Continuous sampling as a pharmacokinetic tool Continuous sampling (CS) of blood through a nonthrombogenic catheter is presented as a tool for determining various pharmacokinetic parameters after a single injection ofa drug. In addition to defining many of the usual parameters used in pharmacokinetic analyses, CS provides an accurate and direct determination
0/ the
total area under the
plasma concentration curve. The theoretic background underlying the CS method is derived, and a practical [ormulation for its use in a clinical setting is described. The aminoglycoside antibiotic, amikacin, was chosen to exemplify the use ofthis technique. The drug was administered to 6 children, and CS was used to define plasma and single organ (kidney) clearance, volume
0/ distribution,
half-life during the final elimination
phase, the shape of the plasma concentration curve, and the exponential factorization of this curve for multicompartmental analysis. The CS method has several theoretical and practical advantages over the usual technique
0/ intermittent blood
sampling; such as accuracy in the determination of the plasma concentration-time curve integral, relative model independence, requirement for few samples, and ease in obtaining samples.
Bert Vogelstein, M.D., A. Avinoam Kowarski, M.D., and Paul S. Lietman, M.D., Ph.D. Baltimore, Md. The Department of Pediatrics and the Division of Clinical Pharmacology of The lohns Hopkins University School of Medicine
In order to describe the metabolism, distribution, and excretion of an injected drug, various pharmacokinetic parameters, such as, clearance, volume of distribution, and half-life, are used. These parameters are estimated from the concentration of the drug in plasma samples drawn at various times after injection. We present an alternative method for obtaining these plasma samples and the estimates that are based on them. The method relies on continu-
Supported by National Institutes of Health Grant RDI-HD-06284. Received for publication Oct, 23, 1976. Accepted for publication March 26, 1977. Reprint requests to: Dr. Paul S. Lietman, Osler 527, The Johns Hopkins Hospital, Baltimore, Md. 21205.
ous, rather than intermittent, sampling of plasma. In addition to defining the usual analytic components used in the analysis of drugs after single injections, continuous sampling (CS) allows a very accurate yet simple determination of the total area under the plasma concentration curve (AUC o_ oo) . This area is a critical factor in the determination of several standard pharmacokinetic parameters. CS has been made feasible because of the development by Kowarski and associates-- of a nonthrombogenic catheter through which blood can be withdrawn at a constant rate. CS has been used to measure the secretion rate of various hormones in humans '": 11, 14, 15, 21, 25 and the clearance-" and bioavailability'" of sul131
Vogelstein. Kowarski, and Lietman
132
Clinieal Pharmaeology and Therapeutics
kel
o
Fig. 1. Multicompartmental pharmacokinetic system.
Theoretic considerations
'0
TIME
Fig. 2. Plasma concentration curve. The pseudoequilibrium phase (PEQ) begins at time t'.
fonamides in dogs. We show that continuous sampling can be used in a clinical setting to determine several pharmacokinetic parameters, including the volume of distribution, plasma and single organ clearances, the half-life during the final elimination phase, distribution and elimination constants in a multicompartmental analysis, and the shape of the plasma concentration curve itself.
Modeling. Some of the pharmacokinetic parameters as defined with continuous sampling are independent of the model chosen. However, in order to simplify the following discussion, a general multicompartmental model will be adopted (Fig. I). There is a central sampled compartment ( 1) from which the drug is eliminated and a number of peripheral compartments (2 to n) in which the drug can be distributed. The processes of elimination and distribution are governed by rate constants as shown in Fig. 1. The drug may be injected into the central compartment (of which blood is assumed to be apart) or into a peripheral compartment, e.g., intramuscularly. The actual number of peripheral compartments is unimportant in this analysis. Instantaneous injection is not required and the drug may be injected over any convenient time period. A theoretic curve of plasma drug concentration vs time, cp(t), is depicted in Fig. 2. The drug is injected beginning at time zero (to) and continued over any desired time period. After some time t', pseudoequilibrium (PEQ) is reached'" and the curve becomes exponential. The term "pseudoequilibrium" is used because even though the concentrations of the drug in the various compartments are still
Volume 22 Number 2
Continuous sampling
changing, the proportion of the total drug contained in any one compartment is constant after this time is reached.? The period (PEQ) is also referred to as the ß-phase. 7 Concentrations of sampIes taken continuously from the central compartment represent average concentrations during the period of collection. For example, if blood is drawn continuously and at a constant rate from time t a to t b, then the plasma concentration of the pooled blood sampIe Cb (the bar denoting average) will satisfy Equation I. Cb
= ( J"cp(t)dt)/(t b
-
tal
(I)
t"
This is simply an expression of the centrallimit theorem of calculus; it is true regardless of the shape of cp(t) and is totally model-independent.
Drug kinetics during the final elimination phase. Equation 2, for the portion of the curve c(t) during the PEQ period, can be determined as folIows. As explained above, the curve is exponential after time t' when the PEQ phase begins. Therefore, for t > t'
(2) for some constants Co and k. With the use of CS to obtain sampIes of concentration C2 and C 3 representing periods of withdrawal from time t l to t 2 and t 2 to t3 , respectively, then from Equation I follows Equation 3. C2(t2 - t 1)
= 1 f'Cp(t)dt, C3(t3 - t2) = 1
I,
f 3Cp(t)dt
(3)
If the periods of collection are equaI, so that
w = t2 - t l = t3 - t2 and t l > t', then from Equations 2 and 3 follow Equation 4a, 4b, and 5.
C2 W =
11
fl'coe- kt = (co/k)(e-ktl - e- kt2) = (co/k)(e kW - l)e- kt2
(4a)
(co/k)(e kw -
(4b)
C3W
=
l)e- kt3
Taking logorithms of both sides of Equation 5 allows solution of k (Equation 6). k = 1.- 1n(c2/c3) W
(6)
Rearrangement of Equation 4a yields Equation
t«
and hence the half-life during the PEQ phase t! is given by Equation 7b: t. = 0.693 w/(ln Cd(3)
Ob)
The analysis above shows that the PEQ phase constants can be found with the use of two continuously collected sampIes. If an estimate of the time at which the PEQ phase begins is not available, several continuous sampIes may be withdrawn and used to identify this phase in the following manner: Let C2, C3 . . . cn be the concentrations of sampIes taken between times t l - t2 , t 2 - t3 , . . • , t n - 1 - t n , respectiveIy, where all collection periods are of equal duration, w. Then from Equation 4 follows Equation 8. CjW
= coe-ktt-t (1
- e-k")/k
(8)
Plotting the values log (Ci) vs (t j _ 1 + t j)/2 will reveal the time at which the PEQ phase begins, since it can be shown that these points will fall on a straight line only when cp(t) is exponential. We can then define a new function, let), by Equation 9. Itt)
=
(co(l - e-kW)/(kw»e- kl
(9)
Since let) is in the form f'(x) = ae-bX, a simple least-squares analysis with the use of the points t j , t j + 10 • • • ,t n - 1o corresponding to In l(t j ) , In l(t j + I)" . . , In l(t n _ I), where t j > t' will yield a statistically accurate estimation of the parameters k and (coO - e-kW)/(kw», from which Co is easily obtained. Plasma clearance. The amount of drug (dQ) that is irreversibly eliminated from the system in a small time interval (dt) is given by Equation 10, dQ
(5)
133
= ke1V . c p (t)dt
(10)
where V is the volume of the central compartment. The constant (ke1V) is equivalent to the plasma clearance rate of the drug (Cl p) 16 . 24 and can be thought of as the volume of plasma totally and irreversibly cleared of the drug per unit time. Since all of the dose injected (Q) will be eventually eliminated, integration of Equation 10 yields Equation 11. Cl, = Q/tofcp(t)dt
(11)
Thus the plasma clearance rate can be found if the area under the plasma concentration curve
134
Vogelstein. Kowarski, and Lietman
Clinical Pharmacology and Therapeutics
is known. This area is diffieult to measure aeeurately by means of intermittent sampling'': 13 but is simply determined with CS. If CS is earried out from time t o - tl> t l - t2 , • • • t n - I - t n and the eoneentrations of the resultant samples are CI> C2' . . . CU> respeetively, then from Equation 1 follows Equation 12, 1 0
r cp(t)dt = t,
10
III cp(t)dt
+
(12)
t,I'cit)dt + ... + In_,fncp(t)dt + t IXcp(t)dt = Clt l + C2(t2 - t l ) + ...
+' cn(tn -
+ coe-kln/k
t n - I)
where the eonstants Co and kare found as in the preeeding seetion. Henee Equation 13: n
Cl., = Q/i~1 Cj(tj
-
ti-I)
+
([coe-kln]/k).
(13)
Cl, has previously been found with the use of
surement of single organ clearanee ean be made with CS in a similar fashion. In these eases, the single organ clearanee (SOC) will be given by Equation 16, SOC = Q/
r cp(t)dt
10
(16)
wh!re Q is the dose injeeted and the integral cp(t)dt is found as in the preeeding seetion. This formula allows aeeurate determinations of renal clearanee without urine collection." Volume 0/ distribution. The amount of drug remaining in all drug-eontaining eompartments (excluding the elimination eompartments), A(t), at any time (t) after injeetion of the dose Q is seen in Equation 17 from Equation 10.
loI
A(t) = Q - Clp,J~ (y)dy
(17)
CS by taking one plasma sample drawn from time 0 until the plasma eoneentration is 0 or nearly 0.J3 It may be impraetieal, however, to perform withdrawal over a long enough time period to ensure that the plasma eoneentration is nearly O. The above analysis permits evaluation of Cl; in these eireumstanees. In addition, it automatieally ensures that the data obtained ean be used even if it is found, after assay, that the plasma eoneentration is still signifieant when CS ends. Single organ clearance. Renal clearanee (RC) ean be defined as the volume of plasma totally cleared of the drug per unit time by the kidneys, i.e., Equation 14,
A time-dependent volume of distribution, V D(t) , ean be defined as the ratio of the amount of drug in the system and the measured eoneentration'" (Equation 18).
dR = RC . cp(t)dt
Single injection pharmacokinetics when baseline is not zero. In a clinieal setting, it is
(14)
where dR is the rate of drug exeretion into the urine. Renal clearanee ean be aeeurately measured by means of CS, sinee CS provides an aeeurate integration of the plasma eoneentration eurve. For example, if CS is performed during a time interval of duration t* and the eoneentration of the resultant sample is e*, then from integration of Equation 14 follows Equation 15, RC = UV/(c*t*)
(15)
where U is the eoneentration of drug in urine eolleeted during the interval and V is the volurne of this eolleetion. If the drug is exclusively eliminated from the plasma by a single organ (e.g., inulin by the kidneys, eertain dyes by the liver), then a mea-
V nCt) = A(t)/c(t)
(18)
During the PEQ phase, it has been shown that V nCt) assurnes a eonstant value whieh has been termed, V Deq19 , V Dß, or VDarea.7 Use of Equations 11 and 17 in Equation 18 shows that, when t > t:", Equation 19 follows. V D,q = Clp/k = Q/(k {cp (t)dt)
r
10
(19)
The integral ep(t)dt is found direetly with t, the use of CS as in the seetion on plasma clearanee.
often desirable to study the behavior of a drug after a subjeet has already reeeived several doses. The eoneepts deseribed in the preeeding see tion allow the determination of pharmaeokinetie parameters when the subjeet has reeeived the drug previously, even when the number of previous doses and amount of drug per dose is unknown. For example, suppose the plasma eoneentration of the drug before injeetion of the test dose is eR and the time sinee the last dose t R is unknown but greater than t'. Then the amount of drug in the body, AR, just before injeetion of the test dose ean be determined by Equations 11 and 17 as Equation 20. AR =
er, IRIX cit)dt
(20)
Continuous sampling
Vo/ume 22 Number 2
Since t R > t", Equation 2 may be used to find Equation 21. (21 )
If the test dose Q is then injected, then all the drug in the body, Q = AR, will eventually be eliminated, and Equations 11 and 21 allow solution of the clearance as Equation 22. Cl, = Q/(lof cp(t)dt - cR/k)
(22)
F'
The integral c(t)dt and k are found with the use of CS a~o in the sections on drug kinetics during the final elimination phase and plasma clearance; CR is the concentration of a "blank" sampIe taken just before the beginning of the injection of the test dose. Similarly, V De" is found in this situation as Equation 23. Y nec = Q/k(t
o
JX cp(t)dt
- cR/k)
(23)
The plasma concentration curve and its components. The shape of the plasma concentration curve as determined with CS can be determined as folIows: SampIes of average concentration cj, c 2 , • • • c n are obtained by CS from time t o - tj, t 1 - t2 , . . . and t n - 1 - t n , respectively. A graph can then be made by plotting the average concentrations Cl, C2, . . . Cn against the times t 1/2, + t2)/2, . . . (t n - 1 + t n)/2, respectively. By making the time intervals (ti) of short duration, this graphic representation can be made as detailed as desired and will approach, in the lirnit, that obtained by intermittent sampling (IS). By use of CS, as with IS, the plasma concentration curve cp(t) can be resolved into exponential components for multicompartmental modeling. For example, with a two-compartment model the concentration curve cp(t) is given by Equation 24. 11
«.
cp(t) = Ae- a t
+ Be-ß t
(24)
The ß-phase constants Band ß can be found with the use of CS as in the section on final elimination kinetics. The o-phase constants may be found as folIows: If the first m sampIes are drawn over equal time periods (v) then from Equation ll Equation 25 is derived.
-- r
CiV -
ti
- 1
B(e-ßti - I
-
Cp (t)dt
= A(e- a l l -
e-ßli)/ ß
1 -
e- a tl) / Cl' +
(25)
135
A new function, J(t), can then be defined as: J(t) = (A(l - e- a v) / ex)e- at Since the values of the function J(t) are known for times 0, tj, . t m _ j, i.e., we have Equation 26.
A least-squares analysis as described in the section on drug kinetics in the final elimination phase will yield values for A and ex. Materials and methods
To illustrate the use of this new approach to pharmacokinetics , arnikacin, a new semisynthetic aminoglycoside antibiotic.v'- 6, 9, 22, 26 was chosen as a model drug. Patients who were already receiving an aminoglycoside drug for an infection or presumed infection at the Children's Medical and Surgical Center of The Johns Hopkins Hospital were chosen for study. The patients' ages ranged from 4 to 15 yr. Patients were taken off their aminoglycoside drug für 24 hr, during which time they were given three doses of amikacin every 8 hr at a dose of 420 mg/rn'vdose. The last dose priortoourkinetic analysis preceded the test dose by approximately 8 hr. Before injection of the test dose, a 20-gauge heparinized catheter, prepared by the method of Kowarski and co-workers;" was placed in an antecubital vein and apredose blood specimen was obtained. The test dose of amikacin was then infused intravenously over 4 to 5 min in the opposite arm. The exact amount infused was found by measuring the infusate concentration and weighing the syringe of amikacin before and after injection. CS was started at the beginning of infusion with the use of a Sigmamotor pump at the rate of either 4 or 8 ml/hr. Five 15-min sampIes were drawn followed by five 30-min sampIes. Sample changes were accomplished simply by changing the glass collection tube at the end of the heparanized tubing. Urine was collected from the beginning of the infusion (after voiding) until the end of CS. Amikacin was measured in plasma and urine by means of a modification of an assay originally described by Haas and Davies." Total clearance, renal clearance, volume of distribution, and other pharmacokinetic parameters
Vogelstein, Kowarski, and Lietman
136
Clinieal Pharmaeology and Therapeuties
Table I. Results 0/ the study Pa-
tient No.
Age (yr)
Weight (kg)
1 2 3 4 5 6
4 6 7 7 12 15
14.0 17.3 15.9
Mean
27.4
27.0 77.0
Surface area
RC (mllminl
TC (mllminl
(m 2)
1.73 m 2)
1.73 m 2)
RCITC (%)
0.63 0.75 0.70 0.96 1.0 2.0
130 94 86 70 103 90 96
164 105 90 108 123 116 118
80 90 96 65 84 78 82
V D eq
tVl (hr)
(% body
1.2
44 40 35 32 29 33 35
1.8 1.7 1.7 1.3
2.2 1.7
weight)
RC: renal clearance; TC: total clearance; V De ,, : volume of distribution.
were calculated by means of the formulas derived in preceding sections. Results
The results of the study are shown in Table I. The total clearance rate was 118 ± 23 mll mini 1.73 m2 (± values represent ± 1 SO). Renal clearance was 96 ± 18 ml/rnin/ 1.73 m2 and averaged 82% of the total clearance. The half-life of amikacin during the PEQ phase was 1.7 ± 0.3 hr. The volume of distribution at PEQ, V De q ' was 35% ± 5.1 % of the body weight. The "apparent volume of distribution"17 was also calculated by extrapolating the exponential part of the curve back to 0 time and then dividing the dose injected by this value. The volume of distribution obtained in this way was 40% ± 5.0% of body weight; it was larger in every case (p < 0.01, Student's t test) than the V D e q ' This suggests that a one-compartment analysis might be inadequate to describe the pharmacokinetics of amikacin. To test the assumption that the plasma decay curve becomes monoexponential during a PEQ phase, the values obtained during the last 2 hr of the experiment were correlated with the straight line obtained from the least-squares analysis of these points. The mean correlation coefficient obtained in this way was 0.98, and in none of the 6 patients was it less than 0.95. The part of the area that was estimated from extrapolation of the PEQ phase of the curve to infinite time (=coe-kt/k) averaged only 17% ± 5.0% of the total area. The integrated plasma concentrations obtained with CS in Patient 1 are shown in Fig. 3.
Each bar represents the average plasma concentration of amikacin during the corresponding collection period. The shape of this bar graph fits weIl with what would be predicted by a two-compartment open model, as suggested for amikacin by Cabana and Taggart. 3 The concentration curve c(t), if analyzed assuming two compartments, would be given by: cp(t)
= 40.6
e- 3 .29 t
+ 40.0 e- O. 5Ht •
The curve cp(t) is plotted together with the bar graph in Fig. 3 for comparison. Discussion
We have chosen amikacin, an experimental synthetic aminoglycoside drug, to serve as an example illustrating the CS method for obtaining pharmacokinetic data. We found that constant withdrawal of blood through the nonthrombogenic catheter was a simple technique to use in our patients, even the younger ones; in other experiments, it has been possible to sustain constant withdrawal for as long as 24 hr. 12, 25 CS for measurement of plasma concentrations of a drug after single injection has several advantages over IS, especially when the area under the plasma concentration curve is one of the pharmacokinetic parameters sought. This area is model-independent, and parameters derived from this area, such as clearance measurements or volume of distribution at PEQ, will be relatively model-independent. 1 Therefore, these results can be determined without the necessity of establishing the mode of distribution. They are unbiased by the number of compartments with which the investigator has
Continuous sampling
Volume 22 Number 2
137
60
40
!E
Cl
'" .3z
0 ~
20
0/ the
total area under the
plasma concentration curve. The theoretic background underlying the CS method is derived, and a practical [ormulation for its use in a clinical setting is described. The aminoglycoside antibiotic, amikacin, was chosen to exemplify the use ofthis technique. The drug was administered to 6 children, and CS was used to define plasma and single organ (kidney) clearance, volume
0/ distribution,
half-life during the final elimination
phase, the shape of the plasma concentration curve, and the exponential factorization of this curve for multicompartmental analysis. The CS method has several theoretical and practical advantages over the usual technique
0/ intermittent blood
sampling; such as accuracy in the determination of the plasma concentration-time curve integral, relative model independence, requirement for few samples, and ease in obtaining samples.
Bert Vogelstein, M.D., A. Avinoam Kowarski, M.D., and Paul S. Lietman, M.D., Ph.D. Baltimore, Md. The Department of Pediatrics and the Division of Clinical Pharmacology of The lohns Hopkins University School of Medicine
In order to describe the metabolism, distribution, and excretion of an injected drug, various pharmacokinetic parameters, such as, clearance, volume of distribution, and half-life, are used. These parameters are estimated from the concentration of the drug in plasma samples drawn at various times after injection. We present an alternative method for obtaining these plasma samples and the estimates that are based on them. The method relies on continu-
Supported by National Institutes of Health Grant RDI-HD-06284. Received for publication Oct, 23, 1976. Accepted for publication March 26, 1977. Reprint requests to: Dr. Paul S. Lietman, Osler 527, The Johns Hopkins Hospital, Baltimore, Md. 21205.
ous, rather than intermittent, sampling of plasma. In addition to defining the usual analytic components used in the analysis of drugs after single injections, continuous sampling (CS) allows a very accurate yet simple determination of the total area under the plasma concentration curve (AUC o_ oo) . This area is a critical factor in the determination of several standard pharmacokinetic parameters. CS has been made feasible because of the development by Kowarski and associates-- of a nonthrombogenic catheter through which blood can be withdrawn at a constant rate. CS has been used to measure the secretion rate of various hormones in humans '": 11, 14, 15, 21, 25 and the clearance-" and bioavailability'" of sul131
Vogelstein. Kowarski, and Lietman
132
Clinieal Pharmaeology and Therapeutics
kel
o
Fig. 1. Multicompartmental pharmacokinetic system.
Theoretic considerations
'0
TIME
Fig. 2. Plasma concentration curve. The pseudoequilibrium phase (PEQ) begins at time t'.
fonamides in dogs. We show that continuous sampling can be used in a clinical setting to determine several pharmacokinetic parameters, including the volume of distribution, plasma and single organ clearances, the half-life during the final elimination phase, distribution and elimination constants in a multicompartmental analysis, and the shape of the plasma concentration curve itself.
Modeling. Some of the pharmacokinetic parameters as defined with continuous sampling are independent of the model chosen. However, in order to simplify the following discussion, a general multicompartmental model will be adopted (Fig. I). There is a central sampled compartment ( 1) from which the drug is eliminated and a number of peripheral compartments (2 to n) in which the drug can be distributed. The processes of elimination and distribution are governed by rate constants as shown in Fig. 1. The drug may be injected into the central compartment (of which blood is assumed to be apart) or into a peripheral compartment, e.g., intramuscularly. The actual number of peripheral compartments is unimportant in this analysis. Instantaneous injection is not required and the drug may be injected over any convenient time period. A theoretic curve of plasma drug concentration vs time, cp(t), is depicted in Fig. 2. The drug is injected beginning at time zero (to) and continued over any desired time period. After some time t', pseudoequilibrium (PEQ) is reached'" and the curve becomes exponential. The term "pseudoequilibrium" is used because even though the concentrations of the drug in the various compartments are still
Volume 22 Number 2
Continuous sampling
changing, the proportion of the total drug contained in any one compartment is constant after this time is reached.? The period (PEQ) is also referred to as the ß-phase. 7 Concentrations of sampIes taken continuously from the central compartment represent average concentrations during the period of collection. For example, if blood is drawn continuously and at a constant rate from time t a to t b, then the plasma concentration of the pooled blood sampIe Cb (the bar denoting average) will satisfy Equation I. Cb
= ( J"cp(t)dt)/(t b
-
tal
(I)
t"
This is simply an expression of the centrallimit theorem of calculus; it is true regardless of the shape of cp(t) and is totally model-independent.
Drug kinetics during the final elimination phase. Equation 2, for the portion of the curve c(t) during the PEQ period, can be determined as folIows. As explained above, the curve is exponential after time t' when the PEQ phase begins. Therefore, for t > t'
(2) for some constants Co and k. With the use of CS to obtain sampIes of concentration C2 and C 3 representing periods of withdrawal from time t l to t 2 and t 2 to t3 , respectively, then from Equation I follows Equation 3. C2(t2 - t 1)
= 1 f'Cp(t)dt, C3(t3 - t2) = 1
I,
f 3Cp(t)dt
(3)
If the periods of collection are equaI, so that
w = t2 - t l = t3 - t2 and t l > t', then from Equations 2 and 3 follow Equation 4a, 4b, and 5.
C2 W =
11
fl'coe- kt = (co/k)(e-ktl - e- kt2) = (co/k)(e kW - l)e- kt2
(4a)
(co/k)(e kw -
(4b)
C3W
=
l)e- kt3
Taking logorithms of both sides of Equation 5 allows solution of k (Equation 6). k = 1.- 1n(c2/c3) W
(6)
Rearrangement of Equation 4a yields Equation
t«
and hence the half-life during the PEQ phase t! is given by Equation 7b: t. = 0.693 w/(ln Cd(3)
Ob)
The analysis above shows that the PEQ phase constants can be found with the use of two continuously collected sampIes. If an estimate of the time at which the PEQ phase begins is not available, several continuous sampIes may be withdrawn and used to identify this phase in the following manner: Let C2, C3 . . . cn be the concentrations of sampIes taken between times t l - t2 , t 2 - t3 , . . • , t n - 1 - t n , respectiveIy, where all collection periods are of equal duration, w. Then from Equation 4 follows Equation 8. CjW
= coe-ktt-t (1
- e-k")/k
(8)
Plotting the values log (Ci) vs (t j _ 1 + t j)/2 will reveal the time at which the PEQ phase begins, since it can be shown that these points will fall on a straight line only when cp(t) is exponential. We can then define a new function, let), by Equation 9. Itt)
=
(co(l - e-kW)/(kw»e- kl
(9)
Since let) is in the form f'(x) = ae-bX, a simple least-squares analysis with the use of the points t j , t j + 10 • • • ,t n - 1o corresponding to In l(t j ) , In l(t j + I)" . . , In l(t n _ I), where t j > t' will yield a statistically accurate estimation of the parameters k and (coO - e-kW)/(kw», from which Co is easily obtained. Plasma clearance. The amount of drug (dQ) that is irreversibly eliminated from the system in a small time interval (dt) is given by Equation 10, dQ
(5)
133
= ke1V . c p (t)dt
(10)
where V is the volume of the central compartment. The constant (ke1V) is equivalent to the plasma clearance rate of the drug (Cl p) 16 . 24 and can be thought of as the volume of plasma totally and irreversibly cleared of the drug per unit time. Since all of the dose injected (Q) will be eventually eliminated, integration of Equation 10 yields Equation 11. Cl, = Q/tofcp(t)dt
(11)
Thus the plasma clearance rate can be found if the area under the plasma concentration curve
134
Vogelstein. Kowarski, and Lietman
Clinical Pharmacology and Therapeutics
is known. This area is diffieult to measure aeeurately by means of intermittent sampling'': 13 but is simply determined with CS. If CS is earried out from time t o - tl> t l - t2 , • • • t n - I - t n and the eoneentrations of the resultant samples are CI> C2' . . . CU> respeetively, then from Equation 1 follows Equation 12, 1 0
r cp(t)dt = t,
10
III cp(t)dt
+
(12)
t,I'cit)dt + ... + In_,fncp(t)dt + t IXcp(t)dt = Clt l + C2(t2 - t l ) + ...
+' cn(tn -
+ coe-kln/k
t n - I)
where the eonstants Co and kare found as in the preeeding seetion. Henee Equation 13: n
Cl., = Q/i~1 Cj(tj
-
ti-I)
+
([coe-kln]/k).
(13)
Cl, has previously been found with the use of
surement of single organ clearanee ean be made with CS in a similar fashion. In these eases, the single organ clearanee (SOC) will be given by Equation 16, SOC = Q/
r cp(t)dt
10
(16)
wh!re Q is the dose injeeted and the integral cp(t)dt is found as in the preeeding seetion. This formula allows aeeurate determinations of renal clearanee without urine collection." Volume 0/ distribution. The amount of drug remaining in all drug-eontaining eompartments (excluding the elimination eompartments), A(t), at any time (t) after injeetion of the dose Q is seen in Equation 17 from Equation 10.
loI
A(t) = Q - Clp,J~ (y)dy
(17)
CS by taking one plasma sample drawn from time 0 until the plasma eoneentration is 0 or nearly 0.J3 It may be impraetieal, however, to perform withdrawal over a long enough time period to ensure that the plasma eoneentration is nearly O. The above analysis permits evaluation of Cl; in these eireumstanees. In addition, it automatieally ensures that the data obtained ean be used even if it is found, after assay, that the plasma eoneentration is still signifieant when CS ends. Single organ clearance. Renal clearanee (RC) ean be defined as the volume of plasma totally cleared of the drug per unit time by the kidneys, i.e., Equation 14,
A time-dependent volume of distribution, V D(t) , ean be defined as the ratio of the amount of drug in the system and the measured eoneentration'" (Equation 18).
dR = RC . cp(t)dt
Single injection pharmacokinetics when baseline is not zero. In a clinieal setting, it is
(14)
where dR is the rate of drug exeretion into the urine. Renal clearanee ean be aeeurately measured by means of CS, sinee CS provides an aeeurate integration of the plasma eoneentration eurve. For example, if CS is performed during a time interval of duration t* and the eoneentration of the resultant sample is e*, then from integration of Equation 14 follows Equation 15, RC = UV/(c*t*)
(15)
where U is the eoneentration of drug in urine eolleeted during the interval and V is the volurne of this eolleetion. If the drug is exclusively eliminated from the plasma by a single organ (e.g., inulin by the kidneys, eertain dyes by the liver), then a mea-
V nCt) = A(t)/c(t)
(18)
During the PEQ phase, it has been shown that V nCt) assurnes a eonstant value whieh has been termed, V Deq19 , V Dß, or VDarea.7 Use of Equations 11 and 17 in Equation 18 shows that, when t > t:", Equation 19 follows. V D,q = Clp/k = Q/(k {cp (t)dt)
r
10
(19)
The integral ep(t)dt is found direetly with t, the use of CS as in the seetion on plasma clearanee.
often desirable to study the behavior of a drug after a subjeet has already reeeived several doses. The eoneepts deseribed in the preeeding see tion allow the determination of pharmaeokinetie parameters when the subjeet has reeeived the drug previously, even when the number of previous doses and amount of drug per dose is unknown. For example, suppose the plasma eoneentration of the drug before injeetion of the test dose is eR and the time sinee the last dose t R is unknown but greater than t'. Then the amount of drug in the body, AR, just before injeetion of the test dose ean be determined by Equations 11 and 17 as Equation 20. AR =
er, IRIX cit)dt
(20)
Continuous sampling
Vo/ume 22 Number 2
Since t R > t", Equation 2 may be used to find Equation 21. (21 )
If the test dose Q is then injected, then all the drug in the body, Q = AR, will eventually be eliminated, and Equations 11 and 21 allow solution of the clearance as Equation 22. Cl, = Q/(lof cp(t)dt - cR/k)
(22)
F'
The integral c(t)dt and k are found with the use of CS a~o in the sections on drug kinetics during the final elimination phase and plasma clearance; CR is the concentration of a "blank" sampIe taken just before the beginning of the injection of the test dose. Similarly, V De" is found in this situation as Equation 23. Y nec = Q/k(t
o
JX cp(t)dt
- cR/k)
(23)
The plasma concentration curve and its components. The shape of the plasma concentration curve as determined with CS can be determined as folIows: SampIes of average concentration cj, c 2 , • • • c n are obtained by CS from time t o - tj, t 1 - t2 , . . . and t n - 1 - t n , respectively. A graph can then be made by plotting the average concentrations Cl, C2, . . . Cn against the times t 1/2, + t2)/2, . . . (t n - 1 + t n)/2, respectively. By making the time intervals (ti) of short duration, this graphic representation can be made as detailed as desired and will approach, in the lirnit, that obtained by intermittent sampling (IS). By use of CS, as with IS, the plasma concentration curve cp(t) can be resolved into exponential components for multicompartmental modeling. For example, with a two-compartment model the concentration curve cp(t) is given by Equation 24. 11
«.
cp(t) = Ae- a t
+ Be-ß t
(24)
The ß-phase constants Band ß can be found with the use of CS as in the section on final elimination kinetics. The o-phase constants may be found as folIows: If the first m sampIes are drawn over equal time periods (v) then from Equation ll Equation 25 is derived.
-- r
CiV -
ti
- 1
B(e-ßti - I
-
Cp (t)dt
= A(e- a l l -
e-ßli)/ ß
1 -
e- a tl) / Cl' +
(25)
135
A new function, J(t), can then be defined as: J(t) = (A(l - e- a v) / ex)e- at Since the values of the function J(t) are known for times 0, tj, . t m _ j, i.e., we have Equation 26.
A least-squares analysis as described in the section on drug kinetics in the final elimination phase will yield values for A and ex. Materials and methods
To illustrate the use of this new approach to pharmacokinetics , arnikacin, a new semisynthetic aminoglycoside antibiotic.v'- 6, 9, 22, 26 was chosen as a model drug. Patients who were already receiving an aminoglycoside drug for an infection or presumed infection at the Children's Medical and Surgical Center of The Johns Hopkins Hospital were chosen for study. The patients' ages ranged from 4 to 15 yr. Patients were taken off their aminoglycoside drug für 24 hr, during which time they were given three doses of amikacin every 8 hr at a dose of 420 mg/rn'vdose. The last dose priortoourkinetic analysis preceded the test dose by approximately 8 hr. Before injection of the test dose, a 20-gauge heparinized catheter, prepared by the method of Kowarski and co-workers;" was placed in an antecubital vein and apredose blood specimen was obtained. The test dose of amikacin was then infused intravenously over 4 to 5 min in the opposite arm. The exact amount infused was found by measuring the infusate concentration and weighing the syringe of amikacin before and after injection. CS was started at the beginning of infusion with the use of a Sigmamotor pump at the rate of either 4 or 8 ml/hr. Five 15-min sampIes were drawn followed by five 30-min sampIes. Sample changes were accomplished simply by changing the glass collection tube at the end of the heparanized tubing. Urine was collected from the beginning of the infusion (after voiding) until the end of CS. Amikacin was measured in plasma and urine by means of a modification of an assay originally described by Haas and Davies." Total clearance, renal clearance, volume of distribution, and other pharmacokinetic parameters
Vogelstein, Kowarski, and Lietman
136
Clinieal Pharmaeology and Therapeuties
Table I. Results 0/ the study Pa-
tient No.
Age (yr)
Weight (kg)
1 2 3 4 5 6
4 6 7 7 12 15
14.0 17.3 15.9
Mean
27.4
27.0 77.0
Surface area
RC (mllminl
TC (mllminl
(m 2)
1.73 m 2)
1.73 m 2)
RCITC (%)
0.63 0.75 0.70 0.96 1.0 2.0
130 94 86 70 103 90 96
164 105 90 108 123 116 118
80 90 96 65 84 78 82
V D eq
tVl (hr)
(% body
1.2
44 40 35 32 29 33 35
1.8 1.7 1.7 1.3
2.2 1.7
weight)
RC: renal clearance; TC: total clearance; V De ,, : volume of distribution.
were calculated by means of the formulas derived in preceding sections. Results
The results of the study are shown in Table I. The total clearance rate was 118 ± 23 mll mini 1.73 m2 (± values represent ± 1 SO). Renal clearance was 96 ± 18 ml/rnin/ 1.73 m2 and averaged 82% of the total clearance. The half-life of amikacin during the PEQ phase was 1.7 ± 0.3 hr. The volume of distribution at PEQ, V De q ' was 35% ± 5.1 % of the body weight. The "apparent volume of distribution"17 was also calculated by extrapolating the exponential part of the curve back to 0 time and then dividing the dose injected by this value. The volume of distribution obtained in this way was 40% ± 5.0% of body weight; it was larger in every case (p < 0.01, Student's t test) than the V D e q ' This suggests that a one-compartment analysis might be inadequate to describe the pharmacokinetics of amikacin. To test the assumption that the plasma decay curve becomes monoexponential during a PEQ phase, the values obtained during the last 2 hr of the experiment were correlated with the straight line obtained from the least-squares analysis of these points. The mean correlation coefficient obtained in this way was 0.98, and in none of the 6 patients was it less than 0.95. The part of the area that was estimated from extrapolation of the PEQ phase of the curve to infinite time (=coe-kt/k) averaged only 17% ± 5.0% of the total area. The integrated plasma concentrations obtained with CS in Patient 1 are shown in Fig. 3.
Each bar represents the average plasma concentration of amikacin during the corresponding collection period. The shape of this bar graph fits weIl with what would be predicted by a two-compartment open model, as suggested for amikacin by Cabana and Taggart. 3 The concentration curve c(t), if analyzed assuming two compartments, would be given by: cp(t)
= 40.6
e- 3 .29 t
+ 40.0 e- O. 5Ht •
The curve cp(t) is plotted together with the bar graph in Fig. 3 for comparison. Discussion
We have chosen amikacin, an experimental synthetic aminoglycoside drug, to serve as an example illustrating the CS method for obtaining pharmacokinetic data. We found that constant withdrawal of blood through the nonthrombogenic catheter was a simple technique to use in our patients, even the younger ones; in other experiments, it has been possible to sustain constant withdrawal for as long as 24 hr. 12, 25 CS for measurement of plasma concentrations of a drug after single injection has several advantages over IS, especially when the area under the plasma concentration curve is one of the pharmacokinetic parameters sought. This area is model-independent, and parameters derived from this area, such as clearance measurements or volume of distribution at PEQ, will be relatively model-independent. 1 Therefore, these results can be determined without the necessity of establishing the mode of distribution. They are unbiased by the number of compartments with which the investigator has
Continuous sampling
Volume 22 Number 2
137
60
40
!E
Cl
'" .3z
0 ~
20
