Influence of Sampling Site and Fluid Flow on the Accuracy of Total Body Clearance Calculation

Journal of Pharmaceutical Sciences 109 (2020) 2079-2089

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Journal of Pharmaceutical Sciences journal homepage: www.jpharmsci.org

Pharmacokinetics, Pharmacodynamics and Drug Transport and Metabolism

Influence of Sampling Site and Fluid Flow on the Accuracy of Total Body Clearance Calculation Brandon LaPorte, Florin Marcel Musteata* Albany College of Pharmacy and Health Sciences, Department of Pharmaceutical Sciences, 106 New Scotland Avenue, Albany, New York 12208

a r t i c l e i n f o

a b s t r a c t

Article history: Received 27 January 2020 Revised 22 February 2020 Accepted 4 March 2020 Available online 10 March 2020

Studies have showed that by assuming arteriovenous drug concentrations are homogenous after intravenous injection, the determination of total body clearance based on venous drug concentrations is often inaccurate. This study considers the use of a fluidic pharmacokinetic profile generator where 28 different profile types were generated corresponding to a physiological model with varying sampling sites, administration locations, and fluid flow rates. Clearance was calculated using established equations, commercial software, and recently proposed models. The results show large differences in clearance values calculated with published equations and commercial software relative to the actual value of clearance. Alterations in sampling site, administration location, and fluid flow rates each influence the extent of calculation errors. The data show that a significant drug concentration gradient exists within the central circulatory system. The results show that the best way to address this issue would be to inject the drug at a peripheral location to allow for sufficient mixing and then sample from a large vein. Extrapolating for missing data can also lead to large errors in clearance calculation; this can be addressed by collecting more samples early after IV bolus administration or by collecting data during steady state conditions for an IV infusion. © 2020 American Pharmacists Association®. Published by Elsevier Inc. All rights reserved.

Keywords: clearance in vitro model pharmacokinetics blood flow nonlinear regression elimination

Introduction The biological nature of drug distribution and disposition is complex, but understanding it is pivotal for establishing proper drug dosing regimens. Clinically, compartment models are predominately used to determine drug dosing regimens for patients. However, the issue is that compartment models assume that once a drug is injected into the body, it is mixed instantaneously and homogenously throughout the circulatory system. Studies have shown errors as high as 70% for metolazone, 81% for cephalexin,

Abbreviations used: 1C, one-compartment; 2C, two-compartment; AC, antecubital; AUC, area under the curve; CO, cardiac output; Cl, clearance; D0, dose; DL, loading dose; DEO, drug eliminating organ; GCF, gradient correction factor (ratio between calculated and physiological clearance); IV, intravenous; MRT, mean residence time; PK, pharmacokinetic; Qa, total fluid flow; PBPK, physiologically based pharmacokinetic; RET, rapidly equilibrating tissue; RSD, relative standard deviation; SET, slowly equilibrating tissue; SS, steady state; Vc, apparent volume of the central compartment. Conflicts of interest: None. Research data: The research data used in preparation of the manuscript is available by contacting the corresponding author. * Correspondence to: Florin Marcel Musteata (Telephone: þ1 518 694 7883). E-mail address: [email protected] (F.M. Musteata).

and 196% for ampicillin in normal patients within the general population when using the 1C model to calculate clearance.1 Clinically, an 8% error is often deemed acceptable, and there are also circumstances where an error of 12% may also be tolerable.1 Calculating the actual value of Cl is a complex task, but noncompartmental analysis is currently considered the best method in PK studies. Noncompartmental analysis is used to determine Cl for new drugs because this method can be applied without knowledge of the intercept of plasma concentration versus time profiles and associated rate constant(s).2 However, there are inaccuracies when using this method to calculate clearance, largely attributed to the location of the sampling site. Significance of Sampling Location Based on the mass balance law, when using noncompartment model analysis to calculate clearance, the total AUC should be identical in arterial and venous blood as long as the drug is not metabolized by the sampling tissue.3 However, there have been multiple studies that have showed variations in drug concentration between arterial and venous blood after IV bolus injection. Depending on the location of the sampling site, the same drug can have completely different PK profiles ultimately leading to

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alterations in parameters such as apparent volume of the central compartment (Vc), apparent volume of the central compartment during infusion (Vc infusion), Cl, and mean residence time (MRT). The gradient that exists between arterial and venous blood is influenced by multiple factors such as physicochemical properties of the drug, blood flow rates, nature of the sampling tissue, other concurrent medications, and pathology. Physical drug properties include the partition between blood and tissue, protein binding, pKa, and complex formation with proteins and nucleic acids.4,5 Drugs with longer half-lives have been shown to have smaller arteriovenous concentration differences during the terminal phase.3 Lower blood flow rates increase the contact time a drug has with tissue, which allows for increased extraction depending on the volume of the tissue and partition of drug between the blood and tissue, and can therefore lead to more marked arteriovenous concentration differences.6 Berezhkovskiy7 suggests that total body Cl is underestimated in some cases since the concentration of drug in arterial blood entering the eliminating organ does not match the venous blood concentration that is used to calculate Cl. The author developed a theoretical equation to calculate Cl that depends on blood flow rates in addition to drug concentrations. Studies have shown that alterations in cardiac output influence the value of Cl by affecting blood flow. Other studies have shown that drug extraction and washout can be much different depending on flow, even in tissues with similar volume and partition coefficients.8 The location of the sampling site is also extremely important because different venous sampling locations have been shown to result in different Cmax and AUC estimates. This is because different tissues receive different blood flows and also may have a different partition coefficient for the same drug.9 Medications that alter blood flow or protein binding can also have effects on this gradient. Many different disease states can influence the PK of a drug and ultimately the concentration gradient between arterial and venous blood. There have been human and animal studies showing that at least 42 compounds have marked dependence of concentration on sampling location.10,11 It takes ~0.5 min for a maximum concentration to be achieved in arterial blood and 1-5 min for maximum concentration to be achieved in venous blood.11 It is obvious that the arteriovenous gradient is most pronounced during the initial interval but can last for many hours and even days after injection.10 Studies have also showed that when the rate of IV infusion increases, there is a greater arteriovenous concentration difference.11 Arterial blood sampling is considered ideal because the pharmacological effect is predominately affected by arterial blood concentration.12 Venous sampling is considered inaccurate because it is representative of the drug concentration in sampling tissue that is poorly perfused.11 In human studies, a drug is typically administered into a vein of the antecubital (AC) fossa and sampled from the same vein on the contralateral side of the body.

Errors From Extrapolation Weiss13 proposed that the calculated Cl errors might be attributed to the failure to recognize the arterial-venous transit time between the entry of the drug into the bloodstream and its appearance at the sampling site. After IV bolus administration, the clinician rarely has the ability to collect a sufficient amount of necessary samples from the individual, which can lead to errors during PK analysis.6 Because sampling often occurs 1-2 min after bolus injection, the initial highly concentrated drug pulse is neglected. Neglect for this initial drug pulse underestimates the AUC and thus results in Cl overestimation. Weiss13 determined that when sampling began 1-min after IV injection, there are significant errors in clearance value, especially for highly extracted drugs.

Curve fitting is a common technique used to address this issue. However, extrapolation still assumes that the drug is mixed instantaneously and homogenously throughout the central compartment, which is not true because there is an initial mixing and distribution phase. By extrapolating the first time point back to the initial concentration, the first time point plays a large role in determining the initial AUC in the PK profile. Multiple studies have found that later sampling can lead to significant overestimation of noncompartmental parameters.14 Furthermore, drug concentration gradients still exist at steady state because drug is added to a blood flow instead of a blood volume. The blood after the infusion site will have a higher concentration of the drug than blood after an eliminating organ. Blood flow to an eliminating organ influences its clearance and therefore can further influence this gradient.15 Changes in temperature, level of activity, certain medications, and pathology can also play a role to some degree.8 In Vitro Devices Data from in vivo experiments can be limited because of the difficulty in obtaining frequent data points and accurate fluid flow rates. Having limited data make it difficult to properly assess the mixing and distribution phases of a drug, thus limiting PK analysis.16 On the other hand, these limitations can be overcome by using a mechanical fluidic device for generating PK profiles, making in vitro models ideal for these types of studies. They allow for rapid, frequent, and accurate monitoring of changing drug concentrations over time leading to correct PK parameter calculations. Furthermore, mechanical devices can be easily adjusted to represent a wide range of experimental conditions. Other advantages include repeated measurements under identical experimental conditions, multiple sampling locations, in vitro testing of various pharmaceutical formulations, and dosing to toxic drug concentrations without risk of harm to animals or patients.17-19 By controlling the fluid flow rates and volume of a fluidic system, parameters can be adjusted to match those of a live organism.18 Certain states with high and low drug clearance can be easily simulated as well. An important advantage of fluidic devices is in knowing the true physiological clearance value, given by the flow rates of the elimination pumps. Recently, fluidic devices were shown to be easy to build, accurate, and cost-efficient for generating realistic PK profiles that can be used in various aspects of PK research such as analytical method development and PK-PD studies with cell cultures. In the present study, a fluidic pharmacokinetic profile generator was used to investigate the accuracy of various methods for calculating drug clearance. Experimental Procedures Reagents and Materials Methyl blue was purchased from Sigma (Atlanta, GA). Stock solutions of methyl blue were prepared in deionized water obtained from a Siemens water purification system. Silicone tubing and tubing connectors were purchased from VWR International, Inc. (Pittsburgh, PA). Equipment The Jenway 7305 UV/Vis spectrophotometer, used for spectrophotometric analysis, was purchased from Bibby Scientific (UK). PD5201 peristaltic pumps were purchased from Heidolph North America (Elk Grove Village, IL). Allegro peristaltic pumps were purchased from KD Scientific (Holliston, MA). The PD5201 pumps contained a single channel pump head for fluid recirculation, whereas the Allegro pumps had a dual channel pump head and

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Figure 1. Overall schematic of the fluid exchange pharmacokinetic profile generator where RET represents rapidly equilibrating tissues, SET represents slowly equilibrating tissues, and DEO represents drug-eliminating organ. Four different sampling locations (1 to 4) were used and 4 different injection locations, where P represents a peripheral (venous) injection and C represents a central (arterial) injection.

were used for fluid exchange. All pumps were calibrated in accordance with manufacturer’s instructions. A digital laboratory magnetic stirrer from Thermo Scientific (Waltham, MA) was used for additional mixing of fluid in the central compartment. Simulation of IV infusion was achieved using a syringe pump from KD scientific (Holliston, MA). Design of the Fluidic PK Profile Generator The overall design of the fluidic device matches that of a physiologically based pharmacokinetic model, as shown in Figure 1. The central compartment was composed of 148 mL of fluid, representing the rapidly equilibrating tissues (RET) and main circulatory loop. The RET compartment contained 50 mL of fluid with a stir bar to promote rapid mixing. The main loop fed the other compartments, such as the slowly equilibrating tissues (SET) and the drugeliminating organ (DEO). The SET compartment was built as a tube with 152 mL fluid with controlled flow, allowing for slower transfer of drug into the compartment to simulate multiexponential PK profiles. The DEO allowed for fluid exchange via a dual channel pump, which simultaneously removed drug-containing fluid and replaced it with an equal volume of drug-free fluid. Four different sampling locations and 4 different injection sites were used. Sampling sites 1, 2, and 4 correspond roughly to venous locations, whereas sampling site 3 approximates an arterial location. With the fluidic device, drug can be injected in the peripheral (“P,” venous) or central (“C,” arterial) side of the circulatory system. Three pumps were used to control circulation. One pump was responsible for total flow, another pump controlled flow to the SET, and a final pump facilitated drug elimination. Normal fluid flow was defined as a total flow of 55.9 mL/min and SET flow of 18.8 mL/min. The volume of distribution and fluid flow rates were chosen to approximately match those of a laboratory rat.18 For every PK profile generated, the flow on the DEO pump (clearance) was set to 8.0 mL/ min. Analysis by UV-VIS Spectrophotometry Methyl blue’s maximum absorbance is between 594 and 610 nm. Absorbance was monitored in real time at 600 nm by passing

the circulatory fluid of the fluidic device through a low volume flow-through cuvette in the spectrophotometer. The time intervals were set so that recordings were more frequent during the initial distribution phase (every 3 s) and less frequent during the elimination phase or at steady state (every 30 s). The spectrophotometer was calibrated with a seven-point calibration curve run in triplicate and checked with quality control samples; recalibration was performed as needed; all calibration curves had an r2 > 0.999. For each pharmacokinetic profile, the concentration of compound was calculated from its absorbance and the most recent calibration equation. The spectrophotometer was equipped with a xenon lamp with a very stable response signal.

Clearance Calculation and PK Analysis The standard noncompartmental method for calculating Cl is represented by Equation 1. This is currently considered the most accurate equation for calculating Cl. Calculations based on Equation 1 were performed using WinNonlin (version 5.3, Pharsight Corp., Mountainview, CA). Equation 2 can be used to determine clearance over a fixed time interval. This equation was the most suitable for calculating clearance with the in vitro model because the amount of drug eliminated and AUC can be accurately measured. The AUC between 2 measurements was determined using the trapezoidal rule in Microsoft Excel and WinNonlin, as per Equation 3. Integration errors were minimized by taking very frequent measurements (interval of 3 s during distribution, followed by 30 s intervals during the elimination phase or at steady state conditions). Finally, Equation 4 was derived by Berezhkovskiy7 to show the dependence of total Cl on blood flow rate.

Cl ¼

Cl ¼

D0 ½AUC∞ 0 ½DE t0 ½AUCt0

(1)

(2)

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Figure 2. PK profiles obtained after 0.75 mL injection of a 5.0 mg/mL solution of methyl blue with (a) peripheral administration and data collection at sampling site 1; (b) peripheral administration and sampling at site 2; (c) peripheral administration and sampling at site 3; (d) peripheral administration and sampling at site 4; (e) central administration and data collection at sampling site 3. Total fluid flow (Qa) and flow to the SET (slowly equilibrating tissues) varied for each set of PK profiles as shown in the legend. DEO (drug-eliminating organ) clearance was set at 8.0 mL/min for all profiles.

½AUCtt21

Cl ¼

c þ c2 ,ðt2  t1 Þ ¼ 1 2

ClE 1  ClE =Qa

(3)

(4)

where Cl is total body clearance, D0 is dose, ½DE t0 is the amount of drug eliminated from time 0 to time t, ½AUCt0 is the area under the concentration versus time curve from time 0 to time t, ½AUCtt21 is the area under the concentration versus time curve from time 1 to time 2, ClE is the clearance of the eliminating organ, and Qa is the total blood flow rate. Chiou 20 and others showed that there can be errors associated with using the trapezoidal rule for calculating AUC; however, the current experiments had many time points obtained in such close proximity to one another that the errors associated

with the trapezoid rule can be considered negligible. The DtE was determined by collecting all the eliminated fluid with the drug and measuring its volume and concentration. The ClE was the flow rate set on the DEO pump, whereas the Qa was the flow rate set on the overall circulation (‘cardiac’) pump. Clearance was calculated using each equation, and the values were compared with the physiological clearance to determine the most accurate method. Physiological clearance was the flow rate set on the elimination pump, and its value was verified by measuring the volume of fluid eliminated over the duration of the experiment. PK data analysis was performed individually for each concentration versus time profile generated. The data points were collected automatically using the spectrophotometer software and then exported to the Microsoft Excel format. Statistical analysis and curve fitting using noncompartmental analysis were performed in WinNonlin, version 5.3.

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Results Twenty-eight different types of PK profiles were generated, with 16 types for IV bolus administration and 12 types for IV infusion. The type of profile varied based on drug administration rate (bolus or infusion), sampling site location, injection point location, and fluid flow rates. For each PK profile, triplicates were performed at a minimum. The concentration-time profiles were found to be highly reproducible, with the relative standard deviation (RSD) of the repeated data points varying between 0.87% and 7.6%, with the vast majority being lower than 5%. There were only 2 profiles where the RSD was up to 7.6% for some of the time points. Bolus Profiles A 0.75-mL dose of methyl blue solution (5.0 mg/mL) was injected either in the peripheral or central location followed by data collection from the 4 sampling sites; the resulting PK profiles are shown in Figure 2. DEO clearance was set at 8.0 mL/min for all profiles. The first set of PK profiles was a peripheral injection with sampling at site 1, shown in Figure 2a. It is evident that alterations in flow rates alter the AUC at this sampling location. The measured physiological clearance value in all 3 sets of profiles was 7.94 mL/ min. Under normal fluid flow conditions (Qa 55.9 mL/min and SET 18.8 mL/min) when using Equation 2 to calculate clearance, the value is 8.10 ± 0.07 mL/min (n ¼ 4), which is slightly higher than the actual value (p < 0.05). When total fluid flow and flow to the SET were cut in half, there was a higher overestimation in clearance at 8.61 ± 0.15 mL/min (n ¼ 4, p < 0.01). When total fluid flow rate was cut in half and flow to the SET was kept normal, clearance was considerably overestimated at 23.84 ± 1.53 mL/min (n ¼ 5, p < 0.001). The location of the sampling site was then changed to site 2 to determine how the AUC will be affected by flow rates at a location immediately after the SET, with the resulting profiles shown in Figure 2b. These profiles are very different from the previous set (Fig. 2a), especially for the initial portion. The physiological clearance value (set by the elimination pump) for these sets of PK profiles was measured to be 7.94 mL/min. Under normal flow conditions, clearance was calculated with Equation 2 to be 6.56 ± 0.23 mL/min (n ¼ 3), which is significantly lower than the physiological value (p < 0.01). When total fluid flow rate was decreased to 28.8 mL/min clearance was still significantly lower at 7.25 ± 0.27 mL/min (n ¼ 3, p < 0.05). When both the total fluid flow and flow to the SET was cut in half clearance was 7.53 ± 0.37 mL/min (n ¼ 3), which did not differ significantly from the set value. An additional set of PK profiles were run at even lower fluid flow rates where total flow was set to 18.6 mL/min and flow to the SET was set to 9.4 mL/ min to determine how this may impact clearance. Under these conditions, clearance was calculated to be 7.61 ± 0.32 L/min (n ¼ 3). Nevertheless, all calculated clearance values for this set of profiles were close to the physiological value, indicating an optimal sampling site. Figure 2c shows the set of PK profiles obtained at sampling location 3. Variation in the AUC is evident for the first 10 min. Under all 4 sets of experimental conditions, calculated clearance was found to be significantly lower than the actual value (p < 0.05). Under normal flow rates, conditions where total flow was halved, conditions where both total flow and SET flow rate were halved, and low flow rate conditions clearance were 6.89 ± 0.31 mL/min (n ¼ 3), 6.82 ± 0.47 mL/min (n ¼ 4), 6.51 ± 0.32 mL/min (n ¼ 3), and 6.56 ± 0.33 mL/min (n ¼ 3), respectively. The final set of PK profiles with a peripheral injection was obtained at sampling site 4, shown in Figure 2d. For all 3

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experimental conditions, there was no significant change in calculated clearance. Under normal conditions, the clearance value was calculated to be 7.59 ± 0.33 mL/min (n ¼ 3). When total flow was cut in half and flow to the SET was kept normal, clearance was 7.76 ± 0.14 mL/min (n ¼ 3). When total flow and flow to the SET was cut in half, clearance was 7.48 ± 0.57 mL/min (n ¼ 4). Intravascular in vivo drug administration is typically performed in the antecubital vein, from where the bolus of the drug rapidly travels to the heart and gets distributed throughout the body. To simulate this injection site using the current PK profile generator, experiments were also performed with the drug injection point located immediately after the RET (to achieve less dilution before distribution). Sets of PK profiles were generated under both normal and low flow rates to see how the injection point location and fluid flow alter clearance in this case. The PK profiles are shown in Figure 2e. Under normal flow conditions, the clearance was calculated to be 8.03 ± 0.56 mL/min (n ¼ 3), practically the same as the physiological set value. However, under conditions where total fluid flow was set to 18.6 mL/min and flow to the SET was set to 9.4 mL/min, calculated clearance was found to be significantly higher at 12.88 ± 0.86 mL/min (n ¼ 3, p < 0.05). For each set of PK profiles, the physiological clearance values measured by collecting the eliminated fluid were compared statistically with the clearance values calculated with Equation 2; physiological clearance values were set on the elimination pump and were very reproducible, with less than 0.5% variation between experiments. The overall results are shown in Table 1. As shown in Table 1, sampling location, injection site locations, and fluid flow all influence the calculated clearance value when using noncompartmental analysis. The error in clearance determination ranged from -18% to 200%. The extent to which the duration of data collection (PK profile) impacted the amount of drug eliminated and the associated clearance values was also evaluated, seen in Table 2. Only sampling location 2 and 3 were evaluated because these are most similar to in vivo conditions. With peripheral drug administration and sampling at site 2 as well as in the case of central drug administration and sampling at site 3, clearance tends to start out high initially and then decrease as the duration of data collection increases for each set of PK profiles. For peripheral injection and collection at site 3, calculated clearance starts out low initially and then increases in increments until 30 min of a profile is collected; however, Table 1 shows that after 90 min of data collection, the calculated clearance value decreases. Infusion Data For IV infusion experiments, a loading dose (DL) was administered for each set of PK profiles at the beginning of the infusion to achieve steady state conditions faster. Most profiles achieved >95% steady state between 10 and 30 min, as shown in Figure 3. The rate of infusion was set to 0.96 mL/h for all profiles. The other experimental conditions were the same as the ones used for IV bolus administration. Calculations were carried out only based on data collected over a duration of 30 min under steady state conditions. Sampling sites 2 and 3 were used to collect data, corresponding to in vivo sampling from a peripheral vein or from a major blood vessel. The physiological clearance value for all PK profiles was set at 8.0 mL/min. The data for the first set of PK profiles were collected at sampling site 2 with peripheral infusion to see how certain fluid flow rates impact the profiles when sampling occurred after the

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Table 1 Clearance Values Obtained for Various Injection Locations, Sampling Locations, and Fluid Flow Rates After IV Bolus Administration. Injection & Sampling Locations

Total Flow Rate (mL/min)

SET Flow Rate (mL/min)

Physiological Cl (mL/min)

Equation 4: Cl (mL/min)

Equation 2: Cl ± SD (mL/min)

Average AUC ± SD (mgminmL1)

Peripheral & Site 1

55.9 28.8 28.8 55.9 28.8 28.8 18.6 55.9 28.8 28.8 18.6 55.9 28.8 28.8 55.9 18.6

18.8 18.8 9.40 18.8 18.8 9.40 9.40 18.8 18.8 9.40 9.40 18.8 18.8 9.40 18.8 9.40

7.94 7.94 7.94 7.94 7.94 7.94 8.00 7.94 7.94 7.94 8.00 7.94 7.94 7.94 7.94 8.00

9.34 11.1 11.1 9.34 11.1 11.1 14.0 9.34 11.1 11.1 14.0 9.34 11.1 11.1 9.34 14.0

8.10 ± 0.07 23.8 ± 1.50 8.61 ± 0.15 6.56 ± 0.23 7.25 ± 0.27 7.53 ± 0.37 7.61 ± 0.32 6.89 ± 0.31 6.82 ± 0.47 6.51 ± 0.32 6.56 ± 0.33 7.59 ± 0.33 7.76 ± 0.14 7.48 ± 0.57 8.03 ± 0.56 12.9 ± 0.86

341 114 325 405 371 349 323 390 389 406 397 349 349 355 318 206

Peripheral & Site 2

Peripheral & Site 3

Peripheral & Site 4

Central & Site 3

± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ±

8.3 9.1 38 30 35 30 27 29 39 37 36 22 15 29 40 24

Two tailed p-Value

GCF

0.0400 0.00002 0.0052 0.0087 0.0471 0.1977 0.1659 0.0269 0.0174 0.0163 0.0169 0.2081 0.1454 0.2033 0.9341 0.0103

1.02 3.00 1.08 0.83 0.91 0.95 0.95 0.87 0.86 0.82 0.82 0.96 0.98 0.94 1.01 1.61

Two-tailed p-value is comparing physiological clearance with the calculated clearance values. p < 0.05 denotes statistical significance; total flow rate represents the overall (“cardiac”) output of the fluidic device; SET flow rate represents the fluid flow through the “slowly equilibrating tissues.” GCF represents the gradient correction factor (ratio of calculated clearance and physiological clearance). Values that deviate more than 25% from the physiological clearance are shown in bold.

SET, (Fig. 3a). Under normal flow conditions, clearance at steady state was calculated to be 7.74 ± 0.10 mL/min (n ¼ 3), which was not significantly different than the physiological clearance value. After changing the total flow rate to 28.8 mL/min, the calculated clearance was found to be 7.82 ± 0.16 mL/min (n ¼ 3), not significantly different. When both the total flow and flow to the SET were halved, calculated clearance was again not significantly different at 7.45 ± 0.41 mL/min (n ¼ 3). However, when total flow was set to 18.6 mL/min and flow to the SET was kept at 9.4 mL/ min, clearance was statistically lower with a value of 7.23 ± 0.27 mL/min (n ¼ 3, p < 0.05). Overall, the various clearances obtained under these conditions were close to the set value. A corresponding set of PK profiles were measured at sampling site 3, as seen in Figure 3b. In this scenario for each experiment the concentration rapidly reaches a peak before declining to SS conditions. Under normal flow conditions, clearance was calculated to be 7.40 ± 0.24 mL/min (n ¼ 3, p > 0.05), not significantly different from the pump-set value. When total flow was cut in half clearance significantly decreased to 7.33 ± 0.21 mL/min (n ¼ 3, p < 0.05). When both total flow and flow to the SET was cut in half clearance was 7.40 ± 0.22 mL/min (n ¼ 3, p < 0.05), which was also significantly different. Under the low flow conditions clearance was calculated to be significantly reduced at 7.47 ± 0.03 mL/min (n ¼ 3, p ¼ 0.001; this set had very little variation when running triplicates). A final set of infusion profiles, shown in Figure 3c, were generated based on sampling from site 3 with the administration site corresponding to a central location. Under normal flow conditions, clearance was calculated to be 7.90 ± 0.09 mL/min (n ¼ 3), with insignificant difference from the set value. When the total flow was changed to 28.8 mL/min, the calculated clearance was 7.65 ± 0.36 mL/min (n ¼ 3), not significantly different from the set value. When both the total flow and flow to the SET were halved, clearance was measured to be 7.85 ± 0.30 mL/min (n ¼ 3), again no significant change. However, when the total flow was set to 18.6 mL/min and flow the SET stayed at 9.4 mL/min, clearance significantly increased to 13.42 ± 1.01 mL/min (n ¼ 3, p < 0.05). For each set of PK profiles, the physiological eliminating organ clearance was compared statistically with the calculated clearance, with the results shown in Table 3. The error in clearance measurement was found to range from -10% to 68%.

Discussion The data from these bolus and infusion experiments show that sampling site location, administration site location, and fluid flow rates all influence total body clearance measurement to a certain degree when using noncompartmental analysis to calculate clearance. The classical assumption when using noncompartmental analysis is that drug concentration in arterial and venous blood is the same, though in actuality, a continuous concentration gradient exists throughout the body. What makes the fluidic device used in this study optimal is the fact that the physiological Cl value can be accurately set on the elimination pump. Furthermore, the physiological clearance can be accurately measured (verified) by collecting the fluid eliminated over the duration of an experiment. Accordingly, the fluidic device provides a means for determining the best approach for calculating Cl and investigating the errors associated with various experimental conditions. The device used for this research was previously shown to be able to generate realistic PK profiles corresponding to various models and routes of drug administration.18 Variation in Clearance Measurement Arterial sampling is currently considered the most relevant in PK studies because arterial blood flow and arterial drug concentration determine the input to the site where a drug exerts its pharmacological effect.9,14,21,22 However, venous sampling is often easier and thus more common in research, which is why 3 different simulated venous sampling locations were included in this study and only 1 arterial location was included. Sampling site 2 and site 3 are the most physiologically relevant because these locations are the most synonymous ones to in vivo sampling sites. By sampling from all 4 locations, the presence of a gradient within the system can be shown. The data also show that this gradient can be altered by changes in fluid flow rates. Based on theoretical models, Berezhkovskiy7 proposed that differences in clearance would be quite pronounced for different flow rates. He has also shown that by considering in greater detail the initial drug distribution and elimination, the mean residence time, volumes of distribution, and clearance can be considerably different from the values calculated by the classical

9.12 ± 0.15 12.7 ± 0.52 7.26 ± 0.060 7.26 ± 0.060 9.50 ± 0.60 20.1 ± 1.30 35 19 47 44 24 6.4 ± ± ± ± ± ± 1735 1799 1760 1885 1683 1979 11.8 ± 0.034 23.7 ± 0.26 6.91 ± 0.24 6.16 ± 0.06 10.9 ± 0.09 31.1 ± 0.77 3.19 15 40 12 10 38 ± ± ± ± ± ± 1118 1346 1163 1340 1181 1518 15.8 ± 0.069 41.9 ± 0.65 6.62 ± 0.054 5.31 ± 0.039 12.5 ± 0.074 47.9 ± 0.87 884 ± 3.8 1060 ± 16 909 ± 7.4 1003 ± 7.4 998 ± 5.9 1359 ± 25 Central & Site 3

Peripheral & Site 3

Durations include 5 Min, 10 Min, 15 Min, and 30 Min. ± Values represent one SD; total flow rate represents the overall (“cardiac”) output of the fluidic device. SET flow rate represents the fluid flow through the “slowly equilibrating tissues.” Values that deviate more than 25% from the physiological clearance are shown in bold.

35.7 ± 0.34 45.7 ± 2.6 5.70 ± 0.025 0.187 ± 0.02 15.6 ± 0.13 343 ± 5.8 547 ± 5.2 61.1 ± 3.5 548 ± 2.4 27.1 ± 2.4 712 ± 6.0 1110 ± 18 7.94 8.00 7.94 8.00 7.94 8.00 55.9 18.6 55.9 18.6 55.9 18.6 Peripheral & Site 2

18.8 9.40 18.8 9.40 18.8 9.40

Amount Eliminated After 30 min (mg) & Calculated Cl (mL/min) Amount Eliminated After 15 min (mg) & Calculated Cl (mL/min) Amount Eliminated After 10 min (mg) & Calculated Cl (mL/min) Amount Eliminated After 5 min (mg) & Calculated Cl (mL/min) Eliminating Organ Cl (mL/min) SET Flow Rate (mL/min) Total Flow Rate (mL/min) Injection & Sampling Locations

Table 2 Average Amount of Drug Eliminated and Noncompartmental Clearance Values Obtained After Certain Durations Based on Various Injection Locations, Sampling Locations, and Organ Fluid Flow Rates via IV Bolus Administration (n ¼ 3)

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pharmacokinetic equations.23 However, the data in this study suggest that changes are quite small for most PK profiles and flow rates. Nevertheless, there are certain conditions where the gradient (concentration difference) can be rather pronounced at certain sampling locations. At sampling site 1 when the total flow rate was set to 28.8 mL/min and flow to the SET was set to 18.8 mL/min, clearance was calculated to be 23.84 mL/min, much higher than the set value of 8.0 mL/min. This was for 3 reasons. First, the drug was injected slightly downstream from the sampling location, so there was more complete mixing in the central compartment before sampling. Second, after the bolus traveled through the RET, roughly two-thirds would be distributed to the SET (dictated by flow rates) leading to further dilution. Finally, one-third of the bolus dose travels to the DEO where it gets further diluted; because the difference between the total flow and flow to the SET is 10 mL/min, the DEO has a 0.80 extraction ratio. By the time the drug reaches the sampling site, the concentration is very small leading to a decrease in AUC and significant overestimation in clearance. A similar result was obtained for central bolus injection and sampling at site 3. The difference between the total flow and flow to the SET caused an extraction ratio of 0.87 when the drug traveled to the DEO to be eliminated. Moreover, low flow rates cause slower mixing, a significant decrease in AUC at sampling locations distant from the injection point and thus overestimation of clearance. The physiological clearance value was compared with clearance values determined experimentally after IV bolus injection for various simulated conditions; the calculations were carried out using Equations 2 and 4; the results are shown in Figure 4a. Of the 16 types of IV bolus profiles for which noncompartmental analysis was used to calculate clearance (Figs. 4a), 10 experimental configurations resulted in a calculated clearance significantly different from the physiological value. These data are consistent with previous findings that fluid flow rates and sampling site location influence the value of clearance calculated using noncompartmental analysis.7 The dependence of clearance on blood flow rates tends to be difficult to show with in vivo experiments, when determining the total amount of drug eliminated from the body by all possible routes can be very difficult. The data generated using the fluidic device showed that even though calculated clearance can vary significantly with experimental conditions, the amount of drug eliminated stayed the same (due to physiological clearance set on the elimination pump being the same). Chiou10,11 claimed that when calculating clearance based on the AUC, the most appropriate term should be “time average clearance.” The calculated clearance can change because it is influenced by the duration of the collected data set. From Table 2, it is obvious that there can be very large deviations in clearance when collecting PK data for short periods of time. However, as the duration of the data set increases, the calculated clearance tends to get closer to the physiological value. This calculation bias is most likely because of initial mixing effects from IV bolus administration and errors in extrapolating for missing data points. The best way to minimize these types of errors is to collect data from continuous IV infusion and calculate clearance based on area under the curve under steady state conditions. When comparing clearance values obtained from IV bolus profiles with values from IV infusion at steady state, the calculated clearance values are very close with a trend for slightly lower clearance values for IV infusions. As discussed, this is because of time-dependent and variable initial mixing that happens in the case of IV bolus administration. Incomplete mixing can result in an elevated initial AUC, and this overestimation makes the clearance value appear lower. Although the clearance values obtained with data collected at steady state are closer to the physiological clearance value, there are still certain experimental conditions where the values are significantly different.

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Figure 3. Pharmacokinetic profiles obtained after a loading dose and simultaneous infusion of stock solution at a rate of 0.96 mL/h with (a) data collection at site 2 and peripheral administration; (b) data collection at site 3 and peripheral administration; (c) data collection at site 3 and central administration. Total fluid flow (Qa) and flow to the SET (slowly equilibrating tissues) varied for each set of PK profiles as shown in the legend. DEO (drug-eliminating organ) clearance was set at 8.0 mL/min for all profiles.

perform initially.3 The clearance values calculated based on IV infusion at steady state for various experimental conditions were compared with physiological clearance as shown in Figure 4b.

Clearance values obtained at steady state are considered more accurate than those obtained after bolus administration. The fact that the data in this research show only small differences in clearance calculated from IV bolus, and IV infusion data indicate that the fluidic device can achieve high accuracy in generating PK profiles. This high accuracy is due at least in part to the ability to sample every 3 s starting at time 0, minimizing the data extrapolation errors due to initial mixing and distribution. In vivo studies have shown larger differences between bolus and infusion data because they are limited with the amount of sampling they can

The Influence of Blood Flow on Calculated Clearance Value Although Berezhkovskiy7 suggested that blood flow rate has a substantial impact on clearance; our results show that the effect is much lower than predicted theoretically. The theoretical model expressed by Equation 4 matched the experimental data only for a

Table 3 Clearance Values Obtained for Various Infusion Locations, Sampling Locations, and Organ Fluid Flow Rates Infusion & Sampling Locations

Total Flow Rate (mL/min)

SET Flow Rate (mL/min)

Physiological Cl (mL/min)

Equation 4: Cl (mL/min)

Equation 2: Cl ± SD (mL/min)

Average AUC ± SD (mgminmL1)

Peripheral & Site 2

55.9 28.8 28.8 18.6 55.9 28.8 28.8 18.6 55.9 28.8 28.8 18.6

18.8 18.8 9.40 9.40 18.8 18.8 9.40 9.40 18.8 18.8 9.40 9.40

8.00 8.00 8.00 8.00 8.00 8.00 8.00 8.00 8.00 8.00 8.00 8.00

9.34 11.1 11.1 14.0 9.34 11.1 11.1 14.0 9.34 11.1 11.1 14.0

7.74 ± 0.10 7.82 ± 0.17 7.45 ± 0.41 7.23 ± 0.27 7.40 ± 0.24 7.33 ± 0.21 7.40 ± 0.22 7.47 ± 0.03 7.90 ± 0.09 7.65 ± 0.36 7.85 ± 0.30 13.42 ± 1.0

192 242 269 267 261 265 246 267 252 258 260 154

Peripheral & Site 3

Central & Site 3

Two-tailed p-value is comparing eliminating organ clearance to the calculated clearance value. p < 0.05 denotes statistical significance. Total flow rate represents the overall (“cardiac”) output of the fluidic device Set flow rate represents the fluid flow through the “slowly equilibrating tissues.” GCF represents the gradient correction factor (ratio of calculated clearance and physiological clearance). Values that deviate more than 25% from the physiological clearance are shown in bold.

± ± ± ± ± ± ± ± ± ± ± ±

5.6 10 12 15 14 18 21 2.4 6.8 11.8 8.3 13.6

Two tailed p-Value

GCF

0.0508 0.1936 0.1476 0.0379 0.0516 0.0308 0.0405 0.0010 0.2138 0.2369 0.4666 0.0113

0.97 0.98 0.93 0.90 0.93 0.92 0.93 0.93 0.99 0.96 0.98 1.68

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Figure 4. Comparison between clearance values obtained with Equation 2 from sampling site 1 to 4 after peripheral or central administration, for IV bolus (a) and IV infusion (b). The physiological clearance set on the elimination pump was 8 mL/min for all cases. Legend: The first letter, P, signifies peripheral injection. S# signifies the sampling site location. Q represents total fluid flow rate and SET signifies fluid flow to the slowly equilibrating tissue. The letter after Q or SET represents the flow, where n means normal flow, h means half normal flow, and t means one-third the normal flow.

few experimental configurations, such as those with low flow rates, central administration, and sampling at site 3 (last bars in Figs. 4a and 4b); for almost all other experimental setups, the calculated clearance was lower than physiological clearance and much lower than the value predicted by Equation 4. At least 1 of the reasons for this discrepancy is the fact that clearance calculation with Equation 2 assumes the same drug concentration in the plasma throughout the body, while Berezhkovskiy’s derivation uses different plasma concentrations for different organs; a full exploration of the differences between the 2 equations is presented in 2 of his research studies.23 A simple way to update Equation 4 consists of including in its derivation the average concentration of drug in plasma (used to obtain Cl from AUC) calculated from the various concentrations in the body:

Cpssh ¼

ss þ C ss Corg p

2

(5)

where Cpss is concentration in the plasma at steady state (except for ss is the plasma steady state concentranoneliminating organs), Corg tion in noneliminating organs, and Cpssh is the average concentration in the plasma at steady state conditions (with the assumption that the amount of plasma in the arteries is equal to that in the veins). This leads to a slightly revised version of Equation 4:

Cl ¼

2 ClE 2  ClE =Qa

(6)

Equation 6 shows that blood flow rates have a lower impact on the value of clearance than previously assumed. The current research shows that the sampling location, drug delivery location, and the ratio of blood flows to various areas of the body have more impact than overall flow on the value of clearance calculated from AUC.

The Influence of Extrapolation on Calculated Clearance Value Curve fitting is a common technique used in PK studies because it is extremely difficult to obtain high time-resolution data from in vivo studies. Sampling often begins 1-2 min after bolus injection, sometimes later than 5 min thereafter. Because clearance is commonly determined using noncompartmental analysis, the initial time points are extrapolated back to time 0. As a result, the drug concentration in blood is considered highest at time 0 when in

actuality that is just when mixing begins. Furthermore, the profile is extrapolated from the last sampling point to infinity. These extrapolations have been shown to lead to large errors in clearance measurement.13,14 For each set of 90 min IV bolus profiles collected, the extent of the extrapolation error was quantified by comparing physiological clearance with clearance calculated by performing noncompartmental analysis on the full data set (starting at time 0), data set without the first 3 min, and data set without the first 5 min. The last 3 columns in Table 4 show the clearance values calculated with WinNonlin under these conditions. Theoretically, each row in Table 4 should show approximately the same clearance values. However, the table shows that even 3 min of missing data after bolus injection can lead to extremely large variations in calculated clearance under certain conditions. Calculated clearance can either increase or decrease based on the time when the first data points are available. This is largely attributable to the shape of the curve when extrapolation and curve fitting begin. Errors as high as 39.4% were evident when comparing profiles with data starting at time zero to profiles with data starting at 5 minutes post injection. When comparing extrapolated values with the physiological clearance value, errors as high as 259% were observed when data were used starting at 3 min after injection and 178% when the first 5 min after injection was ignored. The differences in these values are most likely because of the unpredictable initial mixing of the drug and the presence of concentration profiles different from the ones used to extrapolate for missing data in WinNonlin. This confirms Weiss13 was correct in stating that extrapolating time points may lead to substantial errors in clearance estimation. Gradient Correction Factor To avoid errors because of extrapolation, clearance was calculated in this research based on profiles starting at time zero followed by collecting numerous data points (every 3 s initially). Nevertheless, even without extrapolation errors, calculated clearance was significantly different from physiological clearance for several experimental conditions e because of the concentration gradient that exists within a continuously changing system. A simple way to compensate for this type of error when calculating clearance would be to use a “gradient correction factor” (GCF) defined as

GCF ¼

Calculated clearance Physiological clearance

(7)

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Table 4 Comparison of Physiological Clearance With Values Calculated by Noncompartmental Analysis of Differently Truncated Data Sets Injection & Sampling Locations

Total Flow Rate (mL/min)

SET Flow Rate (mL/min)

Physiological Cl (mL/min)

Cl ± SD (mL/min); Data: 0-90 min

Cl ± SD (mL/min); Data: 3-90 min

Cl ± SD (mL/min); Data: 5-90 min

Peripheral & Site 1

55.9 28.8 28.8 55.9 28.8 28.8 18.6 55.9 28.8 28.8 18.6 55.9 28.8 28.8 55.9 18.6

18.8 18.8 9.4 18.8 18.8 9.4 9.4 18.8 18.8 9.4 9.4 18.8 18.8 9.4 18.8 9.4

7.94 7.94 7.94 7.94 7.94 7.94 8.00 7.94 7.94 7.94 8.00 7.94 7.94 7.94 7.94 8.00

9.20 ± 0.83 28.5 ± 2.8 7.26 ± 0.90 7.77 ± 0.54 8.47 ± 1.3 8.11 ± 1.3 9.25 ± 1.0 8.36 ± 1.0 7.98 ± 1.3 7.92 ± 1.3 8.32 ± 1.6 9.27 ± 0.93 9.01 ± 0.41 8.11 ± 1.5 10.1 ± 2.0 14.4 ± 2.3

6.54 ± 1.4 28.5 ± 2.8 6.68 ± 0.89 7.64 ± 0.52 8.29 ± 1.2 7.93 ± 1.2 9.24 ± 1.0 7.96 ± 1.0 6.06 ± 1.5 6.14 ± 0.98 5.83 ± 1.0 6.04 ± 0.50 10.2 ± 0.57 9.05 ± 1.8 10.1 ± 2.0 14.4 ± 2.3

6.86 ± 1.7 22.1 ± 3.6 4.40 ± 3.2 7.48 ± 0.51 8.01 ± 1.2 7.90 ± 1.2 9.06 ± 1.0 8.79 ± 1.5 7.94 ± 1.7 8.42 ± 2.1 4.49 ± 0.50 9.11 ± 0.81 9.99 ± 0.35 6.04 ± 3.0 9.42 ± 1.9 13.9 ± 2.2

Peripheral & Site 2

Peripheral & Site 3

Peripheral & Site 4

Central & Site 3

Calculations were performed with WinNonlin; n ¼ 3. Total flow rate represents the overall (“cardiac”) output of the fluidic device. SET flow rate represents the fluid flow through the “slowly equilibrating tissues.” Values that deviate more than 25% from the physiological clearance are shown in bold. The last three columns show clearance values calculated from the same data sets, corresponding to the conditions described in the corresponding row.

The value of the GCF depends on fluid flow rates, sampling location, and drug administration location. Calculated clearance and GCF can subsequently be used to calculate physiological clearance for various experimental conditions. The range of GCF values obtained in this research is shown in Tables 1 and 3. The further the value of GCF from 1, the higher the error owing to the gradient in the circulatory system. For IV bolus administration, the largest deviation was observed for peripheral administration and sampling at site 1 (after the drug eliminating organ), though this happened only when the difference between total flow and SET flow was small (10 mL/min difference, for a total flow of 28.8 mL/min). Another notable deviation was observed for both IV bolus and IV infusion in the case of central drug administration and sampling at site 3 (corresponding to central arterial location), for the lowest investigated values of fluid flow rates.

Conclusions It is known that using either noncompartmental analysis or curve fitting can lead to extremely aberrant clearance estimates ultimately impacting a drug’s dosing regimen and most importantly compromising the care of the patient. However, it is unclear how to fully address this clearance calculation issue. The biggest deviations were found when the total fluid flow in the system was reduced and sampling was performed after a DEO or at an arterial location. The current research shows that errors in clearance calculation could be mitigated by using a gradient correction factor or by carefully selecting the drug administration and sample collection sites. For example, by injecting the drug in a peripheral location (e.g. the smallest usable vein) there will be better mixing allowing for a more homogenous arteriovenous concentration. After peripheral administration, sampling should be performed from a large vein (such as vena cava), where the gradient correction factor is close to 1; however, it should be noted that in vivo sampling usually requires cannulation, which can affect blood flow and therefore drug clearance. To avoid errors from extrapolating for missing data points, it is more evident now that samples should be collected frequently and starting as early as possible.

Acknowledgements The authors would like to acknowledge funding from Albany College of Pharmacy and Health Sciences. The authors would also like to acknowledge the School of Graduate Studies for providing additional funding via the Graduate Research Award (for BL). The authors’ work was independent of the funders.

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