journal of Clinical Pharmacy and Therapeutics (1992)17,55-59.
Dosage optimization methods applied to imipramine and desipramine in enuresis treatment M. Tamayo, M. M. Fernindez de Gatta, M. J. Garcia and A. Dominguez-Gil Department of Pharmacy and Pharmaceutical Technology, University of Salamanca, Spain
SUMMARY
Three methods for estimating maintenance dosage requirements of imipramine were compared retrospectively in 146 enuretic patients. The dosing methods evaluated included individual (serum levels data) and/or population (average pharmacokinetic parameter) information. The use of imipramine and desipramine serum concentrations, as opposed to average population parameters only, improved forecast precision and accuracy for dosage individualization.The clinical acceptability of this was achieved through knowledge of a single serum concentration. No significant differences were seen between non-linear regression and the Bayesian method, this is in agreement with the high contribution of the patient's data to the Bayesian fitting (FF= 0.8). When one or two serum level data were available, a better performance was obtained by estimating pharmacokinetic parameters than levekdose ratios. INTRODUCTION
Many methods are available to optimize drug dosing. Among such methods is the so-called a priori method, which defines standard doses based on the mean kinetic behaviour of the drug in a given population, and methods that employ the individual serum levels data of the drug. More recently, Sheiner et al. (I) have introduced the Bayesian estimation method, which, together with the measurement of serum levels, includes population pharmacokinetic parameters. Dose optimization methods are applied to drugs with the following characteristics: a concentration-effect relationship, a narrow therapeutic range and large interindividual differences in disposition. Imipramine (IMI)fulfills the above criteria Correspondence Professor A. Dominguez-Gil, Department of Pharmacy and Pharmaceutical Technology, Avda Camp0 Charro s/n, 37007-Salarnanca,Spain.
and some authors (2-5) have confirmed the usefulness of therapeutic drug monitoring in children who receive the drug for both enuresis and other types of psychiatric disorders. The lack of information about population parameters in paediatrics has limited the application of these methods in such populations. Our previous experience (6, 7) in therapeutic drug monitoring of IMI in enuretic children and the growth of information available on serum levels in such subjects has led us to apply population pharmacokineticanalysis to this population. The aim of the present study was to evaluate the performance of different dosage optimization methods based on individual and/or population data applied to IMI and its active metabolite, desipramine (DMI), in a paediatric population and to compare them with results obtained using a simpler pharmacokinetic model that estimates the 1evel:dose ratios (LID)(7). MATERIALS A N D METHODS
Patients
The study included 146 paediatric patients (5-16 years, 46 girls) treated on an out-patient basis with different doses (12-5-125 mg/day) of TofraniP for enuresis. The doses were administered at bed-time. IMI and DMI serum levels were determined at least once during the course of treatment, and when steady-state was achieved. One hundred and forty-six, 76, 45 and 17 patients had one, two, three and four measured concentrations, respectively. The sampling time employed was 12 h after administration of the last dose. The serum levels of IMI and DMI were analysed by gas chromatography with a specific phosphorus-nitrogen thermoionic detector (8). Pharmacokinetic model
A one-compartment pharmacokinetic model was used for IMI and DMI with first-order appearance and disappearanceprocesses. The equations that describe the
56
M. Tamayo et al.
Table 1. Population phannacolunetic parameters of IMI and DMI in 49 enuretic children estimated by the standard two stage method Parameter
Mean
Standard deviation
€1v,(kg/l)
0.0213 0-0425 0-0359 0.0008
or,,,,
KJh-') K,,,(h-') QNV (kg1l.h) 111
Mefhod 2. The non-linear regression method by weighted least squares is based on the use of the individual serum levels of IMI and DMI. The objective function to be minimized i s
00081 0.0170
OK,
0.0247 0.0007 2.274
OK,
n
ss= 1 yrc; - f(t;, P,)I2
ut = the residual variance.
F:D 1 -exp( - nijXei~z) clij =L. exp( -Kej-tij),
CZii=
1 - exp( -Kei-z)
Fi*D*K6
[eqn I]
I -exp( -nii.Kmi.z)
*[
VmjCKe,-Kmj)
exp( - Kmj.tii)-
[eqn 31
i=l
time-courseof the drug and its metabolite in serum in the post-absorption and post-distribution are as follows:
vd,
Method 1. The average parameter method predicts the serum levels of IMI and DMI for the dose administered by introducing the mean population parameters (Table I) into the equations defining the pharmacokinetic model (eqns 1 and 2).
1 - exp( -Kmj.z)
1 -exp( - nii.Kej.r)
1 -exp( -Kei.z)
1
.exp( - Kei.tij)
kqn 21 where Cliiand CXiare the serum concentrations of IMI and DMI, respectively; Fi is the bioavailable fraction; D is the dose administered; vdj and Vmjare the apparent distribution volumes of IMI and DMI, respectively; Kej and Kmjare the elimination rate constants of the drug and its metabolite; K6 is the formation rate constant of the metabolite; z is the dosage interval nu is the number of doses administered from the start of treatment and tii is the sampling time. The parameters to be estimated are therefore F/vd, K,, Kp F/V, and K,. The mean population parameters and their intra- and interindividual variability, necessary for Bayesian estimation, were previously estimated in 49 enuretic children by the standard two stage method (Table I).
where n is the number of observed serum levels data (C,), W, is the statistical weighting factor, f(tP P,) is the pharmacokineticmodel defined in equations 1 and 2, ti is the time (independent variable) and P, is the vector of pharmacokinetic parameters P ( I ) , P(2), P(3) and P(4). In order to estimate the four pharmacokinetic parameters at least two IMI and DMI serum levels are necessary. Accordingly, when only one serum level of IMI and DMI is available, the mean population values of F/Vd and K, are introduced and the other two parameters are estimated.
Mefhod 3. The Bayesian method combines information about population pharmacokinetic parameters (mean and variance) with the serum levels of the drug and its metabolite in the patient. The objective function to be minimized is: n
ss= FFC[c;+(ti, i=I
Three methods, based on pharmacokineticcriteria, were used for dosage individualization in order to reach a serum level within the therapeutic range established for enuresis (50-150 ng/ml) (5).
m ...
+ (I-FF) Fj-Pi)2/wzi j=1
where rn is the number of parameters, Fj and ojare the mean and standard deviation of the population parameters, Pi is the parameter vector estimated in each individual and FF is the overall weight of patient data. This factor takes a value from 0 to 1 and can be evaluated statistically from the following expression:
FF=Methods of prediction and statistical analysis
P)]~o~
1 1 +d
where d is the residual variance. Estimation of the individual pharmacokinetic parameters was done with the MULTI 2(BAYES) program (9) which has both a non-linear least square regression
Imipramine dosage in enuretic children 57 Table 2. Regression analysis between measured and predicted total serum concentrations by using three methods applied with different amounts of information on drug serum levels (n = 0,1 , 2 and 3)
Method
n
Number of patients
1
0
146
2
1 2 3
76 45 17
1 2 3
76 45 17
3
and a Bayesian algorithm. The program also allows individualization of the dosage schedules and calculation of the serum concentration predicted for a given dosage regimen. Predictions of subsequent IMI and DMI concentrations were made using from 0 to 3 measured concentrations. These predicted concentrations were compared to the serum levels observed later. The performance of the different methods was calculated from the mean prediction error WPE = I / N (Cpred- CObJ and its standard deviation as estimates of accuracy and precision, respectively (10).Linear regressions between predicted and measured serum levels and correlation coefficients were also calculated. Comparison of the mean prediction errors for the different methods was done with the Kruskal-Wallis test. RESULTS AND DISCUSSION
The broad interindividual variability in the disposition kinetics of IMI in the paediatric population justifies the use of dosage optimization methods, based on the use of serum drug levels. The possibility of reducing the number of serum levels data, necessary for a correct dosage optimization demonstrated for some drugs with the Bayesian algorithm, led us to compare this technique with more conventional dosing methods (11-14). Because the drug’s pharmacological activity is related to the total serum levels (IMI DMI), for the different methods, the IMI, DMI and IMI DMI serum levels were predicted. Predicted and observed serum concentrations from the subsequent dose were analysed by linear regression. Table 2 shows the results of this analysis for the three methods using 0,1, 2 and 3 serum levels. Correlation
+
+
Regression equation
cob=50.71 + 0O9Cpred
+089C,,d +097cpred +093Cp, Cobs= 2.77 + 0.93Cpr4 cobs=2.91 + 0.96Cpr4 cobs =6.77
Correlation coefficient 0.27
Cobs=3.54 Cobs = 5.89
0.94 0.92 0.84
cob= 1.41 + 0-97cprd
0.89 0.93 0.86
Table 3. Prediction errors for the three methods using different amounts of information on drug serum levels (n = 0,I, 2 and 3)
Method
n
MPE fSD IMI
MPE fSD DMI
1
0
- 5.72 f 21.95
- 16.50 f 50.65
2
1 2 3
0.55 f 8-45 0 2 2 f 7.88 -2.73k6.31
- 2.04 f 17.45 1.56+ 15.21 2-98 18.81
1
5.75 f6.18 4.88 f 6.15 5.00 f 4.60
10.14 f 14.30 9.03 f 12.27 13.02 f 13.53
3
2 3
+
coefficients are significant in all cases, although the correlation is better with methods that use serum levels data. The clinical acceptability of prediction errors must first be established, thus minimizing the risk of under- or overdosing (I).Assuming that the amplitude of the therapeutic interval (50-150 ng/ml) is 100 ng/ml and that the desired target concentration is close to 100 ng/ ml, for 95% of the population to have serum levels within that range the sum of the mean prediction error and its standard deviation should not exceed 25 ng/ml. Table 3 shows the mean prediction errors and their standard deviations for the three prediction methods evaluated in the study. In general, worse predictions are obtained for the metabolite. This fact is probably due to the greater interindwidual variability in its disposition kinetics relative to those of the parent compound (15, 16). It is clear that methods that use serum drug
58
M. Tamayo et al.
concentrations to individualize dosing are more accurate than other methods that do not use this type of information. The average parameter method ignores the interindividual variability in the disposition kinetics of IMI+DMI and assumes that the patient will show the same kind of behaviour as that reflected by the mean population pattern. The regression analysis for this method points to its pooraccuracy and precision,anaspect also confirmed in the MPE values. The standard deviation of the MPE is far higher than 25 ng/ml and hence this method is unacceptablefrom the clinical point of view. The poor performance of this method, based exclusively on population information, justifies the need to apply methods based on more specific information about the patient. In this sense, determination of the serum levels is an important advance in dosage optimization. The prediction errors for the method that uses only data on the serum levels (method 2) are shown in Table 3. It is seen that regardless of the number of serum level data, accuracy is good because the MPE is close to zero and the precision evaluated by the standard deviation of the MPE is never above 25 ng/ml. Application of the Kruskal-Wallis test revealed no significant differences in either the accuracy or precision of the method with different numbers of serum level measurements ( P < 0-05). The limited number of concentration data normally available in therapeutic monitoring work means a low level of precision in the individual estimated parameters. This loss in precision may specially affect outliers. In order to overcome this problem, a Bayesian algorithm has been introduced in clinical pharmacokinetics (I). This method optimizes the dosage using not only individual serum levels data but also population pharmacokinetic parameters. An important factor to be studied is the contribution of the patient data to the Bayesian fitting (FF). This was calculated in the present study from the value of the residual variance and was found to be 80%.To determine its influence on Bayesian performance, other FF values within the range from 0 to 1 (0.2 and 0 5 ) were employed. Figure 1 shows the Bayesian prediction error for IMI DMI when I, 2 and 3 serum levels data are known for different FF values (0.2;0.5 and 0.8).The discontinuous lines represent the maximum range in which 95% of the errors should be included if the prediction is clinically acceptable. The columns delimit a standard deviation on each side of the MPE, indicated
+
W
a I
-90L
n- I
n=2
n:3
Fig. 1. Prediction errors of the Bayesian method for different levels of information concerning the serum levels (n= 1,2 and 3)using different FFvalues [FF= 0.2(8); 0.5(N)and 0-8(O)].
by the horizontal mid-line. It may be seen that the Bayesian method that uses an FFvalue of 0.8 is the most accurate and precise. This implies that the patient data do not have large experimental errors and that the contribution of the mean population parameters and their variances are small for the Bayesian method. No significant differences were observed on applying the Kruskal-Wallis test either in the accuracy or the precision of the Bayesian method on increasing the number of serum levels used in the analysis. Figure 2 shows the MPE fSD for IMI DMI calculated by optimization methods 2 and 3 when I, 2 or 3 serum levels data are available and using pharmacokinetic parameters and a 1evel:doseratio (7). An important improvement is seen in the accuracy and precision of the results when data on the serum levels are known, an aspect that justifies the monitoring of IMI and DMI serum levels. When one or two serum level data are available, better performance is obtained by estimating the pharmacokinetic parameters using both non-linear regression and Bayesian estimation. For n = 3, no significant differences were obtained, regardless of whether pharmacokinetic parameters or 1evel:dose ratios were estimated. The methods that estimate pharmacokinetic parameters require a smaller number of data on serum levels to be clinically acceptable. This is logical because the model used in the estimation of the leve1:dose ratio is a simplification of that employed in the estimation of pharmacokinetic parameters, which more reliably reflects the time course of serum levels.
+
Imipramine dosage in enuretic children
n:l
n=2
n:3
Fig. 2. Prediction errors of methods 2 (NLR=non-linear regression) and 3 (Bayesian estimation) for different levels of information about drug concentrations,estimating pharmacokinetic parameters (Poand the leve1:dose ratio (LID).(0) NLR (Po; (El)NLR (LID);(El)Bayesian (Po; (El) Bayesian (LID).
The results obtained in the present study point to the usefulness of knowledge of drug serum levels and the application of pharmacokinetic criteria in the dosage individualization of IMI in the treatment of enuresis. According to the results obtained from the present study, the Bayesian method does not display any advantages when compared with the method that only employs serum level data. However, our opinion is that when few data on the serum levels are available the Bayesian method is able to detect outliers from the population due to situations such as non-compliance, analytical errors and non-attainment of steady-state conditions. Nevertheless, when making a therapeutic decision, the serum level data should be considered within the clinical context of the patient. They should not be used as substitutes for clinical observations but rather should be considered as complementary information. REFERENCES
Sheiner LB, Beal SL, Rosenberg B, Marathe W. (1979)Forecasting individual pharmacokinetics. Clinical Pharmacology and Therapeutics, 26,294-305. Jorgensen OS, Lober M, Christiansen J, Gram LF. (1980) Plasma concentration and clinical effect in imipramine treatment of childhood enuresis. Clinical Pharmacokinetics, 5,386-393. Potter WZ, Cali1 HM, Sutfin TA et al. (1982)Active
59
metabolites of imipramine and desipramine in man. Clinical Pharmacology and Therapeutics, 31,393-401. 4. Puig-Antich J, Perel JM, Lupatkin W et al. (1987)Imipramine in prepubertal major depressive disorders. Archives of General Psychiatry, 4481-89. 5. Rapoport JL, Mikkelsen EJ, Zavadil A et al. (1980) Childhood enuresis 11. Psychopathology, tricyclic concentration in plasma and antienuretic effect. Archives of General Psychiatry, 37,1146-1152. 6. Fernindez de Gatta MM, Garcia MJ, Acosta A, Gutierrez JR, Dominguez-Gil A. (1984)Monitoring of serum levels of impramine and desipramine and individualization of dose in enuretic children. Therapeutic Drug Monitoring, 6,438-443. 7. Fernindez de Gatta MM, Tamayo M, Garcia MJ et al. (1989)Prediction of imipramine serum levels in enuretic children by a Bayesian method: comparison with two other conventional dosing methods. Therapeutic Drug Monitoring, 11,637-64 1. 8. Baselt CR. (1980)Analytical Procedures for Therapeutic Drug Monitoring and Emergency Toxicology, pp. 290-291. Biomedical Publications, Davis, CA. 9. Yamaoka K, Nakagawa T, Tanaka H, Yasuhara M, Okumura K, Hori R. (1985)A nonlinear multiple regression program, MULTI2(BAYES), based on Bayesian algorithm for microcomputers. Journal of PharmacobioDynamics, 8,246-256. 10. Sheiner LB, Bed SL. (1981)Some suggestions for measuring predictive performance. Journal of Pharmacokinetics and Biopharmaceutics,9,503-512. 11. Godley PJ, Black JT, Frohna PA, Garrelts JG. (1988) Comparison of a Bayesian program with three rnicrocomputer programs for predicting gentamicin concentrations. Therapeutic Drug Monitoring, 10,287-291. 12. Sheiner LB, Beal SL. (1982)Bayesian individualization of pharmacokinetics: simple implementation and comparison with non-Bayesian methods. Journalof Pharmaceutical Sciences, 71,1344-1348. 13. Vozeh S,Uematsu T, Hauf G.F, Follath F. (1985)Performance of bayesian feedback to forecast lidocaine serum concentration: Evaluation of the prediction error and the prediction interval. Journal of Pharmacokinetics and Biopharmaceutics,13,203-212. 14. Yuen GJ, Taylor JW, Ludden TM, Murphy MJ. (1983) Predicting phenytoin dosages using Bayesian feedback: a comparison with other methods. Therapeutic Drug Monitoring, 5,437-441. 15. Gram LF, Sondergaard I, ChristianseJ, Petersen GO, Bech P, Reisby N. (1977)Steady-state kinetics of imipramine in patients. Psychopharmacology, 54,255-261. 16. Preskom SH, Bupp SJ, Weller EB, Weller RA (1989) Plasma levels of imipramine and metabolites in 68 hospitalized children. Journal of the American Academy of Child and Adolescent Psychiatry, 28,373-375.
Dosage optimization methods applied to imipramine and desipramine in enuresis treatment M. Tamayo, M. M. Fernindez de Gatta, M. J. Garcia and A. Dominguez-Gil Department of Pharmacy and Pharmaceutical Technology, University of Salamanca, Spain
SUMMARY
Three methods for estimating maintenance dosage requirements of imipramine were compared retrospectively in 146 enuretic patients. The dosing methods evaluated included individual (serum levels data) and/or population (average pharmacokinetic parameter) information. The use of imipramine and desipramine serum concentrations, as opposed to average population parameters only, improved forecast precision and accuracy for dosage individualization.The clinical acceptability of this was achieved through knowledge of a single serum concentration. No significant differences were seen between non-linear regression and the Bayesian method, this is in agreement with the high contribution of the patient's data to the Bayesian fitting (FF= 0.8). When one or two serum level data were available, a better performance was obtained by estimating pharmacokinetic parameters than levekdose ratios. INTRODUCTION
Many methods are available to optimize drug dosing. Among such methods is the so-called a priori method, which defines standard doses based on the mean kinetic behaviour of the drug in a given population, and methods that employ the individual serum levels data of the drug. More recently, Sheiner et al. (I) have introduced the Bayesian estimation method, which, together with the measurement of serum levels, includes population pharmacokinetic parameters. Dose optimization methods are applied to drugs with the following characteristics: a concentration-effect relationship, a narrow therapeutic range and large interindividual differences in disposition. Imipramine (IMI)fulfills the above criteria Correspondence Professor A. Dominguez-Gil, Department of Pharmacy and Pharmaceutical Technology, Avda Camp0 Charro s/n, 37007-Salarnanca,Spain.
and some authors (2-5) have confirmed the usefulness of therapeutic drug monitoring in children who receive the drug for both enuresis and other types of psychiatric disorders. The lack of information about population parameters in paediatrics has limited the application of these methods in such populations. Our previous experience (6, 7) in therapeutic drug monitoring of IMI in enuretic children and the growth of information available on serum levels in such subjects has led us to apply population pharmacokineticanalysis to this population. The aim of the present study was to evaluate the performance of different dosage optimization methods based on individual and/or population data applied to IMI and its active metabolite, desipramine (DMI), in a paediatric population and to compare them with results obtained using a simpler pharmacokinetic model that estimates the 1evel:dose ratios (LID)(7). MATERIALS A N D METHODS
Patients
The study included 146 paediatric patients (5-16 years, 46 girls) treated on an out-patient basis with different doses (12-5-125 mg/day) of TofraniP for enuresis. The doses were administered at bed-time. IMI and DMI serum levels were determined at least once during the course of treatment, and when steady-state was achieved. One hundred and forty-six, 76, 45 and 17 patients had one, two, three and four measured concentrations, respectively. The sampling time employed was 12 h after administration of the last dose. The serum levels of IMI and DMI were analysed by gas chromatography with a specific phosphorus-nitrogen thermoionic detector (8). Pharmacokinetic model
A one-compartment pharmacokinetic model was used for IMI and DMI with first-order appearance and disappearanceprocesses. The equations that describe the
56
M. Tamayo et al.
Table 1. Population phannacolunetic parameters of IMI and DMI in 49 enuretic children estimated by the standard two stage method Parameter
Mean
Standard deviation
€1v,(kg/l)
0.0213 0-0425 0-0359 0.0008
or,,,,
KJh-') K,,,(h-') QNV (kg1l.h) 111
Mefhod 2. The non-linear regression method by weighted least squares is based on the use of the individual serum levels of IMI and DMI. The objective function to be minimized i s
00081 0.0170
OK,
0.0247 0.0007 2.274
OK,
n
ss= 1 yrc; - f(t;, P,)I2
ut = the residual variance.
F:D 1 -exp( - nijXei~z) clij =L. exp( -Kej-tij),
CZii=
1 - exp( -Kei-z)
Fi*D*K6
[eqn I]
I -exp( -nii.Kmi.z)
*[
VmjCKe,-Kmj)
exp( - Kmj.tii)-
[eqn 31
i=l
time-courseof the drug and its metabolite in serum in the post-absorption and post-distribution are as follows:
vd,
Method 1. The average parameter method predicts the serum levels of IMI and DMI for the dose administered by introducing the mean population parameters (Table I) into the equations defining the pharmacokinetic model (eqns 1 and 2).
1 - exp( -Kmj.z)
1 -exp( - nii.Kej.r)
1 -exp( -Kei.z)
1
.exp( - Kei.tij)
kqn 21 where Cliiand CXiare the serum concentrations of IMI and DMI, respectively; Fi is the bioavailable fraction; D is the dose administered; vdj and Vmjare the apparent distribution volumes of IMI and DMI, respectively; Kej and Kmjare the elimination rate constants of the drug and its metabolite; K6 is the formation rate constant of the metabolite; z is the dosage interval nu is the number of doses administered from the start of treatment and tii is the sampling time. The parameters to be estimated are therefore F/vd, K,, Kp F/V, and K,. The mean population parameters and their intra- and interindividual variability, necessary for Bayesian estimation, were previously estimated in 49 enuretic children by the standard two stage method (Table I).
where n is the number of observed serum levels data (C,), W, is the statistical weighting factor, f(tP P,) is the pharmacokineticmodel defined in equations 1 and 2, ti is the time (independent variable) and P, is the vector of pharmacokinetic parameters P ( I ) , P(2), P(3) and P(4). In order to estimate the four pharmacokinetic parameters at least two IMI and DMI serum levels are necessary. Accordingly, when only one serum level of IMI and DMI is available, the mean population values of F/Vd and K, are introduced and the other two parameters are estimated.
Mefhod 3. The Bayesian method combines information about population pharmacokinetic parameters (mean and variance) with the serum levels of the drug and its metabolite in the patient. The objective function to be minimized is: n
ss= FFC[c;+(ti, i=I
Three methods, based on pharmacokineticcriteria, were used for dosage individualization in order to reach a serum level within the therapeutic range established for enuresis (50-150 ng/ml) (5).
m ...
+ (I-FF) Fj-Pi)2/wzi j=1
where rn is the number of parameters, Fj and ojare the mean and standard deviation of the population parameters, Pi is the parameter vector estimated in each individual and FF is the overall weight of patient data. This factor takes a value from 0 to 1 and can be evaluated statistically from the following expression:
FF=Methods of prediction and statistical analysis
P)]~o~
1 1 +d
where d is the residual variance. Estimation of the individual pharmacokinetic parameters was done with the MULTI 2(BAYES) program (9) which has both a non-linear least square regression
Imipramine dosage in enuretic children 57 Table 2. Regression analysis between measured and predicted total serum concentrations by using three methods applied with different amounts of information on drug serum levels (n = 0,1 , 2 and 3)
Method
n
Number of patients
1
0
146
2
1 2 3
76 45 17
1 2 3
76 45 17
3
and a Bayesian algorithm. The program also allows individualization of the dosage schedules and calculation of the serum concentration predicted for a given dosage regimen. Predictions of subsequent IMI and DMI concentrations were made using from 0 to 3 measured concentrations. These predicted concentrations were compared to the serum levels observed later. The performance of the different methods was calculated from the mean prediction error WPE = I / N (Cpred- CObJ and its standard deviation as estimates of accuracy and precision, respectively (10).Linear regressions between predicted and measured serum levels and correlation coefficients were also calculated. Comparison of the mean prediction errors for the different methods was done with the Kruskal-Wallis test. RESULTS AND DISCUSSION
The broad interindividual variability in the disposition kinetics of IMI in the paediatric population justifies the use of dosage optimization methods, based on the use of serum drug levels. The possibility of reducing the number of serum levels data, necessary for a correct dosage optimization demonstrated for some drugs with the Bayesian algorithm, led us to compare this technique with more conventional dosing methods (11-14). Because the drug’s pharmacological activity is related to the total serum levels (IMI DMI), for the different methods, the IMI, DMI and IMI DMI serum levels were predicted. Predicted and observed serum concentrations from the subsequent dose were analysed by linear regression. Table 2 shows the results of this analysis for the three methods using 0,1, 2 and 3 serum levels. Correlation
+
+
Regression equation
cob=50.71 + 0O9Cpred
+089C,,d +097cpred +093Cp, Cobs= 2.77 + 0.93Cpr4 cobs=2.91 + 0.96Cpr4 cobs =6.77
Correlation coefficient 0.27
Cobs=3.54 Cobs = 5.89
0.94 0.92 0.84
cob= 1.41 + 0-97cprd
0.89 0.93 0.86
Table 3. Prediction errors for the three methods using different amounts of information on drug serum levels (n = 0,I, 2 and 3)
Method
n
MPE fSD IMI
MPE fSD DMI
1
0
- 5.72 f 21.95
- 16.50 f 50.65
2
1 2 3
0.55 f 8-45 0 2 2 f 7.88 -2.73k6.31
- 2.04 f 17.45 1.56+ 15.21 2-98 18.81
1
5.75 f6.18 4.88 f 6.15 5.00 f 4.60
10.14 f 14.30 9.03 f 12.27 13.02 f 13.53
3
2 3
+
coefficients are significant in all cases, although the correlation is better with methods that use serum levels data. The clinical acceptability of prediction errors must first be established, thus minimizing the risk of under- or overdosing (I).Assuming that the amplitude of the therapeutic interval (50-150 ng/ml) is 100 ng/ml and that the desired target concentration is close to 100 ng/ ml, for 95% of the population to have serum levels within that range the sum of the mean prediction error and its standard deviation should not exceed 25 ng/ml. Table 3 shows the mean prediction errors and their standard deviations for the three prediction methods evaluated in the study. In general, worse predictions are obtained for the metabolite. This fact is probably due to the greater interindwidual variability in its disposition kinetics relative to those of the parent compound (15, 16). It is clear that methods that use serum drug
58
M. Tamayo et al.
concentrations to individualize dosing are more accurate than other methods that do not use this type of information. The average parameter method ignores the interindividual variability in the disposition kinetics of IMI+DMI and assumes that the patient will show the same kind of behaviour as that reflected by the mean population pattern. The regression analysis for this method points to its pooraccuracy and precision,anaspect also confirmed in the MPE values. The standard deviation of the MPE is far higher than 25 ng/ml and hence this method is unacceptablefrom the clinical point of view. The poor performance of this method, based exclusively on population information, justifies the need to apply methods based on more specific information about the patient. In this sense, determination of the serum levels is an important advance in dosage optimization. The prediction errors for the method that uses only data on the serum levels (method 2) are shown in Table 3. It is seen that regardless of the number of serum level data, accuracy is good because the MPE is close to zero and the precision evaluated by the standard deviation of the MPE is never above 25 ng/ml. Application of the Kruskal-Wallis test revealed no significant differences in either the accuracy or precision of the method with different numbers of serum level measurements ( P < 0-05). The limited number of concentration data normally available in therapeutic monitoring work means a low level of precision in the individual estimated parameters. This loss in precision may specially affect outliers. In order to overcome this problem, a Bayesian algorithm has been introduced in clinical pharmacokinetics (I). This method optimizes the dosage using not only individual serum levels data but also population pharmacokinetic parameters. An important factor to be studied is the contribution of the patient data to the Bayesian fitting (FF). This was calculated in the present study from the value of the residual variance and was found to be 80%.To determine its influence on Bayesian performance, other FF values within the range from 0 to 1 (0.2 and 0 5 ) were employed. Figure 1 shows the Bayesian prediction error for IMI DMI when I, 2 and 3 serum levels data are known for different FF values (0.2;0.5 and 0.8).The discontinuous lines represent the maximum range in which 95% of the errors should be included if the prediction is clinically acceptable. The columns delimit a standard deviation on each side of the MPE, indicated
+
W
a I
-90L
n- I
n=2
n:3
Fig. 1. Prediction errors of the Bayesian method for different levels of information concerning the serum levels (n= 1,2 and 3)using different FFvalues [FF= 0.2(8); 0.5(N)and 0-8(O)].
by the horizontal mid-line. It may be seen that the Bayesian method that uses an FFvalue of 0.8 is the most accurate and precise. This implies that the patient data do not have large experimental errors and that the contribution of the mean population parameters and their variances are small for the Bayesian method. No significant differences were observed on applying the Kruskal-Wallis test either in the accuracy or the precision of the Bayesian method on increasing the number of serum levels used in the analysis. Figure 2 shows the MPE fSD for IMI DMI calculated by optimization methods 2 and 3 when I, 2 or 3 serum levels data are available and using pharmacokinetic parameters and a 1evel:doseratio (7). An important improvement is seen in the accuracy and precision of the results when data on the serum levels are known, an aspect that justifies the monitoring of IMI and DMI serum levels. When one or two serum level data are available, better performance is obtained by estimating the pharmacokinetic parameters using both non-linear regression and Bayesian estimation. For n = 3, no significant differences were obtained, regardless of whether pharmacokinetic parameters or 1evel:dose ratios were estimated. The methods that estimate pharmacokinetic parameters require a smaller number of data on serum levels to be clinically acceptable. This is logical because the model used in the estimation of the leve1:dose ratio is a simplification of that employed in the estimation of pharmacokinetic parameters, which more reliably reflects the time course of serum levels.
+
Imipramine dosage in enuretic children
n:l
n=2
n:3
Fig. 2. Prediction errors of methods 2 (NLR=non-linear regression) and 3 (Bayesian estimation) for different levels of information about drug concentrations,estimating pharmacokinetic parameters (Poand the leve1:dose ratio (LID).(0) NLR (Po; (El)NLR (LID);(El)Bayesian (Po; (El) Bayesian (LID).
The results obtained in the present study point to the usefulness of knowledge of drug serum levels and the application of pharmacokinetic criteria in the dosage individualization of IMI in the treatment of enuresis. According to the results obtained from the present study, the Bayesian method does not display any advantages when compared with the method that only employs serum level data. However, our opinion is that when few data on the serum levels are available the Bayesian method is able to detect outliers from the population due to situations such as non-compliance, analytical errors and non-attainment of steady-state conditions. Nevertheless, when making a therapeutic decision, the serum level data should be considered within the clinical context of the patient. They should not be used as substitutes for clinical observations but rather should be considered as complementary information. REFERENCES
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