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Factors Affecting Precise Control of Serum Drug Levels in Patients
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:\I) :\I'TIVE CONTROL OF I)R U(; I)El.lVERY SYSTE~IS
FACTORS AFFECTING PRECISE CONTROL OF SERUM DRUG LEVELS IN PATIENTS R. W. Jelliffe, T. Iglesias, J. Rodriguez, A. K. Hurst and K. A. Foo /.I/I){) m IrJl l ' Sdlf)()/.I
0( ilj'j,lil'fl l'hl/n)/l/('(Jliilll'/ ir,I, (I)/i1'(' l'.Ii/." 01 Sou/hl'l'II CI/lilOl'llil/, 0( ,\1('''irilll' 1/1/11 1'llfInl/(lf.", I ,IJ.I AII,!!,''''I'''. CA WHI]J . (lS ;\
For drugs whose margin of safety is narrow, precise dosage regimens are important, coupled with useful pharmacokinetic models, good serum level monitoring, and good procedures to analyze the data and reconstruct the past model behavior for careful comparison with the patient's clinical behavior over the entire past therapeutic history. In this way, truly individualized therapeutic goals (usually serum levels) can be chosen, and intelligent methods used to compute the new dosage regimen, after which the entire scenario recycles again. This report will review both pharmacokinetic and nonpharmacokinetic factors which play significant roles in this process. Pharmacokinetic Factors 1. The Fitting Procedure and the Pharmacokinetic Model Thirteen patients with infections requiring gentamicin were retrospectively analyzed by fitting their dosage history and a "first cluster" of either 3 (2 patients) or 4 (11 patients) serum levels. Nine of these patients had very significant changes in their renal function during therapy, while 4 did not. A total of 99 subsequent serum levels were then available to be predicted by the various models. The fitting procedures evaluated were the current maximum a posteriori probability (M.A.P.), Bayesian method (B), weighted nonlinear least-squares regression (LS), and linear regression (LR). The pharmacokinetic models used were a 1-compartment model with parameters V and Kel (K model), and a similar model with parameters V, Ki, and Ks (Ks model), where Kel = Ki + (Ks x CCr). Thus six combinations of method and model were available to analyze the patient's data retrospectively, fitting to the first cluster of serum level data, and then predicting the 99 subsequent levels. Predictions were analyzed for bias (mean error, ME), and preCision (mean squared error, MSE). The population model APKs served as the prior in the MAP Bayesian fitting procedure. Its own predictive ability was also evaluated independently. The results were:
The BKs combination was the best predictor, and was slightly, but not significantly, better that the LSKs combination. It was significantly better than the combinations without a Ks feature and the LR methods in genera 1. It was interesting that the APKs model, without any fitting to serum level data at all, was a slightly better predictor than LRK, the current community standard method of monitoring serum levels and individualizing aminoglycoside therapy. The Ks model gave superior prediction over the K model when the Bayesian (paired T = 2.82, P<.003) and LS (paired T = 3.53, P(.0003) procedures were used, but the LR procedure was unable to perceive this difference, (paired T = 0.28, P = .39), due to its less precise prediction (see above). 2. Optimal Monitoring Strategies In a separate study, a Similar, but more renally stable group of 13 patients has been studied to date. Only 1 patient has had significant changes in estimated CCr (over 10ml/min/1.73M2) during therapy. A 1-compartment Kslope model was used. The patients were analyzed retrospectively just as in the first study. However, the "first cluster" of serum levels now also included a true peak (Abs PK), drawn out of the opposite arm at the exact end of a 1/2 hour gentamicin infusion, another more conventional "peak" level (30 min) drawn 30 minutes after the end of the infusion, another level drawn 1.44 estimated half-lives after the end of the infusion (1.44 T 1/2), a trough (TR1) before the dose, a trough (TR2) drawn before the next dose, and another level (21hr) drawn 21 hours after the end of the infusion, usuall y 2 or 3 dose intervals later. These various times were selected because of considerations of optimal monitoring strategies (1,2). The 1-compartment Kslope model and the Bayesian fitting procedure were used to analyze the data. The following strategies were examined with respect to the bias (ME) and precision (MSE) of the resulting predictions. Precision
Bias
ME--SD
METHOD
Bias (ME)
BKs APKs LSKs BK LSK LRKs LRK
.53 -.25 .56 .88* .97* .79* .84*
Precision (MSE) 1.87 3.44 2.01 3.04* 3.38* 3.79* 3.96*
* Different from BKs (paired T, P < .05)
Abs PK, 1.44T1/2 Abs PK, 21 hr
-0.26 + 1.26 -0.05 ~ 1.41
MSE
SD
1.63 ±. 2.69 1.94 ±. 2.81
RMSE
1.28 1.39
Conventional TR " 3Omin, TR2 TR1, 30 min Full cluster A priori, no levels*
-0.74 + 1.23 -0 .62 ~ 1.28 -0.52 ~ 1.18 -1.01 1.85
±
2.02 2.00 1.63 *4.40
*Different from Abs PK, 1.44T 1/2 , (P<'05).
±. ±. ±. ±.
3.41 3.91 3.42 5.89
1.42 1.41 1.28 2.10
146
R. W . .IcllifTc
tI
Ill.
Good ±4.0 mg ±6.0 min ±6.0 min +6.0 min ±0.6 min ±0.6 min
Poor
While the results are still preliminary and no significant differences in precision have yet been found except with regard to the a priori model, the Abs PK + 1.44 T 1/2 strategy currently seems to give slightly better predictions than the other 2 point strategies, and was equal in precision to the full cluster of levels. Because of this, it might possibly be a more cost-effective monitoring strategy, but further work is still required to confirm this initial preliminary impression.
A "good" assay SD was defined to be half that of our hosp i ta l' s SD for its emit assay of gentamicin, and on a "bad" assay to be twice that of our hospital's assay. These SD's were:
Non-Pharmacokinetic Clinical Factors
Concentration (ug/ml)
Precise control of serum drug levels also requires precise preparation (the pharmacy) and administration (the ward care) of doses, a precise assay to measure the levels (the laboratory) and precise recording of when serum levels were drawn (the phlebotomy service). The relative contribution of these factors to precise control of serum tobramycin levels is not yet possible to evaluate in a clinical study, but they can be examined in a careful MonteCarlo simulation of therapy, in which each dose each patient receives is randomly corrupted by an error having a stated standard deviation (SD), and the infusions are also started and stopped with a stated SD in time. The true serum levels are recorded in time with a stated SD, and are also measured with a certain stated SD. With such Monte-Carlo simulations, thoughtful therapeutic scenarios can be created and their pharmacokinetic results examined.
"Good" assay SD (ug/ml)
Such a study was done using a "mul tiplemodel" adapti ve control computer program previously developed in this laboratory (3), which was linked to a Monte-Carlo simulator. This program has a Bayesian prior in which the V and Ks are probability density functions, each of which is divided into 9 parts, resulting in 81 possible "models" of the patient, each with a certain probabil ity of "being" that patient. The user enters values for the SD of 1) the errors with which the doses are prepared (the pharmacy), 2) the discrepancy between when infusions are supposed to start and stop versus when they actually do (the ward care), 3) the serum assay error (the laboratory), and 4) the discrepancy between when serum levels were thought to be drawn and when they actually were obtained (the phlebotomy service). The total squared error between the stated therapeutic goal at its desired times and the actual true serum level at those times was also computed.
Pharmacy - - - - - Phlebotomy Svc. Draw Ward Care ( IV start ( IV stop Smart Pump{ IV start ( IV stop
Dose time time time time time
SD SD SD)SD)SD)SD)-
o
- ±16 mg
- +24 min - ~18 min - ±12 min
2
4
8
.57
.48
.42
.41
.80
"Poor" assay SD (ug/ml) 2.28
1.92
1.68
1.64
3.20
The "mul tipl e-mode 1" computer program was capable of considering the various stated uncertainties in the clinical situations, developing a dosage regimen on day 1, and modifying it for the serum levels found, to develop a new regimen for days 2 through 4. After day 4, all serum level data was again analyzed to develop the regimen for days 5 through 7. After day 7, all serum level data was again analyzed to develop the final regimen for days 8 through 10. Eight groups of 100 truly identical patients each (see below) received such simulated therapy. The total therapeutic squared error and resulting average deviation of serum levels from the desired goals was:
Total Sq. Rx Error All Good - - 39.4 Poor Phleb . Svc . - - 43.0 Poor Lab - - - - - 51.9 Poor Pharmacy - - - 80.8 Poor Ward - - - - - 159.4 All Poor - - - - - 233.8 All Poor+Smart Pump 37.8 All Good+Smart Pump 7.5 Scenario
SD 24.1 29.6 27.3 43.5 84.8 94.0 21.9 3.6
Avg ug/ ml Dev from Rx Goal - - - - - 0.99 1.04 NS 0. 9, 3.4, ( .002 1.12 1.42 8. 3, (.001 2.00 13.6, <'001 2.42 20.0, ( .001 NS 0. 5, 0.97 Q.43 13.1, (.001 T, P, vs All Good
Ten days of tobramycin therapy were simulated for a 70~ patient with a stable CCr of 50ml/min/1.73M • Doses were "given" IV over 1 hour, every 12 hours. The therapeutic goals were serum levels of 7.OUg/ml at the end of the infusion, and 1.5ug/ml at trough. Serum level moni toring was performed by "drawing" and ''measuring" serum levels at the exact end of each infusion and at 1 hour before the trough, on days 1,4, and 7 of therapy.
In this study, accurate administration and recording of dosage were by far the most important. Surprisingly, the "smart pump", by not only providing accurate delivery but also by giving accurate records, contributed the most to precise ser um level control, followed by the ward care, the pharmacy, the laboratory, and the phlebotomy service. When all factors were poor but the smart pump was used, it overcame all the other poor factors, resulting in precision equal to that of good pharmacy, ward care, laboratory, and phlebotomy service combined. The smart pump can also deliver multiple doses from a single bottle and tubing, reducing therapeutic costs, and reducing labor for ward personnel, who now can monitor the pump at their, not its, convenience.
The SD's of "good" and "poor" factors, and of a "smart" programmable infusion pump with buil t-in microprocessor and clock to start and stop the infusions automatically and reliably (4), were:
The above results show that significant improvements in precise prediction (and therefore control) of serum drug levels can be obtained when fundamental mathematical principles are acted upon, as procedures mathematically
Factors A tTe ctinl-{ Prec isc Control of Scrum nrul-{ Lc\'els in Patients
designed to capture more information from the available data now appear empirically to perform better in at least the clinical setting of aminoglycoside therapy. Cl inical software to implement these concepts can now be run on an IBM PC (5). Newer software to calculate the effect of the various clinical uncertainties upon the optimal drug regimen are in development, coupled with programs to store the combined clinical experience in ongoing patient-bypatient, population models for each community or teaching hospitals (6,7,8). References 1. D'Argenio DZ: Optimal Sampling Times for Pharmacokinetic Experiments. J of Pharmacokinetics and Biopharmaceutics, 9(6): 739-755, 1981. 2. Jelliffe RW: unpublished work. 3. lIes D and Schumitzky A: Multiple Model Adaptive Control of Pharmacokinetic Systems. University of Southern California, Laboratory of Applied Pharmacokinetics, Technical Report 84-1, 1984. 4. Jelliffe RW and Nicholson WF: Computer Assisted Therapy and Automated Infusion Apparatus. MEDINFO-83 Seminars, edited by O. Fokkens et al., The Netherlands. 5. Jelliffe RW, D'Argenio DZ, Schumitzky A, Hu L, and Liu M: The USC PC-PACK Programs for Managing Drug Dosage Regimens. Presented at the IUPHAR meetings, Sydney, Australia, August 2328, 1987. 6. The USC PC PACK collection Users Manual. 7. Schumitzky A: A Recursive Method for Maximum Likelihood Estimation of Population Pharmacokinetic Parameters. CPT'86, Stockholm, Sweden, July 27 - August 1, 1986. 8. Schumitzky A: A PC Program for Sequential Analysis of Population Data in Clinical Settings. Presented at IUPHAR meetings, Sydney, Australia, August 23-28, 1987.
147
:\I) :\I'TIVE CONTROL OF I)R U(; I)El.lVERY SYSTE~IS
FACTORS AFFECTING PRECISE CONTROL OF SERUM DRUG LEVELS IN PATIENTS R. W. Jelliffe, T. Iglesias, J. Rodriguez, A. K. Hurst and K. A. Foo /.I/I){) m IrJl l ' Sdlf)()/.I
0( ilj'j,lil'fl l'hl/n)/l/('(Jliilll'/ ir,I, (I)/i1'(' l'.Ii/." 01 Sou/hl'l'II CI/lilOl'llil/, 0( ,\1('''irilll' 1/1/11 1'llfInl/(lf.", I ,IJ.I AII,!!,''''I'''. CA WHI]J . (lS ;\
For drugs whose margin of safety is narrow, precise dosage regimens are important, coupled with useful pharmacokinetic models, good serum level monitoring, and good procedures to analyze the data and reconstruct the past model behavior for careful comparison with the patient's clinical behavior over the entire past therapeutic history. In this way, truly individualized therapeutic goals (usually serum levels) can be chosen, and intelligent methods used to compute the new dosage regimen, after which the entire scenario recycles again. This report will review both pharmacokinetic and nonpharmacokinetic factors which play significant roles in this process. Pharmacokinetic Factors 1. The Fitting Procedure and the Pharmacokinetic Model Thirteen patients with infections requiring gentamicin were retrospectively analyzed by fitting their dosage history and a "first cluster" of either 3 (2 patients) or 4 (11 patients) serum levels. Nine of these patients had very significant changes in their renal function during therapy, while 4 did not. A total of 99 subsequent serum levels were then available to be predicted by the various models. The fitting procedures evaluated were the current maximum a posteriori probability (M.A.P.), Bayesian method (B), weighted nonlinear least-squares regression (LS), and linear regression (LR). The pharmacokinetic models used were a 1-compartment model with parameters V and Kel (K model), and a similar model with parameters V, Ki, and Ks (Ks model), where Kel = Ki + (Ks x CCr). Thus six combinations of method and model were available to analyze the patient's data retrospectively, fitting to the first cluster of serum level data, and then predicting the 99 subsequent levels. Predictions were analyzed for bias (mean error, ME), and preCision (mean squared error, MSE). The population model APKs served as the prior in the MAP Bayesian fitting procedure. Its own predictive ability was also evaluated independently. The results were:
The BKs combination was the best predictor, and was slightly, but not significantly, better that the LSKs combination. It was significantly better than the combinations without a Ks feature and the LR methods in genera 1. It was interesting that the APKs model, without any fitting to serum level data at all, was a slightly better predictor than LRK, the current community standard method of monitoring serum levels and individualizing aminoglycoside therapy. The Ks model gave superior prediction over the K model when the Bayesian (paired T = 2.82, P<.003) and LS (paired T = 3.53, P(.0003) procedures were used, but the LR procedure was unable to perceive this difference, (paired T = 0.28, P = .39), due to its less precise prediction (see above). 2. Optimal Monitoring Strategies In a separate study, a Similar, but more renally stable group of 13 patients has been studied to date. Only 1 patient has had significant changes in estimated CCr (over 10ml/min/1.73M2) during therapy. A 1-compartment Kslope model was used. The patients were analyzed retrospectively just as in the first study. However, the "first cluster" of serum levels now also included a true peak (Abs PK), drawn out of the opposite arm at the exact end of a 1/2 hour gentamicin infusion, another more conventional "peak" level (30 min) drawn 30 minutes after the end of the infusion, another level drawn 1.44 estimated half-lives after the end of the infusion (1.44 T 1/2), a trough (TR1) before the dose, a trough (TR2) drawn before the next dose, and another level (21hr) drawn 21 hours after the end of the infusion, usuall y 2 or 3 dose intervals later. These various times were selected because of considerations of optimal monitoring strategies (1,2). The 1-compartment Kslope model and the Bayesian fitting procedure were used to analyze the data. The following strategies were examined with respect to the bias (ME) and precision (MSE) of the resulting predictions. Precision
Bias
ME--SD
METHOD
Bias (ME)
BKs APKs LSKs BK LSK LRKs LRK
.53 -.25 .56 .88* .97* .79* .84*
Precision (MSE) 1.87 3.44 2.01 3.04* 3.38* 3.79* 3.96*
* Different from BKs (paired T, P < .05)
Abs PK, 1.44T1/2 Abs PK, 21 hr
-0.26 + 1.26 -0.05 ~ 1.41
MSE
SD
1.63 ±. 2.69 1.94 ±. 2.81
RMSE
1.28 1.39
Conventional TR " 3Omin, TR2 TR1, 30 min Full cluster A priori, no levels*
-0.74 + 1.23 -0 .62 ~ 1.28 -0.52 ~ 1.18 -1.01 1.85
±
2.02 2.00 1.63 *4.40
*Different from Abs PK, 1.44T 1/2 , (P<'05).
±. ±. ±. ±.
3.41 3.91 3.42 5.89
1.42 1.41 1.28 2.10
146
R. W . .IcllifTc
tI
Ill.
Good ±4.0 mg ±6.0 min ±6.0 min +6.0 min ±0.6 min ±0.6 min
Poor
While the results are still preliminary and no significant differences in precision have yet been found except with regard to the a priori model, the Abs PK + 1.44 T 1/2 strategy currently seems to give slightly better predictions than the other 2 point strategies, and was equal in precision to the full cluster of levels. Because of this, it might possibly be a more cost-effective monitoring strategy, but further work is still required to confirm this initial preliminary impression.
A "good" assay SD was defined to be half that of our hosp i ta l' s SD for its emit assay of gentamicin, and on a "bad" assay to be twice that of our hospital's assay. These SD's were:
Non-Pharmacokinetic Clinical Factors
Concentration (ug/ml)
Precise control of serum drug levels also requires precise preparation (the pharmacy) and administration (the ward care) of doses, a precise assay to measure the levels (the laboratory) and precise recording of when serum levels were drawn (the phlebotomy service). The relative contribution of these factors to precise control of serum tobramycin levels is not yet possible to evaluate in a clinical study, but they can be examined in a careful MonteCarlo simulation of therapy, in which each dose each patient receives is randomly corrupted by an error having a stated standard deviation (SD), and the infusions are also started and stopped with a stated SD in time. The true serum levels are recorded in time with a stated SD, and are also measured with a certain stated SD. With such Monte-Carlo simulations, thoughtful therapeutic scenarios can be created and their pharmacokinetic results examined.
"Good" assay SD (ug/ml)
Such a study was done using a "mul tiplemodel" adapti ve control computer program previously developed in this laboratory (3), which was linked to a Monte-Carlo simulator. This program has a Bayesian prior in which the V and Ks are probability density functions, each of which is divided into 9 parts, resulting in 81 possible "models" of the patient, each with a certain probabil ity of "being" that patient. The user enters values for the SD of 1) the errors with which the doses are prepared (the pharmacy), 2) the discrepancy between when infusions are supposed to start and stop versus when they actually do (the ward care), 3) the serum assay error (the laboratory), and 4) the discrepancy between when serum levels were thought to be drawn and when they actually were obtained (the phlebotomy service). The total squared error between the stated therapeutic goal at its desired times and the actual true serum level at those times was also computed.
Pharmacy - - - - - Phlebotomy Svc. Draw Ward Care ( IV start ( IV stop Smart Pump{ IV start ( IV stop
Dose time time time time time
SD SD SD)SD)SD)SD)-
o
- ±16 mg
- +24 min - ~18 min - ±12 min
2
4
8
.57
.48
.42
.41
.80
"Poor" assay SD (ug/ml) 2.28
1.92
1.68
1.64
3.20
The "mul tipl e-mode 1" computer program was capable of considering the various stated uncertainties in the clinical situations, developing a dosage regimen on day 1, and modifying it for the serum levels found, to develop a new regimen for days 2 through 4. After day 4, all serum level data was again analyzed to develop the regimen for days 5 through 7. After day 7, all serum level data was again analyzed to develop the final regimen for days 8 through 10. Eight groups of 100 truly identical patients each (see below) received such simulated therapy. The total therapeutic squared error and resulting average deviation of serum levels from the desired goals was:
Total Sq. Rx Error All Good - - 39.4 Poor Phleb . Svc . - - 43.0 Poor Lab - - - - - 51.9 Poor Pharmacy - - - 80.8 Poor Ward - - - - - 159.4 All Poor - - - - - 233.8 All Poor+Smart Pump 37.8 All Good+Smart Pump 7.5 Scenario
SD 24.1 29.6 27.3 43.5 84.8 94.0 21.9 3.6
Avg ug/ ml Dev from Rx Goal - - - - - 0.99 1.04 NS 0. 9, 3.4, ( .002 1.12 1.42 8. 3, (.001 2.00 13.6, <'001 2.42 20.0, ( .001 NS 0. 5, 0.97 Q.43 13.1, (.001 T, P, vs All Good
Ten days of tobramycin therapy were simulated for a 70~ patient with a stable CCr of 50ml/min/1.73M • Doses were "given" IV over 1 hour, every 12 hours. The therapeutic goals were serum levels of 7.OUg/ml at the end of the infusion, and 1.5ug/ml at trough. Serum level moni toring was performed by "drawing" and ''measuring" serum levels at the exact end of each infusion and at 1 hour before the trough, on days 1,4, and 7 of therapy.
In this study, accurate administration and recording of dosage were by far the most important. Surprisingly, the "smart pump", by not only providing accurate delivery but also by giving accurate records, contributed the most to precise ser um level control, followed by the ward care, the pharmacy, the laboratory, and the phlebotomy service. When all factors were poor but the smart pump was used, it overcame all the other poor factors, resulting in precision equal to that of good pharmacy, ward care, laboratory, and phlebotomy service combined. The smart pump can also deliver multiple doses from a single bottle and tubing, reducing therapeutic costs, and reducing labor for ward personnel, who now can monitor the pump at their, not its, convenience.
The SD's of "good" and "poor" factors, and of a "smart" programmable infusion pump with buil t-in microprocessor and clock to start and stop the infusions automatically and reliably (4), were:
The above results show that significant improvements in precise prediction (and therefore control) of serum drug levels can be obtained when fundamental mathematical principles are acted upon, as procedures mathematically
Factors A tTe ctinl-{ Prec isc Control of Scrum nrul-{ Lc\'els in Patients
designed to capture more information from the available data now appear empirically to perform better in at least the clinical setting of aminoglycoside therapy. Cl inical software to implement these concepts can now be run on an IBM PC (5). Newer software to calculate the effect of the various clinical uncertainties upon the optimal drug regimen are in development, coupled with programs to store the combined clinical experience in ongoing patient-bypatient, population models for each community or teaching hospitals (6,7,8). References 1. D'Argenio DZ: Optimal Sampling Times for Pharmacokinetic Experiments. J of Pharmacokinetics and Biopharmaceutics, 9(6): 739-755, 1981. 2. Jelliffe RW: unpublished work. 3. lIes D and Schumitzky A: Multiple Model Adaptive Control of Pharmacokinetic Systems. University of Southern California, Laboratory of Applied Pharmacokinetics, Technical Report 84-1, 1984. 4. Jelliffe RW and Nicholson WF: Computer Assisted Therapy and Automated Infusion Apparatus. MEDINFO-83 Seminars, edited by O. Fokkens et al., The Netherlands. 5. Jelliffe RW, D'Argenio DZ, Schumitzky A, Hu L, and Liu M: The USC PC-PACK Programs for Managing Drug Dosage Regimens. Presented at the IUPHAR meetings, Sydney, Australia, August 2328, 1987. 6. The USC PC PACK collection Users Manual. 7. Schumitzky A: A Recursive Method for Maximum Likelihood Estimation of Population Pharmacokinetic Parameters. CPT'86, Stockholm, Sweden, July 27 - August 1, 1986. 8. Schumitzky A: A PC Program for Sequential Analysis of Population Data in Clinical Settings. Presented at IUPHAR meetings, Sydney, Australia, August 23-28, 1987.
147