Pharmacokinetic Pearls for the Clinical Pharmacist

Abstract

Background: The interpretation of imperfect data is one of the challenges of clinical pharmacokinetics. The equations and theorems used to guide our pharmacokinetic assessment of the patient often assume perfection, while the practicing clinician is faced with many discrepancies from dose to dose, drug to drug, and even institution to institution.

Objectives:  To determine whether there is a clinically relevant difference between a peak vancomycin concentration drawn 1 hour after the end of a 1-hour infusion and a back-extrapolated theoretical concentration.  To determine whether non-steady-state aminoglycoside concentrations drawn around the third dose after small changes in dosing intervals can be interpreted using steady-state equations.

Methodology:  Computer simulations of hypothetical scenarios were performed.

Results: For the majority of patients, there is no clinically relevant difference between peak vancomycin concentrations drawn 1 hour after the end of a 1-hour infusion and back-extrapolated theoretical concentrations.  The only exception is in patients with a half-life of vancomycin shorter than 3.11 hours. These recommendations apply only to patients for whom a 1-hour post vancomycin concentration is appropriate. For the majority of patients (who have aminoglycoside half-lives that are consistent with population estimates), clinicians can use steady-state equations to interpret non-steady-state aminoglycoside concentrations drawn around the third dose after minor derangements in dosing intervals.  The only exception is in patients with a half-life of aminoglycosides longer than approximately 5 hours.

Conclusion: We have discussed two commonly-encountered pharmacokinetic scenarios and provided practical recommendations on how to interpret drug concentrations obtained during each situation.

Introduction

The interpretation of imperfect data is one of the challenges of clinical pharmacokinetics.  The equations and theorems used to guide our pharmacokinetic assessment of the patient often assume perfection, while the practicing clinician is faced with many discrepancies from dose to dose, drug to drug, and even institution to institution.  In this article, we discuss two commonly-encountered pharmacokinetic scenarios and provide practical recommendations on how to interpret drug concentrations obtained during each situation.  The purpose of this investigation is neither to debate the merits of therapeutic drug monitoring nor to examine the relationship between concentration and effect, but rather to highlight the relevance of utilizing imperfect data.

The Clinically Relevant Peak Concentration

Studies are inconsistent with regards to a standard sampling time of peak aminoglycoside concentrations.  With conventional dosing (i.e., multiple-daily dosing), some pharmacists use the back-extrapolated aminoglycoside concentration (i.e., the theoretical concentration attained at the end of an intravenous infusion), while others rely on peak concentrations measured 0-60 minutes after a 30- to 60-minute intravenous infusion. By convention, current practice patterns dictate that the clinically relevant aminoglycoside peak concentration be drawn 30 minutes after the end of a 30-minute infusion, and thus, target ranges for aminoglycoside concentrations (values depend on the indication for treatment) refer to this peak. Aminoglycoside concentration monitoring with once-daily dosing has not been delineated clearly, although if samples are drawn, the clinician must consider the prolonged distribution phase following administration of a once-daily dose. One institution recommends that the first random concentration after a once-daily aminoglycoside dose be drawn 4 hours after completion of the infusion, based on pharmacokinetic studies in trauma patients and healthy volunteers.

The evidence for peak vancomycin concentrations is even more conflicting.  An overview suggested that the optimal sampling time for peak vancomycin concentrations was poorly defined due to its multi-compartmental pharmacokinetic characteristics. As such, it was difficult to compare different nomograms, in which clinicians have measured peak concentrations 1, 2, or 3 hours after the end of a 1-hour intravenous infusion.

Similarly, controversy exists among three local teaching hospitals as to definition of a clinically relevant peak vancomycin concentration.  Guidelines at a 950-bed adult acute care facility recommend drawing vancomycin concentrations 3 hours after the end of a 1-hour infusion, and utilize this 3-hour post as their peak concentration (target 15 to 25 mg/L). In contrast, pharmacists at another adult acute care hospital with 445 beds measure vancomycin concentrations 3 hours after the end of a 1-hour infusion (target 15 to 20 mg/L), and extrapolate back to the theoretical concentration at the end of the infusion. This back-extrapolated concentration is then deemed the relevant peak (target 20 to 30 mg/L). At a 400-bed, tertiary-care institution caring for maternity and pediatric patients, the peak vancomycin concentrations are drawn 1 hour after the end of a 1-hour infusion (target 20 to 40 mg/L). Within each institution, some pharmacists may deviate from institutional protocol and use back-extrapolated theoretical concentrations to guide vancomycin therapy.

Since vancomycin is a time-dependent kill antibiotic, trough concentrations may be a more clinically significant surrogate marker.  However, vancomycin peak concentrations may be of therapeutic value for select patient populations in whom it is useful to confirm that target concentrations have been attained.  Moreover, all three local teaching hospitals have distinct recommendations for monitoring of vancomycin peak levels, creating controversy among practitioners who are faced with discrepancies from institution to institution, and even from pharmacist to pharmacist.

Objective

Our objective was to determine whether there is a clinically relevant difference between a peak vancomycin concentration drawn 1 hour after the end of a 1-hour infusion and a back-extrapolated theoretical concentration.

Methodology

Computer iterations using simulated data were performed using a Microsoft® Excel (Microsoft Corporation, Redmond, WA) spreadsheet.  A series of minimum concentrations (Cmin, defined as the vancomycin concentration immediately before the dose) were matched with a series of maximum concentrations (Cmax, defined as the theoretical vancomycin concentration extrapolated to immediately after the end of a 1-hour infusion) and concentrations 1 hour post end of infusion (C1h post, defined as the vancomycin concentration 1 hour after the end of a 1-hour infusion).  An arbitrary number of simulations were performed to encompass concentrations within and out of the therapeutic range.

Two variables were fixed (i.e., Cmin and either Cmax or C1h post), to calculate the elimination rate constant (K) and elimination half-life (t1/2) using standard pharmacokinetic equations.The remaining concentration (either Cmax or C1h post) was calculated using Equation 1, where C refers to the lower concentration (i.e., C1h post), Co is the higher concentration (i.e., Cmax), K refers to the elimination rate constant, andt is the time of the infusion (i.e., 1 hour).  All possible combinations of Cmin, Cmax and C1h post were tested.

C = Co x e-Kt      (Equation 1)

The final step in the computer simulations involved calculation of the percentage difference between the values of Cmax and C1h post (∆C, defined as percentage change in concentration between Cmax and C1h post) using Equation 2.  Since a total error of 25% associated with a single concentration (i.e., a sum of administration, documentation, rounding-off, sampling, analytical errors, etc.) would not be unexpected, we used a conservative threshold of 20% as our maximum allowable error (i.e., clinically irrelevant ∆C defined as ≤ 20%).

∆C = (Cmax – C1h post) / Cmax x 100%(Equation 2)

The computer simulations were repeated for 12-, 8-, and 6-hour vancomycin dosing intervals.

Results

Tables I, II, and III show the results of the computer simulations for 12-, 8-, and 6-hour vancomycin dosing intervals, respectively.  The grey-shaded boxes represent fixed concentrations, while the unshaded boxes represented calculated values.  The yellow-shaded boxes denote clinically relevant ∆C > 20%.

Table I. Computer Simulations for 12-hour Vancomycin Dosing Interval.

Cmin (mg/L) Cmax(mg/L) C1h post(mg/L) ∆C (%) K (h-1) t1/2 (h)
5.0 20.0 17.6 11.84 0.13 5.50
5.0 22.9 20.0 12.94 0.14 5.00
5.0 30.0 25.5 15.03 0.16 4.25
5.0 35.9 30.0 16.40 0.18 3.87
5.0 40.0 33.1 17.22 0.19 3.67
5.0 49.3 40.0 18.77 0.21 3.33
10.0 20.0 18.8 6.11 0.06 11.00
10.0 21.4 20.0 6.70 0.07 10.00
10.0 30.0 27.2 9.50 0.10 6.94
10.0 33.5 30.0 10.40 0.11 6.31
10.0 40.0 35.3 11.84 0.13 5.50
10.0 46.0 40.0 12.94 0.14 5.00
15.0 20.0 19.5 2.58 0.03 26.50
15.0 20.6 20.0 2.84 0.03 24.09
15.0 30.0 28.2 6.11 0.06 11.00
15.0 32.2 30.0 6.70 0.07 10.00
15.0 40.0 36.6 8.53 0.09 7.77
15.0 44.1 40.0 9.34 0.10 7.07
2.5 30.0 23.9 20.22 0.23 3.07
2.5 38.5 30.0 22.00 0.25 2.79
10.0 50.0 34.2 13.61 0.15 4.74
10.0 58.7 50.0 14.87 0.16 4.31

Cmin = concentration immediately before the dose (i.e., true trough); Cmax = concentration extrapolated to immediately after the end of the 1h infusion; C1h post = concentration 1h after the end of a 1h infusion; ∆C = percentage change in concentration between Cmax and C1h post; K= elimination rate constant; t1/2 = elimination half-life

Table II. Computer Simulations for 8-hour Vancomycin Dosing Interval.

Cmin (mg/L) Cmax(mg/L) C1h post(mg/L) ∆C (%) K (h-1) t1/2 (h)
5.0 20.0 16.4 17.97 0.20 3.50
5.0 25.2 20.0 20.63 0.23 3.00
5.0 30.0 23.2 22.58 0.26 2.71
5.0 40.4 30.0 25.82 0.30 2.32
5.0 40.0 29.7 25.70 0.30 2.33
5.0 56.6 40.0 29.29 0.35 2.00
10.0 20.0 18.1 9.43 0.10 7.00
10.0 22.5 20.0 10.91 0.12 6.00
10.0 30.0 25.6 14.52 0.16 4.42
10.0 36.0 30.0 16.73 0.18 3.78
10.0 40.0 32.8 17.97 0.20 3.50
10.0 50.4 40.0 20.63 0.23 3.00
15.0 20.0 19.2 4.03 0.04 16.86
15.0 21.0 20.0 4.68 0.05 14.45
15.0 30.0 27.2 9.43 0.10 7.00
15.0 33.7 30.0 10.91 0.12 6.00
15.0 40.0 34.8 13.07 0.14 4.95
15.0 47.1 40.0 15.08 0.16 4.24
2.5 30.0 21.0 29.88 0.35 1.95
2.5 45.4 30.0 33.91 0.41 1.67
10.0 50.0 39.7 20.54 0.23 3.01
10.0 65.4 50.0 23.53 0.27 2.58

Cmin = concentration immediately before the dose (i.e., true trough); Cmax = concentration extrapolated to immediately after the end of the 1h infusion; C1h post = concentration 1h after the end of a 1h infusion; ∆C = percentage change in concentration between Cmax and C1h post; K= elimination rate constant; t1/2 = elimination half-life

Table III. Computer Simulations for 6-hour Vancomycin Dosing Interval.

Cmin (mg/L) Cmax(mg/L) C1h post(mg/L) ∆C (%) K (h-1) t1/2 (h)
5.0 20.0 15.2 24.21 0.28 2.50
5.0 28.3 20.0 29.29 0.35 2.00
5.0 30.0 21.0 30.12 0.36 1.93
5.0 47.0 30.0 36.11 0.45 1.55
5.0 40.0 26.4 34.02 0.42 1.67
5.0 67.3 40.0 40.54 0.52 1.33
10.0 20.0 17.4 12.94 0.14 5.00
10.0 23.8 20.0 15.91 0.17 4.00
10.0 30.0 24.1 19.73 0.22 3.15
10.0 39.5 30.0 24.02 0.27 2.52
10.0 40.0 30.3 24.21 0.28 2.50
10.0 56.6 40.0 29.29 0.35 2.00
15.0 20.0 18.9 5.59 0.06 12.04
15.0 21.5 20.0 6.94 0.07 9.64
15.0 30.0 26.1 12.94 0.14 5.00
15.0 35.7 30.0 15.91 0.17 4.00
15.0 40.0 32.9 17.81 0.20 3.53
15.0 51.1 40.0 21.75 0.25 2.83
2.5 30.0 18.3 39.16 0.50 1.39
2.5 55.8 30.0 46.27 0.62 1.12
10.0 50.0 36.2 27.52 0.32 2.15
10.0 74.8 50.0 33.13 0.40 1.72

Cmin = concentration immediately before the dose (i.e., true trough); Cmax = concentration extrapolated to immediately after the end of the 1h infusion; C1h post = concentration 1h after the end of a 1h infusion; ∆C = percentage change in concentration between Cmax and C1h post; K = elimination rate constant; t1/2 = elimination half-life

Discussion

The computer simulations illustrate that for the majority of the hypothetical scenarios, there is no clinically relevant difference between a peak vancomycin concentration drawn 1 hour after the end of a 1-hour infusion and a back-extrapolated theoretical concentration. A 1-hour post concentration would be appropriate in patients who do not have a prolonged distribution phase (e.g., patients without renal impairment or severe congestive heart failure). (2), (10) In these patients, the only situations where ∆C was greater than 20% were cases in which the calculated t1/2 of vancomycin was shorter than 3.11 hours.  This can be mathematically proved using Equation 1, assuming C = C1h post = 0.8, Co = Cmax = 1 (thus ∆C = 0.2 or 20%), and t = 1 hour.

C = Co x e-Kt       (Equation 1)
C1h post = Cmax x e-Kt
0.8 = 1 x e-Kx1
ln (0.8) = -k
k = 0.223 h-1
t1/2 = 3.11 h

It is impossible to predict vancomycin’s t1/2a priori; thus, the threshold 3.11-hour t1/2 cannot be utilized a priori to predict whether the 1-hour post and back-extrapolated concentrations will be clinically different.  However, if a patient does exhibit a calculated vancomycin t1/2 of shorter than 3.11 hours (using the 1-hour post and trough concentrations), the pharmacist should be aware that there is likely a greater than 20% difference between the 1-hour post and back-extrapolated concentrations.  In this scenario, it may be appropriate to use the back-extrapolated concentration rather than the 1-hour post concentration to guide therapy.

One caveat is that a 1-hour post concentration may not be suitable for patients who have a prolonged distribution phase (e.g., patients with renal impairment or severe congestive heart failure). (10)

Recommendations

For the majority of patients, there is no clinically relevant difference between peak vancomycin concentrations drawn 1 hour after the end of a 1-hour infusion and back-extrapolated theoretical concentrations.  The only exception is in patients with a half-life of vancomycin shorter than 3.11 hours; in this case, the difference between peak concentrations drawn 1 hour after the end of a 1-hour infusion and back-extrapolated theoretical concentrations is greater than 20%.  In patients with a suspected half-life shorter than 3.11 hours, it may be appropriate to use the back-extrapolated concentration to guide therapy. These recommendations only apply to patients for whom a 1-hour post vancomycin concentration is appropriate.

The Effect Changing Dose Intervals

The fundamental concept of steady state refers to a situation where administration of multiple doses of a drug at fixed intervals results in drug accumulation until the rate of elimination is equal to the rate of administration. However, initial doses often are given in changing, not fixed, intervals in order to comply with standardized medication administration times; despite this, drug concentrations often are ordered with the third dose, regardless of changing intervals. Therefore, it may be unclear to the clinician whether these non-steady-state drug concentrations can be interpreted.  This scenario will be illustrated using aminoglycosides as a prototype.

Objective

Our objective was to determine whether non-steady-state aminoglycoside concentrations drawn around the third dose after small changes in dosing intervals can be interpreted using steady-state equations for calculation of pharmacokinetic parameters.  Specifically, we assumed that a gentamicin 80 mg IV every 8 hours order was given as follows: first dose administered from 00:00 to 00:30; second dose administered from 10:00 to 10:30; third dose administered from 17:00 to 17:30; and subsequent doses administered every 8 hours.  Also we assumed that our patient weighed 70 kg and had a volume of distribution of 17.5 L (0.25 L/kg).  This exercise applies only if these specific assumptions are made.

Methodology

Computer simulations were performed using a Microsoft® Excel (Microsoft Corporation, Redmond, WA) spreadsheet.  A series of half-lives (t1/2) and their corresponding elimination rate constants (K) were specified, and then the concentration at 12 time points over a 42-hour time period (00:00, 00:30, 10:00, 10:30, 17:00, 17:30, 01:00, 01:30, 09:00, 09:30, 17:00, 17:30) were determined.  Aminoglycoside concentrations immediately before the next dose (Cmin) were calculated using Equation 1, where Co is the concentration at the end of the infusion, and t is the time between the end of the infusion and the next dose.  Theoretical aminoglycoside concentrations extrapolated to immediately after the end of an 0.5-hour infusion (Cmax) were calculated using Equation 3, where Cmin is the concentration immediately before the dose, t is the infusion time (i.e., 0.5 hour), ko refers to dosing rate (i.e., 80 mg / 0.5 hour), V is the volume of distribution (i.e., 17.5 L), and K refers to the elimination rate constant.

Cmax = (Cmin x e-Kt) + (ko/VK [1 – e-Kt])(Equation 3)

Finally, steady-state equations (erroneously assuming steady state) were utilized to calculate t1/2 and K with concentrations drawn around the third dose; these values were then compared with the original assumed t1/2 and K.

Results

Table IV shows the results of the computer simulations for an assumed aminoglycoside t1/2 of 3.00 h and K of 0.23 h-1.  If the steady-state equations had been used (erroneously assuming that the third dose had been at steady state), t1/2 would have been 3.24 h and K 0.21 h-1.  In actuality, true steady state was reached with the fifth dose (i.e., the dose at 09:00 on the second day of therapy).  For a shorter assumed t1/2 of 1.00 h, t1/2 calculated using steady-state equations was 1.15 h.  Steady state was reached sooner in this scenario with the fourth dose (i.e., the dose at 01:00 on the second day of therapy).  In the case of a longer assumed t1/2 of 5.00 h, t1/2 calculated using steady-state equations was 4.88 h.  In contrast, steady state was not reached until the sixth dose (i.e., the dose at 17:00 on the second day of therapy).  Simulations for t1/2 of 2.00 h, 10.00 h, and 20.00 h were also performed.

Table IV. Computer Simulations for Changing Aminoglycoside Dose Intervalsa.

Time Concentration (mg/L)
00:00 0.00
00:30 4.32
10:00 0.48
10:30 4.75
17:00 1.06
17:30 5.26
01:00 0.93
01:30 5.15
09:00 0.91
09:30 5.13
17:00 0.91
17.30 5.13

a Assumptions: 70 kg patient; volume of distribution 17.5 L (0.25 L/kg); dose 80 mg given over 0.5 hours; half-life 3.00 hours.

Discussion

The computer simulations illustrate that for patients with an aminoglycoside t1/2 of 3.00 h and changing dose intervals (first two doses given 10 hours apart and the third dose 7 hours later), non-steady state aminoglycoside concentrations can be interpreted using steady-state equations for calculation of pharmacokinetic parameters.  The same is true with a shorter aminoglycoside t1/2 of 1.00 h, as steady state is reached sooner.

The simulations also show that when aminoglycoside t1/2 was prolonged (≥ 5 hours), the length of time to reach steady state was greater.  In this case, a larger discrepancy between assumed t1/2 and calculated t1/2 (using concentrations around the third dose and steady-state equations) would occur; presumably, the clinician would be administering aminoglycoside doses utilizing a longer dosing interval.

It is impossible to predict an aminoglycoside’s t1/2 a priori; thus, the threshold 5-hour t1/2 cannot be utilized a priori to predict whether the assumed and calculated t1/2 will be clinically different.  However, if it is suspected that a patient will exhibit a calculated aminoglycoside t1/2 of longer than 5 hours, the pharmacist should be aware of the limitations of concentrations drawn around the third dose.  In this scenario, it may be appropriate to obtain concentrations with a later dose, or to use random concentrations (e.g., two concentrations drawn at least one t1/2 apart to calculate K and t1/2) to guide dosing and respond to changes in patient status.

Recommendations

For the majority of patients (who have aminoglycoside half-lives that are consistent with population estimates), clinicians can use steady-state equations for calculation of pharmacokinetic parameters in the interpretation of non-steady state aminoglycoside concentrations drawn around the third dose after minor derangements in dosing intervals.  The only exception is in patients with a half-life longer than approximately 5 hours; in this case, the length of time to reach steady state is greater, and the difference between assumed half-life and calculated half-life (using steady-state equations) is larger.  In patients with a suspected half-life of longer than 5 hours, it may be appropriate to obtain concentrations with a later dose or to use random concentrations to guide dosing.

Conclusion

In this article, we have discussed two commonly-encountered pharmacokinetic scenarios and provided practical recommendations on how to interpret drug concentrations obtained during each situation.

Acknowledgement

Many thanks to the pharmacists at Children’s and Women’s Health Centre of British Columbia for submitting pharmacokinetic issues observed in their clinical practice.  These ideas became the basis for this article.

Authors Competing Interests

None declared.

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