3.2) (Cole Parmer Co., Ltd, USA); microcomputer (Shenzhou Youya Q230B); central chamber; 85-2 magnetic stirrer (Shanghai Meiyingpu Co., Ltd) and the waste collector. The central chamber was the filtration-type compartment [15], and was airtight during the experiment. The volume of medium (water) was 200 mL. The model system was placed in the anaerobic chamber, and the temperature was maintained at 35 °C. Gradually decreasing flow rate was employed to simulate two-compartment kinetics of levo-ornidazole in vitro. Since the maximum segment for flow rate modulation using WinLIN software was fifty, a program for flow rate modulation with eight segments was designed. First, the time point for changing flow rate was determined based on the theoretical F–t curve obtained according to Eq.(xiii) (Table 1). The predicted concentration of levo-ornidazole in vitro was calculated according to Eq. (4). Then the sum of square of residual error between in vitro concentration and in vivo target value was calculated. The optimized value of flow rate was obtained by minimizing the sum of square using Solver macro in Excel software. The dose of levo-ornidazole in vitro was 2.3134 mg. In the experiment, 0.5 mL sample was collected from the sampling port of central chamber at 0, 2, 4, 6, 8, 10, 12, 24, 48 and 72 h post
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dosing, and were stored at -80 °C for drug assay. All experiments were repeated in triplicate. The concentration of levo-ornidazole was determined using liquid chromatography-tandem mass spectrometry (LC–MS/MS) method. The LC–MS/MS system includes Waters 2690 (Waters Co. Ltd., Milford, MA, USA) and the TSQ Quantum Discovery Max (Thermo Finnigan Co. Ltd., San Jose, CA, USA). Chromatography was performed using Symmetry Shield RP C18 column (2.19 50 mm, 5 lm pore size). The mobile phase was 0.1 % formic acid: acetonitrile (82:18), and the flow rate was 0.2 mL/min. The internal standard (IS) was metronidazole. Both the analyte and the internal standard were ionized by atmospheric pressure chemical ionization (APCI) and were detected in the multiple reaction monitoring (MRM) mode. Parent and product ions were monitored at m/z 220.0 ? 128.1 for levo-ornidazole. For metronidazole, parent and product ions were monitored at m/z 172.1 ? 128.1. The pretreatment of the samples were as follows: A 100 lL sample was mixed with 300 lL acetonitrile and 10 lL metronidazole (1.00 lg/mL). After centrifugation (12,000 rpm for 10 min), 350 lL supernatant was transferred into the tube and was extracted with 350 lL ethyl acetate for 10 min on a rotator. Following the same centrifugation, 500 lL supernatant was transferred and evaporated to dryness under a stream of nitrogen gas at 40 °C. The extract was dissolved in 150 lL of mobile phase, and then centrifuged at 12,000 rpm for 5 min. 3 lL of the supernatant was injected into the LC/MS–MS system. The lower limit of quantification was 0.01 lg/mL and linear range was 0.01–5 lg/mL. The ranges of inter- and intra-day accuracy (relative error) for levo-ornidazole were -1.3 * 13.8 % and -1.2 * 13.7 %. The ranges of inter- and intra-day precision (RSD) were 0.6 * 1.1 % and 1.1 * 3.4 %, respectively. PK parameters of levo-ornidazole in vitro were analyzed using WinNonlin software, and were compared with that in vivo. In detail, the mean concentration of levo-ornidazole in vitro was calculated. The non-compartment analysis was performed to obtain Cmax, AUC0–72 and T1/2 of levoornidazole. Then the in vitro mean concentration curve of levo-ornidazole was fitted using two-compartment model (the weight was 1/C). The PK parameters Vd, a, b, A and B were obtained. Finally, the ARD, ME and RMSE were calculated between the in vivo and in vitro PK parameters of levo-ornidazole.
Results In vitro pharmacokinetic simulation using literature data In vitro model with constant flow rate could simulate onecompartment model in vivo. The simulation results of
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Fig. 3 Flow rate for the simulation of one-compartment model using imipenem as an example (a) or zero-order model (d) following iv bolus injection. The predicted c-t curves using equation method (9)
were shown in panels (b) and (e). The ratio of predicted concentration in vitro to the fitting values of in vivo concentration was indicated in panels (c) and (f)
imipenem profile were described in left panels of Fig. 3. As shown in Fig. 3a, the flow rate obtained from equation method was 0.967 mL/min. The simulated concentration was close to the in vivo target (Fig. 3b), and the maximal range of deviation was less than 0.4 %. The estimated PK parameters were
close to the setting values of imipenem (Table 2). The ARD and RMSE were 0.116 % and 0.0004, respectively. In vitro model with decreasing flow rate could simulate two- and three- compartment models in vivo. The simulation results of ceftazidime profile in healthy volunteers
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Fig. 4 Flow rate for the simulation of ceftazidime profile as two- (a) or three-compartment model (d) with iv injection. The predicted c-t curves using equation method (9) were shown in panels (b) and (e).
The ratios of predicted concentration in vitro to the fitting values of in vivo concentration were shown in panels (c) and (f)
were demonstrated in the left panels of Fig. 4. The flow rate decreased rapidly during initial 2 h, and then maintained constant during 2–12 h. With Eq. (4), the flow rate values could be used to reproduce in vivo c-t curve quite well (Fig. 4b). As indicated in Fig. 4c, the deviation between predicted concentration and the fitting values of
in vivo concentration was lower than 6 %. The estimated values of PK parameter were close to the setting values (Table 3). The ARD and RMSE were -0.173 % and 1.27, respectively. The simulation results of ceftazidime profile in patients were indicated in the right panels of Fig. 4. The flow rate
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Fig. 5 Flow rate for the in vitro PK simulation of levo-ornidazole as two-compartment model (a). The modulation time and flow rate values were shown in the table. Panel (b) showed the time profile of
levo-ornidazole, where actual values were represented as mean ± SD (n = 3). Inset graph showed the concentration–time curve within 8 h post dosing
declined when time increased, and maintained at the minimal level at 18 h post-dosing. Figure 4e further proved that the simulation method could reproduce the in vivo time profile of the drug quite well. The deviation between simulated concentration in vitro and the fitting values of in vivo concentration was less than 4 % (Fig. 4f). The estimated values of PK parameter obtained from the simulation were close to the setting values. The corresponding ARD and RMSE were -0.340 % and 0.602 (Table 3). In vitro model with elevating flow rate could simulate zero-order model in vivo. The simulation results of zeroorder model were provided in the right panels of Fig. 3. When time increased, the flow rate elevated and its increasing tendency became more obvious (Fig. 3d). The simulation ratio of predicted concentration in vitro to in vivo concentration maintained constant at one. As shown in Table 2, the estimated values obtained from equation method were identical to the setting values of PK parameters. The ARD, ME and RMSE were extremely low (10-5 order of magnitude).
between in vitro PK parameters of levo-ornidazole and that in vivo were 0.925 % and 3.25, respectively.
Experiment confirmed the feasibility of flow rate modulation method As shown in Fig. 5a, the actual flow rate-time curve was close to the curve calculated from the equation method. The residual fraction between in vitro simulated concentration and in vivo mean curve was 7.36 9 10-5 %. The mean time profile of levo-ornidazole in vitro was consistent with two-compartment model (Fig. 5b). The relative deviation of Vd and b was -1.16 and 1.47 % between in vitro and in vivo. The relative deviation of AUC0-72 was -5.44 %, while the in vitro values of Cmax and T1/2 were the same as that in vivo (Table 4). The ARD and RMSE
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Discussion Accurate simulation of PK model is important for the PK/ PD study in vitro, and the way to perform PK simulation is mostly determined by the type of PK model. At present, the simulation of different types of PK model mainly depends on the modification of the model structure, which always brings inconvenience, even conflict with the PD research. To avoid such dilemma, several papers introduced approximate simulation of PK model by modulating flow rate in several steps [14, 18, 19]. Taking the simulation of two-compartment model as an example, the biexponential decay of c-t curves were approximated to a sequence of ‘pseudo-monoexponential’ profiles with different apparent elimination rate constants [14]. However, these papers did not describe how to set time point for flow rate modulation. There is still no report of flow rate modulation in the simulation of three-compartment model and zero-order model. In our early investigation of flow rate modulation methods, we tried to divide the flow rate into l segments to simulate the linear mammillary model with l compartments, the flow rate maintained constant in each section. The time for flow rate modulation was set empirically based on the change pattern of c-t curve. Although it was quite easy to carry out, the mathematical basis for determining the time point for flow rate modulation could not be found. The proposed method in this paper has settled this problem, the key point is that making the relative change rate of concentration per unit time in vitro [(dC/dt)/C] identical to that in vivo. In contrast, the literature method
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Table 4 In vivo and in vitro PK parameters of levo-ornidazole Parameter In vivo valuea In vitro value
a
Cmax (mg/L)
AUC0-72 (hmg/L)
T1/2 (h)
Vd (L)
a (1/h)
b (1/h)
A (mg/L)
B (mg/L)
11.7
167
12.3
43.2
0.462
0.0543
2.87
8.69
11.7
158
12.3
42.7
0.420
0.0551
3.72
7.99
ARD (%)
ME
RMSE
0.925
1.19
3.25
a
They were estimated from the mean time-profile of levo-ornidazole Cmax maximal concentration AUC area under the concentration–time curve, T1/2 half-life. The equations for calculating ARD, ME and RMSE are shown in Table 2
that only considering the same change rate of concentration in vitro (dC/dt) as that in vivo could not obtain the formula since the expression for flow rate modulation was transcendental equation. The main requisite for the proposed method to be available is that the flow rate of the pump should be automatically and continuously changed. The existence of many smart pumps has provided the convenience for this method to be available. Even the peristaltic pump in which the flow rate can’t change continuously is employed in the model, our method is still valid. According to the calculated flow rate chart, in theory, one can divide the whole process into many stages and determine the flow rate for each stage easily. The first method is to set a series of time points, then calculate the flow rate at each time point. This is shown in the PK simulation experiment of levo-ornidazole (Fig. 5a). The second method is to determine the modulation gradient of flow rate, then calculate the time points for flow rate modulation. The more stages were divided in the permitted frequency for the flow rate modulation, the simulation will be more accurate. If the micropulse pump was employed in the model, the method could also help us to obtain the modulation scheme by transforming the flow rate to frequency data. So the method provided by us is intelligible and useful, and the flow rate could be obtained with the help of commercial software such as Microsoft Excel. It could be observed that the ratio of in vitro concentration to that in vivo is smaller than one for the simulation of two- and three-compartment models (Fig. 4c and Fig. 4f). The main reason is that the calculation of in vitro concentration based on recurrence formula (Eq. (4)) had computational error. Taking the simulation of two-compartment model as an example, the flow rate decreased along with time. However, the flow rate (Fi) was constant at each segment during the calculation using recurrence formula (Eq. (4)). This resulted in lower in vitro concentration compared to the actual values. Hence, the ratio of in vitro concentration to that in vivo was smaller than one (Fig. 4c). The same reason could be used to explain the fact that the ratio below one for the in vitro simulation of threecompartment model (Fig. 4f). When the computing time
gradient (ti?1-ti) is smaller, the ratio will be closer to the value one. Before in vitro PK simulation study, it is necessary to obtain prior in vivo PK information in humans. There are three sources for in vivo PK information: (1) In vivo time profile of the drug: the flow rate could be obtained from equation method following compartment analysis. Using mean value of PK parameter to calculate flow rate is recommended; (2) Type of PK model and value of PK parameters: the flow rate could be obtained by substituting PK parameters in the equation directly; (3) Type of population pharmacokinetic (PPK) model and value of PPK parameters. First, the PK parameter for specified population is obtained from the PPK model using typical values and covariate values. Then the flow rate could be calculated using equation method. Our method could be extrapolated to mimic multiple PK in vitro. The flow rate equations for simulating linear mammillary model and zero-order model with iv injection have been obtained. The in vitro model shown in Fig. 1 could be used within maximum 72 h post dosing. We are making effort to use in vitro model with flow rate modulation to simulate multiple PK within 5–7 days post dosing. This could validate the feasibility of flow rate modulation method following multiple dosing. The main merit of our method is providing a uniform setting for the simulation of linear mammillary model with different compartments. It’s not a method which could be used to simulate the PK models with different administration modes simultaneously. However, if the model structure is modified to simulate the PK model with oral or infusion dosing, by continuous flow-rate modulation, one can accurately simulate the linear mammillary model with different compartments in a uniform model setting. We are now expanding our method for the calculation of flow rate in these situations. The in vitro combination studies of antimicrobials with different mechanisms combating bacterial resistance are increasing [20, 21]. We have great interest on how to perform the simulation of two drugs’ profile when both of them conform to two-compartment model using the in vitro system reported in the paper [22, 23]. Results showed that
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42 Table 5 Summary of the formula for the verification of flow rate modulation equation
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Type Linear mammillary model
Equation of ‘dC/C’ l P
dC C
¼ P
l P
(i)
Ai ki eki t
dt
i¼1 l
Concentration–time equation in vitro
C0 Ai
C¼
l P
Ai eki t
(vi)
i¼1
i¼1
(ii)
c0
C ¼ c0 eke t
(vii)
(iii)
A?B
C = Ae-at ? Be-bt
(viii)
(iv)
A ? B?R
C = Ae-at ? Be-bt ? Re-ct
(ix)
(v)
c0
C = c0 - ke t
(x)
Ai eki t
i¼1
C drug concentration in the central chamber, C0 initial drug concentration, l the number of compartments
Onecompartment model
dC C
¼ ke dt
Twocompartment model Threecompartment model
dC C
þ Bbe ¼ Aae dt Aeat þ Bebt
dC C
¼ Aaeat þ Bbebt
Zero-order model
dC C
ke ¼ c0 k dt et
at
bt
at
Ae
bt
þ Be
the time profiles of two drugs could be simulated approximately when the flow rate decreased along with time based on equation method (data not shown). We are now making effort to improve the proposed method for better PK simulation in vitro. In summary, this report showed that the PK model with iv injection could be accurately simulated in vitro via flow rate modulation. In vitro model with constant flow rate could simulate one-compartment model, while the in vitro model with decreasing flow rate could mimic the linear mammillary model with multiple compartments. Zeroorder model could be reproduced using in vitro model with elevating flow rate. The equation method for flow rate modulation was validated using literature data and PK simulation experiment of levo-ornidazole. The proposed method significantly simplifies the structure of in vitro model and brings convenience to the in vitro PD study. Acknowledgments The work was supported by the National Natural Science Foundation of China (No. 81202582), the Major Research and Development Project of Innovative Drugs, Ministry of Science and Technology (2012ZX09303004-001) and China Postdoctoral Science Foundation (No. 2012M511045). We thank the help of Dr Sheng-li Li for his revision of the manuscript.
þ Rcect ct þ Re
dt
l P
Ai eki t C ln ¼ ln i¼1 l P C0 Ai
ð5Þ
i¼1
Making that
l P
Ai , the Eq. (5) could be simplified as
i¼1
Eq.(vi) in Table 5, which had the same form as c-t equation of linear mammillary model in vivo. For the simulation of one-compartment model in vitro, the Eq.(ii) (Table 5) could be obtained by substituting Eq.(xii) (Table 1) into Eq. (2). The Eq.(vii) (Table 5) could be yielded through the definite integral of Eq.(ii) (Table 5) during 0–t interval when C0 equals to c0. The Eq.(vii) (Table 5) had the same form as Eq.(ii) (Table 1). As for the simulation for two-compartment model in vitro, similar method could be used to verify the formula of flow rate modulation when the initial value (C0) equals A ? B. For three-compartment model in vitro, the formula for flow rate modulation could be demonstrated using the same method when C0 is the sum of A, B and R. The same process could be applied to verify the formula for flow rate modulation mimicking zero-order model when C0 equals to c0.
Conflict of interest We have no conflicts of interest related to the content of this study.
References Appendix For the flow rate modulation formula of linear mammillary model, the Eq.(i) (Table 5) could be obtained by substituting Eq.(xi) (Table 1) into Eq. (2). The following equation could be derived by the definite integral of Eq.(i) (Table 5) at both sides during 0–t interval:
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ORIGINAL PAPER
In vitro simulation of in vivo pharmacokinetic model with intravenous administration via flow rate modulation Yuan-cheng Chen • Wang Liang • Jia-li Hu Gao-li He • Xiao-jie Wu • Xiao-fang Liu • Jing Zhang • Xue-qian Hu
•
Received: 20 July 2014 / Accepted: 21 October 2014 / Published online: 30 October 2014 Ó Springer Science+Business Media New York 2014
Y. Chen W. Liang J. Hu G. He X. Wu X. Liu J. Zhang Key Laboratory of Clinical Pharmacology of Antibiotics, National Population and Family Planning Commission, Shanghai 200040, China
Then an experiment was performed to confirm the feasibility of flow rate modulation method using levo-ornidazole as an example. The accuracy and precision of PK simulations were evaluated using average relative deviation (ARD), mean error and root mean squared error. In vitro model with constant flow rate could mimic one-compartment model, while the in vitro model with decreasing flow rate could simulate the linear mammillary model with multiple compartments. Zero-order model could be simulated using the in vitro model with elevating flow rate. In vitro PK model with gradually decreasing flow rate reproduced the two-compartment kinetics of levo-ornidazole quite well. The ARD was 0.925 % between in vitro PK parameters and in vivo values. Results suggest that various types of PK model could be simulated using flow rate modulation method without modifying the structure. The method provides uniform settings for the simulation of linear mammillary model and zero-order model based on in vitro one-compartment model, and brings convenience to the pharmacodynamic study.
Present Address: W. Liang Gene Tech (Shanghai) Company Limited, Shanghai 200241, China
Keywords In vitro model Pharmacokinetic simulation Flow rate modulation Equation method Intravenous injection
Present Address: J. Hu Department of Pharmacy, Shanghai Ruijin Hospital, Shanghai 200025, China
Introduction
Abstract The aim of this paper was to propose a method of flow rate modulation for simulation of in vivo pharmacokinetic (PK) model with intravenous injection based on a basic in vitro PK model. According to the rule of same relative change rate of concentration per unit time in vivo and in vitro, the equations for flow rate modulation were derived using equation method. Four examples from literature were given to show the application of flow rate modulation in the simulation of PK model of antimicrobial agents in vitro.
Y. Chen W. Liang J. Hu G. He X. Wu X. Liu J. Zhang (&) Institute of Antibiotics, Huashan Hospital, Fudan University, Shanghai 200040, China e-mail: [email protected]
Present Address: X. Liu Department of Pharmacy, Shanghai Tongji Hospital, Shanghai 200065, China X. Hu Department of Pharmacy, Tongren Hospital Affiliated to Medical College of Shanghai Jiaotong University, Shanghai 200336, China
Pharmacokinetic/pharmacodynamic (PK/PD) is an important concept in the clinical pharmacology that proposes to take PK process into account when evaluating PD properties [1, 2]. The primary data for PK/PD analysis are often derived from clinical studies and animal models, which are always limited by the ethics or species difference in PK process. The method of in vitro PK/PD model could overcome these limits by mimicking the PK/PD process in
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34
an in vitro setting. Such an approach is usually adopted in the PD studies of antibiotics because the antimicrobial effects are mainly associated with the interaction between antimicrobial agents and pathogenic microorganisms [3] and most of the pathogenic microorganisms are easy to grow in the in vitro circumstances. The in vitro PK/PD models have been proved successful in the pharmacological study of antimicrobial agents [4, 5]. The basic structure of an in vitro PK/PD model mainly includes a fresh medium reservoir, a central chamber and a waste collector which are usually connected using silicon tubes in tandem [6, 7]. The medium is driven by a peristaltic pump from the fresh medium reservoir to the central chamber then flows out to the waste collector under pressure. According to the structure of central chamber, the in vitro PK/PD models are classified into dilution, dialysis and filtration models [5]. The last two are the modified forms of dilution model with the addition of a cellulose ester membrane or a hollow fiber for preventing bacteria from being washed out by the medium flow [5, 8–10]. It’s no matter which form is used in the model, the mimicking of the PK process is the same, which is realized by setting the medium volume in the central chamber and the medium flow rate. With a fixed flow rate and given a specific medium volume, the one-compartment model with intravenously dosing could be easily mimicked. This is the most popular form in many in vitro PK/PD studies [11–13]. The two- or three-compartment model and the zero-order model are more complex in nature than one-compartment model, which carries difficulties for simulation of PK process in in vitro models. Sevillano D and his colleagues had proximately mimicked the two-compartment model by dividing flow rate profile into two one-compartment PK processes [14], which is a breakthrough for the complex model simulation in vitro. However, the shortcoming of Sevillano D’s method is the inconvenience to extrapolate. This method inspired us to hypothesize that the complex PK processes could be accurately mimicked in a simple model setting by intensively or continuously modulating the flow rate. Here, we described a method for calculation of flow rate scheme, which is applicable for linear mammillary models and zero-order model.
Materials and methods In vitro PK/PD model Our method was based on a basic in vitro PK/PD model which contained a fresh medium reservoir, central chamber and waste collector connected in tandem by silicon tubes. A smart peristaltic pump drives the medium from the reservoir to the central chamber, and with the effect of
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Fig. 1 Schematic illustration of the in vitro one-compartment model. The broth in the chamber and reservoir contained drug and drug-free medium, respectively
pressure an identical volume of medium flows out from the central chamber to the waste collector. The central chamber could be any form by means of dilution, dialysis or filtration. Figure 1 illustrates the structure of a dilutionbased chamber model. When it was used in an in vitro PK/ PD study, a cellulose ester membrane (0.45-lm pore size) was placed at the bottom of the central chamber to prevent bacteria loss [15]. Equation method for flow rate modulation The following criteria should be reached when performing in vitro simulation of in vivo PK: (1) the drug concentration in the central chamber (C) is identical to that in vivo (c); (2) the elimination rate of the drug in vitro (dC/dt) should be the same as that in vivo (dc/dt). Therefore, the equation (Eq) 1 could be obtained as follows: dCðmg=LÞ=dtðhÞ dcðmg=LÞ=dtðhÞ ¼ Cðmg=LÞ cðmg=LÞ
ð1Þ
where (dC/dt)/C and (dc/dt)/c indicate the relative change rate of drug concentration per unit time in vitro and in vivo. The symbol ‘Krel’ and ‘krel’ were used to denote (dC/dt)/ C and (dc/dt)/c, respectively. As for the in vitro model (Fig. 1), the change rate of drug concentration in the central chamber (C) was in positive proportion to the flow rate (F), while it also had inverse correlation with the broth volume in the chamber (VC), which could be described as follows: dCðmg=LÞ FðL=hÞ ¼ Cðmg=LÞ; dtðhÞ VC ðLÞ
C0 ðmg=LÞ ¼
DðmgÞ VC ðLÞ ð2Þ
where D is the drug dose in vitro calculated as c0 9 VC. Hence, the relative change rate of concentration in vitro could be derived by the transformation of Eq. (2):
J Pharmacokinet Pharmacodyn (2015) 42:33–43
35
Table 1 Concentration–time equation, krel formula for in vivo PK model and flow rate (F) for in vitro PK simulation following single dose Type
Concentration-time equation
Linear mammillary model
c¼
l P
Ai eki t
F
Krel l P
(i)
krel ¼ i¼1P l
i¼1
l P
(vi)
Ai ki eki t Ai eki t
i¼1
One-compartment modela Two-compartment model
a
Three-compartment modela Zero-order model
(ii)
c = Ae-at ? Be-bt
(iii)
krel ¼
c = Ae-at ? Be-bt ? Re-ct
(iv)
þ Bbe þ Rce krel ¼ Aae Aeat þ Bebt þ Rect
c = c0 - ket
(v)
krel = - ke
krel ¼
Ai eki t
i¼1
c ¼ c0 eke t
(vii)
at þ Bbebt Aae Aeat þ Bebt at
(xi)
Ai ki eki t
F ¼ VC i¼1 l P
bt
ct
ke c0 k et
F = ke VC
(xii)
Aaeat þBbebt Aeat þBebt
(viii)
F ¼ VC
(ix)
þBbe þRce F ¼ VC Aae Aeat þBebt þRect
(x)
at
F¼
bt
(xiii) ct
VC ke c0 ke t
(xiv) (xv)
krel: relative change rate of concentration per unit time in vivo calculated as (dc/dt)/c a
One-, two- and three-compartment model are the common types of linear mammillary model
Krel ð1=hÞ ¼
FðL=hÞ : VC ðLÞ
ð3Þ
The expression of relative change rate of in vivo concentration was derived based on the type of PK model. Making that the relative change rate of concentration in vitro equal to that in vivo, the formula for the flow rate modulation could be deduced as the following:
Formula of flow rate modulation following single dosing For the one-compartment model with iv injection in vivo, the concentration (c) could be described as Eq.(ii) in Table 1, where ke indicates the elimination rate constant. The relative change rate of concentration could be derived shown as Eq.(vii) (Table 1). Making this equation identical to Eq. (3), the formula of flow rate could be obtained as Eq.(xii) (Table 1). For the two-compartment model with iv injection in vivo, making the relative change rate of concentration defined as Eq.(viii) (Table 1) identical to Eq. (3), the formula of flow rate modulation could be obtained as Eq.(xiii) (Table 1). In this equation, A and B are the intercept of log concentration–time (c-t) curve in the distribution and elimination phases, while a and b are the corresponding slopes. The formula of flow rate modulation for simulating three-compartment model with iv injection could be derived using the same method, and the result was defined as Eq.(xiv) in Table 1. Similarly, the flow rate equation for simulating linear mammillary model with iv injection could be derived as Eq.(xi) (Table 1). In this formula, Ai and ki indicate the intercept and slope of log c-t curve in the ith phase, and l means the total number of phases (i.e., compartments). For the simulation of zero-order model with iv injection, the flow rate could be derived using the same method. The result was shown as Eq.(xv) (Table 1), where c0 represents the initial concentration, and ke is the slope of c–t curve.
Fig. 2 Flow chart representing the application of flow rate modulation method in the in vitro PK simulation. CA, compartment analysis; NCA, non-compartment analysis; PK, pharmacokinetic
Verification of the formula for flow rate modulation Substituting the formula of flow rate modulation into the differential equation of concentration in the chamber (Eq. (2)), then transformed the expression and performed the definite integral of the equation. The formula verification was successful if the integral formula had the same form as the expression in vivo. Detail of the verification was provided in the Appendix. Validation of the method using literature data Four examples were given to show the application of flow rate modulation in the simulation of PK model of antimicrobials in vitro. The process was summarized in Fig. 2. Specifically, the in vivo mean concentration was obtained from the literature using a function of graphical input from mouse in Matlab software (Ver7.0, Mathworks Inc., USA), then the in vivo PK analysis was performed. The flow rate at each time point was calculated according to the specified PK parameters. Predicted concentration in the central
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Table 2 Estimated and setting values of PK parameters for one-compartment model and zero-order model Ke (1/h)
Vd (L)
0.232
14.8
Estimated value
0.233
14.8
Setting value
0.08
60
Estimated value
0.08
60.0
Parameter One-compartment model Zero-order model
Setting valuea
ARDb (10-1 %)
MEb (10-1)
RMSEb (10-1)
1.16
-0.00283
0.00382
-0.0001
-4.51 9 10-6
-6.51 9 10-6
a
Setting values were estimated from the literature data of imipenem [16] P P P Calculation according to: ARD(%) = 100 9 (yi’/yi-1)/m, ME = (yi-yi’)/m, RMSE = [ (yi-yi’)2/m]0.5, where yi and yi’ are the setting value and estimated value, m is the number of data
b
Table 3 Estimated and setting values of ceftazidime PK parameters for two- and three- compartment model with iv injection a (1/h)
Parameter Two-compartment model Three-compartment model
c (1/h)
A (lg/mL)
B (lg/mL)
R (lg/mL)
Vd (L)
Setting valuea
3.22
0.324
\
115
47.5
\
6.14
Estimated value
3.26
0.325
\
117
45.5
\
6.14
Setting value
a
Estimated value a
b (1/h)
3.65
0.286
0.0554
44.4
36.5
2.85
11.9
3.68
0.286
0.0556
45.5
35.4
2.75
11.9
ARD (%)
ME (10-3)
RMSE
-0.173
-0.185
1.27
-0.340
-4.81
0.602
Setting values were estimated from the literature data [17]. The equations for calculating ARD, ME and RMSE are shown in Table 2
chamber was obtained by substituting flow rate values into a recurrence formula (Eq. (4)) derived from Eq. (2) using forward difference method. The computing time gradient (ti?1-ti) was fixed at 0.02 h. Ciþ1 ¼ Ci ½1 Fi ðtiþ1 ti Þ=VC :
ð4Þ
in Table 1. Then the simulated concentration in the central chamber was calculated according to Eq. (4). Simulation of ceftazidime time-profile in healthy subjects as two-compartment model
The simulation ratio of in vitro concentration (C) to the fitting values of mean concentration in vivo (c) was calculated. Then the PK parameters of in vitro model were estimated based on simulated concentration data and compared with the estimated values in vivo. The accuracy and precision of the PK simulations were evaluated using average relative deviation (ARD), mean error (ME) and root mean squared error (RMSE). All these procedures were implemented in the Excel software (Microsoft Co. Ltd, Redmond, WA).
The in vivo concentration was obtained from the mean serum data of healthy volunteers following iv bolus administration of 1 g ceftazidime [17]. The estimated values of PK parameters (a, b, A, B and Vd) were shown in Table 3. The VC was 250 mL. The drug dose in vitro was calculated using the same method as example 1 (40.69 mg). The flow rate was calculated according to Eq.(xiii) (Table 1). Simulated concentrations in vitro were obtained by substituting flow rate values into Eq. (4).
Simulation of imipenem profile in patients as onecompartment model
Simulation of ceftazidime time-profile in patients as threecompartment model
The in vivo concentration of imipenem was obtained from the mean plasma data in the patients with renal insufficiency following iv bolus injection of imipenem-cilastatin (500 mg each) [16]. The estimated value of PK parameters [distribution volume (Vd) and ke] were presented in Table 2. The volume of broth in the central culture chamber (VC) was 250 mL. The drug dose in the central chamber (D) was calculated according to the rule of same initial concentration in vitro as that in vivo (Dose 9 VC/Vd), therefore D was 8.433 mg. The flow rate was calculated according to Eq.(xii)
The in vivo concentration was obtained from the mean serum data of patients with liver cirrhosis following iv bolus injection of 1 g ceftazidime [17]. The estimated values of PK parameter (a, b, c, A, B, R and Vd) were indicated in Table 3. VC was the same as that in example 1. The in vitro drug dose was 20.9 mg, which was calculated using the same method as that in example 1. The flow rate value was calculated according to Eq.(xiv) in Table 1. Simulated concentration was obtained by substituting flow rate value into Eq. (4).
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Simulation of zero-order model The rate of concentration decrease over time (ke) was 0.08 mg/(hL) (Table 2). The Dose and Vd were 240 mg and 60 L, while VC was the same as that in example 1. The drug dose in vitro was 1 mg. The flow rate was calculated according to Eq.(xv) in Table 1. The predicted concentration was obtained by substituting the flow rate values into Eq. (4). Experiment validation using levo-ornidazole as an example The pharmacokinetics of levo-ornidazole (a 5-nitroimidazole drug) was investigated following single infusion of 500 mg for 1 h in twelve healthy Chinese volunteers. PK analysis was performed using WinNonlin software (Ver6.0, Pharsight Co.Ltd, USA), including non-compartment analysis and compartment analysis. During the compartment analysis, model selection was based on the Akaike Information Criterion (AIC), and goodness of fit was assessed by the weighted sum of square of error (the weight was 1/C). Results showed that the mean concentration curve of levo-ornidazole was best described using two-compartment model (AIC was -6.35 for one-compartment model and -21.7 for two-compartment model). The in vitro one-compartment model (Fig. 1) was employed to simulate the mean time profile of levo-ornidazole following the end of infusion. The components of in vitro model system include: reservoir; MasterFlex L/SÒ pump and WinLIN software (Ver3.2) (Cole Parmer Co., Ltd, USA); microcomputer (Shenzhou Youya Q230B); central chamber; 85-2 magnetic stirrer (Shanghai Meiyingpu Co., Ltd) and the waste collector. The central chamber was the filtration-type compartment [15], and was airtight during the experiment. The volume of medium (water) was 200 mL. The model system was placed in the anaerobic chamber, and the temperature was maintained at 35 °C. Gradually decreasing flow rate was employed to simulate two-compartment kinetics of levo-ornidazole in vitro. Since the maximum segment for flow rate modulation using WinLIN software was fifty, a program for flow rate modulation with eight segments was designed. First, the time point for changing flow rate was determined based on the theoretical F–t curve obtained according to Eq.(xiii) (Table 1). The predicted concentration of levo-ornidazole in vitro was calculated according to Eq. (4). Then the sum of square of residual error between in vitro concentration and in vivo target value was calculated. The optimized value of flow rate was obtained by minimizing the sum of square using Solver macro in Excel software. The dose of levo-ornidazole in vitro was 2.3134 mg. In the experiment, 0.5 mL sample was collected from the sampling port of central chamber at 0, 2, 4, 6, 8, 10, 12, 24, 48 and 72 h post
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dosing, and were stored at -80 °C for drug assay. All experiments were repeated in triplicate. The concentration of levo-ornidazole was determined using liquid chromatography-tandem mass spectrometry (LC–MS/MS) method. The LC–MS/MS system includes Waters 2690 (Waters Co. Ltd., Milford, MA, USA) and the TSQ Quantum Discovery Max (Thermo Finnigan Co. Ltd., San Jose, CA, USA). Chromatography was performed using Symmetry Shield RP C18 column (2.19 50 mm, 5 lm pore size). The mobile phase was 0.1 % formic acid: acetonitrile (82:18), and the flow rate was 0.2 mL/min. The internal standard (IS) was metronidazole. Both the analyte and the internal standard were ionized by atmospheric pressure chemical ionization (APCI) and were detected in the multiple reaction monitoring (MRM) mode. Parent and product ions were monitored at m/z 220.0 ? 128.1 for levo-ornidazole. For metronidazole, parent and product ions were monitored at m/z 172.1 ? 128.1. The pretreatment of the samples were as follows: A 100 lL sample was mixed with 300 lL acetonitrile and 10 lL metronidazole (1.00 lg/mL). After centrifugation (12,000 rpm for 10 min), 350 lL supernatant was transferred into the tube and was extracted with 350 lL ethyl acetate for 10 min on a rotator. Following the same centrifugation, 500 lL supernatant was transferred and evaporated to dryness under a stream of nitrogen gas at 40 °C. The extract was dissolved in 150 lL of mobile phase, and then centrifuged at 12,000 rpm for 5 min. 3 lL of the supernatant was injected into the LC/MS–MS system. The lower limit of quantification was 0.01 lg/mL and linear range was 0.01–5 lg/mL. The ranges of inter- and intra-day accuracy (relative error) for levo-ornidazole were -1.3 * 13.8 % and -1.2 * 13.7 %. The ranges of inter- and intra-day precision (RSD) were 0.6 * 1.1 % and 1.1 * 3.4 %, respectively. PK parameters of levo-ornidazole in vitro were analyzed using WinNonlin software, and were compared with that in vivo. In detail, the mean concentration of levo-ornidazole in vitro was calculated. The non-compartment analysis was performed to obtain Cmax, AUC0–72 and T1/2 of levoornidazole. Then the in vitro mean concentration curve of levo-ornidazole was fitted using two-compartment model (the weight was 1/C). The PK parameters Vd, a, b, A and B were obtained. Finally, the ARD, ME and RMSE were calculated between the in vivo and in vitro PK parameters of levo-ornidazole.
Results In vitro pharmacokinetic simulation using literature data In vitro model with constant flow rate could simulate onecompartment model in vivo. The simulation results of
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J Pharmacokinet Pharmacodyn (2015) 42:33–43
Fig. 3 Flow rate for the simulation of one-compartment model using imipenem as an example (a) or zero-order model (d) following iv bolus injection. The predicted c-t curves using equation method (9)
were shown in panels (b) and (e). The ratio of predicted concentration in vitro to the fitting values of in vivo concentration was indicated in panels (c) and (f)
imipenem profile were described in left panels of Fig. 3. As shown in Fig. 3a, the flow rate obtained from equation method was 0.967 mL/min. The simulated concentration was close to the in vivo target (Fig. 3b), and the maximal range of deviation was less than 0.4 %. The estimated PK parameters were
close to the setting values of imipenem (Table 2). The ARD and RMSE were 0.116 % and 0.0004, respectively. In vitro model with decreasing flow rate could simulate two- and three- compartment models in vivo. The simulation results of ceftazidime profile in healthy volunteers
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J Pharmacokinet Pharmacodyn (2015) 42:33–43
39
Fig. 4 Flow rate for the simulation of ceftazidime profile as two- (a) or three-compartment model (d) with iv injection. The predicted c-t curves using equation method (9) were shown in panels (b) and (e).
The ratios of predicted concentration in vitro to the fitting values of in vivo concentration were shown in panels (c) and (f)
were demonstrated in the left panels of Fig. 4. The flow rate decreased rapidly during initial 2 h, and then maintained constant during 2–12 h. With Eq. (4), the flow rate values could be used to reproduce in vivo c-t curve quite well (Fig. 4b). As indicated in Fig. 4c, the deviation between predicted concentration and the fitting values of
in vivo concentration was lower than 6 %. The estimated values of PK parameter were close to the setting values (Table 3). The ARD and RMSE were -0.173 % and 1.27, respectively. The simulation results of ceftazidime profile in patients were indicated in the right panels of Fig. 4. The flow rate
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J Pharmacokinet Pharmacodyn (2015) 42:33–43
Fig. 5 Flow rate for the in vitro PK simulation of levo-ornidazole as two-compartment model (a). The modulation time and flow rate values were shown in the table. Panel (b) showed the time profile of
levo-ornidazole, where actual values were represented as mean ± SD (n = 3). Inset graph showed the concentration–time curve within 8 h post dosing
declined when time increased, and maintained at the minimal level at 18 h post-dosing. Figure 4e further proved that the simulation method could reproduce the in vivo time profile of the drug quite well. The deviation between simulated concentration in vitro and the fitting values of in vivo concentration was less than 4 % (Fig. 4f). The estimated values of PK parameter obtained from the simulation were close to the setting values. The corresponding ARD and RMSE were -0.340 % and 0.602 (Table 3). In vitro model with elevating flow rate could simulate zero-order model in vivo. The simulation results of zeroorder model were provided in the right panels of Fig. 3. When time increased, the flow rate elevated and its increasing tendency became more obvious (Fig. 3d). The simulation ratio of predicted concentration in vitro to in vivo concentration maintained constant at one. As shown in Table 2, the estimated values obtained from equation method were identical to the setting values of PK parameters. The ARD, ME and RMSE were extremely low (10-5 order of magnitude).
between in vitro PK parameters of levo-ornidazole and that in vivo were 0.925 % and 3.25, respectively.
Experiment confirmed the feasibility of flow rate modulation method As shown in Fig. 5a, the actual flow rate-time curve was close to the curve calculated from the equation method. The residual fraction between in vitro simulated concentration and in vivo mean curve was 7.36 9 10-5 %. The mean time profile of levo-ornidazole in vitro was consistent with two-compartment model (Fig. 5b). The relative deviation of Vd and b was -1.16 and 1.47 % between in vitro and in vivo. The relative deviation of AUC0-72 was -5.44 %, while the in vitro values of Cmax and T1/2 were the same as that in vivo (Table 4). The ARD and RMSE
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Discussion Accurate simulation of PK model is important for the PK/ PD study in vitro, and the way to perform PK simulation is mostly determined by the type of PK model. At present, the simulation of different types of PK model mainly depends on the modification of the model structure, which always brings inconvenience, even conflict with the PD research. To avoid such dilemma, several papers introduced approximate simulation of PK model by modulating flow rate in several steps [14, 18, 19]. Taking the simulation of two-compartment model as an example, the biexponential decay of c-t curves were approximated to a sequence of ‘pseudo-monoexponential’ profiles with different apparent elimination rate constants [14]. However, these papers did not describe how to set time point for flow rate modulation. There is still no report of flow rate modulation in the simulation of three-compartment model and zero-order model. In our early investigation of flow rate modulation methods, we tried to divide the flow rate into l segments to simulate the linear mammillary model with l compartments, the flow rate maintained constant in each section. The time for flow rate modulation was set empirically based on the change pattern of c-t curve. Although it was quite easy to carry out, the mathematical basis for determining the time point for flow rate modulation could not be found. The proposed method in this paper has settled this problem, the key point is that making the relative change rate of concentration per unit time in vitro [(dC/dt)/C] identical to that in vivo. In contrast, the literature method
J Pharmacokinet Pharmacodyn (2015) 42:33–43
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Table 4 In vivo and in vitro PK parameters of levo-ornidazole Parameter In vivo valuea In vitro value
a
Cmax (mg/L)
AUC0-72 (hmg/L)
T1/2 (h)
Vd (L)
a (1/h)
b (1/h)
A (mg/L)
B (mg/L)
11.7
167
12.3
43.2
0.462
0.0543
2.87
8.69
11.7
158
12.3
42.7
0.420
0.0551
3.72
7.99
ARD (%)
ME
RMSE
0.925
1.19
3.25
a
They were estimated from the mean time-profile of levo-ornidazole Cmax maximal concentration AUC area under the concentration–time curve, T1/2 half-life. The equations for calculating ARD, ME and RMSE are shown in Table 2
that only considering the same change rate of concentration in vitro (dC/dt) as that in vivo could not obtain the formula since the expression for flow rate modulation was transcendental equation. The main requisite for the proposed method to be available is that the flow rate of the pump should be automatically and continuously changed. The existence of many smart pumps has provided the convenience for this method to be available. Even the peristaltic pump in which the flow rate can’t change continuously is employed in the model, our method is still valid. According to the calculated flow rate chart, in theory, one can divide the whole process into many stages and determine the flow rate for each stage easily. The first method is to set a series of time points, then calculate the flow rate at each time point. This is shown in the PK simulation experiment of levo-ornidazole (Fig. 5a). The second method is to determine the modulation gradient of flow rate, then calculate the time points for flow rate modulation. The more stages were divided in the permitted frequency for the flow rate modulation, the simulation will be more accurate. If the micropulse pump was employed in the model, the method could also help us to obtain the modulation scheme by transforming the flow rate to frequency data. So the method provided by us is intelligible and useful, and the flow rate could be obtained with the help of commercial software such as Microsoft Excel. It could be observed that the ratio of in vitro concentration to that in vivo is smaller than one for the simulation of two- and three-compartment models (Fig. 4c and Fig. 4f). The main reason is that the calculation of in vitro concentration based on recurrence formula (Eq. (4)) had computational error. Taking the simulation of two-compartment model as an example, the flow rate decreased along with time. However, the flow rate (Fi) was constant at each segment during the calculation using recurrence formula (Eq. (4)). This resulted in lower in vitro concentration compared to the actual values. Hence, the ratio of in vitro concentration to that in vivo was smaller than one (Fig. 4c). The same reason could be used to explain the fact that the ratio below one for the in vitro simulation of threecompartment model (Fig. 4f). When the computing time
gradient (ti?1-ti) is smaller, the ratio will be closer to the value one. Before in vitro PK simulation study, it is necessary to obtain prior in vivo PK information in humans. There are three sources for in vivo PK information: (1) In vivo time profile of the drug: the flow rate could be obtained from equation method following compartment analysis. Using mean value of PK parameter to calculate flow rate is recommended; (2) Type of PK model and value of PK parameters: the flow rate could be obtained by substituting PK parameters in the equation directly; (3) Type of population pharmacokinetic (PPK) model and value of PPK parameters. First, the PK parameter for specified population is obtained from the PPK model using typical values and covariate values. Then the flow rate could be calculated using equation method. Our method could be extrapolated to mimic multiple PK in vitro. The flow rate equations for simulating linear mammillary model and zero-order model with iv injection have been obtained. The in vitro model shown in Fig. 1 could be used within maximum 72 h post dosing. We are making effort to use in vitro model with flow rate modulation to simulate multiple PK within 5–7 days post dosing. This could validate the feasibility of flow rate modulation method following multiple dosing. The main merit of our method is providing a uniform setting for the simulation of linear mammillary model with different compartments. It’s not a method which could be used to simulate the PK models with different administration modes simultaneously. However, if the model structure is modified to simulate the PK model with oral or infusion dosing, by continuous flow-rate modulation, one can accurately simulate the linear mammillary model with different compartments in a uniform model setting. We are now expanding our method for the calculation of flow rate in these situations. The in vitro combination studies of antimicrobials with different mechanisms combating bacterial resistance are increasing [20, 21]. We have great interest on how to perform the simulation of two drugs’ profile when both of them conform to two-compartment model using the in vitro system reported in the paper [22, 23]. Results showed that
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42 Table 5 Summary of the formula for the verification of flow rate modulation equation
J Pharmacokinet Pharmacodyn (2015) 42:33–43
Type Linear mammillary model
Equation of ‘dC/C’ l P
dC C
¼ P
l P
(i)
Ai ki eki t
dt
i¼1 l
Concentration–time equation in vitro
C0 Ai
C¼
l P
Ai eki t
(vi)
i¼1
i¼1
(ii)
c0
C ¼ c0 eke t
(vii)
(iii)
A?B
C = Ae-at ? Be-bt
(viii)
(iv)
A ? B?R
C = Ae-at ? Be-bt ? Re-ct
(ix)
(v)
c0
C = c0 - ke t
(x)
Ai eki t
i¼1
C drug concentration in the central chamber, C0 initial drug concentration, l the number of compartments
Onecompartment model
dC C
¼ ke dt
Twocompartment model Threecompartment model
dC C
þ Bbe ¼ Aae dt Aeat þ Bebt
dC C
¼ Aaeat þ Bbebt
Zero-order model
dC C
ke ¼ c0 k dt et
at
bt
at
Ae
bt
þ Be
the time profiles of two drugs could be simulated approximately when the flow rate decreased along with time based on equation method (data not shown). We are now making effort to improve the proposed method for better PK simulation in vitro. In summary, this report showed that the PK model with iv injection could be accurately simulated in vitro via flow rate modulation. In vitro model with constant flow rate could simulate one-compartment model, while the in vitro model with decreasing flow rate could mimic the linear mammillary model with multiple compartments. Zeroorder model could be reproduced using in vitro model with elevating flow rate. The equation method for flow rate modulation was validated using literature data and PK simulation experiment of levo-ornidazole. The proposed method significantly simplifies the structure of in vitro model and brings convenience to the in vitro PD study. Acknowledgments The work was supported by the National Natural Science Foundation of China (No. 81202582), the Major Research and Development Project of Innovative Drugs, Ministry of Science and Technology (2012ZX09303004-001) and China Postdoctoral Science Foundation (No. 2012M511045). We thank the help of Dr Sheng-li Li for his revision of the manuscript.
þ Rcect ct þ Re
dt
l P
Ai eki t C ln ¼ ln i¼1 l P C0 Ai
ð5Þ
i¼1
Making that
l P
Ai , the Eq. (5) could be simplified as
i¼1
Eq.(vi) in Table 5, which had the same form as c-t equation of linear mammillary model in vivo. For the simulation of one-compartment model in vitro, the Eq.(ii) (Table 5) could be obtained by substituting Eq.(xii) (Table 1) into Eq. (2). The Eq.(vii) (Table 5) could be yielded through the definite integral of Eq.(ii) (Table 5) during 0–t interval when C0 equals to c0. The Eq.(vii) (Table 5) had the same form as Eq.(ii) (Table 1). As for the simulation for two-compartment model in vitro, similar method could be used to verify the formula of flow rate modulation when the initial value (C0) equals A ? B. For three-compartment model in vitro, the formula for flow rate modulation could be demonstrated using the same method when C0 is the sum of A, B and R. The same process could be applied to verify the formula for flow rate modulation mimicking zero-order model when C0 equals to c0.
Conflict of interest We have no conflicts of interest related to the content of this study.
References Appendix For the flow rate modulation formula of linear mammillary model, the Eq.(i) (Table 5) could be obtained by substituting Eq.(xi) (Table 1) into Eq. (2). The following equation could be derived by the definite integral of Eq.(i) (Table 5) at both sides during 0–t interval:
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